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Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide range of topics by publishing timely reviews which are more extensive than just literature surveys but normally less than a full monograph. Each Report deals with one speciﬁc subject. These reviews are specialist in nature but contain enough introductory material to make the main points intelligible to a non-specialist. The reader will not only be able to distinguish important developments and trends but will also ﬁnd a suﬃcient number of references to the original literature. Submission In principle, papers are written and submitted on the invitation of one of the Editors, although the Editors would be glad to receive suggestions. Proposals for review articles (approximately 500–1000 words) should be sent by the authors to one of the Editors listed below. The Editor will evaluate proposals on the basis of timeliness and relevance and inform the authors as soon as possible. All submitted papers are subject to a refereeing process. Editors J.V. ALLABY (Experimental high-energy physics), EP Division, CERN, CH-1211 Geneva 23, Switzerland. E-mail: [email protected] D.D. AWSCHALOM (Experimental condensed matter physics), Department of Physics, University of California, Santa Barbara, CA 93106, USA. E-mail: [email protected] J.A. BAGGER (High-energy physics), Department of Physics & Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore MD 21218, USA. E-mail: [email protected] C.W.J. BEENAKKER (Mesoscopic physics), Instituut–Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands. E-mail: [email protected] G.E. BROWN (Nuclear physics), Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11974, USA. E-mail: [email protected] D.K. CAMPBELL (Non-linear dynamics), Dean, College of Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] G. COMSA (Surfaces and thin ﬁlms), Institut fur . Physikalische und Theoretische Chemie, Universit.at Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] S.C. COWLEY (Plasma physics), Plasma Physics Group, Blackett Laboratory, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BW, UK. E-mail: [email protected] J. EICHLER (Atomic and molecular physics), Hahn-Meitner-Institut Berlin, Abteilung Theoretische Physik, Glienicker Strasse 100, 14109 Berlin, Germany. E-mail: [email protected]

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Instructions to authors

M.P. KAMIONKOWSKI (Astrophysics), Theoretical Astrophysics 130-33, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA. E-mail: kamion@tapir. caltech.edu M.L. KLEIN (Soft condensed matter physics), Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. E-mail: [email protected] A.A. MARADUDIN (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] D.L. MILLS (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] H. ORLAND (Statistical physics and ﬁeld theory), Service de Physique Theorique, CE-Saclay, CEA, 91191 Gif-sur-Yvette Cedex, France. E-mail: [email protected] R. PETRONZIO (High-energy physics), Dipartimento di Fisica, Universita" di Roma – Tor Vergata, Via della Ricerca Scientiﬁca, 1, I-00133 Rome, Italy. E-mail: [email protected] S. PEYERIMHOFF (Molecular physics), Institute of Physical and Theoretical Chemistry, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] I. PROCACCIA (Statistical mechanics), Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] E. SACKMANN (Biological physics), Physik-Department E22 (Biophysics Lab.), Technische Universit.at Munchen, . D-85747 Garching, Germany. E-mail: [email protected] A. SCHWIMMER (High-energy physics), Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] R.N. SUDAN (Plasma physics), c/o Physics Reports Editorial Oﬃce, P.O. Box 103, 1000 AC Amsterdam, The Netherlands. E-mail: [email protected] W. WEISE (Physics of hadrons and nuclei), Institut fur . Theoretische Physik, Physik Department, Technische Universit.at Munchen, . James Franck Strae, D-85748 Garching, Germany. E-mail: [email protected] Manuscript style guidelines Papers should be written in correct English. Authors with insuﬃcient command of the English language should seek linguistic advice. Manuscripts should be typed on one side of the paper, with double line spacing and a wide margin. The character size should be suﬃciently large that all subscripts and superscripts in mathematical expressions are clearly legible. Please note that manuscripts should be accompanied by separate sheets containing: the title, authors’ names and addresses, abstract, PACS codes and keywords, a table of contents, and a list of ﬁgure captions and tables. – Address: The name, complete postal address, e-mail address, telephone and fax number of the corresponding author should be indicated on the manuscript. – Abstract: A short informative abstract not exceeding approximately 150 words is required. – PACS codes/keywords: Please supply one or more PACS-1999 classiﬁcation codes and up to 4 keywords of your own choice for indexing purposes. PACS is available online from our homepage (http://www.elsevier.com/locate/physrep). References. The list of references may be organized according to the number system or the nameyear (Harvard) system.

Instructions to authors

vii

Number system: [1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53 (1974) 249–315. [2] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). [3] B. Ziegler, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York, 1986) p. 293. A reference should not contain more than one article. Harvard system:

Ablowitz, M.J., D.J. Kaup, A.C. Newell and H. Segur, 1974. The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53, 249–315. Abramowitz, M. and I. Stegun, 1965, Handbook of Mathematical Functions (Dover, New York). Ziegler, B., 1986, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York) p. 293. Ranking of references. The references in Physics Reports are ranked: crucial references are indicated by three asterisks, very important ones with two, and important references with one. Please indicate in your ﬁnal version the ranking of the references with the asterisk system. Please use the asterisks spar-ingly: certainly not more than 15% of all references should be placed in either of the three categories. Formulas. Formulas should be typed or unambiguously written. Special care should be taken of those symbols which might cause confusion. Unusual symbols should be identiﬁed in the margin the ﬁrst time they occur.

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Instructions to authors

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Available online at www.sciencedirect.com

Physics Reports 389 (2004) 1 – 59 www.elsevier.com/locate/physrep

Light dripline nuclei Bj&orn Jonson Experimentell Fysik, Chalmers Tekniska Hogskola and Goteborgs Universitet, S-412 96 Goteborg, Sweden Accepted 9 July 2003 editor: G.E. Brown

Abstract Experimental studies of light dripline nuclei are reviewed. Progress in the production of very short lived nuclei and the development of radioactive nuclear beams has given this /eld the necessary tools for detailed studies of the most exotic nuclei. A well-known feature for some of the light dripline nuclei is that under certain circumstances they may form a neutron or a proton halo with a dilute mass distribution extending far outside the core of the nucleus. The /rst observation of halo states was made in the middle of the 1980s and it generated an interest in dripline physics, both experimentally and theoretically, that has gone far beyond the study of halo states. The experimental results for halo states are starting to give a fairly complete understanding of their structure in many cases. The data include masses, spins, moments, reaction data over a wide energy range and beta-decays. There are two main classes of halo state: the two-body halos with one nucleon surrounding the core, like the one-neutron halos 11 Be and 19 C and the one-proton halo 8 B; and the Borromean three-body halos with two valence nucleons around the core, the key examples being 6 He, 11 Li and 14 Be. Experimental information about systems lying just outside the dripline play an important rˆole in understanding the structure of the halo states, examples being 10 Li and 13 Be, which form two-body subsystems of 11 Li and 14 Be, respectively. Unbound resonance states that correspond to exotic unbound quantum systems like 5 H, 7 H and 9 He have been identi/ed. There are continuum states existing above the particle separation threshold as well as spectra indicating cluster or molecular structure. The traditional magic numbers valid for more stable nuclei have been found to disappear and be replaced with new ones in the dripline regions. The beta-decays in these regions may give access to halos in excited states and the associated beta-delayed particle decay modes provide information about coupling to the continuum. After a short historical overview, examples on the most recent experimental results from this rapidly growing /eld of nuclear physics will be given. c 2003 Elsevier B.V. All rights reserved. PACS: 27.20.+n; 24.70.+s Keywords: Dripline nuclei; Radioactive beams; Nuclear halos; Unbound nuclei; Reaction experiments; Beta decay

E-mail address: [email protected] (B. Jonson). c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.07.004

2

B. Jonson / Physics Reports 389 (2004) 1 – 59

Contents 1. 2. 3. 4. 5. 6. 7. 8.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physics at the driplines—a brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production of dripline nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiments in the dripline regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Halo states in dripline nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General conditions for halo occurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determinations of the size of halo states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Studies of halo states in reaction experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Two-body halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Three-body halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. The A = 8 isobar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. The /ve-body structure of 8 He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. The proton halo nucleus 8 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Continuum excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Molecular structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Magic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Beyond the driplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. Unbound He isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2. The N = 7 isotones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3. The 13 Be case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4. Beyond the proton dripline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Beta-decays at the driplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Exotica and new possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 6 7 9 11 12 17 17 22 24 25 26 26 29 30 32 32 34 36 36 38 43 46 46 46

1. Introduction The exploration of nuclear matter under extreme conditions, which can be created in modern accelerator laboratories, is one of the major goals of modern nuclear physics. The opportunities oIered by beams of exotic nuclei for research in the areas of nuclear structure physics and nuclear astrophysics are exciting, and worldwide activity in the construction of diIerent types of radioactive beam facilities bears witness to the strong scienti/c interest in the physics that can be probed with such beams [1]. With access to exotic nuclei at the very limits of nuclear stability, the physics of the neutron and proton driplines has become the focus of interest. The driplines are the limits of the nuclear landscape, where additional protons or neutrons can no longer be kept in the nucleus— they literally drip out. In the vicinity of the driplines, the structural features of the nuclei change compared to nuclei closer to the beta-stability line. The normal nuclear shell closures may disappear and be replaced by new magic numbers. The gradual vanishing of the binding energy of particles or clusters of particles may at the driplines give rise to beta-delayed particle emission or even particle radioactivity. In some neutron-rich light nuclei a threshold phenomenon, nuclear halo states, was discovered about 15 years ago. Since then, the halo phenomenon has been studied extensively,

B. Jonson / Physics Reports 389 (2004) 1 – 59

Proton Halo

17 Ne

3

Two-Proton Halo

109.2 109.2 ms ms

17 F 64.8 64.8 ss

13 O 14 O 15 O 16 O 17 O 8.58 8.58 ms ms

70.59 70.59 ss

2.03 2.03 m m

stable stable

stable

11 N 12 N 13 N 14 N 15 N 16 N 17 N 18 N 19 N unbound unbound

9C 126.5 126.5 ms ms

11 11 ms ms

stable stablem 20.38 stablem 20.38

9.96 9.96 m m

7.13 7.13 ss

4.17 4.17 ss

0.63 0.63 ss

329 329 ms ms

10 C 11 C 12 C 13 C 14 C 15 C 16 C 17 C 18 C 19 C 19.3 19.3 ss

8B

stablem 20.38

20.38 20.38 m m

stablem 20.38

5730 5730 aa

2.45 2.45 ss

0.747 0.747 ss

193 193 ms ms

92 92 ms ms

49 49 ms ms

10 B 11 B 12 B 13 B 14 B 15 B 16 B 17 B 18 B 19 B stable

770 770 ms ms

20.2 23.6 ms ms 17.33 17.33 ms ms 13.8 13.8 ms ms

10.4 10.4 ms ms

unbound unbound

5.1 5.1 ms ms

unbound unbound

7Be 8Be 9Be 10 Be 11 Be 12 Be 13 Be 14 Be 6 1.6 106 aa

stable

53.29 d

8.5 ms 13.8 s

7Li

8Li

9Li 10 Li 11 Li

stable --

stable -d

840 ms

179 ms

unbound

3He 4He 5He 6He 7He 8He 9He stable --

stable --

1H

2H

3H

stable ---

-stable

12.3 12.3 yy

23.6 ms

6Li

unbound unbound

808 808 ms ms

unbound unbound

n

119 119 ms ms

unbound

4.35 ms

8.5 ms

Two-Neutron Halo

unbound unbound

α+4n

One-Neutron Halo

10.25 m

Fig. 1. Many new phenomena occur for dripline nuclei. Some of them are illustrated here for the A = 8, 11 and 17 isobars. The A = 8 isobar consists of nuclides, which are all radioactive. The astrophysically interesting nucleus 8 B is a one-proton halo nucleus while 8 He may best be described as an alpha core surrounded by four neutrons. For A = 11 the neutron dripline is reached for 11 Li (T1=2 = 8:3 ms), which is a two-neutron halo nucleus, while 11 Be is a one-neutron halo with an intruder s-state in its ground state. The level order of the ground state and the /rst two excited states in 11 Be is identical to that of its unbound mirror nucleus 11 N. The A = 17 isobar is limited by the two-neutron and two-proton halo nuclei 17 B and 17 Ne, respectively, and the beta-decay of the latter feeds an excited proton-halo state in 17 F at 495 keV.

both experimentally and theoretically, and is now a well established structural feature of many light dripline nuclei. Although there had been much eIort to study exotic nuclei in the past, the real breakthrough came with the observation of halo states. This meant a new paradigm in physics at the driplines. The many and interesting unsolved problems that suddenly appeared attracted a lot of interest; new experimental techniques were developed and new facilities built. Still, the real novelty was perhaps that dripline physics suddenly attracted theory in a way unseen before, and since then there has been a steady interplay between the experimental teams and theory groups. Fig. 1 shows the nuclear chart for the lightest elements and gives a brief illustration of some of the many diIerent phenomena that have been discovered at the driplines in the past 15 years. There have been many reviews of dripline nuclei and halo states in the past decade [2–16]. Therefore, in the present work I will focus on some of the most recent experimental results after a brief discussion of the main developments that have led this /eld to it present status. 2. Physics at the driplines—a brief history Let us /rst mention a few landmarks in the development of this /eld of nuclear physics. The /rst laboratory-produced dripline nucleus was 6 He. This was achieved by bombarding /ne-grained

4

B. Jonson / Physics Reports 389 (2004) 1 – 59

Be(OH)2 with neutrons from a beryllium–radon source in the reaction 9 Be + n → 6 He + . From the measured half-life [17] and energy spectrum of beta particles [18] it could be concluded that the observed radioactivity was from the decay of 6 He. As we shall see later, the nucleus 6 He, today available as a radioactive beam at several laboratories worldwide, continues to attract considerable interest. There are two major problems to overcome when producing and studying exotic nuclei at the driplines. First, they are normally produced in minute cross-sections together with a vast amount of other, less exotic nuclei. In addition, the half-lives are typically very short, so any delay between production and experiment should be kept minimal. The ingenious solution to this problem was provided more than 50 years ago with the /rst successful demonstration of the feasibility of the on-line mass separator method. The experiment [19,20] was carried out in Copenhagen at what is today the Niels Bohr institute. 1 The idea was to direct a neutron beam, produced by bombarding a beryllium target with deuterons, onto an uranium oxide target to produce radioactive krypton and xenon isotopes in /ssion reactions. These should then be transported to the ion source of an electromagnetic isotope separator. The trick used to get the radioactivity to the ion source was to mix the target material with baking powder and to place a cold trap close to the ion source. The decomposition products of the powder (NH3 , H2 O and CO2 ) then served as a pump that swept the produced noble gases towards the cold trap and into the ion source (see also Ref. [22]). The Copenhagen experiment, which was designed to produce new isotopes to test Pauli’s neutrino hypothesis, became a main inspiration for the European nuclear physics community to propose a large-scale on-line mass separator facility. The CERN synchro-cyclotron (SC) with its beam of 600 MeV protons was selected as the driver for the facility. The project, ISOLDE, was proposed and accepted to be built at CERN and the /rst beam for experiments was delivered to the ISOLDE target in 1967 [23]. The new installation performed very well and a number of short-lived isotopes of noble gases and mercury could be identi/ed [24] in the /rst experiments. Since then, the on-line technique has developed tremendously and one can today produce beams of radioactive isotopes from about 70 elements at ISOLDE. The on-line technique was also applied at many laboratories worldwide, such as the OSIRIS Facility at Studsvik [25], the on-line mass separator at the UNILAC at GSI [26], LISOL [27] at Louvain-la-Neuve and IGISOL [28,29] at Jyv&askyl&a. It is fair to say that the on-line technique was the dominant method for producing exotic nuclei well into the 1980s. However, in a pilot experiment performed at the LBL Bevalac in Berkeley a new era in the /eld began when Symons, Westfall and co-workers [30,31] showed the production capacity for exotic nuclei by fragmentation of high-energy heavy ions. In these experiments they did not employ the traditional reaction process to bombard a heavy target with a light projectile and search for spallation products. Instead they inverted the reaction process by bombarding a light target with heavy ions and studied the projectile fragments. In these /rst experiments they used beams of 205 MeV=u 40 Ar and 220 MeV=u 48 Ca and bombarded C and Be targets to produce neutron-rich isotopes of elements from N to Cl. They were able, for example, to show the existence of 28 Ne and 35 Al for the /rst time. In view of the present strong impact of in-Oight facilities it is interesting to read the rather modest comment on their achievement by Symons at the HelsingHr conference in 1

As a tribute to the importance of this early experiment in the development that led to the present-day dripline research, a meeting was held in Copenhagen at the end of 2001 [21], where the present status and the future of this /eld were discussed.

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5

1981 [32] “: : :we questioned the applicability of high-energy heavy-ion accelerators to this 9eld. Our experience at the Bevalac leads us to believe that this question does indeed have a positive answer. If the physics interest justi9es it, then high-energy heavy-ion beams can certainly be expected to play a rˆole in the study of nuclei at the limits of stability: : :”. In passing one should perhaps also mention 11 Li, which over the past 15 years has been one of the most investigated dripline nuclei, both theoretically and experimentally. The /rst observation of this nucleus was made in 1966 [33] in Berkeley and spectroscopic information such as its half-life [34], Pn -value [35] and mass [36] was obtained at the CERN proton synchrotron (PS). The Q-value for beta-decay is 20:61 MeV and this, together with the low break-up thresholds for particles and clusters of particles in its beta-decay daughter 11 Be, makes it a precursor to many diIerent beta-delayed particle channels. In the late 1970s and early 1980s several new beta-delayed decay modes such as ( ; 2n) [37], ( ; 3n) [38] and ( ; t) [39] were observed for 11 Li. An interesting result, which was one of the /rst indications of new physics for dripline nuclei, was obtained for 11 Be. This nucleus is only bound by 504 keV, has ground-state spin and parity I =1=2+ and only one particle-bound excited state, namely at 320 keV with I = 1=2− . In experiments on the reactions 3 H(9 Be; p )11 Be and 9 Be(t; p )11 Be the lifetime of the 320 keV level was measured with the Doppler shift attenuation method [40]. The adopted value of 166 ± 15 fs corresponds to an E1 strength of 0:36 W:u:, which is orders of magnitude larger than in nuclei closer to stability. It was shown that this very large E1 strength could be understood on the basis of shell-model calculations with realistic single-particle matrix elements, but in order to obtain the matrix element for low binding energies they had to integrate out to very large radii. This was thus the /rst observed eIect that we now understand as being due to the 11 Be halo structure. An important series of data was obtained in the Berkeley experiments performed by Tanihata and his group in 1985. In the /rst experiment [41] secondary beams of He isotopes were produced through projectile fragmentation of an 800 MeV=u 11 B primary beam. The produced fragments were separated in a fragment separator and the cross-sections were measured in a transmission-type experiment. The deduced radii of the heaviest bound isotopes 6 He and 8 He were found to have a larger increase of their radii than the normal A1=3 trend. Shortly after this /rst experiment similar results for Li isotopes of 790 MeV=u were published [42]. Here it was obvious that something very interesting had been discovered. The matter radius deduced for the heaviest Li isotope 11 Li showed an increase of about 30% compared to its closest particle-stable neighbour 9 Li. This rather unexpected jump in the matter radius could be explained by a neutron halo formed as a consequence of the low binding energy of the last neutron pair in 11 Li [43]. The halo structure would mean that, in the case of 11 Li, the 9 Li core would be surrounded by a dilute tail of neutron matter. The core should be little aIected by the outer neutrons and one would therefore expect the charge distribution to be similar for these two nuclei. An experimental proof of this was performed at ISOLDE, CERN, where combined optical and beta-decay measurements were used to determine the magnetic dipole [44] and electric quadrupole moments [45]. The spin and magnetic moment for 11 Li were found to be I = 3=2 and I = 3:6673 n:m:, respectively. This latter value is close to the single-particle Schmidt value of sp = 3:79 n:m: for a proton in the 0p3=2 state. This identi/es the 11 Li ground state as a spherical 0p3=2 con/guration like in 9 Li and is thus compatible with the halo picture. From the measured quadrupole splittings of the -NMR signal from 9 Li and 11 Li implanted in a non-cubic LiNbO3 crystal one deduced the ratio of the electric quadrupole moments, Q[11 Li]=Q[9 Li] = 1:14(16) [45]. The similarity of these for 9 Li and 11 Li demonstrates that

6

B. Jonson / Physics Reports 389 (2004) 1 – 59

the charge distribution is similar in the two nuclei and in support of the picture of 11 Li as a 9 Li core surrounded by a neutron halo. It was predicted [43,46] that the presence of a halo should be associated with very large Coulomb cross-sections. This was demonstrated experimentally [47] for 11 Li where the Coulomb dissociation cross-section in a lead target was found to be about 80 times larger than the corresponding Coulomb cross-section for 12 C. Another observable that would be aIected by the halo structure is the momentum distribution of fragments from break-up reactions. The spatially large size of the halo would lead one to expect a narrow width of the momentum distribution of the fragments after breakup of a halo nucleus. This was indeed observed for both 9 Li fragments [48,49] and neutrons [50] after 11 Li break-up reactions. The low two-neutron separation energy in halo nuclei makes them possible precursors to the rare decay mode beta-delayed deuteron emission. The /rst observation of this decay mode was made for 6 He in an experiment at CERN’s ISOLDE facility [51]. 3. Production of dripline nuclei Progress in the experimental studies of exotic nuclear species out to the driplines relies heavily on the methods used to produce beams of them. From the section above it is clear that the two main approaches until now has been the on-line isotope (ISOL) and the in-Oight methods. Fig. 2 gives a brief overview of present production methods and future trends. • The oldest method, though still under continuous development, is the ISOL separation. The radioactive nuclides are produced in reactions with beams of protons or heavy ions from a primary accelerator or by neutrons from a reactor or a neutron converter. The target is directly connected to the ion source of an electromagnetic isotope separator. DiIerent combinations of target matrix and ion source have been developed to produce intense beams of long chains of isotopes from more ISOL post-acceleration

ISOL

ISOL Ion Trap

In-Flight Accelerator

Thick Production Target

Ion Source

Ion Source

Accelerator

Isotope Separator

In-Flight post-accelerator In-Flight

Accelerator Thin Production Target

Gas Stop Separator

Fragment Separator Accelerator

Experiments Fig. 2. The basic methods for producing radioactive nuclei and radioactive nuclear beams. The ISOL and the in-Oight methods are the most used at present. The obvious extension of the ISOL method to post-accelerated beams has already been realized at several laboratories, while the new idea of stopping a high-energy fragment beam in a gas cell followed by post-acceleration belongs to the next generation of projects.

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7

than 70 diIerent elements [52–54]. Most of these elements are ionized with surface or plasma ion sources. A recent development, which gives access to elements that are not reachable with these sources, is a laser ion source [55]. The laser system consists of copper vapour lasers, tunable dye lasers and non-linear crystals for doubling or tripling of the frequency in order to obtain two- or three-step ionization. • An interesting possibility is to use the exotic beams from an ISOL facility and feed them into a post-accelerator. This was pioneered at Louvain-la-Neuve [56] where the production target is irradiated with 30 MeV protons and the radioactive nuclei then post-accelerated in a K = 110 cyclotron, which also acts as an isobaric mass analyser. The resulting radioactive beams have energies in the range 0.65 –12 MeV=u and intensities up to 2 × 109 ions=s. The ISAC post-accelerator at TRIUMF uses 500 MeV protons to produce the radioactive isotopes which are then accelerated up to energies of 1:5 MeV=u. An upgrade to 6:5 MeV=u is expected in the coming years [57]. Two new post-accelerators have recently been brought into operation in Europe. The /rst is the CIME cyclotron at GANIL’s SPIRAL facility [58]. It came into operation in spring 2001 and can deliver radioactive beams between 2 and 25 MeV=u. The second is the REX ISOLDE post-accelerator at ISOLDE, which was delivering its /rst beams at the end of 2001 [59]. The post-accelerator at REX ISOLDE is a linac, which at present is delivering radioactive beams with the maximal energy of 2:2 MeV=u. • The second main method for producing radioactive nuclei is the in-Oight method (for a review see [60]). Here an energetic heavy-ion beam is fragmented or /ssioned while passing through a thin target and the reaction products are subsequently transported to a secondary target after mass, charge and momentum selection in a fragment separator. Since the reaction products are generated in Oight, no post-acceleration is required. This method involves no chemical processes and results in short delay times and high-intensity beams. After the pioneering work in Berkeley, mentioned above, a number of in-Oight facilities have been built at diIerent laboratories. Examples in diIerent energy domains are the LISE3 spectrometer [61] at GANIL giving beams of typically 20 –50 MeV=u, RIPS at RIKEN [62] with energies up to 135 MeV=u, the A1900 at MSU [63] up to 150 MeV=u and the FRS at GSI [64] with beam energies ranging up to about 1 GeV=u. • It has also been proposed to construct a hybrid version of the two methods in which the beam from an in-Oight facility is stopped in a gas cell [65,66] and then post-accelerated. It is planned that the proposed Rare Isotope Accelerator (RIA) [67] in the United States will employ this technique. The isotopes are produced in projectile fragmentation or /ssion, followed by in-Oight separation. The fast-moving exotic isotopes are then stopped in a helium gas-cell, ionized, and re-accelerated. The time for the whole process, from target to gas cell and /nally to the postaccelerator, is a matter of milliseconds. This new separation technology, in combination with a powerful new driver and eVcient post-accelerators is expected to give high-quality beams of exotic isotopes of all elements from lithium to uranium. 4. Experiments in the dripline regions The development of techniques for the production of exotic radioactive nuclei and making beams of them, has been of key importance for the development of the /eld. In parallel, there has been a need for novel experimental techniques to be able to perform meaningful experiments under conditions

8

B. Jonson / Physics Reports 389 (2004) 1 – 59

Inelastic Scattering

Elastic Scattering Mass

Knockout Reactions Spins Moments

Transfer Reactions

Beta Decay

Reaction Cross Sections Momentum Distributions

Correlations

Beta-Delayed Particles

Fig. 3. Experimental data used to get information about dripline nuclei. The ground state properties mass, spin and moments as well as beta-decay experiments are mainly obtained at ISOL type facilities. Interaction and reaction cross-sections and elastic scattering give information about radii and density distributions. Fragment momentum distributions at diIerent energies can show the presence of halo states. Angular and energy correlations provide information that is generally used to determine details of the nuclear structure. Knockout reactions have turned out to be a sensitive tool for determination of spectroscopic factors. Unbound nuclei can be studied in knockout reactions, transfer reactions and elastic resonance scattering experiments.

where the very short lived nuclides are produced in extremely low yields. Fig. 3 serves as an illustration of the variety of information that is collected and combined to extract properties of nuclides in the dripline regions. The ground-state properties such as mass, spin and moments are mainly measured with stopped low-energy beams at ISOL facilities [68]. Good mass data are needed for theoretical calculations and the precision mapping of the nuclear mass surface has progressed enormously with the cooling and storing the nuclides in traps and storage rings [68] allowing the determination of the mass from measurements of the cyclotron frequency. The ISOLTRAP spectrometer [69] and the RF Smith spectrometer MISTRAL [70] have both been able to give high-accuracy data for isotopes with half-lives in the millisecond region. The ingenious methodological developments [71] for measurements of ground-state spins, moments and isotope shifts have given such data out to the dripline for many elements. Beta-decay and the associated process of beta-delayed particle emission that occur for the more exotic nuclear species provide information that is diVcult to obtain in reaction studies [13]. The energetic radioactive beams obtained with the in-Oight technique are, however, the main tool for studies of nuclei at the driplines. The measurements of interaction and reaction cross-sections have been an important source of information since these relatively simple experiments can be performed with beam intensities as low as 0:01 ions=s and there are extensive amounts of data available [72] for dripline nuclei. Elastic proton scattering at intermediate energy to obtain information about nuclear matter distributions for short-lived nuclei becomes possible by using the inverse kinematics technique [73] where a radioactive beam is incident on a hydrogen target. Momentum distributions of particles or fragments have been a very rich source of information, especially about halo nuclei. The narrow momentum distribution provides one signature of the presence of a halo, but the data

B. Jonson / Physics Reports 389 (2004) 1 – 59

9

have to be treated with some precaution since the reaction mechanism and /nal-state interactions may inOuence the results considerably [5,9]. Knockout reactions have proven to be an excellent tool for getting spectroscopic information about states in exotic nuclei and the work at MSU on spectroscopic factors is a clear breakthrough [14]. There is considerable interest in unbound nuclear systems close to the driplines, both in themselves and as subsystems of Borromean 2 halo nuclei. Knockout reactions, transfer and stripping with radioactive beams [74] as well as resonance scattering in inverse kinematics [75] have provided a wealth of new results for nuclei beyond the driplines. 5. Halo states in dripline nuclei When approaching the driplines the separation energy of the last nucleon or pair of nucleons decreases gradually and the bound nuclear states come close to the continuum. The combination of the short range of the nuclear force and the low separation energy of the valence nucleons results, in some cases, in considerable tunnelling into the classical forbidden region and a more or less pronounced halo may be formed. As a result the spatial structure of the valence nucleons is very diIerent from the rest of the system and the valence and the core subsystems are to a large extent separable [76]. Therefore, halo nuclei may be viewed as an inert core surrounded by a low density halo of valence nucleon(s). They may therefore be described in few-body or cluster models [2,15]. The formation of halo states is characteristic especially for light nuclei in the dripline regions, although not all of these can form a halo. There is a large sensitivity of the spatial structure and the separation energy close to the threshold. The increase in size, which is due to quantum mechanical tunnelling out from the nuclear volume, will only take place if there are no signi/cant Coulomb or centrifugal barriers present. There are at present many well-established halo states for light neutron-rich dripline nuclei consisting of a core plus one or two neutrons. On the neutron-de/cient side the Coulomb barrier sets a rather stringent limit on proton halos, and the only case which is reasonably well-established is 8 B. There are also more complicated structures such as 8 He, which may be described as a /ve-body + 4n system, but no well-developed multi-nucleon halo states have been observed as yet. As an example of a one-neutron halo nucleus we may select 11 Be, which can be described as a two-body system with one valence neutron outside the 10 Be core. The neutron in the ground state is mainly an intruder s-state with binding energy Sn = 504 keV. There is only one bound excited state at 320 keV with I = 1=2− . The radius of the 10 Be core is 2:30 fm, while the rms distances between the 10 Be core and the valence neutron are 6.64 and 5:41 fm for the ground state and the /rst excited state, respectively (see Fig. 4). The unnormalized external part of the s-state wave function √ is asymptotically (r) = exp(−r), with a reciprocal decay length given by = 2Sn =˝ in terms of the reduced mass and the neutron separation energy Sn . The core-neutron distance is of the order of 1= and, with the low neutron-separation energy, this means that it is so large that many of the properties of 11 Be are determined by the tail of the wave function. The large spatial extension of the wave functions results in very large cross-sections for Coulomb and nuclear dissociation reactions and in particular in narrow widths of the resulting break-up fragments. Several light nuclei with 2

The term Borromean was coined in Ref. [2] to denote a bound three-body system for which no binary subsystem is bound.

10

B. Jonson / Physics Reports 389 (2004) 1 – 59

χ(r) (fm)−1/2

χ 2 (r) (fm ) -1

4.0 3.5 3.0 2.5

0.5

0.0

-0.5 0

10

2.0

20

r, fm

1.5

ps 1.0 0.5

Potential, MeV

0 -10

5

10

15

20

25

r, fm

-30 -50 -70

Fig. 4. The halo properties of the I = 1=2+ ground state and the I = 1=2− state at 320 keV in 11 Be are illustrated by showing the square of the radial wave functions ((r) = r (r)) single-particle components of the 1s1=2 (full-drawn line) and 0p1=2 (hatched line) levels in 11 Be. As comparison the same quantity for the deep bound 0s1=2 (dotted line) orbit is shown. The normalized wave functions for the 0s1=2 , 1s1=2 and 0p1=2 levels in 11 Be, bound by 31 MeV, 0:504 MeV and 0:183 MeV, respectively, are shown in the inset. The potential for the s waves is shown in the lower part of the /gure. The rms distances between the 10 Be core and the valence neutron for the 1s and 0p states are 6.64 and 5:41 fm, respectively [77].

a one-neutron halo in their ground states have been identi/ed, such as 15 C, 19 C and 23 O, and some recent experimental studies of them will be discussed in the following sections. The neutron-dripline nucleus 11 Li has been the subject of considerable interest ever since its ground-state halo structure was revealed. This is because it may be described as a three-body system consisting of a 9 Li core surrounded by two valence neutrons. Since the two subsystems of 11 Li, the di-neutron and 10 Li, are unbound it belongs to the group of halo nuclei referred to as Borromean [2]. Such nuclei are limited to a restricted region of neutron–neutron (Vnn ) and core–neutron (VCn ) potential space (see Fig. 3 in Ref. [5]). For the three-body description of Borromean systems it is convenient to use Jacobi coordinates, de/ned as the normalized relative coordinates between two particles (x) and between their centre-of-mass and the third particle (y). The corresponding hyperspherical coordinates are (; ; x ; y ), where 2 =x2 +y2 is the hyperradius, =arctg(x=y), and x and y are the directions of x and y (see Fig. 5). The exact asymptotic form of the ground-state wave function is known [78] and is a natural generalization of the Yukawa-type wave function in √ the two-body case. Its analytic form is () = exp(−)=5=2 , where = 2mS2n =˝ in terms of the nucleon mass m and the two-neutron separation energy S2n . Besides 11 Li the best example of a Borromean nucleus is 6 He. Recent measurements of the reaction cross section of 83 MeV=u 16 C [79] gave indications of a large (1s1=2 )2 component in the ground state. Based on this it was suggested [79] that a halo state was present in the ground state of the non-Borromean nucleus 16 C. Recent measurements of the longitudinal momentum distributions of 14 C and 15 C fragments after breakup in a beryllium target seem, however, to contradict this conclusion [80].

B. Jonson / Physics Reports 389 (2004) 1 – 59

Vnn

n

x n

11

y

Bound

VCn

VCn

core

Unbound

Fig. 5. Borromean nuclei are bound whereas the binary subsystems are unbound since the two-body potentials, Vnn and VCn , are not strong enough to bind them. One set of Jacobian coordinates used to describe Borromean nuclei in a three-body model is shown to the √ left. The hyperradius is de/ned as = x2 + y2 and the asymptotic wave function is () = exp(−)=5=2 , where = 2mS2n =˝ in terms of the nucleon mass m and the two-neutron separation energy.

At the proton dripline the formation of proton halos becomes possible, although as previously mentioned they are limited to the lightest elements because of the Coulomb barrier and are therefore, in general, less developed. The only known example of a nucleus with a pronounced ground-state proton halo is 8 B. The astrophysically interesting [81] excited state at 495 keV in 17 F has an extended proton wave function with halo properties and the beta-decay of the Borromean nucleus 17 Ne into this state is clearly aIected by this structure [13]. Proton halo states with ‘ = 0 have been predicted in the 1s0d shell [82] starting at light phosphorus isotopes but they are not expected to be very pronounced because of the high Coulomb barrier [83].

6. General conditions for halo occurrence A halo state consists of a veil of dilute nuclear matter that surrounds the core. This is in contrast to the nuclear skin [84], which essentially is a diIerence in the radial extent of the proton and neutron distributions. A loose de/nition of a halo would be that the halo nucleon(s) spend about 50% of the time outside the range of the core potential and thus in the classically forbidden region. The necessary conditions for the formation of a halo have been investigated [85–87] and it was found that, besides the condition of a small binding energy for the valence particle(s), only states with small relative angular momentum may form halo states. Two-body halos can thus only occur for nucleons in s- or p-states, while three-body halos are restricted to states with hyperspherical quantum numbers K = 0 or 1. Proton-halo states have the same restrictions, and in addition to this, the Coulomb barrier suppresses halo formation in elements with Z ¿ 10. Jensen and Riisager [88] give as necessary and suVcient conditions for the occurrence of halo states that the binding energy should be B ¡ 2=A2=3 MeV for both s- and p-states, while p-states have the additional limitation in charge Z ¡ 0:44A4=3 . Dimensionless, universal scaling plots of radii versus binding energies of two- and three-body systems were constructed [89] to characterize and classify halo states. A halo state can be recognized by knowledge about the mean square distance r 2 and the binding energy B between the clusters. For two-body systems, the dimensionless quantities are (r 2 =R2 ) and BR2 =˝2 , where is the reduced mass and the scaling radius R is chosen as the radius of the equivalent square-well potential. For

12

B. Jonson / Physics Reports 389 (2004) 1 – 59 10 2

10 2

10

< ρ 2 > / ρ02

< r 2> /R 2

l =0

l =1

10

l =2 1

1 10

(a)

-2

10

2

µBR /h

-1

_2

10

1

(b)

-2

10

-1

2 _2 mBρ0/h

1

Fig. 6. Scaling plots for two-body and three-body halos from Ref. [89]. In the plot for two-body halos (a) the dashed line corresponds to a pure s-wave Yukawa wave function, the solid lines are results from square-well potentials and the dash-dotted line results for an r −2 potential. For three-body halos (b) the full drawn lines are from calculations with diIerent hypermomenta K. The dashed lines show the E/mov states for a symmetric system (# = 1:01251) and for minimum attraction (# = 0). Filled symbols are from experimental data.

three-body halos the quantities (2 =20 ) and mB20 =˝2 [89] are used, where is the hyperradius [2], 0 a scaling length and m an explicit mass unit. These universal scaling plots are shown in Fig. 6 and they provide a quick evaluation of possible halo candidates. Such scaling plots are also useful for molecular and atomic halo states. A more complete discussion may be found in the recent review by Jensen et al. [90]. A simple geometrical classi/cation scheme has been suggested [91], where the ratio of the valence nucleon to core nucleon radii = rh =rc is used as a gauge for the halo. Here rh = (AC =A)rCn and 2 rc = R2rms (C) + (1=A2 )rCn . For light nuclei close to beta stability we have typically ∼ 1:2–1.25, 6 11 while for He and Li, where the valence neutrons are in a halo state, = 1:8 and 2.17 [92] are obtained, respectively. One may also note that the one-proton halo nucleus 8 B has = 1:75. In a recent paper [93] skins, halos and surface thickness were analysed in self-consistent Skyrme–Hartree–Fock–Bogoliubov and relativistic Hartree–Bogoliubov theories. For nuclei with large neutron excess the analysis, in terms of nucleonic density form factors, makes it possible to de/ne a quantitative measure of the halo size. 7. Determinations of the size of halo states Fig. 7 gives a schematic illustration of the sizes involved in the case of the two-neutron halo nucleus 11 Li. The binding energy for the two halo neutrons is only about 300 keV 3 and they are 3

The most recent S2n -value, obtained with the MISTRAL spectrometer at ISOLDE, is 302(18) keV [94].

B. Jonson / Physics Reports 389 (2004) 1 – 59 208

Pb

13

48

Ca

Li

7 fm

12 fm

9

11

Li

Fig. 7. The size and granularity for the most studied halo nucleus 11 Li. The matter distribution extends far out from the nucleus such that the rms matter radius of 11 Li is as large as 48 Ca, and the radius of the halo neutrons as large as for the outermost neutrons in 208 Pb.

mainly in s- and p-states and can therefore tunnel far out from the core. It turns out that the rms matter radius of 11 Li is similar to the radius of 48 Ca while the two halo neutrons extend to a volume similar in size to 208 Pb. As mentioned, the /rst series of measurements of interaction cross-sections using radioactive beams was performed by Tanihata and coworkers in 1985 [41,42]. The &I were measured with transmission-type experiments. Their classical results for He and Li isotopes were one of the main experimental hints of the existence of halo states in nuclei. The measured interaction cross sections were used to extract rms radii using Glauber-model analysis. This type of experiment has been continued at the Fragment Separator (FRS) at GSI and there exists an extensive quantity of measured interaction and reaction cross-sections for isotopes ranging from 3 He to 32 Mg [95]. The measured cross sections have been used to deduce rms matter radii by a Glauber-model analysis in the optical limit [72]. Fig. 8 shows the systematics of deduced radii. The theoretical method, which assumes static density distributions, has some problems for the loosely bound halo systems. For such nuclei the granular structure of the nucleus, with a compact core and widely dispersed halo neutrons, has to be taken into account [97,98]. In such a treatment the calculated cross sections are reduced considerably, giving increased values for the rms radii. For 11 Li, for example, the value of 3:12 fm (Fig. 8) is adjusted up to 3:55 fm. Some examples of cross-sections and deduced rms radii are given in Table 1. Proton elastic scattering data for dripline nuclei has been obtained in experiments using the so-called inverse kinematics method where a radioactive beam of about 700 MeV=u is directed towards a proton target. Data from such experiments have been obtained at GSI with the hydrogen-/lled IKAR multiple ionization chamber, which served both as target and as recoil-proton detector [99]. From the diIerential scattering cross-sections at small momentum transfer both the overall size and the shape of the radial nuclear matter distribution were obtained [73,99–102] for isotopes of He and Li. Fig. 9 shows the data for Li isotopes. For 11 Li the extracted radius is 3:62(19) fm which is close to the value obtained in the reanalysis of the interaction cross-section data [97]. The results for He and Li are given in Table 1. Integral measurements like the total reaction cross-sections and the elastic scattering cross-sections, measured only in a small momentum transfer region, are only sensitive to the overall size of the system. In order to explore the single particle and collective structures continuum excitations play an important rˆole. The three-body breakup 6 He → 4 He + n + n with a 240 MeV=u secondary 6 He beam

B. Jonson / Physics Reports 389 (2004) 1 – 59

1/2

(fm)

14

Charge radii (stable nuclei)

A Fig. 8. Nuclear matter radii for light isotopes obtained in Glauber-type analysis of interaction cross-section and reaction cross-section data. The smooth solid line represents charge radii obtained in electron scattering experiments on stable isotopes. From [96]. Table 1 Interaction cross-sections and size parameters for selected dripline nuclei 4

&I b rrms c rrms d rrms e rc−nn(n) f rms g rnn

Hea

503(5) 1.54(4) — — — —

6

He

722(5) 2.48(3) 2.71(4) 2.30(7) [101] 3.36(39) [103] 5.9(1.2)

8

He

8

B

749(6) 798(6) 2.53(3) 2.38(4) — 2.50(4) 2.45(7) [101] — — — — —

11

Li

1060(10) 3.12(16) 3.55(10) 3.62(19) [102] — 6.6(1.5)

10

Bea

813(10) 2.30(2) — — — —

11

Be

942(8) 2.73(5) 2.90(5) — — 5.4(1.0)

14

Be

1109(69) 3.16(38) 3.20(30) — 5.7(3) [107] —

19

C

1231(28) 3.13(7) — — — —

a

Included for comparison. Beam energy 790 MeV=u, carbon target, from compilation in Ref. [72]. c Glauber model analysis in the optical limit [72]. d Analysis in a few-body Glauber model [97,98]. e From elastic proton scattering in inverse kinematics. f From E1 sumrules. g From intensity interferometry [109]. b

was studied at the ALADIN-LAND set-up at GSI [103]. The excitation energy spectrum obtained with a lead target, shown in Fig. 10, was used to deduce the E1-strength distribution. There was good agreement between data, theoretical E1 strength [104] and the sum rules in the energy interval up to 10 MeV excitation energy. Then the non-energy-weighted (NEW) cluster sum rule may be

B. Jonson / Physics Reports 389 (2004) 1 – 59

15

10 7 ×1000 11

p

dσ/dt, mb/ (GeV/c)2

10 6

Li, E = 697 MeV

×100 9

p Li, E = 703 MeV

10 5 ×10

p 8 Li, E = 698 MeV

10 4

6

10 3

p Li, E = 697 MeV

10 2 0.01

0.02

0.03

0.04

0.05

-t, (GeV/c)2

Fig. 9. Absolute diIerential cross-sections d&=dt versus the four momentum transfer squared (−t) for proton elastic scattering on 6;8;9;11 Li in inverse kinematics. The curves through the data points are the results of /ts performed with Glauber multiple scattering theory. The rms radius for 11 Li was obtained as 3:62(19) fm [102].

dB(E1)/dE* (e2fm2 /MeV)

dσ/dE* (mb/MeV)

150

100

50

0 1

(a)

2

3

4

E* (MeV)

5

6

(b)

E* - Ethr (MeV)

Fig. 10. (a) DiIerential cross-section d&=dE ∗ as a function of the excitation energy E ∗ of 6 He deduced from the invariant mass in the +n+n decay channel obtained with a lead target [103]. The curves are the calculated diIerential cross-section from Ref. [104] (dotted) and the result after convolution with the experimental response. (b) The experimentally derived E1-strength distribution is shown as a line and the broad shaded band corresponds to the errors. This distribution was obtained by starting with a trial E1 distribution from which the cross-section was calculated in a semi-classical approximation, convoluted with the detector response. The E1-distribution was then modi/ed in an iterative procedure until the experimental data were reproduced. The dotted and dashed curves are from calculations in Refs. [106] and [104], respectively.

used to get information about the geometry of the ground-state wave function [105]: 3 2 2 2 3 2 2 Nh 2 NEW Z e rc = Z e SClus = r ; 4 c 4 c Ac h

(1)

mass of the whole nucleus where rc and rh describe the distance from the centre of to that of the core and halo, respectively. The deduced rms values were rc2 = 1:12(13) fm and rh2 = 2:24(26) fm.

B. Jonson / Physics Reports 389 (2004) 1 – 59

14

60

4

Be

10

3 2

C

dN/dq [counts]

16

2 4 6 8

40

5 20 0

0

50

100

150

0

50

100

150

0

q [MeV/c] Fig. 11. The panel to the left shows the measured two-particle distribution (/lled triangles with error bars) and the successively reconstructed denominators in Eq. (2) (dotted, dashed and solid lines for i=1; 2 and 8 iterations, respectively). The right-hand panel shows the correlation function C for 14 Be. The line shows a /t to the function assuming a Gaussian source and the inset shows the evolution of the source variance with the number of iterations. From this analysis a value rms of rnn = 5:4(1:0) fm is extracted [109].

The distance between the -particle and the centre of mass of the two valence neutrons is then r−2n = 3:36(39) fm. A similar analysis was performed for the one-neutron halo nucleus 19 C. In a Coulomb dissociation experiment of 67 MeV=u 19 C into 18 C+n a large B(E1) strength of 0:71(7) e2 fm2 was observed at low energies [107]. From the E1 sum rule strength for a decoupled neutron [108] this correspond to an rc−n distance of 5:5(3) fm and rms radius for 19 C of 3:0(1) fm. Information about the spatial con/gurations of the halo neutrons in Borromean nuclei may be explored by using the technique of intensity interferometry [109]. This is an interesting new idea based on the pioneering work in stellar interferometry by Hanbury-Brown and Twiss [110], which has been extended to measurements of source sizes in high-energy collisions [111–113]. A two-particle correlation function de/ned as the ratio between the measured two-particle distribution and the product of the independent single-particle distributions C(p1 ; p2 ) =

d 2 n=dp1 dp2 (dn=dp1 )(dn=dp2 )

(2)

is used to describe the inOuence of /nal-state interaction and quantum-statistical symmetry on two identical particles with momenta p1 and p2 . The problem with using this method for two-neutron halo systems is that both neutrons are liberated together in the dissociation process so that only the two-particle distribution can be obtained experimentally. MarquXes et al. [109] solved this problem by generating the single-particle distributions in the denominator using an event-mixing technique, which washes out the correlations in the data set. Fig. 11 shows, as an example, the application of the iterative method to the measured 12 Be + n + n dissociation events from 14 Be. The deduced n–n rms distances are rnn = 5:9 ± 1:2, 6:6 ± 1:5 and 5:4 ± 1:0 fm for the three Borromean nuclei 6 He, 11 Li 14 and Be, respectively. These results are in agreement with those predicted from three-body models [2,114,115]. Table 1 summarizes some of the size data obtained with techniques described above.

B. Jonson / Physics Reports 389 (2004) 1 – 59

17

8. Studies of halo states in reaction experiments The main body of information about halo states comes from reaction experiments performed with radioactive beams. The /rst breakup experiments revealed narrow momentum widths, which were interpreted as corresponding to the wide spatial size of the halo ground state. As already mentioned the interpretation is, however, not entirely straightforward and the reaction mechanism, /nal-state and the experimental /lter, must also be included in the analysis before a meaningful comparison with theory can be made. Most experiments are compared to theoretical calculations in order to extract the physics, but techniques have also been developed [116] that allow model-independent characterization e.g. of the shape of the momentum distribution to be made. These techniques are based on robust and descriptive statistics and have been applied to break up from 11 Li and 11 Be measured at GANIL and it was found that the neutron momentum distributions are broadened and shifted as a function of angle and also that the line shape is changed. The experimental trend at present is to set up complete kinematics experiments where all particles and fragments are measured and where also gamma rays from excited fragments may be detected. An example of a typical set-up used for investigations of breakup reactions of dripline nuclei is given in Fig. 12. In the following some recent reaction experiments on two- and three-body halos are discussed. 8.1. Two-body halos The ground state of 11 Be is an intruder s-state and the theoretical understanding of this parity inversion in the ground state needs in most models a contribution from a coupling between the CHARGED PARTICLES

NEUTRONS DIPOLE

TARGET DETECTOR (γ, p, n)

25

m

IC OT EX MS A BE

Fig. 12. Experimental arrangement for complete kinematics experiments on dripline nuclei. The incoming beam as well as the fragments emerging from reactions in the target are identi/ed by position, time-of-Oight and energy-loss measurements. The charged particles from the reactions are bent in a dipole magnet and the neutrons are detected in forward direction. The target is surrounded by gamma, proton and neutron detectors.

B. Jonson / Physics Reports 389 (2004) 1 – 59

1-→ 2+

2-→ 2+ 2 →0 +

+

Counts / (40 keV)

10 3

2

-

6.26

1

-

5.96

2

+

3.37

0

+

0.012

0.011

0

0.010

(MeV) -

+

1 → 0

10 2

β-asymmetry

18

0.009

0.008

0.007 7.82

10

7.83

7.84

7.85

7.86

7.87

7.88

Frequency [MHz] 2000

4000

6000

Energy (keV)

Fig. 13. Left: Doppler-corrected gamma spectrum measured in coincidence with 10 Be fragments after one-neutron knock out reactions from 60 MeV=u 11 Be in a beryllium target (From [119]). The inset shows the 10 Be level scheme. Right: The -NMR signals from 11 Be in a beryllium host crystal. The measurements gave a Lamour frequency of 7:8508(6) MHz, yielding a magnetic moment of −1:6816(8) N for 11 Be [121].

/rst excited 2+ state at 3:34 MeV in the quadrupole-deformed 10 Be core. In order to get an experimental determination of the relative weights of its [10 Be(0+ ) ⊗ 1s1=2 ]1=2+ and [10 Be(2+ ) ⊗ 0d5=2 ]1=2+ components, Fortier et al. [117,118] employed a radioactive beam of 35:3 MeV=u 11 Be from the SISSI device at GANIL and studied the p(11 Be; 10 Be)d reaction. An analysis of the 10 Be nuclei in the energy-loss spectrometer SPEG placed at 0◦ gave diIerential cross-sections, which were compared to DWBA calculations with bound-state form factors from coupled-channel calculations in a particle-vibration coupling model. The result was that the calculated cross-sections could reproduce the experimental data with 16% core excitation admixture in the 11 Be ground-state wave function. A new technique based on in-Oight separated beams from fragmentation reactions, where the projectile residues from single-nucleon removal are observed in inverse kinematics with a high-resolution spectrograph used in the energy-loss mode and identi/ed by their gamma decay, has been developed [14] at the NSCL at MSU. This technique was employed for the 9 Be(11 Be; 10 Be + )X reaction [119] with a 60 MeV=u 11 Be beam. The Doppler-corrected spectrum of gamma rays detected in an array of 38 cylindrical NaI(Tl) detectors [120] from this experiment is shown in Fig. 13. The spectrum shows clearly a peak from the 2+ → 0+ transition. From this the core excited admixture was determined to be 18%, which is in close agreement with the /ndings in Refs. [117,118]. The gamma spectrum also reveals contributions from the 1− and 2− levels at 5.96 and 6:26 MeV, respectively. The population of these states originates in knockout reactions from the core while the halo neutron remains at the fragment. Of great relevance for the 11 Be ground-state structure is the recent measurement of its magnetic moment [121]. The experiment was performed at the ISOLDE facility at CERN where the 11 Be isotopes were produced by fragmentation of uranium in a hot UC2 target bombarded with 1 GeV protons from the CERN PS-Booster. The experimental method is very sophisticated and worth a few

B. Jonson / Physics Reports 389 (2004) 1 – 59

19

words of description. The produced Be atoms evaporate from the target matrix into a tungsten cavity [122] where two laser beams (234.9 and 297:3 nm) excite the atoms from the 2s21 S0 atomic ground state to an autoionizing state via the 2s2p1 P1 state. The 11 Be+ beam is then optically polarized in-beam by a collinear frequency-doubled CW dye laser beam with a frequency corresponding to an ultraviolet resonance line. The ions are then implanted in a beryllium crystal placed in the centre of an NMR magnet. The /rst-forbidden beta-decay to 11 B of the polarized nuclei is detected with two scintillators and the beta asymmetry measured. The -NMR signal is shown in Fig. 13. From the observed Lamour frequency the magnetic moment (11 Be) = −1:6816(8) N was obtained. This value is in good agreement with the theoretical value predicted by Suzuki et al. [123] if a core polarization admixture of the magnitude given in the two experiments described above [117–119] is assumed. A two-body halo is also found in 19 C. Experiments on the one-neutron breakup in carbon and lead targets have been performed [95,124–127] at diIerent beam energies. The data reveal a large value for the one-neutron removal cross-section (&−1n = 233(51) mb) and a narrow momentum width of the 18 C breakup fragments (69(3) MeV=c FWHM) compared to those for the lighter carbon isotopes. From this it was concluded that 19 C is a one-neutron halo nucleus. The 1s1=2 and the 0d5=2 states are expected to lie close to each other and the ground-state spin has been proposed to be either I =1=2+ with [1s1=2 ⊗ 18 C(0+ )] or I = 5=2+ with [0d5=2 ⊗ 18 C(0+ )]. In both cases there may be a component from the core excited state 18 C(2+ ) coupling either to the 0d5=2 or the 1s1=2 state. The presence of the 0d5=2 component in the wave function would result in a less pronounced halo [128,129] for 19 C. Nakamura et al. [107] studied Coulomb dissociation in a lead target of a 67 MeV=u 19 C beam into the 18 C + n channel and measured the relative energy spectrum, which showed a strong E1 strength of 0:71(7) e2 fm2 at low energies. They also analysed the diIerential cross-section d&=d(*) as a function of the centre-of-mass deOection angle, which led to a value of the neutron separation energy of Sn = 530(130) keV. (One should note that this derived value is much larger than the value Sn = 160(110) keV given by the Atomic Mass Data Center compilation [130], where the separation energy is based on the two most recent mass measurements.) The results given in Ref. [107] were analysed in an adiabatic treatment of the projectile excitation [131] with diIerent con/gurations of the ground-state wave function. The conclusion was that the dominant con/guration would be 18 C(0+ ) ⊗ 1s1=2 . A Glauber model analysis of the 18 C momentum distribution and the one-neutron removal cross-section at 910 MeV=u [126] favours the dominance of a 1s1=2 component in the ground-state wave function [132,133] but the analysis fails to reproduce either the measured cross-section or the width of the momentum distribution. The measured charge-changing cross-sections at 910 MeV on a carbon target [134] were combined with the interaction cross-sections [95] to give the neutron-removal cross-sections (see Fig. 14). The data for the heavier carbon isotopes show an increase starting at 16 C and peaking at 19 C. A narrow width of the 14 C longitudinal momentum distribution in the fragmentation of 15 C [125,135] has shown a one-neutron halo structure of 15 C. The result shown in Fig. 14 seems to contradict this but if the interaction cross-section is scaled up from 790 to 910 MeV=u [136] it is found that the data point for 15 C is underestimated and there is thus no evidence against the 15 C one-neutron halo. In a recent Coulomb breakup experiment of 15 C [137], where the 14 C fragments were studied in coincidence with gamma rays, one found that about 90% of the breakup cross section leaves the 14 C in its ground state. The main ground-state con/guration of 15 C is thus 14 C(0+ ) ⊗ 1s1=2 .

20

B. Jonson / Physics Reports 389 (2004) 1 – 59

2000

dσ/dP [mb/(GeV/c)]

σ−xn (mb)

500 400 300

=

200 100

S n= 0.8 ± 0.3 MeV

1000

0 12

14

16

18

20

A 0 5.75

5.8

5.85

5.9

5.95

=

P (18 C (g.s.)) [GeV/c]

Fig. 14. Left: The neutron removal cross-section, &−xn , for carbon isotopes [134]. Right: Longitudinal momentum distribution of 18 C ground-state residues after one-neutron removal from 19 C on a beryllium target. The calculated momentum distribution for an s-state with Sn = 800 keV is shown as the full drawn line (the dotted lines are calculations with Sn ± 300 keV) while the dash-dotted line corresponds to an assumed d-state [139].

Maddalena et al. [138,139] used knockout reactions to study the 19 C with an approach similar to the one used for 11 Be [119]. The theoretical analysis, extended to be used also for removal of non-halo nucleons from the projectile, used the expression for the theoretical cross-section &th (I ) = C 2 S(I ; nlj)&sp (Sn ; nlj) ; (3) j

where C 2 S is the spectroscopic factor for removal of a nucleon with single-particle quantum numbers (nlj) and &sp is the single-particle removal cross-section. In spite of low beam intensity (order of one 19 C atom per second) they were able to tag the events with gammas to obtain the ground-state momentum distribution. In a black-disc calculation [140] they showed that the momentum distribution is consistent with an s-state halo structure and, together with the large partial cross-section, indicates a ground-state spin of 1=2+ for 19 C. The measured momentum width is also very sensitive to the assumed neutron-separation energy (see Fig. 14) and they found the value Sn =0:8(3) MeV supporting the result presented in Ref. [107]. An analysis of the permitted values in an Sn -C 2 S plane gave a spectroscopic factor in the range 0.5 –1.0 [139]. There have been theoretical discussions about the possible doubly-magic nucleus 28 O [141], however, it turns out that this nucleus lies beyond the dripline. Further, experimental attempts to produce 26 O have failed [142,143] and it was /nally shown that 26 O is unbound [144]. The conclusion is thus that the heaviest particle-stable oxygen isotope is 24 O. 4 4

31

One may here mention that the most neutron-rich isotopes observed for the three elements above oxygen are F [145], 34 Ne and 37 Na [146,147].

(0- ,1- )

(5.8 MeV)

3+

4.5 MeV

2+

3.2 MeV

0+

dσ-1n/dp [mb/(MeV/c)]

B. Jonson / Physics Reports 389 (2004) 1 – 59

21

0.4

0.3

l=2 0.2

l=0

0.1

0

22

O

-120

-40

40

120

plong (MeV/c)

Fig. 15. Left: Proposed level scheme for 22 O based on the information given in Refs. [155–158]. Right: Ground state exclusive momentum distribution for 22 O fragments after one-neutron knock-out from 23 O. The momentum distribution was obtained by subtracting the distribution corresponding to excited states by gating the coincidences with 3:2 MeV gamma rays. The curves are theoretical calculations in an Eikonal model for the knockout process assuming ‘ = 0 and 2. From [155].

In a systematic study of one-neutron removal reactions in neutron-rich psd-shell nuclei [148,149] a narrow momentum distribution of the 22 O fragments from 23 O was observed. This together with a large interaction cross-section [95] suggests a 1s1=2 ground state with a one-neutron halo for 23 O. Measurements of the longitudinal momentum distributions of the one- and two-neutron removal fragments from 72 MeV=u 23 O [150] showed narrow widths for both distributions. It was argued [150,151] that the data gave evidence for a modi/cation of core nucleus 22 O. However, the results may also be explained in an sd shell-model calculation [152], but this is still under debate [153]. In a recent experiment at GSI the one-neutron removal reaction from a 938 MeV=u 23 O beam in a carbon target was studied. The 22 O fragments were detected in coincidence with gamma rays [154,155] and three gamma energies were observed. The transitions corresponding to the 2+ and 3+ levels are consistent with the earlier observed gamma rays from excited states in 22 O [156–158] and interpreted as a 0d5=2 hole coupled to an 1s1=2 neutron. A third gamma ray with E = 2:6 MeV was interpreted as stemming from a state at 5:8 MeV, which could be the 0− or 1− state as proposed in Ref. [152]. The level scheme is shown in Fig. 15. The inclusive momentum distribution showed a width slightly larger than presented in Refs. [148,149]. The longitudinal momentum distribution leaving the 22 O fragment in its ground state (see Fig. 15) was obtained by subtracting the measured distribution in coincidence with 3:2 MeV gamma rays from the inclusive distribution and the FWHM width was found to be 127(20) MeV=c. The data were compared to calculations in an Eikonal model, which favour the angular momentum ‘ = 0 (see Fig. 15), indicating that the 23 O ground state most probable is an 1s1=2 neutron coupled to the 22 O(0+ ) core giving I = 1=2+ . The measured cross section [155] to the ground state, &−1n = 50(12) mb gives a spectroscopic factor of 0.97(23), which is in good agreement with the shell model prediction of 0.797 [152]. The experimentally deduced spectroscopic factors for the excited states are, however, much smaller than the shell-model prediction.

B. Jonson / Physics Reports 389 (2004) 1 – 59

dσ/dpx (mb/MeV.c)

22

5 He

0.8 0.6 0.4 0.2 0.0 -200 -150 -100 -50

0

50

100 150 200

px (MeV/c) Fig. 16. Transverse momentum distribution in one dimension of 5 He after one-neutron knockout from 240 MeV=u 6 He in a carbon target [159]. The dotted line shows the result of a calculation in the transparent limit of the Serber model with a microscopic three-body 6 He ground-state wave function [160]. The solid line is the result of a calculation using an asymptotic single-particle wave function with a cylindrical cut of radius 3:1 fm.

8.2. Three-body halos Of the halo systems, two-neutron halo nuclei have received most attention. This is due to their Borromean character [2] where the three-body system is bound with its pairwise subsystems unbound (see Fig. 5). The most studied nuclei of this type are 6 He, 11 Li and 14 Be. In experiments on three-body halo systems, reactions in which the core and one of the neutrons are detected are referred to as one-neutron knockout (or stripping) reactions. In the sudden approximation, the momentum transfer to the (A-1) system can be neglected in experiments with high beam energy. In the projectile rest frame we have pn1 + pn2 + pC = 0. The momentum of the (A-1) system is then equal to the momentum of the ejected neutron pn1 with opposite sign and will therefore directly reOect the internal neutron momentum distribution. This is, however, only valid to a certain extent. An example is provided by an experiment with a beam of 240 MeV=u 6 He where the 5 He fragments after one-neutron knockout were studied [159]. The Fourier transform of the wave function from a three-body cluster model [160] showed a too large momentum width. The reason for this is that the knockout process inOicts a ‘wound’ on the wave function, and only the remaining part of it should be taken into account in the calculation. This was done by using a cylindrical shape of the cut [179] and in this way the momentum distribution could be reproduced as illustrated in Fig. 16. Further, in the same experiment a large spin alignment of the 5 He fragment was observed [161]. The angular distribution of the pn vector on polar angles in a coordinate system with the z-axis parallel to the direction of the p5 He momentum shows an anisotropy, which can be described with a correlation function W (*n ) = 1 + 1:5 cos2 (*n ) (see Fig. 17). The correlation coeVcient was used [162] to show that the ground-state wave function of 6 He can be described as a (0p3=2 )2 con/guration with a 7% admixture of (0p1=2 )2 . The 6 He case has a very simple structure from the theoretical point of view and the core and neutrons may to a great degree of con/dence be treated as structureless. The ground-state wave function has been described either as a di-neutron coupled to the alpha particle core or two neutrons

B. Jonson / Physics Reports 389 (2004) 1 – 59

dσ/dΩ (mb/sr)

5

15

He

θαn

23

n2

α

10 n1

cos (θαn) Fig. 17. Distributions of the angle *n between the momentum of the recoiling 5 He system (sum of fragment () and neutron (n2 ) momenta) and the relative momentum between fragment and neutron (diIerence of their momenta). The solid lines represent polynomial /ts in cos(*n ) to the data points, including corrections for experimental eIects [161].

on either side of the alpha in a cigar-like con/guration [2]. In an experiment with a gaseous helium target bombarded with 25 MeV=u 6 He beam the 4 He(6 He; 6 He) reaction was studied [163]. The measured diIerential scattering cross-section showed large values in the backward direction. Both DWBA calculations [163] and an analysis in a realistic four-body model [164] showed that the n − n − con/guration has a spectroscopic factor close to unity in 6 He and that the di-neutron component of this three-body con/guration dominates in the 2n transfer reaction. The ground-state structure of 11 Li has been the subject of much discussion. Early theoretical calculations [165] showed that an admixture of approximately equal contributions of (1s1=2 )2 and (0p1=2 )2 components gave the best /t to the experimentally measured narrow momentum distribution of 9 Li recoils after breakup of 11 Li [48,49]. The relative contributions of s- and p-components were determined in a one-neutron knockout experiment from 264 MeV=u 11 Li where the recoil momentum p(10 Li) = p(9 Li + n) was measured in a complete kinematics experiment [166]. The transverse component px is displayed in Fig. 18a. The data were /tted using /rst spherical Hankel functions for the s- and p-neutrons, with the result that the 11 Li ground state contains a 45 ± 10% (1s1=2 )2 component. In the paper by Simon et al. [166] an additional and model-independent proof of the presence of mixed parity states was given in an analysis of the distribution of decay neutrons from 10 Li similar to that performed for 5 He after one-neutron knockout from 6 He [161]. The distribution showed a skew shape which could be /tted with a polynomial of second order in cos(*nf ) as W (*nf ) = 1 − 1:03 cos(*nf ) + 1:41 cos2 (*nf ) ;

(4)

where the linear term shows the presence of both s- and p-states in the ground state wave function of 11 Li. The heaviest bound beryllium isotope 14 Be was identi/ed early on as a halo nucleus based on radii deduced in measurements of interaction and reaction cross-sections [167,168], and the rms radius has been deduced as 3:20(30) fm [97]. In a complete kinematics experiment Labiche et al. [169] studied reactions with 35 MeV=u 14 Be in carbon and lead targets at the LISE3 spectrometer at GANIL. The data were compared with three-body calculations [115] from which it is suggested the ground state wave function in 14 Be contains a large (1s1=2 )2 component. A more quantitative determination

24

B. Jonson / Physics Reports 389 (2004) 1 – 59

15

0p 1/2 0.1

1s 1/2

-100

(a)

13

Li dσ/dΩ (mb/sr)

dσ/dpx (mb/(MeV/c))

10 1

0

px (MeV/c)

100

10

-0.5

(b)

Be

cos (θnf)

0.5

Fig. 18. (a) Transverse momentum distribution of 10 Li. The solid line represents the best /t to the data obtained with a 45% (1s1=2 )2 contribution [166]. (b) Distribution of the decay neutrons from 13 Be formed in 14 Be neutron knockout reactions [170]. The distribution asymmetry is due to a linear term in cos(*nf ) and shows that there are contributions from interfering s- and p-states.

of this component was obtained in reactions with a 14 Be (287 MeV=u) beam on a carbon target [170]. The diIerential cross-section in the one-neutron knockout channel was measured and the correlation function of the angle *nf between the 12 Be fragment and the neutron was found to have an asymmetric shape akin to the one found for 11 Li [170] as shown in Fig. 18b. The skewness of the angular correlation is again a sign of mixed parity in the 14 Be ground state. The angular correlation function in terms of cos(*nf ) shown in Fig. 18b give the leading-order terms W (*nf ) = 1 − 0:37 cos(*nf ) + 0:44 cos2 (*nf ) :

(5)

An expected (d5=2 )2 component in its ground-state wave function would result in terms with power up to 4, but with the limited statistics no signi/cant coeVcients for such terms could be extracted. The analysis in Ref. [170] gives the following components in the ground state of 14 Be: 80% (1s1=2 )2 with the remaining part consisting of equal amounts of (0p1=2 )2 and (0d5=2 )2 components. One should note here that a picture of 14 Be as an inert 12 Be core with two halo neutrons most likely is too simple. Recent theoretical work [171,172] indicates that its structure is much more complicated than the other Borromean two-neutron halo nuclei. More experimental data is clearly needed here. 9. The A = 8 isobar The A = 8 isobar consists of four nuclei that all have some peculiar properties and can all be described with diIerent cluster structures as shown in Fig. 19. The neutron-rich dripline nucleus 8 He is Borromean in the sense that the subsystem 6 He is bound, but still not a ‘classical’ halo nucleus. It is rather an + 4n system. The astrophysically interesting nucleus 8 Li, which is relevant for the formation of 12 C in the primordial nucleosynthesis via its 8 Li(; n)11 B reaction [173], has been described as an ( + t + n) system. The very short-lived 8 Be nucleus shows a low-lying excitation spectrum that can be understood as a rotational band built by two touching -particles. The 8 Li mirror nucleus, 8 B, is the only known case that has been shown to have a proton halo as its ground

B. Jonson / Physics Reports 389 (2004) 1 – 59

α+3 He+p

8

25

B 8

Be

α+α 8

α+t+n

Li 8

He

α+4n

Fig. 19. Illustration of possible cluster structures in the A = 8 isobar. The two dripline nuclei 8 B and 8 He are discussed in the text.

state. The proton halo state in 8 B is not viewed as a 7 Be core with one valence proton but as a three-body ( + 3 He + p) system [91,174–176] similar to the proposed 8 Li structure. The cluster character of the entire A = 8 isobar makes it especially appealing as a test bench for few-body and cluster models. Below some of the experimental results for the two dripline nuclei, 8 He and 8 B, are given. 9.1. The 9ve-body structure of 8 He The heaviest bound He isotope 8 He has a matter radius similar to that of 6 He [72], while its two-neutron separation energy is much larger (S2n (8 He) = 2:139 MeV, S2n (6 He) = 0:973 MeV). The /rst experimental indication of a predominant + 4n structure came from a compilation of cross-section data at 790 MeV=u [177] where it was found that the following relation is valid &−2n (8 He) + &−4n (8 He) = &I (8 He) − &I (4 He) :

(6)

This suggests that 8 He may be described as an -particle surrounded by four valence neutrons. This conclusion is also supported by experiments on the beta-decay of 8 He [178] where it was found that almost half of the GT sum-rule strength is concentrated to a state at 9:3 MeV in 8 Li, which decays with the emission of tritons together with 5 He ( + n). This shows that the ground state of 8 He has a large overlap with ( + t + n) in 8 Li [174]. In a recent study of 227 MeV=u 8 He breaking up in a carbon target [179], the sum of the decay channels &in + &−1n + &−2n was found to be 190(27) mb. Since the diIerence in interaction cross-section between 8 He and 4 He at this energy is 292(8) mb one could attribute the excess to the + 4n breakup, which again shows the /ve-body character of the 8 He ground state. The angular distribution of the decay neutron from 7 He populated in a one-neutron knockout reaction from 8 He could be described with a correlation function W (*nf )=1+0:7 cos2 (*nf ). The anisotropy coeVcient is less than half of the same parameter observed for 5 He. This has been attributed [179] to the more complicated structure for 8 He as compared to 6 He. The spectroscopic factors for the diIerent con/gurations in the ground state of 8 He was in Ref. [179] deduced from the partial cross sections in the one-neutron knockout channels. The strongest contribution was observed from the 6 He(2+ ) + 2n con/guration. Evidence for such a con/guration has also been obtained in backward 8 He(; ) scattering [180] and in a p(8 He; t) reaction [181]. The /ve-body cluster structure of 8 He has been treated theoretically in Refs. [182–184]. The simplest approach where the ground state wave function is described analytically was obtained using

26

B. Jonson / Physics Reports 389 (2004) 1 – 59

α

(a)

α

(b)

α (c)

Fig. 20. Three con/gurations in the 8 He(0+ ) ground-state wave function with maximal probability for the angular part of the spatial correlation function. Con/guration (a) is the most symmetrical one while (b) resembles a 4 n con/guration and (c) a pair of di-neutrons. From [182].

the cluster orbital shell model approximation [182] where the four valence neutrons are occupying the p3=2 shell. The angular part of the calculated correlation function showed pronounced maxima and minima. Some of the most probable spatial con/gurations are shown in Fig. 20. 9.2. The proton halo nucleus 8 B The proton-dripline nucleus 8 B with Sp = 137 keV has been shown to be a one-proton halo nucleus. The /rst experiments performed at the FRS at GSI [185,186] revealed a narrow momentum distribution of the 7 Be fragments (FWHM value of 91 ± 5 MeV=c) and a large one-proton removal cross-section, &−1p = 98 ± 6 mb, in breakup reactions of 1440 MeV=u 8 B in a carbon target. The data were reproduced in a theoretical calculation [91,176] where the 8 B wave function was obtained from an extended three-body model ( + 3 He + p) with explicit inclusion of the binary 7 Be + p channel. The model [176] predicts a sizeable fraction of core excitation of the 429 keV, I = 1=2− state ([7 Be∗ (1=2− ) ⊗ 0p3=2 ]2+ ) in the ground state wave function. The magnitude of the core excited component was determined in experiments with 936 MeV=u 8 B impinging on carbon and lead targets where the de-excitation and the longitudinal momentum distribution of the emerging fragments were detected in coincidence. The result for the carbon target, where the one-proton knockout reactions are mainly of nuclear origin, gave the relative probability of core excitation as 13(3)% [187] (see Fig. 21). With the lead target, Coulomb dissociation dominates and the relative probability for core excitation was measured to be 8.5(2.1)% [188]. The combined data from the two targets were analysed using the 8 B wave function from the extended three-body model ( + 3 He + p) [91,176]. The Eikonal approximation of the Glauber model was used for calculations of the momentum distributions and breakup cross-sections. From the combined data the weight of the [7 Be∗ (1=2− ) ⊗ 0p3=2 ]2+ component was deduced as 13.3(2.2)%. 10. Continuum excitations The dripline nuclei are characterized by very low binding energy and therefore have very few or no bound excited states. In addition the low binding energy gives rise to strong eIects associated with coupling to the continuum. Therefore it is essential to study both resonant and non-resonant continuum transitions and decompose their multipole strength. In experiments studying inelastic

B. Jonson / Physics Reports 389 (2004) 1 – 59 250

429 keV

Counts

200

27

8B 429 keV

1/2

-

M1/E2

150 0 keV

3/2

-

137 keV

Sp

7 Be

100

2+

8B

50

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Eγ (MeV) Fig. 21. Spectrum of gamma rays, detected in an array of NaI detectors, in coincidence with 7 Be fragments after one-proton knockout from 8 B in a carbon target. The peak in the spectrum at 429 keV corresponds to the M1/E2 transition from the /rst excited state in 7 Be. The inset shows the relevant level scheme of the nuclei involved. From [187].

channels one can easily reach continuum states. Continuum excitations are in some sense similar to the discrete-level spectroscopy for nuclei closer to stability and they play an essential rˆole in exploring the single-particle and collective structures of weakly bound nuclei in the dripline regions. Theoretically there are predictions of a considerable low-lying multipole strength for dripline nuclei, which has been con/rmed experimentally. Nuclear and Coulomb inelastic scattering in light and heavy targets into continuum states may be investigated by selecting events in which the dissociated halo is measured in coincidence with the core. The invariant mass spectra for the + n + n system after three-body breakup of 240 MeV=u 6 He in carbon and lead targets were studied by Aumann et al. [103]. The data for the carbon target show a narrow resonance at low energies, corresponding to the known 2+ state in 6 He at 1797 keV, and a broad distribution extending towards higher energies with no pronounced structure. The calculations of the continuum spectrum in a three-body model using the method of hyperspherical harmonics, which reproduces the known 2+ state, also predict a 2+ or a 1+ resonance around E ∗ = 4:3 MeV excitation [189], but no sign of such a structure could be observed experimentally. A certain precaution has to be taken when assigning narrow structures in low-energy continuum spectra since a shake-oI mechanism as proposed in Ref. [190] or a correlated background [191,192] may enhance the intensity in the low-energy part of the spectrum and mimic a real resonance. The proof of the correct assignment in the 6 He case was obtained from the angular distribution of the -particle in the decaying 6 He∗ rest frame, which proved that the observed resonance at about 1:8 MeV corresponds to the 2+ state [193]. For heavy targets electromagnetic dissociation at large impact parameters dominates the breakup and is responsible for the main part of the inelastic cross-section. It was found that about 80% of the nuclear contribution for 6 He (at 240 MeV=u) on a lead target was of electromagnetic origin [103]. The electromagnetic cross-section may have components from various multipolarities but it was shown that the major part is due to dipole strength. The derived E1 strength for 6 He is shown in the right part of Fig. 10. The dashed and dotted curves show the theoretical predictions from

28

B. Jonson / Physics Reports 389 (2004) 1 – 59

8 He

15

dσ/dθ (mb/mrad)

dσ/dEcnn (mb/MeV)

8 He

10

5

0

1

0.1 0

(a)

10

1

2

3

4

Ecnn (MeV)

5

6

0

(b)

10

20

30

θcm (mrad)

40

50

Fig. 22. (a) Excitation energy spectrum reconstructed from the measured momenta of the two neutrons and 6 He after dissociation of 227 MeV=u 8 He in a carbon target. The solid line is the result of a Monte Carlo simulation assuming resonances at 2:9 MeV (1 = 0:3 MeV) and 4:15 MeV (1 = 1:6 MeV). From [179]. (b) DiIerential cross-section for inelastic scattering of 227 MeV=u 8 He in a lead target as a function of the center-of-mass scattering angle. The dotted line represents pure Z‘ = 1 Coulomb excitation while the dashed line is the result of a DWBA calculation under the assumption of both nuclear and Coulomb contributions. The solid line shows the result after correction for resolution and acceptance of the detector system. From [196].

Refs. [104,105] where the diIerence between the theoretical results reOects the diIerent interactions being used. A simple estimate of the E1-strength function at low energies has been given as dB(E1)=dE ∼ E 3 =(E +1:5B)11=2 [194,195], where E is the energy above the three-body threshold and B the two-neutron separation energy. The maximum strength for 6 He is thus expected at 1:75 MeV, which is close to the maximum observed in Fig. 10. Experiments with detection of two neutrons in coincidence with 6 He in the breakup of 8 He correspond to a quasi-sequential process where 8 He is excited as a single system to the continuum followed by its decay. The reaction mechanism might either be diIractive dissociation or inelastic scattering to separate resonance states. The excitation energy spectrum is shown in Fig. 22a for a 227 MeV=u beam interacting with a carbon target. The broad energy distribution was interpreted to contain a 2+ state but is too broad to be due to one single state [179]. The suggestion was that the distribution corresponds to an overlap between a relatively narrow 2+ state at 2:9 MeV excitation energy and a broad peak from a state centered around 4:15 MeV. The diIerential cross-section as a function of centre-of-mass angle gave an indication of an I = 1− assignment of the 4:15 MeV state. This assignment is further supported by the observation of Coulomb-nuclear interference in the diIerential cross section measured with a lead target [196] (see Fig. 22b). Iwata et al. [197] studied dissociation of 8 He in tin and lead targets. The neutron momentum distribution was measured and the dipole strength function was deduced. A comparison with calculations in the COSMA model and with the cluster sum rule favours the well established + 4n structure of the 8 He ground state. The dipole strength distribution for 11 Li was deduced from the measured 9 Li + n + n invariant mass spectrum [198] after breakup in carbon and lead targets. The data were interpreted in terms of excitation of 11 Li prior to dissociation. The excitation spectra showed a concentration of the cross-section at excitation energies below 4 MeV, where there are no other open channels than 9 Li + n + n. For the lead target the extracted multipole strength distribution, d&=dE ∗ , was analysed in terms of the energy-weighted sum rule and it was concluded that the entire strength can be

B. Jonson / Physics Reports 389 (2004) 1 – 59

dσ/dE* (mb/MeV)

11

29

14

Li

Be

16

14

8

7

2

4

6

2

6

10

Excitation Energy E*, MeV

Fig. 23. Excitation energy spectra reconstructed from the measured momenta of the two neutrons and 9 Li and 12 Be after dissociation of 11 Li and 14 Be in a carbon target. The solid lines are the results of a Monte Carlo simulation assuming resonances at 1:3 MeV (1 = 0:8 MeV) and 2:1 MeV (1 = 2:5 MeV) for 11 Li and 2:7 MeV (1 = 1:9 MeV), 5:1 MeV (1 = 2:0 MeV) and 7:1 MeV (1 = 2:2 MeV). From [199]. The thresholds above which there are other open channels than core + 2n are indicated with vertical lines.

interpreted to be of E1 character. Fig. 23 shows recent results for the excitation energy spectra from dissociation of 264 MeV=u 11 Li and 287 MeV=u 14 Be in carbon targets. The vertical lines show the position of the break-up thresholds above which there are other open channels than 9 Li + n + n and 12 Be + n + n, respectively. The results of Monte Carlo simulations with assumptions of resonances in the excitation energy spectra are also shown in the /gure. 11. Molecular structure The halo structure of nuclei like 11 Be, 6 He and 11 Li is well described in few-body and cluster models as a core with one or two valence neutrons forming the halo. The success of such models for describing halo structures is, however, not an isolated phenomenon. It is well known that clustering in light nuclei has an important inOuence on their structure [200]. The well known − -structure of the 8 Be ground state [201] and the rotational band built on it has a moment of inertia which is similar to that of two touching alpha particles. It has been suggested that molecular-like structures could be a more general feature of light nuclei especially at the driplines where an −Xn structure may describe many of the observed data. As an example the ground state of 9 Be could be described as a three-body − n − structure. An extensive systematic study of evidence for molecular structure in light nuclei has been made by von Oertzen [202,203]. Theoretical work where the nuclear densities are calculated using antisymmetrized molecular dynamics description also reveal molecular-like shapes in many light nuclei [204,205]. The one-neutron halo nucleus 11 Be has been studied in the two-neutron transfer reaction 9 Be(13 C; 11 C)11 Be and states up to 25 MeV excitation energy were observed [206]. An energy-spin systematics built on the K = 3=2− con/guration up to spin 19=2− shows a linear dependence on I (I + 1), which could be evidence for a molecular-type structure. The highest spin state with I = 19=2− cannot be formed from the three valence neutrons since single-particle orbits cannot generate such a high spin.

30

B. Jonson / Physics Reports 389 (2004) 1 – 59 40

(6 )

+

(6 )

Counts

20

+

(8 )

30 +

15

(4 )

25 20

10 15 10

5

Excitation Energy (MeV)

+

35

5 0

(a)

10

15

20

25

E x (MeV)

30

0

(b)

20

40

60

I (I+1)

Fig. 24. (a) Excitation energy spectrum for states in 12 Be that break up into two 6 He nuclei. (b) Energy and spin systematics of the 6 He breakup states (dots). The line shows a /t to the data points, giving ˝2 =2J = 150 keV, and the dotted line is the result of an extrapolation assuming a ground state band with a rotational energy of ˝2 =2J = 350 keV. From [207,208].

In a molecular model one could imagine that the two alpha particles of the 8 Be core are apart a large distance and rotate in an ‘ = 4 resonance state, which could generate the last four units of angular momentum [208]. Such an interpretation is consistent with a picture of two alpha particles stabilized by three neutrons as proposed in Ref. [203]. There are, however, some doubt about the spin assignments since several of the observed states also have been seen in the beta decay of 11 Li (see Section 14) and this interpretation of the transfer data awaits con/rmation in detailed spectroscopic measurements. The picture of 12 Be as an + 4n + chain state [203] was tested in experiments on the breakup of 31:5 MeV=u 12 Be into 6 He + 6 He and 8 He + 4 He on carbon and (CH2 )n targets [207]. These measurements indicated that the breakup takes place from rotational states at energies between 10 and 25 MeV. The results from the 6 He+6 He channel are shown in Fig. 24 where the spin assignments have been made from angular correlation measurements. The /gure to the right shows the energy– spin systematics and the linear /t to the data points gives ˝2 =2J = 150 keV, which is close to the value for two touching 6 He nuclei with R = 1:3A−1=3 . As pointed out in Ref. [208] the extrapolated band head is at too high energy as compared with the prediction in Ref. [202] and this may indicate that either there is a weakening of the molecular interaction or that the observed states do not belong the − 4n − classi/cation. More exotic molecular structures have also been searched for, such as the possible triple chains. One candidate for this is 16 C, which is predicted to have an − 2n − − 2n − structure, but experiments searching for evidence of such states have been unsuccessful up to now [209,210]. 12. Magic numbers One issue that has been addressed in several theoretical and experimental papers is the validity of the traditional magic number when one approaches the driplines. The increase in pairing correlations and the shallow single-particle potentials for nuclei close to the driplines may result in a more uniformly spaced spectrum of single particle states [211,212]. Some of the magic numbers vanish

B. Jonson / Physics Reports 389 (2004) 1 – 59

31

and new ones appear. The classical example of the vanishing of magicity is given by the experiments performed at the CERN PS where the /rst excited 2+ state in 22 Mg at an energy of 0:885 MeV [213] was observed. The data thus revealed a sudden onset of deformation for N =20. The vanishing of the N = 20 magic number has also been observed in Coulomb excitation of 32 Mg [214] where the B(E2: 0+ → 2+ ) value was found to be much larger than that expected from a shell closure. The sd orbitals are relatively uniformly spaced so that the N = 14 and 16 subshells should be pronounced [215,216]. The famous parity inversion in the 11 Be ground state shows that the N = 8 magic number is disappearing. The large gap between the 0p shell and the sd shell expected from the shell model disappears and the ground state of 11 Be is an intruder 1s1=2 state. Another indication of the disappearance of the N = 8 shell was given from inelastic proton scattering of 12 Be in inverse kinematics, exciting the /rst excited 2+ state [217]. The deformation length deduced in a coupled-channel analysis showed an indication of a strong quadrupole deformation. Iwasaki et al. [218] also studied inelastic scattering of 12 Be on carbon and lead targets and measured gamma rays in coincidence with 12 Be. They observed a strong gamma transition from a state at 2:68 MeV which was interpreted as an I = 1− state and its low excitation energy is consistent with the disappearance of the N = 8 magic number. A 0+ isomer in 12 Be was recently identi/ed by measuring gamma rays from the in-Oight decay of 12 Be produced by projectile fragmentation of 100 MeV=u 18 O [219]. The two gamma rays from the decay 12m Be → 12 Be∗ (2+ ) → 12 Beg:s: were detected and the angular correlations was used to determine the spin and parity of the isomeric state as I = 0+ . The excitation energy of this 0+ state is 2:24 MeV and such a low energy is a strong evidence against the N = 8 magic number for 12 Be. Another evidence for the non-magicity of N = 8 in 12 Be was obtained in an experiment performed at MSU where the one-neutron knockout reaction (12 Be, 11 Be + ) on a 9 Be target at 78 MeV=u [220] was studied. It was possible to separate the 1s1=2 ground state in 11 Be from the 0p1=2 excited state at 320 keV and to determine the spectroscopic factors, which showed that N = 8 is not a good closed shell in 12 Be. The deduced s and p spectroscopic factors are about equal in magnitude, which is similar to the s=p ratio observed for the N = 8 isotone 11 Li [166]. The systematics of neutron separation energies, Sn , and interaction cross-sections, &I , for neutronrich nuclei in the p and sd regions show evidence for a new neutron magic number with N =16 [221]. A clear break at N = 16 for Sn and a large increase of &I at N = 15 both support the assignment of a new magic number and it was suggested to be due to halo formation. The calculated single-particle energies for A=Z = 3 nuclei, shown in Fig. 25a, were used as evidence for this. In a recent paper [222] magic numbers were discussed in terms of eIective single-particle energies and it was found that in neutron-rich exotic nuclei the magic numbers N = 8, 20 may disappear, while N = 6; 16 may arise. The magicity in light neutron-rich nuclei has also been discussed in terms of the monopole interaction in S = 0 neutron-proton pairs, which can account for the new magic numbers [223]. Fig. 25b shows the eIective 1s1=2 –0d5=2 gap for the N = 16 isotones. A systematic study also including Q-values and energy of the /rst excited state in even–even nuclei showed indications of several new magic numbers [224]. There are at present several proofs of the disappearance of magic numbers and the indications from systematics are that new ones will appear at the driplines. Experiments at the future radioactive beam facilities will certainly address questions of this type so that /nal con/rmations will become available.

32

B. Jonson / Physics Reports 389 (2004) 1 – 59 N for A/Z=3 20

30

40

50

0 Binding energy (MeV)

16

f5/2 p1/2

s -5 p3/2 -10

0

(a)

g d3/2

p1/2 8

d5/2

p3/2

f7/2 20

6

s ∆E (MeV)

10

4

2

28

SDPF USD Kuo

0

50

A

(b)

8

16

12

20

Z

Fig. 25. (a) Spectrum of single-neutron orbitals for A=Z = 3 nuclei from [221]. (b) The eIective 1s1=2 –0d5=2 gap in N = 16 isotones calculated with diIerent shell-model Hamiltonians [222].

13. Beyond the driplines The light dripline nuclei are in many cases just marginally bound and have no particle-bound excited states. An example is 11 Li, which has no excited state below the two-neutron threshold energy. One may, however, reach states or resonances above the particle emission thresholds that are long lived enough to be observed. We saw examples of such continuum states in Section 10. There are, however, certain (Z; N ) combinations just outside the driplines that cannot form a particle bound state but may still be produced and observed as a resonance. If we look at the sequence of He isotopes, as an example, we /nd that 4 He cannot bind one neutron to form 5 He but two, to form 6 He (T1=2 = 806:7 ms). The next bound isotope is then 8 He (T1=2 = 119 ms) since 7 He is unbound. In this section we shall discuss some aspects of these unbound nuclei, their relevance in understanding the structure of some of the bound dripline nuclei and also the physics interest they may have in themselves. 13.1. Unbound He isotopes One-neutron knockout reactions from 240 MeV=u 6 He in a carbon target were studied at GSI [159] and the data revealed a rather narrow peak in the relative energy spectrum. The 5 He (I = 3=2− ) ground-state resonance is comparatively long lived (1=600 keV [225], corresponding to a lifetime of more than 300 fm=c) and therefore decays far away from the reaction zone. This makes the knockout reaction a very eVcient tool to study resonance states in general. The shape of the observed resonance could be reproduced in a 5 He → + n sequential model based a Monte Carlo technique using known 5 He ground-state resonance parameters. An additional proof of the observation of 5 He was obtained in a correlation function analysis with an event mixing method, which agreed with the correlation function obtained in the sequential fragmentation model. The ground state of 7 He is known [225] to have spin and parity I =3=2− , corresponding to a 0p3=2 orbital according to the standard shell-model prediction. The spin–orbit partner of the ground state was only recently observed in an experiment where 7 He was produced in a one-neutron knockout reaction from 8 He [179,226]. The relative energy spectrum of 6 He+n showed a distribution that,

dσ/dEcn (mb/MeV)

B. Jonson / Physics Reports 389 (2004) 1 – 59 α+ n

150

6

He + n

33

150

100

100

50

50

1

0

2

1

2

Ecn (MeV)

Fig. 26. Relative energy spectrum of + n and 6 He + n after neutron knockout from 240 MeV=u 6 He and 227 MeV=u 8 He, respectively. The curve for 5 He shows the result of a Monte Carlo calculation in a sequential fragmentation model with known resonance parameters for 5 He. The dashed curve for 7 He is the result of a calculation with the known 7 He ground-state parameters only, while the solid curve is the result of a calculation assuming an excited state (dotted curve) at 1:0(1) MeV (1 = 0:75(8) MeV) in addition to the ground state (from [179,226]).

(MeV) 3.3 (3) 5/2

1.87 1.8 1.0(1)

1/2 3/2

-

6 He( 2 + )+n

5 He+2n 0.975 4 He+3n

0.43(2) 7 He

0 6 He+n

Fig. 27. Proposed level scheme of 7 He and the relative energy of the 6 He + n, 5 He + 2n and 4 He + 3n thresholds.

in a sequential fragmentation model, could be /tted with the known ground-state resonance parameters and an additional resonance at Er = 1:0(1) MeV (1 = 0:75(8) MeV), which is interpreted as the 0p1=2 spin–orbit partner of the ground state (Fig. 26). A second excited state in 7 He at an energy about 3:3 MeV above the threshold is also known [227–229]. The decay of the 3:3 MeV resonance goes mainly into +3n, which indicates that it is a 5=2− state with probable structure [6 He∗ (2+ )⊗0p1=2 ]5=2− [227]. The level scheme of 7 He is shown in Fig. 27. The observed spin–orbit splitting of the 3=2− –1=2− states of about 0:6 MeV is in fair agreement with recent quantum Monte Carlo calculations [201,230]. A resonance at an energy of 1:2 MeV above the 8 He+n threshold (see for example Ref. [231]) was observed and was interpreted as the 9 He ground state. An assignment of ‘ = 1 for it was made [232] based on the narrow width of the state. It turned, however, out that this resonance is not the ground

34

B. Jonson / Physics Reports 389 (2004) 1 – 59

state of 9 He. In an experiment at MSU the 9 He resonance was produced with a 28 MeV=u 11 Be beam incident on a 9 Be target [233]. With the method of sequential neutron decay spectroscopy at 0◦ [234] measuring the fragment-neutron velocity-diIerence spectrum after the (11 Be; 8 He + n) reaction, a narrow peak was observed and interpreted as an 1s1=2 ground state in 9 He with a scattering length of as 6 − 10 fm. The deduced scattering length can be translated into an excitation energy with the relation E = ˝2 =2ma2s , where m is the reduced mass, resulting in an excitation energy of less than 200 keV. The T = 5=2 states in 9 Li, which are isobaric analogue states of those in 9 He were recently studied in an resonance scattering experiment [235] 8 He + p, with a thick gaseous target in inverse kinematics. These kind of data are of relevance for the 9 He case and three T = 5=2 resonances were identi/ed in 9 Li [235]. None of these could, however, give any additional information concerning the ground state structure of 9 He. The nucleus 10 He would according to the shell model be doubly-magic, but the breakdown of the magic numbers discussed above make this less obvious. There have been several attempts to observe a 10 He resonance and the most feasible reactions would probably be proton or knockout reactions from 11 Li and 14 Be, respectively (see Section 15). A proton knockout from 11 Li in CD2 and carbon targets was done at RIKEN [236]. The measured triple coincidences 8 He + n + n showed evidence for a resonance at 1:2 MeV with a width less than 1:2 MeV. The reaction 10 Be(14 C; 14 O)10 He [237] shows a peak at 1:07(7) MeV, while the p(11 Li; 2p)10 He reaction gives and energy of 1.7(4) [238]. None of these results is, however, conclusive and more data are clearly needed here. 13.2. The N = 7 isotones The ‘ = 0 ground state of 9 He [233] is the third light N = 7 isotone that shows a parity inversion in its ground state. The ground state of 11 Be is well known and was studied early [40] while the interest in 10 Li was mainly triggered by its rˆole as one of the unbound binary subsystems in the Borromean halo nucleus 11 Li [165]. Theoretical models for 11 Li need as input the structure of the low lying states in 10 Li. The current experimental situation is that the ground state of 10 Li is clearly a 1s1=2 neutron coupled to the I = 3=2− ground state of 9 Li to give a 2− and a 1− state, where the 2− state is expected to be the ground state [239]. The /rst excited state is the 1+ state from the coupling of the 0p1=2 neutron to the I = 3=2− ground state of 9 Li. There have been several diIerent experiments, all of which have contributed to the present picture of 10 Li. In experiments at GSI the momentum distribution of neutrons in coincidence with 9 Li fragments in proton- and neutron-removal reactions from 11 Be and 11 Li, respectively, revealed narrow widths that could only be understood if the ground state of 10 Li was an s state [240]. The relative velocity distribution between 9 Li and the neutron [241,242] in the decaying 10 Li produced with an 18 O beam was found to peak at zero relative velocity, which may be interpreted as an ‘ = 0 ground state in 10 Li. A similar velocity spectrum could originate in the possible decay of an excited state in 10 Li to the /rst excited state in 9 Li at 2:7 MeV. This was, however, ruled out in an experiment studying proton stripping of a radioactive 11 Be beam in a beryllium target [243]. It was found that only 7% of the 9 Li residues were in coincidence with the 2:7 MeV gamma ray, showing that the observed low-energy neutrons from the 10 Li decay originated in a direct ‘ = 0 transition to the 9 Li ground state. The low-energy peak in the neutron-fragment velocity distribution from the decay of 10 Li was also observed when it was produced with a 11 Be beam [233]. Since the neutron originates in a dominant ‘=0 state a selection-rule argument allows a /rm ‘=0 assignment of lowest odd-neutron

B. Jonson / Physics Reports 389 (2004) 1 – 59

2

15 O

14 N

13 C

12 B

11 Be

10 Li

35

9 He

Eex. - Sn (MeV)

0 -2 -4 -6 -8

1/2 + 1/2 -

-10 -12

Z

-14 8

7

6

5

4

3

2

Fig. 28. Systematics of the diIerence between the energy of the 1=2+ and 1=2− states and the neutron separation energy for the N = 7 isotones.

state in 10 Li. The scattering length obtained from the diIerent experiments mentioned above gives values around as 6 − 20 fm, corresponding to an excitation energy of less than 50 keV. From these results it is clear that the valence neutron corresponding to the ground state of 10 Li is a 1=2+ intruder state like the ground states of 11 Be and 9 He. The systematics of the 1=2+ and 1=2− levels in the N = 7 isotones is shown in Fig. 28. The level crossing has been interpreted as the result of neutron–proton monopole interaction [244] and there are also contributions from quadrupole deformation and pairing blocking [245]. The relative energy spectrum of 9 Li + n after one-neutron removal [246] reveals a structure that is consistent with a 2− state as the ground state. A more recent result [199] con/rms the low-energy state and also shows a state at about 0:7 MeV, which is interpreted as the [0p1=2 ⊗ 3=2− ]1+ state. The relative-energy spectrum of 9 Li+n is shown in Fig. 29. The low-energy part corresponds to the ‘ = 0 state and the cross-section is /tted with an R-matrix expression √ A E d&=d = ; (7) (E − E )2 + E ∗ (G=2)2 where E ; G and A are parameters of the /t. The roots of the denominator are real for a scattering state and complex for a resonance state. For 10 Li the /t is consistent with a scattering state with a scattering length of as ¿ − 40 fm, which is in agreement with the value given above. In addition to the low-energy state there is a resonance at 0:68(10) MeV (1 = 0:87 MeV) in fair agreement with the energy of 0:54 MeV as given in Ref. [242]. An interesting approach to study 10 Li is via the reaction 9 Li(d; p)10 Li in inverse kinematics with a 9 Li beam. New data for this reaction has been obtained at MSU with a 20 MeV=u 9 Li beam [247] and at REX-ISOLDE with 2:3 MeV=u 9 Li beam [248]. The MSU data may be /tted with either one resonance at about Sn = −0:35 MeV or with two resonances with energies similar to the GSI result [199]. The experiment at REX-ISOLDE seems to be of slightly better resolution and with higher statistics though it is limited to the region around the ground state of 10 Li with a cutoI at ∼ 0:5 MeV above the 9 Li + n threshold.

36

B. Jonson / Physics Reports 389 (2004) 1 – 59

dσ/dE (mb/MeV)

10

Li

100

55%

50

45% 0

1

2

Relative Energy E(9Li+n) [MeV]

Fig. 29. Relative energy spectrum between 9 Li and a decay neutron after the one-neutron knockout reaction of a 264 MeV=u 11 Li beam. The solid line represents an R-matrix /t to the data points. For the ground state the neutrons are assumed to be in an ‘ = 0 (dotted) motion relative to the core. The dashed line shows the ‘ = 1 contribution with a resonance energy of 0:68(10) MeV. The experimental resolution and acceptance are included in the /t functions. From [199].

13.3. The

13

Be case

Some early work attempting to observe the ground state of 13 Be failed to observe any low-lying structure [249,250]. However, theory predicts a low-lying ‘ = 0 state to lie close to the neutron threshold [251,252] but it has also been proposed that the ground state of 13 Be would be an I =1=2− state at an energy of about 0:3 MeV [253]. The presence of a low-lying structure close to the neutron threshold was /rst observed in neutron knockout reactions from 287 MeV=u 14 Be [199] and further evidence [254] was given with the sequential-neutron-decay-spectroscopy method. Thoennessen et al. [254] interpret their data as a scattering state with as ¡ − 10 fm. A similar result was obtained in relative velocity and invariant mass analyses after one-neutron knockout from a 35 MeV=u 14 Be beam [255]. The relative energy spectrum between 12 Be and the decay neutron after one-neutron knockout from 14 Be (see Fig. 30) rather favours an s-wave resonance since the /t to the data results in complex roots in Eq. (7) [256]. A similar conclusion was obtained from the analysis of an experiment using a single proton removal reaction from 41 MeV=u 14 B [257] to produce 13 Be. These data indicate a broad s-wave state as the ground state. 13.4. Beyond the proton dripline The proton dripline for the A=11 isobar is 11 C, while the next member, 11 N, is unbound. There is, however, considerable interest in the low-lying structure of 11 N since it is the mirror nucleus of 11 Be and then in particular if the level sequence is the same with an intruder s-state as the ground state. Theory favours the spin and parity 1=2+ for the ground state [258,259]. The /rst experiment on 11 N [260] used the three-neutron pickup reaction 14 N(3 He; 6 He)11 N and observed a state at 2:24 MeV above the 10 C + p threshold, which was interpreted as the analogue of the /rst excited 1=2− state in 11 Be. LXepin et al. [261] used the reaction 12 C(14 N;15 C)11 N and observed well de/ned resonances

B. Jonson / Physics Reports 389 (2004) 1 – 59

dσ/dE [mb/MeV]

13

37

Be

60 80% 30

10% 0

10% 2

4

12

counts

Relative Energy E( Be+n) [MeV]

100

120

50

60

-0.5

0.5

cos(θnf)

-0.5

0.5

cos(θnf)

Fig. 30. Relative energy spectrum between 12 Be and a decay neutron after the one-neutron knockout reaction of a 287 MeV=u 14 Be beam in a carbon target. The solid line represents an R-matrix /t to the data points. For the ground state the neutrons are assumed to be in an ‘ = 0 (dotted) motion relative to the core. The dashed and dash-dotted lines show the assumed 0p1=2 and 0d5=2 resonances. The experimental resolution and acceptance are included in the /t functions. The two lower panels show the angular distribution of decay neutron from 13 Be. The low-energy part, which is a pure s-state is as expected isotropic, while the region of overlapping s- and p-states shows a skew distribution, characteristic for mixed parity states. From Ref. [256].

in the spectrum of the 15 C ejectiles. They could easily observe the 1=2− and 5=2+ mirror states but failed to detect the ground state. In an experiment [262,263] where 11 N was populated in the reaction 9 Be(12 N; 11 N) followed by proton decay of 11 N, a state around 1:45 MeV above the 10 C + p threshold was identi/ed. This is close to the predicted position of the 1=2+ state [258,259]. A similar result with a resonance energy of 1:63 MeV has also been reported [264]. An elastic resonance scattering technique [75] with an 11 MeV=u radioactive beam of 10 C was used at GANIL to study 11 N [265]. A clear identi/cation of the ground state and the /rst two excited states in 11 N could be made [265,266]. The target consisted of a CH4 gas and when the beam was gradually stopped in the gas it scanned through the 11 N resonances in the reaction 10 C + p. The measured excitation function (see Fig. 31) was then analysed in a potential model and the best /t to the experimental data was obtained with a sequence of partial waves in the order 1s1=2 , 0p1=2 and 0d5=2 . The /rst three excited states in 11 N could then be identi/ed as 1=2+ , 1=2− and 5=2+ with energies (widths) 1:27 MeV (1 = 1:44 MeV), 2:01 MeV (1 = 0:84 MeV) and 3:75 MeV (1 = 0:60 MeV). In a recent experiment using a 14 N(3 He; 6 He)11 N reaction [267] and the energies and widths as well as the angular distributions of the observed levels were measured. The previous spin assignments were con/rmed in a distorted-wave Born approximation analysis. The mirror nucleus of 10 Li, 10 N, was recently identi/ed in a multi-nucleon 10 B(14 N; 14 B)10 N reaction [268]. A small peak could be observed in the 14 B spectrum and it was /tted by an ‘ = 0

38

B. Jonson / Physics Reports 389 (2004) 1 – 59

11

dσ/dΩ (mb/sr)

1000

N

800 600 400 200 0 1

3

total fit s wave p wave d wave

800

dσ/dΩ (mb/sr)

2

4

E r (0d5/2)

600

400 E r (1s1/2) E r (0p1/2)

200

0 0

1

2

3

4

Ecm (MeV) Fig. 31. (a) Experimental excitation function of 11 N where the /lled circles are from experiments at GANIL and the open squares from MSU. The curve shows the /t to the data with a potential model. (b) The /gure shows the decomposition of the full drawn curve in (a) into partial waves s1=2 , p1=2 and d5=2 . From [266].

resonance with Er =2:6(4) MeV. This is an interesting result and more information about the structure of 10 N will most likely become available from the resonance scattering reaction 9 C + p [269]. A resonance scattering experiment with an 8 MeV=u 14 O beam on a C2 H4 target was used to study the unbound nucleus 15 F at MSU [270]. The 1=2+ ground state, unbound with 1:51(11) MeV, and a 5=2+ state at an excitation energy of 1:34(15) keV could be identi/ed in this experiment. 14. Beta-decays at the driplines As we have seen the main body of information about dripline nuclei has over recent years been collected from diIerent types of reaction experiments with radioactive beams. However, nuclear beta-decay is a well-proven probe of nuclear structure as well as of weak interactions and it is

B. Jonson / Physics Reports 389 (2004) 1 – 59

39

clearly also of interest at the driplines. Beta-decays of the exotic nuclei in the dripline regions diIer in many respects from those closer to stability. As we move towards the driplines the continuum nuclear structure becomes more and more important. This is particularly valid for the beta-delayed particle emission processes that in many nuclei close to the dripline will dominate over decays to bound states [271]. For exotic nuclei emitted particles are important experimental observables, but also -rays often remain interesting. Another example is the halo nuclei where the continuum degrees of freedom start to play a rˆole in understanding the decaying state. The halo structure may have a direct inOuence on the nuclear beta-decays. One eIect is directly linked to the large spatial extension of the halo state, which might reduce the overlap with the daughter state, after the beta-decay. Another feature is that the halo might decay more or less independently from the core, which might give speci/c patterns as in the decays of 6;8 He and 9;11 Li [272], or could lead directly into the continuum [51,273]. The patterns in the beta-decay might also be used to establish details of the structure of the halo states. The very high energy available for the beta-decay, together with the low separation energy for nucleons or clusters in the daughter nuclei, give rise to a variety of diIerent beta-delayed particle processes. The Q-value for − -delayed emission of one or several neutrons from a nucleus A Z can be written as Q − xn = Q − − Sxn (A (Z + 1)) = Q − (A−x Z) − Sxn (A Z) :

(8)

The /rst expression involves the Q-value of the mother nucleus and the separation energies of the beta-decay daughter, but as seen it can also be written in terms of the Q-value of a lighter isotope and the separation energies of the mother nucleus. Beta-delayed neutron and multi-neutron emission are important for the predictions of abundances of elements from the r-process [274], but the data for the relevant isotopes in the r-process path are as yet unreachable except for some waiting-point nuclei [275]. The heaviest neutron-dripline nuclei where beta-decays have been studied are 15 B [276], 17 B [277], 18 C [278] and 19 C [279], which have all been identi/ed as beta-delayed neutron emission precursors. Beta-delayed one- and two-neutron emission was recently reported for 19 B, 22 C and 23 N [280]. The Q-value for diIerent delayed particle-emission processes was rewritten in a generalized form in Ref. [13] as QX = c − S ;

(9)

where the parameter c and the ‘separation energy’ S for the diIerent processes are collected in Table 2 (all separation energies refer to the mother nucleus A Z and for delayed -emission a Q-value for the /nal nucleus enters). The c-parameter for beta-delayed deuteron emission is only 3007 keV and since S2n in most nuclei exceed this value means that there are very few nuclei that may show this decay mode. The three Borromean nuclei 6 He, 11 Li and 14 Be all have two-neutron separation energies low enough to give a positive Q-value for this decay mode, and it is fair to say that beta-delayed deuteron emission is typical for Borromean halo nuclei. The /rst experimental observation of beta-delayed deuteron emission was made at ISOLDE in an experiment on 6 He [51]. This experiment was mainly aimed at observing the new decay mode; later an improved set-up [178] gave the energy spectrum of deuterons shown in Fig. 32. Beta transitions

40

B. Jonson / Physics Reports 389 (2004) 1 – 59

Table 2 Parameters of the Eq. (9) for the nucleus A Z in − - and electron-capture-delayed particle emission (from [13]) X

c (keV)

S

− p − d − t − ECn ECd EC3 He EC

782 3007 9264 29860 −782 1442 6936 26731

Sn S2n S3n S4n + Q (A−4 (Z − 1)) Sp S2p S3p S4p + QEC (A−4 (Z − 3))

β

Intensity (decay-1 MeV-1)

6

He

-4

+d

10

6

Li

10

10

-5

-6

0

500 Ed (keV)

1000

Fig. 32. The beta-delayed deuteron spectrum from 6 He. The data points are from [178], the lines are theoretical calculations. Two of these, the solid line [281] and the dashed line [282], assume the decays to proceed directly to continuum states, the dotted line is from an R-matrix calculation (a = 3:0 fm, internal contribution larger than external) [283].

are normally assumed to feed states or resonances in a daughter nucleus, but this assumption may break down for halo nuclei. The decay of 6 He is one such example. This decay seems to take place directly to continuum deuteron states, and is caused by the large spatial extension of the initial halo state and by speci/c correlations, see Refs. [281,282]. In a recent experiment [284] on the beta-decay of 6 He performed at the TISOL facility at TRIUMF, the beta-delayed deuteron spectrum was measured with very good statistics. The deduced branching ratio for the beta-delayed deuteron emission in this experiment is about a factor three smaller than the one given in Ref. [178] while the spectral shape remains similar. The reason for the discrepancy in the branching ratio is not clear since both experiments were performed under almost ideal conditions. Additional data seems to be needed here. The Borromean nucleus 11 Li has a two-neutron separation energy of S2n = 302 keV, which gives a relatively large window for d emission. Theoretical calculations [273,285] could not give a unique

B. Jonson / Physics Reports 389 (2004) 1 – 59

41

prediction of the branching ratio since the d-9 Li interaction is not known, but a branch in the order 10−4 is expected. With the very large Q -value of 20:61 MeV and low separation energies for particles or clusters of particles in 11 Be, the particle spectrum after the 11 Li beta-decay becomes very complicated. It is the beta-delayed triton spectrum in particular that disturbs the observation of deuterons that have much lower energies. This problem was solved [286] at ISOLDE by using the fact that beta-delayed deuterons and tritons give the residual nuclei 9 Li and 8 Li, respectively. From the observation of the beta-decay half-lives typical of these two isotopes at mass position 11 in the isotope separator, a solid proof of the presence of both these decay modes for 11 Li [286] could be given. A beta-delayed triton branch of 8:0(5) × 10−3 has been observed for 8 He [178]. R-matrix calculations [178,287] suggest that the triton emission proceeds via a single narrow 1+ state at 9:3 MeV excitation energy with a reduced Gamow–Teller (GT) transition probability of BGT = 5:18, which is almost half of the GT sum rule strength. This result indicates that the ground state of 8 He has a large overlap with an -particle and a neutron cluster. The con/guration of 8 He is expected to be an -particle surrounded by four neutrons and the neutron con/gurations proposed in Ref. [182] (see Fig. 20) might explain the large beta-delayed triton branch. The unusually large spatial extent and near single-particle structure of halo states will, of course, also be noted in other beta-transitions. Provided reliable calculations of the transition matrix elements can be made, beta-decays can be used for tests of halo structure that add to the information derived from nuclear reactions. An example is given by 11 Li where the ground state has I = 3=2− and the allowed beta-decay proceeds only to one bound state, the p-wave halo state at 320 keV in 11 Be. The branching ratio for this transition is sensitive to the (p1=2 )2 admixture in the 11 Li ground state [288]. There are three experimental determinations of the beta-decay branch to the 320 keV state [289–291] with a weighted average of 7.0(4)%. Shell-model calculations can reproduce this feeding if there is about 50% of the (p1=2 )2 component in the ground-state wave function in agreement with the result obtained from neutron knockout data [166]. The /rst-forbidden beta-decay of the Borromean nucleus 17 Ne to the 495 keV state in 17 F has a branching ratio of 1.59(17)% [292–294]. This value is about two times larger than expected from a comparison with the mirror decay of 17 N into the 871 keV state in 17 O [295]. This very large mirror asymmetry was explained in shell-model calculations in Ref. [292] to be due to the unusually large spatial extent of the 1s1=2 proton orbit. Another suggestion is that the large asymmetry is due to charge-dependent s-occupancy for the initial state [296]. An early study of the beta-decay of 9 Li revealed a rather peculiar pattern [297], with very large BGT strength to a region around the known excited states at 11:18 MeV and 11:8 MeV in 9 Be. The BGT is 5.3(1.0) and it was recently found [298,299] that the main part of this strength populates the 11:8 MeV state. This BGT value is similar to the one found for 8 He [178] and belongs to a general pattern observed for 6;8 He and 9;11 Li with a large BGT strength to states with excitation energies around 2 MeV below the ground state of the mother nucleus [272]. A recent study [300] of the 9 Li mirror nucleus 9 C showed a BGT to a state at an excitation of 12:2 MeV in 9 B, which is the analogue state of the 11:8 MeV state in 9 Be, with BGT = 1:20(15). The results give the largest asymmetry parameter observed for any mirror system, (ft)+ =(ft)− − 1 = 3:4(1:0). Beta-delayed neutrons from 14 Be were studied at MSU [301] and RIKEN [302], and in the RIKEN experiment a neutron peak with energy 287 keV was observed indicating that much of the beta-decay feeds a state at 1:28 MeV in 14 B. It was shown at ISOLDE that close to 100% of the beta-decay

42

B. Jonson / Physics Reports 389 (2004) 1 – 59

3/2 11

-

3/2

18.15

9

Li

9

11.80

Li

Li+d

8 Li+t 10.59

6

3/2

He+ α+n

9

9 Be+2n

0.32 11

Be

1/2 +

8

Be+n

Be

1/2

10

Be+n

0+ 14

-

0+ 12

Be

12

Be+d

11

Be+t

11

B+3n

12 1.28

B

11

B+2n

13

B+n

1+ 12

1+ 2-

14

Be

B

B+n

(MeV)

Fig. 33. Schematic decay schemes of the two Borromean nuclei 11 Li and 14 Be and their core nuclei 9 Li and 12 Be. The decay patterns with strong feeding to low and high energy states are similar for the core nucleus and the halo nucleus decays in the two cases. In the decay of 11 Li a strong GT strength is found to a state at 18:15 MeV with a beta-decay energy of about 2:3 MeV. The 11:8 MeV state in 9 Be is fed by a beta-decay with similar energy and a large GT strength. The decay patterns in 14 Be and 12 Be are both dominated by a strong feeding of a low-lying 1+ state in the daughter nucleus.

feeds this state [303] and that there were no signi/cant multi-neutron emission branches. With double and triple coincidences between rays, delayed neutrons and rays Aoi et al. [304] found a 91(9)% branch to the 1+ state. The schematic level scheme for the 14 Be beta-decay is shown in Fig. 33. A search for beta-delayed charged particles that could be due to the halo decay was recently performed [305] and evidence for beta-delayed tritons was found. The beta-strength distribution could be extracted from the data and much lower strength was found at high energy than expected from shell-model calculations.

B. Jonson / Physics Reports 389 (2004) 1 – 59

43

Nilsson et al. [12] discussed the possible factorization of the halo wave function into core (c) and halo (h) parts. If the beta-decay operator is O one then obtains O |halo = O (|c |h ) = (O |c )|h + |c (O |h ) ;

(10)

where both terms on the right-hand side are needed to have the correct isospin in the /nal state. This equation will only describe the beta-decays of halo nuclei well if the right-hand part is approximately an eigenstate of the daughter nucleus. If we now look at the decay patterns of 9 Li [297] and 11 Li, shown in Fig. 33, we observe some similarities. The strong feeding to the 11:8 MeV excitation region in 9 Li is a dominant feature in the decay. For 11 Li the study of charged particles showed a strong feeding to a state at 18:15 MeV with BGT ¿ 1:6 [306]. If we compare the decay schemes of 12 Be and 14 Be we again observe that the patterns are similar, with a strong feeding to the 1+ ground-state in 12 B in the beta-decay of 12 Be and a similar strong feeding to a 1+ state at 1:8 MeV in 14 B in the beta-decay of 14 Be. Both these decay patterns /t well into the factorization hypothesis [12].

15. Exotica and new possibilities The discussed experiments, at the dripline and beyond, in the preceding sections stand on relatively safe ground. The experiments are, however, in some cases extremely diVcult to perform and to analyse and there are many traps making it possible that an interpretation based on one single experiment might be leading to wrong conclusions. Several of the diIerent approaches for studying the dripline nuclei depicted in Fig. 3 are in most cases a must in order to make the correct conclusions. One may note that it has been a long way to /nd the solution of the ‘10 Li puzzle’ from the early attempts [307] to the present understanding—and it seems still to be much more to say about this challenging unbound system. In this section some selected examples of very interesting new result will be presented. A search for the presence of bound neutron clusters was performed at GANIL using a new production and detection approach by measuring the recoil protons in a liquid scintillator. In an experiment [308] on the break-up of medium-energy beams of the very neutron rich nuclei 11 Li, 14 Be and 15 B this method was tested with the DEMON detector array. The expected distribution of the proton energy to the neutron energy for single and multiple neutron events was compared to experimental data. For single neutrons and an ideal detector this ratio should be less or equal to one. With a real detector the /nite resolution gives a higher maximum and for DEMON the ratio is Ep =En ∼ 1:4. The measured distributions for (11 Li; X + n) and (15 B; X + n) both showed events within such a ratio. The 14 Be data, however, also showed a number of events with a ratio above 1.4 (see Fig. 34). Possible background eIects that could create such events have been discussed very carefully [308] but none of them seems to be able to explain the data. The origin of these events is not yet completely clear and here one needs much more information before a /nal conclusion may be drawn. The possibility of a bound neutron cluster, 4 n, would be an exciting explanation of the events with excess energy. However, a bound tetra-neutron would be very diVcult to explain theoretically [309,310] and it is therefore very important to get more data that may shed additional light to this result.

44

B. Jonson / Physics Reports 389 (2004) 1 – 59 E n =11-18 MeV/nucleon

10

3

10

2

n target

10 Si-CsI

Be

Si

1

PPAC

2 PPAC

1

14

PID [arb. units]

14

Be 0 0

1

2

3

Ep/ En

Fig. 34. Scatter plot and projections of the particle identi/cation parameter PID versus Ep =En for the reaction C(14 Be; X+n). The dotted lines correspond to Ep =En = 1:4 and to the region centered around the 10 Be peak. The six events with Ep =En ¿ 1:4 exhibit the characteristics of possible multi-neutron clusters. The experimental arrangement with the beam tracking detectors, the carbon target and the telescope for detection of charged particles is shown. The neutrons from the break-up of 14 Be were detected at a distance of 3.5 –6:5 m downstreams the target using 90 modules of the DEMON array. From [308].

Another example of an intriguing and very interesting new result is the observation of the ‘super-heavy’ hydrogen isotope 5 H [311]. In principle this isotope would be bound—if there is a bound 4 n—but the present data only show an indication of a resonance. In an experiment at the fragment separator ACCULINA at JINR in Dubna [311] the reaction 1 H(6 He; pp)5 H was studied with a 36 MeV=u 6 He beam. By detecting the two protons emitted in the decay of 2 He from the reaction a peak was observed at an energy 1:7(3) MeV(1 = 1:9(4) MeV) above the t+n+n threshold (see Fig. 35a). The two-neutron transfer reaction t(t; p)5 H gave a resonance at 1:8(1) MeV [312] and the same resonance energy was obtained in the reactions 3 He(t; p)5 H and 2 H(6 He; 3 H)5 H [313]. An experiment where 5 H was produced in reactions induced by stopped − in a 9 Be target, 9 Be( − ; pt)5 H and 9 Be( − ; dd)5 H [314] did not give a narrow low-lying resonance but rather a broad structure and a resonance energy of 5:5(2) MeV with a width of 5:4(5) MeV. A recent analysis of data from one-proton knockout reactions from 240 MeV=u 6 He impinging on a carbon target and reconstruction of invariant mass spectra for the t+n and t+n+n channels was able to reproduce the known 4 H resonance but failed to /nd any narrow resonance in the t+n+n spectrum corresponding to 5 H [315–318]. The t+n+n data show a broad distribution (Fig. 35b) around 3 MeV, which may be described in a three-body microscopic calculation as t+n+n in a I = 1=2+ state [319]. From the measured angular and energy correlations it was shown that the neutrons to a large extent occupy the p-shell [315]. The situation is thus not completely clear in the 5 H case either, and more data are needed. Proton knockout from a beam of 61:3 MeV=u 8 He was recently studied [181] in the reaction p(8 He; pp)7 H and evidence for the existence of a 7 H resonance close to the t+4n threshold was found.

B. Jonson / Physics Reports 389 (2004) 1 – 59

45

60 120

4

40 30

dσ/dEtnn

Counts

50

1

20 10

2

0 -4

1/2+

80

40

II

3

I 0

-2

0

2

4

6

8

0

10

E5H (MeV)

(a)

(b)

1

2

3

4

5

6

7

8

Etnn (MeV)

Fig. 35. (a) The 5 H spectrum measured in coincidence with tritons from the decay of the residual 5 H system. The curves 1–3 are from diIerent assumed sources of background and curve 4 is a combination of the possible background curve and a Breit–Wigner expression folded with the experimental resolution. From [311]. (b) Relative energy spectrum in the t+n+n system after a proton knockout from 6 He. The solid line is the result of a three-body calculation assuming I =1=2+ [319]. The two dashed curves illustrate the shape of a possible background assuming the excitation of a resonance in 6 He close to the t+2n+p threshold (I) and from initial state correlations in the relative energy distribution arising from the 6 He ground-state wave function [192] (II). From [318].

p

θp (deg)

85

knockout

80

75

θp n knockout

70

0

1

θα(deg)

θα

Beam direction

2

Fig. 36. Density of coincidence events between proton recoils and fragments from knockout reactions of 717 MeV=u 6 He in a liquid hydrogen target measured in inverse kinematics. The angles are de/ned relative to the beam direction as shown in the right part of the /gure. The n-knockout region is due to non-correlated -neutron events from the decay of 5 He. The upper (red) region shows correlations between -particles and protons from knockout leading to a di-neutron in the /nal state [320].

As an example of an interesting new possibility to reach exotic states beyond the dripline we may mention selective knockout reactions. It was shown in an experiment with 717 MeV=u 6 He at GSI [320] that and proton knockout reactions from exotic beams at relativistic energies may be an interesting new approach to reach unbound nuclei in the dripline regions (Fig. 36). As an example one may reach 10 He in proton a knockout reaction 11 Li → p + 10 He or in an knockout reaction 14 Be → + 10 He. Another example is 18 B which may be reached in either of the two reactions 19 C → p + 18 B or 22 N → + 18 B. Another interesting new development is the two-proton knockout reaction observed in 9 Be(28 Mg; 26 Ne+ )X [321]. It was shown that this process is a direct reaction. The two-proton knockout process

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for neutron-rich systems and the corresponding two-neutron knockout from proton-rich systems give access to extremely rare systems far away from stability and gives a sensitive probe for excited states and correlations in the many-body nuclear wave function [322]. 16. Outlook In this review, I have given some examples of recent results from a sub/eld of nuclear physics that is remarkably active at present. It is clear that, to a large extent, the discovery of halo states sparked oI this interest. But the halo phenomenon is only one of many new elements that have added to the nuclear paradigm for experiments with radioactive nuclear beams. We are still only in the initial stages of exploring the outer parts of the nuclear landscape, and the next generation of experiments with radioactive nuclear beams will undoubtedly provide new possibilities for research with very good chances of discovering unexpected phenomena. High priority should be given to systematic investigations of nuclei spanning the region from stability towards the edges of the nuclear landscape. At the same time there is a need for strong interaction with theory, so that further steps can be taken in carrying out calculations on as fundamental a level as possible. A few of the burning issues that may be addressed in the years to come can be mentioned. It is clear that continued investigations of the structure of halo states not only need better detection techniques and higher beam intensity, but also access to heavier systems. The continuum structure of neutron-rich nuclei in particular is important for a full understanding of these nuclei. The rˆole of the binary subsystems in Borromean nuclei has to be understood in more detail. In this context, unbound nuclei in the vicinity of the driplines could provide essential information. The exotic unbound few-nucleon systems, like the heavy hydrogen resonances, need more data before /nal conclusions about them can be made The structural changes in the dripline regions that have already been observed in several cases need more investigation and further mapping. The rˆole of fusion reactions to reach far from stability with high cross-sections should also be investigated. Other important subjects that RNB physics should address are an exploration of the position of the neutron dripline for heavier elements, the exploration of exotic nuclei with large isospin, the N = Z line up to 100 Sn, exotic processes as the recent observation of 2p radioactivity [323,324], superheavy elements and studies of nuclei and nuclear reactions of relevance for nuclear astrophysics [325]. On the theory side, better understanding is needed how cluster and few-body models relate to shell-model and mean-/eld theories. Acknowledgements The author would like to thank all those who provided information about their recent work, and in particular, I would like to thank L.V. Chulkov, C. ForssXen, M. Meister, G. Nyman, K. Riisager, G. Schrieder, H. Simon and M.V. Zhukov for suggestions and remarks on the manuscript. References [1] R. Bennett, P. Van Duppen, H. Geissel, K. Heyde, B. Jonson, O. Kester, G.-E. K&orner, W. Mittig, A.C. Mueller, G. M&unzenberg, H.L. Ravn, K. Riisager, G. Schrieder, A. Shotter, J.S. Vaagen, J. Vervier, NuPECC Report on Radioactive Nuclear Beam Facilities, April 2000.

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[310] N.K. Timofeyuk, J. Phys. G 29 (2003) L9. [311] A.A. Korsheninnikov, M.S. Golovkov, I. Tanihata, A.M. Rodin, A.S. Fomichev, S.I. Sidorchuk, S.V. Stepantsov, M.L. Chelnokov, V.A. Gorshkov, D.D. Bogdanov, R. Wolski, G.M. Ter-Akopian, Yu.Ts. Oganessian, W. Mittig, P. Roussel-Chomaz, H. Savajols, E.A. Kuzmin, E.Yu. Nikolsky, A.A. Ogloblin, Phys. Rev. Lett. 87 (2001) 092501. [312] M.S. Golovkov, Yu.Ts. Oganessian, D.D. Bogdanov, A.S. Fomichev, A.M. Rodin, S.I. Sidorchuk, R.S. Slepnev, S.V. Stepantsov, G.M. Ter-Akopian, R. Wolski, V.A. Gorshkov, M.L. Chelnokov, M.G. Itkis, E.M. Kozulin, A.A. Bogatchev, N.A. Kondratiev, I.V. Korzyukov, A.A. Yukhimchuk, V.V. Perevozchikov, Yu.I. Vinogradov, S.K. Grishechkin, A.M. Demin, S.V. Zlatoustovsky, A.V. Kuryakin, S.V. Fil’chagin, R.I. Il’kayev, F. Hanappe, T. Materna, L. Stuttge, A.H. Ninane, A.A. Korsheninnikov, E.Yu. Nikolskii, I. Tanihata, P. Roussel-Chomaz, W. Mittig, N. Alamanos, V. Lapoux, E.C. Pollacco, L. Nalpas, Phys. Lett. B 566 (2003) 70. [313] S.I. Sidorchuk, D.D. Bogdanov, A.S. Fomichev, M.S. Golovkov, Yu.Ts. Oganessian, A.M. Rodin, R.S. Slepnev, S.V. Stepantsov, G.M. Ter-Akopian, R. Wolski, V.A Gorshkov, M.L. Chelnokov, M.G. Itkis, E.M. Kozulin, A.A. Bogatchev, N.A. Kondratiev, I.V. Korzyukov, A.A. Korsheninnikov, E.Yu. Nikolaiski, I. Tanihata, Nucl. Phys. A 719 (2003) 229c. [314] M.G. Gornov, M.N. Ber, Yu.B. Gurov, S.V. Lapushkin, P.V. Morokhov, V.A. Pechkurov, N.O. Poroshin, V.G. Sandukovsky, M.V. Tel’kushev, B.A. Chernyshev, JETP Lett. 77 (2003) 344. [315] M. Meister, L.V. Chulkov, H. Simon, T. Aumann, M.J.G. Borge, Th.W. Elze, H. Emling, H. Geissel, M. Hellstr&om, B. Jonson, J.V. Kratz, R. Kulessa, K. Markenroth, G. M&unzenberg, F. Nickel, T. Nilsson, G. Nyman, V. Pribora, A. Richter, K. Riisager, C. Scheidenberger, G. Schrieder, O. Tengblad, Nucl. Phys. A 723 (2003) 13. [316] M. Meister, Ph.D. Thesis, G&oteborg University, G&oteborg, 2003, ISBN 91-628-5688-X. [317] L.V. Chulkov, in: Proceedings of the 10th International Conference on Nuclear Reaction Mechanisms, Varenna, June, 2003. [318] M. Meister, L.V. Chulkov, H. Simon, T. Aumann, M.J.G. Borge, Th.W. Elze, H. Emling, H. Geissel, M. Hellstr&om, B. Jonson, J.V. Kratz, R. Kulessa, K. Markenroth, G. M&unzenberg, F. Nickel, T. Nilsson, G. Nyman, V. Pribora, A. Richter, K. Riisager, C. Scheidenberger, G. Schrieder, O. Tengblad, Phys. Rev. Lett. 91 (2003) 162504. [319] N.B. Shul’gina, B.V. Danilin, L.V. Grigorenko, M.V. Zhukov, J.M. Bang, Phys. Rev. C 62 (2000) 014312. [320] L.V. Chulkov, Nuclear Physics Spring Meeting, M&unster, 2003. [321] D. Bazin, B.A. Brown, C.M. Campbell, J.A. Church, D.C. Dinca, J. Enders, A. Gade, T. Glasmacher, P.G. Hansen, W.F. Mueller, H. Olliver, B.C. Perry, B.M. Sherrill, J.R. Terry, J.A. Tostevin, Phys. Rev. Lett. 91 (2003) 012501. [322] P.G. Hansen, J.A. Tostevin, Annu. Rev. Nucl. Sci. 53 (2003). [323] J. Giovinazzo, B. Blank, M. Chartier, S. Czajkowski, A. Fleury, M.J. Lopez Jimenez, M.S. PravikoI, J.-C. Thomas, F. de Oliveira Santos, M. Lewitowicz, V. Maslov, M. Stanoiu, R. Grzywacz, M. P&utzner, C. Borcea, B.A. Brown, Phys. Rev. Lett. 89 (2002) 102501. [324] M. Pf&utzner, E. Badura, C. Bingham, B. Blank, M. Chartier, H. Geissel, J. Giovinazzo, L.V. Grigorenko, R. Grzywacz, M. Hellstr&om, Z. Janas, J. Kurcewicz, A.S. Lalleman, C. Mazzocchi, I. Mukha, G. M&unzenberg, C. Plettner, E. Roeckl, K.P. Rykaczewski, K. Schmidt, R.S. Simon, M. Stanoiu, J.-C. Thomas, Eur. Phys. J. A 14 (2002) 279. [325] M. Terasawa, K. Sumiyoshi, T. Kajino, G.J. Mathews, I. Tanihata, Astrophys. J. 562 (2001) 470.

Available online at www.sciencedirect.com

Physics Reports 389 (2004) 61 – 117 www.elsevier.com/locate/physrep

Mesons beyond the naive quark model Claude Amslera;∗ , Nils A. T,ornqvistb b

a Physik-Institut der Universitat Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland Department of Physical Sciences, University of Helsinki, P.O.Box 64, Fin-00014, Helsinki, Finland

Accepted 1 September 2003 editor: J.V. Allaby

Abstract We discuss theoretical predictions for the existence of exotic (non-quark-model) mesons and review prominent experimental candidates. These are especially the f0 (1500) and f0 (1710) mesons for the scalar glueball, fJ (2220) for the tensor glueball, (1410) for the pseudoscalar glueball, f0 (600); f0 (980); a0 (980), the still to be 7rmly established (800) and the f2 (1565) for q2 q82 or two-meson states, and 1 (1400); 1 (1600) for hybrid states. We conclude that some of these states exist, o9er our views and discuss crucial issues that need to be investigated both theoretically and experimentally. c 2003 Elsevier B.V. All rights reserved. PACS: 12.39.Mk; 12.39.Jh; 13.25.Jx; 14.40.Cs Keywords: Quark model; QCD; Scalar mesons; 4-quark states; Deuteronlike states; Gluonium; Hybride

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The light meson spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Four-quark mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Ja9e’s four-quark states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Deuteronlike meson–meson bound states (or deusons) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. One-pion exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Predictions for deuteronlike meson–meson bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Are the scalars below 1 GeV non-qq8 states? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The hadronic widths of the a0 (980) and f0 (980) mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. widths of the a0 (980) and f0 (980) mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Corresponding author. E-mail addresses: [email protected] (C. Amsler), [email protected] (N.A. T,ornqvist). URL: http://unizh.web.cern.ch/unizh/

c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.09.003

62 63 66 66 70 70 72 74 75 76

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3.1.2. Radiative widths of the (1020) to a0 (980) and f0 (980) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. The f0 (980) produced in Ds → 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. A possible interpretation of the nature of a0 (980) and f0 (980) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Is the f0 (600) a non-qq8 state and does the (800) exist? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. The f0 (600) (or ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. The (800) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ 3.4. Observation of a charm-strange state DsJ (2317) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Do we have a complete scalar nonet below 1 GeV? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Theoretical predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Is the f0 (1500) meson the ground state scalar glueball? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Hadronic decay width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. 2 -decay width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Mixing with qq8 states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The tensor glueball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. The pseudoscalar glueball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Hybrid mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Theoretical predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. A 1−+ exotic meson, the 1 (1400) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Another 1−+ exotic meson, the 1 (1600) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Other hybrid candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76 79 80 81 81 84 85 87 88 88 90 92 95 96 97 99 102 102 104 108 110 111 113 113

1. Introduction The nearly 40 years old naive or constituent quark model (NQM), including many generalizations, has been since the pioneering work of Gell-Mann and Zweig [1,2] the basic framework within which most of the hadronic states could be understood, at least qualitatively. The NQM was very successful in describing the observed spectrum, especially for the heavy (c and b) Navour sector. As expected, there are very well established heavy quark–antiquark S-wave vector (3 S1 ) and pseudoscalar (1 S0 ) mesons, as well as P-wave states (3 P2 ; 3 P1 ; 3 P0 and 1 P1 ) which can be identi7ed in the observed spectrum without ambiguities. No clearly superNuous and well established heavy meson state has been reported. The success of the NQM can be understood within QCD from the fact that the bound system is approximately non-relativistic for heavy constituents, and from the fact that the e9ective couplings become suOciently small, so that higher order or non-perturbative e9ects can be neglected as a 7rst approximation. In particular, the scalar cc8 and bb8 states behave as expected for 3 P0 states, whose axial and tensor siblings are the heavy 3 P1; 2 mesons. Their production in radiative transitions from the 23 S1 states and decays into 13 S1 or light hadrons are as expected. Nothing appears to be “exotic” (suggesting a composition di9erent from qq) 8 for these heavy scalar mesons. However, the situation is quite di9erent both theoretically and experimentally for the light meson spectrum. Since the e9ective coupling within QCD becomes large, higher order graphs cannot be

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neglected and there may be non-perturbative e9ects which cannot be described within the NQM nor by tree graphs within a phenomenological e9ective Lagrangian. The quark model has to be unitarized, requiring from the formalism the right analytic properties. Also, crossing symmetry should be at least approximately imposed. In addition, the qq8 system becomes inherently non-relativistic and one should allow the amplitudes and the spectrum to be consistent with the (almost exact) chiral symmetry of QCD for the light u and d quarks. All this clashes with the NQM assumptions, and one should not a priori believe any simple results from the NQM without criticism. From the experimental side one should devote most e9orts to look for states that cannot be described within the NQM, but which are consistent with QCD and con7nement, such as gluonium (composed of only glue), multiquark states (such as qqq 8 q), 8 hybrid states (gqq, 8 composed of qq8 and a constituent gluon), or meson–meson bound states. Such states are expected especially among the light hadrons, where the NQM must eventually break down. For most of the ground state light NQM qq8 nonets one can, with reasonable Navour symmetry breaking and binding assumptions, easily associate well established experimental candidates [3]. This is remarkable indeed. The main exception is the scalar (3 P0 ) nonet, for which there are too many observed candidates. On the other hand, if mesons with “exotic” quantum numbers (that do not couple to qq8 and therefore cannot appear in the qq8 NQM) were observed, this would give clues as to how the NQM should be generalized. The light scalar mesons stand out as singular and their nature has been controversial for over thirty years. There is still no universal agreement as to which states are mainly qq, 8 as to how a glueball would appear among the light scalars, and whether some of the too numerous scalars are multiquark, or meson–meson states, such as K K8 bound states. Since the NQM performs rather well for heavy constituents, the predicted mass spectrum of heavy mesons might be more reliable, hence non-qq8 states are easier to identify. For example, the recently ∗ (2317) [4], the mass of which lies far below predictions, is likely discovered (presumably scalar) DsJ to throw new light also on the light scalar sector. These are fundamental questions of great importance in particle physics. In particular, the scalar mesons have vacuum quantum numbers and are crucial for a full understanding of the symmetry breaking mechanisms in QCD, and presumably also for con7nement. The structure of this review is as follows. In the next section we brieNy review the current status of the qq8 spectrum. For a recent comprehensive review on light quark spectroscopy we refer to Ref. [5]. We then discuss the theoretical predictions for the existence of diquark–antidiquark states and qq8 − qq8 meson–meson bound states (Section 2). In Section 4 we present the predictions for the existence of mesons without quark constituents, the glueballs, and discuss current candidates for the scalar (4.2), the tensor (4.3) and the pseudoscalar (4.4) states. Section 5 is entirely devoted to hybrid mesons which are made of qq8 pairs with vibrating gluons. In the last section we summarize the status of non-qq8 mesons and o9er our views on critical issues that need to be investigated theoretically and experimentally. 1.1. The light meson spectrum In the NQM mesons made of the light quarks u; d; s are classi7ed in qq8 nonets of SU(3)-Navour. Theoretical predictions for their mass spectrum can be found in Ref. [6]. Fig. 1 shows the current

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C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117 2S+1 n

ν ≈ m[GeV] ++ 3 2P2=2

3 P =1++ 2 1

a 2 (1700)

a 1 (1640)

f2 (2010) f2 (1950)

K(1830)

3 -1D2=2

ρ3 (1690) K 2 (1820)

φ 3 (1850)

ρ(1700) K *(1680)

π2 (1670)

2

K 2 (1770) η 2 (1870)

1 D =2-+ 3 P =1++ 1 2 1 1

++ 3 1P2=2

π(1300) K(1460) η(1295) η(1440)

ρ(1450) K *(1410) ω(1420) φ(1680)

3 S =1-2 1

-+ 1S 2 0 =0

π K η η'

ρ(770) K *(892)

ω(1650)

η 2 (1645)

-3 3 S1=1

-+ 1 3S 0 =0

0

3 -1D3 =3

ω 3 (1670)

η(1760)

1

PC

J

K *3 (1780)

K 2 *(1980) π(1800)

L =J

a 2 (1320)

a 1 (1260)

K 2 *(1430)

K 1a

f2 (1270) f2 '(1525)

f1 (1285) f1 (1420)

a 0 (1450)

b1 (1235)

K 0 *(1430)

K 1b

f0 (1370) f0 (1710)

h 1 (1170)

++ 3 1P 0=0

+1 1P1=1

3D =1-1 1

n

h 1 (1380)

ω(782) φ(1020)

L

-3 1 S1 =1

1 -+ 1S 0=0

0

1

2

L

Fig. 1. Tentative quark–antiquark mass spectrum for the three light quarks, according to SU(3). The states are classi7ed according to their total spin J , relative angular momentum L, spin multiplicity 2S + 1 and radial excitation n. The vertical scale gives the radial number =n+L−1, the horizontal scale the orbital excitation L. Each box represents a Navour nonet containing the isovector meson, the two strange isodoublets, and the two isoscalar states. The mass scale is approximate. The shaded assignments are clear and de7nitive.

experimental status of light quark mesons. The ground state (angular momentum L=0) pseudoscalars (J PC = 0−+ ) and vectors (1−− ) are well established, but many of the predicted radial excitations (n ¿ 1) and orbital excitations (L ¿ 0) are still missing. Among the 7rst orbital excitations (L = 1), consisting of the four nonets 0++ ; 1++ ; 2++ ; 1+− , only the tensor (2++ ) nonet is complete and unambiguous. A nearby additional tensor meson, the f2 (1565) [7] could be the 7rst radial excitation (n = 2) of the f2 (1270). However, this state is observed in proton-antiproton annihilation only, which suggests a rather di9erent nature, a four-quark state or a pp 8 (baryonium) state [8]. In the 1++ nonet two states compete for the ss8 assignment,

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the f1 (1420) shown in Fig. 1 and the f1 (1510), which is not well established [9]. There are too many scalar (0++ ) mesons to 7t in the ground state nonet: the f0 (600) (or ); a0 (980); f0 (980) and K0∗ (1430) are well established, but the former three are generally believed not to be qq8 states. This issue will be discussed in detail below, as well as the nature of the three isoscalar states, f0 (1370); f0 (1500) and f0 (1710), for which signi7cant progress was made recently in pp 8 annihilation and in pp central collisions. These states are believed to mix with the ground state scalar glueball. In the 1+− nonet the ss8 meson is not established, although a candidate, h1 (1380), has been reported [3]. The identi7cation of the 7rst radial excited pseudoscalars (n = 2) would be crucial to resolve the now 20 years old controversy on the existence of a pseudoscalar glueball around 1440 MeV. The (1440) meson (previously called E=–) is produced in radiative J= decay, a channel traditionally believed to enhance gluonic excitations. As we shall discuss below, there is now evidence for two pseudoscalar mesons in the 1400 MeV region, one of which could be non-qq. 8 Only overall colour-neutral qq8 con7gurations are allowed by QCD. However, additional colourless states are possible, among them multiquark mesons such as q2 q82 or q3 q83 states. Bag model predictions for 0+ ; 1+ and 2+ q2 q82 states have been presented in Ref. [10]. For q2 q82 mesons one predicts a rich spectrum of isospin 0, 1 and 2 states in the 1–2 GeV region, most of which have not been observed so far. This casts doubt on whether multiquark states really bind or are suOciently narrow to be observed. However, we shall show that the low lying scalar states a0 (980); f0 (980) and f0 (600) are prime candidates for such states. Four-quark states were also searched for, in particular in pp 8 annihilation. The so-called “baryonia” [8] are bound states or resonances of the antiproton–proton system. The short range nucleon–nucleon interaction is repulsive, presumably due to heavy meson t-channel exchanges (e.g. ! exchange). Through G-parity transformation the interaction becomes attractive for various partial waves of the proton–antiproton system, and a rich spectrum of bound states and resonances was predicted [11]. With the possible exception of the f2 (1565) candidate [7], none of these states was actually observed, perhaps because they easily decay into two mesons and are therefore very broad. Also, the predictions for bound states relies on the short range attraction of the nucleon–nucleon interaction which may instead be mediated by one-gluon exchange spin–spin contact interaction, in which case pp 8 and pp are not related by G-parity transformation. A remarkable prediction of QCD is the existence of isoscalar mesons which contain only gluons, the glueballs (to be discussed in Section 4). They are a consequence of the non-abelian structure of QCD which requires that gluons couple to themselves and hence may bind. Lattice gauge calculations predict the existence of the ground state glueball, a scalar, at a mass between 1500 and 1700 MeV [12]. The 7rst excited state is a tensor and has a mass of about 2200 MeV. One expects that glueballs will mix with nearby qq8 states of the same quantum numbers [13]. We shall show below that the f0 (1500) is a prime candidate for the ground state scalar glueball, possibly mixed with nearby qq8 states. Mesons made of qq8 pairs bound by an excited gluon g, the hybrid states, are also predicted [14]. We shall show in the section on hybrids below that the quantum numbers 1−+ do not couple to qq. 8 We shall refer to meson states with these quantum numbers as exotic states. Hence the discovery of mesons with such quantum numbers would prove unambiguously the existence of exotic (non-qq) 8 − + mesons. There are so far two prominent candidates for exotic states with quantum numbers 1 , the 1 (1400) and 1 (1600).

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In contrast with glueballs, exotic hybrids do not mix with qq8 states.

2. Four-quark mesons 2.1. Ja@e’s four-quark states Already back in 1977 Ja9e [10], using the bag model [15] in which con7ned coloured quarks and gluons interact as in perturbative QCD, suggested the existence of a light nonet composed of four quarks with masses below 1 GeV. The essential result from that work was more recently reformulated by Ja9e in Ref. [16] which we follow here. To lowest order, the dominant graph is that of one gluon exchange in Fig. 2. The e9ective Hamiltonian is He9 ˙ − (1) %˜i · %˜j˜ i · ˜ j ; i=j

where ˜ i and %˜i are the 2 × 2 Pauli spin and 3 × 3 Gell–Mann colour operators for the ith quark normalized in the usual way, Tr( k )2 = 2 for all three spin components k, and Tr(%a )2 = 2 for all eight gluons a. This is a simple generalization of the Breit spin–spin interaction to include a similar colour–colour piece. It is also known as the “colour-spin” or “colour-magnetic” interaction of QCD, and was 7rst discussed in the pioneering work of De Rujula et al. [18]. The sum runs over all pairs of quarks in the state. For the light spectrum one takes only into account the light quarks u; d; s for which the masses are small or comparable to the QCD scale, and looks for states with the lowest energy. Of course con7nement, strong renormalization (higher twist), 7nite width e9ects, etc., are assumed not to completely distort the 7rst order results obtained from the e9ective Hamiltonian in Eq. (1). Most of the results follow from 7rst applying Eq. (1) to the simple case of a diquark q1 q2 . The spins can be combined either to a singlet antisymmetric spin S = 0 state |0, or a triplet symmetric spin S = 1 state |1. The eigenvalues of ˜ 1 · ˜ 2 are ˜ 1 · ˜ 2 = 1 = −3

for S = 1 ; for S = 0

Fig. 2. One gluon exchange between two quarks following Ja9e [16,17].

(2)

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67

states. Thus de7ning a spin exchange operator S P12 =

1 1 + ˜ 1 · ˜ 2 ; 2 2

(3)

one has S S P12 |0 = −|0; P12 |1 = +|1 ;

(4)

S tells whether the spin part of the state is antisymmetric or symmetric under the exchange i.e., P12 of the quarks. One can de7ne a similar operator for the Navour and colour SU(3) degrees of freedom. For colour this is C P12 =

1 1 ˜ ˜ + %1 · %2 ; 3 2

(5)

and for Navour one obtains a likewise expression, but the Gell–Mann matrices (here denoted )˜ i ) operate in Navour instead of colour: F P12 =

1 1 ˜ ˜ + )1 · )2 : 3 2

(6)

C F S These operators P12 and P12 for colour and Navour have similar properties as P12 for spin in that their eigenvalues and eigenvectors tell whether the state is symmetric or antisymmetric. Now, coupling two triplets (u; d; s for Navour) one obtains 3×3=9 states, of which 6 (uu; dd; ss; ud + du; us + su; ds + sd) are symmetric and form the six dimensional representation 6F , while three are antisymmetric (ud − du; us − su; ds − sd) and form the three dimensional representation 38F of SU(3)F (see Fig. 3).

3F

3F

6F Fig. 3. Weight diagrams for the fundamental representation of SU(3)F (denoted 3F ), for the antisymmetric diquarks 38F and the symmetric diquarks 6F .

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The same applies to colour SU(3)C with u; d; s replaced by red, blue, green. One then obtains instead of Eqs. (4): F 8 |3F = −|38F ; P12

F P12 |6F = +|6F ;

(7)

C 8 P12 |3C = −|38C ;

C P12 |6C = +|6C :

(8)

Hence, the antisymmetric combinations of the two quarks in a diquark (for Navour: ud − du; 8 s). us − su; ds − sd) behave just as the anti-triplet composed of the anti-quarks (u; 8 d; 8 Now if the two quarks are in a relative S wave the spatial part of wave function is symmetric. The remaining spin-Navour-colour part of the wave function must be antisymmetric for fermions. This leads to a simple relation between the three exchange operators C F S P12 P12 = −1 P12

(9)

C S F P12 P12 = −P12 :

(10)

or Now using Eqs. (3) and (5) one can rewrite the colour-spin interaction (1) as 4 S 2 C S C P12 + P12 + 2P12 − : %˜i · %˜j˜ i · ˜ j = −4P12 He9 ˙ − 3 3

(11)

i=j

Inserting relation (10) the 7rst term can be written in terms of the Navour exchange operator leading to 4 S 2 F C He9 ˙ − (12) + P12 + 2P12 − : %˜i · %˜j˜ i · ˜ j = 4P12 3 3 i=j

Surprisingly, although the original relation does not depend on Navour, Cavour exchange has the largest weight (namely 4) in Eq. (12). Therefore the colour-spin interaction leads to large mass splitting between multiplets of di9erent Navours. It is now easy to evaluate Eq. (12) for the four possible totally antisymmetric systems of a diquark in the combined Navour-colour-spin variables. With obvious notation one gets He9 |qq; 38F ; 38C ; 0 ˙ −8|qq; 38F ; 38C ; 0 ;

(13)

He9 |qq; 38F ; 6C ; 1 ˙ −4=3|qq; 38F ; 6C ; 1 ;

(14)

He9 |qq; 6F ; 38C ; 1 ˙ +8=3|qq; 6F ; 38C ; 1 ;

(15)

He9 |qq; 6F ; 6C ; 0 ˙ +4|qq; 6F ; 6C ; 0 :

(16)

Hence the channel with the strongest attraction is in the con7guration which is separately antisymmetric in all three variables, Navour, colour and spin. On the other hand, attraction or repulsion between the two quarks is weaker for symmetry in two variables and antisymmetry in the third. This singles out the con7guration |qq; 38F ; 38C ; 0, Eq. (13), as the lightest diquark con7guration. It behaves much like an anti-quark under Navour and colour, but is a spin-singlet state. We shall denote this state by the shorthand (qq)38. Thus in a multi-quark environment one expects large binding between two quarks in this particular Navour-colour-spin state.

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This basic observation has several consequences for spectroscopy, which agree with experiment. The most obvious one is that the lightest baryons composed of three quarks are in a Navour octet with spin 1/2. Since baryons are colour singlets each quark must be coupled to a colour 38C diquark. The octet can be obtained by adding a quark to the lightest diquark state |qq; 38F ; 38C ; 0. On the other hand, the baryon decuplet 10F with total spin 3/2 can only be constructed by adding a quark to the heavier |qq; 6F ; 38C ; 1. For qq8 mesons Eq. (1) reduces to the simple Breit spin–spin interaction and leads to pseudoscalar mesons that are lighter than vectors, much like para- and orthopositronium. But Eq. (1) also leads to less obvious predictions such as the absence of light Navour-exotic states, e.g. + + or K + + resonances, or more generally higher Navour multiplets than octets and singlets not excluded by con7nement. For Navour-exotic q2 q82 states the colour-spin interaction Eq. (1) predicts that the colour-spin force is repulsive. To build a colour singlet but Navour exotic q2 q82 state, either the diquark or the anti-diquark (or both) must be a Navour sextet which is less tightly bound than the anti-triplet, see Eqs. (15), (16). Such exotic states should therefore be heavier and broader. However, Navour non-exotic light q2 q82 states can be formed with both the diquark and the anti-diquark in spin singlets, colour triplets and Navour triplets (as in Eq. (13)). These states, (qq)38(q8q) 8 3 would be light, and if they exist could be misinterpreted as or mixed with qq8 states, since they also form an SU(3)F nonet. Ja9e’s suggestion [10] was that the lightest scalar mesons (today named f0 (600) or ; a0 (980); f0 (980), and the uncon7rmed (800)) build up such a nonet. Then the qq8 0++ states would lie higher in the 1.2–1:7 GeV region, as shown in Fig. 1. The most striking prediction of such a (qq)38(q8q) 8 3 model is the inverted mass spectrum shown in Fig. 4. This is simply obtained by letting the number of strange quarks determine the mass splitting. It is then tempting to identify the lightest state, an isospin singlet, with the f0 (600), and the heaviest states which form a isospin triplet and a singlet, by the a0 (980), and f0 (980). Then the mesons with strangeness would lie in between, forming the isospin doublet (800). Although such a (800) was claimed earlier and more recently by the E791 experiment [19], its existence remains controversial [20,21]. This will be discussed in more detail in Section 3.3. Within the heavy meson sector Gelman and Nussinov [22] considered recently the possible 8 3 four-quark state and argued that such a 1+ isosinglet state may exist existence of a (cc)38(u8d) ∗ near the DD8 threshold, possibly mixed with a deuteronlike state of same quantum numbers (see Section 2.2).

Fig. 4. The inverted mass spectrum (left diagram) expected in Ja9e’s four-quark model when the s-quark is heavier than the u and d quarks, compared with a similar qq8 nonet (right diagram).

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C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

One can also raise objections to the four-quark scheme of Ja9e [10]. Obviously, an essentially nonrelativistic zero width model without chiral symmetry is used in a regime where the e9ective couplings to hadronic decay channels are very large. In such a case, mass shifts from e.g. → → ; a0 → K K8 → a0 and Navour related loops should be large and distort any naive bare spectrum [23,24]. Furthermore, crossed channel e9ects can generate scalar bound states between two pseudoscalars [25], and even unitarization can generate new bound states in addition to those introduced, as we discuss in Section 3.3. 2.2. Deuteronlike meson–meson bound states (or deusons) The deuteron and heavier nuclei being in fact multi-quark states, it is natural to ask whether a similar mechanism which binds the deuteron could also bind two mesons and produce four-quark states with given quantum numbers (Fig. 5). This question has been studied surprisingly sparsely in the literature. It is generally mentioned only in passing within general phenomenological models for meson–meson bound states (e.g. Refs. [27,28]). Some special attention to this problem was given in Refs. [29–31]. When assessing whether pion exchange binds two hadrons, the deuteron is certainly the prime reference state. There one knows that the dominant binding energy comes from pion exchange between two colourless qqq clusters—a proton and a neutron (see Refs. [32–34]). For heavy enough constituents, one can follow the approach which was so successful for the deuteron, generalized to meson–meson states. Hence a rather simple nonrelativistic potential model can be used, and one looks for the bound states by solving the Schr,odinger equation. For states similar to the deuteron, i.e. with small binding energies and comparatively heavy constituents, a nonrelativistic treatment should already provide a very good approximation. The results are then easily understood without too many theoretical assumptions. In particular, if the constituent meson mass is assumed to be in7nite and the interaction is attractive, the potential term dominates the kinetic energy term. Then a bound state exists with a mass just below the sum of the two constituent masses. In other words one expects deuteronlike states to exist if pion exchange is attractive and strong enough. 2.2.1. One-pion exchange The broken chiral symmetry of QCD predicts that the pion is singled out as the by far lightest hadron and one knows since long ago that the pion plays a crucial role in the dynamics of hadrons, u u

q

q

d Meson

Proton Pion and meson cloud d d

Neutron

Pion and meson cloud q

q

Meson

u

Fig. 5. The deuteron as neutron–proton bound state (left) and a loosely bound state of two mesons, called a deuson (right).

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71

whenever long distance e9ects are important. In nuclear physics pion exchange was traditionally the 7rst mechanism proposed for understanding nuclear binding. The 7rst step in studying the e9ects of virtual pions is to 7nd the one-pion exchange potential. The modern way of deriving this is from the QCD Lagrangian using chiral perturbation theory (see e.g. Ref. [31]). One can derive an e9ective quark pion interaction g Lint = q(x) 8 5˜.q(x)9- (x) ; (17) F where F is the pion decay constant ≈ 132 MeV and g is an e9ective pion quark–pseudovector coupling constant. In a non-relativistic approximation, and using SU(6) wave functions for the hadrons, one solves the Schr,odinger equation with a potential that depends on the spin-isospin quantum numbers of the constituents and the bound state. Deuteronlike states can be expected when the potential is attractive and strong enough to make a bound state like the deuteron. To describe the coupling between a qq8 pair and the pion (Fig. 6) we follow Ref. [29]. It is useful to introduce the constant V0 =

m3 g2 ≈ 1:3 MeV ; 12 F2

(18)

2 =4 = 0:08 = 25 (g2 =F2 )m2 the numerical value of which is 7xed by the N coupling constant (fN 9 from which g ≈ 0:6). The quantity V0 is a measure for the e9ective potential between the two quarks, when the total spin and the isospin are unity, Sqq = Iqq = 1 (see Fig. 6). This is a good approximation for heavy spectator quarks Q. A good test would be a measurement of the partial width of the D∗ meson to D. One of us [29] predicted

4(D∗+ → D0 + ) =

1 g2 3 p3 p = 2V ; 0 6 F2 m3

(19)

which gave 63:3 keV, in excellent agreement with the recent experimental result of 65 keV [35]. The same formula gives for the K ∗ width 37 MeV, in reasonable agreement with the experimental result, 51 MeV. This agreement is also in accord with heavy quark symmetry, that the pion couples to the light quarks only, and the coupling does not depend on the heavy spectator quark mass. These relations then lead to a one-pion exchange potential between two hadronic constituents which can be written compactly in the form V (r) = − V0 [D · C(r) + S12 (r) ˆ · T (r)] ;

d

Q

(20)

V0 g

g

Q

V0 u Fig. 6. One-pion exchange coupling to qq. 8 The quark Q is assumed to be a spectator.

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Here D is diagonal (often a unit matrix) and S12 is the tensor operator of matrix form which connects di9erent spin-orbitals. The quantity measures the strength of the one-pion exchange potential, as built up by pion exchange between the constituent quarks. For such a quark pair depends on a spin-isospin factor—( 1 · 2 )(.1 · .2 ), where 1 · 2 (and .1 · .2 ) is +1 (for triplet) or −3 (for singlet) depending on the qq spin and isospin. Taking furthermore into account the internal symmetry wave functions of the two hadrons one 7nds for example that is 25/3 for the deuteron and 6 for D∗ D8 ∗ in the I = 0; S = 0 state. The larger the , the stronger is the attraction, and for negative values of the interaction is repulsive. In the equal mass case of the constituents (like D∗ D8 ∗ ) the radial dependence of V (r) is given by the functions e − m r ; (21) C(r) = m r 3 3 T (r) = C(r) 1 + ; (22) + m r (m r)2 where the tensor potential T (r) contains singular r −2 and r −3 terms. These make the latter dominate at small r like an axial dipole–dipole interaction. Therefore this singular behaviour of the tensor potential must be regularized at small distances, which introduces some model dependence. 2.2.2. Predictions for deuteronlike meson–meson bound states Since parity forbids three-pseudoscalar couplings, two pseudoscalar mesons cannot be bound by one-pion exchange. The lightest expected bound states are pseudoscalar (P)-vector (V) states. For V the pion is too light to be a constituent, because the small reduced mass of the V system would give a too large kinetic energy, which cannot be overcome by the potential. Thus the lightest bound states in which pion exchange can play a dominant role are K K8 ∗ systems which lie around 1400 MeV. For Navour exotic two-meson systems (I = 2, double strange, charm or bottom)—such as B∗ B∗ — pion exchange is always either weakly attractive or repulsive ( small or negative). Calculations do not support such bound states to exist from pion exchange alone, and shorter range forces are expected to be repulsive. Should they exist, however (cf. [31]), they would be quite narrow since they would be stable against strong decays. On the other hand, for non-exotic systems such calculations 7nd that deuteronlike meson–meson bound states should exist [29]. The 12 expected states for D∗ D8 ∗ and B∗ B8 ∗ are given in Table 1. In the bottom sector these states are bound by about 50 MeV from one-pion exchange only. In the charm sector binding from pion exchange is weaker but states near threshold could also bind with small contributions from shorter range attraction. The widths are expected to be quite narrow, provided that heavy quark annihilation is not too strong. Such annihilation should be suppressed if the states are, like the deuteron, much larger in size than heavy quark qq8 states. A search for the heavy deuteronlike mesons predicted in Table 1 could be conducted with pp8 annihilation in Night, and possibly with 8 decay into open charm-anticharm states. Light mesons are much harder to bind since the attraction from one-pion exchange is not suOcient to overcome the large kinetic energy of the constituents. This is especially true for the pion itself as a constituent, but also for K K8 ∗ systems for which the potential term is only half as strong as

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Table 1 Predicted heavy isoscalar loosely bound two-meson states, or deusons, with masses in MeV, close to the DD8 ∗ and the D∗ D8 ∗ thresholds, and about 50 MeV below the BB8 ∗ and B∗ B8 ∗ thresholds [29] Composite

J PC

Mass (MeV)

Composite

J PC

Mass (MeV)

DD8 ∗ DD8 ∗

0−+ 1++

≈ 3870 ≈ 3870

BB8 ∗ BB8 ∗

0−+ 1++

≈ 10545 ≈ 10562

D∗ D8 ∗ D∗ D8 ∗ D∗ D8 ∗ D∗ D8 ∗

0++ 0−+ 1+− 2++

≈ 4015 ≈ 4015 ≈ 4015 ≈ 4015

B∗ B8 ∗ B∗ B8 ∗ B∗ B8 ∗ B∗ B8 ∗

0++ 0−+ 1+− 2++

≈ 10582 ≈ 10590 ≈ 10608 ≈ 10602

As discussed in the text, the mass values are obtained from (a rather conservative) one-pion exchange contribution only.

Table 2 Meson–meson channels in the light meson sector for which one-pion exchange is attractive. The nearby known mesons, some of which could be deuteron like (or mixed with deuteron-like states) are listed in the last column Composite

I

J PC

Threshold (MeV)

Nearby states

K K8 ∗ K K8 ∗ K ∗ K8 ∗ K ∗ K8 ∗ K ∗ K8 ∗ K ∗ K8 ∗ √ (99 + !!)= √2 (99 − !!)=√2 (99 + !!)= 2√ (K ∗ 9 − K ∗ !)= 2

0 0 0 0 0 0 0 0 0

0−+ 1++ 0++ 0−+ 1+− 2++ 0−+ 0++ 2++ 0++

1390 1390 1790 1790 1790 1790 1540 –1566 1540 –1566 1540 –1566 1665 –1678

(1410)1 f1 (1420) f0 (1710) (1760)

1 2

(1480)2 f0 (1500) f2 (1565)

The (1410) (1 ) denotes the low mass region and the (1480) (2 ) the high mass region of the (1440).

needed (this conclusion depends somewhat on how the tensor potential is regularized). Table 2 shows the predicted states with the largest attractive channels and their quantum numbers, together with the nearby experimental candidates. We shall discuss below other more likely assignments for the f0 (1500) and f0 (1710) mesons. Further bound states could exist with additional attraction of shorter range. For Navour exotic systems or for states with exotic quantum numbers (such as 1−+ ; 0−− ; 0+− ) pion exchange is generally repulsive or very weakly attractive and hence these states do not bind. One pion exchange is generally a factor three weaker for I =1 systems than for I =0. Such states are therefore not expected within the light meson sector. Deuteron-like vector mesons are not expected either since the attraction from pion exchange is too small. Also, such states should have been seen in e+ e− annihilation.

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Events/5 MeV/c

2

30

15

0 3820

3860 + -

3900 2

M(π π J/ψ) (MeV/c )

Fig. 7. + − J=

invariant mass distribution in B± decays to K ± X (X → + − J= ) (from Ref. [36]).

A lower bound for the widths of the predicted light states in Table 2 is given by the widths of the constituents. For example, the width of a (K K8 ∗ ) deuteronlike meson should be at least that of the K ∗ (51 MeV), while the width of a (K ∗ K8 ∗ ) state near threshold should be around 100 MeV. Annihilation amplitudes would of course increase these lower bounds. The BELLE Collaboration [36] reported a new narrow charmonium state in B± decay to K ± + − J= at 3871:8 ± 0:7 MeV with a width 4 ¡ 3:4 MeV, smaller than the experimental resolution. This is 60-100 MeV above the expected spin 2 cc( 8 3 Dc2 ) state [37,38]. The new state (Fig. 7) is + − observed in the J= invariant mass distribution with a signi7cance of 8:6 . This looks very much like one of the two deuteron-like DD8 ∗ states at 3870 MeV listed in Table 1, and as was predicted in Table 8 of Ref. [29] over 10 years ago. Its spin is, however, not determined yet. If deuteron-like, its spin-parity would be 0−+ or 1++ according to Table 1. It would be an isosinglet with a mass very close to the DD8 ∗ threshold. But, as the binding energy of such a deuteron-like state is of the same order as the isospin mass splittings one should expect large isospin breaking. In fact, the observed peak is almost exactly at the D0 D8 ∗0 threshold (3871:2 MeV). The main decay mode of the new state, if deuteron-like, should thus be D0 D8 0 0 since the charged modes lie about 2 MeV above resonance. One can then, using isospin and D∗ width measurements [35], estimate the width to be of the order of 50 keV. For further comments on this state see Ref. [39].

3. Are the scalars below 1 GeV non-qq! states? An essential experimental input for understanding the nature of the lightest scalar mesons comes from their couplings to two pseudoscalars, their -widths and their radiative widths which we now discuss in detail.

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3.1. The hadronic widths of the a0 (980) and f0 (980) mesons The a0 (980) meson decays mainly into while f0 (980) decays mainly into . However, their partial widths to K K8 are still rather large: 4[a0 (980) → ] = 0:85 ± 0:02 ; 8 4[a0 (980) → + K K]

(23)

4[f0 (980) → ] = 0:78 ± 0:02 ; 8 4[f0 (980) → + K K]

(24)

where we have averaged over some listed data [3]. Indeed, one would naively expect the K K8 modes to be strongly suppressed as the nominal meson masses lie below K K8 threshold. Thus only the 8 This is shown in Fig. 8 for couplings to high end of the resonance peak can decay to K K. and K K8 derived from pp 8 annihilation at rest into a0 (980) (see Ref. [40]). In fact, without phase space correction the f0 (980) couples much more strongly to K K8 than to . Models give e.g. gf2 0 K+K − =gf2 0 +− = 4:00 ± 0:14, see below. √ 8 (uu8 + dd)= 8 Now, if the isovector a0 (980) is pure qq8 it must be nn, 8 i.e. ud; 2 or ud. 8 Then the degeneracy of masses would suggest that the isosinglet f (980) is also composed of nn, 8 i.e. 0 √ 8 8 (uu8 − dd)= 2. But, this clearly contradicts the above large K K= coupling ratio of 4, since in that case that ratio would be 1/4 instead (assuming Navour symmetry and the OZI rule). The K K8 coupling can be large only if the f0 (980) and a0 (980) wave functions contain a signi7cant fraction 8 or within a multiquark structure. of ss, 8 either in the form of pure qq, 8 or in the form of K K, Therefore other structures have been suggested such as four-quark states (a0 (980) ≡ ss(d 8 d8 − uu) 8 and f0 (980) ≡ ss(d 8 d8 + uu)) 8 [10], or K K8 molecular states [41–43]. An issue which complicates things somewhat is that any unitarization, together with the fact that the K K8 threshold is nearby, always introduces a large K K8 component into the physical wave function if the coupling to K K8 is large.

14 12

Intensity

10 8 6 4 2

KK 0 0.8

1.0

1.2

1.4

m [GeV]

8 (in arbitrary units, Fig. 8. A qualitative and K K8 mass distribution for the a0 (980) resonance in pp 8 → X and K KX assuming that no other resonance is produced). The dashed line shows the intensity in the absence of K K8 decay mode (from Ref. [40]).

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The bare states (“the seeds”), be they qq8 or q2 q82 , must thus be dressed with a cloud of K K8 around the core [23,26]. 3.1.1. widths of the a0 (980) and f0 (980) mesons It is generally argued that molecular states should have much smaller widths than qq8 states [44]. The expected widths for qq8 mesons can be estimated using the measured widths of 2++ mesons and the (relativistic) formula [45] 3 m0 40 242 : (25) m2 One 7nds ∼ 0:8 keV for the a0 (980) and ∼ 0:1 keV for the f0 (980), assuming a pure ss8 nature of the latter. More sophisticated calculations lead to 0:64 keV [46] and 0.3–0:5 keV [46,47], respectively, which are comparable to the widths of molecular states (∼ 0:6 keV for the a0 (980) [48]). Hence measurements of the widths cannot distinguish between molecular and qq8 states. The experimental values are 0:30 ± 0:10 keV for the a0 (980) and 0:39+0:10 −0:13 for the f0 (980) [3], somewhat smaller than predicted. The comparison with theory and experiment is summarized in Table 3. 3.1.2. Radiative widths of the (1020) to a0 (980) and f0 (980) Radiative (1020) decay into a0 (980) and f0 (980) has been proposed as a sensitive reaction to distinguish between qq8 states, q2 q82 and K K8 molecules: K K8 molecules should be produced with a branching ratio of 10−5 while four-quark states (qqq8q) 8 would be produced with a much larger rate of 10−4 [44]. However, this has been criticized because of over-simpli7ed modelling of the decay process [49,50]. On the other hand, the a0 (980) as an isovector qq8 state could not be produced in radiative decay (1020) → a0 (980) in the limit of ideal mixing in the vector nonet. This is due to the OZI rule which prevents a pure ss8 (1020) to decay into an nn8 state. Therefore one would expect the rate (driven by a K K8 loop) to be quite small, much smaller than the corresponding

(1020) → f0 (980) , assuming a dominant ss8 structure for the f0 (980). Indeed, Achasov and Gubin [44] predict rates of about 10−5 for (1020) → a0 (980) and 5 × 10−5 for (1020) → f0 (980) , much smaller than for q2 q82 states. The 7rst measurements of these radiative decays were performed at the e+ e− VEPP-2M ring at Novosibirsk by the SND and CMD-2 collaborations. The reactions (1020) → 0 0 and 0 , leading to 7ve 7nal state photons, were reconstructed from a sample of 2×107 decays. The f0 (980) and a0 (980) appear to dominate the 0 0 and 0 mass spectra, respectively. The data samples were rather small (a few dozen a0 (980) and a few hundred f0 (980) events) and the expected contribution from the continuum S-wave (e.g. in ) could not be determined precisely, leading to a large systematic error. SND reports the branching ratios given in Table 4. The corresponding results for Table 3 Theoretical predictions of widths in KeV compared to experiment State

Eq. (25)

Refs. [46,47]

Experiment [3]

a0 (980) f0 (980)

∼ 0:8 ∼ 0:1

0.64 0.3– 0.5

0:3 ± 0:1 0:39+0:10 −0:13

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Table 4 Experimental and estimated theoretical branching ratios in units of 10−4 on → f0 (980) → 0 0 and → a0 (980) → 0 Experiment

→ f0 → 0 0

→ a0 → 0

Ref.

SND CMD2 KLOE

1:17+0:44 −0:58 0:97 ± 0:68 1:49 ± 0:07

0:88 ± 0:17 0:90 ± 0:26 0:74 ± 0:07

[51,52] [53] [54,55]

Theory qq8 state qqq8q8 state K K8 molecule

∼ 0:5 ∼ 1:0 ∼ 0:1

∼ 0:1 ∼ 1:0 ∼ 0:1

[44] [44] [44]

Fig. 9. Left: Di9erential branching ratio for (1020) decay into 0 0 as a function of 0 0 mass (from Ref. [54]). Right: event distribution for (1020) decay into 0 for → (top) and → + − 0 (bottom) as a function of 0 mass. The curve shows the theoretical prediction 7tted to the data (from Ref. [55]).

CMD-2 agree, see Table 4. The errors for f0 (980) are large, but at least the branching ratio for a0 (980) (∼ 10−4 ) appears to be consistent with expectations for q2 q82 states. More precise branching ratios were measured recently at the DAYNE factory with larger data samples (5 × 107 decays) and better reconstruction eOciency. The 0 0 mass distribution in 5 events from the KLOE collaboration [54] is shown in Fig. 9. The low mass tail is determined by the

contribution to the S-wave. For the the authors choose (somewhat arbitrarily) the Fermilab results from D+ → + + − decay [60] in which a mass of 478 MeV and a width of 324 MeV were reported. The 7nal state appears to interfere destructively with f0 (980) . The contribution from

→ 90 0 (90 → 0 ) is negligible. The net results are consistent with the Novosibirsk ones, but are more precise: multiplying the 0 0 rate by an isospin factor of three to take into account the

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Table 5 Coupling constants in units of GeV2 or GeV extracted by KLOE (7t B with ) [54,55] from → f0 (980) → 0 0 and → a0 (980) → 0 data gf2 0 K+K − =(4) gf2 0 K+K − =gf2 0 +− g

ga20 K + K − =(4) ga0 =ga0 K + K − gf2 0 K K8 =ga20 K K8

2:79 ± 0:12 4:00 ± 0:14 0:060 ± 0:008 0:40 ± 0:04 1:35 ± 0:09 7:0 ± 0:7

Table 6 Experimental values of the coupling gf2 0 K + K − =(4) in units of GeV2 extracted from → f0 (980) → 0 0 assuming a virtual K K8 loop compared with some theoretical predictions Experiment

gf2 0 K + K − =(4)

KLOE 7t A, (no ) [54,55] KLOE 7t B (with ) [54,55] CMD-2 [53] SND [51,52]

1:29 ± 0:14 2:79 ± 0:12 1:48 ± 0:32 2:47+0:73 −0:51

Theory Linear sigma model [62] QCD sum rules [63]

∼ 2:2 ∼ 4:0

unobserved f0 (980) → + − decay mode, KLOE obtains the branching ratio B[ (1020) → f0 (980) ; (f0 (980) → )] = (4:47 ± 0:21) × 10−4 ;

(26)

which is an order of magnitude larger than expected for a dominantly ssf 8 0 (980). The KLOE result for the a0 (980) channel [55] also agrees with the Novosibirsk one. Here the meson from a0 (980) decay is detected in both its + − 0 and decay modes, leading to consistent results. This is an important check that systematical errors are under control. The contribution from

→ 90 0 (90 → ) is negligible. The 0 mass distribution is similar to the one for 0 0 in f0 (980) decay and is also shown in Fig. 9. The branching ratio B[ (1020) → a0 (980) ; (a0 (980) → )] = (0:74 ± 0:07) × 10−4 ;

(27)

is again much larger than expected for a qq8 state. The data are summarized in Table 4. The coupling constants extracted from KLOE are given in Table 5. Their value for ga20 K + K − =(4)=0:40±0:04 agree well with a di9erent determination [56] in a coupled channel framework who get ga20 K + K − =(4) = 0:356. Furthermore in Table 6 the values of gf2 0 K + K − =(4) obtained by di9erent experiments are listed and compared with some theoretical predictions. As q2 q82 states the f0 (980) and a0 (980) are assumed to be produced in radiative decays through the emission of a from a K + K − loop (Fig. 10). The radiative partial widths are therefore equal,

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

φ

79

γ

K+ K-

π a0, f0 η, π

Fig. 10. Radiative decay into a0 (980) or f0 (980).

in the range of ∼ 2 × 10−4 [42,43,64]. From Eqs. (26) and (27) one obtains, however, the ratio

(1020) → f0 (980) = 6:7 ± 0:7 ;

(1020) → a0 (980)

(28)

taking the K K8 decay modes into account according to Eqs. (23) and (24). An argument for the much larger f0 (980) yield was presented in Refs. [57,220] as due to large isospin mixing arising from the nondegenerate K + K − and K 0 K80 loops, and the near degeneracy of a0 (980) and f0 (980). However, the fact that the decay channels f0 → and a0 → are open complicates the issue. Since both f0 → and a0 → are open one must use a coupled channel framework, and not 8 This reduces the possible isospin breaking substantially [58]. (A large isospin breaking only K K. would in fact imply also large f0 → , which is not observed.) Achasov and Kiselev [59] have also done an independent analysis of the KLOE a0 data and (mainly by varying the a0 mass) found a considerably larger value than KLOE in Table 5 for ga20 K + K − =(4) between 0.55 and 0:82 GeV2 . From the large production rates one could argue that the a0 (980) and f0 (980) are four-quark states, although not everybody agrees as, for example, the authors of Ref. [47] using the linear sigma model (L M). However, the L M for the light scalars does not necessarily imply that the states need to be qq. 8 A four quark nonet as obtained from e.g. Ja9e’s model could equally well be used as scalars in the L M [65]. For a theoretical analysis of the → f0 (980) using a model with the K + K − loop which includes both the L M and chiral perturbation theory in a complementary way see Ref. [66]. The coupling constants and their ratios in Table 5 above are useful to understand the nature of the scalar mesons. 3.1.3. The f0 (980) produced in Ds → 3 The meson f0 (980) is strongly produced in Ds+ (cs) 8 → + + − decay [67,61]. Fig. 11 shows the + − mass distribution and D → 3 Dalitz plot in which a prominent peak is observed just below 1 GeV. The f0 (980) contributes about 50% to the Ds+ decay Dalitz plot. From the Cabibbo favoured c → s decay one would expect the f0 resonances such as f0 (980) to be produced mainly as an ss8 state, which then decays to through a virtual K K8 loop (Fig. 12), see Refs. [68,69]. Note that for the f0 (980) only the channel is open, hence the OZI rule must be violated in the decay even if it is produced as an ss8 pair at short distances. This

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Fig. 11. (a) m2 (+ − ) invariant mass distribution in Ds+ → + + − decay which is dominated by the f0 (980). The broader peak around 1:9 GeV2 is due to the f0 (1370). The hatched area is the background distribution; (b) Dalitz plot (from Ref. [61]).

π c

W

s

K π

f0(980)

Ds

π s

K

Fig. 12. W -emission in the Cabibbo allowed c → s transition leading to the formation of the f0 (980).

explains its narrow width. However, the nearness of the K K8 threshold and the large gf0 K K8 generates through unitarity large virtual K K8 clouds in the f0 (980) wave function. However, for the f0 (1370) which is also seen in Ds+ (cs) 8 → + + − decay (Fig. 11) the same argument poses a problem, since f0 (1370) is known to be mainly an nn8 state. For instance, f0 (1370) 8 [70] which suggests that this state is an nn8 and not a naive ss, is not observed in Ds → K K 8 since 8 then one would expect the K K mode to be large. This indicates that the annihilation contribution Ds → W → is also present, or more generally, that the production process is more complicated than for f0 (980). Another experimental fact is that the f0 (980) is observed in hadronic Z 0 decays with cross sections similar to those for nn8 states [71]. This argues in favour of a similar two-quark nature of the f0 (980). However, the production cross section for four-quark states is not known. 3.2. A possible interpretation of the nature of a0 (980) and f0 (980) Since the radiative widths of both a0 (980) and f0 (980) are large and agree with theoretical expectations for compact four-quark states (Table 4), this suggests that they have substantial four-quark components.

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81

On the other hand, these states lie just below the K K8 threshold and the resonance couplings to K K8 are therefore very large (see Tables 5 and 6). Hence their cores must by unitarity be surrounded by a rather large cloud of virtual K K8 pairs [23], which extends further out from the central four-quark core. From Section 2.1 one expects from QCD the strongest bound four-quark states to be those (qq)38(q8q) 8 3 where the two quarks (and the two anti-quarks) are in a triplet state of both colour and Navour. Thus the following picture of the a0 (980) and f0 (980) emerges, which was discussed previously by Close and T,ornqvist [72] and is consistent with present data and theory: The central core is composed predominantly of a four-quark (qq)38(q8q) 8 3 state a[ la Ja9e, whose quarks recombine at larger radii to two colour singlet qq’s, 8 and then form a standing wave of virtual K K8 at the periphery of the state. The K K8 component also partly explains the narrowness of the a0 (980) and f0 (980) mesons: in order to decay the K K8 component must 7rst annihilate near the origin to , respectively . 3.3. Is the f0 (600) a non-qq8 state and does the (800) exist? Apart from the f0 (980) and a0 (980) there are two additional light scalar meson candidates below 1 GeV, the f0 (600) and the (800). It is natural to ask whether these altogether four scalars could be related and form an SU(3)F nonet made of two- or four-quark mesons, or meson–meson bound states. Let us 7rst discuss the experimental evidence for the f0 (600) and the (800) mesons. 3.3.1. The f0 (600) (or ) There is now a rather widespread agreement that a light and very broad f0 (600) or f0 (600) pole exists in the scattering data, although di9erent views prevail on its nature and its importance as a physical state in the nonperturbative regime of QCD. The PDG [3] cites numerous determinations of the pole mass in the neighbourhood of 600 MeV. One of the perhaps most precise determinations of the pole position was achieved using chiral perturbation theory together with constraints of analyticity, unitarity, crossing symmetry and the Roy equations [73], leading to the result m − i4=2 = (470 ± 30) − i(295 ± 20). Many analyses (see e.g. [25,74]) also generate the f0 (600) from crossed channel exchanges. Such results do certainly not disprove the f0 (600) as a true resonance as it is well known from the duality arguments of the 1970s that s and t channel resonances generally come together in hadronic amplitudes. The E791 Collaboration [60] observes in D+ → + + − a rather clear resonance bump which dominates their 3 Dalitz plot (Fig. 13). They quote a (Breit–Wigner) resonance of mass +43 478+24 −23 ± 17 MeV and width 324−40 ± 21 MeV, which they interpret as the light f0 (600). In their 7t they, however, assume a Breit–Wigner shape, which means that the phase of the S-wave amplitude should reach 90◦ at the peak mass. This is not easily compatible with the known phases (Fig. 14) and the Watson 7nal state theorem, which states that the phase should be (up to some possible constant production phase) the same as the S-wave phase, which is only 45◦ at 600 MeV. This problem certainly needs further study. In Fig. 14 we show as an example the results of a recent new 7t to the → elastic data in the scalar–isoscalar channel below 1 GeV by the Krakow group [75]. This group uses all well known theoretical constraints (e.g. crossing symmetry and Roy’s equations) in their 7t, and 7nd the unique ‘down-Nat’ solution shown in Fig. 14a. Together with chiral symmetry constraints,

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Fig. 13. (a) m2 (+ − ) invariant mass distribution in D+ → + + − decay, which is dominated by a broad low-mass f0 (600). The hatched area is the background distribution; (b) Dalitz plot (from Ref. [60]).

(a)

(b)

Fig. 14. Isoscalar S-wave phase shifts from Refs. [75,76]. The left 7gure (a) shows the preferred ‘down-Nat’ solution (full circles) while (b) shows the ‘up-Nat’ data (open circles) [77]. The low-energy diamonds show the Ke4 data from Ref. [78]. The solid lines represent 7ts to Roy’s equations and to data.

the slow increase of the phase shift can be interpreted as due to the presence of a very broad f0 (600) meson pole. Chiral symmetry requires an Adler zero for the f0 (600) in the → amplitude near s = m2 =2. This suppresses the low energy tail of the f0 (600) as a naive Breit–Wigner resonance. Without that proper low energy behaviour one may easily miss the pole in the data analysis. A simple way to see this is to note [73] that current algebra predicts that (in the chiral limit, when the pion mass vanishes, and s is small) the scalar–isoscalar amplitude, t00 = s=(16F2 ). Although this amplitude vanishes at the two-pion threshold, one soon reaches a very strong 7nal state interaction which 0 0 2 violates unitarity. An √ easy way to unitarize t0 is to write instead t0 = s=(16F − is). This expression contains a pole at s = 463 − i 463 MeV, which is not far from the region where most f0 (600) pole determinations are. Shifting the Adler zero sA from 0 and taking into account the 7nite pion mass one can slightly 0 2 improve the expression to t0 = −Im @(s)=[16F + Re @(s) + iIm @(s)], where Im @(s) = −(s − sA ) 1 − 4m2 =s and Re @(s) is determined from Im @(s) using a dispersion relation

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

83

(subtracted at s ≈ 0 to agree with the previous expression near s = 0). But since |spole |m2 the pole position is not changed signi7cantly. It should, however, be emphasized that such unitarized expressions, although utterly simple, cannot be approximated by a single Breit–Wigner amplitude. One must include other terms, or at least a speci7c constant “background”, as e.g. the four-point contact term in the linear sigma model. This background then interferes destructively with the resonance at the low energy side to give the Adler zero. Another way to see that for a very broad resonance (say of width 4 ≈ 500 MeV) the pole parameters (M − i4=2) are quite di9erent from the Breit–Wigner mass MBW and width 4BW is to consider the following simple but instructive nonrelativistic (Flatt]e) form for the inverse propagator: 2 2 P −1 (s) = MBW − s − iMBW 4BW (s) = MBW −s−i

√ g2 MBW s − sth 8

(29)

2 where sth is the threshold. For this propagator the phase shift passes 90◦ at s = MBW , and the 2 Breit–Wigner width would be 4BW (MBW ). Note that s appears twice in the above expression. On the other hand the pole position which is obtained from a zero in the inverse propagator

P −1 (spole ) = 0

(30)

is very di9erent from the narrow width approximation MBW − 2i 4BW (MBW ), when g2 =(4) is large and the resonance is above threshold. In general MBW is much larger than the pole mass M obtained √ from spole = (M − 2i 4)2 or M = Re( spole ). For a broad resonance it is important to give the pole position rather than the Breit–Wigner values, since the pole is independent of the reaction under study. Only at the pole does the amplitude factorize, and the pole is independent of the “background”. Therefore a pole can lead to resonances with di9erent Breit–Wigner masses and widths in di9erent reactions. On the other hand, determining the pole position requires a reliable theory for the amplitudes and this has unfortunately been for a long time a source of much confusion, especially when broad resonances were involved. We have shown in Section 3.1.2 above that for the f0 (980) and a0 (980) mesons the couplings to the channels K + K8 − and K 0 K8 0 were very large (gf2 0 K + K − =4 ∼ 1–4 GeV2 , cf. Table 6). Yet the resonances appear narrow in the and channels, respectively. This is due to the K K8 threshold opening at the resonance masses. If the latter were increased far above decay threshold (i.e. 1 − sth =M 2 of order 1) the widths would become very large, easily reaching 500 MeV [79]. Now, if the f0 (600) and (800) indeed belong to the same family as the f0 (980) and a0 (980) mesons (say if the f0 (600) were composed of 2 or 4 u and d type quarks) then no such mechanism would suppress the decay f0 (600) → or (800) → K. Thus if f0 (600) and (800) belong to the same multiplet as a0 (980) and f0 (980) one would expect their widths to be very large. Therefore the broad f0 (600) could well belong to the same family of light mesons as the narrow a0 (980) and f0 (980), as for example in the Ja9e four-quark model or in the U (3) × U (3) linear sigma model [65,80]. Suggestions that the f0 (600) could be a glueball have been made [81,82]. However, as we shall show later, the best estimates from lattice QCD locate the glueball in the 1500 –1700 MeV region. The partial width (4 = 3:8 ± 1:5 keV [83]) of the f0 (600) can be understood from general gauge invariance requirements [84]. We shall deal with widths of glueballs in Section 4.

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

1.0

0.8

0.6

Kπ phase shift

0.4

K η threshold 0.2

Kπ absorption parameter*100 0.6

(a)

|A Kπ|

Phase shift , η parameter

84

0.8

1.0 1.2 √s = m(K π ) GeV

K η' threshold

0.0 1.4

0.6

1.6

(b)

0.8

1.0 1.2 √ s = m(K π ) GeV

1.4

1.6

Fig. 15. K S-wave phase shift (a) and magnitude of the S-wave partial wave amplitude (b) measured by LASS [85] (dots) and 7tted in the unitarized quark model of T,ornqvist [23,24].

Fig. 16. The D+ → K − + + Dalitz plot. A broad is reported under the dominating K ∗ (892) bands (from Ref. [19]).

3.3.2. The (800) Fig. 15 shows the LASS elastic K S-wave phase shifts [85]. The phase shift does not pass through 90◦ until 1350 MeV and hence there is no Breit–Wigner resonance behaviour below 1 GeV. Nonetheless several theoretical models arguing in favour of a light around 800 MeV [10,47,65, 86–89] have been presented. However, in some of the analyses (especially the experimental analyses), one does not make a clear distinction between pole and Breit–Wigner mass. For instance, no distinction is made if only tree level graphs are included without the loops required by unitarity. +43 The E791 Collaboration reported a light with mass 797+19 −43 MeV and width 410−87 MeV, but uses a Breit–Wigner amplitude [19]. Their Dalitz plot is shown in Fig. 16. This claim was, however, not con7rmed by the CLEO Collaboration [20,90]. In fact, Cherry and Pennington [21] argued that

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

85

Fig. 17. (a) K S-wave squared running mass m2 (s) = m20 + Re @(s) and the corresponding width function m(s)4(s) = −Im @(s) which 7ts the phase shift in Fig. 15 (from Ref. [23]); (b) Running mass2 = Re @(s) and Im @(s) when the overall coupling of the model in (a) is increased from its physical value (from Ref. [26], see also [24]).

the mass cannot exceed ∼ 825 MeV assuming the LASS phase shifts (Fig. 15) to be correct. A lighter and very broad pole is nonetheless possible and should be looked for in future data analyses. In unitarized models with correct analytic behaviour at thresholds one must add a running mass Re @(s) to the constant mass term m20 in the Breit–Wigner resonance amplitude. This is illustrated in Fig. 17a. When the running mass (top curve in Fig. 17a) crosses the K mass curve (s=m(K)2 ) the phase shift passes through 90◦ . Note in particular the strong cusps at the K and K thresholds. For a naive Breit–Wigner resonance the Re @(s) with cusp behaviour would be replaced by a constant. The nonlinear form of @(s) can produce two poles in the amplitude, although only one seed state (qq8 or 4-quark state) is introduced. In order to clarify this point one can, within a model, increase the e9ective coupling from its physical value. Then with increasing coupling one 7nds a virtual bound state near the K threshold. For suOciently large coupling even a bound state in K would appear (for more details see Refs. [24,26]). An example is shown in Fig. 17b, where the 7rst crossing between Re @(s) and s would be a bound state and the third crossing a resonance like the K0∗ (1430). (The second crossing corresponds to a slow anti-clockwise movement in the Argand plot which is not a resonance.) In conclusion, there are theoretical arguments for why a light and broad pole can exist near the K threshold and many phenomenological papers support its existence [10,47,65,86–89]. But the question of whether a (800) exists near the K threshold is not yet conclusive, since Breit–Wigner 7ts have been used. We believe that experimental groups should look for pole positions in their data analysis, which also include the aforementioned nonlinear e9ects from S-wave thresholds. ∗ (2317) 3.4. Observation of a charm-strange state DsJ ∗ (2317) The BABAR Collaboration has recently reported the observation of a very narrow meson DsJ (4 ¡ 10 MeV, smaller than the experimental resolution) in the heavy-light sector, which apparently decays through isospin violation to Ds+ 0 (Fig. 18) [4]. This state was soon con7rmed by CLEO

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C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

∗ Fig. 18. Ds± 0 mass distribution for Ds± → K + K − ± (a) and Ds± → K + K − ± 0 (b) showing the DsJ (2317) (after Ref. [4]).

2.8 -

1 2.73 0 2.67 +

2.6

2 2.59 + 1 2.55, 2.56

M [GeV]

+

0 2.48

+ DsJ (2573) + Ds1 (2536)

2.4

DK 2.32

+ o

D K

o +

DK

2.2 -

+ D*s (2112)

-

+ Ds (1969)

1 2.13

2.0

0 1.98

1.8

Fig. 19. Experimental (solid) and theoretical (dashed) [91] cs8 mass spectrum. The long horizontal dotted lines show the D+ K 0 and D0 K + thresholds, below which the BABAR state [4] is shown (after Ref. [92]).

[93]. Its spin is still uncertain, but the quantum number J P = 0+ are preferred. This discovery may well turn out to be crucial also for the light scalars. It is not easy to identify this state with the 3 P0 cs8 state because its mass is about 160 MeV below the expected value near 2480 MeV [91] (see also Fig. 19). Such a large deviation from

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87

these predictions seems at 7rst unusual. This expected 3 P0 cs8 state would lie above the DK threshold ∗ (2317) lies and would therefore [91] be several hundred MeV broad. However, the observed DsJ ∼ 45 MeV below this threshold (in fact the average of two isospin related thresholds, 2358 MeV for D0 K + and 2367 MeV for D+ K 0 ). The only open charm and strangeness conserving threshold is ∗ (2317) had isospin 0 the narrow width could be understood, Ds+ 0 where it was observed. If the DJs ∗ at least qualitatively, since DJs (2317) → D+ 0 would violate isospin. A small isospin breaking is expected from the rather large 9 MeV di9erence in the D0 K + and D+ K 0 thresholds, and from − mixing. Educated guesses of the total width are of the order 10 keV [94,95]. Numerous discussions on this state can already be found in the literature with various interpre∗ (2317) could be a state tations [92,94–98]. We 7nd the possibility especially intriguing that the DsJ related by Navour symmetry to the light scalars below 1 GeV. It is unavoidable that this state (be it composed of two quarks, four quarks or just a DK molecule) should couple strongly to the closed S-wave threshold DK. Then the hadronic mass shift (or running mass) due to DK loop would be important and could lead to a 160 MeV downward mass shift. The e9ect is similar to the (800) running mass of the previous section. Remember also that two states are often generated by one input seed state, one near the strong threshold and the other higher in energy. In fact Ref. [97] supports such a picture with a model rather similar to the one used in the discussion above [23,24]. CLEO also found another new, also very narrow, charm-strange state at 2460 MeV [93]. This peak is seen in the Ds∗ 0 channel and lies about 40 MeV below the D∗ K threshold, which would be the nearest strong (but closed) S-wave threshold, assuming that this state is an axial charm-strange meson, for which DK is forbidden by parity. Similar strong cusp e9ects in the running mass would also here be expected, which could explain the low mass, 100 MeV below predictions [91]. In fact, even among the heavy 8 states similar threshold e9ects (there due to the opening of the strong BB8 thresholds) explain why the 8(4S)–8(5S) mass splitting is so large, about 80 MeV larger than in naive potential models [99]. A better understanding of these two narrow states is thus likely to throw new light also on the enigmatic light scalars. 3.5. Do we have a complete scalar nonet below 1 GeV? From the previous discussion, the a0 (980); f0 (980) and f0 (600) could belong to the same Navour nonet, since the large di9erence in widths could be understood by the wide open phase space for the

, and by the K K8 threshold distortions for the a0 (980) and f0 (980) states. The con7rmation of a light and equally broad (800) would lead to a light scalar nonet below 1 GeV. The large radiative widths of the a0 (980) and f0 (980) mesons favour large four-quark components. This would not only agree with Ja9e’s original predictions for four-quark states, but would also be consistent with the conventional wisdom that meson–meson forces are attractive in octet and singlet channels, but repulsive for Navour exotic quantum numbers. Thus the formation of bound (or nearly bound) meson–meson bound states should be expected only in octet and singlet channels. Some years ago, one of us [23,24] was able to interpret many of the light scalars as originating from a nearly degenerate nonet of bare states which were strongly shifted in masses by unitarization. In that scheme a “resonance doubling” would appear, even though only one bare state was introduced. The lower state usually appeared near the 7rst strongly coupled threshold. Flavour symmetry breaking

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C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

in the output spectrum came mainly from the large splitting between the two-pseudoscalar thresholds 8 ; and ). (i.e., for I = 0 : ; ; K K; In an educated guess [72], the lightest scalars are composed of a central core with four quarks. Following Ja9e’s QCD arguments this central core would consist predominantly of a four-quark (qq)38(q8q) 8 3 state. At larger distances the quarks would then recombine to a pair of colour singlet qq’s, 8 building two pseudoscalar mesons as a meson cloud at the periphery. 4. Glueballs 4.1. Theoretical predictions QCD predicts the existence of isoscalar mesons which contain only gluons, the glueballs. These states are a consequence of the non-abelian structure of QCD which requires that gluons couple to themselves and hence may bind. Fig. 20 shows results from lattice gauge calculations as a function of lattice spacing a. When the scale parameter r0 (estimated from the string tension in heavy quark mesons) is taken to be about 0:5 fm, one 7nds by extrapolation to a=0 a mass of 1611±30±160 MeV for the ground state glueball, a scalar (the 7rst error is statistical while the second error reNects the uncertainty on r0 ). The 7rst excited state is a tensor and has a mass of 2232 ± 220 ± 220 MeV [12]. Further mass predictions from the lattice can be found in Refs. [100,101]. Hence the low mass glueballs lie in the same mass region as ordinary isoscalar qq8 states, that is in the mass range of the 13 P0 (0++ ) and 23 P2 ; 33 P2 ; 13 F2 (2++ ) states, see Fig. 1. This is presumably the reason why they have not yet been identi7ed unambiguously.

Fig. 20. Predictions for the mass m of the ground state glueball (0++ ) and for the 7rst excited state (2++ ); a is the lattice spacing and r0 is a scale parameter, see text (from Ref. [12]).

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117 12

0+ 2+ 2' + 0' +

3++

r0mG

8 0

4

3+

2 + ++

3 2 1

3

1+

0 +

6 2++

2

mG [GeV]

10

4

89

0++

1 2

0

++

+

+

0

PC

Fig. 21. Predicted quenched glueball spectrum from the lattice (from Ref. [103]).

For pure gluonium one expects couplings of similar strengths to ss8 and uu8 + dd8 mesons because gluons are Navour-blind. This leads to the Navour “democracy” for the glueball decay rates into , 8 and of 3:4:1:0. In contrast, ss8 mesons decay mainly to kaons, and uu8 + dd8 mesons K K; 8 and can be used to distinguish glueballs mainly to pions. Hence decay rates to ; K K; from ordinary mesons. Therefore a detailed understanding of the qq8 nonets is mandatory. Unfortunately, lattice calculations predict that glueballs with the exotic quantum numbers J PC = −− 0 ; 0+− ; 1−+ ; 2+− , etc., lie far above 2 GeV [102,103]. This is in the diOcult region of radial and orbital excitations, where the states become increasingly broad and overlap. Fig. 21 shows the glueball spectrum from lattice QCD. The lightest glueball with exotic quantum numbers (2+− ) has a mass of about 4 GeV. The lattice calculations assume that the quark masses are in7nite and therefore neglect qq8 loops. Nonetheless, one expects that glueballs will mix with nearby qq8 states of the same quantum numbers [13]. There are indications that the predicted mass of the scalar glueballs decreases slightly in the unquenched approximation, at least with two quark Navours, while the mass of the tensor does not change signi7cantly [104]. On the other hand, mixing with nearby qq8 states will modify the decay branching ratios and obscure the nature of the observed state. However, one would still 7nd three isoscalar states in the regions of the 0++ and 2++ nonets, instead of only two. As we shall discuss below, signi7cant progresses have been made recently to identify the 0++ glueball, while much uncertainty remains for the 2++ assignments. As discussed before, lattice gauge calculations place the ground state glueball, an isoscalar 0++ state, in the 1400 to 1800 MeV mass interval, that is in the mass region where the 13 P0 (0++ ) and 23 P0 isoscalar qq8 mesons are also expected. In the charmonium system, the Cc0 (1P) cc8 meson lies

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140 MeV below the Cc2 (1P). A similar mass splitting is also predicted for light quark mesons in the relativized quark model with chromodynamics of Ref. [6]. The 13 P0 isoscalar mesons are expected somewhat below the corresponding 13 P2 mesons f2 (1270) and f2 (1525) while their 7rst radial 3 excitations √ 2 P0 are predicted around 1900 MeV. The isoscalar qq8 mesons have the quark structure 8 and ss, nn8 ≡ 1= 2(uu8 + dd) 8 or a mixture thereof. One expects that glueballs will mix with nearby qq8 states of the same quantum numbers [13,105] but, nonetheless, one would still 7nd three nearby isoscalar states, instead of only two. 4.2. Is the f0 (1500) meson the ground state scalar glueball? Five isoscalar resonances are well established: the very broad f0 (600) (or ), the f0 (980), the broad f0 (1370), and the comparatively narrow f0 (1500) and f0 (1710). We have dealt with the f0 (600) and the f0 (980) in a previous section and discuss now the three upper mass states f0 (1370); f0 (1710); f0 (1500). In the following we shall show that the data suggest that f0 (1370) is largely nn; 8 f0 (1710) mainly ss, 8 and f0 (1500) mainly glue. Experimental evidence for a 100 MeV broad isoscalar state at 1527 MeV, decaying into two pions, was 7rst reported in pp8 annihilation at rest into three pions [106,107]. A spin 0 assignment was cautiously suggested and no K K8 decays were observed. A somewhat broader scalar meson at 1592 MeV, named G(1590) and decaying into [108], [109], but not [108] was reported in high energy pion induced reactions. Clari7cation and 7rm evidence for the existence of a 100 MeV broad isoscalar scalar meson at 1500 MeV came with the high statistics data from Crystal Barrel in pp8 annihilation at rest (for a review see Ref. [40]). The newly baptized f0 (1500) meson was reported to decay into [110], [111], [112], K K8 [113] and 4 [114,115]. It was con7rmed in many experiments, e.g. in pion induced reactions [116], by the Obelix collaboration also in pp 8 annihilations [117], in central collisions [118–120], in J= radiative decays [121], and perhaps in Ds decays [67]. It was, however, not observed in collisions [122,123]. A sketch of the Crystal Barrel apparatus, a large solid angle high granularity -detector, is shown in Fig. 22. Details can be found in Ref. [124]. The Dalitz plots for pp 8 annihilation at rest into 30 ; 0 and 0 0 were analysed by replacing the usual Breit–Wigner amplitudes describing two-body 0 0 or 0 resonances by T -matrices [125]. This is the recommended procedure for coupled channels and overlapping resonances of the same quantum numbers which ensures that unitarity is ful7lled. Masses and widths were derived by searching for poles of the T -matrices in the complex energy plane. The S-wave scattering data from the CERN-Munich collaboration [126] were included. The 30 and 0 channels demand two high mass isoscalar scalar mesons, f0 (1370) and f0 (1500), decaying into 0 0 and , while annihilation into 0 0 also requires a high mass isovector decaying into 0 , the a0 (1450). Consistency between the three data sets was obtained by performing a simultaneous coupled channel 7t [127]. Fig. 23 shows the resulting 0 0 and S-wave intensities for the three annihilation channels, apart from multiplicative phase space factors 2p=m (where p is the daughter momenta in the rest frame of a resonance with mass m). At low masses one observes a strong contribution from f0 (980) producing a dip in the 30 channel (due to destructive interferences in this channel) and a broad enhancement attributed to the f0 (600) meson.

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

91

4 7 p

6 5 1

3

2

Fig. 22. Sketch of the crystal barrel detector at the low energy antiproton ring (LEAR) at CERN. 1,2–magnet yoke; 3–magnet coil providing a longitudinal 7eld of 1.5 T; 4 – detection barrel made of 1380 CsI(Tl) crystals with photodiode readout; 5 –jet drift chamber; 6 –proportional wire chambers; 7–4 cm long hydrogen target.

The K K8 decays of the f0 (1370) and f0 (1500) mesons were also observed by Crystal Barrel in pp 8 annihilation at rest into 0 KL KL [113], where one of the two KL ’s interacted in the CsI barrel and the other one was missing. The decay kinematics was reconstructed from the measured directions and energies of the two ’s from 0 decay and from the direction of the interacting KL . The contributions from f0 (1370) and f0 (1500) to the 0 KL KL Dalitz plot were found to be small, although they could not be determined precisely as a0 (1450) also decays into KL KL since KL KL has both isospin 0 and 1. Therefore one must subtract the a0 (1450) amplitude from the f0 (1370) and f0 (1500) amplitudes. The a0 (1450) contribution was determined by using isospin conservation and analysing the reaction pp 8 → KL K ± ∓ in which the isoscalar S-wave is absent [129]. The a0 (1450) mass and width, determined from the annihilation channel pp 8 → KL K ± ∓ are M = 1480 ± 30 MeV and 4 = 265 ± 15 MeV, respectively, in excellent agreement with the result from the annihilation channel pp 8 → 0 0 ; M =1450±40 and 4=270±40 MeV [130], respectively. This argues against the low mass (M 1300 MeV) and narrow (4 80 MeV) a0 reported by the Obelix collaboration in the annihilation channel KS K ± ∓ [131]. A high mass but somewhat narrower a0 (1450) is also found in the annihilation channel pp 8 → !+ − 0 , where a0 (1450) decays to !9 [132]. The f0 (1370) and f0 (1500) mesons were also observed by the WA102 collaboration in pp central collisions at 450 GeV. Signals for these states and the f0 (1710) were observed in the + − and K + K − S-waves [119]. Fig. 24 shows the K + K − S-wave from a coupled channel analysis of + − and K + K − data, using the T-matrix formalism. Signals from the f0 (1500) and f0 (1710) are clearly seen and pole positions for the f0 (1370) and f0 (1500) mesons are in excellent agreement with Crystal Barrel data. The f0 (1500) was also observed by WA102 in its [133], [134] and 4 [137] decay modes.

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C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

1600 f0(1500)

Intensity

1200

f0(1370) 800

400

f0(600)

f0(980) 0

400

800 1200 m [MeV]

1600

Fig. 23. Isoscalar 0 0 S-wave intensities in 30 (solid curve) and 0 0 (dashed curve) and S-wave intensity in 0 (dotted curve), apart from phase space factors. The vertical scale is arbitrary (from Ref. [128]).

Events / 0.04GeV

3000

2000 f0 (1500)

1000

0

f0 (1710)

1

1.5

2.0

2.5

M (K+K ) [GeV]

Fig. 24. K + K − S-wave in central production (from Ref. [119]).

4.2.1. Hadronic decay width For the f0 (1500) meson the crystal barrel and WA102 ratios of measured decay branching ratios into two pseudoscalar mesons are listed in Table 7. They are in good agreement. Ref. [135] quotes somewhat smaller = and = ratios but data from older less precise experiments are 7tted

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

93

Table 7 Ratios of decay branching ratios into pairs of pseudoscalar mesons for the f0 (1500) Ratio

pp 8 annihilation

Ref.

Central production

Ref.

4()=4()

0:226 ± 0:095a 0:157 ± 0:062b 0:066 ± 0:028a 0:042 ± 0:015b 0:186 ± 0:066a 0:119 ± 0:032b

[110,111] [127] [110,112] [127] [110,113] [127]

0:18 ± 0:03

[133]

0:095 ± 0:026

[134]

0:33 ± 0:08

[119]

4( )=4() 8 4(K K)=4() a b

Measured in pp 8 annihilation and in central production from the single channel analyses. Measured from the coupled channel analysis of 30 ; 20 and 0 2.

Table 8 Ratios of decay branching ratios into 4 for the f0 (1500) measured in pp 8 annihilation and in central production Ratio

pp 8 annihilation

Ref.

Central production

Ref.

4(2[]S )=4(4) 4(99)=4(4) 4((1300))=4(4) 4(a1 (1260))=4(4) 4(99)=4(2[]S )

0:26 ± 0:07 0:13 ± 0:08 0:50 ± 0:25 0:12 ± 0:05 0:50 ± 0:34

[115] [115] [115] [115] [115]

— — — — 2:6 ± 0:4a 3:3 ± 0:5b

[137] [137]

a b

From 2+ 2− . From + − 20 .

simultaneously. We note that the K K8 signal for the f0 (1500) is much larger than for the f0 (1710) (see Fig. 24), even though the latter couples more strongly to K K8 [119]. Hence the production of f0 (1500) appears to be enhanced in central collisions, in accord with the conventional wisdom that gluonic states should be enhanced in Pomeron exchange reactions. 8 For the f0 (1500) the ratio K K= is much smaller than one. We recall that a pure ss8 meson does not decay to pions and, therefore, the f0 (1500), if interpreted as qq8 state, cannot have a large ss8 content. A more quantitative statement will be given below. The 4 decay modes of the f0 (1500) were observed in pp 8 annihilation at rest into 50 [136], in pn 8 annihilation into − 40 [114,115], + 2− 20 [115] and in central collisions [137]. The partial width for annihilation into 4 is about half of the total width (44 = 0:55 ± 0:05 4tot , following Ref. [115]). The ratios of decay branching ratios are given in Table 8. The 2[]S mode refers to the decay into two S-wave pion pairs. The S-wave parameterization was taken from the phase shift analyses of Ref. [126]. The relative strengths of 99 decay to 2[]S decay is of interest to understand the internal structure of the f0 (1500) meson. In the Nux tube simulation of lattice QCD one expects a glueball to decay in leading order into gluon pairs [13]. On the other hand, if the f0 (1500) is a mixture of the ground state glueball with nearby qq8 scalars, 99 decay dominates 2[]S , at least in the framework of the 3 P0 model [138]. However, the experimental situation is still unclear, since the 99=2[]S ratio from crystal barrel and WA102 disagree (Table 8).

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C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

8 For the f0 (1370) the K K= ratio is diOcult to determine precisely due to the large width of this state which cannot be easily disentangled from the f0 (600). Ref. [115] quotes values between 0.2 and 1.4 while Ref. [119] reports 0:46 ± 0:19. The 4 decay mode of the f0 (1370) is dominant [114] which also indicates that this meson cannot have a large ss8 content. Hence we are left with two nearby isoscalar mesons, both with a dominant nn8 structure. The 2002 issue of the Review of Particle Physics [3] quotes for the masses and widths f0 (1370) : M = 1200 − 1500 MeV; f0 (1500) : M = 1507 ± 5 MeV;

4 = 300 − 500 MeV ;

(31)

4 = 109 ± 7 MeV :

(32)

Let us now deal with the f0 (1710) meson. This state was 7rst observed by Crystal Ball in radiative J= decay into [139]. The spin of the f0 (1710) (J = 0 or 2) remained controversial for many years. The issue was 7nally settled in favour of 0++ by the new data from WA102 in central collisions at 450 GeV [140]. This f0 (1710) meson was discovered long before the f0 (1500) and the 2++ assignment arose from the assumption that the 1500 MeV region was dominated by the f2 (1525). As mentioned before, the K K8 coupling of the f0 (1710) is much larger than the . WA102 reports the ratio of partial widths [119] 8 4(K K)=4() = 5:0 ± 1:1 ;

(33)

2.5

120

2.25 100

m 2( πη ) [GeV 2]

2 1.75

80

1.5 60

1.25 1

40

0.75 20

0.5 0.25 0.25 0.5 0.75

0

1

1.25 1.5 1.75

m 2( πη ) [GeV 2]

2

2.25 2.5

number of events / 13.3 MeV

which, assuming a qq8 state, clearly points to a dominant ss8 structure. Nonetheless no signal for this state was reported earlier in the amplitude analysis of K − p → KS KS D interactions [141]. However, the assumption was that its spin would be 2. As we have seen, scalar mesons are strongly produced in pp 8 annihilation but the OZI rule forbids the production of pure ss8 states. The f0 (1710) was searched for in pp 8 annihilation into 0 and 0 3 with 900 MeV=c antiprotons [142]. Fig. 25 shows the Dalitz plot for pp 8 → 0 and the corresponding mass projection. The a0 (980) and a2 (1320) → and f0 (1500)=f2 (1525) →

700

D

600 500 400 300 200 100 0 1000

1200

1400

1600

1800

2000

m ( πη ) [MeV]

Fig. 25. Left: Dalitz plot for pp 8 → 0 with a0 (980) (A), a2 (1320) (B), f0 (1500)/f2 (1525) (C). The arrow (D) shows the expected location of the f0 (1710). Right: mass projection showing the f0 (1500)/f2 (1525). The shaded histogram is the 7t.

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117 102 101

0.020

100

R2

f0(1710)

80

140

0=180

10-1 10-2

40

Γγγ (f0) [MeV]

70 60

100

10

WA102 CB

f0(1500)

10-3 10-4 10-4

0.000

R1 10-3

10-2

95

10-1

100

k=15/4

k=2

0

101

30

60

90

120 150 180

α [o]

8 Fig. 26. Left: relative branching ratio R2 = B(K K)=B() vs. R1 = B()=B() as a function of mixing angle F (in deg.); right: predicted -width for the f0 (1500). The experimental upper limit is shown by the box (from Ref. [143]).

are clearly seen. However, no signal for the f0 (1710) → is observed. This is prima facie evidence that f0 (1710) cannot have a large nn8 component. 8 Fig. 26 (left) shows the SU(3) predictions for the ratio of branching ratios R2 = B(K K)=B() vs. R1 = B()=B() for scalar mesons, apart from phase space factors. Details can be found in Ref. [143]. The boxes show the data from Crystal Barrel and WA102 (2 boundaries) on the f0 (1500) and f0 (1710). The angle F describes the mixing of the two nonet isoscalar mesons, |f0 = cos F|nn 8 − sin F|ss 8

with |nn 8 ≡

uu8 + dd8 √ : 2

(34)

Hence for F = 0, f0 is pure nn8 and for F = 90◦ , pure ss8 (ideal mixing). Note that SU(3) predictions for two-body decay branching ratios of tensor mesons (F = 82◦ ) are in excellent agreement with data [13]. If one would assume that f0 (1500) and f0 (1710) are the isoscalar qq8 states, one would conclude from Fig. 26 (left) that the former is mainly nn8 and the latter mainly ss. 8 However, for f0 (1500) we shall see in the next section that this conclusion is at variance with its 2 width. 4.2.2. 2 -decay width Let us now deal with two-photon processes which are useful to probe the charge content of mesons through their electromagnetic couplings. Glueballs do not couple directly to photons and their production should therefore be suppressed in -processes. New data in -collisions have been presented by the LEP collaborations. L3 observes three peaks below 2 GeV in the KS KS mass distribution [122] (Fig. 27, left): f2 (1270) (interfering with a2 (1320)) and f2 (1525), but the spin 0 f0 (1500) is not seen. The spin of the third peak, fJ (1710) around 1765 MeV, is determined to be mainly 2 but a large spin 0 component is also present [144]. Since f0 (1500) does not couple 8 its absence in Fig. 27 (left) is perhaps not surprising. However, ALEPH studying strongly to K K, the reaction → + − , does not observe f0 (1500) either [123] (see Fig. 27, right). An upper limit of 1:4 keV (95% CL) can be derived for its -width from the ALEPH result [123], using the known decay branching ratio of the f0 (1500) [3].

96

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

Fig. 27. Left: KS KS mass distribution in -collisions at LEP/L3 (from Ref. [122]); right: + − mass distribution from LEP/ALEPH showing only the f2 (1270) (from Ref. [123]).

The -width of a qq8 state can be predicted from SU(3). Apart from an unknown nonet constant C and for a meson of mass m: √ (35) 4 = C(5cos F − 2 sin F)2 m3 : The -width of a scalar meson is related to that of the corresponding tensor by 3 m0 ++ 4 (2++ ) ; 4 (0 ) = k m2

(36)

with obvious notations. Here the factor k = 15=4 arises from spin multiplicities in a non-relativistic calculation, while relativistically k 2. Data on the charmonium states Cc2 and Cc0 are in excellent agreement with Eq. (36). The -width for scalar mesons can now be predicted as a function of F by 7rst calculating the constant C in Eq. (35) for tensor mesons, using their measured partial widths [3] and then introducing into Eq. (36). Fig. 26 (right) shows the prediction for the partial width of the f0 (1500) as a function of F, together with the ALEPH upper limit [143]. Assuming a qq8 structure, one concludes that f0 (1500) is dominantly ss, 8 at variance with the hadronic results discussed above. This contradiction indicates that f0 (1500) is not qq8 and the lack of -coupling points to a large gluonic content. For the f0 (1710), the ALEPH data are consistent with an ss8 state, although its decay branching ratio is not known. In Ref. [143] we argued that the spin 0 component in the fJ region of Fig. 27 (left) was consistent with an ss8 f0 (1710), while the spin 2 contribution arose from the (isovector) a2 (1700) radial excitation of the a2 (1320). 4.2.3. Mixing with qq8 states The most natural explanation is that f0 (1500) is the ground state glueball predicted in this mass range by lattice gauge theories. However, one would expect a pure glueball to decay into ; ; and K K8 with relative ratios 3 : 1 : 0 : 4, in contradiction with the ratios in Table 7. Mixing of the pure glueball G with the nearby two N = nn8 and S = ss8 isoscalar scalar mesons was 7rst introduced to explain the 7nite rate and the small K K8 rates observed for the f0 (1500) meson [13]. In 7rst

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

order perturbation one 7nds, assuming that the quark–gluon coupling is Navour blind, √ |G + G( 2|N + !|S) ; |f0 (1500) = 1 + G2 (2 + !2 ) where ! is the ratio of mass splittings m(G) − m(N ) : != m(G) − m(S)

97

(37)

(38)

If G lies between the two qq8 states, ! is negative and the decay to K K8 is hindered by negative interference between the decay amplitudes of the nn8 and ss8 components in Eq. (37). Conversely, the two isoscalars in the qq8 nonet acquire a gluonic admixture. In Ref. [13] G is at most 0.5 so that the nn8 state is essentially the f0 (1370) and the ss8 state the (then not yet well established spin 0) f0 (1710), both with a small glue admixture, while f0 (1500) is dominantly glue. This model was extended in Ref. [105] and applied to both Crystal Barrel and WA102 data. The mass matrix √   MG f 2f    f 0  MS (39)   √ 2f 0 MN was diagonalized and the eigenstates f0 (1710); f0 (1500) and f0 (1370) expressed as superposition of the G; N and S bare states. Here f = G|M |S and we have assumed Navour independence for simplicity (see Ref. [105] for a generalization). The best 7t to the two-pseudoscalar decay branching ratios led to the dominantly ss8 f0 (1710), while f0 (1500) and f0 (1370) share roughly equal amounts of glue ( 40%). The pure glueball was found at a mass M (G) 1440 MeV, while for pure nn8 and ss8 M (N ) 1380 MeV and M (S) 1670 MeV, respectively. Mixing with nearby qq8 isoscalar 0++ states is hence probable but not necessarily required. In fact, as much as 60% of qq8 admixture in the f0 (1500) wave function can hardly be accommodated by the ALEPH upper limit. Judging from Fig. 26 (right) one could tolerate an nn8 fraction of at most 25%. More accurate data in -collisions are needed for a more quantitative statement on mixing. Also, a systematic study of the so far not observed decay branching ratios of the f0 (1710), in particular 4 8 or K K would have to be conducted, e.g. with the COMPASS experiment at CERN. We have discussed in Section 2 the nature of the low mass scalar mesons and have concluded that they are compatible with four-quark states and meson–meson resonances. From the present discussion we suggest that the ground state qq8 nonet lies in the 1200 –1700 MeV range. Table 9 then shows the resulting classi7cation scheme for scalar mesons. 4.3. The tensor glueball The ground state 2++ nonet is well known. In this nonet the isoscalar 13 P2 mesons f2 (1270) and f2 (1525) are well established. At higher masses three to four isoscalar states appear to be solid: (i) the f2 (1565) (or AX ) observed at LEAR in pp 8 annihilation at rest [145] is perhaps the same state as f2 (1640) also reported to decay into !! [146,147]; (ii) the rather broad f2 (1950) decaying to 4 and is observed by several experiments, e.g. in central production [148] and in

98

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

Table 9 A likely classi7cation of the low-mass scalar mesons showing the scattering resonances below 1 GeV and the ground state qq8 nonet (13 P0 ). The supernumerary f0 (1500) (not shown) is dominantly glue State

4 (MeV)

Isospin

Dominant nature

a0 (980) f0 (980) f0 (600) (800)?

∼ 50 ∼ 50 ∼ 800 ∼ 600

1 0 0 1/2

8 q2 q82 K K; 8 q2 q82 K K; ; q2 q82 K; q2 q82

a0 (1450) f0 (1370) f0 (1710) K0∗ (1430)

265 ∼ 400 125 294

1 0 0 1/2

8 du; ud; 8 dd8 − uu8 8 dd + uu8 ss8 us;8 ds;8 su; 8 sd8

pp 8 annihilation at 900 MeV=c [142]; (iii) a broad structure (of perhaps several states) decaying to

was reported around 2300 MeV in − N reactions [149,150] and in central collisions [151]. The JETSET Collaboration at LEAR measuring the cross section for pp 8 →

also reported a broad enhancement at 2:2 GeV, just above the

threshold [152], a channel that should be suppressed by the OZI rule. Since the 2++ glueball is expected around 2200 MeV we 7rst discuss the experimental evidence for the narrow structure fJ (2220) (previously called G) reported around 2230 MeV. The observation of a ∼ 20 MeV broad state around 2230 MeV dates back to Mark III at SPEAR. It was seen in radiative J= decay to K + K − and KS KS [153]. The latter implied that J PC = (even)++ . More recently, this state was reported by BES at the e+ e− collider in Beijing with a mass of 2231:1 ± 3:5 MeV and a width of 23 ± 87 MeV [3]. It was observed by BES to decay into + − , K + K − ; KS KS ; pp 8 [154] and 0 0 [155] with statistical signi7cance of about 4 in each decay mode. Several features made this state an attractive candidate for the 2++ glueball: (i) its mass which agrees with lattice predictions (although the 2++ assignment has not really been proven); (ii) its unusually narrow width for a qq8 excitation; (iii) its observation in the gluon rich environment of J= radiative 8 in line with Navour independence; (v) its decay; (iv) its comparable partial widths to and K K, non-observation in collisions [156]. According to BES the fJ (2220) meson decays to pp 8 at BES and hence should be observed in pp 8 formation experiments. However, all searches in pp 8 → G → 2; K K8 and

have been negative so far. Crystal Barrel at LEAR has searched for narrow states decaying to 0 0 and (leading to 4 ) as a function of p8 momentum [157]. Fig. 28 shows the cross sections for nine momenta in the mass range of the G. The resolution was about ±0:6 MeV in the c.m.s. system. No structure was observed. Using the product of branching fractions B(J= → G)B(G → pp; 8 0 0 ) measured by BES and the 95% CL upper limit of 6 × 10−5 for B(pp 8 → G)B(G → 0 0 ) measured by Crystal Barrel, one 7nds that the observed decays amount to at most 4% of all G decays, hence most G decay channels have not been observed yet. Furthermore, B(J= → G) ¿ 3 × 10−3 which is comparable to the branching ratio for the known decay J= → . A striking is observed in the inclusive J= decay spectrum, while, however, G is not seen [158]. Hence the data are inconsistent: the pp 8 decay width measured at BES appears to be too large or the narrow G simply does not exist.

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

99

Fig. 28. Cross section for pp 8 → 20 (top) and 2 (bottom) for cos H ¡ 0:85). The curves are straight line 7ts (from Ref. [157]).

Above the f2 (1525), none of the nine reported isoscalars [3] can be de7nitely assigned to the expected radial or orbital excitations in the expected 23 P2 ; 33 P2 or 13 F2 nonets. Therefore, the identi7cation of the tensor glueball is premature. A systematic study of the two-body channels 8 and , similar to the one performed for scalar mesons at lower energy, would have ; K K; to be conducted. 4.4. The pseudoscalar glueball The evidence for a 0−+ state around 1400 MeV dates back to the sixties. The then called E-meson 8 mass spectrum of pp 8 was observed in the K K 8 annihilation at rest into K K3 [159]. This state was reported to decay through a0 (980) and K ∗ (892)K8 with roughly equal contributions. The quantum numbers of the E-meson (now called (1440)) remained controversial as the experimental evidence from − p peripheral reactions led to a 1++ state, the f1 (1420). The (1440) was later 8 [160]. Since radiaobserved as a broad structure around 1400 MeV in radiative J= decay to K K tive J= decay to light quark proceeds through an OZI forbidden process, namely the annihilation of both (cc) 8 quarks, the rather large production of (1440) (then called –) was indicative of a strong gluon–gluon interaction, presumably leading to the formation of a glueball. 8 through the intermediate a0 (980) channel and In J= radiative decay the (1440) decays to K K hence a signal was also to be expected in the a0 (980) → mass spectrum. This was indeed observed by Mark III, reporting a signal at 1400 ± 6 MeV (4 = 47 ± 13 MeV) [161] and also in pp 8 annihilation at rest. Crystal Barrel observed the (1440) in the reaction pp 8 → (+ − )0 0 0 0 + − and ( ) [162]. Fig. 29 shows the two mass distributions containing together roughly 9000 (1440) decays. The average mass between the neutral and charged channels was found to be 1409 ± 3 MeV and the width 4 = 86 ± 10 MeV. The quantum numbers were determined to be 0−+ and the observation of the 0 0 decay mode proved that the (1440) was indeed an isoscalar.

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

Events / 20 MeV

x 103

x 103 η(1440)

η’

12

8 η(1440)

4

0

800

(a)

1200 m (π0π0η) [MeV]

Events / 20 MeV

100

6

2

0

1600

η’

4

800

(b)

1200 m (π+π-η) [MeV]

1600

Fig. 29. 0 0 mass distribution (left) and + − (right) in pp 8 → 20 + − . The dashed line shows the result of the partial wave analysis (from Ref. [162]).

Table 10 The (1440) splits into two pseudoscalar mesons, L and H . The slightly di9erent masses for two decay modes of the L are not considered to be signi7cant State

Mass (MeV)

Width (MeV)

Decays

L L H

1405 ± 5 1418 ± 2 1475 ± 5

56 ± 6 58 ± 4 81 ± 11

(a0 (980) and ()S ) 8 (a0 (980) dominant) K K 8 (K ∗ K8 dominant) K K

There is now evidence for the existence of two pseudoscalars in the (1440) region which are called L and H by the Particle Data Group [3]. The L around 1410 MeV decays into (through a0 (980) or ()S , where ()S is an S-wave dipion). The H around 1480 MeV decays mainly to 8 In addition, the axial f1 (1420) also contributes to the K K 8 7nal state. The simultaneous K ∗ (892)K. observation of the two pseudoscalars L and H is reported with three production mechanisms [3]: peripheral − p reactions, radiative J= (1S) decay, and pp 8 annihilation at rest. All of them give values for the masses, widths and decay modes in reasonable agreement, with the exception of DM2 8 above the H → K ∗ K8 [163]. which 7nds the L → K K The 1400 MeV region is extremely complicated, due to the presence of both the K ∗ K8 thresh8 channel. The average of all old at 1390 MeV and the a0 (980) at the K K8 threshold in the K K measurements, following Ref. [3], is given in Table 10. Systematic e9ects and model dependence are probably important. Therefore the error scaling factors between di9erent experiments analysing di9erent reactions are large, which is presumably the reason why the two states are not yet explicitly divided into (1410) and (1480) in Ref. [3]. However, the presence of a pseudoscalar doublet is highly suggestive. Axial (1++ ) states, such as the f1 (1420) meson, are diOcult to observe in pp 8 annihilation at rest, because annihilation proceeds mainly through the atomic S-states which are dominantly populated in liquid hydrogen targets. However, these states can be observed using gaseous targets in which annihilation from atomic P-states is enhanced [164]. The Obelix Collaboration at LEAR has analysed

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117 0-+ a0(980)π

0-+KK*

101

1++KK*

Events / 0.03GeV

400 300 200 100 0

1.4

1.5

1.4 1.5 m (KKπ) [GeV]

1.4

1.5

8 mass distribution for pp Fig. 30. Partial wave analysis of the K K 8 annihilation into K ± K 0 ∓ + − showing the contri8 and from the 1++ wave to K ∗ K8 (adapted from Ref. [167]). butions from the 0−+ wave to a0 (980) and K ∗ K,

8 mass spectrum in pp 8 the K K 8 annihilation at rest into K K3 in liquid and in gas. They observe the L and H [165] but also the f1 (1420) recoiling into an S-wave dipion [166–168]. The three resonating contributions in the partial wave analysis are shown in Fig. 30. One of the two pseudoscalars could be the 7rst radial excitation of the , with the (1295) being the 7rst radial excitation of the . Ideal mixing is suggested by the nearly equal masses of the (1295) and (1300) which then implies that the high mass isoscalar in the nonet is mainly ss8 8 in agreement with the H . Assuming that H is mainly ss8 and (1295) and hence couples to K ∗ K, mainly nn, 8 one furthermore predicts from the mass formula that the mass of the strange member in the nonet would be about 1400 MeV, in agreement with the mass of the so-called K(1460) [3]. The mass of the latter is, however, poorly established. Finally, we note that the H width is in accord with expectation from the 3 P0 model for the radially excited ss8 state [169,170]. The 2 -width of the H was observed by L3 at LEP in the reaction → KS K ± ∓ [171]. Fig. 31 shows the KS K ± ∓ mass distribution for various transverse momenta. At small transverse momenta the photons are quasi-real and therefore the production of spin 1 states are forbidden by the Yang theorem. A high mass (but not low mass) pseudoscalar is observed. The H is observed with a mass of 1481 ± 12 MeV and a width of 48 ± 9 MeV. At high transverse momentum the photons become virtual and the distribution is dominated by the axial f1 (1285) and f1 (1420) mesons (see 8 (212 ± 60 eV) is in agreement Ref. [171]). The partial width for 2 production and decay into K K with H being the 7rst radially excited state of the (958) [172]. The L state therefore appears to be supernumerary. An exotic interpretation was proposed, perhaps gluonium mixed with qq8 [169] or possibly a bound state of gluinos [173]. Note that the L is also not observed in → + − [171]. This, however, does not argue in favour of a gluonium nature for the L since the (1295) is not seen either. Also, the radiative decay partial width into + − width has been measured to be rather large for a gluonium candidate: crystal barrel reports a ratio of + − to + − widths of 0:111 ± 0:064 [174]. Finally, the gluonium interpretation is also not favoured by lattice gauge theories, which predict the 0−+ ground state glueball to lie above 2 GeV. However, a low mass pseudoscalar glueball is possible in gluonic Nuxtubes [175]. To summarize this section, there is strong evidence for the presence of two overlapping pseudoscalar isoscalar mesons around 1400 MeV, separated in mass by about 50 MeV. One of them,

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Fig. 31. KS K ± ∓ mass distribution in 2 -collisions for di9erent intervals in transverse momentum PT . At low PT the 0−+ H is produced while at large PT the 1++ f1 (1285) and f1 (1420) dominate (from Ref. [171]).

probably the low mass state, is of a di9erent nature than qq, 8 but the two states overlap and hence are likely to mix. This situation is reminiscent to that of the scalar spectrum around 1500 MeV, here with the additional complication from axial vector mesons contributing to the same 7nal states. The 8 and , but its precise decay lower mass pseudoscalar state around 1405 MeV decays into K K 8 branching ratios into a0 (980); ()S and direct K K have not been established unambiguously. 8 A comprehensive study of this compliThe higher mass state around 1480 MeV decays into K ∗ K. cated spectrum will require large statistical samples in J= radiative decays such as those expected from CESR running at the (2S). 5. Hybrid mesons 5.1. Theoretical predictions According to the Nux tube model, hybrid mesons should lie in the 1:9 GeV region. Eight nearly mass degenerated nonets with quantum numbers J PC =0±∓ ; 1±∓ ; 2±∓ and 1±± have been predicted [176–178]. Lattice QCD also predicts the lightest hybrid, an exotic 1−+ , at a mass of 1:9 ± 0:2 GeV [179,180]. However, the bag model predicts the four nonets 0−+ ; 1−− ; 2−+ and the exotic 1−+ at a much lower mass, around 1:4 GeV [14,181]. Hybrids have distinctive decay patterns. They are expected to decay mainly into pairs of S- and P-wave mesons (for example the 1−+ state into

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

103

m1 JPC

π, K

L

X

m2

Y p

p

Fig. 32. Peripheral p or Kp reaction leading to the production of a resonance X with quantum numbers J PC which in turn decays into two mesons m1 and m2 .

f1 (1285); b1 (1235)), while the decay into two S-wave mesons is suppressed [182]. Most hybrids are rather broad but some can be as narrow as 100 MeV [183]. In contrast to glueballs, hybrids can have isospin 0 and 1. A state with quantum numbers 1−+ does not couple to qq: 8 for J PC = 1−+ the angular momentum ‘ between the quark and the anti-quark must be even, since P = −(−1)‘ . The positive C-parity then requires the total quark spin s to be zero, since C = (−1)‘+s . This then implies J = ‘ and therefore excludes J = 1. Likewise, it is easy to show that the quantum numbers 0−− ; 0+− and 2+− do not couple to qq8 either. The discovery of a state with such quantum numbers would prove unambiguously the existence of exotic (non-qq) 8 mesons. We brieNy describe the nomenclature used in the framework of the isobar model for the partial wave analysis of peripheral p or Kp reactions of the type ab → Xc, where a is the incident or K and X a resonance with quantum numbers J PC decaying into two mesons m1 and m2 (Fig. 32). Details can be found in the literature [184,185]. Since parity P is conserved in strong interaction physics, so is the reNection R = P exp(−iJy ) :

(40)

The quantization axis z is chosen in the direction of the incident particle a, seen from the rest frame of the decaying resonance X . The y-axis around which the rotation is performed in Eq. (40) is chosen orthogonal to the plane spanned by z and the direction of the resonance decay daughters in the rest frame of X , This is the so-called Gottfried–Jackson reference frame. To avoid negative values of the spin projection M , it is convenient to express the eigenstates of R in the so-called reNectivity basis. They are |jM = M(M )(|M − jP(−1)J −M | − M ) ;

√

(41)

where P is the parity of the resonance X; M = 1= 2 or 1/2 for M ¿ 0 or M = 0, respectively. The reNectivity j is +1 for natural parity and −1 for unnatural parity exchanges of the meson Y . 1 For M = 0 the reNectivity j = P(−1)J IS excluded, since the eigenstate (41) vanishes. For peripheral reactions of the type shown in Fig. 32 and for incident spin zero particles like or K the projection M is 0 or 1. If one now assumes that, say m1 , is a meson resonance, one can 1

For natural parity exchange the meson Y has quantum numbers J P = 0+ ; 1− ; 2+ , etc., while for unnatural parity exchange J P is 0− ; 1+ ; 2− , etc.

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characterize the partial waves by their quantum numbers J PC [m1 ]LM j , where L is the relative angular momentum between m1 and m2 . A particular case arises when both m1 and m2 are pseudoscalar mesons, in which case J (X ) = L and P = (−1)L . The eigenstates of R are then |jM = M(M )(|M − j(−1)M | − M ) :

(42)

Obviously, for M = 0 j = +1 is excluded. The contributing partial waves Lj are then P+ ; D+ ; F+ ; etc:; for M = 1 and natural parity exchange ; S0 ; P0 ; D0 ; etc:; for M = 0 and unnatural parity exchange ; P− ; D− ; F− ; etc:; for M = 1 and unnatural parity exchange :

(43)

For example, for an exotic 1−+ meson decaying into (L = 1), the contributing partial waves are P+ for natural parity exchange, P0 and P− for unnatural parity exchanges. 5.2. A 1−+ exotic meson, the 1 (1400) The decay channel is a favourable one to search for 1−+ hybrid, in which case the two pseudoscalar mesons would be in a relative P-wave. The state would be isovector and hence could not be confused with a glueball. Both neutral and charged decays (0 and ± ) should be observed. The J PC = 1−+ exotic meson decaying to − , called 1 (1400), was reported in the reaction − p → − p at 18:3 GeV=c by the E852 collaboration using the Multi-Particle Spectrometer (MPS) at the AGS [186,187]. A sketch of the MPS, a large angle magnetic spectrometer, is shown in Fig. 33. The was detected in its decay mode (47,235 events). Results for the → + − 0 decay mode (2,235 events) are statistically less signi7cant but consistent [186]. The 1 (1400) was observed

Fig. 33. Sketch of the MPS spectrometer used by the E852 Collaboration at the AGS (after Ref. [187]). 1 – cylindrical drift chambers surrounding the 30 cm long hydrogen target; 2 – array of 198 CsI(Tl) crystals; 3 – multiwire proportional chambers; 4 – drift chambers; 5 – array of 3054 lead-glass crystals; 6 – lead-scintillator veto; 7 – beam veto scintillation counters.

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

105

Fig. 34. Acceptance corrected forward–backward asymmetry for emission in the Gottfried–Jackson frame as a function of mass (after Ref. [187]).

Fig. 35. D+ (a) and P+ (b) intensities for the − system in − p → − p at 18 GeV (after Ref. [187]).

as an interference between the angular momentum L = 1 and L = 2 amplitudes, leading to a forward/backward asymmetry in the angular distribution (Fig. 34). The (natural parity exchange) D+ and the exotic P+ intensities are shown in Fig. 35. The peak in the D+ amplitude is due to the a2 (1320) meson, the peak in the P+ amplitude is due to the exotic 1 (1400). The exotic intensity is a small fraction (about 3%) of the dominating a2 (1320) contribution. There is an eightfold ambiguity in the 7t (central error bars in Fig. 36), which was already noticed earlier in the partial wave analysis of this reaction [188,189]. Contributions from unnatural parity exchanges were found to be small. Fig. 36 shows the phase movement as a function of mass. The mass and width of the 1 (1400) are given in Table 11. The 1 (1400) state (called 9(1405) ˆ previously), was reported earlier by the GAMS collaboration in − p reactions at 100 GeV=c [190]. Mass and width are given in Table 11. However, the enhancement was observed in the (unnatural parity exchange) P0 wave. Ambiguous solutions in the partial wave analysis were pointed out in Refs. [188,189]. Clear enhancements in the P+ wave were also reported at 6:3 GeV=c [191] and 37 GeV=c [192], although the evidence for an actual resonance was not deemed to be conclusive.

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Fig. 36. Phase movements of the D+ and P+ waves, as a function of mass; 1–D+ phase, 2–P+ phase, 3–relative phase (after Ref. [187]).

Table 11 Mass and width (in MeV) of 1−+ exotic mesons. The 7rst error is statistical, the second represents the systematic uncertainties State

Reaction

Mass (MeV)

Width (MeV)

Ref.

1 (1400)

− p → − p − p → 0 n pn 8 → 0 − pp 8 → 0 0

1370 ± 16+50 −30 1406 ± 20 1400 ± 28, 1360 ± 25,

385 ± 40+65 −105 180 ± 20 310+71 −58 220 ± 90

[186,187] [190] [195] [196]

1 (1600)

− p → 90 − p − p → 90 − p − p → − p − p → [b1 (1235); ; 9]− p

1593 ± 8+29 −47 1620 ± 20 1597 ± 10+45 −10 1560 ± 60

168 ± 20+150 −12 240 ± 50 340 ± 40 ± 50 340 ± 50

[198,199] [201] [202] [200]

On the other hand, an analysis of the reaction − p → 0 n at 18:3 GeV=c was performed [193]. The data were also collected by the E852 collaboration at the MPS. The and 0 were both reconstructed from their observed decays to 2 (45,000 events). An exotic P-wave similar to the one reported for − p [186,187] was found. However, the resonance behaviour was not compelling. The authors of Ref. [193] pointed out that the inclusion of the M = 0 and 1 contribution, i.e. unnatural parity exchange, did not lead to a consistent set of Breit–Wigner parameters. The crystal barrel collaboration at LEAR also searched for a 1−+ resonance in the P-wave in low energy pp 8 annihilation into . For pp 8 → 0 0 (isospin I =0) and with stopping antiprotons, annihilation proceeds through the initial pp 8 atomic states. In liquid hydrogen and for the intermediate 1 (1400) → 0 this is mainly 1 S0 with some, presumably small contribution from 3 P1 , since density e9ects enhance S-wave annihilation. For pn 8 → − 0 (isospin I = 1) and with stopping antiprotons in liquid deuterium, the dominating initial states are 3 S1 and 3 P1 . Here the spectator proton may remove angular momentum and hence the inclusion of initial P-waves becomes mandatory.

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

107

Fig. 37. Dalitz plot of pn 8 → − 0 (after Ref. [195]).

A weak 1−+ structure with poorly de7ned mass and width was 7rst reported in pp 8 → 0 0 1 [194]. However, the production of 1 (1400) could be suppressed from S0 but enhanced from 3 S1 . The reaction pn 8 → − 0 was therefore studied in deuterium [195]. The increasing complexity in the amplitude analysis due to P-waves is compensated by the absence of 0++ isocalars which do not contribute to − 0 . Events were selected with a single − and 0 → 4 . Spectator protons of less than 100 MeV=c were required, hence not escaping from the deuterium target. The channel − 0 could thus be treated as quasifree, thereby avoiding 7nal state rescattering with the proton. The Dalitz plot (52,567 events) is shown in Fig. 37. The accumulation of events in the mass regions around 1300 MeV above the 9 band indicates the presence of interferences between a2 (1320) and some other amplitude. The 7t could not describe the observed interference pattern without the inclusion of a resonant P-wave. The accumulation of events above the 9 (visible in Fig. 37) also leads to a forward=backward asymmetry in the rest frame along the a2 (1320) band. Mass and widths given in Table 11 are in good agreement with the results from E852 [186,187]. The contribution of 1 (1400) to the − 0 channel was 34% of the dominating 3 S1 contribution, hence much larger than for the − p → − p reaction of Refs. [186,187]. The channel pp 8 → 0 0 was studied again with annihilation in liquid hydrogen (280,000 events) but now using also annihilation in high pressure gas (270,000 events), in which the contribution of P-waves was enhanced [196]. Both data sample were 7tted simultaneously, but the relative contribution from S- and P-waves was 7xed by atomic cascade calculations [197]. The 1 (1400) was observed dominantly from the 3 P1 atomic state with a small contribution from 1 S0 which explained the weak signal reported earlier in liquid [194]. The mass and width are given in Table 11. It appears that 1 (1400) is mostly produced from pp 8 spin triplet states (3 S1 or 3 P1 ).

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1

+ρ [ ] P 0 ,1

Intensity

4000

1 (a)

+ ρ [ ] P 1+ (b)

1500 1000

2000

500 0

1.0

1.5

0

2.0

1.0

1.5

2.0

Mass [GeV]

Fig. 38. Exotic 1−+ contribution to 9 in − p → − − + p at 18 GeV for (a) unnatural (M = 0 and 1) and (b) natural (M = 1) parity exchanges. The dark histogram shows the background contribution (after Ref. [199]).

3

ϕ [rad]

2

1

1

2 0

1.5

1.6 1.7 Mass [GeV]

1.8

Fig. 39. Phase motion of the natural parity exchange 1−+ 9(770) wave (curve 1) and 2−+ f2 (1270) wave (curve2, after Ref. [199]).

5.3. Another 1−+ exotic meson, the 1 (1600) Another 1−+ state, 1 (1600), was reported to decay into 9 [198,199]. It was observed by the E852 collaboration in the peripheral reaction − p → − − + p at 18:3 GeV=c. Contaminating reactions involving excited nucleons (e.g. O++ → p0 ) could be removed with the arrays of CsI(Tl) and lead-glass calorimeters vetoing s from 0 -decay. The partial wave analysis was based on 250,000 reconstructed events. Apart from the known mesons (f2 (1270); a2 (1320), 2 (1670), etc.) a resonating partial wave was found in the exotic waves 1−+ [9(770)]P0− and 1−+ [9(770)]P1− (unnatural parity exchange) and 1−+ [9(770)]P1+ (natural parity exchange). This is shown in Fig. 38. Accordingly, this resonance was named 1 (1600). No statement on the 1 (1400) → 9 could be made below 1500 MeV due to leakage from other partial waves. A rapid phase movement was observed for the M j = 1+ wave with respect to all other significant natural parity exchange waves. Fig. 39 shows for example the 1−+ [9(770)]P1+ phase motion, resonating around 1600 MeV, and that of the 2−+ [f2 (1270)]S0+ , resonating at the 2 (1670).

C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61 – 117

109

No statement could be made on a phase advance of the unnatural parity contribution to the formation of a 1−+ resonance since contributions from unnatural parity exchanges (f1 or b1 ) were small for all partial waves. The mass and width of the 1 (1600) meson are given in Table 11. The VES collaboration at IHEP studied the same reaction − A → − − + A but at 36:6 GeV=c, using a beryllium target and a large aperture magnetic spectrometer [200]. Preliminary results on this channel were reported earlier by VES [201]. In the − 90 channel they reported a broad shoulder around 1:6 GeV for the (natural parity) 1−+ [9(770)]P1+ wave while the contributions from unnatural parity exchanges remained negligible. The mass and width from Ref. [201] are given in Table 11. However, the author of Ref. [200] warns that the intensity and width of the 1:6 GeV enhancement is quite sensitive to details of the partial wave analysis. Also, the 1−+ contribution is small, about 2% to the total intensity. Note that the 3 system is rather complicated with 27 contributing partial waves in the simplest 7ts [199]. A much simpler channel is the one for which 1 (1600) decays into two pseudoscalar mesons. The 1 (1600) decaying into was reported by the E852 collaboration in the reaction − p → − p at 18:3 GeV=c [202]. The was reconstructed through its decay mode → + − with → . The photons were detected in the lead-glass array. A sample of 6,040 − p events were collected. The main contributing amplitudes were the (natural parity) P+ ; D+ and G+ waves. Fig. 40 shows the exotic P+ intensity and D+ contributions. The former intensity is dominant in the 1500 –1800 MeV range, reaching a maximum around 1600 MeV, while the latter peaks at the a2 (1320). There is a weak indication of the 1 (1400) in the P+ intensity, although the 7t does not require it. Table 11 also gives the reported mass and width of 1 (1600) in the decay mode. The VES collaboration observed earlier a broad enhancement in the exotic 1−+ wave at 37 GeV [192] but mass and width were not given. Fig. 41 shows the P+ wave for and (contributions from P− and P0 are negligible). The contribution exceeds in the 1600 MeV region, although phase space favours the latter. This would favour hybrids over q2 q82 states [203] in this mass region. A signal for 1 (1600) → b1 (1235) was also reported by VES [200] and a combined 7t to the b1 (1235), and 9(770) data was performed [200]. Mass and width of the 1 (1600) are compatible with the results of E852 (Table 11). Furthermore, the 1 (1600) decay branching ratios to the three 7nal states b1 (1235); and 9(770) are of compatible strength, 1: 1:0 ± 0:3 : 1:5 ± 0:5. The experimental errors are quite large, and the predicted dominance of b1 (1235) for hybrid states cannot be excluded.

Events / 0.05GeV

1500 P+

D+

1000

500

0

1.5

2.0

2.5 1.5 M (η'π ) [GeV]

2.0

2.5

Fig. 40. Intensity of the P+ (left) and D+ (right) partial waves in the reaction − p → − p (after Ref. [202]).

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|T|

2

6

η'π ηπ

4

2

0

1.00

1.25 1.50 Mass [GeV]

1.75

2.00

Fig. 41. Intensity of the P+ wave in − A → − A and − A → − A at 37 GeV (after Ref. [192]).

Evidence for the 1 (1600) decaying to b1 (1235) is also reported from pp 8 annihilation at rest into !+ − 0 [132]. Summarizing this part, we now have evidence for two 1−+ exotics, 1 (1400) and 1 (1600) from peripheral reactions and antiproton annihilation. However, and rescattering e9ects appear to be large and a non-resonant interpretation for the 1−+ wave has been suggested [204]. This a9ects especially 1 (1400). The 1−+ signals are rather small in peripheral reactions and an imperfect description of the experimental acceptance, or of the dominating a2 (1320) meson, could mimic a resonance at 1400 MeV. However, the signal for 1 (1400) is rather strong in pn 8 and pp 8 annihilation and directly visible in the Dalitz plots. In Ref. [205] it is suggested that a Deck generated background from 7nal state rescattering in 1 (1600) decay could mimick 1 (1400). However, this mechanism is absent in pp 8 annihilation. The data require 1 (1400) and cannot accommodate a state at 1600 MeV [206]. Hence antiproton annihilation data argue for the existence of 1 (1400). As isovectors, 1 (1400) and 1 (1600) cannot be glueballs. The coupling to of the former points to a four-quark state while the strong coupling of the latter is favored for hybrid states [207,208]. As mentioned already, the Nux tube model and lattice calculations concur to predict a mass of about 1:9 GeV. The 1 (1600) mass is not far below these predictions. Note that a 1−+ structure around 2 GeV decaying to f1 (1285) was reported by one experiment [209]. 5.3.1. Other hybrid candidates Hybrid candidates with quantum numbers 0−+ ; 1−− , and 2−+ have also been reported. The (1800) decays mostly to a pair of S- and P-wave mesons [210]), in line with expectations for a 0−+ hybrid meson, although recent data contradict this, indicating a strong 9! decay mode [211]. This meson is also rather narrow if interpreted as the second radial excitation of the pion. The evidence for 1−− hybrids required in e+ e− annihilation and in . decays was discussed in Ref. [212]. A candidate for the 2−+ hybrid, the 2 (1870), was reported in interactions [213], in pp 8 annihilation [214] and in central production [215]. The near degeneracy of 2 (1645) and 2 (1670) suggests ideal mixing in the 2−+ qq8 nonet and hence the second isoscalar should be mainly ss. 8 Data for K ∗ K8 decay are unfortunately not available for such high masses. However, 2 (1870) decays into a2 (1320) and f2 (1270) with a relative rate of 4:1 ± 2:3 [214] or 20:4 ± 6:6 [216]. These large numbers are compatible with a predicted ratio of 6 for a hybrid state [178].

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111

6. Conclusions and outlook The light scalars below 1 GeV have for long been considered as good candidates for non-qq8 states. The measured radiative decay widths to and from DA NE [55,54] are not compatible with f0 (980) and a0 (980) being qq8 states [44]. The f0 (600) could belong to the same Navour nonet, since the large di9erence in width compared to f0 (980) and a0 (980) might be due to the wide open phase space for the former, and to the K K8 threshold distortions for the latter states. There are theoretical arguments in favour of a light and broad (800) pole near the K threshold. However, the experimental evidence is not conclusive. A more detailed analysis of experimental data using the K-matrix formalism [125] rather than Breit–Wigner amplitudes will be required. This remark also applies to the f0 (600) and, in general, to all overlapping broad resonances. In a recent topical review [72], it was suggested that the lightest scalars are at the central core composed of a four quarks. Following Ja9e’s QCD arguments this central core would consist predominantly of a four-quark (qq)38(q8q) 8 3 state. At larger distances from the core the four quarks would then recombine to a pair of colour singlet qq’s, 8 building two pseudoscalar mesons as a meson cloud at the periphery. Better experiments with adequate theoretical treatments of the f0 (600) pole and the likely (800) pole in D and Ds decays are important. A better understanding of the nature of the recently discovered ∗ (2317) by BABAR [4] and the D (2460) by CLEO [93], which lie slightly below very narrow DsJ sJ the 7rst allowed strong S-wave (DK, respectively D∗ K) threshold, is likely to throw new light, also on the light scalar sector. There is now strong experimental evidence that the lightest qq8 scalar nonet (13 P0 ) consists of the mesons a0 (1450); K0∗ (1430); f0 (1370) and f0 (1710). From hadronic reactions the f0 (1710) appears to be made dominantly of ss8 quarks [105]. The nonet mixing angle is not far from ideal mixing [143]. However, the decay branching ratios of the f0 (1710) to two pseudoscalar mesons have been measured by one experiment only [119,133]. They should be checked, e.g. with the COMPASS 8 experiment at CERN. Furthermore, the strongest decay channels, presumably into 4 and K K, have not been observed yet. Branching fractions are important to (i) determine the nonet mixing angle and (ii) to establish the partial width of the f0 (1710). The K K8 and the upper limit for the partial widths of the f0 (1500) are not compatible with a qq8 state [143]. The absence of signal in suggests that f0 (1500) contains a large fraction of glue. Mixing with qq8 is likely [13]. On the other hand, data in collision are statistically limited. Much better data in → K K8 and are required to pin down the fraction of glue in the wave function. The a0 (1450) was observed so far in pp 8 annihilation only, presumably because the production branching ratio is rather small. Experimental data [127,129] argue for a high mass a0 (1450) and against a low mass state around 1300 MeV. The a0 (1450) should also be observed in collisions, in particular decaying to KS KS . However, the branching ratio for a0 (1450) → K K8 is not known. An upper limit of 33% can be deduced from SU(3), in accord with data [129]. Data from LEP (see Fig. 27) are consistent with a0 (1450) for a branching ratio to K K8 of about 10% or less. Thus, a search for many body decays of the a0 (1450) e.g. to 3 or 5 would be useful. In fact, the decay branching ratio into !9 seems rather large [132]. These channels are diOcult but data in collision to would be very useful, since the decay branching ratio of a0 (1450) is comparable to that for K K8 [129].

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The f0 (1500) should be observed in J= radiative decay, which is traditionally believed to enhance gluonium production. A scalar state is indeed observed in this region [121,218] but the data are statistically limited and do not allow a proper treatment with the K-matrix formalism. Large statistical samples will hopefully become available with the commissioning of CLEO-III at Cornell. For a complete theoretical treatment of the light scalars one would need to include the qq8 states, the possible four-quark states, and a glueball as seed states, and treat all Navour related states simultaneously in order to constrain the parameters. Nonperturbative e9ects from the strongly coupled two-pseudoscalar channels would be included through unitarization, in a way consistent with at least unitarity, analyticity, Navour and chiral symmetry. They would mix the bare resonance states (seeds) with themselves and with the meson–meson continuum and distort any naive mass spectrum and tree-level resonance widths. This would be clearly an ambitious program. Quoting Ja9e [217]: “It would be wonderful to have a uni7ed quark/hadron description of scattering with (i) open (uncon7ned), (ii) closed (kinematically forbidden), and (iii) con7ned channels”. Such a program should not be too diOcult to realize with modern computers. In the tensor sector, the identi7cation of the 2++ glueball is premature. The 7rst radially excited nonet (23 P2 ) is not established, although candidates exist (see Fig. 1). A high statistics systematic investigation akin to the one performed in pp 8 annihilation at rest [40] is called for in high energy − − p or K p interactions between 1500 and 2300 MeV, e.g. with the COMPASS experiment at CERN. The nature of the f2 (1565) is unclear. It could be a 99 + !! molecule (see Table 2). However, it is observed only in pp 8 annihilation [3] and as such could be a deeply bound nucleon– antinucleon state [11]. Here also, good data in radiative J= decays will hopefully settle the issue on the existence and quantum numbers of a narrow state around 2200 MeV. The nature of the (1440) is unclear. In the eighties, this state observed in the gluon rich radiative J= decay process was considered a prime candidate for the ground state scalar glueball. However, lattice gauge theories later predicted the 0−+ glueball to lie around 2:5 GeV (see Fig. 21). On the experimental side little progress was made due to the lack of good statistics data, but the experimental evidence now points to the existence of two pseudoscalar states in this mass region, one around 1410 MeV, the other around 1480 MeV [3,167]. This situation is likely to improve with the commissioning of CLEO-III. The high mass state in the (1440) mass region decays mainly to K ∗ K8 and is hence consistent with being the radially excited ss8 pseudoscalar state. On the other hand, the 7rst radially excited pseudoscalar nonet (21 S0 ) is not well established (see Fig. 1). The other isoscalar, the (1295) was reported so far only in peripheral − p reactions [3]. It is not observed in pp 8 annihilation, in contrast to (1440). Also the K(1460) → K is poorly established. This calls for new attempts to investigate the strange meson sector. The (1440) structure is furthermore complicated by the presence of an axial vector, the f1 (1420). 8 ∗ molecular state (see Table 2) is unclear. Whether this state is the ss8 state of the 1++ nonet or a KK In the latter case the elusive f1 (1510) [9] could be the ss8 state. Experiments are called for in which both the f1 (1420) and the f1 (1510) are observed simultaneously. In the hybrid sector, the exotic 1 (1600) seems well established. Good data for the channel in pd 8 or pp 8 would be very valuable. The 1 (1400) is strongly observed in antiproton annihilations but its resonant nature is currently being debated. On the other hand, according to theoretical predictions, 1−+ hybrids should be observed at a mass of about 1:9 GeV [176,179]. The issue of 1−+ exotics will be addressed further in photoproduction at CEBAF.

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113

In the non-exotic sector the nature of the 2 (1870) is unclear. Its decay fractions into a2 (1320) and f2 (1270) are consistent with expectations for a 2−+ hybrid. However, it could be the ss8 member of the 2−+ nonet in which case a strong signal should appear in the KK ∗ decay channel. On the other hand, the mass region above 1500 MeV is very complicated due to the overlapping of broad resonances and the opening of two-body thresholds. Charmed hybrids should be easier to identify since predictions for the qq8 charmonium spectrum are believed to be quite reliable. Furthermore, charmed hybrids are expected to be narrow if they lie below the DD1 (2420) threshold, since decay into a pair of S-wave mesons is suppressed. For instance, in the bag model one expects the sequence m(0++ ) ¡ m(1−+ ) ¡ m(1−− ) ¡ m(2−+ ) [219]. However, complications due to threshold e9ects and surprises can be expected. Charmed hybrids will be investigated at the planned GSI facility. Acknowledgements N.A.T. acknowledges partial support from the EU grant HPRN-CT-2002-00311 (Eurodice). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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Physics Reports 389 (2004) 119 – 159 www.elsevier.com/locate/physrep

Dynamics of ionization in atomic collisions S.Yu. Ovchinnikova; b; c;∗;1 , G.N. Ogurtsovc;2 , J.H. Maceka; b;1 , Yu.S. Gordeevc a

Department of Physics and Astronomy, 401 Nielson Bldg., University of Tennessee, Knoxville, TN 37996-1200, USA b Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA c A.F.Io0e Physico-Technical Institute, St.-Petersburg 194021, Russia Accepted 22 September 2003 editor: J. Eichler

Abstract The present state of the theoretical study of ionization in ion–atom and atom–atom collisions is reviewed on the basis of quantum mechanical approaches to the solution of the time-dependent Schr7odinger equation. Perturbative theories as well as the methods employing exact numerical solutions of the Schr7odinger equation, expansion of wave functions on atomic and molecular bases and Sturmian expansions are considered. Advantages and limitations of these methods are assessed for colliding systems with one “active” electron, e.g. H+ –H and He2+ –H. Comparison of calculations with available experimental data is given in a broad collision energy range. Perspectives for further developments are discussed. c 2003 Elsevier B.V. All rights reserved. PACS: 34.50.Fa; 34.50.Pi; 34.80.Dp; 34.20.Mq; 34.10.+x Keywords: Atomic collision; Ionization; Electron spectra

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2. Time-dependent theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3. Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 ∗

Corresponding author. Department of Physics and Astronomy, 401 Nielson Bldg., University of Tennessee, Knoxville, TN 37996-1200, USA. E-mail addresses: [email protected] (S.Yu. Ovchinnikov), [email protected] (G.N. Ogurtsov), [email protected] (J.H. Macek), [email protected] (Yu.S. Gordeev). 1 Supported by INTAS under Grant No 2001-0155. 2 Supported by the Chemical Science, Geosciences and Biosciences Division, OIce of Basic Energy Science, OIce of Science, US Department of Energy under Grant No. DE-FG02-02ER15283. c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.09.005

120 4. 5. 6. 7.

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Distorted wave theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupled channel approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct solutions of time-dependent Schr7odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabatic approximations and hidden crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. General formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. S-ionization and superpromotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. T-ionization and top-of-barrier electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. D-ionization and radial decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Sturmian theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Scale transformation of Solov’ev and Vinitsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Sturmian basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Wave functions and transition amplitudes in Fourier space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Calculation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Total ionization cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. DiGerential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126 130 135 138 138 139 141 143 144 144 146 147 150 151 151 153 155 156

1. Introduction Ionization processes in atomic collisions are still an active research Meld. These processes, apart from their fundamental signiMcance, have important applications in fusion reactors [1,2], the modelling of ionizing radiation in biological material [3,4], transmission of heavy ions through gases and solid targets [5–7], and a host of other applications too numerous to mention. In many of these applications, details may turn out to be unimportant for many of the averaged quantities that enter into the modelling, and simple, reasonably reliable, cross sections are required. In this spirit, the Mrst Born approximation forms the basis for computations of the penetration of ions in matter. For ionization in the independent particle approximation, more advanced distorted wave approximations still give relatively simple expressions and have proved useful at intermediate and high impact velocities. We briePy review these theories in Sections 3 and 4. At impact velocities below 1 a:u. the advanced adiabatic approximation and the hidden crossing theory have provided closed form expressions for ionization that are theoretically well founded [8,9]. The formulae can be used for modelling purposes provided precise details of energy and angular distributions are not needed. These theories and applications to proton impact ionization are described in Section 7. Our discussion of approximate methods omits purely classical approaches, such as the classical trajectory Monte Carlo (CTMC) method, even though this method is widely used to extract cross sections that are diIcult to compute in the quantum theory thus it is frequently used to obtain data not otherwise readily calculated [10,11]. Closely related quantum theories based on Feynman or Van Vleck’s propagator [12,13] are also omitted since they have not been widely used to study ionization by heavy particle impact. We do, however, employ the now standard semiclassical approximation where the relative motion of target and projectile species are treated classically while the electron motion is treated according to the quantum theory.

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In addition to reliable approximate formula, applications also require calculations that can be benchmarked against detailed experimental measurements. In recent years, for example, the COLTRIMS [14] technique has provided some of the most detailed energy and angular distributions of electrons ejected from rare gas targets. These measurements have identiMed new features that provide stringent tests for ab initio computations of ionization [15]. On the theoretical front many methods based on the semiclassical approximation have been developed for excitation and charge transfer processes, but only in recent years have computer resources become suIciently powerful and readily available to reliably model ionization in one electron species over a broad energy range. These methods all employ the semiclassical approximation which we briePy review of in Section 3. Methods to solve the semiclassical time dependent Schr7odinger equation are reviewed in Sections 5 and 6. Basis set expansions are a standard way to solve dynamical Schr7odigner equations, but for collisions of atomic species the need for target and projectile states often requires overcomplete sets. Diatomic Mxed nucleus molecular basis states are exceptional in this regard in that they are complete, orthonormal, and include both target and projectile channels. Unfortunately, allowance for electron translation greatly complicates the use of these otherwise mathematically convenient sets [16,17]. A way to use these sets for dynamical calculations was found by the present authors based on the work of Solev’ev [8]. An essential ingredient is the use of molecular Sturmian functions instead of Mxed nucleus energy eigenstates [18]. The resulting theory is referred to here simply as the Sturmian theory, although it is recognized that atomic Sturmian functions are also employed in ab initio calculations [19,20]. The molecular Sturmian theory is described in Section 8. In Sections 5, 6 and 8 we emphasize diIculties inherent in ion–atom collisions, namely, the need to include target excitation, electron transfer to projectile states and ionization processes where the competition between attraction to target nuclei on the one hand and to the projectile nuclei on the other complicates the electron distributions. This competition brings new features such as projectile cusps and dynamical molecular orbital distributions in cross sections diGerential in the electron momentum. These problems are being overcome, thus we present results of calculations and experimental measurements of total cross sections in Section 9. Some concluding remarks are given in Section 10. 2. Time-dependent theory Ion–atom collisions are usually treated theoretically in approximations which take advantage of the small ratio of electron to nuclear masses. Owing to the small mass ratio, heavy particle motion is only weakly inPuenced by electrons. In addition, wavelengths of the de Broglie waves for the heavy particles are usually suIciently small that motion in the interatomic coordinate in the initial channel Ri can be treated classically. In this case, some simple approximations have become fairly standard [21,22], namely, a plane wave factor of relative motion exp(iKi · Ri ) is factored out of the total wave function and the amplitude of this plane wave is computed in approximations where second derivatives with respect to Ri are ignored. Then a “time” is deMned according to vt =Ki ·Ri = where Ki is the initial wave vector of relative motion, is the reduced mass of the target (T ) and projectile (P) system, and v is the initial velocity. To Mrst approximation, this velocity is taken to be constant and the nuclei move along straight line trajectories with Ri = Ct + b, where b is the

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impact parameter vector perpendicular to C. These approximations yield a time-dependent Schr7odinger equation (TDSE) for the amplitude (t; r) 9 (1) i − H (t; r) (t; r) = 0 9t where H (t; r) = H0 + VP (t; r) + VT (t; r) + VPT (t) + Vee (r) ;

(2)

and where the atomic electrons, whose coordinates are denoted collectively by r, move in the Melds of the target and projectile nuclei. In equation (1) H0 is the kinetic energy operator and the potentials VP (t; r), VT (t; r), VPT (t), and Vee (r) refer to electron–projectile, electron–target, projectile–target, and electron–electron interactions, respectively. Note, that the derivative with respect of time in (1) is taken holding the electrons coordinates r constant in the initial frame of reference where we set Ri = Ct + b. To keep the form of TDSE the same in all frame of references a phase transformation and a change in deMnition of time t is required [22]. This phase factor is called the translational factor fa (r; t) introduced by Bates and McCarroll [23]. The initial conditions associated with an electron that is in the bound atomic states i (ri ) with the energies Ei are t →−∞

i (t; r) −→ i (ri )e−iEi t eifi (r; t) ; while for large positive times we have t →+∞ afi (b)f (rf )e−iEf t eiff (r; t) :

i (t; r) −→

(3) (4)

f

The sum in (4) goes over an asymptotically complete set of states that include bound target and projectile eigenstates and continuum states representing unbound electrons, weighted according to the impact parameter dependent transition amplitudes afi (b). It is understood that the amplitudes also depend upon the velocity v, however this dependence is usually not noted. For purposes of interpretation one often replaces the variable v with an equivalent variable Q = (Ei − Ef )=v where Q is the semiclassical approximation to the momentum change in directions parallel to C. The approximations are usually grouped together and are referred to as “the SemiClassical Approximation” (SCA). This manuscript uses a coordinate system with the z-axis along C and the x-axis along b. A general vector A will often be resolved into vectors parallel A and perpendicular A⊥ to C. Atomic units are used throughout. Almost all computations at low and intermediate energies use the semiclassical approximation. Alternatively, high energy theories such as the Born approximation usually work in the wave representation although Born approximations in the semiclassical representation are also deMned. The two representations are connected by a Fourier transform on the impact parameter [22,24]. Thus if a(b) is the amplitude in the semiclassical picture then the amplitude A(Q) in the wave picture is given by −1 A(Q) = (2) a(b; Q )exp[iQ⊥ · b] d 2 b ; (5) where Q=Ki −Kf is the momentum change of the projectile. It has variable perpendicular components Q⊥ and Mxed parallel components Q = SE=v. In the remainder of this paper, the dependence of

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a(b; Q ) on Q will be understood and not noted explicitly. We have already suppressed the indices fi for conciseness. The wave representation will be used for the high energy Born and distorted wave Born (DWB) theories discussed in the next sections. Impact parameter amplitudes can always be obtained by the Fourier transform inverse to (5). This is not to say that the physical content of the two representations is identical. Indeed, only total cross sections are identical in the two pictures since angular distributions require the wave representation of (5). If the right-hand side of (5) is evaluated in the stationary phase approximation, one often has that the angular distribution is given by the impact parameter amplitude a(b) with b given in terms of the scattering angle Kf through a classical trajectory relation. This is often more accurate than the plane wave Born amplitude, so that the impact parameter version of the Born approximation is sometimes used even at high energies. For that reason stationary phase relations are commonly used to compare computed amplitudes a(b) with experiment for bound Mnal states. Similar relations for ionization have become relevant only recently. In this case the stationary phase expression for the diGerential cross sections is d5 Mv db = |a(b(Kf ))|2 b(Kf ) ; 3 Q⊥ dKf d k d Kˆf

(6)

b = b(Kf )(iˆ cos Kf ) + (jˆ sin Kf ) :

(7)

where With this relation we have that electron energy distributions are given by d3 d5 ˆf = |a(b)|2 d 2 b; = d K d k3 d k 3 d Kˆf

(8)

where a(b) is the ionization amplitude in the semiclassical approximation. Our discussion focuses upon ionization of a target by a structureless charged particle. This processes will be considered at high energies as a transition between target eigenstates. Of course, other processes such as electron capture also occur. The high energy theory for electron transfer will not be discussed, since such rearrangement reactions require special considerations. 3. Born approximation When incident energies are suIciently high, probabilities for all transitions, except elastic scattering, are much less than unity. If, in addition, the initial state is not much changed during the collision, the Born approximation, which employs unperturbed initial and Mnal states, applies. The exact transition amplitude for the transition from eigenstate i to eigenstate f is given by Tfi = f(−) |Vf | i(+) ;

(9)

where | i(+) is the exact initial state for the combined system of target and projectile with an incoming wave in the initial channel i and outgoing waves in all other channels, and |f(−) is an “unperturbed” Mnal state. The potential Vf represents the interaction between T and P in the Mnal state.

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The Born approximation replaces the exact initial state | i(+) by the eigenstate of the full Hamiltonian minus the potential Vf . For a bare ion projectile P of charge ZP incident on a one-electron ion with nuclear charge ZT we obtain T (B1) = Kf

(− ) f (r)|Vf |Ki i (r)

;

(10)

where the K denotes a plane wave normalized on the momentum scale; K (Ri ) = (2)−3=2 exp[iK · Ri ] ;

(11)

and where Ri is the coordinate of P relative to the center of mass of the target atom, and i; (f) represents the initial (Mnal) electronic wave function of the target. If f represents a continuum state then f is usually represented by the wave function for a continuum electron normalized on the energy scale E − ;kˆ (r), where Ek = 12 k 2 . In the theory of ion–atom collisions it is often more k conventional to normalize on the momentum scale since it gives a Galilean invariant cross section d 6 =d kf3 dKf3 directly. Wave functions normalized on the momentum scale are denoted by k . With this normalization the cross section is given by Kf d5 = 2 (2)4 |Tfi |2 ; 2 3 ˆ K d k d Kf i

(12)

where = MP (MT + m)=(MP + MT + m) is the reduced mass of the target and projectile, and m is the mass of the electron. This mass is unity in atomic units, however, we will frequently indicate the electron mass explicitly when needed for clarity. An important feature of atomic cross sections is their continuity across various thresholds [25,3]. In the present case continuity across the ionization threshold provides a useful check on theory and measurement. Such continuity is best expressed in terms of ionization states normalized on the energy scale. We can readily transform the Galilean invariant cross section to the electron energy ˆ scale using d k 3 = k dEk d 2 k. The Born approximation employs plane waves for initial and Mnal motion in Ri . Upon substituting plane waves from (11), and explicit expressions for Vf into the Mrst Born matrix element and integrating over the coordinate Ri we obtain   Ne (B1) 2 − 1 ZP i ;   exp i (13) = (2 ) Q · r T f j Q2 j=1 where Ne is the number of electrons in the target. Here it is assumed that the incident projectile is a bare ion. The Born approximation for the more general case where the projectile also carries electrons is discussed in Ref. [26]. If the target is in a bound state, then it is apparent that Q is equal to the recoil momentum of the target. For ionization processes the momentum transfer Q is shared between the ejected electron k and the target ion in the Mnal state. In this case the recoil momentum Kr diGers from Q, and a complete determination of Mnal state requires measurement of Kf and k in coincidence, or the coincident measurement of Kf and Kr . The later measurement is often called recoil ion momentum spectroscopy or RIMS. Since any unknown initial momentum of the target severely restricts RIMS measurements, it is desirable to cool the target. Such measurements with cold targets are called COLd Target Recoil Ion Momentum Spectroscopy (COLTRIMS) [14]. When combined

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with imaging techniques they have proved to be an eIcient way to measure fragmentation processes where several momentum variables must be speciMed in order to make measurements, sometimes called complete measurements, where initial and Mnal eigenstates are completely speciMed. In the remainder of this review we will concentrate on processes involving one “active” electron. The corresponding theory will therefore employ only one-electron wave functions. The simplest such functions are those for a single electron moving in the Meld of a single positive ion, or in a collision, in the Meld of two positive ions. In this case the B1 ionization amplitude when the target electron is initially in a 1s state is found to be [21,27] √ 2 2ZT5=2 ZP 1 exp[ − $T =2]%(1 + i$T ) (B1) Tfi = k3 Q2 (Q − k)2 + ZT2 2

−i$T Q + (ZT − ik)2 i$T − 1 i$T + 1 × − 2 : (14) Q + (ZT − ik)2 (Q − k)2 + ZT2 (Q − k)2 + ZT2 Expressions for arbitrary initial bound states nlm are given by Omidvar [28]. The corresponding 1s ionization cross section is 28 2 Kf ZT6 ZP2 1 d 5 B1 = 2 d k 3 d Kˆf kKi Q [1 − exp(−2$T )] [(Q − k)2 + ZT2 ]4 ˆ 2 ˆ 2 + ZT2 (Qˆ · k) (Q − k · Q) [(Q + k)2 + ZT2 ][(Q − k)2 + ZT2 ]

2kZT : ×exp −2$T arctan Q2 − k 2 + ZT2

×

(15)

The cross section has two maxima as a function of k for Mxed Q, namely, one when k = Q and a small maximum at k = −Q. The former peak corresponds to a “binary” collision between the fast projectile and the relatively stationary target electron. In this case all of the momentum lost by the projectile is transferred to the electron and none to the residual target ion, i.e. the target does not recoil. The binary encounter peak represents quasi-elastic scattering of target electrons from the projectile. In the elastic scattering model (ESM) [29] this part of the electron distribution is determined by the electron–projectile elastic scattering cross section weighted with an electron velocity distribution determined by the Compton proMle of the target. The B1 cross section agrees with the ESM only for bare projectiles where the Rutherford cross section is exact in Mrst order. Even for bare projectiles the B1 amplitude is deMcient since it does not include the Rutherford phase factor, which we denote by exp[iR ]. For projectiles carrying some bound electrons where the Mrst order amplitude does not give the correct e–P elastic scattering cross section, the B1 cross section is inaccurate near the binary encounter peak, whereas the ESM model is fairly reliable [29–32]. In any event the B1 amplitude is incorrect near the binary encounter region, even for bare projectiles, since it does not have the expected Rutherford phase. The second maximum at k = −Q corresponds to a recoil ion momentum approximately equal to Kr ≈ 2Q = −2k. Here electrons ejected by collisions with projectiles scatter backwards from target nuclei on the way out. This peak is therefore referred to as the recoil peak. The Mrst Born amplitude

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with target eigenstates does indeed allow for this possibility since it incorporates interactions with projectiles to Mrst order, and interactions of electrons with targets to all orders. It is expected that target eigenstates give correct cross sections in the limit of vanishing outgoing electron momenta. In this limit one sees that the Galilean invariant cross section of (15) diverges as 1=k when k → 0. This divergence is readily traced to the Coulomb interaction in the Mnal state. It implies that the cross section per unit electron energy d5 d5 =k d 3 k d Kˆf dEe d kˆ d Kˆf

(16)

is Mnite and non-zero in the limit as k → 0. The Mnite value at k = 0 integrated over electron solid angles kˆ exactly matches the cross section for excitation of states with large n according to the connection formula n3 d 2 n; ‘; m d3 = lim ; (17) lim ˆf n→∞ k →0 dEk d K ZT2 d Kˆf ‘; m

where we have used that dEk =dn = ZT2 =n3 . The connection between excitation and ionization is a general feature of processes involving attractive Coulomb potentials in the Mnal state [25]. In this regard we immediately note a deMciency of the conventional Mrst Born approximation, namely that there is an attractive interaction between electron and P as well as T in the Mnal state. One manifestation of this attraction is electron capture to bound states nP ‘P mP of P. According to the connection argument, the capture cross section for nP → ∞ must connect with the ionization cross section in the projectile or primed frame. That is, one must have n3 d 2 ncap‘ m d3 P P P P = lim lim 2 ˆ nP →∞ kP →0 dEk d Kˆf Z d K P P ‘P ; m P

(18)

Since kP = k − C so that kP dEkP d kˆP = d 3 kP = d 3 k it follows that the ionization cross section in the lab frame cap d3 k n3P d 2 nP ‘P mP = kP Z2 dEk d Kˆ d Kˆ ‘P ; m P P

(19)

must diverge as 1=kP near kP = 0. Because this divergence connects with capture to bound states it is often referred to as the “charge transfer to continuum states” (CTC) peak. This peak is absent in the B1 approximation. 4. Distorted wave theories To include both the Rutherford phase and continuum capture one may employ continuum states of the projectile rather than the target in the Mnal state [33,34]. As Mrst discussed by Briggs [35] in the context of electron capture by highly charged ions, a consistent theory emerges by considering expansions of the full amplitude in powers of the small parameter ZT =ZP . This theory has been

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developed over the years [37,38] and is known as the distorted wave strong potential Born (DSPB) approximation. This theory incorporates eikonal distorted waves and multiple scattering terms in the initial state. For a particular choice of eikonal distortion the resulting wave functions are identical to the impulse approximation (IA) initial state functions. In either (DSPB) or (IA) the physical picture that emerges is one where the collision of a bare charged nucleus ZP with a one electron ˜ target species (ZT ; e) has the initial state represented by a momentum distribution (p) where the electron moves in an eigenstate of the target p−C (rP ). Owing to the electron-target interaction V (rT ), the electron makes a transition to the Mnal eigenstate of the target which could be a bound state n; l; m (rP ) (charge transfer) or a continuum state kP (rP ) (ionization). Because this approximation employs projectile eigenstates it describes the Thomas Double Collision [36] peak, where electrons collide with the projectile in a Mrst collision then with target before being “captured” into continuum states of the projectile. This process is represented since multiple collisions with the projectile are included in the Mnal target state wave function. Alternatively, because multiple collisions with the target nucleus are not included in projectile eigenstates, the recoil peak is only described in Mrst order. While DSPB or IA theories do have the correct Rutherford amplitude near the binary encounter peak and accurately describe charge transfer to continuum states, they are inaccurate for the low energy electrons that represent the major part of the ionization cross section. Alternatively, the Mrst Born approximation with target eigenstates represents this region very well. Somehow, a correct theory must incorporate motion of the electron in both the Meld of the projectile and target in the Mnal state. The simplest way to do this is to multiply the Mnal state IA wave function by the factor common to projectile continuum states of the B1 theory, namely, DT = N ($T )1 F1 [i$T ; 1; i(krT + k · rT )] ;

(20)

where $T = ZT =k, rT is the coordinate of the electron relative to the target nucleus and N ($) = exp($=2)%(1 + i$). The resulting Mnal state wave function is called a “continuum distorted wave” or CDW wave function. It was Mrst introduced to obtain a high energy theory of charge transfer [39] but was subsequently used for ionization [40,41]. The CDW-B1 diGerential cross section is d 5 CDW−B1 d 5 B1 = |N ($P )|2 : d 3 k d Kˆf d 3 k d Kˆf

(21)

The CDW Mnal state function with a B1 initial state modiMes the Mrst Born T matrix element by the phase factor of the Rutherford amplitude and a normalization factor N ($P ) where $P = ZP =kP . Since |N ($P )|2 = → 2ZP =kP as kP → 0, the resulting amplitude connects with electron capture, as expected. While the CDW–B1 amplitude corrects the known deMciencies of the Mrst Born amplitude, it introduces a normalization factor |NP ($P )|2 that diGers signiMcantly from unity unless 2ZP =kP 1. This strong departure from unity even extends to the binary encounter region so that the CDW–B1 amplitude disagrees with the ESM. For that reason the CDW–B1 theory is no longer used. If the CDW Mnal state is employed together with the IA (or DSPB) initial state the resulting amplitude is diIcult to compute, although those few studies which have employed the CDW–IA amplitude Mnd fairly good agreement with experiment [42].

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A further approximation to the IA initial state yields the CDW initial state, i.e. a B1 function multiplied by the projectile distortion factor DP = N ($P )1 F1 [i$P ; 1; i(kP rP + kP · rP )] : The resulting CDW–CDW amplitude is known in closed form [41] and is fairly accurate for much of the electron spectrum but it diverges at certain values of k and Kf , most notably at the Thomas double collision peak. Because this region makes only a very small contribution to total cross sections, or to cross sections integrated over Kˆf , the divergence is often ignored. The CDW–CDW cross section is given by d 5 CDW−CDW d 5 B1 = |N ($P )|2 A|N (ZP =v)|2 d 3 k d Kˆf d 3 k d Kˆf ×|F(i$P ; iZP =v; 1; z) − i$P !F(1 + i$P ; 1 + iZP =v; 1; z)|2 ;

(22)

where F(a; b; c; z) is the gaussian hypergeometric function and ! = !0 =(. + /) ; z = z0 =(. + /) ;

0 C + ; !0 = . 1 B z0 = / − .0=1 ; . = q2 =2 ; / = −(k 2 + ZT2 )=2 ; 1 = [Q − k2 + ZT2 ]=2 ; 0 = k P · C − kP v + / ; 1 if q2 ¡ k 2 + ZT2 ; A= e−2$P if q2 ¿ k 2 + ZT2 ; B = Q2 − (1 + iZP =v)Q · k ;

v v 2 C = [ − Q · k + k (1 + i$T )] − 1 + [ − Q · C + k · C(1 + i$T )] : kP kP

(23)

One sees that the cross section is singular when the denominator . + / vanishes [43]. The singularity is of suIciently low order that distributions integrated over Kˆf , for example, are well deMned. For this reason the theory can be used where fully diGerential cross sections are not needed. The divergence, however, is conceptually important. It is not present in the CDW–IA theory, thus its presence can be traced to approximations made in going from the IA to the CDW initial wave function. The divergence is avoided if the initial distortion factor is replaced by its eikonal asymptotic form to give the eikonal initial state or EIS wave function. The resulting CDW–EIS amplitude is similar to the CDW–CDW amplitude, but has no unphysical singularities.

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The CDW–EIS cross section has the same form as the CDW–CDW expressions in (23) except that !, z and A are replaced by !EIS = !0 =/ ; z EIS = z0 =/ ; AEIS = e−2$P :

(24)

Since the denominator / never vanishes, the CDW–EIS amplitude is everywhere well deMned. Its main deMciency is the lack of the Thomas peak but that peak gives only a small contribution to the total ionization cross section. A diGerent approach to a general theory of ionization at high energies was taken by Taulbjerg and co-workers [44–46]. They noted that, when the DSPB function is approximated by a B1 initial state and a phase factor arising from the projectile–target interaction Ui (R), one obtains an amplitude similar in form to the CDW–B1 amplitude but with the normalization factor N (kP ) replaced by a factor 8kP (kP − Q) that depends upon kP and Q. This amplitude is called the D2C amplitude, but in a nomenclature where both the initial and Mnal states are referenced it would be called the CDW–DB1 amplitude. In any event it is given by T (CDW−DB1) (Q) = T (B1) (Q) exp(iR )8kP (kP − Q) :

(25)

Various expressions for 8k (p) are given in Ref. [44]. The expressions depend upon the somewhat model dependent interaction Ui (R). For most reasonable choices of Ui (R), the factor 8kP approaches unity near the binary encounter peak and N ($P ) near the continuum capture peak. It therefore gives a high energy approximation that correctly represents the three main features of one-electron ionization, namely a peak at small values of electron energy in the target frame, the CTC peak at low electron energies in the projectile frame, and the binary encounter peak at Q = k or equivalently at kP = v in the projectile frame. For completeness we give the CDW–DB1 cross section for a cut-oG channel potential for which the cross section is known in closed form [44]. It is found to be d 5 CDW−DB1 d 5 B1 = |8k+ (kP − Q)|2 ; d 3 k d Kˆf d 3 k d Kˆf P

(26)

where |8k+P (p)|2 = |%(1 + i($P − ZP =v); iSR0 =p)eZP =2v + i(2b=e:)iZP =v (a=:)i$P −1 ×[1(1 + i$P ; iaR0 =p) + i(ZP =v)(p⊥ =2R0 a)2 1(3 + i$P ; iaR0 =p)]|2 ;

(27)

and where %(c; d) and 1(c; d) = %(c) − %(c; d) are incomplete Gamma functions as deMned by Abramowitz and Stegun [47] and : = (p2 − kP2 )=2 ; a = : − ZP p =vR0 :

(28)

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The singly diGerential ionization cross sections d3 = |T (CDW–DB1) (Q)|2 d 2 Kˆf d k3

(29)

have three main features. They are the slow electron peak, the CTC peak and the binary encounter ridge. These same three features appear for all approximations which employ the CDW Mnal state. The CDW–DB1 and CDW–EIS generally give good overall agreement with experimental data even though they omit multiple scattering responsible for the Thomas double collision mechanism. They are relatively simple to compute since essentially closed form expressions are available. For that reason they are widely used when it is desired to go beyond the B1 theory. Except for the region near the CTC peak, the cross sections B1 agree fairly well. This is quite important since the B1 theory forms the basis for theories of the penetration of charged particles in matter [3,48]. Such applications are beyond the scope of this review. Also not reviewed here are developments based upon more elaborate distorted wave approximations. These are not needed at suIciently high energy, but one of the aims of theory is develop methods that are applicable over a broad range of incident energies. High energy theories may be used as a guide for extrapolation into intermediate energy region. We have already mentioned the CDW-IA approximation. Further developments based upon this theory have recognized that the CDW wave function generally poorly represents electron motion in the Meld of two charged particles [49], and that a better representation is needed at intermediate energies. Such representations have been developed and do indeed improve upon the CDW function. This improvement, however, comes at the expense of considerable complexity in actually evaluating transition matrix elements. For this reason, the theory has not been widely used and is not reviewed here, despite its promise.

5. Coupled channel approximations As pointed out in the previous sections, perturbative approaches, like the Born approximation, provide accurate enough description of ionization processes in fast ion–atom collisions. However, even at Ep ¡ 200 keV=amu Born calculations considerably overestimate total ionization cross sections, and the diGerence between calculations and experimental data sharply increases with decreasing collision energy. Apart from this quantitative disagreement, the Born approximation fails to give an adequate explanation of such features of ejected electron energy spectra as charge transfer to the continuum (CTC) [25] and top-of-barrier promotion [10]. For this reason, much eGort has been directed to extend calculations to intermediate and low energies. The quantum mechanical eGorts can be divided into two groups: direct numerical solution of the time-dependent Schr7odinger equation and coupled channel approximations. The coupled channel method is widely used in the study of collision processes in the intermediate and low energy regions where multiple scattering eGects are signiMcant. It takes into account strong couplings between exchange, excitation and ionization channels. In general, coupled channel approximations are based on expansion of total wave functions of collision systems in complete basis sets, so that the time-dependent Schr7odinger equation is replaced by systems of diGerential equations for the coeIcients of the expansions [50].

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The theory is usually formulated within the semi-classical approximation in which the nuclear motion is described by the classical trajectory R(t). Such approaches are valid for a wide range of collision velocities, v ¿ 10−2 v0 [51] (v0 is the orbital velocity of atomic electron). Then the time-dependent Schr7odinger equation for electron motion; namely,

9

(r; t) = 0 ; (30) He − i 9t where He =− 12 ∇2r −Z1 =r1 −Z2 =r2 =H0 +VT +VP is the electronic Hamiltonian, is solved by expanding the electronic total wave function (r; t) in a complete basis set {
(r; t) = an (t)
with initial conditions (r; t)t →−∞ →
(32)

and the transition probability is Pm (b) = |am (b)|2 :

(33)

The functions
(34)

where n (rj ; t) are eigenstates of certain Hamiltonians (e.g. (H0 + VT ), (H0 + VP ) or (H0 + VT + VP )), and Fn (rj ; Rj ) are translation factors describing the electron motion relative to the chosen coordinate system. For example, if an origin of the system is chosen at the center-of-mass of two nuclei moving along a straight-line trajectory with the relative velocity v then this factor has the form [23] 1

2 2

Fn (rj ; Rj ) = eipj C·rj − 2 pj v t ;

(35)

where Rj = pj R = pj (b + Ct), pj = Mi =(Mi + Mj ), Mi and Mj are the nuclear masses of the colliding particles “j” and “i”. The factor Fn ensures that (He −i 99t )
(r; t) = ck (t)=k (r; t) : (36) k

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Then the coupled set of equations for the coeIcients ck (t) can be written in a matrix form as iS

d c = Mc ; dt

(37)

where Skl = =k |=l and Mkl = =k |He − i 99t |=l . The functions =k should be chosen such that =k (rj ) → k (rj )Fk (rj ; Rj ) as t → +∞, and the boundary condition at t → −∞ is satisMed. From the computational point of view, a compromise should be found between the two desired features of the basis {=k }, namely, maximal mathematical completeness and inclusion of all physically relevant channels. The basis set {=k } can be represented by square-integrable radial functions centered on the target, or the projectile, or both centers. These may be hydrogenic [19], Gaussian [53], even-tempered [54], etc. and include both bound and continuum states. Diagonalization of corresponding Hamiltonians in the underlying bases gives sets of radial eigenfunction pseudo-states [55,56]. The eigeneneries can be checked to insure that the appropriate bound states are well represented. The ionized electron spectrum is represented by a Mnite set of discrete pseudocontinuum states. The basis set {=k } can be modiMed in many ways. For example, if the energy range of the pseudo-states does not extended high enough then the length scale can be shortened, or more basis states included [57]. The former choice decreases the density of the states per unit energy interval while the latter increases it. The eGect of state densities, as well as the choice of the translation factors Fn (r; R) will be discussed below in considering particular versions of the coupled channel method. These versions can be classiMed in terms of basis sets used in calculations. In the one-center atomic orbital (1CAO) coupled channel calculations, the eigenstates are obtained by diagonalization of the atomic Hamiltonian (He = H0 + VT ) in a basis of target-centered orbitals. This method is quite reliable for excitation and ionization at suIciently high energies, where electron capture to projectile discrete and continuum states does not introduce essential perturbations. With decreasing collision energy the number of basis states needed for obtaining converged results increases drastically. For example, Ford et al. [55] obtained reliable results for the total cross sections for electron loss in proton–hydrogen collisions using the 1CAO method with 1040 basis functions at Ep ¿ 15 keV. Morishita et al. [58] have formulated the 1CAO method to describe ionization in full detail, including electron energy distributions and parameters of autoionization proMles. However, an enormously large atomic basis is needed to account for the two-center nature of the collision event, thus the use of 1CAO method is limited to the high-energy region. Another drawback of the single-center expansion is that one cannot determine the individual capture and ionization cross sections, but only their sum. An advantage of 1CAO method is that for basis functions centered on a single nucleus the proper electron translation factors are uniquely and simply deMned by (35). In the two-center atomic orbital (TCAO) coupled channel calculations, the basis set {=iT (rT ; t)} of functions centered on the target is added to the basis set {=jP (rP ; t)} of functions centered on the projectile. The use of two-center expansions makes it possible to estimate the probability of electron transfer to the continuum of the projectile and, consequently, to improve the accuracy of calculations. In [59], in addition to the orbitals of separated atoms, some united atom orbitals were explicitly included in the basis set (AO+expansion) allowing one to reproduce accurately the molecular correlation diagram of the collision system. The TCAO coupled channel approximation has been used in many theoretical studies of ionization processes in H+ –H collisions [53,54,56,59,60]. The calculated results agree well with experiment

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133

in the region of the maximum of ionization cross sections (EP ≈ 50 keV). However even the best TCAO coupled channel calculations yield cross sections which are about 20% higher than the experimental ones. This discrepancy might be connected with some intrinsic problems inherent to the large-size TCAO coupled channel calculations, in particular, to the construction of continuum states. Conceptually, the TCAO coupled channel expansion should include bound states on targets, and projectiles and continuum states of both species. Really, the two sets of pseudo-continuum states (one set for each center) are used which can span the same continuum space, especially when the two nuclei are close to each other, thus giving rise to over-completeness of the basis set. This can lead to instability of individual excitation and capture cross sections exhibiting unphysical oscillatory structures as a function of collision energy [61]. Kuang and Lin [56] attributed such instability to the simultaneous use of pseudo-continuum states on two centers per se and proposed to use them only on one center. However, it was not possible to make both excitation and capture cross sections stable simultaneously. When pseudo-continuum states are used only on the target, excitation cross sections are stable and well behaved, but capture cross sections remain unstable. Just the opposite situation occurs when pseudo-continuum states are used only on the projectile. At the same time, the ionization cross section for proton-hydrogen atom collisions calculated using the one-center pseudo-continuum states converges to a diGerent value from that obtained with full two-center calculations. Toshima [60] showed that the unphysical oscillatory structures were caused by strong coupling between bound and pseudo-continuum states belonging to diGerent centers. The number of pseudocontinuum states needed for stabilizing the cross sections increases drastically with decreasing collision energy. Some problems are related to the choice of the electron translation factors. In the TCAO coupled channel calculations these factors, determined by (35), are attached to a particular center (target or projectile) and are equal for all basis states. Really, in the close approach of the colliding particles, “the degree of attachment” of a basis state to a particular center can be diGerent from the asymptotic one and the molecular nature of the collision system should be taken into account. This problem can be solved by using “switching functions” which will be discussed below, in considering molecular basis. However even the large-size TCAO coupled channel calculations cannot adequately describe saddle-point electron promotion, which plays an important role in low-energy regions [62]. To consider these eGects, the triple-center atomic orbital expansion has been used [20,63,64]:

(r; t) =

3 k

where Fk. (r; t)

a.k (t)

. . . k (r. (r; t))gk (r; t)exp(−i@k t)Fk (r; t)

;

(38)

.=1

1 1 = exp i f. C · r − if.2 v2 t 2 2

;

(39)

  −1; . = Target ;   f. = +1; . = Projectile ;    2p − 1; . = Saddle Point ;

(40)

p = ZT1=2 =(ZT1=2 + ZP1=2 ) ;

(41)

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with each function k. being a bound or continuum atomic wave function centered on the nucleus . with corresponding eigenvalue @k . Relation (40) Mxes the third center to be the equiforce (saddle) point. The available calculations of ionization in the collision systems H+ –H [63,64] and He2+ –H, H+ –He+ [20] show that triple-center AO methods yield stable ionization cross sections at energies as low as several keV which are considerably larger than those of two-center methods and agree better with the experimental data. This indicates the importance of electron ejection near the top of the barrier. Though the triple-center close-coupling method possesses the same problems with over-completeness of basis sets, much faster convergence allows one to overcome these diIculties with much less eGort. An alternative to atomic expansions is the molecular orbital coupled channel formalism. This method employs a basis of molecular adiabatic wave functions, modiMed by electron translation factors. In this case the choice of these factors is not a trivial problem since it is not clear to which center the pseudo-continuum states should be attached. One of the approximations employs an electron translation factor common for all basis states [65,66]. Then the molecular coupled channel expansion can be written as t

(r; t) = F(r; t) ai (t)=i (r; R)e−i Ei (t ) dt ; (42) i

Fk (r; t) = exp i 12 f(r; R)C · r − 12 iv2 t ;

(43)

where =i (r; R) are molecular adiabatic wave functions, F(r; t) is the common translation factor and f(r; R) is the switching function. These common translation factors include switching functions [67], since F(r; t) → F . (r; t) and f(r; R) → f. as t → ∞ with .=T; P. Because these switching functions require r → r. , they are only deMned for bound or pseudo-states centered on P or T asymptotically. The switching functions are not unique and are normally chosen semi-empirically. The calculations [65] show that the common translation factor approach accurately reproduces excitation and electron loss cross sections at low and intermediate collision energies. Another approximation for the electronic translation factors employs switching functions which are diGerent for each molecular state [68]. In this approach the molecular orbitals = are expressed as linear combinations of atomic orbitals .n (LCAO–MO) [69]: . . = = Un n ; (44) n;.

. where the transformation Un is invertible, and the switching function fMO for the state = is deMned by the relation fMO = = Un fn. .n ; (45) n;.

where fnT = −1 (38) if n is centered on the target (. = T ), and fnP = +1 if n is centered on the projectile (. = P). The switching functions fMO rePect a local “degree of attachment” to one or the other center and they are diGerent for each molecular state. The molecular coupled channel methods simultaneously reproduce cross sections for excitation, capture and ionization in slow H+ –H and He2+ –H collisions and provides good agreement with

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135

the best two-center atomic orbital calculations at intermediate energies thus keeping the same disagreement with experimental data. At low energies (1–10 keV), good agreement of the triple-center molecular orbital calculations with experimental data has been achieved, although with extensively enlarged of the basis sets [70]. The coupled channel methods are generally the methods of choice for computations of excitation and electron transfer. For ionization, however, they are incomplete owing to the translation factor problems. The methods can be used to compute total ionization, however, electron energy and angular distributions are rarely reported, probably due to ambiguities in extracting continuum channels uniquely. This is especially true for molecular basis sets, thus it is diIcult to compute ionization at low impact energies by such methods. 6. Direct solutions of time-dependent Schr%odinger equation Progress in modern computation techniques have made it possible to solve the time-dependent Schr7odinger equation directly without evaluating the large number of matrix elements present in the coupled channel methods. Formally,the solution proceeds as follows: Divide the whole time space into a great number of small but Mnite intervals 0t. If the Hamiltonian of the system is known and the electron wave function is deMned at t = t0 then the time derivative of the wave function at t = t0 is determined from the time-dependent Schr7odinger equation. Using the wave function and its time derivative, the solution is propagated over intervals 0t to obtain the value of wave function at t = t1 . Successive application of this procedure determines the wave function for all time. Calculations are performed either in conMguration space [71] or in momentum space [72]. The formal solution of the time-dependent Schr7odinger equation is given by

(t) = U (t; t0 ) (t0 )

(46)

for small time interval 0t diGerent methods [73] may be used to approximate the inMnitesimal time-evolution operator U (t; t0 ), U (t; t0 ) = exp{−H [ 12 (t + t0 )]0t} :

(47)

In the lattice time-dependent Schr7odinger equation (LTDSE) method, the electronic wave function represents solutions of the time-dependent Schr7odinger equation in conMguration space obeying initial and boundary conditions. Typically, this method employs a numerical lattice of Mnite spatial extent that has a Mnite (though large) number of grid points or basis elements. Therefore, when choosing a particular representation, a compromise should be found between the spatial extent of wave functions and the spatial resolution with which they are to be described. The lattice solution of the time-dependent Schr7odinger equation (LTDSE) [74,75] 9 (r; t) i = H (r; t) (48) 9t involves a discrete representation of the electronic wave function (x; y; z; t) → (xi ; yj ; zk ; t) and all coordinate-space operators on a three-dimensional Cartesian mesh. Local operators such as potentials become diagonal matrices composed of their values at the lattice points V (x; y; z) → V (xi ; yj ; zk )0i; i 0j; j 0k; k . Derivative operators, such as the kinetic energy, have lattice representation in terms of matrices 9=9x → Di;(x)i 0j; j 0k; k .

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In this case, representing the Coulomb potential is diIcult since it is singular at the charged centers. This problem is solved by introducing the so-called “soft-core” Coulomb potential [74] to describe electron interactions with the target and projectile: ZT ; c + r2 ZP VP (r; t) = − ; c + |r − rP (t)|2 VT (r) = − √

(49) (50)

where c is the real, positive soft-core parameter and rP is the projectile position vector. The introduction of the soft-core potential allows one to avoid the singularities of the Coulomb potential at the charged particle centers. The value of c is adjusted to obtain reliable binding energies. In LTDSE methods it is important to overcome the so-called “multiscale problem” when the scale of wave function changes as a function of time. This especially relates to highly excited or continuum states whose spatial extents exceed the boundaries of the numerical grid. During a collision, the part of the wave function describing emission to the high-energy continuum leaves the space spanned by the lattice, so that information regarding this part is lost when the boundary is passed. Thus, the time propagation must be stopped before the corresponding part of the wave function leaves the grid. If this propagation time is large enough, then the fraction of the wave function associated with ionization can be evaluated by subtracting the projection of the wave function onto all bound states:

ion (r; t) = (r; t) − <jT (r)<jT | (t) − <jP (r − R)<jP | (t) ; (51) j

j

where <jT and <jP represent bound states of the target and projectile, respectively. The spectrum of ejected electrons is simply given by the momentum distribution (the Fourier transform ˜ ion ) of escaping electrons: Bion (k) = lim | ˜ ion (k; t)|2 ;

(52)

t →+∞

where k is the momentum of the ejected electron in the laboratory frame. In the two-center momentum space discretization method (TCMSD) [76] the electronic wave function is expanded in momentum space: 1 2 <(p; t) = T˜ lm (p; t)Ylm (p) P˜ lm (q; t)Ylm (q) ˆ + e−i(p·R− 2 v t) ˆ ; (53) l; m

l; m

where q=p−C ;

(54)

and where T˜ , and P˜ are radial functions in momentum space which depend on time and the magnitude of the momentum with respect to the target and projectile, respectively. The phase factor in front of the projectile-centered expansion is the plane wave translation factor [23] in momentum space. The two-center expansion reduces the number of spherical harmonics Ylm necessary for conversion thereby reducing the computational eGort. The main advantage of the expansion in momentum space is that the wave function is conMned, i.e. it vanishes for large momentum.

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137

The radial functions are expanded in B-splines up to the momentum after which asymptotic forms of the functions are achieved; T˜ lm (p; t) =

N −1

cilm (t)B˜ i (p) + cNlm =p4+l ;

(55)

lm 4+l ˜ dlm : i (t)Bi (p) + dN =p

(56)

i=1

P˜ lm (p; t) =

N −1 i=1

The second terms in (55) and (56) are proportional to the asymptotic form each partial waves. Substituting (53) to the time-dependent Schr7odinger equation gives a set of Mrst-order coupled equations for expansion coeIcients cilm (t) and dlm i (t). The expansion coeIcients and their time derivatives are propagated by the Runge–Kutta method. After propagation to some large time after the collision, the resulting momentum space wave function is analyzed for bound-state amplitudes and ejected electron distributions. Ionization amplitudes are calculated by subtracting away the bound component: T (P) ˜ ion ˜ ˜ T˜ (P) (p; t) = T ( P) (p; t) − anlm (t)Fnl (p) ; (57) lm lm n

where T (P) anlm (t)

=

0

∞

˜ lm (p; t)p2 dp Fnl (p)T˜ (P)

(58)

and Fnl (p) are the hydrogenic radial functions in momentum space. The TCMSD method was used for calculations of total ionization cross sections [77] and momentum distributions of electrons ejected in H+ –H collisions [78,79]. It was shown that electron densities in momentum space migrate from target atoms to a region between the two centers as proton velocities decrease from 5 to 1 a.u. As in the LTDSE calculations [80], the total ionization cross sections obtained by the TCMSD method [78] are about 30% higher than the experimental data [81] at the maximum of the cross sections. However, at lower energies agreement with experimental data is better than for the LTDSE calculations with Fourier collocation method. Direct solutions of the time-dependent Schr7odinger equation were partly motivated by the failure of coupled channel methods to adequately describe the full complexity of ionization processes. The direct solutions, however, also have inherent diIculties related to the size of the box that can be used. In practice the box cannot be made too large since then the electron density is low and the number of points is large leading to numerical inaccuracies. However, if the box is not large enough then the electron wave function unphysically rePect from the boundaries. This in turn is compensated for by using absorptive boundaries conditions thereby introducing new parameters into calculations. Only when the results are independent of these parameters can it be said that they converge. It should be noted that rePection from the boundaries is, in principle at least, a problem for basis set expansion methods. Expansion in term of square integrable basis functions introduces boundaries set by the spatial extend of the basis functions. If the basis is not suIciently extended, rePection from the boundaries can also occur leading to inaccurate results. The requirements of integration to large times and adequate representation of the positive energy states over a suIciently large spatial

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region places severe demands upon computational techniques. How to overcome requirements related to large time and space intervals is discussed in Section 8 below. 7. Adiabatic approximations and hidden crossings 7.1. General formalism The adiabatic approximation is widely used in calculations of inelastic processes in very slow atomic collisions where the relative velocity of colliding particles is much less than orbital velocities of atomic electrons. According to the adiabatic theorem [82], in the limit of zero collision velocity electron non-resonance transitions cannot occur. With changing internuclear distance the electron wave function adapts smoothly and remains in its adiabatic eigenstate. For this reason, it is convenient to expand the electronic wave function in molecular basis functions, which contain the internuclear distance R as a parameter. The adiabatic representation is usually combined with a semi-classical approach, in which the motion of nuclei is treated classically and the motion of electrons quantum mechanically. If the collision velocity is low, then transitions are most likely to occur at internuclear distances where the two levels have smallest energy separation. The Neumann–Wigner theorem states that exact crossing of two levels of the same symmetry is impossible, and they exhibit an “avoided crossing”. Smallest energy separations usually occur near such avoided crossing. An elegant method of solving the Schr7odinger equation is obtained by considering quasi-molecular eigenenergies (potential energy curves) as functions of complex internuclear distance. In that case the potential energy curves cross exactly at a certain complex distance RC in the vicinity of which the complex energy diGerence is given by the relation E1 (R) − E2 (R) ˙ R − RC : (59) Eq. (59) indicates that RC is a square root branch point. The presence of branch points has very important consequences. The expectation value of the transition matrix element depends on the value of the complex coordinate R. At R = RC the transition matrix element has a pole. This means that the actual transition probability can be derived by solving the couple equations in the neighborhood of this pole. This transition probability [27] is given by

2 Pij = exp − :ij ; (60) v where :ij is the so-called Massey parameter deMned as RC E (R) − E (R) i j :ij = dR ; vR Re RC

(61)

where vR is the radial velocity. In the case of ionization, the situation is complicated since the level crossings often occur far from the real R axis, so that they have a little eGect on the behavior of energy levels at real internuclear distances (“hidden crossings”) [83,84]. Moreover, for ionization the promoted initial term must cross an inMnite number of Rydberg levels. For this reason, the conventional adiabatic approximation predicts negligible probability for ionization, in contradiction with the experimental Mndings.

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139

The solution of this problem was suggested by Demkov [85] who put forward the idea that quasi-molecular eigenenergies @(R) could be considered as analytical functions deMned on multisheeted Riemann surface. The analytical features of @(R) in the complex plane R are closely related to the probabilities for transitions between an inMnite succession of the quasi-molecular levels. This approach has been extended [83] to develop the theory of hidden crossings and establish the main mechanisms governing the quasi-molecular level coupling with the continuum. The study of direct coupling with the continuum (direct ionization) was largely stimulated by the experimental work of Woerlee et al. [86] who measured energy spectra of electrons ejected in collisions of ions and atoms of noble gases in the keV ion energy range. It was found that the high-energy parts of the doubly diGerential cross sections were well described by a rather simple empirical formula: d2 R20 .0 (Ee − E0 ) ; (62) = exp − dEe dDe 4Ee v where Ee is the ejected electron energy, De is the electron ejection solid angle, R0 , E0 and .0 are constants. This relation holds in a very broad electron energy range corresponding to variation of the diGerential cross sections by Mve orders of magnitude. The right-hand side of (62) resembles very much the generic result of the adiabatic approximation (see (60) and (61)) since the exponent has a sense of the Massey parameter. Thus it was a challenge to interpret (62) within the framework of the adiabatic approximation and to explain the physical meaning of the empirical constants. Such an explanation was given in terms of series of the branch points associated with diGerent mechanisms of direct ionization, which are discussed in next sections. 7.2. S-ionization and superpromotion (n+1)lm Solov’ev [87] discovered branch points Rnlm connecting the terms Enlm (R) and E(n+1)lm (R) (n+1)lm successively for all n ¿ l + 1. The set of branch points Rnlm with diGerent n values but Mxed set {lm} forms an inMnite series of points localized in a small region D of the complex R plane and has a limit point (n+1)lm Rlm = lim Rnlm : n→∞

(63)

The branch points connect all terms of the given series to form a unique analytical function Elm (R). In the domain D, the energy surface resembles a corkscrew, so that a single turn around the branch point Rlm promotes the state Enlm (R) to the neighboring state E(n+1)lm (R) (Fig. 1). The series {lm} was designated S(l+1)lm , and the mechanism of electron production due to promotion of a “diabatic” term to the continuum through a succession of these points was called S-ionization (from the word “superpromotion”). Analytical expressions for the coordinates of the limit branch points were obtained in [88], namely,  

2

2  2 2 1 1 (m + 1) 1 (m + 1) Rlm = l+ ± i(m + 1) 2 l + ; (64) − −  Z 2 2 2 4 where Z is the united atom charge. In addition to the branch points R(l+1)lm connecting neighboring levels, branch points R(l+1) ˜ ˜ connecting the lowest state of the series with quasistationary and virtual lm

140

S.Yu. Ovchinnikov et al. / Physics Reports 389 (2004) 119 – 159 0 4p

3p

ReE

-0.25 .25

2p

-0.5

-0.75 0.75

ReR

00 0.5

-1

1

0.5

ImR

1

1.5

1.5

Fig. 1. Riemann surface associated with S-ionization in H+ –H collisions.

˜ states {lm} were found in [89]. This discovery made it possible to compute ionization within the framework of the advanced adiabatic approximation [8]. Classical interpretation of S-ionization is based on the topology of the electron motion. At close approach of the colliding particles, a united atom centrifugal barrier appears. This barrier keeps the electron out of the region between nuclei, so that the electron trajectory along the line between nuclei becomes unstable. Oscillation of the electron along this unstable trajectory transfers energy from nuclear to electron motion until the electron acquires enough energy to escape. In terms of the quantum theory S-promotion corresponds to rearrangement of quasimolecular wave functions to united atom wave functions. In the advanced adiabatic theory the electron energy distribution integrated over the electron direction is given by the probability P(E) [8]: E 1 dR(E) 2 2i ; P(E) = C (E) exp R(E) dE (65) 2v dE v where R(E) is the function reciprocal of E(R) and C(E) is the normalization coeIcient of the adiabatic wave function averaged over the angular distribution. At high enough electron energies the following approximations can be used: dE 2 (66) C (E) = 4 Im R(E) ; dR

S.Yu. Ovchinnikov et al. / Physics Reports 389 (2004) 119 – 159

Im R(E; b) = Im R(E) 1 +

b2 2|R(E)|2

141

:

(67)

Integration of (65) over the impact parameter b gives the following expression for the diGerential cross section for electron ejection [90]: ∞ d .(E) = 2 ; (68) P(E; b)b db = A(E) exp − dE v 0 where 4|R(E)|2 Im R(E) A(E) = ; .(E)

.(E) = 2

E

E0

R(E ) dE :

(69)

Using the theorem of the mean for the integral in (69) and comparing (68), (69) with the empirical formula (62) we Mnd the physical meanings of the constants in that formula, namely, R0 corresponds to the absolute value of the coordinate Rlm , .0 = 2 Im Rlm to the stability exponent of the unstable trajectory and E0 to the energy of the ionized quasi-molecular level. 7.3. T-ionization and top-of-barrier electrons The T-type branch points (from the words “top of barrier”) were discovered by Ovchinnikov and Solov’ev [62] and used to compute transitions between bound states. It was later recognized that these transitions were closely related to the ionization mechanism now called top of barrier promotion or T-promotion. This mechanism is eGective at large internuclear distances as the colliding particles separate. At such distances the Coulomb potential barrier between the nuclei becomes comparable to the electron energy. When the particles approach, the Coulomb potential barrier decreases, so that the atomic electrons begin to move in the Meld of both centers. When the particles recede from each other, the barrier increases, and the electrons can be captured on the top of the barrier (the saddle point of potential) and, Mnally, promoted into the continuum. The motion of the electrons on the top of the barrier is unstable therefore energy is transferred from nuclear to electron motions. The branch points RTn1 n2 m (n1 and n2 are the parabolic quantum numbers) connect states with the same n1 but diGerent n2 . Electrons ejected due to T-ionization are often called “saddle point electrons”. In the case of equal charges, Z1 = Z2 , the velocity of such an electron equals one half of the incident ion velocity. In terms of quantum theory T-promotion corresponds to rearrangement of the quasi-molecular wave functions to the separated atom wave functions. The Riemann surface calculated for T-ionization in H+ –H collisions [91] is shown in Fig. 2 as a plot of the function n(R)=@−1=2 (R) vs. S =R1=2 . The branch points connecting the 1s−3d−5g sheets are clearly visible. These are the Mrst of an inMnite series of branch points connecting diGerent sheets with the same n1 but diGerent n2 quantum numbers. To the left of the inMnite series the Riemann surface (Fig. 2) is remarkably Pat and structureless. This sloping Pat region represents harmonic oscillator behavior in the vicinity of the saddle point [91]. On the real R axis the surfaces represent the Rydberg states. The integral in the Massey parameter (61) for T-promotion always pass through the harmonic oscillator region. The probability of T-ionization from the initial 1s orbital can be written as [92] (T ) P = (1 − P000 )P00 ;

(70)

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5g

3d

2

Re n(S)

3

1s 4 1

Re

2

0

S

1 0

2

ImS

Fig. 2. Riemann surface associated with T-ionization in H+ –H collisions. The function n(R) = @−1=2 (R) vs. S = R1=2 .

where (T ) P0m

=

∞

P0n2 m = e

−(2=v):T0m

;

:T0m

=

n2 =0

∞

:0n2 m ;

n2 =0

P0n2 m is the probability for transitions between the terms E0n2 m and E0(n2 +1)m connected by the branch point RT0n2 m , and :0n2 m is the relevant Massey parameter. After integration over the impact parameter the cross section for 1s-ionization is obtained: vR2000 T Q(:000 )e−(2=v)(:00 −:000 ) ; (71) (T ) ∼ = :000 where Q(:) = e−2:=v (1 − e−:=v ) − 12 e−4:=v (1 − e−2:=v ), R000 and :000 are the coordinate of the branch point and the Massey parameter for the transition 1s − 3d, respectively, and :T00 is the sum of the S Massey parameters for the superpromotion T00 . For another succession of transitions, via 2p − 2p S rotational coupling followed by T01 promotion, one gets: T P(T ) = P2p−2p P01 ;

(72)

T ; (T ) = 2p−2p P01 ∞ P2p−2p (b)b db ; 2p−2p = 2

(73)

0

(74)

and the total cross section for ionization due to S- and T-promotion [93] is ion = (S) + T + T :

(75)

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143

T

1

+

H -H 4 keV

dσ/dE, 10

-18

2

cm /eV

S

0.1

0.01 0

2

4

6

8

10

12

14

E, eV

Fig. 3. DiGerential cross sections for ionization in H+ –H collisions: contributions of S- and T-ionization.

The contributions of S- and T-ionization to diGerential cross sections for electron ejection in H+ –H collisions are shown in Fig. 3. It is seen that T-ionization gives the major contribution to the low-energy part of the energy spectrum. 7.4. D-ionization and radial decoupling D-ionization (from the word “decoupling”) was discovered by Ovchinnikov and Macek [94] and studied using the zero-range model potentials [95]. Within the framework of quantum mechanics, this mechanism occur at very small internuclear distances because the electron wave function remains unchanged as the internuclear distances decreases. This decoupling of electron and nuclear motion implies that the electronic state becomes non-adiabatic and transitions to other bound states and continuum take place. A given classical orbit of an electron moving in the Meld of two nuclei scales with R. As R → 0, the classical orbit contracts accordingly. However, in quantum mechanics an electronic wave function cannot be reduced in size without limit because at R=0 it has to coincide with the united atom wave function, which has a Mxed spatial extension. Therefore, as the nuclear distance decreases, at some very small distance R0 (which is well within the united atom limit) the electron will eGectively move in the Coulomb Meld of only one (united) nucleus, and the quasi-molecular wave function will become independent of R: The radial decoupling mechanism is especially important at low collision energies, since its contribution fundamentally alters the adiabatic limit. In the adiabatic limit, the velocity dependence of inelastic transition cross sections is not exponential, as predicted by the hidden crossing theory and other models of non-adiabatic coupling. Instead, the cross sections display a power law dependence on the collision velocity, D ˙ v n . In recent works of Krstic et al. [96,97] devoted to the study of non-adiabatic transitions between bound states this problem is treated in terms of complex branch points of the radial velocity, which

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are deMned by the relation vtv = ±ib :

(76) 4

The authors obtained the dependence ˙ v for inelastic transitions in heteronuclear systems and ˙ v8 in homonuclear (e.g. H+ –H) systems. These results are valid irrespective of the particular form of coupling matrix elements and adiabatic energies. The possibility of providing detailed information about the mechanisms responsible for ionization and the ranges of internuclear distances where they act is an important merit of the theory of hidden crossings. In general, the adiabatic representation gives reliable results on cross sections for inelastic processes in slow atomic collisions. The molecular basis used in such calculations provides an adequate description of one topology of the electron motion at small internuclear distances. However, extension of this approach to higher energy ranges and ionization meets principal diIculties connected with Galilean non-invariance of adiabatic eigenfunctions and the translated motion of electrons at t → ±∞. As a result, the non-adiabatic coupling matrix elements do not vanish as R → ∞ leading to unphysical transitions between the adiabatic states. A rigorous method for constructing of Galilean invariant basis functions was developed by Solov’ev and Vinitsky [98] who used a time-depending scaling of the electron coordinates, q = r=R(t), and an additional transformation of the wave function to preserve the form of the Schr7odinger equation. However, practical calculations in this representation were very diIcult due to the divergence of some matrix elements at R = 0. Further development of the scaling theory employs the Sturmian method discussed below. This method circumvents the divergence problem and exploits the Galilean invariance for the scaling approach to two center collision problem in a broad energy range. 8. Sturmian theory 8.1. Scale transformation of Solov’ev and Vinitsky The Sturmian method is based on the following three transformations: (i) the scale transformation of Solov’ev and Vinitsky, (ii) the expansion of the total wave function in eigenfunctions of the Sturm–Liouville problem, and (iii) the representation of the wave function in the form of the Fourier integral. The transformation of Solov’ev and Vinitsky [98] allows one to take into account translational and rotational eGects. It includes the change of variables: r q= ; (77) R(t) t dt E= (78) 2 −∞ R (t ) and the transformation of the wave function: ˙ 1 iR(E) 2

(r; t) = 3=2 q <(q; E) : exp R (E) 2R(E)

(79)

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145

Then in a reference frame rotating around the vector n = C × b with the frequency D we obtain a new Schr7odinger equation [18,98]: 1 2 1 2 2 9<(q; E) ˆ − ∇q + D q + DLn + R(E)V (q) <(q; E) = i : (80) 2 2 9E In the scaled space {q; E} the nuclei are Mxed, and the potential in the transformed coordinates does not depend on the direction of R(t). The dynamical eGects are described by a scalar function R(E) and two additional terms in the new Hamiltonian, namely, the isotropic harmonic oscillator potential 1 2 2 D q and the angular momentum operator Lˆ n . 2 In the scaled representation solutions for V = 0 are Galilean invariant in the limit t → ±∞. This implies that solutions for V = 0 have correct asymptotic behavior and explicit translation factor are not needed. In eGect the exponential factor in (79) is a translation factor with a switching factor that is appropriate for bound and continuum states. Because it is appropriate for continuum states it is possible to compute electron spectra in this representation. In contrast other translation factors only allow one to compute total ionization probabilities. For straight-line trajectories the nuclei rotate with the constant angular velocity D=vb in the scaled representation, therefore it will be convenient to introduce the angle of rotation = DE, 0 6 6 , as a new “time” in the Schr7odinger equation and R() = b=sin . In scaled coordinates {q; } the initial conditions are
(81)

|
(82)

The initial condition (81) ensures that there is no continuum in the initial state and condition (82) normalizes the wave function. To obtain the spectra of the ejected electrons one projects the time-dependent wave function onto plane waves, A(k) = lim ’∗k (r; t) (r; t) d 3 r ; (83) t →+∞

where ’k (r; t) =

1 2 eik·r−i(k =2)t : 3=2 (2)

(84)

It is more convenient to evaluate the projection in the scaled space. Since the scale transformation conserves wave-function normalization and the exponential factors cancel, one has A(k) = lim ’∗k (q; )<(q; ) d 3 q ; (85) →−0

where the function ’k (q; ) is the transformed plane wave,

3=2 iD k 2 k 2 2 ·q+ q + cos b v ’k (q; ) = e 2 sin v 2 sin 3=2 i = KD∗ (q; − ; k=v; 0) v

(86)

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and is proportional to the propagator KD for the Schr7odinger equation with the isotropic harmonic oscillator potential 12 D2 q2 and the angular momentum operator Lˆ n . Taking into account that KD (q; ; k=v; 0) → 0(q − k=v) as → +0 and interchanging the order of the integration and the limit in (85), one can obtain [9] 3=2 i A(k) = <(k=v; ) : (87) v At Mrst glance it might seem that unitarity would imply |A(k)|2 dk = 1, i.e. the initial state is always completely ionized. This conclusion is incorrect because one cannot interchange integration over k and the limit → . This is necessary to insure unitarity. 8.2. Sturmian basis In most cases basis of wave functions are eigenenergies of certain model potentials. The Sturmian basis is diGerent in that the Sturmian eigenfunctions do not correspond to the eigenenergies. In this representation the parameter !=E(R)R2 is introduced instead of the energy E(R). A Sturm–Liouville problem is furnished by the system of diGerential equations: ! 2 1 1 2 2 ˆ n + B$ (!)V (q) − !D S$ (!; q) = 0 ∇ + D q + D L (88) q 2 2 with proper boundary conditions. The values of internuclear distance at Mxed potential energy are taken as new eigenvalues B$ (!). They are solutions of the equation @(B)B2 = !D = const: The Sturmian eigenfunctions are normalized according to S$ (!)| − V |S$ (!) = − S˜ $ (!; q)V (q)S$ (!; q) d 3 q = 0$; $ ;

(89)

(90)

where the dual functions S˜ $ (!; q) are obtained from S$ (!; q) by inverting the axis of rotation. The normalization with the dual function avoids complex conjugation, and allows us to use the analyticity of Sturmian eigenfunctions in the complex plane of !. The corresponding Sturmian eigenfunctions S$ (!; q) are deMned for all values of !, including negative, positive and even complex values. In contrast to the adiabatic functions, the Sturmian functions do not depend, even parametrically, upon the internuclear distance. The Sturmian bases for particular cases of zero-range potentials and two Coulomb centers have been discussed in [18,99,100]. To compute the Sturmians it is often convenient to write (88), as the homogeneous Fredholm integral equations of the second kind [73] S$ (!; q) = B$ (!) G(!; q; q )V (q )S$ (!; q ) d 3 q ; (91) where G(!; q; q ) is the Green function for the Schr7odinger equation with the isotropic harmonic oscillator potential 12 D2 q2 and the angular momentum operator Lˆ n . The Green function has the close

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147

analytic form [100,101]; %(1=2 − !=D) ei[q ×]·q √ G(!; q ; q ) = 2|q − q q − qxz | D √ √ 9 9 −I D!=D−1=2 ( 2D I)D!=D−1=2 (− 2DH) ; × H 9I 9H

(92)

where I = 12 (|q − q | + |q − qxz |) ; |) ; H = 12 (|q − q | − |q − qxz

(93)

and where D. (z) is the parabolic cylinder function [102], %(z) is the Gamma function [102], and qxz = (x; −y; z) is the rePection of q from the xz-plane. 8.3. Wave functions and transition amplitudes in Fourier space The total time-dependent wave function expanded in the Sturmian basis functions in Fourier space is written as [103] " c$i (!) bD ∞+ia i!(=2−)
Re !→−∞

S$ (!; rj =B$ (!)) ˙

i (rj )0v; i

(95)

insures that the coeIcients c$i (!) are elements of a square matrix c(!). The matrix c(!) are solutions of the three term matrix recurrence relations (80). c(! − 1) = M −1 (!)[2b−1 (!)c(!) − M T (! + 1)c(! + 1)] ;

(96)

where Mkl (!) = Sk (!)|V |Sl (! − 1) and Bkl (!) = 0kl Bk (!). To choose from the many solutions of the recurrence relation (96), one requires that the integral in (94) must converge. The Sturmian functions S(!; q) rapidly decrease at large negative Re ! on the contour of integration, but oscillate with slowly changing amplitude at Re ! ¿ 0. Therefore, to ensure convergence of integration, the coeIcients c$i (!) must decrease as Re ! → ∞, i.e., limRe !→∞ c(!) = 0. There are an inMnite number of solutions with this convergence property. They are obtained by starting at asymptotically large Re ! with arbitrary initial matrices, then stepping to lower value of ! using (96). Such solutions diGer by periodic matrices f (!) = f (! + 1). Conversely for a given initial matrix all other asymptotically decreasing solutions can be found by multiplication of the matrix c(!) and some periodic matrix f (!). The periodic matrix f (!) is chosen to satisfy initial conditions (81) and (82). Reference [18] gives a method applicable when atom are initially in bound states. The periodic matrix is found by analyzing the asymptotic behavior of c(!) at large negative Re !. To do this one writes c(!) = J + (!)c L+ (!) + J − (!)c L− (!) ;

(97)

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where c$L± (!) are two solutions of (96) deMned by their asymptotic behavior [104] at large negative Re !;

! b L± d! ± i ! : (98) arcsin lim c$ (!) = exp ±i Re !→−∞ B$ (! ) 2 0 The periodic matrices J + (!) and J − (!) are given by i (99) J ± (!) = ∓ W {c(!); c L∓ (!)} ; 2 where W {A(!); B(!)} ≡ A(! + 1)B(!) − A(!)B(! + 1). Asymptotic evaluation of (94) at → 0 and → by the stationary-phase method shows that the appropriate periodic matrix f (!) is f (!) = [J − (!)]−1 ;

(100)

so that c(!) becomes c(!) = [J − (!)]−1 J + (!)c L+ (!) + c L− (!) :

(101)

The S matrix of transitions between the bound states is the constant term in the Fourier expansions of the periodic functions [J − (!)]−1 J + (!): 1 [J − (!)]−1 J + (!) d! : (102) S= 0

It must be emphasized that (96) only pertains to bound states. For that reason this reduced S matrix is not unitary. One way to calculate the ionization probability Pion from the initial state “i” is by using unitarity |Si$ |2 : (103) Pion = 1 − $

An important feature of this method is that excitation, electron transfer and total ionization are computed from three term recurrence equations directly. There is no need to actually compute the time dependent wave function. For that reason the theory is accurate and computationally eIcient. To obtain momentum distributions, however the actual wave function is needed. The use of the Sturmian basis in the Fourier space has evident advantages, which allow one to provide accurate quantitative description of ionization processes in a broad range of velocities and internuclear distances. First, the Sturmian functions form a complete discrete set for all values of !, i.e. they describe in a unique way both discrete and continuum states of the colliding system. Integration over ! from −∞ to +∞ ensures that the total wave function and each term of the expansion contain the whole energy spectrum, including both the discrete states and the continuum. Second, since the Sturmian functions do not depend on the internuclear distance, the matrix elements for dynamical interaction do not contain singularities at R = 0. Third, since positions of nuclei are Mxed in the Sturmian basis, Eq. (96) are Galilean invariant and additional translation factors are not necessary. Finally, sums over $ in (94) converge very rapidly, therefore a small number of Sturmian functions is usually suIcient to describe particular collision processes. Momentum distributions are given by (87) which becomes in the Sturmian theory: b ∞+ia −i!=2 c$i (!) A(k) = √ e S$ (!; k=v) d! : (104) BP (!) v i $ −∞+ia

S.Yu. Ovchinnikov et al. / Physics Reports 389 (2004) 119 – 159

149

|A |2 0.0 4

0.02 0 2

-2 -1

1 0 -1 2

k⊥ /

1

υ

0

k|| / υ

-2

Fig. 4. Electron distributions |A(˜k)|2 vs. k=v for H− –H collisions at v = 10 a:u: in the center-of-mass frame.

|A |2 0.015 0.010 0.005 0 10

-10 -5

10

k⊥ /

k|| / υ

0 -5

5

υ

5 0

-10

Fig. 5. Electron distributions A(˜k)2 vs. k=v for H− –H collisions at v = 0:1 a:u: in the center-of-mass frame.

It has been shown [100,105] that the main features of the ejected electron spectra can be obtained using only one Sturmian function. This single function connects high-energy and low-energy regions. Calculation results for the system H− –H are presented in Figs. 4 and 5. At high velocity, v=10 a:u: (Fig. 4), there are two peaks, one centered at the target (˜k = 0) and another one at the projectile (˜k = 1), separated from an oval-shaped binary encounter ridge at |˜k − ˜v| = v. These features rePect those observed in high-energy ion–atom collisions. At low velocity, v = 0:1 a:u: (Fig. 5), there is only one peak for g states centered at ˜k =˜v=2, and a small ridge at k ≈ 7 a:u: In comparison with the high-velocity distributions one notes that the binary encounter peak completely disappears, and separate target and projectile peaks merge into one peak centered between target and projectile. The situation is similar for u states where there is a node at the midpoint owing to symmetry requirements. In this case there is also no binary encounter peak,

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and the separate target and projectile peaks have merged into a peak with a node exactly midway between target and projectile. The distribution with “atomic” initial conditions exhibits the peak at the center-of-mass velocity and a series of interference features. This expression has been used to compute electron spectra for two Coulomb centers with b = 0 [99,103]. At high velocities the soft electron peak, the binary encounter ridge and the continuum capture cusp dominate the energy spectrum, while at low velocities S-, T- and D-ionization mechanisms are revealed. Evaluating the transition amplitudes by expanding in inverse powers of v gives the high-velocity distribution, while evaluating them in a stationary phase approximation gives the low-energy advanced adiabatic theory [8] for S-promotion [99]. 8.4. Numerical solutions The main numerical diIculties come from computation of the Sturmian eigenfunctions, matrix elements Mik , and inversion of the matrix M . Since this must be done for all ! the computations can be quite time consuming. This is especially true when calculations require a large number of Sturmian functions. To overcome these diIculties one can replace integrals by sums using any convenient integration quadrature [73], where u(q) d 3 q = u(qk )wk : (105) k

Then one can deMne the unitary matrix U (!), where Unk (!) = Sn (!; qk ) V (qk )wk :

(106)

The unitary matrix U(!) diagonalizes the matrix G(!): −1 (!) = U (!)G (!)U −1 (!) :

(107)

Then using U (!) one can transform the three term matrix recurrence relations to a new representation. Taking into account that M (!) = U (!)U −1 (! − 1) one can write this three term matrix recurrence relations in the form B(! − 1) = 2bG (!)B(!) − B(! + 1) ;

(108)

B(!) = U (!)c(!)U −1 (!)

(109)

where and Gnn (!) =

# V (qn )wn G(!; qn ; qn ) V (qn )wn :

(110)

The transitions amplitudes between the bound states are determined by (102), where one uses lim

Re !→−∞

B(!) =

lim

Re !→−∞

J + (!)c L+ (!) + J − (!)c L− (!) :

Finally, electron momentum distributions in this representation are b ∞+ia −i!=2 e G(!; k=v; q$ )V (q$ )B$i (!) d! : A(k) = √ v i $ −∞+ia

(111)

(112)

S.Yu. Ovchinnikov et al. / Physics Reports 389 (2004) 119 – 159

151

Propagation of the matrix B in ! by three term recurrence equation (108) is fast because the matrix G is easily calculated, and only matrix multiplication is needed. In this connection note that wk cancels the singularities of Coulomb potentials in spheroidal coordinates. Thus there are no need to introduce the ad hoc soft core potential [74].

9. Calculation results 9.1. Total ionization cross sections Most of the available calculations are related to the total ionization cross sections in H+ –H collisions. Comparison of the typical calculation results and experimental data is given in Fig. 4. The experimental cross sections are represented by the recommended data [81] corrected with account of the recent data of Shah et al. [106]. This comparison reveals the main advantages and limitations of diGerent approximations used in the calculations. The Born approximation [107] gives reliable results at the proton energies Ep ¿ 200 keV. At lower energies disagreement with experimental data progressively increasing with decreasing impact energy. The CDW-EIS approximation [108] gives good agreement with experimental data at intermediate and high energies, but underestimates the cross sections at low energies (Ep 6 30 keV). This is explained by the impossibility of the CDW wave functions to represent adequately the electron motion in the Meld of two charged particles, that is important at low energies. Some curves in Fig. 6 display the results of calculations using diGerent versions of the coupled channel approximation. The use of the two-center atomic basis [53] gives good results at high energies, but overestimates the cross sections at intermediate energies, that is caused by the convergence problem discussed in Section 5. The use of three-center atomic bases [63,64] improve the agreement with experimental data in the region of the cross section maximum, but underestimates the cross sections at higher energies, that can be caused by insuIcient number of orbitals centered on the saddle point, which could be used in the calculations. Both AO versions of the close-coupling approximation give unreliable results at low energies. The molecular orbital versions of the coupled channel approximation [65,70] allow improving agreement with experimental data at low energies, though this requires considerable extension of the basis size. The calculations using direct numerical solution of the time-dependent Schr7odinger equation [75,77] yield cross sections which are considerably higher than the experimental data at low and intermediate energies, that may be connected with diIculties in propagation of the computations to suIciently large internuclear distances. Our Sturmian calculations provide excellent agreement with experimental data in a very broad energy range, from 1 keV up to 200 keV. Fig. 7 shows the similar comparison of the experimental data [109,110] and calculations for the colliding system He2+ –H. One can see a very large dispersion of the calculation results obtained using diGerent approximations. The upper curve is that of the Born calculations, while the lower one is that of the CDW–EIS calculations. The 3CMO approximation gives good agreement with experimental data in the projectile energy range 100 –400 keV, and again the Sturmian calculations provide good agreement with experimental data in a broad energy range.

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10

-17

σ, 10 cm

2

D E F G H I J K L M B

1

0.1

0.01 10

EH+, keV

100

1000

Fig. 6. Total cross sections for ionization in H+ –H collisions: B, recommended experimental data [81]; D, Sturmian calculations; E, Mrst Born approximation [107]; F, CDW–EIS approximation [108]; G, 482-state two-center AO calculations [53]; H, 36-state triple-center AO calculations [63]; I, 40-state triple-center AO calculations [64]; J 362-state triple-center MO calculations [70]; K, 20+ Gaussian set triple-center MO calculations [65]; L, direct solution by LTDSE method [75]; M, direct solution by TCMSD method [77].

-17

σ, 10 cm

2

100

D E F G H B

10

1 10

100

1000

EHe2+, keV Fig. 7. Total cross sections for ionization in He2+ –H collisions: B, experimental data [109,110]; D, Sturmian calculations; E, Mrst Born approximation [107]; F, 34-state triple-center AO calculations [20]; G, 41+Gaussian set triple-center MO calculations [65]; H, CDW–EIS approximation [108].

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153

H0 − He Experiment

σ (cm2)

•

10−16

•

•

•

• •

•

•

Theory •

•

10−17 1

10

100

E H + (keV)

Fig. 8. Cross sections for stripping in H0 –He collisions. Solid curve—calculations [111], points—recommended experimental data [112].

Fig. 8 shows calculations [111] and recommended experimental data [112] on the cross sections for stripping in H0 –He collisions. In this case the stripping is caused by promotion of a single 2p electron and no Coulomb potential barrier is formed during the collision. For this reason, S-ionization can be considered as the dominant mechanism of stripping. One can see that even this simpliMed approximation gives quite reasonable agreement with the experimental data, that provides promising perspectives of extending the hidden crossing theory to the colliding systems with one “active” electron and non-Coulomb interaction. 9.2. Di0erential cross sections The study of diGerential cross sections for electron ejection is an eGective tool for examining ionization dynamics in atomic collisions. For this purpose, many attempts have been undertaken to measure and calculate energy and angular distributions of electrons ejected in collisions of protons with simple targets, such as H2 and He. Most of the studies has been performed at high enough proton energies, where the Born approximation and classical approach give satisfactory results (e.g. see [57]). In the course of the studies two problems have arisen which are not resolved up to now. One of them is connected with the role of the saddle-point ionization mechanism. Much of the debate centers on a model proposed by Olson [10,11] which oGers an explanation for the bulk of ionization in near-matching velocity ion–atom collisions—the saddle point mechanism. His calculations for H+ –H collisions by the classical trajectory Monte Carlo (CTMC) method claimed that saddle point ionization was the dominant mechanism at intermediate energies of 40 –60 keV. Classical calculations of Bandarage and Parson [113] supported Olson’s conclusion and even predicted that the importance of the saddle point mechanism should decrease at lower energies. Illescas et al. [114] extended the classical theory, but concluded that the saddle point ionization was important only at low energies. This point of view is shared by Winter and Lin [63] who calculated the total ionization cross sections

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in H+ –H collisions using 3CAO approximation, while the T-ionization calculations of Pieksma and Ovchinnikov [92] support Olson’s prediction. Many experiments have been carried out to search for saddle-point electrons, also with conPicting results. The recent TCMSD and CTMC calculations of Sidky et al. for the H+ –H system [115] have led the authors to the conclusion that presence of the v=2 peak in the longitudinal velocity distribution of ejected electrons is not an unambiguous signature of the saddle point mechanism. At higher energies, this peak can appear in direct ionization, due to elimination of electrons that remain bound to the target (Ce = 0) or become captured to the projectile ion (Ce = C). The contribution of the saddle point mechanism per se decreased rapidly with energy to about 1% at 50 keV. Some doubt in the importance of the saddle point mechanism is aroused due to the experimental Mndings of Abdallah et al. [116] who studied collisions of bare nuclei with He and Ne atoms at a Mxed projectile velocity varying only the projectile charge state. It was found that increasing the projectile charge state caused the ejected electron velocity distribution to shift toward the projectile center in velocity space, while the saddle point velocity shifted toward the target center. However, it should be noted that in the experiment [116] both the projectile charge and the projectile species (and, consequently, the kind of promoted orbitals) were varied, so that the studied processes need a special theoretical consideration. Thus, the role of the saddle point mechanism is not Mrmly established yet, and more elaborate calculations of diGerential cross sections are necessary. This especially relates to the detailed Sturmian calculations of electron velocity distributions in ion–atom collisions which have been performed only for the case of b=0 up to now. Another problem that still causes animated discussion is connected with the experimental observation [14] of rapid oscillations in the momentum distributions of electrons ejected in H+ –He collisions. The experiment was performed using the COLTRIMS technique. The measured distributions showed a two-peak structure, the heights of the peaks changed rapidly in varying the proton energy from 5 to 15 keV. The Sturmian theory for the H+ –H system [9] implies that such oscillations are associated with T-ionization and, in particular, with behavior of the real part of the potential energy @(R) in the harmonic oscillator region. In this case the diGerential cross section for electron ejection can be written as d3 ˙ |A (k=v) + a exp[i=v]A (k=v)|2 ; (113) d k3 ∞ where = −Re R0 [@ (R) − @ (R)] dR, A and A are the amplitudes of ionization from g and u orbitals, a is the expansion coeIcient. The amplitude A is nodeless, but A has a node when k⊥ = 0. For this reason, the electron distribution is not symmetric about the primary beam axis and changes rapidly with velocity. The electron distributions for 5, 10 and 15 keV H+ –H collisions at an impact parameter b=1:2 a:u: are shown as density plots in Fig. 9. In these plots, the z-axis is taken along the ion velocity so that k + kz and the z-axis lie in the scattering plane along the direction of the impact parameter vector. The phase changes by nearly 2 over the 5 –15 keV energy range, that gives rise to drastic change in the electron distributions, in accordance with the experimental Mndings. However, the TCMSD calculations [117,118] performed at the same impact parameter did not reveal rapid oscillations in the transverse momentum distributions of electrons ejected in H+ –H collisions in the proton energy range 5 –15 keV. The calculations showed two peaks maintaining nearly the same relative height. The authors [117] came to the conclusion that the rapid

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Fig. 9. Density plot of the electron distributions, ˜k=v, for proton impact on atomic hydrogen at Mxed impact parameter b = 1:2 a:u: and ion energies of 5, 10 and 15 keV.

oscillations were not a general feature of ion–atom collisions at low energies, but were speciMc to the H+ –He system. On the other hand, the more recent LTDSE calculations for H+ –H system at smaller impact parameter [119], with propagation of the wave function to signiMcantly larger Mnal distances, showed a drastic change in the form of electron transverse momentum distribution in the energy range 1–25 keV. Thus, additional theoretical and experimental studies are necessary to put the eGect on the quantitative basis. This is of particular importance, because analysis of the experimentally observed structure in the electron momentum distribution [120] can give information about the parameters of potential energy curves for complex values of the coordinate R. 10. Conclusions In this paper, we have given a brief review of the present state of theoretical study of ionization in atomic collisions. We have considered the main quantum mechanical approaches restricting ourselves

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to the simplest colliding systems with one “active” electron. By their basic concepts, these approaches can be divided into three groups. One of them is aimed at direct solution of the time-dependent Schr7odinger equation. In this case, the main outlook is connected with rapidly extending possibilities of up-to-date computers. However, comparison of the available calculation results shows that the time interval in which the time-dependent wave function can be propagated now is still insuIcient to provide the necessary accuracy of the data obtained, and great achievements in this Meld are far ahead. The second group uses expansions of the total wave function in atomic or molecular bases, remaining within the concept of the real values of eigenenergies. In this case, obtaining satisfactory agreement with experimental data requires the use of a great number of bound states and pseudostates. For example, 362 states were taken in [70] to obtain a good agreement with the experimental data on the total ionization cross sections for the H+ –H system in the proton energy range 1–10 keV. Penetration to the region of complex internuclear distances (the hidden crossings theory) facilitates calculations at low energies. Furthermore, the use of the derived relations makes it possible to determine the parameters of quasimolecules from the analysis of experimental data on energy spectra of ejected electrons [90,120]. Internal problems connected with convergence of results and Galilean invariance do not allow highly accurate calculations in a broad energy range. The third group uses the Sturmian expansion, Solov’ev–Vinitsky scaling and Fourier transformation of the wave functions. This method ensures solution of the problem by using small number of basis states, avoiding the diIculties connected with the lack of Galilee invariance, singularities of matrix elements and non-convergence of expansions. Up to now, the Sturmian calculations provide the best agreement with experimental data in a very broad energy range. However, the quantitative data on the total and diGerential cross sections for ionization obtained by the Sturmian method are still very scarce. Some inconvenience is also introduced by the fact that the Sturmian basis functions have no resemblance to the energy eigenstates. The present review shows that even for the simplest colliding system, H+ –H, the problems in understanding the ionization process remain that are needed to be solved in the future. This is to be done by improving the accuracy of calculations and by calculating and measuring ionization process in greater detail. In particular, the experimental data on energy distribution of electrons ejected in H+ –H collisions are urgently needed. Finally it is important to extend theory to ionization processes in collisions of many-electron atomic systems. References [1] R.K. Janev, H.W. Drawin, Proceedings of the IAEA Committee Meeting on Atomic and Molecular Data for Fusion Reactor Technology, Cadarache, France 1993, p. 130. [2] C.J. Joachain, D.E. Post, Atomic and Molecular Processes in Controlled Thermonuclear Fusion, Plenum Press, New York, 1983, p. 320. [3] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297. [4] U. Fano, Ann. Rev. Nucl. Sci. (1961). [5] R. Itikawa, Adv. At. Mol. Phys. 33 (1996) 253. [6] R.E. Johnson, Energetic Charged Particles Interactions with Atmospheres and Surfaces, Springer, Berlin, 1990. [7] M.Ya. Marov, V.I. Shematovich, D.V. Biscalo, Space Sci. Rev. 76 (1996) 1. [8] E.A. Solov’ev, Sov. Phys. Usp. 32 (1989) 228. [9] J.H. Macek, S.Yu. Ovchinnikov, Phys. Rev. Lett. 80 (1998) 2298.

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Physics Reports 389 (2004) 161 – 261 www.elsevier.com/locate/physrep

The covariant-evolution-operator method in bound-state QED 0 en Ingvar Lindgren∗ , Sten Salomonson, Bj/orn As2 Department of Physics, Chalmers University of Technology and Goteborg University, SE-41296 Goteborg, Sweden Accepted 17 September 2003 editor J. Eichler

Abstract The methods of quantum-electrodynamical (QED) calculations on bound atomic systems are reviewed with emphasis on the newly developed covariant-evolution-operator method. The aim is to compare that method with other available methods and also to point out possibilities to combine that with standard many-body perturbation theory (MBPT) in order to perform accurate numerical QED calculations, including quasi-degeneracy, also for light elements, where the electron correlation is relatively strong. As a background, the time-independent many-body perturbation theory (MBPT) is brie9y reviewed, particularly the method with extended model space. Time-dependent perturbation theory is discussed in some detail, introducing the time-evolution operator and the Gell–Mann–Low relation, generalized to an arbitrary model space. Three methods of treating the bound-state QED problem are discussed. The standard S-matrix formulation, which is restricted to a degenerate model space, is discussed only brie9y. Two methods applicable also to the quasi-degenerate problem are treated in more detail, the two-times Green’s-function and the covariant-evolution-operator techniques. The treatment is concentrated on the latter technique, which has been developed more recently and which has not been discussed in more detail before. A comparison of the two-times Green’s-function and the covariant-evolution-operator techniques, which have great similarities, is performed. In the appendix a simple procedure is derived for expressing the evolution-operator diagrams of arbitrary order. The possibilities of merging QED in the covariant evolution-operator formulation with MBPT in a systematic way is indicated. With such a technique it might be feasible to perform accurate QED calculations also on light elements, which is presently not possible with the techniques available. c 2003 Published by Elsevier B.V. PACS: 12:20: − m; 31.10.+z; 31.15.Ar; 31:25: − v; 31.30.Jv; 32.10.Fn Keywords: Quantum electrodynamics; Many-body perturbation theory; Time-evolution operator; Gell–Mann–Low formula; S-matrix; Electron self-energy; Vacuum polarization; Screened self-energy; Two-photon exchange; Quasi-degeneracy; Two-times Green’s function; Covariant time-evolution operator

∗

Corresponding author. E-mail address: [email protected] (I. Lindgren).

c 2003 Published by Elsevier B.V. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.09.004

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Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Time-independent many-body perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Perturbation theory. Extended model space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Second quantization. The electron-Held operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The linked-diagram expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. All-order procedures. The coupled-cluster approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Coupled-cluster approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. Pair correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3. Numerical evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Relativistic MBPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1. The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2. No-virtual-pair approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Time-dependent MBPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The time-evolution operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Adiabatic damping. The Gell–Mann–Low relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Nondegenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Extended model space. The generalized Gell–Mann–Low relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The reduced time-evolution operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Wave operator and eJective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Time-independent interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3. Time-dependent interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4. Generalization to all orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Comparison with time-independent MBPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. S-matrix formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Single-photon exchange. The photon propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. S-matrix for single-photon exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The electron propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The Lamb shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. The electron self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Self-energy renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. The vacuum polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Covariant-evolution-operator formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Single-photon exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Single-photon exchange. Alt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Nonradiative two-photon exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Separable ladder diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Nonseparable ladder diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Electron self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Two-electron radiative eJects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Screened self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Fourier transform of the covariant evolution operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The two-times Green’s-function formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. The Fourier transform of the two-times Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Extended model space. (Quasi)degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Screened self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1. Irreducible part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 165 165 167 169 172 173 174 175 176 177 177 177 179 179 181 182 182 185 187 188 190 193 196 198 198 199 200 202 204 204 206 207 208 208 210 211 212 214 214 215 215 218 219 219 221 222 228 228

I. Lindgren et al. / Physics Reports 389 (2004) 161 – 261 6.4.2. Reducible part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. General comparison between the Green’s-function and the evolution-operator methods . . . . . . . . . . . . . . . . . . . 7. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Applications on hydrogenlike ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1. Lamb shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2. HyperHne structure and Zeeman eJect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Applications on heliumlike ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Applications on lithiumlike ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Possibilities of merging of QED with MBPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Comparison of QED with MBPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. The Bethe–Salpeter equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Pair functions with ‘uncontracted’ photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Evaluation of one and two-photon evolution-operator diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Evaluation of the single-photon exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Evaluation of the two-photon ladder diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3. Evaluation of the screened self-energy diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Evaluation of time-ordered diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1. Two-photon ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1. Separable and nonseparable parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. General evaluation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1. General rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.1. Two-photon cross . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.2. Screened self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 229 230 232 232 232 232 233 235 235 236 237 239 242 242 243 243 246 249 250 250 252 254 254 255 255 256 257

1. Introduction The theory of quantum electrodynamics (QED), i.e., the theory of interactions between electrons and electromagnetic radiation, was developed largely in the 1940s, but it is only during the last two decades or so that it has been possible to test the theory to a high degree of accuracy. The theory has been extremely successful for the simplest systems that are free from strong interaction, like the free electron and the exotic systems positronium and muonium. For the g-factor of the free electron the QED contribution has been experimentally veriHed with the amazing accuracy of a few ppb (parts per billion), and for positronium and muonium the agreement between theory and experiment is of the order of ppm (parts per million). The same order of agreement is also obtained for the Hne structure of neutral helium. In these cases the analytical approach is used in the theoretical evaluation, i.e., a double power expansion in and Z, starting from free particles. The QED theory is less well tested in strong 3elds, for instance, in the neighborhood of a highly charged nucleus. During the last decade particularly interesting information has been accumulated concerning very highly charged few-electron systems—up to hydrogenlike uranium—mainly from the SIS/ESR facility at GSI in Darmstadt and the SuperEBIT ion trap at the Lawrence Livermore Nat. Lab. This has stimulated further development of the numerical QED approach, which starts from electrons generated in the Held of the nucleus (Furry picture), thereby eliminating the Z part of

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the expansion. This technique has now reached a high degree of sophistication, and good agreement with experimental data have been attained in a number of cases (Mohr, 1982; Mohr et al., 1998). Since the QED eJects increase rapidly with the nuclear charge, the heavy few-electron systems are of particular interest in testing the theory. One big diPculty in the theoretical treatment is here the nuclear e6ect, which in many cases is at least comparable to the QED eJect. This eJect can to some extent be eliminated by comparing, for instance, hydrogenlike and lithiumlike systems with the same nucleus. In the heaviest systems also new physical phenomena may occur, when the Held reaches the ‘supercritical’ level (SoJ et al., 1996). Also the intermediate region, with nuclear charges in the range Z = 5–30, say, is of great interest. Here, very accurate data is now appearing from laser and X-ray experiments, but so far there has been only limited comparison with QED theory. The most accurate test has been performed for the atomic g-factor of hydrogenlike carbon, where the bound-QED contribution is veriHed to the order of one part in 1000. Accurate experimental information is available also for heliumlike ions, but a major problem here is to treat the electron correlation properly within the QED formalism. This problem will be of major concern in the present article. For atomic and molecular problems in general the many-body perturbation theory (MBPT) has proven to be quite successful, particularly in the form known as the linked-diagram expansion (Lindgren and Morrison, 1986). By means of various iterative techniques, such as the coupled-cluster approach (CCA), the electron correlation can be treated essentially to all orders of perturbation theory, and this is widely used in quantum chemistry. This scheme can be used also in the relativistic case, using the so-called no-virtual-pair approximation (Sucher, 1980). However, as higher accuracy is required, it is necessary to take also QED eJects more properly into account. According to present knowledge, iterative procedures used in MBPT cannot be used in QED calculations, and therefore correlation eJects have to be treated perturbatively order by order. Since the complication of a QED calculation increases very rapidly with the order of perturbation, a strict QED treatment of strong electron correlation is presently not feasible. Mainly two techniques have so far been applied to QED calculations of few-electron systems in the intermediate Z region. One technique is the application of (relativistic) MBPT with the QED corrections added in the lowest order, i.e., lowest order in as well as Z (Plante et al., 1994). The other technique, which is limited to two-electron systems, is the use of correlated, nonrelativistic wavefunctions of Hylleraas type with low-order relativity as well as QED corrections from the power expansion (Drake, 1988). These techniques work relatively well in the intermediate region, but the restriction to low-order corrections limits the accuracy. Particularly in the low-intermediate region, Z = 5–10, say, it will be necessary to develop new numerical techniques in order to match the accuracy of the experimental data that is presently becoming available. Here, the new experimental techniques can determine, for instance, Hne-structure splittings to ppm accuracy—an accuracy that seems out of reach for the presently available numerical as well as analytical techniques. An approach to improve the situation might be to ‘merge’ the MBPT and numerical QED techniques in some systematic fashion, as will be discussed in the present article. Another serious problem in bound-state QED is the treatment of the quasi-degeneracy, appearing, for instance, in evaluating the Hne-structure separations of light elements in the relativistic formalism. In MBPT this problem can readily be handled by means of an extended model space, which is not possible with the standard S-matrix procedure. Two techniques for handling this problem in QED are available and will be discussed in the present work—the two-times Green’s function and the more

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recently developed covariant-evolution-operator method. Particularly the latter has a structure which largely resembles MBPT, and for that reason it is likely that this new technique may form the ground for merging the MBPT and QED procedures in a more systematic way than what has previously been possible. The vision is that it would then be possible to combine the QED and MBPT eJects in such a way that—in addition to important MBPT eJects to all orders of perturbation theory—also QED eJects would be included and combined with MBPT eJects to all orders. Some ideas in that direction will be presented. The outline of this paper is as follows. In Section 2 we summarize the time-independent MBPT— including relativistic MBPT—as an introduction, emphasizing the method with extended model space. In the next chapter we treat time-dependent MBPT in some detail, since this forms a natural link between MBPT and QED. In that section we introduce the Held-theoretical form of the interaction between electrons and photons, which makes it possible to work also with time-dependent (retarded) interactions between the electrons. We derive the Gell–Mann–Low theorem for the energy shift for an arbitrary model space, and show that it is valid also for interactions of Held-theoretical type. In the following sections we treat the current methods for bound-state QED calculations, starting with the standard S-matrix formulation. Next, we treat the recently developed covariant-evolution-operator method and the two-times Green’s-function method, which are capable of treating also quasidegenerate states. A comparison of these two methods is also made. A simple procedure for expressing the covariant-evolution diagrams of arbitrary order is derived in the appendix. In the Hnal section we sketch an extension of the covariant-evolution-operator method to include also instantaneous interactions to arbitrary order, thereby making it possible to evaluate QED eJects with correlated wavefunctions. When developed, this may hopefully improve the accuracy of numerical QED calculations signiHcantly, particularly in the low-intermediate Z region. 2. Time-independent many-body perturbation theory 2.1. General As an introduction to the general bound-state problem, we shall brie9y review the time-independent many-body perturbation theory. This is well documented in the literature, and we refer to the book of Lindgren and Morrison (1986) for further details. 1 The time-dependent Schr/odinger wave function for an N -electron system satisHes the timedependent Schrodinger equation 2 9 S (x) = HS (x) ; (1) 9t where x = (t; x1 ; : : : ; xN ) is the space–time coordinate, xi being the space coordinate of the individual electron, and H is the Hamiltonian of the system. This representation is known as the Schrodinger picture (SP). i

1

The book is now out of print, but a number of copies is available and can be obtained upon request from the senior author: [email protected] 2 Throughout this article we use relativistic units: ˝ = m = c = 0 = 1; e2 = 4.

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We assume here that the Hamiltonian is time independent, which means that there are stationary solutions of the form 3 S (x) = (x; : : : ; xN )e−iEt :

(2)

The space part of the wave function then satisHes the time-independent Schrodinger equation H(x; : : : ; xN ) = E(x; : : : ; xN ) :

(3)

The eigenfunctions of the Hamiltonian Hi = Ei i

(4)

deHne a Hilbert space, where the number of particles (electrons and photons) is a constant of the motion. 4 In nonrelativistic MBPT for atomic and molecular systems we start from the N -electron Hamiltonian H=

N i=1

(− 12 ∇2i + vext (ri )) +

N e2 ; 4rij i¡j

(5)

where vext (r) is the external (normally nuclear) potential. As usual, we partition the Hamiltonian into a zeroth-order Hamiltonian and a perturbation, H = H0 + H ;

(6)

where we assume that the eigenfunctions and eigenvalues of H0 are known. The modiHcations due to the perturbation are in standard perturbation theory treated order by order. We assume here that the operators (6) are of the form H0 =

N

hS (i) =

i=1

N i=1

(− 12 ∇2i + vext (ri ) + u(ri )) ;

N N e2 u(ri ) + : H = − 4 rij i¡j i=1

(7)

The additional single-electron potential, u(r), is hermitian but otherwise optional and can be chosen to improve the convergence rate. The perturbation H may also contain other (time-independent) interactions, such as interaction with a static magnetic Held. The eigenstates of H0 form our spectrum of basis functions, H0 M = E0M M :

3

(8)

We do not consider the spontaneous decay of excited states here. Later, in the Held-theoretical approach we shall work in the more general space, where these numbers are not necessarily conserved (see e.g., Schweber, 1961). 4

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Since H0 is assumed to be of single-particle type, the basis functions can be expressed in the form of antisymmetrized products of single-electron functions—or Slater determinants— 1 M = √ det{1 (x1 )2 (x2 ) · · · N (xN )} : N!

(9)

The single-electron functions satisfy the single-electron Schr/odinger equation hS i (x) = i i (x) :

(10)

2.2. Perturbation theory. Extended model space In MBPT we are interested in one or several eigenstates of the Hamiltonian H with the eigenfunctions , H = E

( = 1; 2; : : : ; d) ;

(11)

which we refer to as target functions, representing target states. For each target function, , we assume that there exists a zeroth-order approximation—or model function—0 , which, for instance, can be a wave function of the independent-particle type. If there are no states with the same or nearly the same energy that can be mixed by the perturbation, then a perturbation expansion can easily be generated in the standard way. In the more general case, on the other hand, the situation can be more complicated. Closely lying—or quasi-degenerate—states can lead to serious convergence problems. This can be the case, for instance, when studying the atomic Hne-structure of light elements in the relativistic formalism. This problem can usually be remedied by extending the model space and including closely lying states in that subspace. Also completely degenerate states that are mixed by the perturbation are conveniently treated with this formalism, which we shall brie9y review. (For more details, we refer to the book by Lindgren and Morrison (1986).) The model functions deHne a model space, which can contain an arbitrary number of eigenvalues of the unperturbed Hamiltonian. All unperturbed functions of the same energy must be either completely inside or completely outside the model space. In other words, no degeneracy is allowed between states in the model space and states in the complementary space. In the general case, we cannot Hnd directly an expansion for the wave function as in the nondegenerate case, since the zeroth-order or model function is not generally known from the start. Instead, it is convenient to introduce a wave operator or MHller operator (MHller, 1945; L/owdin, 1965), which transforms all model functions into the corresponding target functions = 0

( = 1; 2; : : : ; d) :

(12)

The model functions are solutions of a secular equation HeJ 0 = E 0 ;

(13)

where HeJ is an e6ective Hamiltonian, operating within the model space. The eigenvalues of this operator are the exact energies (11) of the target states. Also this operator is in general unknown at the start of the calculation.

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The wave operator satisHes the generalized Bloch equation (Bloch, 1958a, b; Lindgren, 1974; KvasniUcka, 1977; Lindgren and Morrison, 1986) P : [; H0 ] P = H P − HeJ

(14)

is here the e6ective interaction, deHned by HeJ ; HeJ = PH0 P + HeJ

(15)

and P is the projection operator for the model space. A condition for the theory to work is that the model states are linearly independent and, thus, span the entire model space. We assume now that the model functions are the projections of the target functions onto model space, 0 = P ;

(16)

which we refer to as the intermediate normalization (IN). The wave operator then satisHes the condition PP = P ;

(17)

and the eJective Hamiltonian and the eJective interaction have the forms HeJ = PHP;

HeJ = PH P :

(18)

Then the Bloch equation assumes the frequently used form [; H0 ]P = Q(H − PH )P :

(19)

Here, Q=I −P

(20)

is the projection operator for the complementary space and I is the identity operator for the Hilbert space we operate in. By expanding the wave operator perturbatively = 1 + (1) + (2) + · · · ;

(21)

the Bloch equation can be solved order by order. This leads to the generalized Rayleigh–Schrodinger expansion, valid also in the quasi-degenerate case, [(1) ; H0 ]P = QH P ; [(2) ; H0 ]P = Q(H (1) − (1) PH )P ; [(3) ; H0 ]P = Q(H (2) − (1) PH (1) − (2) PH )P :

(22)

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We note that with the intermediate normalization all components of the wave operator—beyond the trivial zeroth order—have their Hnal state in the complementary space, which is also a consequence of condition (17). The general procedure of MBPT with an extended model space can be summarized in the following rules: • Evaluate the wave operator to the desired accuracy, using the Bloch equation; • Evaluate the matrix elements of the e6ective Hamiltonian; • Diagonalize the matrix of the e6ective Hamiltonian to obtain the exact energies of the target states and the model functions; • Evaluate the wave function of the target states if needed. 2.3. Second quantization. The electron-3eld operators In many-body theory it is convenient to work in second quantization (see, for instance, Schweber, ˆ 1961, Chapter 5 or Lindgren and Morrison, 1986, Chapter 11). A quantum-mechanical operator, O, 5 can then be expanded as 1 1 Oˆ = C + ci† di; j cj + ci† cj† dij; kl cl ck + · · · = Oˆ 0 + Oˆ 1 + Oˆ 2 + Oˆ 3 + · · · ; 2! 3!

(23)

where the terms on the right-hand side represent the zero-, one-, two-,: : : body parts of the operator. cj and cj† are electron annihilation/creation operators, which satisfy the usual anti-commutation relations {ci† ; cj† } = ci† cj† + cj† ci† = 0 ; {ci ; cj } = ci cj + cj ci = 0 ; {ci† ; cj } = ci† cj + cj ci† = $ij ;

(24)

where $ij is the Kronecker delta factor. The coePcients in expansion (23) can be expressed as ˆ di; j = i|O1 |j = d 3 x1 †i (x1 )Oˆ 1 j (x1 ) ; dij; kl = ij|Oˆ 2 |kl =

5

d 3 x1 d 3 x2 †i (x1 )†j (x2 )Oˆ 2 k (x1 )l (x2 ) ;

(25)

We shall use a ‘hat’ to indicate operators in second quantization, apart from the creation/annihilation operators. We employ the summation convention with implicit summations over repeated indices that appear only on the r.h.s.

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etc. {j (x)} is a set of single-electron functions, which are solutions to the Schr/odinger equation (10) in the Held of the nucleus and possibly other electrons. This is usually referred to as the Furry picture (Furry, 1951), although in his original work Furry only considered the potential from the (point) nucleus. It should be noted that we here let the bras and kets represent straight products of single-particle functions. 6 An antisymmetric product of single-particle functions (Slater determinant) (9) can be expressed M = c1† c2† · · · cN† |0 ;

(26)

where |0 represents the vacuum state. The nonrelativistic Hamiltonian (5) has one- and two-body parts and can be expressed in second quantization as Hˆ = ci† i|H1 |jcj + 12 ci† cj† ij|H2 |klcl ck ;

(27)

where H1 = − 12 ∇2 + vext (r)

and

H2 =

e2 : 4r12

We deHne the electron 3eld operators in the Schr/odinger representation by ˆ S (x) = cj j (x);

ˆ † (x) = c† † (x) ; j j S

(28)

which are time independent in this representation. The Hamiltonian (27) can then be expressed 1 † 3 ˆ ˆ ˆ d 3 x1 d 3 x2 ˆ †S (x1 ) ˆ †S (x2 )H2 ˆ S (x2 ) ˆ S (x1 ) : (29) H = d x1 S (x1 )H1 S (x1 ) + 2 In an alternative to the Schr/odinger picture, the Heisenberg picture (HP), the wavefunctions are time independent and the time-dependence is transferred to the operators, ˆ

H = S (t = 0) = eiH t S (x);

ˆ ˆ Oˆ H = eiH t Oˆ S e−iH t :

(30)

In perturbation theory it is often convenient to work in an intermediate picture, known as the interaction picture (IP). Here, the operators and wavefunctions are related to those in the Schr/odinger picture by ˆ

I (t) = eiH 0 t S (t);

6

ˆ ˆ Oˆ I (t) = eiH 0 t Oˆ S e−iH 0 t ;

The true two-body matrix elements, using antisymmetric wavefunctions, then becomes {ij}|Oˆ 2 |{kl} = 12 0|ci cj ci† cj† di j ; k l cl ck ck† cl† |0 = 12 (dij; kl + dji; lk − dji; kl − dij; lk ) = dij; kl − dij; lk ;

assuming the operator to be symmetric with respect to interchange of the coordinates 1 ↔ 2, etc.

(31)

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partitioning the Hamiltonian in the same way as before (6). The relation between the Heisenberg and the interaction pictures is ˆ

ˆ ˆ Oˆ H (t) = eiH t Oˆ I e−iH t :

H = eiH t I (t);

(32)

The wave function of time-independent MBPT corresponds in all pictures considered here to the time-dependent wave function with t = 0, = H = S (0) = I (0) :

(33)

In the Heisenberg picture (30) the electron-Held operators (28) become ˆ H (x) = eiHˆ t ˆ S (x) e−iHˆ t ;

ˆ † (x) = eiHˆ t ˆ † (x)e−iHˆ t ; H S

(34)

and in the interaction picture (IP) (31) ˆ I (x) = eiHˆ 0 t ˆ S (x)e−iHˆ 0 t = eiHˆ 0 t cj j (x)e−iHˆ 0 t = cj j (x)e−ij t = cj j (x) ˆ † (x) = c† † (x)eij t = c† † (x) : j j j j I

(35)

We now introduce the time-dependent creation/annihilations operators in the IP by cj (t) = cj e−ij t ;

cj† (t) = cj† eij t ;

(36)

which gives ˆ I (x) = cj (t)j (x);

ˆ † (x) = c† (t)† (x) : j j I

(37)

The creation/annihilation operators are said to be in normal order, if all creation operators appear to the left of the annihilation operators. A contraction of the operators is deHned as the di6erence between the ordinary (time-ordered) product and the normal-ordered product, x y = xy − {xy} ;

(38)

where we use the curly brackets to denote the normal product. From this deHnition it follows that ci† cj† = ci cj = ci† cj = 0

and

ci cj† = $ij :

(39)

Normal order and Wick’s theorem. The handling of operators in second quantization is greatly simpliHed by Wick’s theorem (Wick, 1950), which states that a product of creation and annihilation operators Aˆ can be written as the normal product plus all single, double : : : contractions with the uncontracted operators in normal form, or symbolically ˆ : Aˆ = { Aˆ } + {A}

(40)

A particularly useful form of Wick’s theorem is the following. If Aˆ and Bˆ are operators in normal form, then the product is equal to the normal product plus all normal-ordered contractions between

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ˆ or formally Aˆ and B, ˆ + { Aˆ Bˆ } : Aˆ Bˆ = {Aˆ B}

(41)

This forms the basic rule for constructing the MBPT diagrams. 2.4. The linked-diagram expansion By using second quantization and Wick’s theorem, the perturbation expansion can conveniently be expressed in terms of diagrams—see, for instance (Lindgren and Morrison, 1986, Chapter 12). By means of the theorem in the form (41), the Rayleigh–Schr/odinger expansion can easily be transformed into normal-ordered products. Each such product is represented by a (Goldstone) diagram, and this leads to the diagrammatic expansion of the many-body wave function. The corresponding energy diagrams are obtained by ‘closing’ the wave function diagram by a Hnal perturbation, so that the Hnal state lies in the model space. It is then found that such an expansion can contain diagrams that are referred to as unlinked, i.e., contain one or several disconnected, closed parts (with the initial and the Hnal state in the model space). The remaining diagrams are known as linked. It can be shown that all unlinked terms cancel in the Rayleigh–Schr/odinger perturbation expansion, provided the model space is complete, i.e., contains all conHgurations that can be formed from the valence electrons. This is the linked-diagram theorem, Hrst shown for closed-shell systems by Brueckner (1955) and Goldstone (1957) and later extended to open-shell systems (Brandow, 1967) as well as to quasi-degenerate model space (Lindgren, 1974; Lindgren and Morrison, 1986). The Bloch equation (14) can then be written as ) [; H0 ]P = (H − HeJ linked P ;

(42)

= PH P in the intermediate normalization (17). The second term on the r.h.s. is rewhere HeJ ferred to as folded and is usually interpreted in a special way. The denominators of the two parts, and H eJ , are independent. For that reason the corresponding time-ordered (Goldstone) diagrams are often drawn as ‘folded’ with all possible time orderings between the interactions of the two parts. By using the standard Goldstone evaluation rules and the ‘factorization theorem’, the denominators of the two parts can then be factorized (Goldstone, 1957; Lindgren and Morrison, 1986, Chapter 13). In the formalism we shall develop, the ‘folded’ diagrams need not be drawn in a folded way. The factorization of the denominators follows directly from the Bloch equation. If drawn in a ‘stretched’ way, the folded diagrams have an intermediate state in the model space, and we shall refer to such contributions as model-space contributions (MSC). Later, in dealing with time-dependent interactions, we shall Hnd that there is an additional type of MSC. In second order, the linked-diagram form of the Bloch equation (42) leads instead of the Rayleigh–Schr/odinger expression (22) to

[(2) ; H0 ]P = Q(H (1) − (1) PH )linked P :

(43)

As an illustration we consider a two-electron system, where the electron orbitals are solutions of the Schr/odinger equation in an external (nuclear) Held (10). The solution to the equations for (1) and

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Fig. 1. Diagrammatic representation of the two lowest orders of the wave operator for a two-electron system. The heavy vertical lines represent electron states in the nuclear potential (10) and the dotted horizontal lines the Coulomb interaction. The last diagram, originating from the second term in the Bloch equation (42), is a type of model-space contribution (MSC) with the intermediate state in the model space. This is also referred to as folded and often drawn in a folded way.

(2) can then be expressed as rs|H |ab ; a + b − r − s  rs|H |tu tu|H |ab rs|(2) |ab =  (a + b − r − s )(a + b − t − u )

rs|(1) |ab =

|tu∈Q

−

|tu∈P

rs|H |tu tu|H |ab (a + b − r − s )(t + u − r − s )

 

:

(44)

linked

This is illustrated in Fig. 1. The Hrst diagram of (2) P represents QH (1) P. It follows from Wick’s theorem (41) that only the fully contracted term can contribute in this case. Here, the intermediate state (tu) lies in the complementary space, Q. The second diagram represents the term (1) PH P, and this is a model-space contribution with the intermediate state in the model space, P. This diagram is here drawn in the conventional way as folded, so that the Goldstone evaluation rules can be used. 2.5. All-order procedures. The coupled-cluster approach A great advantage of the many-body procedure of the type presented here is that important perturbative eJects—i.e., most of the electron correlation—can be treated iteratively to essentially all orders of perturbation theory. This can be achieved by separating the wave operator in second quantization into one-, two-,: : : body eJects (23), = 1 + 1 + 2 + · · ·

(45)

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—which should not be confused with the perturbative expansion (21). Here, the n-body eJects can be expanded as (see, for instance, Lindgren and Morrison, 1986, Chapter 15) 1 = ci† xji cj ; ij cl ck ; etc: 2 = 12 ci† cj† xkl

(46)

ij : : : are the expansion coePcients or ‘amplitudes’ for the particular ‘excitation’. The where xji ; xkl Bloch equation in the linked-diagram form (42) can then be separated into a set of equations for n = 1; 2; : : :

)n; linked P : [n ; H0 ]P = Q(H − HeJ

(47)

The equations for diJerent orders n are coupled and have to be solved iteratively. The most important component is normally n = 2, which corresponds to pair correlation (Bartlett and Purvis, 1978; Ma0 rtensson, 1980). For open-shell systems also n = 1 can be quite important, but less so for closed-shell systems. The latter contributions represent one-body eJects that can be included in the single-electron orbitals. With such orbitals there are no single excitations in a conHguration-interaction (CI) expansion, and the zeroth-order wave function has maximum overlap with the exact one. These orbitals are known as Brueckner orbitals or maximum overlap orbitals (Brenig, 1957; L/owdin, 1962; Lindgren et al., 1976; Lindgren, 1985; Lindgren and Morrison, 1986, p. 260). 2.5.1. Coupled-cluster approach An improved iterative technique can be obtained by expressing the wave operator in exponential form, = exp S = 1 + S + 12 S 2 + · · · ;

(48)

a technique Hrst developed in nuclear physics in the late 1950s (Hubbard, 1957; Coster, 1958; Coster and K/ummel, 1960; K/ummel et al., 1978) and later further developed and extensively applied in U zek, 1966; Paldus and CiU U zek, 1975; Bartlett and Purvis, 1978; Pople et al., quantum chemistry (CiU 1978). For open-shell systems form (48) leads to ‘spurious’ terms, which are eliminated by choosing the normal-ordered form of the exponential (Ey, 1978; Lindgren, 1978; Lindgren and Morrison, 1986) = {exp S} = 1 + S + 12 {S 2 } + · · · :

(49)

The normal-ordering, denoted by curly brackets, implies that there are no ‘contractions’ between the cluster operators, which eliminates the spurious terms of the straight exponential (48). It can be shown that with a complete model space the cluster terms are connected, which is a stronger condition than linked. 7 7 A disconnected diagram is still termed ‘linked’, if all the separate pieces are open. If the model space is incomplete, then disconnected cluster diagrams may appear with the formalism described here. By modifying the procedure, it is possible to maintain the connectivity also for incomplete model space, as discussed particularly by Mukherjee (1986), (Lindgren and Mukherjee, 1987).

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In analogy with the wave-operator expansion (45), we expand the cluster operator S in terms of one-, two; : : : body clusters S = S1 + S2 + · · ·

(50)

with S1 = ci† sji cj ; ij S2 = 12 ci† cj† skl cl ck ; etc:

Inserted in the Bloch equation (14), this leads to the coupled-cluster equations [Sn ; H0 ]P = Q(H − HeJ )n; conn P ;

(51)

where ‘conn’ stands for terms/diagrams that are connected. As before, this leads to a set of coupled equations for n = 1; 2; : : :, which are solved iteratively. One essential advantage of this approach over the simpler approach of the previous subsection (47) is that important four-body e6ects are automatically included in the pair-correlation approach via the {S 2 } term. For quantum-chemistry applications the approach furthermore has the advantage of satisfying the separability condition (Pople et al., 1976), which implies that the wave function of the system separates correctly upon fragmentation. 2.5.2. Pair correlation As before, the pair term, S2 , in the cluster expansion (50) is the most important, followed by the S1 term. A frequently used approximation is the ‘coupled-cluster-singles-and-doubles approximation’ (CCSD), where the coupled equations for S1 and S2 are solved to self-consistency (Purvis and Bartlett, 1982). Here, the wave operator is approximated by (Lindgren and Morrison, 1986, Chapter 15) 1 1 (52) {S1 }3 + {S1 }4 : 3! 4! (The eJect of the last three terms with three or more disconnected clusters is usually quite small and often omitted.) Inserted into the cluster equation (51), the pair approximation yields the equations = 1 + S1 + S2 + 12 {S1 }2 + {S1 S2 } + 12 {S2 }2 + 12 {S12 S2 } +

)1; conn P ; [S1 ; H0 ]P = (H − HeJ [S2 ; H0 ]P = (H − HeJ )2; conn P ;

(53)

= PH P in the intermediate normalization. The CCSD approximation normally represents with HeJ 95 –98% of the electron correlation. In more elaborate calculations also connected triple and quadruple excitations are (partially) included (see, for instance, Kucharski and Bartlett, 1992, for a review). As a simple illustration of the pair equation we shall consider a two-electron system (He-like system) with the zeroth-order Hamiltonian and the perturbation (7) 2 Ze2 e2 2 1 H0 = − 2 ∇i + ; H = : (54) 4ri 4r12 i=1

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Fig. 2. The pair function for a two-electron system (with no core electrons) is equivalent to an inHnite sequence of ladder diagrams (including the folded diagrams).

The pair equation (53) then becomes [S2 ; H0 ]P = Q(H (1 + S2 ) − S2 H eJ )2; conn P :

(55)

Since there are no core electrons in this case, there are no S1 clusters. With expansion (50) this becomes rs tu rs = rs|H |ab + rs|H |tusab − sab ab|HeJ |ab ; (a + b − r − s )sab

(56)

where the last folded term should also include an exchange contribution. The pair (r; s) is here diJerent from the pair (a; b). This equation is graphically illustrated in Fig. 2. By introducing the pair function rs |rs ; |.ab = sab

(57)

we obtain the following pair equation: (a + b − h0 (1) − h0 (2))|.ab =|rs rs|H |ab + |rs rs|H |.ab − |.ab ab|HeJ |ab :

(58)

Solving this equation self-consistently, is equivalent to generating an inHnite sequence of ladder diagrams—in addition to the folded diagrams—as indicated in the second row of the Hgure. This corresponds to solving the two-particle equation exactly. The corresponding contribution to the energy—or generally the eJective interaction (18)—is then obtained by ‘closing’ the pair function by a Hnal interaction rs |ab = cd|HeJ |rssab = cd|HeJ |.ab cd|HeJ

(59)

depicted in Fig. 3. Here, the Hnal state (cd) lies in the model space. 2.5.3. Numerical evaluation For atomic problems we primarily consider here, it is convenient to separate the MBPT diagrams into spin-angular and radial parts. This is based upon the standard expansion of perturbation (54) in

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177

Fig. 3. Closing the pair function by a Hnal interaction yields the contribution to the energy—or generally the eJective interaction. The Hnal state (c; d) lies here in the model space.

spherical waves, using the relation ∞

rl 1 ¡ = C l (1) · C l (2) ; l+1 r12 r¿

(60)

l=0

l

where C is a spherical tensor, closely related to the spherical harmonics (Lindgren and Morrison, 1986). The spin-angular part can be evaluated using angular-momentum diagrams, and only the radial part has to be evaluated numerically. For the numerical evaluation essentially two schemes have been developed. One scheme is based upon the use of B splines and used particularly by the Notre Dame group (Johnson et al., 1988). The other scheme is based upon a discretization of the / radial space and matrix inversion. This is developed by Salomonson and Oster (1989a, b) and used mainly by the G/oteborg group. (See also the review by Mohr et al., 1998) 2.6. Relativistic MBPT 2.6.1. The Dirac equation According to Dirac’s relativistic electron theory, the equation for a single electron in an external (nuclear) potential vext is 9 (61) (x) = ( · p + 0 + vext )(x) : 9t Here, (x) represents a four-component wave function, p = −i∇ is the momentum operator and ; 0 are the 4 × 4 Dirac matrices. The stationary states are of the form i (x) = i (x)e−ii t , where the space part satisHes the corresponding time-independent equation i

hD i (x) = i i (x);

hD = · p + 0 + vext :

(62)

2.6.2. No-virtual-pair approximation Formally, relativistic many-body problems have to be treated in the framework of QED. There exists no relativistic Hamiltonian corresponding to the nonrelativistic one (5). However, various approximations can be constructed, which have been found to work quite well. The Hrst natural choice for a relativistic many-body Hamiltonian might be to replace the Schr/odinger single-electron operator of the nonrelativistic Hamiltonian (5) by the Dirac operator (62), which leads to the Hamiltonian HDC =

N i=1

hD (i) +

N e2 ; 4rij i¡j

(63)

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known as the Dirac–Coulomb Hamiltonian. Due to the negative-energy continuum of the Dirac equation, the eigenvalues of this Hamiltonian are not bound from below, and it is therefore, as it stands, not suitable for many-body calculations. This is known as the Brown–Ravenhall disease (Brown and Ravenhall, 1951). Nevertheless, the Hamiltonian has been used for a long time in practical works, particularly in self-consistent Dirac–Fock and multi-conHgurational Dirac–Fock calculations (Desclaux, 1975). It turns out that by choosing appropriate boundary conditions, the appearance of negative energy states can be strongly suppressed. Formally, this can be expressed as a projected Dirac–Coulomb Hamiltonian (Sucher, 1980) N

N e2 HProjDC = 1+ 1+ ; hD (i) + (64) 4rij i¡j i=1 where 1+ is the projection operator for the positive energy spectrum of the Dirac equation. When relativity is considered, there is, in addition to the electrostatic interaction between the electrons, a magnetic interaction of order 2 , where is the Hne-structure constant ( ≈ 1=137; 0600). This leads to an additional term in the Hamiltonian, Hrst formulated by Gaunt (1929), and the so-called Coulomb–Gaunt interaction, HCG =

N e2 (1 − i · j ) : 4rij i¡j

(65)

The Coulomb and the Gaunt interactions above are instantaneous. It was Hrst shown by Breit (1930, 1932) that also the retardation of the Coulomb interaction gives rise to eJects of the same order. This leads together with the magnetic interaction to the so-called Breit interaction and the Coulomb–Breit interaction

N · r )( · r ) ( e2 i ij j ij HCB = 1 − 12 i · j − : (66) 2 4r 2r ij ij i¡j Replacing the instantaneous Coulomb interaction in the projected Hamiltonian by this operator, leads to N

HNVPA = 1+ hD (i) + HCB 1+ ; (67) i=1

known as the no-virtual-pair approximation (NVPA) (Sucher, 1980). The Breit interaction is instantaneous, although it compensates for the leading eJect of the retardation of the Coulomb interaction. In a proper QED treatment, there is an additional retardation eJect of the Breit interaction of order 3 . The Coulomb interaction, on the other hand, is strictly instantaneous in this model, which is the Coulomb gauge. In an alternative gauge, frequently used in QED, the Feynman gauge, the instantaneous interaction is identical to the Coulomb–Gaunt interaction. This interaction does not contain any retardation, and therefore the retardation correction to this interaction is of the order 2 , i.e., an order of 1= larger than in the Coulomb gauge. This implies that when the Feynman-gauge is used in the NVPA for heavy elements, considerable errors may be introduced (Gorciex and Indelicato, 1988; Lindroth and Ma0 rtensson-Pendrill, 1989; Sucher,

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1988; Lindgren, 1990). In QED calculations, on the other hand, when the retardation is properly taken care of, this error is eliminated, and the Feynman gauge is often used due to its simplicity. The NVPA in the Coulomb gauge is normally a very good starting point for relativistic MBPT. The Hamiltonian is partitioned as before (6) with H0 =

N

(hD (i) + u(ri ));

N H =− u(ri ) + HCB :

i=1

i=1

Then the linked-diagram expansion and the coupled-cluster approach can be generated in a straight/ forward manner (Salomonson and Oster, 1989a). This yields very good results also for quite heavy elements.

3. Time-dependent MBPT 3.1. General In this section we shall consider the time-dependent form of MBPT, which will form a link between time-independent MBPT and quantum electrodynamics (QED) for bound states to be discussed in the following sections. In QED the interaction of electrons/positrons with the photon Held is in the interaction picture (IP) (31) represented by ˆ I (x) ; ˆ H I (t) = d 3 xH (68) where ˆ I (x) = −e ˆ † (x)3 Aˆ 3 (x) ˆ (x) H is the interaction Hamiltonian density (Schweber, 1961). Here, ˆ † (x); ˆ (x) are the electron-Held operators in the IP (37), and 3 represents the four-component Dirac matrices, related to the standard Dirac matrices (61) by 3 = (1; ) : (These are related to the Dirac 4 matrices by 3 = 40 43 .) Aˆ 3 are the electromagnetic Held operators Aˆ 3 ˙ 3j (k)(a†j (k)ei5x + aj (k)e−i5x ) ;

(69)

where 3j (k) are the four-component polarization vectors, a†j (k) and aj (k) the photon creation and annihilation operators, respectively, and x = (t; x) and 5 = (!; k) the four-component k vector. With the metric we use, the four-component scalar product is 5x = !t − k · x. The only nonvanishing commutation relation for the photon operators is (Mandl and Shaw, 1986, Eq. (5.28)) [ai ; a†j ] = ai a†j − a†j ai = ±$i; j ; where the upper (lower) sign is for the space (time) part of the operators.

(70)

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Perturbation (68) commutes with the number operator for the electrons, † N= c i ci ;

(71)

i

which means that the electronic charge (number of electrons minus positrons) is conserved. The electromagnetic-Held operator, on the other hand, contains unpaired creation and annihilation photon operators, which implies that the number of photons is not conserved by the perturbation. Therefore, this perturbation operates in a more general space (see, e.g., Schweber, 1961, Chapter 6), which we can write as H = H0 ⊗ H+1 ⊗ H−1 :

(72)

Hx represents here a ‘restricted’ Hilbert space, where the number of photons is conserved. H0 is the ‘central’ space, where the model functions are located, while H+1 and H−1 represent the corresponding spaces with one photon more and less, respectively. This will be further discussed in Section 8. With a perturbation of type (68), the interaction between the electrons is formed by two perturbations with contracted photon operators. This contraction (38) deHnes a photon propagator, DF83 , by iDF83 (x2 − x1 ) = A8 (x2 )A3 (x1 ) = 0|TD [A8 (x2 )A3 (x1 )]|0 : TD is here the Dyson time-ordering operator, A(x1 )B(x2 ) (t1 ¿ t2 ) ; TD [A(x1 )B(x2 )] = B(x2 )A(x1 ) (t1 ¡ t2 ) ;

(73)

(74)

and |0 represents the vacuum state. Since the vacuum-expectation value of the normal-ordered product vanishes, the contraction is given by the time-ordered product. The Fourier transform of the photon propagator is deHned by dz DF83 (x2 − x1 ; z)e−iz(t2 −t1 ) ; DF83 (x2 − x1 ) = (75) 2 which in the Feynman gauge becomes ∞ g83 d 3 k eik · (x2 −x1 ) k d k sin(kr12 ) =− 2 ; DF83 (x2 − x1 ; z) = −g83 2 3 2 2 r12 0 z 2 − k 2 + i; (2) z − k + i; where k = |k|. The interaction between the electrons then becomes ∞ 2kd kf(k) 2 3 8 I (x2 ; x1 ; z) = e 1 2 DF83 (x2 − x1 ; z) = ; 2 z − k 2 + i; 0 where f(k) = −

e2 (1 − 1 · 2 ) sin(kr12 ) : 42 r12

(76)

(77)

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181

Performing the k integration yields I (x2 ; x1 ; z) =

e2 (1 − 1 · 2 ) ei|z|r12 : 4r12

(78)

This is the retarded Gaunt interaction. When z = 0, this becomes the corresponding instantaneous interaction (65). In the Coulomb gauge the corresponding interaction becomes

1 ei|z|r12 e2 ei|z|r12 − 1 IC (x2 ; x1 ; z) = ; (79) − 1 · 2 + 1 · ∇1 ; 2 · ∇2 ; 4 r12 r12 z 2 r12 which is the retarded form of the Coulomb–Breit interaction (66). For numerical work it is often convenient to expand interaction (77) in spherical waves, in analogy with expansion (60), ∞

sin kr12 = (2l + 1)jl (kr1 )jl (kr2 )C l (1) · C l (2) ; kr12

(80)

l=0

where jl (kr) are spherical Bessel functions, and to perform the radial integrations before the k integrations. 3.2. The time-evolution operator We consider now a general time-dependent perturbation, of which the QED perturbation (68) is one example. We assume further that the operators involved are expressed in second quantization and that the states are represented by state vectors in the generalized Fock space (72). A state represented by the function (x) will then be represented by the vector |(t). The time-dependent Schr/odinger equation (1) then takes the form i

9 |(t) = Hˆ (t)|(t) 9t

(81)

and in the interaction picture (31) i

9 |I (t) = Hˆ I (t)|I (t) : 9t

The Schr/odinger equation (82) has the solution t |I (t) = |I (t0 ) − i dt Hˆ I (t )|I (t ) ; t0

(82)

(83)

and we introduce the time-evolution operator in the IP, deHned by 8 |I (t) = Uˆ (t; t0 )|I (t0 ) ;

8

This operator does not preserve the intermediate normalization (17).

(84)

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which satisHes the equation i

9 ˆ U (t; t0 ) = Hˆ I (t)Uˆ (t; t0 ) : 9t

(85)

This leads to the expansion (Fetter and Walecka, 1971, Eq. (6.23); Itzykson and Zuber, 1980, Eq. (4.56)) t tn t2 ∞ Uˆ (t; t0 ) = 1 + (−i)n dtn dtn−1 · · · dt1 Hˆ I (tn ) · · · Hˆ I (t1 ) t0

n=1

=1+

∞ (−i)n n=1

n!

t0

t0

t

t0

dtn · · ·

t0

t

dt1 TD [Hˆ I (tn ) · · · Hˆ I (t1 )]

t dt Hˆ I (t) ; = TD exp −i t0

(86)

where TD is the time-ordering operator (74). Using the interaction density (68), the evolution operator can then be expressed t ∞ (−i)n t 4 ˆ I (x n ) · · · H ˆ I (x1 )] ˆ d xn · · · d 4 x1 TD [H U (t; t0 ) = 1 + n! t0 t0 n=1 t ˆ I (x) ; d4 x H (87) = TD exp −i t0

where the space integration is performed over all space and the time integration as indicated. 3.3. Adiabatic damping. The Gell–Mann–Low relation 3.3.1. Nondegenerate case In time-dependent perturbation theory for bound-state problems an ‘adiabatic damping factor’ is normally added to the perturbation, Hˆ I (t) → Hˆ I (t; 4) = Hˆ I (t) e−4|t | ;

(88)

where 4 is a small, positive number. We assume that the damping is the only time dependence of the perturbation in the Schr/odinger picture. With the damping, the time-dependent Schr/odinger equation (81) is still valid, but there are no stationary solutions for Hnite 4. In order to return to the original problem, the damping factor is adiabatically ‘switched oJ’ at the end of the calculation, and we shall now study this limiting process. We consider Hrst the case with a single target function, which in the IP evolves according to (84) |I4 (t) = Uˆ 4 (t; t0 ) |I4 (t0 ) : The evolution operator satisHes now Eq. (85) with the damped perturbation, 9 i Uˆ 4 (t; t0 ) = Hˆ I (t) e−i4|t | Uˆ 4 (t; t0 ) ; 9t

(89)

(90)

I. Lindgren et al. / Physics Reports 389 (2004) 161 – 261

which leads to expansion (86) Uˆ 4 (t; t0 ) = 1 +

∞ (−i)n n=1

n!

t0

t

dt n : : :

t0

t

dt1 TD [Hˆ I (tn ) : : : Hˆ I (t1 )]e−4(|t1 |+|t2 |:::+|tn |) :

183

(91)

The damped perturbation (88) vanishes, when 4t → ±∞, and the perturbed (target) wave function approaches in these limits an eigenfunction of Hˆ 0 , |I4 (t) ⇒ |0 :

(92)

We can expect this function to be identical to the unperturbed model function of time-independent MBPT, Hˆ 0 |0 = E0 |0 :

(93)

The target function in the IP at arbitrary time for Hnite 4 is then according to (89) |I4 (t) =

Uˆ 4 (t; −∞) |0 ; 0 |Uˆ 4 (t; −∞) |0

(94)

using intermediate normalization (17). This function will depend on the parameter 4, but we shall show that |I4 (0) satisHes the time-independent Schr/odinger equation in the limit 4 → 0. Note that it is not possible to let 4 → 0 in the unnormalized form (89), since the evolution operator will then be singular. In order to study the limit 4 → 0, we shall follow essentially the treatment of Gell–Mann and Low (1951) (see also Fetter and Walecka, 1971, p. 61; Schweber, 1961, p. 336). We consider one term in expansion (91) t (−i)n t (n) ˆ U 4 (t; −∞) = dtn dtn−1 · · · TD [Hˆ I (tn )Hˆ I (tn−1 ) · · · ] e4(t1 +t2 :::+tn ) : (95) n! −∞ −∞ (As long as t does not approach +∞, we can leave out the absolute signs in the damping factor.) Using the identity [H0 ; ABC · · · ] = [H0 ; A] BC · · · + A[H0 ; B] C · · · + · · · we obtain

9 9 [Hˆ 0 ; H I (tn )H I (tn−1 ) · · · ] = −i + + · · · Hˆ I (tn )Hˆ I (tn−1 ) · · · : 9tn 9tn−1 ˆ

ˆ

(96)

(We note that Hˆ is assumed to be time independent in the SP.) This gives t (−i)n+1 t (n) ˆ ˆ dtn dtn−1 · · · [H 0 ; U 4 (t; −∞)] = n! −∞ −∞ 9 9 ˆ ˆ ×TD + + · · · H I (tn )H I (tn−1 ) · · · e4(t1 +t2 :::+tn ) : 9tn 9tn−1

When integrating by parts, each term yields the same contribution, and the result can be expressed −1) (t; −∞) + in4 Uˆ 4(n) (t; −∞) : [Hˆ 0 ; Uˆ 4(n) (t; −∞)] = −Hˆ I (t) Uˆ (n 4

(97)

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Introducing an order parameter, ?, Hˆ = Hˆ 0 + ? Hˆ I (t) ;

(98)

the result can be expressed [Hˆ 0 ; Uˆ 4 (t; −∞)] = −Hˆ I (t) Uˆ 4 (t; −∞) + i4?

9 ˆ U 4 (t; −∞) : 9?

By operating with this commutator on the unperturbed function (92), we obtain for t = 0 9 ˆ U 4 (0; −∞) |0 ; (Hˆ 0 − E0 + Hˆ ) Uˆ 4 (0; −∞) |0 = i4? 9? where Hˆ = Hˆ I (0), and using (94) this yields (9=9?) Uˆ 4 (0; −∞) |0 ; (Hˆ 0 + Hˆ − E0 ) |4 = i4? 0 |Uˆ 4 (0; −∞) |0

(99)

(100)

(101)

where |4 = |I4 (0). The r.h.s. is here (9=9?) Uˆ 4 (0; −∞) |0 9 |4 = [E4 |4 + i4? i4? 9? 0 |Uˆ 4 (0; −∞) |0 with [E4 = i4?

0 | (9=9?) Uˆ 4 (0; −∞)|0 ; 0 |Uˆ 4 (0; −∞)|0

(102)

which yields 9 (Hˆ 0 + Hˆ − E0 − [E4 ) |4 = i4? (103) |4 : 9? Provided that the perturbation expansion of |4 converges, the r.h.s. will vanish as 4 → 0. Then Uˆ 4 (0; −∞)|0 ˆ 4 (0; −∞) |0 4→0 0 |U

| = lim |4 = lim 4→0

(104)

will be an eigenfunction of the original, undamped Hamiltonian of the system and satisfy the time-independent Schr/odinger equation (3) (Hˆ 0 + Hˆ ) | = E |

(105)

with the energy eigenvalue E = E0 + [E. The energy shift due to the perturbation is given by [E = lim i4? 4→0

0 | (9=9?) Uˆ 4 (0; −∞) |0 : 0 |Uˆ 4 (0; −∞) |0

(106)

Relations (104) and (106) represent the Gell–Mann–Low theorem, which is the basis for timedependent perturbation theory.

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Generally, the evolution operator contains singularities, due to unlinked terms—in the graphical representation corresponding to unlinked diagrams. These terms do not appear in the ratios (104) and (106), which are regular. This is the linked-diagram theorem, mentioned in Section 2.4, and Hrst shown by Goldstone (1957), using time-dependent perturbation theory. Goldstone thereby showed that the limits (104) and (106) do exist and are represented by linked diagrams only. In its original formulation the relation is valid only for a single reference function, 0 , i.e., for a one-dimensional model space, but it can be extended to more general cases, as we shall demonstrate below. We have assumed here that the perturbation is of general time-dependent form. If it is of form (68), then the photon number is not a constant of the motion. This implies that the eigenfunctions are superpositions of functions with diJerent photon numbers. This is necessary in order to be able to handle time-dependent interactions between the electrons, which are formed by contracting the Held-theoretical perturbation at diJerent times. We shall discuss that further in the following sections. In the nondegenerate case, singularities of the evolution operator appear when the initial or reference state appears as an intermediate state. The singularities are eliminated in the Gell–Mann–Low expressions, such as (106). When the perturbation is time or energy dependent, the elimination of such a contribution is incomplete, and there is a residual contribution, usually known as the reference-state contribution. In the more general situation we shall consider below, we shall refer to this contribution as the model-space contribution (MSC). To determine this contribution, the limiting process 4 → 0 has to be carried out. 3.3.2. Extended model space. The generalized Gell–Mann–Low relation The time-dependent MBPT was in the 1960s and 1970s further developed by several groups (Morita, 1963; Kuo and Brown, 1967; Brown and Kuo, 1967; Tolmachev, 1969; Jones and Mohling, 1970; Oberlechner et al., 1970; Kuo et al., 1971), mainly in connection with nuclear calculations. We shall summarize and extend this treatment here. In particular, we shall prove a generalization of the Gell–Mann–Low theorem for an arbitrary model space. Following Tolmachev (1969), we choose the parent states to be the limits of the target states (89) for Hnite 4 as t → −∞, | I4 ⇒ |

( = 1; 2 · · · d) :

(107)

The parent functions are then eigenfunctions of H0 , Hˆ 0 | = E0 | ;

(108)

but we cannot say which eigenvalue a speciHc target state will converge to in the general case. In analogy with (94) we construct the states |4 =

N Uˆ 4 (0; −∞) | = N |˜ 4 : ˆ |U 4 (0; −∞)|

(109)

The states |˜ 4 are normalized to the parent states, |˜ 4 = 1, and hence regular as 4 → 0. In the intermediate normalization (16) we normalize against the projection of the target functions on the

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model space, |0 = P| , and then an additional normalization constant, N , is generally needed. Below we shall show that N Uˆ 4 (0; −∞) | ˆ 4 (0; −∞)| 4→0 |U

| = lim

(110)

is an eigenfunction of the original Hamiltonian of the system for all values of , (Hˆ 0 + Hˆ )| = E |

( = 1; 2; : : : ; d) :

(111)

This is a generalization of the Gell–Mann–Low relation (104), and it holds for an arbitrary model space, i.e., also when this is quasi-degenerate with several energy levels. In the one-dimensional model space, singularities appear in Uˆ for unlinked terms. In the general multi-dimensional case, singularities can appear also for linked diagrams, which have an intermediate state in the model space. We refer to such diagrams as reducible. 9 The remaining irreducible diagrams are regular. In addition, so-called quasi-singularities can appear—i.e., very large, but Hnite, contributions—when an intermediate state is quasi-degenerate with the initial state. All singularities and quasi-singularities are eliminated in the ratio (110)—in analogy with the original Gell– Mann–Low theorem. The elimination of these quasi-singularities represent the major advantage of the procedure using an extended model space. In the next subsection we shall see that this procedure can be applied also in QED, thus eliminating a major shortcoming of the standard S-matrix formulation. In order to show that functions (110) are eigenfunctions of the original Hamiltonian, we shall mainly follow the procedure used in the previous case. We start from the identity (99) at t = 0 (Hˆ 0 + Hˆ )

Uˆ 4 (0; −∞)Hˆ 0 | (9=9?) Uˆ 4 (0; −∞)| Uˆ 4 (0; −∞)| = + i4? ; |Uˆ 4 (0; −∞)| |Uˆ 4 (0; −∞)| |Uˆ 4 (0; −∞)|

(112)

and in analogy with (103), using (109), we obtain

ˆ 4 (0; −∞)| | (9=9?) U |4 Hˆ 0 + Hˆ − i4? |Uˆ 4 (0; −∞)| =

N Uˆ 4 (0; −∞)Hˆ 0 | 9 | : + i4? 9? 4 |Uˆ 4 (0; −∞)|

(113)

Since the parent functions are assumed to be eigenfunctions of Hˆ 0 (108), we see that the Hrst term on the r.h.s. becomes E0 |4 , and we retrieve the relation (103) for a general model space,

(Hˆ 0 + Hˆ − E0 − [E4 )|4 = i4?

9 | : 9? 4

(114)

As before, we can assume that the second term on the r.h.s. vanishes as 4 → 0, which demonstrates that functions (110) are eigenfunctions of the original Hamiltonian. An important observation is here 9

See footnote in Section 3.4.3.

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that a necessary condition for the wave function (110) to satisfy the time-independent Schr/odinger equation is that the parent state is an eigenfunction of H0 . 10 The energy of the target states are given by

| (9=9?) Uˆ 4 (0; −∞)| : (115) E = lim E0 + i4? 4→0 |Uˆ 4 (0; −∞)| This expression is not very useful for evaluating the energy, since the eigenvalue E0 of the parent state is generally not known. The procedure is here used mainly to demonstrate that the functions satisfy the Schr/odinger equation. Instead we shall derive an expression for the eJective Hamiltonian (13), which is the natural tool for a multi-level model space. 3.4. The reduced time-evolution operator In order to Hnd more useful expressions for actual evaluations, we introduce a new operator, the reduced evolution operator, U˜ 4 , by the relation (Lindgren et al., 2001) U 4 (t; −∞)P = P + U˜ 4 (t; −∞)PU 4 (0; −∞)P :

(116)

(We leave out the ‘hat’ on the evolution operator.) This leads to the expansion U (t)P = P + U˜ (t)P + U˜ (t)P U˜ P + U˜ (t)P U˜ P U˜ P + · · · ; where we temporarily leave out the initial time t0 = −∞ and the Hnal time t = 0 in the factors P U˜ P as well as the subscript 4. This can also be expressed U˜ (t)P = U (t)P − P − U˜ (t)P U˜ P − U˜ (t)P U˜ P U˜ P − · · ·

(117)

which is a very useful expression that we shall use frequently in the following. Expanding this operator perturbatively U˜ (t) = U˜ (1) (t) + U˜ (2) (t) + U˜ (3) (t) + · · · ; we obtain in the lowest orders U˜ (1) (t)P = U (1) P ; U˜ (2) (t)P = U (2) (t)P − U (1) (t)PU (1) P ; U˜ (3) (t)P = U (3) (t)P − U˜ (2) (t)PU (1) P − U (1) (t)P U˜ (2) P − U (1) (t)PU (1) PU (1) P :

10

(118)

This observation is in con9ict with the assumption of Kuo et al. (1971), who state that—for the ground state—the parent state can be any state in the model space with nonzero overlap with the Hnal wave function. If the model space contains several energies, the results are con9icting.

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These relations will be used below (Section 3.4.4) to show that the ‘open’ part of the reduced evolution operator is regular—or, in other words—that the counterterms U˜ P U˜ P, U˜ P U˜ P U˜ P · · ·, eliminate the single, double· · · (quasi)-singularities. We recall that with the Held-theoretical perturbation (68), the evolution operator does not conserve the number of photons and therefore operates in the extended Fock space, H (72). The P operator is the projection operator for the model space, which is a part of the Hilbert space H0 , where the photon number is conserved. The Q operator is the projection operator for the complementary part of this space (20). We now introduce a generalized projection operator Q Q=I −P

(119)

for the extended space H, where the number of photons is not necessarily conserved. The general evolution operator can now be expressed U 4 (0; −∞)P = PU 4 (0; −∞)P + QU 4 (0; −∞)P;

(120)

which with (116) leads to the generalized factorization theorem, U 4 (0; −∞)P = [1 + Q U˜ (0; −∞)] PU 4 (0; −∞)P :

(121)

We shall demonstrate below that the Hrst factor on the r.h.s. is regular in the limit 4 → 0, and consequently all (quasi)singularities are contained in the second factor. This is a generalization to the more general Fock space (72) of the factorization theorem, demonstrated in nuclear theory (Morita, 1963; Tolmachev, 1969; Oberlechner et al., 1970; Kuo et al., 1971). The fact that the reduced evolution operator is regular has important implications. This implies that in that part each adiabatic-damping factor 4 can be turned oJ individually, in contrast to the situation with the original Gell–Mann–Low relation, as discussed above. The sign of the 4 term, though, is normally important, since that determines the position of the pole in the integration process. The model-space contribution is obtained by means of the expansion (118) without the need of any limiting process. 3.4.1. Wave operator and e6ective Hamiltonian The model states corresponding to the target states (110) are in intermediate normalization given by the projection onto the model space (16), N P U 4 (0; −∞) | ; 4→0 |U 4 (0; −∞)|

|0 = P | = lim

(122)

and wave function (110) can then be expressed, using the factorization theorem (121), | = [1 + Q U˜ (0; −∞)] |0 :

(123)

This leads to a generalized wave operator (12) = 1 + Q U˜ (0; −∞) ;

(124)

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189

operating in the extended space H. In a ‘restricted’ Hilbert space H0 , where the number of photons is conserved, this operator is identical to the standard MBPT wave operator (12). The e6ective Hamiltonian is deHned by (13) Hˆ eJ |0 = E |0 ;

(125)

which in the extended space leads to Hˆ eJ = PH P = PH [1 + Q U˜ (0; −∞)]P

(126)

and to the eJective interaction (18) Hˆ eJ = PH P = PH [1 + Q U˜ (0; −∞)]P :

(127)

Hˆ and Hˆ are the Hamiltonian and the perturbation, respectively, at t = 0. An alternative form of the eJective interaction can be obtained in the following way. From (85) we have

i

9 U4 (t; −∞)P = Hˆ (t) U4 (t; −∞)P ; 9t

and using deHnition (116) and the factorization theorem (121) this yields for t = 0 9 ˜ PU 4 (0; −∞)P = H [1 + Q U˜ (0; −∞)] PU 4 (0; −∞)P i U 4 (t; −∞) 9t t=0

(128)

(129)

or 9 Hˆ eJ = P i U˜ (t; −∞) P : 9t t=0

(130)

This is a generalization of the energy-shift formula given by Jones and Mohling (Jones and Mohling, 1970), and it is the form we shall mainly use in the following. The form (130) of the eJective interaction can also be derived in an alternative way. We start now from the time-dependent Schr/odinger equation (1) at t = 0, 9 i |S (x) = Hˆ | : (131) 9t t=0 The eigenfunctions of the system at t = 0, | , are given by the generalized Gell–Mann–Low relation (110) and satisfy the time-independent Schr/odinger equation (111). This gives 9 i |S (x) = E | (132) 9t t=0 and

9 P i |S (x) = E |0 ; 9t t=0

(133)

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where P| = |0 is the model function (122). With relation (31) this leads to 9 = E |0 ; PH0 | + P i | 9t t=0

(134)

where | is the wave function in the IP. From the Gell–Mann–Low relation (110) we can also obtain the wave function in the interaction picture at arbitrary (Hnite) time N U4 (t; −∞) | ; 4→0 |U4 (0; −∞)|

| (t) = lim

and using relations (122) and (116) we Hnd that 9 9 = i U˜ (t; −∞) |0 : i | (t) 9t 9t t=0 t=0

(135)

(136)

This leads with relation (134) to the secular equation HeJ |0 = E |0 ; where the operator

9 ˜ ˆ ˆ P H eJ = P H 0 P + P i U (t; −∞) 9t t=0

(137)

is the e6ective Hamiltonian (13) and the second term is the eJective interaction (130). We recall that we have assumed here that the perturbations can be of general time-dependent form. All forms of the eJective Hamiltonian/interaction given here are therefore valid for interaction between the electrons that are time- or energy dependent, including the Held-theoretical perturbation (68). 3.4.2. Time-independent interactions We shall now apply the formalism presented here to atomic systems with interactions that are time independent in the Schr/odinger picture, like the instantaneous Coulomb interaction. Time- or energy dependent interactions will be treated in the following subsection. When using the Held-theoretical perturbation, the time-independent interactions between the electrons correspond to contractions at equal time. Therefore, only perturbations of even order of the evolution operator will appear. We can now work in the restricted Hilbert space H0 (72) and replace the general projection operator Q by the traditional operator Q. The wave operator then becomes = 1 + QU˜ (0; −∞)

(138)

and the eJective interaction Hˆ eJ = P Hˆ P = P Hˆ [1 + QU˜ (0; −∞)]P :

(139)

As a Hrst illustration of the evolution-operator technique, we consider the second-order Coulomb interaction between two electrons illustrated in Fig. 4. The evolution operator (87) can then be

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191

Fig. 4. The second-order evolution-operator diagram for the Coulomb interaction between two electrons.

expressed U (2) (t ; −∞) = −

t

−∞

dt2

t2

−∞

dt1 VI (t2 ) VI (t1 ) e4(t1 +t2 ) ;

(140)

where VI is the Coulomb interaction in the interaction picture, VI (t) = eiH0 t V e−iH0 t ;

(141)

and V = e2 =4 r12 is the time-independent interaction in the Schr/odinger picture. This gives t2 t (2) rs|U (t ; −∞)|ab = − dt2 dt1 rs|VI (t2 )|tu tu|VI (t1 )|ab e4(t1 +t2 ) −∞

−∞

(142)

after inserting a complete set of intermediate states, 11 which leads to the time integral t2 t −it2 (t +u −r −s +i4) dt2 e dt1 e−it1 (a +b −t −u +i4) −∞

−∞

e−it (a +b −r −s +2i4) : (a + b − r − s + 2i4)(a + b − t − u + i4)

=−

The result then becomes rs|U (2) (t ; −∞)|ab =

rs|V |tu tu|V |ab e−it (Ein −Eout +2i4) ; (Ein − Eout + 2i4)(Ein − Eint + i4)

(143)

using the notations Ein = a + b , Eout = r + s and Eint = t + u . In the limit 4 → 0, this becomes (quasi)singular, when Eint ≈ Ein or Eout ≈ Ein . In the former case we include the quasi-degenerate state(s) in the model space. From expansion (118) we then have U˜ (2) P = U (2) P − U (1) PU (1) P ;

11

(144)

As before, we employ the summation convention with implicit summation over repeated indices that do not appear on the l.h.s.

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Fig. 5. For time-independent interactions the open, reducible two-photon–photon ladder diagram with the corresponding counterterm corresponds to a folded diagram (147) in standard MBPT, cf. Fig. 1. The intermediate state |tu lies in the model space.

where the second term is the counterterm. This case is illustrated in Fig. 5. This is a model-space contribution with the intermediate state in the model space, and such a diagram is also referred to as reducible. In the same way as before we obtain for the counterterm 0 t (1) (1) rs|U PU |ab = − dt2 dt1 rs|VI (t2 ) |tu tu|VI (t1 )|ab e4(t1 +t2 ) ; (145) −∞

−∞

which yields rs|U (1) PU (1) |ab =

rs|V |tu tu|V |ab e−it (Eint −Eout +i4) : (Eint − Eout + i4)(Ein − Eint + i4)

(146)

Subtracting this from the main term (143), gives for the reducible or MSC part of evolution operator U˜ (2) at time t = 0 rs|U˜ (2) (0; −∞)|abRed = −

rs|V |tu tu|V |ab : (Ein − Eout + 2i4)(Eint − Eout + i4)

(147)

When the outgoing state lies in the Q space, this is according to the deHnition (123) a contribution to the wave operator. We see that the (quasi)singularity for Eint ≈ Ein is here eliminated. This model-space contribution is identical to the folded diagram obtained in time-independent MBPT (Fig. 1, Eq. (44)). The eJective interaction (139) is in second order (2)

H eJ = PHI (0) QU (1) (0; −∞)P ;

(148)

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193

and this yields for the example considered here (2)

rs|H eJ |ab =

rs|V |tu tu|V |ab : Ein − Eint

(149)

The intermediate state is here conHned to the Q space, and there is no (quasi)singularity and no MSC or folded diagram in the second-order eJective Hamiltonian. In third order we have (3)

H eJ = PH QU˜ (2) (0; −∞)P ;

(150)

and here there is a contribution from the folded diagram (147) in U˜ (2) . We have now shown that the reducible part of QU˜ (2) is regular, and since the irreducible part is always regular, it follows that QU˜ (2) is completely regular for time-independent interactions. We shall generalize this proof to higher orders in the next subsection in connection with time-dependent interactions. 3.4.3. Time-dependent interactions We have seen that when the interactions between the electrons are time independent, there is a model-space contribution to the eJective interaction and the wave operator, normally represented by so-called folded diagrams, which appear in the energy and eJective interaction of third and higher orders. We shall now consider time or energy dependent interactions and show that this leads to an additional form of MSC, appearing also in the second-order energy or eJective interaction. As an illustration we consider the second-order diagram shown in Fig. 6. We assume that the interaction is of the form dz V (t2 − t1 ) = V (z) e−iz(t2 −t1 ) ; (151) 2 where V (z) is the Fourier transform and z is the energy parameter. In the interaction picture this becomes dz V (z)(eiH0 t2 e−izt2 e−iH0 t2 )(eiH0 t1 eizt1 e−iH0 t1 ) : VI (t2 − t1 ) = (152) 2

Fig. 6. Second-order diagram with time-dependent interactions.

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We assume the time orderings to be t ¿ t3 ¿ t1 and t ¿ t4 ¿ t2 , and the matrix element of the evolution operator, corresponding to the time-independent result (142), is then rs|U (2) (t ; −∞)|ab t4 t dt4 dt2 = −∞

−∞

t

−∞

dt3

t3

−∞

dt1 rs|VI (t4 − t3 )|tu tu|VI (t2 − t1 )|ab :

(153)

The time dependence is here (in the limit 4 → 0) eit4 (s −u −z ) eit3 (r −t +z ) eit2 (u −b −z) eit1 (u −a +z)

or e−it4 (q −p +z ) e−it3 (q−p−z ) e−it2 (p +z) e−it1 (p−z) ;

using the notations p = a − t ; p = b − u ; q = a − r ; q = b − s . The time integrations then yield e−it (q+q ) : (q + z + z )(p + z)(q − z − z )(p − z)

(154)

If the interactions do not overlap in time, as in Fig. 6, the diagram is said to be separable. 12 We can then have the time orderings t4 ¿ t3 ¿ t1 ; t2 , which leads to the integration ordering t3 t4 t2 t1 t3 t (155) dt4 dt3 dt2 dt1 + dt1 dt2 : −∞

−∞

−∞

−∞

−∞

−∞

Considering also the time ordering t3 ¿ t4 ¿ t1 ; t2 , the integral becomes e−it (q+q ) 1 1 1 1 1 : + + q + q q + p − z q + p + z p + p p − z p + z The matrix element (153) can then be expressed as rs|U (2) (t ; −∞)|abSep = where

V (A; B) =

12

dz V (z) 2

rs|V (q + p ; q + p)|tu tu|V (p; p )|ab −it (q+q ) e ; (q + q )(p + p )

1 1 + A−z B+z

(156)

:

A diagram is here said to be separable, if it can be separated into two legitimate diagrams by cutting all orbital lines at a certain time. In the older literature (see, for instance, Jones and Mohling, 1970) the term reducible was normally used for this type of diagram. We have, however, adopted the terminology developed mainly in recent years, where the term ‘reducible’ is used for separable diagrams with the intermediate state is in the model space. We have therefore introduced the term separable for the wider group in order to avoid confusion (Lindgren et al., 2001). Note that a reducible diagram must always be separable.

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195

Using the previous notations (143), this becomes rs|U (2) (t ; −∞)|abSep =

rs|V (q + p ; q + p)|tu tu|V (p; p )|ab −it (Ein −Eout ) e : (Ein − Eout )(Ein − Eint )

(157)

When Ein − Eint ≈ 0 we have a (quasi)singularity and a corresponding counterterm in analogy with the previous result (146) rs|V (q − p; q − p )|tu tu|V (p; p )|ab −it (Eint −Eout ) e rs|U (1) P Uˆ (1) |ab = : (Eint − Eout )(Ein − Eint )

(158)

With the notations V (q + p ; q + p) = V (Ein − r − u ; Ein − t − s ) = V2 (Ein ) ; V (q − p; q − p ) = V (Eint − r − u ; Eint − t − s ) = V2 (Eint ) ; the main ‘ladder’ term (156) becomes rs|U (2) (t ; −∞)|abLad =

rs|V2 (Ein )|tu tu|V (p; p )|ab −it (Ein −Eout ) e (Ein − Eout )(Ein − Eint )

(159)

rs|V2 (Eint )|tu tu|V (p; p )|ab −it (Eint −Eout ) e : (Eint − Eout )(Ein − Eint )

(160)

and the counterterm (158) rs|U (1) PU (1) |abCounter =

Applying relation (130), the time derivative eliminates the last (leftmost) denominator, and the corresponding reducible contribution to the eJective interaction becomes (2)

rs|H eJ |abRed =

rs|V2 (Ein ) − V2 (Eint )|tu tu|V (p; p )|ab : Ein − Eint

(161)

With [E = Ein − Eint this becomes in the limit [E → 0 (2) rs|H eJ |abRed = rs|

9 (V2 (E))E=Ein |tu tu|V (p; p )|ab : 9E

(162)

This shows that the (quasi)singularity is eliminated also when the interactions are time-dependent, but that there is an additional 3nite model-space contribution also in second order due to the time dependence. In order to obtain the corresponding contribution to the wave operator (123), we set the time t = 0, and expressions (159) and (160) yield the contribution rs|V2 (Ein )|tu tu|V (p; p )|ab rs|V2 (Ein − [E)|tu tu|V (p; p )|ab ; − (Ein − Eout ) [E (Ein − [E − Eout ) [E

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Fig. 7. Reducible diagram of higher order and the corresponding counterterm.

which we can write as 1 rs|2 (Ein )|tu − rs|2 (Ein − [E)|tu tu|V (p; p )|ab [E 9 = rs| (2 (E))E=Ein |tu + · · · tu|V (p; p )|ab 9E

(163)

by including the last denominator in 2 . This shows that the (quasi)singularity is eliminated also in the second-order wave operator. The remaining part is the model-space contribution, which in this case contains a folded part, present also for time-independent interactions, as well as an additional part due to the time dependence. 3.4.4. Generalization to all orders The previous treatment can be generalized to higher orders. Let us consider a reducible U diagram of the form U (m) PU (n) , as indicated in the left diagram in Fig. 7, where the two parts represent irreducible m- and n-fold interactions, respectively, and the intermediate state lies in the model space. This is regarded as a single diagram of ‘ladder’ type, which implies that all denominators are evaluated from the bottom. All energies are then functions of the initial energy, Ein = a + b , and we can represent the contribution to the wave operator by rs|W2 (Ein )|tu tu|W1 (Ein )|ab rs|2 (Ein )|tu tu|W1 (Ein )|ab : = (Ein − Eout ) [E [E Here, Eout = r + s and [E = a + b − t − u , and W1 =W2 represent the m=n-fold interactions. This diagram is (quasi)singular, due to the denominator [E. From expansion (117) it follows that there is a counterterm of similar form, represented by the second term in the Hgure. This diJers from the leading term only in the fact that the denominators of the left part are evaluated from the intermediate state (t; u) and that the time of the right part is set to zero. The denominators of the left part are the same as in the ladder with Ein replaced by Ein − [E. Assuming as before that the interactions depend on the initial energy, the counterterm can be expressed rs|W2 (Ein − [E)|tu tu|W1 (Ein )|ab rs|2 (Ein − [E)|tu tu|W1 (Ein )|ab = ; (Ein − [E − Eout ) [E [E

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197

Fig. 8. Doubly reducible diagram with the corresponding counterterm.

and the sum of the reducible ladder and the counterterm can be expanded in analogy with the second-order case (163) 9 (2 (E))E=Ein |tu tu|W1 (Ein )|ab + · · · : 9E This shows that the (quasi)singularity is eliminated also in this higher-order case. Next we consider in a similar way a diagram that is doubly reducible, i.e., with two intermediate model-space states, as illustrated in Fig. 8. With obvious notations we can then express the ladder diagram, representing the wave operator, as rs|

rs|3 (Ein )|vw

1 1 vw|W2 (Ein )|tu tu|W1 (Ein )|ab [E2 [E1

and the counterterm as rs|3 (Ein − [E2 )|vw

1 1 vw|W2 (Ein − [E1 )|tu tu|W1 (Ein )|ab : [E2 [E1

In the limit when the [E s → 0, the latter becomes 9 3 (E) rs|3 (Ein ) − [E2 |vw 9E E=Ein 1 9 1 W2 (E) × vw|W2 (Ein ) − [E1 |tu tu|V1 (Ein )|ab : [E2 9E [E1 E=Ein The double singularity is eliminated by the counterterm, and the diJerence becomes 9 9 3 (E) W2 (E) − rs| |vw vw| |tu tu|V1 (Ein )|ab ; 9E 9E E=Ein E=Ein in addition to the single singularities, introduced by the counterterm, 1 9 rs|3 (Ein )|vw vw| |tu tu|V1 (Ein )|ab W2 (E) [E2 9E E=Ein 1 9 3 (E) |vw vw|W2 (Ein )|tu tu|V1 (Ein )|ab : rs| 9E [E1 E=Ein

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These single singularities are eliminated by the terms −U˜ (m) P U˜ (n) P − U˜ (n) P U˜ (m) P of the expansion (118). In a similar way the cancellation of (quasi)singularities for triply ... reducible diagrams can be shown. This veri3es that the wave operator (124) and the e6ective interaction (127) are regular in all orders for a two-electron system. 3.5. Comparison with time-independent MBPT We shall now summarize our observations regarding the relation between the time-dependent and time-independent forms of MBPT. In time-independent MBPT, generated by means of the Bloch equation in the linked-diagram form (42), there are two types of contributions to the wave operator. The Hrst type originates from the Hrst term on the r.h.s., and in the case of a two-electron system this gives rise to diagrams of ‘ladder’ type. The second term on the r.h.s. of the Bloch equation gives rise to ‘folded’ diagrams. In the ladder type of diagrams all intermediate states lie in the Q space and in the folded diagrams one or several intermediate states lie in the P space. The folded diagrams are therefore a type of model-space contribution (MSC). In time-dependent MBPT with time-independent interactions the wave operator can be expressed by means of the reduced evolution operator (138). Here, states of the model space can appear as intermediate states, which leads to a (quasi)singularity—so-called reducible contributions—and then there is a corresponding counterterm, which eliminates the singularity (117). The combination of the singular ladder diagram and the corresponding counterterm leads to a MSC that exactly corresponds folded to the diagram of time-independent MBPT. In time-dependent MBPT with time- or energy-dependent interactions there is an additional MSC, which in the case of complete degeneracy leads to a contribution involving the energy derivative of interaction (162). 4. S-matrix formulation In the present and the following main sections we shall consider diJerent schemes for bound-state QED calculations. We shall begin with a brief review of the standard S-matrix formulation, which is well documented in the literature. (For further details we refer to the recent extensive review by Mohr et al., 1998). Then we shall consider two more general methods, which have been developed more recently and which can be applied also to the quasi-degenerate situation. First we shall describe in some detail the covariant-evolution-operator method, developed by us (Lindgren 0 en 2000; Lindgren et al., 2001), and the present report represents—together with the thesis of As2 (2002)—the Hrst more detailed account of this new method. Next we shall more brie9y describe the two-times Green’s-function method, developed by Shabaev et al., which has recently been extensively reviewed (Shabaev, 2002), and we refer to this article for further details concerning this method. The two methods will be intercompared, and the possibility of combining the former with MBPT in a systematic fashion will be indicated. We assume now that the perturbation is of the Held-theoretical form (68) ˆ I (x) = −e ˆ † (x) 3 Aˆ 3 (x) ˆ (x) : H

(164)

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199

The electron-Held operators are in the IP given by (37), and we assume that the orbitals are solutions of the Dirac equation (62) in the Held of the nucleus (nuclei). We have in the previous section discussed this type of perturbation and derived the corresponding Gell–Mann–Low relations in the nonrelativistic case. This theory, however, is not covariant in the relativistic sense, and the relativistic problem with negative energy states cannot be handled. The simplest way to remedy the situation is to let the time integrations run over all times, which leads to the S-matrix formulation. This we shall consider in this section. Another way is to modify the standard time evolution operator to make it covariant, which we shall consider in the next main section. The Sucher energy formula Sucher (1957) has shown that the energy shift can as an alternative to the Gell–Mann–Low formula (106) be expressed as 0 | 9=9? U 4 (∞; −∞)|0 i : 4? 4→0 2 0 |U 4 (∞; −∞)|0

[E = lim

(165)

Uˆ 4 (∞; −∞) is the scattering matrix or S-matrix, primarily used in scattering theory. Like the Gell–Mann–Low formula, the Sucher formula is valid also when the interaction between the electrons is time- or energy dependent, but in contrast to the former it is also valid in the relativistic case. The Gell–Mann–Low–Sucher procedure has been the standard approach in bound-state QED for a long time (see, for instance Mohr et al., 1998) and will be discussed brie9y in the next subsection. 4.1. Single-photon exchange. The photon propagator The Held-theoretical form of the evolution operator, representing multi-photon exchange between the electrons, is given by expansion (86) t 1 U (t; t0 ) = 1 − d 4 x1 d 4 x2 TD [( ˆ † (x) e3 Aˆ 3 (x) ˆ (x))1 ( ˆ † (x) e8 Aˆ 8 (x) ˆ (x))2 ] 2 t0 + ··· ;

(166)

where, as before, the space integration is performed over all space and the time integration as indicated. We consider Hrst the exchange of a single photon between the electrons, as indicated in Fig. 9. As in the previous section we consider the limit, where the initial time t0 → −∞, which implies that we start from an eigenstate of the unperturbed Hamiltonian Hˆ 0 . The evolution operator,

Fig. 9. The single-photon exchange between the electrons, compared with potential scattering.

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including the adiabatic damping (88), is then given by U (2) (t ; −∞) t 1 d 4 x1 d 4 x2 TD [( ˆ † (x)e3 Aˆ 3 (x) ˆ (x))1 ( ˆ † (x)e8 Aˆ 8 (x) ˆ (x))2 ] e−4(|t1 |+|t2 |) : =− 2 −∞

(167)

The contraction between the electromagnetic Held operators leads to the photon propagator (73), and disregarding for the moment other possible contractions, this yields t 1 U (2) (t ; −∞) = − d 4 x1 d 4 x2 ˆ † (x1 ) ˆ † (x2 ) iI (x2 ; x1 ) ˆ (x2 ) ˆ (x1 ) e−4(|t1 |+|t2 |) (168) 2 −∞ with 13 I (x2 ; x1 ) = e13 DF83 (x2 − x1 ) e28 :

(169)

The Fourier transforms of I (x2 ; x1 ) is deHned by dz I (x2 ; x1 ; z) e−iz(t2 −t1 ) ; I (x2 ; x1 ) = 2 where I (x2 ; x1 ; z) is given by deHnition (77). 4.1.1. S-matrix for single-photon exchange The scattering matrix (S-matrix) is deHned by S = U (∞; −∞), and as a Hrst illustration we shall study the S-matrix for a single-photon exchange. Eq. (168) then yields 1 (2) ˆ S =− d 4 x1 d 4 x2 ˆ † (x1 ) ˆ † (x2 ) iI (x2 ; x1 ) ˆ (x2 ) ˆ (x1 ) e−4(|t1 |+|t2 |) 2 1 d 4 x1 d 4 x2 ci† †i (x1 )cj† j (x2 ) iI (x2 ; x1 ) cl l (x2 )ck k (x1 ) e−4(|t1 |+|t2 |) ; =− (170) 2 where the integration is performed over the entire space–time volume. This is a two-body operator and becomes according to the second-quantization expression (23) 1 † † S (2) = ci cj ij|S (2) |kl cl ck : (171) 2 i; j; k;l

IdentiHcation then yields the expansion coePcients rs|S (2) |ab = − d 4 x1 d 4 x2 †r (x1 )†s (x2 ) iI (x2 ; x1 ))b (x2 )a (x1 ) e−4(|t1 |+|t2 |) =−

4

4

d x1 d x2

dz † (x1 )†s (x2 ) iI (x2 ; x1 ; z))b (x2 )a (x1 ) 2 r

×e−it1 (a −r −z) e−it2 (b −s +z) e−4(|t1 |+|t2 |) : 13

I (x2 ; x1 ) corresponds to iI21 of Lindgren (2000) and Lindgren et al. (2001).

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After the time integrations this becomes dz (2) ˆ 2 E4 (q − z) 2 E4 (q + z) rs|I (x2 ; x1 ; z)|ab ; rs|S |ab = −i 2 where q = a − r and q = b − s . The E function is here deHned ∞ 24 dt eiqt e−4|t | = 2 = 2 E4 (q) ; q + 42 −∞

201

(173)

(174)

which has the following properties 14 lim E4 (q) = $(q) ;

4→0

lim 4 E4 (q) = $q; 0 ;

4→0

∞

−∞

d z E4 (z − a) E5 (z − b) = E4+5 (a − b) :

Here, $(q) is the Dirac delta function and $q; r is the Kronecker delta factor (=1 for q = r and zero otherwise). Using the last relation above we can replace expression (173) by rs|Sˆ(2) |ab = −2i E24 (q + q ) rs|I (x2 ; x1 ; q)|ab : The single-photon exchange can be compared with the S matrix for the potential scattering from a time- or energy-dependent potential, Veq (x2 −x1 ) with the Fourier transform Veq (x1 ; x2 ; z), as indicated by the rightmost diagram in Fig. 9. Since two times are involved also in this process, it has to be regarded as a second-order process, yielding 1 (2) ˆ d 4 x1 d 4 x2 ˆ † (x1 ) ˆ † (x2 ) Veq (x2 − x1 ) ˆ (x2 ) ˆ (x1 ) e−4(|t1 |+|t2 |) : S pot = − (175) 2 After time integration, the matrix element becomes as in the previous case rs|Sˆ(1) pot |ab = −2i E24 (q + q ) rs|Veq (q)|ab :

(176)

This implies that the single-photon exchange is equivalent to potential scattering by an equivalent potential given by Veq (q) = I (x2 ; x1 ; q) = e2 13 28 DF83 (x2 − x1 ; q) :

14

The Hrst two relations are obvious, and the third can easily be shown by means of the identity 1 1 1 1 1 E4 (z − a) E5 (z − b) ≡ − − : (2i)2 z − a − i4 z − a + i4 z − b − i5 z − b + i5

Here, only the cross products, which have one pole on each side of the axis, contribute to the integral.

(177)

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In the Feynman gauge the equivalent potential becomes, using deHnition (77), ∞ e2 2k d k f(k) F ; f(k) = − (1 − 1 · 2 ) sin(kr12 ) Veq (q) = q2 − k 2 + i; 42 r12 0

(178)

or after integrating over the k space F (q) = Veq

e2 (1 − 1 · 2 ) ei|q|r12 ; 4 r12

(179)

which is the retarded Coulomb–Gaunt interaction (78). The energy shift is given by the Sucher formula (165), which in the lowest order (n = 2) yields [E = lim i4 rs|Sˆ(2) |ab : 4→0

(180)

In the present case this gives, using (173), [E = $q; −q rs|Veq (q)|ab :

(181)

The Kronecker delta factor implies here that the result is nonvanishing only for q + q = 0 or a + b = r + s , which means that in the S-matrix formalism energy must be conserved between the initial and 3nal states. This has the disadvantage that those elements of the eJective Hamiltonian that are nondiagonal in energy cannot be evaluated. Therefore, the procedure is not applicable to the procedure of an extended model space, discussed above for the treatment of quasi-degeneracy. In the following two main sections we shall discuss two methods that do not have this serious shortcoming, but Hrst we shall develop the S-matrix formulation a little further. 4.2. The electron propagator In relativistic problems we must also allow for time running backwards, which represents antiparticle creation. For that purpose the so-called Feynman electron propagator, SF (x; x0 ), is introduced, deHned by iSF (x; x0 ) = 0|T [ ˆ (x) ˆ † (x0 )]|0 = 0|F(t − t0 ) ˆ (x) ˆ † (x0 ) − F(t0 − t) ˆ † (x0 ) ˆ (x)|0 :

(182)

Here, F(t) is the Heaviside step function (equal to unity for t ¿ 0 and zero for t ¡ 0) and T is the Wick time-ordering operator (t1 ¿ t2 ) ; A(x1 )B(x2 ) T [A(x1 )B(x2 )] = (183) −B(x2 )A(x1 ) (t1 ¡ t2 ) ; not to be confused with the Dyson time-ordering operator (74). The expression (182) represents the contraction between the electron Held operators (35). Separating the Held operators (35) into particle ( ˆ + ) and hole ( ˆ − ) parts, corresponding to electrons with positive and negative energy,

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respectively, the electron propagator can be expressed iSF (x; x0 ) = 0|F(t − t0 ) ˆ + (x) ˆ †+ (x0 ) − F(t0 − t) ˆ †− (x0 ) ˆ − (x)|0 = F(t − t0 )p (x) †p (x0 ) e−ip (t −t0 ) − F(t0 − t)†h (x0 ) h (x) e−ih (t −t0 ) :

(184)

Here, p ; h represent the single-electron wavefunctions (10), with positive energy (‘particle states’) and negative energy (‘hole states’), respectively. By analytical continuation the electron propagator can be expressed as an integral in the complex plane d! j (x) †j (x0 ) −i!(t −t ) 0 SF (x; x0 ) = e ; 2 ! − j + i;j

(185)

where j runs over all states (with positive as well as negative energy), and ;j is an inHnitesimally small quantity with the same sign as j , indicating the position of the pole. The Fourier transform (with respect to time) of the electron propagator is j (x) †j (x0 ) : SF (x; x0 ; !) = ! − j + i;j

(186)

Regarding the space part of the single-electron functions as coordinate representations of the corresponding Dirac states, j (x) = x|j;

†j (x0 ) = j|x0 ;

we can express the electron propagator (186) SF (x; x0 ; !) = x|SˆF (!)|x0

(187)

or as the coordinate representation of the electron-propagator operator SˆF (!) =

|j j| : ! − j + i;j

(188)

We also introduce the four-dimensional coordinate representations of the Dirac states, x|j = j (x) = j (x) e−ij t ;

j|x = †j (x) = †j (x) eij t :

(189)

Then the Held operators (35) become ˆ (x) = x|j cj ;

ˆ † (x) = c† j|x ; j

(190)

and the electron-propagator (184) can be expressed as iSF (x; x0 ) = F(t − t0 ) x|p p|x0 − F(t0 − t) x|h h|x0 ;

(191)

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and the form (185) d! x|j j|x0 : SF (x; x0 ) = 2 ! − j + i;j

(192)

Operating with the electron propagator (191) on the Held operator, ˆ (x0 ), and integration over the space coordinates, then yields d 3 x0 iSF (x; x0 ) ˆ (x0 ) = F(t − t0 ) x|p cp − F(t0 − t) x|hch = F(t − t0 ) ˆ + (x0 ) − F(t0 − t) ˆ − (x0 ) and similarly d 3 x ˆ † (x) iSF (x; x0 ) = F(t − t0 ) ˆ †+ (x) − F(t0 − t) ˆ †− (x) :

(193)

(194)

4.3. The Lamb shift In the second-order evolution operator (167) we can also have contractions between the electronHeld operators in various ways. Two equivalent contractions are indicated below: ( ˆ † (x)e3 A3 ˆ (x))1 ( ˆ †(x)e8 A8 ˆ (x))2

( ˆ † (x)e3 A3 ˆ (x))1 ( ˆ † (x)e8 A8 ˆ (x))2 ;

(195)

and together with the photon Held contraction this represents the electron self-energy, depicted for the S-matrix in Fig. 10 (left). Contracting the electron Held operators at the same vertex ( ˆ † (x)e3 A3 ˆ (x))1 ( ˆ † (x)e8 A8 ˆ (x))2

( ˆ † (x)e3 A3 ˆ (x))1 ( ˆ † (x)e8 A8 ˆ (x))2

(196)

represents the vacuum polarization, shown in the right diagram of the Hgure. 4.3.1. The electron self-energy By considering only one of the electron-Held contractions in (195), we can eliminate the factor of 12 , and the S-matrix for the electron self-energy becomes in analogy with the single-photon

Fig. 10. The S-matrix diagrams representing the Hrst-order electron self-energy and vacuum polarization.

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exchange (170) (2) SSE = − d 4 x2 d 4 x1 ˆ † (x2 )iSF (x2 ; x1 )iI (x2 ; x1 ) ˆ (x1 )e−4(|t1 |+|t2 |) =−

d 4 x2 d 4 x1 ci† †i (x2 )iSF (x2 ; x1 )iI (x2 ; x1 )cj j (x1 )e−4(|t1 |+|t2 |) :

(197)

This is a one-body operator, and identiHcation with expansion (23) then yields the ‘matrix element’ (2) r|SSE |a = − d 4 x2 d 4 x1 †r (x2 )iSF (x2 ; x1 )iI (x2 ; x1 )a (x1 )e−4(|t1 |+|t2 |) : (198) Using the Fourier transforms of the propagator and the interaction, this becomes dz † d! (2) (x2 )SF (x2 ; x1 ; !)I (x2 ; x1 ; z)a (x1 ) r|SSE |a = d 4 x2 d 4 x1 2 2 r × e−it2 (!+z−r ) e−it1 (a −!−z) e−4(|t1 |+|t2 |) :

(199)

The time integrations yield here in analogy with the single-photon exchange (173) the factors 2E4 (! + z − r ) and 2E4 (a − ! − z), and after integration over ! this becomes (2) r|SSE |a = 2E24 (a − r ) d 3 x2 d 3 x1 ×

dz † (x2 )SF (x2 ; x1 ; a − z)I (x2 ; x1 ; z)a (x1 ) : 2 r

(200)

This can be expressed as (2) r|SSE |a = −2iE24 (a − r ) r|G(a )|a ;

deHning the self-energy operator by dz † 3 3 r|iG(a )|a = d x2 d x1 (x2 )iSF (x2 ; x1 ; a − z)iI (x2 ; x1 ; z)a (x1 ) 2 r d z tr|I (z)|at =− 2 a − t − z + i;t

(201)

(202)

and using form (187) of the electron propagator. With the photon interaction in the Feynman gauge (77) this becomes (see Appendix A.1) tr|2kf(k)|at dz r|G(a )|a = i dk 2 (a − t − z + i4t )(z 2 − k 2 + i;) tr|f(k)|at = dk ; (203) a − t − (k − i4)t where (·)t has the same sign as t . The energy shift due to the electron self-energy is then given by the Sucher formula (180) $ESE = $a ;r r|G(a )|a :

(204)

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Fig. 11. The free-electron self-energy (205).

As in the previous case, the energy must be preserved between the initial and Hnal states in this procedure. A more general treatment is given in Section 5.3. 4.3.2. Self-energy renormalization The electron self-energy, represented by the emission and absorption of the same photon, is a process that corresponds to an inHnite energy or mass. For the free electron this is in analogy with the bound-state result (203) pr; qs|f(k)|qs; pr free $ESE (pr ) = pr|G(pr )|pr = d k ; (205) pr − qs − (k − i4)q illustrated in Fig. 11. We use here the momentum representation—p; q denote the momentum and r; s components of the Dirac spinor. The factor (k − i4)q is positive for electrons and negative for positrons. The free-electron self-energy represents a part $m of the physical mass of the electron and should be subtracted from the self-energy of the bound electron. This renormalization process eliminates the singularity. For an electron in the bound state |a the renormalized self-energy is then given by renorm $ESE = a|G(a )|a − a|$m|a :

(206)

The renormalization term—also referred to as the mass counterterm—is the average of free-electron self-energy in the state |a, as illustrated in Fig. 12.

Fig. 12. The mass counterterm is the average of the free-electron self-energy in the bound state |a. The thick vertical line represents a bound-electron and the thin line a free-electron state.

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Fig. 13. The bound-electron self-energy can be expanded into a zero-, a one- and a many-potential term.

Fig. 14. The bound-state vacuum polarization can be expanded in a zero-, one-, and many-potential term as in the self-energy case (Fig. 13).

A bound-electron propagator can be expanded in a free-electron propagator with zero, one, two,... interactions of the external (nuclear) Held. Applied to the self-energy diagram this leads to the expansion in Fig. 13. Here, the Hrst two terms are inHnite, while the last ‘many-potential term’ is Hnite. In the method introduced by Brown et al. (1959) and later modiHed by Mohr (1982) the zeroand one-potential terms are combined with the mass counterterm, which leads to a Hnite quantity that can be evaluated analytically. The Hnal result is then obtained by evaluating numerically the Hnite many-potential term. 4.3.3. The vacuum polarization The second part of the Lamb shift (Fig. 10), the vacuum polarization (VP), is also singular and has to be renormalized. The bound-state VP can be expanded into a zero-potential, a one-potential and a many-potential term, as in the self-energy case (Fig. 14). The zero-potential term is zero, due to the Furry theorem (Mandl and Shaw, 1986). The one-potential term is singular but can be renormalized analytically, as Hrst shown by Uehling (1935) and Serber (1935). The last term, known as the Wichmann–Kroll term (Wichmann and Kroll, 1956), is Hnite and evaluated numerically. Some applications of the S-matrix formulation are brie9y discussed in Section 7. For further information the reader is referred to the review article by Mohr et al. (1998).

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5. Covariant-evolution-operator formalism 5.1. Single-photon exchange As mentioned previously, the S-matrix formalism cannot handle the quasi-degeneracy problem, due to the energy-conservation condition (181), which is caused by the integration over all times. A possibility to circumvent this problem might therefore be to consider instead of the S-matrix the original evolution operator (86) with a limited time integration. As mentioned, however, the evolution operator in its original form is not relativistically covariant, implying that the relativistic problem cannot be handled. By generalizing the operator, so that time can evolve forwards as well as backwards, it can be shown that the relativistic covariance can be restored. This method—which we refer to as the covariant-evolution-operator method—has recently been developed and success0 en, 2002) fully applied to the quasi-degenerate situation (Lindgren, 2000; Lindgren et al., 2001; As2 and is illustrated in Fig. 15 for single-photon exchange. In the covariant-evolution-operator method we use for the single-photon exchange between two electrons—instead of the standard time-evolution operator (168)—the expression 1 (2) UCov d 4 x1 d 4 x2 [F(t − t1 ) ˆ †+ (x1 ) − F(t1 − t ) ˆ †− (x1 )] (t ; −∞) = − 2 ×[F(t − t2 ) ˆ †+ (x2 ) − F(t2 − t ) ˆ †− (x2 )]iI (x2 ; x1 ) ˆ (x2 ) ˆ (x1 )e−4(|t1 |+|t2 |) : (207) Here, we integrate over all times. For integration times smaller than the time t of the evolution operator, which corresponds to positive-energy states, we integrate in the positive direction from

Fig. 15. The one-time evolution operator for single-photon exchange between the electrons, including forward and backward time evolution, represented by three time-ordered (Goldstone) diagrams (top) and a single Feynman diagram (bottom). The wavy line represents the photon propagator, the open, solid lines the electron-Held operator and the straight line between dots the electron propagator. The subscript of the electron Held operators indicates positive- and negative-energy part, respectively.

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209

the negative inHnity, and correspondingly for integration times larger than t , which corresponds to negative-energy states, we integrate in the negative direction from the positive inHnity. With this operator, positive- and negative-energy states can be handled in analogous ways. Generally, the evolution operator (86) is a two-times operator, with an initial as well as a Hnal time. However, in perturbation theory, using the adiabatic damping (91), it is convenient to set the initial time t0 = −∞, which directly leads to a perturbation expansion starting from an unperturbed state. We shall normally apply that in the following. Using relation (194), we can replace the square brackets in (207) by space integration over the electron propagators, yielding 1 (2) 3 3 ˆ† ˆ† d x1 d x2 (x1 ) (x2 ) UCov (t ; −∞) = − d 4 x 1 d 4 x2 2 ×iSF (x1 ; x1 )iSF (x2 ; x2 )iI (x2 ; x1 ) ˆ (x2 ) ˆ (x1 )e−4(|t1 |+|t2 |)

(208)

with xi = (t ; xi ). (Note that x1 and x2 have the common time t .) In analogy with the single-photon exchange (172) the ‘matrix elements’ become (2) rs|UCov (t ; −∞)|ab = − d 3 x1 d 3 x2 †r (x1 )†s (x2 ) d 4 x1 d 4 x2 ×iSF (x1 ; x1 )iSF (x2 ; x2 )iI (x2 ; x1 )a (x1 )b (x2 )e−4(|t1 |+|t2 |) =− dt1 dt2 rs|x1 x2 x1 x2 |iSF (x1 ; x1 )iSF (x2 ; x2 )iI (x2 ; x1 )|x1 x2 x1 x2 |ab ×eit (r +s ) e−it1 a −it2 b e−4(|t1 |+|t2 |) ;

(209)

where we have explicitly shown the coordinates for the bra rs| and the ket |ab. Result (209) is illustrated by the bottom diagram in Fig. 15. The integral is evaluated in Appendix A.1, and the result becomes (2) (t ; −∞)|ab rs|UCov

V (q; q )

=

e−it (q+q ) q + q

=

rs|V (q; q )|ab

1 1 d kf(k) + q − (k − i4)r q − (k − i4)s

(210)

;

(211)

where (A)x = (A)sgn(x ) and f(k) is given by (77) f(k) = −

e2 (1 − 1 · 2 ) sin(kr12 ) : 42 r12

(212)

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When the Hnal state |rs lies in the model space, the contribution to the e6ective interaction becomes, using (130), (1) rs|HeJ |ab = rs|V (q; q )|ab :

(213)

We can now compare the result above with the S-matrix result obtained in the previous main section. When |rs has the same energy as |ab, this agrees with result (181). Then potential (211) reduces to 2k d kf(k) F = Veq V (q; −q) = (q) ; q2 − k 2 + i4 which is the same as the S-matrix result (178). In the evolution-operator result (213), however, the initial and Hnal states do not have to have the same energy, which makes the formalism applicable also to the quasi-degeneracy problem, using an extended model space. 5.1.1. Single-photon exchange. Alt. We shall now derive the expression for the single-photon exchange in an alternative way, using the one-photon covariant evolution-operator method, which will be useful in demonstrating more clearly the analogy with the Green’s-function method to be discussed later. The matrix element (209) is with the notations in Fig. 16 (2) rs|UCov (t ; −∞)|ab = − dt1 dt2 rs|iSF (x1 ; x1 ) ×iSF (x2 ; x2 )iI (x2 ; x1 )|abeit (r +s ) e−it1 a −it2 b e−4(|t1 |+|t2 |) rs|I (z)|ab =i dt1 dt2 (!3 − r + i;r )(!4 − s + i;s )

×eit (r +s −!3 −!4 ) e−it1 (a −z−!3 ) e−it2 (b +z−!4 ) e−4(|t1 |+|t2 |) ;

(214)

integrated over z and all the !’s. In analogy with the previous case, the time integrations yield the delta factors $(a − z − !3 ) and $(b + z − !4 ), and integrations over z to $(a + b − !3 − !4 ).

Fig. 16. The covariant evolution operator for single-photon exchange between the electrons.

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211

Integrations over !4 then yield (2) (t ; −∞)|ab rs|UCov d!3 rs|I (a − !3 )|ab =i e−it (q+q ) : 2 (!3 − r + i4r )(a + b − !3 − s + i4s )

(215)

This is equivalent to integral (A.3) with the substitution a − !3 → z. Rewriting the denominators in analogy with (294) and (A.5), we obtain d!3 e−it (q+q ) rs|I (a − !3 )|ab q+q 2 1 1 : + × !3 − r + i4r a + b − !3 − s + i4s

(2) (t ; −∞)|ab = i rs|UCov

(216)

This can be compared with the phantom-particle equation (IV.22) in (Le Bigot, 2001). The contribution to the eJective Hamiltonian is then obtained by means of (130), which yields d!3 1 1 (1) rs|I (a − !3 )|ab : (217) + HeJ = i 2 !3 − r + i4r a + b − !3 − s + i4s This is identical to result (302), obtained below with the nonhermitian form of the eJective Hamiltonian in the Green’s-function method. 5.2. Nonradiative two-photon exchange The QED eJects can be separated into two categories, which we refer to as nonradiative and radiative eJects. The radiative eJects are characterized by having at least one self-energy or vacuumpolarization loop, while the nonradiative eJects are free from such parts. The nonradiative two-photon eJect for a two-electron system is of the type shown in Fig. 17, the two-photon ladder and the two-photon crossed diagram. The ladder diagram has a substantial MBPT part in it. The crossed and the radiative diagrams have no MBPT counterpart.

Fig. 17. The nonradiative two-photon exchange diagrams, the ladder diagram (left) and the crossed-photon diagram (right).

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The fourth-order evolution operator is according to expansion (87) with interaction (68) t 1 d 4 x4 d 4 x3 d 4 x2 d 4 x1 TD [( ˆ † (x)eH AH ˆ (x))4 ( ˆ † (x)eI AI ˆ (x))3 4! t0 ×( ˆ † (x)e8 A8 ˆ (x))2 ( ˆ † (x)e3 A3 ˆ (x))1 ] :

(218)

In order to form the two-photon exchange diagrams in Fig. 17—the ‘ladder’ and the ‘crossed-photon’ diagrams—the contractions can be performed in 12 distinct ways, all leading to equivalent diagrams. The covariant evolution operator for the ladder diagram is then in analogy with the single-photon exchange (208) 1 (4) 3 3 ˆ† ˆ† d x3 d x4 (x3 ) (x4 ) UCov (t ; −∞)Ladder = d 4 x3 d 4 x4 iSF (x3 ; x3 ) 2 ×iSF (x4 ; x4 )iI (x4 ; x3 ) dt1 dt2 iSF (x3 ; x1 )iSF (x4 ; x2 )iI (x2 ; x1 ) ˆ (x1 ) ˆ (x2 ) ×eit (r +s ) e−it1 a −it2 b e−4(|t1 |+|t2 |+|t3 |+|t4 |) :

(219)

The ‘matrix element’ then becomes after identiHcation with the second-quantized expansion (23) (4) d 4 x3 d 4 x4 iSF (x3 ; x3 )iSF (x4 ; x4 )iI (x4 ; x3 ) rs|UCov (t ; −∞)|ab = rs| ×

dt1 dt2 iSF (x3 ; x1 )iSF (x4 ; x2 )iI (x2 ; x1 )|ab

×eit (r +s ) e−it1 a −it2 b e−4(|t1 |+|t2 |+|t3 |+|t4 |) ;

(220)

and similarly for the crossed diagram. 5.2.1. Separable ladder diagram We consider Hrst the two-photon ladder diagram. Here, we distinguish between two situations, whether the two photons overlap with each other in time or not. If the photons do not overlap in time, we refer to the diagram as being separable, and in the opposite case as being nonseparable. The separable two-photon ladder is illustrated by the leftmost diagram in Fig. 18. The Heldtheoretical evaluation is given in the Appendix A.2 (B.10), assuming all states to be positive energy states, (4) rs|UCov (t ; −∞)|abSepLadder = rs|V (q + p ; q + p)|tu tu|V (p; p )|ab

e−it (q+q ) ; × (q + q )(p + p )

(221)

where V (q; q ) is the eJective one-photon interaction (210), (211). The corresponding contribution to the eJective interaction then becomes, using (B.10), rs|HeJ |abSep =

rs|V (q + p ; q + p)|tu tu|V (p; p )|ab : p + p

(222)

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Fig. 18. Graphical representation of the reducible two-photon–photon ladder diagram and the corresponding counterterm. The dotted line represents a time with no uncontracted photon, i.e., a time after the Hrst photon has been absorbed and before the second has been created.

When the diagram is reducible, i.e., separable with the intermediate state in the model space, there is a counterterm (118), U (2) PU (2) P, rs|UCounter |ab = rs|U (2) |tu tu|U (2) |ab e−it (q+q −p−p ) : (223) = rs|V (q − p; q − p )|tu tu|V (p; p )|ab (q + q − p − p )(p + p )

Using the notations of (147), Ein = a + b ;

Eout = r + s ;

[E = p + p = Ein − t − u ;

we can express the separable ladder (221) diagram as (4) (t ; −∞)|abSepLadder rs|UCov

e−it (Ein −Eout ) = rs|V2 (Ein )|tu tu|V (p; p )|ab (Ein − Eout )[E

(224)

e−it (Ein −[E −Eout ) ; (Ein − [E − Eout )[E

(225)

and the counterterm as

rs|UCounter |ab = rs|V2 (Ein − [E|tu tu|V (p; p )|ab

where V2 (X ) = V (X − r − u ; X − s − t ). The contribution to the eJective interaction then becomes, using (13), rs|V2 (Ein )|tu − rs|V2 (Ein − [E)|tu tu|V (p; p )|ab : [E

(226)

The leading term is here given by the energy derivative of the interaction, rs|

9 (V2 (E))E=Ein + · · · |tu tu|V (p; p )|ab ; 9E

(227)

which demonstrates that the counterterm removes the (quasi)degeneracy of the reducible ladder diagram. This result is quite analogous to the expression for the second-order diagram, derived

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Fig. 19. The covariant-evolution-operator diagrams representing the Hrst-order electron self-energy and vacuum polarization.

with time-dependent MBPT (163). A more detailed comparison with MBPT will be made in Section 8. 5.2.2. Nonseparable ladder diagram The nonseparable ladder diagram is evaluated in Appendix B, Eq. (B.16), and the result becomes, assuming only positive-energy states are involved, 1 rs|HeJ |abNonsep = d k d k rs|f(k )|tu tu|f(k)|ab (q + p − k )(q − k − k )(p − k) 1 : (228) + (q + p + k )(q + k + k )(p + k) 5.3. Electron self-energy Next, we consider the radiative eJects and start with the single-electron eJects, treated also in the previous main section with the S-matrix formulation, Section 4.3. The Hrst-order radiative eJects are illustrated in Fig. 10 for the S-matrix formulation. The corresponding evolution-operator diagram are shown in Fig. 19. Here, we shall evaluate the electron self-energy diagram as an illustration. In analogy with the single-photon exchange (209) the matrix element becomes (2) r|USE (t ; −∞)|a = − d 3 x2 †r (x2 ) ×

d 4 x2 d 4 x1 iSF (x2 ; x2 )iI (x2 ; x1 )iSF (x2 ; x1 )a (x1 )e−4(|t1 |+|t2 |) :

(229)

Using the Fourier transform of the electron propagator (187) and of the interaction (169), this yields dz tr|I (z)|at d!2 d!1 r|USE (t ; −∞)|a = i 2 2 2 (!2 − r + i;r )(!1 − t + i;t ) × dt2 dt2 e−it (!2 −r ) e−it2 (!1 −!2 +z) e−it1 (a −!1 −z) e−4(|t1 |+|t2 |) : (230)

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215

Fig. 20. The two-electron radiative eJects in lowest order. The Hrst two diagrams represent the two-electron self-energy (screened self-energy and vertex modiHcation) and the last two represent the two-electron vacuum polarization (screened vacuum polarization and the photon self-energy).

Using the deHnition of the self-energy operator (202), the time and ! integrations yield in analogy with the single-photon exchange, treated in Appendix A.1, d z rt|I (z)|ta e−it (a −r +i4r ) r|USE (t ; −∞)|a = i a − r + i4r 2 a − t − z + i4t e−it (a −r +i4r ) r|G(a )|a : a − r + i4r

=

(231)

This leads to the contribution to the eJective Hamiltonian, using (130), (1) |a = r|G(a )|a : r|HeJ

(232)

The result is the same as in the S-matrix formulation (204), when r =a . The present result, however, is valid also when the initial and Hnal energies are diJerent. When needed, we shall assume that the self-energy expressions are renormalized (see Section 4.3.2). 5.4. Two-electron radiative e6ects The covariant-evolution-operator diagrams for the two-electron radiative eJects in lowest order are depicted in Fig. 20. Here, we shall treat the Hrst of these diagrams, the screened self-energy (leftmost diagram) in some detail. 5.4.1. Screened self-energy The covariant evolution operator for the screened self-energy, depicted in Fig. 21, is in analogy with the two-photon exchange (220) (4) rs|UCov (t ; −∞)|ab = dt1 dt2 rs| d 4 x3 d 4 x4 iSF (x4 ; x4 )iSF (x4 ; x3 )iI (x4 ; x3 )iSF (x3 ; x1 ) ×iSF (x2 ; x2 )iI (x2 ; x1 )|abeit (r +s ) e−it1 a −it2 b e−4(|t1 |+|t2 |+|t3 |+|t4 |) :

(233)

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Fig. 21. The covariant-evolution-operator diagram representing the screened self-energy. (There is also a hermitian adjoint diagram, which is not necessarily identical in the nonhermitian formulation we use.)

Introducing the electron propagators (192), this becomes after time integrations (4) (t ; −∞)|ab rs|UCov d!4 d!5 dz r|G(!5 )|t ts|I (z)|ab d!3 =i 2 2 2 2 (!3 − r + i;r )(!5 − t + i;t )(!4 − s + i;s )

×e−it (!3 +!4 −r −s ) $(a − z − !5 )$(b + z − !4 )$(!5 − !3 )

(234)

and after integration over the !’s (4) (t ; −∞)|ab rs|UCov r|G(a − z)|t ts|I (z)|ab dz =i e−it (q+q ) 2 (q − z + i4r )(p − z + i4t )(q + z + i4s )

(235)

with q = a − r ; p = a − t ; q = b − s . Rewriting two of the denominators as before, leads to d z r|G(a − z)|t ts|I (z)|ab eit (q+q ) (4) rs|UCov (t ; −∞)|ab = i (q + q ) 2 p − z + i4t 1 1 : (236) × + q − z + i4r q + z + i4s The contribution to the eJective Hamiltonian is then, using (130), 1 d z r|G(a − z)|t ts|I (z)|ab 1 (2) + rs|HeJ |ab = i 2 p − z + i4t q − z + i4r q + z + i4s ru|I (z )|ut ts|I (z)|ab 1 1 d z d z + =− 2 2 (p − z − z + i4u )(p − z + i4t ) q − z + i4r q + z + i4s (237)

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217

with p = a − u and with expression (202) for the self-energy operator. The integral is evaluated in Appendix A.3 in the Feynman gauge (77), assuming only positive-energy states are involved, which yields (2) rs|HeJ |ab = d k d k ru|f(k )|ut ts|f(k)|ab 1 1 1 + × (p + q − k )(p + q ) p − k + i4 q − k + i4 1 1 1 + : (238) + (p − k − k + i4)(p − k + i4) p + q − k + 2i4 q − k + i4

The Hrst term corresponds to the separable part, where the photons do not overlap in time, and the second term to the nonseparable part. The separable part can also be expressed, using expression (203), 1 1 ts|f(k)|ab (2) rs|HeJ |abSep = r|G(a + q )|t d k + : (239) p + q (p − k + i4) (q − k + i4) Reducible part The separable part of the screened self-energy (239) has a (quasi)singularity when E = p + q = a + b − t − s ≈ 0. This is eliminated by the counterterm in the reduced evolution operator (117) U˜ (4) P = U (4) P − U (2) PU (2) P :

(240)

The counterterm U (2) PU (2) P, illustrated in Fig. 22, is a product of an electron self energy (232) and a single-photon exchange (211) (2) rs|HeJ |abCounter = r|G(t )|t × d k ts|f(k)|ab

1 1 1 + : p − k + i4 q − k + i4 p + q

Fig. 22. The counterterm for the screened self-energy in the covariant evolution-operator method).

(241)

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The diJerence yields the reducible part or the model-space part of the eJective Hamiltonian (2) rs|HeJ |abRed =

r|G(t + E) − G(t )|t E 1 1 + × d k ts|f(k)|ab p − k + i4 q − k + i4

(242)

with E = p + q = a + b − t − s . In the limit of complete degeneracy, the Hrst factor becomes the derivative of the self-energy with respect to the energy parameter (2) rs|HeJ |abRed

9 1 1 : (243) = d k ts|f(k)|ab r|G(!)|t + 9! p − k + i4 q − k + i4 !=t

5.5. Fourier transform of the covariant evolution operator The Fourier transform of the evolution operator U (t ; −∞) with respect to the time is 1 dt eiEt U (t ; −∞) : U (E) = 2

(244)

If U (t ; −∞) is of the form U (t ; −∞) = F(E )e−iE t ;

(245)

U (E) = $(E − E )F(E ) :

(246)

then

Similarly, we deHne the Fourier transform of the reduced evolution operator (116) 1 dt eiEt U˜ (t ; −∞) : U˜ (E) = 2

(247)

It follows from form (130) that the energy-dependent e6ective interaction is related to the Fourier transform of the reduced evolution operator by = HeJ

Ed E U˜ (E) :

(248)

The Fourier transform of the single-photon matrix element (210) is (2) rs|UCov (E)|ab = $(E − (q + q ))

rs|V (q; q )|ab ; q + q

(249)

and (248) yields the eJective interaction rs|HeJ |ab = rs|V (q; q )|ab

in agreement with (213).

(250)

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219

Applying the same rule to the separable two-photon matrix element (224), yields the contribution to the eJective interaction rs|HeJ |abLadder =

1 rs|V2 (Ein )|tu tu|V (p; p )|ab ; [E

(251)

and when the diagram is reducible the counterterm (225) yields |abCounter = rs|HeJ

1 rs|V2 (Ein − [E)|tu tu|V (p; p )|ab : [E

(252)

This agrees with the previous result (226) in Section 5.2. Some applications of the covariant-evolution-operator technique are discussed in Section 7. 6. The two-times Green’s-function formalism 6.1. General We shall now consider the two-times Green’s-function method, mainly for the purpose of making comparison with the covariant-evolution-operator method, discussed in the previous section. For further details regarding the two-times Green’s-function method, the reader is referred primarily to the recent review article by Shabaev (2002) and to the thesis of LeBigot (2001). In Held theory the single-particle Green’s function is usually deHned (Fetter and Walecka, 1971, Eq. (7.1)) 15 iG(x; x0 ) =

0H |T [ ˆ H (x) ˆ †H (x0 )]|0H ; 0H |0H

(253)

where T is the Wick time-ordering operator (74) and ˆ H ; ˆ †H are the electron Held operators in the Heisenberg representation (34). |0H is the lowest eigenstate of the Fock-space Hamiltonian, Hˆ , in this representation—or the ‘Heisenberg vacuum’—which is time independent. This state satisHes the condition ˆ H (x)|0H = 0 :

(254)

In the interaction picture (IP) the vacuum evolves in time according to the deHnition (84), |0I (t) = U (t; t0 )|0I (t0 ) :

(255)

The ‘unperturbed’ vacuum in the IP is |0 = |0I (−∞), assuming an adiabatic damping (88), and is related to the Heisenberg vacuum by |0H = |0I (t = 0) = U (0; −∞)|0 :

(256)

Often the Green’s function is deHned using ]ˆ H = ˆ †H 40 instead of ˆ †H and sometimes without the imaginary unit; see e.g. Itzykson and Zuber (1980, Eq. (6.1)). 15

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The relation between operators in the HP and the IP is given in Eq. (32)

ˆ ˆ Oˆ H (t) = eiH t OI e−iH t = U (0; t)OI U (t; 0) ;

and we can then transform the Green’s function (253) to the interaction picture 0|U (∞; 0)T [U (0; t) ˆ (x)U (t; 0)U (0; t0 ) ˆ † (x0 )U (t0 ; 0)]U (0; −∞)|0 ; iG(x; x0 ) = 0|Uˆ (∞; −∞)|0

(257)

(258)

which, using (87), can be transformed into (Fetter and Walecka, 1971, Section 8; Shabaev, 2002, Eq. (3)) iG(x; x0 ) =

∞

ˆ I (L)] ˆ (x) ˆ † (x0 )}|0 d 4 LH : ∞ ˆ I (L)]|0 0|T exp[ − i −∞ d 4 LH

0|T {exp[ − i

−∞

(259)

It can be shown that the denominator in expression (259) has the eJect of removing all unlinked (unconnected) diagrams, and the result can be expressed (Fetter and Walecka, 1971, Eq. (9.5)) ∞ 4 † ˆ ˆ d LH (L) (x) (x0 ) |0conn iG(x; x0 ) = 0|T exp −i =

∞ (−i)n n=0

n!

−∞

∞

−∞

4

d x1 · · ·

∞

−∞

d 4 x n 0|T [H (x1 ) · · · H (x n ) ˆ (x) ˆ † (x0 )]|0conn : (260)

This leads to the expansion iG0 (x; x0 ) = 0|T [ ˆ (x) ˆ † (x0 )]|0 ; ∞ iG1 (x; x0 ) = −i 0| d 4 x1 T [H (x1 ) ˆ (x) ˆ † (x0 )]|0conn ; −∞

1 iG2 (x; x0 ) = − 0| 2

∞

−∞

4

d x1

∞

−∞

d 4 x2 T [H (x1 )H (x2 ) ˆ (x) ˆ † (x0 )]|0conn ;

etc:

(261)

The n-particle Green’s function is deHned in an analogous way 0H |T [ ˆ H (x1 ) · · · ˆ H (xn ) ˆ †H (x10 ) · · · ˆ †H (x n0 )]|0H iG(x1 ; x2 · · · xn ; x10 ; x20 · · · x n0 ) = ; 0H |0H

(262)

which leads to the expansion in the interaction picture iG(x1 ; x2 · · · xn ; x10 ; x20 · · · x n0 ) ∞ ∞ (−i)n ∞ 4 = d x1 · · · d4 x n n! −∞ −∞ n=0 ˆ (x1 ) · · · H (x n ) ˆ H (x1 ) · · · ˆ H (xn ) ˆ † (x10 ) · · · ˆ † (x n0 )]|0conn : × 0|T [H

(263)

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221

If we set all incoming times ti0 = t0 and all outgoing times ti = t , we have the two-times Green’s function, extensively discussed by Shabaev et al. (2002, 1993, 1994), Shabaev and Fokeeva (1994) and Artemyev et al. (2000). 6.2. The Fourier transform of the two-times Green’s function Assuming the vacuum state is normalized, we have from deHnition (253) iG(x; x0 ) = 0H |T [ ˆ H (x) ˆ †H (x0 )]|0H = F(t − t0 ) 0H | ˆ H (x) ˆ †H (x0 )|0H − F(t0 − t) 0H | ˆ †H (x0 ) ˆ H (x)|0H :

(264)

Considering t ¿ t0 , we have, from deHnition (34), ˆ ˆ ˆ ˆ iG+ (x; x0 ) = 0H | ˆ H (x) ˆ †H (x0 )|0H = 0H |(eiH t ˆ S (x)e−iH t )(eiH t0 ˆ †S (x0 )e−iH t0 )|0H :

(265)

We insert a complete set of positive-energy eigenstates of the Hamiltonian Hˆ (27) between the Held operators, Hˆ |n = En |n ;

(266)

which yields the Lehmann representation ˆ ˆ iG+ (x; x0 ) = 0H |eiH t ˆ S (x)|ne−iEn (t −t0 ) n| ˆ †S (x0 )e−iH t |0H n

=

n

0H | ˆ S (x)|ne−iEn (t −t0 ) n| ˆ †S (x0 )|0H ;

(267)

setting the energy of the vacuum to zero. We can now perform a Fourier transform of the Green’s function, including the adiabatic damping e−4H (see Section 3.3), yielding (H = t − t0 ¿ 0) ∞ 0H | ˆ S (x)|n n| ˆ † (x0 )|0H S ; (268) G+ (x; x0 ; E) = dH eiEH G+ (x; x0 ; H) = E − E + i4 n 0 n using

0

∞

dt eit e−4t =

i : + i4

(269)

We then see that the poles of the Green’s function represent the true eigenvalues of the system. Assuming no degeneracy, the eigenvalues can be obtained from the formula (Shabaev, 2002, Eq. (44)) En = Mn Mn

dEEG+ (x; x0 ; E) dEG+ (x; x0 ; E)

;

(270)

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where the contour Mn encircles the pole in question (and no other). This formula can be compared with the corresponding formula for the covariant evolution operator (248). The eigenstates |n in the eigenvalue equation (266) are Fock states, and the functions n (x) = n| ˆ †S (x)|0H

(271)

are the corresponding wavefunctions in conHguration space (in the Schr/odinger representation). [Formally, these functions can be expressed as eigenfunctions of a hypothetical Hamiltonian in conHguration space (H ) that corresponds to the Fock-space Hamiltonian (Hˆ ), Hn (x) = En n (x):] We can then express the Fourier transform (268) of the Green’s function n (x)† (x0 ) n : G+ (x; x0 ; E) = E − E n + i4 n

(272)

(273)

Note, that this is the exact single-particle Green’s function (positive-energy or retarded part), since the states are eigenstates of the exact Hamiltonian (cf. (186)). With no degeneracy, the numerator in (270) then becomes dEEG+ (x; x0 ; E) = n (x)En n† (x0 ) (274) Mn

with no summation over n, and since the denominator is then n (x)n† (x0 ), the result (270) follows directly. The retarded Green’s function (273) can also be written as x|n n |x0 ; (275) G+ (x; x0 ; E) = E − En + i4 n which is the coordinate representation of a (retarded) ‘Green’s-function operator’ (cf. Eq. (188)) |n n | Gˆ + (E) = : (276) E − E + i4 n n The single-particle Green’s function depends on time through a single time variable H = t − t0 , as follows from the Lehmann representation (267). The procedure above can easily be generalized to many particles, if we set all 3nal times equal to t and all initial times equal to t0 . 6.3. Extended model space. (Quasi)degeneracy Essentially following Shabaev (2002, Section 2.5.8, 1993, 1994) we shall now extend the treatment of the two-times-Green’s-function formalism to the case of degeneracy or quasi-degeneracy in the model space by means of an extended model space, in close analogy with the treatment of time-independent and time-dependent MBPT in the previous sections (Sections 2.2 and 3.3.2) (see also LeBigot, 2001). As in Section 2.2, we introduce a model space (D) of dimensionality d, which contains the model states of all degenerate or quasi-degenerate states. The model space is spanned

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223

by eigenfunctions of the unperturbed Hamiltonian H0 i = E0i i

(i = 1; 2 · · · d) :

The matrix of the retarded Green’s-function operator (276) in this basis is then i|n n |j : i |Gˆ + (E)|j = i|Gˆ + (E)|j = E − En + i4 n

(277)

(278)

i|n is the projection of the state |n onto the model-space state |i, and the entire projection, ˆ n = P|

d

|i i|n = |n0 ;

(279)

i=1

is the zeroth-order or model state, corresponding to the target state |n in intermediate normalization (12). We now construct the P matrix with the elements 1 ˆ Pij = i|P|j = dE i|Gˆ + (E)|j = i|n n |j = i|n0 n0 |j (280) 2i MD D D and the analogous K matrix 1 ˆ Kij = i|K|j = dEE i|Gˆ + (E)|j = i|n En n |j = i|n0 En n0 |j : (281) 2i MD D D Here, the integration is performed around all poles corresponding to the target states, and the sumˆ are the corresponding operators ˆ K mations are then restricted to these states. P; ˆ = |n0 n0 |; K |n0 En n0 | : (282) Pˆ = D

D

The model states |n0 are not necessarily orthonormal. For that reason we introduce a ‘dual set’ of states in the model space, |˜ 0n , deHned by m0 |˜ 0n = ˜ 0n |m0 = $mn :

(283)

It then follows that ˆ ˜ 0n = |n0 P|

Pˆ −1 |n0 = |˜ 0n :

and

With these notations |˜ 0n ˜ 0n | ; Pˆ −1 =

(284) (285)

D

and the standard projection operator for the model space (15) becomes |n0 ˜ 0n | = |˜ 0n n0 | : Pˆ = D

(286)

D

It also follows that ˆ Pˆ −1 |n0 = En |n0 ; K

(287)

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I. Lindgren et al. / Physics Reports 389 (2004) 161 – 261

Fig. 23. The two-times Green’s function for single-photon exchange between the electrons.

which implies that ˆ Pˆ −1 = K

D

|n0 En ˜ 0n |

(288)

is an e6ective Hamiltonian, which operates within the model space and generates the exact energies of the corresponding target states. This is completely equivalent to the eJective Hamiltonian introduced in the MBPT section (13). In both cases the operator is nonhermitian, and the eigenstates are the model states, which are in general nonorthogonal. As shown by Shabaev (2002, 1993, 1994), Shabaev and Fokeeva (1994) and Artemyev et al. (2000), it is possible to express the eJective Hamiltonian (288) in a hermitian form ˆ Pˆ −1=2 )|Pˆ −1=2 n0 = En |Pˆ −1=2 n0 : (Pˆ −1=2 K

(289)

This is equivalent to the hermitian form of the MBPT eJective Hamiltonian introduced by des Cloizeaux (des Cloizeaux, 1960; Lindgren 1974). The two-times Green’s function for single-photon exchange, represented in Fig. 23, is obtained from expansion (263), considering only relevant contractions for this case, iG(x1 ; x2 ; x10 ; x20 ) 1 ˆ (x1 )H ˆ (x2 ) ˆ (x1 ) ˆ (x2 ) ˆ † (x10 ) ˆ † (x20 )]|0 d 4 x1 d 4 x2 0|T [H =− 2 1 d 4 x1 d 4 x2 iSF (x1 ; x1 )iSF (x2 ; x2 )iI (x2 ; x1 )iSF (x1 ; x10 )iSF (x2 ; x20 ) ; =− 2

(290)

ˆ I (x) = −e ˆ † (x)3 A3 ˆ (x) and (169) I (x2 ; x1 ) = e3 DF83 (x2 − x1 )e28 . We shall now using (68) H 1 evaluate this expression in some detail. Using form (187) of the electron propagator, we have x1 |r r|x1 x2 |s s|x2 1 x1 |a a|x10 G(x1 ; x2 ; x10 ; x20 ) = − d 4 x 1 d 4 x2 I (x2 ; x1 ; z) 2 !3 − r + i;r !4 − s + i;s !1 − a + i;a ×

x2 |b b|x20 −i!3 (t −t1 ) −i!4 (t −t2 ) −iz(t2 −t1 ) −i!1 (t1 −t0 ) −i!2 (t2 −t0 ) e e e e e ; !2 − b + i;b

(291)

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225

integrated also over z and the !’s. The time integrations of t1 and t2 , performed over all times, yield according to formula (174) the factors $(!1 − z − !3 ) and $(!2 + z − !4 ), respectively, and the integration over z leads to $(!1 + !2 − !3 − !4 ). As in the covariant-evolution-operator method, the adiabatic damping can here be performed individually for each vertex, and we can therefore directly replace the time integrations (174) by Dirac delta functions. The Fourier transform of G(x1 ; x2 ; x10 ; x20 ) with respect to the times t and t0 is 1 dt dt0 eiE t e−iEt0 G(x1 ; x2 ; x10 ; x20 ) (2)2 ⇒ $(E − !3 − !4 )$(E − !1 − !2 )G(E; E ) : Integrations over !2 and !4 lead to the delta function $(E − E), which can be eliminated together with the delta function $(!1 + !2 − !3 − !4 ) above, yielding the matrix element (cf. (209)) d!1 d!3 rs|G(E)|ab = − 2 2 ×

rs|I (!1 − !3 )|ab : (292) (!3 − r + i4r )(E − !3 − s + i4s )(!1 − a + i4a )(E − !1 − b + i4b )

The eJective Hamiltonian is given by (289) HeJ = P−1=2 KP−1=2 ; 1 K= 2i 1 P= 2i

(293a)

M

Ed EG(E) ;

(293b)

M

dEG(E) ;

(293c)

where the integration M should be performed in the positive direction and enclose the unperturbed 0 energies of the initial (Ein0 = a + b ) and Hnal states (Eout = r + s ) but no other unperturbed energies. The denominators in (292) can be rewritten as 1 1 1 + 0 (!3 − r + i4r ) (E − !3 − s + i4s ) E − Eout + i4r + i4s 1 1 1 + × ; 0 (!1 − a + i4a ) (E − !1 − b + i4b ) E − Ein + i4a + i4b

(294)

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I. Lindgren et al. / Physics Reports 389 (2004) 161 – 261

which corresponds to the phantom-particle diagrams, discussed by LeBigot (2001, Eq. (IV.24)). In 0 the K integral (293b) the poles Ein0 and Eout contribute. The former yields 1 Ein0 1 + 0 0 0 (!3 − r + i4r ) (Ein − !3 − s + i4s ) Ein − Eout 1 1 ; × + (!1 − a + i4a ) (a − !1 + i4b ) where the last bracket leads to the delta function −2i$(!1 − a ) (indicated by !1 → a in Fig. 23). Similarly, the other pole yields 0 1 1 Eout + 0 −2i$(!3 − r ) 0 : Eout − Ein0 (!1 − a + i4a ) (Eout − !1 − b + i4b ) Integrating the Hrst contribution over !1 and the second over !3 , gives the matrix elements ˆ rs|K|ab 1 1 Ein0 d!3 rs|I (a − !3 )|ab 0 + =i 0 2 (!3 − r + i4r ) (Ein0 − !3 − s + i4s ) Ein − Eout 0 1 1 Eout d!1 rs|I (!1 − r )|ab 0 + 0 +i ; 2 Eout − Ein0 (!1 − a + i4a ) (Eout − !1 − b + i4b ) (295) which is identical to Eq. (25) in Artemyev et al. (2000). In a similar way we obtain the P integral (293c) ˆ rs|P|ab 1 1 1 d!3 rs|I (a − !3 )|ab 0 + =i 0 2 (!3 − r + i4r ) (Ein0 − !3 − s + i4s ) Ein − Eout 1 d!1 1 1 + 0 +i ; rs|I (!1 − r )|ab 0 2 Eout − Ein0 (!1 − a + i4a ) (Eout − !1 − b + i4b ) (296) which is the same as Eq. (26) in Artemyev et al. (2000). Expanding (293a) yields in Hrst order (1) HeJ = K(1) − 12 P(1) K(0) − 12 K(0) P(1) ; 0 where K(0) ij = $ij Ei . This yields the contribution to the matrix element (1) 0 |ab = rs|K(1) |ab − 12 (Ein0 + Eout ) rs|P(1) |ab rs|HeJ

(297)

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227

d!3 1 1 rs|I (a − !3 )|ab + 2 (!3 − r + i4r ) (Ein0 − !3 − s + i4s ) d!1 1 i 1 rs|I (!1 − r )|ab + 0 + : 2 2 (!1 − a + i4a ) (Eout − !1 − b + i4b )

i = 2

(298) The photon interaction I (z) has in the Feynman gauge form (77) e2 13 23 2kf(k) ; f(k) = − sin(kr21 ) : I (z) = d k 2 z − k 2 + i; 42 r12 Assuming r and s to be positive-energy states, we integrate over !3 in the negative half plane (poles at r − i4 and a + k − i;), which gives 2k 1 1 1 d k rs|f(k)|ab 2 + + 2 q − k 2 + i4 q + k + i4 q − k + i4 1 1 1 d k rs|f(k)|ab + ; (299a) = 2 q − k + i4 q − k + i4 where q = a − r and q = b − s . Similarly for !1 1 1 1 d k rs|f(k)|ab + : − 2 q + k − i4 q + k − i4 This gives the Hnal result 2k 2k 1 (1) rs|HeJ |ab = d k rs|f(k)|ab 2 + ; 2 q − k 2 + i4 q 2 − k 2 + i4

(299b)

(300)

which agrees with result (29) of Artemyev et al. (2000). This is identical to the Mittleman potential (Mittleman, 1972). If we instead use the nonhermitian form of the eJective Hamiltonian (288), we have in place of (297) (1) (1) HeJ = K(1) − K(0) P(1) ⇒ rs|HeJ |ab = rs|K(1) |ab − Eout rs|P(1) |ab ;

(301)

which becomes (1) rs|HeJ |ab

=i

d!3 1 1 rs|I (a − !3 )|ab : + 2 (!3 − r + i4r ) (Ein0 − !3 − s + i4s )

(302)

This is identical to the result of the evolution-operator method with the substitutions !3 → a − z and Ein0 = a + b and leads with the Feynman gauge to the result of the covariant-evolution-operator

228

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Fig. 24. The two-times Green’s-function diagram representing the screened self-energy.

method (213), (A.8), where also a nonhermitian eJective Hamiltonian is used, 1 1 (1) + : rs|HeJ |ab = d k rs|f(k)|ab q − k + i4 q − k + i4

(303)

6.4. Screened self-energy 6.4.1. Irreducible part As a second example we consider the two-times Green’s function for the screened self-energy, depicted in Fig. 24, i G(x1 ; x2 ; x10 ; x20 ) = d 4 x1 d 4 x2 d 4 x 3 d 4 x4 2 ×iSF (x4 ; x4 ; !3 )iSF (x2 ; x2 ; !4 )iG(x4 ; x3 ; !5 )iSF (x3 ; x1 ; !5 )iI (x2 ; x1 ; z) ×iSF (x1 ; x10 ; !1 )iSF (x2 ; x20 ; !2 ) ×e−it (!3 +!4 ) e−it4 (!5 −!3 ) e−it1 (!1 −z−!5 ) e−it2 (!2 +z−!4 ) eit0 (!1 +!2 )

(304)

(leaving out the ! and z integrations). Here, G represents the self-energy operator (202). After time integrations this becomes 1 x |r r|x4 x2 |s s|x2 d 3 x1 d 3 x2 G(x1 ; x2 ; x10 ; x20 ) = d 3 x3 d 3 x4 4 2 !3 − r + i4r !4 − s + i4s ×G(x4 ; x3 ; !5 )

x3 |t t|x1 x1 |a a|x10 x2 |b b|x20 I (z) !5 − t + i4t !1 − a + i4a !2 − b + i4b

×e−it (!3 +!4 ) eit0 (!1 +!2 ) $(!1 − z − !5 )$(!2 + z − !4 )$(!5 − !3 ) :

(305)

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229

Integration over z and !5 gives as before $(!1 + !2 − !3 − !4 ). The Fourier transform leads in the same way as the single-photon exchange (292) to rs|G(E)|ab d!1 d!3 = 2 2 ×

r|G(!3 )|t ts|I (!1 − !3 )|ab ; (!3 − r + i4r )(E − !3 − s + i4s )(!3 − t + i4t )(!1 − a + i4a )(E − !1 − b + i4b ) (306)

which is equivalent to the results of LeBigot et al. (2001) and LeBigot (2001, Eq. (IV.9)). The treatment is then quite analogous to the single-photon exchange, and we obtain instead of expression (298) (1) rs|HeJ |ab i =− 2 i − 2

d!3 r|G(!3 )|t ts|I (a − !3 )|ab 1 1 + 2 !3 − t + i4t (!3 − r + i4r ) (Ein0 − !3 − s + i4s ) d!1 r|G(!3 )|t ts|I (!1 − r )|ab 1 1 + 0 : 2 !3 − t + i4t (!1 − a + i4a ) (Eout − !1 − b + i4b ) (307)

Using instead the nonhermitian form of the eJective Hamiltonian, leads in analogy with single-photon result (302) to the simpler expression (1) |ab rs|HeJ 1 1 d!3 r|G(!3 )|t ts|I (a − !3 )|ab : + =−i 2 !3 − t + i4t (!3 − r + i4r ) (Ein0 − !3 − s + i4s ) (308)

This is identical to the evolution-operator result (237), if we make the substitution !3 → a − z. This expression contains a (quasi)singularity, when the intermediate state is (quasi)degenerate with the initial one, Ein0 = a + b ≈ s + t . In the evolution-operator method this singularity is eliminated by the counterterm (117), and in the Green’s-function method it will be eliminated by a similar counterterm, as will be shown below. 6.4.2. Reducible part In order to evaluate the reducible part of the screened self-energy diagram, i.e., when the intermediate state lies in the model space, one has to consider also products of Hrst-order contributions to the K and P integrals (LeBigot et al., 2001; LeBigot, 2001), shown in Fig. 25. The contribution to the nonhermitian eJective Hamiltonian (288) is −K(1) P(1) .

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Fig. 25. The counterterm for the screened self-energy in the Green’s-function method (cf. Fig. 22).

From (296) we have d!3 1 (1) rs|P |ab = i rs|I (a − !3 )|ab 0 0 2 Ein − Eout 1 1 × + (!3 − r + i4r ) (Ein0 − !3 − s + i4s ) d!3 1 rs|I (a − !3 )|ab =i 2 (!3 − r + i4r )(Ein0 − !3 − s + i4s ) and together with the self-energy K part this yields ts|I (a − !3 )|ab d!3 (1) : rs|HeJ |abCounter = i r|G(t )|t 2 (!3 − t + i4t )(a + b − !3 − s + i4s )

(309)

(310)

This removes the singularity of the eJective-interaction result (308). 6.5. General comparison between the Green’s-function and the evolution-operator methods We shall now compare the two methods for bound-state QED discussed above, the two-times Green’s-function and the covariant-evolution-operator methods, and we take the single-photon exchange between the electrons as an example. As pointed out before, both these methods are, in principle, two-times methods, although in the covariant-evolution-operator method the initial time is normally set to t0 = −∞, which simpliHes the handling considerably (Section 5.1). In order to make the comparison with the two-times Green’s-function method more transparent, however, we shall use the original two-times form also of the evolution-operator method. The two-times Green’s-function expression for the single-photon exchange (290) iG(x1 ; x2 ; x10 ; x20 ) 1 ˆ (x1 )H ˆ (x2 ) ˆ (x1 ) ˆ (x2 ) ˆ † (x10 ) ˆ † (x20 )]|0 d 4 x1 d 4 x2 0|T [H =− 2 1 d 4 x1 d 4 x2 iSF (x1 ; x1 )iSF (x2 ; x2 )iI (x2 ; x1 )iSF (x1 ; x10 )iSF (x2 ; x20 ) =− 2

(311)

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Fig. 26. The two-times Green’s function for single-photon exchange between the electrons is represented by the diagram on the left, and the corresponding two-times covariant evolution operator by the diagram on the right.

is represented by the Hrst diagram in Fig. 26. This we shall compare with the corresponding two-times covariant-evolution operator, which by a straightforward generalization of the single-time result (208) is given by 1 (2) 3 3 ˆ† ˆ† d x1 d x2 (x1 ) (x2 ) U Cov (t ; t0 ) = − d 4 x1 d 4 x2 iSF (x1 ; x1 ) 2 d 3 x10 d 3 x20 iSF (x1 ; x10 )iSF (x2 ; x20 ) ˆ (x10 ) ˆ (x20 ) (312) ×iSF (x2 ; x2 )iI (x2 ; x1 ) and represented by the second diagram in the Hgure. This comparison yields in the present case the following relation between the two-times Green’s function and the two-times covariant evolution operator U (2) Cov (t ; t0 )

=

d 3 x1

d 3 x2

d 3 x10 d 3 x20 ˆ † (x1 ) ˆ † (x2 )iG(x1 ; x2 ; x10 ; x20 ) ˆ (x10 ) ˆ (x20 ) (313)

—a relation that holds for any two-particle Green’s function/evolution operator and can easily be generalized to the n-particle case. It should be noted that the evolution operator is an operator, acting in the Fock space, while the Green’s function is a function of the time and space coordinates. It is now interesting to compare the two methods in some more detail. Starting with the singlephoton exchange in the GF method, we see that in the nonhermitian case (302) it is the pole E = Ein0 in the ‘phantom-particle’ expression (294) that contributes. The denominators, originating from the propagators of the incoming lines, lead here to the delta factor $(!1 − a ). In the evolution-operator method, the initial time is set to t0 → −∞ and !1 to a from the onset. We also see that the 0 ˆ (1) is eliminated in the expression for the eJective Hamiltonian, H (1) = denominator Ein0 − Eout of K eJ (1) (0) (1) K − K P . In the evolution-operator method the corresponding denominator is eliminated (217) by means of the time derivative. The situation is similar for the screened self-energy. The observations above are quite general. The two-times Green’s-function and covariant-evolutionoperator methods are quite analogous. After time integrations both expressions depend generally on the initial and Hnal time (although the initial time is in the latter method normally set to t0 = −∞). In the GF method a Fourier transform is performed and the eJective Hamiltonian is constructed by integrations over the energy. In the evolution-operator method the same expression is obtained by

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means of time derivation. In the GF method with nonhermitian eJective Hamiltonian the combined denominator from the propagators of the outgoing lines is eliminated by the energy integration and in the evolution-operator method by the time derivation. In the GF method the energy integration has the eJect that the energy parameters of the propagators are replaced by the orbital energies. In the (one-time) evolution-operator method this is set from the onset. (Cf. Table IV.1 in the thesis of LeBigot, 2001). Some applications of the two-times-Green’s-function technique are brie9y described in Section 7. For further information the reader is referred to the review article by Shabaev (2002). 7. Applications The bound-state techniques described here can be applied to various problems in QED. Here, we shall summarize some applications on stationary problems. For dynamical problems, like photoionization and radiative electron capture (REC) we refer to the current literature (Klasnikov et al., 2002; Yerokhin et al., 2000; Beier et al., 1999). 7.1. Applications on hydrogenlike ions 7.1.1. Lamb shift Pioneering works on the problem of bound-state QED calculations were carried out by Brown et al. (1959) and Desiderio and Johnson (1971) within the framework of the S-matrix formulation. Later the numerical technique was developed to a high degree of sophistication mainly for the Hrst-order self-energy of hydrogenlike ions by Mohr (1975, 1982, 1985, 1992). This technique was originally best suited for heavy ions, but a technique was later developed and applied also to low-Z ions (Jentschura et al., 1999), and this represents the most accurate result at present for neutral hydrogen and singly ionized helium. Accurate calculations of the Hrst-order vacuum polarization on these ions, including the Wichmann–Kroll term, have been performed by Persson et al. (1993) and Sunnergren (1998). In order to reach a numerical accuracy for light elements that can match the analytical approach ( − Z expansion) for light elements, it is necessary to consider also the two-photon contributions. This is computationally quite challenging and has only recently been possible to attack in a more comprehensive way. A number of more or less complete calculations have appeared during the last years (Mallampalli and Sapirstein, 1998; Labzowsky et al., 2000; Yerokhin, 2000; Yerokhin and Shabaev, 2001; Jentschura and Pachucki, 2002). 7.1.2. Hyper3ne structure and Zeeman e6ect The S-matrix formalism has been used also for accurate calculations for hydrogenlike ions of the eJect of an ‘external’ perturbation, like the hyperHne structure or the Zeeman eJect (atomic g-factor) (Persson et al., 1996b, 1997; Sunnergren et al., 1998; Blundell et al., 1997; Beier et al., 2000; Artemyev et al., 2001; Yerokhin et al., 2002). The diagrams in lowest order are depicted in Fig. 27. The hyperHne structure of some heavy hydrogenlike ions has been studied with the SuperEBIT at Livermore and at GSI in Darmstadt. The QED eJects are here of the order of 0.5% and clearly

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233

Fig. 27. The diagrams representing the lowest-order QED corrections to an additional perturbation like the hyperHne structure or Zeeman eJect for hydrogenlike ions.

observable within the experimental accuracy. This eJect, however, is normally overshadowed by the nuclear eJect. Therefore, comparison between theory and experiment is here mainly used to extract information about the nucleus, particularly the nuclear magnetization (Beiersdorfer et al., 2001; Gustavsson, 2001). The atomic g-factor has been measured with extreme precision for some light hydrogenlike ions at the university of Mainz, and the agreement with the theoretical predictions is very good. Here, the comparison between theory and experiment can actually be used to improve the atomic value of the electron mass (H/aJner et al., 2000; Beier et al., 2002; Yerokhin et al., 2002). 7.2. Applications on heliumlike ions The nonradiative part of the two-photon interaction for the ground states of heliumlike ions (Fig. 17) has been evaluated using the S-matrix formulation by Blundell et al. (1993) and Lindgren et al. (1995). The radiative eJects (Fig. 20) for the same systems have been evaluated by Persson et al. (1996a), using the S-matrix formalism, and by Yerokhin et al. (1997) using the two-times Green’s-function method. The results obtained are in good agreement with the experimental results obtained with the SuperEBIT at Livermore (Marrs et al., 1995), although the QED eJects are barely detectable. The nonradiative diagrams for the excited 1s2s states of heliumlike ions have also been evalu0 en ated by means of the S-matrix formulation (Mohr and Sapirstein, 2000; Andreev et al., 2001; As2 et al., 2002), as well as of the 1s2p states, excluding the quasi-degenerate J = 1 states (Mohr and Sapirstein, 2000). Recently, the covariant-evolution-operator technique has been applied to the 1s2p states of some lighter elements, including the quasi-degenerate J = 1 states, and the results obtained 0 en, 2002) The agree well with the experimental Hne-structure results (Lindgren et al., 2001; As2 screened-self-energy diagrams for these states of some heavier elements have also been evaluated using the two-times-Green’s-function technique by Indelicato et al. (LeBigot et al., 2001; LeBigot, 2001) and the vacuum-polarization screening corrections by Yerokhin et al. (Artemyev et al., 2000). The experimental results for the Hne structure of some heliumlike ions together with the theoretical results are given in Table 1. As discussed in the Introduction, the results of Plante et al. (1994) are obtained by means of relativistic MBPT with the QED corrections added in lowest order in − Z, and the results of Drake (1988) with nonrelativistic Hylleraas-type wave 0 en (2002), function and relativistic as well as QED corrections to lowest order. The results of As2

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Table 1 The 1s2p Hne-structure separations of some heliumlike ions. Values for Z = 2; 3 given in MHz and for Z ¿ 9 in H (1 H = 27:2 eV) 3

Z

P 1 − 3 P0

3

P2 − 3 P 0

2

29616.9509(9) 29616.9496(10)

3

155704.27(66) 155703.4(1,5)

9

701(10) 680 690 690

5050 5050 5050

10

1371(7) 1361(6) 1370 1370

8458(2) 8455(6) 8469 8460

18

3

P2 − 3 P 1 2291.1759(10) 2291.1736(11) −62678.41(66) −62679.4(5)

124,960(30) 124,810(60) 124,942 124,940

Experimentala Theoryb Experimentalc Draked

4364,517(6) 4362(5) 4364 4364

Experimentale Draked Plantef 0 eng As2

265880 265860 265880

Experimentalh Draked Plantef 0 eng As2 Experimentali Draked Plantef 0 eng As2

a

George et al. (2001); Castillega et al. (2000). Pachucki and Sapirstein (2000); George et al. (2001). c Riis et al. (1994). d Drake (1988). e Myers et al. (1999). f Plante et al. (1994). g 0 en (2002). Lindgren et al. (2001); As2 h Curdt et al. (2000). i Kukla et al (1995). b

(Lindgren et al., 2001) are obtained by means of the covariant evolution-operator method to second order with higher-order MBPT corrections added. Only the nonradiative QED parts are fully calculated and the remaining eJects taken from the power expansion. Full QED calculations are now under way. It can be seen from the comparison in the table that the diJerence between the QED eJects to leading order and the all-order result is hardly noticeable with the present numerical accuracy. For argon there is a signiHcant diJerence between the result of Drake and the other theoretical results, which is expected to be due to the approximation of the relativistic eJect in the method of Drake. It would be a challenge to try to reproduce with the evolution-operator method the accurate result for the separation 3 P2 − 3 P1 in heliumlike 9uorine, which would most likely test higher-order (in Z) QED corrections. It is presently unclear if this accuracy can be reached with the present technique. The experimental accuracy obtained for single ionized lithium and, in particular, for neutral helium,

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is deHnitely out of reach at present. An improved technique, which might be applicable in these cases, will be discussed in the next main section. 7.3. Applications on lithiumlike ions Lithiumlike ions can to a large extent be treated as a single-electron system with nonCoulombic potential. Early calculations with this approach were performed in order to calculate the Lamb shift of the 1s2p transitions of Li-like uranium (Blundell, 1992, 1993; Lindgren et al., 1993), and the results were in excellent agreement with the accurate experimental results of Schweppe et al. (1991). More elaborate calculations, including the two- and three-electron interactions, have now been performed particularly by Yerokhin et al. (1999, 2001) and Artemyev et al. (1999).

8. Possibilities of merging of QED with MBPT We have in the previous main sections considered three diJerent methods for bound-state QED calculations, the S-matrix, the covariant-evolution-operator and the two-times-Green’s-function methods. The latter two methods have the advantage compared to the S-matrix formulation that they can be used with an extended model space and thereby be applicable also to a quasi-degenerate situation. All three methods, however, have the shortcoming that in practice electron correlation can only be evaluated to relatively low order. This limits the accuracy, for instance, for simple systems with low nuclear charge, for which the electron correlation is comparatively strong. We know that in MBPT the electron correlation can be treated to essentially all orders, as discussed in Section 2.5. In the present section we shall consider the possibility of introducing some of these ideas into bound-state QED. In principle, all electromagnetic interactions between electrons could be treated entirely within the QED framework by considering one-, two-, three-: : : photon interactions. In practice, however, it is presently hardly possible to go beyond two-photon interactions in any reasonably complete manner. For that reason, it would be highly desirable to be able to supplement the QED calculations to second order, say, with higher-order eJects using MBPT methods. A simple and straightforward way that has been applied to heliumlike ions is to add eJects of third and higher orders from MBPT 0 en, 2002). In order to the second-order QED results (Persson et al., 1996a; Lindgren et al., 2001; As2 to achieve higher accuracy, however, particularly for very light elements, it is necessary to combine the two eJects in a more complete way, which would imply that the QED eJects are evaluated by means of correlated wave functions, rather than with simple hydrogenic ones. In the method developed by Drake, very accurate non-relativistic two-electron, correlated wavefunctions are constructed, using the method of Hylleraas, where the interelectronic distance r12 is explicitly used. The disadvantage with this technique when applied to QED calculations is that the QED eJects—as well as relativistic eJects—have to be evaluated analytically, using (the lowest-order) analytical expressions. Such an approach is superior to other available methods for very light elements, where electron correlation is relatively strong and the QED eJects quite small. For heavier elements, on the other hand, the approach cannot compete with available numerical QED approaches. By combining the numerical QED technique with the MBPT technique, as will be outlined in the present

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main section, it is expected that the QED eJects can be accurately evaluated by means of correlated wave functions, thus combining the advantages of the two approaches. The covariant-evolution-operator method is particularly suited as a basis for the combined approach, because of its formal analogy with MBPT, as demonstrated, for instance, in the two-photon case. This analogy remains also in higher orders. One possibility could therefore be to restrict the full QED calculations to the lowest orders and to evaluate the remaining (smaller) terms by a combination of QED and (relativistic) MBPT. This can be done by modifying the coupled-cluster equations, particularly the pair equation, to include also QED eJects. Below we shall Hrst demonstrate the close analogy between the QED treated by the covariant evolution-operator method and the traditional MBPT. Then we shall see how this analogy can be used to derive two-electron or pair equations to generate certain QED eJects to all orders. Eventually, this will lead to the complete Bethe–Salpeter equation (1951, 1957). Finally, we shall discuss some practical schemes for generating combined QED–MBPT eJects of high order. 8.1. Comparison of QED with MBPT In standard MBPT the second-order contribution to the energy or the eJective interaction due to the electron–electron interaction, V , is rs|V |tu tu|V |ab ; (314) [E |tu∈Q

where the denominator is equal to the negative of the excitation energy of the intermediate state, [E = a + b − t − u , and the summation runs over states in the complementary space (Q). This can be compared with the contribution due to the separable two-photon diagram (222) rs|V (q + p ; q + p)|tu tu|V (p; p )|ab + MSC : (315) [E |tu∈Q

where

V (q; q ) =

1 1 d kf(k) + q − k + i4 q − k + i4

;

assuming only positive-energy states are involved. The Hrst term in (315) is here very similar to the MBPT expression and represents the irreducible part for which the intermediate state lies in the Q space. The term ‘MSC’ represents the model-space contributions, introduced in Section 3.3.1, i.e., contributions due to the reducible part, for which the intermediate state lies in the model space. The lowest-order contributions to the eJective interaction due to multi-photon exchange then become (1) rs|HeJ |ab = rs|V (q; q )|ab (2) rs|HeJ |ab =

rs|V (q + p ; q + p)|tu tu|V (p; p )|ab [E

|tu∈Q

+MSC + rs|V2 |abNonsep :

(316)

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237

Fig. 28. The pair equation for a two-electron system, using the full QED one-photon interaction between the electrons, in analogy with the MBPT pair function in Fig. 2. This generates an inHnite sequence of ladder single-photon diagrams in addition to model-space contributions (MSC).

The diJerence from the corresponding MBPT results is here that the interactions are time-dependent (retarded), which also leads to the appearance of the model-space contribution (MSC) and the nonseparable part, represented by the last two terms of the second equation. We shall now utilize this close analogy between the QED and the MBPT results in order to indicate how the schemes can be combined in a systematic fashion. 8.2. The Bethe–Salpeter equation The pair equation with instantaneous Coulomb interactions, discussed in Section 2.5, can straightforwardly be generalized to include the full QED photons. In analogy with the expression for the separable two-photon ladder above, we can set up a pair equation by replacing the interaction in the MBPT equation (56) by the corresponding two-photon expressions rs tu (q + q )sab = rs|V (q; q )|ab + rs|V (q + p ; q + p)|tusab + MSC ;

(317)

where q = a − r ; q = b − s ; p = a − t ; p = b − u . The folded term is constructed in analogy with the corresponding MBPT expression in (55) and model-space contribution, as described in Section 5. Eq. (317) will generate an inHnite sequence of single-photon ladders (including folded diagrams and MSC), as indicated in Fig. 28. The iteration scheme of the single-photon exchange (V1 ) above can in principle be applied also to the nonseparable two-photon exchange (V2Nonsep ), etc. Including the nonseparable interactions to all orders V Nonsep = V1 + V2Nonsep + V3Nonsep + · · ·

(318)

leads to the complete Bethe–Salpeter equation (Fetter and Walecka, 1971, p. 562) (see Fig. 29) rs|V Nonsep |tu tu|VBS |ab + MSC : (319) [E The contribution to the energy—or the eJective interaction—is then obtained by closing the function by a Hnal interaction, in analogy with the MBPT case in Fig. 3. The two-particle interactions contain rs|VBS |ab = rs|ab +

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Fig. 29. Graphical representation of the complete Bethe–Salpeter equation in the form of a Dyson equation. The solid area represents the complete two-particle interaction, including no interaction, and the dashed area the nonseparable part (318). The intermediate state lies in the Q space. The MSC represents the model-space contribution of the reducible part.

Fig. 30. Graphical representation of the complete Bethe–Salpeter equation in the form of the Green’s-function equations (320) and (321).

here also radiative parts, with self-energy and vacuum polarization loops, which, of course, have to be properly renormalized. In principle, also the one-particle radiative eJects, can be iterated in a similar way by means of a single-particle equation. This can then be coupled to the two-particle equation in the same way as in the MBPT case (53). Result (319) can also be represented in the form of a Green’s-function equation G(x1 ; x2 ; x10 ; x20 ) = iSF (x1 ; x10 )iSF (x2 ; x20 ) 4 ] 1 ; x2 ; x1 ; x2 )G(x1 ; x2 ; x10 ; x20 ) + MSC ; + d x1 d 4 x2 K(x

(320)

where K] represents a kernel of all nonseparable interactions. This can be illustrated by the same Hgure, if we interpret the lines as electron propagators (see Fig. 30). The Green’s function depends only on the relative times, and by setting the initial times equal, t10 = t20 = t0 , as well as the Hnal times, t1 = t2 = t and t1 = t2 = t , we can make a Fourier transform with respect to the time di6erences H = t − t0 and H = t − t0 , which leads to d! iSF (x1 ; x10 ; !)iSF (x2 ; x20 ; E − !) G(x1 ; x2 ; x10 ; x20 ; E) = 2 ] 1 ; x2 ; x1 ; x2 ; E)G(x1 ; x2 ; x10 ; x20 ; E) + MSC ; + d 3 x1 d 3 x2 K(x (321) where E is the energy parameter. The procedure indicated here represents a generalization of the all-order procedures, discussed in Section 2.5. It is clear that the nonseparable multi-photon interactions can be handled very much like the interactions in standard MBPT—but will obviously be considerably more time consuming.

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239

Fig. 31. The pair function with an uncontracted photon (left) and with a completed photon exchange (right).

It is important that only the nonseparable parts of the interactions are iterated, in order to avoid double-counting. The nonseparable interactions are free from singularities. The intermediate states between the interactions are restricted to the Q space, as in ordinary MBPT, while inside the interactions all intermediate states (Q as well as P space) should be included. In addition, there will be Hnite model-space contributions (MSC) of the reducible part, which can be obtained as indicated earlier. 8.3. Pair functions with ‘uncontracted’ photons We shall now consider an alternative approach for generating higher-order diagrams, based upon a combination of the MBPT and QED approaches. We have seen in Section 3 that the Held-theoretical perturbation (68) due to the interaction between the electrons and the photon Held can create or destroy a virtual photon. A contraction between two such operators is needed to form an interaction between the electrons. We consider now a standard MBPT pair function (57) which is perturbed by a single perturbation (68). This leads to a pair function with what we shall refer to as an uncontracted tu photon, depicted in Fig. 31 (left). Assuming the MBPT pair functions is |.ab = sab |ab, the modiHed + ru+ function with an uncontracted photon can be expressed |.ab; k = sab (k)|ab, where

ru+ sab (k)

ˆ |tstu r|H ab = ; a − r + b − u − k + i4

(322)

ˆ is here the interaction (68), operating on assuming that only positive-energy states are involved. H a single electron. The denominator above is obtained using the general scheme derived in Appendix C. The pair function then satisHes the equation

ˆ (a + b − h0 (1) − h0 (2) − k).+ ab (1; 2; k) = Q H (1).ab (1; 2) ;

(323)

where h0 is the single-electron Schr/odinger (7) or Dirac (62) Hamiltonian. The Q projection operator assures that the r.h.s. is orthogonal to the initial state |ab. In order to complete the photon exchange, as indicated in the second diagram in Fig. 31, we ˆ (2), which after contraction leads to the electron–electron operate with a second interaction, H interaction (77). The function f(k) f(k) = −

e2 (1 − 1 · 2 )sin(kr12 ) 42 r12

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Fig. 32. The pair function with an uncontracted photon can also be iterated with instantaneous interactions before closing the photon.

Fig. 33. Closing the uncontracted photon in the pair function illustrated in Fig. 32 can yield a new pair function including the eJects indicated.

is expanded in spherical waves according to (80) ∞ sin kr12 = (2l + 1)jl (kr1 )jl (kr2 )C k (1) · C k (2) ; kr12

(324)

k=0

and then it is essentially the Bessel function jl (kr) that appears in the radial part of the equation ˆ . This procedure requires evidently one pair function for each value of the above in the place of H photon momentum k. With the denominator in (322), the contribution to the interaction from the full photon exchange becomes d k f(k) ; q + p − k + i4 corresponding to the Hrst part of the interaction in expressions (211) and (222). The second part of the interaction corresponds to a photon that is emitted from the second electron. The pair function with an uncontracted photon can also be iterated further with instantaneous interactions (V ), before closing the photon and before making the k-integration. This leads to eJects depicted in Fig. 32 and corresponds to the pair equation + ˆ (a + b − h0 (1) − h0 (2) − k).+ ab (1; 2; k) = Q(H (1).ab (1; 2) + V.ab (1; 2; k)) :

(325)

ˆ Then the pair function can be ‘closed’ by a second interaction, H(2), as before, and performing the k integration leads to the corresponding contribution to the energy or the eJective interaction, depicted in Fig. 33. By solving the corresponding pair equation, we obtain a new pair function with contracted photons only, which can then be iterated in the same way as the standard MBPT pair function, as indicated in Fig. 34. A new input for the whole scheme above can also be used,

I. Lindgren et al. / Physics Reports 389 (2004) 161 – 261

241

Fig. 34. The pair function in Fig. 33 can be iterated further with instantaneous interactions, as well as with an uncontracted photon, leading to eJects of the type indicated. In the last diagram we have also included the vacuum-polarization part, which can be obtained by modifying the orbitals and the Uehling part of the photon self-energy by modifying the photon propagator.

Fig. 35. By means of pair functions with two uncontracted photons eJects of the type indicated can be evaluated.

yielding repeated eJects of the type illustrated in the last diagram in the Hgure. In that diagram we have also indicated that the vacuum-polarization (Uehling part) can be included, by modifying the orbitals, and the photon self-energy by modifying the photon propagator, as discussed in Section 4.3.3. The eJects obtained with the procedure indicated here include the entire eJect due to the exchange of a single QED photon as well as the completely separable parts of two-, three-,: : : photon exchange. In addition, it contains most of the eJect of nonseparable two-, three-,: : : photon exchange. For instance, the diagrams in Figs. 33 and 34 contain the eJects of two crossed photons, the vertex correction, and the screened electron self-energy, where one of the photons is retarded and the other is instantaneous. Also much of the vacuum-polarization eJects, including the Uehling part of the photon self-energy, can be included, as indicated in Fig. 34. (Of course, the self-energy and vertex parts have to be properly renormalized.) Most importantly, however, these eJects are evaluated by means of correlated wave functions instead of pure hydrogenic ones. When the eJects are iterated, a good approximation to the full Bethe–Salpeter equation would be achieved. Work in realizing this scheme is now under way at our laboratory. In order to include the full two-photon eJects with correlated wavefunctions, it will be necessary to generate pair functions with two ‘uncontracted’ photons. Then also eJects of the type shown in Fig. 35 could be included. This would then represent the next step towards the solution of the

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full Bethe–Salpeter equation. Although straightforward in principle, this step does not seem to be computationally feasible, however, for the time being. 9. Conclusions and outlook In this work we have concentrated on two-electron ions for several reasons. Firstly, hydrogenlike ions have been extensively treated in the review article on the S-matrix formulation by Mohr et al. (1998). Secondly, there has been a rapid development concerning heliumlike ions lately— experimentally as well as theoretically—a development we expect to continue for quite some time to come. Heliumlike ions represent the simplest systems where the interplay between QED eJects and electron correlation can be studied, and here several interesting and challenging problems will emerge. For light elements the electron correlation is so strong that it cannot be handled to a sufHcient degree of accuracy with the currently available methods for bound-state QED. Furthermore, these systems contain levels which are very close in energy, which represents another theoretical challenge. Experimentally, some Hne-structure separations in light-to-medium-heavy heliumlike ions can now be measured with an accuracy up to 1 ppm, as in heliumlike 9uorine (Myers et al., 1999) (see Table 1 in Section 7). Calculations are now under way at our laboratory in order to try to reproduce this value. It is unclear, though, whether this can be achieved with the current technique. Under way are also some eJorts to realize the modiHed scheme, presented in the previous main section, where pair functions with an uncontracted photon are generated. It is expected that this technique will improve the accuracy considerably in cases where the electron correlation plays a major role. The Hne-structure separation in neutral helium is of particular interest. Here, the experimental accuracy is as high as 30 ppb, and it is anticipated that the accuracy could be improved by another order of magnitude (George et al., 2001). Since the Hne-structure is due entirely to relativity and QED (proportional to 2 in leading order), a comparison between theory and experiment may yield a value of the Hne-structure constant with an accuracy comparable to (in principle half) the experimental uncertainty. The Hrst evaluation of this constant from the experimental data and available theoretical estimates yielded a value with an uncertainty of 23 ppb, which however deviated four standard deviations from the accepted, and more accurate, value obtained mainly from the free-electron g-factor (Mohr and Taylor, 2000). According to newer estimates, the theoretical uncertainty had been underestimated, and the new value agrees with the accepted value but with a larger uncertainty of about 200 ppb (Pachucki and Sapirstein, 2002). Hopefully, a combination of the analytical and numerical approaches for some light ions might here improve the situation. Acknowledgements The authors want to express their thanks for stimulating discussions with our colleagues Eric-Olivier LeBigot, Paul Indelicato, Peter Mohr, Vladimir Shabaev and Gerhard SoJ, as well as to our former and present collaborators Thomas Beier, Martin Gustavsson, Ann-Marie Pendrill, Hans Persson, and Per Sunnergren, who have contributed considerably to the works presented here. The support from the Swedish Research Council and the Alexander von Humboldt Foundation is gratefully acknowledged.

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Appendix A. Evaluation of one and two-photon evolution-operator diagrams In this appendix we shall Hrst evaluate some covariant-evolution-operator diagrams in the standard way. In Appendix B we shall consider time-ordered diagrams and evaluate each time-ordering separately. This will lead to a general scheme for diagram evaluation, described in Appendix C, which is utilized in the merging procedure of QED and MBPT in Section 8. A.1. Evaluation of the single-photon exchange The covariant evolution operator for single-photon exchange, illustrated in Fig. A.1, is (209), rs|Uˆ (2) Cov (t ; −∞)|ab =− dt1 dt2 rs|iSF (x1 ; x1 )iSF (x2 ; x2 )iI (x2 ; x1 )|abeit (r +s ) e−it1 a −it2 b e−4(|t1 |+|t2 |)

=i

dt1 dt2 rs|

dz 2

d!1 |t t| 2 !1 − t + i;t

|u u| d!2 I (z)|ab 2 !2 − u + i;u

×eit (r +s ) e−i!1 (t −t1 ) e−i!2 (t −t2 ) e−iz(t2 −t1 ) e−it1 a −it2 b e−4(|t1 |+|t2 |) ;

(A.1)

using form (192) of the electron propagators and the Fourier transform (169) of the electron– electron interaction. The quantities ;t ; ;u are inHnitesimally small quantities with the same sign as t and u , respectively, with the purpose of determining the poles of the electron propagator. The time integration over t1 becomes, using the E function (174), 24 = 2E4 (!1 + z − a ) : (A.2) dt1 eit1 (!1 +z−a ) e−4|t1 | = (!1 + z − a )2 + 42 The !1 integral then becomes 1 24 d!1 e−it (!1 −r ) ; 2 2 2 !1 − r + i;r (!1 + z − a ) + 4 using the fact that only the terms t = r and u = s survive. Here, the poles appear at !1 = r − i;r and !1 = a − z ± i4. If r ¿ 0 (;r = ;) we may integrate over the positive half plane with the pole

Fig. A.1. Graphical representation of the covariant-evolution operator for single-photon exchange (Fig. 15).

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!1 = a − z + i4, yielding 1 e−it (q−z+i4) q − z + i; + i4 with q = a − r . Similarly, when r ¡ 0 we may integrate over the negative half-plane (pole !1 = a − z − i4), and the integration yields 1 e−it (q−z−i4) : q − z − i; − i4 In the integrals here there are two imaginary parts, one (;) associated with the electron propagator and one (4) with the adiabatic damping. The purpose of the former is to indicate the position of the poles of the propagator, while the latter is a parameter that is going to zero in the adiabatic process. It should be noted that these quantities are of di6erent character—4 is a 3nite quantity, which is eventually switched oJ, while ; is an inHnitesimally small quantity. Therefore, we can omit ;, when it appears together with 4, and the results above can be summarized as 1 e−it (q−z+i4r ) ; q − z + i4r where 4r = 4 sgn(r ). In the same way the integrations over t2 and !2 yield 1 e−it (q +z+i4s ) q + z + i4s with q = b − s and 4s = 4 sgn(s ). After the integrations above, expression (A.1) becomes rs|I (z)|ab dz (t ; −∞)|ab = i rs|Uˆ (2) e−it (q+q ) Cov 2 (q − z + i4r )(q + z + i4s ) 1 dz e−it (q+q ) 1 rs|I (z)|ab : =i + q + q 2 q − z + i4r q + z + i4s

(A.3)

Eventually, all 4:s will go to zero, and they are needed only to determine the position of the poles. Since the factor (q + q ) is not involved in any integration, we can leave out the imaginary part of that factor. The interaction I (z) is in the Feynman gauge given by (77) ∞ 2kd kf(k) 2 3 8 I (z) = e 1 2 DF83 (x2 − x1 ; z) = 2 − k 2 + i; z 0 f(k) = −

e2 (1 − 1 · 2 )sin(kr12 ) : 42 r12

The z integral is here 1 1 dz 1 : + 2 2 q − z + i4r q + z + i4s z − k 2 + i;

(A.4)

(A.5)

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245

We can rewrite the last denominator as (z − k + i;)(z + k − i;) = z 2 − k 2 + 2ki; : Since ; is an inHnitesimally small positive quantity, 2k; is equivalent to ; for positive k. The Hrst term in (A.5) has poles at z = q + i4r and z = ±(k − i;). When 4r = 4 ¿ 0, there is one pole in the negative half-plane, z = k − i;, and the integral becomes −

i : 2(k − i;)(q − k + i; + i4)

As before, we can omit the ; term in comparison with the 4 term, but we keep for the moment the ; term in the Hrst factor, −

i : (2k − i;)(q − k + i4)

When 4r = −4 ¡ 0, there is one pole in the positive half-plane, z = −k + i;, and the integral becomes similarly −

i ; (2k − i;)(q + k − i4)

and the result can be summarized as −

i ; (2k − i;)(q − (k − i4)r )

where (A)x = (A)sgn(x ). Similarly, the integration of the second term in (A.5) yields −

i ; (2k − i;)(q − (k − i4)s )

and the complete integral becomes i 1 1 − : + (2k − i;) q − (k − i4)r q − (k − i4)s When including the interaction (A.4), there is a factor of 2k in the numerator, and then it follows that the pole at k = 0 does not contribute. Therefore, the matrix element of the covariant evolution operator for single photon exchange becomes rs|Uˆ (2) Cov (t ; −∞)|ab = rs|V (q; q )|ab

V (q; q )

=

e−it (q+q ) q + q

1 1 d kf(k) + q − (k − i4)r q − (k − i4)s

(A.6) ;

(A.7)

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Fig. A.2. Graphical representation of the covariant-evolution operator for the two-photon ladder diagram (Fig. 17).

where q = a − r and q = b − s . According to the expression (130) for the eJective interaction, V (q; q ) is the Hrst-order contribution to the eJective Hamiltonian, (1) rs|HeJ |ab = rs|V (q; q )|ab :

(A.8)

A.2. Evaluation of the two-photon ladder diagram 0 en, 2002.) (See also As2 The matrix element of the two-photon ladder diagram, shown in Fig. A.2, is given in Eq. (220) (4) ˆ rs|U Cov (t ; −∞)|ab = rs| d 4 x3 d 4 x4 iSF (x3 ; x3 )iSF (x4 ; x4 )iI (x4 ; x3 ) dt1 dt2 iSF (x3 ; x1 )iSF (x4 ; x2 )iI (x2 ; x1 )|ab

×

×eit (r +s ) e−it1 a −it2 b e−4(|t1 |+|t2 |+|t3 |+|t4 |) ;

(A.9)

which in analogy with the single-photon case can be expressed rs|Uˆ (4) Cov (t ; −∞)|ab d!3 |r r| |s s| d!4 d z =− dt3 dt4 rs| I (z )|tu 2 2 !3 − r + i;r 2 !4 − s + i;s d!1 |t t| |u u| d!2 dz I (z)|ab × dt1 dt2 tu| 2 2 !1 − t + i;t 2 !2 − u + i;u

×eit (r +s ) ; e−i!3 (t −t3 ) e−i!4 (t −t4 ) e−iz (t4 −t3 )

×e−i!1 (t3 −t1 ) e−i!2 (t4 −t2 ) e−iz(t2 −t1 ) e−it1 a −it2 b e−4(|t1 |+|t2 |+|t3 |+|t4 |) :

(A.10)

The time integrations yield here, using Eq. (174), E4 (!1 − !3 − z )E4 (!2 − !4 + z )E4 (a − z − !1 )E4 (b + z − !2 ) (leaving out the factors of 2). If r is a positive-energy state, we integrate !3 over the positive half plane with the pole !3 = !1 − z + i4, which yields 1 : !1 − z − r + i4 + i;

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247

The !1 integrand now becomes 2E4 (a − z − !1 ) d!1 2 (!1 − z − r + i4 + i;)(!1 − t + i;t ) and the poles appear at !1 =t −i;t , !1 =r +z −i4r and !1 =a −z ±i4. If also t is a positive-energy state, we integrate over the positive half-plane with the pole !1 = a − z + i4, yielding (q − z −

z

1 + 2i4)(p − z + i4)

with q = a − r and p = a − t . Similarly, if both t and r are negative-energy states, we obtain (q − z −

z

1 : − 2i4)(p − z − i4)

If we assume that t is a negative-energy state and r still a positive-energy state, then there are two poles in each half-plane—in the negative half-plane !1 =a −z −i4 and !1 =z +r −i4−i;—yielding (q − z − −

z

1 + i;)(p − z − i4 − i;)

2i4 : (q − z − z + 2i4 + i;)(q − z − z + i;)(q − p − z + i4 + 2i;)

Here, we see that it is important to keep the ; term, since the 4 term vanishes in the Hrst denominator. The last term vanishes as 4 → 0. The corresponding result is obtained when the signs of r and t are reversed. Generally, the result of the !1 integration can then be expressed (in the limit) (q − z −

z

1 ; + i4r + i4t + i;r )(p − z + i4t + i;t )

but since the imaginary parts are here only used to indicate the position of the poles, this is equivalent to (q − z −

z

1 : + i4r )(p − z + i4t )

The results after the complete ! integrations can now be summarized as follows: rs|I (z )|tu d z d z (4) rs|Uˆ Cov (t ; −∞)|ab = − 2 2 (q − z − z + i4r )(q + z + z + i4s ) ×

tu|I (z)|ab e−it (q+q ) (p − z + i4t )(p + z + i4u )

(A.11)

with q = a − r ; q = b − s ; p = a − t and p = b − u . As before, we leave out the imaginary part in factors not involved in any integration. The last two denominators of (A.11) can be written as 1 1 1 1 = + (p − z + i4t )(p + z + i4u ) p − z + i4t p + z + i4u p + p

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and the Hrst two z

1 + i4r )(q + z + z + i4s )

(q − z − 1 1 1 = + ; q − z − z + i4r q + z + z + i4s q + q

(A.12)

which gives rs|Uˆ (4) Cov (t ; −∞)|ab (A + B + C + D)e−it (q+q ) d z d z rs|f(k )|tu tu|f(k)|ab ; = 2 2 (q + q )(p + p ) d z d z A=− d k d k 2 2

(A.13)

4kk ; (q − z − z + i4r )(p − z + i4t )(z 2 − k 2 + i;)(z 2 − k 2 + i;) d z d z d k d k B=− 2 2 ×

4kk ; (q − z − z + i4r )(p + z + i4u )(z 2 − k 2 + i;)(z 2 − k 2 + i;) d z d z d k d k C =− 2 2 ×

4kk ; (q + z + z + i4s )(p − z + i4t )(z 2 − k 2 + i;)(z 2 − k 2 + i;) d z d z d k d k D=− 2 2 ×

×

4kk : (q + z + z + i4s )(p + z + i4u )(z 2 − k 2 + i;)(z 2 − k 2 + i;) (A.14)

As a consequence of the generalized factorization theorem (121) and the regularity of the reduced evolution operator, the adiabatic-damping parameter 4 can be switched oJ individually for each vertex in the evolution-operator method—in contrast to the situation in the S-matrix method, using the Gell–Mann–Low–Sucher method (165). The 4’s are needed, though, for the pole integrations, and therefore the sign of 4 is important (but not its size). Then it is possible to apply a simpliHed method, where the time integrations will directly lead to Dirac delta functions, and the ! integrations will be trivial. It has to be observed, though, as illustrated above, that the 4 term might disappear when negative-energy states are involved, and then the ; term from the propagator will determine the position of the pole.

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249

We now evaluate the z; z integrals when t; u (as well as r and s) are positive-energy states (4t = 4u = 4 ¿ 0). Then A has one z pole z = k − i; and one z pole z = k − i; in the negative half planes (cf. (A.5)), yielding A=

(q − k −

k

1 : + i4)(p − k + i4)

B has two z poles z = k − i; and z = −p − i4 and one z pole z = k − i; in the negative half-planes, which yields similarly B=

(q − k −

k

2k 1 + : + i4)(p + k + i4) (q + p − k + i4)((p + i4)2 − k 2 )

Similarly, C has the poles z = −k + i;, z = p + i4 and z = −k + i; in the positive half plane, and integration yields C=

(q

−k −

k

2k 1 + : + i4)(p + k + i4) (q + p − k + i4)((p + i4)2 − k 2 )

D has the poles z = −k + i; and z = −k + i; in the positive half-plane, yielding D=

1 : (q − k − k + i4)(p − k + i4)

The B term can be rewritten as 1 1 1 + B= q + p − k + i4 q − k − k + i4 p − k + i4 and the C term 1 C= q + p − k + i4

1 1 + q − k − k + i4 p − k + i4

;

which eliminates an apparent pole in the k integration. A.3. Evaluation of the screened self-energy diagram Next, we shall evaluate the covariant evolution-operator diagram for the screened self-energy, given by expression (237), assuming all states being positive-energy states (see Fig. A.3) ru|I (z )|ut ts|I (z)|ab d z d z (2) rs|HeJ |ab = − 2 2 (p − z − z + i4)(p − z + i4) 1 1 + : (A.15) × q − z + i4 q + z + i4 For the Hrst term in the square brackets we integrate over the negative half-plane with the pole z = k − i; from the photon propagator, yielding i

ru|I (z )|ut ts|f(k)|ab (p − k − z + i4)(p − k + i4)(q − k + i4)

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Fig. A.3. The covariant-evolution-operator diagram representing the screened self-energy.

and after integration over z ru|f(k )|ut ts|f(k)|ab : (p − k − k + i4)(p − k + i4)(q − k + i4) The second term has two poles in the negative half plane, z = k − i; and z = −q − i4, and yields similarly ru|f(k )|ut ts|f(k)|ab (p − k − k + i4)(p − k + i4)(q + k + i4) 1 1 ru|f(k )|ut ts|f(k)|ab − : + (p + q − k + i4)(p + q + i4) q − k + i4 q + k + i4 After some algebra the denominators can be rewritten, eliminating an apparent pole, 1 1 1 + (p + q − k + i4)(p + q ) p − k + i4 q − k + i4 1 1 1 + : + (p − k − k + i4)(p − k + i4) p + q − k + i4 q − k + i4

(A.16)

(A.17)

Appendix B. Evaluation of time-ordered diagrams In this appendix we shall consider the evaluation of time-ordered diagrams, which, as we shall see, will lead to a general scheme for expressing the covariant-evolution-operator diagrams at arbitrary order. This procedure will form the basis for the model of merging QED with MBPT, discussed in Section 8. B.1. Two-photon ladder As an illustration we consider the two-photon ladder, treated in Appendix A, for which two time-orderings are shown in Fig. B.1. Using expression (184) of the electron propagators we can

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251

Fig. B.1. Two time-ordered two-photon-ladder diagrams, representing the separable part (left) and nonseparable part (right) of the two-photon ladder diagram.

write the two-photon ladder (A.9) as rs|Uˆ (4) Cov (t ; −∞)|ab d z 4 4 [F(t − t3 )†r+ (x3 ) − F(t3 − t )†r− (x3 )] =− d x3 d x4 2

×[F(t − t4 ) †s+ (x4 ) − F(t4 − t )†s− (x4 )]I (x4 ; x3 ; z ) dz 4 4 [F(t3 − t1 )t+ (x3 )†t+ (x1 ) − F(t1 − t3 )†t− (x1 )t− (x3 )] × d x1 d x2 2 ×[F(t4 − t2 ) u+ (x4 )†u+ (x2 ) − F(t2 − t4 )†u− (x2 )u+ (x4 )] ×I (x2 ; x1 ; z)a (x1 )b (x2 ) ×e−it3 (t −r −z ) e−it4 (u −s +z ) e−it1 (a −t −z) e−it2 (b −u +z) e−4(|t1 |+|t2 |+|t3 |+|t4 |) :

(B.1)

For simplicity we introduce the following short-hand notations: d1 = a − t − z = p − z;

d2 = b − u + z = p + z;

d 3 = t − r − z = q − p − z ; d12 = d1 + d2 = p + p ; d13 = d1 + d3 = q − z − z ;

d4 = u − s + z = q − p + z ;

d34 = d3 + d4 = q + q − p − p ; d24 = d2 + d4 = q + z + z ;

d123 = d1 + d2 + d3 = q + p − z ;

d124 = d1 + d2 + d4 = q + p + z ;

d1234 = d1 + d2 + d3 + d4 = q + q ; and the notations d1± = d1 ± i4, etc. to indicate the sign of the imaginary part.

(B.2)

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We assume Hrst that all states are positive-energy states. Then we have the time-ordering t ¿ t3 ¿ t1 and t ¿ t4 ¿ t2 , and the time integrations yield t3 t e−it d13+ −it3 d3+ −it1 d1+ dt3 e dt1 e =− ; d13+ d1+ −∞ −∞ t t4 e−it d24 −it4 d4+ dt4 e dt2 e−it2 d2+ = − : (B.3) d24+ d2+ −∞ −∞ The total time integration then becomes 1 1 1 e−it d1234 e−it d1234 1 1 = + + d13+ d24+ d1+ d2+ d1234 d13+ d24+ d12+ d1+ d2+ and with the notations above 1 1 1 e−it (q+q +4i4) + q + q + 4i4 q − z − z + 2i4 q + z + z + 2i4 p + p + 2i4 1 1 + ; × p − z + i4 p + z + i4

(B.4)

(B.5)

in agreement with (A.4). Here, also the magnitude of the imaginary parts come out correctly, although we do not need them in our method. If the intermediate state t is a negative-energy state—and r still a positive-energy state—then the time-ordering becomes t ¿ t3 ¡ t1 , and the time integration over t1 and t3 becomes t3 t e−it d13+ dt3 e−it3 d3+ dt1 e−it1 d1− = − : (B.6) d13+ d1− −∞ ∞ Here, we have an example where the 4 contribution cancels, and the + sign is due to the ; term, as discussed in Appendix A.2. This leads to the change in (B.5) 1 1 ; ⇒ p − z + i4 p − z − i4 in agreement with (A.14). B.1.1. Separable and nonseparable parts We consider next the separable part of the ladder diagram. Assuming Hrst that all states are positive-energy states, the separable diagram in Fig. B.1 corresponds to the time-ordering t ¿ t4 ¿ t3 ¿ t2 ¿ t1 , and the time integration yields t4 t3 t2 t −it4 d4+ −it3 d3+ −it2 d2+ dt4 e dt3 e dt2 e dt1 e−it1 d1+ −∞

=

−∞

e−it d1234+

d1234+ d123+ d12+ d1+

−∞

:

−∞

(B.7)

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253

The remaining time-orderings are obtained by means of the exchanges 1 ↔ 2 and 3 ↔ 4, which leads to e−it d1234+ 1 1 1 1 1 (B.8) + + d1234+ d123+ d124+ d12+ d1+ d2+ or e−it (q+q +2i4) q + q + 2i4

1 1 + q + p − z + 2i4 q + p + z + 2i4 1 1 1 + : × p + p p − z + i4 p + z + i4

(B.9)

This leads to the contribution to the e6ective interaction due to the separable ladder, using (130), rs|HeJ |abSep =

rs|V (q + p ; q + p)|tu tu| V (p; p )|ab ; p + p

(B.10)

where V is given by (A.7). The nonseparable diagram in Fig. B.1 corresponds to the time-ordering t ¿ t4 ¿ t2 ¿ t3 ¿ t1 , and the time integral is obtained from (B.7) by the exchange 3 ↔ 2, e−it d1234+ : d1234+ d123+ d13+ d1+

(B.11)

Similarly, the opposite time-ordering, t ¿ t3 ¿ t1 ¿ t4 ¿ t2 , yields e−it d1234+ : d1234+ d124+ d24+ d2+

The total time integration for the nonseparable part of the ladder diagram then becomes e−it d1234+ 1 1 ; + d1234+ d123+ d13+ d1+ d124+ d24+ d2+

(B.12)

(B.13)

when all states are positive-energy states. As a corollary we may add the separable (B.8) and nonseparable (B.13) parts of the ladder diagram, 1 1 1 1 1 + + + d123+ d124+ d1+ d2+ d123+ d13+ d1+ d124+ d24+ d2+ =

1 1 + ; d13+ d1+ d2+ d24+ d1+ d2+

(B.14)

which agrees with (B.4). From (B.13) the nonseparable contribution to the eJective interaction contains I (z )I (z) I (z )I (z) + ; (q + p − z )(q − z − z )(p − z) (q + p + z )(q + z + z )(p + z)

(B.15)

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and integrations over z and z yield (q +

p

f(k )f(k) f(k )f(k) + : − k )(q − k − k )(p − k) (q + p + k )(q + k + k )(p + k)

(B.16)

Appendix C. General evaluation procedure C.1. General rules The diagram evaluation discussed above using time-ordered diagrams can be generalized to higher orders. When the involved states are positive-energy states, we Hnd that we can construct the energy denominators in the following way. Inserting a horizontal line above each vertex, the corresponding denominator is given by • the orbital excitation energies counted from the bottom • a term −z + i4 for each photon cut by the line. as illustrated in Fig. C.1. (The direction of the photon line is immaterial, but we have here assumed that it is directed upwards, which yields the minus sign of z.) If a photon line is cut by only one horizontal line, considering Hrst positive-energy states, then the denominator is of the type 1 ; A − z + i4 and the poles for the z integration are located at z = A + i4 and z = ±(k − i;) from the photon propagator. We then integrate over the negative half-plane with the pole z = k. (As discussed in Appendix A.1 the ; term can be omitted in relation to the 4 term.) The result of the integration is then obtained by replacing z by k and multiplying by −i=2k, i.e., −i 1 ⇒ : A − z + i4 2k(A − k + i4)

Fig. C.1. Two time-ordered versions of the two-photon-crossed diagram.

(C.1)

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255

Fig. C.2. Two time-ordered versions of the two-photon-crossed diagram.

If there are several photon lines cut by several vertical lines, then the denominators are of the type 1 1 ··· : A − z + i4 B − z − z + i4 Here, each z always appears with the same sign, and we can integrate over the z’s as before, yielding 1 1 i i ··· ⇒ ··· : (C.2) A − z + i4 B − z − z + i4 2k(A − k + i4) 2k (B − k − k + i4) The rules given here hold with minor modiHcation also when there are negative-energy states involved. The only diJerence is that certain time integrations are performed to t = +∞ and the sign of the corresponding imaginary part is reversed. C.2. Application C.2.1. Two-photon cross We shall Hrst apply the rules given above to evaluate the evolution-operator diagram for the two crossed photons, shown in Fig. 17. Two time-ordered variants are shown in Fig. C.2. With the time-ordering of the Hrst diagram in the Hgure, the evaluation yields (for simplicity leaving out the imaginary parts) 1 1 1 ; q + p − k p − k + p − k p − k evaluating the denominators from the bottom and leaving out the Hnal denominator. Reversing 1 → 4, leads to the replacement p − k → p − k in the last factor, and 3 → 2 to q + p − k → q + p − k. Adding these eJects together, yields 1 1 1 1 : + q +p−k q+p −k p −k p−k Finally, we can reverse the order of 3 and 4, which leads to the second diagram in the Hgure. The denominators then become 1 1 1 q+p −k q−k −k p−k

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Fig. C.3. Time-ordered diagram representing the one-time covariant-evolution-operator for the screened self-energy.

and reversing the direction of the photons yields the Hnal contribution 1 1 1 : q + p − k q − k − k p − k 0 en (2002). This agrees with the result of As2 C.2.2. Screened self-energy Next, we consider the screened self-energy diagram with this general procedure. Starting with the time-ordering shown in Fig. C.3, t ¿ t4 ¿ t3 ¿ t2 ¿ t1 , and using the notations d1 = a − t − k = p − k;

d 2 = b − s + k = q + k ;

d3 = t − u − k = p − p − k ; d12 = p + q ;

d4 = u − r + k = q − p + k ;

d13 = p − k − k ;

d123 = p + q − k ;

the denominators become 1 1 : = d123 d12 d1 (p + q − k )(p + q )(p − k) Reversing 1 and 2 yields 1 1 : = d123 d12 d2 (p + q − k )(p + q )(q − k) Reversing 2 and 3 of the Hrst expression yields 1 1 = d123 d13 d1 (p + q − k )(p − k − k )(p − k) and Hnally reversing 2 and 4 of the last expression 1 1 : = d134 d13 d1 (q − k)(p − k − k )(p − k) This agrees with the previous result (A.17).

d134 = q − k ;

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Physics Reports 389 (2004) 263 – 440 www.elsevier.com/locate/physrep

Nuclear spinodal fragmentation Philippe Chomaza , Maria Colonnab , J*rgen Randrupc;∗ a

GANIL (DSM-CEA/IN2P3-CNRS), B.P. 5027, F-14076 Caen Cedex 5, France b Laboratori Nazionali del Sud, Via S. So-a 44, I-95123 Catania, Italy c Nuclear Science Division, Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720, USA Accepted 22 September 2003 editor: G.E. Brown

Abstract Spinodal multifragmentation in nuclear physics is reviewed. Considering 1rst spinodal instability within the general framework of thermodynamics, we discuss the intimate relationship between 1rst-order phase-transitions and convexity anomalies in the thermodynamic potentials, clarify the relationship between mechanical and chemical instability in two-component systems, and also address 1nite systems. Then we analyze the onset of spinodal fragmentation by various linear-response methods. Using the Landau theory of collective modes in bulk matter as a starting point, we 1rst review the application of mean-1eld methods for the identi1cation of the unstable collective modes and the determination of their structure and the associated dispersion relations yielding their growth rates. Subsequently, the corresponding results for 1nite nuclei are addressed and, within the random-phase approximation, we establish the connection between unstable modes in dilute systems and giant resonances in hot nuclei. Then we turn to the temporal evolution of the unstable systems, discussing 1rst how the dynamics changes its character from being initially linear towards being chaotic and then considering the growth of initially agitated instabilities within the framework of one-body dynamics. We review especially the body of work relating to the Boltzmann–Langevin model, in which the stochastic part of the residual two-body collisions provides a well-de1ned noise that may agitate the collective modes. We seek to assess the utility of various approximate treatments, including brownian one-body dynamics, and discuss the many possible re1nements of the basic treatment. After these primarily formal or idealized studies, we turn to the applications to nuclear multifragmentation and review the various investigations of whether the bulk of the collision zone becomes spinodally unstable. Fragmentation studies with both many-body and stochastic one-body models are discussed and we address the emerging topic of isospin fractionation. We then make contact with experimental data which indicates that the spinodal region is being entered under suitable conditions and we discuss in particular recent results on multifragment size correlations that appear to present signals of spinodal fragmentation. It is demonstrated how various aspects of the data can be understood both qualitatively and quantitatively within the stochastic one-body framework, thus strongly suggesting that nuclear ∗

Corresponding author. E-mail address: [email protected] (J. Randrup).

c 2003 Published by Elsevier B.V. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.09.006

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spinodal fragmentation indeed occurs. We 1nally outline perspectives for further advances on the topic and make connections to current progress on related issues. c 2003 Published by Elsevier B.V. PACS: 24.10.Cn; 24.60.−k; 05.60.−k

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Spinodal instability in thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. General features of thermodynamic stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Phase coexistence and spinodal instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Uniform matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Van der Waals Duid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Spinodal instability in classical many-body systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Two-component systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Mechanical and chemical stability in asymmetric nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Two-component nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Finite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Isochore canonical ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Spinodal instability in molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Experimental evidence and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Concluding remarks about thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Onset of spinodal fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Hydrodynamical instabilities in classical Duids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Collective motion in Fermi Duids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. The unstable response of Fermi liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4. Linear response in semi-classical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5. Re1ned analysis of the linear-response treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6. Role of the damping mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.7. Evolving systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.8. Linear response in quantum approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.9. Instabilities in asymmetric nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Finite nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Thomas–Fermi dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Expanding nuclear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Diabatic eHects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Conclusions from studies with Duid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Quantal description of instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6. Concluding remarks about instabilities in 1nite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Dynamics of spinodal fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. From the linear regime towards chaotic evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Non-linear eHects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Chaos and collective motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Exploratory dynamical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Mean-1eld studies of fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

266 268 268 270 272 273 274 276 277 279 281 285 287 288 290 290 292 293 294 295 297 300 305 306 307 308 310 314 314 317 318 321 321 330 331 331 332 332 337 338

P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440 4.2.2. Nuclear Boltzmann dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Boltzmann–Langevin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Basic features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Linearization of collective stochastic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. Lattice simulations of Boltzmann–Langevin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. Re1nements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Approximate Boltzmann–Langevin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Reliability of one-body treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Brownian one-body dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Concluding remarks about spinodal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Applications to nuclear fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Is the spinodal region reached in the calculations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Entering the spinodal zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Role of heating and compression in semi-classical expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3. Expansion and dissipation in TDHF simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4. Role of the Ductuations on the expansion dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5. Investigations with many-body approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Fragmentation studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Fragmentation with molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. First stochastic one-body simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3. BOB simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Isospin dependence of spinodal fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Fragmentation of dilute isobars with A = 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Confrontation with experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Source characteristics: is the spinodal region reached? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Radial collective Dow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2. Correlation between particles and fragments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3. Temperature of the emitting source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4. Negative speci1c heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Comparison with the INDRA data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Fragment velocity correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Partition correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Seyler–Blanchard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1. Statistical weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2. Hot nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.3. Multifragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3. Landau parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1. Density ripples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4. Collective modes in unstable nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Boltzmann–Langevin transport treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1. Expansion around the mean trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2. Lattice simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3. Simpli1ed Boltzmann–Langevin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.1. Quality of the simpli1ed model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.2. Minimal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4. Memory eHects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 339 341 342 343 347 350 352 354 355 356 357 357 357 360 362 364 366 369 370 371 373 376 377 382 382 382 383 384 385 385 389 391 392 394 396 397 397 398 399 400 401 401 402 403 404 407 408 411 412 412 414

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B.5. Relativistic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. Analytical approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1. Fermi surface moments of higher degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2. Angular averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.1. Legendre expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3. Overlap matrix and dual basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4. Source terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D. Expanding bulk matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1. Comoving variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2. Linear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix E. Spinodal fragmentation in FMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1. Static properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2. Early fragmentation dynamics in dilute systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2.1. AMD framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2.2. FMD framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.3. Phase transitions in 1nite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.4. Dynamical evolution of excited and dilute 1nite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.5. Further developments of molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

417 418 418 420 421 422 423 424 424 425 427 427 428 429 430 431 431 432 433 434

1. Introduction Spinodal decomposition is intimately related to 1rst-order phase transitions and the mechanism plays an important role in many areas of both pure and applied physics, including the inDationary stage of the early universe, the hadronization of a quark–gluon plasma, and binary Duids or solids (such as alloys or glasses). Spinodal decomposition may also play a role in nuclear physics as a mechanism for multifragmentation, the breakup of an agitated nuclear system (usually produced in a collision between two heavy nuclei) into several massive fragments. Nuclear multifragmentation has been studied extensively over the past two decades. A review of the topic was given about 10 years ago [1] and a more recent impression of this vigorous and diverse 1eld can be gained from Ref. [2], for example. The phenomenon oHers the prospect of providing unique experimental information about the equation of state as well as other equilibrium and non-equilibrium properties of nuclear matter far from its ordinary state [3,4]. Yet, the basic physics underlying the multifragmentation process has proven to be rather elusive, as a variety of models based on diHerent mechanisms have been able to reproduce many aspects of the data to a comparable degree. Recently an overview of the various microscopic approaches to fragmentation of nuclei and phase transitions in nuclear matter was presented in Ref. [5], while the occurrence of critical phenomena in nuclear fragmentation was reviewed in Ref. [6]. A few recent perspectives on the topic can be found in Refs. [7–9]. However, an accumulating amount of experimental evidence suggests that spinodal decomposition plays a central role for multifragmentation, in a suitable range of bombarding energies for which an expanding composite system is formed. The bulk of the matter might then acquire densities and temperatures that correspond to spinodal instability and clusterization would ensue as the system seeks to separate into the corresponding coexisting liquid and gas phases. It therefore appears timely to review our understanding of spinodal nuclear fragmentation.

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Before embarking on this task, we wish to stress that spinodal decomposition is but one particular mechanism for multifragmentation and that many alternative scenarios have been advocated. While it is not within the scope of the present review to discuss those, we do wish to mention a few alternative scenarios. For example, one may treat multifragment production as a generalization of either sequential fragment emission from a compound nucleus [10] or the transition-state description of binary 1ssion [11]. Such scenarios assume that multifragmentation is a fully equilibrated process and there is thus no need for evoking any instabilities. Alternatively, it has been suggested that multifragmentation happens on a very short time scale at either high or low density. The former scenario receives some support from molecular-dynamics simulations in which it is possible to recognize preformed fragments already during the early stage of high compression, far outside the liquid–gas coexistence region [12]. The low-density scenario is visualized as a rapid transition to an assembly of massive clusters in statistical equilibrium [13], a picture that is supported by the fact that many features of the nuclear multifragmentation data can be well reproduced by purely statistical event generators, such as FREESCO [14–16], MMMC [17,18], and SMM [19]. In fact, most observed features of nuclear multifragmentation display a large degree of equilibration. For example, the fragment size distribution can to a large degree be understood in a rather universal manner as a result of a percolation process [20,21] as well as a manifestation of Fisher condensation [21,22]. Obviously, to the degree that the quantities measured exhibit such universal features, they cannot provide information about the speci1c dynamical process underlying the particular fragmentation phenomenon. Therefore, as we shall keep in mind throughout this review, it is of key importance to identify observable signals that are unique to a speci1c fragmentation mechanism, such as spinodal decomposition. The presentation is organized as follows. Within a general thermodynamic framework, we discuss in Section 2 how spinodal instability is intimately related to the occurrence of a 1rst-order phase transition, as signalled by a convex anomaly in the thermodynamic potential. The onset of the associated spinodal decomposition is then discussed in Section 3, where the early stage of the phase separation is elucidated by analyzing the linear response of matter within the phase region of spinodal instability. The collective dynamics may then be described by simple feed-back equations and the associated dispersion relation identi1es the modes that are ampli1ed most rapidly and which therefore will tend to become dominant in the course of time. The theoretical tools for studying the further increasingly complex evolution are discussed in Section 4 and we show how the key physics is naturally contained within the Boltzmann-Langevin treatment, in which the stochastic outcome of individual two-body collisions between the constituent nucleons are propagated and ampli1ed by the self-consistent eHective one-body 1eld. Subsequently, in Section 5, we review the applications of the various dynamical treatments to nuclear fragmentation. These applications suggest that the nuclear collision zone indeed enters the spinodal phase region, under suitable conditions of bombarding energy and impact parameter. Finally, we address the confrontation of the theoretical expectations with the relevant experimental data. Especially important is the INDRA observation of a small but signi1cant non-statistical component consisting of multifragmentation events in which the intermediate-mass fragments have very similar charges [23]. We conclude by brieDy sketching the perspectives for further work on this evolving topic. Various useful and instructive details have been relegated to the appendices in order to facilitate the Dow of the presentation.

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2. Spinodal instability in thermodynamics Spinodal instabilities are intimately related to phase equilibria and phase transitions. Although it consists of unstable states, the spinodal region of the phase diagram can be addressed by standard thermodynamics, as this 1rst section illustrates. After 1rst recalling the thermodynamical aspects of the instability problem, we discuss the key issues in nuclear spinodal decomposition, 1rst in two-component matter and then in 1nite nuclear systems. 2.1. General features of thermodynamic stability The usual discussion of phase transitions is carried out in the thermodynamical limit of bulk matter, in which the extensive variables are additive. This limit is well de1ned for saturating forces, such as the strong interaction which has a 1nite range. (The limit is also sensical when the Coulomb repulsion between protons is included, provided that a suitable counterbalancing negative charge density is added as well, as is done in studies of neutron stars.) We give here a brief presentation of the key thermodynamical features and refer the reader to the standard textbooks for more thorough discussions (see, for example, Ref. [24]). Consider now such a thermodynamical system and let it be composed of a number of subsystems {i}, each of which by itself is in the thermodynamic limit. Let each such subsystem be characterized by a set of extensive variables, Xi = {Xi‘ }, where ‘ labels the speci1c attributes (such as the energy E, the particle number N , or the volume V ). The values of the corresponding extensive variables for the combined system are then given by X = i Xi . Let us 1rst consider the case when the statistical ensemble is characterized by a de1nite value of X . (This is the standard microcanonical situation occurring when the combined system is isolated from its surroundings so that {X ‘ } are constants of motion.) The relative probability P([X1 ; X2 ; : : : ]) that the system exhibits a speci1c partition {Xi } is then given by the product of the respective phase-space volumes Wi , each of which expresses the number of elementary states accessible in the subsystem i for the speci1ed value of Xi . Accordingly, the corresponding total entropy, S([X1 ; X2 ; : : : ]), is given as the sum of the individual subsystem entropies, S([X1 ; X2 ; : : : ]) = Si (Xi ) ; (2.1) i

where Si = ln Wi is the entropy of the subsystem i. (We employ units in which the Boltzmann constant k is unity.) The stable thermodynamic equilibria can thus be determined by maximizing the entropy. The 1rst requirement for having a local maximum at Xi = XO i is that the entropy be stationary, i.e. the variation of the entropy must vanish, 9Si : 0= Si (Xi = XO i ) = Xi‘ = i‘ Xi‘ ; (2.2) ‘ 9X i Xi =XO i i i‘ i‘ =9Xi‘ which are intensive. Since the where we have introduced the conjugate variables i‘ ≡ 9Si : variations preserve the total value of X , we must have X ‘ = i Xi‘ = 0 for each of the attributes ‘, and then Eq. (2.2) can only be satis1ed if all the corresponding intensive variables are equal,

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i‘ = i‘ . For example, for a thermodynamic system with given energy E, particle number N , and volume V , the intensive variables are iE ≡

9Si 1 = ; 9Ei Ti

iN ≡

9Si i =− ; 9Ni Ti

iV ≡

9Si Pi = ; 9Vi Ti

(2.3)

so all the subsystems must have the same values of the temperature T , the chemical potential , and the pressure P. It is often more convenient to employ intensive variables for the characterization of a thermodynamic ensemble. (This is especially true when the system is not entirely isolated so some of its attributes Ductuate; for example, a system in contact with a heat reservoir has a Ductuating energy E whose average value is determined by the associated conjugate variable E = 1=T , where T is the reservoir temperature.) The thermodynamic equilibria can then be obtained by maximizing a correspondingly constrained entropy S Si (Xi ) → Si (Xi‘ ; ‘ ) ≡ Si (Xi ) − ‘ Xi‘ ; (2.4) ‘

where the attributes ‘ are those speci1ed by their intensive variables ‘ , while ‘ refer to all the others. The stationarity condition (2.2) then becomes S = 0. (It may be noted that the constrained entropy, being the logarithm of the ensemble partition sum, is simply the negative of the corresponding thermodynamic potential divided by the temperature.) The stationary points identi1ed by the above 1rst-order variation of the entropy function S([X1 ; X2 ; : : : ]) encompass both maxima and minima of the partitioning probability function P([X1 ; X2 ; : : : ]). In order to ensure that a given extremum con1guration be (locally) stable, one must insist that the entropy have a maximum, which in turn requires the second-order variation to be negative de1nite, 92 S : 2 i O 0¿ Si (Xi = X i ) = Xi‘ Xi‘ : (2.5) ‘ ‘ 9Xi 9Xi Xi =XO i i i“

This condition is met if and only if the curvature matrix 92 Si =9Xi‘ 9Xi‘ has only negative eigenvalues. We note that when all the normal curvatures are negative, then all the double derivatives (which are the entropy curvatures in the X ‘ directions) are negative as well, 92 S=9X ‘2 ¡ 0. This feature imposes constraints on the equations of state that relate the intensive and extensive variables since it implies 9‘ =9X ‘ ¡ 0 (where we recall the de1nition ‘ ≡ 9S=9X ‘ ). This general feature is of particular interest for the heat capacity C, the compressibility , and the chemical susceptibility ,

EE : 0 ¿

9E 1 9T 9E =− 2 ⇒C≡ ¿0 ; 9E T 9E 9T

(2.6)

VV : 0 ¿

1 9P 9V 9P = ⇒ −1 ≡ −V ¿0 ; 9V T 9V 9V

(2.7)

NN : 0 ¿

9N 9 1 9 =− ⇒ −1 ≡ ¿0 : 9N T 9N 9N

(2.8)

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These relations expresses thermal, mechanical, and chemical stability, respectively, all of which must be present in thermodynamic equilibrium. If we employ the density = N=V as a state variable, we can see that in fact the last two conditions are intimately related. (In asymmetric nuclear matter there are two chemical potentials present and the situation is correspondingly more complex, see Section 2.3.) It is important to recognize that the above conditions on the diagonal curvature terms are necessary but not suPcient stability conditions, since the principal directions of the curvature matrix need not be aligned with the adopted thermodynamic variables. Therefore, the condition for stability involves the entire curvature matrix, including oH-diagonal terms. In particular, there may well be instabilities present even if all of the double derivatives are negative. Generally, then, when the curvature matrix associated with a given stationary con1guration is not negative de1nite the given stationary con1guration is unstable. In that situation, there exist variations of the attributes [X1 ; X2 ; : : : ] away from their stationary values [XO 1 ; XO 2 ; : : : ] that cause the (constrained) entropy to increase (or the corresponding thermodynamic potential to decrease) and the system will seek to escape from the stationary con1guration, rather than to return.

2.1.1. Phase coexistence and spinodal instability Of particular relevance to the present review is the possible occurrence of phase coexistence and spinodal instability, two intimately related phenomena that arise when the entropy function S(X ) is locally convex. To bring out the key features as clearly as possible, we consider in the following only one single extensive attribute X . The convexity anomaly then occurs when 92 S=9X 2 ¿ 0 and it implies that the conjugate intensive thermodynamical variable, (X ) = 9S=9X , is globally non-monotonic. (For example, if X is the volume V then is related to the pressure P and convexity occurs if the compressibility is negative.) The situation, illustrated generally in Fig. 2.1, is further discussed below. Consider now a system for which the entropy function S(X ) exhibits a convex region, as illustrated by the top curve in Fig. 2.1, with the curvature anomaly, 92 S=9X 2 ¿ 0, occurring for Xmin ¡ X ¡ Xmax . Due to the local convexity, there exists a common tangent to S(X ) between X1 (below Xmin ) and X2 (above Xmax ). Since thus (X1 ) = (X2 ), the two phases can coexist in thermodynamic equilibrium. A linear interpolation along the common tangent then yields a globally ˜ ), lying above S(X ) for X1 ¡ X ¡ X2 , thus signaling that the system concave envelope function, S(X is unstable against separation into the phases 1 and 2 throughout that region. To understand this central feature in more detail, it is convenient to eliminate the trivial size dependence of the extensive variables. Towards this end, we may generally introduce reduced extensive variables, x ≡ X =N , where N is the number of particles. In particular, the entropy per particle is =S=N . The common division by N does not aHect the conjugate variables since ≡ 9S=9X =9=9x. Let us now prepare a uniform thermodynamic system with a value of x = X=N lying within the interval (x1 ; x2 ). It has the total entropy S = N(x). Alternatively, we may split this system into two thermodynamic subsystems for which the reduced extensive variables have the values x1 and x2 associated with the two coexisting stable phases. (As noted above, the corresponding intensive variables match, (x1 )=(x2 ), since the local tangent to S(X ) is common.) Let the respective particle numbers be N1 and N2 , respectively, with N1 + N2 = N . The resulting total entropy of the combined

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2

S(X) 1 convex Maximum slope

Minimum slope

spinodal

λ(Χ)

λ0

X1

X min

X max X 2

X

phase coexistence

Fig. 2.1. Phase coexistence. The 1gure illustrates the relationship between entropy convexity, spinodal instability and phase coexistence. The entropy function S(X ) (top) is convex between the inDection points at Xmin and Xmax , at which the conjugate variable (X ) = 9S=9X (bottom) has a local minimum or maximum, respectively. Through this region of spinodal instability, (X ) increases so the system is mechanically unstable. The region of phase coexistence extends from X1 to X2 , where the phases 1 and 2 are determined by the common tangent to S(X ) (which exists due to the local convexity); obviously, X1 ¡ Xmin and X2 ¿ Xmax . Since the two slopes are the same, these two stable phases can coexist in thermodynamic equilibrium. Entropy can then be gained by separating the uniform system into these two phases, since ˜ ), moves along the common tangent as X is increased from X1 to the entropy of the corresponding mixed system, S(X ˜ ) maintains a constant value 0 which can be determined by the familiar Maxwell construction (2.11) X2 . Meanwhile (X requiring that the two hatched areas be equal.

mixed-phase system would then be N1 N2 S(X1 ) + S(X2 ) ¿ S : S˜ = N1 (x1 ) + N2 (x2 ) = N N

(2.9)

This key feature is readily visualized by noting that S˜ lies on the common tangent which is above S (see Fig. 2.1). Thus, the separation of the initial uniform state into the two coexisting phases maximizes the total entropy and such a mixed-phase con1guration is therefore thermodynamically favored. The above is true for any value of X between X1 and X2 . If the value of X is increased steadily through this interval, the relative preponderance of the two phases evolves correspondingly, with N1 decreasing from N to zero as N2 increases from zero to N . Their ratio follows immediately from the additivity of the extensive state variable, X1 + X2 = X , N 1 x2 − x = : N2 x − x 1

(2.10)

The corresponding value of remains constant through this gradual transformation, as is evident ˜ , from the fact that the system moves along the common tangent. This constant value, 0 = d S=dX

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can also be determined directly from the function (X ) by the familiar Maxwell construction, X2 : dX ((X ) − 0 ) = 0 ; (2.11) X1

which amounts to demanding that the area enclosed by (X ) below 0 be equal to the area enclosed above (the hatched areas on Fig. 2.1). Because of the convexity in the entropy function S(X ), the function (X ) is invariably nonmonotonic through the phase coexistence region, decreasing 1rst to its (local) minimum at xmin , then rising to its (local) maximum at xmax , and 1nally decreasing again to its starting value. These two extrema delineate the spinodal region, within which (X ) is thus a steadily rising function, as the (local) slope of (x) changes from its minimum to its maximum value. Thus, within the spinodal region, the system responds to an increase in volume by increasing its pressure, thus accelerating the disturbance. Equivalently, a local variation of the density will become ampli1ed and the state may then spontaneously undergo a phase separation. It is this anomalous behavior that characterizes the spinodal region and, as the above discussion brings out, spinodal instability occurs whenever the entropy function exhibits local convexity. In contrast to the local instability of the states inside the spinodal region, the remaining states within the coexistence region are locally stable (since the local curvature of S(X ) is negative). These states are thus globally metastable and are protected by a 1nite barrier against spontaneous phase separation. These key features are also evident from the basic diHerence between concavity and convexity: When the system is situated within the convex region, entropy can be gained by transforming it into a mixture of two neighboring systems (on opposite sides). Thus, it is possible to transform the system in a continuous manner into a mixture of the two coexisting phases 1 and 2 with the entropy increasing steadily until it reaches its maximum possible value S˜ when the transformation is complete and the system lies on the common tangent. On the other hand, when the system is situated in a concave region then any such local transformation into a mixture of neighboring states would result in an entropy decrease. Consequently, the transformation of the system into the mixture of the coexisting phases must necessarily involve a temporary reduction of the entropy and can thus not occur spontaneously. 2.2. Uniform matter The previous discussion was not restricted to any speci1c case but considered the stability of general thermodynamic states by analyzing the curvature of the relevant thermodynamical potential. We shall now focus on the instabilities of uniform nuclear matter. Systems can be considered as being uniform with speci1c values of the thermodynamic quantities, such as energy density, particle density, and isospin density, if they are devoid of macroscopic correlations. (Throughout this review, we use the term “isospin” rather loosely to denote the neutron excess degree of freedom which is simply related to the third component of the isospin.) For macroscopic systems, this can be experimentally achieved by quenching the system suPciently deep into the coexistence region of the phase space through a rapid variation of a state variable (as in the rapid cooling of binary alloys) or through the tuning of external control parameters. (In nuclear physics such experiments can be performed only on small systems and the associated complications will be addressed in Section 2.4.) In the

Temperature T (Tc )

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1.0

0.5

Phase coexistence Spinodal boundary

Pressure P (Pc )

0.0

01

23

1

0

−1

−2

0

1

2

3

Density ρ (ρc)

Fig. 2.2. Van der Waals equation of state. Bottom panel: The pressure P (in units of the critical pressure) as a function of the density (in units of the critical density) for 14 equidistant temperatures T =0–1.3 (in units of the critical temperature) for a Van der Waals Duid as given in Eq. (2.12). The border of the spinodal zone is delineated by the long-dashed curve, which passes through the maxima and minima of P(); uniform matter situated below this curve is mechanically unstable. Also indicated is the phase coexistence line (short dashes) as obtained with the Maxwell construction. The critical point is indicated by the solid dot. Top panel: The corresponding (; T ) phase-plane representation of the spinodal boundary and the phase coexistence line.

present section, the general properties of uniform nuclear matter will be elucidated with the aid of various simple models. It is an important simplifying feature that mean-1eld approaches are excellent tools for describing the stability of such systems and that the observed spinodal instability is not an artefact of this approximation but a general property. 2.2.1. Van der Waals @uid Perhaps the best known example of a system exhibiting spinodal instabilities is the Van der Waals Duid and it is well suited to illustrate the principal features. Since the state variables are (T ; N; V ), the associated constrained entropy (2.4) is S = S − E=T and the appropriate thermodynamic potential is then the free energy F(T ; N; V ) = −TS = EO − TS = Vf(T; ). The key quantity is the equation of state describing how the pressure in macroscopically uniform matter depends on the state variables, 2 N 9F NT T −a − a2 ; P(T ; N; V ) = − = = (2.12) 9V TN V − bN V 1 − b where a is the strength of the mean-1eld attraction, while b governs the short-range repulsion. It is depicted in Fig. 2.2 (following the most common form of presentation, we use the density = N=V as the abscissa rather than the volume V employed above for pedagogical reasons). It is seen that when the temperature T exceeds a certain critical value, Tc = 8a=27b, the pressure rises steadily with

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the density and we are faced with a normal thermodynamical situation in which there is precisely one thermodynamic state for each speci1ed P and T . By contrast, for subcritical temperatures (0 ¡ T ¡ Tc ) the function P() has a maximum followed by a minimum and the situation is thus similar to that discussed generally above (see Fig. 2.1). Then the speci1cation of a pressure P below the critical value Pc , determines three possible densities. The smallest density, G , lies in the gas phase below the spinodal region, while the highest density, L , lies in the liquid phase above the spinodal region. The coexisting phase points can be determined by a Maxwell construction, dV (P −P0 )=0, and they are shown in Fig. 2.2. While these thermodynamic states have positive compressibilities, the intermediate density is situated inside the spinodal zone and it has thus a negative compressibility and is, accordingly, mechanically unstable. The region of spinodal instability is determined by the extrema of P() and this boundary is also shown in the 1gure. As a general feature, the boundaries for phase coexistence and spinodal instability both start at the origin, = 0, T = 0, P = 0. Equally generally, the coexistence region extends up to the ground-state density 0 (which is equal to the maximum density of 3c for the Van-der-Waals Duid), while spinodal instability usually disappears well before that density is reached. However, for the somewhat pathological case of the Van-der-Waals Duid spinodal instability remains present at all densities. 2.2.2. Nuclear matter Since nuclear forces have an attractive long-range part and a repulsive hard core, the nuclear system resembles the Van der Waals Duid. The thermodynamics of the (in1nite) nuclear medium can be studied by omitting the Coulomb repulsion between the protons, leading to the concept of nuclear matter consisting of neutrons and uncharged protons (see, for example, Ref. [25]). Such idealized systems are useful for understanding the bulk properties of large nuclei. For the understanding of the structure of neutron star crusts it is important to include the Coulomb force which can be done by simultaneously introducing a compensating smooth negative charge density resulting from the electrons present. A convenient framework for thermodynamic studies of nuclear matter is provided by mean-1eld treatments employing phenomenological eHective interactions (which may, in principle, be derived through BrUuckner–Hartree–Fock regularization of the repulsive hard core [26]). The oldest and perhaps most commonly used eHective interactions employed in nuclear physics are of the Skyrme type [27–29]. Such interactions are particularly convenient for studies of the equation of state [30,31]. In its simplest form, the isoscalar part of the eHective two-body Skyrme interaction is a function of the local density (r), r1 + r2 1 V12 = (r1 − r2 ) t0 + 6 t3 ; (2.13) 2 where the parameters t0 ; t3 ; are 1tted to nuclear properties at zero temperature. The 1rst term represents the long-range attraction (t0 ¡ 0), while the second term provides a short-range repulsion (t3 ¿ 0), thus ensuring saturation at a certain density 0 . The associated curvature of the energy, i.e. the nuclear compressibility, is governed by the parameter . The resulting equation of state is then akin to the Van der Waals form shown in Fig. 2.2 and a speci1c example is shown in Fig. 2.3 (left).

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Fig. 2.3. Nuclear matter isotherms. Typical equations of state for nuclear matter obtained within a mean-1eld treatment using either a Hartree–Fock approach with an eHective Skyrme force (left) (from Ref. [32]) or the Thomas–Fermi approximation with a modi1ed Seyler–Blanchard force (right) (adapted from Ref. [35]). The pressure P is plotted as a function of the nucleon density for temperatures in the range T = 0–20 MeV. The spinodal boundary (dashed) and the coexistence curve (solid) are indicated.

The above prototype interaction (2.13) has two notable shortcomings, it has zero range and there is no medium eHect on the nucleon mass, and numerous re1nements have been made. A simple interaction that remedies both of these problems and leads to a good overall description of many macroscopic nuclear properties (such as sizes, masses, and barriers) is the Seyler and Blanchard interaction [33] as modi1ed by Myers and Swiatecki [34] (see Appendix A for details),

2 p12 (r1 ) 2=3 e−r12 =a s (r2 ) 2=3 1 − O 2 − V12 = −C : (2.14) + r12 =a b s s s In addition to the original Seyler–Blanchard parameters a, b, and C, the parameter controls the balance between momentum and density dependence and thus allows an independent speci1cation of the eHective mass. The corresponding equation of state [35] is shown on the right in Fig. 2.3. The similarity of the two equations of state displayed in Fig. 2.3 is typical of this general type of approach. Thus, for cold symmetric nuclear matter, the spinodal region extends up to about 23 s , while the critical density is roughly c ≈ 13 s , with the corresponding critical temperature being Tc ≈ 14–18 MeV. However, it should be noted that the mean-1eld approaches have certain inherent shortcomings. One is the fact that the associated Fermi-gas level density g(j) signi1cantly underestimates the level density of actual nuclear systems, thus yielding a correspondingly erroneous statistical weight; the use of a more realistic g(j) will lead to a lower critical temperature [36]. Also notable is the inadequacy of mean-1eld treatments at the critical point (and for other second-order phase transitions), as reDected in incorrect values of the critical exponents. Fortunately, this local failure has little bearing on the rest of the phase diagram and the treatments may then be quite suitable for the study of 1rst-order phase transitions, such as those underlying the spinodal instability. This will be illustrated below with exact molecular dynamics.

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2.2.3. Spinodal instability in classical many-body systems The above illustrations of spinodal phase separation in uniform matter have been made within the framework of mean-1eld treatments. In order to bring out the important fact that the spinodal phenomenon is not limited to this approximative framework, we recall here some relevant studies for classical many-particle systems for which exact numerical calculations can be performed. Such treatments consider A particles within a box of volume V (for which periodic boundary conditions are usually imposed to eliminate surface eHects). In1nite matter is then well approximated when the dimensions of the box are large in comparison with the interaction range. For a given model Hamiltonian function H ({ri }; {pi }), the temporal evolution of the microscopic state of the system, ({ri (t)}; {pi (t)}), is governed by the associated Hamilton equations of motion, r˙i =

9H 9pi

p˙i = −

9H ; 9ri

i = 1; : : : ; A :

(2.15)

For any given dynamical state ({ri }; {pi }), it is then possible to extract an eHective temperature [37] 9H 1 1 pi · = pi · r˙i : (2.16) T= 3A i 9pi 3A i Furthermore, a convenient expression for the stress tensor T, which is directly related to the momentum transport, can be obtained by replacing the time average in the Virial Theorem by an instantaneous average over all the particles in the system [37], 9H 9H pi − ri = (pi r˙i − Fi ri ) : (2.17) 3TV = 9pi 9ri i i The eHective pressure is then given by 1 Fij rij ; P = trc T = T + A i¡j

(2.18)

where = A=V is the density and Fij = −9H=9rij is the radial force between the particles i and j, with their separation being rij = |ri − rj |. These relations, which also hold for momentum-dependent interactions [37], make it possible to extract the location of the system in the TP phase diagram. (The accuracy can be improved inde1nitely by averaging the result over an ever larger number of systems that have been initialized by a suitable statistical sampling, using for example the Metropolis method.) A simple way to numerically investigate spinodal instability in such many-body systems is to 1rst prepare the system at a density and temperature corresponding to a phase point (; T ) outside the spinodal region, where the system is stable and thus prefers to have a macroscopically uniform density = A=V . Subsequently, the system is brought to a phase point of interest inside the unstable region by a sudden cooling and/or expansion which can be readily accomplished by rescaling the velocities and/or positions of the individual constituent particles. (Such quenching can also be carried out experimentally and is in fact routinely performed in various areas of physics, as well as in industry; a well-known example is the rapid supercooling of binary alloys—for an early review of phase separation experiments, see Ref. [38].)

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277

Fig. 2.4. Equation of state for 40 Ar. Isotherms of the pressure P versus the speci1c volume v ≡ V=A = 1= for a system of 20 (charged) protons and 20 neutrons which are constrained to be approximately uniformly distributed within a sphere of volume V , as obtained by means of quasi-classical molecular dynamics with a Pauli potential. (From Ref. [37].)

An early example in nuclear physics was the study by Dorso et al. [37] who considered approximately spherical con1gurations by means of a two-body interaction with three terms: the standard Coulomb repulsion between protons, a nuclear interaction of Lennard–Jones form, and a repulsive “Pauli” potential which depends on both the spatial separation and the diHerence in momentum and was intended to emulate the eHect of the Pauli exclusion between fermions. Fig. 2.4 shows the resulting equation of state, displayed in the form of the general illustration in Fig. 2.1: the pressure plotted against the speci1c volume v for a given temperature T . It is evident that for temperatures below ≈ 3 MeV there is a density region where the compressibility is negative, −1 = (−v9P=9v)−1 , thus signaling the spinodal instability against phase separation. A similar study was undertaken later on by Jacquot et al. [39] who employed a Lennard–Jones potential with a hard core for classical particles in a box with periodic boundary conditions. This force was also chosen so as to approximately reproduce the key nuclear features (size and binding). The resulting phase diagram is shown in Fig. 2.5, with the spinodal boundary delineated. It was demonstrated that the exact dynamics and thermodynamics are well reproduced by mean 1eld approaches using the local density approximation with a Skyrme-type energy functional. The above two examples serve to bring out the fact that spinodal instability is a general phenomenon in many-body systems, rather than merely an artifact of the mean-1eld approximation. Thus we may expect that the occurrence of a convex entropy function signals a preference for the system to undergo phase separation and that the mechanically unstable spinodal region is characterized by a negative compressibility modulus. 2.3. Two-component systems The preceding discussion has been phrased as if the systems under consideration were composed of just a single particle species, nucleons. However, the nuclear medium contains two nucleonic species,

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Fig. 2.5. Phase diagram from molecular dynamics. Phase diagram obtained in Ref. [39] for a system of classical particles having the interaction shown in the insert. The “data points” indicate the onset of instability as obtained by numerical simulation of the microscopic equations of motion. For low densities there is good agreement with the mean 1eld spinodal boundary (the drawn curve), while the near vertical behavior at higher densities reDects the presence of a solid phase in a classical particle system, an artefact that is unimportant for the present discussion.

neutrons and protons, and the analysis therefore needs to be correspondingly two-dimensional. Instructive discussions of this situation have been made in Refs. [40–46]. In the two-component system, the particle number is represented by two observables that commute ˆ Alternatively, with the Hamiltonian Hˆ , for example the baryon number Aˆ and the charge number Z. ˆ ˆ ˆ ˆ ˆ we may use the neutron and proton numbers N n = A − Z and N p = Z, thus preserving the symmetry between the two species. For systems characterized by temperature and particle number(s), it is convenient to consider the (Helmholtz) free energy, ˆ

F(T; V; Np ; Nn ) = −T log Tr[(Nˆ p − Np )(Nˆ n − Nn )e−H =T ] Vf(T; p ; n ) :

(2.19)

The last expression pertains to the thermodynamical limit where F is proportional to the volume V , so we may consider the free energy density f = F=V and use only the proton and neutrons densities as state variables, n =Nn =V and p =Np =V , in addition to the temperature T . The chemical potentials then follow, n ≡

9F 9f = ; 9Nn 9n

p ≡

9F 9f = : 9Np 9p

Furthermore the stability condition (2.5) requires that the curvature matrix, 2 9 f=9n 9n 92 f=9p 9n 9n =9n 9n =9p C= = ; 9p =9n 9p =9p 92 f=9n 9p 92 f=9p 9p

(2.20)

(2.21)

have only positive eigenvalues. For a 2 × 2 matrix this requirement amounts to det[C ] = c− c+ ¿ 0

and

tr[C ] = c− + c+ ¿ 0 ;

where c− is the smallest eigenvalue and c+ is the largest one.

(2.22)

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279

Thus, if it is possible to calculate (or estimate) the free energy for uniform con1gurations (for example by use of the mean-1eld approximation or by quenching, as discussed above), then the above criterion (2.22) makes it possible to determine if a given uniform system is unstable against phase separation and, if so, the global equilibrium can be obtained by a Maxwell-type construction as that combination of the stable phases that minimizes the free energy at the speci1ed values of the temperature, pressure, and chemical potentials. The system is stable as long as both eigenvalues are positive and the spinodal boundary is reached when det[C ] turns from positive to negative. There is then one negative eigenvalue, c− ¡ 0, and the phase separation will initially proceed in the corresponding principal direction. Further into the spinodal region, the sum of eigenvalues may become negative, tr[C ] ¡ 0, as the magnitude of the negative eigenvalue exceeds the positive eigenvalue, |c− | ¿ c+ . Finally, both eigenvalues may become negative. However, we need not discuss this case since it is not expected to occur in dilute nuclear matter. 2.3.1. Mechanical and chemical stability in asymmetric nuclear matter In asymmetric matter, the variation of the pressure is given by P = n n + p p . However, it is common to replace the individual densities n and p by the total density, = n + p , and the proton concentration, y = Z=A = p =(n + p ). The usual discussion then proposes two corresponding criteria, usually referred to as “mechanical” and “chemical” instability conditions, 9P ¡0 ; (2.23) “Mechanical” : 9 T;y “Chemical” :

9p 9y

T;P

¡0 :

(2.24)

However, this formulation of the instability problem is misleading because it might be taken to suggest that there are two independent instabilities: a mechanical instability conserving the proton concentration y and a chemical instability occurring at constant density . Such an interpretation would not be correct. Not only do the chemical instabilities involve changes in the density and the mechanical instabilities produce changes in the proton concentration, but the two situations are in fact associated with one single instability. The above “mechanical” and “chemical” instability conditions do not provide any directional information and the best way to understand the two-dimensional instability features is to recall the basic stability criterion, that the curvature matrix of the free energy in the n –p plane, C , be positive de1nite. Thus, as pointed out above, one needs to consider its eigenvalues and eigenvectors. − In the typical case, there is one negative eigenvalue, c− , and the associated eigenvector, (− p ; n ), − gives the direction of the instability. If − p : n = p : n then the instability preserves the ratio − between protons and neutrons so it is a purely mechanical disturbance. Conversely, if − p = −n then the total density is constant and we are facing a pure chemical instability. Except for these two very special situations, a disturbance along the unstable eigendirection conserves neither nor y, but has a mixed character with both chemical and mechanical contents. Nevertheless, if − p and − n have the same sign then protons and neutrons are aHected similarly and the disturbance thus

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has a predominantly isoscalar character, while it may be characterized as isovector-like if − p and − have opposite signs [47]. n It is also noteworthy that the two derivatives in the instability conditions (2.23) and (2.24) are intimately related [48], 9p 9P = (1 − y)2 |C | : (2.25) 9y T;p 9 T;y Since the determinant |C | turns negative at the spinodal border, when the smallest eigenvalue passes through zero, the “chemical” instability condition (2.24) is always met 1rst, irrespective of the actual direction of associated eigenvector. Furthermore, (9P=9)T; y becomes negative only when the instability is strong enough to aHect the curvature in the direction of constant y. When this happens (if there is only one negative eigenvalue, as is typically the case), then the above relation shows that (9p =9y)T; p must again become positive. Only in the case when both eigenvalues of C are negative (which never happens in dilute nuclear matter) will both conditions (2.23) and (2.24) be violated. These features are illustrated below for a simple example of two independent Duids. 2.3.1.1. Instability analysis for two independent @uids The instabilities of a two-component system can be elucidated by the simple case of two independent (i.e. non-interacting) Duids. The curvature matrix is then diagonal, 0 9 p p ; (2.26) C= 0 9 n n and so it is easy to derive the following expressions, 9Pp 9P 9Pn =y + (1 − y) ; 9 Ty 9p T 9n T 9p 9Pp 9P 9Pn y= ; 9 Ty 9y TP 9p T 9n T

(2.27) (2.28)

where the individual pressures are Pi , with i = p; n, and the spinodal region for the component i is where 9i Pi = i 9i i is negative. Consider now the situation when one component enters its spinodal region. One of the factors on the right in Eq. (2.28) has then turned negative but, since the other is still positive, it is evident from Eq. (2.27) that the total compressibility is still positive. Indeed, since Eq. (2.27) can be rewritten in terms of the compressibilities, 1 9P 1 1 ≡ = + ; (2.29) 9 Ty p n the total compressibility becomes negative only if the magnitude of the negative compressibility of the unstable component becomes smaller than the compressibility of the stable one. When this happens, the quantity (9p =9y)TP , which turned negative simultaneously with the entry of the 1rst component into the spinodal region, reverts to being positive (see Eq. (2.28)).

P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440

940

Pa

Pb

281

µn µp

Pc

Pd Liquid

920

Gas

µ [MeV]

930

Pe

Pd

910

Pc

900

Pb 890 0.0

T = 10MeV

Pa 0.1

0.2

0.3

0.4

0.5

y Fig. 2.6. Isobaric contours. Isobaric contours in the y– plane (with the speci1ed pressure decreasing from a to e, with Pb being the critical pressure), as obtained with the relativistic mean-1eld approach for nuclear matter at T = 10 MeV. For subcritical pressures (contour c), both n (dashed) and p (solid) exhibit anomalous behavior (n increases and p decreases as the proton concentration y is increased) leading eventually to backbending (contour d), signaling the presence of spinodal instability. The boundary of the coexistence zone can be determined by demanding that the chemical potential for each species be equal in the liquid and the gas phases, for given values of the common pressure P and temperature T , as indicated by the rectangle which represent a two-dimensional extension of the Maxwell construction. (Adapted from Ref. [42].)

In this example we are clearly facing a single instability, namely the mechanical instability of one component, but an analysis in terms of Eqs. (2.23) and (2.24) would suggest that we pass from a “chemical” to a “mechanical” instability, since we 1rst encounter (9p =9)TP ¡ 0 and subsequently (9P=9)Ty ¡ 0. Such an interpretation would be misleading, since the instability retains its direction along the same axis with only its magnitude changing. Thus, no special meaning should be attached to the two instability conditions (2.23) and (2.24); only the border of the instability is meaningful. Rather, to ascertain the character of an instability, one should consider the eigenvalues and the associated eigenvectors of the curvature matrix C . 2.3.2. Two-component nuclear matter After the above general discussion, we now turn to the nuclear case. As a framework for our discussion, we shall employ the results obtained with the relativistic mean 1eld by Serot et al. [42,49,50]. Other nuclear models yield qualitatively similar results, so the features are rather universal. We start by considering the relationship between the proton concentration y and the chemical potentials n and p , for given values of temperature and pressure. This is illustrated in Fig. 2.6. As noted above, when the pressure is reduced below the critical value, it is the “chemical” stability condition which is 1rst violated. This important general feature is brought out more clearly in Fig. 2.7 which shows the corresponding equation of state with the various boundary curves added. Entering from the one-phase region of the –P phase plane, one 1rst crosses the coexistence curve

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P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440 0.4

16 CP

0.2

Coe

xist

0.0

y = 0.3

T=0

l

-0.3

nica

-0.2

al mic

-0.1

enc 4 e

Che

0.1

ha Mec

p MeV/fm 3 ]

12 8

0.3

-0.4 0.00 0.02 0.04 0.06 0.08 0.10 0.12 ρ [fm-3]

Fig. 2.7. Instability boundaries. The pressure as a function of the total density in nuclear matter with a proton concentration of y = 0:3 for various temperatures T (in MeV), as obtained with the relativistic mean-1eld approach. The critical point (CP), the coexistence line, and the “mechanical” (2.23) and “chemical” (2.24) spinodal boundaries are indicated. (Adapted from Ref. [42].)

and subsequently the boundary for chemical instability determined by Eq. (2.24) (at the critical point these two coincide), while the so-called mechanical instability determined by Eq. (2.23) is encountered only deeper into the spinodal zone. However, since relation (2.25) causes 9p =9y to change sign at the same time as the compressibility, the “mechanical” instability boundary does not represent the onset of a second instability. Rather, this arti1cial boundary marks only a quantitative modi1cation of the instability properties, either an increase of the negative curvature or a rotation of the instability direction, but not a qualitative modi1cation of the instability nature. As a consequence, one should not attach a strong signi1cance to this “mechanical” spinodal line. This important point was noted already over 1fteen years ago [47] (and has been further stressed recently [48,51]), but apparently it remains unrecognized in many discussions of asymmetric nuclear matter. In order to achieve a deeper insight into the two-dimensional nature of the spinodal instability, one should determine the principal direction of the curvature matrix C in the n –p density plane, as illustrated in Fig. 2.8. Using an eHective interaction of either Skyrme or Gogny form, Margueron et al. [48] diagonalized the curvature matrix as a function of n and p . The 1gure shows a contour plot of the associated speed of sound c, where c2 = (1=18m)c− . The outer contour, corresponding to c = 0, is then spinodal boundary, while the instability grows steadily stronger as one moves inside. This kind of plot brings out very clearly that we are facing a single instability. Furthermore, the direction of the corresponding eigenvectors are indicated along the contours. While they are generally directed neither along constant density nor along constant proton concentration, they do have a predominantly mechanical (isoscalar) character. Nevertheless, the result of the “mechanical” instability criterion (2.23) bears little resemblance to the actual results. One might also note that the instabilities have a chemical character as well, since they always seek to reduce rather than enlarge isosymmetry in the dense regions. This local tendency toward the restoration of the isospin symmetry is also present in the global phase equilibrium, as we shall now illustrate. As already discussed in connection with Fig. 2.6, the coexistence curve and the conditions for phase equilibrium can be extracted from the various

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283

Fig. 2.8. Two-dimensional spinodal boundary. Spinodal instability in the n –p density plane as calculated with an eHective interaction of either Skyrme (left) or Gogny (right) form. Each contour corresponds to a 1xed value of the sound velocity c = (c− =18m)1=2 (c− is the smallest eigenvalue of the curvature matrix), the outer one being the spinodal boundary corresponding to c = 0. The direction of the corresponding eigenvectors are indicated along the contours. The result of the “mechanical” instability criterion (2.23) is also shown (dashed boundary). (From Ref. [48].)

isotherms. Thus, given the relationship between the chemical potentials and y for given temperature and pressure, the matching conditions for the chemical potentials, nL = nG and pL = pG , makes it possible to determine the respective proton concentrations of the liquid and gas phase, yG and yL . In order to discuss in some detail how the phase transformation proceeds, we consider a system with an overall proton concentration of y = 0:3 which is held at a 1xed temperature of T = 10 MeV as the pressure is gradually increased. This situation is illustrated in Fig. 2.9. We may note that generally the gas phase of a neutron-rich system has a smaller proton concentration than the corresponding liquid phase. Indeed, when the gas phase has been reached (point D in the 1gure) its proton concentration is rather small, yD ≈ 0:1. This is a typical fractionation process, akin to the distillation of alcohol from water. In nuclear matter the liquid tries to get as close as possible to equal concentrations of neutrons and protons, since symmetric matter is energetically favorable, and the gas is then left with the excess. This mechanism, 1rst noted in Ref. [14], can be an important signal of the occurrence of phase transitions. It was thoroughly investigated in a lattice gas model [52] and has been investigated experimentally through the study of isotopic ratios [53–57]. As we have seen, for a two-component system the pressure is not constant during the phase transformation. The underlying reason for this feature is that the coexisting gas and liquid phases have diHerent chemical compositions which provides an additional degree of freedom. Furthermore, the relative abundances of the phases change as well. Therefore, the calculation of the phase transformation by use of the Gibbs criteria (the generalized Maxwell construction) requires knowledge of the isotherms for a range of concentrations y. The resulting equation of state is shown in Fig. 2.10. As noted in Refs. [42,49,50], the phase transition is continuous. However, the continuity of the phase transformation as a function of the applied pressure should not be taken as evidence that the transition is of second order. In fact, any phase point inside the coexistence region is associated with

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CP

3

0.16

D

0.12

MA

T = 10MeV

C

uid

liq

p [MeV/fm ]

0.20

0.08

B

ga

s

EC

A

0.04 0.0

0.1

0.2

0.3

0.4

0.5

y

Fig. 2.9. Phase transformation in a two-component system. Illustration of the phase transformation in two-component nuclear matter, as induced by increasing the pressure at 1xed temperature, T = 10 MeV. Having a 1xed overall proton concentration (y = 0:3 in this illustration) and starting in the gas phase at zero pressure, P = 0, the system responds to an increase in P by initially moving vertically upwards, until it reaches the coexistence boundary at A. From this point on, an ever increasing fraction of the system is transformed into the liquid phase, so a second trajectory branch appears at B. The liquid has generally a higher proton concentration than the gas. As the pressure is increased further, both branches develop upwards along the respective boundaries of the coexistence region, with the liquid phase growing steadily more prominent as ever more gas is condensed into liquid. Because the condensing gas has a lower isospin fraction than the liquid, the isospin contents in the liquid is being diluted and the liquid trajectory is moving to the left. This evolution continues until the gas has been entirely exhausted and the gas branch terminates (D). At this time, the liquid has recovered the overall isospin fraction (C) and from there on the trajectory is again vertical. (Adapted from Ref. [42].)

yD C

D

0.10

3

p [MeV/fm ]

0.15

0.05

B

A

y = 0.3

0.00 -0.05 -0.10

yB T = 10MeV

0.00 0.02

0.04

0.06

0.08

0.10

0.12

-3

ρ [fm ]

Fig. 2.10. EHective equation of state. Pressure as a function of the total density in nuclear matter at T = 10 MeV and for various y: yD ≈ 0:1, y = 0:3, yB ≈ 0:45. The result of the Maxwell construction is also shown (heavy curve): when the pressure reaches the coexistence boundary A (determined by using the previous 1gure) the system becomes a mixture of coexisting gas and liquid phases having two diHerent proton concentrations. Point A and point B satisfy the Gibbs criteria, but A belongs to the isotherm with y = 0:3, while B belongs to another isotherm (having y = yB ). When all the gas has been transformed into liquid (C), the path returns to the original isotherm having y = 0:3 but the pressure has increased. The resulting eHective equation of state is thus obtained by interpolating between the appropriate isotherms having diHerent values of y. (From Ref. [42].)

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285

2

S(X) mixed

Minimum slope

uniform

X1

X min

Maximum slope

1

X max

X2

X

phase coexistence

Fig. 2.11. Isolated 1nite system. When the entropy function for a uniform 1nite system (lower curve) has a local convexity region, the isolated system may gain entropy by reorganizing itself into a mixture of the two coexisting phases, but the resulting equilibrium entropy function (upper curve) will always lie below the common tangent (dashed line).

a 1rst order phase transition. Indeed, the order of the phase transition does not depend on how the state has been reached but is an inherent state property determined by the topology of the surface of the thermodynamic potential. (The same situation occurs in the familiar water-vapor system: the phase transition is manifestly of 1rst-order but the phase transformation is continuous if the volume is kept constant.) Thus it is possible to have at the same time continuous phase transformations and spinodal decomposition. 2.4. Finite systems We have seen above that the concept of spinodal instability applies in general to macroscopically uniform systems that are suddenly brought deep into the coexistence region of their phase diagram. The principal characteristic of this out-of-equilibrium phenomenon is the instability against local disturbances of the order parameter. This instability occurs when the entropy function for the uniform system has a local convexity. Indeed, in the thermodynamical limit the entropy is additive and can thus be increased in the region of convexity by splitting the system into independent subsystems. The thermodynamical limit then guarantees the ensuing mixed-phase system acquires an entropy function that is nowhere concave (see Fig. 2.1). For 1nite systems the above features are modi1ed by the fact that the energy associated with the interface between the coexisting phases is no longer negligible. Then the energy is no longer additive. By the same token, the entropy of the combined system is not merely a sum of the individual subsystem entropies (but is somewhat smaller than that) and the Maxwell construction thus loses its validity. The situation is illustrated in Fig. 2.11. As for in1nite systems, it is possible to prepare a 1nite system in a macroscopically uniform state and one may consider the corresponding entropy function S(X ). (One may, for example, think of X as controlling the geometrical size of the system through a scaling of the position r.) When a convexity is present in S(X ), the uniform state

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2

Seq(X)

X1

X min

Maximum slope

1

Minimum slope

λX

X max

ln P(X)

X2

X

Fig. 2.12. Canonical ensemble of 1nite systems. When the 1nite system is brought into contact with a reservoir, it may explore the entire range of X values and the resulting bimodal statistical equilibrium distribution is given by P(X ) ∼ exp(S(X ) − X ). The 1gure shows the case when the Lagrange multiplier equals the slope of the common tangent. The two peaks in P(X ) have then the same height, its points of inversion coincide with those of S(X ), (so ln P(X ) has positive curvature in between), and its minimum lies where the slope of S(X ) also equals .

is out of equilibrium as entropy can be gained by reorganizing it into a phase mixture. However, the resulting maximal entropy for the given value of X falls below the common tangent, due to the 1nite interface energy. Thus, the resulting equilibrium entropy function lies below the Maxwell line and, consequently, it will still exhibit a convexity. The above considerations apply to a microcanonical situation, as the system considered is isolated, with a 1xed value of X , and is initially out of thermodynamic equilibrium. If situated inside the region of convexity, the isolated system may then spontaneously undergo a spinodal decomposition into a mixture of two coexisting phases in an equilibrium state conditioned by the speci1ed value of X . If now such a system is subsequently brought into contact with a reservoir, the value of X may Ductuate as the system now explores the entire phase space. The associated equilibrium distribution is P(X ) ∼ exp (S(X ) − X ), where is the Lagrange multiplier controlling the average. This situation is illustrated in Fig. 2.12. It is seen that when a convexity is present, the resulting X distribution acquires a bimodal character. When is adjusted so the two peaks in P(X ) have the same height (the common tangent to S(X ) has then the slope ), the curvature of ln P(X ) is positive throughout the region of entropy convexity and P(X ) has a minimum at the value of X corresponding to the intermediate coexisting unstable phase (where the slope of S(X ) equals that of the common tangent). The fact that 1rst-order phase transitions in 1nite systems can be associated with the occurrence of a curvature anomaly in the appropriate thermodynamic potential has been discussed in a considerable amount of recent work [18,58–71]. The entire coexistence region may be explored by varying the associated extensive state variables, which present the relevant order parameters. In the region of negative curvature, the system will acquire conditional stability when kept in isolation, while its instability will be revealed as a bimodal distribution of the order parameter when it is put into contact with a reservoir. Furthermore, the occurrence of a 1rst-order phase-transitions is often accompanied

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Fig. 2.13. Lattice gas. Left panel: The free energy (minus a linear term) of a canonical lattice gas in a 1xed total volume as a function of the particle number (bottom) and its derivative, the chemical potential (top left). The concave intruder in the free energy produces a backbending of the chemical potential, thus leading to a negative chemical susceptibility. The pressure (top right), being the variation of the free energy with respect to the volume, exhibits a simultaneous anomaly reDecting a negative compressibility throughout the phase transition region. The mean-1eld result (dotted curve) associated with a uniform density is unstable against phase separation (solid points), as illustrated in Fig. 2.11. (Adapted from Ref. [62].) Right panel: Volume and energy distribution in the 1rst-order phase-transition region (after the system has also been put into contact with a volume reservoir), with three associated projections: on the energy (bottom left), on the volume (top right) and on the line which connect the two maxima which can be seen as the best order parameter (bottom right). The convex intruder in the probability distribution (which is responsible for the observed bimodality) corresponds to a convex intruder in the corresponding entropy. Therefore it signals the occurrence of negative speci1c heat and compressibility. (From Ref. [64].)

by a plateau-like behavior of the caloric curve, or even a “backbending” corresponding to a negative speci1c heat. However, since the caloric curve is aHected by the speci1c dependence of the volume on the excitation energy it may not provide direct information on the character of the phase transition [63,72,73]. 2.4.1. Isochore canonical ensembles Let us 1rst illustrate the general de1nition of a 1rst-order phase transition as the occurrence of a curvature anomaly of the thermodynamical potential as a function of one order parameter. For the liquid–gas phase transition, the density can be taken as an order parameter. Since the density is related to both particle number and volume, one may consider an ensemble in which these two extensive quantities are state variables. This is the case, for example, for the canonical lattice gas in a constant volume where a phase transition is signaled by a concavity anomaly in the free energy F = −T ln Z = E − TS (which follows from the partition sum Z(T; V; N )), as shown in Refs. [62,65] and illustrated in Fig. 2.13 (left). For the chemical equation of state, =9N F, the concavity anomaly appears as a back bending of the chemical potential as a function of the particle number N . This back bending implies that the chemical susceptibility −1 = 9N has a negative branch. Furthermore, the mechanical equation of state, P = 9V F, exhibits a similar back-bending behavior as a function of the density, thus leading to a negative compressibility. It should be stressed that these two anomalies are found for a 1nite system in an equilibrium characterized by a 1xed particle number and volume.

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These features can be further elucidated by imagining that the 1nite system in equilibrium is brought into contact with a reservoir for particles (or volume). Then any arbitrarily small mismatch between the intensive variables (chemical potential or pressure) will become ampli1ed, since the negative chemical susceptibility (or compressibility) will cause a change of particle number (or volume) that will further increase the diHerence. The 1nite system will then evolve away from the anomalous region of negative curvature and approach a stable equilibrium state that has again the same value of the intensive variable as the reservoir but with a normal curvature. In this sense, the original 1nite system 1nds itself in an unstable equilibrium con1guration and it will therefore seek to escape from the spinodal region when oHered an opportunity in the form of a suitable perturbation. To further illustrate this point, let us now consider the liquid–gas phase transition in a system of N particles for which the volume is not 1xed but may Ductuate. In such a case, we can de1ne an observable Vˆ as a measure of the volume of the system and we can characterize a statistical ensemble of realizations of this system by its average volume Vˆ . As usual in statistical physics, one may then introduce a corresponding Lagrange multiplier v (which has dimension of a pressure divided by a temperature). As shown in Fig. 2.13 (right), in such an isochore canonical ensemble it is possible to de1ne the −1 −1E −v V distribution P1v (E; V ) = WO (E; V )Z1 e , which contains the complete information since the v events are sorted according to the two thermodynamical variables E and V . It leads to the density O V )). One can see that of states, which is simply the exponential of the entropy, WO (E; V ) = exp (S(E; in the 1rst-order phase transition region the probability distribution is bimodal, as expected since the entropy presents a convex intruder. Moreover, for a 1xed v , there exists a temperature for which the two maxima have equal height, corresponding again to the Maxwell construction. It is clear from the 1gure that for such a system in contact with a reservoir (of energy and volume in the present case) the anomalous concavity region tends to become depopulated, since the probability distribution displays a minimum, and this region can thus be characterized as unstable. Nevertheless, contrary to the thermodynamical limit, the probability density remains 1nite throughout the unstable region. The joint distribution of energy and volume and at its two projections show that the system has simultaneously a negative heat capacity and a negative compressibility. Thus, in liquid–gas phase transitions, spinodal instability is intimately related to the occurrence of negative speci1c heat. 2.4.2. Spinodal instability in molecular dynamics The clustering caused by the spinodal instability of dilute nuclear matter was studied by Peilert et al. [74] who performed canonical Metropolis simulations with a quasi-classical many-body model. For this purpose, the QMD model [75], which includes two- and three-body Skyrme-type interactions as well as a 1nite-range Yukawa force, was augmented with a momentum-dependent Pauli-potential [76] that serves to keep the nucleons apart in phase space, VP ∼ ij exp(−rij2 =2q02 − pij2 =2p02 ). Furthermore, since both 1nite nuclei and in1nite nuclear matter were considered, the Coulomb interaction was screened, using a suitably large range. Fig. 2.14 illustrates how the dilute matter prefers to cluster, the eHect becoming increasingly pronounced as the volume is increased. The associated gain in energy can be considerable for temperatures below ≈ 8 MeV, as is illustrated in Fig. 2.15. As the authors pointed out, it is important to take this phenomenon into account when seeking to determine the nuclear equation of state from data. However, due to the irregular and varied appearance of the clustered system, the inclusion of

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Fig. 2.14. Clustered matter. Typical manifestations of nuclear matter at two densities inside the spinodal region of the phase diagram, as obtained with quasi-classical molecular dynamics (the positions of 254 nucleons are projected onto the xy plane). (From Ref. [74].)

Fig. 2.15. Clusterization energy. The average energy per nucleon as obtained for either random positions (higher values) or a canonical (Metropolis) ensemble (lower values), for either a 1xed density as a function of temperature (left) or for a 1xed temperature as a function of density (right), as obtained with quasi-classical molecular dynamics. (From Ref. [74].)

the subsaturation clustering into calculations of the nuclear equation of state poses a considerable challenge which has not yet been overcome. An instructive numerical experiment has been performed in Ref. [77] within classical molecular dynamics. The particles interact through a Lennard–Jones type potential, V (r)=(r −8 −r −4 )3(r−Rmax ), and are prepared at a given density and temperature within a 1xed box (with periodic boundaries). They are then evolved until they have equilibrated, at which time the pressure and the temperature are extracted. The resulting equation of state (isotherms of the pressure versus density) is shown on the left in Fig. 2.16. Although the statistics is rather poor, it can be clearly seen that there is a region where 9P=9 is negative. It is thus possible to identify a spinodal region and the associated liquid–gas co-existence region. The critical temperature can also be extracted as the temperature of the isotherm having an inDection point.

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Fig. 2.16. Spinodal instability in classical molecular dynamics. Left panel: The equation of state (isotherms of the pressure P versus density ) obtained for a system of 512 Lennard–Jones particles in a box with periodic boundary conditions. Right panel: Contour plot in the T – plane of the power 5 characterizing the equilibrium mass distribution of the clusters formed within the spinodal region. (From Ref. [77].)

When the system is prepared inside the liquid–gas coexistence region it becomes clumpy and an associated “fragment” mass distribution can be extracted. Its density and temperature dependence was also studied in Ref. [77] and it was found that the mass distribution in the range 4 ¡ A ¡ 30 exhibits a power falloH characterized by the exponent 5 shown on the right in Fig. 2.16. Large values of 5 indicates that the mass distribution is evaporation-like (U-shaped) (at small T ) or vaporization-like (at high T ). The lowest value of 5 was obtained near the critical point, 5min ≈ 2:2. The occurrence of critical behavior inside the coexistence region is a generic result for systems with a very small number of constituents. It has been shown in Ref. [62] that for a lattice–gas simulation the fragment distribution presents a critical behavior with power laws and scaling not only at the critical point but at all densities along a line passing by the critical point but diving deeply inside the coexistence and even the spinodal region. This is illustrated in Fig. 2.17. 2.4.3. Experimental evidence and perspectives Both bimodal energy distributions and negative microcanonical heat capacities have been recently reported for open 1nite systems undergoing a phase transition. In the case of nuclear multifragmentation the heat capacity has been reconstructed using partial energy Ductuations [78]. The 1rst results, obtained for peripheral strongly-damped Au+Au collisions [79], were followed by results for central “fusion-like” processes [80–82] and for pion-induced reactions [83]. The same general phenomenon has been reported recently in another 1eld of physics: metallic cluster melting [84], where the canonical distribution of events was demonstrated to be bimodal. It should be noted that all these results were obtained for open systems in which the volume may Ductuate and the observed heat capacity is then related to CP . Using the relation between CP , CV and T , one can see that when CP diverges then T approaches zero, which demonstrates the equivalence between the negative heat capacity region and the mechanical spinodal region. 2.5. Concluding remarks about thermodynamics In this 1rst section, we have seen that spinodal instability of a system is a general feature that reDects the particular structure of the associated phase diagram. Generally, spinodal instability is associated with a convex anomaly of the entropy function. In macroscopically uniform systems,

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Fig. 2.17. Critical fragment distribution. Top: Fragment distributions at various densities and temperatures in a lattice gas. At each density, a typical U-shaped distribution emerges below a critical temperature (dark histograms) with both light fragments and heavy residues, while only small fragments remain above (light histograms). Around the critical temperature the distribution is resembles a power law (dots). Bottom: The density dependence of the corresponding critical temperature (dots) and the boundary of the coexistence region (light curve). The critical-temperature curve intersects the coexistence boundary close to its top (the thermodynamic critical point, but critical behavior appears both above (along the so-called KertZesz line) and below (inside the coexistence region). (Adapted from Ref. [62].)

the entropy can then be increased by in1nitesimal Ductuations of the local order parameter and the system decomposes itself into two coexisting phases in equilibrium, thereby eliminating the convexity. In an isolated 1nite system, such a spontaneous transformation may not be possible and the system may establish a conditional equilibrium inside the anomalous region. However, if such a system is brought into contact with a suitable reservoir, it will explore the entire range of the order parameter which will then acquire a bimodal equilibrium distribution exhibiting a minimum in the unstable concave region. In the nuclear case, where two interacting components are present, protons and neutrons, the discussion of spinodal instability is more complicated because of the presence of two independent chemical potentials, p and n . Although it has some intuitive appeal to discuss the phenomenon in terms of separate “mechanical” and “chemical” instabilities (representing changes in the nucleon density or the proton concentration, respectively), such an approach can be misleading since those are in fact associated with a single instability. Indeed, the relevant variables are determined by the corresponding normal modes, as obtained by diagonalizing the curvature matrix of the thermodynamic potential, and they generally involve changes in both total density and concentration. We have seen that the direction of the unstable normal mode is predominantly isoscalar (i.e. of mechanical character), which shows that the underlying mechanism is in fact a liquid–gas phase transition rather than a chemical (i.e. isovector) separation of protons from neutrons (so it is not a ferromagnetic phase transition in the isospin space). Nevertheless, a spinodal decomposition of the nuclear system

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generally induces isospin fractionation, since the high-density (liquid) phase is closer to symmetric matter than its low-density (gas) partner phase.

3. Onset of spinodal fragmentation In the previous section, we have discussed the concept of spinodal instability within the framework of thermodynamics, where it appears when a uniform system in equilibrium loses stability against phase separation. Since these systems are unstable, they will evolve in time and a dynamical point of view must therefore be adopted. We generally wish to investigate the fate of systems that have been prepared in an unseparated con1guration situated within the region of spinodal instability. This can typically be achieved by 1rst preparing the system in a suitable equilibrium con1guration, such as hot uniform matter, and then making the system unstable by performing a quench, a sudden reduction of the temperature or/and the density. A basic theoretical framework for treating spinodal decomposition was developed by Hillard, Cahn, and coworkers [85–91]. Further important developments were made by Langer et al. [92,93] and a good overall discussion of the kinetics of 1rst-order phase transitions can be found in Ref. [94]. In order to bring out the main features, we start by considering the phenomenological Landau–Ginzburg model [92,95] within which the time evolution of the density (r) is governed by a simple diHusion equation, 9 F[(r; t)] (r; t) = v∇2 : 9t (r; t)

(3.1)

Here the free energy F[(r)], which is a functional of the density (r), may be expressed in a simple approximate form, (3.2) F[(r)] = dr[f((r)) + 12 c(∇(r))2 ] ; where f() is the free-energy density for matter with a uniform density and where the gradient term seeks to account for the 1nite range of the interaction. The pressure is given by P = 2 9 6, where 6=f= is the free energy per particle in the uniform system, and the compressibility modulus is −1 = 9 P = 2 92 f. The onset of spinodal decomposition can then be studied by linearization of the time-dependent Ginzburg–Landau equation (3.1). One thus considers small variations of the density around a uniform value, (r; t) = (r; t) − 0 , and 1nd

9 92 f 2 2 (r; t) = v∇ (3.3) − c∇ (r; t) : 9t 92 =0 For plane-wave disturbances, (r; t) ∼ exp (ik · r − i!k t), the dispersion relation for the associated frequency !k can then be obtained, 1 2 2 !k = −ivk : (3.4) + ck 2

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The system is situated inside the spinodal region when the compressibility is negative, ¡ 0, and a 2 mode k is then unstable if its wave number is below a maximum value, k ¡ kmax , where 2 kmax c= −1. For these modes, the frequency is imaginary, !k =±i8k , so the disturbance evolves exponentially, k ∼ exp (±8k t). The growth rate tends to zero at the two boundaries, 8k → 0 for k → 0 and k → kmax , and it thus attains a maximum value at a certain preferred wave number, k0 , which in the simple Landau–Ginzburg model is given by half the maximum value, k0 = 12 kmax . It may therefore be expected that density irregularities with wave numbers near k0 will come to dominate the spatial√pattern of the spinodal decomposition. The corresponding characteristic wave length, 0 = 290 −c, is determined by the ratio of the compressibility modulus −1 and the parameter c which reDects the 1nite range of the interaction. If the system is suPciently dilute, the result of the spinodal decomposition will be the transformation of the unstable uniform system into an assembly of fragments. It then follows from the above analysis that the fragments should exhibit a corresponding preferential size, Af ≈ 03 0 . Those characteristics, the ampli1cation of small Ductuations with growth rates that exhibit a preference for a certain length, are generic properties of spinodal decomposition. In macroscopic systems, such as binary alloys, this phenomenon has long been known and is now a standard part of text books on statistical physics (see Ref. [95], for example). Its relevance to nuclear multifragmentation was 1rst pointed out by Bertsch and Siemens [3] over 20 years ago. However, although the general features discussed above will also be present in the spinodal decomposition of nuclei, phenomenological descriptions of the above type are clearly inadequate for the treatment of the nuclear problem. Of particular importance are the following features: 1. Nuclear matter is Fermi liquid with speci1c quantum properties. Furthermore, the uncertainty principle plays a strong role in nuclear physics because of the relatively small size of nuclei. Moreover, up to several MeV of excitation, speci1c nuclear structure such as shell eHects may play a role, as we will see in the complete RPA treatment of the nuclear instabilities. 2. Nuclei are not large in comparison with the range of the strong interaction, so surface eHects are signi1cant and any description must have a correspondingly 1ne spatial resolution. In addition, nuclei are charged and the long-range Coulomb repulsion renders the thermodynamical limit problematic. 3. The time scales for the macroscopic nuclear dynamics occurring in heavy-ion collisions are fairly short and the system is typically out of local equilibrium. The transport description must therefore be correspondingly detailed. 4. Linear treatments apply only to the early development of instabilities. Once the density disturbances grow large the dynamics becomes non-linear and a more complete treatment is required. Moreover, the phenomenological treatment does not concern itself with how the system has become unstable or how the disturbances originated. These important questions will be addressed in the next section. In the present section, we will focus on the early growth of the disturbances. 3.1. Nuclear matter To make the connection with thermodynamics, we 1rst consider spinodal instability in bulk matter. As we shall see in this section, and in contrast to the time-dependent Ginzburg–Landau

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approach which only treats the diHusion of the density Ductuation, the density Ductuations generally propagate and thus form sound waves. Since nucleons are fermions, the density Ductuations should be investigated in Fermi-liquid theory. Then the sound properties are strongly aHected, since only particles near the Fermi surface participate in the collective motion. However, before embarking on the speci1c description of Fermi liquids, let us 1rst recall what can be expected for a classical Duid. We shall see in this section that spinodal instabilities in in1nite matter can be seen as ordinary sound waves whose speed has turned imaginary upon entry into the spinodal region. At suPciently long wave lengths, where the situation is the simplest since only the bulk properties and the Coulomb force play a role, the speed of sound ultimately approaches zero. In the opposite extreme of small wave numbers the speed of sound also tends to zero, due to the 1nite range of the interaction. Thus there is a length scale for which the growth rate has a maximum, as is characteristic of spinodal decomposition. 3.1.1. Hydrodynamical instabilities in classical @uids Let us 1rst consider the dynamics of density Ductuations in a classical Duid. The evolution of a Duid element caused by the pressure gradients is governed by the Navier–Stokes equation, d 9 C ≡ C + C · ∇C = ∇P ; dt 9t

(3.5)

while the continuity equation, which expresses the conservation of matter, yields 9 + ∇ · C = 0 : 9t

(3.6)

If we linearize around the initial uniform density 0 , (r; t)=(r; t)−0 , the 1rst equation becomes 0 9C=9t = ∇P, while the second reduces to 9=9t = 0 ∇ · C. With a corresponding linearization of the pressure, P(r; t) = P0 + (9P=9)0 (r; t), these two equations lead to a single closed equation for the density disturbance, 92 9P 2 (r; t) = ∇ P = ∇2 (r; t) : (3.7) 2 9t 9 =0 Taking the Fourier transform and introducing the speed of sound vs = (−m)1=2 , where −1 = 9 P is the compressibility modulus, we obtain the dispersion relation, !k2 = vs2 k 2

or

k 2 = m!k2 :

(3.8)

This expression brings out the fact that the dynamics of the sound waves is directly related to the thermodynamics. In a perfect gas, P = T , the compressibility is always positive and the speed of sound is always real. In this regime, called -rst sound, the density Ductuations propagate. The situation remains similar after the increase of the pressure caused by hard collisions has been taken into account. However, the attractive long-range part of the interaction tends to reduce the pressure. (In the Van der Waals Duid, this eHect is emulated by the attractive term containing the parameter a.) Then a new type of sound arises, zero sound, for which the propagation is due to the attractive tail of the interaction. This attractive part is responsible also for the condensation of a gas into a liquid,

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a phenomenon associated with the occurrence of a negative compressibility. The speed of sound is then imaginary and the system is unstable against density undulations. It should be noted that in the nuclear case the fermionic nature of the nucleons modi1es the above discussion because the exclusion principle suppresses collisions, thus reinforcing the role of the mean 1eld, and reduces the number of active particles to those near the Fermi surface, thus aHecting the propagation of waves. 3.1.2. Collective motion in Fermi @uids In the standard treatment, the Duid dynamical equations are deduced from the Boltzmann equation by taking the zeroth and the 1rst moments of the local momentum distribution. This approach is also possible for Fermi Duids. However, as an alternative treatment, we shall discuss here the derivation of the Duid dynamics from a variational approach [96,26], which will allow us to derive equations for collective modes, thus simplifying the extraction of the region of instability. In the variational formulation of quantum Duid dynamics, the starting point is the action integral, I = dt;|i˝9=9t − Hˆ |;, from which the many-body SchrUodinger equation can be obtained. The expectation value of the Hamiltonian, ;|Hˆ |;, is calculated by making a speci1c ansatz for the many-body wave function, ;(r1 ; r2 ; : : : ; rA ; t) = <(r1 ; r2 ; : : : ; rA ; t) e(i=˝)S(r1 ;r2 ;:::;rA ;t) ;

(3.9)

where the amplitude <(t) and the phase S(t) are real functions of the positions. With the assumption that the phase is additive, S(t) = An=1 S(rn ; t), and that <(t) is a Slater determinant characterized by its one-body density , it is possible to express the action [26,97], t2 (r) 9 ∇S(r) · ∇S(r) ; (3.10) I= dt E[] + d 3 r S(r) (r) − 9t 2m∗ (r) t1 where m∗ (r) is the eHective nucleon mass and E[] is the energy functional. A local density approximation is often performed so that E[] = d 3 r E((r)), where E() is the energy density of bulk matter at the density . (Such a local approximation is used for the kinetic energy in the Thomas–Fermi treatment and for the interaction energy in Skyrme-type models.) Within these approximations, the density (r) plays the role of the coordinate while the phase S(r) is the conjugate momentum. The requirement of stationarity under variations in then leads to the continuity equation, 9 (r) + ∇ · (u(r)(r)) = 0 ; 9t

(3.11)

with the velocity 1eld being u = (1=m∗ )∇S. Thus the phase function S(r) may be interpreted as a velocity potential and the Lagrange equation for S(r) then yields an Euler-type equation for the velocity 1eld u(r; t). Small-amplitude collective motion can now be studied by linearizing the above continuity equation around a given reference state having the uniform density 0 . Thus, for a given collective mode =, the velocity potential may be parametrized as S(r; t) = q˙= (t)S= (r), where q= is the collective amplitude. The corresponding distortion of the density is given by (r; t) = q= (t)= (r), where the speci1c density distortion can be obtained from the continuity equation (3.11), = = −0 ∇((1=m∗ )∇S= ).

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The expectation value of the Hamiltonian then assumes a normal form, |H | = E0 + 12 M= q˙2= + 12 C= q=2 ;

(3.12)

where E0 is the energy of the equilibrium state. Furthermore, the inertia and stiHness coePcients for the mode are 0 M= = d 3 r ∗ ∇S= (r) · ∇S= (r) ¿ 0 ; (3.13) m (r) 2 E[] C= = d 3 r d 3 r = (r )= (r) ; (3.14) (r)(r ) 0 and the collective frequency != is then given by !=2 = C= =M= . When C= becomes negative the collective mode turns unstable, != = ±i8= , where 8= is the growth rate of the amplitude q= . We 1nally note that all the input quantities for the above treatment are 1xed by the standard parameters of the considered interaction, so in that sense the analysis is parameter free. However, we wish to stress that the reduction of the density dependence of the kinetic term to the diagonal part (r) neglects the deformation of the Fermi surface which might be important for the collective motion [96,98]. 3.1.2.1. Nuclear matter. After the above general discussion, we now turn to the study of collective motion in nuclear matter. Thus the reference con1guration has a uniform density 0 and the distortion can be expanded on plane waves. Accordingly, we may characterize the collective mode by its wave vector k and the associated velocity potential becomes Sk (r) = 12 (eik·r + h:c:) = cos k · r :

(3.15)

The corresponding distortion in density then is determined by the continuity equation (3.11), k (r; t) = 2Mk cos k · r, where the collective inertia is Mk = (0 =2m∗ )k 2 . The restoring force, Ck =Mk2 , is simply related to the Fourier transform of the eHective interaction V (r; r )=2 E=(r)(r) and since the Skyrme interaction is local with a quadratic momentum dependence it is a simple polynomial, Ck =Mk2 = (A + Bk 2 ) where A and B depend on the speci1c Skyrme interaction employed. The frequencies of the collective modes are thus given by the dispersion relation, 0 (A + Bk 2 )k 2 : (3.16) !k2 = 2m∗ The 1rst term is the compressibility, A = 9P=9. When this quantity is negative, the sound velocity vc = (A0 =2m∗ )1=2 is imaginary and we are within the spinodal zone where the disturbances evolve exponentially. For small values of k (gentle undulations), the growth rate is linear, 8k ≈ |vs |k. As k is increased the quadratic term gains signi1cance, as the non-local eHects become important. It generally counteracts the 1rst term and causes to√growth rate drop to zero at kmax = (−A=B)1=2 . The maximum growth rate, 80 , occurs for k0 = kmax = 2. As in the linear Ginzburg–Landau model, the problem is thus characterized by two quantities: the magnitude of the sound velocity |vc |, which is the initial slope of 8k (k), and the typical length scale 0 = 29=k0 giving the wave length of the most strongly ampli1ed modes.

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00

ρ=0.30ρ0

ρ=0.50ρ0 ρ=0.55ρ0

0 0.2 0.4 0.6 0.8

1 1.2 1.4

Wave number k (fm-1)

Growth rate γ (c/fm)

Growth rate γ (c/fm)

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297

ρ=0.30ρ0

ρ=0.50ρ0 ρ=0.55ρ0

0 0.2 0.4 0.6 0.8

1 1.2 1.4

Wave number k (fm-1)

Fig. 3.1. Dispersion relation for unstable sound waves in nuclear matter. The growth rate 8k = Im(!k ) of unstable modes as a function of their wave number k, as obtained either without (left) or with (right) the Coulomb interaction. (From Ref. [99].)

3.1.2.2. Coulomb interaction. Up to now we have only considered the nuclear attraction, since it is the source of the condensation phenomenon. However, protons are charged so the Coulomb interaction must also be taken into account. Using the Hartree–Fock expression, with the exchange term evaluated in the local-density approximation, we may write the Coulomb energy density as [100] 1=3 p (r ) 3 3 − e2 eC (r) = 12 e2 p (r) d 3 r p (r)4=3 ; (3.17) |r − r| 4 9 where p (r) is the proton density. Of course, the Coulomb energy diverges in in1nite matter, so it must be counterbalanced by a suitable negative background charge density. Alternatively, one may employ a simple screening by modulating the 1rst term by the factor exp (−|r − r|=a). When the screening length a is large compared to the nuclear radius it should not aHect the properties of 1nite nuclei while making the matter calculation possible (the quantitative reliability of the results can be checked by changing a). Since the restoring force in in1nite matter is the Fourier transform of the interaction, the inclusion of the screened Coulomb energy density adds repulsive term having a Lorentzian shape, [Ck =Mk2 ]C ∼ a2 =(a2 + k 2 ). This term is positive and thus reduces the growth rate. It is important for long wave lengths k, but becomes negligible for small , as illustrated in Fig. 3.1. For wave lengths of the order of the nuclear radius, the reduction of the growth rate induced by the Coulomb interaction appears to be insigni1cant. Thus, for the length scales accessible in nuclear systems, the Coulomb interaction induces only a small reduction of the spinodal growth rate. These results agree well with those of obtained by Fabbri and Matera [101] with the linearized Vlasov equation (which will be discussed in Section 3.1.9). 3.1.3. The unstable response of Fermi liquids We now discuss how the fermionic nature of the nucleons can be taken into account in a more detailed manner. The propagation of density waves in Fermi liquids has been considered already a long time ago, in particular by Landau [102,103]. With a view towards nuclear systems formed in heavy-ion collisions, Pethick and Ravenhall studied the growth of instabilities within the spinodal region of a normal Fermi liquid [104]. Restricting their considerations to wave lengths that are long compared with both the interparticle spacing and the

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Landau parameters F0 and F1

0.5 m* = m

0.0 m* = 0.7m

-0.5

m* = 0.4m

-1.0 -1.5 -2.0 -2.5 0.0

T=0 λ=∞ 0.2 0.4 0.6 0.8 Average relative density ρ/ρs

1.0

Fig. 3.2. Landau parameters. The standard Landau parameters F0 (bottom) and F1 (top), as functions of the density (in units of the saturation density s ), as obtained with the Seyler–Blanchard model (described in Appendix A) for various degrees of momentum dependence as characterized by m∗ , the value of the eHective nucleon mass at saturation.

(strong) interaction range, as well as to frequencies and temperatures that are low compared with the Fermi kinetic energy, the authors carried out their study within the framework of Landau Fermi-liquid theory. Simple analytic results emerge in the opposite extremes when residual collisions are either very frequent or very rare, as we shall now brieDy recall. When the collision time is short on the macroscopic time scale and the mean free path is short on the macroscopic length scale, the hydrodynamical limit is approached and, as we have seen in Section 3.1.1, the dispersion relation is simple, !k2 = vs2 k 2 where vs is the speed of sound, vs2 = (1=m)9 P ∼ 1 + F0 . For conditions under which vs2 becomes negative (F0 ¡ − 1), the modes have imaginary frequencies. Their amplitudes then evolve exponentially with growth rates given by 1=2 ˝ 1 m∗ ∞ 8k = −i!k = − = − kVF ; (3.18) (1 + F0 ) tk 3 m where the eHective mass is m∗ = (1 + F1 =3)m. The quantities F0 and F1 are the standard Landau parameters characterizing the medium. They are illustrated for nuclear matter in Fig. 3.2 and are discussed further below and in Appendix A. The opposite extreme occurs when the residual collisions occur only rarely, so the collision rate is small on the macroscopic time scale and the mean free path is large on the macroscopic length scale. In nuclear physics, this dilute limit occurs at low excitations where the Pauli principle suppresses direct collisions and thus renders the optical potential purely real. One may then employ the Landau kinetic equation for a disturbance in the quasiparticle phase-space density f(r; p; t),   9 9f0 (p) f(r; p) + C · ∇f(r; p) − C · ∇ U (r) + Fpp (r)f(r; p ) = 0 : (3.19) 9t 9j p An analysis of the pole structure for various values of F0 then yields the dispersion relation for the corresponding physical scenarios. For example, in the stable regime the collective modes are of zero sound form when F0 + (m=m∗ )F1 ¿ 0. We note that this approach assumes that

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299

f(p + 12 q) − f(p − 12 q) ≈ C · q9f(p)=9j. This semi-classical approximation may not be accurate at the low densities where spinodal instabilities appear [105], as will be discussed in Section 3.1.8. The modes are unstable when F0 ¡ − 1. When F0 is only slightly below threshold, the growth rate is linear in F0 , 2 F0 ≈ −1 : 8k ≈ − (1 + F0 )kVF : 9 It increases as F0 grows increasingly negative and approaches another simple form, 1=2 1 m∗ 9 + F0 kVF ; F0 − 1 : 8 k ≈ − 3 m 5

(3.20)

(3.21)

which is seen to be smaller than the hydrodynamic growth rate 8∞ k given in (3.18). In the intermediate regime, −1 ¡ F0 ¡ 0, the hydrodynamical limit yields undamped oscillations while the collisionless limit leads to Landau damping. This particular regime can play an important role in nuclear collisions, since F0 is expected to change from the positive values characteristic of hot compressed matter to values below −1 inside the spinodal region, as the collision evolves. Adopting the relaxation-time approximation for the collision integral, the authors of Ref. [106] bridge the above opposite limits where the mean free path is either very short (hydrodynamics) or very long (Knudsen scenario) and for which analytic results may be derived. In this manner it is possible achieve a better understanding of the somewhat counterintuitive result that the collisions tend to increase the growth rates. The key reason is that the collisions act to reduce the distortions of the Fermi surface (except for the monopole and dipole which are 1xed by the continuity equation, thus reducing the stiHness and facilitating the growth of the instabilities. This eHect is further examined in Section 3.1.6 and shown to be relatively small. In a subsequent study within the same framework, Heiselberg et al. [107] calculated the density response and derived the dispersion relation in hot Fermi liquids, including in particular modes that are unstable with respect to density waves. 3.1.3.1. Growth factors. In a more quantitative study, Heiselberg et al. [106] treated the instabilities with Landau’s kinetic equation. Ignoring the small eHect of dissipation and employing the relaxation-time approximation, they obtained the growth rate for density ripples of a given wave number k, 8k (; T ) = −i!k (; T ), in the manner described above. In order to estimate the factor by which a given unstable mode k can grow, uniform nuclear matter was prepared at a given point in the phase plane and its subsequent phase evolution was then determined, ((t); T (t)) (assuming that there is no dissipation). If the total energy is negative, the phase point oscillates around the equilibrium density while a steady expansion results when the energy is positive. In either case, the growth rate was integrated over the time spent inside the spinodal region, yielding Gk = 8k ((t); T (t)) dt. The factor by which the amplitude of the mode k is expected to grow is then given approximately by eGk . By employing the thermal equilibrium Ductuation of the mode for the initial density Ductuation, it was possible to subsequently obtain an estimate of the absolute magnitude of the resulting density undulation. This analysis suggested that fragmentation may be induced for growth factors in the range of G ≈ 1–3. The resulting regions of initial phase conditions are shown in Fig. 3.3.

300

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Fig. 3.3. Growth factors. Contours of initial conditions in symmetric nuclear matter for which the largest growth exponent G for the most rapidly growing mode falls within a speci1ed range. The zero-pressure line (P = 0), which gives the saturation density as a function of temperature T , is shown together with adiabats, selected instability lines, and the zero-energy contour (E = 0). (From Ref. [106].)

In this type of description, the dynamical phase trajectory depends on the equation of state over a broad range of densities, whereas the spinodal growth rates depend only on the equation of state in a limited subsaturation regime where it is relatively well known. Therefore, as the authors of Ref. [106] point out, experimental information on the high-density part of the equation of state can be obtained if the initial conditions can be extracted from the data. As it happens, for the cases of actual interest, conditions are close to the collisionless limit. It would therefore appear that the adiabatic spinodal line, which is relevant for instabilities in the hydrodynamic limit, plays no role in the present context. (This was con1rmed by the molecular-dynamics studies by Lopez et al. [108,109] to be discussed in Section 5.1.5.) 3.1.4. Linear response in semi-classical approaches After the above general discussion, we now turn towards more speci1c aspects of instabilities in nuclear systems. The analytic structure of the energy of nuclear matter in the subsaturation metastable region was studied by Friman et al. [110], who obtained the imaginary part of the energy as a non-perturbative sum of ring diagrams. For the present review, it is particularly interesting to consider the pure Vlasov approach [111–113], in which the residual collisions are entirely neglected. We thus assume that the individual nucleon is governed by an eHective Hamiltonian of the form h(r; p)=p2 =2m+U [](r). (We ignore for simplicity the possible medium modi1cation of the mass; this more general case is described in Appendix A.) The propagation of a small disturbance in uniform matter, f(r; p; t) = f(r; p; t) − f0 (p; t) is then obtained by linearizing the Vlasov equation Df=dt = 0, p 9 9f0 9 9 9 f + · f − · U = 0 : (3.22) 9t m 9r 9p 9 9r

P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440

Here

(r; t) = g

d3 p f(r; p; t) h3

301

(3.23)

is the associated density disturbance, with g being the (four-fold) spin–isospin degeneracy. Performing a Fourier expansion on plane waves, fk (p; t) eik·r ; (3.24) f(r; p; t) = k

we 1nd that the diHerent wave numbers k are decoupled in uniform matter. Thus, for each value of k, collective solutions of the form fk (p; t) = fk (p)e−i!k t are determined by the transform of the Vlasov equation (3.22), 9f0 9U (3.25) (−!k + C · k)fk (p) = C · k k ; 9j 9 where C = 9h=9p = p=m is the nucleon velocity, leading directly to the corresponding dispersion equation, 3 C·k 9f0 9U d p (3.26) 0 = C(!k ) ≡ 1 − g h3 C · k − !k 9j 9 3 d p (C · k)2 9f0 9U : (3.27) =1− g h3 (C · k)2 − !k2 9j 9 The solutions to the dispersion equation are thus the roots of the susceptibility C(!). The second relation follows by eliminating odd terms in the integrand and it shows that the solutions come in real or imaginary pairs of opposite sign. Introducing the sound speed in units of the Fermi speed VF , sk ≡ !k =kVF , the dispersion equation (3.27) becomes independent of k. Thus, the speed of sound, vs = !k =k = sk VF , is the same for all wave numbers k. Contact with the Landau treatment can be made by means of the relation 9 U ≈ 23 jF F0 = (see Appendix A). 3.1.4.1. Zero temperature. In order to elucidate the collective dynamics described by the dispersion relation (3.27), we 1rst look for real solutions at zero temperature, T = 0. The momentum integral is then restricted to the Fermi surface (since 9j f0 = −(j − jF )) so it reduces to an elementary angular average (see Appendix C), yielding 1 s s+1 =1+ : (3.28) ln 2 s−1 F0 This dispersion relation is completely analogous to the one obtained in Fermi liquid theory [102,103,114] and it has real solutions with s2 ¿ 1 when F0 ¿ 0. These disturbances thus propagate with a speed that exceeds VF . The corresponding roots are shown in Fig. 3.4 as functions of F0 . It is equally simple to look for imaginary solutions, s = ±i8, with 8 ¿ 0. The corresponding dispersion equation, 1 1 8 arctan = 1 + ; (3.29) 8 F0

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Fig. 3.4. Dispersion relation in nuclear matter at zero temperature. Roots (solid curves) of the dispersion relation (3.27) as a function of the Landau parameter F0 for both stable (F0 ¿ 0) and unstable (F0 ¡ − 1) modes, together with the analytical continuation (dotted curve) into the zero-sound regime (−1 ¡ F0 ¡ 0). (From Ref. [113].)

has the desired solutions for F0 ¡ − 1 and they correspond to exponential growth and decay, respectively. Since the derivative of the pressure is given by 9U 2jF 2 jF 9P = + = (1 + F0 ) ; (3.30) 9 9 3 3 it is readily seen that the condition F0 ¡ − 1 corresponds to the usual stability condition 9 P ¡ 0. Thus, the analysis of the density dynamics yields the same conditions as the thermodynamical analysis of Section 2. We note that on the positive half of the imaginary axis, s = i8 with 8 ¿ 0), the dispersion equation (3.29) is equivalent to 9 1 8 : (3.31) − arctan 8 = 1 + 2 F0 If this relation is analytically continued into the negative half of the imaginary axis (8 ¡ 0), we may obtain a solution also when −1 ¡ F0 ¡ 0. While this root is not a solution of the dispersion equation Eq. (3.26), it does provide a means for understanding the response of the system in this intermediate regime −1 ¡ F0 ¡ 0. Since these solutions have 8 ¡ 0 they act as a damping, the phenomenon is usually called Landau damping and occurs when the interaction (as measured by F0 ) is not attractive enough to produce an instability. 3.1.4.2. Finite range. The above analysis has been carried out for the idealized case when the interaction is local so that the potential depends only on the density at the speci1ed position, U [](r) = U ((r)). To be realistic, one must take account of the 1nite interaction range and 9 U should then be replaced by the appropriate Fourier component which we denote by (9 U )k . This is most easily seen by imagining that the potential is obtained through a convolution of the density by a suitable kernel. For example, if the kernel is of Gaussian form, ∼ exp (−r 2 =2a2 ), then (9 U )k = 9 U exp (− 12 k 2 a2 ) and the Landau parameter acquires a k dependence, F0 (k) = F0 exp (− 12 k 2 a2 ). Studies using 1nite-range Gogny forces have been carried out by Ventura et al. [115,116]. In order to give a quantitative impression of this 1nite-range eHect, we show in Fig. 3.5 (left) how the function F0 () depends on the wave length of perturbation. It is seen that F0 () steadily moves

P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440 0.0

-0.5 -1.0 λ=4 fm 5 6

-1.5

T=0

8 12

-2.0

m* = 0.7

Landau parameter F0 (τ,k)

Landau parameter F0 (k)

0.0

-0.5

T = 10 MeV

-1.0

λ=8 fm m* = 0.7

-1.5

λ=∞

-2.5 0.0

303

0.1

0.2 0.3 0.4 0.5 0.6 0.7 Average relative density ρ/ρs

T=0

0.8

-2.0 0.0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 Average relative density ρ/ρs

0.8

Fig. 3.5. Dependence of F0 on wave length and temperature. The eHective Landau parameter F0 (k; T ) is displayed as a function of the density (relative to the saturation density s ) for either various wave lengths at T = 0 (left) or for various temperatures T for the wave length = 8 fm (right). The results were obtained with the Seyler–Blanchard model for an eHective mass of m∗ ≈ 0:7m (see Appendix A).

upwards as k is increased. Thus, for 1xed thermodynamic conditions ( and T ) inside the spinodal region, the 1nite range of the force generally reduces the magnitude of F0 and hence stabilizes the system. While the eHect is negligible for long wave lengths, it may be signi1cant for the most rapidly ampli1ed wave lengths which are of nuclear dimensions and therefore are of practical interest. It follows that spinodal instability occurs only for perturbations having a suPciently long wave length, ¿ min , while the Landau regime, −1 ¡ F0 ¡ 0, contains more rapid undulations. For the Gaussian form of the interaction (see above), the minimum wave length for which spinodal instability 2 is encountered is determined by min = 49a2 ln (1=|F0 (0)|), while a kernel of Yukawa form, ∼ 2 2 2 2 1=(1 + a k ), would yield min = 29a =|1 + F0 (0)|. 3.1.4.3. Finite temperature. The above type of analysis can readily be extended to 1nite temperatures as well. Fig. 3.5 illustrates how F0 depends on the temperature T for a given wave length = 8 fm, which is in the range of the most rapidly ampli1ed values (see below). As T is increased, F0 generally moves towards positive values, thus shrinking the unstable region and reducing the growth rates (see below). The dependence of the spinodal boundary in the –T plane on the wave length = 29=k is illustrated in Fig. 3.6 (left). The thermodynamic spinodal boundary corresponds to the limit of in1nitely long wave lengths, = ∞. As the wave length of the mode is decreased, the corresponding spinodal boundary shrinks steadily and 1nally, for suPciently short wave lengths (below ≈ 3:5 fm), any perturbation is spinodally stable. Further insight into the spinodal instabilities can be gained from Fig. 3.6 (right) which, for the particular wave length = 29=k = 8 fm, shows the dependence of the growth times tk on density and temperature in the form of a contour plot. The corresponding spinodal boundary corresponds to tk = ∞ and lies well inside the thermodynamics boundary determined by tk=0 = ∞. The region within which the growth rate exceeds a certain value shrinks steadily towards a point on the axis which thus identi1es the condition for fastest growth of perturbations of the speci1ed wave length. In the present case, = 8 fm, the shortest growth time is about 0:6 × 10−22 s ≈ 20 fm=c and it occurs for a density of about 0:25s (and T = 0).

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P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440 18.0

16.0

λ=12 fm

14.0

Temperature T (MeV)

Temperature T (MeV)

18.0

λ=∞

16.0

λ=8 fm

12.0 λ=6 fm

10.0

λ=5 fm

8.0 6.0

λ=4 fm

4.0

(λ = ∞)

14.0 12.0

tλ = ∞ tλ = 4

10.0

tλ = 2

8.0 6.0

tλ = 1·10-22s

4.0

2.0

2.0

0.0 0.0

0.0 0.0

0.1

0.2 0.3 0.4 0.5 0.6 Relative relative density ρ/ρs

0.7

λ = 8 fm 0.1

0.2 0.3 0.4 0.5 0.6 Relative relative density ρ/ρs

0.7

3.0

10

2.5 2.0

Energy h/tk (MeV)

Growth time t k (10-22 s)

Fig. 3.6. Spinodal boundaries. Left panel: Spinodal boundaries in the –T phase plane for density undulations with a speci1ed wave length . The thermodynamic spinodal boundary corresponds to = ∞. Right panel: Contours of the growth times t for a speci1ed wave length = 8 fm, with the thermodynamic spinodal boundary indicated as well. [These results have been obtained with the Seyler–Blanchard model described in Appendix A (using = 0:75).

T=10 MeV

1.5 1.0

T=0

0.5

m*=0.7m

0.0 0.0

0.1

λ=8 fm 0.2

0.3

0.4

0.5

Average density ρ (in units of ρs)

0.6

8

m *=0.7m

ρ=0.3ρs

6 4 2

T=0 T =10 MeV

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Wave number k (fm-1)

Fig. 3.7. Dependence of growth rates on density, temperature and wave length. Left panel: The collective growth time tk for modes with a wavelength of = 8 fm, for various temperatures T as a function of the density . Right panel: The characteristic energy Ek = ˝=tk for various temperatures T , as a function of the wave number k in matter at the density = 0:3s . (Adapted from Ref. [117].)

More typically, for systems produced in a nuclear collision at intermediate energy, the temperature is several MeV, so the optimal situation does not occur dynamically. Taking as an example T =5 MeV and = 0:4s , which could likely occur, we see that spinodal instability may occur for ¿ 5 fm and that disturbances with = 8 fm have a characteristic time of tk ≈ 10−22 s ≈ 30 fm=c. Finally, we illustrate the resulting growth rates in Fig. 3.7. The left panel shows the density dependence of the growth time tk for modes of wave length = 8 fm, which is close to the most rapidly ampli1ed value. For each temperature below critical, the growth rate rises from zero at the gas-like spinodal boundary as the density is increased, reaches a maximum value, and then drops to zero at the liquid-like spinodal boundary. At moderate temperatures, the growth rate changes little with density. The right panel shows the dependence of the growth rate on the wave number k at the density = 0:3s , which lies in center of the spinodal region. In such a plot, for each given temperature T , the growth rate rises linearly from the origin, exhibits a maximum that depends inversely on T , and then drops to zero at the appropriate maximum value. It should be noticed that there is little temperature dependence below a few MeV.

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305

The model dependence of the instability growth time was investigated by Idier et al. [118] who studied the evolution of instabilities in nuclear matter using a large variety of interactions within the framework of a pseudo-particle treatment. The decomposition times were found to be in the range of 60 –200 fm=c. Hence, in addition to the expected strong dependence on density and temperature, the growth times were shown to also depend signi1cantly on the interaction itself. Therefore, it is important to use realistic forces and in particular realistic ranges (or momentum dependence) in order to have reliable results. It is important to recognize that the numerical method used to produce a smooth density may increase the eHective range of the force, so the eHective interaction should be appropriately adapted to ensure that the numerical implementation has the speci1ed properties. Thus, the range of the force should be reduced in order to compensate for the 1nite lattice spacing and it is important to verify that the numerically obtained dispersion relation in fact correspond to the analytically expected one. 3.1.5. Re-ned analysis of the linear-response treatment The above linear response treatment has been reconsidered by Bo˙zek [119] who demonstrated that the standard solution (3.25) based on the Fourier time resolution of the linearized Vlasov equation (3.22) is incomplete when an initial perturbation at t = 0 is employed. The additional contributions (see below) are usually damped and may be neglected at large times, but their presence early on is essential to recover the correct initial density perturbation. Because of the principal importance of this analysis, we brieDy summarize the main points below. Consider a density perturbation with the wave number k and denote its initial support by gk (p) = fk (p; t =0). Because the system is thus initialized at t =0, the Fourier resolution should include only positive times. This restriction introduces an additional step function in the integrand which produces an additional term in the Fourier resolution of the time derivative. (The usual treatment presented above, in which the Fourier integral extends over all times, may thus be understood as propagating an in1nitesimal perturbation introduced at t =−∞.) With the additional term, the transformed Vlasov equation (3.25) is modi1ed, (−!k + C · k)fk (p) − C · k

9f0 9U k = igk (p) : 9j 9

The solution may be written as an integral in the plane of the complex frequency, ∞+ij d! Gk (!)Ck (!)e−i!t ; k (t) = −i −∞+ij 29 where Ck (!) is the susceptibility de1ned in Eq. (3.26) and where d3 p gk (p) : Gk (j) = 3 (29) k · C − !

(3.32)

(3.33)

(3.34)

In the standard treatment, the integration of (3.33) retains only the two pole contributions from the zeros of the susceptibility, Ck (!± ) = 0, which are the solutions of the dispersion equation. These contributions yield pole

k (t) = −

Gk (!+ ) −i!+ t Gk (!− ) −i!− t e e − : 9! Ck (!+ ) 9! Ck (!− )

(3.35)

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In the spinodal region, where the roots !± are imaginary, k (t) represents the growing and decaying collective solutions. However, Bo˙zek pointed out that the zeros of the susceptibility Ck (!± ) = 0 are not the only singularities to be considered, since the susceptibility has a cut on the real ! axis leading to an additional contribution, ∞ d! Gk (! + ij) Gk (! − ij) −i!t cut − e k (t) = −i : (3.36) Ck (! + ij) Ck (! − ij) −∞ 29 pole

The standard solution k (t) having only growing and decreasing components, diHers at t = 0 from the speci1ed initial condition and one needs to include the additional contribution cut k (t). When this additional contribution to k (t) is included, the decaying component falls oH more rapidly, especially for small values of the growth rate. Thus, the unstable modes gain earlier prominence, while the long term eHect can be expressed as a renormalization of the overlap between the initial perturbation and the unstable mode. Rather than Fourier resolving the time-dependent equation of motion, many studies employ an expansion over a complete set of normal modes in order to propagate the disturbance. Clearly, if a truly complete set of normal modes is used, then the results are not aHected by the present discussion. However, if only the collective states are considered, as is most often the case, then the initial Ductuation cannot be fully represented and the problem remains present. 3.1.6. Role of the damping mechanism If we add the collision integral to the Vlasov equation, i.e. we have an additional damping due to nucleon–nucleon collisions. For example, in the Boltzmann equation of motion, the average eHect of the residual two-body collisions, IO[f], is to provide a dissipative term in the evolution of the phase-space density f(r; p; t). In so far as this term can be treated in the relaxation-time approximation, it is fairly straightforward to include its eHect in the above treatment, as already discussed in Ref. [104]. In the relaxation-time approximation, the collision term takes the form ˜ 0 , where t0 is the local relaxation time and f(r; ˜ p) is the local equilibrium density. IO[f] = −(f − f)=t The former can be estimated as [120] 2 2 −1 −1 2T 2T ; (3.37) t0 ≈ 9 2 VF 0 0 1 + 9 2 jF jF where 0 ≈ 4 fm2 is the nucleon–nucleon scattering cross section and T is the temperature. The local equilibrium density f˜ diHers from the initial density f0 due to the change in the local density, (r), and the presence of a local current density j(r). Exploiting the continuity equation, we obtain the relation m! k = k · j, and then 1nd, in the limit of small temperatures T jF [104], ! 2 jF 9f0 ˜ : (3.38) 1+3 2 2 k·C f ≈ f0 − 3 0 9j k VF Accordingly, the equation for the Fourier component fk (p) becomes !k i i 2jF 9f0 k : fk = − k · C + 1 + 3 2 2k · C !k − k · C + t0 t0 30 9j k VF

(3.39)

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307

3.0

Growth time tk (10

-22

s)

Effect of dissipation on t k λ=8 fm

2.0

t0

1.0

T=0

t0 T = 8 MeV

0.0 0.0

0.1

T = 4 MeV

0.2

0.3

0.4

0.5

0.6

Average relative density ρ/ρs

Fig. 3.8. EHect of dissipation. The eHect on the growth times tk caused by including the dissipative BUU collision term in the relaxation-time approximation, for modes having a wave length = 8 fm in nuclear matter with an average density = 0:3s . In deriving the modi1ed dispersion relation, the temperature has been assumed to be small, for simplicity, and the strength of the dissipative term has been calculated from the simple approximation (3.37) using T = 0 (solid), 4 MeV (dashed), and 8 MeV (dotted), leading to the relaxation times t0 indicated by the arrows on the right. (From [117].)

The form of the eigenfunctions fk (p) then immediately follows and the eigenfrequencies !k are determined by the corresponding modi1ed dispersion equation. Within the spinodal region, the addition of the dissipative term changes the eigenfrequencies, but they remain purely imaginary, !k = i=tk . This is illustrated in Fig. 3.8 for a typical scenario having = 0:3s and = 8 fm. The 1gure shows the growth times tk calculated in the above manner for various strengths of the dissipative term, as governed by the relaxation time t0 . This quantity depends mostly on temperature (see Eq. (3.37)) and the results for 5 = 0; 4; 8 MeV are displayed; the corresponding values of the relaxation time t0 are indicated on the right. Perhaps somewhat counterintuitively, the inclusion of the damping term increases the growth rate [104,117], but the eHect is negligible as long as t0 is larger than the undamped growth time tk . This is because the damping reduces the distortions of the Fermi surface which tend to stabilize the mode; this feature is also manifested by the hydrodynamic growth rates being larger than the collisionless ones. For the fastest modes, the growth times are reduced by only about 10% for T = 6 MeV, which is near the upper limit of applicability of the analytical approximations employed in subsequent developments. Therefore, it is justi1ed to ignore the eHect of the average collision term. Of course, when high accuracy is called for, the dissipation-induced shifts in the characteristic times should be included, and the 1nite temperature should be taken into account. 3.1.7. Evolving systems Spinodal instability is an out-of-equilibrium concept. Up to now we have discussed this phenomenon in the context of uniform systems having an equilibrated kinetic energy distribution. Since spinodal instability may be derived within transport theory, the treatment can be extended to more general situations. In fact, spinodal instability may be more generally de1ned as a collective instability of the density distribution relative to any (static or dynamic) state.

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3.1.7.1. Counterstreaming matter. With a view towards the conditions created early on in energetic nuclear collisions, Larionov et al. [121] considered the development of instabilities of spinodal and two-Dow types arising in collisions of two slabs of nuclear matter using the Vlasov equation with a self-consistent nuclear potential. The expansion of the matter formed following the collision of the slabs leads to a strong longitudinal contraction of the Fermi surface which in turn increases the growth rate for the unstable longitudinal modes after the system has entered the region of spinodal instability. Instabilities occurring at the initial counter-streaming collision stage was also investigated in an approximate manner. The principal conclusion was that the development of spinodal instability is signi1cantly enhanced if the initial counter-streaming form of the distribution function is still maintained at the expansion stage. The occurrence of instabilities in matter with an anisotropic momentum distribution has long been known in electromagnetic plasma physics [122] and the mechanism may induce color 1lamentation in the quark-gluon plasma created in a high-energy nuclear collision [123–125]. 3.1.7.2. Spinodal modes in expanding bulk matter. The growth of spinodal modes in expanding nuclear matter has been studied within a Vlasov framework in Ref. [126], as described in Appendix D, and very recently in Ref. [127]. As a main result, it appears that the eHect of the expansion can be accounted for by employing time-averaged inertial and Landau parameters, as well as a decreasing range for the interaction. While the eHective values of the key quantities determining the dominant collective modes in the fragmentation pattern are thus modi1ed by the expansion dynamics, as one would intuitively expect, the essential characteristics of the spinodal instabilities are preserved in expanding systems and thus the associated fragmentation follows the same phenomenology as in static matter. 3.1.8. Linear response in quantum approaches The above derivations have been performed within a semi-classical description of spinodal instabilities in nuclear matter. It turns out that the fastest growing collective modes, which are those that will become predominant, have wave numbers comparable to those of nucleons at the Fermi surface. For example, for densities ≈ 0:30 and typical temperatures T = 4–5 MeV, the wave numbers of the fastest growing modes are in the order of k ≈ 0:8 fm−1 , while the Fermi wave number for the same density is kF ≈ 1:0 fm−1 . This suggests that the quantal eHects associated with the mean-1eld evolution can inDuence the growth of spinodal instabilities. Therefore, the time-dependent Hartree–Fock equation for the single-particle density matrix (t) ˆ provides a more suitable framework for studying the dynamics of density Ductuations in nuclear systems than the semi-classical Vlasov treatment [105], i

9(t) ˆ ˆ ]; − [h[ ˆ (t)] ˆ =0 : 9t

(3.40)

This equation describes the mean-1eld evolution in terms of an eHective one-body Hamiltonian ˆ ] h[ ˆ = −(1=2m)∇ˆ 2 + Uˆ []. ˆ A perturbation ˆ of the single-particle density matrix away from the 1nite-temperature equilibrium characterized by ˆ0 is propagated by the linearized TDHF equation, i

9ˆ ˆ − [Uˆ ; ˆ0 ] = 0 ; − [hˆ0 ; ] 9t

(3.41)

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where h0 is the mean-1eld Hamiltonian at the equilibrium state and U denotes the Ductuating part of the one-body mean 1eld. This response treatment provides a quantal framework for describing the early evolution of the spinodal instabilities in a 1nite nuclear system, as we will see later. In the context of the present review, we are interested in the quantal eHects on the growth of instabilities in nuclear matter. Furthermore, we assume that the mean-1eld Hamiltonian in the equilibrium state, hˆ0 , is uniform so hence the equilibrium single-particle density matrix is diagonal in the momentum representation, p|ˆ0 |p = (p − p )0 (p), where 0 (p) is a 1nite-temperature Fermi–Dirac function. Carrying out a Fourier transform with respect to time, the mean-1eld part of the linearized TDHF equation in the plane wave representation becomes (! − jp1 + jp1 )(p1 ; p2 ; !) = U (p1 ; p2 ; !)[0 (p1 ) − 0 (p2 )] ;

(3.42)

where (p1 ; p2 ; !) denotes the Fourier transform of the Ductuating part of the single-particle density matrix, (p1 ; p2 ; !) = dt ei!t p1 |(t)|p ˆ (3.43) 2 ; and analogously for U (p1 ; p2 ; !). The situation is relatively simple when the eHective mean 1eld is generated from the local density (r) by convolution with a 1nite-range kernel g(r); U [] = g ∗ U˜ , where U˜ ((r)) is a function of the local density. Then the disturbance of the mean 1eld can be expressed as U (p + 12 k; p − 12 k; !) =

9Uk () (k; !) ; 9

(3.44)

where Uk () = g(k)U˜ () is the Fourier transform of U [](r) and (k; !) is the Fourier transform of the local density disturbance in space and time, d3 p (k; !) = d 3 r dt ei(!t −k·r) (r; t) = (p + 12 k; p − 12 k; !) : (3.45) (29)3 Insertion of Eqs. (3.42) and (3.44) into Eq. (3.45) then yields a quantal dispersion equation for the frequency of the collective mode over the wave number k, d 3 p 0 (p) − 0 (p − k) 9Uk () 1= : (3.46) 9 (29)3 ! − jp−k + jk The roots appear in opposite pairs. They are real for the stable modes and imaginary for the unstable ones, for which we may then write !k = ±i=tk , where tk ¿ 0 is the characteristic growth time of the mode k. When the background distribution is isotropic, 0 (p) = 0 (p), as is most often the case, the eigenfrequencies depend only on the magnitude k. Furthermore, at zero temperature the momentum integral in the dispersion equation can be evaluated exactly, 1 1 1 s + s0 + 1 s − s0 + 1 2 2 + = − [(s − s0 ) − 1] ln ; (3.47) [(s + s0 ) − 1] ln F0 (k) 2 8s0 s + s0 − 1 s − s0 − 1

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Fig. 3.9. Quantal versus semi-classical dispersion relation. Comparison between the semiclassical dispersion relation (3.27) (dashed curves) and its quantal form (3.46) (solid), as obtained with a 1nite-range interaction. The growth rate is shown as a function of the wave number for two phase points well inside the spinodal region. Left panel: Symmetric nuclear matter for 0 = 0:3s and T = 3 MeV. (Adapted from [105].) Right panel: Three degrees of isospin asymmetry, I = 0; 0:3; 0:6 (from top), for 0 = 0:4s and T = 5 MeV. (From Ref. [101].)

where s = !k =kVF and s0 = k=2kF . For small wave numbers, kkF , this expression reduces to the semi-classical zero-temperature dispersion equation (3.28). However, for large k the quantal dispersion relation diHers from the semi-classical one. At 1nite temperatures, the dispersion equations (3.46) and (3.27) may be solved numerically. As an illustration, Fig. 3.9 shows the growth rates obtained with a 1nite-range interaction that has been adjusted to reproduce the ground-state properties of 1nite nuclei [128]. It can be seen that the quantal treatment modi1es the dispersion relation substantially, with the most rapidly ampli1ed wave length being increased from 0 ≈ 5:5 to 8:5 fm. It is interesting to note that the eHect of the quantal treatment is similar to the eHect of increasing the interaction range in the semi-classical treatment. This can be intuitively understood by recalling the Heisenberg uncertainty principle, which states that in order to maintain a spatial resolution of the system must exhibit a momentum dispersion of ]p=˝=k =h=. The associated energy cost, which amounts to ]E = ˝2 k 2 =2m = h2 =2m2 , will act as a k-dependent potential energy, thus constituting an additional 1nite-range eHect. Therefore, for unstable systems, quantal eHects can be included by a suitable rede1nition of the interaction range. 3.1.9. Instabilities in asymmetric nuclear matter As discussed in Section 2, the nuclear case is more complicated than the usual Fermi liquid because of the simultaneous presence of two components, neutrons and protons. A two-component system can readily be investigated within semi-classical mean-1eld transport framework, which then provides two coupled Vlasov equations for the individual phase-space densities fq (r; p; t) [44,101,129–131], 9fq (r; p; t) p 9fq 9U (r; t) 9fq + · − · = 0; 9t m 9r 9r 9p

q = p; n :

(3.48)

In the simplest treatment, the momentum dependence is neglected (so the eHective nucleon mass equals the free value) and one employs a local mean-1eld potential Uq (r; t) with a Skyrme-like

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form [130], Uq = A

0

+B

0

D+1

+C

0

5q +

1 dC() ( )2 − D + D 5q ; 2 d 0

(3.49)

where ≡ n + p and ≡ n − p . Furthermore, the sign of the isospin is 5q = +1(−1) for q = n(p). The parameters A; B; D, and D can be adjusted to reproduce the saturation properties of symmetric nuclear matter as well as the nuclear surface energy [132–134]. The 1nite range of the isovector interaction is governed by the coePcient D ≈ 13 D, which is usually chosen according to Ref. [135], yielding a value close to that used in the SKM∗ interaction [136]. We may now discuss the linear response of the coupled Vlasov equations (3.48). Thus we consider small periodic perturbations, fq (r; p; t) ∼ exp(−i!t), relative to the stationary Fermi–Dirac distributions, fq(0) (p)=1=[exp((jq −q )=T )+1], where jq =p2 =(2m)+Uq(0) and q denote the energy and chemical potential of type-q nucleons. Linearizing Eq. (3.48), we then 1nd − i!fq +

9Uq(0) 9fq 9Uq 9fq(0) p 9fq · − · − · =0 ; m 9r 9r 9p 9r 9p

(3.50)

where Uq (r) is the time-dependent part of the mean-1eld potential. Since we are in matter, ∇Uq(0) = 0 and fq ∼ exp(−i!t + ik · r) in Eq. (3.50). Following the standard Landau procedure [104,130], we can then derive the following coupled equations for the proton and neutron density disturbances, np

(3.51)

pp

(3.52)

[1 + F0nn n ]n + [F0 n ]p = 0 ; pn

[F0 p ]n + [1 + F0 p ]p = 0 ; where C = p=m is the nucleon velocity and 3 9fq(0) d p k·C 2 q (!; k) = ; Nq (T ) h3 ! + i0 − k · C 9jqp

(3.53)

is the long-wave limit of the Lindhard function [104]. Furthermore, the thermally averaged level density is

3 (0) d p 9fq 92 T 2 ; (3.54) Nq (T ) = −2 ≈ Nq (0) 1 − h3 9jqp 12 jqF q

where the Fermi momentum for the specie q is given by pF = (392 q )1=3 ˝) and the corresponding q q Fermi energy is jF = (pF )2 =2m. The zero-temperature single-particle level density for the species q q is then Nq (0) = mpF =(92 ˝3 ). Finally, the quantities q q

F0 1 2 (k) = Nq1 (T )

Uq1 ; q2

q1 ; q2 = n; p

(3.55)

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Fig. 3.10. Spinodal boundaries for various asymmetries. Spinodal boundaries in the –T plane for diHerent isospin asymmetries I , as obtained by solving the dispersion equation (3.57). (From Ref. [44].)

represent the generalization of the usual Landau parameter F0 (k) to a multi-component system. For the particular choice of potential given by Eq. (3.49), these parameters are readily obtained, A D C q1 q2 2 2 + (D + 1)B D+1 + Dk + − D k 5 q 1 5q 2 F0 (k) = Nq1 (T ) 0 0 0

dC d 2 C 2 : (3.56) + (5q + 5q2 ) + d 0 1 d2 20 The eigenfrequencies are the roots of the determinant of the coupled equations (3.51)–(3.52), so the dispersion equation is pp

np

pn

(1 + F0nn n )(1 + F0 p ) − F0 F0 n p = 0 :

(3.57)

Since the present focus is on the unstable modes, the Lindhard function Eq. (3.53) is evaluated for ! = i8 where 8 ¿ 0 is the growth rate of the mode, yielding [104] 1 q (sq ) = 1 − sq arctan ; (3.58) sq q

where sq = 8=(kvF ). The resulting thermodynamic spinodal boundary can be obtained by solving the dispersion equation (3.57) for k = 0 and the result is illustrated in Fig. 3.10. For each value of the nuclear matter asymmetry I = =, the region under the boundary curve is unstable and we have pp

np

pn

(1 + F0nn )(1 + F0 ) − F0 F0 ¡ 0 :

(3.59)

Furthermore, for each phase point (; T ) inside the thermodynamic instability region there exists a wave number k for which the associated spinodal boundary passes through that point. The asymmetry leads to shrinking of the spinodal region, reducing both the critical temperature and the critical density [42]. This is a quite general eHect arising from the fact that the eHective neutron–proton interaction is attractive while the interaction between like nucleons is repulsive [129], thus strengthening the symmetry term in the equation of state for asymmetric nuclear matter and thereby making the instability region smaller. For distortions with a 1nite wave length, k ¿ 0, the dispersion equation Eq. (3.57) can be solved for speci1ed values of the temperature T and the densities p and n (usually given in terms of

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313

Fig. 3.11. Growth rates in asymmetric nuclear matter. Growth rates in two-component nuclear matter as functions of the wave vector k, calculated from the dispersion relation (3.57). Left panel: Three phase points inside the spinodal region for two asymmetries of the initial uniform system, I ≡ =. The insert shows the asymmetry of the perturbation, I ≡ =, as a function of I for the most unstable mode, in the case where = 0:250 and T = 5 MeV. (From Ref. [44].) Right panel: Three degrees of asymmetry, I = 0; 0:3; 0:6 (from top) at T = 5 MeV and 0 = 0:4s , calculated either with (solid) or without (dashed) the Coulomb interaction. (From Ref. [101].)

the total density and the isospin asymmetry I ). The eHect of the asymmetry on the growth rate 8k is illustrated in Fig. 3.11. The appearance of 8k for symmetric matter is as expected from our earlier discussion: The growth rate exhibits a maximum and then drops oH as a result of the 1nite interaction range. From the left panel it can be seen that the region of instability shrinks when the relative density =s is increased from 0.2 to 0.4 and it then shrinks further when the temperature T is increased from 5 to 10 MeV. As both panels bear out, an increase in the isospin asymmetry generally reduces the growth rate, as one would expect already from the shrinkage of the instability region shown in Fig. 3.10. However, while the eHect is signi1cant, it is not large in the region of most rapid growth, where it amounts to around ≈ 30%. It should be noted that the overall reduction of the growth rate is accompanied by a shift of the maximum towards longer wave lengths. While this eHect appears to be only slight at the lower temperature, it is considerable at the higher temperature (where the system is closer to the spinodal boundary) and one may expect a corresponding increase in the size of the resulting (pre)fragments. It is also interesting to note the eHect of the Coulomb interaction, which was also considered in the study by Fabbri and Matera [101]. In general, as seen from the right panel of Fig. 3.11, the Coulomb force stabilizes the modes. In particular, perturbations with a wave length above max ≈ 30 fm become stable, a value that is rather long on the nuclear scale. Thus, the main Coulomb eHect amounts to an overall reduction (by about ≈ 25%) in the growth rates of the most relevant modes. Further understanding of the spinodal decomposition process in a two-component system can be achieved by studying the chemical composition of the evolving instability. For any given k inside the associated spinodal region, a solution of Eqs. (3.51)–(3.52) yields the corresponding unstable eigenvector (p ; n ) (as already discussed in Section 2.3.1, only one of the two eigenmodes is unstable in dilute nuclear matter). One may then consider the associated asymmetry I , i.e. the ratio of ] = n − p and = n + p , which generally deviates from the corresponding ratio I = ]0 =0 in the original uniform system. As is illustrated in the insert of Fig. 3.11 for the most

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unstable mode in the case of = 0:25s and T = 5 MeV, the asymmetry of the eigenmode is smaller than the original asymmetry (one obtains I ≈ 0:5I ). This chemical eHect arises from the fact that the symmetry energy per nucleon increases with density, in the dilute density region considered here, and it therefore grows stronger with increasing total density. An important consequence of this general feature is that the growth of an unstable mode produces high-density regions (liquid phase) that are more symmetric and low-density regions (gas phase) that are less symmetric. Hence, the spinodal fragmentation causes a collective migration of protons from low-density regions to high-density regions (and a reverse migration of neutrons). (This isospin fractionation eHect was already discussed in connection with the liquid–gas phase transition in two-component systems in Section 2.3.) As already stressed in Section 2, the spinodal decomposition occurring in asymmetric nuclear matter at low density arises from a single instability having both mechanical and chemical features. The calculations reported by Li et al. [137] are particularly instructive in this regard, as they show very clearly that the instabilities behave similarly throughout over the spinodal region, irrespective of whether they are being characterized as mechanical or chemical. In conclusion, we note that the fast spinodal decomposition mechanism in neutron-rich matter leads to a mixture of intermediate-mass fragments that are more symmetric than the overall system and a gas of light fragments with an enhanced neutron excess. This general expectation appears to be borne out by recent fragmentation experiments with neutron-rich nuclei [53,54,138], where the light fragments were found to be very neutron-rich. However, it is important to recognize that this feature is expected already on statistical grounds [14]. Thus, the results suggest that the spinodal fragmentation mechanism tends to drive the system towards an assembly of fragments that are in equilibrium with regard to their neutron and proton contents (often referred to as chemical equilibrium), as we shall discuss further in Section 5.3. 3.2. Finite nuclei The above treatments of spinodal instability in nuclear matter can be adapted to the more realistic problem posed by 1nite nuclei. 3.2.1. Thomas–Fermi dynamics The growth of instabilities of 1nite nuclear systems at low densities has been investigated by a number of authors [139–143]. Common to these approaches is the adoption of a suitable energy functional E[] yielding the energy of the nuclear system in the geometry speci1ed by the matter density (r), at a certain temperature T . For simplicity, a Skyrme form is usually employed, with a gradient term to account for the 1nite interaction range. (Appendix A gives the energy functional for the 1nite-range Seyler–Blanchard interaction.) For a Duid dynamical treatment of small distortions around a spherical density 0 (r), it suPces to perform a multipole expansion of the velocity potential. Thus one may consider the linear response to distortions of an elementary form, ˆ = q˙kLM SkLM (r) : S(r; t) = q˙kLM jL (kr)YO LM (r)

(3.60)

Here YO LM = [YLM + YLM ∗ ]=[2(1 + M 0 )] are the real spherical harmonics and the radial form factor jL (kr) is assumed to be a Bessel function. 1=2

0.10 0.75 0.05 0.25 0.00

A = 50 t=0.2 fm t=0.8 fm

k=1

2

k=.1

4 6 Radius r (fm)

8

Growth rate γ (c/fm)

ρ k (fm -3 ) 0

0.10 0.08 0.06 0.04 0.02

δρ k (fm -3 )

P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440 0.10 0.08 0.06 0.04 0.02

L=2

0.10 0.08 0.06 0.04 0.02

L=3

0.10 0.08 0.06 0.04 0.02

L=4

0

315

A = 50

0.5 1.0 1.5 Wave number k (fm-1)

Fig. 3.12. Dilute spherical nucleus. Left panel: The density pro1le of a dilute spherical nucleus containing A=50 nucleons, as obtained by either scaling the equilibrium density (thin curve marked t = 0:8 fm) or by minimizing the energy with respect to the thickness parameter t (thick curve marked t = 0:2 fm) (top), together with the corresponding octupole transition densities for two values of the radial wave number k (bottom). Right panel: The corresponding dispersion relation for three multipolarities L, for either the scaled pro1le (thin curves) or the optimized pro1le (thick curves). (From Ref. [99].)

From this point, the various approaches diHer in their treatment of the nuclear surface. Let us 1rst discuss Ref. [140] which is using a realistic shape for the nuclear density pro1le of Wood–Saxon form, 0 (r) = c =[1 + exp(r − R)=t]. The central density c is chosen well inside the spinodal zone and the radius R = RA is then 1xed by the nucleon number A. Two cases were considered for the surface pro1le: (1) a simple scaling of the surface thickness parameter t in proportion to the expansion factor and (2) a minimization of the energy E0 with respect to t. Fig. 3.12 (left, top) shows the resulting density pro1les 0 (r) for A = 50. Given the pro1le, the continuity equation yields the transition density, 0 kLM (r; t) = qkLM ∇ · : (3.61) ∇S kLM m∗ It should be noted that the wave number k is a continuous variable, contrary to the sharp-surface approximation. For large wave numbers, kR ¿ 1, the velocity 1eld (3.60) approximates those of the bulk instabilities, while the opposite limit, kR ¡ 1, describes surface modes. Fig. 3.12 (left, bottom) shows the radial part of the transition density L (r) for A = 50, and for L = 3 and for radial wave numbers k = 1:0 fm−1 (volume modes) and k = 0:1 fm−1 (surface mode). By proceeding as described in Section 3.1.2, it is straightforward to compute the inertial coePcient ML and the stiHness CL . If the kinetic term is expressed by the usual Fermi integrals, the resulting dispersion relation is neglecting the contribution arising from the deformation of the Fermi surface [96,98,105]. The validity of this approximation can be ascertained in the case of in1nite matter by comparing the Duid dynamical calculations with the result of the semi-classical

P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440 Growth rate γ (c/fm)

316

0.10 0.08 0.06 0.04 0.02

A = 200

ρ = 0.075 fm -3

2

4

0.06 0.04 0.02

Infinite matter

0.5 1.0 k (fm -1)

ρ = 0.06 fm -3

6 8 10 Multipolarity L

12

Fig. 3.13. Dispersion relations. Fastest growth rates for unstable modes in dilute spheres as functions of the multipolarity L, obtained with the optimized surface pro1les for central densities 0 of either 0:075 fm−3 (thick curves) or 0:060 fm−3 (thin curves). The insert shows the corresponding dispersion relation in uniform matter. (From Ref. [140].)

RPA dispersion relation which includes eHects due to distortion of the Fermi surface. It can be seen that the Duid dynamical calculations provide a good approximation, within about 10 –20%, for the semi-classical dispersion relation in the unstable zone. However, it is well known that in the case of high-frequency surface vibrations at normal density, such as the giant-quadrupole mode, the dominant contribution to the restoring force arises from the deformation of the Fermi surface. At lower densities, this eHect is still important and tends to stabilize the surface modes, but it does not aHect the volume instabilities appreciably. Consequently, well inside the spinodal region, where the instabilities are dominated by the unstable volume modes, the eHect of the deformation of the Fermi surface is expected to be small. We will discuss in the next subsection recent Duid dynamics approaches which consider the relaxation toward a spherical Fermi distribution [141–143] and then we shall consider quantal RPA solutions which naturally include the deformation of the Fermi sphere. Fig. 3.12 (right) shows the growth rates tk−1 = |!k | for three multipolarities for a source having A = 50 nucleons expanded to about one third of saturation, 0 = 0:06 fm−3 . (For the system with the optimized surface thickness, only the modes with L = 2 and L = 3 are unstable.) The fact that the growth rate exhibits a maximum at a 1nite k demonstrates the volume nature of the instability. It is interesting to note that the scaled system is generally more unstable than the optimized one. In fact, when the surface pro1le has been optimized, the surface modes occurring for k → 0 are stable. Thus the properties of the surface modes, including their possible instability, depend upon the details of the nucleus surface. Naturally, larger systems have more unstable modes. For example, for A = 200 the most unstable modes have a radial wave number in the range of k0 ≈ 0:7–1:0 fm−1 and a corresponding typical growth rate of t0 ≈ 0:05–0:06 c= fm, as illustrated in Fig. 3.13. As seen from this 1gure, the growth rates of the most unstable modes of the 1nite system are nearly the same for diHerent multipolarities up to a maximum multipolarity Lmax and they are only slightly reduced compared to the bulk instability. However, large multipolarities are suppressed. This result suggests that the unstable spherical system can develop into fragmentation channels of diHerent multipolarity (hence IMF multiplicity) with nearly equal probability (apart from the factor 2L + 1), with the production of small fragments being inhibited. These 1ndings are in agreement with the recent results obtained from Boltzmann–Langevin simulations [144–146] to be discussed in Section 5.2.2. This behavior of the 1nite system is qualitatively diHerent from the development of instabilities in nuclear matter, in which the most unstable mode is characterized by a single characteristic length corresponding to the

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optimal wave number k0 . For 1nite systems, the limited size is the most important restriction, in eHect limiting the magnitude of the multipolarity L. However, once the condition on L, the radial wave numbers tries to adjust itself to the optimum value obtained for in1nite nuclear matter. It should be noted that the Coulomb potential (included in the calculation) has, in general, a very small eHect on the growth rates of the unstable collective modes, except close to the border of the spinodal zone, where it stabilizes unstable modes of very long wave-length. 3.2.2. Expanding nuclear systems For an irrotational and non-viscous Duid [147], it was demonstrated that the dynamical evolution of density waves in an expanding nuclear system diHers markedly from their evolution in an expanded but static system. In particular, statically stable modes may become dynamically unstable, thus in eHect enlarging the spinodal region. This general eHect of expansion on the stability of density waves was con1rmed in a further study that took account of both 1nite size and non-linearities [148]. That study also demonstrated that the fraction of expansion trajectories that lead to multifragmentation changes from zero to one only gradually as the initial conditions are changed from the stable to the unstable region of the –T phase diagram. The expansion of a spherical nucleus was also considered by Csernai et al. [149]. Imposing an overall scaling expansion, they studied in particular the properties of bubble instabilities, both by analytical means and by numerical solution of the Vlasov equation with a self-consistent 1nite-range interaction. On this basis, they investigated the dependence of the spinodal multifragmentation phenomenon on such quantities as the rate of expansion, the radius and diHuseness of the bubble, and the nuclear forces as manifested in compressibility and interaction range. As we shall discuss later (Section 5), numerical simulations of an expanding nuclear system with either the Vlasov equation or the nuclear Boltzmann equation often yield hollow unstable con1gurations [150–153]. The properties of such systems can be investigated within the method described above by employing a monopole distortion of the form (r) = −∇ · (0 ∇j0 (kr)), with its amplitude q0 chosen to produce the desired central depletion. Proceeding in this manner for a system with a central depletion of about 30%, the authors of Ref. [140] have shown that the hollow con1guration is slightly more unstable than the uniformly diluted system. However, this eHect is suPciently small to render the bulk instabilities robust against variations in the pro1le. The Vlasov equation can be considered as a more suitable framework for studying the early development of instabilities in nuclei. Indeed, important eHects, such as the deformation of the Fermi sphere, are automatically taken into account within such a framework. Moreover the eHect of two-body collisions can be included by solving the BUU equation, which includes the average eHects of the two-body collisions. In such a context, the early dynamical behavior of dilute 1nite nuclei has been studied numerically [144–146]. It appears that the early fragmentation process is dominated by relatively few unstable modes and by the lack of light-cluster production, due to the suppression of short wave length perturbation. The maximum multipolarity L that can be accommodated by a 1nite system is found to dependent on the range of the eHective interaction used and on the size of the system considered. The details of the eHective interactions also inDuence the instability growth rates. A detailed discussion of the features of the early fragmentation pattern in spinodal decomposition of nuclei can be found in Section 5.

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Fig. 3.14. Density distortions of a sphere. Density distributions in a sharp sphere (cut across the (x; y) plane) for the lowest collective modes having n = 0 and n = 1 (left and right column, respectively) and multipolarities L = 1 − 4 (from top to bottom). The grey color in the centers and at the borders correspond to the density in the initial undistorted sphere, while the darker (lighter) shades indicate larger (smaller) densities. (From Ref. [142].)

3.2.3. Diabatic eCects The studies described above rely on the assumption that the local momentum distribution maintains a Fermi–Dirac form throughout, as would be true in the adiabatic limit of very slow macroscopic motion. Since this is hardly realistic, it is important to study the eHect of diabatic evolution in which the macroscopic motion brings the microscopic degrees of freedom out of equilibrium. This phenomenon has been studied within the framework of Fermi Duid dynamics [141–143], as we shall now brieDy recall. These studies employ sharp boundary conditions, so that the radial wave number k is discrete and can then be replaced by the node number n. Fig. 3.14 shows the density distributions associated with perturbations of the form nLL (r; t).

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319

Let us for simplicity consider only one mode and let us include damping in the relaxation-time approximation. The Duid dynamics can then be recast into the simple form of a damped oscillator [142], t M qU + C dt e−(t −t )=trelax q(t ˙ ) + Cq = 0 ; (3.62) 0

with the inertia M and the stiHness C being computed as above, i.e. neglecting the contribution from the deformation of the Fermi sphere. (So C represents the adiabatic stiHness coePcient.) The 1nite relaxation time produces the additional restoring term proportional to C , which will eventually be damped out. Using the diabatic single-particle basis {|D} associated with the occupation number nD and the energy CD , we obtain 9 9CD 92 CD 9CD 9n˜D ; C= n˜D = n˜ − C ; (3.63) C =− 2 D 9q 9q 9q 9q 9q D D D where n˜D is the local equilibrium occupation number. (The commutator terms arising from the energy or the occupation number are neglected but, as has been often discussed, these non-diagonal terms may be important and should be investigated.) The various terms in the coePcients C and C can easily be understood, since the 1rst one in C is the restoring force caused only by the energy variation at constant occupation, while C is the modi1cation of this energy due to the variation of both the single-particle energies and the occupation numbers. Considering early times, ttrelax , we obtain the dispersion relation, − M!2 + C

! +C =0 : ! + i=5

(3.64)

The authors of Ref. [142] consider a Skyrme-type eHective interactions, so that they can explicitly derive all terms of the above equation. First, they split the energy into its various components, e[] =

˝2 [] ˆ + eV [] + eW [] + eC [] + eS [] ; 2m∗

(3.65)

where [] ˆ represents the kinetic energy density, eV is the volume term related to the momentumindependent part of the interaction (the parameters t0 ; t3 ; x0 , and x3 in the Skyrme force), while eW is the WeizsUacker term related to density gradients due to the interaction range (associated with the density-dependent terms proportional to t1 ; t2 ; x1 , and x). When the kinetic energy is treated semi-classically, then the WeizsUacker term can also include quantal corrections proportional to the Laplacian of the density [141]. The Coulomb energy is eC [] (was already discussed for in1nite matter). The last term eS [] is new and takes account of the surface around the nucleus. All these diHerent terms will directly contribute to the restoring force. As an example, let us consider the adiabatic limit of an in1nitely small relaxation time, trelax → 0. Then the dispersion relation simply leads to !2 = C=M . Since C is directly related to the energy, the above decomposition can be used to analyze how the mode frequency depends on the various parts. These are illustrated in Fig. 3.15 (left), and one can see that indeed the origin of the instability is the volume term, which is reduced by the kinetic and the 1nite-range. Since a 1nite wave number

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Fig. 3.15. Fermi Duid dynamics. Fermi Duid dynamics for nuclear droplets obtained with the SkM* Skyrme force. Left panel: Various contributions to the restoring force for a quadrupole mode (L = 2) with no node (n = 0) in a Au nucleus. Also shown is the square of the frequency, which diHers slightly from C=M because of coupling to higher node numbers. Right panel: Adiabatic spinodal boundaries in the –T phase plane for various nucleon numbers A. The dashed lines follow isentropic expansions. (From Ref. [142].)

is needed to 1t at least one oscillation inside the drop, the 1nite-range (WeizsUacker) reduces the degree of instability in the 1nite system. As already discussed in connection with in1nite matter, it was pointed out in Ref. [141] that the WeizsUacker term leads to a k 2 correction in the sound speed, thus explaining the high-k suppression in the dispersion relation. All the other contributions, including Coulomb eHects, are almost negligible. Only near the spinodal boarder might they become a more important, since the other terms tend to zero. The spinodal regions obtained with this adiabatic approximation are shown in Fig. 3.15 (right). It can be seen that indeed the spinodal region is reduced (mainly by the WeizsUacker term) as compared with in1nite matter. It should be noted that the above treatment yields so-called adiabatic spinodal which is already much reduced relative to the isothermal spinodal. This feature arises from the speci1c approximations made. In fact, the system is supposed to remain on an adiabat (see the constant entropy lines in Fig. 3.15 (right)), so it can only explore instabilities along this line. Therefore, the curvature anomaly of the thermodynamical potential has to lie on this line for spinodal instability to occur. This restriction leads to a reduction of the instability region. However, on this adiabat the density Ductuations induce a temperature variation so that the system will become inhomogeneous in temperature. A more complete treatment should then also solve the heat propagation coupled to the collective dynamics. Considering the smallness of the typical most unstable wave length and of the nuclear dimension in most cases, the heat propagation cannot be safely neglected. In fact, it is probable that during the typical instability time thermalization of the density irregularities will occur. In such a case, the present local adiabatic treatment will not be valid. Rather, if the thermalization is fast, the isothermal spinodal will determine the sound propagation.

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3.2.4. Conclusions from studies with @uid dynamics In the Duid dynamic approaches it is possible to study many aspects of the instabilities, such as the competition between bulk and surface, the deformation of the Fermi sphere, and the most unstable wave length and multipolarity. However, since rather drastic approximations have been made in order to bring the problem onto a Fermi droplet form, caution should exercised when interpreting the results. In the following, we shall point out when results from more re1ned methods appear to conDict with conclusions drawn from the simpli1ed models described above. The following lessons from Duid-dynamical studies appear to be robust and should not be modi1ed in more complete treatments: • The Coulomb force has little inDuence on the instabilities. • The bulk instabilities are not very sensitive to the nuclear shape but the possible surface instabilities are (for example, a very diHuse density may induce surface instabilities and lead to hollow con1gurations). However, as we shall see, a full quantal treatment of the problem reduces the role of surface instabilities signi1cantly. • The main eHect of the 1nite size is the shrinkage of the spinodal region, caused primarily by the limitation in which wave lengths can be 1tted inside the nucleus. • The radial structure of modes with various L favors a zero-node transition density, since the associated k is already suPciently large and larger node numbers are suppressed. • The role of the damping is two-fold. On the one hand it slows down the collective motion, but on the other hand it allows the system to explore more unstable con1gurations by rearranging the single-particle occupations. Therefore, we may expect that it will enlarge the instability region but at the same time slow down the growth. However, this is still an open question which calls for a complete quantal calculation of the unstable response, including the eHects of non-diagonal terms arising from the non-commutation of density and energy. Finally, we saw that the suppression of large wave numbers imposes a maximum multipolarity of the instabilities but that the semi-classical treatment render the allowed L about equally unstable. This feature will be strongly modi1ed by the introduction of quantal eHects, since a speci1c collective mode must be built out of transitions between diHerent single-particle states. Then the speci1c single-particle structure will directly inDuence the stability of the collective states. In particular, we shall see that the low-lying collective 3− state is always the 1rst to become unstable in a quantum treatment, while it is more often the quadrupole mode that becomes unstable 1rst in a Duid dynamic approximation. 3.2.5. Quantal description of instabilities In 1nite dilute systems quantal eHects are expected to have an even more important role, relative to what is obtained in nuclear matter. Indeed, in addition to the matter properties, the presence of boundary conditions introduces shell eHects which will not subside until the temperature is comparable to the shell gaps (i.e. several MeV). Moreover, the nuclear response and so the instability are strongly inDuenced by approximate symmetries of the single-particle states, such as those arising from the similarity between the nuclear mean-1eld and the harmonic-oscillator potentials. In this section, we extend the quantal RPA treatment presented in the nuclear matter Section 3.1.4 in order to investigate the early evolution of the unstable collective modes for such

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1nite nuclei that may be formed in nuclear collisions. In the 1rst part, in order to make contact with the semi-classical treatments discussed above, we calculate the growth rates of the unstable collective modes by solving a quantal dispersion equation. In the second part we will solve the complete RPA directly, thus going beyond the approximations that make it possible to recast the quantal RPA into a simple dispersion equation. We will also address the question of isospin fractionation in 1nite systems. 3.2.5.1. RPA equations. In order to study systems out of equilibrium, such as those produced in nuclear reactions, the initial density 0 = (t = 0) can always be seen as the solution of a constrained Hartree–Fock equation ˆ 0 ] = 0 ; [h[0 ] − Q;

(3.66)

where Qˆ is a suitable constraining operator for preparing the system in the desired state (e.g. expanded) and is the associated Lagrange multiplier. The stability of such a state against density perturbations is more conveniently studied in the “comoving frame”, by introducing the associated boost transformation [154], e.g. ˆ

ˆ

(t) ≡ e(i=˝)Qt (t)e−(i=˝)Qt :

(3.67)

The TDHF equation is then correspondingly modi1ed, i˝

9 ˆ (t)] ; (t) = [h (t) − Q; 9t

(3.68)

where h is the boosted mean-1eld Hamiltonian. In order to investigate the early evolution of instabilities, we study a perturbation (t) around the reference solution, 0 (t), of the TDHF equation (3.68), with the initial condition 0 (0) determined by the constrained Hartree–Fock equation (3.66). The evolution of (t) ˆ is determined by the linearized TDHF equation in the comoving frame, i˝

9 = [h0 (t) − Q ; ] + [U (t); 0 (t)] = M(t) · (t) ; 9t

(3.69)

where h0 (t) is the comoving mean-1eld Hamiltonian and U (t) represents the distortion of the mean-1eld potential in the comoving reference system. Furthermore, M(t) denotes the instantaneous RPA matrix. The formal solution of this equation can be expressed as (t) = U(t) · (0) ; (3.70) t where U(t)=T(exp[−(i=˝) 0 dt M(t )]) denotes the linearized evolution operator with T being the time ordering operator. However the construction of U(t) is generally a very diPcult task. Usually, only the early evolution of the instabilities in the vicinity of the initial state 0 is considered and so the RPA problem associated with M(0) is solved. We will go beyond this approximation in the next section, where also large-amplitude motion will be discussed. Performing a Fourier transform with respect to time, the RPA equation becomes (˝!= − ji + jj )i|= |j = (j − i )i|U= |j ;

(3.71)

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where i and ji are the occupation number and the energy associated with the constrained Hartree– Fock state |i, respectively. The temperature dependence enters into the calculations through the occupation number i , which is assumed to be given by the Fermi–Dirac function in terms of the single-particle energies ji . The RPA equation (3.71) can be solved using standard techniques [154,26], but to provide a better insight into the problem of instabilities, let us 1rst recast it into an approximate dispersion relation. 3.2.5.2. RPA dispersion relation. The RPA problem can be highly simpli1ed if we can guess a parametrization of the transition density associated with the collective mode under study. As far as spinodal instability is concerned, as in the Duid dynamics approaches, we can use the multipole expansion of sound waves to modulate the spherical reference density 0 (r), = (r) = r|= |r = D= jL (kr)0 (r)YO LM (3; 6) ;

(3.72)

where D= is the amplitude of the collective excitation. Inversion of this relation yields an expression for the amplitude, D= = KL d 3 r FL (r)= (r) ; (3.73) where FL (r) = FL (r)YO LM (3; 6), with FL (r) being a smooth pro1le function that is not orthogonal to the collective transition density jL (kr)0 (r). The normalization factor KL , which is then 1nite, is determined by 1 = d 3 r FL (r)jL (kr)0 (r)YO LM (3; 6) : (3.74) KL A dispersion relation for the frequencies of the collective modes can be deduced from the selfconsistency condition that is obtained by inserting the solution of the RPA equation (3.71) for = , j − i = = |ii|= |jj| = |ii|U= |jj| ; (3.75) ˝!= − ji + jj i; j i; j into the right-hand side of Eq. (3.73), in order to factorize the collective amplitude DL (!). This gives i|9U=9D= |jj|FL |i 1 = ; (3.76) KL ˝!= − ji + jj i; j where the transition 1eld U= is written in terms of the collective amplitude D= as U= = D= 9U=9D= . In principle, this dispersion equation holds for any choice of FL (r), provided that parametrization (3.72) is a good approximation for the density distortions in a multipole mode. Usually one selects FL (r) = 9U=9D= because it yields a symmetric dispersion equation, |i|9U=9D= |j|2 1 = (j − i ) : (3.77) ˝!= − ji + jj KL i; j It is instructive to note that this dispersion equation is equivalent to the RPA result for a separable interaction with the coupling constant given by the normalization factor KL , V (r; r ) = 12 KL FL (r)FL (r ) :

(3.78)

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Fig. 3.16. RPA results for A = 40. Spherical systems with 40 nucleons treated in the harmonic-oscillator representation (left) or the constrained Hartree–Fock representation (right) at zero-temperature. Bottom panels: The density pro1les (r). Top panels: Minimum values of !L2 =|!L | for modes with L = 0; 2; 3 as a function of the root-mean-square size r 2 1=2 . (From Ref. [155].)

The dispersion equation (3.77) makes it possible to determine the frequencies != associated with the collective modes of the nucleus. For a spherically symmetric system, collective frequencies depend on the multipole L and the radial wave number k. In the unstable region, the collective frequencies are imaginary, != ≡ !L (k) = ±i=tL (k). In the calculations reported in Ref. [155], a Skyrme-like parametrization for the eHective mean-1eld potential was used. In order to solve the dispersion equation (3.77), 1rst we need to determine the single-particle representation of the constrained Hartree–Fock problem (CHF). However, it is diPcult to guess a ˆ suitable form of a constraining operator Q(r) that will yield the single-particle representation for a ˆ wide range of densities in the unstable region. An operator of Gaussian form Q(r) = r 2 exp(−r=r0 ) provides a reasonable constraint for preparing the system at low densities. However, in the spinodal region, as soon as the monopole mode L = 0 is unstable, the CHF calculations can not be carried out. For this reason, we take here a more schematic approach and solve the dispersion relation by employing the harmonic-oscillator wave functions and the wave functions of a Wood–Saxon-like potential, instead of the CHF wave functions (see Ref. [155] for more details). We present calculations carried out for systems containing A = 40 and 140 nucleons by including 100 and 120 orbitals, respectively. For the Skyrme force we have used = 1; t0 = 1000 MeV fm3 , and t3 = 1500 MeV fm6 [155]. This force gives a saturation density of s = 0:16 fm−3 and a stiH compressibility of K = 350 MeV. The range parameter c = −126 MeV fm5 is the same as in the corresponding term in the Skyrme-III force. The results for A = 40 are illustrated in Fig. 3.16. Bottom panels show the density pro1les of obtained in either the harmonic-oscillator representation or the Hartree–Fock representation with a Gaussian constraint. The density pro1les are shown in the bottom panels, while the top panels show minimum value of the !L2 =|!L | for various multipolarities, plotted as a function of the root-mean-square size of the system, RRMS = r 2 1=2 . The mutual agreement between the two treatments is rather good. As the size is increased, the collective modes grow steadily softer

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Fig. 3.17. Coulomb eHect on RPA modes. Minimum values of !L2 =|!L | for collective modes of various multipolarity L in spherical systems with either A = 40 (left) or A = 140 (right) as a function of the root-mean-square size. Calculated in the harmonic-oscillator representation without (solid) and with (dashed) the Coulomb force. (From Ref. [155].)

and they turn unstable when !L2 becomes negative. For the octupole mode, this occurs at RRMS ≈ 3:7 fm. As reported in Ref. [155], the radial wave numbers associated with the most unstable modes calculated in the harmonic oscillator representation display a crossover from a surface (small k) to a volume character (large k) at RRMS ≈ 4:0–4:5 fm for all modes. Thus, while the diHerent multipolarities become unstable at diHerent degrees of dilution, they all tend to change acquire a volume character at the same point of dilution. This this feature appears to be governed by the overall geometry of the con1guration (the decisive property presumably being the ratio between the surface thickness and the overall bulk radius). The eHect of the Coulomb force is illustrated in Fig. 3.17 for the most important multipolarities. The Coulomb force decreases the degree of spinodal instability, but the eHect is generally seen to be relatively small. Furthermore, as one would expect, it increases with the number of nucleons. To illustrate the dependence on temperature, results for T = 0 and 3 MeV are compared in Fig. 3.18 for the same two values of A. As generally expected, a hot system is spinodally more stable. Furthermore, the hotter the system the lower the density at which it turns unstable. This behavior is similar to that exhibited byspinodal instabilities in uniform matter. In the bottom panel of Fig. 3.19, the dispersion relations for various multipolarities are plotted as a function of the radial wave number k for A=40 and 140, as obtained for a temperature T =3 MeV in both the harmonic-oscillator and the Woods–Saxon representations. The density pro1les are also displayed. As can be seen, the dispersion relation is not very sensitive to the diHerences in initial conditions, except for the lowest unstable mode. In both cases the growth rates for large values of the radial wave number are suppressed due to quantal and surface eHects, reDecting the fact that the system admits only one radial oscillation in the unstable modes. In the case of A=40, the calculations show that the system is unstable only against quadrupole and octupole deformations. In particular, in the calculations with the Woods–Saxon representation the octupole is the dominant unstable mode. This is consistent with the CHF calculations presented in Fig. 3.16, in which the 1rst mode to become unstable is the low-lying octupole mode. In the larger system (A = 140), several multiple

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Fig. 3.18. Temperature dependence of RPA modes. Minimum values of !L2 =|!L | for collective modes of various multipolarity L in spherical systems with either A = 40 (left) or A = 140 (right) as a function of the root-mean-square size. Calculated in the harmonic-oscillator representation for either T = 0 (dashed) or T = 3 MeV (solid). (From Ref. [155].)

Fig. 3.19. RPA for various multipolarities. RPA results at T = 3 MeV obtained in either the harmonic-oscillator representation (solid) or the Woods–Saxon representation (dashed) for dilute spherical nuclei with A = 40 (left) and A = 140 (right), Top panels: Density pro1les (r). Bottom panels: The growth rates |!L | for various multipolarities L as functions of the radial wave number k. (From Ref. [155].)

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modes up to L = 5 become unstable but, due to quantal and surface eHects, modes with L larger than 5 are strongly suppressed. The unstable modes are predominantly volume modes, with the associated wave numbers in the range of k = 0:6–0:9 fm−1 . In fact, as noted above for low temperatures, when the system expands into the region of spinodal instability, some of the unstable modes 1rst appear as surface instabilities with small radial wave numbers. However, at higher temperatures (T ≈ 3 MeV), the instabilities associated with surface Ductuations are suppressed and the local instabilities of the mean-1eld trajectories are dominated by the volume modes. It is important to note that the maximum value of the growth rate, 8L (k0 ), is nearly independent of the multipole order for L = 2–5, indicating that these modes are ampli1ed at comparable rates. Therefore, provided that they are equally agitated by the Ductuation source, they grow equally prominent in the fragmentation pattern, apart from the geometrical weight 2L + 1. These results are in agreement with the Duid-dynamic calculations of spinodal instabilities (see above). It is also interesting to note that the maximum of the growth rate for a typical multipole mode in a 1nite source is comparable to what is obtained in nuclear matter [155]. 3.2.5.3. Full RPA calculations for dilute nuclei. A more accurate treatment can be found in Ref. [51], where a fully self-consistent comoving solution of the 1nite-temperature RPA problem is given. We shall brieDy summarize this approach here. In terms of the RPA functions ij= = i|= |j and the residual interaction Vil; kj = i|9U=9lk |j, the RPA equation (3.71) takes on its standard form, != ij= = (ji − jj )ij= + (nj − ni )Vil; kj kl (3.79) = : kl

Given an initial density matrix 0 , this equation can be directly diagonalized. To mimic conditions observed in dynamical simulations, 0 can be taken as a self-similar scaling of a hot Hartree–Fock density. Then, the Hartree–Fock equation is 1rst solved for the ground state, [hHF ; HF ] = 0, yielding the single-particle wave functions |’i and the associated energies Ci . Subsequently, a 1nite-temperature density matrix is introduced as HF (T ) = [1 + exp(hHF − CF (T ))=T )]−1 , where CF (T ) is the Fermi level adjusted to the 1xed particle number A. Then one applies a scaling transformation, R(D), which stretches the wave functions in the radial direction by the dilution factor D, r | R(D)’ = D−1=3 r | ’, and the density matrix for hot and dilute system is 1nally given as 0 (D; T ) = R(D)HF (D2 T )R† (D). The corresponding constrained Hamiltonian is thus hO0 (D) = D2 R(D)hHF R† (D), so that the constraint can be identi1ed as −QD = hO0 (D) − h[0 (D; T )]. By construction, hO0 (D) and 0 (D; T ) commute, since they can be diagonalized simultaneously. The eigenstates of the constrained Hamiltonian are given by |i = R(D)|’i and the corresponding energies and occupation numbers are ji = D−2 ji and ni = [1 + exp((Ci − CF (D2 T ))=D2 T )]−1 , respectively. In Ref. [51], such Hartree–Fock calculations have been performed for Ca and Sn isotopes in the coordinate representation using the Skyrme force SLy4 [156,157]. The particle states were obtained by diagonalizing the Hamiltonian in a large harmonic-oscillator representation that includes 12 major shells for Ca and 15 for Sn. Then the RPA equation (3.79) was solved by a direct diagonalization using a discrete two-quasi-particle representation. The top panel of Fig. 3.20 shows the isoscalar strength function calculated for 40 Ca as a function of the dilution parameter D. We observe that the frequencies associated with the dominant modes decrease as dilution becomes larger. At a critical dilution they cross through zero and become

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Fig. 3.20. Isoscalar strength functions. For 40 Ca and 120 Sn are shown contour plots of the isoscalar strength functions associated with the multipolarity L = 2–5 (top panels) and minus the growth rates for unstable modes, −|!= | (bottom panels), as functions of the dilution parameter D. (From Ref. [51].)

imaginary. The (negative of) the associated growth rates |!= | are included in the 1gure. It can be seen that when several branches of the strength function exist, each one may develop towards instability. Consequently, for suPciently large degrees of dilution, there may be more than one unstable mode with a given multipolarity (see L = 2 for Ca and L = 2; 3 for Sn). At moderate dilutions, D ≈ 1:2, only the octupole mode is unstable. In general, density Ductuations with odd multipolarity become unstable at smaller degrees of dilution than those of even multipolarity. This is a genuine quantum eHect: the majority of the particles have to jump only one major shell in order to produce an odd natural-parity particle–hole excitation, while twice this energy is required for an even one. In nuclei at normal density this makes the 3− a strongly collective state at low energy and this state is the 1rst to turn unstable as the system is diluted. The second important feature is that distortions of large multipolarity are hardly becoming unstable. This is due to 1nite-range and quantum eHects that prevent break-up into the correspondingly small fragments. As a consequence, the fastest growth time, tmin ≈ 28 fm=c, occurs for L = 2 at a dilution of D = 1:8. However, deep inside the instability region the octupole mode is almost as unstable as the quadrupole. The 1gure shows clearly that only a small dilution suPces to bring the system well inside the spinodal region, where then the mean-1eld treatment should be as valid as at normal density. Results of similar calculations performed for 120 Sn are shown in the bottom part of Fig. 3.20. Also in this case is the octupole mode the 1rst to turn unstable, which happens at D ≈ 1:1. Moreover, large multipoles are more unstable in 120 Sn than in 40 Ca. This should be expected since L must

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Fig. 3.21. Full RPA calculations for unstable octupole modes in Sn isotopes. Results for unstable octupole modes in Sn isotopes obtained with the full RPA calculation: (a) radial dependence of the form factor at the dilution D = 1:5 for neutrons (solid), protons (dotted), and nucleons (dashed); (b) contour plots of the perturbed neutron density; (c) contour plots of the perturbed proton density. (From Ref. [51].)

increase in proportion to R in order to maintain the wave length of the associated density undulation at a constant value. The behavior of charge-asymmetric systems has also been investigated. In order to gain a deeper insight into the instability properties, it is useful to study the behavior of the RPA solution in coordinate space and consider n (r) and p (r) separately. For diHerent Sn isotopes, Fig. 3.21 shows the radial dependence of the form factor associated with the unstable octupole mode at the dilution D = 1:5, both for neutrons and protons separately for the sum. Contour plots of the perturbed neutron and proton densities are also shown. We observe that neutrons and protons move mostly in phase, which demonstrates that unstable modes have mostly an isoscalar character. Moreover, the collective modes try to restore the isospin symmetry in the dense phase. In fact, in neutron-rich systems (e.g. 132 Sn), protons oscillate even more than neutrons as a means for forming more symmetric fragments and thus reduce the symmetry energy. This eHect is related to the isospin fractionation that occurs in unstable asymmetric nuclear matter, as discussed above. Moreover, proton oscillations are located at the surface of the system, which helps to minimize the repulsive Coulomb energy. However, in the full treatment, the surface and volume instabilities do not appear to be in competition but combine to a unique mode. However, the RPA approach is a local diabatic approach which considers only particle–hole excitations around a given state. If the instability needs a larger rearrangement of the single-particle states, TDHF and RPA approaches will miss it. So this problem would certainly deserve more study. The RPA studies at 1nite temperature and dilutions de1ne the border of the instability region for diHerent unstable modes. In order to draw a more usual phase diagram, the dilution factor can be replaced by a density, = 0 =D3 . Fig. 3.22 shows the resulting phase diagrams for octupole instabilities in Ca and Sn isotopes. Also shown are the boundaries corresponding to a speci1ed growth time tL . The size of the instability region generally increases with A but it still remains

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Fig. 3.22. Border of the instability region. Boundaries of the instability region (solid cures) associated with L=3 collective modes in Ca and Sn isotopes. Also delineated are phase points having the same growth time tL equal to either 100 fm=c (dashed) or 50 fm=c (dots). (From Ref. [51].)

signi1cantly smaller than that for nuclear matter. For example, the limiting temperatures for Ca and Sn are about 4.5 and 6 MeV, respectively, while it is about 16 MeV in symmetric nuclear matter. Moreover, more asymmetric systems are spinodally more stable. For example, in spite of its larger mass, 132 Sn is more stable than 120 Sn. This behavior is also in agreement with the nuclear matter calculations, which indicate that the instability region shrinks in asymmetric nuclear matter [47]. 3.2.6. Concluding remarks about instabilities in -nite systems In this section, we have discussed in detail various aspects of the spinodal instabilities in nuclei and we have seen that the essential features of spinodal instability are not strongly aHected by the 1niteness of the systems. First we showed that the suppression of large wave numbers to quantal corrections and the 1nite size of the range of nuclear forces is manifested in 1nite nuclei by the fact that only low multipoles are unstable and that the radial form factor of the instability does not have any node. Because of the fact that the wave length of the instability should 1t into the nuclear radius, the spinodal region is reduced compared with the in1nite system. Another important factor is the quantum nature of the nucleons. In particular the parity sequence of the various single-particles orbital implies that at moderate temperature the 1rst mode to grow unstable is the low-lying 3− collective state. All those eHects may have experimental consequences, such as the possible importance of ternary 1ssion in the onset of multifragmentation. The various calculations have also shown that when the system is suPciently well inside the spinodal region the various unstable modes of diHerent L have very similar growth rates, so that one might expect that the various multi-IMF channels become populated in a fairly democratic fashion. This feature may lead to a scale invariant Ductuation distribution over a broad range of multipolarities, something that could be investigated experimentally. Furthermore, the various calculations suggest that the Coulomb force plays a marginal role. By contrast, the isospin degree of freedom is important. Since the instabilities are essentially of isoscalar nature, they try to form liquid drops close to 1 stability.

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With regard to the competition between surface and volume modes, full RPA calculations have shown that this phenomenon is merely an artefact of the considered approximation. Indeed, in general only a single mode turns unstable and it is a mix of surface and volume transitions, with the volume character gaining in relative prominence as the system grows. It should also be noted that mean-1eld approaches do not allow a fast rearrangement of many single-particle orbitals and may therefore miss some instabilities. In the case of the opening of such many-particle many-hole channels the dissipation may help the system to explore the new degrees of freedom. More work is needed on this point, in particular in the framework of extended quantal mean-1eld approaches. Finally, all the approaches discussed in this section are restricted to small-amplitude motion. Moreover, they do not address the origin of the irregularities that from the seeds for ampli1cation by the dynamics. Thus, in order to achieve a complete understanding of an instability, one would need to go beyond these limitations and study the source of the Ductuations and propagate them to large amplitudes. This is the subject of the next section. 4. Dynamics of spinodal fragmentation Spinodal fragmentation may occur when the bulk of the evolving nuclear system enters the phase region of spinodal instability. Irregularities in the density are then ampli1ed and, if given suPcient time, this mechanism may cause the system to break up into separate fragments. In the previous section, we focussed on the understanding of the involved instabilities in the linear regime of small amplitudes. In the present section, we go beyond the small-amplitude regime, allowing us to consider the actual formation of fragments. We will thus discuss the conceptual framework and the calculational tools for treating and analyzing spinodal multifragmentation. In particular, the non-linear regime of the dynamics will be carefully investigated and the seeds for the fragment formation elucidated. 4.1. From the linear regime towards chaotic evolution The above discussion of spinodal multifragmentation is based on the basic physical picture advanced in Ref. [106] which utilizes a simple linear analysis of the instabilities. However, a more accurate treatment may aHect the results in important ways, especially for 1nite nuclear systems. In particular, the presence of the self-consistent one-body 1eld renders the mean-1eld equations non-linear and the ensuing dynamics is therefore expected to exhibit features characteristic of chaotic processes [158,159]. At the early stage the unstable modes are independent and their amplitudes evolve exponentially. As the disturbances increase in magnitude, the modes become progressively coupled and the evolution grows correspondingly complicated as the non-linearities gain importance. Concurrently, the overall growth of the density irregularities is being attenuated due to the fact that the density must remain within zero and (in practice) the saturation density. The system then acquires a lumpy appearance and may be roughly described as an assembly of prefragments (into which the system would split if it were free to expand). Subsequently, over a longer time scale, this lumpy structure will proceed to equilibrate. This process consists predominantly in the fusion of the smaller prefragments into ever larger ones, as the system seeks to organize itself into the two coexisting liquid and gas

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phases, leading ultimately towards a few large fragments surrounded by a dilute gas of very light clusters. It is an important feature of the systems formed in nuclear collisions that they are generally endowed with an overall expansion which will then cause the evolution of the “prefragment gas” to be truncated before phase coexistence has been reached, thus eHectively providing a snapshot of the non-equilibrium dynamics. 4.1.1. Non-linear eCects The emergence of a non-linear behavior was studied for a spherical nucleus by Donangelo et al. [160]. Analyzing the instability growth, they demonstrated that higher-order terms have a stabilizing eHect, thus eHectively shrinking the spinodal region. Non-linear evolution cannot be described as merely a quantitative modi1cation of linear dynamics but has a qualitatively diHerent phenomenology. In particular, nuclei can be considered as soliton solutions of the non-linear mean-1eld dynamics. The problem of dynamical instability and clusterization in the breakup of nuclear systems was discussed from such a perspective by Kartavenko et al. [161]. Noting that the existence of solitary waves is essentially determined by the interplay between non-linearity and dispersion, the authors showed that the non-linear terms are associated with the volume (i.e. spinodal instability), while the dispersion terms arise from the surface (i.e. Rayleigh– Taylor instability). Within the illustrative framework of a simple one-dimensional three-level system, it was then found that clusterization may appear in the form of stable solutions (solitons) that are formed as a result of a mutual compensation between the two types of instability. 4.1.2. Chaos and collective motion Generally, the presence of non-linearities makes the nuclear mean-1eld dynamics a candidate for chaotic behavior. The occurrence of chaos would be of great importance since it might provide a justi1cation for the apparent success of statistical approaches in the description of multifragmentation events [162]. Accordingly, the possibility of disorder and chaos in nuclear fragmentation has been discussed by many authors [17,19,163–174]. Chaos may be especially expected for spinodal fragmentation since the density irregularities acquire large amplitudes. However, the studies in Refs. [112,113,126,145,175–177] suggest that spinodal decomposition, simulated through full mean-1eld calculations, is signi1cantly inDuenced by the regular regime of ampli1ed unstable collective modes. In order to elucidate this central issue, considerable eHort has been devoted to the understanding of the dynamical character of the nuclear Vlasov dynamics in the presence of instabilities [162,172,173,175,178–184]. The analysis of chaos in presence of instabilities is a delicate task. In particular, the extraction of the Lyapunov exponent, a standard diagnostic tool in chaos studies, is complicated by the fact that trajectories exhibit an exponential divergence already in the early regular regime. In order to separate the linear regime from the chaotic dynamics, Jacquot et al. [184] have introduced new methods that have also been applied (and extended) by Baldo et al. [182,183]. These studies have con1rmed that any initial disturbance evolves through a regular stage, which is well described by the linear analysis and persists up to ≈ 12 0 , before reaching the ultimate irregular (possibly chaotic) stage [184]. Because of their instructive value, we recall these results in some detail below.

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Fig. 4.1. Non-linear versus linearized dynamics. Snapshots of the density, (x) as obtained by either the complete (non-linear) dynamical Vlasov evolution (left) or the corresponding linearized approximation (right), starting from a random perturbation around an average of 0 = 0:4s . (From Ref. [184].)

The studies in Ref. [184] have been performed in two spatial dimensions, with the potential being always averaged along the y direction, so that the evolution occurs only along the x axis. In order to avoid numerical noise, which may interfere with the chaos analysis, the Vlasov equation has been solved on a spatial lattice (rather than with the more common pseudo-particle method), using the code developed in Ref. [178]. Most studies have employed a zero-range interaction [162,172,178,180,182,183], but since this leads to an inde1nite growth of the dispersion relation, Ref. [184] employed a 1nite-range interaction yielding a more realistic dispersion relation. 4.1.2.1. Vlasov dynamics of randomly initialized @uctuations. In the study by Jacquot et al. [184], an ensemble of initial con1gurations was prepared by perturbing a constant density located well inside the spinodal region, 0 = 0:4s , and the resulting exact Vlasov evolution was then compared with the linear dynamics where each unstable mode evolves independently in an exponential manner. In this comparison, care was taken to match the interaction-range employed in the linear analysis to the eHective range resulting from the 1nite lattice width used in the full calculation. For a typical trajectory, Fig. 4.1 shows the resulting evolution of the density pro1le, (x; t), for the two treatments. The two evolutions are almost identical, thus demonstrating the regularity of the early unstable dynamics and the quantitative validity of the linear treatment. Insofar as the dynamics can be described as a simple exponential evolution of the unstable collective modes, an initial density (x; 0), with vanishing time derivative, (x; ˙ 0) = 0, will evolve as follows, (x; t) = k (x) cosh 8k t ; (4.1) k

where each individual term approaches a pure exponential once t exceeds the corresponding characteristic time tk = ˝=8k . Thus, the density will become increasingly dominated by the fastest-growing mode k0 . Once the local density amplitude grows comparable to the saturation value s , the dynamics

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begins to deviate progressively from the linear behavior, as it ceases to be regular, and this might indicate the onset of chaos. However, a more careful analysis is needed to investigate this aspect. 4.1.2.2. Time correlation and the onset of chaos. A standard method for analyzing complex dynamics is to study the relation between the initial value of a given degree of freedom and its value at a later time. At the early, linear stage of the evolution the relevant degrees of freedom are the normal modes characterized by the wave number k. The associated amplitudes can be obtained by L a Fourier analysis of the density, Ak(n) (t) = L−1=2 0 d x e−ikx (n) (x; t), where n labels an individual dynamical history (an “event”), and its square provides a suitable measure of the agitation of the mode, L (n) (n) 2 Ck (t) ≡ |Ak (t)| = d x12 (n) (x1 ) e−ikx12 (n) (x2 ) : (4.2) 0

The corresponding ensemble average, Ck = ≺ Ck(n) is simply the Fourier transform of the density–density correlation function C(x12 ) ≡≺ (n) (x1 )(n) (x2 ) , L Ck ≡≺ |Ak(n) (t)|2 = d x12 e−ikx12 C(x12 ) : (4.3) 0

The onset of chaos can then be quantitatively analyzed by considering the evolution of the dimensionless ampli1cation coePcient Dk(n) (t) ≡ Ck(n) (t)=Ck(n) (0), which is initially unity. Furthermore, its ensemble dispersion, ]Dk (t), remains vanishing within the linear regime, so this quantity presents a suitable diagnostic tool. Thus, as long as ]Dk (t) remains small (compared to unity) the system is dominated by the regular ampli1cation dynamics. Conversely, when it exceeds unity the correlation with the initial state is lost and the chaotic regime has been reached. The results of such an analysis are illustrated in Fig. 4.2. From the left panels, which display a sample of individual histories, one may see how the modes grow exponentially during the early evolution, at the particular rate 8k , but then start diverging strongly. The early exponential growth is also evident from the center panel showing the ensemble average ampli1cation coePcient DOk . During this stage, the associated ensemble dispersions ]Dk remain very small. It can be seen that the most rapidly ampli1ed mode (the 1fth one having k0 ≈ 0:6 fm−1 ) maintains a regular behavior throughout the time period considered, which amounts to about 1ve times its growth time t0 ≈ 40 fm=c, while the slower modes lose their regularity, and the sooner so the smaller their growth rate 8k . It thus appears that the most unstable collective modes are fairly robust against chaos and remain only weakly coupled to the other degrees of freedom. The regularity of the 1rst stage of spinodal decomposition was also found in Refs. [182–184] from studies of the two-time correlation function. Such a gradual onset of chaos in the presence of robust (unstable) zero-sound modes is reminiscent of the survival of (stable) collective motion at high temperature, as in the case of the hot Giant Dipole Resonance [185,186], and it illustrates the general feature that fully regular and fully chaotic regimes are merely two extremes and generally one should expect to encounter an intermediate situation, the weaker modes becoming chaotic sooner than those that are more robust. 4.1.2.3. Lyapunov exponents. With the above results in mind, one may acquire a deeper insight by studying the Lyapunov exponents associated with the spinodal mean-1eld dynamics, as has been

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335

Fig. 4.2. Mode ampli1cation. The evolution of the ampli1cation coePcients extracted in Ref. [184] for two-dimensional matter with a 1nite-range interaction for various modes, labeled by their node number n. In the upper panel the thick lines correspond to n = 5 (the fastest-growing mode), the dashed lines to n = 3 and the thin grey lines to n = 2; in the lower panel the thick lines correspond to n = 5, the dashed lines to n = 6 and the thin grey lines to n = 7. The left panels present a sample of 100 events, {Dn(i) (t)}, the central panels show the corresponding ensemble averages, DOn (t) and the right panels show the ensemble dispersions, ]Dn (t).

reported in Refs. [180,172,187]. The Lyapunov exponents provide a measure of the rate of trajectory divergence. As a general measure of the distance between two trajectories, one may utilize the diHerence in the spatial densities [180,172], N 1 L 1 (1) d12 (t) = d x|(1) (x; t) − (2) (x; t)| = | (t) − (2) (4.4) c (t)| ; L 0 N c=1 c where N is the number of lattice cells c along the x axis. Then it is possible to de1ne a time-dependent Lyapunov exponent as [180,172], 1 d12 (t) ln ; (4.5) K(t) = t d12 (0) d12 (0)→0 where the two initial densities contain white noise and one must consider the limit where their diHerence tends to zero (see also Ref. [175]). The Lyapunov exponent thus extracted during the 1rst stage of the dynamics is compared in Fig. 4.3 with the expectation based on the linear response approximation. It can be seen that the Lyapunov exponent coincides with the maximum growth rate, 80 , over the entire range of temperatures and densities considered. This result can readily be understood on the basis of the discussion in Section 4.1.2.2. Indeed, since the regularly ampli1ed evolution will become dominated by the exponential growth of the most unstable modes (those having growth rates near the largest value 80 ), the distance between two given trajectories has then the same time dependence, d12 (t) ∼ exp(80 t), once 80 t1, and so the Lyapunov exponent becomes K (1=t) ln exp(80 t) = 80 . Thus, insofar as the 1rst stage of the

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Fig. 4.3. Lyapunov exponents. Lyapunov exponents (squares) extracted with Eq. (4.5), as a function of density (left) and temperature T (right), together with the largest growth rate of the linear response, 80 , as obtained from the associated dispersion relation (curves). (From Ref. [180].)

spinodal decomposition is dominated by the regular ampli1cation of the most unstable modes, the associated Lyapunov exponent should equal the largest imaginary RPA frequency. It is therefore evident, as discussed in Ref. [184], that the Lyapunov exponents cannot be used to signal the presence of chaos. Rather, in order to investigate the post-linear chaotic regime, it is necessary to employ more intricate measures, such as the one discussed in Section 4.1.2.2. 4.1.2.4. Fragment formation and chaos. As discussed above, the early stage of spinodal decomposition is dominated by an almost regular ampli1cation of the most unstable modes and is well described by a linear response analysis. This 1rst stage holds up to rather large density Ductuations of the order of ≈ 12 0 . Chaos then occurs only subsequently, after the ampli1cation of the most unstable modes has saturated, when the local density has largely moved out of the spinodal region and become close to either the associated liquid density, L ≈ s , or the coexisting gas density G s . Thus, at this point, the density is very lumpy, as prefragments have already been formed, and it is tempting to associate the onset of chaos with the start of the coalescence stage. The coalescence process is driven by the preference of the system to reorganize itself into the two coexisting liquid and gas phases. Since this mechanism increases the fragment sizes, the small-k amplitudes are increased and become comparable to those of the most unstable spinodal modes k0 that initially dominated the evolution. On the other hand, the high-k components, which are not populated much by the spinodal instability, tend to become further depleted by the coalescence. The above observations imply that even at large times, when disorder is present, the dynamics is not yet fully chaotic, in the sense that a hierarchy is kept between the various modes as a memory of the instabilities present during the early stage of the fragmentation. These 1ndings may have important consequences since they show that if the fragmentation process is fast enough (such as in the case of open expanding systems) the phase space cannot become fully populated in a statistical manner. In particular, spinodal decomposition does not lead to a population of the small-size region of the primary fragment distribution. Furthermore, the fragment-size distribution contains a large-mass tail arising from mode beating, large-wavelength instabilities, and the late-stage coalescence.

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∆t

∆t

∆t

∆t

∆t

time

Vlasov

Boltzmann

Langevin

Fig. 4.4. Characterization of dynamical models. The various semi-classical treatments of microscopic nuclear dynamics can be characterized by the manner in which the single-particle phase-space density is being propagated from one time step to the next. In the Vlasov treatment, the particles experience only the self-consistent eHective 1eld, leading to a single dynamical history f(r; p; t). At the Boltzmann level, the various possible outcomes of the residual collisions are being averaged at each step, leading then to a diHerent but still single dynamical trajectory. Finally, the Boltzmann–Langevin model allows the various stochastic collision outcomes to develop independently, thus leading to a continual trajectory branching and a corresponding ensemble of histories.

4.2. Exploratory dynamical simulations The recognition of the key role played by instabilities in the dynamics of nuclear fragmentation has led to a variety of model studies and we start this section with a brief discussion of several diHerent microscopic dynamical simulations that illustrate some of the key features of the spinodal fragmentation phenomenon. An early guide to the various microscopic models for intermediate-energy nuclear collisions was presented in Ref. [188]. The various models discussed here are all deterministic in principle and the Ductuations in the outcome arise solely from the use of a sample of diHerent initial conditions or from numerical noise associated with the speci1c implementation of the deterministic equations. Subsequently, we will discuss how this lack of dynamical Ductuations can be remedied by introducing stochastic approaches. However, as far as the fragment formation is concerned, the investigation of systems with Ductuating initial conditions is of key importance. In order to elucidate the key characteristics of the various models, let us consider the class of models that describes the nuclear system at the level of the reduced one-body phase-space density, f(r; p; t). (There is, in principle, one such entity for each spin–isospin component of each particle specie considered but we ignore this straightforward complexity here.) It is instructive to group the various dynamical treatments according to the level of re1nement to which they take account of the residual interactions, as illustrated in Fig. 4.4. Within the semi-classical framework, the equation of motion can be expressed on the following form, 9 O f˙ ≡ f − {h[f]; f} = K[f] = K[f] + K[f] ; 9t

(4.6)

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where the left side describes the collisionless propagation of the individual particles in their common self-consistent one-body 1eld, while the right side expresses the eHect of the residual binary collisions. At the simplest level, the Vlasov treatment, the nucleons feel only the common eHective one-body 1eld, described by the self-consistent one-body Hamiltonian h[f](r; p). The total time derivative then vanishes, f˙ = 0. The corresponding quantal version of this level of approximation is the time-dependent Hartree–Fock treatment. The eHect of residual interaction among the nucleons (and any other hadronic species considered) is included on the right-hand side as a collision term, K[f](r; p). This term has a stochastic character. (For example, the distance a particle travels in the medium before colliding is stochastic, as is the resulting scattering angle.) The next level of re1nement includes only the average part of the collision O term, K[f](r; p), along the lines introduced by Boltzmann. Usually, in nuclear systems, the quantum statistics in taken into account by adding suitable Fermi blocking or Bose enhancement factors in the single-particle 1nal states. This re1nement was 1rst made for electronic gases by Nordheim [189] and Uehling and Uhlenbeck [190]. It was adapted to uniform gases of nucleons, pions, and L resonances about 25 years ago [191] and was subsequently augmented by the mean 1eld [192] to provide a description of collisions between 1nite nuclei. The resulting nuclear Boltzmann equation exists in many implementations that diHer with respect to both the physics input (such as the types of constituents included, the form of their eHective Hamiltonian, and their diHerential interaction cross sections) and the numerical methods employed (whether of pseudo-particle or lattice type) and various names have been employed in the literature, including BUU (for Boltzmann–Uhling– Uhlenbeck), VUU (for Vlasov–Uhling–Uhlenbeck), and Landau–Vlasov. It is important to recognize that these models have still a deterministic character, since only the average outcome of each residual collision is being pursued dynamically. Ensembles of 1nal states may be obtained by propagating suitable ensembles of initial con1gurations. The highest level of re1nement, the nuclear Boltzmann–Langevin treatment, includes also the stochastic part of the collision term, K[f](r; p). This leads to a continual splitting of the dynamical trajectories, as all possible outcomes of the residual collisions are being allowed to develop independently, each one with its own self-consistent 1eld. Thus, even a single initial state, as speci1ed by the one-body phase-space density f(r; p; 0), leads to an entire ensemble of diHerent dynamical histories. 4.2.1. Mean--eld studies of fragmentation Let us 1rst recall the early work by Knoll et al. [193–195]. In that approach, the system is represented as a statistical ensemble of Slater determinants whose initial ensemble average yields the initial one-body density matrix. Each of these pure states is then propagated by standard self-consistent Hartree–Fock dynamics using a Skyrme-type interaction with proper saturation properties (in fact this exploratory study neglects the antisymmetrization and thus employed simple Hartree product wave functions, being then entirely analogous to the semi-classical Vlasov treatment). As a consequence, the Ductuations inherent in the initial one-body 1eld are propagated self-consistently, thus enabling the system to explore any symmetry-violating instabilities of the type leading to fragmentation. Although the authors of Refs. [193,195] made no analysis of the speci1c instabilities, whether spinodal or of another character, their results do illustrate the importance of considering an ensemble of dynamical evolutions when instabilities are present, as initially very similar states may then develop

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339

time

STABLE

UNSTABLE

Fig. 4.5. Ensemble propagation. These sketches illustrate the eHect of propagating an ensemble of macroscopically similar initial states, each one experiencing an independent self-consistent dynamical evolution. When the system is stable, the trajectories remain closely bundled throughout and the states are macroscopically similar (left panel), while instabilities lead to divergences that result in distinct trajectory bundles describing qualitatively diHerent 1nal states (right panel).

into qualitatively diHerent 1nal con1gurations. It should be noted that in this type of approach, the only stochastic element enters when the initial states are selected; from there on, the evolutions are fully deterministic, as each initial state evolves in its own self-consistent one-body 1eld. The approach can be considered as an extension of the familiar 1nite-temperature TDHF treatment which propagates a single one-body density matrix representing the entire ensemble of trajectories in the single common (ensemble averaged) one-body 1eld and which is therefore expected to be accurate only when the ensemble consists of macroscopically similar con1gurations. This latter feature is illustrated in Fig. 4.5, which shows that the trajectories for an ensemble of macroscopically similar initial con1gurations remain closely bundled when the system is stable and they may thus be well approximated by a single average trajectory, whereas instabilities may split the ensemble into several subensembles that represent macroscopically diHerent channels. 4.2.2. Nuclear Boltzmann dynamics The nuclear Boltzmann model was used by Batko et al. [150] to study the role of instability growth for fragment formation in expanding nuclear geometries. They considered the development of initially compressed or heated spherical nuclei. On the basis of the nuclear BUU model, it was found that a heavy nucleus initially compressed to 2–3 times normal density expands to a bubble-like quasi-stationary unstable con1guration that then has time to coalesce into massive bound fragments. This spontaneous breaking of the spherical symmetry is due to the numerical implementation of the semi-classical transport. Indeed, Batko et al. used the pseudo-particle method which is inherently endowed with numerical noise because of the associated irregular coverage of the phase space. Such noise breaks the macroscopic symmetries and may thus trigger the formation of clusters. It was demonstrated that the character of the outcome depends on the employed number of pseudo-particles per nucleon N, a purely numerical parameter in the model: The larger the value of N, the smaller the numerical Ductuations in the density distribution and, consequently, the better the initial spherical symmetry is preserved, thus requiring longer time for the instabilities to grow, ultimately allowing the bulk of the system to recontract without fragmenting (provided that the initial compression is suPciently modest).

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Fig. 4.6. EHect of instabilities on the trajectory bundle. The central density as a function of time for 70 MeV=A Ca + Ca for bundles of dynamical trajectories resulting from diHerent initial placings of N pseudo-particles per nucleon, with either N = 40 (a) or N = 100 (b). (From Ref. [196].)

That study suggests the existence of a speci1c nuclear multifragmentation process by which an initially compressed system disassembles into several massive fragments. The characteristic feature is a decomposition leading to an unstable structure whose global evolution is suPciently slow to allow the instabilities to manifest themselves, resulting in a clusterization of the structure into disjoint prefragments. In this calculation the instabilities are continually triggered by the numerical noise, whereas the symmetry breaking in the treatments discussed in Section 4.2.1 has arisen from the irregularity of the initial con1guration. This novel multifragmentation process displays an intricate interplay between the time scale for the global expansion dynamics and those for the triggering and ampli1cation of the various unstable modes and it was noted that its qualitative identi1cation and quantitative exploration could provide important new experimental information on nuclear dynamics. The occurrence of instabilities along the dynamical evolution of nuclear reactions and the sensitivity of the 1nal outcome to the number of pseudo-particles employed was subsequently investigated by Colonna et al. [196]. For an ensemble of similarly prepared collision systems, such as 70 MeV=N 40 Ca + 40 Ca, the authors of extracted the central nucleon density as a function of time and studied the resulting bundle of curves. As illustrated in Fig. 4.6, at early times the curves are all very similar, but they then rather suddenly start to exhibit a growing divergence. This general behavior is interpreted as follows: early on the systems remain well away from any instabilities and they therefore all have very similar evolutions, while they diverge after entering the region of instability, where clusterization is triggered. The rate of the trajectory divergence (the growth of the bundle width) is related to the growth time of the instabilities present. While it is rather robust against modi1cations of the numerical implementation, thus con1rming that the eHect reDects an inherent property of the system, the actual magnitude of the bundle width depends signi1cantly on the number of pseudo-particles used, i.e. on the level of numerical noise. Further studies have elucidated the competition between the expansion of the system and the spinodal growth of the Ductuations [197].

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In order to better understand such spinodal multifragmentation and its interplay with the global dynamics, it is important to compare the growth times for the dominant unstable modes with the other relevant time scales. In the linear regime the growth times vary from 30 to 200 fm=c or more, as discussed in the previous section, and in the next section we will compare this time scale with that of the global dynamics of the system. However, it is important to recognize that the growth rates are not the only important quantities, since a given instability will develop only if is being triggered. So the magnitude and spectral distribution of the irregularities provide the seeds of the spinodal decomposition and are thus important, as we shall see below. 4.3. Boltzmann–Langevin model An essential shortcoming of the nuclear Boltzmann model is the fact that the propagation of the one-body density is, in principle, entirely deterministic. This lack of stochasticity precludes the spontaneous appearance of Ductuations and thus renders the description inadequate when bifurcations and instabilities are encountered in the dynamics. The principal diHerences between the various types of dynamical model were illustrated in Fig. 4.4. The description of nuclear dynamics by means of stochastic models has become a rich and vigorous 1eld which was reviewed several years ago by Abe et al. [198]. In the present review, we seek to include only those contributions that have speci1c relevance to the developments of the models used to treat spinodal multifragmentation. An important early advance was made by Bixon and Zwanzig [199] who extended the theory of hydrodynamical Ductuations to non-equilibrium scenarios by deriving a Boltzmann–Langevin equation in which the stochastic behavior of the Duid arises from the Ductuating force incorporated into the standard Boltzmann equation. The characteristics of this force can then be explicitly related to the collision kernel and, furthermore, it was shown that the Boltzmann–Langevin equation yields a correct description near equilibrium, where it reproduces the standard hydrodynamical results for the Ductuations in pressure and heat current (a review is given in Ref. [200]). The application of this approach to nuclear dynamics was pioneered by Ayik [201] who derived the statistical properties of the residual interaction by the general projection technique of quantum statistics. Subsequently, Ayik and Gregoire [202,203] applied that general scheme to the particular case when the residual interaction is approximated by the Uehling–Uhlenbeck collision term. This approach considers the evolution of the one-body density as a generalized Langevin process akin to the motion of a Brownian particle, with the appropriate dynamical variable being the entire one-body density f(r; p) in place of the momentum of the Brownian particle. From the same starting point, Randrup and Remaud [204] derived the equivalent Fokker–Planck transport equation for the distribution of one-body densities, 6[f], and they presented expressions for the associated transport coePcient functionals. This type of treatment was subsequently developed further by Chomaz et al. [178,179] in connection with the development of a lattice simulation method for the solution of the Boltzmann–Langevin transport equation. A more formal derivation of the Boltzmann–Langevin equation was undertaken by Reinhard et al. [205] using Green’s function techniques in the real-time path formalism, the key point being to allow the free Green’s function to contain two-body correlations, thus providing a source of stochasticity. This study also presents a careful discussion of time scales and order counting and, in particular, it is shown that the derivation requires the Ductuating part of the residual interaction to be weak and

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that the associated collisions must complete before essential changes can occur in the mean 1eld. It may be interesting to note that the Boltzmann–Langevin equation can alternatively be derived from the stochastic time-dependent Hartree–Fock equation of motion by averaging over subensembles with small Ductuations [206,207]. 4.3.1. Basic features As noted above, the Boltzmann–Langevin (BL) equation of motion for the reduced one-body phase-space density can be written on compact form as f˙ = K, see Eq. (4.6). The left-hand side represents the collisionless propagation of f(r; p) in its self-consistent mean 1eld described by the eHective single-particle hamiltonian h(r; p); this part of the problem was discussed in Section 3 and need not concern us here. The right-hand side of the Boltzmann–Langevin equation represents the eHect of the residual interaction. In the present context it is associated with the two-body collisions between the constituent nucleons. The collision term, K(r; p), can be decomposed into two parts. O p), is the familiar collision term entering in the nuclear Boltzmann equation; it The 1rst part, K(r; represents the average eHect of the residual collisions and is thus deterministic in character. The remainder, K(r; p), is a qualitatively new term which Ductuates in a stochastic manner. In the simple physical scenario where the residual interaction can be considered as binary collisions that are well localized in space and time, the average part is given by the familiar Uehling–Uhlenbeck modi1cation of the Boltzmann collision term [189,190], O p1 ) = g K(r; W (12; 34)[fO 1 fO 2 f3 f4 − f1 f2 fO 3 fO 4 ] ; (4.7) 234

where fi is a short-hand notation for f(r; pi ; t) and fO ≡ 1−f is the associated Fermi blocking factor. For simplicity, we consider here only spin and isospin saturated systems and the associated spin– isospin degeneracy factor is then g = 4. The basic transition rate is simply related to the diHerential cross section for the corresponding two-body scattering process, d (p1 + p2 − p3 − p4 ) ; (4.8) W (12; 34) = v12 d` 12→34 with v12 ≡ |C1 − C2 |, and it thus has corresponding symmetry properties, W (12; 34) = W (21; 34) = W (34; 12) :

(4.9)

Since it arises from the same elementary two-body processes, the stochastic part of the collision term is fully determined by the basic transition rate as well, a manifestation of the Ductuation– dissipation theorem. With the collisions assumed to be local in space and time, the correlation function for the Ductuating part of the collision term is of the following form, ≺ K(r; p; t)K(r ; p ; t ) =C(p; p ; r; t)(r − r )(t − t ) ;

(4.10)

where ≺ · denotes the average with respect to the ensemble of possible trajectories resulting from the current one-body density f. Furthermore, for elastic scattering, the correlation kernel is given

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by [202] C(pa ; pb ; r; t) = ab

343

W (a2; 34)F(a2; 34)

234

+

[W (ab; 34)F(ab; 34) − 2W (a3; b4)F(a3; b4)] ;

(4.11)

34

with the short-hand notations ab ≡ h3 (pa − pb ) and F(12; 34) ≡ f1 f2 fO 3 fO 4 + fO 1 fO 2 f3 f4 . The symmetry properties (4.9) of the transition rate ensure that the following sum rules hold, C(p1 ; p2 ; r; t)p1 = C(p1 ; p2 ; r; t)p1 = 0 ; (4.12) 1

2

C(p1 ; p2 ; r; t)p1 =

1

C(p1 ; p2 ; r; t)p2 = 0 ;

(4.13)

C(p1 ; p2 ; r; t)j2 = 0 ;

(4.14)

2

C(p1 ; p2 ; r; t)j1 =

1

2

where ji = pi2 =2m is the kinetic energy for a speci1ed momentum. These sum rules express the fact that each of the elementary binary collisions conserves particle number, momentum, and energy, respectively. 4.3.2. Linearization of collective stochastic dynamics The study of the small-amplitude response in stochastic one-body theories can yield considerable insight into the dynamics of the collective modes. The present discussion is based on the original treatment given in Ref. [112] and the subsequent further analysis in Ref. [208]. We thus wish to study the evolution of small amplitude deviations from the average phase-space trajectory, f(s; t) = f0 (s; t) + f(s; t), where s denotes the phase-space point (r; p) for D spatial dimensions, with ds=h−D d D r d D p being the associated dimensionless volume element. The reference O 0 ]. To density f0 (s) satis1es the non-Ductuating Boltzmann equation, 9f0 =9t = {h[f0 ]; f0 } + K[f leading order, the deviation f(s) then follows the linearized equation of motion, 9 f = −iM[f0 ]f + K[f0 ] ; 9t

(4.15)

where K[f0 ] represents the Ductuating part of the collision term and where the extended RPA O matrix M is de1ned by −iM[f0 ]f = {h[f0 ]; f} + {h[f]; f0 } + K[f]. Any general disturbance can be expanded on the eigenvectors f= of the associated RPA matrix operator, f(s; t) = A= (t)f= (s) : (4.16) =

Here f= (s) solves the equation Mf= = != f= , where the frequency != is in general complex, != = E= − iM= . When the mode is stable, M= is positive and represents the damping width due to the action of the collision term. Conversely, when the mean-1eld treatment renders the mode unstable, its frequency is purely imaginary.

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Substitution of expansion (4.16) into the linearized equation (4.15) yields the following solution for the amplitude A= , t −i!= t i != t A= (0) + ; (4.17) A= (t) = e dt B= (t )e 0

where A= (0) is the initial amplitude of the disturbance, O= f ; f(0) ; A= (0) =

(4.18)

with g; h ≡ ds g(s)∗ h(s) being the scalar product in the one-body density Liouville space. In Eq. (4.17), the noise term B= is the projection of the Ductuating term K(s; t) onto the particular eigenmode, O== f ; K(t) ; (4.19) B= (t) =

with O being the inverse of the overlap matrix, (O−1 )= = f= ; f . Since the Ductuations are (assumed to be) Markovian, the noise correlation is local, ≺ B= (t)B (t )∗ =2D= (t − t ) ;

(4.20)

where the diHusion coePcient in the eigenmode representation is O== L= O ; D= =

(4.21)

=

with L= being its matrix elements with respect to the eigenmodes, L= = f= ; Df = ds ds f= (s)∗ D(s; s )f (s ) 1 = 2

d=12;34 {f= (1)∗ [f (1) + f (2) − 2f (1 )]

+ f= (1 )∗ [f (1 ) + f (2 ) − 2f (1)]} :

(4.22)

In the last relation, L= has been expressed directly in terms of the basic two-body collision processes. Note that in equilibrium the two terms in the integrand are equal, since detailed balance ensures that then d=12;34 = d=34;12 . With the above quantities de1ned, it is now possible to express the general temporal correlation function for the amplitudes of the disturbance, = (t1 ; t2 ) ≡ ≺ A= (t1 )A (t2 ) −i!= t1 +i!∗ t2 = (0; 0) + =e

0

t¡

dt 2D= (t)e

i(!= −!∗ )t

;

(4.23)

where the upper limit is the earliest of the two speci1ed times, t¡ = min(t1 ; t2 ). When one of the time arguments vanishes, = (t; 0) = exp(−i!= t)= (0; 0), the information carried concerns the eigenfrequencies only. In particular, the correlation function for f(s) can be

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expressed as f (t) ≡ (s; s ; t)= ≺ f= (s; t)f (s ; t) f= (s; t) = (t; 0) f (s ; t) = e−iE= t=˝−M= t=˝ f= (s) = (0; 0) f (s ) : = =

(4.24)

=

Thus, for stable systems f attenuates in time due to the collisional damping of the modes, while for unstable systems f will soon become dominated by the exponential growth of the unstable modes. The expression also displays the Landau damping (which is due to the beating of close frequencies) and, if the collisional damping is suPciently small, the PoincarZe recurrence phenomenon. Of special interest is the equal-time correlation function, = (t) ≡ = (t; t), which satis1es a simple feedback equation of motion (the Lalime equation [112]), d = = −i!= = + 2D= ; dt with != ≡ != − !∗ . Thus, when the noise term D= is constant in time, we have 2D= = (t) = −i (1 − e−i!= t ) + = (0)e−i!= t : != The diagonal Ductuations (the variances) then evolve as follows (with D=var ≡ D== ), var D= t= (1 − e−2t=t= ) + == (0)e−2t=t= → 2D=var t= ; == (t) = D=var t= (e2t=t= − 1) + == (0)e2t=t= → (2D=var t= + == (0))e2t=t=

(4.25)

(4.26)

(4.27)

for stable (M= ¿ 0) and unstable (M= ¡ 0) modes, respectively, and their behavior is characterized by the time constant t= = ˝=|M= |. In the former case, the equilibrium variance equals the amount of Ductuation generated by the source over a time interval equal to the relaxation time t= (and it thus independent of the initial situation, as it should be). In the latter case, the asymptotic behavior amounts to an exponential ampli1cation of the initial Ductuations plus the amount of Ductuation generated by the source over a time interval equal to the growth time t= . While the above developments are quite general, we now focus on unstable matter prepared with a phase-space density that is uniform in position and rotationally invariant in momentum. The reference system described by f0 is then stationary in time and uniform in space and, accordingly, the eigenmodes are of plane-wave form, fk= (r; p) = fk= (p) exp(ik · r), so we need consider only one wave number k at a time. Although the eigenfunctions {fk= (p)} form a complete set in momentum space, they are not mutually orthogonal and it is useful to introduce the matrix o= k as the inverse of their overlap matrix, −1 = = (ok ) = fk ; fk = dp fk= (p)∗ fk (p) : (4.28) It depends only on the magnitude k =|k| due to the rotational invariance. The amplitudes in expansion (4.16) of the disturbance fk (r; p) are then given by = ok fk ; fk (t) = qk ; fk (t) ; (4.29) A=k (t) =

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where the functions {qk= (p)} constitute the dual basis, qk= ; fk = = . (They are given explicitly as qk= (p) = Qk= (p)=Qk= ; fk , where Qk= (p) = 1=(k · C − !=∗ ).) Furthermore, the noise term (4.21) is simply the projection of the basic diHusion coePcient onto the dual basis, = o== : (4.30) Dk= = qk= ; Dqk = k fk ; Dfk ok =

When the system is situated within the spinodal boundary associated with the magnitude k of the considered wave number, it has two modes with imaginary frequency, !=±i=tk , exhibiting exponential growth and decay, respectively. Since the ampli1ed mode grows dominant at times exceeding tk , it may suPce to retain only the two collective modes in the treatment. This approximation (which was made in the original analysis [112]) amounts to replacing the full matrix o= by the 2 × 2 matrix o˜= involving only the two collective modes, fk± (p). This approximation receives some support from the linear-response analysis made by Bozek ˙ [119] (see Section 3.1.4) which revealed that there are two distinct contributions, one coming from the zeroes of the susceptibility, i.e. from the collective states, and a second related to the singularity of the susceptibility, i.e. the non-collective states. The fact that the former is dominant at large t supports suggests that it may suPce to retains only the collective modes. When this approximation is adopted, the expansion of the disturbance with the wave number k contains then only the two collective terms, − − + fk (p; t) = A+ k (t)fk (p) + Ak (t)fk (p) :

(4.31)

and the correlations between the amplitudes, k (t) ≡≺ Ak (t)Ak (t) , evolve as follows, 2 d ++ = 2Dk++ + k++ : k++ (t) = Dk++ tk (e2t=tk − 1) ; dt k tk

(4.32)

d +− = 2Dk+− : k+− (t) = 2Dk+− t ; dt k

(4.33)

2 d – k = 2Dk– − k++ : k– (t) = Dk– tk (1 − e−2t=tk ) ; dt tk

(4.34)

where it should be noted that Dk++ = Dk– and Dk+− = Dk−+ . A comparison of the approximate treatment, which includes only the collective modes, with the result of performing a full projection onto the dual basis, Eq. (4.30), was carried out in Ref. [208] on the basis of a realistic interaction [35]. The result for the most important source term, Dk++ , is shown in Fig. 4.7. As it turns out, the two treatments yield very similar results, except for an overall larger magnitude of the full treatment. For the most rapidly ampli1ed mode, having a wave length of 0 ≈ 8 fm, the approximate source term is about 30% below the exact result. In order to compare the two methods with regard to their predictions for the density Ductuations, we consider the Fourier component of the density variance, 2 k (t) ≡ ≺ |k | = d 3 r12 e−ik·r12 ≺ (r1 )(r2 ) 2t ≈ k++ (t) + k– (t) + 2k+− (t) = 2Dk++ tk sinh + 4Dk+− t ; tk

(4.35)

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347

-1.0

Log10(D++) (1022s fm3)

-1.5

λ=8 fm

T=6 MeV

-2.0

T=4 MeV

-2.5

-1.5 -2.0

λ=6 fm

T=4 MeV λ=8 fm

-2.5 -3.0 0.0

0.1 0.2 0.3 0.4 0.5 Average density ρ (in units of ρs)

0.6

Fig. 4.7. Comparison of source terms. The noise term Dk++ for the ampli1ed collective mode having a wave length 29=k, in nuclear matter prepared with a uniform density and with a speci1ed temperature T , using either the complete dual basis (solid) or its collective truncation (dashed). The calculations have been done with the approximate formulas developed in Ref. [117]. The lower panel considers the typical temperature T = 4 MeV and illustrates the dependence on the wave length , while the upper panel keeps the wave length 1xed at = 29=k = 8 fm, near which the most rapid ampli1cation occurs, and illustrates the temperature dependence.

where only the two collective modes have been considered and it has been used that Dk++ = Dk– and Dk+− = Dk−+ . At the spinodal boundary, where the collective frequency tends to zero, the two collective modes become identical. Within the approximate projection method, both Dk++ and Dk+− then diverge, but, as it happens, Dk++ +Dk+− ≈ tk−1 as tk → ∞. So the resulting density variance k tends to zero at the boundary, which is physically reasonable. By contrast, with the exact projection method the mixed source term remains regular as the boundary is approached and so it cannot cancel the divergence of the diagonal term. The density variance then diverges at the boundary where tk → ∞, a physically unreasonable result. (The failure of the exact projection near the spinodal boundary results from the fact that it is expected to be accurate only at late times, ttk , but this situation is never reached at the boundary where tk → ∞.) It would obviously be of interest to cure this problem, so that the exact projection method would be reliable also near the spinodal boundary and, in general, in all cases where tk diverges. The results obtained for k with the two methods are compared in Fig. 4.8. It may be noted that k is a rather Dat function of k at the very early times, t ¡ tk0 , before the ampli1cation of the instabilities has manifested itself. This feature reDects the fact that the modes are being agitated almost equally by the Langevin Ductuation term K, which thus has the character of white noise. At later times, t ¿ tk0 , this democratic behavior is changing, as those modes that have the shortest ampli1cation time grow progressively dominant, making k ever more narrowly peaked around k0 . 4.3.3. Lattice simulations of Boltzmann–Langevin dynamics The dynamical evolution of unstable nuclear matter has been elucidated by lattice simulations of the Boltzmann–Langevin dynamics [162,181]. In this method the stochastic part of the collision number is simulated directly on a lattice in phase space. The mean-1eld evolution in treated by a standard matrix technique. So far calculations have been made only in two dimensions due to the

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log10 σk(t ) (fm-3)

1.0

t =4

ρ =0.3 ρ s t =3

0.0 t =2

-1.0 t =1

-2.0 -3.0 0.0

0.2

0.4

0.6 0.8 1.0 1.2 Wave number k (fm-1)

1.4

1.6

Fig. 4.8. Dependence of density Ductuations on wave number. The variance of the density Ductuations associated with the collective modes, k , after a given time t = 1; 2; 3; 4 × 10−22 s has elapsed, as a function of the wave number k = 29=, for the density = 0:3s at the temperature T = 4 MeV, using either the complete dual basis (solid) or its collective truncation (dashed), as in Fig. 4.7. (From Ref. [208].)

Fig. 4.9. Evolution of the density pro1le in a box. An initially uniform 2D system at half the saturation density, = 12 s , and with a temperature of T = 3 MeV, is evolved with the Boltzmann–Langevin lattice method (see text) and examined at four subsequent times t. Left panel: The density pro1le associated with one particular dynamical history. Right panel: The variance k as a function of the mode number K = kL=29, extracted from an ensemble of histories. (Adapted from Ref. [176].)

heavy numerical eHort required. We summarize here the studies reported in Ref. [162], in which the system is con1ned within a square, with periodic boundary conditions imposed. Furthermore, in order to simplify the analysis, the Ductuations along the y direction are averaged out so that only the evolution along the x direction is interesting. Fig. 4.9 (left) shows the time evolution of the spatial density for a system initialized at half the saturation density, = 12 s , with a temperature of T = 3 MeV. The various stages from the

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Fig. 4.10. Growth of the most unstable mode. The time evolution of the most unstable mode, k0 , as given by k0 (t), the Fourier component of the spatial correlation function, as a function of the time elapsed since starting from a uniform system having = 12 s and T = 3 MeV. (Adapted from Ref. [176].)

uniform density towards clusterization are reDected in the snapshots shown in the 1gure: Since the collision term only produces rearrangements in the local momentum distribution and therefore remains rather uniform early on (1rst frame). The initial uniformity of the spatial density is then destroyed as these rearrangements are being propagated; the resulting density Ductuations have an irregular appearance, in reDection of the random nature of the momentum rearrangements (second frame). The subsequent self-consistent response by the eHective 1eld then starts the ampli1cation process and the magnitude of the density undulations grow, with the most favored wave lengths gaining prominence (third frame). The system then acquires a clusterized appearance, in which the size distribution of the prefragments is not statistical but rather reDects the most rapidly growing modes (fourth frame). (Because of the 1xed size of the box, the further development towards multifragmentation is hindered and the density Ductuations will ultimately reDect the corresponding statistical equilibrium.) As noted in Eq. (4.35), the strength distribution of the density Ductuations, k (t), is simply related to the Fourier components of the spatial density and it can thus be readily extracted from a (suitably large) sample of histories. The result is shown in Fig. 4.9 (right). (The higher harmonics seen at the earliest time, when the undulations are very small, result from the 1nite lattice employed and are numerically unimportant.) It can be seen that the dynamical evolution leads to a predominance of the most unstable modes, causing k (t) to have a peaked character. It is an important result of the study in Ref. [176] that an alternative treatment based on the pseudo-particle treatment leads to a very similar result for k (t), thus ensuring that the observed features are not artifacts of the particular numerical method. Further insight into the features of the behavior of the system can be gained by considering the time evolution of the fastest growing mode, k0 (t), displayed in Fig. 4.10. we expect that the evolution of this quantity should be approximately described by Eq. (4.35), obtained by a linear-response treatment. Thus, at asymptotic times, tt0 , we expect that k (t) ≈ Dk++ t0 exp(2t=t0 ) and we may then extract the key quantities Dk++ and t0 from the behavior: On a logarithmic plot, the slope of 0 a linear 1t is 2=t0 and its value at t = 0 is ln(Dk++ t0 ). This procedure yields values in very good 0 agreement with those obtained from the exact projection method for Dk++ and the dispersion relation

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for tk . So one may conclude that the early fragmentation dynamics is dominated by the growth of few important unstable modes, whose characteristics (amplitude and growth time) agree with those derived by a linear response analysis. This feature completes the discussion in Section 4.1, as it con1rms the persistence of the regular behavior up to large amplitudes and the associated lateness of the entry into the non-linear (and possibly chaotic) regime. 4.3.4. Re-nements In order to bring out the key features, the above discussion has been carried out within the simplest framework, in which the system is described in terms of a semi-classical phase-space density subject to a collision kernel that is local in space and time. Numerous studies of more re1ned treatments have been carried out, as will now be brieDy summarized. 4.3.4.1. Memory and quantum eCects. In the one-body transport treatment discussed so far, the collision term is assumed to be local in both space and time, in accordance with Boltzmann’s original treatment. This simpli1cation is usually justi1ed by the fact that the interaction range, as measured by the residual scattering cross section NN ≈ 4 fm2 , is relatively small on the scale of a typical nuclear system, and the duration of a two-body collision is short on the time scale characteristic of the macroscopic evolution of the system. The resulting collective motion has then a classical character, as is the case also in TDHF. However, when the system possesses fast collective modes whose characteristic energies are not small in comparison with the temperature, quantum eHects are important and the treatment needs to be appropriately improved. Quantum eHects on the stochastic dynamics in nuclear matter were considered by Kiderlen and Hofmann [111] within Landau theory. Using the quantal Ductuation–dissipation theorem, suitably generalized to unstable modes, they deduced the properties of the stochastic force and found sizable quantum eHects both inside and outside the spinodal regime. Further work on the nuclear Ductuation dynamics has been carried out by Kiderlen [209]. When the collision term has a non-Markovian form, then the evolution of the single-particle density matrix depends on the (recent) past. This problem was addressed by Ayik [210] who derived a transport equation for the single-particle phase-space density by performing a statistical averaging of the Boltzmann–Langevin equation. In analogy with Brownian motion, the Ductuating part of the collision term gives rise to a memory time in the collision kernel which, in turn, leads to a dissipative coupling between collective modes and single-particle degrees of freedom. This approach was adapted by Ayik and Randrup [211] to nuclear matter inside the spinodal zone, as is brieDy recalled in Appendix B.4. a == responsible for the agitation of the unstable collecThe key result is that the usual source terms D k a == == (t). tive modes in nuclear matter may be replaced by eHective coePcients of the form Dk== (t)=D k k The Lalime equation (4.25) governing the collective correlation coePcients is then modi1ed accordingly, d == a == == (t) + = + = == (t) : k (t) = 2D k k k dt tk

(4.36)

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It follows that the solutions can be expressed on a simple form, ◦ ==

k== (t) = k (t) O== k (t) ; ◦ ==

(4.37)

where k (t) is the solution for the Markovian case. The memory eHect is thus expressed by means of the renormalization coePcients O== k (t) which in turn can be obtained as suitable time averages of the collective correlation functions (see Appendix B.4). These correction factors ultimately attain constant values and the evolution is then similar to what the standard treatment would give, except for the time-independent renormalization of the source terms. But this limiting simplicity emerges only relatively slowly, particularly for the mixed factor, as is evident from Fig. B.7. Although the analysis in Ref. [211] was carried out for the idealized scenario of initially uniform nuclear matter, the conclusions are expected to hold for more complicated dynamical scenarios, such as may be encountered in nuclear collisions. Since the correction factors can deviate signi1cantly from unity, particularly in the domain where the fastest growth occurs, it appears necessary to re1ne the treatment to take account of the memory time in an appropriate manner, if quantitatively reliable results are to be obtained from numerical simulations based on the Boltzmann–Langevin model, especially when instabilities are present. 4.3.4.2. Collective quantum @uctuations. The development of density Ductuations associated with collective modes in the presence of expansion was studied by Wen et al. [212] by means of classical and quantal transport equations. Considering a quadrupole giant resonance or an low-lying octupole vibration, the authors study the dynamical symmetry breaking as the global expansion drives the mode from its usual stable state towards instability. The time evolution of the collective variable Q is then determined by either classical Langevin dynamics or a quantal transport model containing initial quantal Ductuations. It is found that the 1nal Ductuations of Q arise primarily from the ampli1cation of the initial Ductuations during the unstable stage. Since the growth rates are the same in the two treatments, the quantum treatment yields larger 1nal Ductuations due to the contribution from the initial quantum Ductuations which add to the (common) statistical Ductuations. This feature is particularly important at low temperatures where the quantum Ductuations may exceed the statistical Ductuations. In this situation, their inclusion in the treatment will enhance the symmetry breaking. These results suggest that, as a 1rst approximation, it may be possible to mimic the quantum Ductuations in a classical approach by a suitably tuned arti1cial increase of the initial temperature, as it is well known in the purely harmonic case. Clearly, though, there is a need for developing a quantum Langevin treatment that quantizes the collective motion. Such an approach would go beyond the Stochastic TDHF framework which does not quantize the collective degrees of freedom. 4.3.4.3. Relativistic treatment. Nuclear spinodal fragmentation occurs at energies that are suPciently low to permit the use of non-relativistic kinematics and, moreover, to justify the neglect of mesonic degrees of freedom. For applications of the Boltzmann–Langevin model to nuclear collisions at higher energies, it is of interest to recast the treatment in a proper relativistic form. This undertaking is brieDy reviewed in Appendix B.5.

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4.4. Approximate Boltzmann–Langevin dynamics The numerical implementation of the exact Boltzmann–Langevin dynamics has been feasible only in 2D, so the introduction of approximate treatments is essential to deal with realistic 3D problems. However, typically, such attempts introduce the Ductuations by 1at in a manner that is inconsistent with the general relaxation properties of the one-body density, as expressed through the Ductuation–dissipation theorem. We discuss here in turn those various approaches. An attempt to introduce spontaneous Ductuations in a practically realizable manner was made by Bauer et al. [213]. Their method can be implemented relatively easily into standard BUU codes that use the pseudo-particle method of solution and it consists essentially in forcing similar two-body collisions to occur for neighboring pseudo-particles so that eHectively two entire nucleons are involved in each particular collision event. Employing an idealized two-dimensional nucleon gas as a test case, Chapelle et al. [214] examined this intuitively appealing method. They found that it is able to produce Ductuations of the correct general magnitude, provided that a suitable coarse graining of the phase space is performed, and that these display some of the correlation features expected from the basic characteristics of the two-body collision process. These features can be improved by suitable tuning of the phase-space metric (the concept of a distance in phase space is required for the selection of the “neighboring” pseudo-particles). However, for any tuning, the detailed momentum dependence of the variance in phase space occupancy deviates signi1cantly from what is dictated by quantum statistics. Therefore this simple prescription may be unsuitable for problems in which these properties are important. On a more formal basis, Ayik and Gregoire [215] proposed an approximate method for numerical implementation of the Boltzmann–Langevin theory. The method reduces the Boltzmann– Langevin equation for the microscopic one-body phase-space density f(r; p) to stochastic equations for a set of macroscopic variables, namely the local or global quadrupole moment of the momentum distribution. A random change of the quadrupole moment is then made at each time step and a suitable stretching of f(r; p) is performed subsequently in order to reconstruct the entire phase-space density. This method was also examined by in Ref. [214] and, although several variations of the proposed scheme were examined, it was generally found that the results were far from satisfactory, since the resulting correlations associated with the Ductuating one-body density will tend to reDect the symmetries and other characteristics of the employed reconstruction procedure rather than those of the underlying physical Ductuations. Therefore this method appears unsuitable for calculating quantities that depend sensitively on the details of the momentum distribution. For the purpose of addressing catastrophic phenomena in nuclear dynamics, such as multifragmentation, Colonna et al. [176] explored the possibility of simulating the stochastic part of the collision integral in the Boltzmann–Langevin model by the numerical noise k (0) associated with the 1nite number of pseudo-particles N employed in the ordinary BUU treatment. This idea is based on the observation that for large times, ttk , the Ductuation of density undulations of a given wave number k is given by k2 (t) = Dk tk e2t=t= in the Boltzmann–Langevin treatment, whereas it is k2 (t) = (Dk tk =N + k (0))e2t=t= in the BUU pseudo-particle treatment. Since k (0) also scales as 1=N, the matching of those two asymptotic Ductuations yields a relation determining the value of N. For idealized two-dimensional matter, which presents a suitable test case, as it is here

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practical to simulate the Boltzmann–Langevin equation directly, they demonstrated that N can be adjusted so that the corresponding BUU calculation yields a good reproduction of the spontaneous clusterization occurring inside the spinodal region. This approximate method may therefore provide a relatively easy way to introduce meaningful Ductuations in simulations of unstable nuclear dynamics. This method was subsequently extended to 3D nuclear matter, allowing the direct extraction of the growth times tk of the unstable modes and the associated diHusion coePcients Dk [113]. Guarnera et al. [145] studied the spinodal fragmentation of a hot and dilute nucleus by 1rst expanding the system into a spinodally unstable con1guration and then adding a stochastic density Ductuation that is carefully tuned to reDect the degree of Ductuation in the most unstable mode, as determined by the corresponding linear-response analysis of the unstable sphere. They found that the early clusterization appears to be dominated by unstable modes whose spatial structure is similar to the fastest growing spinodal modes in in1nite matter at similar density and temperature. They followed the development of the instabilities until multifragmentation had occurred and then made an analysis of the resulting fragment size distribution. As expected from the fact that only a few modes dominate, the clusterization pattern has a large degree of regularity which in turn favors breakup into nearly fragments of nearly equal size, with a corresponding paucity of small clusters. Subsequently, Colonna et al. [216] introduced a method that roughly approximates the Boltzmann–Langevin model by adding a suitable noise to the collision term in the usual BUU treatment. The noise employed corresponds to the thermal Ductuation in the local phase-space occupancy, f2 (r; p) = f(1 − f), where f(r; p) is the local Fermi–Dirac equilibrium distribution. By performing such a local momentum redistribution at suitable intervals in the course of the evolution, the inherently stochastic nature of the two-body collision processes is mimiced. The method has the advantage that it is readily tractable and it applies equally well to both stable and unstable parts of the phase diagram. Recently a diHerent approach was taken by Matera and Della1ore [217] who applied white noise to a Vlasov system. The noise term was determined self-consistently by invoking the Ductuation–dissipation theorem and, within the linear approximation, the time evolution of the density Ductuations was found to be given by the same closed form as was found in Ref. [112]. The authors showed that while a white-noise form of the stochastic 1eld is in general not consistent with the Ductuation–dissipation theorem, it may provide a good approximation when the free response function is suPciently peaked. It is important to note that all of the methods described above employ an ad hoc procedure to generate Ductuations. Therefore the microscopic structure in phase space of the produced correlations is typically very diHerent from the prediction of the Boltzmann Langevin theory. However, as stressed 1rst in Ref. [176], in situations where the dynamics is dominated by only a few modes (such as the fastest growing spinodal modes) it may suPce to require equivalence with the exact Boltzmann–Langevin approach for only those few degrees of freedom. As a consequence, several approaches have carefully designed the Ductuation source so as to mimic the eHects of the stochastic Boltzmann–Langevin term on the dynamics of the most unstable modes [113,145,176]. In such a way the dynamics of the spinodal decomposition can be simulated. The comparison with exact many-body dynamics presented in the next section illustrates the power of such instability projected stochastic approaches.

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Fig. 4.11. One-body versus many-body: dispersion relation. The dispersion relation extracted from either the many-body (molecular-dynamics) calculation (open squares) or the stochastically initialized one-body (BUU) simulation (solid dots). The linear-response result for the one-body treatment is shown by the shaded area covering the range of uncertainty in the one-body parameters. The insert shows the corresponding temporal growth of the most unstable mode k0 , as measured by the corresponding correlation strength, k0 . (From Ref. [39].)

4.4.1. Reliability of one-body treatments The quantitative reliability of one-body treatments of spinodal fragmentation was examined by Jacquot et al. [39]. Considering a two-dimensional gas of classical particles subject to a Lennard– Jones interaction, which presents a many-body problem that can be solved exactly by direct numerical solution of the equations of motion, the authors compared these exact results to those obtained by using the approximate one-body method introduced in Ref. [145]. The mean-1eld dynamics was derived from a local-density approximation of the energy functional using a Skyrme-type parametrization, adjusted so as to approximately reproduce the features of the many-body model. The collision cross section is taken as coming from the repulsive hard core in the inter-particle potential. The approximate Boltzmann–Langevin treatment was tested against the exact many-body model for two-dimensional uniform matter inside the phase region of spinodal instability. The emerging 1lamentation patterns exhibited a large degree of qualitative similarity. In order to make the comparison more quantitative, the dispersion relations were extracted from the early dynamical evolutions and they are shown in Fig. 4.11 together with the linear-response result for the one-body model. The two dynamical treatments are seen to yield the same dispersion relation to within about 10% and the corresponding wave numbers for the most rapidly growing density undulations are k0 = 0:399 and 0:359 fm−1 , respectively. As shown in the insert, the temporal evolution of the corresponding correlation strength for the most rapidly ampli1ed mode, k0 (t), are very similar. It was also found that, well inside the spinodal region, the exact early fragmentation follows closely the growth of the most unstable collective modes as obtained by a linear-response analysis of the associated mean-1eld propagation. The good correspondence between the exact many-body result and the approximate one-body treatment is illustrated in Fig. 4.12, which shows two snapshots of the correlation strength distributions k . They are very similar, especially for the most rapidly growing modes that ultimately dominate the fragmentation pattern. Moreover, it was shown that the

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Fig. 4.12. One-body versus many-body: density Ductuations. The spectral distribution of the spatial correlation function, as measured by the variance k of modes with wave number k, for either the many-body (molecular-dynamics) calculation (open squares) or the stochastically initialized one-body (BUU) simulation (solid dots), as extracted at two successive times for a sample of ten events. (From Ref. [39].)

fragmentation pattern obtained with the exact many-body model can be reproduced by the one-body treatment even at late when the system exhibits large many-body correlations. This study thus brings out the fact that one-body treatments, when endowed with a suitable degree of stochasticity, may be quantitatively useful for studies of catastrophic dynamics, such as may occur when 1rst-order phase-transitions are present. 4.4.2. Brownian one-body dynamics A novel method for introducing Ductuations in one-body dynamics was proposed by Chomaz et al. [144]. It consists of employing a brownian force in the kinetic equations and it was therefore denoted as the brownian one-body (BOB) treatment. The basic idea the method is to replace the actual stochastic collision term K by a suitable brownian force F (with ≺ F =0) in such a manner that the novel equation of motion is obtained by making the following replacement in the Boltzmann–Langevin equation (4.6), 9f ˜ K[f] → K[f] = −F · : (4.38) 9p In order to ensure that the resulting brownian one-body dynamics mimic the BL evolution, the stochastic force F is assumed to be local in space and time. Moreover, since nuclear matter is isotropic, the force may also be taken to have rotational invariance. Its correlation function can then be written as follows, ≺ F(r1 )F(r2 ) =2D˜ 0 (r) I (r12 )(t12 ) ;

(4.39)

where I is the unit tensor in position space. The resulting dynamics is then qualitatively similar to that resulting from the Boltzmann–Langevin equation, but the associated diHusion coePcient for the evolution of the phase-space density f(s) is modi1ed, ˜ 1 ; s2 ) = 2D˜ 0 (r) 9f(s1 ) · 9f(s2 ) (r12 ) : 2D(s (4.40) 9p1 9p2

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Fig. 4.13. Test of the brownian one-body simulation method. The evolution of the strength k0 of the most unstable mode (k0 = 0:6 fm−1 ) in idealized two-dimensional nuclear matter, prepared at half the saturation density with T = 3 MeV: The expected BL result taking into account the actual growth time tk0 obtained in the BUU pseudo-particle propagation (solid curve), the actual result of the BOB dynamics (points), and the result of the corresponding linearized BOB dynamics (dashed curve). (From Ref. [144].)

The local strength, as given by the coePcient D˜ 0 (r), is determined by the demand that the dynamics of the fastest unstable modes be as given by the Boltzmann–Langevin theory, in uniform with a density given by the local value (r). This amounts to requiring that the projection D˜ ++ be equal to k Dk++ (see Section 4.3.2). There are thus no adjustable parameters involved in the BOB approximation. The brownian one-body simulation method was tested in Ref. [144] by considering the evolution of the density variance for the most unstable mode, k0 , in idealized two-dimensional nuclear matter prepared at half the saturation density with a temperature of T = 3 MeV, as illustrated in Fig. 4.13. As can be seen, the BOB results are very well reproduced by the analytical linear-response treatment of the Boltzmann–Langevin model, using the actual growth time tk0 as extracted from a sample of pseudo-particle simulations. In the course of time, the growth obtained with the brownian one-body dynamics exhibits an exponential approach to that reference evolution. Moreover, the numerical BOB results are well reproduced by the corresponding linear-response prediction (using the extracted actual growth time tk0 ). Finally, it was found that the evolutions of neighboring modes are similarly well approached [146]. Thus, the BOB method does indeed emulate the corresponding Boltzmann– Langevin dynamics fairly well and it thus appears that it may provide a practical quantitative means for addressing catastrophic nuclear processes. 4.5. Concluding remarks about spinodal dynamics It has been elucidated how the inclusion of Ductuations is essential in the dynamical description of the evolution of unstable systems, such as those undergoing spinodal fragmentation. We have discussed the Boltzmann–Langevin treatment which presents the natural extension of the deterministic nuclear Boltzmann. The Ductuations then arise from the stochastic part of the two-body collision integral and there are thus no new adjustable parameters introduced. Furthermore, we have described powerful approximate tools that make it possible to address multifragmentation processes in realistic 3D systems, such as nuclei. We wish to particularly note

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the brownian one-body (BOB) model which is an approximation to the full Boltzmann–Langevin treatment that involves no adjustable parameters, thus making it practical to carry out essentially parameter-free numerical transport simulations of nuclear reactions. The quantitative utility of this kind of model is well illustrated by the recent detailed study by Matera et al. [127] who considered spinodal decomposition of expanding nuclear matter with a BOB-type model, constraining the stochastic 1eld by the Ductuation–dissipation theorem for the expanding system. The spinodal growth rates, the evolution of the density–density correlation function and the emerging liquid domains were determined. Furthermore, the fragment-size distribution were found to agree well with data and the associated the critical exponents were extracted. In the currently tractable treatments, the Ductuations are introduced at the semi-classical level. However, as we have discussed, quantal eHects appear to be of quantitative importance in the evolution of unstable nuclear systems. Hence it would be of interest to go beyond the semi-classical level and develop a suitable quantal transport treatment where such features as quantum Ductuations and memory eHects are included.

5. Applications to nuclear fragmentation In recent years many experimental and theoretical eHorts have been devoted to the study of reaction mechanisms in nuclear collisions at intermediate energies and, in particular, to the understanding of the observed copious production of intermediate-mass fragments (IMF), one of the most challenging issues in the 1eld. Insofar as the phenomenon of spinodal multifragmentation can be directly related to the properties of nuclear matter around and below the saturation density, those studies have a direct bearing on the experimental observability of the nuclear liquid–gas phase transition. In order to make such a connection, it must 1rst be established that the bulk of the involved system in fact enters the phase region of spinodal instability. This is the topic of the 1rst part of this section. Subsequently we address the resulting multifragmentation and we make contact with the data in the next section. 5.1. Is the spinodal region reached in the calculations? In order to ascertain whether nuclear multifragmentation, as observed in nuclear reactions, can in fact be related to the spinodal decomposition, it is necessary to carry out numerical studies of the collision dynamics and we review below various key results in this regard. 5.1.1. Entering the spinodal zone Over the past decades, many calculations have predicted that the systems formed in nuclear collisions may expand to reach dilute con1gurations containing spinodal instabilities. Using a microscopic quasi-particle transport model, Boal [218] sought to determine the temperature and density regions in which the nuclear liquid–gas phase transition is expected to occur. He found that the mechanical instability region should be easily accessible and that the associated change in entropy appears to be in agreement with that observed experimentally. Further studies of multifragmentation with this microscopic many-body model have been made by Boal and Glosli [219,220].

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On a more macroscopic level, Schultz et al. [221] considered the isentropic expansion of a blob of nuclear matter towards the two-phase instability region under two extreme conditions: instantaneous development of the phase transition and spinodal decomposition. They found as well that the experimentally observed entropy values support the onset of a liquid–gas phase transition. Papp and NUorenberg [222] investigated the path of hot nuclei towards multifragmentation by considering an expanding and evaporating spherical source. Assuming that the expansion is isentropic, they demonstrated that the initial compression and temperature needed for entering the spinodal region depend signi1cantly on the stiHness of the equation of state. For well over a decade, quantitatively useful simulations of nuclear collisions have been made on the basis of semi-classical mean-1eld treatments that include the average eHect of the residual two-body collisions, as discussed in Section 4.2. A particularly thorough and instructive study was carried out by Morawetz [223] who was able to locate instabilities and points of phase transition in the dynamical system. He found an early surface-dominated instability and a later volume-dominated spinodal instability. While the latter occurs only if the incident energy is around the Fermi energy, the former occurs over a much wider range of scenarios. Such studies are carried out with the so-called BUU, BNV or Landau–Vlasov equation of motion for the one-body phase-space density f(r; p; t). It may be considered as a semi-classical approximation to TDHF theory with Pauli-suppressed residual collisions included and it describes the average evolution and generally yields a satisfactory description of the reaction mechanisms in the regimes of low and intermediate energy. However, as discussed in Section 4, this kind of treatment is inadequate for processes exhibiting bifurcations, such as nuclear fragmentation processes, where a small Ductuation in the spatial density may be ampli1ed, thus producing an irreversible divergence of the possible trajectories, as a variety of qualitatively diHerent density con1gurations may develop. In such cases, the average trajectory is no longer physically informative and it becomes necessary to treat an entire ensemble, which is most conveniently done by stochastic methods. This general limitation not withstanding, the average trajectory may still be used as a tool for identifying the onset of instabilities, using various stability criteria [223–225] or the sudden growth of a suitable observable as an indicator (see Ref. [196]). (Of course, once instabilities are encountered, the subsequent average evolution of the system is no longer reliable and must be replaced by a suitable stochastic treatment.) Such BUU-like simulations have been used as exploratory calculations for studying the spinodal crossing and a suitable illustration is provided by the study of the central reactions La+Al and La+Cu at 55 MeV=A carried out by Colonna et al. [226] using a soft equation of state. In order to make comparison with experiment possible, any compound nuclei remaining after the BUU simulation should be subjected to a “afterburner” producing statistical particle and fragment emission. As it turns out, the incomplete fusion residues produced in the La+Al reaction yield evaporation products in good agreement with the data [227], while the calculations fail to produce the abundant IMF emission observed for the energetic La+Cu reaction. This result suggests that the fragments produced in the latter case do not come from an evaporative process. To understand this sudden transition to an apparently prompt fragmentation process, the authors inspected the time evolution of the density and excitation energy in the participant zone, and analyzed them in a thermodynamic framework making use of the nuclear matter equation of state. The central nucleon density was extracted and the corresponding thermal excitation energy per nucleon was then assumed to be of the form j∗ = jkin − 35 jF (), where jF () is the local Fermi energy and jkin

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Fig. 5.1. Phase trajectories for 55 MeV=A La on Al and Cu. Dynamical trajectories in the –T (left) and –P (right) phase planes for the reactions 55 MeV=A La on Al (solid dots) and Cu (open dots) at an impact parameter of 1 fm. The density , temperature T and the pressure P have been extracted in the interaction zone at time intervals of 10 fm=c. The short-dashed curve delineates the spinodal region. The –P plot includes the pressure evaluated at a constant entropy of S = 0:6 (dashed), S = 1:0 (solid) or S = 0 (bottom curve). (Adapted from Ref. [226].)

is the kinetic energy per nucleon computed from the actual momentum distribution. The temperature was then extracted by use of the Fermi gas model, T 2 = j∗ =a, with the level density parameter being a() = 92 =4jF , and the corresponding entropy per nucleon is calculated as s() = 2aT = 2(j∗ a)1=2 . These approximate relations are accurate for T jF , as was the case for this analysis where j∗ = 1–4 MeV. It should be noted, though, that these idealized relations employ the free nucleon mass m rather than the eHective mass m∗ ≈ 0:6 m which would change the values signi1cantly. The resulting dynamical evolution of these thermodynamic quantities display several instructive features, as shown in Fig. 5.1 (left). While the central density for the La+Al reaction experiences an oscillatory behavior, with the formation of one excited compound system, the La+Cu reaction has more available energy and it therefore exhibits a larger initial compression and the subsequent dilution then suPces to bring the system well within the region of spinodal instability, resulting in prompt disassembly. The extracted entropy values remain remarkably constant, s ≈ 0:6 and s ≈ 1:0, respectively. Fig. 5.1 (right) shows the corresponding evolutions in the –P plane, with the pressure P being obtained from the equation of state given by the dynamical model, using the extracted temperature and density. It is clearly seen how the systems evolve isentropically. Even though the La + Al system enters the spinodal region temporarily, it reexpands out of the region before any fragmentation can occur, contrary to La + Cu. The occurrence of volume instabilities may also be revealed by the time evolution of suitable collective variables related to the density. Of particular interest is the monopole moment Q, where Q2 = r 2 , as studied by Cussol et al. [228]. Using the extracted local density (r; t) and j(r; t), these authors extracted the monopole inertial-mass parameter B, and the stiHness C, 2 2 d m 3 j(r; t) 1 Q(t) d ; (5.1) B(t) = E (t) = r coll 2 dt 2 (r; t) C(t) =

d2 d2 E (t) = − Ecoll (t) ; intr dQ2 dQ2

(5.2)

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Fig. 5.2. Evolution of the monopole mode. The time dependence of the square of the monopole oscillation frequency P for the reaction 40 Ar + 50 Ti (b = 0) at 44 MeV=A (solid curve) and 20 MeV=A (dashed curve). Negative values of P2 reveal an excursion into the spinodal region. (From Ref. [228].)

since the total energy Etot =Ecoll +Eintr is conserved, so he collective frequency can be obtained, P2 = C=B. The evolution of this quantity is shown in Fig. 5.2 for the reaction 40 Ar + 50 Ti at two energies, as obtained with the BUU treatment using a soft equation of state. It is seen that while 20 MeV=A leads to compound formation with an oscillatory behavior of Q, the higher beam energy brings the system well into the unstable region characterized by P2 ¡ 0. Since these results depend sensitively on the equation of state employed [229,229a], the analysis of the onset of multifragmentation may provide important information on the stiHness of the nuclear equation of state in regions far from normal density. Employing a relativistic transport calculation, Fuchs et al. [230] investigated thermodynamic properties and instability conditions in intermediate-energy heavy-ion reactions. The thermodynamic variables (density, pressure and temperature) are calculated directly from the phase-space distribution. Instabilities are spotted using the criterion that the eHective compressibility becomes negative. Thus, in the case of semi-central Au+Au reactions at 600 MeV=nucleon, clear indications of instability are found in the center of the spectator matter, with conditions of density and temperature that are consistent with experimental determinations. As the above exposition illustrates, the possible entry of the system into an unstable region can be quantitatively ascertained already on the basis of average dynamics described by mean-1eld dynamical models. 5.1.2. Role of heating and compression in semi-classical expansion In nuclear reactions, the expansion required for the system to enter the spinodal region may occur as a result of the initial compression and/or the thermal pressure. Therefore, we discuss here in some detail the role of heating and compression on the prospects for a nuclear system to enter the spinodal region, an issue of great relevance to fragmentation studies. A detailed study of the dynamical evolution of compressed and/or heated spherical nuclear systems, within the framework of the BUU model, was carried out by Batko and Randrup [150]. As stressed before, the BUU model yields a mean-trajectory description and as such it should preserve the initial symmetries of the system. However, in most existing numerical implementations, the discretization introduces irregularities that act as a numerical noise which may thus be ampli1ed when the system becomes unstable. In order to preserve the overall spherical symmetry even in the presence of

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Fig. 5.3. Collective Dow for 197 Au with spherical BUU . The degree of compression in the central cell =s (upper row), the average radial velocity v = C · r

ˆ (middle row), and the root-mean-square radius Rrms = r 2 1=2 (lower row) as functions 197 of time, for Au, calculated by imposing spherical symmetry in the BUU code. The results corresponding to three diHerent initial values of the compression parameter and the temperature T are indicated by the solid and dashed curves, respectively. (From Ref. [150].)

such instabilities, the authors enforced an angular average of the mean 1eld at each time step, thus obtaining a more reliable approximation to the true average solution. As a main result of this study, it was found that, under suitable conditions, the nucleus expands into a hollow, quasi-stationary unstable con1guration that evolves suPciently slowly to enable the instabilities to develop. Here we review calculations performed for 197 Au nuclei that have been excited to a certain degree either by uniform compression or by a corresponding degree of heating. The resulting evolutions are illustrated in Fig. 5.3. At relatively moderate initial compressions ( ≡ =s =2), the nucleus exhibits an oscillatory motion, while slowly radiating its excess excitation away by emission of individual pseudo-particles. The oscillatory character of the motion is clearly reDected in the periodic fall and rise of the central density, as well as in the periodic behavior of the radial velocity. Oscillations are less evident in the rms radius Rrms , due to the signi1cant and steadily growing contribution from the emitted particles. The continual loss of energy carried oH by the emitted particles appears to be the main source of the damping of the collective radial motion.

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As the initial compression is increased, the oscillatory period grows longer and the rate of particle emission increases. At some point, the initial compression exceeds a critical value and the systems keeps expanding steadily. This qualitative change in behavior occurs for ≈ 2:5, corresponds roughly to the compression where the monopole mode becomes unstable. At the dilute turning point, the density pro1le becomes drastically distorted into a bubble-like con1guration with the local density and temperature corresponding to a phase point well within the spinodal region of mechanical instability, as illustrated in Fig. 5.4. This con1guration is quasi-stationary for ≈ 2:5, with the time spent under these unstable conditions being as large as 100 fm=c, which exceeds the growth times for density irregularities (see Section 3). Therefore, if density irregularities are introduced, the system will undergo a spontaneous transformation leading to condensation of the dilute matter in the shell into a number of prefragments. Thus, on the basis of such studies, one is led to the expectation that compression may be an eHective agency for expanding the system into an unstable con1guration. It is instructive also examine the alternative scenario in which the same degree of initial agitation has been achieved by thermal excitation (also shown in Fig. 5.4). The resulting hot systems generally exhibit an evaporation-like behavior, radiating pseudo-particles while gradually shrinking and cooling. Thus, the thermal initialization appears to be much less eHective in generating collective radial motion. (However, it should be noted that this feature may well be an artifact of semi-classical approaches, since quantum mean-1eld treatments render heating almost as eHective as compression in driving the system to low density [231], as we discuss below.) We 1nally emphasize the important fact that the possible entry of the system into the unstable zone is a feature determined by the average dynamics and thus it can be studied within the framework of mean-1eld treatments. On the other hand, the subsequent evolution within the unstable region is highly dependent on the details of both the physical model and the numerical implementation, as we shall discuss in more detail in Section 5.2. 5.1.3. Expansion and dissipation in TDHF simulations The above type of analysis is based on a semi-classical mean-1eld treatment and therefore the extracted results, such as the value of the critical compression , may change quantitatively if a more quantal approach is employed. In order to investigate this issue, Lacroix et al. [231] performed TDHF simulations for 1nite nuclei prepared under extreme conditions of temperature and pressure together with the corresponding semi-classical approximation, the Vlasov dynamics. (Both of these treatments neglect the residual two-body collisions included in the study discussed above.) They found that the diHerences in the results are primarily due to the wave nature of the nucleons which generally renders the system less dissipative, so that the initial compression required for achieving a given degree of density oscillation is lower in TDHF than in the Vlasov treatment. The faster expansion towards lower densities obtained with TDHF simulations stems primarily from the slower damping and cooling processes. In particular, while a unbound nucleon would be emitted when described as a particle, its description as a wave in TDHF allows it to be partly reDected at the surface, thus reducing the particle evaporation rate and increasing the pressure on the nuclear surface. The replacement of particles by waves in eHect then converts evaporation into expansion.

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Fig. 5.4. Density pro1le for 197 Au with spherical BUU . Time evolution of the nuclear density pro1le for 197 Au obtained by imposing spherical symmetry in the BUU code. The three sets of 1gures display results obtained from diHerent compressed (solid curves) and thermal (dashed curves) initializations, corresponding to pairs of values of the temperature T and the compression parameter = =s that yield the same excitation energy. (From Ref. [150].)

The diHerence in the evaporation process aHects not only the amplitude of the monopole motion but it also changes the size of the remaining residual nucleus. As we have noted, the evaporation in TDHF is reduced by reDections from the surface, so a large part of the system may survive the expansion stage and reach the low density region. By contrast, the fast evaporation in the classical treatment removes more particles and radial momentum, so the residue is smaller and cannot expand as much.

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Fig. 5.5. TDHF versus Vlasov expansion. The central density at the time of maximum dilution (the turning point in the monopole motion), min , as a function of the initial central density init , for a 40 Ca nucleus at various initial temperatures, either in TDHF (left and center) and the corresponding Vlasov treatment (right). The initial excitations correspond to the following values of the entropy per nucleon : 0 (crosses), 1.1 (diamonds), 2.35 (squares) and 3.28 (circles) which correspond to T = 0, 5, 10 and 15 MeV for the uncompressed heated nuclei. (From Ref. [231].)

A quantitative illustration of the diHerences between TDHF and Vlasov is shown in Fig. 5.5. It presents the correlation between the initial compression init and the resulting minimum density min , as obtained in either TDHF calculations, without (left) and with (center) gradient terms in the eHective interaction, or semi-classically (right). It is evident that TDHF yields a larger degree of expansion. For example, for an entropy per nucleon of ≈ 2:35 (corresponding to T =10 MeV for the uncompressed heated nucleus) a small compression of =s = 1:3 suPces in the quantum dynamics since the thermal pressure is the essential agency for producing a very considerable expansion, min =s = 0:1. By contrast, in the semi-classical dynamics the essential agency for expansion is the compression which should exceed about double density, =s = 2, to achieve an equivalent degree of dilution. In summary, when the wave mechanical nature of the nucleon is taken into account, as in TDHF, both compression and heat are as eHective in bringing a larger part of the system to lower density, as a larger part of the initial excitation energy is converted into dilution. In particular, even a system starting at saturation density may expand into the spinodal region if suPciently hot. 5.1.4. Role of the @uctuations on the expansion dynamics When instabilities are encountered in dynamical simulations, irregularities may be drastically ampli1ed. Thus, an ensemble of initially very similar systems, each with its own individual small deviations from the ensemble average (arising from the initial Ductuations or/and from some stochasticity on top of the deterministic dynamics), may display a relatively sudden branching into qualitatively diHerent subensembles, thus rendering the overall average phase-space density meaningless (see Fig. 4.4). Therefore, when the instabilities are present, it is essential to acquire a good understanding of the Ductuation sources and to the development of appropriate treatments is required for reaching reliable conclusions on even the average dynamics. A detailed study of the disassembly of 1nite nuclei was performed by Guarnera et al. [145] using the simpli1ed approach developed in Refs. [113,176] (see discussion in Section 4). They evolved central collisions of Xe and Sn at 50 MeV=A with the BUU model until instabilities were

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Fig. 5.6. Evolution of the radial density pro1le. Time evolution of the radial density pro1le for a hot and expanding nuclear system (with 210 nucleons) initialized inside the spinodal region. (From Ref. [145].)

encountered, leading typically to a relatively heavy compound system (A ≈ 200) with the bulk conditions being inside the spinodal region ( ≈ 12 0 ) and T ≈ 3 MeV. This system was then endowed with a self-similar radial expansion as occurring in the average dynamics (reaching a maximum speed of v ≈ 0:1c at the surface) and a noise that was carefully tuned to reproduce the complete Boltzmann–Langevin dynamics for the most unstable mode in nuclear matter at the considered density and temperature, k0 . (The magnitude of the introduced irregularity is derived from the linearized Langevin dynamics.) In agreement with the calculations performed within the BUU model [150] (described in Section 4.2.2), the matter concentrates at the surface of the system, as shown in Fig. 5.6. This leads to the formation of bubble-like con1gurations when the initial source is spherical (as in the case considered), or torus-like fragmenting systems when the source happens to be Datter (see Ref. [151] for an early exploration of this phenomenon). In either case, the formation of a hollow structure is favored and even stabilized by the fragmentation of the system. The results obtained with this simpli1ed dynamics are important because they show that the instabilities are helping the system to remain in the low-density region long enough to permit the instabilities to fully develop. Indeed, the development of density Ductuations results in a reduction of the restoring force associated with the monopole breathing mode. This slows down the recontraction of the system, thus allowing more time for the instabilities to develop. This important feature deserves a more re1ned analysis of the time scales involved in the fragmentation dynamics. The main disadvantage of the simple method of Ref. [145] is that it introduces noise only at the initial time and thus does not take account of the Ductuations arising from the stochastic nature of the dynamical evolution. A signi1cant improvement of this shortcoming is provided by the brownian one-body (BOB) dynamics model [144,146] which provides a both simple and powerful approximate calculational tool. (We recall that the BOB dynamics introduces noise in the one-body 1eld whenever the local phase conditions correspond to spinodal instability, with the noise being adjusted so that it would produce the same growth rate as the full Boltzmann–Langevin model for the most unstable mode in nuclear matter prepared at that local density and temperature.) A good illustration is provided by the disassembly of a gold nucleus having been prepared in a suitably compressed con1guration.

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Fig. 5.7. Evolution of the density pro1le. Contour plot of the evolving radial density pro1le, as obtained with the BOB treatment of a compressed gold nucleus. The density contours are separated by ] = 0:005 fm−3 and dashed contours are employed in the regions of compression. The insert shows the angular distribution of the fragments formed in a typical event. (From Ref. [146].)

This system is also close to the one discussed above (A ≈ 200) and can be considered as an approximate realization of con1gurations obtained along the path of central nuclear collisions at intermediate energy. The resulting evolution of the radial density pro1le (averaged over many stochastic trajectories) is shown in Fig. 5.7. As in the pure mean-trajectory dynamics [150], if the gold nucleus is initially compressed to twice the normal density it will expand into a quasi-stationary hollow con1guration that is unstable against multifragmentation. It is an important feature of the BOB treatment that the strength of the stochastic force is larger than the numerical noise. The results are therefore no longer sensitive to the latter and thus this non-physical source of Ductuation may be neglected. This resulting robustness of the calculated results is obviously a great advantage of the treatment. After the formation of a hollow con1guration, the matter in the shell condenses into a number of prefragments as shown in the insert of Fig. 5.7. This condensation reduces the restoring force so the system is prevented from relapsing into a single compound nucleus. The stochastic approaches are thus essential to correctly predict the evolution of the system as soon as an instability has been encountered. 5.1.5. Investigations with many-body approaches In the previous sections the dynamical evolutions of highly excited systems have been discussed within mean-1eld approaches (including stochastic mean-1eld models), and it was showed that the spinodal region of the nuclear matter phase diagram is indeed reached under suitable conditions. We have also seen that the development of the instabilities weakens the overall restoring force and thus helps to keep the system within the instability region. To investigate the generality of these key

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Fig. 5.8. Phase evolution of 40 Ar. The evolution in the P–v phase plane of an ensemble of systems prepared at double density, with the temperature T = 10 MeV. The dynamical trajectory joins points at time intervals of 3 fm=c. (From Ref. [37].)

features, it is interesting to perform the same kind of study for scenarios where exact calculations can be carried out. While this cannot yet be done for nuclear systems, it is quite feasible for classical particle systems. Nuclear spinodal fragmentation was studied with molecular dynamics by LZopez et al. [108,109]. Making comparison with Cahn’s theory, they found that isothermal spinodal decomposition plays a dominant role in the breakup, while neither nucleation of bubbles in the two-phase region nor adiabatic spinodal decomposition contribute to the fragment production, consistent with the results of Ref. [106]. The phase evolution of a fragmenting nucleus was studied by Dorso and Randrup [37] who applied a quasi-classical simulation model to systems that have been compressed and heated. Systems consisting of 80 “nucleons” were prepared in states corresponding to a compression by a factor 2, with respect to the saturation density, and several initial temperatures T0 . It was observed that at low temperature the temporal evolution is fairly slow and the system remains essentially as a single entity. On the other hand, at high temperature the system quickly explodes into individual particles and loosely bound clusters, with the spatial density acquiring a 1lamentary character. As already stressed above, this corresponds to the situation where the competition between the radial Dow (resulting from the compression and thermal pressure) and the cohesion (resulting from the inter-particle attraction) results in the break-up of the systems into several clusters. The behavior of the pressure P as a function of the speci1c volume v = 1= along the trajectory of the system (20 events were considered) is depicted in Fig. 5.8, for the initial conditions (T =10 MeV) that lead to the disassembly of the system. As a reference, the isothermal curves of pressure with respect to density are also represented. It can be observed that the trajectory crosses into the spinodal region. Moreover, it should be noted that the emergence of bound clusters is reDected in the fact that the extracted dynamical pressure falls below the coldest isotherm of the P–v diagram.

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Fig. 5.9. Fragmentation of a hot drop. The time evolution of a hot spherical assembly of N interacting particles. Left: Trajectories followed the T – phase plane for the three energies considered and N = 251 (see text). Right: Time evolution of pressure, density, temperature and multiplicity Ductuations obtained in the case yielding maximum IMF production (N = 485). (From Ref. [77].)

More recently, Belkacem et al. [232] performed molecular-dynamics calculations indicating that a hot spherical drop may expand into the spinodal region and disassemble into many fragments. For the speci1c two-body force employed, the trajectory yielding the maximum production of intermediate-mass fragments enters the coexistence region close to the critical point. Furthermore, Pratt et al. [77] have studied the disassembly of a system of particles interacting with a Lennard–Jones type saturating force. They are initially con1ned within a spherical container at the saturation density and their momenta are sampled micro-canonically to yield a given kinetic energy. The hot systems were then allowed to expand according to the classical equations of motion and the phase path followed by the system was extracted. The dynamical trajectories followed by the system in the T – phase plane are shown in Fig. 5.9 (left) for the three energies considered and the results are similar to what is observed in mean-1eld calculations (see e.g. Ref. [150]). At low excitation the system expands only slightly and then retracts to saturation density after evaporating some light fragments, while the system explodes into many small fragments at high excitation. At a suitable intermediate excitations a balance is reached where the drop expands but neither explodes nor retracts. Rather, the matter attains a metastable con1guration where the collective velocity and the pressure are both near zero, see Fig. 5.9 (right). This situation yields a maximum number of intermediate-mass fragments. It should also be noted that the T – trajectory penetrates deeply into the spinodal region, where the liquid and gas phases can coexist, and the fragment formation can be interpreted as a manifestation of the associated phase separation. The occurrence of fragmentation is signalled by the relative variance particles near the center of mass of the system, (]N )2 =N , which measures the degree of spatial clustering relative to random

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particle positions. This quantity exhibits an accelerating growth to values signi1cantly in excess on one while the system is inside the spinodal region, as shown in Fig. 5.9 (right). Based on the various simulations, we may conclude that many initial conditions, from systems formed in nuclear collision to hot or compressed nuclei, evolve in such a manner that their bulk regions enter the spinodal region. From there on those approaches that do not correctly take account of the Ductuations and their dynamics cannot be trusted. It appears that the fragmentation dynamics, which can be regarded as the development of large-scale many-body correlations, reduces the restoring force acting to bring the system back to a single compound nucleus near saturation. As a consequence, the system may remain longer in the spinodal region thereby allowing the instabilities to develop suPciently for fragmentation to occur. We now turn to the more detailed discussion of the implications of such a spinodal decomposition on the properties of the fragmentation pattern. 5.2. Fragmentation studies Many studies have been directed towards fragment formation in nuclear collisions. Already in 1987, Sneppen and Vinet [224], using both Vlasov and BUU one-body dynamics, observed that the expansion of nuclear systems is generally followed by the formation of fragments. Still based on BUU-like studies, Moretto et al. [233] found that the head-on collision of two Mo nuclei at typically 60 MeV=A would lead to relatively thin oblate disks which would then cluster as a result of Rayleigh–Taylor type instabilities. Gross et al. [163] considered the central collision of two Mo nuclei at 55 MeV=A and reported that condensation into fragments occurs throughout the entire spherical volume, once the system has expanded suPciently. (Subsequently, a more detailed study was carried out [225].) Bauer et al. [151] considered Nb+Nb at 60 MeV=A and observed the development of a transient bubble-like con1guration that subsequently turns into a toroidal structure, before breaking up into fragments. The formation of ring-shaped systems, that break up into fragments, was also observed in the BUU simulations of Pb + Pb collisions at 50 MeV=A carried out by Norbeck et al. [152]. Borderie et al. [153] studied central collisions between very heavy nuclei around 30 MeV=A and found Coulomb instabilities, which also lead to the formation of unstable bubble con1gurations. Finally, the inDuence of the employed equation of state on the spatial geometry of the fragmenting system was discussed by Xu et al. [234], who found that a stiH equation of state leads to the prompt formation of several nearly equal fragments positioned in a ring-like structure, due to the formation of metastable toroids. By contrast, a soft equation of state produces a prompt condensation into several nearly equal fragments isotropically distributed on the surface of a metastable bubble. However, in all of those scenarios, the resulting fragmentation pattern depends sensitively not only on physical quantities, such as the equation of state or the dissipation, but also on the speci1c numerical implementation of the one-body dynamics (especially the number of pseudo-particles per nucleon, N). So the fragmentation geometries predicted by BUU-like approaches are not reliable. This can be understood as an inherent feature of deterministic dynamics which cannot explore the various possible branchings of the trajectory and should therefore generally not lead to the multifragmentation. However, the discretization required for the numerical treatment of the deterministic model in eHect presents a source of noise and this agency may break the imposed macroscopic symmetries, thus providing seeds for ampli1cation by any instabilities encountered in the course of the evolution.

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Fig. 5.10. Mass distributions from molecular dynamics. Left panel: Fragment size distributions for three initial temperatures: T0 = 1:44 (diamonds), T0 = 2:10 (circles) and T0 = 4:0 (squares), for N = 251 particles. Right panel: Fitted value of 5 as a function of the initial temperature T0 N = 93 (diamonds), N = 251 (circles) and N = 485 (squares). (From Ref. [77].)

As we have already remarked above, when instabilities are encountered by the system, a reliable description of its further evolution requires a treatment containing the appropriate physical Ductuations. This is the case when fragments are produced at some stage of the reaction. In Section 4, we have presented stochastic mean-1eld approaches and we have shown that they provide a good basis for the understanding of unstable dynamics. We will review the application of those approaches to the problem of the fragment formation in this section. A diHerent type of treatment that is suitable for fragmentation processes is stochastically initialized molecular dynamics where the diHerent break-up channels can be explored by diHerent members of the ensemble of many-body states considered. While the utility of such approaches for providing insight into the collision dynamics has already been illustrated repeatedly, the quantitative validity of classical molecular dynamics in nuclear physics is still being debated. Moreover, the role of the speci1c spinodal decomposition mechanism has not yet been completely elucidated, as will be discussed in Appendix E. 5.2.1. Fragmentation with molecular dynamics The fragmentation of a hot spherical drop with classical molecular dynamics was studied by Pratt et al. [77], who found that the IMF production is enhanced for trajectories that lead the system inside the spinodal region (see above). In the case of maximal IMF yield, the corresponding mass distribution is quite broad, and a power-law 1t, P(A) ∼ A−5 , yields 5 = 1:89 (the value expected at the critical point is 2:2). The maximum IMF production occurs at larger temperatures for larger systems and their mass distribution is also broader, as illustrated in Fig. 5.10. The 1nal mass distribution diHers from what would result in the fragmentation of an equilibrated system having the density and temperature reached in the dynamical evolution. Those “equilibrium” mass distributions (discussed in Section 2.4.2) are characterized by a power 5 that appears to be larger (see Fig. 2.16), i.e. the equilibrium partitions contain fewer intermediate-mass fragments than those obtained in the actual fast dynamics. Hence one can speculate that the IMF excess (relative to equilibrium), as signalled by a small 5, might signal the disassembly of systems that have penetrated deeply into the spinodal region. Other molecular-dynamics studies [232] 1nd explosion-like evolutions that yields a mass distribution represented by a power of 5 ≈ 2:2. However, these systems fragment

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close to the critical point rather than deep inside the spinodal region, so the scenarios may not be comparable. It is interesting to note that a power behavior, with the value 5 signi1cantly below 2:2 have been found experimentally for systems which, according to the mean-1eld calculations, are expected to reach the nuclear spinodal region. This feature certainly deserves a more systematic study, from point of view of general physics as well as in the nuclear context. The utility of modeling nuclear dynamics by means of classical many-particle systems is far from obvious because of their fundamental diHerences in especially two respects: Since they are quantum particles, the nucleons cannot be sharply localized in phase space, and since they are fermions, they are subject to the Pauli exclusion principle. Consequently, over the past 20 years, various extensions of the standard molecular approximation have been proposed in order to partially remedy these problems. The simplest introduce various forms of momentum-dependent forces that serve to mimic the eHect of the Pauli phase-space exclusion [37,76,235–237]. The Time-Dependent Cluster model forms a more elaborate treatment using classical Euler–Lagrange equations of motion and is particularly aimed at collisions with light nuclei [238–240]. Several multifragmentation studies have been performed within the so-called quantum molecular dynamics (QMD) approaches [74,75,241–244] which attempt to take account of the wave-mechanical nature of the nucleons by representing them by Gaussian wave packets (with the particle dynamics being is still purely classical). In these studies, the reactions have a binary character, even for central events, and the fragments are formed early on and pass through the reaction zone without being destroyed [245]. At a more re1ned level, the basic anti-symmetry of the wave function is taken into account by representing the A-body state as a Slater determinant of Gaussians with a complex width that is either kept constant, Anti-Symmetrized Molecular dynamics [246], or allowed to evolve dynamically, Fermionic molecular dynamics (FMD) [247,248]. A set of coupled classical equations of motion then result for the wave packet centroids (and, for FMD, their width). Since these approaches provide a good description of many features, especially the properties of nuclear matter and the structure of light fragments, it is of great interest to employ this framework for studying the fragmentation dynamics and, in particular, for investigating whether spinodal decomposition occurs. For such an undertaking, it is important to calculate the phase diagram and identify the associated spinodal instabilities. Some recent studies along that line will be discussed in Appendix E. 5.2.2. First stochastic one-body simulations Among the 1rst studies based on stochastic one-body approaches were the eHorts to study fragment production by means of momentum-space quadrupole Ductuations [249–252]. But, as already discussed in Section 4.4, it was shown in Ref. [214] that this schematic treatment of the Ductuation may be unsuitable for quantitative calculations. In particular, for central collisions of 40 Ca and 40 Ca at 90 MeV=A, this method predicts that a sizable fraction of the available energy is released from the system by fast pre-equilibrium nucleon emission, and the explosion of the system is caused by a rapid but cold expansion. However, since the appearance of the fast nucleons could be an artifact of the approximation used to implement the Ductuations, the resulting description may not be reliable.

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Fig. 5.11. Final fragmentation con1guration. A typical con1guration after the formation of the primary fragments in the collision of Xe+Sn at 50 MeV=A studied in Ref. [145] with the simpli1ed Boltzmann–Langevin dynamics developed in Refs. [176,113]. The drawing has been rescaled according to the distance from the viewer in order to provide a three-dimensional impression. (From Ref. [254].)

Guarnera et al. [145,253] have applied the simpli1ed approach using a stochastic initialization proposed in Refs. [113,176,177] to the disassembly of nuclear systems that have been brought inside the spinodal region in the course of a collision. The spherical system studied is intended to approximate the compound systems formed in central collisions of Xe+Sn at 50 MeV=A (see the previous section), which provides an expanding heavy system starting at half normal density with a moderate temperature. As discussed above, the spinodal instabilities lead to the formation of a hollow structure which subsequently condenses into large fragments. A typical 1nal fragment con1guration is illustrated in Fig. 5.11. It shows that 1ve fragments of approximately equal size have been formed at the periphery of the system, while there is no production of small clusters. A more quantitative analysis of these features is given below. For spherical systems, the collective modes have the form f(r)YLM (r) ˆ and, insofar as the bulk region resembles nuclear matter, the radial part can be obtained from the expansion of a plane wave on spherical Bessel functions, f(r) ≈ jL (kr). One may therefore perform a multipole analysis of the density Ductuations (n) (r) and consider 2 3 ∗ (n) L (k; t) = ≺ d r jL (kr) YLM (r) ˆ (r; t) ; (5.3) M

which provides the multipole strength distribution of the spatial correlation function given by r(12) = ≺ (n) (r1 )(n) (r2 ) . For each multipolarity L, the growth rate depends on the radial wave number k. Its optimal value is determined by the requirement that the radial density pro1le 1t within the system, so it is given by kL ≈ xL =R, where R is the radius of the sphere and xL is the argument for which jL (x) has its 1rst maximum. Fig. 5.12a shows the corresponding multipole strength distribution, L (kL ), at the time t = 100 fm=c when the fragments are formed. The dominant mode has L ≈ 5 for which the distance between bulges, ≈ 29R=L ≈ 10 fm. For this multipolarity, Fig. 5.12b shows the dependence on wave number (in terms of the reduced variable kred ≡ k=xL ) and the strong preference for the indicated optimal value kL ) is clearly seen. This wave length admits only one radial undulation, so, consequently, the fragments appear around the center of the system thus making the global matter distribution hollow. It should

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(a)

(b)

(c)

Fig. 5.12. Fragmentation of a compressed sphere. An initially compressed sphere is evolved with the BOB model and the following quantities are extracted from a sample of 400 events: (a) the dependence of L (k) on L for kred = 0:11 fm−1 and t = 100 fm=c; (b) the dependence of L (k) on k for L = 5 and t = 100 fm=c; (c) and the time evolution of L (k; t) for L = 5 and kred = 0:11 fm−1 . (From Ref. [145].)

also be noted that the optimal wave length, ≈ 10 fm, is close to the value in nuclear matter (see Fig. 3.7). Finally, Fig. 5.12c shows the time evolution of 5 (k). The initial growth appears to be exponential, with a characteristic growth time of tL ≈ 35 fm=c. This is close to the fastest growth time found in RPA calculations (where t0 ≈ 30 fm=c for 0 ≈ 10 fm [105,155]). This growth rate and the initial amplitude of the irregularities together determine the fragment formation time, which in the present case is ≈ 100 fm=c. In summary, as a key result, it is found that spinodal instability leads to a characteristic time scale for the fragmentation process and a typical size of the resulting fragments. For in1nite matter, we have discussed how the 1nite range of the interaction, which is related to the cost in surface energy of the fragments formed, is responsible for the suppression of instabilities with a short wavelength, thus ensuring the existence of a most favored wave length, 0 ≈ 10 fm. In the spherical system this feature translates into a suppression of high-multipolarity distortions, leading to a preferred value of L0 ≈ 5 in heavy systems. 5.2.3. BOB simulations The results presented above demonstrate the important fact that multifragmentation processes can be described within the framework of stochastic one-body treatments. Therefore it is interesting to discuss the corresponding results of the more accurate BOB treatment [144,146] (see Section 4.4.2). As brought out by the average dynamics discussed earlier, the expanding system forms a hollow structure that condenses into a number of fairly similar prefragments. Since the matter distribution

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Fig. 5.13. Multipolarity strength distribution. The multipolarity strength distribution L as obtained with BOB at successive times during the evolution of the expanding Au system. (From Ref. [146].)

is concentrated near a spherical surface and the radial structure of the density irregularities was examined already within the simple model, it suPces to analyze the density Ductuations in terms of spherical harmonics. Fig. 5.13 shows the multipolarity strength distribution L , extracted at successive times during the evolution, and the time dependence of the most dominant modes is shown in Fig. 5.14 (left). Since the physical noise is approximately white [120], the diHerent multipolarities are about equally agitated at early times. During the early expansion stage, the nuclear bulk is above the critical density for spinodal decomposition, but the expanding surface region is unstable, as the dilute matter seeks to increase its binding. This leads to an ampli1cation of the surface irregularities and determines the initial growth of the multipolarity strengths observed in Fig. 5.13. As the bulk of the system subsequently descends into the region of spinodal instability, the unstable bulk modes are ampli1ed as well. It should be noted that, as already pointed out, the monopole mode (L = 0) is agitated from the outset, while the higher multipolarities are agitated by the stochastic force only when the local spinodal instability occurs. Once the system becomes dilute, a rapid growth of the monopole mode is favored and a hollow con1guration develops promptly. Meanwhile, the various multipoles are being agitated by the brownian force and ampli1ed by the one-body 1eld, so the associated amplitudes start to grow, with the most unstable modes increasing most rapidly. Once the hollow con1guration

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Fig. 5.14. Spinodal fragmentation dynamics in Au. Left panel: The time evolution of the multipolarity strength L of the unstable modes having L = 2– 6 for the expanding Au system. Right panel: The associated time evolution of the average fragment multiplicity obtained by using various values of the density cut-oH cut , as indicated. (From Ref. [146].)

Fig. 5.15. Fragment charge distribution. The fragment charge distribution resulting from the fragmentation of initially compressed gold nuclei, as extracted from the density pattern after the condensation has occurred, using a density cut-oH of cut = 0:05 fm−3 . (From Ref. [146].)

has been formed, there is little change in the strength distribution, except for a continued overall growth, as the ampli1ed irregularities relatively rapidly condense into a number of intermediate-mass fragments. The multiplicity and the characteristics of the fragments obtained are determined primarily by the dominant multipolarities. Similarly to what was seen in Fig. 5.12c, the early exponential growth characteristic of unstable modes is apparent, and the eventual leveling oH at the time when the fragments are fully formed is observed. The 1nal multipole strength is concentrated around L ≈ 4, so that elementary geometrical considerations suggest a 1nal fragment multiplicity of N ≈ 6 [146]. It is possible to identify distinct prefragments at each time step during the evolution by employing a density cut-oH cut . Fig. 5.14 (right) shows the resulting prefragment multiplicity as a function of time. The results obtained with the three diHerent cut-oH values cut converge at t ≈ 150 fm=c when the condensation is completed. Fig. 5.15 displays the resulting 1nal fragment charge distribution extracted at t = 200 fm=c with the lowest cut-oH value, cut = 0:05 fm−3 . The distribution is rather broad, with a concentration near carbon-like fragments. The emergence of these primary fragments is a reDection of the fact that certain unstable multipoles dominate the dynamics. Also, as is evident from the multipolarity strength distribution, the dynamics

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eHectively suppresses the development of high multipolarities, thereby preventing the formation of very small primary fragments. However, the mass distribution is broad, reDecting the competition between several unstable modes and the importance of non-linear dynamics, such as coalescence eHects among the prefragments. Since the wavelengths and the growth times of the most important modes are not very dependent on the particular density and temperature achieved inside the spinodal region, the resulting fragment mass decreases with the dilution of the system. The resulting fragment sizes thus eHectively depend on the expansion velocity at the stage when the fragments are formed. Thus, as illustrated using the simple model of Ref. [145], larger fragments are obtained for systems that expand only slowly. We 1nally stress that we have discussed only the formation of prefragments. Since these are still excited, their subsequent de-excitation must be taken into consideration before any comparison to experimental data would be meaningful. For example, the de-excitation generally produces an abundance of very light fragments (Z 6 2) and reduces the mass of the primary fragments. It is therefore important to carefully ascertain whether the 1nal fragmentation pattern retains a signi1cant memory of the original spinodal fragmentation dynamics. For example, the light-fragment population, which is suppressed in the primary IMF yield, is largely being replenished as a result of evaporation, so that no low-Z hole is apparent in the 1nal yield curve. As shown by these simulations, the presence of spinodal decomposition is signalled by the formation of nearly equal-sized fragments, due to the dominance of a few collective unstable modes in the linear regime. Therefore, one may search for events that keep the memory of this early con1guration, typical of spinodal decomposition, even after the secondary decay. With this aim, one may employ the correlation analysis proposed by Moretto et al. [255] which reveals the presence of such events even if they are only relatively few. It basically consists of producing a two-dimensional histogram (occasionally called a LEGO plot) in terms of the extracted mean and dispersion of the IMF charge distribution, Z and ]Z. Fig. 5.16 presents the results of such an analysis made for the BOB simulations of central Xe + Sn collisions at 32 MeV=A [256]. It can be seen that a peak at very small values of ]Z, i.e. approximately equal-size fragments, stands out quite clearly against a rather structureless background, even though it contains only a few per cent of the yield. The background can be well accounted for by statistical considerations (e.g. it can be reproduced from arti1cial events constructed by sampling the fragments randomly from diHerent events). Thus, the emerging peak cannot be of statistical origin but must be regarded as a very speci1c signal of the spinodal decomposition scenario. This analysis presents a powerful tool for identifying the spinodal disassembly scenario in experiments, as will be illustrated in the next section. 5.3. Isospin dependence of spinodal fragmentation The presence of the isospin degree of freedom enriches the physics considerably and a number of studies have sought to elucidate how nuclear spinodal fragmentation depends on the neutron and proton fractions [44,130,137,257,258]. Neutron-rich nuclear system can be created in the central collision of two heavy nuclei at intermediate energies (Elab ∼ 100 MeV=A) or by bombardment of a heavy target nucleus with a light projectile at high energy (Elab ∼ 1 GeV=A). In general, the N=Z ratio of the primary fragments is

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Fig. 5.16. Fragment size correlations. The IMF size correlations displayed as a two-dimensional histogram where the axes represent the average fragment charge Z and the associated dispersion ]Z, as obtained with BOB simulations. The peak at very small ]Z contains events having approximately equal-size fragments, while the background can be well reproduced by event mixing. (From Ref. [256].)

largely determined by the interplay of symmetry and Coulomb energies. As already discussed in Section 2, the spinodal decomposition process drives the system towards thermodynamic liquid–gas phase coexistence, where the high-density phase (liquid) becomes isotopically more symmetric while the low-density phase (gas) accumulates most of the excess neutrons. This eHect is related to the increase of the symmetry energy with the density at subsaturation densities and is included in all realistic eHective interactions. 5.3.1. Fragmentation of dilute isobars with A = 197 In order to elucidate the characteristic features we review here results obtained by Larionov et al. [257] for the multi-fragment break-up of systems containing A0 = 197 nucleons with a variable number of protons, Z0 = 63; 79; 95. The source temperature is taken as T0 = 3 MeV. The systems are prepared with a density of half the saturation value and are endowed with a radial Dow of 3 MeV=A (to prevent a recontraction of the source). The total excitation energy is E0∗ =A = 8 MeV including compression, Dow, and thermal energy. We shall discuss especially various key observables, namely charge yields, IMF (Z ¿ 3) multiplicity distributions, N=Z ratio versus IMF charge (isotopic ratios), and seek to elucidate their sensitivity to the neutron excess of the source system (i.e. on its proton fraction y0 = Z0 =A0 ). We shall concentrate on the primary fragments, which are generally hot and subsequently deexcite by sequential emission of light fragments. The evolution of the fragmentation process was calculated with the stochastic mean-1eld (SMF) treatment [145], which represents a rough approximation to the Boltzmann–Langevin model and can be regarded as a simple version of the brownian one-body BOB model discussed above. The radial distribution the resulting fragments at the time t =150 fm=c is shown in Fig. 5.17 (left). As already seen in various earlier studies (see in particular the original study of the breakup of an expanding 197 Au system [150]), the fragments form a bubble-like structure where the fragments are situated at R = 7–13 fm, with the larger fragments being closer to the center. At this point, the

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Fig. 5.17. Fragmentation of an expanding 197 Eu system. The fragmentation of a dilute and expanding spherical source containing 63 protons and 134 neutrons is calculated with the stochastic mean-1eld model. Left: The radial density pro1le of the hot fragments at the time t = 150 fm=c. Solid, short- and long-dashed curves refer to fragments with charge numbers Z = 1–2, Z = 3–10 and Z ¿ 11, respectively. Right: Kinetic energies as a function of the fragment charge number Z, as obtained with SMF (histograms) as well as with the statistical model SMM before (crosses) and after (squares) Coulomb acceleration. (From Ref. [257].)

dynamical evolution is stopped, the fragments are accelerated by their mutual Coulomb repulsion, and the secondary deexcitation process is carried out (see later). The authors of Ref. [257] discussed their dynamical results relative to those obtained within a statistical model, for which they used SMM [19]. (Instructive application of SMM was also made by Botvina and Mishustin [259].) These reference results were calculated for a freeze-out density equal to one-third of the saturation value so the corresponding volume is approximately that enclosed by the bubble con1guration displayed in Fig. 5.17 (left). Finally, the overall excitation energy is taken as E ∗ =A=8 MeV, with no radial Dow. (For inclusion of radial Dow into statistical multifragmentation, see Ref. [16].) It should be noted that in the dynamical treatment the source is expanding as the fragments are being formed, so once their formation is complete (i.e. they no longer feel their mutual nuclear attraction) they take up a volume larger than that of the initial unstable source. Fig. 5.17 (right) shows the average IMF kinetic energy at the breakup time as a function of the fragment charge Z as obtained either dynamically or statistically, both before and after the Coulomb acceleration. The mutual Coulomb repulsion causes the fragment kinetic energies to increase with Z, except for the heavier fragments which are formed closer to the center where the Coulomb force is weaker. This eHect is already visible in the dynamical spectra at the breakup time. Since the statistical model considers fragments that are distributed uniformly within the freeze-out volume, their Coulomb interaction energy is larger than that of the dynamical fragment con1gurations of the same RMS size. This qualitative diHerence in the fragment con1gurations should then be reDected in the subsequent Coulomb acceleration. Indeed, from Fig. 5.17 (right) it can be seen that while the kinetic energy of the heavy fragments are very similar in the two treatments prior to the Coulomb acceleration, (and they are largely independent of Z), Coulomb acceleration is more

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(b)

(c)

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Fig. 5.18. Charge yields from various sources with A = 197. The fragment charge yields from both hot (a,b,c) and cold (d) sources with A=197 and various values of Z as indicated, as obtained with either SMF (histograms) or SMM (diamonds). (From Ref. [257].)

eHective for the uniform distribution of the statistical treatment than for the hollow con1guration produced dynamically. The fragment charge distributions obtained for the various sources are shown in Fig. 5.18. The results are overall very similar in the two treatments, suggesting that a high degree of equilibration occurs during the dynamical evolution. However, it is important to that the dynamical production of primary light clusters is signi1cantly reduced relative to the statistical yield. This characteristic feature arises because the ampli1cation of unstable modes with short wave lengths is suppressed by the 1nite range of the nuclear interaction. The corresponding IMF multiplicity distributions are shown in Fig. 5.19. It is interesting to note that the statistical IMF multiplicity is roughly proportional to the charge of the fragmenting source, primarily as a consequence of the Coulomb interaction, while the dynamical IMF multiplicity is fairly constant. This latter feature can be understood from the fact that the wave lengths of the dominant unstable modes are rather independent of the proton fraction y0 . Furthermore, Fig. 5.20 shows the average ratio of the produced primary fragments, N=Z, as a function of the fragment charge number Z. For the almost symmetric system having Z = 95 (a), the fragments have practically the same N=Z. For Z = 79 (b), the fragments have slightly lower N=Z ratio relative to the source due to the enhanced neutron loss during the multi-fragment breakup. This reduction of N=Z for the primary fragments is more pronounced for the most neutron-rich system having Z = 63 (c). In this latter case we observe also a clear eHect of the density dependence of the symmetry energy: the dynamical calculations (solid histogram) yield a decrease of N=Z with Z, for 3 6 Z 6 10. Light fragments are more likely produced in regions of lower density while the heavier fragments arise from higher densities. Since low-density regions tend to be more neutron

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Fig. 5.19. IMF multiplicity distributions from A = 197. The multiplicity distributions of intermediate-mass fragments for the sources with 197 nucleons considered in Fig. 5.17, as obtained with either SMF (histograms) or SMM (diamonds). (From Ref. [257].)

(a)

(b)

(c)

(d)

Fig. 5.20. Neutron-to-proton ratio. The average neutron-to-proton ratio N=Z of the primary fragments versus their charge number Z, as obtained with either SMF (histograms) or SMM (diamonds) for the same four scenarios as considered in Fig. 5.17. The arrows indicate the N=Z ratio of each source. The arrows show the N=Z ratios of initial sources. (From Ref. [257].)

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(a)

(b)

(c)

(d)

381

Fig. 5.21. Dispersion in N=Z. Dispersion N=Z = [ (N=Z − N=Z )2 ]1=2 as a function of the fragment proton number Z. The cases displayed are the same as in Fig. 5.18. (From Ref. [257].)

rich (because of the isospin fractionation eHect), this explains the decreasing behavior of N=Z as a function of Z. The last observable considered is the dispersion in the N=Z ratio of the primary fragments, shown in Fig. 5.21 as a function of their charge number Z. This quantity reveals a signi1cant diHerence between the two treatments at the primary stage, even when the mean values of N=Z are the same (cf. Figs. 5.20b and 5.21b). Generally, dynamically formed fragments have N=Z ratios that are closely tied to that of the source and therefore exhibit little variability, while the statistical model produces fragments whose N=Z ratios Ductuate to a degree constrained only by phase space. As it turns out, many of the diHerences are washed out by the sequential de-excitation process, to which the primary fragments are subject. For example, the relative paucity of light fragments in the dynamical calculation is being largely eliminated by the feed-down from the decay of heavier fragments, while many of the additional fragments obtained in the statistical calculation are to light that their decay converts them into light particles, thus reducing the 1nal IMF multiplicity. Concerning the N=Z of the 1nal products, a universal decrease of N=Z with Z is obtained, mostly due to a larger Coulomb barrier for the emission of light charged particles by heavier hot fragments, thus increasing neutron emission relative to charged-particle emission. The fact that the two calculations yield a similar dependence of N=Z on Z for the 1nal fragments suggests that the excess neutrons of the hot fragments are very weakly bound, so they can escape from the compound nucleus at the very early stage of the deexcitation process. Such prompt neutron emission changes the excitation only little and the subsequent de-excitation steps then proceed almost unaHected. A similar smearing occurs for the N=Z dispersion. As can be seen from Fig. 5.21, the large diHerence present for the hot primary fragments (c) is washed out entirely by the subsequent deexcitation (d). However, the calculations depend to some degree on structure properties of very exotic nuclei that are largely unknown, so one may expect a quite large uncertainty in the predictions of the sequential decay by the method of Ref. [19]. Nevertheless, this inherent uncertainty notwithstanding, it seems

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safe to conclude that any suitable candidate signal of the spinodal nuclear fragmentation mechanism must be carefully designed to survive the eroding eHects of the sequential deexcitation process. 6. Confrontation with experimental data In recent years considerable experimental eHort has been devoted to the exploration of nuclear multifragmentation with the primary focus on the possible connections between this phenomenon and the occurrence of phase transitions in nuclear matter. Of particular relevance to this issue is multifragmentation occurring in dissipative heavy ion collisions at intermediate energies. Recently developed detector arrays permit an accurate selection of events, thus enabling detailed studies of intermediate-mass fragment (IMF) production and a number of experimental 1ndings have provided support for the occurrence of spinodal multifragmentation. This has stimulated vigorous further activity on both the experimental and the theoretical fronts. For example, the observations have revealed a relatively sudden onset of radial expansion and an associated shortening of the disassembly time [83,260]. Furthermore, the ALADIN group [261] has reported that the nuclear caloric curve shows evidence for a phase transition at an almost constant temperature over a broad range of excitation energies and, recently, a negative speci1c heat was reported in the fragmentation of excited projectile-like sources [79]. In one line of research, multifragmentation data was analyzed within Fisher’s droplet model [262–264] and the associated critical exponent was extracted [21,82]. In this connection, it is important to note that one expects, on the basis of studies of the various critical signals in 1nite systems made in exactly solvable models of the liquid–gas phase transition [62], that “critical” behavior occurs along an entire line in the phase diagram and so is compatible with a 1rst-order phase transition. In this section, we 1rst review the experimental evidence for the occurrence of spinodal multifragmentation. Then we will present a detailed comparison of stochastic one-body simulations of multifragmentation events with the multifragmentation data obtained in the very central collisions studied by the INDRA collaboration [23,265]. 6.1. Source characteristics: is the spinodal region reached? When trying to evaluate the experimental evidence for spinodal fragmentation, the 1rst issue is whether the bulk of the system really reaches conditions of density and temperature that are inside the spinodal instability region. Several diHerent experimental observables indicate that the system may have reached these conditions: the occurrence of a radial collective Dow, the size of the emitting source, the associated fragment emission time, and the temperature of the primary fragments. We shall now discuss these in turn. 6.1.1. Radial collective @ow The existence of a radial collective Dow in fragmenting nuclear sources, 1rst observed by FOPI [266], implies that the bulk of the system may reach low density values inside the spinodal region. The nuclear expansion sets in at around 5 –7 MeV=A of excitation energy and the systematics is illustrated in Fig. 6.1.

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Fig. 6.1. Radial Dow. A compilation of experimental data on radial Dow demonstrating the onset of the nuclear explosion around 5 –7 MeV per nucleon of excitation energy. (From Ref. [267].)

As a typical example, it was reported that the reaction 129 Xe +nat Sn at 50 MeV=A leads to the formation of a single isotropic source at an excitation energy of 12 MeV=A and with fragment kinetic-energy spectra indicating a fast disintegration of the system with a radial collective energy of about 2 MeV=A [268]. The value of the radial Dow can be inferred from the experiment by assuming that at the freeze-out con1guration consists of a number of fragments located inside a given volume and endowed with a certain amount of radial collective velocity in addition to the thermal kinetic energy. To the degree that Coulomb eHects can be ignored, the radial Dow can be extracted by exploiting the diHerent dependence of the collective and the statistical energies on the fragment mass [266]. More elaborate analyses make use of statistical multifragmentation models that include Coulomb eHects, such as WIX [16] or SMM [19]. The freeze-out density, which inDuences the interfragment Coulomb energy, and the value of the radial Dow are then 1tted to the observed fragment energy spectra [82,268], leading to remarkably good reproduction of the data. 6.1.2. Correlation between particles and fragments Evidence for bulk fragmentation, which could be related to spinodal decomposition, has recently been reported in ISIS multifragmentation studies with 8–10 GeV=c 9− and p on 197 Au [269]. Making use of two-fragment correlation functions for events sorted in excitation energy, the authors found that the fragment emission time decreases rapidly as the excitation energy increases. In the interval E ∗ =A = 2–4 MeV the time decreases by an order of magnitude. Then above 3–4 MeV per nucleon it becomes compatible with a simultaneous production of the various fragments. The correlation can also be used to infer the volume of the emitting source which is found to increase by a factor of 3–5 in the same energy range. Moreover, it was found that the decrease in emission time is strongly correlated with the onset of multifragmentation and thermally induced radial expansion [83,260]. These 1ndings, which are summarized in Fig. 6.2), provide strong evidence for a multifragmentation mechanism with the character of spinodal decomposition, i.e. a simultaneous breakup of the system into several IMFs.

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Fig. 6.2. Energy dependence of source characteristics. The dependence on the excitation energy E ∗ =A of the source lifetime (bottom), its thermally driven expansion energy jr (center) and the probability of observing a given IMF multiplicity (top). In the bottom panel, the shaded area indicates the range of possible space-time solutions consistent with IMF observables. The solid line is an exponential 1t to the ISIS results. (From Ref. [269].)

Evidence for a transition from surface emission, akin to a gentle evaporative process, to a sudden breakup of the entire system (bulk fragmentation) was recently found also by the Medea-MultiCS collaboration [270] who observed that the production of thermal bremsstrahlung photons is in anti-coincidence with the fragment production. These photons are emitted from proton–nucleon collisions occurring in an equilibrated system and their rate increases strongly with density because of the associated increase of the Fermi speed. Therefore they can be used to measure the density reached by the system during the dynamics. An anticorrelation was then observed between IMFs and thermal photons for central collisions of Ni + Au at 45 MeV=A, while this eHect is not present for the same reaction at 30 MeV=A. The interpretation is that the fragments arise from prompt multifragmentation during the expansion phase following the initial collisional shock, thus preventing the system from forming a compound source for thermal photons. 6.1.3. Temperature of the emitting source Information about the temperature of multifragmenting systems was reported for example, by Marie et al. [268,271]. They applied correlation techniques to multifragmentation events of the reaction 129 Xe+ nat Sn at 50 MeV=A, thereby deducing the multiplicities of light particles (hydrogen and helium isotopes) that were emitted by the hot primary fragments, as well as their kinetic-energy spectra.

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From the knowledge of the secondary light charged-particle multiplicities and kinetic energies, it is possible to reconstruct the average charges of the hot primary fragments and to estimate their mean excitation energies. In this manner, the fragment excitation energies were found to be ≈ 3:0 MeV=A for the full IMF range, indicating a common temperature around 5 MeV. This global constancy indicates that, on the average, thermal equilibrium has been achieved at the disassembly stage of the source. It should be noted that this value of the temperature is quite close to the value associated with the “plateau” observed in the nuclear caloric curve reported by the ALADIN collaboration [261] and in other experiments with emitting sources in the same mass range, as shown in Ref. [272]. It is also consistent with the limiting nuclear temperature imposed by the Coulomb 1eld [273–275] as recently discussed in Ref. [276]. 6.1.4. Negative speci-c heat From the observations reviewed above, one may conclude that the maximum IMF production is accompanied by a dilution of the system together with a shortening of the emission times. The temperature of the hot primary fragments is found to be around 5 MeV, in the case of fragmenting sources with A ≈ 200. These pieces of evidence are all indicating that fragmentation happens because the nuclear system enters the coexistence region of the liquid–gas phase diagram. Further evidence of multifragmentation as a process happening inside the co-existence region of the nuclear matter phase diagram is provided by the recent observation of negative speci1c heat in the fragmentation of excited projectile-like sources [79]. In Ref. [78] it has been proposed to use the kinetic-energy Ductuations of events having the same total energy to directly derive the ensemble heat capacity. In fact, from a classical point of view, for an ensemble at constant energy, the sharing of energy between the kinetic and the interaction parts should be governed by the respective entropies and the associated Ductuations depend upon the respective heat capacities. It is possible to show that negative heat capacity is signalled by the microcanonical kinetic-energy Ductuation becoming larger than the expected canonical limit. Hence the observation of negative speci1c heat indicates the presence of a 1rst-order phase transition. However, the utility of this signal is still being debated [73]. From the experimental point of view, the total energy can readily be split into two Ductuating parts by considering the thermal agitation and Ductuation versus the partition Q-value plus the Coulomb energy. Fig. 6.3 shows the 1rst experimental evidence of negative speci1c heat reported in Ref. [79]. It is interesting to note that the ISIS data [263] have been found to exhibit a Fisher-type scaling, which is typical of second order phase transitions (in in1nite systems), suggesting the occurrence of critical behavior. The reason for the concurrent observation of critical behavior and negative speci1c heat appears to be associated with the 1niteness of the nuclear system [62]. 6.2. Comparison with the INDRA data Multifragmentation events in 129 Xe + nat Sn reactions at 25 –50 MeV=A have been recently studied by the INDRA collaboration [23,256,265,278]. From 32 MeV=A and up, compact single source events (which are associated with very central impact parameters) can be isolated making use of a Dow angle selection.

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Fig. 6.3. Speci1c heat. The kinetic-energy Ductuation (left) and the speci1c heat (right), versus the excitation energy, as inferred from event-by-event energy Ductuations in quasi-projectile sources formed in peripheral Au+Au collisions at 35 MeV=A, using two diHerent freeze-out hypotheses. (Adapted from Ref. [277].)

At 32 MeV=A, the available excitation energy per nucleon for the total system is the same as for the reaction 155 Gd + nat U at 36 MeV=A, which has also been investigated with the INDRA array. Comparisons of the properties of central multifragmentation events obtained in these two reactions have revealed that the charge distribution (normalized by the average IMF multiplicity) coincide over three orders of magnitude, while the average fragment multiplicities are in the ratio 1.49, i.e. almost exactly the ratio of the total charges of the systems (156/104). Such behavior may reveal that bulk eHects play a major role in the multifragmentation of these systems and suggests that spinodal decomposition may provide the onset of the disassembly of the system. Moreover, as already pointed out, the presence of radial collective Dow has been experimentally observed, thus indicating that the system may expand into the unstable low-density region [279]. The kinematical properties of single-source events, as analyzed experimentally, are compatible with the assumption that, at a given point along the reaction dynamics, a freeze-out con1guration is reached, i.e. the system can be described as an ensemble of hot products in thermal equilibrium, free from mutual nuclear interaction. This implies that the total system at this instant occupies a signi1cantly larger volume than would a nucleus with the same mass, or, in other words, that it has reached a low average density. Hence these systems are good candidates for our analysis of the possible occurrence of spinodal instabilities and for a careful comparison with the predictions of the brownian one-body (BOB) model. The model simulations show that central collisions of the two reactions considered (Xe + Sn at 32 MeV=A and Gd + U at 36 MeV=A) lead to the formation of a dense and nearly spherical composite source that subsequently expands and enters the instability region. In the BOB simulations the Ductuations are turned on already along the composite source formation path, in order to ensure that they are eHective as soon as instabilities are encountered. Table 1 lists the source properties at the time of maximum compression as well as when the bulk of the system is well inside the spinodal region, while the characteristics of the system at the end of the BOB simulations are shown in Table 2. The presence of a radial Dow aHects the kinematical properties of fragments and particles. As discussed above, the calculations yield several IMF’s and

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Table 1 For the two reactions considered, the properties of the combined system are listed at both t = 40 fm=c (left), when the maximum compression occurs and the radial velocity vanishes, and at t = 100 fm=c (right), when the bulk is well within the spinodal region (vmax is the radial velocity at the surface and the central temperature T is given in MeV) (from Ref. [278]) Maximum compression

129 155

Xe + Gd +

nat

Sn U

nat

Spinodal zone

A

Z

=0

T

A

Z

=0

T

vmax =c

247 389

103 154

1.25 1.27

8.3 8.3

238 360

100 142

0.41 0.41

4.0 4.0

0.09 0.10

Table 2 Characteristics of the two systems at the end of the BOB simulations, when the de-excitation stage begins. The total mass Atot and charge Ztot are shared between Nf fragments (each having Z ¿ 5) that have an average charge of Zf . The thermal and energies C∗ and Crad are in MeV per nucleon, while the Coulomb energy ECoul is in MeV (from Ref. [278])

129 155

Xe + nat Sn Gd + nat U

Atot

Ztot

Nf

Zf

C∗

Crad

ECoul

194.0 320.0

76.1 120.8

5.1 8.1

13.4 12.6

3.2 3.3

0.81 1.54

175.2 430.0

Fig. 6.4. Comparison of charge distributions. Charge distribution obtained for Xe + Sn at 32 MeV=A and Gd + U at 36 MeV=A. Experimental data are compared to the predictions of the BOB model. (From Ref. [278].)

since these are still excited, the subsequent de-excitation stage is treated using the standard code SIMON [280]. In Fig. 6.4 the charge distributions for the two systems are presented and compared with the experimental data. The calculated charge distributions are similar for the two systems and agree

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Fig. 6.5. Charge distribution of the largest fragments. Charge distributions of the three largest fragments emitted from the single multifragmenting sources formed in 32 MeV=A Xe + Sn and 36 MeV=A Gd + U, as obtained either experimentally (points) or in BOB simulations (histograms). (From Ref. [278].)

with the data. Moreover, the IMF multiplicity is close to the corresponding experimental value for both reactions. Furthermore, very light fragments are not present in the primary distributions. The rather broad charge distribution obtained for the primary fragments is due to the beating of several unstable modes and to coalescence among prefragments. After secondary de-excitation has been taken into account, the charge distribution grows even broader as the light-mass region is being populated. A 1t with a power law, P(Z) ≈ Z −5 , will typically yield 5 ¡ 2. It is interesting that a power behavior is found also in the disassembly of excited classical systems [77] (with the value 5 signi1cantly smaller than 2:2), as discussed in Section 5.2.1. Hence one may speculate that this kind of mass (or charge) distribution characterizes the disassembly of systems that are well within the spinodal region. Fig. 6.5 shows the charge distribution of the three largest fragments detected in each event. These distributions are in excellent agreement with data, with not only the average value being well reproduced, but the entire shape as well. The geometrical shape of the events is also well reproduced by the simulations, as one can recognize by inspecting Fig. 6.6 (left), which displays the observed isotropic ratio (which is related to the isotropy of the fragmentation events) as extracted from either the data and or the BOB events. The comparison is less satisfactory for the fragment angular correlations, as shown in Fig. 6.6 (right). The 1gure represents the fragment relative angle distribution, as obtained in the simulations, divided by the distribution observed experimentally. Small relative angles occur less frequently in the simulations than in the data, in the case of Gd + U, while the occurrence of both of small and large relative angles is underestimated in the Xe + Sn case. These diHerence may be due to the mixing of diHerent central impact parameters in the data, while the calculations were performed only at zero impact parameter. It would obviously be interesting to clarify this point by performing BOB simulations also for 1nite impact parameters.

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389

Fig. 6.6. Event geometry. The reactions 32 MeV=A Xe + Sn (top) and 36 MeV=A Gd + U (bottom) are studied either experimentally (points) or with BOB simulations (histograms). Left: The mean isotropic ratio of the IMF fragments versus the fragment multiplicity. Right: The calculated distribution of relative fragment angle, PBOB (312 ) divided by the experimental distribution, Pexp (312 ), plotted against cos 312 ; equal distributions would yield unity. (From Ref. [278].)

The average kinetic energy, as a function of the fragment Z number, is compared with experimental data in Fig. 6.7. The nearly linear rise in the data for small Z may be attributed to the presence of a collective radial Dow. The simulations provide a good description of the fragment energies for the Gd + U system, for which the Coulomb contribution to the radial Dow is larger; for Xe + Sn the calculated energies fall within ≈ 20% of the measured values. The simulations underestimate the fragment kinetic energies (and hence the radial Dow), but this discrepancy is also below 20%. As indicated in the previous section, this underestimation could presumably be remedied by a quantum dynamical treatment since TDHF simulations have shown that semi-classical calculations tend to underpredict the expansion velocity by about 50%, as compared to the quantum calculations. However, considering that no parameters have been adjusted to reproduce the data, the present level of agreement is already impressive. It may thus be concluded that the BOB simulations yield a good global reproduction of the measured fragment properties, thus supporting spinodal decomposition as a candidate mechanism for the onset of multifragmentation. 6.2.1. Fragment velocity correlations To achieve a deeper insight into the structure of the fragmentation pattern, more exclusive eventby-event analyses have been performed on both the experimental and theoretical side. Correlation functions for the relative fragment velocity are shown in Fig. 6.8 for the two INDRA reactions considered above and compared with the BOB predictions. Also shown are the statistical model

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C(Vred )

C(Vred )

Fig. 6.7. Fragment kinetic energies. Comparison between measured (points) and calculated (histograms) average fragment kinetic energies, either total (left) or per nucleon (right), for Xe + Sn at 32 MeV=A (top) and Gd + U at 36 MeV=A (bottom). (From Ref. [278].)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Gd + U 36 A.MeV

data BoB SMM

Xe + Sn 32 A.MeV

data BoB SMM 0

10

20

30

40

50

60

70

80

1/2

1000×βrel / (Z1 + Z2 )

Fig. 6.8. Fragment velocity correlations. Correlation functions for the relative fragment velocity, as obtained in the INDRA data and compared with BOB and SMM predictions. (From Ref. [281].)

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391

predictions for the disassembly of dilute and excited spherical sources under the hypothesis of statistical equilibrium. The agreement between the data and the BOB calculations is very good. On the other hand, the comparison with SMM reveal a signi1cant discrepancy. This seems to indicate that the dynamical simulations provide a fairly reliable description of the fragmentation geometry and the observed rather weak velocity correlation suggests that the eHective freeze-out volume is relatively large. By contrast, the idealized 1lling of a spherical volume employed in the statistical calculation tends to overemphasize the fragment velocity correlations. 6.2.2. Partition correlations Recently it has been investigated whether it may be possible to identify a speci1c signal reDecting the particular features of the early fragmentation pattern that are associated with spinodal decomposition. As extensively discussed in the previous sections, the onset of spinodal decomposition is characterized by the presence of nearly equal-sized fragments, due to the dominance of few unstable modes in the linear regime. Therefore, one may search for events that keep the memory of this initial con1guration, typical of spinodal decomposition. Of course, these events are expected to appear in very small quantity, also because the experimental analysis is obviously done on the 1nal fragment con1gurations. Following the analysis proposed by Moretto et al. [255], that reveals the presence of such events, even if they are very few, this study has been performed in [256], still on the INDRA data. This has revealed the presence of a few percentage of events having equal-size fragments. The streaking feature is that very similar results are obtained when applying the same procedure to the BOB simulations, while statistical models do not predict at all the productions of such events [282]. As seen in Fig. 6.9, the peak at very small Ductuation ]Z is an indication of a production of equal-size fragments, unambiguously larger than the background value, constructed considering partitions that mix randomly fragments taken from diHerent events. This observation can be related to the particular features of the early partitions produced by the spinodal decomposition. Given the relative weakness of the signal and its possible sensitivity to the detailed procedures (see Ref. [283]), much work has been done recently to improve the quality of the considered correlation function. In this connection, Desesquelles et al. [282] have developed a method for suppressing 1nite-size eHects and express the correlation data relative to a purely statistical population of the phase space, thus obtaining a vastly improved representation of the physical information. Fig. 6.10 illustrates the corresponding result for the INDRA data. The spinodal signal now appears as a peak over a Dat background, which is obviously much easier to interpret. These authors also showed that the fragment production in both percolation models and standard statistical multifragmentation is eHectively independent, whereas the spinodal decomposition mechanism yields strong correlations characteristic of breakup into fragments of nearly equal size. Altogether, these results provide strong evidence for the existence of a non-statistical fragmentation process leading to intermediate-mass fragments of almost equal size (the average IMF charge is Z = 12–14 and the dispersion is only about one charge unit). This novel event class is present within a limited range of bombarding energy and it is associated with small impact parameters. Although it represents only a few percent, it can be identi1ed very clearly over a purely statistical background. The features of these events correspond very well to what is expected from a spinodal decomposition, as demonstrated particularly by BOB simulations which yield a quantitatively good

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Fig. 6.9. Fragment size correlations. The IMF size correlations displayed as a two-dimensional histogram where the axis represent the average fragment charge Z and the associated dispersion Z , as obtained either in the INDRA experiment or with the BOB simulations. The peak at very small Z contains events having approximately equal-size fragments. (From Ref. [256].)

reproduction of the data with respect to both the small non-statistical population component and the dominant background which can be well described statistically (see below). 6.3. Discussion From the above theoretical and experimental results it appears that the nuclear systems formed in central collisions at a suitable energy are compressed and/or heated suPciently to ensure that their

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32 A.MeV

39 A.MeV 3

1+R(< Z > ,σZ )

1+R(< Z > ,σZ )

3 2.5 2 1.5 1 0.5 0

393

5

σZ

40 10

20

M

×

<

2 1.5 1 0.5 0

60

0

2.5

Z

>

60

0 5

σZ

40 10

20

M

×

<

Z

>

Fig. 6.10. Fragment size correlations. The IMF size correlations displayed as a two-dimensional histogram against the average fragment charge Z and the associated dispersion Z , as obtained by a re1ned analysis of the INDRA data for two diHerent bombarding energies. The peak at very small Z contains events having approximately equal-size fragments. (From Ref. [284].)

bulk regions expand into the phase region of spinodal instability. Therefore, spinodal decomposition presents a plausible mechanism for the formation of multiple intermediate-mass fragments and, indeed, it seems that all the characteristics observed in various experiments can be understood as a result of spinodal decomposition. This is clearly illustrated by the ability of the brownian one-body dynamics (BOB) to reproduce the various data, including the multi-fragment size correlations. This success is especially remarkable in view of the fact that the calculations do not have any adjustable parameters, BOB being a well-de1ned approximation to the Boltzmann–Langevin equation in which the stochastic term is fundamentally related to the elementary scattering cross section in the medium. The relation between the spinodal fragmentation mechanism and statistical multifragmentation deserves special discussion. Over the past 20 years, statistical multifragmentation models have been introduced and applied to multifragmentation data. Among these models, the most commonly employed assume that the partitions of the system are given by the statistical weights for an ensemble of hot products that are spatially con1ned, as would be the case if a uniform system within the liquid–gas coexistence phase region were allowed to equilibrate. (This general class of models include in particular FREESCO [14–16], MMMC [17,18], and SMM [19].) By suitable tuning of the parameters, including the freeze-out density, these models can generally be brought to render a good reproduction of the experimental inclusive fragment yields. For example, the observed INDRA IMF yields discussed above can be well reproduced by employing a freeze-out density that is one third of the saturation value and using a total excitation energy of about 6 MeV=A [285]. In fact, apart from the correlations between fragment speed and size, the statistical approaches are reproducing the various observables very well. Therefore, since the thermodynamic con1gurations deduced from such analyzes lie well inside the phase coexistence zone, this agreement may be considered as a further evidence for multifragmentation being associated with a 1rst-order phase transition. This general success of statistical models notwithstanding, it should be recognized that the underlying mechanism need not be statistical. Indeed, whenever the underlying dynamics is suPciently complicated, one may generally expect a statistical appearance of reduced quantities, such as single-fragment observables. In particular, we have seen that a speci1c microscopic dynamical model,

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namely the BOB approximation to the Boltzmann–Langevin equation, which contains no assumptions of thermal or chemical equilibrium (nor any tunable parameters), can 1t the same single-fragment data equally well as the statistical models (and in fact does a better job for the fragment correlations), a feature noted already in the discussion of spinodal instabilities in asymmetric nuclear matter (Section 3.1.9). Thus, while it may at 1rst seem surprising that the results of a dynamical description are so close to those of a statistical model, it should be recalled that thermodynamic equilibrium corresponds to an unbiased population of the available phase space. Therefore, one may take the similarity between dynamics and statistics as an indication that the dynamics is eHective in 1lling the phase space during the relatively brief time available. In the microscopic dynamical description, a fast democratic 1lling of the fragment partition phase space is in fact facilitated by the coupling of several important unstable modes, as discussed earlier. Thus, the spinodal decomposition mechanism provides the seeds of fragment formation and, even though this is typically a non-equilibrium process, it is possible for the single-fragment features to display a large degree of equilibration. Furthermore, the inequities in the population of the available phase space are reduced by the 1niteness of the system as well as by the secondary decay processes. In this manner, the spinodal fragmentation dynamics provides justi1cation for the application of statistical models. However, it is to be expected that the success of the statistical models in accounting for the data depends on the degree of inclusiveness in the observables considered and that the character of the underlying dynamics may be revealed by more exclusive measurements. In particular, as the INDRA analysis of the fragment size correlations has shown, even a relatively small deviation from the statistical predictions may provide an unambiguous signal of the speci1c fragmentation mechanism. In addition to accounting for the observed small non-statistical component consisting of nearly equal-mass fragments, the spinodal mechanism also provides a quantitative understanding the dominant statistical multifragmentation component as the combined result of the complex dynamics of the unstable modes, with the associated non-linearities and chaos leading to a fast population of the accessible phase space. Thus, for the vast majority of the events, their non-statistical origin in the spinodal decomposition mechanism is lost through the complicated fragmentation dynamics, while a few percent retain the spinodal characteristics to a degree suPcient for identi1cation through correlation analysis. These can be considered as signature events for the spinodal multifragmentation mechanism. 7. Perspectives Spinodal instability is a general phenomenon of relevance in many areas of physics. While the present review has focussed on the relationship between spinodal instability and fragmentation within the context of nuclear collisions, many of the methods and lessons are quite general and, as such, may be of utility in a broader context as well. Within a general thermodynamic framework, we discussed in Section 2 how spinodal instability is intimately related to the occurrence of a 1rst-order phase transition, as signalled by a convex anomaly in the thermodynamic potential. The onset of the associated spinodal decomposition was discussed in Section 3, where we saw how the onset of the phase separation can be understood by analyzing

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the linear response of matter within the phase region of spinodal instability. The collective dynamics may then be described by simple feed-back equations. The associated dispersion relation identi1es the modes that are ampli1ed most rapidly and these will then tend to become dominant in the course of time. The theoretical tools for studying the further, increasingly complex, evolution were then discussed in Section 4 and we showed how the key physics is naturally contained within the Boltzmann–Langevin treatment, in which the stochastic outcome of individual two-body collisions between the constituent nucleons are propagated and ampli1ed by the self-consistent eHective one-body 1eld. In particular, we discussed a relatively simple approximate treatment of the Boltzmann– Langevin equation, the Brownian One-Body model, in which the stochastic part of the two-body collision term is emulated by a suitably tuned stochastic one-body term, without the introduction of any adjustable parameters. Then, in Section 5, we reviewed the applications of various dynamical treatments to nuclear fragmentation. These applications suggest that the nuclear collision zone indeed enters the spinodal phase region, under suitable conditions of bombarding energy and impact parameter. Furthermore, because the most unstable spinodal modes grow dominant, the resulting fragmentation pattern develops some regularity, leading to a signi1cant enhancement of partitions into intermediate-mass fragments of nearly equal size, over what a purely statistical breakup would suggest. Finally, we addressed the confrontation of these theoretical expectations with the relevant experimental data. Especially important is the INDRA observation of a small but signi1cant non-statistical component consisting of multifragmentation events in which the intermediate-mass fragments have very similar charges. The agreement of the dynamical calculations with these data is remarkable, for the non-statistical component as well as for the dominant but featureless part of the events which can also be well accounted for statistically. Since it is hard to account for such a result by other means, it appears that spinodal decomposition is a major mechanism behind the multifragmentation phenomenon. These intriguing 1ndings suggest further work both experimentally and theoretically. At the breakup stage, spinodal fragmentation leads to the dominance of few fragmentation modes and an associated paucity of light fragments. However, as discussed in Section 5, these signature features may be degraded by the secondary deexcitation of the excited primary fragments. It is therefore essential to identify suitably robust observables. Even though some experimental analyzes have already been performed with the aim of extracting information about to the primary fragment partitions, most notably the IMF charge partition analysis of the INDRA data, additional work still remains. Indeed, the charge correlation results are still under debate, with regard to both the de1nition of the uncorrelated reference that determines the normalization and the statistical signi1cance of the extracted signal. This very important issue requires more work on the analysis methods and probably additional data with higher statistics. In addition to the need for disentangling the eHect of the secondary deexcitation processes, it is important to explore the dynamical path of the fragmentation process since it is likely to depend on the speci1c mechanism operating. For this task new types of analysis must be developed. Nowadays the possibility of studying fragmentation reactions with neutron-rich nuclei opens up entirely new perspectives. As discussed, the spinodal decomposition in asymmetric nuclear matter involves also a chemical component which is sensitive to the symmetry term in the nuclear matter equation of state and the fragmentation produces a corresponding chemical fractionation between the

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liquid and gas phases. This type of information provides additional constraints that can be used to elucidate more precisely the multifragmentation path and thus help to test our understanding of the spinodal fragmentation process. From the results reviewed in Section 5.3, it appears that spinodal decomposition in neutron-rich nuclei leads to isospin fractionation as a quite selective mechanism and it is encouraging that this phenomenon was recently observed [53,54]. Isospin fractionation may in fact be an even better signal of spinodal decomposition than the production of equal-size fragments, since coalescence eHects and the competition between several unstable modes render the dynamical distributions rather similar to the statistical ones, apart from the de1cit of light clusters in the spinodal dynamics. By contrast, the N=Z composition of the intermediate-mass fragments (which represent the liquid phase) does not depend much on the particular unstable modes involved and is therefore more robust. Thus, in the spirit of the fragment-size analysis performed on the INDRA data [23], one might seek analyses designed to reveal the N=Z values of the primary fragments. Such an approach could provide a more stringent test of spinodal decomposition and, at the same time, establish a more direct link between the observed results and the employed nuclear equation of state. On the theoretical side, it is desirable to improve our description of spinodal decomposition dynamics. Even though the agreement between calculations and data is very good, there is a need for calculations that are theoretically more accurate. (We recall that the predictions for the INDRA case were made with the stochastic one-body model which is merely a rough approximation to the underlying Boltzmann–Langevin treatment, which itself is only a rather simplistic semi-classical model.) Only when the model is accurate and reliable can one hope that a comparison with data may yield quantitatively useful information about such key quantities as the nuclear equation of state. In this regard it seems important to go beyond the semi-classical level and take account of various quantal eHects, such as memory time. Another aspect deserving further theoretical investigation is the relation between the dynamical path and the population of the available phase space. It might be instructive to determine the degree of equilibration achieved in the course of the nuclear spinodal decomposition process and, in association with this, to identify signals reDecting the non-equilibrium features that are speci1c to the spinodal mechanism. Concurrent with the advances in understanding nuclear spinodal fragmentation, discussed here in detail, there have been exciting developments on related topics, speci1cally regarding the occurrence of anomalies in the nuclear speci1c heat, which provides another phase-transition signal. This has helped stimulate renewed interest in nuclear multifragmentation as a means for probing nuclear matter away from the ordinary equilibrium. We hope that the present review will provide a valuable resource for anyone interested in this topic.

Acknowledgements This work was supported by the Director, OPce of Energy Research, OPce of High Energy and Nuclear Physics, Nuclear Physics Division of the US Department of Energy under Contract No. DE-AC03-76SF00098.

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Appendix A. Seyler–Blanchard model Here we give some speci1c expressions for the generalized Seyler–Blanchard model, a selfconsistent microscopic model based on a 1nite-range two-body interaction whose strength is modulated by a momentum and density dependence. When solved in the semi-classical (Thomas–Fermi) approximation it yields a good quantitative reproduction of a variety of macroscopic nuclear properties, such as the binding energies, 1ssion barriers, and density distributions of nuclei [34,286]. In addition, for our particular parameter choice, the energy dependence of the optical potential is also quite well reproduced. Further details can be found in Ref. [35]. A.1. Basics For spin–isospin saturated systems, the basic two-body interaction is given by −r12 =a 2 (r1 ) 2=3 e p12 s (r2 ) 2=3 V (r1 ; r2 ) = −C 1 − O 2 − ; + b s s s r12 =a

(A.1)

where r12 ≡ |r1 − r2 | is the separation between the two nucleons, p12 ≡ |p1 − p2 | is the diHerence between their momenta, and O ≡ 1 − . Furthermore, s denotes the saturation value of the matter density (A.6) and s is the corresponding value of the zero-temperature kinetic density (A.7). There are a total of four parameters in the model, a; b; C; . The 1rst three characterize the original Seyler–Blanchard model [134], which emerges for = 0. For those we employ the same values as in Ref. [35], a = 0:557 fm, TSB = b2 =2m = 89:274 MeV, and C = 435:1 MeV, leading to a saturation density of s = 0:153 fm−3 , a compressibility of K = 294:60 MeV, and a surface-energy coePcient of a2 = 18:06 MeV [35]. (For two-component systems, the strength parameter C is replaced by Cl and Cu governing the interaction strength between like and unlike nucleons, respectively [134].) However, the energy dependence of the optical potential is then too strong, as reDected in the value m∗s = 0:3817m for the eHective nucleon mass at = s . This problem can be largely remedied by replacing part of the momentum dependence by a density dependence [34,286]. The generalized interaction (A.1) was introduced in Ref. [35] for this purpose. It has been constructed so any value of the additional parameter leads to the identically same results for the average nuclear properties mentioned above, at zero temperature. Furthermore, as is increased from zero, the eHective nucleon mass m∗ increases steadily and reaches its free value for = 1. A realistic value, m∗s = 0:7118m is obtained for = 0:75 [35]. When employing the semi-classical mean-1eld approximation, the motion of the individual nucleons is governed by an eHective one-body Hamiltonian having a simple standard form, h[f](r; p) =

p2 2m∗ [](r)

+ U [](r) ;

(A.2)

due to the quadratic momentum dependence of V12 . The eHective mass m∗ (r) and the eHective potential U (r) are generally functionals of the entire one-body phase-space density f(r; p), but for the particular interaction (A.1), m∗ is a functional of the matter density (r) only and U is

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a functional of (r) and the kinetic density (r), C m = 1 + O (r) ˜ ; m∗ (r) TSB 8(r) 5 (r) ˜ ; U (r) = −C (r) ˜ − O(r) ˜ − 8(r) ˜ −3 (r)

(A.3) (A.4)

where we have introduced the auxiliary kinetic density 8(r) = ((r)=s )5=3 s and the tilde denotes a convolution with the normalized form factor g(r) = exp(r=a)=49a2 r, ˜ ≡ g ∗ , ˜ ≡ g ∗ , and 8˜ ≡ g ∗ 8. In the mean-1eld approximation, the state of the system is characterized by its one-particle phase-space density distribution f(r; p). We shall assume that the system is in local thermal equilibrium at a given temperature T , 2

f(r; p) = [1 + e1(h(r; p)−(r)) ]−1 = [1 + ep =2m

∗

(r)T −D(r) −1

]

;

(A.5)

with D(r) = ((r) − U (r))=T . If the density (r) is speci1ed, then the local chemical potential (r) follows. Conversely, speci1cation of the chemical potential determines the density. This latter quantity and the associated kinetic density (r) can be expressed in terms of the Fermi–Dirac moments Fn (D) = d x xn =[1 + exp(x − D)], dp f(r; p) = F1=2 (D(r))T (r) ; (A.6) (r) ≡ g h3 2m∗ (r) dp p2 f(r; p) = TF3=2 (D(r))T (r) ; (A.7) (r) ≡ g h3 b2 b2 with g = 4 and T (r) = 29[2m∗ (r)T ]3=2 =h3 . For a given temperature T , the above system of equations can be solved self-consistently for either uniform matter [35] or for less trivial con1gurations, such as slabs, rods, and spheres [36]. A.2. Thermodynamics The key thermodynamic quantity is the free energy F = E − TS. For the present discussion of uniform matter, it is more convenient to consider the corresponding intensive quantities, namely the densities e = E=V , s = S=V , and f = F=V = e − Ts, e = TSB − 12 Cn + CneH ; Ts = T

5 3

F3=2 (D) − DF1=2 (D) T = 53 TSB − Cn + 83 CneH − ;

f = − 23 TSB + 12 Cn − 53 CneH + ;

(A.8) (A.9) (A.10)

where n ≡ 49a3 is the number of nucleons within the interaction range. The energy per nucleon, j ≡ e=, and entropy per nucleon, = s=, are displayed in Fig. A.1. While j has a minimum for all temperatures (which moves outwards as T is increased), is steadily decreasing due to the decrease in the available volume.

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10

χ=0.75 0

Entropy per nucleon σ

Energy per nucleon ε (MeV)

399

20 15 10

-10

5 τ=0

τ =20 MeV τ =15 MeV τ =13 MeV τ =10 MeV τ = 5 MeV

4 3 2 1

m*=0.7 -20 0.00

0.10

0.20

0 0.00

0.30

0.05

0.10

0.15

0.20

Nucleon density ρ (fm-3)

Nucleon density ρ (fm-3)

Fig. A.1. Energy and entropy of nuclear matter. The energy per nucleon j = E=A (left) and the entropy per nucleon, =S=A (right), in uniform isosymmetric nuclear matter as a function of the nucleon density , for a range of temperatures, T = 0; 5; 10; 15; 20 MeV. Results are also shown for the limiting temperature Tlim = 13 MeV at which nuclear matter ceases to be self-cohesive. (From Ref. [35].)

4

m*=0.7

0

Reduced probability ln(W)/A

Free energy per nucleon φ (MeV)

10

-10 -20 -30

τ = 0 MeV τ = 5 MeV τ =10 MeV τ =13 MeV τ =15 MeV τ =20 MeV

-40 -50 -60 0.00

0.10

0.20

Nucleon density ρ (fm-3)

0.30

m*=0.7 3

2

1

0 0.00

τ τ τ τ τ

= 20 MeV = 15 MeV = 13 MeV = 10 MeV = 5 MeV

0.10

0.20

0.30

Nucleon density ρ (fm-3)

Fig. A.2. Free energy and statistical weight of nuclear matter. The free energy per nucleon j=E=A (left) and the associated reduced statistical weight, ! = (1=A)ln W = −16 = − 1j (right), in uniform isosymmetric nuclear matter as a function of the nucleon density for a range of temperatures. Nuclear matter ceases to be self-cohesive at the limiting temperature Tlim = 13 MeV. (From Ref. [35].)

The thermodynamic equilibrium state is determined by the free energy per nucleon, 6 = j − T, which is shown in Fig. A.2 (left). This quantity exhibits a minimum that moves inwards with T , since the subtraction of the entropy introduces a steady bias towards lower densities. The corresponding phase diagram, shown in Fig. 2.3, is most easily determined from the pressure, P = −9V F = 2 9 6. A.2.1. Statistical weight For in1nite matter the free energy diverges in proportion to A and the statistical weight W [] becomes a singular distribution sharply peaked at the most probable con1guration. However, it is possible for 1nite systems to make excursions away from the most probable con1guration. (Such systems can be studied by imposing periodic boundary conditions.) It is therefore instructive to consider the quantity ! = (1=A)ln W = − 1j, which is the logarithm of the statistical weight divided by the number of nucleons in the system [35]. It is shown in Fig. A.2 (right).

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Symmetric Multifragmentation

-4 Potential Energy VN (q)/A (MeV)

Coefficients (MeV)

30

aV

20 FA (T) = -aV A + aS A

2/3

10 aS

0

N=12

A=120

-5

N=8

-6

N=6 N=4

-7

N=3

-8

N=2

-9 0

5

10 Temperature T (MeV)

15

20

4

5

6 7 8 9 Root-Mean-Square Size (fm)

10

Fig. A.3. Hot nuclei and multifragmentation. Left: Temperature dependence of the volume and surface coePcients aV and aS in the liquid-drop expansion of the free energy FA of a hot nucleus embedded in a nucleon vapor held at a given temperature T , see Eq. (A.11). Right: A system with 50 (charged) protons and 70 neutrons is represented as a symmetric con1guration of N spherical fragments of equal size. The 1gure shows the energies of the corresponding N -fragment systems, as functions of the rms size of the total matter distribution, calculated at zero temperature. The vertical bars indicate the point at which the N spheres touch. For N = 12 the open circles are obtained for an alternate symmetric arrangement of the fragments. The dashed curve shows the result of a monopole distortion of 120 Sn. (Adapted from Ref. [36].)

For temperatures below the limiting value Tlim , the statistical weight has two peaks, reDecting the bimodal nature of the phase-coexistence equilibrium. The liquid-type peak is near the saturation density, as expected, and the most favored density shifts gradually towards more dilute values as the temperature is raised. In concert with this shift, the peak broadens and the vapor spike in the dilute region near zero density grows increasingly prominent. Thus, the system is metastable, in accordance with the fact that a nucleus held at a constant temperature will gradually disassemble into a dilute gas through sequential evaporation. When the temperature reaches a certain limiting value, there is no longer a maximum at 1nite density and the statistical weight increases monotonically with decreasing density, basically reDecting the increase in the entropy. This limiting temperature is thus the highest one for which a self-cohesive metastable nucleus can exist. Above Tlim and up to Tcrit , metastability can be maintained only if an external pressure is applied. A.2.2. Hot nuclei Nuclear multifragmentation involves hot nuclei embedded a nucleon vapor [100,287,288] and the framework described above makes it possible to calculate the thermostatic properties of such systems. In particular, the dependence of the level density parameter and the free energy on mass number and temperature can be determined [36]. For example, if only the volume and surface terms are considered, the free energy of an uncharged hot nucleus may be written on liquid-drop form, FA (T ) = −aV (T )A + aS (T )A2=3 :

(A.11)

The resulting volume and surface coePcients aV and aS are shown in Fig. A.3 (left). As the temperature is raised from zero, the volume coePcient increases steadily from its normal saturation value, aV (0) ≈ 16:0 MeV, until T reaches the critical temperature, Tcrit ≈ 16:7 MeV, above which there is only a single phase. At the same time, the surface coePcient, aS (T ), decreases from its normal

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value of aS (0) ≈ 18:0 MeV towards zero. The associated surface tension is very well reproduced by a simple analytic approximation, 8(T ) ≈ 8(0)(1 + 32 T=Tc )(1 − T=Tc ) [36]. A.2.3. Multifragmentation An advantage of the treatment is that once the density distribution (r) has been speci1ed, the associated free energy can be determined for any temperature and it yields directly the corresponding statistical weight (see above). Thus it is possible to discuss an equilibrium ensemble of diHerent manifestations of the same system. This is of particular interest in connection with nuclear multifragmentation where a given system may transform itself into macroscopically diHerent con1gurations. As an illustration of this utility, the authors of Ref. [36] considered the breakup of a system with A = 120 into various multifragment channels, each one consisting of equal-size spherical fragments arranged symmetrically. The associated potential energy barrier is shown in Fig. A.3 (right). These are particularly relevant for the regular kind of fragmentation favored by the spinodal mechanism (see Sections 5.2.2 and 5.2.3) and they typically amount to a few MeV per nucleon, increasing with the number of fragments. A.3. Landau parameters We shall now consider small deviations from uniform matter, (r) = 0 + (r). If k denotes the Fourier component of (r), the induced change in the eHective Hamiltonian, h = 49a3 (9h=9) ∗ , then has Fourier components of the form hk = (9hk =9)k , where 9hk =9 denotes the Fourier component of 9h=9. The Landau parameters of interest can then be readily obtained [117], 0 9hk O 0 − 2TF 3 TSB − 2T m∗0 n0 C TF − ; (A.12) F0 (k; T ) ≡ 60 ≈ T 9 m TF TSB 2 1 + a2 k 2 ∗ 3n0 C=T O m0 SB −1 =− F1 ≡ 3 ; (A.13) m 1 + n0 C=T O SB with n0 = 49a3 0 . Furthermore, TF = PF2 =2m ∼ 02=3 is the zero-temperature Fermi kinetic energy associated with the speci1ed average density 0 , while the temperature dependent “eHective Fermi energy” is T0 = TSB 9=9 = m∗0 61 =m60 ≈ 3m∗0 T=2m60 , with 6n being the Fermi-surface moments de1ned in Appendix C.1. For k; T → 0, corresponding to in1nitely gentle density undulations in cold matter, the usual Landau parameter F0 is recovered. For any value of the wave number k, the spinodal region is characterized by F0 (k; T ) ¡−1, and within this region the system is unconditionally unstable against harmonic density distortions having a wave number smaller than k. Because of the 1nite range of the interaction, such instabilities can occur only for wave numbers below a certain maximum value k0 depending on the density and temperature considered. For positive values of F0 the system is unconditionally stable against density undulations, whereas it is subject to Landau damping for F0 values in the interval between 0 and −1. The above standard Landau parameters F0 and F1 for cold uniform nuclear matter (i.e. k; T = 0) were displayed in Fig. 3.2 as functions of the density , for diHerent values of the parameter (which governs the eHective mass at saturation, m∗s ). The dotted curves correspond to = 0, the standard Seyler–Blanchard model in which all the modulation arises from the quadratic momentum

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dependence [134]. The dashed curves are for = 0:75, for which the eHective mass has a realistic value, m∗s ≈ 0:7m [35]. The solid curves show what would result if the interaction had no momentum dependence at all, = 1, as is most often the case in eHective Skyrme-type interactions. It should be noted that since all values of the parameter lead to the same static properties at zero temperature, all curves pass through the value F0 = −1 at the same density, ≈ 0:66s , thus agreeing on the delineation of the spinodal zone at T = 0. Interestingly, for our preferred model having = 0:75, the Landau parameter F0 changes from negative to positive just below the saturation value of the density, so that all density undulations are unconditionally stable for ¿ s . The dependence of F0 on the wave length of the density undulations, = 29=k, was illustrated in Fig. 3.5 (left), for zero temperature. The value of F0 generally increases (i.e. the curve moves up) as the wave number k grows, since more rapidly changing undulations are more costly in energy. The spinodal zone then shrinks and the smallest unstable wavelength is min ≈ 3:48 fm and occurs at ≈ 0:12s . The temperature dependence of F0 was illustrated in Fig. 3.5 (right) for the wave length = 8 fm, near which the most rapid growth occurs. It is qualitatively similar to the dependence on the wave number k, and thus F0 grows steadily as the temperature is increased (or, conversely, the collision system becomes increasingly unstable as it expands and cools). As the temperature is increased from zero (solid curve, corresponding to the short-dashed curve in the left panel) in increments of 2 MeV up to T = 10 MeV (dotted curve), the value of F0 (k; T ) steadily increases, in accordance with the fact that the spinodal density zone shrinks as a function of the temperature. Fig. 3.6 showed the corresponding spinodal boundaries associated with various wave lengths . As the wave number of the mode is increased, the region of instability shrinks. The most noticeable eHects are the steady reduction of both the maximum temperature and the maximum density, whereas the minimum density only grows signi1cantly close to the maximum unstable wave number. The standard spinodal boundary obtained in the thermodynamic analysis corresponds to = ∞. A.3.1. Density ripples We 1nally note that for a harmonic density distortion, (r) = 0 (1 + c cos k · r), the average free energy per nucleon is of quadratic form, 6 = 60 + 12 62 c2 , for c1. The stiHness coePcient 62 is simply related to the Landau parameter, 62 ∼ 1 + F0 , and for zero temperature it is given by 6T2 =0 = 13 TF

1 + n0

C TSB

+

n0 C 1 + a2 k 2

TF − TSB

1 2

= 13 TF [1 + F0 (k; T = 0)] :

(A.14)

Thus it is readily seen that the restoring coePcient 62 is a steadily decreasing function of the wavelength of the ripples, = 29=k. Whenever it is negative, the system is unstable against in1nitesimal oscillations of the speci1ed wavelength. In the intermediate region, outside the spinodal region but still inside the region of phase coexistence, the system is stable with respect to small undulations, but there exist 1nite redistributions of the matter that have a more favorable statistical weight. (Indeed, as an example, the possibility of phase coexistence implies that cold uniform matter slightly below saturation prefers to condense into large separated regions of matter of normal density, interspersed with a dilute nucleon gas.) This method of analysis was employed in Ref. [35] to determine the maximum wave length for which spinodal instability exists at a given point in the –T phase plane.

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A.4. Collective modes in unstable nuclear matter The phase diagram of nuclear matter exhibits a spinodal zone within which the system is mechanically unstable, so that small density irregularities may be ampli1ed by the eHective 1eld. We apply here the formal framework described in Section 3.1.4 to the Seyler–Blanchard model in order to obtain speci1c quantitative results for the growth rates of the unstable collective modes in nuclear matter. The collisionless evolution of the phase-space density f(r; p) is described by the Vlasov equation, 9f=9t ={h[f]; f}. We wish to consider situations in which the system exhibits small deviations from uniform matter at thermal equilibrium. The phase-space density is then of the form f=f0 +f, where ff0 and f0 (r; p) = [1 + exp((h(r; p) − )=T )]−1 is the Fermi–Dirac equilibrium distribution. For uniform matter the eHective 1eld U (r) and the eHective mass m∗ (r) are independent of position, and so f0 depends only on the magnitude momentum. Moreover, the corresponding eHective Hamiltonian is h0 (r; p) = j = p2 =2m∗0 + U0 , where the subscript 0 refers to the speci1ed uniform density 0 . To leading order in the small distortion f(r; p), the temporal evolution is then governed by the following equation: 9f 9f 9h0 9f0 9h + · − · =0 : 9t 9r 9p 9p 9r

(A.15)

We note that 9f0 =9p = (9f0 =9h0 )(9h0 =9p) where 9h0 =9p = C. Since 9f0 =9h0 ≈ (j − jF ), it follows that h needs to be evaluated only near the Fermi surface, as already anticipated in Eq. (A.12). The corresponding equation of motion for the associated Fourier component dr fk (p; t) = √ e−ik·r f(r; p; t) (A.16) P is given by 9 9f0 9hk fk + ik · Cfk − ik · C k = 0 ; 9t 9j 9

(A.17)

fk± (p; t) = fk± (p)e∓i!k t ;

(A.18)

where 9hk =9 is given in (A.12) and k = (g=h3 ) dp fk (p). As shown in Ref. [112], there are two collective modes for each wave number k,

and inside the spinodal zone the frequencies are imaginary, so we may write !k =i=tk . The momentum dependence is given by fk± (p) =

k·C F0 9hk 9f0 ≈− f0 fO 0 : − 1 9 k · C ∓ itk 9j 60 ∓ i8k

(A.19)

In the last relation we have used the fact that T 9f0 =9j = −f0 fO 0 . Moreover, since the eigenmodes receive their support from states near the Fermi surface, we have that k · C ≈ kVF , so the angular and energy dependencies eHectively decouple. Finally, we have introduced the dimensionless quantity 8k = 1=kVF tk . This quantity is generally smaller than unity, being about one-third for the most rapidly ampli1ed mode ( = 8 fm) and at the typical temperature (T = 4 MeV), and it therefore provides

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a meaningful expansion parameter. Expansions in terms of 8k are especially useful near the spinodal boundary where this quantity tends to zero, i=8k 1 =1+ ≈ − → − 12 98k + 82k + · · · ; (A.20) − i=8k − i=8k F0 where · denotes the angular average over the direction pˆ of the momentum p, evaluated in the Fermi surface where p = PF . Thus 8k → 0 at the spinodal boundary where F0 → −1, i.e. the growth times diverge, tk → ∞, as one would expect. Integration of the equation of motion (A.17) over the momentum leads to a dispersion equation from which the characteristic time tk can be determined, k·C 9f0 dp 9hk (A.21) 1=g h3 9 k · C ∓ itk−1 9j 2 1 ≈ −F0 (k; T ) = −F0 (k; T ) 1 − 8k arctan : (A.22) 8k 2 + 82k The dispersion equation (A.21) implies that the normalization of eigenfunctions (A.19) is such ± 3 that ± ≡ (g=h ) dp f (p) = 1. In deriving approximation (A.22), it has been exploited that the k k factor 9f0 =9j is peaked in the Fermi surface, and so the energy dependence of the angular average has been ignored, which is a very good approximation [117]. It should be emphasized, though, that it is important to employ an accurate value of the surface moment 60 . However, because the relevant densities are so low, the standard Sommerfeld expansion is inadequate for temperatures in excess of a few MeV and it would in fact be better to employ the rough exponential approximation 60 ∼ exp(−(9T=jF )2 =12); at T = 4 MeV the fastest growth rates would then be overestimated by typically 10% and the slower ones by considerably more, and the error increases rapidly with temperature. For the results presented here, 60 is generally calculated numerically to ensure good accuracy. Since the function 8 arctan(1=8) increases steadily from 0 to 1 as 8 is changed from 0 to ∞, it is readily seen that the above dispersion equation (A.22) has real roots ±8k when F0 (k; T ) ¡ − 1. It is straightforward to determine 8k by iterating the relation 8−1 = arctan(8−1 )F0 =(1 + F0 ), starting from the value 8 = 1, for example. It is an excellent approximation to ignore the energy average and just perform the angular average at the Fermi surface [117]. The resulting growth rates were illustrated in Fig. 3.7. Appendix B. Boltzmann–Langevin transport treatment In this appendix, we discuss the most important developments regarding the transport treatment of the Boltzmann–Langevin model. We consider an ensemble of identical many-body systems, labelled by n = 1; : : : ; N , each of which has its own reduced one-particle phase-space density, f(n) (r; p). In order to facilitate the discussion, we shall, for the time being, ignore the spin–isospin degrees of freedom so that we eHectively have a one-component system with a degeneracy of g = 4. The incorporation of spin and/or isospin is straightforward but of course makes the formulas more complicated. Moreover, it simpli1es the

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405

notation to use a single symbol to denote a location in phase space, s = (r; p). It is convenient to normalize f(n) (s) so it expresses the occupancy of an elementary phase-space cell of volume hD , D D d r d p (n) (n) f (r; p) = d D r (n) (r) = A ; (B.1) g ds f (s) = g hD where (n) (r) is the spatial density of the system n and A is the total number of nucleons in the system. It is important to note that although the physical system consists of individual nucleons (and thus has ascertain coarseness), its one-particle phase-space density appears as a continuous (and incompressible) Duid. Assume now that the N systems have all been prepared in the same manner, so their initial one-particle phase-space densities are all equal, f(n) (s; t = 0) = f0 (s), while the systems generally diHerent the many-body level. Each of the N systems now evolve independently under the combined actions of its individual self-consistent eHective one-body hamiltonian (its “mean 1eld”) and the residual interactions. Due to the stochastic nature of the residual interaction, the individual systems evolve diHerently at the many-body level and, generally, their reduced one-body densities will therefore also start to diHer. The phase-space density of an individual system may then be written in the form f(n) (s; t) = f(s; t) + f(n) (s; t) ;

(B.2)

where the 1rst term is the average density, N 1 (n) f (s; t) : f(s; t)= ≺ f (s; t) = N n=1 (n)

(B.3)

The second term, f(n) (s; t), denotes the deviation of the individual one-body density from the ensemble-averaged density (obviously ≺ f(n) (s; t) vanishes). The individual evolutions of the one-body densities f(n) (r; p) will in turn cause the individual mean 1elds h[f(n) ] to become diHerent. As a result, in the abstract space of the one-body phase-space densities, the ensemble will evolve from the single initial point f0 into a ever spreading bundle of individual trajectories {f(n)(t) }. When the spread is relatively moderate, as it will be initially and as it tends to be for stable scenarios, it is useful to characterize the Ductuations of f(n) away from the ensemble average f by means of the correlation function for the one-particle phase-space occupancy, (s; s ) ≡ ≺ f(n) (s)f(n) (s ) =

N 1 (n) f (s)f(n) (s ) N n=1

= 2 (s)(s − s ) + cov (s; s ) ; (s − s ) = hD D (r − r )D (p − p ).

(B.4)

where In the last line, the correlation has been decomposed into its diagonal part expressing the local variance in occupancy and the non-local remainder expressing how Ductuations at diHerent phase-space locations are correlated. The two parts are related by unitarity, 2 (s) = − ds cov (s; s ). Once the above quantity is known, then the correlation between any two one-body observables can readily be extracted. The transport treatment makes it possible to derive a closed set of approximate equations of motion for the mean evolution of the phase-space occupancy at given points, f(r; p; t), and the associated correlations between the Ductuations in those occupancies [204,179] (see later).

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For notational convenience we note that the dynamical variable, the reduced one-particle phasespace density, can be represented by its values {fi } on a discrete lattice {si } in phase space, f(s) = (f1 ; f2 ; : : :). The stochastic nature of the two-body interactions causes each individual many-body system to evolve diHerently. The ensuing ensemble of N dynamical histories, {f(n) (r; p; t)}, can be described by the following (evolving) distribution of one-body phase-space densities, N N 1 1 [f(n) − f] = lim (fi(n) − fi ) ; N →∞ N N →∞ N n=1 n=1 i

6[f] = 6(f1 ; f2 ; : : :) = lim

(B.5)

which represents the probability density that a randomly selected system n has the speci1ed occupancy function f(r; p). In the Fokker–Planck approximation, the dynamical evolution of the distribution 6[f] is given by a diHusion equation, 9 92 ˙ 6[f] =− Vi [f]6[f] + Dij [f]6[f] : 9fi 9fi 9fj i ij

(B.6)

(As noted in [162], it is straightforward to carry the expansion to higher order, at the cost of signi1cantly increased complexity, but it is not of practical interest.) The Fokker–Planck equation (B.6) contains two sets of transport coePcients, the drift coeGcients Vi that govern the average change at the phase-space lattice site i and the diCusion coeGcients Dij that govern the degree to which the Ductuations created at two sites i and j are correlated. In order to derive expressions for the transport coePcient functionals V and D, it is useful to note that the number of elementary transitions expected to occur during a brief time interval ]t from two given phase-space elements around (r1 ; p1 ) and (r2 ; p2 ) into two other given phase-space elements around (r3 ; p3 ) and (r4 ; p4 ) is given by d=O12;34 =

d D r1 d D p1 d D r2 d D p 2 f(r ; p ) f(r2 ; p2 ) D (r1 − r2 ) 1 1 hD hD

(B.7)

d D r4 d D p4 O d D r3 dp3D O ; p ) f(r f(r4 ; p4 )D (r3 − r4 ) 3 3 hD hD

(B.8)

×

× D (r1 − r3 )w(p1 ; p2 ; p3 ; p4 )]t :

(B.9)

It has here been used that the two-body collisions are assumed to be local in both space and time. The f factors represent the phase-space occupancies at the initial sites while the availabilities at the 1nal sites are denoted by fO ≡ 1 − f. The above expression holds in any dimension D, but the elementary transition rate w depends on D, for example, D=2

w(p1 p2 ↔ p3 p4 ) =

D=3

w(p1 p2 ↔ p3 p4 ) =

d(12 ↔ 34) h4 (4) (p1 + p2 − p3 − p4 )v12 ; 2m d334

(B.10)

d(12 ↔ 34) h6 (4) (p1 + p2 − p3 − p4 ) : 2 m dP34

(B.11)

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Since the drift coePcient measures the average rate of change, it can be obtained by combining the expected changes from all the possible elementary processes, + O V [f](s) = f(s)W (s) − f(s)W − (s) ;

W − (s)

+

where the W (s) and s, respectively, + O f(s)W (s)]t =

1 2

f(s)W − (s)]t =

1 2

(B.12)

denote the rates of transition into and out of the phase-space location

d =O1; 2;3; 4 (s1 − s) ;

(B.13)

d =O1; 2;3; 4 (s1 − s) ;

(B.14)

and V is just simply the diHerence between the rate of growth and the rate of depletion at the speci1ed phase-space location s. To express the diHusion coePcient, it is instructive to introduce the rates W (s; s ) for the joint occurrence of transitions to or from the phase-space locations s and s , −− 1 f(s)f(s )W (s; s )]t = 2 d =O1; 2;3; 4 (s1 − s)(s2 − s ) ; (B.15) O f(s O )W ++ (s; s )]t = 1 d =O1; 2;3; 4 (s1 − s)(s2 − s ) ; f(s) (B.16) 2 O f(s)f(s )W +− (s; s )]t = 12 d =O1; 2;3; 4 (s1 − s)(s1 − s ) ; (B.17) with W −+ (s; s ) = W +− (s ; s). The oH-diagonal diHusion coePcient measures the accumulated covariance between the changes at the two indicated sites and, accordingly, it can be expressed as follows, O f(s O )W ++ (s; s ) Dcov [f](s; s ) = f(s)f(s )W − − (s; s ) + f(s) O )W −+ (s; s ) + f(s)f(s O + f(s)f(s )W +− (s; s ) :

(B.18)

The diagonal part can be obtained by subsequently applying the unitary relation, or by simply adding the rates of growth and depletion, yielding the expected result, var + O 2D [f](s) = − ds Dcov [f](s; s ) = f(s)W (s) + f(s)W − (s) : (B.19)

B.1. Expansion around the mean trajectory The Fokker–Planck transport equation (B.6) governs the temporal evolution of the distribution 6[f] of one-body phase-space densities f(s), and the expressions above makes it possible to calculate the transport coePcients for any given density f(s). Generally, even when starting from a common density f0 (s), the dynamics has the character of an ever widening bundle of trajectories (as illustrated in Fig. 4.4). When the trajectories are suPciently well concentrated, as they will be at early times after a common initialization and as they may be in a stable scenario, it is useful to characterize the distribution 6[f] by its 1rst and second moment, corresponding to the mean trajectory and the dispersion around it.

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It is possible to derive a closed set of equations for these quantities [204,179]. To derive these equations of motion, one 1rst makes a linear expansion of the drift coePcient around the average one-body density ≺ f(s) , using 9W (s) = W − (s; s ) − W + (s; s ) ; 9f(s )

=± :

(B.20)

The equation of motion for the average is density is then d ≺ f (s)= ≺ V [f](s) ≈ V [ ≺ f ](s) ≡ V (s) ; dt

(B.21)

since the linear corrections average out to zero. In the last line the underscore indicates that the quantity should be evaluated on the basis of the mean density. By contrast, in the expression for the second moment the linear corrections combine with the deviations f(s ) to produce feedback terms proportional to the width of the distribution. The resulting expression for the variance of 6[f] is then quite simple, d 2 (s) = 2Dvar (s) − 2(W − (s) + W + (s))2 (s) : dt

(B.22)

It can be obtained by applying unitarity to the expression for the covariance, d + O (s; s ) + ds [Z(s; s )cov (s ; s) cov (s; s ) = f(s)Z − (s; s ) + f(s)Z dt + Z(s ; s )cov (s; s ) + Z(s ; s )cov (s; s ) + (s ↔ s )] ;

(B.23)

where O )W (s; s ) − f(s )W O (s; s )] ; Z (s; s ) = [f(s

(B.24)

and Z = Z − + Z + , with O = −. The form of Eq. (B.22) for the variance shows that 2 (s) will O seek to relax towards f(s)f(s) which is recognized as the equilibrium variance of the Fermi–Dirac occupancy. Moreover, once the relation 2 = ffO has been obtained it will remain in eHect thereafter. B.2. Lattice simulation method The mean-trajectory treatments have the inherent limitation of being inadequate when instabilities and bifurcations occur, such as in nuclear fragmentation. In order to make such more general processes practically tractable, a phase-space lattice simulation was developed [178,179]. It is based on the assumption that the elementary two-body scattering processes can be regarded as Markovian (though a 1nite memory time can be included as well). Then, within a given suitably small time interval ]t, the number of transitions actually occurring, =, is governed by the corresponding Poisson distribution having the mean given in Eq. (B.7), d =. O It therefore follows in particular that the variance of the number of transitions is equal to the mean number, d=2 = d =O : This basic feature can be exploited for the numerical treatment.

(B.25)

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409

Fig. B.1. Test of the lattice simulation method: mean occupancy. The evolution of the ensemble-averaged phase-space occupancy, f(p), as obtained with the lattice simulation method (histograms) for a uniform two-dimensional fermion gas prepared as a sharp hollow sphere in momentum space. The corresponding Fermi–Dirac equilibrium pro1le is shown by the dashed curve. (From Ref. [178].)

The lattice simulation method can now be brieDy described as follows: The Vlasov evolution can readily be made on a phase-space lattice. (In practice, to achieve suPcient accuracy it is necessary to employ phase-space bins that are signi1cantly smaller than hD .) The two-body collisions then redistribute probability between the space–space bins. The time step ]t is chosen so that the expected number of nucleons moved in a particular elementary process, =O12;34 , is small compared to unity. The actual number of nucleons, =12;34 , moving from the two given initial bins to the two speci1ed 1nal bins deviates from the expectation by a stochastic amount, =12;34 , =12;34 = =O12;34 + =12;34 :

(B.26)

The deterministic part, =, O is obtained by integrating the expected diHerential change d =O12;34 in (B.7) over the speci1ed lattice bins, while the stochastic part, =, is sampled from a normal distribution with the variance =2 = =. O This procedure guarantees that the resulting Ductuations have the physically correct relationship with the average changes, thus ensuring that the Ductuation–dissipation theorem is upheld. However, since the basic bin size is typically smaller than hD , the method causes the phase-space occupancy f to make local excursions into the unphysical regions (either f ¡ 0 or f ¿ 1). Fortunately, this is of no import since physical observables automatically smear the information over volumes of at least hD . This lattice simulation method was tested for a two-dimensional gas prepared in suitable non-equilibrium states [177–179]. Fig. B.1 shows the evolution of the average phase-space

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Fig. B.2. Test of the lattice simulation method: variance in occupancy. The evolution of the variance in the phase-space occupancy, 2 (p), as obtained with the lattice simulation method (histograms) for a uniform two-dimensional fermion gas prepared as a sharp hollow sphere in momentum space. The corresponding value of ffO is shown by the thin histogram, while the dashed curve is the quantum-statistical equilibrium variance. (From Ref. [178].)

density, f(p; t), for a uniform gas prepared with an initially sharp hollow Fermi sphere. It is seen how the momentum pro1le gradually approaches the corresponding 1nitetemperature Fermi–Dirac distribution fFD (p) (which is fully determined by the initial conditions). Fig. B.2 shows the associated variance in the phase-space occupancy, f2 (p; t) = f(p; t)2 − f(p; t)2 . Being initially zero, the variance in f grows initially fastest inside the hollow sphere to where most scatterings occur early on. Subsequently, it gradually evolves towards the appropriate quantum-statistical form given by ˜ 2f (p) = fFD (p)fO FD (p). It is also interesting to note that O the instantaneous equilibrium value, f(p; t)f(p; t), presents a very good approximation to the 2 evolving non-equilibrium variance, f (p; t). These and other more complicated (non-isotropic) test cases have shown that the lattice simulation method is numerically valid. Thus, in principle, it can be used to solve the Boltzmann–Langevin transport equation for general dynamical scenarios, in particular fragmentation in nuclear collisions. However, as of yet, its application for non-uniform geometries in three spatial dimensions is rather computer intensive and therefore still impractical. There is thus a need for methods that, on the one hand, provide reasonable approximations to the full lattice simulation while, on the other hand, are computationally practical.

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411

B.3. Simpli-ed Boltzmann–Langevin model For the subsequent discussion it is convenient that the diHusion coePcient can be decomposed into diagonal and oH-diagonal parts [289], cov D D[f](s1 ; s2 ) = [D1var 12 + D12 ] (r12 ) :

(B.27)

Here the subscripts provide a convenient means of indicating the momentum values and 12 means hD D (p12 ). Since the diHusion coePcient is local in space, we need only be concerned with one position at a time, the coupling between diHerent spatial points being provided by the Vlasov propagation in the eHective 1eld. It is therefore convenient to introduce the contracted diHusion coePcient which is obtained from the full one above by integrating over the spatial separation r12 ; this obviously yields the expression in the bracket. From this starting point, it is possible to develop simpli1ed descriptions of the Boltzmann– Langevin dynamics [290,120]. In particular, adapting a method developed in connection with Fermi liquids [291,292], Randrup and Ayik [120] derived simple approximate expressions for the transport coePcients for a free Fermi–Dirac gas,

1 0 0 O0 0 Drift : V1 ≈ − f1 − f1 − f1 f 1 C12 (f2 − f2 ) ; (B.28) t0 2 DiHusion :

D12 ≈

1 0 O0 var cov [f f 12 − f10 fO 01 C12 f20 fO 02 ] = D12 12 + D12 ; t0 1 1

(B.29)

where the local Fermi–Dirac equilibrium distribution is f0 (p) = [1 + exp(j=T )]−1 with j = p2 =2m. The overall transport rate is governed by the local relaxation time t0 , 2 2 −1 2 T 2 T (B.30) t0 ≈ 9 2 1 − 9 2 0 VF ; jF jF and the coePcient C12 expressing the correlation between the Ductuations induced at p1 and p2 is given by j1 − j0 j2 − j0 60 p1 p2 ‘ + [2 − (−1)‘ ]P‘ (cos 312 ) ; (B.31) 60 C12 = T T 6‘ 2m ‘ ¿0

where 312 is the angle between the two momenta. Furthermore, T2 ≈ 13 92 T 2 is the variance of the Fermi–Dirac pro1le function, −T 9f0 =9j=f0 fO 0 , and the associated energy moments (see Eq. (C.2)) are 1 n 0 O0 D T n T2 1 2 1 (B.32) jf f ≈ j 1 + 6 9 (n − 1)(n − 1 + 2 D) 2 : 6n = 1 1 1 1 2 jF F jF It is important to note that these approximate expressions conserve particle number, energy, and D D momentum exactly, as borne out by the fact that 1 V1 F1 and 2 D12 F2 vanish, with F D representing either the particle number A (F1A = 1), the energy E (F1E = j1 ), or the momentum P (F1P = p1 ).

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P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440

Moreover, it was shown in Ref. [120] that the resulting distribution indeed relaxes towards the appropriate Fermi–Dirac equilibrium form, f1 (t) → f10 ;

(B.33)

12 (t) → f10 fO 01 12 − f10 fO 01 S12 f20 fO 02 ;

(B.34)

where S12 consists of the 1rst two multipole terms in C12 , j1 − j0 j2 − j0 p 1 · p2 +D : 60 S12 = 1 + T T 2mj0

(B.35)

The three terms reDect conservation of particle number, energy, and momentum, respectively. B.3.1. Quality of the simpli-ed model The derived expressions for the transport coePcients provide quite reasonable approximations to those given by the full theory. To illustrate this, one may 1rst consider the overall diHusion rate, as obtained by integrating D1var in (B.29) over p1 . Comparisons of the simple expression (B.30) with numerical evaluations of based on the exact expression (B.19) and approximation (B.29) yield agreements to within one per cent for temperatures up to 5 MeV. cov , One may next consider the energy dependence of D12 1 cov D(j; j ) ≡ (j1 − j)(j2 − j )D12 (B.36) 60 W0 12 √ j − j0 j − j 0 jj f0 fO 0 f0 fO 0 1+ : (B.37) ≈− T T T T jF The 1rst expression can be calculated by direct numerical (Monte Carlo) evaluation of the integrals cov , while the approximate expression follows from the analytical approximation in Eq. (B.18) for D12 (B.29). The quality of the above approximations to D(j; j ) is illustrated in Fig. B.3. cov , Finally, it is also instructive to consider the angular dependence of D12 2 1 1 cov D(3) ≡ (cos 312 − cos 3)D12 ≈ − : (B.38) 1 60 W0 12 2 cos 2 3 sin 12 3 The approximate expression is implied by the approximation (B.29). Fig. B.4 shows D(3) as obtained by numerical calculation of the rate functions in (B.18) (dots) and as given by the simple trigonometric expression (C.14) (curves). The results have been decomposed into their positive and negative parts and the approximation is seen to be quite good. B.3.2. Minimal model While the simple approximate expressions (B.28) and (B.29) for the Boltzmann–Langevin transport coePcients derived above may be quantitatively useful for both analytical and numerical studies of dynamical properties of Fermi liquids, such as nuclear matter, further simpli1cation can be achieved by retaining only those correlations that are required by conservation laws. Such extreme simpli1cation can be achieved by discarding those terms in the expressions for the transport coePcients that have multipolarity higher than ‘ = 1, corresponding to putting C12 = S12 ,

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413

70

D (ε,ε’)

50

Conditional mean <ε’> (MeV)

Projection D(ε) (×103)

60 Analytical Simplified

40

Exact (MC)

30 20 10 0

Diffusion coefficient D (ε,ε’)

60 50 40 30 20

<ε’>|ε = ε

10

Exact (MC)

Simplified

0 0

10

20

30 40 50 Energy ε (MeV)

60

70

80

0

10

20

30 40 50 Energy ε (MeV)

60

70

Diffusion coefficient D(θ12)

cov Fig. B.3. Energy dependence of the diHusion coePcient. The energy dependence of the mixed diHusion coePcient D12 can be expressed by the function D(j; j ) de1ned in Eq. (B.36). This 1gure illustrates D(j; j ) for a Fermi–Dirac gas with a Fermi energy of jF = 37 MeV and a temperature of T = 4 MeV. On the left is shown the projection D(j) as obtained by integrating D(j; j ) over j . The solid dots results from a numerical (Monte Carlo) integration of the exact cov D12 (given in Eq. (B.19)). The open dots follow by integrating the approximation (B.29) numerically, and the dashed curve is the simple approximation (B.37). On the right is shown the corresponding conditional mean energy j j , i.e. the centroid of D(j; j ) considered as a function of j for a speci1ed value of j. (From Ref. [120].)

4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -1.0

Angular dependence of D12

Simplified Exact (MC)

-0.5

0.0 cos(θ12)

0.5

1.0

cov Fig. B.4. Angular dependence of the diHusion coePcient. The angular dependence of the mixed diHusion coePcient D12 can be expressed by the function D(3) de1ned in Eq. (B.38). This quantity is shown as a function of cos 312 , where 312 is the relative angle between the two speci1ed momenta p1 and p2 , for the same case as displayed in Fig. B.3. The exact result (obtained by numerical Monte-Carlo evaluation) is shown by the solid dots, with the open dots indicating its positive and negative parts. The curves show the corresponding quantity for the trigonometric approximation included in Eq. (B.38), with the dashed curves showing separate parts. (From Ref. [120].)

yielding Drift :

V1 = W0 (f10 − f1 );

DiHusion :

D12 = W0 [f10 fO 01 12 − f10 fO 01 S12 f20 fO 02 ] = W0 ˜ 12 :

(B.39)

This corresponds to the simple relaxation-time approximation in which the evolution is governed by the global relaxation time t0 = W0−1 . The average occupancy of a given cell at s1 then evolves

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P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440 Relaxation of anisotropy

Fluctuations of anisotropy 0.3 Variance of anisotropy σq2

Average anisotropy

0.3

0.2

0.1

0.0 0.0

0.5

1.0 1.5 Time t (in units of t 0)

2.0

0.2

0.1

0.0 0.0

0.5

1.0 1.5 Time t (in units of t 0)

2.0

Fig. B.5. Relaxation of the quadrupole moment. Left: For a uniform gas of free fermions in two dimensions is shown the relaxation of the average anisotropy, q = Qxx − Qyy =Q, when starting from a deformed momentum distribution. Right: The growth of the variance of the anisotropy, q2 , for the same case as well as when starting from a Fermi–Dirac equilibrium density, which is isotropic. The corresponding analytic behaviors are indicated by the solid curves. (From Ref. [120].)

according to f˙ 1 = W0 (f10 − f1 ) and so relaxes in an exponential fashion towards the appropriate Fermi–Dirac equilibrium value f10 with the characteristic time constant t0 . Moreover, since 9V1 =9f2 = −W0 12 , the covariance matrix satis1es ˙12 =2W0 (˜ 12 −12 ) and so each element relaxes exponentially towards its proper equilibrium value ˜ 12 at twice that rate. This maximally simple approximate version of the Boltzmann–Langevin transport model still contains the essential features of the full model so that it may provide a useful framework for testing numerical methods, as well as provide a quick impression of the results. As a test illustration the methods developed, the authors of Refs. [290,120] considered the relaxation of the quadrupole moment of the momentum distribution in a uniform Fermi–Dirac gas. Results obtained with the simpli1ed model described above are shown in Fig. B.5 and it is apparent that the stochastic simulation leads to a good reproduction of the analytical expectations for both the ensemble average distortion and the associated variance. B.4. Memory eCects In the one-body transport treatment discussed so far, the collision term is assumed to be local in both space and time, in accordance with Boltzmann’s original treatment. This simpli1cation is usually justi1ed by the fact that the interaction range, as measured by the residual scattering cross section NN ≈ 4 fm2 , is relatively small on the scale of a typical nuclear system, and the duration of a two-body collision is short on the time scale characteristic of the macroscopic evolution of the system. The resulting collective motion has then a classical character, as is the case also in TDHF. However, when the system possesses fast collective modes whose characteristic energies are not small in comparison with the temperature, then quantum eHects are important and the treatment needs to be appropriately improved. When the collision term has a non-Markovian form, then the evolution of the single-particle density matrix depends on the (recent) past. The resulting memory eHects on the spreading width of

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415

a collective state was discussed by Ayik and Dworzecka [293,294]. Deriving an extended RPA equation for the normal modes, these authors demonstrated that the memory eHects in the collision term play an important role in determining the spreading width of the collective mode, in particular through the enforcement of energy conservation between the collective mode and the states responsible for its decay. The collisional damping of collective nuclear vibrations was subsequently studied by Ayik and Boilley [295] on the basis of a Boltzmann-type transport equation with memory eHects and it was shown that the incorporation of memory eHects into the collision term is essential for obtaining the proper damping of small-amplitude high-frequency collective vibrations. In a further development, Ayik [210] derived a transport equation for the single-particle phase-space density by performing a statistical averaging of the Boltzmann–Langevin equation. In analogy with Brownian motion, the Ductuating part of the collision term gives rise to a memory time in the collision kernel which, in turn, leads to a dissipative coupling between collective modes and single-particle degrees of freedom. Within this framework, Ayik et al. [296–298] have made important further advances. In particular, Ivanov and Ayik [296] showed that the coupling between mean-1eld Ductuations and single-particle motion provides a coherent damping mechanism which has an especially large eHect on the relaxation times in the vicinity of the spinodal boundary where the equilibration rate signi1cantly exceeds that resulting from the standard (i.e. Markovian) Boltzmann treatment. This approach is also suitable for studying the role of memory eHects on the spinodal fragmentation process and it was adapted by Ayik and Randrup [211] to nuclear matter inside the spinodal zone. As shown in Ref. [112], the (early) evolution of the collective modes is governed by the simple Lalime feed-back equation (4.25), in which the modes are agitated by a source term arising from the Ductuating part of the collision term and ampli1ed exponentially by the unstable selfconsistent eHective 1eld. The characteristic ampli1cation time tk corresponding to a given wave number k is determined by the associated dispersion relation. As it turns out [113,299], the fastest-growing collective modes, which are those that will become predominant, have fairly high characteristic energies Ek = ˝=tk . For example, for densities ≈ 0:30 and wavelengths of 7–8 fm, for which the fastest ampli1cation occurs, we have Ek ≈ 8 MeV. This is clearly not small in comparison with the temperature in the system, which is T ≈ 4 MeV, typically. Consequently, one must expect quantum-statistical eHects to be important in the dynamics of the collective modes and the standard BL treatment may therefore be inadequate, since it treats the collective modes as classical. We therefore brieDy recall the results obtained in Ref. [211]. The term K(r; p; t) in equation the BL equation is a stochastic function governed by a certain distribution function which depends on the phase-space density f, is zero on the average, and has a correlation function of the form ≺ K(r; p; t)∗ K(r ; p ; t ) =C(p; p ; t − t )(r − r ) :

(B.40)

In the standard BL model, the collisions are assumed to be instantaneous but the 1nite duration of a two-body collision generally modulates the corresponding frequency spectrum, d! −i!(t −t ) ˜ C(p; p ; t − t ) = e C(p; p ; !) ; (B.41) 9

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Correlation functions χ ν(t )

2.0

2.0

T =4 MeV

1.5

T =6 MeV

1.5

1.0

1.0

0.5

0.5

0.0

0.0 χ+(t ) k χ-- (t k

-0.5

χ+(t ) k

-0.5

)

χ-- (t ) k

-1.0

-1.0 0

5

10 15 20 Time t (fm/c)

25

30

0

5

10 15 20 Time t (fm/c)

25

30

Fig. B.6. Temporal correlation functions. The collective correlation functions k+ (t) (solid) and k− (t) (dashed) for an unstable mode, for either T = 4 MeV and tk = 24 fm=c (left) or T = 6 MeV and tk = 36 fm=c (right). (From Ref. [211].)

where ˜ p ; !) = C˜ 0 (p; p ; !)G(!tc ) ; C(p;

(B.42)

with C˜ 0 (p; p ; !) being the kernel entering in the standard treatment. A rough estimate of the duration time yields tc ≈ 2a=v ≈ 5–6 fm=c which is a fairly brief length of time as compared with the typical free travel time for a nucleon, tmfp ≈ =vF ≈ 20–30 fm=c. One would therefore expect that the treatment developed in Ref. [112] will still be applicable, but with suitably modi1ed transport a == responsible for the agitation of the unstable collective coePcients. Indeed, the source terms D k a == == (t). modes in nuclear matter may be replaced by eHective coePcients of the form Dk== (t) = D k k The modulation factor is given by k== (t) ≡ k= (t) + k= (t), where the collective correlation functions for the given collective mode, k± (t), are illustrated in Fig. B.6. It is now possible to see that when a memory time is included the collective correlation coePcients satisfy the following modi1ed equation of motion, d == a == == (t) + = + = == (t) : k (t) = 2D k k k dt tk

(B.43)

This equation is of the same general form as the standard Lalime equation (4.25), but it has a a == == (t), and the solution is then given by == (t) = time-dependent diHusion coePcient, Dk== (t) = D k k k ◦ ==

k (t)O== k (t), where the renormalization coePcients can be expressed as suitable time averages of the collective correlation functions, t t == −(=+= )8k t = = dt e [k (t ) + k (t )] dt e−(=+= )8k t : (B.44) Ok (t) = 0

0

These correction factors are illustrated in Fig. B.7. The 1nite memory time introduces a certain gentleness into the source term which is reDected in the suppression of high-frequency components, as expressed by the modulation function G(!tc ). However, since tc tk , this has little eHect on the agitation of the collective modes. Indeed, the factor G(!k tc ) produces a reduction of only about two per cent for the fastest mode, so it is apparent that this eHect is not essential.

P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440 2.0

T=4 MeV

2.5

Correction factors χ νν’ (t )

Correction factors χ νν’ (t )

3.0

417

ρ=0.3ρ0 2.0 1.5 1.0

χ ++(t ) k

χ +--(t )

0.5

T=6 MeV ρ=0.3 ρ0

1.5

1.0 χ ++(t ) k

0.5

χ +--(t k

k

0.0

)

0.0 0

10

20

30

40

50

60

0

Time t (fm/c)

10

20 30 40 Time t (fm/c)

50

60

Fig. B.7. Correction for memory time. The correction factors O== determining the time-dependence of the covariance k coePcients describing the agitation of collective modes in unstable nuclear matter, for the same two cases as in Fig. B.6. The arrows indicate the asymptotic values. (From Ref. [211].)

Much more important is the quantum-statistical enhancement expressed by the factor F (˝!=2T ). [In the case of stable collective modes, the factor F guarantees that the appropriate quantumstatistical equilibrium is approached [210], whereas the standard Boltzmann–Langevin treatment leads to a classical (Boltzmann) equilibrium occupation of the collective modes, Pk ∼ exp(−˝!k =T ).] Since the characteristic energy Ek =˝=tk exceeds the temperature T , the factor F causes a signi1cant enhancement of the collective source terms Dk and, consequently, the density undulations will grow correspondingly larger in the course of a given time interval. The eHect depends strongly on the temperature T and the growth time tk , but for the fastest mode and the most typical temperatures, the enhancement factor is 50 –100%. B.5. Relativistic formulation The Boltzmann–Langevin model has been adapted to nuclear collisions at higher energies on the basis of Walecka-type 1eld theory by Ayik [300]. This treatment considers spin–isospin degenerate nucleons, L resonances and pions and thus describes the associated one-body phase-space density distributions, fJ (p; x), where J = N; L; 9 denotes the particular particle species and p = (E; p) and x = (t; r). Their dynamical evolution is governed by a set of coupled relativistic Boltzmann–Langevin equations, 9EJ 9 9EJ 9 9 + · − · fJ (p; x) = KJ (p; x) + KJ (p; x) : (B.45) 9t 9p 9r 9r 9p Here KJ (p; x) represents the (deterministic) average eHect of collisions on particles of the species J and KJ (p; x) is the stochastic part characterized by KJ (p; x)KJ (p ; x ) = CJJ (p; x; p ; x )(4) (x − x ) :

(B.46)

The various quantities, EJ , KJ , and CJJ , can be evaluated in a consistent manner on the basis of Walecka-type 1eld theory at diHerent levels of approximation. In Ref. [300] the weak-coupling limit

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was considered by including only binary collisions and the L decay vertex (L ↔ N9) in the lowest order. The interactions are mediated by scalar and vector 1elds, 6(x) and V (x) = (V0 (x); V (x)), having masses mS and mV . They modify the momentum, pJ∗ = p − gJV V (x), as well as the mass, MJ∗ = MJ − gJS 6(x), and the dispersion relation for the baryons is then EJ∗ (p; x) = [(pJ∗ )2 + (MJ∗ )2 ]1=2 ;

J = N; L ;

(B.47)

while the pions remain on shell, E92 = p92 + M92 . The resulting general Boltzmann–Langevin equation, a more complicated stochastic non-linear integro-diHerential equation, has not yet been implemented into a tractable form but, as an illustration, Ref. [300] considers the Ductuations of the L multiplicity in a spatially uniform NL mixture close to thermal equilibrium. It is shown that the L population is described by a Langevin equation and that the corresponding Fokker–Planck equation for the multiplicity distribution has an equilibrium solution of the form Pequil (NL ) ∼

1 −]G=T e ; ML

(B.48)

where ML is the total L production rate and ]G is the diHerence between the free energy of the actual NL mixture and that of the corresponding pure nucleon gas. As a further development of the treatment in Ref. [300], Ayik et al. [301] reduced the relativistic Boltzmann–Langevin equation to a stochastic multi-Duid model. With a view towards relativistic energies, they studied especially the case of two counter-streaming systems of nuclear matter (of the form 1rst considered in Ref. [191]) and extracted the interDow friction force and the associated diHusion tensor.

Appendix C. Analytical approximations In this appendix we derive a number of simple analytical expressions for the various quantities of special interest. C.1. Fermi surface moments of higher degree When treating the transport properties of fermion systems, it is useful to introduce the following moments, dp n 1 29 2m 3=2 ∞ (k) k O O k ; n ≡ j (f(p)f(p)) = dj jp (ff) (C.1) hD h2 0 where f(j) is the Fermi–Dirac distribution corresponding to a given (small) temperature T . In the present investigation, the dimension of the physical space is D = 3, so the power of the energy is p = n + 12 . This quantity must exceed minus one in order for the moment to be 1nite, although the singularity at j = 0 is suppressed by the factor exp(−nj=T ) which rapidly tends to zero for small temperatures.

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419

For k = 1 these moments are those employed in Refs. [302,120] and in Eq. (B.32), (1) n

2 92 3 T n 3 T n 1 T = 6n ≈ jF 1 + (n − 1) n + 2 2 ≈ j ; 2 jF 6 2 jF F jF

(C.2)

where an expansion has been made in the (small) ratio T=jF . In the present study we also need to consider moments of higher degrees, k = 2; 3. At 1rst sight, this may appear to be a problematic task, due to the singular character of the integrand, ffO = T 9j f ≈ T(j − jF ). (We use 9j to denote 9=9j for brevity.) However, it is possible to evaluate these moments by elementary means, performing a suitable number of partial integrations with respect to the energy j, as brieDy outlined below. We are only interested in powers n ¿ − 1, for which the moments n(k) are well de1ned, and we then need not be concerned with the boundary contributions when performing the partial integrations. First note that for an arbitrary function F(j) we have dj Ff9j f =

1 2

dj[(1 − T 9j )F]9j f ;

(C.3)

by 1rst performing a partial integration and then utilizing that f2 − f − ffO = f + T 9j f. Performing another partial integration, and using that T 92j f = 2T (9j f)2 + f9j f, we then 1nd Sn(2)

≡T

2

T =− 6

T dj j (9j f) = − 3 p

2

dj [(1 + T 9j )jp ]f9j f

1 (1) 1 T 2 (1) 2 dj [(1 − T 9j )j ]9j f = Sn − n − S ; 6 4 6 n− 2 2 2

p

(C.4)

where the last relation has been obtained by utilizing the relation (C.3) with F(j) = (1 + T 9j )jp . Consequently, to second order in T=jF we have (2) n ≈

2 9 1 T2 1 1 T n 1 T n (n − 1) − n + ≈ jF 1 + n + j ; 4 jF 2 6 2 j2F 4 jF F

(C.5)

where the last relation holds to leading order in T=jF . So to this order one may make the replacement O 2 → 1 ff, O for any dimension D. (ff) 6 By employing similar techniques, we can also calculate the third-degree Fermi-surface moment, (3) n . First we perform a partial integration and 1nd Sn(3) ≡ −T 3

dj jp (9j f)3 =

T2 5

dj ([(T 9j + 2)jp ]f(9j f)2 + 4jp (9j f)3 ) ;

(C.6)

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where it has been used that f9j (9j f)2 = (2=T )(9j f)2 − 4(9j f)3 . Using this relation again, in conjunction with a second partial integration, we 1nd the result T (3) dj[(1 + T 9j )(2 + T 9j )jp ]f2 9j f Sn = − 20 T2 T p 2 dj[(1 + T 9j )(2 + T 9j )j ](9j f) − dj[(1 − T 2 92j )(2 + T 9j )jp ]9j f =− 20 40 1 (1) 1 T 2 (1) 2 S + n − S = + ··· ; (C.7) 30 n 4 24 n−2 where we have again used that f2 = T 9j f + f and relation (4.39) with F = (1 + T 9j )(2 + T 9j )jp . Consequently, we have 2 2 9 5 1 T 1 1 T n 1 T n (3) (n − 1) − n− ≈ jF 1 + n + j : (C.8) n ≈ 2 20 jF 2 6 4 2 20 jF F jF For any dimension D, and to leading order in T=jF , we may then make the replacement O 3 → 1 ff. O (ff) 30 The approximate expressions (C.2), (C.5), and (C.8) help to simplify the treatment considerably, both in the present context and for a variety of other problems. The second-order approximations are quite accurate for k = 1; 2; 3 and in the temperature range of interest, leading to slight overestimates. For n = 0, which is the moment needed in the present investigation, the error is well below one per mille at the standard temperature of T = 4 MeV, and it has grown to only 1–2% at T = 10 MeV. In the same temperature range the lowest-order expressions, which are the ones employed throughout the present study, have errors growing from about 1% to 6 –8%. C.2. Angular averages In this appendix we derive a number of simple analytical expressions for the various quantities of special interest, utilizing the dispersion relation, dp 9hk k·C 9f0 1=g (C.9) h3 9 k · C ∓ itk−1 9j 1 2 ≈ −F0 (k; T ) = −F0 (k; T ) 1 − 8k arctan ; (C.10) 8k 2 + 82k and taking advantage of the fact that the temperature is (assumed to be) small relative to the Fermi energy. Since the integrals over momentum space contain the modulation factor 9f0 =9j, the integrands are con1ned to a relatively narrow interval around the Fermi surface so, consequently, the energy integration eHectively decouples from the directional average. We can then readily obtain the following angular averages, 1 9 i=8k − =1+ = 1 − 8k + 82k + · · · ; ≈ (C.11) F0 − i=8k − i=8k 2

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421

which follow directly from the dispersion relation. The expansion in terms of 8k shows the behavior near the spinodal boundary where F0 → −1 so 8k tends to zero. Moreover, the auxiliary quantities gn ≡ (2 + 82k )−n can also be evaluated, yielding 82k g1 = 8k arctan(1=8k ) = 1 + 1=F0 and 282k g2 = g1 + 1=(1 + 82k ). We can then evaluate the following angular averages, 1 − 82k g1 1 1 1 = ==O ; =− − + (C.12) + i =8k − i= 8k F0 F0 1 + 82k 1 − 382k g1 + 284k g2 1 + F0 1 1 1 = g1 + (== − 1)82k g2 = 2 == − ==O ; (C.13) − i=8k + i= 8k 8k F 0 1 + 82k 2 1 12 22 2 8k = = == 12 ; (C.14) 1 + i=8k 2 − i= 8k 1 + i=8k 2 − i= 8k F02 where =O ≡ −=. In the last quantity the directional average is with respect to both pˆ1 and pˆ2 and we have used the trigonometric relation 12 ≡ cos 312 = cos 31 cos 32 + sin 31 sin 32 cos 612 . Since 612 only the 1rst term contributes which eHectively allows the replacement 12 → 1 2 and thus causes the two integrations to decouple. C.2.1. Legendre expansion It is instructive for the analysis, as well as calculationally helpful, to evaluate the angular averages by performing multipole expansions of the functions involved and we indicate here the most important relations. We 1rst note that the directional part of the dual basis functions can be expanded in terms of the Legendre polynomials Pn (), i= = Nk (2n + 1)qn= Pn () ; (C.15) qk= (p) ≈ Nk + i=8k n ¿0 where the expansion coePcients are given by qn= = −i=Qn (−i=8k ) [303]. The complex Legendre polynomials Qn (z) satisfy the same recursion relation as the ordinary Legendre polynomials, (n + 1)Qn+1 (z) = (2n + 1)zQn (z) − nQn−1 (z) ;

(C.16)

starting from Q0 = −i=6 and Q1 = −8k 6 − 1, where 6 = arctan(8k ). We note that (qn= )∗ = qn=O = (−)n qn= . The expansion for the eigenfunctions is equally simple, F0 f0 fO 0 (2n + 1)fn= Pn () ; (C.17) fk= (p) ≈ 60 n¿0 where fn= = n0 − qn=O , since =( − i=8k ) = 1 + i==( − i=8k ). The elements of the overlap matrix are then given by 1 == ≈ (o− k )

1 F02 (2n + 1)(fn= )∗ fn= : 6 60 n¿0

(C.18)

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It follows from the symmetry properties that the terms are all positive for the diagonal elements, whereas they alternate in sign for the oH-diagonal elements. The above relations suPce for calculating the variance part of collective source terms, as well as the covariance part in the minimal model where the coupling term C12 contains only monopole and dipole terms. In the more general case, the covariance part can be evaluated by taking advantage of the simple multipole expansion derived in Ref. [120], C12 ∼ ‘ c‘ P‘ (12 ). The key relation is −i= i= P‘ (12 ) 1 − i=8k 2 + i= 8k d!1 d!2 = ∗ = Pn (!1 )P‘ (!12 )Pn (!2 ) = (q‘= )∗ q‘= : = (2n + 1)(2n + 1)(qn ) qn (C.19) 49 49 nn C.3. Overlap matrix and dual basis The eigenfunctions fk= (p) are not orthogonal in momentum space. It is therefore of interest to 1 consider the 2 × 2 overlap matrix o− between the collective modes. It has the elements k 1 1 + 92 8k == dp = ∗ = F0 −1 == → f (p) f (p) ≈ − : (C.20) (ok ) ≡ k F0 h3 k 660 2 + 1+8 660 2 k

We have here used the above formula (C.12) and also employed the expression (C.5) which eHectively replaces (f0 fO 0 )2 by 16 f0 fO 0 . On the right is indicated the behavior as the spinodal boundary is approached. The overlaps may also be evaluated by means of a Legendre expansion, which is sometimes instructive. We note that the overlaps are positive and both tend to 1=(660 ) at the spinodal boundary. It was shown in Ref. [208] that the functions qk= (p) ≡

= o== k fk (p) =

=

i=kVF Nk k · C + i=tk−1

(C.21)

form the dual basis, qk= |fk= = == , with the normalization constant Nk given by 2 9 F0 9 1 −1 − 2 8k + · · · : 1+ = + Nk ≈ − 8k 2 4 1 + 82k

(C.22)

The normalization coePcient Nk remains fairly constant, exceeding its limiting value of 9=2 by at most about 7%, while the inclusion of the next term in the expansion (C.22) brings the accuracy within 1%, except for the most rapidly ampli1ed modes at zero temperature, where 8k is largest. In the above de1nition (C.21) the summation extends over all the states =, not merely the collective ones. The original treatment in Ref. [112] corresponds to including only the two collective modes in the expansion (C.21) of the dual basis, leading to the approximate dual states q˜± k (p) [208].

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423

C.4. Source terms

The 2 × 2 matrix of the collective source terms Dk== can be obtained by projecting the diHusion coePcient onto the dual basis given in Eq. (C.21) [208]. This yields

60 2 1 + F0 ==O 1 + F0 2 3== == : (C.23) Nk == + − − 2 Dk ≈ t0 8 k F0 82k F0 1 + 82k F0 The 1rst two terms arise from the variance part of the diHusion coePcient, Dˆ var (p1 ; p2 ), while the two last terms are the odd and even contributions from the covariance part, Dˆ cov (p1 ; p2 ). We 1 note that Dk== ∼ 8− near the spinodal boundary, while the mixed source term remains 1nite (see k Ref. [208]). The initial growth rate of collective density undulations having the wave number k is then proportional to 2 F0 − 2 8k 4 60 1 260 (1 + F0 )(2 + F0 ) == Dk ≡ 1+ → Dk ≈ − ; (C.24) 2 2 2 t 9 t0 8 k 1 + 8 1 + 8 F 0 0 k k == where the behavior near the boundary has been indicated. By similar means it is possible to derive expressions for the approximate source terms employed in Ref. [112]. In that initial analysis, the source term was written as a matrix product, D˜ k = ok Lk ok , where o== k are the elements of the inverse overlap matrix (C.20), where the inverse overlap matrix is given by −1 F0 F0 F0 6== 60 1+ 3+ 1+ 1+ → =6== 60 : (C.25) o== == O k 2 2 2 F0 98k 1+8k 1+8k 1+8k In deriving this result, we have employed the formula (C.8) which makes it possible to replace 1 0 O0 f f in the integrand. Moreover, (f0 fO 0 )3 by 30 dp = ∗ ˆ dp 1 −1 == 1 + 3== 82k = ≡ f (p ) )f (p ) ≈ (o ) − : (C.26) D(p; p L== k k k hD hD 5t0 k 36t0 60 It then follows that D˜ == k

o== 60 F0 = k − 5t0 t0

F0 3+ 1 + 82k

−2

+ ==

F0 1+ 1 + 82k

−2

:

(C.27)

1 ˆ and it diverges as 8− containing ok arises from the variance part of D, k . The two last the even and odd contributions from the covariance part, respectively. Near the spinodal 2 the latter quantity diverges as 8− k . However, these singularities cancel when the various contributions are combined,

1 6 1 60 2 6 2 0 → D˜ == + ; (C.28) D˜ k ≡ k ≈ − F0 F0 t0 3 + 1+82 5 F0 3 + 1+82 5 t0 ==

The term terms are boundary collective

k

k

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so the collective density Ductuations remain regular at the spinodal boundary. It should also be noted that the covariance terms are of the same order of magnitude as the variance part (their relative magnitudes being approximately 5:6), but of opposite sign, so a signi1cant degree of cancellation occurs when the various contributions are all combined. Appendix D. Expanding bulk matter Since the systems for which spinodal fragmentation occur are generally in a state of expansion, it is important to ascertain how this complication may aHect the character of the process. The eHect of expansion on spinodal fragmentation was investigated in [126] and we summarize the results here. This study treated a macroscopically uniform system (with periodic boundary conditions) within the framework of the self-consistent Vlasov equation. Since the geometry of the space is that of a multidimensional torus, the overall expansion can be introduced by prescribing a suitable time dependence of the torus radii. Although such a rescaling can readily be done in each dimension separately, a single common scaling factor is employed in order to describe an isotropic uniform expansion of nuclear matter. D.1. Comoving variables It is convenient to replace the usual position and momentum variables (q; p) by the corresponding angle and action variables (Q; P), q Q= ; (D.1) R(t) ˙ P = R(t)p − mR(t)q ;

(D.2)

where the common torus radius has a prescribed time dependence, R(t). The single-particle Hamiltonian of the expanding system then acquires a standard form, 2 ˙ 1 R(t) P2 h(q; p) = p−m q + u(q) = + U (Q; t) = H (Q; P) ; (D.3) 2m R(t) 2M (t) where the mass is the associated moment of inertia, M (t) ≡ mR(t)2 , and the potential energy is U (Q; t) ≡ u(q = R(t)Q). Consequently, the Vlasov equation retains its familiar form, 9f(Q; P; t) 9U (Q; t) 9f(Q; P; t) P 9f(Q; P; t) · − · : (D.4) = 9t 9Q 9P M (t) 9Q To obtain the physical matter density from the above phase-space distribution of the angle-action variables, one must take account of the increased volume, D d P g −D f(Q; P; t) : (D.5) (q; t) = R(t) %(Q) = D R(t) hD Furthermore, if the eHective potential is generated by convoluting the density with a smearing function of a physical range a, a correspondingly smaller range A(t) = a=R(t) must be employed when working only with the stretched variable Q.

P. Chomaz et al. / Physics Reports 389 (2004) 263 – 440

It is instructive to note that the above Vlasov equation has the following solution, −1 P2 − (t) f0 (Q; P; t) = 1 + exp 1(t) ; 2M (t)

425

(D.6)

where 1(t) = 1(0)=(t)2 and (t) = (0)=(t)2 , with (t) ≡ R(t)=R(0) being the expansion factor. When written in terms of the physical variables, −1 2 ˙ R(t) 1 f0 (q; p; t) = 1 + exp 1(t) p−m q − (t) ; (D.7) 2M (t) R(t) this solution is recognized as a Fermi–Dirac distribution boosted by the local comoving frame and that the Hubble-type expansion causes the temperature T = 1=1 and the chemical potential to decrease in time as ∼ (t)−2 . D.2. Linear response We now adopt the above Fermi–Dirac solution as the background distribution and obtain the RPA equation for the small disturbance f(Q; P; t) = f(Q; P; t) − f0 (P; t), 9f 9U [%(Q)](t) 9%(Q) 9f 9f ·V − =0 ; + · 9t 9Q 9P 9%(Q) 9Q where V (t) ≡ M (t)P. For the Fourier transform, dQ −iK ·Q dP fK (P; t) = f(Q; P; t); %K (Q) = g fK (P; t) ; e 29 hD the RPA equation then reads as follows, 9 9f0 9UK (t) i fK (P; t) = K · V fK (P; t) + %K (t) ; 9t 9C(t) 9%

(D.8)

(D.9)

(D.10)

where we have introduced the kinetic energy C(t) = P 2 =2M (t). This equation can be recast, dP 9fK (P; t) =g i KK ; K (P; P ; t)fK (P ; t) ; (D.11) 9t hD where the RPA kernel K is diagonal in the mode index K, D h 9f0 9UK (t) KKK (P; P ; t) = KK K · V (P − P ) − : g 9C(t) 9%

(D.12)

This kernel is time dependent since the inertia, the potential, the temperature and the chemical potential are evolving with time. However, it can be easily seen that at a given time it corresponds to the linear response kernel of the static expanded system. Therefore, the instantaneous eigenfrequencies of this kernel are not aHected by the expansion. However, in order to study properties such as the stability of the system, one needs to follow the propagation of f over a 1nite time t. Since K is

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diagonal, this problem can be solved independently for each K , ˆ −i fK (t) = Te

t 0

d t K(t )

fK (0) ≡ MK (t)fK (0) ;

(D.13)

where Tˆ is the time-ordering operator. The monodromy matrix MK propagates fK from 0 to t and the time evolution can thus be studied by considering its eigenvalues EK= and eigenfunctions fK= , =

fK= (t) = MK (t)fK= (0) = eiEK (t)t fK= (0) :

(D.14)

In the usual case of a non-expanding system, where the kernel K is independent of time, the solutions of Eq. (D.14) are the eigenstates of the RPA kernel and the eigenfrequencies PK are obtained from the associated dispersion relation (3.26). In the case of an expanding system, applying the Magnus transformation [304], it is possible to rewrite Eq. (D.13) as follows, fK (t) = eB(t)−i

t 0

d t K(t )

fK (0) ;

(D.15)

where B is a sum of integrals depending on the two-time commutators of the RPA kernel. For relatively slow expansion speeds, or for a very short time interval t, one may retain only the 1rst term of (D.15) so that it is possible to express the time dependence of fK (t) as in Eq. (D.14) where EK (t) is now given by new dispersion relation, K · VO dP 9f0 9UO K (t) ; (D.16) 1=g D O h K · V + EK (t) 9C(t) 9% where we have introduced the time-averaged inertial parameter, 1 t −1 −1 O dt M (t ) M (t) = t 0 and the time-averaged potential, O (t) 1 t M dt UK (t ) ; UO K (t) = M (t) t 0

(D.17)

(D.18)

and VO (t) ≡ P= MO (t). The dispersion equation (D.16) shows that the expanding scenario is very similar to the familiar static case. Indeed, if we introduce a renormalized Landau parameter, dP 9f0 9UO K (t) K O g ; (D.19) F 0 (t) = − 9% hD 9C(t) O and a renormalized free response function V, K · VO 9f0 OV(EK ; t)g dP 9f0 = g dP ; D D O h 9C(t) h K · V + EK (t) 9C(t)

(D.20)

then the dispersion equation can be recast into the standard form, O K (t); t) = −FO K (t)−1 : V(E 0

(D.21)

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427

D.3. Discussion The above analysis shows that during a given time, over which the commutator between the RPA kernel taken at the time t and the initial one is small compared with the eigenvalues of any of these two kernels, the system follows the standard phenomenology and so it will present instabilities, provided that FO K0 (t) ¡ − 1. In such a case the expansion of Fermi liquids will lead to a modi1cation of the sound velocity associated with long wave lengths. For short wave lengths we know that the range of the force associated with the potential U is reducing or even suppressing the instability by eHectively introducing an ultraviolet cut-oH. The simplest way to include these eHects is to expand the Fourier transform of the potential in powers of K and consider only the lowest orders, UK (t) = U (t)(1 − 12 K 2 Q2 (t)) ;

(D.22)

where Q is related to the range of the interaction in the variable Q and therefore is inversely proportional to R(t). In such a case UO becomes UO K (t) = UO (t)(1 − 12 K 2 O 2Q (t)) ; with the renormalized range being given by t t O 2Q (t) dt U (t ) = Q2 (t) dt U (t )2 (t)2 (t ) : 0

0

(D.23) (D.24)

If we consider only the 1rst order terms in the expansion, ≈ 1, we obtain O 2Q (t) ≈ Q2 (t)(t) :

(D.25)

The associated interaction range in the q variable evolves according to the same equation. Thus the expansion mimics a force with a longer range. This can be understood by considering that particles being too far apart to interact at the time t might have been initially closer and so in interaction. Therefore, the memory of this interaction enlarges the eHective range, as described in Eq. (D.25). From this study of expanding matter, one can conclude that the main eHect of the expansion can be taken into account by introducing time-averaged inertia and Landau parameters as well as the rescaled interaction range. Thus spinodal instabilities are present also in expanding systems and follow the same phenomenology. Just as this review was being completed, a paper appeared in which density Ductuations in expanding matter was studied within a one-body model augmented by a stochastic 1eld constrained by the Ductuation–dissipation theorem for the expanding medium [127]. An analysis of the coupling between the evolving Ductuations and the expanding Duid yielded the distribution of the liquid domains resulting from the spinodal decomposition and the associated fragment size distribution compared favorably with the experimental multifragmentation data. Appendix E. Spinodal fragmentation in FMD The occurrence of spinodal fragmentation is a general feature of 1rst-order phase transitions and not merely an artifact associated with the use of mean-1eld treatments. In order to illustrate how spinodal fragmentation occurs in many-body approaches as well, we summarize here a study made

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Fig. E.1. FMD energy. Energy per nucleon as a function of a, variance of the Gaussian width parameter, for a two-dimensional stationary system having a lattice mesh of d = 5 fm for a system with (solid curve) and without (dashed curve) D packaging. The dotted line shows the binding energy of the uniform system, which is approached when a grows large. (From Ref. [305].)

with in the framework of fermionic molecular dynamics (FMD) [305]. For studies of spinodal fragmentation within classical molecular dynamics, see Refs. [108,109,220]. E.1. Static properties We consider idealized two-dimensional nuclear matter at zero temperature (using periodic boundary conditions). The nucleons are represented by Gaussian wave packets and they interact through a Skyrme potential having t0 = −243:5 MeV fm2 and t3 = 846 MeV fm4 [162]. Since the eHective interaction is independent of spin and isospin, the single-particle wave functions have a four-fold degeneracy at zero temperature and this symmetry is preserved along the entire time evolution. The wave packets form initially a regular lattice and two cases are considered: (1) each lattice site has a four-fold degeneracy or (2) each site has only one nucleon whose spin and isospin alternate (to achieve the same density, the mesh size d should then be halved). Fig. E.1 shows how the corresponding energy depends on the width parameter a, the variance of the Gaussian wave packet. When the width is large, the system resembles uniform matter and the energy value approaches the value expected at the considered density, =4d −2 =0:16 fm−2 (the saturation density is s =0:55 fm−2 for this force). Since the system may gain energy by forming D clusters, the spin–isospin saturated system has optimum width minimizing the energy, a0 = 2 fm2 . Such a minimum is observed for all dilute systems and the corresponding density has bumps at the lattice sites. By contrast, when the spin–isospin symmetry is broken by -at, energy is gained only by spreading, as the uniform limit is approached. To study the phase diagram and the possible coexistence of two phases, and at the same time to learn about the fragmentation dynamics, one may perturb the initially uniform system and follow its response. In this way one may see whether instabilities occur (and hence de1ne the spinodal region) and, at the same time, one may study the most important features of the early fragment formation.

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E.2. Early fragmentation dynamics in dilute systems For a given Hamiltonian H , the equations of motions for the various parameters may be derived from the time dependent variational principle (see [247,248]). Since, to a good approximation, the early dynamics is described by a linear response treatment, we summarize an RPA study of fermionic molecular dynamics [305]. We use sn to denote the four complex Gaussian wave packet parameters (the three phase-space centroids and the complex variance a) for the nucleon located at the lattice site n. By linearizing the equations of motion [247] around a stationary solution, sn = sa n + sn , we obtain the equations of motion [305], 92 H 92 H ∗ ∗ ∗ (E.1) Ann · s˙n = · sn + · sn ; 9sn 9sn 9sn 9sn∗ n n The matrix multiplying the time derivative s˙∗n is de1ned as Ann =

92 L0 92 L0 − ; ∗ 9sn 9s˙n 9s˙n 9sn∗

(E.2)

where L0 = L + H, with L being the Lagrange function for the system and H = Hˆ . The centroids of the Gaussian wave packets are placed on a lattice of a given mesh d, so that the index n runs over the lattice sites. Due to the imposed translational symmetry of the unperturbed system, the above relations do not depend on the speci1c lattice site n. The imaginary part of the centroid position (proportional to the momentum) vanishes in the unperturbed stationary state, as does the imaginary part of the width parameter. Furthermore, due to the isotropy of the system, we need to consider perturbations z along only one of the lattice directions, but we shall give the general form of the equations. The above linear equation (E.1) then yields two coupled equations of motion, Ann · s˙∗n = Bnn · sn + Cnn · sn∗ ; (E.3) n

n

where Bnn =

92 H ; 9sn 9sn

Cnn =

92 H : 9sn 9sn∗

(E.4)

Takingthe Fourier transform with respect to time and considering the plane wave representation, sn = k sk eikdn , we obtain the dispersion equation [305], ∗ ∗ i!k Ak · s− k = Bk · sk + Ck · s−k :

(E.5)

Information about the evolution of the system is given by determining eigenvalues and eigenvectors of this system of complex equations. The four eigenvalues come in pairs of solutions with opposite sign, so that there are two independent solutions for !k . Negative solutions for !k2 signal the presence of instabilities. Moreover, the value of !k informs us about the early dynamical development of the perturbation introduced in the system and hence on the path towards fragmentation.

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Fig. E.2. FMD dispersion relations. Dispersion relations obtained within AMD (left) or FMD (right), using several values of the wave packet variance: a = 2 fm2 (dot-dashed), a = 4 fm2 (dashes), a = 7 fm2 (dots). Also shown is the result obtained with linearized TDHF (solid). (From Ref. [305].)

E.2.1. AMD framework Let us 1rst consider the case where the Gaussian widths are kept 1xed, as in AMD calculations [246]. The only time-dependent parameter is then the Gaussian centroid. The electric isoscalar modes are de1ned by the unique complex equation for the displacement of the centroid positions of the wave functions, zn = (x n ; yn ; zn ; 0), ∗ ∗ i!k Ak · z− k = Bk zk + Ck z−k

(E.6)

Then only two opposite solutions appear, ±!k , for each wave vector k. For example, let us consider a mesh having d = 3:8 fm corresponding to an average density of 0 = 12 s . In this case, !k2 may be negative and the system is then unstable. The resulting growth time tk ≡ |!k |−1 is shown in Fig. E.2 (left) for various values of the width a, to which it is very sensitive. For values of around a ≈ 2$ fm2 the growth times are close to those predicted by linearizing the quantal mean-1eld treatment TDHF (quantal RPA) (solid curve). However, in such a case, due to the small width, the observed instabilities can be understood as spinodal instabilities of a gas of D particles. When the width of the Gaussians is increased, the instabilities are being quenched. In particular, in the case of a uniform initial density (large values of the width), the instabilities are almost fully suppressed, in contrast to the quantal RPA calculations (see Fig. E.2). This eHect originates from the fact that the forces acting on the wave packet centroids are very weak due to the smoothing resulting from the large widths. Then the motion of the centroids is inhibited and so are the instabilities. The vertical cutoH seen in Fig. E.2 arises because modes with wave-lengths shorter than twice the lattice mesh cannot be represented on the lattice. Conversely, in a full TDHF calculation sound waves can be propagated by the co-operative contribution of several single-particle wave functions, which are not constrained to have any particular form. However, the RPA dispersion relation also yields a cutoH, but at larger wave numbers, k ¿ 2 fm−1 , due to quantal eHects [105]. When the width is kept 1xed, as in AMD, it is well-known that particles can experience a spurious dynamics, due to conDicts between the 1xed width of the wave packets and the need to have anti-symmetric Slater determinants. In particular, particles of the same type exhibit spurious

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scattering. This problem, which could aHect the results presented above, is removed in calculations of the FMD type, where also the widths of the wave packets evolve. E.2.2. FMD framework In the FMD case, we consider a mesh size of d = 3:8 fm, which is typical. The initial stationary state then has minimal energy when the width value a = 2 fm2 is employed. We shall also discuss a nearly uniform case having a = 7 fm2 , which is approximately stationary since the derivative of the energy is small for large width values (see Fig. E.1). Large widths mimic also stationary states for more disordered systems (where the spin–isospin symmetry is explicitly broken). With the width as a time-dependent parameter, one obtains two independent solutions for !k . They are associated with diHerent coupled motions of the centroids and the widths. In Fig. E.2 (right) we display the solution that most corresponds to a displacement of the centroid positions. The inclusion of the width as a dynamical parameter slightly modi1es the dispersion relation only slightly from its AMD form. Hence also in the framework of FMD calculations, instabilities are suppressed when the stationary width is large. In the case of a nucleon gas (i.e. when the spin–isospin symmetry is broken), the minimum energy is reached asymptotically for large values of the width a (see the dashed line in Fig. E.1). Therefore, one can conclude that in an excited situation, where the D-package is broken, the natural tendency of the nucleons will be to present a large width. This is in fact the well-known eHect related to the wave packet spreading. Then, according to the previous discussion about dispersion relations, one should expect an important quenching of the spinodal instabilities when the width is considered as a dynamical variable (FMD). From this study it follows that when the D packing is broken the FMD evolution of a nuclear reaction allows clusterization to happen only at the dense stage, when correlations among the particles are strong and the value of the width is still close to the initial one. This, of course, would not correspond to spinodal decomposition. Conversely, in approaches using a 1xed width, spinodal decomposition may happen if the system enters low density regions, even in the absence of D packing [306]. Furthermore, for appropriate values of the width, one may obtain features similar to those obtained in stochastic mean 1eld treatments as well as in classical systems interacting through long-range forces [39]. E.3. Phase transitions in -nite systems In the context of FMD dynamics, it has been shown that, for systems of interacting nucleons in the 1eld of an external-container, it is possible to observe a liquid–gas phase transition of 1rst order [307]. Nuclei in their ground state and con1ned by an external 1eld are excited by displacing all wave packets randomly from their ground-state positions. The system then relaxes into a coexistence between large clusters (the liquid phase) and very light particles (the vapor phase). As the excitation energy is increased, a 1rst-order phase transition appears, as inferred from the analysis of the associated caloric curve (temperature versus excitation energy). Increasing the amplitude of the external 1eld (which can be considered as an external pressure acting on the system), the plateau in the caloric curve 1nally disappears and one can in this manner identify the critical temperature. For 16 O nuclei, the value of the critical temperature obtained is Tc = 8 MeV. From this analysis, together with the analysis made above, one can deduce that the coexistence between liquid and gas

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Fig. E.3. FMD fragmentation pattern. Fragmentation pattern exhibited by a dilute system of 160 nucleons with D symmetry enforced. (From Ref. [305].)

phases may be due to the resilience of the D-like structure present in the ground state of the 1nite nuclei considered. In fact, we have seen that, in absence of the D packing, the width acquires large values when the density is low and cluster formation is inhibited. E.4. Dynamical evolution of excited and dilute -nite systems To illustrate the dynamical evolution of excited 1nite systems, we discuss 3D simulations for 160 nucleons based on the FMD approach [305]. The parameters chosen for the Skyrme-like force are t0 = −1033 MeV fm3 and t3 = 14 687:5 MeV fm6 . To account for 1nite-range eHects (important when surfaces are presents), the one-body density has been convoluted with a Gaussian of width g = 0:4 fm. The system is initialized as a piece of a 3D lattice at half normal density, using the D packing. The D clusters are then perturbed randomly around their initial positions to allow possible ampli1cation of unstable modes. The widths of the wave packets are also perturbed. It is seen that the dynamical evolution induces oscillations of the widths around the equilibrium value (2 fm2 ) for D particles. When these oscillations are suPciently small, so that the widths remain small during the entire evolution, one observes the formation of “fragments” (clusters of D particles), as illustrated in Fig. E.3. As discussed before, this mechanism resembles the spinodal decomposition of a gas of D particles. (It should be noted that if the widths were kept 1xed (as in AMD) at a rather small value (like a = 2 fm2 ), the occurrence of instabilities and cluster formation would have been observed as well.) Another case worth considering is the nucleon gas. Here the 160 nucleons are initially placed randomly within a sphere of radius R = 8 fm (the average density is then 12 s ) and the system is endowed with a small self-similar expansion. Two cases are considered for the initial widths: a=2 fm2 (corresponding to an energy of 2 MeV=A) and a = 4 fm2 (corresponding to E=A = −2 MeV). The time evolution of these systems are shown in Fig. E.4. In both cases the early dynamics is dominated by an increase of the width. This is the behavior expected for a disordered system since the minimum energy is obtained for large widths (see Fig. E.1). The more excited system undergoes a vaporization into individual nucleons, while the other system collapses into a single source, evaporating few nucleons from its surface. Almost no clusterization is observed. This absence of spinodal instabilities in the simulations con1rms the conclusion drawn from the linear response treatment. Because of the appearance of large widths the residual interaction between centroids is smoothed out so that spinodal instabilities are suppressed.

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Fig. E.4. FMD evolution. Time evolution of a nucleon gas initialized at half the saturation density, for two diHerent values of the excitation energy. (From Ref. [305].)

However it should be noticed that clusterization could occur also in the case of a nucleon gas if the width is kept 1xed, as in AMD calculations. E.5. Further developments of molecular dynamics We 1nally brieDy recall certain further developments of molecular dynamics that appear to provide an improved framework for understanding nuclear dynamics. In particular, in order to alleviate problems associated with the use of a 1xed wave packet, various forms of stochastic branching have been introduced into many-body approaches of the FMD and AMD type. The resulting extended treatments have been applied to nuclear fragmentation, though the speci1c spinodal fragmentation mechanism has yet to be investigated with these re1ned approaches. In order to take account of the energy Ductuations in a system described by a wave packet [308], Ohnishi and Randrup [309,310] augmented the AMD propagation by a stochastic term, leading to the Quantum Langevin model. The Langevin term enables the Gaussian wave packet system to explore its entire energy spectral distribution, rather than being restricted to its average value. In this manner, it is ensured that the system relaxes towards the appropriate quantum-statistical equilibrium, contrary to what happens in the usual deterministic FMD and AMD treatments. The inclusion of quantum Ductuations via the Langevin term improves the nuclear equation of state and inDuences the outcome of nuclear reactions [311]. (This general quantum mechanical eHect also plays a role for the critical properties of noble gases [312].) In particular, the quantum Langevin term is signi1cant in nuclear multifragmentation, as was demonstrated by the investigation of Au+Au collisions at 100 – 400 MeV per nucleon [313]. It appears that the inclusion of quantum Ductuations enhances the average multiplicity of IMF, especially in central collisions, with respect to standard QMD or AMD simulations, thus improving the

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441

CONTENTS VOLUME 389 B. Jonson. Light dripline nuclei C. Amsler, N.A. To¨rnqvist. Mesons beyond the naive quark model

1 61

S.Yu. Ovchinnikov. G.N. Ogurtsov, J.H. Macek, Yu.S. Gordeev, Dynamics of ionization in atomic collisions

119

I. Lindgren, S. Salomonson, B. A˚se´n. The covariant-evolution-operator method in bound-state QED

161

P. Chomaz, M. Colonna, J. Randrup. Nuclear spinodal fragmentation

263

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