Advances in Atomic and Molecular Physics, Volume 5

Advances in ATOMIC AND MOLECULAR PHYSICS VOLUME 5

CONTRIBUTORS TO THIS VOLUME A. BEN-REUVEN C. D. H. CHISHOLM R. J. S. CROSSLEY A. DALGARNO H. G. DEHMELT HOLLY THOMIS DOYLE F. C. FEHSENFELD E. E. FERGUSON F. R. INNES

ROY H. NEYNABER A. L. SCHMELTEKOPF 0. SCHNEPP

ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited by

D. R. Bates DEPARTMENT OF APPLIED MATHEMATICS THE QUEEN'SUNIVERSITYOF BELFAST BELFAST, NORTHERN IRELAND

Immanuel Estermann DEPARTMENT OF PHYSICS THE TECHNION ISRAEL INSTITUTE OF TECHNOLOGY HAIFA, ISRAEL

VOLUME 5

@) 1969 ACADEMIC PRESS New York London

COPYRIGHT 0 1969, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W l X 6BA

LIBRARY OF CONGRESS CATALOG CARDNUMBER : 65-18423

PRINTED IN THE UNITED STATES OF AMERICA

List of Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.

A. BEN-REUVEN, The Weizmann Institute of Science, Rehovot, Israel (201) C. D. H. CHISHOLM, Department of Chemistry, The University, Sheffield, Yorkshire (297) R. J. S. CROSSLEY, Department of Mathematics, University of York, York, England (237) A. DALGARNO, Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts (297) H. G. DEHMELT, Department of Physics, University of Washington, Seattle, Washington (109) HOLLY THOMIS DOYLE, Harvard College Observatory, Cambridge, Massachusetts (337)

F. C. FEHSENFELD, Environmental Science Services Administration, Boulder, Colorado (1) E. E. FERGUSON, Environmental Science Services Administration, Boulder, Colorado ( I )

F. R. INNES, Air Force Cambridge Research Laboratories, Bedford, Massachusetts (297)

ROY H. NEYNABER, Space Science Laboratory, General Dynamics/ Convair, San Diego, California (57) A. L. SCHMELTEKOPF, Environmental Science Services Administration, Boulder, Colorado (1)

0. SCHNEPP, Department of Chemistry, University of Southern California, Los Angeles, California (1 55) V

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Foreword This serial publication is intended to occupy an intermediate position between a scientific journal and a monograph. Its main object is to provide survey articles in fields such as the following: atomic and molecular structure and spectra, masers and optical pumping, mass spectroscopy, collisions, transport phenomena, physical and chemical interactions with surfaces, gas kinetic theory. It is the aim of the editors to select articles which combine a rigorous yet understandable introduction to their subject, a summary of past work with emphasis on progress in the last five years, and a sufficiently complete bibliography to enable the reader to locate quickly relevant details in the literature. Suggestions for topics to be covered in the future will be appreciated.

D. R. BATES I. ESTERMANN

Belfast, Northern Ireland Haifa, Israel

vii

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Contents V

LISTOF CONTRIBUTORS FOREWORD CONTENTS OF PREVIOUS VOLUMES

vii xiii

Flowing Afterglow Measurements of Ion-Neutral Reactions E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf I. Historical Introduction 11. General Experimental Aspects of the Flowing Afterglow Technique 111. The Flow Analysis IV. Data Reduction V. Production of Reactant Species VI. Optical Spectroscopic Studies VII. Temperature-Variable Flowing Afterglow Studies VIII. Some Miscellaneous Results IX. Summary References

1 4 14 35 36 46 46 50 52 55

Experiments with Merging Beams Roy H . Neynaber I. 11. 111. 1V. V. VI. VII. VIII.

51 59 62 80 89 100 105 106 107

Introduction General Principles Ion-Neutral Reactions Ion-Ion Reactions Neutral-Neutral Reactions Electron-Ion Reactions Current or Very Recent Studies Concluding Remarks References

Radiofrequency Spectroscopy of Stored Ions II: Spectroscopy H . G . Dehmelt 3. Manipulation and Investigation of Stored Charge 4. Spectroscopic Experiments Relying on Spin Exchange with Polarized

Atomic Beam

5 . Spectroscopic Experiments Based on other Collision Reactions

6. Spectroscopic Line-Shifts and -Broadening ix

109 124 142 149

CONTENTS

X

7. Conclusion Errata for Part I References

152 153 153

The Spectra of Molecular Solids 0. Schnepp I. Lattice Vibrational Spectra 11. Intramolecular Vibrational Spectra 111. Spectra of Solid Hydrogen

References

155 176 187 197

The Meaning of Collision Broadening of Spectral Lines: The ClassicalOscillator Analog A . Ben-Reuven I. Introduction 11. The Fourier-Transform Method 111. Impact Damping

IV. Complex Oscillators V. Statistical Broadening VI. Resonance Broadening References

201 204 210 217 221 228 234

The Calculation of Atomic Transition Probabilities R. J . S . Crossley I. Introduction 11. General Formulas for the Dipole Approximation

III. Approximate Wave Functions: General Considerations IV. V. VI. VII. VIII.

Criteria for Calculation Variational Wave Functions Semiempirical Methods Perturbation Treatments Sum Rules, Bounds, and Variational Principles IX. Summary References

231 243 248 255 257 267 273 279 28 1 288

Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sAs'lfpq C . D. H . Chisholm, A . Dalgarno, and F. R. Innes I. Introduction 11. The Calculation of cfp 111. Applications of Two-Particle cfp

IV. Description of Tables References

297 301 308 309 3 34

CONTENTS

xi

Relativistic Z-Dependent Corrections to Atomic Energy Levels Holly Thomis Doyle I. Introduction 11. The Relativistic Z-Dependent Theory 111. Irreducible Tensor Expansions of the Electrostatic and Breit Interaction

Operators IV. Antisymmetrization V. Reduction of Matrix Elements to Matrix Elements between One- and TWOElectron States VI. Results, Comparisons, and Conclusions References

AUTHORINDEX SUBJECTINDEX

331 342 352 357 363 369 412 415 425

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Contents of Previous Volumes Volume 1

Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. T. Amos Electron Affinities of Atoms and Molecules, E. L. Moiseiwitsch Atomic Rearrangement Collisions, E. H. Eransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Partly and J . P. Toennies High Intensity and High Energy Molecular Beams, J. E. Anderson, R. P. Andres, and J . B. Fenn AurnoR INDEX-SUBJECT INDEX

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W . D. Davison Thermal Diffusion in Gases, E. A . Mason, R. J. Munn, and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W . R. S. Carton The Measurement o f the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . de Heer Mass Spectrometry of Free Radicals, S.N. Foner AUTHOR INDEX-SUBJECT INDEX

Volume 3

The Quanta1 Calculation of Photoionization Cross Sections, A. L. Srewarr Radiofrequency Spectroscopy o f Stored Ions. I: Storage, H. G. Dehmelr Optical Pumping Methods in Atomic Spectroscopy, E. Eudick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Worf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood AUTHOR INDEX-SUBJECT INDEX xiii

xiv

CONTENTS OF PREVIOUS VOLUMES

Volume 4 H. S. W.Massey-A Sixtieth Birthday Tribute, E. H . S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D . R. Bates and R . H. G . Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P . A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I . C. Perciual Born Expansions, A . R. Holr and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionization, C. B. 0 . Mohr Recent Measurements on Charge Transfer, J. B. Hasred Measurements of Electron Excitation Functions, D. W. 0 . Heddle and R . G . W. Keesing Some New Experimental Methods in Collision Physics, R. F. Srebbings Atomic Collision Processes in Gaseous Nebulae, M. J. Searon Collisions in the Ionosophere, A. Dalgarno The Direct Study of Ionization in Space, R.L. F. Boyd AUTHORINDEX-SUBJECTINDEX

FLOWING AFTERGLOW MEASUREMENTS OF ION-NEUTRAL REACTIONS E . E . FERGUSON. F. C . FEHSENFELD. and A . L . SCHMELTEKOPF Environmental Science Services Administration Boukfer. Cotorado

. .

1 I Historical Introduction ............................................ I1 General Experimental Aspects of the Flowing Afterglow Technique ........ 4 A Overall Description ............................................ 4 B Ion Sources.................................................... 5 C The Afterglow .................................................. 7 D Measurement and Calibration of Flow and Pressure ................ 9 E Ion Sampling and Detection ..................................... 10 111 The Flow Analysis................................................. 14 A . The Simple Model ............................................. 15 B Radial Diffusion ............................................... 16 C Nonuniform Velocity Profile ..................................... 18 D Axial Diffusion ................................................ 22 E Axial Velocity Gradient and Slip Flow ............................ 24 F. Inlet Effects ................................................... 26 G Conclusions concerning the Analysis .............................. 32 H . Test of the Assumptions......................................... 33 IV Data Reduction ................................................... 35 V Production of Reactant Species...................................... 36 A Reactant Ion Production ........................................ 36 B Neutral Reactant Production .................................... 40 C. Role of Impurities ............................................. 44 VI Optical Spectroscopic Studies ....................................... 46 VII. Temperature-Variable Flowing Afterglow Studies ..................... 46 VIII Some Miscellaneous Results ........................................ 50 IX . Summary ........................................................ 52 References ....................................................... 55

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I Historical Introduction This article presents a discussion of the flowing afterglow technique . This technique has been developed by the authors in the U.S. Department of Commerce Research Laboratories (formerly NBS. now ESSA) in Boulder. Colorado. starting in 1963. primarily for the quantitative measurement of ion-molecule reaction rate constants . 1

2

E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf

Flowing systems had been widely used in the measurement of neutral reactions by chemists for many years (Schiff, 1964, Kaufman, 1964 and references cited therein). These flow systems ordinarily had slower flow rates and smaller tube diameters. Consequently, charged species did not persist very far downstream and accordingly were not observed in these experiments. The precurser to the first ESSA flowing afterglow system was actually a large diameter, fast-pumped, glass afterglow tube utilized for optical spectroscopic studies in the NBS-Washington Laboratory of H. P. Broida (Schmeltekopf and Broida, 1963). Visual examination of the recombination light clearly showed that charged species existed for a meter or more down a 10-cm diameter tube with gas velocity lo4 cm/sec. It was a logical step to place a mass spectrometer at the end of such a tube and then add neutral reactants into the flowing stream in order to study reactions. The first quantitative reaction rate measurements in the ESSA flowing afterglow system were for atomic helium ions reacting with 0,and N, . They werereportedin 1964(Fergusonetal., 1964).In thecourse ofthesemeasurements we learned that Sayers and Smith (1964), at a Faraday Society Discussion in Edinburgh, reported measurements on the same reactions using stationary afterglow techniques. The flowing afterglow results (for which a factor of 2 accuracy was claimed at that time) agreed within 24 and 31 % of Sayers and Smith's results. Therefore, it was clear from the beginning that the flowing afterglow technique had sufficient capability for quantitative measurements to merit further development. In other words, the initial direct comparison of flowing afterglow measurements with measurements obtained by other methods indicated the general validity of the flowing afterglow method prior to the analysis of the rather complex hydrodynamics. Subsequent analysis has also established this validity. In addition to the example just cited, a number of other comparisons of the flowing afterglow results with those obtained by other techniques have followed, e.g., by Warneck (1967) for a number of positive ion reactions, and by Moruzzi et al. (1968) for several negative ion reactions. It was not required that the flowing afterglow technique yield very accurate (e.g., 10%) rate constants in order to be useful. In the field of ionospheric chemistry, which has been the primary area of application by the present authors, there were many important reactions whose rate constants were entirely unknown. Order-of-magnitude rate constant data could quite drastically alter aeronomical theories and factor of 2 rate constants were as accurate as could be effectively utilized in most cases due to the comparable, or greater, uncertainties in neutral atmospheric composition, ionizing fluxes, ion compositions, and other factors entering ionospheric analysis. We still encounter situations in which yes-no answers on reactions lead to valuable chemical data, e.g., relative electron affinities of molecules, bond dissociation energies, etc.

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FLOWING AFTERGLOW MEASUREMENTS

3

It was quickly apparent that the flowing afterglow technique had a substantially wider versatility than other techniques for measuring ion-neutral reactions at thermal energies. For example, ions could be reacted with unstable neutral species. In 1965, the reaction N2+ 0 -+ NO+ + N was reported (Ferguson et af., 1965a). This is still the only reported measurement of this important reaction, which is a major N,’ loss process in the Earth’s ionosphere. Most aeronomists believed in 1965 that this very fast reaction did not occur, and aeronomy has been significantly influenced by this single measurement. Also, more control of reactant states was possible using the flowing afterglow technique than with other experimental methods. As an example, it is possible in many cases to guarantee that all reactants are in their ground electronic and vibrational states. It has even been possible to measure reactions of several ions with controlled vibrationally excited N, . These are the first such laboratory measurements of ion-molecule reaction rate constants as a function of neutral reactant vibrational temperature (Schmeltekopf et al., 1968). Recently rate constants for reactions of ions with electronically excited oxygen molecules have been measured (Fehsenfeld er af., 1968). Because of the above advantages, sufficient incentive existed to develop and exploit the flowing afterglow technique for ion-molecule reaction studies. There have been steady advances in the experimental technology of the flowing afterglow technique and also in the rather detailed analyses of some of the problems associated with extracting quantitative rate constants from observed changes in ion concentrations. The present article is intended to be a fairly comprehensive compilation of these experimental aspects and to present the detailed mathematical analysis of the system. Several reviews have concentrated more on the experimental results obtained, the consequent theoretical implications and generalizations, and the applications to ionospheric physics problems (Ferguson et al., 1965b; Ferguson, 1967, 1968, 1969). In Section 11, experimental details are discussed in sequence down the flow tube: from the “ nonchemical” aspects of ion production (B), the afterglow (C), the flow measurements (D), and the ion sampling (E), all of which are preceded by a brief overall description (A) designed to give the broad picture before immersing the reader in detail. In Section 111, a detailed discussion of the gas flow and hydrodynamic analysis is given. More space is given to this topic then any other. In part this is because much of the analysis is recent and has not been presented elsewhere, in part because the subject matter unfortunately is somewhat complex and hard to condense, and perhaps most importantly because the tractability of the hydrodynamics is the essential key to the utility of the fast-flowing afterglow system for accurate quantitative rate measurements. Indeed, it seems likely that the potential complexity of the hydrodynamics discouraged previous workers from

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E. E.Ferguson,F. C.Fehsenfeld, and A. L.Schmeltekopf

attempting to use fast-flow systems for quantitative reaction rate studies. Section IV is a brief discussion of the data analysis, including our recent addition of on-line computing capability which speeds data acquisition and reduces operator error. Section V is a discussion of the chemical aspects of ion production (differentiated from direct electronic ionization, as in Section I1,B) and neutral production, including excited states and unstable species. Further sections describe the application of the flow tube to spectroscopic studies of reactions and a temperature variable system which obtains rate constants from 80 to 600°K and which is proving to be useful for three-body reaction studies. 11. General Experimental Aspects of the Flowing Afterglow Technique A. OVERALL DESCRIPTION

The earliest ESSA flow tube was a Pyrex tube about 1 m long and 8 cm inside diameter. The glass tube allowed some estimation of the flow properties of the afterglow by visual observation. This was extremely helpful in the early period. The spatial distribution of certain light-producing reaction products was qualitatively apparent to the eye and gave an indication of such parameters as the distance needed for the added neutral reactant to fill the tube, etc. This point is illustrated by the color photographs of Bass and Broida (1963). A recent version of the flow tube, constructed of stainless steel, is shown in Fig. 1. The metal tube is substantially more versatile.

FIG.1. Pictorial representationof the flowing afterglow tube.

FLOWING AFTERGLOW MEASUREMENTS

5

-

A fast gas flow (u lo4 cmlsec) is established in the tube by a Roots-type pump backed by a mechanical forepump. This pumping system can maintain a 500 liter/sec pumping speed at pressures from to 10 Torr. The principal, or buffer, gas normally has been helium. With a helium flow of 180 atm cm3/sec and at maximum pumping speed, the helium pressure in the tube is typically about 0.4 Torr. This pressure has been varied from 0.1 to 5 Torr by varying the helium flow or the pumping speed. Ions are produced by an excitation source and are carried down the tube in the flowing gas. A wide range of reactant ion species, both positive and negative, can be produced by the addition of a relatively small concentration (compared to the helium) of a suitable gas. This gas can enter with the helium through the excitation region or be injected into the helium afterglow immediately after the excitation source. These ions are carried down the tube past a neutral reactant port where this reactant is added in measured amounts. In the region following the neutral addition port, reactions between the reactant ions and the neutrals take place. The reaction region is terminated by the sampling orifice of a mass spectrometer. The ions are sampled, mass analyzed, and counted. The variation of an ion signal as a function of the reactant concentration leads quantitatively to the reaction rate constant. When the reaction leads to a product ion, it is also observed in the mass spectrometer. Reaction rates may be determined in a flow tube by ways other than variation of the neutral gas addition. One alternative is to establish a fixed neutral reactant gas flow and vary the gas velocity.

B.

ION

SOURCES

In the first tubes a microwave discharge was used as the primary ionization source. Small microwave cavities of the Evenson type (Fehsenfeld et al., 1965a) were slipped over a converging-diverging (deLaval nozzle) section of a small quartz tube preceding the large glass flow tube. While the microwave source was clean and easy to use, it had one distinct disadvantage. The ratio of ultraviolet photons to ions produced by the microwave discharge was very high. These photons are mainly the n'P+ 1 'S series and principally the (2'P + 1 ' S ) 584 A line. This line is resonantly absorbed by neutral helium. Since the lifetime of the resulting excited atom is much longer than the time between the emission and absorption of the photon, these resonant photons spend most of their time absorbed. The resulting excited atoms can be handled experimentally like metastable atoms, as will be detailed later. Since the discharge was produced in the deLaval nozzle (approximately Mach one), the doppler shift given to the photons by the sonic gas flow moved the center of the emission line one doppler half-width to the blue side of the

E. E.Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

6

absorption line. A fraction of the photons were then nonresonant and could thus traverse the length of the flow tube and produce reactant ions near the mass spectrometer sampling hole. This could lead to the deduction of erroneously low reaction rate constants, since the reactant ions produced at this location would not have the full time to react with the neutral reactant. However, pulsing the discharge and gating the ion detector later (-6-12 msec) would eliminate this prompt spurious photoionization signal. Ion arrival-time spectra illustrating these phenomena were published by Goldan et al. (1966). These phenomena would only arise in reaction studies for which the reactant ion was the ion of a neutral species present in a significant concentration in the system. The only instance where ultraviolet photons introduced an error in a published rate constant concerned the reaction N 2 + + O2 + 02+ N, and this was rather quickly discovered and corrected by the pulse technique (Fehsenfeld et al., 1965b). In an attempt to decrease the photon to ion ratio, the microwave discharge was replaced by a dc discharge consisting of a large cylindrical cathode near the glass tube wall and a small grounded wire anode in the center of the tube. This type of discharge produces a lower photon to ion ratio. Moreover, this discharge was in the slow-flowing gas stream, so that the 584 A photons could now be absorbed by the background gas. For this type of discharge however, higher-lying states are excited. Now, the photons from excited states of the He+, primarily 304 A, became a problem, since the He' density was so low that little absorption took place. Therefore, the pulsed mode was still used with the dc discharge but was not as important as it had been for the microwave discharge. Compared to the microwave discharge, the cylindrical cathode resulted in a factor of 20 reduction in the downstream photoproduction of reactant ions. The dc discharge had other advantages over the microwave discharge as well. Its geometrical configuration was more compatible with the detailed flow analysis, and it was more easily incorporated in metal flow tubes, which were soon found to be advantageous. The dc discharges had some serious drawbacks, however. Observable quantities of cathode material were sputtered during operation. Positive ions of the cathode metals and negative ions of the cathode metal oxides were detected in the mass spectrometer. The sputtered metals also had deleterious effects on optical windows, electrical insulation, etc. A rather substantial improvement in excitation sources became possible with the development of the barium zirconium oxide filament by MacNair (1967). This filament can be operated in an oxidizing atmosphere for many hours and is ideally suited to flpw tube use. By employing electron energies at which the electron impact ionization cross section is a maximum ( x 100 eV), the downstream photoproduction of reactant ions is about a factor of 200 less than that produced by the microwave discharge. With this ion source,

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FLOWING AFTERGLOW MEASUREMENTS

7

practically no photon effects have been detected of ion effects); consequently, the pulse method need no longer be employed. The use of an electron emitting filament has greatly increased the versatility of the system. By varying the emission current, the electron density can be varied between wide limits. By varying the gun voltage the electron energy can be controlled. This, in turn, can be used to selectively control ion production. For example, if an electron energy less than 24.6 eV is used in helium, metastable helium atoms will be produced, but no helium ions. If then N, is added to this afterglow, only N,' ions will be produced for further reaction studies (with no N + ions). In addition, the electron emission filament has a much higher efficiency (ions per watt) than the discharges and thus supplies less heat to the gas. Finally, the electron gun is much cleaner than the dc discharge.

C. THEAFTERGLOW Over the years of operation of the flowing afterglow a certain amount of investigation has been carried out on the nature of the helium afterglow, since this was the initial medium for most of the ion-molecule studies. The electron density, electron temperature, gas temperature, and helium metastable concentrations were determined under a variety of conditions. For this purpose, a quartz tube replaced the Pyrex tube, so that ultraviolet spectroscopic and microwave interferometric studies could be carried out. The electron density and temperature were determined by absolute intensity measurements of the He I n 'P + 2 ' S series up to levels near the continuum at 2600 A (e.g., n 19). At levels close to the ionization limit (those within ~ 5 / kT, 2 of the limit), a Boltzmann distribution of levels is maintained by electron collisions, so that the relative line intensities yield T,, which is generally close to 300°K a few centimeters after the excitation region. Absolute intensities were used in conjunction with the Saha equation to determine the electron density n, , The electron density determined was verified by microwave interferometer measurements at two frequencies, X band (9 GHz) and V band (50 GHz). These measurements represent the lowest electron densities and electron temperatures measured by this spectroscopic technique as far as we know. Since the measurements did not enter our ion reaction rate determinations in any critical way we have never refined or published these results. A typical Saha plot is shown in Fig. 2. The neutral gas temperature has also been measured spectroscopically, both by doppler line width and He, rotational intensity distribution measurements and found to be equal to the wall temperature. The He(2 'S) and He(2 'S)densities were measured by absorption of the helium 3889 and 5016 A lines. Measurements have shown that molecular helium ions are formed primarily from the atomic helium ions in the tube, leading to a He,+/He+ ratio

8

E. E. Fergiison, F. C.Fehsenfeld, and A. L. Schmeltekopf

FIG.2. Sample of the results obtained for the determination of electron densities and temperatures in a flowing helium afterglow. The statistically weighted upper state density of the n3P states of helium is plotted versus the separation in reciprocal centimeters, from the series limit, where P = 0.5 Torr, flow = 220 atm cm3/sec, and t = 600 psec.

of -0.1 at the mass spectrometer in very pure helium at 0.4 Torr and 300°K. This is in good agreement with theoretical prediction using the known rate He (Beaty and Patterson, 1965). At high pressures for He' + 2He + He,' this ratio rapidly increases, and Hez+ becomes the dominant ion. Under such conditions He,' reactions with neutrals have been measured. At 80"K, in the ESSA temperature-variable afterglow tube, He,' is found to be the dominant ion in the helium afterglow at pressures greater than -0.2 Torr. This weakly bound ion is not observable at 300"K,consistent with its properties as reported by Patterson (1968). The primary ion loss process in the helium afterglow is ambipolar diffusion to the tube walls, except possibly at very low electron densities (n, < lo7 cm-j). Argon has also been extensively used as a buffer gas in the flowing afterglow tube. Its afterglow properties are similar to those of helium, but have been much less studied. However, its hydrodynamic properties are different and must be given due consideration.

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FLOWING AFTERGLOW MEASUREMENTS

9

The use of the filament source gives the following conditions in the helium afterglow. For 100-eV electrons the only species significantly present are He+, He(2 3S),and He,'. The photon density is very low, as discussed above, and He(2 'S)atoms are not present because superelastic collisions by electrons convert the He(2'S) to He(Z3S). The ions and electrons thermalize to near gas temperature in the first few centimeters. When high currents are used in the ionizer, the electron density midway down the tube can be as high as 2 x 10" ~ m - In ~ .this case photons, metastable states, and highly excited Rydberg states are produced by recombination, as well as He' by He(2 'S) + He(2 3 S )+ He' + He + e. Usually much lower ionization currents are used so that the electron density is =lo9 cm-3, where the above effects are not large enough to be important. In fact, even electron densities less than lo7 cm-3 can be used a t which point one must begin to worry about the plasma condition still holding. At sufficiently low ionization voltages one produces only metastables and, since there are then no electrons present for superelastic collisions, both 2 ' S and 2 3 S metastables are present. One of the most serious problems associated with the use of the flowing afterglow for quantitative measurements can be a definite knowledge of where the reactant ions are being produced. It must be established in some way that there are no significant sources of reactant ions past the point at which the neutral reactant gas is added. Since the introduction of the electron impact ion source the concern here is no longer photoionization, but reactant ion production by secondary processes such as ion-neutral reactions, electron attachment, .Penning ionization, etc. The actual determination of whether there are sources of reactant ions in the reaction zone is a problem specific to the ion. Ordinarily this determination is made by varying either the flow of the reactant ion parent gas or the electron density and observing whether the rate constant is affected by these changes. In the absence of downstream sources of reactant ions, a semi-log plot of the reactant ion current versus added reactant gas flow is fairly linear, as will be noted later, and a strong curvature is usually an indication of reactant ion production in the reaction zone. D. MEASUREMENT AND CALIBRATION OF FLOW AND

PRESSURE

As will be shown in Section 111, the reaction rate constant is dependent on the reactant gas flow Q and the square of the buffer gas flow velocity u. The latter is calculated from the measurements of the downstream pressure and the buffer gas flow. Consequently, if accurate reaction rate constants are to be obtained, these two gas flows, as well as the downstream pressure, must be measured accurately. Finally, in many instances it is convenient to be able to measure the flow of the reactant ion parent gas.

10

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

The downstream pressure is measured by a Baratron capacitance manometer. The calibration of this gage is checked periodically by an untrapped McLeod gage. Three Baratron gages in use in this laboratory have been compared and the overall agreement at 0.376 Torr is 1 % . The flow of the buffer gas, the reactant ion parent gas, and the reactant gas are measured by a method which depends on gas viscosity. In this technique the flow elements are so designed that within the specified range of operation the flow through the elements is viscous. Thus, the flow of gas through the element is proportional to the pressure difference across the element and the average pressure in the flow element. In actual practice the pressure at the input to the flow element is used instead of the average pressure. The slight nonlinearity thereby introduced is negligible. The input pressure is maintained at 760.0 Torr. The three channels of gas addition are shown in Fig. 3. In the buffer gas channel the flow element is designed to accommodate a maximum flow of 300 atm cm3/sec of helium. Typical buffer gas flows are on the order of 180 atm cm3/sec. In the other two channels, a combination of three elements can be used to allow gas flows to be measured over a range extending from 0.001 to 10 atm cm3/sec of helium in full scale ranges of 0.1, 1.0, and 10.0 atm cm3/sec. In order to relate the output of the pressure transducers to absolute gas flows, calibration procedures have been devised. The input pressure transducers PIand P, are calibrated by a water manometer (range: 780 to 740 Torr & 0.05 Torr). The relatively large buffer gas flows permit calibration of the AP3 transducer by a commercial precision wet-test meter (0.2%). The smaller flows of the reactant ion parent gas and reactant gas are calibrated by measuring the rate of pressure decrease in known volumes V , and V, . The details of these calibration procedures are too lengthy to be discussed here, but can be extracted from Fig. 3 by careful study. After calibration, the accuracy of the flow measurements is better than 0.5%. The short-term stability of the apparatus (during a set of measurements) can be as good as 0.02% . Periodic checks of the calibration are routine.

E.

ION SAMPLING AND

DETECTION

The sampling orifice is located in the center of the blunt end of a truncated cylindrical cone. Other nose-cone geometries were tried with less success. The distance between the front of the nose cone and the end of the flowing tube is about equal to the flow tube diameter. Because of this positioning of the sampling port, it is expected that the flowing plasma near the front of the nose cone will characterize the central portion of the steady state flowing plasma just upstream from the end of the flow tube. The flowing gas is

760.0 Torr Lme

FIG.3. Detailed schematic representation of the flowing afterglow system indicating the equipment used to measure and calibrate the flows, the mass spectrometer, the timing system for pulsed operation, and the on-line data scanning system.

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12

E. E.Ferguson,F. C.Fehsenfeld, and A. L.Schmeltekopf

smoothly separated by the conical sides of the nose cone and subsequently exhausted into the pump. This has been qualitatively confirmed by the appearance of the nose cone following a series of experiments on metal ions. The metal ions were produced by ionizing metal atoms evaporated from a furnace located at the input end of the flow tube. Because of the very high accommodation coefficient of the atoms on room temperature surfaces they are deposited on the walls of the flow tube and surface of the nose cone. The metal atoms on the nose cone were distributed in such a way that when it was illuminated concentric interference rings appeared on the surface of the nose cone. The heaviest coating was located on the blunt end of the nose cone and decreasing concentrations were found on the conical walls. The nearcoaxial nature of these rings indicated that the flowing gas was uniformly intercepted. To reduce possible sampling problems arising from the edges of the sampling orifice, this hole is made large, typically 0.5-1.5 mm. The thickness of the edges of the sampling orifice has not been found to be critical for thicknesses between 0.025 and 0.25 mm. Generally, the thickness of the orifice has been made less than one-fourth the diameter of sampling orifice. The electron density in the afterglow region near the sampling orifice is, in general, lo7 cm-3 or greater. Accordingly, the Debye length of the ions is, in most instances, less than the diameter of the sampling orifice. Since the pressure in the afterglow region is of the order of 0.4 Torr, the mean free paths for neutrals and ions are also less than the diameter of the sampling orifice. For these reasons we expect ions to be carried convectively by the buffer gas through the sampling hole, in much the same way as they are carried down the main flow tube. All our experience supports this view. The pressure in the chamber following the sampling orifice is approximately Torr. In this region the motions of the ions are determined by the local fields rather than by collisions. We have found that the most severe sampling problems have been due to deflection of the ions by surface charge on the inside surface of the nose cone and on the front surface of the quadrupole case. To minimize this, these surfaces are cleaned periodically, depending on system use. This cleaning consists of electrocleaning followed by a distilled water rinse. It has been our experience that neither ultrasonic cleaning nor vapor degreasing effectively cleans the surface contamination on these metal surfaces. In the heated system cleaning is supplemented by baking. It was found that reliable sampling, especially when oxygen atoms were in the system, could not be obtained when the material of the sampling orifice was gold, stainless steel, copper, brass, or rhodium. The only two materials which have been found to give reliable results are molybdenum or carbon, the latter in the form of a coating prepared from an alcohol suspension of graphite. Because it is more stable, molybdenum is now used. We do not

FLOWING AFTERGLOW MEASUREMENTS

13

understand why molybdenum works better than other metals. For example, when the sampling orifice was in a gold disk, a drop in the sampling efficiency of lo4 has been observed in the presence of oxygen atoms. With clean molybdenum no change is observed. The nose-cone potential En,is typically near ground potential while the potential of the quadrupole case E,,, is 2-6 V. Variations in these potentials are found not to affect the measured reaction rate constants. The quadrupoles have usually been 10 cm long, typically giving a resolving power of about 120 for full width at half-peak height. They are scanned from 1 to 200 amu, using a variable frequency, fixed voltage, power supply, or from 1 to 400 amu, using a fixed frequency, variable voltage power supply. The frequency-scanning quadrupole has the advantage that the resolving power and mass peak height are independent of mass. The voltage-scanning power supply with fixed ac to dc voltage ratio allows a rapidly scanned display of the mass spectrum on an oscilloscope. The rapid scan facilitates the determination of the qualitative behavior of ions in the system under varying conditions. Moreover, the quadrupole spectrometer using the voltage scan has a resolving power as large as 900 for a full width at half-peak height. With very high resolving power, Eqc must be reduced to less than 0.5 V to prevent some peak broadening due to the velocity acquired by the ions before their entrance into the mass spectrometer. The mass spectrometer chamber is differentially pumped by a 6 in. oil diffusion pump separated from the chamber by a water-cooled baffle and a sorbent trap. The multiplier chamber is pumped by a 4 in. oil diffusion pump and is similarly trapped. No evidence of oil backstreaming has been observed. to The detection system, which has input currents ranging from lo-'' A, normally utilizes pulse counting techniques. Pulse counting has the following advantages over the conventional electrometer detector : (1) easier to obtain a given accuracy; (2) facilitates use of digital analysis on the output; (3) readily adaptable to repetitive pulsed operation; and (4) simplifies detection circuitry to measure positive or negative ions. Points (3) and (4) are the principal reasons for using pulse counting on the ESSA system. The pulse amplifiers and counters have a pulse pair resolution of lo-* sec so that counting rates of lo6 counts/sec can be used with essentially no correction for pulse overlap. The discriminator is set for the plateau condition so that amplifier noise is rejected but essentially all of the pulses from the multiplier cathode are accepted. It has been noticed that nitrogen and perhaps other gases tend to reduce the gain of the multiplier slightly. Therefore, it is important to operate in the plateau condition so that slight changes in gain which could result from the reactant gas addition are not reflected in signal changes. The cathode of the multiplier is operated 1500 V attractive

14

E. E.Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

above the exit of the quadrupole. Since a further increase in this voltage has no effect on the signal, it is felt that all ions leaving the quadrupole are collected and counted. The most important concern with regard to the sampling and detection system is that the sampling efficiency be constant, that is, independent of the reactant ion density. Since the calculated rate constant is derived from the decline of the reactant ion signal as a function of the addition of reactant gas, it must be known that no change in overall sampling is occurring while the reactant gas is added. (As discussed earlier changes did occur when oxygen atoms were added before the molybdenum sampling hole was used.) Certain tests may be used to establish that the sampling efficiency is constant. For positive ions, when no further reactions of the product ions occur and when recombination and diffusion of the product ions are taken into account, the product ion signal should balance the initial reactant ion signal. The same is true for negative ion reactions with negative ion products. For associative-detachment reactions of negative ions, persistent, non reactive impurities (e.g., C1- and NOz-) have been monitored for constancy during experiments. Since the molybdenum sampling orifice has been used, no evidence of change in sampling has been observed except when the system is dirty, and a change in sampling is now taken as evidence of a dirty system. 111. The Flow Analysis

The analysis used to compute a reaction rate constant from the variation of ion signals as a function of the addition of a reactant gas is described in this section. To be reasonably accurate, this analysis must take into account many properties o f the flow. In order to describe the complete analysis it is instructive to first discuss an extremely simple model and then describe modifications of this simple model by the inclusion of more realistic flow properties. The analysis of the simple model is described in Section III,A. In Section III,B, the simple model is extended to include radial diffusion, Also contained in this section is a brief discussion of the analytical similarities of the various experimental approaches to the afterglow experiment. Section III,B concludes with the introduction of a generalized form of the simple solution obtained in Section III,A which will be used for the inclusion of the various flow properties into the analysis. Section III,C describes the effect produced by the inclusion of a parabolic velocity profile. In Section IKD, axial diffusion is introduced with a short discussion of the diffusion coefficients which are used in these calculations. Section III,E considers the pressure and velocity gradients of the flowing gas and incorporates these effects into the rate calculation. Section Il1,F is concerned with the effects

FLOWING AFTERGLOW MEASUREMENTS

15

associated with the perturbation of the reactant ion distribution by the localized addition of reactant gas. Section III,G reviews the conclusions derived from the analysis. Finally, Section III,H describes an experimental test of the accuracy of the assumed flow properties of this system.

A. THESIMPLEMODEL The original analysis of the flowing afterglow data used an oversimplified but mathematically tractable model. This model makes a suitable starting point for the discussion since it is easy to visualize and the further detailed analysis can be viewed as a series of modifications of the simple model. Neglecting diffusion, the decrease in the reactant ion A' in the reaction zone as a consequence of the addition of neutral reactant B, due to the reaction A+

+B +C+ +D

is given by

-dCA+l - - k[A'][B], dt where k is the binary rate constant for the reaction and [A'] and [B] are the concentrations of A+ and B. If the axis of the tube is designated as the z axis and the flow velocity is assumed to be constant and directed along the z axis, then the independent variable t in (2) may be changed to z. Equation (2) may be rewritten as uo -= - k[A+][B], a2

(3)

where z is measured from the reactant gas inlet and uo is the constant flow velocity. If [B] p [A'] for all z 2 0, then (3) may be integrated to yield

where [Ao'] is the concentration of A' prior to the introduction of the reactant gas and 1 is the length of the reaction zone. Now assume that B is initially introduced uniformly over the cross section of the tube so that

CB1 = 0, z
=

- Qkl/nu2uO2.

(8)

16

E. E. Ferguson,F. C.Fehsenfeld, and A. L.Schmeltekopf

Since I and a are dimensions of the flow tube and uo is the independently measured average gas flow velocity, the rate constant can be calculated from the reduction of the reactant ion signal by the reactant gas. Thus

k, = -(na'vo2/Ql) ln([A']/[A,+]) (9) where k, is used to designate the rate constant calculated assuming this simple model. B. RADIAL DIFFUSION To include radial diffusion, a loss term is added to (3) v O X.4' 7 1= D A -1 -a r- a[A'] r ar ar

(

) - (k[A'][B]

0

for z < O for z 2 0

(10)

where D A is the diffusion coefficient for A'. As was the case in the preceding section, the neutral concentration is assumed to be uniform. No azimuthal variations are considered; [A'] is allowed both radial and axial variations. In the solution of (10) it is assumed that [A'] = 0 at r = a, that all modes of [A+ J are continuous at z = 0, and that the interaction region is of infinite extent in either direction along z. When z < 0, A' is attenuated by diffusion alone and the ion distribution decays according to the relation

[A"]

CC

J,(li r/U) eXp[ - &'DA Z/a'Uo]

i= 1

(1 1)

where Jo(ljr/a) is the ith mode of the zero order Bessel function and ,Ii is the ith root of Jo . Since A1 is much smaller than the other Aithe higher modes will be rapidly attenuated compared to the fundamental mode. Thus, an initial distribution composed of a combination of modes will quickly relax to the fundamental mode as it proceeds down the tube. Therefore, the distribution of [A'] will be assumed to be fundamental mode at z = 0, i.e., by the time the ions have reached the reaction zone. The solution of (10) for z 2 0 yields the following expression for the concentration of ions on the axis of the flow tube, attenuated by both radial diffusion and reaction

where [Ai+] is the axial concentration of A + at z = 0. If [Ao'] is the concentration of A' at the end of the reaction region when attenuated by diffusion alone, then the relative loss of ions by reaction would be identical to that from (8) and the rate constant would still be determined according to (9).

FLOWING AFTERGLOW MEASUREMENTS

17

Equation (12) provides a useful basis for the discussion of the various ways in which rate measurements are currently obtained in flowing afterglows. In the ESSA experiment the only parameter which is usually varied is the rate of addition of reactant gas Q. In this method of performing the experiment, the first term representing the ion diffusion loss remains constant. Thus, at the end of the reaction zone ( z = I), the relative decline in the ion signal due to the introduction of the reactant gas is given by (9). In another method, the reactant gas flow is fixed and the reactant gas inlet is made movable, thus z is varied. It is possible to design a movable inlet so that its movement does not significantly change the diffusion losses in the tube. Therefore, the diffusion-loss term in (12) remains a constant. The reaction-loss term is a function of the reaction zone length, the now-movable distance between the reaction zone inlet port, and the sampling orifice. Hence, the relative change in the reaction ion signal due to the movement of the gas inlet is given by (9). Since the change in pumping speed alters the background gas pressure and thereby the gas flow velocity, a reaction rate may also be measured by changing the pumping speed. The diffusion coefficient of A + is inversely proportional to the background gas pressure which, so long as the buffer gas flow is held constant, is inversely proportional to the flow velocity. Thus, the diffusion coefficient of A + is directly proportional to the flow velocity. The net result is that the diffusion-loss term in (12) is once again a constant. Consequently, as in the two preceding cases, the relative change in the ion signal due to the variation in the flow velocity is still given by (9). Certainly, as more details of the gas flow dynamics are taken into account, these three methods are no longer strictly equal. However, even from the solution of the complex transport equations presented later there is little to a priori recommend one method of measurement over another. There are other ways to perform these experiments. These methods combine variations in both the diffusion and the reaction attenuations. It is probably advisable, however, to make reaction rate measurements in such a fashion that diffusion may in the first approximation be neglected. As more realistic flow dynamics are taken into account it is no longer possible to calculate a rate constant from the simple expression of (9). It is still possible, however, to present the results of the analysis in a relatively simple form. To do this, (9) is rewritten in a form which is modified to include corrections obtained from calculations using more realistic gas dynamical models:

k=

+ Ci a i > ( D ~ a 2 0 0 2 / QW[A+I/[&+l), 0

(13)

where the cli and jl are rate correction factors. Each correction factor represents an individual process which separates the results of the sophisticated

18

E. E. Ferguson,F. C.Fehsenfeld,and A. L.Schmeltekopf

analysis from those of the simple case. The physical processes associated with the various terms in the a series of correction factors are: (1) ctl is the correction for the nonplanar velocity profile of the flowing gas; (2) a2 is the correction for axial diffusion; and (3) a3 is the correction for inlet effects, i.e., a3 refers to the correction associated with the initial source distribution of reactant gas and the perturbation produced on the initial ion distribution by the presence of the neutral gas inlet port. The correction factor pertains to the pressure gradient produced by the flowing gas. This particular correction does not originate from the transport equation which shall be solved later ; therefore, it is considered separately from the ct corrections. In flowing afterglow experiments, reaction rate constants are determined by the reduction of the relative ion signal as a result of the variation of one parameter on the right side of (9). In the ESSA experiment the variable parameter is usually the reactant gas flow. Accordingly, in the following discussion, the values of other parameters will be fixed and the reactant flow will be variable. General expressions have not been derived for the correction factors. All the terms in the ct expansion depend on the velocity profile of the flowing gas. Thus, all the higher-ordered cli are functions of a l . Likewise, due to coupling of diffusion and reaction, all of the aiwill be interdependent to a certain extent. Therefore, the ai should be taken only as indications of the magnitude of the respective effect and must be applied together.

C. NONUNIFORM VELOCITYPROFILE The gross gas flow properties of the flowing afterglow system are determined by the buffer gas, since the relative concentration of the reactant gas introduced into the system is typically only of the order of a few tenths of a percent or less. Under typical operating conditions the Knudsen number (the ratio of the mean free path of the buffer gas atom to the radius of the flow tube) is about 0.01 and the Reynolds number is 17. [ R = 1.41 x MQb/uq,where M is the atomic weight, Qb is the flow (atm cm3/sec), and q is the viscosity (poise) of the buffer gas.] Accordingly, the developed flow should be almost laminar with no more than a few percent slip at the wall due to molecular flow. The distance required for the buffer flow to fully develop the characteristic parabolic profile is given by d = 0.227 aR (Dushman and Lafferty, 1962). For helium at room temperature with a flow rate of 180 atm cm3/sec, d is 14 cm. As an experimental check, the pressure differential measured between points in this tube has been found to agree quite well with calculations based on laminar flow [see (24)]. As a consequence, the form of the velocity profile can be expressed:

u(r)= 2u0[1

- (r/a)’],

(14)

19

FLOWING AFERGLOW MEASUREMENTS

where oo is the average velocity. This average flow velocity (or bulk flow velocity) is defined as 00

=

Qb(atmcm'lsec) 760(Torr/atm) T("K) P(Torr) 273.16("K) ' na2(cmz)

(15)

where P is the pressure in the flow tube. This is an obvious definition; nevertheless, the velocity plays such an important role in determining the rate constant that an explicit statement seems worthwhile. To include the radial velocity profile, (10) can be altered slightly to read

where v(r) is expressed by (14) for the case at hand. Equation (16) is still separable, but because of u(r), the radial part is no longer Bessel's equation. Consequently, its solution is obtained by using both analytical and numerical methods. In the absence of reaction (z < 0), A + is attenuated by diffusion alone. It is no surprise that this solution is similar to that of the zero-order Bessel equation, which was discussed in connection with diffusion in the plane flow in Section II1,B. The similarity can be seen in Fig. 4, which plots the I .o

0.9

0.8 FIG.4. The computed fundamental mode ion distribution in the flowing afterglow for two different assumed velocity distributions in the absence of reactions. The relative ion density is plotted against tube radius. The curve labeled normal Bessel distribution is obtained by assuming a plane velocity profile while the curve labeled fundamental mode distribution in flowing system was obtained assuming the characteristic velocity profile for the flowing gas.

0.7 v)

z

#

0.6

z

0 0.5

5

W

Fundamental Mode

0.4

0.3 0.2 0.I 0

xis

RADIUS (cm)

4 cm

fundamental mode with plane flow and the fundamental mode with parabolic flow (with 10% slip at the walls) against tube radius.

20

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

In any event, neglecting reaction, each of the modes will be attenuated by diffusion following the relation [A']

=

1ai(r)exp[ -yi2D,z/a2u,], i

(17)

where the ai(r)are the profiles of the various modes and the y i are characteristic constants, analogous to the ,Ii. The values of the yi2 for the first five modes are listed in Table I along with the corresponding values of the Ai2 TABLE I

DIFFUSION ATTENUATION CONSTANTS

CHARACTERISTIC

Y'2 ArZ Mode (Parabolic flow) (Plane flow) ~

1 2 3 4 5

3.66 22.30 56.96 107.59 174.20

5.78 30.47 74.89 139.03 222.93

associated with Eq. (1 1). Two observations can be made concerning these values. In the first place, since y1 is much smaller than the other coefficients, the higher-order modes will be attenuated quite rapidly compared to the lowest mode. Thus, a distribution composed of a combination of modes will quickly relax to the fundamental mode as it proceeds down the tube. In the second place, the coefficients for the diffusion attenuation of a particular mode in parabolic flow are smaller than the corresponding term for plane-flow modes. This difference originates from the fact that the peak ion distribution is located in regions where the velocity is maximum. Thus, the ions are transported more rapidly and suffer less diffusion loss. This latter effect is also reflected in the reaction loss. The chief difference between this case for parabolic flow and the plane-flow case, however, lies in the fact that the eigenmodes which make up the radial profile are not the same for z > 0 as for z < 0. Thus, the radial profile may differ in the two regions. Physically this means that when the velocity profile is parabolic, even though the reactant density is uniform, the reaction can distort the ion distribution. In other words, coupling between reaction and diffusion is thereby introduced. Taking these facts and keeping in mind the continuity condition on A' at z = 0, the following description of the flowing afterglow is reached. For z < 0 (no reaction), a fundamental mode distribution compatible with a parabolic velocity profile exists for A'. For z > 0, the fundamental mode

FLOWING AFTERGLOW MEASUREMENTS

21

profile for A' is changed due to the inclusion of reactions. The greater the degree of reaction loss of A', compared to diffusion loss, the more acute will be this distortion. Since the fundamental modes in the two regions are not identical, higher-order modes will be required to maintain continuity at z = 0. The larger the reaction-loss term, the larger the amplitudes of the higher-order modes must be to maintain continuity. However, these higher-order modes are rapidly damped by diffusion relative to the fundamental mode so that the .A' profile in the reaction zone soon relaxes to a fundamental mode compatible with both the velocity profile and reaction. In order to put these observations in a proper perspective, it is useful to introduce the ratio of the rate of loss of ions per unit distance down the flow tube due to reaction to that due to diffusion:

First, consider the case where 5 4 1, i.e., diffusion dominated. In this limit, reaction is not capable of distorting the ion profile, and the numerical solution of (1 6) is

Thus, the reaction rate constant is given by (13) with a1 = 0.626. The correction factor is positive in this case because the parabolic velocity distribution effectively transports the ions down the tube at a more rapid rate. Thus, the reactant gas has less effective reaction time with the ion. Now consider the situation where the reaction attenuation becomes comparable or greater than the diffusion loss ( 5 b 1). The fundamental mode in the reaction zone is now significantly different from that upstream as is shown in Fig. 5. Here a1 has been calculated as a function of [A']/[Ao']. For a set of typical values (helium flow = 167.5 atm cm3/sec, D , = 2580 cm2/sec, a = 4 cm, and I = 42 cm), the results are shown in Fig. 6. Note that a1 increases only slightly as [A']/[Ao'] declines. This is due to the combination of two opposed effects. On the one hand, the reaction introduces higher-order diffusion modes which enhance diffusion loss. This enhanced loss would, tend to reduce al. The source of the other effect can be seen from Fig. 5. Note that as reactions are included, the fundamental mode distribution becomes even more peaked at the center where the flow velocity is high. These ions are transported more rapidly down the tube and therefore have less reaction time. This tends to increase a l . The combination of these two effects almost balance; consequently, the variation in a, is not large. In fact, for a 2 order of magnitude decline in [A']/[Ao+] the change in the rate correction, 1 + tll, is only about 1 % .

E. E.Ferguson,F. C.Fehsenfeld,and A. L. Schmeltekopf

22 I.o

0.9 0.8

>

0.7

5

z

0.6

FIG.5. Plot of relative ion density versus tube radius with and without reaction. The curve with no reaction assumes diffusive loss alone, while in the other curve the reactive loss is four times greater than the diffusive loss.

z

'

0.5

W

L

0.4

J

w

a 0.3 0.2 0.I

C

4cm

RADIUS (cm)

Ql

0.63 0.6211 I I I

0.7 0.5

I

'

0.3 0.2

1 1 1 1 I I

I

I

0.1 0.07 0.05 0.03 0.02

I 0.01

FIG.6. Plot of alversus [A+]/[A,+].

Besides this small reaction dependence, a1 also depends on pressure. As the pressure is reduced the slip flow due to molecular diffusion increases. Thus, the velocity profile more nearly resembles the plane profile and this reduces al. In practice, however, the pressures used in the ESSA tubes are large enough so that the pressure dependence of 1 + a1 is less than 1 % .

D. AXIALDIFFUSION In the absence of reaction, radial diffusion produces a gradient in the

axial direction. This gradient produces a transport of ions by diffusion. As

FLOWING AFTERGLOW MEASUREMENTS

23

reactions are added, this gradient becomes steeper so that the rate of axial diffusion increases. It is the purpose of this section to investigate the contribution of axial diffusion. In the solution of (lo), a unique diffusion coefficient DA is assumed to exist for A'. Since the flowing afterglow experiments are performed with moderate electron densities (n, 5 lo'), the diffusion of the positive ions and electrons will be ambipolar. Now, in general, there will be a mixture of positive ions present in the actual experiment. This is certainly true after a reactant gas has been added and the product ion is substituted for the reactant ion. It has been shown that in the first approximation when there is a mixture of charged particles composed of several types of positive ions and electrons, all of which are at gas temperature, the'diffusion of one type of positive ion is not influenced by the presence of the other types (Oskam, 1958). Thus, for a situation in which there are several types of positive ions and electrons, it is permissible to use a unique ambipolar diffusion coefficient for each species. The ambipolar diffusion coefficients of the ions are taken to vary as T for a constant gas density. This temperature dependence corresponds to a pointcharge, induced-dipole interaction. A detailed discussion of how the temperature dependencies originate from the nature of the interaction is given ' , by McDaniel(l964). For all ions diffusing in helium, except for He' and H the product of the ambipolar diffusion coefficient and pressure at 300°K and 1 Torr has been taken to be 900 cm2 Torr/sec. This very simple expression for the diffusion coefficient arises because of the low mass and nonreactivity of helium and argues strongly for its use as a buffering agent. In order to include axial diffusion, the transport equation becomes 0,

z
r (20) where u(r) is given by (14). As in the preceding sections, [B] is assumed to be uniform in the reaction region. A numerical solution is obtained for (20). For corresponding conditions this solution is compared with the solution of (16). By this comparison the rate correction factor u2 for axial diffusion is obtained. The values of a2 as a function of [A']/[Ao'] are shown in Fig. 7. As noted, the various curves correspond to variation in the values of the diffusion parameter y1 DA/uoa' and to differing reaction lengths 1. A typical operating condition would be one in which D A = 2580 cm2/sec, P = 0.350 Torr, helium flow = 167.3 atm cm3/sec, and a = 4 cm. This corresponds to Y1DA/a2v0 = 0.075 in Fig. 7. A different value of the diffusion parameter might be obtained by varying the pumping speed or the buffer gas flow.

24

E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf

[A*l/[A:l

FIG.7. Plot of a2 versus [A+]/[A,+]. Solutions were obtained for several values of the diffusion parameter, yIDA/uO d,and for reaction lengths of 40 and 85 crn.

Since an increase in either the diffusion attenuation or the reaction attenuation increases the axial ion density gradient, the axial diffusion correction factor must be positive and must increase as [A+]/[A,'] decreases. Moreover, for a fixed [A']/[Ao'], the ion density gradient is larger for a shorter reaction length; thus, a2 must be larger. The ratio a2/cll is typically 0.1 or less.

E. AXIALVELOCITY GRADIENT AND SLIPFLOW As discussed earlier in Section HI$, the buffer gas flow in the ESSA tube is predominantly viscous with a few percent slip (molecular). The viscous conductance is F,(cm3/sec) = (103na4/8q1)P, (21) where q is the gas viscosity in poise and P is the average gas pressure (in Torr) over the length 1. The molecular conductance is F,(cm3/sec) = 30.48 x 103(a3/l)(T/M)'/2, (22) where T is the gas temperature in degrees Kelvin and M is the atomic weight of the buffer gas atom (Dushman and Lafferty, 1962). The total conductance can be obtained by combining (21) and (22) in the fashion

F = F,,+ZF, (23) where Z is a pressure-dependent term varying between 0.81 and 1.00, as tabulated by Dushman and Lafferty (1962). The axial pressure gradient follows from (23) AP 0.760Qb - -Torr (24) Az cm - (na4P(z)/8q) 30.48Za3(T/M)'/2'

( )

+

FLOWING AFTERGLOW MEASUREMENTS

25

The background gas pressure Po is measured at a point in the reaction zone. The pressure at all other positions, i.e., the axial pressure profile, is then obtained from (24). This axial pressure profile and (15) lead to the average axial velocity ij(z) at any point along the tube. To combine the contributions of slip and viscous flow on the radial dependence of the axial velocity, it is assumed that these two contributions act independently; therefore, P(T,

z) = E(Z){(2F"/F)(I- r'/u2)

+ Z(F,/F)}.

In order to check (25) pitot-tube measurements have been made to determine the velocity profile. Only relative velocity profiles were measured because the pitot tubes which could be used for these measurements had Reynolds numbers less than one. In the low-pressure flowing gas of the afterglow tube, the proper dimensions required by pitot tube theory ( R > 1) were not possible. Thus, an absolute velocity determination from the measurements made with the pitot tube would require large corrections which are not known with sufficient accuracy. However, the results which were obtained, and which should be correct insofar as measuring the profile, show a parabolic profile with an indication of some slip flow at the wall, as predicted by (25). The solution of the transport equation which results from the above discussion is discussed in the next section. It is worthwhile, however, a t this point to discuss how these variations will affect the physical properties of the solution. In the solution of (20) the diffusion terms are inversely proportional to velocity. However, the diffusion coefficients which appear in the solution of this equation are inversely proportional to pressure. This means that, except for effects associated with the small changes in the velocity profile, the diffusion properties are not changed by the pressure gradient. However, the variation of gas pressure (hence average flow velocity) is reflected in the rate of loss of the reacting ions, since A[A+]/Az cc l / v o 2 . Referring to (15), this means that if the average velocity is computed from a pressure measured at the beginning of the reaction region, the average velocity obtained will be too small. Accordingly, the rate constant calculated on this basis will be too small. The opposite result would follow if the pressure were measured at the downstream end of the reaction zone. In practice to avoid this problem, the pressure measurement is made at a point centrally located in the reaction zone. Consequently the p correction represents a " reference point " correction factor to be used only when the pressure is measured at some point other than the center of the reaction region. Figure 8 shows the values for p which are required if points other than the center are used in the measurement. In these calculations the assumed buffer flow was 180 atm cm3/sec, the viscosity of the buffer gas is 1.877 x lov4P. This viscosity corresponds to that of helium at 300°K.For the curve labeled

E. E. Ferguson,F. C.Fehsenfeld, and A. L.Schmeltekopf

26

P 0.900

0

I

1

10

20

I

1

30 40

Z

I

FIG8. The /3 correction factor is plotted as a function of the location at which the background gas pressure is measured. The correction factor is plotted for two pumping speeds (i.e., two background gas pressures). Buffer gas flow = 180 atm cm3/sec; gas viscosity = 1.877 x P; curve I, P0=0.390 Torr; curve 11, Po = 1.560 Torr.

50 60

I, the pressure at the center of the reaction region was 0.390 Torr, while for I1 the pressure was 1.560 Torr. The smaller pressure is a typical operating pressure, and at this pressure there is a considerable gradient across the reaction zone. Thus, the average velocity changes by about 5 % from the midreaction zone velocity. This results in about a 10% variation in fl about the reference value of 1.00. As the pressure is increased the correction is reduced. For the higher pressure shown in Fig. 8, the variation in is less than 1 %. F. INLETEFFECTS In the actual flowing afterglow experiment the initial neutral concentration is not uniform. The neutral reactant is introduced through an inlet port, diffuses into the flow stream, and eventually achieves a uniform distribution. Since the concentration of reactant neutral varies down the tube, concentration gradients in A + are produced as a result of the reaction with B. These gradients modify the transport of ions by diffusion and consequently further couple diffusion and reaction. This process depends on both the distribution of B and its concentration. Thus, these effects vary with the amount of reactant gas introduced. In order to correct for these “inlet ” effects, the ci3 factor is computed. With the introduction of this inlet effect, the model is essentially a complete description of the flowing afterglow. The family of transport equations for this model is

where u(r,z) is given by (25), R i is the number density of each reactant (positive ion, negative ion, electron, or neutral), and D i is the appropriate diffusion

FLOWING AFTERGLOW MEASUREMENTS

coefficient for Ri . The term 6, contains all reaction " sinks '' are two-body reactions of the form

"

27

sinks" for Ri . These

or three-body reactions of either the form

where M is the density of the buffer gas, or

such that ai = ai, + ai, + ai, 3 . Since a sink for one reactant is a source for another, we require ai = 0. The diffusion of all reactant species is determined by the interaction with the buffer at operating pressure. Thus, all the transport properties in (26) are determined for all points in the tube and are used as a table of values for the point-by-point solution of (26). The solution of (26) is done on a computer and is based on an explicit finite difference method. The reader will note that the axial diffusion term D i a2Ri/az2is missing in (26). This is because axial diffusion gives rise to a second difference term in the axial coordinate which must be dropped in order to obtain a " well-posed question " for the difference calculation. In analytical solutions of partial differential equations with given initial conditions the solutions which are unbounded are neglected as unreasonable. However, in the difference approach, these solutions cannot be selectively suppressed. Thus, the terms which give rise to them must be modified or dropped. Since in Section III,D it was shown that the correction factor for axial diffusion was small, it is expected that the magnitude of this type of correction will not significantly change in this case. Therefore, the axial diffusion correction a2 will simply be added on here as a perturbation correction. In the solution, a net size in the z direction is determined by the ,,,,Ju(r, z),,,~"and where Ar is stability criterion A z cc ( A r 2 / 2 0 , where o = Di, the net size in the radial direction. The term Di, is the maximum diffusion coefficient for any reactant; u(r, z),,,~" is the minimum value assigned to the velocity. In order to reduce computation time, one chooses u(r, z),~" as large as possible without exceeding a tolerable error. It was found that u(r, z),,,~" could be as large as 10% of uo before an error of 1 % was reached. The net size in the r direction was obtained by successively decreasing the net size until the maximum total error between the difference calculation and the exact solution for a simple model where an exact solution is possible, was less than

28

E. E. Ferguson, F. C.Fehsenfeld, and A. L.Schmeltekopf

2 %. It was found in practice that in the first 2 cm following the introduction

of the reactant gas a net size of 1/90th the tube radius was required. Following this region, a net size of 1/18th the radius of the tube was found to be adequate. The following list of assumptions have been made in these calculations. 1. The concentration of ions, electrons, and metastables are zero at the tube wall, while neutral reactant concentrations have a vanishing normal derivative at the wall. 2. All particles are isothermal at the wall temperature. 3. A unique diffusion coefficient exists for each reactant. 4. The neutral diffusion coefficients in helium vary as T112at constant gas density. [This corresponds to a hard-sphere scattering model (McDaniel, 1964).] The product of the diffusion coefficient and pressure for all neutral reactants (except H2 and atomic hydrogen) is taken to be 570 cm2 Torr/sec at 300°K and 1 Torr.

Most experimental conditions lead to a simplified form of (26) in which only one type of neutral reactant R, is present and is made to greatly exceed the density of all other reactants in the reaction zone. Thus, 6, = 0; i.e., the loss of reactant molecules in reaction with the low concentration reactants (ions, metastable atoms, etc.) can be neglected. Moreover, by virtue of their small concentrations, the binary reaction losses between any combination of the other reactants usually can be neglected so that ki, Ri R j = 0, i, j # 1. However, more complex systems than this have been analyzed, for example, where dissociative recombination is important or where metastable atoms react with metastable atoms. These calculations, however, will not be described since the present discussion is intended to point out some of the more important conclusions concerning the dependence of the computed rate constant on the assumed transport properties of the flowing afterglow. Only a simple reaction of the form A + B + C + + D will be discussed. To initiate the calculation, the fixed parameters of the system ( I , a, QHe, P o , D A , D , , Qk), are supplied along with the initial distribution of the reactant ion and the initial distribution of the reactant neutral source. In practice, it is impossible to know these initial experimental distributions exactly. For this reason, two combinations of initial ion and neutral distributions are considered. These represent extreme cases. The actual experimental distributions should be somewhere between these extremes. In the first case, the reactant neutral issues from a point source and the initial ion distribution is in the fundamental mode for the parabolic flow. This case represents the extreme situation in which the initial distribution of reactant neutral produces the maximum attenuation of the axial ion distribution and the reactant inlet tube produces no perturbation in the initial ion concentration. The initial

+

29

FLOWING AFTERGLOW MEASUREMENTS

ion distribution associated with this case is plotted in Fig. 9 and is labeled “ fundamental mode distribution in flowing system.” In the second case, the reactant neutral issues from an axial source 1 cm in diameter while the initial Inner Boundory for

FIG.9. Initial ion distributions used in the explicit finite difference solution of (26). Relative ion density is plotted against tube radius. The curve labeled fundamental mode distributionin the flowing system pertains to the fundamental mode assuming a characteristic velocity profile while the curve labeled 1-8 coaxial ion distribution pertains to a fundamental coaxial mode between coaxial cylinders of 1- and 8-cm diameter.

3

0.7

0.6

5’ P

0.5 Fundanentot Mode

0.4

RADIUS (cm)

ion distribution is a fundamental coaxial mode between the inner wall of the flow tube and the outer wall of the reactant inlet. This initial ion distribution is also plotted in Fig. 9 and is labeled “1-8 coaxial ion distribution.” A l-cmdiameter inlet was chosen because this represents the maximum inlet tube size which is used in the ESSA experiments. This second arrangement thus represents the other limit where the reactant inlet tube produces the maximum possible perturbation while the reactant source represents the maximum initial dispersal of reactant neutrals. The distortion of the reactant ion distribution by the point source injection of the reactant neutral, and the subsequent relaxation of this perturbation, is shown in Fig. 10. The rate of addition of reactant neutrals was taken to be 7.56 x 1019 molecules/sec, and the reaction rate constant 5 x lo-’’ cm3/sec. In these calculations uo = 8 x lo3 cm/sec, D,P = 900 cm2 Torr/sec, D , P = 570 cm2 Torr/sec, and Po = 0.390 Torr. Upon introduction of the reactant gas from a point source, the ion distribution in the center of the tube is drastically attenuated as shown in the profile plot which corresponds to 0.1 cm. The subsequent curves indicate, .however, that as the reactant neutral is dispersed the ion profile fills in until at 10 cm there is no longer a depression in the center of the ion distribution. At 10 cm, diffusion averaging has almost

E. E. Ferguson, F. C.Fehsenfeld, and A. L.Schmeltekopf

30 IO ,W

Init iol Distribution

cm

Axis

FIG.10. Diffusion redistribution of the ions following the point source introduction of the reactant neutral. The curves were obtained from a solution of Eq. (26). In these calculations the characteristic velocity profile for the flowing system was used and the initial ion distribution was assumed to be fundamental mode compatible with this velocity profile. For comparison the ion profile at 10 cm in the presence of diffusion attenuation alone is also plotted.

4 cm

RADIUS (crn)

rectified the enhanced reaction depletion of the axial ions. The profile corresponding to the ion distribution at 10 cm with noisu'id alone is included for comparison. The persistence of the perturbation with distance down the tube would be increased by lowering the ion or neutral diffusion coefficient or increasing the flow velocity. The resulting t13 correction for the nonperturbing point source of reactant gas is plotted in Fig. 11. It was assumed that the flow profile was parabolic andthat the initial ion distribution was fundamental mode with uo = 8 x lo3 cm/sec, DAP = 900 cm2 Torr/sec, D, P = 570 cmz Torr/sec, and Po = 0.390 Torr. This set of calculations was done for reaction lengths of 10, 30, and 60 cm. In each case t13 is negative. This is due to the fact that the reactant gas injected in this manner is more effective at quenching the axial ion distribution than is the uniformly distributed reactant. The physical reasons for this increased efficiency of a point source are twofold. In the first place, the point source produces larger than average concentrations of reactant gas in regions of larger than average ion concentration. Second, the reactions distort the fundamental mode distribution of the reactant ion and thereby increase diffusion losses. A second feature of the curves in Fig. 11 is that the absolute value of the correction coefficient associated with the point source decreases with a decreasing ratio of [A']/[Ao'] and approaches zero in the limit. This

31

FLOWING AFTERGLOW MEASUREMENTS

-

00'111

1

1.0 0.7 0.5

1

I

0.3

1 1 1 1

I

I

I

I

0.1 0.070.05 0.03

[A+]

I

QOI

/ [Ad

FIG.1 I . The aj correction factor for point source reactant gas addition plotted against [A+]/[A,+] for three reaction lengths. The characteristic velocity profile of the flowing gas was assumed along with an initial ion distribution compatible with this velocity profile.

decrease is due to the fact that the initial addition of reactant produces a large attenuation of the axial ion distribution. Additional reactant gas is consequently less effective in attenuating the ion signal. Finally, it is to be noted that a3 increases as the reaction length is decreased. This arises because diffusion is less able to smooth the initial perturbations prior to ion sampling for the shorter lengths. The calculations of the last paragraph were repeated for the other inlet condition limit, namely the 1-8 coaxial reactant ion distribution. The experimental conditions were otherwise unchanged. The correction factor computed for this case is plotted for'reaction lengths of 20 and 60 cm, and the results are shown in Fig. 12. A comparison of Figs. 11 and 12 indicates that

-0 -0.4

-0.3b

-O.*r

-o.L--L--J [!GI 01.0 0.7 0.5

0.1 0.07 0.05 0.03

0.3

[A+]

0.01

/

FIG.12. The a3 correction factor associated with a 1-cm-diameter reactant gas addition port plotted against [A+]/[A,+] for two reaction lengths. The characteristic velocity profile of the flowing gas was assumed along with an initial ion distribution which is the fundamental coaxial mode between the reactant gas inlet and the main flow tube.

32

E. E. Ferguson,F. C.Fehsenfeld,and A. L.Schmeltekopf

for the shoyter reaction lengths, the clj correction for the nonperturbing point source is larger than for the extended coaxial inlet and is about the same for the longer lengths. This is what one would expect. Since the presence of the coaxial inlet has quenched the axial ion distribution, the reactant neutral is initially added into regions of low ion density. Thus, a larger than average neutral distribution occurs in a region in which the ion distribution is initially smaller than average. Consequently, for the same addition of reactant gas, the attenuation of ions reaching the sampling port of the mass spectrometer is relatively less for the extended coaxial inlet than for the nonperturbing point source. For the longer reaction lengths, this effect has damped out.

G. CONCLUSIONS CONCERNING THE ANALYSIS The reader should be cautioned that the correction factors of the preceding parts are not universal correction factors, but apply to the specific conditions of P o , a, I, D, , D , , and v,, which were assumed. No general expressions have been derived which could be applied to every system. To show the reader the net effect of these calculations Fig. 13 is presented

FIG.13. Relative decrease in the reactant ion signal as a function of Qkl for four cases. The realistic model contains all the corrections of Section 111. For these two examples a fundamental mode initial ion distribution compatible with the characteristic velocity profile and a point source introduction of reactant neutral were assumed.

and shows the relative decrease in the reactant ion signal as a function of Qkl for four cases. In each of these curves uo = 8 x lo3 cmlsec, Po = 390 Torr, a = 4 cm. The simple model given by (8) is linear in this semilog plot. For the simple model it is not necessary to specify the reaction length since in the absence of coupling between reaction and diffusion the curves for all reaction lengths are identical The curve for the parabolic velocity profile

FLOWING AFTERGLOW MEASUREMENTS

33

model computed from (16) appears to be linear in this plot; nevertheless, there is a slight curvature due to the coupling of reaction and diffusion, as discussed in Section 111,C. This curve was calculated assuming D,P = 900 cm2 Tom/ sec and 1 = 40 cm. Because of the coupling of reaction and diffusion, the curves for other reaction lengths would be different. However, the difference would be extremely slight. Finally two curves, labeled the realistic models, were calculated for reaction lengths of 20 and 40 cm and included all of the effects mentioned in Sections II1,B-E. These curves were calculated for the point source inlet condition of Section III,F and for D,P = 570 cm2 Torr/ sec. Neither of these two curves is linear, although the curvature for the 20-cm reaction length is much greater. The shorter reaction zone lengths place a much more severe requirement on the accurate specification of both ion and neutral initial conditions since for such short lengths diffusion averaging has not taken place. In addition, for shorter reaction lengths the axial diffusion correction becomes so large that the approach used here to solve the transport equation is no longer valid. Because of these difficulties, the ESSA reaction rate measurements have never been made with I less than 40 cm. The reaction rate constant for the simple model is interpreted from the slope of the line. Due to the curvature obtained for the realistic model, no unique slope exists. Consequently, there is no straightforward method of obtaining a rate constant, as in the simple model. For large values of Q k and in the absence of axial diffusion the curves for the realistic model will parallel the curve for the model of Section III,C (parabolic velocity profile). As a result of the calculations it may be concluded that the two curves for the simple model and the parabolic velocity profile model represent the limits between which all practical cases lie, i.e., 15 35 cm and R 5 5 (a x 4 cm). Finally, a point which should be reemphasized is the fact that the initial reactant ion distribution at the reactant neutral inlet port is not precisely known and hence cannot be incorporated into a model. However, the calculations in Section III,F show that, for reasonable reaction zone lengths (1 > 40 cm), the error thereby introduced is small (a few percent, at most). H. TESTOF THE ASSUMPTIONS To test the correctness of the model for the hydrodynamics and diffusion, the following analysis was made and was compared to an appropriate experiment. The analysis was used to calculate the behavior of a diffusing ion sheet in the afterglow tube. A thin sheet of ions was generated at z = 0 and t = 0 by a pulsed beam of electrons incident perpendicular to the axis of the tube. The pulse width was characteristically 100 psec. For ions other than those of helium a suitable source gas was added with the flowing buffer. Thus, the

34

E. E.Ferguson, F. C.Fehsenfeld, and A. L.Schmeltekopf

diffusive properties of various ions, negative as well as positive, were tested in helium. The diffusing ion sheet was carried downstream by the flowing gas. The sampling port of a mass spectrometer was located on the axis downstream. The detector was gated with a width of 100 psec; thus, an ion arrivaltime spectrum could be recorded. In the analysis (Jarvis, 1968) it is assumed that the ions obey the transport equation XA'1 at

XA'1 + u(r) aZ =

DA

V2[A']

- k[A+][B],

and it is assumed that the distribution of B is uniform. Equation (27) is solved using integral transforms. Detailed calculations have been made only for uniform radial initial ion distributions, although other initial distributions can be evaluated as easily. The reader will note that (27) does not include the axial pressure gradient. However, the p correction calculated in Section II1,E can be directly applied. Figure 14 shows a comparison of an experimental arrival-time spectrum

FIG. 14. Plot of an experimental arrival-time spectrum (solid points) and a calculated spectrum (open points) for a flow length of 113 cm. The normalized He+ intensity is plotted against arrival time.

ARRIVAL TIME (m sec)

(solid points) and the calculated spectrum (open points) for the values DA = 2400 cm*/sec, uo = 8 x lo3 cm/sec, and flow distance of 113 cm. The agreement shown was typical for all gases and flow conditions tested. Thus, it is felt that diffusion and hydrodynamics have been handled correctly.

FLOWING AFTERGLOW MEASUREMENTS

35

IV. Data Reduction

This section expresses the mechanics with which a reaction rate constant is determined from raw experimental data using the analysis of Section 111. An on-line computer facilitates this reduction, as well as quick and accurate data acquisition. In a typical reaction rate study the fixed parameters are entered: the viscosity of the buffer gas, the viscosity and flow range of the reactant ion parent gas, the viscosity and flow range of the reactant gas, the calibration factors of the various flow elements, the downstream pressure, the length of the reaction region, the diameter of the flow tube, and the system temperature. During a typical run, the flow rate of the neutral reactant is varied and the count rates of reactant, product, and impurity ions are recorded. In addition to these variables, the flow rates of the reactant ion parent gas and the buffer gas, although nominally fixed during the typical run, are recorded because of the sensitivity of the results to their variations. Additional digital channels are available for recording such parameters as optical absorption, optical emission, and special temperatures and pressures, which are needed in certain measurements. The outputs of all digital channels (see Fig. 3) are coded and sent into a teletypewriter, which makes both a typed copy of the data and a punched paper tape. Upon completion of a run, the latter is used to feed the data into the computer for reduction. The computer automatically converts the raw input data into the appropriate physical parameters. The rate of addition of reactant and source gas, and the flow of the buffer gas are computed in standard units. The gas temperature and pressure, together with the computed buffer gas flow, are used to obtain the average flow velocity. Moreover, certain corrections to the detected ion signal can also be made. These corrections include removal of background counts and compensation for drift and changes in the diffusion losses of ions. The background count is not associated with the ion signal. The subtraction of these extraneous pulses (generally from 0.1 to 1 count/sec for positive ions and from 1 to 10 counts/ sec for negative ions) is only important for small ion signals. To make the correction for drift, the reactant ion current that exists before the introduction of the reactant gas is used as a reference, and during the course of a run the reactant gas will be occasionally removed to check for drift. If changes of 5 % have occurred, the run is aborted and the source of the drift is found and eliminated. For small changes, a linear interpolation of the signal is made. A change in the diffusion loss may occur in certain slow two-body reactions (k 2 cm3/sec) or in three-body reactions because relatively large quantities of reactant gas are required to produce a significant change in the reactant ion signal. Under these circumstances, the assumption that the

36

E. E. Ferguson, F. C, Fehsenfeld, and A . L. Schmeltekopf

reactant ions are diffusing only in helium may not be valid. This produces data from which the deduced reaction rate constant is too small. In certain cases, it is possible to circumvent this problem by monitoring the behavior of an ion which is nonreactive and whose intensity then reflects the perturbation of the ion transport properties by the added gas. The computer now has the reactant ion signal (corrected as described above, if necessary) as a function of the rate of addition of the reactant neutral (particles per second). Also available to the computer is a curve like that in Fig. 13 (realistic model), which predicts the decline in the reactant ion signal as a function of Qk (particles cm3/sec2). There are several ways to obtain the reaction rate constant from these two curves. The simplest and most frequently used is the following. First, a specific decline in the reactant ion signal is chosen (in practice, a decade) and the measured Q associated with this decline is noted. Then, for the same decline, the value of Qk is obtained from the calculated curve. The reaction rate k follows. The dependence of k on the extent of the specified decline can, of course, be tested easily. For an ideal case, this simple method fully utilizes all of the analysis of Section 111. However, experimental data is often less than ideal, e.g., fluctuations in reactant flow (particularly for unstable neutrals) and reactant ion signal (especially for low signal levels). Consequently, it is often difficult to determine accurately the Q at which the specific decline of the ion signal occurs. Thus, it is useful to have a more general method for determining the reaction rate constant. Perhaps the best such method is to let the computer select the value of k which gives the best fit of the analytical curve (Section 111) to the experimental data. In addition to the possibility of a more accurate value of k , this method allows easy comparison of the curvatures of the experimental data and analytical curve, thereby providing a check for possible downstream reactant ion sources, etc. V. Production of Reactant Species A. REACTANT ION PRODUCTION 1. Electron Impact The most straightforward method of ion production is direct ionization of atoms or molecules by electron impact or discharge source. This has the advantage that the ions are produced in a localized region well removed from the reaction zone. For some purposes, this has the relative disadvantage that large flows of reactant ion parent gas (- 10 atm cm3/sec) are required because

FLOWING AFTERGLOW MEASUREMENTS

37

of relatively low values of electron impact ionization cross sections. For example, 0' and 0,' can be produced by electron impact on O,, or 0' can be produced by electron impact on CO or other gases, if 0,' would be a confusing ion to have present during 0' reaction studies. In a similar manner, many negative ions can be produced by dissociative attachment, e.g., NH3+e+NHz- +H, (28) H2 + e+ . H- +H,

(29)

or dissociative ionization, COz + e - + O - +CO+.

(30) Although electron impact is the most straightforward method of ion production, the following techniques are most commonly used. 2. Secondary Reactions A substantial gain in efficiency can be achieved when reactions between the reactant ion parent gas and primary excited or ionized species of the buffer gas can be used to produce the ions of interest. Examples are, He(2w

+ Nz

+.

+ + He

N2+ e

He++CO+C+ +O+He Ar++02+02++Ar.

Penning reactions, such as (31), can be used to produce the parent ions of most gases. On the other hand, helium ions dissociatively ionize most small molecules as in (32): N + from N, and NO, 0' from 0, , C' from CO, etc. Argon is sometimes used as a buffer gas and (33) produces 0,' without O', when that is desirable. With fast reactions [such as (31)-(33) where k 2 lo-'' cm3/sec], only small quantities of the reactant ion parent gas are required, e.g., 0.1 to 1 atm cm3/sec. It is essential that sufficient quantities be added, however, to assure that reactant ion production is completed before the gas flow enters the reaction zone. It is usually possible to ascertain experimentally that this condition is satisfied. When ions are produced in the reaction zone, they do not of course have the full reaction time which is assumed in the data analysis, as discussed in Section 11. Negative reactant ions can be produced in several ways. For example, the most convenient method for producing NO- has involved the addition of N,O through the discharge with the helium buffer gas. The reaction sequence is

+

+

N 2 0 e +. 0- N2

(34)

0-+NzO+NO- +NO.

(35)

followed by

38

E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf

Alternatively, NO- can be formed by direct electron attachment, NO + e

+ M -+NO-+ M,

(36 ) as can 02-and most negative ions of stable neutrals. Negative ions can also be formed by charge transfer, e.g. and When substantial uncertainties exist, such as the location of ion formation or its vibrational state, it is useful to produce the ion several ways and compare rate constants so obtained. 3. Metal Zon Production

The flowing afterglow system has proven to be useful for reaction studies involving metal ions. The ions Fe', Mg", Ca', K",and Na" have all been produced and their two-body reactions with O3 and three-body reactions with O2 studied. These are the first rate constant measurements of metal ion reactions at thermal energies. The metal ions are produced by vaporizing the corresponding metals into the buffer gas stream from a small, electrically heated, furnace placed in the afterglow tube (Ferguson and Fehsenfeld, 1968). The buffer gas was argon (rather than the usual helium). The argon was bombarded by 14 V electrons producing argon metastable atoms. These, in turn, were used to Penning ionize the metals, i.e., N

ArM+M+Ar+M++e.

(39)

In order to produce MgO" for reaction studies, the reaction

+ Mg

+0 +

MgO+

(4W +Mg+ 0 3 (40b) was utilized. About one-fifth of (40) went into channel a. In a like manner, Si' and SiO' have been made. 02+

-+

4. Reactant Zon States

It was shown quite early in the flowing afterglow studies that certain atomic ions, e.g., O', could safely be assumed to be in their ground electronic states. This is because the time for superelastic electron collision deexcitation (i.e., O"*+ e + 0"+ e + KE) can be made short compared to the flow time between ion formation and ion reaction (Fehsenfeld et al., 1965b). While theoretical deexcitation cross sections are available only in special cases (such

FLOWING AFTERGLOW MEASUREMENTS

39

as O+*), the magnitudes of these cross sections are not expected to ever be so low that this experimental situation cannot be achieved. With the electron impact ion source, this point can be checked experimentally by varying the emission current (and thereby the electron density) and ascertaining that measured rate constants are independent of electron density. This has been ' + N, + NO+ + N. done for certain critical reactions, for example 0 In the case of molecular ions, the question of vibrational state distribution arises, since collisions cannot be depended upon to be sufficiently effective to cause rapid vibrational relaxation. In the case of N,', it has been possible to determine that the reactant ion is in its ground electronic and ground vibrational states. This is due to the fact that in the flow tube N,' ions are produced principally in a radiating state by the Penning process, He(2 3S)+ N, + N2+B'Xu+ + e + He(1 '9. The radiation B 'Xu+ +X 2Xg8+ (first negative system) has been observed spectroscopically and analysis shows that the resulting ions largely radiate into the ground vibrational and electronic states before reaching the reaction zone. (Spectroscopic studies will be discussed further in Section VI.) This situation is of course not general and ordinarily one cannot determine the vibrational distribution spectroscopically. (N,' has been studied more extensively in the ESSA system because of its importance in the ion chemistry of the Earth's atmosphere.) The FranckCondon factors for N, are such that electron impact ionization would also yield ground state N,'. Clearly, it is possible to produce ground state molecular ions alone by electron impact by keeping the ionizing electron energy sufficientlylow. In practice, however, this may run into severe intensity problems. In the case of molecular negative ions, the question of reactant ion states is more severe. Most molecular negative ions do not have metastable electronically excited states, so that the electronic state will not often be a question; however, the vibrational state distribution has so far been unknown. For none of the negative ion production processes is the vibrational distribution of the resulting ion well known. Even if it were, unless the ion is formed in the ground state, the unknown extent of vibrational deexcitation between the time of formation and reaction would be a complication. There are several possible procedures for studying effects of the vibrational state of the ion on reaction rate constants. For example, in some cases reactant ions can be produced several different ways. This has been done for the reaction 0,- + 0 + 0, e, which was found to be independent of the manner of 0,- formation. Such a finding strongly implies either that the ions are largely ground state in both cases, or that the reaction rate is relatively insensitive to the vibrational state of the negative ion. In either case, one has a reasonably confident knowledge of the ground state ion reaction rate constant.

+

40

E. E. Ferguson, F. C. Fehsenfeeld, and A . L. Schmeltekopf

Alternatively (or additionally) one could vary the flow tube length or gas pressure in such a way as to increase the negative ion lifetime between formation and reaction. This would cause more vibrational relaxation, unless the vibrational relaxation efficiency were unexpectedly low for the negative molecular ion (i.e., -g for reasonably attainable circumstances). The corresponding variation (or lack of variation) in a particular rate constant would then suggest dependence (or independence) on mode of ion formation. B. NEUTRAL REACTANT PRODUCTION

I . Neutral Reactant States As mentioned earlier, the neutral reactant is added downstream in the flowing afterglow system; therefore, it is not subject to excitation or dissociation. As a consequence, the neutral reactant is known to be in a Boltzmann distribution of electronic, vibrational, and rotational states at the tube wall temperature. This situation is not generally true of other experimental methods. The flowing afterglow system is not plagued by the confusion of concurrent reactions, such as

which occur in mass spectrometer ion source reaction rate studies and are difficult to unravel. By the same token, charge-transfer reactions are easily studied, e.g., co2++ 0 2 + 0 2 + + coz. (43) In conventional methods, such as mass spectrometer ion sources or stationary afterglows in which both the 0, and CO, are subject to the same ionizing conditions, the lower ionization potential of 0, would cause it to be selectively ionized. Therefore, most of the observed 0,' would be due to primary ionization rather than the charge-transfer reaction (43). For this reason very few thermal energy charge-transfer rate constants had been determined by conventional methods prior to the flowing afterglow development. Also, it has been possible to measure several ion reactions with excited state neutral reactants. For example, rate constants for the ions He' and 0' have been measured in reaction with N,, where the N, vibrational temperature is varied from 300 to 6000°K (Schmeltekopf el a/., 1968). The neutral N, is vibrationally excited up to 6000°K by a microwave discharge in a quartz side tube through which the N, is added. The vibrational temperature is controlled by varying the distance between the microwave cavity and the exit end of the tube. The resulting vibrational temperature is measured

41

FLOWING AFTERGLOW MEASUREMENTS

spectroscopically as described below in Section VI and in detail in the paper of Schmeltekopf et al. (1968). (The 0 ' + N,(T,) + NO' + N result is shown in Fig. 15.) Other ions could similarly be studied in reaction with N, and

1

1 1000

I

I

I

2000

3000 Tv

I 4000

I

I SO00

I

I 6000

(OK)

FIG. 15. Rate constant for the reaction O+ nitrogen vibrational temperature.

+ Nz

-+ NO+

+ N as a function of the

E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf

42

it is expected that similar measurements could be made on other vibrationally excited molecules. Recently, negative ion reactions with electronically excited 0, molecules have been carried out. Specifically the reactions and have been measured (Fehsenfeld et al., 1969). A microwave discharge in oxygen is known to produce up to 20% 02('A8), more or less independently of small variations in certain impurities. A discharge also produces a somewhat lesser concentration of atomic oxygen, which must be removed for the purposes of studying reactions (44) and (49, since 0,- 0 + O3 + e and 0- 0 40, e have large rate constants (Fehsenfeld et al., 1966). This is accomplished by having a glass wool plug coated with mercury oxide in the oxygen tube after the microwave discharge. It is known that the mercury oxide surface destroys atomic oxygen without destroying 02('Ag) molecules (Elias et al., 1959). The O,('AJ concentrations are determined by measuring the intensity of the O,('A$ -+ 02(3q,-) transition at 1.27 p with a PbS detector. This technique is limited to ions which do not react rapidly with ground state 0,, since the concentration of ground state 0,greatly exceeds that of 02('A8). Reactions (44) and (45) are endothermic for ground state 0, in place of 02('Ag).

+

+

+

2. Chemically Unstable Neutrals Another of the marked advantages of the flowing afterglow system is the thus-far unique capability for measuring rate constants for reactions of ions with chemically unstable species at thermal energy. Atoms such as N, 0, and H, and radicals such as OH, and molecules like O3and NO,, which pose some problems in measurement and handling, have been studied. Such measurements have been especially useful for the understanding of certain geophysical and astrophysical situations where such reactions occur in nature. Nitrogen atoms are produced by the discharge of N, and are measured by the titration reaction N + N O + N, + 0 so that the quantitative measurement of NO flow yields the quantitative N flow. The reactions with N atoms so produced are necessarily carried out in the presence of a large excess of N, ([N,] 102[N]) so that only ions which do not effectively react with N, can be studied. An example is the reaction 0,' N + NO' + 0. It is easy to demonstrate (simply by not discharging and dissociating the N,) that the NO does not occur at an observexothermic reaction 02'+ N, 4NO' able rate. N

+

+

43

FLOWING AFTERGLOW MEASUREMENTS

The same NO titration scheme can be used to generate a known concentration of 0 atoms, i.e., titration of the N + NO + N, + 0 reaction to the end point where the 0 atom production then equals the measured NO addition. Again the reaction is carried out in an excess of N, , so that only reactions of ions not reacting with N, can be measured. An important example is N2++ 0 + NO' + N, the main N,+ loss process in the earth's ionosphere. The symmetric charge transfer N2' N, + N, Nb+ has no observable effect in the afterglow system. The H atom production is by a microwave discharge of H, and measurement is by the titration reaction H + NO, + OH NO. The reaction H - + H 4 H, + e has been measured in this way (Schmeltekopf et ul., 1967). This reaction is of interest since it is the controlling loss process for H- ions in the solar photosphere; H- controls the sun's opacity in a wide region of the visible and ultraviolet spectrum (Branscomb, 1968). This reaction also has the only rigorously calculable rate constant of any ion-molecule reaction, and it is gratifying to note that experiment and theory are in agreement here. In this method excess H2 is present so that only ions which do not readily react with H2 can be studied. This eliminates such interesting reactions as 0 ' + H + H + + 0 from consideration, since 0 ' H, +OH+ H has a rate constant of 2 x lo-' cm3/sec at 300°K (Fehsenfeld et al., 1967). The H + NO2 + OH NO titration has been used to produce OH as a neutral reactant for certain ion reactions (Ferguson et ul., 1969a). Ozone plays a key role in the earth's negative ion chemistry and metal ion (meteor ion) chemistry and both metal and negative ions have been reacted with ozone in the ESSA flowing system afterglow. Ozone is produced in a commercial ozonizer which discharges oxygen to produce up to 5 % ozone in the oxygen stream. In the metal ion studies molecular oxygen does not interfere, since the metal ions studied do not react exothermically with oxygen in binary reactions. The same procedure can be applied for many negative ion reactions as well, since molecular oxygen does not often interfere. For example, a very important process for controlling the earth's negative ion chemistry is

+

+

+

+

+

+

-

0 2 - +03+0,-

+ 0 2

(46)

which has been measured in the ESSA flow tube and for which 0,would not interfere. It is generally true in problems of nature that for ions which can , the react with 0,,there will be little interest in their reactions with 0 3 since O3 to 0,ratio in nature is ordinarily small. The ozone-oxygen mixture was fed from the ozonizer, through an optical absorption cell, and thence into the reaction zone of the flow tube. The ozone flow was measured by determining the oxygen flow rate, cell pressure, and ozone concentration, the latter by absorption of the 2537 A Hg line in a double-beam absorption device.

44

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

The addition of O2 can be avoided when it is necessary or desirable to study 0, reactions without 0, present, and our earliest ozone studies were carried out this way, The ozonizer effluent was carried through a U-shaped tube filled with silica gel which was maintained at dry ice and acetone temperature, The ozone was trapped on the silica gel where it was readily visible due to its bright blue color. Oxygen is not appreciably trapped on the silica gel and it was pumped off. When the ozone was to be used for reaction studies it was eluded off of the silica gel by an inert carrier gas; in our case helium was used. The direct use of the oxygen-ozone mixture from the ozonizer is not only a great deal more convenient and time saving, but also somewhat safer than storing of ozone in the laboratory. As is well known, ozone is quite corrosive to some substances. We found it necessary to replace rubber 0 rings (or their fragmentary remains) in our pump lines with synthetic Viton 0 rings, for this reason.

c. ROLEOF IMPURITIES The possible role of impurities in confusing or falsifying flowing afterglow data merits discussion here. It is perhaps surprising to many members of the current generation of experimentalists in this field, schooled in ultrahigh vacuum technology, that impurity problems associated with such large gas flows are not completely ruinous. Why this is indeed not the case is discussed below. There are two ways that impurities can be deleterious to a rate constant measurement in the flowing afterglow technique. The first concerns reactant ion production, while the second directly enters the rate constant measurement. With regard to reactant ion production, the relevant impurities are those which are introduced with the buffer gas and the reactant ion parent gas and those released from the walls of the flow tube. It is apparent that these impurities can produce a problem if their concentration is sufficient to seriously deplete the reactant ions before addition of the neutral reactant gas. As an example, with the helium buffer sufficiently contaminated by nitrogen, the helium ions would be destroyed before reaching the mass spectrometer sampling port. A reactant ion source is not considered satisfactory unless the reactant ion is the dominant ion in the afterglow. This is generally not a problem because sufficiently pure buffer gases and reactant ion parent gases can be obtained in the required bulk quantities. Furthermore, the impurities from the polished, stainless steel tube walls are relatively quickly purged to tolerable levels by the flowing gases. Another problem which may arise from these impurities is the production of an impurity ion with the same mass as the reactant ion. Fortunately, this has not proven to be a problem so far.

FLOWING AFTERGLOW MEASUREMENTS

45

By far the most serious impurity problem is in the case where impurities are added with the neutral reactant gas. Such impurities, if present in a constant mixing ratio in the neutral reactant gas, could lead to a systematic decrease in the reactant ion signal which would mimic a reaction with the added neutral reactant. Thus, this impurity source has the potential of leading to error in the measured rate constant. The difficulties posed by these impurities are highly dependent on the reaction involved. For fast reactions, the problem of purity is not serious. For reactions which occur on nearly every collision (as many ion reactions do) a 1 % impurity in the neutral reactant could at most lead to a 1 % error in rate constant. For extremely slow reactions, the impurity problem can however be quite serious. The ability to measure an extremely slow reaction depends, obviously, on the limit of purification of the neutral reactant gas and this property of gases varies widely. It also depends on the nature of the impurity. If a trace impurity reacts with the ion but produces a clearly distinguishable product ion, as would usually be the case, then the interference may possibly be allowed for. As an illustration of the measurement of small rate constants, it has been possible to establish that several exothermic reactions have rate constants less than IO-" cm3/sec, e.g., Oz+ + N z + N O + + N O

(47)

and 0s-

+ Nz +NOz- + NO.

(48) This is a demonstrated dynamic range of lo6 for measured rate constants in the flowing afterglow system, which exceeds that of any other experimental technique so far for rate constant measurements. However, the ability to cover this wide range is not general. A positive result on (47) could have been accepted only if the concentration of NO in the N, was less than 1 ppm because NO reacts rapidly with 0,' to produce NO'. Aside from the specific disadvantages noted above, impurities may be simply a nuisance or, on the other hand, may be quite helpful. An example of the former is the general contamination of commercial sodium with potassium so that the product ion NaO' of the reaction

(49) is masked by the K + impurity of the same mass. An advantage of a nonreactive impurity has been mentioned earlier. An example was C1- in negative ion reaction studies. A final point to make in connection with the role of impurities in the flowing afterglow system is that a mass scan of the ions is always carried out before and after neutral reactant addition as a part of the experiment. Thus, difficulties associated with impurities are not likely to go undetected and lead Na+ + 0 3 + N a O + +02

46

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

to error. We know of no case in which an impurity has caused an erroneous rate constant to be deduced. There have been a few instances where the nature of the product ion was obscured, but this of course can also happen by virtue of secondary reactions in completely pure gases. VI. Optical Spectroscopic Studies The flowing afterglow system is rather well suited to spectroscopic investigations since an ion-molecule reaction can be caused to take place in a fairly large volume, fixed in time, in front of a spectrometer slit. Spectroscopic studies are necessarily much more complicated than ion-molecule reaction rate measurements but of course yield much more detailed information about reaction processes, in favorable circumstances. As mentioned earlier, one practical utilization of spectroscopy has been the determination of the state of the reactant ion. An example is the production of N,+ by He (2 3S)+ N, + N 2 + B+ e + He, followed by N2+B-+ N 2 + X hv (first negative system). Absolute intensity measurements of the first negative system show that most of the N2+ ions are produced in the B state. Study of the vibrational distribution shows that most of the product N2+X ground state ions are in the v = 0 vibrational level (Schmeltekopf et al., 1968). Hence N,+ reactions, using the N2+ so produced, can be safely assumed to be largely reactions of N,+X 'Zg+, u = 0. Nitrogen can be vibrationally excited by a microwave discharge and its temperature measured spectroscopically. The vibrationally excited N, can be ionized into the N2+B state by either high energy electrons or He(23S), both of which give vertical transitions. Observations of the relative intensities of the bands of the subsequent N,+B -+ N2+Xtransition, the transition probabilities of these bands, and the Franck-Condon factors connecting N,X to N,+B yield the vibrational temperature of the N, (Schmeltekopf et al., 1968). One particularly interesting spectroscopic result obtained so far (Albritton et al., 1968) involved simultaneous study of the second negative system (N,+C2C,+ -+ N,+X2Cg+)and the N+/N2+ ratio produced in the reaction He+ + N, --t N,+C 'Xu+ He, followed by either N 2 + C+ N,+X + hv or N,+C --t N + N + . By making this study as a function of N, vibrational temperature, it was possible to determine in which vibrational level of the C state predissociation occurs and to determine the ratio of predissociative to radiative lifetimes for this vibrational level.

+

+

VII. Temperaturevariable Flowing Afterglow Studies Recently a flowing afterglow system has been constructed in which the gas temperature is variable from 82 to 600°K.This allows the determination of

FLOWING AFTERGLOW MEASUREMENTS

47

reaction rate constants as a function of temperature (Dunkin et al., 1968). This information is of interest in ionospheric physics and also adds additional dataupon which thermal energy reaction rate theory can build. Also three-body reactions, whose rate constants are greatly enhanced at low temperatures, have been studied as a function of temperature. The marked tendency of ions to cluster at low temperature has led to the observation of many previously unobserved ion clusters in this system, at -82"K,e.g., He3+, Ar3+, N5+, N6+,N 7 + , N8+,N9+,Ar202+,Ar04+, etc. A schematic diagram of this apparatus is shown in Fig. 16.This stainless Downstream Pressure Measurement Line

&Copper

Heat Shields

Stainless Steel TLbe with Copper Heal Sink

Ceramic Insulators

4" Diff. PU-mP

6" Diff. Pump Gas Lines: Healer, Thermocouple. . and Filament Leads

FIG.16. Temperature-variableflowing afterglow apparatus.

steel reaction tube is also approximately 8 cm inside diameter and 100 cm long. The reaction tube is surrounded by, and is, in intimate contact with a copper heat sink on which are mounted heaters, cooling lines, gas lines, and thermocouples. A thin polished copper heat shield is wrapped over the tube and heat sink. Polished copper heat shields are also mounted from the outside wall of.the insulating vacuum jacket. This vacuum jacket is pumped by a trapped 4 in. oil diffusion pump. The tube temperature is measured with three chromel-alumel thermocouples which are placed near the ion source, the reactant inlet, and the mass spectrometer sampling port. Each thermocouple goes to a separate proportional servo temperature controller which controls heaters in its respective region to eliminate temperature gradients which would otherwise occur due to nonuniform heat losses. Figure 17 shows the ions present in a relatively clean helium afterglow at 80°K.This is the first mass spectrometric observation of He3+ whose existence was inferred earlier by Patterson (1968). Figure 18 shows the same afterglow ions, sampled with increasing drawout potential on the sampling

E.E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

48

+/+-+

Mass 16

Mass 28

'O3I

L

lo*o

80

160

240

L

lo*o

1

2

3

4

5

6

7

NOSE CONE VOLTAGE (volts)

HELIUM FLOW (atm d s e d

FIG.17. Variation of ion signals present in a rrlatively clean helium afterglow at 80°K as a function of the rate of helium flow. The experiment was performed at maximum pumping speed. Consequently this is essentially a plot of the ions as a function of pressure from 0.17 to 0.42 Torr.

FIG.18. Variation of ion signals present in a relatively clean helium afterglow at 80°K as a function of the sampling port potential. The experiment was performed at a helium flow of about 90 atm cm3/sec. The decline of the He3+ signal with the increase in potential is taken as an indication of the weak binding of the He3 cluster. +

port. The relative decrease of the He,' signal shows that it is only weakly bound. This ion did not exist in measurable concentrations at 300°K. The existence of many cluster ions at low temperature is due in part to the increased three-body rate constants for ion association at these temperatures. This has the consequence that such three-body reactions are readily measured in the flowing afterglow and indeed as a function of temperature. Table I1 gives some data of this kind recently obtained. Figure 19 shows the saturations of two ion association reactions (Bohme et af., 1968) in which the reactions N2+ + N 2 + H e + N 4 + + H e

(50)

and Oz+-t02fHe+04++He (51) become independent of helium pressure at 80'K. These are the first observations of ion association reactions carried out through the region from third order to second order kinetics. This data leads to valuable kinetic information

49

FLOWING AFTERGLOW MEASUREMENTS

TABLE I1 THREE-BODY RATECONSTANTS MEASURED IN THE ESSA FLOWING AFTERGLOW SYSTEM

Reaction N2++N2+He+N4+ +He N + + N Z + H e + N 3 + +He Oz++ 0 2 + H e + 0 4 + + H e O+ + N 2 + H e + N 2 0 + * + H e 0- COz +He + C 0 3 He Ar+ +Ar+He-,Arz+ + H e

+

+

+ + + + + +

+

0- Nz He + N 2 0 - + He Mg+ + 0 2 + A r + M g 0 2 + +Ar Fe+ O2 Ar + F e 0 2 + + Ar Ca+ 0 2 Ar + C a 0 2 + Ar

Temp. (“K)

Rate constant (cm6/sec)

82 280 82 280 82 82 280 82 290 82 300 300 300

1.2(-28) 1.9( -29) 7.2(-29) 8.6(-30) 3.1( -29) 5.4( -29) 1.5(-28) 1.6( - 30) -1.3(-31) 1.3(- 30) -2.5(-30) 1.0(-30) 6.6( -30) N N

Helium Pressure (Torr)

FIG.19. Plot of the apparent two-body rate constant versus helium pressure for two ion association reactions. The change from three-body kinetics to two-body kinetics shown is taken as an indication of the saturation of these reactions.

for example, the lifetimes of the initially formed Na* and 0 : . (Bohme et al., 1968). The “saturation” has not yet been observed at 300”K, presumably because N i * and 0: * lifetimes become much shorter at higher temperatures so that the saturation pressure is higher than we have so far been able to attain. Figure 20 shows the rate constants for several reactions as a function of temperature (Dunkin ef a/., 1968; Ferguson ef al., 1969b). The decrease in rate constant for O++Nz+NO+ + N (52)

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

50

-

fs

10'0

c

c

2 u)

FIG.20. Rate constants for several reactions as a function of temperature.

c

0

V

c 0)

16"

0

a

4

IO-~~~

200

400

Temperature

600

(OK)

with increasing temperature has been explained as being due to the shorter NzO+*intermediate complex lifetime at higher temperatures (Ferguson et al., 1969b). The products NO' and N provide the most exothermic channel for NzO'* decomposition, The longer the NzO'* lifetime, the more complete will be the mixing of internal vibrational modes, and mode mixing favors the exothermic decomposition channels because of their greater phase space volume. The temperature independence of the He' N2reaction rate constant from 300 to 600°K is consistent with other studies in which the He+ kinetic energy is varied as well as a reasonable (albeit naive) model for such fast reactions. Accordingly, it seems unlikely that systematic errors are introduced in the flow tube behavior or analysis due to the temperature variation. Also, the 0 ' + 0, -+ Oz+ + 0 measurements in Fig. 20 are in excellent agreement with measurements of Smith and Fouracre (1968) for the temperature range 185-567°K.

+

Vm. Some Miscellaneous Results One kind of negative ion reaction has no positive ion analog, namely, the process of associative detachment A-+B+AB+e (53)

FLOWING AFTERGLOW MEASUREMENTS

51

where either A or B may be diatomic or polyatomic. The measurement of such reactions raises a possibility of a new kind of experimental error which must be considered. The reaction (53) changes the electron density. Accordingly, during a reaction rate measurement which involves measuring the A- ion signal sampled from the weak afterglow plasma as a function of [ B ] , it is conceivable that the ion sampling efficiency might depend on electron density and therefore might also vary. Since the change in A- signal due to addition of B is interpreted as being due entirely to reaction (53), such an effect would obviously introduce error. In cases where (53) did not occur at all and the entire change in the A- current was due to sampling efficiency variations, the error could be a very gross one indeed. Ironically, the presence of certain impurity ion signals have demonstrated the absence of this error. It has been typical, especially with the early glass tubes, to find persistent impurity ion signals, particularly C1- and, when the glass had been cleaned with HF, F-. When these impurity ions cannot react exothermically with the reactant gas, the observed constancy of their signals demonstrates that no change in sampling occurs. For example, Fig. 21

-z P 0 V W

u)

\

u)

L

i

3

FIG.21. Ion current versus nitric oxide flow reaction 0-+ NO NO, + e. --f

8

v

+ z

W

a a

3

V

z

0

NO FLOW (RELATIVE SCALE)

illustrates the data for the reaction 0-+NO+N02+e.

(54)

Note that the C1- signal does not significantly vary, while the 0-signal drops by a factor of 600 due to (54).

52

E. E. Ferguson,F. C.Fehsenfeld,and A. L.Schmeltekopf

Another peculiarity of the associative-detachment reactions is that no detectable charged product results (in the experimental arrangements so far employed). Very often, as in (54), associative detachment is the only exothermic reaction possible and there is no ambiguity in the identity of the products. When two or more exothermic channels exist, e.g., 02-fH-tOH-fO

+Hot

+ e,

care must be taken in specifying the products. In this example, the presence or absence of OH- will in principle settle this question. However, the branching ratio cannot be accurately determined because the products of only one of the branches are detected. A further complication arises when these products can undergo further reactions. For example, if OH-

+ H-+H20+e

(56)

is fast, one might incorrectly infer that only (55b) had occurred. We have discussed the comparison of flowing afterglow rate constant measurements with other experimental techniques. This, of course, is desirable no matter how carefully the analysis of an experiment is done, because one should always be concerned that some relevant factor has been overlooked in the analysis. Alternatively, one can compare experiment and theory if cases exist where theory is well grounded. One ion-neutral reaction does appear to exist, the associative-detachment reaction H-

+ H +H, + e,

(5 7)

for which theory appears to be very secure and fairly accurate (Dalgarno and Browne, 1967). Our preliminary measurement of (57) (Schmeltekopf et af., 1967) agrees with the theoretical calculation. Unfortunately, a factor of 2 uncertainty was considered possible because of uncertainty in the measurement of the H atom concentration. This measurement will be eventually improved. IX. Summary

We have described the experimental and analytical techniques which have been developed for flowing afterglow applications to the quantitative study of ion-neutral reaction processes. The discussion has concentrated on the experience and results obtained in the authors’ laboratory in Boulder. Similar flowing afterglow tubes are either under construction, or are now being operated in other laboratories and it can be expected that the broad potential of this technique for supplying ion-neutral reaction rate data will be well exploited in the future.

FLOWING AFTERGLOW MEASUREMENTS

53

A flowing afterglow system in the laboratory of H. I. Schiff at York University in Toronto, Canada, has recently been used to measure a number of reactions of H3” with a variety of gases (Burt, 1968).Anotherflowingafterglow system has recently been constructed at the University of Pittsburgh and reaction rate constants obtained for the charge-transfer reaction of N,’, 0, +, and NO’ with sodium atoms (Farragher et al., 1969). These results are of great importance for meteor ion chemistry. The present article has concentrated on experimental and analytical methods rather than the application of results to the understanding of ionneutral chemistry. This emphasis was chosen because most of the results have been, or are being reported, in the literature; whereas, the rapidly developing methodology has not been adequately described previously. Tables I1 and 111 give a sampling of flowing afterglow rate constant results as an illustration of the method’s versatility. In order to complete this discussion with a proper perspective, we would like to very briefly summarize some of the broad trends in ion-neutral reaction behavior which have been discovered in the past few years, primarily as a consequence of the application of the flowing afterglow technique. It has been found on the basis of very many examples, with almost no exceptions, that exothermic charge-transfer from atomic or molecular ions to molecular neutrals at thermal energy is invariably fast. Most other techniques for the measurement of ion-molecule reaction rate constants are inherently unsuited for the study of an ion reacting with a neutral where the neutral has the lower ionization potential, and sufficient data for this generalization did not exist prior to the flowing afterglow results. Theoretical prediction, at least for the small molecules of aeronomical interest, was that thermal energy chargetransfer would often be slow. Several dozens of reactions of this type have been measured in the ESSA flowing afterglow system. Furthermore, the same generalization is true for negative ion charge-transfer reactions. This generalization is valid when no competing atom-rearrangement reactions occur and might be stated: “When charge transfer to a molecular neutral is exothermic at 300”K, a fast reaction occurs (fast meaning k > lo-” to lo-’’ cm3/sec).” Charge transfer reactions of negative ions have sometimes been useful in establishing relative electron affinities of molecules, which are often difficult to measure. Positive ion charge-transfer reactions have, on occasion, been useful in establishing relative ionization potentials of molecules, generally known or better measured in more direct ways. Associative-detachment reactions were first measured in the flowing afterglow system and most of the presently available data come from this source. This exploits the flowing afterglow capability of reacting unstable neutrals with negative ions, which is the most common situation in which associative detachment is exothermic. Certainly, reactions with unstable neutrals have

E. E. Ferguson,F. C.Fehsenfeld, and A. L. Schmeltekopf

54

TABLE 111 EXAMPLES OF BINARY ION-NEUTRAL REACTIONS MEASURED IN FLOWING AFTERGLOW SYSTEMS

Temp. ("K)

Reaction

+

O+ Nz 4NO+

+N

80 300 600 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300

O+ +NZ(v=O)+NO+ + N v=l v=2 v=3

Oz-

+ 0 + 0 3+ e

03-

f

0 2 -

+ + +

+

coz + co3- + 0 0 3 +o,- + oz

2

+ +

Fe+ 0, + FeO+ 0, MgO+ 0 + Mg+ 0 2 SiO+ O+Si+ 0, O z ++ N a + N a + + 0 2 NO+ + N a + N a + + N O Nz+ + N a + N a + + N z H3+ +CO+COH+ +Hz NOH+ + N O + N O + + N O H H3+ Hz0 + H30+ Hz Hez+ NZ+ N z + + H e Oz- Oz('A,) -+ 2O2 e 0- Oz('A,) -+ O3 e H- H +Hz e Oz+ Nz +NO+ + N O

+

+ +

+ + +

+

+

+ + +

600 300

a

Rate constant (cm3/sec) 1%-

12) 1 2 - 12)

Source a U

5.0( - 13)

U

12-12) 1.2(-12) 5.0( - 1 1 ) 12-10) 2.S( - 10) 4.0(- 10) 4.0(- 10) I .S( - 10) 1 .O( - 10) -2.0(-10) 6.7( - 10) 7.0( - I I ) 5.8( - 10) 1.4( -9) 7.0( 10) 2.9(- 9) -6.0(-10) -2.O(-10) 3.0( - 10) 13-9) < l.O(- 15) < 1 .O(- 15) < 1.O( - 15)

a a a a a

-

~

N

11

a U

a a b b b c c C

a a n U

a a

a

ESSA results.

' Farragher et al. (1969), University of Pittsburgh results. Burt (1968), York University results.

the most applied interest. Examples are Oz-

+ 0 + 0 3+ e

in aeronomy and perhaps gas discharges, and H-

+H+Hz +e

(59)

in astrophysics. Based on a little over a dozen cases so far, the above generalization can be extended to include associative-detachment reactions. Only two exceptions to this are known: 0- + Nz - + N z O+ e + 0.21 eV, k< cm3/sec (60)

FLOWING AFTERGLOW MEASUREMENTS

55

and OH-

+N

HNO

+ e + 2.4 eV,

k < lo-" cm3/sec.

(61) Qualitatively, this generalization is readily understandable in terms of a model involving rapid autodetachment of the negative ion complex formed in the collision (Ferguson, 1968). This has the implication that studies of associative-detachment reactions can be expected to lead to useful information on molecular negative ion potential curves, which are generally rather inaccessible to study. It has clearly been of value in understanding ion-neutral reactions, to simply have a great deal of reaction rate constant data available. The capability of providing a large amount of data (and the application of this data to a variety of applied problems) has probably been the major contribution of the flowing afterglow technique in these first few years of its application. It is apparent, however, that the more detailed studies now possible (e.g. temperature dependence of rate constants, dependence on vibrationally and electronically excited states, emission spectra of ion-molecule reactions, etc.) will yield an increasingly detailed picture of ion-neutral reaction mechanisms. In this way, the flowing afterglow technique should complement other methods such as crossed-beam studies which lead rather directly to reaction mechanism details. --f

ACKNOWLEDGMENTS The authors would like to express appreciation to their colleague Dr. D. L. Albritton for a very substantial contribution to the writing of this manuscript. Thanks are also due to Mrs. Alice Keysor for typing the manuscript. This work has been supported in part by the Defense Atomic Support Agency.

REFERENCES Albritton, D. L., Schrneltekopf, A. L., and Ferguson, E. E. (1968). Bull. Am. Phys. SOC.13, 212. (To be published.) Bass, A. M., and Broida, H. P. (1963). J . Res. Narl. Bur. Std. A 67, 319. Beaty, E. C., and Patterson, P. L. (1965). Phys. Rev. 137, A346. Bohrne, D. K., Dunkin, D. B., Fehsenfeld, F. C., and Ferguson, E. E. (1968). J. Chem. Phys. 49, 5201. Branscornb, L. M. (1968). Intern. Cont. Electron. At. Collisions, Sth, Leningrad, 1967, Joint Inst. Lab. Astrophys., Univ. of Colorado, Boulder, Colorado. Burt, J. A. (1968). Ph. D. Thesis, York Univ., Toronto, Canada. Dalgarno, A., and Browne, J. C. (1967). Astrophys. J. 149, 231.

56

E. E.Ferguson,F. C.Fehsenfeld,and A. L.Schmeltekopf

Dunkin, D. B., Fehsenfeld, F. C., Schmeltekopf, A. L., and Ferguson, E. E. (1968). J. Chem. Phys. 49, 1365. Dushman, S., and Lafferty, J. M. (1962). “Scientific Foundation of Vacuum Technique,” 2nd ed. Wiley, New York. Elias, L., Ogryzlo, E. A., and Schiff, H. I. (1959). Can. J . Chem. 37, 1680. Farragher, A. L., Peden, J. A., and Fite, W. L. (1969). J. Chem. Phys. 50,287. Fehsenfeld, F. C., Evenson, K. M., and Broida, H. P. (1965a). Rev. Sci. Znstr. 36,294. Fehsenfeld, F. C., Schmeltekopf, A. L., and Ferguson, E. E. (1965b). Planetary Space Sci. 13, 919. Fehsenfeld, F. C., Ferguson, E. E., and Schmeltekopf, A. L. (1966).J. Chem. Phys. 45,1844 Fehsenfeld, F. C., Schmeltekopf, A. L., and Ferguson, E. E. (1967). J. Chem. Phys. 46, 2802. Fehsenfeld, F. C., Albritton, D. L., Burt, J. A., and Schiff, H. I. (1969). Can. J. Chem. 47, No. 8. Ferguson, E. E. (1967). Reo. Geophys. 5, 305. Ferguson, E. E. (1968). Advan. Electron. and Electron Phys. 24,l. Ferguson, E. E. (1969). Can. J. Chem. 47, No. 8. Ferguson, E. E., and Fehsenfeld, F. C. (1968) J. Geophys. Res. 73, 6215. Ferguson, E. E., Fehsenfeld, F. C., Dunkin, D. B., Schmeltekopf, A. L., and Schiff, H. I. (1964). Planetary Space Sci. 12, 1169. Ferguson, E. E., Fehsenfeld, F. C., Goldan, P. D,, Schmeltekopf, A. L., and Schiff, H. I. (1965a). Planetary Space Sci. 13, 823. Ferguson, E. E., Fehsenfeld, F. C., Goldan, P. D., and Schmeltekopf, A. L. (1965b). J. Geophys. Res. 70, 4323. Ferguson, E. E., Fehsenfeld, F. C., and Schmeltekopf, A. L. (1969a). Adoun. Chem. Ser. 80, 83. Ferguson, E. E., Bohme, D. K., Fehsenfeld, F. C., and Dunkin, D. B. (1969b).J.Chem.Phys. (in press). Goldan, P. D., Schmeltekopf, A. L., Fehsenfeld, F. C., Schiff, H. I., and Ferguson, E. E. (1966). J. Chem. Phys. 44,4095. Jarvis, S. (1968). J. Res. Natl. Bur. Std. B 72, 135. Kaufman, F. (1964). Ann. Geophys. 20, 106. MacNair, D. (1967). Rev. Sci. Znstr. 38;124. McDaniel, E. W. (1964). “Collision Phenomena in Ionized Gases.” Wiley, New York. Moruzzi, J. L., Ekin, J. W., and Phelps, A. V. (1968). J. Chem. Phys. 48, 3070. Oskam, H. J. (1958). PhilZips Res. Repts. 13, 401. Patterson, P. L. (1968). J. Chem. Phys. 48, 3625. Sayers, J., and Smith, D. (1964). Discussions Furaday SOC. 37, 167. Smith, D., and Fouracre, R. A. (1968). Planetary Space Sci. 16, 243. Schiff, H. I. (1964). Ann. Geophys. 20, 115. Schmeltekopf, A. L., and Broida, H. P. (1963). J. Chem. Phys. 39, 1261. Schmeltekopf, A. L., Fehsenfeld, F. C., and Ferguson, E. E. (1967). Astrophys. J. 148, L155. Schmeltekopf, A. L., Ferguson, E. E., and Fehsenfeld, F. C. (1968). J. Chem. Phys. 48, 2966. Warneck, P. (1967). J. Geophys. Res. 72, 1651.

EXPERIMENTS WITH MERGING BEAMS RO Y H . N E YNABER Space Science Laboratory, General DynamicslConuair San Diego, California I. Introduction.. ................................................... 11. General Principles ................................................

111. Ion-Neutral Reactions ............................................ A. Symmetric-Resonance Charge Transfer .......................... B. Charge Rearrangement ........................................ IV. Ion-Ion Reactions ................................................ A. Symmetric-Resonance Charge Transfer .......................... B. Mutual Neutralization ......................................... C. Radiative Recombination. ...................................... V. Neutral-Neutral Reactions ........................................ A. Charge Transfer and Ionization ................................. B. Rearrangement.. ............................................. VI. Electron-Ion Reactions. .......................................... A. Theory ...................................................... B. Dissociative Recombination ................................... VII. Current or Very Recent Studies ................................... VIII. Concluding Remarks. ............................................ References .....................................................

57 59 62 62 72 80 80 83 87 89 90 ,100

.lo0 100 .lo1 .lo5 .lo6 .lo7

I. Introduction The merging beams (mb) technique was originally developed to study twobody collisions in the energy range from a few tenths to several electron volts (henceforth called the low energy region) because other experimental methods could not be used in this range. Typical problems with these other methods include low ion intensity because of space charge effects, poor energy resolution, and a low upper limit to the energy (i.e., about 0.3 eV) of neutrals generated from thermal sources. The mb technique consists of two beams traveling in the same direction along a common axis. Various other names that have been given to the method include superimposed beams, overtaking beams, and confluent beams. The laboratory energy of each beam is typically several kiloelectron volts with an energy spread of a few electron volts, whereas the relative energy of the beams in the c.m. system is in the low energy region. The energy spread 57

58

Roy H. Neynaber

at this relative energy is generally just a few percent. The use of high laboratory energies minimizes space charge difficulties. The method can also be used to study collisions at energies outside of the low energy region, e.g., at thermal energies and at several hundred electron volts. The mb technique is also of interest in the low energy region and above because it can be used as a method, with good energy resolution, for studying reactions between two, general labile species. In ion-neutral studies up to the present, crossed beams have been used where the neutral species is labile. This conventional method is advantageous when the neutral species can be obtained from a thermal source (e.g., 0 from an rf source or H from a high temperature oven) since beam fluxes from such a source are relatively large. However, a more comprehensive series of free radicals can be obtained by the more general technique of neutralizing the parent ion by charge transfer. For reactions involving many of these species, the mb technique may be the only method for measuring cross sections because larger signals (arising from the availability of more intense beams and longer interaction regions) can be obtained for a given interaction energy. Other merits of the mb technique include easy collection of products for total reaction cross section measurements, and relative ease in detecting all products of reaction. Davis and Barnes (1929a,b, 1931), Barnes (1930), and Webster (1930) merged beams of electrons and alpha particles many years ago. More recently, Cook and Ablow (1959, 1960) suggested the use of superimposed beams for the study of collisions between two heavy particles. The possibility of a very low interaction energy, coupled with a high energy resolution was first pointed out by Trujillo et al. (1963, 1965). The technique can be used to investigate ion-neutral, ion-ion, neutralneutral, and electron-ion reactions. Merging beams studies that have been conducted of such processes will be discussed. Since the method (employment of the concept of low relative energy coupled with high energy resolution is assumed) is in its infancy, the total number of such experiments is small. An earlier review of the subject has been written by the author (Neynaber, 1968). It will be noted, as the discussion proceeds, that an mb apparatus is relatively complicated. Its configuration is tied closely to the type of reaction for which it is to be applied, and this configuration is generally quite inflexible. Although some of the machines to be discussed have been used to study more than one type of process, it would be safe to say that none of them is a universal apparatus for the study of all the above types of reactions. The development of merging beams has ushered in the concept of the study of two-body collisions by the intersection of two beams at a small angle. This method is characterized by properties, as one might expect, that are in between those of crossed and merging beams. The method has been called inclined beams.

EXPERIMENTS WITH MERGING BEAMS

59

There will be no further discussion of inclined beams except to comment briefly on the contemplated work of two groups of scientists. M. F. Harrison and R. Rundel of the Atomic Energy Laboratory at Culham are constructing a machine in which two beams will intersect at 20”. They plan to study H + + H- + H + H at interaction energies of about 200 eV to 30 keV. An advantage of using inclined instead of merging beams for this experiment is that the parent ion of each product can be identified. However, for the same laboratory energy of the reactants, lower interaction energies could be achieved in an mb experiment. Harrison and Rundel also plan to study H + + H- + H(2S) + H and H C + H - + H + H + e using the inclined beams method, J. Schutten and colleagues of the FOM Institute for Atomic and Molecular Physics, Amsterdam, are also constructing an inclined beams machine in which the angle of intersection will be 10”. Included in their plans are experiments to measure cross sections for multiple ionization in neutralneutral, ion-neutral, and ion-ion collisions. The inclined beams method will also enable them to identify the parent ion of each product for all reactions studied. In addition, the mechanics of bringing about the interaction of two charged beams of the same polarity is easier for inclined than for mb systems.

+

11. General Principles

A low interaction energy (i.e., relative energy in c.m. coordinates) can be obtained with an mb system because the difference in beam velocities can be made small even though high energy beams are employed and because the difference of the laboratory energies of the beams is many times larger than the interaction energy. Analytical expressions are presented below. If we assume that all particles in the two beams move along parallel lines (ideal case), then the interaction energy W is

w = t p ( c , - u$,

(1)

where p is the reduced mass of the system, and u1 and u2 are the laboratory velocities of the particles in each beam. We may also express W a s follows:

where m, and m 2 are the masses, and El and E, are the laboratory energies of the particles in each beam. For simplicity, let m, = m, = m. The general conclusions reached for this case apply equally well when m, # m 2 .For equal masses, Eq. (2) becomes

60

Roy H. Neynaber

When the absolute value of the difference in beam energies, AE, is small compared to these energies and E is the average of the two beam energies, it can be shown that W w AE ' / 8 E .

(4)

Now define D = AE/ W. The term D is an energy deamplification factor. We have (again for small A E ) D = A E / W z 8 E / A E + 1.

(5)

Equation ( 5 ) indicates that the interaction energy is small compared to the difference of the laboratory energies of the particles for large laboratory energies. As an example, let El = 5000 eV and E, = 5100 eV. Then AE = 100 eV, and from Eq. (5), D w 400 and W w 0.25 eV. The concept of a deamplification factor can also be applied to the more realistic case of two beams each having an energy distribution rather than being monoenergetic. From Eqs. (4) and (5) it can be shown that a perturbation 6 E in AE gives approximately the following perturbation 6 Win W :

6 W/W x 2 6E/ W D =2 6E/AE.

(6)

Assuming a reasonable energy spread, i.e., full width at half-maximum, of 1.5 eV for each beam (6E = & 1.5 eV), we obtain an approximate spread in W of 6 W = 10.0075 eV, or an uncertainty in W of A 3 %. Therefore, because

of the deamplification factor for an mb experiment, good resolution for the interaction energy can be obtained. Similarly, small, random fluctuations in El or E, result in negligible perturbations in W. Equation (6)rigorously applies to rectangular energy distributions. If the distributions are Gaussian, the expression for 6 W/W should be divided by Therefore, for Gaussian distributions, Eq. (6) conservatively estimates the fractional perturbation in W . We have previously assumed that the particles in the two beams have parallel paths. In a realistic case, there is nonparallelism, and transverse velocities exist. This could result in an average energy of interaction Wfor the realistic case that is different from W , and could cause an uncertainty in the measured cross section. Thus it is necessary to take proper account of transverse velocity components, or to ensure that the largest of these is insignificant. The mathematical formalism for extracting a cross section from an mb measurement in which transverse velocities exist appears formidable (Cook and Ablow, 1959, 1960). To circumvent this task it seems desirable to eliminate particles with large transverse velocity components and to keep only those particles with sufficiently small transverse velocity components. The undesirable particles can be removed by passing the beams through

fi.

EXPERIMENTS WITH MERGING BEAMS

61

two collimating apertures. Small, residual transverse components exist after the passage of the beams through these apertures. For the case when the distance I between the collimating apertures is equal to the length of the interaction region (i.e., that region in which cross section measurements are made), when AE is small compared to the beam energies, and when m, = m2, we have derived the following expression for the upper bound of the fractional difference between Wand W : ( W - W ) /W c 0.54(Ed/AEl)2,

(7)

where W is the energy of interaction averaged over all possible angles of intersection of the beams and over all points in the interaction region, and d is the diameter of the collimating apertures. Applying Eq. (7) to the previous numerical example and choosing the values d = 0.25 cm and 1 = 25 cm, we obtain an upper bound on the percentage difference of approximately 13.5%. Similar studies have been made for the effect of the residual transverse components on the measured cross section. An energy dependence of the cross section must be assumed to make the computations. For a resonant charge transfer process at low energy the effect has been computed and, for the conditions above, the cross section for the realistic case is different from that of the ideal case by less than 6 % . As W increases, these percentage differences due to the residual transverse components become even smaller for a given E. An experiment might consist of determining the number of particles per second S which has been generated in the interaction region by the desired reaction. The expression relating the cross section Q in square centimeters per particle to S is (again assuming equal masses)

Q = (S/I)(EiEz/mV1",

(8)

where

s,

I = J1J 2 d x d y d z , and the integral is performed over the volume of the beams in the interaction region; z is along the axis of the beams in the interaction region; J1 and J, are the beam flux densities in particles per square centimeter per second for the respective particle energies El and E2 in ergs. Since the integrand is nonzero only where the beams overlap, I is called the overlap integral. Also, W is in ergs, and rn in grams. The mb technique should be useful for measuring total reaction cross sections even when the collisions are accompanied by appreciable c.m. scattering since the maximum solid angle for reaction products is much smaller than that for conventional experiments. This is because the kinetic

62

Roy H . Neynaber

energy in the center of mass is much smaller than the laboratory energy of the reactants. For example, with a kinetic energy in the center of mass of several electron volts and initial reactant energies of several kiloelectron volts, the products are confined to a cone whose vertex angle is a few degrees. Finally, the reaction products are easily detected by secondary electron emission because they have large laboratory kinetic energies.

III. Ion-Neutral Reactions A. SYMMETRIC-RESONANCE CHARGE TRANSFER 1. Double Source Experiment Symmetric-resonance charge transfer experiments in Ar have been conducted with a double source apparatus (Neynaber et al., 1967a,b). Relative cross-section measurements were made in a range of W from 0.1 to 20 eV. In addition, absolute cross section measurements were made for W = 0.3 and 100 eV. In Section III,A,2, similar experiments in H and D will be discussed in which only a single source was used. a. Description of Apparatus. The double source apparatus (Trujillo et al., 1966) was designed to study the Ar reaction by observing product ions. The process is Arc(El) Ar(Ez)--+Ar(EA +Art(.&) (9) where the E's represent the laboratory energies of the particles in a given interaction region. Outside of the interaction region, the laboratory energy of the particles in each primary beam was Eo = E, = 3000 eV. For E, > E l , as was the case, El = (3000 - A E ) eV, where AE was defined in Section 11. A schematic diagram of the apparatus is shown in Fig. 1. The sources were of a low-pressure, oscillating-electron-bombardment type (Carlston and Magnuson, 1962) which produced beams with a spread of about 1.5 eV. An Ar' (3000 eV) beam from source 1 was merged with a mechanically chopped Ar (3000 eV) beam. The Ar beam was obtained by passing Ar' from source 2 through a charge-transfer cell containing H, . It was assumed that negligible energy was lost in the near-resonant charge transfer process and that the energy and energy spread of the Ar beam emerging from the cell were the same as that of the Ar' beam entering the cell. An electric field between the condenser plates that follow the cell was used to remove Ar ' which did not undergo charge transfer. The superimposed beams were then collimated to eliminate large transverse velocities and were passed into a decelerating-accelerating system containing the interaction region. The energy difference between Ar and Ar' that was necessary to

+

MOVABLE DETECTOR APERTURE

FIG.1. Schematic diagram of douole source, mb apparatus for studying Ar+

+ Ar -+ Ar + Ar+. Apertures are not to the scale shown.

64

Roy H . Neynaber

obtain the desired interaction energy W was established by raising the potential of the interaction region by an appropriate amount AE/e, where e is the magnitude of the electronic charge. The primary ion beam was thereby decelerated to (3000 - A E ) eV at the entrance to the interaction region and accelerated to 3000 eV at its exit. A schematic diagram of the apparatus for achieving this is shown in Fig. 2. Argon ions produced by the desired process GRID ,4

LFIRST COLLIMATING APERTURE

FIG.2. Schematic diagram of deceleration-acceleration apparatus for double source, mb system. For details of the geometry see Neynaber et al. (1967a). The grid structure and spacing between grids were chosen to achieve a compromise between maximum transmission, minimum electric field penetration, and uniformity of electric fields. Grids 2 and 3 are electrically connected together. Grids 3 and 4 and the electrostatic screen are attached to the movable detector assembly (see Fig. 1 ) and could be translated in all directions.

in the interaction region were accelerated at its exit to an energy equal to (3000 + AE) eV. After the merged beams left the interaction region, they entered the demerging magnet. The function of the magnet was to separate fast neutrals from primary and product ions. The magnet was not designed with a resolution that could separate the primary from the product ions. Other undesired particles were prevented from reaching the detector, a Bendix electron multiplier, by employing several components of a detector assembly (shown in Fig. 1 within the dotted box). One of these components was a retarding grid which was at a potential that would allow product ions to pass through but which would retard both the primary ions and ions formed by other processes. Another component was a low-resolution, hemispherical, electrostatic energy analyzer. Its purpose was reduction of ion noise, which is defined later. The sweep plates following the hemispherical analyzer were not incorporated in this experiment. Finally the output of the multiplier was fed into a lock-in amplifier. The detector assembly was supported on a carriage which could be moved along a set of rails parallel to the axis of the interaction

EXPERIMENTS WITH MERGING BEAMS

65

region and in directions perpendicular to this axis. The purpose of the movable assembly was to permit profiles of each primary beam to be taken along the interaction region to obtain information for calculating Zof Eq. (8). b. Absolute Cross Section Determinations. To obtain an absolute value of the neutral current for determining absolute cross sections, the secondary electron emission coefficient y of the “dirty” A1 plate (part of the neutral beam monitor shown in Fig. 1) for neutrals must be known. It was assumed that y was the same for ions and neutrals of the same energy. Utterback and Miller (1961) have obtained experimental evidence that the y’s are equal to within 20 % at energies above several hundred electron volts for N, on gold surfaces that were not clean. Haugsjaa et al. (1968) have found similar evidence for Ar above 140 eV on a “dirty” gold surface. Other investigators have also reported equal coefficients for energies above 100 eV (Berry, 1948; Rostagni, 1934). With this assumption, an absolute measurement of the neutral current was obtained. The y for ions was measured using ions from source 2, but without gas in the cell, with grounded condenser plates at the exit of the cell, and with the merging and demerging magnets turned off. The ratio of ion currents at the A1 plate with extraction and with suppression of secondary electrons was equal to y + 1. For absolute cross section measurements, J1 and J2 were measured at a number of points in space to permit I to be evaluated numerically. The flux densities were determined by measuring the current under investigation after passing it through an aperture of known diameter. c. Noise. A typical signal arising in the interaction region was about A (the primary beams were each several hundred nanoamperes). Two sources of noise were important. One of these depended only on the presence of the primary ion beam while the other depended only on the presence of the primary neutral beam. These are designated as ion noise and neutral noise, respectively. The ion noise was partially dependent, whereas the neutral noise was totally dependent, on background pressure. Since only the primary neutral beam was chopped, the output from the lock-in amplifier for neutral noise was coherent while that for ion noise was random. It was not surprising that these sources of noise existed since the ratio of desired signal to primary beam current was The effect of ion noise was reduced by using suitable integrating times. The coherent, or neutral, noise was probably due to stripped neutrals from the primary beam. The least energy the primary particles could lose in stripping reactions was their ionization energy. The voltage of the retarding grid could thus be set to minimize the neutral noise without reducing the desired signal. The voltage was (2985 + AE/e) V. The existence of excited neutrals in the beam would probably result in more neutral noise than for a ground state primary beam since excited particles would have larger stripping

66

Roy H . Neynaber Neynaber

cross sections. Since the neutral noise could be obtained separately, it could be subtracted from the signal which appeared with both beams present. There was evidence that excited species existed in the Ar beam when the gas in the charge transfer cell was Ar, since for a given neutral intensity and grid potential the neutral noise was larger than when the gas in the cell was H, . Accordingly, H, was used. d. Results and Discussion. The potential of the interaction region was raised to that AE/e appropriate for the desired W . From the total output of the lock-in amplifier, neutral noise and an anomalous output that was dependent on the presence of both beams when AE/e was zero were subtracted. The net result S was proportional to a relative Q. Each cross-section measurement at W was accompanied by a measurement at 1 eV. From these measurements and Eq. (8), the square root of the ratio of the cross section at W , Qw, to the cross section at 1 eV, Q , , was computed. These ratios, (Qw/Q1)''2, are shown 1.3

I

I

I

r

I

I 1 1 1 1

I

I

l

1

l

l

l

l

I

FIRSOV ;RAPP AND FRANCIS

: *z+s-d: ;-

PRESENT EXPERIMENT

OI 0.9o.8-

0.7

'

>

-

a..

-_

POPESCU IOVITSU AND IONESCU -PALLAS 1

1

I

I

1

I l l

1

I

I

I

1

1 1 1

-2

1 I

FIG.3. (a) (QW/QI)'12vs W for symmetric-resonance charge transfer in Ar. Each dot represents a value obtained for a single measurement; a dot accompanied by a number 2 means that two measurements resulted in the same quantity. Crosses indicate arithmetic averages of dots. The standard deviation of the slope of the experimental line confines the extremities of that line to the space between the arrows. (b) Square root of the absolute cross section for symmetric-resonance charge transfer in Ar.

EXPERIMENTS WITH MERGING BEAMS

67

in Fig. 3(a). The theoretical results of Firsov (F) (1951), Popescu-Iovitsu and Ionescu-Pallas (PI) (1960), and Rapp and Francis (RF) (1962) are included in the figure. The slopes of the F and R F lines are almost equal. The error bar is about f6 % and represents the standard deviation for measurements at 10 eV. Uncertainty in (Qw/Q1)1/2at W = 0.1 eV due to the anomalous signal, end effects of the decelerating-accelerating system, and residual transverse velocity components of the beams in the interaction region is less than 8 % . At higher W the uncertainty is even smaller. The experimental line is made to pass through ( Qw/Q,)”* = 1 at 1 eV. Its slope was obtained from the crosses (properly weighted) by the method of least squares. From Fig. 3a it is noted that a straight line appears to fit the crosses. The extremities of the F and RF lines fall well within the limits of the extremities of the experimental line. The end points of the PI line fall outside these limits. Absolute Q measurements at 0.3 eV were made using the technique described above and also a slightly different one (Trujillo et al., 1966). (This latter method was also used to measure an absolute Q at 100 eV.) The results were compatible and gave an average Q at 0.3 eV of 47.1 A’ with an estimated total error of + 25 and - 21 % . The error excludesthe possibility of metastable reactants, about which little can be said. This value of Q together with the experimental line in Fig. 3a were used to obtain the experimental line in Fig. 3b. The R F line is in agreement with the experimental results, the F and PI lines are not. The average value of the cross-section measurements at 100 eV is 18.1 A2 with a total error (again excluding the possibility of metastable reactants) of + 18 and - 13% . It can be shown that this cross section is compatible with the results in Fig. 3b. Table I is a compilation of some representative theoretical and experimental cross sections for the resonant charge transfer of Ar at W = 100eV. The TABLE I CROSSSECTIONSAT W = 100 eV FOR RESONANT CHARGE TRANSFER OF Ar Experimental or theoretical

Result

Reference Potter (1954) Ghosh and Sheridan (1957) Popescu-Iovitsu and Ionescu-Pallas (1960) Cramer (1 959) Rapp and Francis (1962) Hasted (1951) Kushnir ef al. (1959)

E E T E T E E

19.9 20.9 23.0 23.2 24.0 27.0 30.8

(A2)

68

Roy H. Neynaber

values for the mb experiment agree with Potter’s and Ghosh and Sheridan’s within the mutual errors of the experiments. The experimental cross sections listed in Table I were obtained using beam-gas techniques. Their accuracy depends linearly upon absolute pressure measurements performed with a McLeod gage. These measurements were made prior to the discovery of a systematic error in the normal use of McLeod gages, i.e., the Ishii-Nakayama effect (Ishii and Nakayama, 1962; Rothe, 1964). This error would result in measured cross sections that are too large by, perhaps, 10-30 % . Systematic errors in absolute Q determinations for the mb experiment could be due to excited species in either primary beam (the magnitude of this error is difficult to estimate), the determination of y for neutrals (-5 to + lo%), and the determination of the overlap integral I (+ 10 %). An estimate of the composite systematic error (excluding errors due to excited states) is then - 5 to + 14 ”/,. Excited species could be more easily controlled through the use of electron-bombardment ion sources which operate at lower pressure and in which only single collisions between electrons and atoms occur (Stebbings e t a / . , 1966). The determination of y for neutrals could be made by the alternate method of measuring the heat input to a thermopile from the neutral beam; then the assumption that the y for ions and neutrals is the same would not be necessary. Finally, the overlap integral could be more accurately and readily determined by incorporating automatic or semiautomatic equipment to permit J , and J , to be measured at many more points in the interaction region.

2. Single Source Experiment Belyaev er a/. (1966, 1967a,b) have conducted mb studies of the symmetricresonance charge transfer processes H f + H -+ H + H + and D + D -+ D + D + in a range of W from 5 to 100 eV using a single source apparatus. A mixed atom-ion beam was obtained through partial charge transfer of a single ion beam from the source. In the interaction region the ion beam passed completely inside the atomic beam. a. Description ofApparatus. Figure 4 is a schematic diagram of the apparatus. The anode of the ion source (which is described as an oscillating type operating in a longitudinal magnetic field) was maintained at a potential E/e = + 1 kV with respect to the system. Protons (deuterons) were extracted from this source and focused by lens 2 into a 90“, second-order spatial focusing, monochromator magnet 3. About 10 % ofthese ions were neutralized when the beam passed through the charge transfer cell 5, which contained COz . Some of the resultant atoms were left in the metastable 2 s state. After collimation of the mixed atom-ion beam, the metastable atoms were con-

+

EXPERIMENTS WITH MERGING BEAMS

69

FIG.4. Schematic diagram of single source, mb apparatus. (1) ion source; (2) extracting lens; (3) monochromator magnet; (4) quadrupole lenses; (5) charge transfer cell; (6, 9-13, 15) circular apertures; (7) condenser array; (8) region of high vacuum; (14) collision chamber; (16, 23) electron suppressors; (17) Faraday cylinder; (18) electrometer; (19) analyzing sector magnet; (20,22) adjustable holes; (21 ) cylindrical condenser; (24) calibrating Faraday cylinder; (25) galvanometer; (26) detector; (27-30) diffusion pumps; (31) titanium pump; (Sl, S,) switches.

verted to the ground state during their passage through a n electrostatic field produced by the condenser array 7. The electrodes of this array were divided into three sections, the field in the center section being opposite to that in the end sections ( S , in position a). The ratio of the fields in the sections was chosen so that the ion beam, after passing through the electrodes, returned to the extension of its previous trajectory. The angular divergence of the beam entering the collision chamber (interaction region) 14 was determined by the circular apertures 6 and 9 and was 1'20'. Aperture 9 was placed at the focus of magnet 3. In the collision chamber the energy spread of the ion beam was about 4 eV, and the ion current was approximately A. Because of the discharge voltage of the source, the energy spread of the ion beam at the exit of the source was considerably larger than it was in the interaction region. The laboratory energy and energy spread of the atom beam just outside the collision chamber were presumably about the same as for the ion beam since near-resonance charge transfer took place in cell 5. The collision chamber was cylindrical. At the ends were circular apertures I 1 and 12. The curved surface consisted of fine wires parallel to the axis of the cylinder. The chamber was maintained at an appropriate potential, AE/e, and was used in much the same way as the decelerating-accelerating system previously described for the double source experiment.

70

Roy H. Neynaber

The weakly focusing electrostatic lens formed by apertures 9-11 at the entrance to the collision chamber guaranteed passage of the ions inside the atomic beam along the entire length of interaction. Ions formed in the collision chamber (including stripped primary neutrals) passed through an energy analyzer consisting of a sector magnet 19 and a cylindrical condenser 21, which were tuned for ions of energy E + AE. Primary particles were rejected, but the resolution of the analyzer was not sufficiently good to eliminate neutral noise. The ions that passed through the analyzer were measured at the detector by counting scintillations induced by secondary electrons. These electrons were ejected from a metal plate by the impinging ions and accelerated to 15 keV. To minimize neutral noise, the Torr. pressure in the collision chamber was maintained at about 2 x 6 . Measurements. Cross sections were determined by applying Eq. (8); S was equal to the ion counts per second when the primary ion beam was not deflected away from the collision chamber by the condenser array 7 ( S , in position a ) minus the counts per second when the beam was deflected ( S , in position 6). Neutral noise was eliminated by taking this difference. The ion current corresponding to this difference was in the range lo-’’ to A. In determining the overlap integral, it was assumed that the flux density of the neutral component of the mixed beam was uniform everywhere in the interaction volume, i.e., that volume of space swept out by the narrower ion component of the beam. Under this assumption, the integral was equal to the product of the currents of both components in the interaction region, the length of the region, and the inverse of the cross sectional area of the neutral beam in the region. The ion current through the collision chamber was measured at the Faraday cup 17 ( S , in position a). The bottom of this cup was removable. The neutral current was determined by (1) measuring the current of secondary electrons from the bottom of the cup ( S , in position b) produced by the ion current and that produced by the neutral current and (2) measuring, in a separate experiment, the ratio of the secondary electron emission coefficients of the same metal as the bottom of the cup for ions and atoms at an energy of 1 keV. The ratio was approximately unity for protons and hydrogen atoms and for deuterons and deuterium atoms. An average cross sectional area of the atomic beam was determined from geometrical considerations. The length of the interaction region was obtained from the potential distribution along the axis of the collision chamber. This distribution was measured using an electrolytic bath. c. Results and Discussion. Cross sections as a function of Ware shown in Fig. 5. Included are the results of the single source mb experiments, of other experiments (Fite et al., 1960; Fite et al., 1962; McClure, 1966), and of theoretical studies (Smirnov, 1964; Dalgarno and Yadav, 1953). Belyaev

71

EXPERIMENTS WITH MERGING BEAMS INTERACTION ENERGY, wd (eV)

10

10 6

n

5

N

ms

4

rl I

0

a

3 U

2

1

loo

101

lo2

10

lo4

INTERACTION ENERGY, Wp(eV)

FIG.5 . Cross sections for charge transfer of protons in hydrogen atoms (1, 3-7) and deuterons in deuterium atoms (2) as a function of interaction energy (W, for proton experiment and W, for deuteron experiment). ( I , 2) single source, mb results; (3) Fite et al. (1960); (4) Fite et al. (1962); ( 5 ) McClure (1966); (6) theoretical curve of Smirnov (1964); (7) theoretical curve of Dalgarno and Yadav (1953). The values of W, and W dassociated with any point on the graph correspond to the same relative velocities of particles in the H H and D D reactions.

+

+

et al. (1967a,b) indicate that the uncertainty in the mb cross sections was caused mainly by statistical error in the measurement of S of Eq. (8) and by the presence of energy and angular spread in the interacting beams. It is noted that for the same relative velocity, the mb cross sections for the H + + H and D + + D reactions are the same. This is expected on theoretical grounds. Using crossed beams to study the same reactions (where proton energies are in the range from about 200 to 1500 eV), Fite et al. (1958) do not achieve this result. Figure 5 shows quite good agreement between the various experimental results. Agreement of experiment with theory is best at the higher energies. Resolution in W could be improved in this experiment by using a source that emitted ions with a smaller energy spread. A source like those used in the double source experiment would probably suffice. The aperture arrangement at the entrance and exit of the collision chamber is not the most effective means of achieving uniform decelerating and accelerating fields and of minimizing field penetration into the chamber. Grids such

72

Roy H . Neynaber

as those used in the system shown in Fig. 2 would be better and would probably allow measurements to be made at considerably smaller W. A disadvantage of the single source experiment (in addition to the fact that the reactants must be of the same species) is the inability to use modulation techniques effectively. Much of the statistical error in the measurements probably arises from ion noise. This could be largely eliminated as a source of trouble if the neutral beam were modulated and phase and frequency-sensitive detection methods were used. The difficulty of modulating the neutral beam without modulating the ion beam is obvious. Finally, a better determination of the overlap integral could be obtained if profiles of the atom and ion components of the beam could be determined in the interaction region. The single source experiment is another good example of the potential of the mb technique. With the method, cross sections for the H + + H and D + + D reactions have been obtained at lower W than have been achieved previously.

B. CHARGE REARRANGEMENT

+ H,+H,+ + H study of H,+ + H, H 3 + + H has been made with merging beams

1. H2+

A + (Neynaber and Trujillo, 1968) and represents the first application of the technique to charge rearrangement (ion-molecule) reactions. a. Experimental. Except for one modification, the apparatus used for this study is identical to that employed in the investigation of symmetric-resonance charge transfer in Ar (see Fig. 1). An H2+beam from source 1 was merged with a mechanically chopped H, beam. The energy of the particles in each beam was 3000 eV. The H, beam was obtained by passing H2+ from source 2 through a charge-transfer cell The energy difference between H2+and H, that was necessary containing H2. to obtain the desired interaction energy W was established by raising or lowering the potential of the interaction region by an appropriate amount AE/e. The product ions H,+ were then separated from the reactants by the demerging magnet. Other undesired particles were prevented from reaching the detector by employing the retarding grid and the hemispherical electrostatic energy analyzer tuned for the passage of the H,'. Before the addition of a set of sweep plates after the electrostatic analyzer (these plates were not used in the apparatus employed for charge-transfer studies and represent the one modification of that apparatus), the multiplier output was fairly sensitive to small changes in the fields of the demerging magnet and electrostatic analyzer. Such changes could cause small movements

EXPERIMENTS WITH MERGING BEAMS

73

of the H3+ beam over the multiplier face, and assuming gain irregularities over the face, could result in output variations. The plates were designed to sweep the H,' beam over an area on the multiplier face and so smooth output variations caused by gain irregularities. The use of these plates did solve the above problem. A retarding potential curve for the product ions could be obtained for each W. From the plateaus of these curves relative cross sections could be derived. For positive and negative AE/e's associated with the same W, ion currents at plateaus had the same value, within experimental error. The energy resolutions of the demerging magnet and electrostatic analyzer were such that, for a given AE/e, and therefore, W , H3+ of all possible energies would be passed for fixed settings of these detector components. Because of transverse velocities, H3+ was formed along path lengths of the merged beams where the reactants had the same energy. Therefore, from this mechanism, H 3 + was continually being generated outside the interaction region and, when AE/e = 0, inside as well. If the potential of the retarding grid in the detector were sufficiently low, these ions would be detected. When AE/e # 0, the detector signal was due to H3+ formed inside the interaction region at energy W, and, if the retarding grid were at a sufficiently low potential, to H3+ formed outside. The contribution from H3+ outside could be measured independently by applying any AE/e that would result in no measurable signal from inside the interaction region. For this purpose it was determined that the AE/e associated with W = 10 eV, which will be labeled AE'/e (AE'/e = +470 or -510 V), was satisfactory. The contribution associated with W due to H 3 + formed inside the interaction region was the difference between signals measured with the potential of this region at AE/e and at AE'/e. The retarding potential curve for a given W could be derived from this difference. Each cross-section measurement at W was accompanied by a measurement at 1 eV so that ratios of cross sections, Q,/Q,, were obtained. It was not necessary to take measurements for complete retarding potential curves (i.e., curves for a succession of retarding potentials beginning with potentials sufficiently large to result in no measurable signal to potentials well into the region of the plateau) to derive a given ratio. Only a single measurement on each plateau was required, and, in general, this was the technique followed. Data for a given cross-section ratio could be obtained in a few minutes. Absolute cross-section measurements could be made by the technique discussed for the Ar experiment. b. Results and Discussion. The mb results for Qw/Ql are shown in Fig. 6. Included are data for positive and negative AE/e's associated with the same W. To give an idea of the accuracy of ratios designated by crosses, the error for the ratio at 0.1 eV is estimated to be - 11 to + 6 % , At 3 eV the standard

74

Roy H . Neynaber

0.1

-

1

INTERACTION ENERGY W (aV) 0.25 0.5 I 1

--

1

1

2 3 4 7 I

1 ,

v

PRESENT EXPERIMENT

w-l/z

-

-

( e v I-'''

+

+Hz+ H 3 + H. Each dot represents a FIG.6. Qw/Ql vs W-'/' and W for H2+ value obtained for a single measurement; a dot accompanied by a digit means that number of measurements resulted in the same quantity. Crosses indicate arithmetic averages of dots.

deviation is about 18%. Also shown in Fig. 6 is a W - ' / 2 straight-line fit to the data for 0.1 eV < W < 1 eV. This line has the energy dependence predicted by Gioumousis and Stevenson (GS) (1958). For W 2 1.5 eV the ratios fall below the line. Perhaps a spectator-stripping model (Henglein, 1966) could be used to explain this fall-off. The values of the crosses in Fig. 6 are given in Table 11. Also included are the ratios Qw/Ql for a tandem mass spectrometer experiment by Giese and Maier (GM) (1963), and for a single-stage mass spectrometer experiment by

75

EXPERIMENTS WITH MERGING BEAMS

TABLE I1 RATIOOF CROSS SECTION AT

W (ev) 0.1 0.2 0.3 0.4 0.6 1.o 1.5 2.0 2.5 3.0 4.0 5.0 7.0 10.0 12.0 13.5

mb experiment QwlQi

3.22 2.45 2.04 1.45 1.oo 0.69 0.53 0.33 0.20 0.15 0.07 0 0

w,Qw, TO CROSS SECTION AT 1 ev, Ql Giese and Maier (1963) (QwlQih

1.63 1.39 1.oo 0.65

0.44 0.29 0.19 0.08 0.04

-

Reuben and Friedman (1962) (QwlQih~

’

1.35 1.oo 0.78 0.63 0.52 0.45 0.35 0.29 0.28 0.27 0.18 0

Ratios taken from curve labeled Q = 0 of Giese and Maicr (1963).

Ratios are the result of an analysis of the Reuben and Friedman (1962) data as outlined by Giournousis (1966).

Reuben and Friedman (RF)(1962). The mb results are in better agreement with the GM ratios than with the RF values. An absolute measurement was made at 1 eV with the result that Q , = 12 A2 with an estimated error of -26 to + 37 %. Considerably more reliability should be placed on the ratios Q,/Q, than on the absolute value of Q , as evidenced by the estimated errors indicated above. This is primarily because the absolute value of Q , depended upon measurements of the transmissions of the grids in the decelerating-acceleratingsystem, the secondary electron coefficient of the ion collecting plate in the neutral beam monitor, the primary ion and neutral beam shapes, the relative positions of the primary beams, and the gain of the detector assembly (Trujillo et al. 1966). These quantities were subject to rather large errors. The determinations of ratios did not depend on these quantities. The value of Q,, subject to its rather large uncertainty, overlaps the value of 15AZpredicted bytheGS theory. The experimental errors associated with Q,/Q, became larger as W increased because the Q , became smaller and were more difficult to measure.

Roy H. Neynaber

76

Experience with crossed beams, on the other hand, has indicated that the cross section errors become larger as beam energies decrease. This is attributed to beam intensities (which largely dominate such errors) becoming smaller with decreasing energy. From Fig. 6 it appears that the best fit to the data between 0.1 and 1 eV is a smooth curve that is concave downward. The possibility that this curve represents an actual departure of the cross-section ratio from a linear dependence on W-'12 should not be discounted. There is concrete evidence from signals at an extremely low energy (i.e., W 6 0.03 eV) that the quantity (QW/Ql)W'I2 is from 15 to 30% less than the corresponding value given by the slope of the W -'I2 fit in Fig. 6 . In other words, the W fit shown does not apply at this low energy. These signals arise when AE/e = 0. Transverse velocities account for the interaction energy. An upper bound to the energy has been calculated as 0.03 eV. Uncertainty in the energy prevents a determination of the cross-section ratio. Wolf (1968) has used a modified phase-space theory to calculate cross sections for the process under discussion. He predicts a curve of cross-section ratio versus W that has the same general shape as a smooth curve that fits the mb data. It should be noted that the states of the mb reactants are unknown. If excited states did exist, cross sections for ground state reactants could be obtained, in principle, by applying corrections to the quoted values. The percentage correction to the value of Ql would probably be considerably larger than the percentage correction to the cross-section ratios. Chupka et al. (1968) find that for W less than a few electron volts the reaction cross section decreases with increasing vibrational excitation of H,' ; whereas at higher W a mechanism for which the cross section increases with increasing vibrational energy of the ion becomes more prominent. The existence of such excitation in the mb experiment would therefore result in a smaller Q, than would be obtained from just ground state H,'. If the GS theory applies at 1 eV, electronically excited H, primaries would be expected to result in a Q, larger than would be extracted from measurements with a pure, ground state neutral beam. This follows from the larger polarizability of an excited neutral.

-'',

2. Na

+ 02++ NaO' + 0

The existence of NaO' has been established (Rol and Entemann, 1968) through the use of merging beams of Na and 0,'. The failure to observe NaO' in a flowing afterglow experiment is mentioned by Fite (1968) in a paper in which he discusses the merits of the mb technique. In the mb experiment, a Na beam of 2810 eV and an 02' beam with an

EXPERIMENTS WITH MERGING BEAMS

77

energy ranging from 3953 to 4354 eV were merged, resulting in W's ranging from 0.05 to 5 eV, respectively. The NaO' ions were formed with nearly the same velocity in the laboratory system as the reactant particles, corresponding to a laboratory energy near 4765 eV, which was much higher than the energy of any primary beam particles. The cross section Q for the reaction Na 02+--t NaO+ + 0 was determined through the use of Eq. (8). This Q as a function of Wand W-l" is shown in Figs. 7 and 8, respectively. For each W the maximum of the NaO' distribution corresponded to a considerably higher relative c.m. energy W of the products than calculated from the spectator-stripping model. According to this model, W' =kW, where k is a constant. For this system, k = 0.3. The apparatus used for these measurements was similar to that shown in Fig. 1. However, the energy distribution of product ions was measured directly with an electrostatic hemispherical condenser instead of indirectly with a retarding grid. Good resolution was obtained by passing the ions through a decelerating lens system before they entered the hemispherical analyzer. Figure 9 is a schematic diagram of the apparatus. The Na beam was produced by symmetric charge transfer of Na' in a vapor cell. The Na' was formed from source 1 by emission from a heated glass (prepared from a mixture of NaOH, SO,, and A120,) and was subsequently focused and collimated. The 0 ' was produced in an electron

+

INTERACTION ENERGY, W (eW

FIG.7. Q versus W for Na + O2

--f

NaOt + 0.

78

Roy H. Neynaber

w-1/2, (e") - 1 0

FIG.8. Q versus W-'/'for Na

+ 02+-+NaO+ + 0.

bombardment-type ion source 2 using approximately 40-eV electrons. Only single collisions between electrons and 02+ occurred in the source. A fraction (Turner et al., 1968a) of the OZf ions was formed in a metastable electronic state (411,) which has a lifetime longer than the transit time through the apparatus. In order to determine whether the measured Q was for the reaction of Na with ground state or excited state 02+, the fraction of metascurrent) by lowering the energy of tables was lowered (as was the total 02+ bombarding electrons. The Q was negligibly affected, and therefore may be considered as characteristic of the reaction of Na with ground electronic state 02+. The vibrational and rotational states of the Oz+ were unknown. The intensity of the 2810-eV Na beam was determined by measuring the secondary electron current in a Faraday cup and assuming that the secondary electron emission coefficient for neutral atoms is the same as that measured for ions at the same energy. The NaO+ signal was measured with a Bendix Model 306 electron multiplier. The current was calculated assuming that the where gain for NaO' was equal to the gain for Na" plus half the gain for OZt, the Na+ and Oz+gains were each measured at the NaO+ velocity (Kaminsky, 1965). A transverse component of relative velocity introduced a mean relative energy of about 0.03 eV when the beam velocities had the same magnitude. Thus the fractional uncertainty in the energy, A W/W , becomes rather large for small W.

SOURCE 1

SOURCE 2

FIRST COLLIMATING APERTURE

SECOND COLLIMATING APERTURE

DEFLECTION PLATES

\

\

NEUTRAL BEAM MONITOR ELECTROSTATIC DEMERGER

I

MOVABLE

I RETARDING

._ I

I

I I

I

I I I

I

I DECEL-ACCEL I

I LOCK-IN MAGNET

-

PHOTOCELL LAMP

PLATE

I

A

I

I 1

i

ELECTRON MULTIPLIER

I

WHEEL BEAM FLAG

10 cm

/

FOCUSING LENS

FIG.9. Schematic diagram of rnb apparatus for studying Na

/

HEMISPHERICAL ANALYZER

I

~ARADAY CUP

+ 02+ NaO+ + 0. Apertures are not to the scale shown. --f

80

Roy H . Neynaber

IV. Ion-Ion Reactions

A. SYMMETRIC-RESONANCE CHARGE TRANSFER Brouillard and Delfosse (1967, 1968) and Brouillard (1968) used an mb method in conjunction with delayed-coincidence techniques to make a preliminary cross section measurement for the symmetric-resonance charge transfer process He+ + HeZ+-+ He2+ + He+. Their measurements were made at W’s in the kiloelectron volt range and show the usefulness of merging beams for studying reactions at considerably higher energy than in experiments previously discussed. Figure 10 is a schematic diagram of the apparatus. The beam, a composite of He’ and HeZ+,was generated in the PIG source 1 with an energy spread of about 100 eV. It was accelerated at 2 by a voltage in the range 5 to 20 kV,

FIG.10. Schematic diagram of apparatus for studying He+ He2++ H e 2 + He+. ( I ) source; (2) accelerating and focusing region; (3) energy analyzer (electric deflector); (4-7) collimators; (8, 9) current probes and valves; (10-14) basic vacuum (diffusion pumps); (15, 16) high vacuum (sorption); (17) ultra high vacuum (ion pump); (18) ultra high vacuum (titanium sublimation); (19) collision chamber; (20) magnetic shielding; (21) magnetic analyzer; (22) Faraday cups for primary beam currents; (23) “cleaning ” deflectors; (24-25) magnetic quadrupole lenses; (26) phosphor; (27) 30-kV detector electrode.

+

+

EXPERIMENTS WITH MERGING BEAMS

81

focused at 2, and analyzed at 3 by a low-resolution electric deflector. After collimation, it entered the collision chamber 19. The diameter of the beam in this chamber was 1 mm and the current was about lo-' A (He'). The collision chamber was a 5-cm-long cylinder whose potential was negative with a magnitude equal to one-tenth of the accelerating voltage. The pressure in the chamber was lo-'' Torr. Collimators 6 and 7 were chosen so as to accept collisions with a deflection angle of less than about 5 x rad. After approximately 90" deflection in the magnetic analyzer 21, the primary ions were collected on Faraday cups 22. The exchanged ions left the magnetic analyzer through slits, passed through " cleaning" deflectors 23 and quadrupole lenses 24-25, and finally impinged on the negative 30-kV detector electrodes 27. The detectors were photomultipliers whose light input was obtained from scintillating phosphors 26. The phosphors, in turn, were excited by secondary electrons emanating from the detector electrodes. The " cleaning" deflectors swept out ions formed from primary ions whose energies had been slightly degraded by collisions with the residual gas in the region of acceleration after the source. Figure 11 shows ion trajectories in the magnetic analyzer. The source of the composite H e f , He2' beam is labeled S. The collision chamber, at negative potential V , , is shown by the dotted box. The primary beams followed trajectories 3 and 4 in the analyzer and were collected on Faraday cups P+ and P+ +. Ions created by the desired charge transfer process in the collision chamber at A , for example, followed trajectories 1 and 6 and were collected on detectors D and D . Ions created by charge transfer (at B ) or ionization (at C ) through interaction of the primary beam with residual gas followed paths 5 and 2, respectively, and did not reach the detectors. The potential on the collision chamber permitted discrimination between ions created by charge transfer within the chamber and those created outside. It also accurately defined the length of the interaction region. This region was maintained at very high vacuum to reduce noise, i.e., ion current produced from residual gas. Since noise was still larger than the desired signal, delayedcoincidence techniques were used. The desired He' ion triggered the $art input of a time-to-pulse-height converter; the He2+ triggered the stopfIn a subsequent pulse height analysis, the desired signal appeared in well-defined channels, whereas the noise was spread over all channels. The channel address yielded an unambiguous identification of the desired event. Bates and Boyd (1962) and Brouillard (1968) have calculated the cross section for

+

He+

++

+ He2++He2+ +He+

using the impact parameter version of the Coulomb-Born approximation. They neglected momentum transfer effects. Their results, which are naturally

Roy H . Neynaber

82

.'. '.

'\

I

D++ PULSE HEIGHT ANALYZER

t TIME-TO-PULSEHEIGHT CONV. START A

P+

STOP

t

+

FIG.11. Ion trajectories in the magnetic analyzer of the apparatus for studying He+ HeZ++ H e Z + +He+. (S)source at positive potential V,; (dotted box) interaction region at negative potential V,; (+) primary He+ beam; (++) primary Hezf beam; (3, 4) trajectories of undisturbed primary beams; (1, 6) trajectories of HeZ+and He+, respectively, created at A by the desired charge transfer process; (2) typical trajectory of He2 created by ionization of primary He+ in residual gas at C ; ( 5 ) typical trajectory of He+ formed by charge transfer of primary HezC in residual gas; (P++,P+) Faraday cups to collect primary beams; ( D + + , D + ) product ion detectors. The curved, dotted lines associate trajectories in the magnetic analyzer with the process responsible for ion production. +

FIG.12. Q versus 2 W.Circles (solid curve) are predicted values from a semiclassical treatment by Brouillard (1968). Crosses are values from a theory by Bates and Boyd (1962).

EXPERIMENTS WITH MERGING BEAMS

83

in accord, are shown in Fig. 12. Brouillard also calculated the differential cross section for the process. A preliminary, experimental determination of the cross section Q has been made at W = 1.65 keV with the result that Q = 1 x 10-16cm2 (-30%, +SO%). It is noted from Fig. 12 that the theoretical Q is about 5.6 x cm2. This value is outside the limits of error for the experimental Q . The quoted error includes estimates of the uncertainty in the cross sectional area of the primary beam (which is assumed to be uniform in the collision chamber), in the angular acceptance and transmission of the system for the desired species, and in the detector efficiencies. The errors associated with angular acceptance and transmission seem to be the most serious of those included in the estimate. There are plans to modify the apparatus in order to reduce these errors (Brouillard and Delfosse, 1968). Another error, which is not included in the above estimate, arises from the existence of H 2 + in the primary beam. It is not sufficiently resolved from the primary He2+ and would result in an underestimation of Q . This problem is being attacked by using He3 instead of He4 (Brouillard and Delfosse, 1968). Additional results with improved accuracy are expected in the near future from this experiment.

B. MUTUALNEUTRALIZATION The ion-ion mutual neutralization cross section, Q, was measured for N + + 0- + N + 0 by an mb technique (Aberth et a/., 1968). Preliminary studies of similar reactions have also been made by Aberth et al. (1967). The measurements were made over a range of W from 0.1 to 86 eV with an uncertainty in W of 0.1 eV due to transverse velocities. The results represent the first direct cross-section measurements for this type of reaction in which the energy and particle identity are specified. A schematic diagram of the complete apparatus is shown in Fig. 13. The duoplasmatron sources introduced an energy spread in the primary beams (whose laboratory energies were in the kiloelectron volt range) of about 3 eV (Aberth and Peterson, 1967). The beams were superimposed with a merging magnet and then entered the interaction chamber (see Fig. 14). After traveling 30 cm in superposition, the beams were separated by electrostatic deflection. The neutral particles formed by ion-ion neutralization, as well as those formed by electron stripping and capture reactions of the beam ions with the background gas, continued along the superimposed beam direction and were detected by secondary electron emission from a stainless steel surface. The ion-ion neutralization products were separated from those due to beambackground interaction by chopping the primary beams at different frequencies and using a lock-in amplifier to detect the difference frequency.

84

Roy H. Neynaber

FIG.13. Schematic diagram of mb apparatus for studying mutual neutralization.

FIG.14. Schematic diagram of interaction chamber of mb apparatus for studying mutual neutralization.

The secondary electron emission coefficient, y, for the products of the desired reaction had to be obtained for absolute cross section determinations. It was considered equal to the average measured values of the secondary electron emission coefficients of the N + and 0-primaries. The overlap integral in Eq. (8) was obtained by assuming that the flux

85

EXPERIMENTS WITH MERGING BEAMS

density of the broad 0 - beam was uniform everywhere in the interaction volume, i.e., that volume of space swept out by the narrow N + beam. This flux density was determined with the use of the variable irises shown in Fig. 14. Because the flux density was not uniform in the interaction volume, an error was introduced into the results. These results are shown in Fig. 15. Aberth et al. (1968) chose to represent 0.1

1

5

10

INTERACTION ENERGY, W (eV) 20 30 40

80

60

100

+

FIG.15. Mutual neutralization rate measurements vs u and W for N C 0-. The solid curve represents an approximate least-squares fit of the data points to an eleventh degree polynomial. Energies associated with points above 30 eV reflect slight corrections to those in the similar figure of Aberth et al. (1968).

their results by a least-squares fit of the data points to an eleventh degree polynomial. This fit (except for some rapid oscillations, which were deemed insignificant and therefore smoothed out, between about 70 and 100% of the maximum speed at which measurements were made) is shown by the solid curve. The absolute accuracy of this curve is estimated to be & 50 % , except at energies below 0.5 eV where QU(u is the relative speed of the reactants) is rising rapidly with decreasing speed and the uncertainty in u becomes relatively large. Much of the error is attributed to uncertainty in y and the 0- flux density. Not only does the assumption of the uniformity of the flux density introduce an error, but uncertainty in the measurements of the density are introduced by hysteresis in the variable iris mechanism (Aberth and Peterson, 1968). Possible major sources of scatter in the data points are beam alignment and focusing problems. The state of excitation of Nf was not determined. On

86

Roy H . Neynaber

the basis of work done by Turner et al. (1968b), however, Aberth et al. are of the opinion that N + was in the ground state. An excited state of an atomic negative ion has never been observed. The general structure of the curve is similar to that calculated for H + H- neutralization by Bates and Lewis (1955) using a Landau-Zener approximation, and suggests that if this approximation is good then the neutralization cross section is relatively insensitive to the details of the atomic structure. Very recently (summer 1968) Aberth and Peterson (1968) measured the + 0,-in a mutual neutralization cross sections for N2+ + 02- and 0 2 + manner similar to that described above. The neutral species resulting from these reactions were not identified. The results are shown in Figs. 16 and 17. The state of excitation of the primary ions and the effect of excitation on the mutual neutralization rates are unknown. The standard deviations of the data from the plotted curves for N + 0-, N,+ + 02-,and 02+ + 02-are 1.56, 1.66, and 2.72 x lo-' cm3/sec, respectively. The general shape of the curve for N2+ + 02-is like that for N f + 0-. The rather sharp increase in Qo for these two curves is not observed for the curve of 02+ +O,-. This characteristic may be unobservable in the latter curve because of the large scatter of the data. There is also the possibility that vibrational and rotational excitation of the ions would obscure this rise. A

+

+

INTERACTION ENERGY, W (eV)

20

0.1

1

5

10

60

80

I

t 0

40

20

0 .o

0.5

1 .o

1

1.5

1

2 .o

RELATIVE SPEED, v(10

2.5 6

100

I

3.0

3.5

m/KC)

+

FIG. 16. Mutual neutralization rate measurements versus v and W for N2+ 02-. The solid curve represents an approximate least-squares fit of the data points to an eleventh degree polynomial.

87

EXPERIMENTS WITH MERGING BEAMS

20

0.1

1

--

20

10

5

40

I

I 0

0

L

OO

-

-

-

-

-

-

60

3

I

1

1

2

-3

comparison of the N + + 0-and N,' + 0,-curves indicates that the low energy rise is more suppressed for the molecular reaction. An improvement in this mb system, which has been recognized by Aberth and Peterson (1968), would be the adaptation of a collision chamber whose potential could be varied and used to create the desired energy difference between the primary beams. The mutual neutralization experiments are fine examples of the use of merging beams for obtaining measurements that could not be made by other techniques.

C. RADIATIVE RECOMBINATION Wiener and Berry (1 968) have constructed an mb system to be used principally for studying radiative recombination of two ions. The system consists of twin accelerating ion sources and a single analytical magnet (TAISSAM) with approximately 180" focusing for both beams. Initial experiments will be conducted with Li' emitted from a surface ionization source. The source of negative ions is similar to that used by Aberth and Peterson (1967). Figure 18 is a schematic diagram of the apparatus. There are two sets of accelerating

88

Roy H . Neynaber

I M

IwLrr, I

!

W

FIG. 18. Schematic diagram of mb apparatus for studying radiative recombination. (S+, S - ) positive and negative ion sources; ( A + , A _ ) accelerating electrodes; ( E + ,E - )

focusing einzel lenses; ( C + ,C - ) collimating apertures; ( M ) magnet pole pieces; ( D ) detector (movable, shown in position to collect both beams); ( W )window above beam area; ( T + ,T - ) typical positive (Li+) and negative (0-) ion trajectories.

electrodes, focusing lenses, and collimating apertures (one set for each beam). The detector is movable for monitoring and determining characteristics of the ion beams. A large window is provided for optical monitoring, which will be done spectrographically in the initial stages of the experiments. Initial experiments are being conducted to test the utility of optical monitoring rather than to study radiative recombination. To this end, mutual neutralization of Li' and 0- will be studied at relative energies in the center of mass of 1 eV or less. Observations will be made of the emission of the lithium red resonance line at 6707 A which immediately follows the reaction Li+('S) + O - ( 2 P )+ Li(2P) + O(3P).Weiner and Berry (1968) estimate that the cross section for this reaction is reasonably large (at least cm2)on the basis of predictions for well-studied mutual neutralization processes. (The 6707 A line should be considerably stronger than emission from radiative recombination.) Since typical ion velocities are about lo7 cm/sec in the laboratory system, while the lifetime of Li(2P) is less than lo-' sec, most of the emission occurs in the immediate vicinity of the interaction region. The

EXPERIMENTS WITH MERGING BEAMS

89

Li+.O- system is advantageous for testing purposes because of a known, large oscillator strength and a sharp, concentrated, and easily identified spectrum. Li+ + H - has been selected as the first system for which radiative recombination will be studied (again at low relative energy in the center of mass). Since several potential curves of this system are now known, it is feasible to calculate the expected emission spectrum as a function of internuclear distance. Therefore, experimental spectra can be compared with theoretical, at least in the adiabatic approximation. Weiner and Berry (1968) explain that, in essence, the energy-selected pair of ions can recombine and emit radiation at any point along the collision trajectory. To a first approximation, emission from an ion pair at a given internuclear distance R, is restricted to the energy separating the right-hand turning point of the lower state at R, and the relative energy of the colliding pair. (This crude description neglects contributions from the left-hand turning points. These may also be important and account can be taken of them but to do so now merely complicates the present discussion.) In this approximation, each wavelength corresponds to a unique internuclear separation, so that the measured spectral intensities only need to be rescaled in order to give the oscillator strength or transition dipole amplitude as a function of internuclear distance. Thus, the continuous free-bound emission provides one probe for the study of electronic wave functions at large internuclear distances, which are in the range where the system is making its transition between behaving like separated particles and like a molecule. V. Neutral-NeutralReactions Neynaber et al. (1969) have used mb techniques to investigate some twobody collisions in each of which both reactants were neutrals. These reactions were Na + 0,+ N a + + 0,(charge transfer) (a) (ionization) Na + 0,+ Na+ + 0,+ e (b)

+ 0, Na+ + 0 + 0Na + 0, + NaO+ + 0Na + 0, NaO + 0 + e Na

+

(dissociative charge transfer)

(4

(4 (el For ground state reactants and products all of the above reactions are endothermic. The heat of the reaction, AH, is positive for an endothermic process. The values of AH for (a), (b), and (c) are 4.71, 5.14, and 8.78 eV, respectively, assuming the electron affinity of 0,is 0.43 eV (Pack and Phelps, 1966) and of 0 is 1.48 eV (Berry et al., 1965). The magnitude of A H is unknown for reactions (d) and (e). The discussion will be largely limited to the first three reactions with only a few words at the end devoted to the last two. +

+

(rearrangement) (rearrangement).

90

Roy H . Neynaber

A. CHARGE TRANSFER AND IONIZATION 1. Apparatus

To study reactions (a), (b), and (c) a retarding potential curve of product ions was obtained for each of several interaction energies, W. These curves yielded information on the laboratory energy distribution of the product ions and cross sections for the processes. To investigate (a) alone, attempts were made to detect 0,-. Although some measurements were made on 0,-, in general the signal-to-noise ratio was too small (for unknown reasons) to extract good data. Most of the information for (a) and all of it for (b) and (c) was obtained by observing Na', and further discussion will be confined to experiments involving detection of this ion. The apparatus used for these measurements was very much like that emH, --f H3' + H and described previously. ployed in the study of H,' Figure 19 is a schematic diagram of the apparatus. An Na' beam from source 1 was obtained by emission from a heated glass made from Na,O, S O , , and A1,0,, The energy spread of particles in the Na' beam was 1.5 eV or less. The Na' beam was then merged in the merging magnet with ,' at 4000eV from an 0,beam. The 0, beam was obtained by passing 0 source 2, which was a low pressure, oscillating electron bombardment source, through a charge transfer cell containing 0,. An electric field between the condenser plates that follow the cell was used to remove 0,' which did not undergo charge transfer. The energy spread of the 0, 'was I .5 eV or less. The energy of Na' was adjusted to give the desired W. After leaving the merging magnet, the superimposed beams passed through a collimating aperture and then a charge transfer cell containing Na vapor. The temperature of this cell was adjusted so that the vapor pressure of Na was optimum for neutralization of the Na' beam. Under these conditions the 0, beam suffered about 35 % attenuation. The Na vapor had negligible effect on the energy distribution of particles in the 0, beam. This conclusion was reached by measuring the energy distribution of 0,- particles in a beam formed by interaction of the 0, beam with background gas. The energy distribution of 0,-was the same for the Na cell at room temperature and at the optimum temperature for neutralization of Na'. The merged beams then passed through an electric field between a set of condenser plates. Charged particles were eliminated by this field. After passage through a second collimating hole, the resultant neutral beams entered the interaction region. This region was surrounded by a device used in previous ion-molecule experiments (Neynaber and Trujillo, 1968; Neynaber et al., 1967a,b) to establish the desired energy difference between the primary

+

FIG. 19. Schematic diagram of apparatus for studying Na-0, collisions. The collimating apertures are 2.5 mm in diameter.

92

Roy H . Neynaber

beams and to accelerate product ions when they left the region. In the present experiment the device was used in conjunction with the retarding grid in the detector assembly to allow product ions formed inside the interaction region to reach the detector but to prevent those formed outside from doing so. This was accomplished by applying an appropriate potential, P, to the device in order to accelerate ions formed inside the region. The potential of the retarding grid R allowed transmission of these ions but prohibited the passage of slower product ions formed outside the region. Since deexcitation of excited states of Na to the ground state can occur by direct, optically allowed transitions or by cascading, it is assumed that the Na beam in the interaction region consisted of only particles in the ground state. The general conclusions of this experiment, however, are independent of this assumption. In this region the O2 beam presumably contained some excited particles. After leaving the interaction region, the reactants and products passed through a 0.874-cm-diameter hole in the aperture plate. Na’ was separated from the reactants by the demerging magnet. Other undesired particles were prevented from reaching the detector (a Bendix multiplier) by employing the retarding grid and a hemispherical electrostatic energy analyzer tuned for the passage of the Na’. The output of the multiplier was fed into a Cary 31 electrometer and then displayed on a strip chart recorder. The experiment was conducted using dc techniques. 2. Kinematics Kinematics for reaction (a) are shown by the Newton diagram of Fig. 20. In this case, the magnitude of the laboratory velocity of Na, Iv, 1, is less than that of 0 2 ,IvzI. General expressions for the magnitude of the c.m. velocity of Na before the collision, JV11, and that of Na’ after the collision, lV31, are given by Eqs. (10) and (11): IVII = ( 2 W ~ )”z/ml, IV3J= [2p( W - AU - A H ) ] 1 / 2 / m l

(10)

(1 1)

where p is the reduced mass before and after the collision, m, is the mass of sodium, and AU is the internal energy of the products minus that of the reactants. If W ’ is defined as the relative kinetic energy in the center of mass after the collision, then W ’ = W-AU-AH. (12) For the “ after collision ” case, three circles are shown for the loci of the tip of the c.m. velocity of Na’. Therefore, points on a given circle indicate

93

EXPERIMENTS WITH MERGING BEAMS

BEFORE COLLISION

'AU
+

FIG. 20. Newton diagram for N a +Or + N a + 02-. Subscripts 1, 2, 3, and c refer to Na, O r , N a + , and the center of mass, respectively. (v) laboratory velocity; (V) velocity in c.m. system; ( A U ) internal energy of products minus that of reactants; ( v ~minimum ~) laboratory velocity of N a + for AU = 0. (Symbols with arrows in the figure are equivalent to boldface symbols in the text.)

different angular scattering of Na'. Each circle represents one of the three possible cases for the conversion of c.m. energy from internal to translational modes and vice versa. For AU > 0, translational energy of the reactants is converted into internal energy of 0,-. Sufficient energy to excite Na' was never available. When AU = 0, all of the internal energy of O2is converted into internal energy of 0,- or all of the reactants and products are in the ground state. When AU < 0, there is a conversion of internal energy of 0, into translational energy of the products. For the case of AU = 0, two directions are shown for the c.m. velocity of Na'. When the direction is the same as that of the c.m. velocity of Na before the collision, we define the c.m. scattering angle to be zero. Note that the minimum laboratory velocity of Na', vim, exists for this condition. The minimum laboratory velocity of Na' for AU = 0 is less than or greater than that for AU > 0 or
94

Roy H. Neynaber

been a source of noise in the study of the AU = O case (which was important because it included the case when all of the reactants and products were in the ground state), then JvlI had to be less than I v , ~ , where v3 is the laboratory velocity of Na'. Figure 20 and Eqs. (10)-(1 1) show that this was the case since IvlI c lv21 and JV31< IVII. Experiments for which IvlI > lvzl could not be conducted because of the noise introduced by stripped Na. Newton diagrams similar to Fig. 20 can be drawn for reactions (b) and (c). The diameter of a circle is proportional to W"". For a given AU the circle diameters are different for reactions (a), (b), and (c) because the AH'S are different. The diameter for reaction (a) is the largest. Kinematic considerations of reaction (b) show that Eq. (11) gives the magnitude of the c.m. velocity of Na' if the kinetic energy of the electron in the center of mass is considered as a contribution to the internal energy of the products of the reaction. With this interpretation, Eq. (12) gives the relative kinetic energy in the center of mass of the products Na' and 02. For reaction (c) and a given AU, Eq. (1 1) gives only the maximum IV, I, which occurs when the velocities (magnitude and direction) of 0 and 0-are the same. Equation (12) then gives the relative kinetic energy in the center of mass for Na' and the 0, 0-complex. 3. Method

As mentioned previously, the experiment primarily consisted of measuring retarding potential curves of Na'. It should be noted that the retarding grid only retards normal components of the Na' velocity. The desired Na' signal, or current I,, was the detector output with both primary beams on minus the sum of the outputs due to each beam separately. The output from a single beam was similar to ion noise, which has been described previously. Extraneous signals caused by one primary beam modulating (attenuating or enhancing) the noise associated with the other beam were investigated by making P = O . This had the effect of eliminating the real signal without altering the noise significantly. For this case, the detector output with both beams on minus the sum of the outputs for each beam by itself was negligible, indicating the absence of any significant modulation. An absolute cross section could be obtained by a technique discussed previously. This technique included measuring the profiles of each primary beam in the interaction region, the gain of the detector assembly, and determining the secondary electron emission coefficient of the neutral beam monitor for both Na and O2. Typical primary beam currents in the interaction region were 0.5 pA for both Na and 0 2 .

EXPERIMENTS WITH MERGING BEAMS

95

4. Results and Discussion

Figure 21 shows Z3 as a function of R for W = 12 eV. (For this case, the laboratory energy of Na was E, = 2408 eV.) The solid curve (except the kink at 2995 V, which is assumed to exist and will be discussed later) is drawn on the basis of statistical evidence for the existence of inflections. These inflections suggest the existence of discrete energy distributions of Na'. The

3.0

-

v)

f 2.0 a K >

2

cm

K

4

(CI

CI

1 .a

0 RETARDING POTENTIAL, R(W

FIG.21. Nat current (I3) vs R for Na-02 collisions at W = 12 eV (El = 2408 eV; P = 476 V). (E,) laboratory energy of Na; (P)potential of interaction region; (E.) minimum

+

+

) laboratory energy of Na+ from reaction Na O2+ Na+ 02- for AU = 0 ; ( E ~ minimum laboratory energy of Nat from reaction Na O2.+Na+ 0 2 e for AU = 0; (E,) minimum laboratory energy of Nat from reaction Na O2 Nat 0 0-for AU = 0. Each dot represents a value obtained for a single measurement; the dot accompanied by the number 2 means that two measurements resulted in the same quantity. The basis for drawing the solid curve is described in the text.

+

+

+ + --f

+ +

data are not sufficiently good to deduce fine details of these distributions. The inflections together with the data points and confidence limits (not shown) only give some indication of energy limits and integrals of the distributions. The solid curve of Fig. 21, which consists of straight lines, is used to represent this information. The curve is visually drawn consistent with the data and

96

Roy H . Neynaber

with plateaus located in the regions of the inflections. The J3's of the plateaus represent the integrals of the distributions. It should be noted that the resolution of the retarding grid is such that a monochromatic beam will appear to have an energy spread of 4 or 5 eV and peak at about 4 eV less than its actual energy. The arrow labeled E, is at Re = 2942 eV (where e is the magnitude of the electronic charge), which is equal to the laboratory energy of Na' associated with v3"' for reaction (a). We will designate this energy as 8, , the minimum laboratory energy for the case AU = O . The arrows labeled &b and E, are analogous to E , , and they refer to reactions (b) and (c), respectively; &b is at Re = 2948 eV and E, at Re = 301 1 eV. The energy E, is calculated under the assumption that the velocities of 0 and 0 - are the same. It can be shown (Neynaber et al., 1969) that scattering of Na' in the center of mass after the collision was predominantly confined to angles less than 90" and that essentially all scattered Na' was detected. At R values less than 2995 V, there is evidence of two Na' distributions. One of these is between the two plateaus and includes the energy E , . This distribution is very narrow, i.e., a few electron volts or less. An analysis of these data (Neynaber et al., 1969) suggests that this energy distribution is associated solely with reaction (a). In addition, it seems likely that AU is very close to zero (Lee,AU < 0.6 eV), and therefore almost all the excess translational energy of the reactants (i.e., W - AH = 7.29 eV) is converted into translational energy of the products with very little going into internal energy. Finally, because of the very narrow energy distribution about E,, the products in the center of mass must be confined to a very small angle about the axis of the interaction region. In particular, Na' in the center of mass appears to be scattered almost completely in the 0" direction. Apparently the process proceeds via a direct, rather than a complex mechanism. The cross section for reaction (a) leading to the formation of Na' in this narrow energy distribution will be called Q, . Consider, now, the second distribution, which is between the plateau to the right of E, and the kink in the high energy tail. It will be assumed that this distribution is the result of reaction (b), that the scattering of Na' in the center of mass and in the vicinity of the peak of the distribution is at 0", and that AU 2 0. Under these assumptions, the lowest energy in this distribution would be equal to & b . Since measurements with the retarding grid shift an actual energy by an Re z 4 eV to the left in such a curve as Fig. 21, E~ would effectively come very close to or actually merge with the energy distribution associated with reaction (a). Figure 21 invalidates these assumptions since it indicates that the distribution is well removed from that of reaction (a) and that its lowest energy is effectively near 2960 eV. However, compatibility could be achieved within the rather flexible limitations imposed by the data

EXPERIMENTS WITH MERGING BEAMS

97

by departing from straight lines between about 2944 and 2963 eV and drawing an inflection between these energies. The introduction of another inflection point at Re 2 2963 eV would result in a well-defined peak at the point. From Eq. (1 1) as applied to reaction (b), the AU associated with this peak (if it were at Re = 2963 eV) would be about 1.3 eV. This means that at 12 eV, reaction (b) is most probable when 1.3 eV, or 19%, of the excess translational energy [i.e., W - AH = 6.86 eV) is converted into either kinetic energy of the free electron, internal energy of 0,, or both. The significance of such a partition of energy is not known. The high energy tail preceding the kink could be explained by the conversion of translational energy into kinetic energy of the electron and internal energy of 0, (the largest AU for the distribution would be about 3.5 eV) and/or by angular scattering of Na'. The cross section for reaction (b) leading to the formation of Na' in the energy distribution discussed above will be called Qb. There is no evidence for the kink at 2995 V in Fig. 21. Small signal-tonoise ratios in the vicinity of the kink make it extremely difficult to obtain such evidence. The straight lines meeting at the kink could be smoothed to form one continuous curve that would fit the data. This could be interpreted in two ways: (1) as simply a longer high energy tail associated with reaction (b) than that described above; and (2) for R > 2995 V, as adistributionresulting from reaction (c) with the peak of this distribution at about E,. The high energy side of this distribution could be explained by assuming a difference in the velocities of 0 and 0 - . The cross section for reaction (c) leading to the formation of Na' in this distribution will be called Q , . If the second interpretation is assumed, then from Fig. 21, Q, : Qb : Q, E 5 : 3 3 : 1. The retarding potential curves for W = 6 , 8, and 10 eV can be interpreted in a fashion similar to that used for W = 12 eV, and at these energies QJQb 1. At W = 10 eV, there were no observable signals for Re greater than or equal to the associated 8 , . If the high energy tail at W = 12 eV for R > 2995 V results from reaction (c), then at 10 eV the absence of signals for Re 2 E, could be due to very small Q , at such energies. At W = 6 and 8 eV there were no high energy tails analogous to those at 12 eV. At 6 and 8 eV there is insufficient energy for the onset of reaction (c). Therefore, for W = 6 , 8, and 10 eV, it will be assumed that Q, = 0. At W = 6 eV (El = 2540 eV and P = 234 V), the retarding potential curve was taken not only for Re > E, = 2877 eV but also for Re down to 2780 eV. Below E , there were two plateaus other than those associated with processes that have been discussed. These indicate the existence of two other processes. From Fig. 20 and Eq. (1 l), it is concluded that reaction (a) with AU < 0 is responsible for this part of the retarding curve. This means that some 0, is

Roy H . Neynaber

98

excited and that there is a conversion of some of this internal energy into translational energy of the products. For W = 4.71 eV (El = 2579 eV and P = 275 V), which is the threshold for reaction (a), E, = 3025 eV. If, indeed, this is the threshold for the process, then no Na' should be observed for Re 2 3025 eV. The largest R for which there was a nonnegligible Na' current was 2970 V. Two plateaus at smaller R were observed; these are further evidence for excited states of O2 and reactions for which AU < 0. Figure 22 shows I3 vs R for W = 6,8,10, and 12 eV. Only the data for those 1.0

I

I

. @

I

- 2.0

I

I

I

I

I

1

I

I

2

a

013% .5-

0

I

I

1

W = 1 2 eV

W = l O eV

2.0 I3 1.0-

1.00

I

2905

I

I

2915

I

2925 R(V)

2920

2930 R(V)

2940

FIG. 22. Na+ current ( I , ) versus R for W = 6, 8, 10, and 12 eV. Symbols have the same meaning as in Fig. 21. At W = 6, El = 2540 eVand P = 234V; at W = 8 eV, El = 2490 eV a n d P = 343 V; at W = lOeV, El = 2448 eV and P = 416V; at W = 12 eV, El = 2408 eV and P = 416 V.

Re values in the vicinity of E, and the plateau immediately to the left of E, are shown. No plateaus were observed for W = 15 and 25 eV. Figure 23 shows relative Po’sas a function of W where Q , is defined as Q, + Q,, Q,. These were obtained by measuring the 13’s of the appropriate plateaus (see Fig. 22) and normalizing the measurements to the same primary beam currents. The Z,s’ for each W were taken one after another as rapidly as possible. Source conditions (except E l ) for all measurements were kept the same. At W = lOeV, an absolute Q , was calculated as 0.1 A’ with an estimated error of - 50 to + 35 % . The primary beam currents used in this calculation were those measured at the neutral beam monitor.

+

EXPERIMENTS WITH MERGING BEAMS I

I

I

I

1

1

99 I

I

0 X

1.

a

If

0

a

-

4

k

a

v)

z 2 > a a

a

+

I 0..

8

-a a

d

0

”=:

t,

; Q d

1;

1 :

1’2

INTERACTION ENERGY, W (eV)

FIG.23. Relative Qoversus W. Qo is the cross section for the formation of Na+ in Na-02 collisions for AU 2 0. Each dot represents a value obtained for a single measurement. Crosses are arithmetic averages of dots. Qo was chosen as unity for the cross at W = 8 eV.

Relative Q, , Qb, and Q, versus W can be obtained from this figure and the known ratios (at given W’s) Q, : Qb : Q, . At 4.71 eV, Qo = Q, x 0.Since the ratios Q, : Qb : Q, are approximately 1 : 1 : 0 at W = 6, 8, and 10 eV, the relative Q, and Qb are the same as the measured relative Q, . The ratio of Q, at 10 eV to Q, at 12 eV is approximately 1.2. The corresponding ratios of Q,’s and Qb’s are about 1.1 and 1.6, respectively. Crossing of potential curves could contribute to the fall-off of Q, and Q b with increasing W above 8 eV. It is not known whether quantitative agreement with the data could be achieved through such a consideration. The threshold energy in Fig. 23 could not be obtained with less than 0.5 eV uncertainty since the signal-to-noise ratio was so small in the vicinity of

Roy H. Neynaber

100

threshold. As a result, the electron affinity for 0, could not be obtained accurately. As mentioned previously, Qo w 0.1 A’ at W = 10 eV. Therefore, at this same energy Q, w Qb w 0.05 A’. (These cross sections apply, of course, when the specific states of the reactants and their abundances are the same as exist in this experiment.) Since the states of the molecular reactant and product are unknown, the experimental results only shed light on the AU of a process. From the results it appeared that AU = 0 for the process with cross section Q, . This reaction is a charge transfer process, and presumably the FranckCondon principle applies. Since the equilibrium internuclear distance for the ground state of 0,and 0,-are nearly equal (Gilmore, 1965), the conversion of ground state 0,to ground state 0,-has a favorable Franck-Condon factor in addition to satisfying the condition that AU = 0. If it is reasonably assumed that ground state 0,to ground state 0,-was the only significant process occurring, then the cross section would be 0.05A2if all the O2 had been in the ground state. If only a fraction of the 0, had been in the ground state, the cross section would be proportionately larger.

B. REARRANGEMENT Reactions (d) and (e) were studied using the same method and procedures as for reactions (a), (b), and (c) except that the components of the detector assembly were tuned for the passage of NaO+ (by using a K + impurity beam from source 1). A signal attributable to NaO+ was obtained. This signal could have resulted from reaction (d) and/or (e). The first experimental evidence for the existence of NaO+ was recently reported by Rol and Entemann (1968). In the present experiment, the curve of the cross section for the formation of NaO+ as a function of Wrose from zero at about 5.5 eV, reached a maximum at 7eV, and became zero again at about 8.5eV. The cross section at W = l eV was estimated to be about 0.004AZ.

VI. Electron-Ion Reactions A. THEORY Because of the mass difference between an electron and an ion, the mb theory for electron-ion studies warrants special consideration. Subscript 1 will refer to the ion and 2 to the electron. From Eq. (2), it can be shown that for a reasonable percentage uncertainty in the ion laboratory energy (i-e., about 0.1 %) and a reasonable uncertainty in the electron laboratory energy (i.e., a few tenths of an electron volt), an approximate expression

EXPERIMENTS WITH MERGING BEAMS

101

for the fractional uncertainty in W, 6 W/ W,depends only upon the uncertainty in the electron energy and is 6 Wl W x 6E,/(E, W)'I2,

(13)

where 6E, is the uncertainty in E2. Of major importance in merging beams studies of electron-ion systems is the achievement of W considerably less than E, . Having done this, one can use a relatively large E, (and still have a small W) and obtain a concomitantly large electron intensity. Therefore, for these systems a deamplification factor, D,, might reasonably be defined as :

D, = E2/W.

(14)

From Eqs. (13) and (14), the following expression is derived :

6 W/W N D','' 6E,/E2.

(15)

In the following examples it will be assumed that the uncertainty in E, is due to the energy spread which we will choose as 0.3 eV, i.e., 6E2 = k0.15 eV. This relatively large spread is compatible with high beam currents of electrons. Such currents will be required for mb studies. Now, if E2=10eV and W = l eV, then GW/Wx k0.05 and 6 W x +0.05eV. If the ion is N 2 + , El w240keV. If W=O.1 eV, then 6 W/W x 50.15, 6 W x kO.015 eV, and El ~ 4 2 keV. 0 Finally, if E, = 1 eV and W=O.leV, then 6WIWwkO.5, 6 W x f 0 . 0 5 e V , and Elw24keV. Therefore, for best resolution, E2should be relatively large. The resolution is quite good for E2 = 10 eV and rather poor for E2 = 1 eV. An upper limit to to E, would be partially dictated by the high voltage requirements for the associated El. It should be noted that the laboratory velocities of the reactants would be relatively high in an electron-ion, mb experiment compared with those for studies involving two heavy particles. Equation (8) indicates that such high velocities would result in relatively small signals. There would be some mitigation from the rather large flux density that presumably could be achieved for the high velocity ions. B . DISSOCIATIVE RECOMBINATION

Hagen (1967, 1968) has measured cross sections for the dissociativerecombination process N 2 + + e + N* +N** (where the stars refer to either an excited or the ground state of N) in a range of W from 0.1 to several electron volts. The experiment consisted of superimposing a beam of N2+ at 1 keV with a beam of electrons whose energy was varied between about 0.1 eV and about 4 eV. Throughout most of the electron energy range, the

Roy H . Neynaber

102

electron velocity was considerably greater than that of N 2 + and, to a first approximation, N2+could be considered at rest with respect to the electron. Except for the fact that the laboratory velocity of N2+was sufficiently high to permit easy detection of the products, the principles of merging beams were not applied in this experiment. It is discussed because, with a slight modification of the apparatus, a true mb experiment could be conducted. In the experiment, N2+ was generated in a directly heated hot cathode bombardment source (see Fig. 24). The extracted ion current was about

SECTION1

j

SECTION

II

SECTION

IlI

NEUTRAL BEAM DETECTOR' SECTION I ION SOURCE

II lII

MASS F I L T E R INTERACTION REGION

PUMP 1500 I l s e c D P

1000 I I s c c 0 P TITANIUM SUBLIMATION PUMP

PRESSURE

2 a IO-'Torr 2 x 10-6Torr

2

x IO-~TOII

FIG.24. Schematic diagram of apparatus for dissociative recombination experiment.

100 PA. The ions were focused and then passed through an rf quadrupole mass filter. Charge transfer of N 2 + with residual gas resulted in a large fast neutral current which was collected in a titanium sublimation pump. The N2+ beam was bent 45" by an electrostatic analyzer and collimated before it entered the interaction region. Figure 25 is a schematic diagram of the interaction region. It consisted of a hollow, cylindrical permanent magnet. Typical magnetic field lines are shown in the figure. The field was uniform at the center of the region. The

103

EXPERIMENTS WITH MERGING BEAMS ELECTRON REPELLER COLLIMATING APERTURE

\

CUP PROBE

.ELECTRON .LECTOR

P

,EXIT APERl‘U RE

ION BEAM

/

OXIDE CATHODE

I

.- -- ,

POLE. P . lFrF

ALNICO V MAGNET

MAGNETIC FIE1.D IC I IUCS 8 . L

FIG.25. Schematic diagram of interaction region of apparatus for dissociative recombination experiment.

electron emitter was an indirectly heated oxide cathode in the shape of a ring and at a potential equal to the desired electron energy. The electrons followed the magnetic field lines and were collected at the other end of the region. The 1.5-mm-diameter electron beam was contained within the collinear ion beam, which was typically 5 mm in diameter. Beam profiles were measured with a small scanning probe. The electron current was about 1 pA per electron volt of energy with an energy independent spread for the electron beam of about 0.2 eV. The perturbation of the ion beam by the magnetic field was negligible. After leaving the interaction region, the N2+ beam was deflected into a collector. Fast neutrals were detected by an electron multiplier. To discriminate against fast neutrals generated by charge transfer of N2+ with the background gas, the electron beam was modulated. The electron energy was continually swept and data were obtained automatically. The electron energy scale was calibrated by measuring the appearance potential of N, . Figure 26 shows the experimentally measured recombination coefficient, u, as a function of Wand the rms deviations associated with random errors. For any W in the range from 0.1 to 0.6 eV the average u = 2 x cm3/sec within f0.5 x lo-’ cm3/sec. Kasner (196?) measured the same u in an afterglow experiment over a range of temperatures from 205 to 480°K. Over this range, u =(2.7 f 0.3) x lo-’ cm3/sec fits the data. The systematic error for the experiment is estimated by Hagen (1968) as + 10 to - 15% . Included in the systematic error are uncertainties arising from the assumption that the ion flux density in the interaction region was always uniform everywhere within the confines of the electron beam, from

104

Roy H . Neynaber

0

1

2

3

4

5

INTERACTION ENERGY, W (eV)

FIG.26. a. versus W for dissociative recombination of N2+-ecollisions. The solid curve is considered the best visual fit to data points obtained at 0.05-eV intervals over the entire range of the curve. Each data point represented an average of continuous measurements over a 0.025-eV interval. Vertical distances between the solid and dashed lines give rms deviations.

absolute determinations of the desired fast neutral current, and from spiraling of the electrons around the magnetic field lines. The spiraling resulted from components of the thermal contribution of the electron velocity and, to a smaller extent, from components of the electric field each being transverse to the magnetic field at the cathode. The primary result of this was an effective increase in the length of the interaction region and a consequent overestimate of a, because this increase was not considered in its evaluation. The existence of a stronger magnetic field at the cathode than along the axis of the interaction region minimized the effect. As an electron proceeded from the cathode through the magnetic field gradient, the ratio of electron velocity transverse to the magnetic field to velocity parallel to the field decreased. From Section VI,A, it is observed that the Hagen system could be converted to a true mb apparatus by the substitution of a several hundred

EXPERIMENTS WITH MERGING BEAMS

105

kiloelectron volt ion-source power supply for the 1 keV supply. Hagen's 6E2= kO.1 eV. This is smaller than that chosen for the merging beams examples, and would result in smaller 6 W's. As shown by these examples, for the same W, better resolution could be obtained in an mb experiment. In addition, W's less than 0.1 eV would be practical in an mb experiment. As a result, a range of W could be achieved which would overlap the W's of the Kasner and Hagen experiments. VII. Current or Very Recent Studies

To make this review of merging beams experiments as up-to-date as possible, a few brief remarks will be made about some very recent studies. Belyaev et al. (1968) have measured cross sections Q for the symmetric C + C C + and N + N N resonance charge transfer processes C' + N + in a range of W from 7 to 100 eV. They used a method similar to their single source, mb technique described in Section 111,A,2. As expected, for each of the processes a straight line will fit their experimental points on a vs In Wplot. Examples of the Q values for the C + + C reaction are about cm2 and 3.5 x cm2 for W=7 and 100 eV, respectively. For 7.5 x the N + + N reaction, Q z 3 . 6 ~ cm2 and 2.3 x cm2 for W = 9 and 100 eV, respectively. Rol and Entemann (1 969) have studied the ion-molecule reactions Na + N O + --* NaO' + N and Na NO' -+ NaN" + 0 at W' s down to 0.05 eV. The method was similar to that described in Section 111,B,2, except that increased sensitivity of the detector to product ions was obtained by using counting techniques. It appears that only metastable states of NO+ were responsible for the observed NaO+ and NaN'. There were no measurable signals due to these product ions when NO+ was entirely in the ground state. Rol and Entemann have also measured W' as a function of W for the Na +NO+ reactions and find that W' is considerably higher than can be explained, for example, by the spectator-stripping model. The high W' effect is more pronounced for the N a + N O + reactions than for the process -+ NaO+ 0 (see Section III,B,2). Na 0 2 + Rozett (1968) is engaged in developing a merging beams system to study collisions of hydrocarbon ions with free radicals. Neynaber and colleagues are studying the reactions Na 0 +Na+ + 0and Na 0 -+ Na' 0 + e in a manner similar to that described in Section V. The results should be easier to interpret than those for the Na + O2 reactions since there will be fewer excited states of 0 than there were of 0 2 .

+

+

+

+

+

+

+

+

+

-+

106

Roy H . Neynaber

VIII. Concluding Remarks From the foregoing discussion, we have seen that the merging beams technique has been successfully used to study several types of reactions in an energy range heretofore inaccessible. The fact that chemical bond and activation energies are in an energy range in which mb experiments can be conducted should stimulate use of the method for studying collision dynamics. In the opinion of Fite (1968) the method can be uniquely used at low energies to investigate reactions having small rate coefficients. He observes that flowing afterglows, for example, while useful in determining large rate coefficients, cannot be used to determine coefficients smaller than about lo-" cm3/sec. This limitation is the result of competing processes and of impurity effects in flowing afterglow experiments. There are, of course, limitations to the mb method and problems associated with it. Obtaining differential cross sections by measuring laboratory angular distributions would be difficult because the maximum laboratory solid angle for reaction products is relatively small in mb experiments. Identification of the states of the reactants and products poses a problem with mb techniques as well as with many other methods for studying two-body collisions. However, excited species in the primary beam can be partially controlled and their abundance studied through the use of electron-bombardment ion sources in which only single collisions between electrons and molecules occur (see Section III,B,2). Turner et af. (1968b) have developed a method for measuring the abundances of specific states of ions in a beam. This method has been used for several ionic species. The ion beam is attenuated by passing it through a cell of gas; abundances are determined from an analysis of the attenuation as ? function of gas pressure in the cell. It is conceivable that a generalization of this method for other ions and neutrals could be applied in the analysis of excited states of particles in an mb experiment. Additional problems with mb experiments include, for example, those associated with making absolute measurements of fast neutral currents and determining the overlap integral It is hoped that solutions to these and other problems will be found with increasing use of the mb technique. ACKNOWLEDGMENTS I wish to thank those people whose work I discussed in this chapter for their assistance in sending me up-to-date reports of their research. I also want to acknowledge useful discussions with E. A. Entemann, R. K. B. Helbing, P. K. Rol, E. W. Rothe, and S. M. Trujillo. I give special thanks to B. F. Myers who kindly read the manuscript and gave many helpful suggestions during its preparation. My work on merging beams has been supported by the Advanced Research Projects Agency through the Office of Naval Research.

EXPERIMENTS WITH MERGING BEAMS

107

REFERENCES Aberth, W., and Peterson, J. R. (1967). Rev. Sci. Instr. 38, 745. Aberth, W., and Peterson, J. R. (1968). Stanford Res. Inst., Palo Alto Calif., Final Rept. for AFCRL Project No. 5710. Aberth, W., Peterson, J. R., Lorents, D. C., and Cook, C. J. (1967). Intern. Conf Phys. Electron. At. Collisions, 5th, Leningrad, 1967, pp. 162-1 65. Nauka, Leningrad. Aberth, W., Peterson, J. R. ,Lorents, D. C., and Cook, C. J. (1968). Phys. Rev. Left. 20,979. Barnes, A. H. (1930). Phys. Rev. 35, 217. Bates, D. R., and Boyd, A. H. (1962). Proc. Phys. SOC.(London) 80, 1301. Bates, D. R., and Lewis, J. T. (1955). Proc. Phys. SOC. (London) A68, 173. Belyaev, V. A., Brezhnev, B. G., and Erastov, E. M. (1966), Zh. Eksperim. i Teor. Fiz. Pis’ma u Redaktsiyu 3, 321 [JETP Letters (English Transl.) 3, 207 (1966)l. Belyaev, V. A., Brezhnev, B. G., and Erastov, E. M. (1967a). Zh. Eksperim. i Teor. Fiz. 52, I170 [Soviet Phys. JETP (English Transl.) 25, 777 (1967)l. Belyaev, V . A,, Brezhnev, B. G., and Erastov, E. M. (1967b). Intern. Conf. Phys. Electron. At. Collisions, 5rh, Leningrad, 1967, pp. 156-158. Nauka, Leningrad. Belyaev, V . A., Brezhnev, B. G., and Erastov, E. M. (1968). Zh. Eksperim. i Teor. Fiz. 54, 1720 [Soviet Phys. JETP (English Transl.) 27, 924 (1968)l. Berry, H. W. (1948). Phys. Rev. 74, 848. Berry, R. S., Mackie, J. C., and Taylor, R. L. (1965). J. Chem. Phys. 43, 3067. Brouillard, F. (1968). Ph.D. Thesis, Univ. of Louvain, Belgium. Brouillard, F., and Delfosse, J. M. (1967). Intern. Conf. Phys. Electron. At. Collisions, 5th, Leningrad, 1967, pp. 159-162. Nauka, Leningrad. Brouillard, F., and Delfosse, J. M. (1968). Private communication. Carlston, C. E., and Magnuson, G. D. ( I 962). Rev. Sci. Instr. 33, 905. Chupka, W. A., Russell, M. E., and Refaey, K. (1968). J. Chem. Phys. 48, 1518. Cook, C. J., and Ablow, C. M. (1959). Stanford Res. Inst., Palo Alto, Calif., AFCRC-TN59-472. Cook, C. J., and Ablow, C. M. (1960). Natl. Meeting Am. Chem. SOC.,137th, Cleveland, 1960, p. 28R (Abstracts). Cramer, W. H. (1959). J. Chem. Phys. 30, 641. Dalgarno, A., and Yadav, H. N. (1953). Proc. Phys. SOC.(London) A66, 173. Davis, B., and Barnes, A. H. (1929a). Phys. Rev. 34, 152. Davis, B., and Barnes, A. H. (1929b). Phys. Rev. 34, 1229. Davis, B., and Barnes, A. H. ( I 931). Phys. Rev. 37, 1368. Firsov, 0. B. (1951). Zh. Eksperim. i Teor. Fir. 21, 1001; as shown in Fig. 12.5 of Hasted, J. B. (1964); “Physics of Atomic Collisions,” p. 417. Butterworth, London and Washington, D.C. Fite, W. L. (1968). Intern. Assoc. Geomagnetism Aeronomy Symp. Lab. Meas. Aeronomic Interest, Toronto, 1968, pp. 59-82. Fite, W. L., Brackmann, R. T., and Snow, W. R. (1958). Phys. Reo. 112, 1161. Fite, W. L., Stebbings, R. F., Hummer, D. G., and Brackmann, R. T. (1960). Phys. Rev. 119, 663. Fite, W. L., Smith, A. C. H., and Stebbings, R. F. (1962). Proc. Roy. SOC.(London) A268, 527. Ghosh, S . N., and Sheridan, W. F. (1957). Indian J. Phys. 31, 337. Giese, C. F., and Maier, 11, W. B. (1963). J. Chem. Phys. 39, 739. Gilmore, F. (1965). J. Quant. Spectry. Radiative Transfer 5, 369. Gioumousis, G. (1966). Lockheed Res. Lab., Palo Alto, California, Rept. No. 2-12-66-4.

108

Roy H . Neynaber

Gioumousis, G., and Stevenson, D. P. (1958). J . Chem. Phys. 29, 294. Hagen, G. (1967). Intern. Conf. Phys. Electron. At. Collisions, 5th, Leningrad, 1967, pp. 165-1 67. Nauka, Leningrad. Hagen, G. (1968). Private communication. Hasted, J. B. (1951). Proc. Roy. SOC.(London) A205,421. Haugsjaa, P. O., McIlwain, J. F., and Amme, R. C. (1968). J . Chem. Phys. 48, 527. Henglein, A. (1966). Advan. Chem. Ser. 58, 63-79. Ishii, H., and Nakayama, K. (1962). Trans. Natl. Vacuum Symp., 8th, Washington, D.C., 1961, Vol. 1 , p. 519. Pergamon Press, Oxford, 1962. Kaminsky, M. (1965). “Atomic and Ionic Impact Phenomena on Metal Surfaces,” p. 319. Springer, New York. Kasner, W. H. (1967). Phys. Reo. 164, 194. Kushnir, R. M., Polyukh, B. M., and Sena, L. A. (1959). Bull. Acad. Sci. U S S R Phys. Ser. (English Transl.) 23, 995. McClure, G. W. (1966). Phys. Rev. 148,47. Neynaber, R. H. (1968). In ‘‘ Methods of Experimental Physics” (L. Marton, ed.-in-chief), “Atomic and Electron Physics: Atomic Interactions” (B. Bederson, and W. L. Fite, eds.), Vol. VII, Part A, pp. 476-486. Academic Press, New York. Neynaber, R. H., and Trujillo, S. M. (1968). Phys. Rev. 167, 63. Neynaber, R. H., Trujillo, S. M., and Rothe, E. W. (1967a). Phys. Rev. 157, 101. Neynaber, R. H., Trujillo, S . M., and Rothe, E. W. (1967b). Intern. Conf Phys. Electron. At. Collisions, 5rh, Leningrad, 1967, pp. 158-159. Nauka, Leningrad. Neynaber, R. H., Trujillo, S. M., and Myers, B. F. (1969). Phys. Rev. 180, 139. Pack, J. L., and Phelps, A. V. (1966). J . Chem. Phys. 44, 1870. Popescu-Iovitsu, I., and Ionescu-Pallas, N. (1960). Zh. Techn. Fiz. 29, 866 [Soviet Phys.Tech. Phys. (English Transl.) 4, 781 (196O)l. Potter, R. F. (1954).J. Chem. Phys. 22, 974. Rapp, D., and Francis, W. E. (1962). J. Chem. Phys. 37,2631. Reuben, B. G., and Friedman, L. (1962). J. Chem. Phys. 37, 1636. Rol, P. K., and Entemann, E. A. (1968). J . Chem. Phys. 49, 1430. Rol. P. K., and Entemann, E. A. (1969). To be submitted for publication. Rostagni, A. (1934). Z. Physik 88, 55. Rothe, E. W. (1964). J. Vacuum Sci. Techol. 1, 66. Rozett, R. W. (1968). Natl. Meeting Am. Chem. Soc. 156th, Atlantic City, 1968, No. 41 (Abstracts). Smirnov, B. M. (1964). Zh. Eksperim. i Teor. Fiz. 46, 1017 [Sooiet Phys. JETP (English Transl.) 19, 692 (1964)l. Stebbings, R. F., Turner, B. R., and Rutherford, J. A. (1966).J . Geophys. Res. 71,771. Trujillo, S. M., Neynaber, R. H., and Rothe, E. W. (1963). Gen. Dyn. Astronautics, San Diego, California, GDA-63-1296. Trujillo, S . M., Neynaber, R. H., Marino, L. M., and Rothe, E. W. (1965). Intern. Con$ Phys. Electron. At. Collisions, 4th, Quebec, 1965, p. 210. Science Bookcrafters, Hastingson-Hudson, New York. Trujillo, S. M., Neynaber, R. H., and Rothe, E. W. (1966). Rev. Sci. Instr. 37, 1655. Turner, B. R., Rutherford, J. A., and Compton, D. M. J. (1968a). J . Chem. Phys. 48, 1602. Turner, B. R., Mathis, R. F., and Rutherford, J. A. (1968b). Conf. Heavy Particle Collisions, Belfast, 1968, pp. 126-130. Utterback, N. G., and Miller, G. H. (1961). Reo. Sci. Instr. 32, 1101. Webster, H. C. (1930). Proc. Cambridge Phil. SOC.27, 116. Wiener, J., and Berry, R. S. (1968). Private communication. Wolf, F. H. (1968). BUN.Am. Phys. SOC.13, 213.

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS 11: SPECTROSCOPY* H . G. DEHMELT Department of Physics. University of Washington Seattle. Washington

.

3 Manipulation and Investigation of Stored Charge ...................... 3.1 Excitation of Ion Oscillations .................................... 3.2 Interaction with Tuned Circuit .................................. 3.3 Ion Cooling .................................................. 3.4 Counting of Ions after Ejection .................................. 4 . Spectroscopic Experiments Relying on Spin Exchange with

Polarized Atomic Beam ............................................ Basic Detection Scheme ........................................ ,He and H2 Magnetic Resonance .............................. Hyperfine Structure of the 3He+ Ion ............................ Electron Spin Resonance ...................................... 5 . Spectroscopic Experiments Based on Other Collision Reactions .......... 5.1 rf Spectra of H, Aligned by Selective Photodissociation .......... 5.2 Bolometric” Detection of the eCyclotron Resonance. ............. 6. Spectroscopic Line-Shifts and -Broadening ............................ 6.1 Doppler Effects ................................................ 6.2 Electric Field Effects .......................................... 6.3 Magnetic Field Effects .......................................... 6.4 Collision Effects .............................................. 7. Conclusion ...................................................... Errata for Part I .................................................. References ........................................................ 4.1 4.2 4.3 4.4

+

.

+

+

109 109 112 119 124 124 124 127 129 140 142 142 148 149 149 150 150 151 152 153 153

.

3 Manipulation and Investigation of Stored Charge 3.1 EXCITATION OF IONOSCILLATIONS Oscillations of an ion contained in any three-dimensional harmonic potential can be excited by homogeneous electric fields alternating at the ion frequency w. and in addition at one of the harmonics 20.. 3w.. 401.. .... if

* Part I. Sections 1 and 2 of this article appear in Volume 3 of this series. 109

H. G. Dehmelt

110

the field is sufficiently inhomogeneous, as is well known. Oscillations along the x and y axes, of course, can be obtained in an analogous manner. Under

excitation by an homogeneous radio-frequency field at o,giving rise to a force f = f o cos(w, t + p) the oscillation amplitude increases linearly with time,

z

= z1 cos(o,

t

+ a) + [ f o / ( 2 m o z ) ]sin(w, t t + p)

(3.1)

neglecting dephasing effects of any kind. It grows exponentially when a harmonic, e.g., 2w, is used, for the field must at least have a nonvanishing gradient. We show this by obtaining an expression for the power P absorbed by the oscillating ion under the action of such a force of amplitude&,

P = (;if,o sin 201, t),, which, withf,,

=foo

(3.2)

+ z dfzo/dzand z = zo sin o,t, zo cos o,t , reduces to

P = dW,/dt cc k zo2

or

dW,ldt = f W z / t Z w .

(3.3)

Since P turns out to be proportional to the energy of the z motion of the ion W,, the quantity t,, denotes the time constant of the corresponding exponential growth or decay also providing a measure of the strength of excitation. If the cage contains a number of noninteracting ions, their center of mass will carry out the same motion as a single particle for excitation at o,. However, in the 20, case the center of mass of the ion cloud will remain at rest even though after an excitation interval t,,, ,under the conditions underlying (3.3) the energy component W,of an ion will have changed by a factor e",

with X = t e x c / t 2 ,

(3.4)

again neglecting dephasing. This can be understood by realizing that two ions oscillating 180" out of phase inside the cage at o,can both still be in phase with the exciting field at 20,. This kind of excitation can also be explained as parametric. Compare, e.g., Landau and Lifshitz (1960). We now apply (3.4) to an ion gas in thermodynamic equilibrium contained in a rf quadrupole trap when subjected to an intense 2E,-excitation pulse short enough so that dephasing effects such as collisions may be neglected. With Videnoting the =3 Wi . The total final total initial average energy we have for the z motion WZi energy may then be expressed as

-

W' = ( 2

+ $e+X + +e-")

F,

v / 3 N &ex

X B 1,

(3.5)

where the actual random phase case has been crudely approximated. Since this energy multiplication process directly affects only the z motion, one does have to rely on ion-ion collisions for redistribution of the energy over the other degrees of freedom. Parametric excitation at 20, would have

111

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

the advantage of directly multiplying jFVi and the total average energy in all degrees of freedom, Wi, may be multiplied directly by simultaneous axial and radial excitation. For the higher harmonics in the presence of corresponding higher terms in the expansion of the inhomogeneous rf field a procedure quite analogous to the one above may be adopted. Besides these very general modes of excitation, another class specific to the rf quadrupole trap must now be discussed. Here excitation by a homogeneous rf fields can, according to (2.27) in Part I of this article, also be expected to occur at frequencies R f W,,2R Wz,etc., of which the first is of special practical interest. As we will show, the motion of the center of charge (2.26) having its dominant component at W,can be effectively excited by a homogeneous rf field El cos(R k GJr, which is very useful when detecting the stored

+= A (

Excitation of coherent ion oscillation

x2+ y2 - 2 ~ ~ )

To ion charge monitor

Atcinic beom

FIG.4. The rf quadrupole-ion trap showing schematicallyelectricalpotential distribution associated with the trapping voltage Uo Vo cos at. The voltage uo cos wot is induced in the resonant circuit by the cooperative motion of the ions and allows their detection. The ion oscillation is excited by an approximately homogeneous alternatingfield at the side-band frequency wo R ; from Schuessler et al. (1969).

+

+

charge by a circuit tuned to W, (see Fig. 4). In our discussion we use the pseudopotential approach (2.14), (2.23):

H . G . Dehmelt

112

For E, we take E, = (E,Z/z,) cos Rt + El cos(R & G,)t where Em is the amplitude of the field on the electrode at x = y = 0. Averaging over one period 2n R,it follows that

+

+

(Ez2),, = Em’Z’/(2zO2) E I 2 / 2 (22ElEm/zo) x (cos2 Rt cos 0,t T cos Rt sin Rt sin 0,t),,

.

(3.6)

Carrying out the average over the last term and taking sin G,t, cos 0 , t as quasiconstants, gives finally, with (2.29),

Here, the first term is the unchanged restoring force while the appearance of the second term demonstrates that except for the amplitude reduction by G,/(RJ?), this case is completely equivalent to sinusoidal excitation at 0,. 3.2

INTERACTION WITH

TUNEDCIRCUIT

3.2.1 Coherent Signals When a charge moves between arbitrarily shaped grounded electrodes, the distribution of field lines that originate at the charge and end on the various electrodes changes. This must be accompanied by a change in the surface charges on a given electrode at which these field lines end, and consequently result in currents to and from the electrode. Since the number of field lines connecting the charge and a given electrode is not affected by the presence of other charges on the electrode or elsewhere in the system, the displacement current reaching the electrode appears to come from an infinite impedance current source provided the motion of the charge is rigidly maintained. The magnitude of the current reaching an electrode is conveniently evaluated from a formula given in the literature (Shockley, 1938; Sirkis and Holonyak, 1966) i = qvE,

(3.8)

where E, is the component of the electric field in the direction of the velocity v with which the charge q moves that would result if the electrode of interest were placed at unit potential, all other electrodes being grounded. Applying (3.8) to a parallel plate capacitor whose plate 1, located at - z o , is connected by a conductor to plate 2 at +z, , we find for motion parallel to the z direction i = (q/2zO)(dz/dt)

(3.9)

neglecting end effects. Formula (3.8) may be derived with the help of energy conservation arguments which we demonstrate here for the simple configuration described by (3.9). Assume plates 1 and 2 to be connected to the negative

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

113

and positive poles of a battery of voltage V . If the positive charge q now moves inside the capacitor at a speed u in the direction of the positive z axis the field will do work on it at a rate uqV/(2zo).Because of energy conservation the battery must furnish an equal power

iv = uqV/(2z,) involving a positive current i = qu/(2zo)to plate 2 in agreement with (3.9). So far then, the moving charge inside the capacitor represents an infinite impedance current source paralleled by the capacity of the plates C. If the motion of the charge is harmonic at w, of amplitude z o , as will be the case in our applications, and it is desirable to develop as high as possible a voltage across the current source, it is advantageous to tune out C by a suitable inductance L, oZzLC= 1. In steady state situations the current source then simply appears to be paralleled by the shunt impedance of the tuned circuit, R, across which it develops a signal voltage of amplitude uo = Rqzo w,/(2zo)*

(3.10)

Measurements of uo , in conjunction with (3.10), may now be used to determine q. However, if more than one ion is in the cage as in almost all practical situations, it is necessary to bear in mind that the energy in an oscillation of the center of mass of the ion cloud once excited will decay and be degraded to disordered motion approximately exponentially with a characteristic phase memory time td = Q,/w,. This is due to various causes such as imperfections of the trap, ion-ion interactions, collisions with residual gas atoms, etc. In praxis, small-signal relative ion resonance linewidth Aw,/w, = Q;' of about 1 % may be realized, and this value will be assumed in the following. In order to obtain a strong response under these circumstances which reflects as closely as possible the absolute magnitude of q and as little as possible the side effects due to variable line shape, anharmonicity, and charge-dependent deformation of the potential well, etc., it appears advantageous to subject the ion cloud to a step function rf pulse at w, which is intense enough to drive the center of mass of the ion cloud against the electrodes at k z o in a time t, 4 Q,/w,, effecting complete neutralization of the ion cloud. If the ion cloud is concentrated in a region small compared to zo , as we will assume for the time being, the forced ion motion is accompanied by an rf current pulse of triangular envelope, duration t,,, and maximum amplitude qw,/2. The response of the tuned circuit now depends strongly on its Q value, Q , . For Q, 4 t,,w,, the signal voltage u faithfully follows the current pulse assuming a peak value uof = Rqw,/2 from which the absolute value of q now may be obtained with fair accuracy. At the same time it is clear that the bandwidth of the detection apparatus must be made the wider the smaller t, is, admitting more and more thermal noise while the peak signal uof remains constant. Consequently, the

H . G . Dehmelt

114

above procedure need not be very suitable when small changes of q are to be detected. Under the constraints inherent in the experiment, it is most convenient to analyze the crucial optimum signal-to-noise conditions on the basis of an energy approach. A constant resonant drive excites a steady state center of mass oscillation of the ion cloud in anrf trap ofenergy W, 4 neD. Degrading of this oscillation leads to an increase in the incoherent ion energy Wi’ at the rate dWi/dt = W,/td.

(3.1 1)

Defining the neutralization time t,, as the interval in which Wi grows to the value neD and the ions hit the electrodes, it holds that

V,’= neDtd/t,,,

(3.12)

n from here on denoting the total ion number (change from notation used in earlier sections). On the other hand, the coherent ion cloud oscillation induces a tuned circuit oscillation of energy V,given by

w,= ~cnt,o/tzt

(3.13)

where t,, = Q,/G, and tZt stands for the chracteristic single-ion z-motion damping time due to coupling with the LC circuit [cf. (3.19)]. Denoting the signal voltage by S and the rms noise voltage by N, both averaged over the observation internal t,, we have Sz cc W , , and using the longest possible observation interval t, = t,, results in N2 a kTt,,/t,, . It follows then that

(S/N)’

= n2ebtd/(kTt,,)

(3.14)

for the optimum signal-to-noise ratio independent of the t, chosen, provided t,, % t d , f,,. Underlying this formula is the following signal behavior. When the amplitude of the resonant excitation pulse is increased the average signal amplitude S first increases while the optimum observation interval t, = t,, decreases, thus resulting in a larger noise N in such a way that S/N remains constant. Later, when the condition t,, % t d , ttOis beginning to get violated t , approaches the minimum value t,, , the persistence time of the signal in the tuned circuit induced by the ions, and N becomes constant. Simultaneously, S goes through a maximum and then decreases rapidly as the shorter and shorter ion current pulses of constant peak amplitude excite the tuned circuit less and less and S/N deteriorates. Using the expressions (2.24) and (3.19) for e n and t,, , (3.14) may be rewritten (S/N)’

= n2e2QiQJ(8kTC).

(3.15)

It will often be possible to use full traps where, according to (2.39), n cc Dz, . Since it also holds that C a zo , one may now write (3.16)

115

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

which favors deep and, to a lesser degree, large traps. In the actual experiments carried out in the past the ions have not been subjected to a constant frequency-resonant excitation pulse as assumed in the analysis, but rather their resonance frequency 0,has been swept through that of a continuously applied excitation field at oo. Assuming that the traversal is quasistationary and that during its course the ions are driven against the electrodes, we take here for r,,, the distance between half-maximum points of the observed response pattern on a time scale, i.e., the dwell time in the resonance line. This is done more or less automatically by the data processing equipment when the lower part of the rectified response pattern of Fig. 5a together with

2 V/Cm

I

l

l

1

1

1

1

1

1

1

1

1

1

1

(b)

FIG5. (a) Ion number rf signal for = 8 x lo6 3He+ions in the Major trap and (b) thermal noise. Coherent excitation of the ion macromotion at the sideband frequency R wocauses the signal voltage to be induced in the fixed circuit tuned to wo when sweeping the macromotion frequency Gzthrough resonance. Experimental parameters: Vo = 137 peak, Uo= 0 V, i, = 25 mA, re = 20 msec, sensitivity is 2 V/cm, amplification is 2000. For the noise trace the ion macromotion frequency is adjusted to an off-resonance value. Sensitivity: 1 mV/cm, amplification: 2000. The time axis runs from left to right, linear sweep duration 32 msec. From Schuessler ef al. (1969).

+

the noisy base line is chopped off by suitably biasing the diode output and the resulting wave form run into an analog-to-digital converter accepting only a single polarity whose output is stored in a memory.

116

H . G. Dehmelt

The whole question of tuned circuit noise (see Fig. 5b) now denoted by

N,,becomes, important in our experiments only when it is larger than the ion noise Ni associated with the Poisson fluctuations ,/. Good design should aim for

n,

= 8kTC/(e2QiQ,) + n.

in the ion number n. (3.17)

If (3.17) is satisfied, the observed signal-to-noise ratio will be essentially SIN, N

A.

(3.18)

Here Ni refers to the signal fluctuations associated with the Poisson fluctuations expected to occur in the number of ions contained in samples created under identical experimental conditions, and n, is that ion number for which these fluctuations equal those due to thermal noise. 3.2.2 Low-Noise Detection Circuits The implications of formula (3.15) for the design of practical low-noise detection circuits, namely small electrode capacity C and as large as possible Q values for ions and tuned circuit, are clear enough. Here values C = 10 pF, Qi = 100, Q, = 200 may be attainable without undue difficulty yielding unity signal-to-noise ratio for n = 20 ions. To this corresponds the value n, = 400 which should be sufficiently low for many experiments. In addition a number of other problems arise, some of them similar to those encountered in NMR circuits. One of them is the removal of the carrier on which the small ion signal appears when excitation by an rf field of the same frequency as that of the tuned circuit is used. This carrier is often contaminated with noise and hum, and its presence also creates difficulties when small rf ion signals have to be amplified before rectification. A standard way of carrier reduction is by means of bridge circuits (Rettinghaus, 1967), see Fig. 6, which, however, have a tendency to become complex. In the case of ions in an rf quadrupole trap the more elegant procedure of excitation by an rf field at the frequency R k 0,,not W,,as already described in Section 3.1 may be employed, thus completely eliminating any carrier at 8,; see Fig. 5a (Dehmelt, 1968; Schuessler et al., 1969). Other difficulties arise in connection with the application of the trapping rf voltages to the trap electrodes. The high inductance coil connecting the cap electrodes to form the tuned detection circuit acts as a choke to the trapping frequency, and, due to the finite capacity between the caps and the ring electrode usually held at a high rf voltage, causes undesirable voltages at R to appear at the cap. These voltages have been reduced in earlier asymmetric circuits (Major and Dehmelt, 1968) by series resonant shunt circuits which add some undesirable capacity to the detection circuit. The symmetric circuit of Rettinghaus (1967), employing a ferrite core coil of high internal

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

117

Detection clrcuil

1 Amplitude modulolion

FIG.6. Bridge circuit for low-noise detection of stored ionic charge. This circuit employs

excitation of the ion macromotion at the frequency of the center-groundeddetection circuit, w o ;after Rettinghaus (1967).

flux linkage, solves this problem in a more elegant way. However, this coil now

must be able to accommodate the considerable currents at R flowing across the interelectrode capacities. The symmetric arrangement also minimizes the interelectrode capacity appearing across the tuned circuit. Further problems arise in connection with the electron beam used for the ion creation from the residual gas as it may load the R-drive circuit and also cause noise- and dynatron-type excitation of the detection circuit at least during the filling phase of the trapping cycle. The simple asymmetric circuit of Fig. 7 developed at the University of Washington constitutes a workable compromise solution. The side-band excitation voltage at R + coo is applied to a very low impedance circuit resonant at this frequency connecting the cap electrode near the electron gun to ground. The trapping rf voltages at R and the dc bias are applied to the ring electrode. The capacity of the other cap electrode in this configuration to ground is about 36 pF for the Major trap. At the expense of an increase of this capacity by one-third, a low impedance path to ground for the trapping frequency is provided. Thus the effective capacity in the parallel resonant circuit of frequency wo formed with the choke coil mounted immediately adjacent to the trap tube amounts to about 50 pF. The ion signal is

Tuned amplifier 134 kHz

cable '

4- -

CJ

Rectifier

8 11

RF regulator

FIG.7. Asymmetric circuit for the low-noise detection of stored ion charge. The ion macromotion at (3, but excited at the sideband frequency R w o ;from Schuessler et al. (1969).

+

= wo is detected at this frequency

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

119

transmitted to the commercial low-noise amplifier by a second LC circuit, which is also resonant at oo, and is formed by the double-shielded cable of several feet in length and a choke, and has a capacity of about one-tenth the impedance of the elements of the principal circuit. In this way, the electric field energy is concentrated in the trap where the field may interact with the ions, loading of the high impedance circuit by the input impedance of the amplifier is minimized, and undesirable signals at R and R coo are strongly rejected. The necessary dc path from cap to ground is provided by a choke. For this circuit with the values C = 50 pF, Q,= 60, Qi = 100, the value S/N= 8 x lo3 was observed for 8 x lo6 ions in a bandwidth about 10 times as wide as the optimum value (see Fig. 5), which agrees reasonably well with the value S/N= 8 x lo4 given by (3.15). For the critical ion number below which thermal noise dominates the Poisson noise, formula (3.17) gives here n, N lo4.

+

3.3 IONCOOLING 3.3.1 Adiabatic Reduction of Well Depth

The question of the minimum temperature attainable for a small ion sample in a perfect vacuum is closely related to how far geometrical and contactpotential imperfections affecting the trap structure may be reduced in an experimental situation. This has already been touched upon in Sections 2.4 and 2.5. Another approach is to look upon these imperfections as fixed macroscopic soft collision centers which cause heating of the ion gas by the rf drive at R in a similar fashion as considered in Section 2.6. Afurtherimperfection of this type becomes important when the amount of stored charge is large enough to distort the well shape. Specifically, patchiness of the contact potential over the electrodes may be expected to set a lower limit to the well depth D at which practical traps may be operated. Rather than trying to achieve low ion temperatures simply by making D small it becomes of some interest to consider ways by which the energy of an ion gas contained in a relatively deep well may be reduced. Adiabatic reduction of the well depth is one possibility since the quotient WZ/az for a single ion is invariant. This implies that W’/D should also be invariant. As an example we consider a small ion sample stored in a trap of initial depth Di = 100 V whose temperature has been reduced to room temperature by coupling it to a tuned LC circuit (see 3.3.3). When the well depth is now slowly reduced to Df = 1 V the ion temperature should fall to 30°K temporarily. This method is obviously much better adapted to the (U,= 0) rf trap since in order to simultaneously reduce the energy in the x, y , and z motions it suffices to reduce V , , while for the Penning trap both U, and the magnetic field H, have to be reduced in proportion.

-

120

H. G . Dehmelt

3.3.2 Collisional Cooling The initial average energy of ions created in a trap by electron bombardment may be expected to be of the order of the well depth eD. Concentrating on situations in which only a single species is stored and the ion lifetime in the trap T is much longer than t, , the characteristic time for momentum transfer, we expect that considerable cooling of the ion gas should occur due to a process analogous to the evaporation cooling of ordinary liquids. The continuous loss of ions in the energetic tail of the Boltzmann distribution by neutralization at the electrodes has already been touched upon in Section 2.5 as the cause of this process. The equilibrium value to which this evaporation cooling reduces the ion gas temperature T,, may be considerably lower than the trap depth e n in practical cases. For a small trap, 22, = 0.32 cm, containing about 3 x lo3 protons and depth values D = 2.6,4.6, and 10 V it was observed that eD/kT,, N 10 (Church and Dehmelt, 1969). Measurements further showed t , c 300 msec with theoretical estimates from (2.45) as low as 20 msec for D = 2.6 V in the face of experimental lifetimes T = 100 sec. The observation that an ion is able to survive 5000 collisions with its partners provides some justification of our assumptions in Section 2.5 that Q rf heating in the trap due to ion-ion collisions is not very important. Unfortunately, this form of cooling loses its effectiveness very rapidly when e b / k T i becomes too large though it may be possible to counteract this by brief periodic adiabatic reductions of b. Another major weakness of evaporation cooling is the continuous loss of ions. Viscous drag cooling by a very light inert buffer gas briefly mentioned in Section 2.5 does not have this deficiency, but involves, of course, a certain perturbation of the ion state. No temperature measurements on such systems have been reported so far but increases of Hg' ion lifetimes from T =2.5 sec at lo-' Torr to T = 200 sec upon introduction of the ,He buffer gas at pressures around Tom have been observed (Huggett and Menasian, 1965 ; Menasian, 1968). The use of velocity-selected slow beams for viscous drag cooling may result in very low ion gas temperatures, In other favorable situations the atomic beam used for ion polarization may also effect this cooling.

3.3.3 Radiative Cooling and Thermometry Assuming that collisions between the stored ions establish very quickly a thermal distribution of ion energies over all motional degrees of freedom it is also possible to cool the ions by coupling their disordered, e.g., axial motion to a suitably tuned (cold) LC circuit of shunt resistance R which may also act as the thermometer. The following simple model exhibits many of the important features of this process. Consider a harmonically bound ion with chargeq

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

121

and mass m oscillating with frequency w,along thezaxis whichis perpendicular to the faces of an infinite parallel plane capacitor in which it is confined. The capacitor is assumed to have negligible capacitance; its plates, separated by a distance d, are connected together by a large resistor R. The oscillating ion according to (3.9) induces a current i = qu,/d to flow through the resistor, where u, is the instantaneous z component of velocity of the ion. The Joule heating of the resistor exponentially damps the ion’s motion with a time constant t,, = md2/(q2R). (3.19)

This formula follows from the expression for the average loss rate of the energy in the z motion, -dVz/dt

w,.

= (i2R>,, = [q2R/(md2)lm(v,2),, = (l/tzo)

(3.20)

The energy of n harmonically bound ions moving with random phases is damped at the same rate. The effect of replacing the infinite plane capacitor by a quadrupole ion trap, d = 2z,, is to increase t,, somewhat. The condition of negligible capacitance is satisfied when the z oscillation frequency of the ions equals the resonant frequency of the tank circuit. For the purpose of obtaining a measure of the ion temperature, the noise voltage across the tank circuit is amplified, square law detected, and filtered yielding a dc voltage proportional to the noise temperature T, of the tank circuit. When the ions are externally heated or cooled the tank circuit temperature and hence the dc level are changed accordingly. For the purpose of discussing the system in greater detail it is convenient to refer to the block diagram shown in Fig. 8. Grouping together x and y motions of the ion cloud stored in an rf quadrupole trap, a temperature T, and a heat capacity c, is assigned to this perpendicular or p motion and analogous quantities are introduced for the axial or z motion. Applied heoting

1 01

I

wp

Porornetric heating 01 ZW,

Number calibration white mse

Oven

I

p motion

u Noise

thermometer

FIG.8. Block diagram for the analysis of radiative cooling of and temperature measurements on a gas of stored ions. Heat capacities and temperatures for the respective heat reservoirs are listed as well as characteristic times reflecting the coupling between them.

H . G. Dehrnelz

122

The LC circuit is treated as a loss-free harmonic oscillator at temperature T, shunted by a resistance R at the same temperature as the infinite heat reservoir, To. Introducing the three characteristic times t,, , t,, , and t,, reflecting the coupling between the four heat reservoirs, the system may be described by the set of coupled linear differential equations, taking t,, = t,,,

cPdT,/dt = - c,(T, - T,)/t,

+ C p V p - T,)/t,,

- TJ/L k dT,/dt = +nc,(T, -T$)/Z,, - k(T, - T,)/z,, .

c, dTzldt =

- c,(T,

(3.21)

In case of external heat inputs appropriate terms will of course have to be added. The symbols n and k refer to the number of stored ions and Boltzmann's constant. We use these equations here to obtain an expression for q, the relative steady state temperature increase of the tuned circuit which results when the p-motion heat reservoir is forcibly held at a temperature T, # To,

v = (T, - To)/(Tp - To) = (nlncYC1 + (n/nc)l*

(3.22) Here the critical ion number n, and the dimensionless quantities a, /? are defined by (3.23) /? = c, t,,/kt,, . n, = (1 a)/(a/?), c1 = cpt,,/c, f,,, In practically important limit i,, < z,, this simplifies to

+

n, N kt,,/(c, z,,,)

T, N T, N Ti, (3.24) where Ti now stands simply for the ion temperature. For temperature measurements on a cloud containing a reasonably constant number of ions it is most convenient to measure q directly. To this end the ions are brought up to a known high temperature T, by coupling them long enough to the LC circuit to assume its temperature which by excitation with white noise has been raised to T, . Since the ions will retain their temperature long after the noise excitation of the LC circuit has been switched off, and it has cooled down to a value T, near the equilibrium value appropriate for Ti = Th, relations (3.22), (3.24) will then apply. Now n may be determined from q as t,, is simply the characteristic damping time of the LC circuit and t,, may conveniently be obtained by analyzing T, transients following pulse heating of the ion motion. In the already cited experimental work of Church and Dehmelt (1969), values of t,, = 3 sec have been realized. This also made it possible to lower ion temperatures so that values eb/kTi N 30 were reached. The lowest value so far attained in this way was Ti = 900"K, with the LCcircuit at room temperature, but the residual causes of ion heating indicated hereby are not well understood. These values were observed in a background gas of undetermined composition and at the lowest pressure achievable, 3 x lo-" Torr. The lifetime of the cooled ions under these conditions was about 3000 sec and appeared to be limited by ion-molecule reactions. Once this residual ion and

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

123

heating is eliminated the way may be open for cooling stored ions down to liquid helium temperatures simply by cooling the elements of the LC circuit. Our analysis applies essentially also to experiments on electrons stored in a Penning trap (Dehmelt and Walls, 1968), see Fig. 9, where no great experimental difficulties were encountered in bringing down the ion temperature to that of the tuned circuit at room temperature.

Vorion 12” electromcqnet

-

L

‘ Q

Aquadag electrostatic shield

r ,

Electron gun Pyrex

’

-

Recorder

I cm

TOW

detector

miw

pussbond 1-23 kHz OSCll lotor

FIG.9. Apparatus for temperature measurements on a gas of radiatively cooled electrons contained in a Penning trap; from Dehmelt and Walls (1968).

3.3.4 Incoherent Signal-to-Noise Ratio

As in the case of coherent detection discussed earlier it is of crucial importance to analyze the optimum signal-to-noise ratio available in these experiments. We do this for the steady state case in which the ions are held at the practical maximum temperature 3kT, = e B (see Section 2.5). Assuming that the noise voltage of the tuned circuit is amplified by a noiseless amplifier and then rectified by a square-law detector the output will be proportional to the tuned circuit noise temperature T , . The rms fluctuations in the detector output when averaged over the measurement time t,,, will be, taking the proportionality constant as unity, (3.25) In the limit n 4 n, it will be T, 1: To, and the signal S = T , - To will be small and given by kS N neBtf0/(3t,,), (3.26)

H. G. Dehmelt

124 yielding

(S/N)’

N

n2(efjlkT0)’t,,,rto/(9t2J.

(3.27)

This may be compared with the corresponding expression for coherent excitation given earlier in (3.14) to form

(S/N)t,/(S/N>L,N ( e W h ) t mtt0/(9td tzt). (3.28) It is very amusing to point out that this quotient may easily approach unity or even exceed it since it usually holds that elfilkT, N 100, f r o 2! fd and while t , certainly cannot exceed the ion lifetime in the trap, it may in favorable cases approach tzt.Also, when the ion temperature is kept low for the purpose of reducing ion loss, 3kT,/eb -4 1, the corresponding loss in S/Nmay be recouped by increasing f,,, . This mode of operation allows essentially non-destructive observation on the ion gas. The recent experiments on protons by Church and Dehmelt (1969) seem to prove this. For a discussion of an experiment in which the temperature of a stored electron gas is continuously monitored, see Section 4.2.2. Since it holds that T i - T o = S/q, (3.27) also determines the accuracy with which measurements of the ion temperature may be carried out. 3.4 COUNTING OF IONS AFTER EJECTION A very direct way of counting the stored ions is by electron multiplier after they have been ejected from the trap. Ejection through a perforated cap electrode by quickly pulsing it to a negative voltage exceeding the well depth has been used by Dawson and Whetten (1968). These authors claim a collection efficiency of 10% of the ions stored in the trap. Earlier a procedure selectively ejecting only a single species had been developed by Jefferts (1968). Here, the center of the storage well is slightly displaced toward one of the cap electrodes which is built from mesh. A short rf pulse of the appropriate frequency may now be used to excite a large amplitude oscillation of the ionic species of interest and drive them through the mesh electrode whereafter a simple ion optic focuses them on the multiplier (see Fig. 24). As evident from formula (3.17) ejection and counting of all the ions has decided noise advantages when a very small number of ions, n 5 lo3, is to be detected. This type of counting procedure is basically destructive, however, while the inducedcurrent approach may be made nondestructive (see Section 3.3).

4. Spectroscopic Experiments Relying on Spin Exchange with Polarized Atomic Beam 4.1 BASICDETECTION SCHEME The basic detection scheme (Dehmelt and Major, 1962; Major and Dehmelt, 1968) will now be described in its original form developed for the rf

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

125

spectroscopy of He’ ions. Briefly, it relies on orienting a sample of He’ ions, contained in an rf trap (Fischer, 1959; Wuerker et al., 1959), by spin-exchange collisions with a beam of optically pumped Cs atoms and subsequently monitoring changes in the He+ orientation induced by an hfs transition through its effect on the rate of He’ neutralization. The latter depends on simultaneous spin-dependent He+-Cs charge-exchange interactions, and is reflected in the residual number of ions still present at the end of the interaction cycle. In more detail the He’ ions were created in the rf trap from the background gas by electron pulses of 80 msec duration and a repetition rate of 1 sec-’ which was fast compared to the ion lifetime of 2 8 sec in the trap. The stored ions were continuously subjected to a beam of Cs atoms which had been polarized by optical pumping to an electron polarizationP = (s,)/s = 0.5. The resulting collisions caused the stored ions to quickly (in -100 msec) assume a polarization p equal to that of the beam by the reaction

(4.1) and also to be lost from the trap by various processes with a characteristic time of 400 msec. The principal spin-dependent loss process is assumed to be the charge-exchange reaction Cst+He+&-+Csl+He+T

C s + H e + + C s + +He*+AE. (4.2) The predominant channels of this reaction are assumed to be leading to the excited He* levels 2s ‘ S and 2p 3P. The corresponding energy defects are quite small, AEl = 0.1 eV and AE, = -0.25 eV. Approximate values for the corresponding cross sections may be obtained from curves published by Rapp and Francis (1962); see Fig. 10. At an ion energy of 4 eV, they amount to cm2 and QY N 0.3 x cm’. A lower limit for Q,, the QY N 6 x spin-exchange cross section, may also be obtained from these curves by taking the charge-exchange cross section for AE = 0, Q, 2 170 x cm2. In an approximation we consider in the following the ions to be instantaneously polarized by spin exchange and slowly analyzed by the charge-exchange interaction. A charge-exchange collision of a He’ ion with a Cs atom in the m, = + 3 state may be considered as an “observation” of the He+-ion spin state just as if an atom were traversing a Stern-Gerlach magnet, which can only give the result m, = + 3 or -3 for the atom. The angular splittingof the atomic beam in the Stern-Gerlach experiment is here replaced by a differencein the ion loss rate for ions with spins parallel and antiparallel to that of the atoms. To relate these ion loss rates, which may be denoted vp and v, , respectively, to the cross sections Q, and Q 3 introduced earlier, we note first that parallel spins correspond to a pure triplet state, while the wave function for the antiparallel case must be expanded, according to

t.l = Htl + tl) + 3(tl - tt),

(4.3)

126

H . G . Dehmelt

-E

12-

01

lo3

Cs + He+ +Cs+

I

2

+ He*,

I

I

4

7 I$

Q(tJ)

1

-

Q,/2;

1

2

I

I

4 7 ’01 v (crnkec)

Q(ft)= Q3

1

-

1

I

2

4

0, $He+)

I

J

7 loe

-

2 x 10“ cm/sec

FIG.10. Theoretical curves of Rapp and Francis (1962) for charge-exchange cross section Q as a function of the velocity in near-resonant collisions involving Cs, with the energy defect A E as the single essential parameter, and energy levels important in the Cs He+ chargeexchange reaction.

+

into equal parts of triplet and singlet states. Assuming nearly identical translational orbitals for these states, it must follow that vp ~c Q 3

and

Va

( ~ ( Q I+

Q3).

(4.4)

Also, for an ion sample of n ions, a description in terms of spin-state populations is readily obtained for ions having a spin polarization p interacting with atoms of polarization P # p , on recalling that the fractions of the ions having m, = ++ and -4, which may be denoted n , / n and n-In, respectively, are given by

n, =(Nl + A n ,

n- = W ( l

-An.

(4.5)

Similar equations hold for N + and N - of the N atoms in the trap. Now the rate equations for these population numbers are given by

Nn,

=

-n, N , vp - n , N - v,,

Nn- N - vp = -n- - n- N , v,. (4.6)

127

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

Rewriting the population numbers in terms of the spin polarizations p and P, one finds, on adding the two equations,

- dn/dr = (n/To)Cl -pP(AQ/Q)/41

(4.7)

where

T1= (Nv,+ v,).

e

=

(NQ, + 3Q3h

and

AQ = Q,- Q 3 .

This analysis remains valid if Q , and Q3 are interpreted to refer to ion loss from all experimentally important processes, not only charge exchange. The polarization signal S ( p P ) may now be defined as =

Cn(pP) - n(O)I/n(O),

(4.8)

where nbP) and n(0) refer to the number of ions remaining in the trap after an interaction time t i . One may now adjust tito optimize the signal-to-noise ratio, assuming the Poisson fluctuations n1/’(0)of the ion number to be the only important noise source. Maximizing S(pP)n’/’(O) as a function of time yields for the optimum signal S * ( p P ) S*(PP) = (+lpP(AQ/Q),

S* G 1

(4.9)

for t i = 2T0. This result is a very useful relationship which is already applicable to such cases of interest as transitions between the two magnetic substates of ,He+ and the double-quantum transition between the (1, + 1) and (1, - 1) hfs levels of the ,He+ ion. In such experiments, when using intense rf, one obtains p = 0 and the vanishing of the signal. Without rf, a signal S* corresponding to p = P is observed. (See the center resonance of Fig. 16.) 4.2 ,He+

AND

H,+ MAGNETIC RESONANCE

Polarization and magnetic resonance of ,He+ ions were first demonstrated with the vertically mounted glass apparatus housing the trap structure and the Cs beam shown in Fig. 11 (Dehmelt and Major, 1962). The sealed-off system was continuously evacuated by two 15-liter/sec Vacion pumps. The intense Cs beam emanated from a refluxing oven. The beam was polarized by irradiating it with circularly polarized resonance radiation emitted by homemade Cs lamps. The lamps were filled with 1 Torr of argon, contained a few grams of Cs in the seal-off tips at the bottom, and were constant-current excited from a 500-W dc-input oscillator. The whole apparatus was immersed in a magnetic field created by a 150-cm-diam Helmholtz pair providing afield of 5.5 G parallel with the axis of the trap. Dimensions and operating parameters of the trap in this experiment have been discussed in Section 2.8. After an interaction period with the polarized Cs beam of 0.8 sec the residual ions were counted in the manner described in Section 3.2.1 using excitation at wo = 6,.

H . G. Dehmelt

128

Cold trop

rf

auadruDole

Electron gun

I

Cesium resonance

polarizers

lamps and r f excitation coils

Ceslum oven-

To He quartz leak and vaclon pumps

0 10 cm bl-duUJ

FIG.11. Basic setup used in the experimental demonstration of He+ orientation and magnetic resonance showing sealed-off glass apparatus housing ion trap and Cs atomic beam. Also indicated is the optical pumping arrangement for polarizing the Cs atoms. In the hfs experiment on 3He+ the glass apparatus was rotated 45" around its axis; from Dehmelt and Major (1962).

The polarization of the optically pumped Cs beam could be destroyed at will by magnetic resonance disorientation. For this purpose, a coil wound around the tube immediately above the pumping region supplied a rf field at 1.93 MHz. This feature made it easy to experimentally obtain the ion polarization signal S * simply by destroying the beam polarization in alternate interaction

129

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

cycles and comparing the respective residual ion numbers. Numerical values S * = 0.04 were measured by direct observation on the oscilloscope in the face of statistical fluctuations in the counting signal amounting to 0.01. The test pattern shown in Fig. 12, which was obtained by summing up about 40 n

1 I I ! I I I' I ! ! ! 1 ! ! ! ' ! ! i

Pumping light onloff

I

Hcfield on/offl Cs - resononce

On

On

I I

Off

I On I

On

On

[Off

I

On

I I

Off

On

Time

-

FIG.12. He+ polarization as demonstrated by pulse difference integrator output as function of Cs spin polarization. The Cs polarization was on-off modulated by a resonant rf field applied in the beam region during alternate interaction cycles. The essential element of the integrator was a RC circuit with a 100-sec time constant into which alternate signals monitoring the residual ion charge were fed with opposite polarity. A division corresponds to about 30 sec; from Major and Dehmelt (1968). N

individual S* signals by means of an RC circuit, established the absence of spurious effects. The experimental S* signal obtained in the above manner could further be destroyed by applying a rf field at 15.44 MHz in the trap region perpendicular to Ho and the trap axis which destroyed the 4He+ polarization by magnetic resonance. An analogous magnetic resonance experiment has also been carried out on the rotational ground state K = 0, I = 0 of H2+ which is sufficiently similar to ,He+ so that the same apparatus could be used though with greatly improved data handling procedures (Schuessler, 1968). 4.3 HYPERFINE STRUCTURE OF

THE

,He+

ION

The hfs levels of ,He+ in a magnetic field are split according to the BreitRabi formula (Fig. 13), and the following transitions between sublevels are possible :

H.G . Dehmelt

130

I

E

FIG.13. Breit-Rabi diagram of 3He+;from Schuessler et al. (1969).

(a) (b) (c) (d) (e) (f)

00-11 00010 00-1 - 1 11-10 11 er 1 - 1 10-1 - 1

at v, = Av - v, + 6, at v,, = Av 26, at v c = Av v, + 6, at vd = v, 6, at v, = v, (double quantum), at vf = v, - 6.

+ + +

Here Av zz 8665.649 MHz is the absolute value of the zero-field hfs separation, v, is defined by (4.10) 2hv, = 19, + g.rl Po Ho 9

and 6 may be expressed as

6 = V,(V,/AV)C~

- (2gr/g.r)12.

(4.1 1)

All listed transitions were observed in a magnetic field Ho corresponding to V, =

10,010.8 f 0.7 kHz

(about 7.23 G) and

6 = 11,510 HZ which was stabilized by a potassium optical pumping magnetometer. While the field-dependent transitions showed linewidths 6 v of about 2 kHz, due to residual field inhomogeneities and hum, it was possible to reduce that of the only weakly field-dependent b transition to as low as 10 Hz.

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

131

In general, when discussing how transitions between the hfs levels of an ion with nonvanishing nuclear spin affect the polarization signal, it is clearly sufficient, within the approximations of the present treatment, to consider only how these transitions affect the electron spin polarization p . This quantity may be expressed in terms of the population numbers nu,np ,n, , nd , adding up to n for the hfs states of the ,He' ion, a = (1, +I), p = (1, 0), y = (1, - l), 6 = (0, 0), and the electron spin polarization values, for which a look at the expanded wave functions shows p.=+l,

ps=o,

&=-I,

pa=o,

(4.12)

leading to

P

=

+ ~ np p

+ P, n, + pa n a b = (nu - n,)/n.

(4.13)

In the absence of rf transitions and on the basis of the assumed ordering of the cross sections, neglecting at this point the relatively slow ion loss processes, the following rate equations may be obtained for the population numbers : ria = rip =

n y= rid =

-2(1 - P)n, + (1 + P ) n p (1 - P)n, +

(1 - P ) n ,

+ (1 + P h , - 3 9 + (1 + P)n, +n6, (1 -P)rtp-2(1 + P ) n , + ( I -P)n.j, + n p + (1 +P)n,+ -3na.

(4.14)

This result may be arrived at in an elementary way (allowing some insight into the spin-exchange process) which will be outlined in the following. During the spin-exchange collision in which electronic angular momentum is transferred to the ,He+ ions, the nuclei are not immediately affected. Nevertheless, in the long collision-free interval afterwards, the hfs precession in the ,He+ ion does transfer angular momentum from the electron to the ,He nucleus. Since the collision frequency T i 1 of a He' ion with Cs atoms is largely determined by the spin-independent polarization forces, we schematize the collision process as follows. At average intervals T p , the exchange interaction between pairs of fixed He' ions and Cs atoms is switched on for a statisticiilly varying time, long compared with the spin-exchange period. During the interaction period, the ,He+ nuclear wave functions remain unchanged, while the electron spin wave functions evolve in time because they are a superposition of two stationary states, singlet and triplet, with different energies (Purcell and Field, 1956). Thus, for example, an initial wave function ' P i K tfl must be written

yi ~ 2 t t J + C(tJ +JT)

+ e'*(tJ- J t ) L = o t.

where Q=

-(E3

- El)t/h.

(4.15)

132

H . G . Dehmelt

In the term 2 f f l the first two arrows refer to the electronic spins of Cs and 3He+, while the third denotes the nuclear spin of the He ion. An analogous position convention is used in the following. After a time t this becomes, bracketing out the Cs electron spin function,

Y f fC2I-l + lt(1 + e'")l

+ 1m1 - e'">l.

(4.16)

Transforming back to the coupled representation for the He ion, one finds Y, cc ~.[Jz(l - e'">al

+ f[(3 + e")j + (1 - ei")61.

(4.17)

The effect of the hfs interaction which we now switch on is to multiply 6 in quite analogous to the procedure leading to (4.15). this expression by ei4(hfs), However, this phase factor has no effect on the average populations in the states a, p, 6 which we obtain by averaging the squares of the absolute values of the expansion coefficients in (4.17) over CP. We see that as a result of the collision, an ion initially in the ffi state has the relative probabilities 1/4, 5/8, and 1/8 of appearing afterwards in the c(, p, 6 , states, respectively. An analogous procedure is applicable to the other ,He+-ion hfs states and the rates at which ions in an ensemble are scattered in and out of the hfs levels for a general Cs spin polarization P can be written down using the (+)(1 + P ) and (+)(1 - P ) weighing factors for the two Cs electron spin states. In this way the system of Eq. (4.14) may be constructed. It should be noted that, as defined, T, is the mean time between collisions which randomize 0 and for which the probability of spin exchange is only 3. Since Qe is defined in terms of a mutual spin flip, the corresponding T, must be related to T, by T, = 2T,. For simplicity of appearance, Eqs. (4.14) have been written with the time measured in units of 4Te. To demonstrate the usefulness of these equations, we derive the relative signal strength for saturated rf transitions between the hfs levels, e.g., ci and p, under slow passage conditions. Since S a p and p = P in the absence of rf transitions, we define the corresponding relative depolarization signal as (4.18) SLs 1 - (Paa/P)= [n(PP) - n(p,jjP)]/[n(PP)- n(O)]. The transition-reduced ion polarization pas is found from Eq. (4.13) and the solution of the rate equations (4.14)for A,, Ad under the conditions n, = ns and A? = rid = 0. The numerical results for the signals SAP for the limiting cases P e 1 and P = 1 are listed in Table 111, and formula (4.19) is applicable for general P . The equilibrium value p = P postulated earlier also follows easily from (4.14). According to Table 111 it is necessary to take special steps in order to make the system sensitive to the b-transition of primary experimental interest. This is possible by simultaneously saturating the a and c transitions. It is clear that additional application of an intense rf field saturating the b-transition will lead to a reduced ion polarization pabc= 0. We obtain a

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

133

TABLE I11 RELATIVESIGNAL STRENGTH s' OF SATURATED rf TRANSITIONS BETWEEN hfs LEVELS OF 3He+ a Transition

Pg 1

P= 1

(1, +1)-(1, 0) ( I , + l t ( l , -1) (1, O)-(l, -1) ( I , +1)-(0, 0) (1, 0)40, 0) (1, -1)-(0, 0)

0.4 1.0 0.4

417 1.o

0.4 0.0

0.0 4/'7

0.4

0.0 0.0

From Schuessler et a/. (1969).

measure of the expected effect of interest by evaluating the corresponding quantity pOc= 3 for P = 1. This shows that the signal due to the b transition alone may be made as large as one-third of that due to the saturated e transition which exhibits the maximum signal obtainable. In order to avoid frequency-pulling and line-broadening effects believed to affect the b transition in this basic scheme it was modified into the pulse scheme sketched in Fig. 14. (F. rn,)

lr

180' pulse d tronsition

(0,O)

bl Ic

t Field; independent

b tronsition

I I

H, = Constant

Tb 2 T>,>

Td and T, ( Not to %ole

1

FIG. 14. Pulse scheme developed for the purpose of making the ( I , 0)- (0, 0) transition in -,He+ directly observable. The quantities HI,p, and T,are, respectively, the amplitude of the rf field, the ion polarization, and the average time for spin exchange with the polarized Cs beam; from Schuessler et al. (1969).

134

H . G. Dehmelt

The time Tb during which the ions were only excited by an rf field near the b transition under investigation was chosen nearly equal to the spin-exchange time T, introduced earlier. In sufficient approximation the average ion polarization indicated in the figure may now be used in the theoretical expression for the signal S(pP). For the experiments on ,He+, the old 4He+ apparatus was modified in the following ways. In order to feed microwave energy for the IAFI = 1 transitions into the trap a A/2 slot antenna was cut radially into one of the cap electrodes. This antenna was excited by means of a dielectrically loaded wave guide butting against it via a reentrant glass tube. Trap dimensions had been chosen thus that the TE013 mode providing an axial rf field had the appropriate frequency. This field was able to induce both nand CT transitions since the trap axis also had been rotated 45"against the H, field, which was increased to 7.23 G and stabilized by an optically pumped potassium magnetometer (Bloom, 1962). The sensitivity of ,He+ to depolarizing charge-exchange collisions with the parent gas necessitated operation at a lower pressure and the increase of the pumping capacity to 400 liters/sec. This in turn forced an increase in the intensity of the ionizing electron beam by an order of magnitude. Somewhat later the signal-to-noise ratio of the ion-counting signal was appreciably improved by incorporating excitation at w, R instead of at w, ,as described in Section 3.2. Essential experimental parameters are listed in Table IV. In order to carry out the various processes required for the observation of the hfs spectrum according to the preceding analysis experimentally, it became necessary to develop a rather complex machinery, especially since the maximum experimental signal S * obtainable (for the e transition) turned out to be only 0.02 in the presence of background fluctuations of about 0.005 and the weak signals corresponding to the various rf transitions had to be extracted from the background fluctuations. A block diagram of the complete experimental setup is shown in Fig. 15. Its most important features only will be touched upon here. The sequencer provided the basic 1-sec cycle and controlled the various pulses necessary for the handling of the stored ions. The ion-counting signal was transformed into a pulse train by an analog-todigital converter and fed into a multichannel memory whose address system was advanced by one channel after each basic cycle and recycled after a predetermined number of channels. By obtaining the rf power for the hfs transitions from a frequency synthesizer (0-50 MHz) containing a voltage-tunable oscillator controlled by a voltage derived from the address system, it was possible to traverse the transition many times and thereby improve the signalto-noise ratio. While it was feasible to obtain the weak power sufficient for the narrow b transition by direct multiplication from the synthesizer output, a phase-locked klystron operating in the 8-GHz range had to be employed for the a and c transitions. The auxiliary circuit simply had the purpose of locking

+

10

m 4

I36

H. G.Dehmelt TABLE IV EXPERIMENTAL PARAMETERS IN 3He+ hfs EXPERIMENT' Axial dimension Radial dimension dc bias ac trapping amplitude Frequency of trapping rf Detection frequency Axial oscillation frequency Axial well depth Radial well depth Stored charge (3He+) Electron current Electron acceleration voltage Electron pulse duration Coherent excitation amplitude Duration of excitation pulse Excitation frequency Detection sweep amplitude Duration of traversal through 8, resonance Quality factor of detection circuit Quality factor of stored ion cloud Ion lifetime 3He density Cs density at site of trap Ultimate vacuum Ratio of ion number signal to total rms fluctuations with Cs beam present at the end of interaction interval (0.8 sec) and comparing consecutive signals Ratio of ion number signal to total fluctuations without Cs beam Ratio of ion number signal to thermal noise

z o = 25.5 mm

ro = 36.0 mm

u,=ov

Vo = 137 V peak !2=27rx 1 MHz W O = 277 x 134 kHZ G Z = 2 x 126kHz D,= 6.1 V Dr = 3.1 V q = 8 x 106e i, = 10 mA ue=500v 1. = 80 msec Vn+- = 3 v Tn+o= 40 msec ~ 0 + ! 2 = 21134kHz ~ U d = O to -3.6 V t,,,= 4 msec

Q,= 50 100 T = 25 sec n, = 2 x lo8 atoms/cm3 N = 1 x 10' atoms/cm3 p = 1 x 1O-lOTorr

Qi =

From Schuessler et al. (1969).

the klystron to the ultrastable vb frequency 10 MHz. The basic frequency of the synthesizer was provided by a high-quality quartz oscillator whose frequency was continuously compared with WWVB. The low-frequency d, e, and f transitions first seen unresolved by Major (1962) were relatively easily observable. In Fig. 16 is shown a digital-analyzer display of the rf-saturated AF = 0 spectrum of ,He+ obtained by Schuessler et al. (1969), with the Cs optically pumped into the m, = -4 state resulting

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

10.013. I

10,002.3

137

9990.8

kHz -Y

FIG. 16. Digital analyzer display of low-frequency 3He+ spectrum in a field of 7.23 G obtained in a 2-hour run. The amplitude of the rf field for this recording was continuously increased to such a value that no further changes in the heights of the d, e, or f peaks occurred; from Schuessler et a/. (1969).

in a depolarization signal S' greater for the lower frequency (1, - 1)-(1, 0) transition than the (1, O)-(l, + 1) transition. The relatively narrow line in the middle is the double-quantum transition (1, - 1)-(1, + 1). The magnetic field had the value H = 7.23 G, for which the spectrum was centered about 10.00 MHz, and the Paschen-Back-Goudsmit shift amounted to 6 = 1 1.5 kHz. The frequency increases from left to right, and the quantity displayed (plotted downward) is simply n (0.8 sec), the number of stored ions remaining after the interaction interval, averaged over 80 traversals of the spectrum. From these data, and with the help of the expression

P = (2 - 5S/')/(2 - 2S/'),

(4.19)

the valuelPl = IpI = 0.5 may be derived for the Cs and ,He+ polarizations. Analysis of the experimental signal S * = 0.02 on the basis of the formula for S * ( p P ) and the theoretical values of the cross sections for charge exchange show that other ion loss processes were about 10 times more effective than the latter. The field-dependent a- and c-microwave transitions were observed with intensities comparable to those of the d and f transitions as expected. A

H.G. Dehmelt

138

display of the much weaker b transition, first investigated by Fortson et al. (1966) with lower resolution, obtained in an 8-hour run by Schuessler et al. (1969), is presented in Fig. 17. Here the individual data points have been (1.0)4 0 . 0 )

I

FIG.17. Digital analyzer display of the field-independent b transition in 3He+ obtained in an 8-hour run. For clarity the individual data points have been arbitrarily connected by straight lines; from Schuessler et al. (1969).

\

8,665.672:887Hz - Y

arbitrarily connected by straight lines. A study of the theoretical line shape which is expected to exhibit weak side bands because of the pulse scheme employed is still in progress. During all runs undertaken for the determination of the hfs separation Av, vb, and v, were measured simultaneously. Data collected in eight runs are given in Table V together with Av values derived from them by means of formula (4.1 1) for 6. An error larger than twice the maximum deviation has been assigned to the data to allow for possible systematic errors. The final result is AvlS = 8,665,649,867 f 10 Hz. The b-transition is expected to be shifted principally by the second-order Doppler effect, spin exchange with the Cs atoms, and the Stark effect (Schwartz, 1959; Fortson et al., 1964). The following expressions may be obtained for the relative absolute values of these shifts, all being negative, 50.6eD/mc2,

5flQg/'

5(1 5a04Vo2)/(Z6e2z02).(4.20)

and

The corresponding numerical values for the present experiment are - 0.7 x - 3 x 10-13, and - 1 x lo-'' which indicates that at present only the second-order Doppler shift requires further consideration. Magnetic fields arising from the motion of the ions through the electric fields in the trap also need not be considered at this stage. Using a somewhat lower estimate for the average ion energy than assumed above, a correction of + 4 Hz has been applied to the data. The experimental values for the hfs separation AveXpmay first be used to obtain a value for the hfs anomaly which is primarily caused by nuclear structure effects. Here 6 is defined by (4.21) hveXplAvF= 1 6.

+

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

139

TABLE V

SUMMARYOF 3He+hfs €ZI?SIJLTS"*~ Field-independent transition vb: (O,O)-(l, 0) 8 8 8 8 8 8 8 8

a

665 665 665 665 665 665 665 665

612 612 612 612

Line width

881,3 881,2 886,5 886,l

612 881,8

612 884,O 612 883,9 612 880,O

22 28 14 14 10 11 13 10

Double quantum transition v.: (1, l)ti(l, -1)

hfs-splitting

Av

10 011 43 10 010 93 10 010 77 10 010 61 10 010 48 10 010 44 10 010 28 10 010 17 Weightedmeanvalue number

8 8 8 8 8 8 8 8 8

861 864 864 865 861 863 863 860 665 649 863

665 665 665 665 665 665 665 665

649 649 649 649 649 649 649 649

From Schuessler et ul. (1969). Frequencies in Hz.

Considering nucleus and electron as point dipoles it holds that

+

+

+

AvF = (y)Z3azcR, n-3(pN/po)[1 (rn/M)]-3[l (a/27c) 0.328(a2/n2)] x [1 ~(n)(za)']{ 1 - a'Z [($) - In 23) [I r(a3 ;n, Z) r(a2rn/M; n, z ) ] , with B(2) = 17/8. B(1) = 3/2,

+

+

+

In this expression the five brackets reflect, respectively, the reduced mass correction, the anomalous electron magnetic moment correction, the relativistic Breit correction, the radiative correction of Kroll and Pollock, and finally n-dependent higher radiative correction of order a3 and a2m/M.Comparing the numerical value obtained with only the first four corrections

Av,

= 8,667,513,000 &

50,000

HZ

with the experimental value yields

6 = -186 k 6 ppm, in agreement with the value obtained by Novick and Commins (1958) from their experimental result for Avzs. That the nuclear structure corrections are identical in the 1s and 2s states is a consequence of a general property of the radial electronic wave functions R,,(r). In the limit r - 0 the Schroedinger equation h2d2(rR,,)/dr2 - 2rn[(Zez/r)+ E,,]rRn= 0 (4.22)

H . G. Dehmelt

140 reduces to

dR,/dr = ZR,/ao,

(4.23)

Now the Fermi contact term is proportional to the average of Rn2(r)over the nucleus. Expanding this quantity around r = 0 yields, with the above result, Rn2(r)= Rn2(0)(1

+ 2Zr/uo +

*

*).

(4.24)

This shows that the ratio Avls/Av2s should be independent of nuclear structure effects within error limits of order (rnuc,eus/u0)2. This makes it possible to use this experimental ratio to check out the (n-dependent) radiative corrections mentioned earlier and the underlying quantum electrodynamics. Defining

+

~ A v ~ ~ =/ 1A v R~ ~

the latest theoretical value (Zwanziger, 1961 ; Sternheim, 1963) is R = 137.21 f 0.03 ppm of which R, = 133.12 ppm is due to the relatively well-established relativistic Breit correction. Combining the Novick and Commins experimental results with the above yields R = 137.33 & 0.19 ppm where the error is mostly due to that in the experimental AvZs value. Subtracting R , from both values yields the theoretical value for the remaining (now purely quantum electrodynamic) corrections Rqed = 4.09 & 0.03 ppm and the corresponding experimental value Rqed = 4.21 f 0.19 ppm which agree within less than 3%.

4.4 ELECTRON SPIN RESONANCE Graeff et al. (1968) have carried out a spin-resonance experiment' on elmtrons stored in a Penning trap (see Section 2.2) with D, = 15 V, ro = 0.9 cm and employing a magnetic field Ho N 2 kG. Their apparatus, which uses a Na beam polarized to P = 0.93 by a 6-pole-magnet state selector, is shown in Fig. 18. Similarly, as in the He' experiment described in Section 4.1.2, the electrons are polarized quickly by spin exchange with the alkali atoms. However, for the analysis of changes in the electron polarization p a different process is relied upon. Cooling of the electron cloud by the spin-dependent excitation process (Salmona, 1965) e + Na(3S) -+ e Na*(3P) - AE, AE = 2.1 eV, (4.25)

+

' This technique has now also been applied to a direcf determination ofg-2; see Graeff e t a / . (1969).

141

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS Beam

Zero field coil

Electron trap Hot wire detector

Homogeneous mognet

FIG.18. Schematic picture of the atomic-beam apparatus used to polarize trapped electrons; from Graeff et al. (1968).

is used here. Just as the neutralization rate for unpolarized reaction partners in the He' experiment is multiplied by the factor

(Q - PP AQ/4)/Q

(4.26)

when the partners are polarized [see formula (4.7)] thus the excitation and, consequently, the cooling rate associated with the process (4.25) is, for polarized partners, multiplied with the same factor (4.26), the symbols now of course referring to cross sections for excitation. Experimentally, spindependent changes in the cooling rates were observed as follows. The trap, initially held at D, = 15 V, is filled by a short pulse of a 1-pA electron beam with electrons of a presumable initial average energy 5 15 eV. The electrons are allowed to interact with the beam for a cooling period of 1 sec, during which thermalization due to e-e collisions according to formula (2.42), which should be applicable in a rough approximation, may be appreciable since T,* N 1 msec. Assuming the latter, an electron temperature T, may be introduced which drops appreciably during the interaction period. The trap depth is then reduced for 0.5 sec to a lower value, presumably D,' N 4 V which, according to Section 3.3.1, should reduce the electron temperature by less than 20% since only the z motion is affected. How many of the electrons will now be lost in this period will be determined by the size of the Boltzmann tail with electron energies exceeding eD,';cf. formula (2.48). Clearly the number of electrons remaining will reflect the electron temperature T,, and thereby the polarization p . The electrons were counted by means of the procedure outlined in Section 3.2.1 relying on excitation at 0,.A signal reflecting changes induced in p by spin resonance is shown in Fig. 19. It has also been proposed to exploit the spin dependence of elastic collisions between exchange-polarized slow electrons and Na atoms for the analysis of the polarization of the former. However, subsequent experiments in which the

H . G . Dehmelt

142 loo

I-----+----( b )

p\

i

Cyclotron line

\,

r \

\

I)

1

/

\

7' Spin - resonance line

5453.43

0.47

051

5459.74 078

C 32

Frequency (MHz)

FIG.19. Variation of electron-detection signal height as a function of applied microwave frequency showing the cyclotron- and spin-resonance lines of the trapped electrons; from Graeff et al. (1968).

electrons were to be boiled out of a flat Penning trap, D, 5 0.1 V, in a spindependent way by rf heating of the type described in Section 2.6 caused by elastic collisions with the atoms in an optically pumped Na beam were not successful, presumably because of fast spin relaxation due to Majorana flops induced by the cyclotron motion through the insufficiently homogeneous and too-low magnetic field, H , N 50 G, then employed (Dehmelt, 1961).

5. Spectroscopic Experiments Based on Other Collision Reactions 5.1 rf

SPECTRA OF

Hz+ ALIGNEDBY

SELECTIVE PHOTODISSOCIATION

Jefferts (1962) has carried out an experiment in which H2+ ions created from H2 by electron impact and stored in an rf quadrupole trap (Paul et al., 1958) were dissociated by bombarding them with linear polarized photons, the survivors becoming aligned. The rate for the reaction hv

+ H,+ -+ H + H + + kinetic energy

(5-1)

is porportional to 60s' (E,R) where E and R are vectors referring to the electric light vector and the (fixed) molecular axis, respectively. This follows from the fact that the photodissociation arises from an electric dipole transi-

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

143

tion between the 'C, ground state I/I, and the 'Xu first excited repulsive state @u (see Fig. 20). The superposition of the two states occurring during the transition

+

Y ( t )= gI/19 uI/Iuexp( - i AEt/h)

(5.2)

cm

FIG.20. Energy levels of the H2-H2+ system. The vibrational levels are unlabeled but to scale; from Jefferts (1968).

corresponds to an electron cloud pulsating along the @xed) molecular axis whose associated electric dipole moment can only interact with an electric field having a component in this direction (Mulliken, 1939). It is now convenient to represent the broad spectrum of light used for the photodissociation by pulses of radiation much shorter than vibrational or rotational periods.

144

H . G . Dehmelt

When taking into account the molecular rotation one will then expect that for large rotational quantum numbers K, the dissociation rate R will vary with sin2(K, E). The actual molecule will be primarily in states with small K values, and one also has to take into account electronic and nuclear angular momenta which couple with the rotational angular momentum. The resulting hfs states with total angular momentum F, on the basis of the vector modes depicted in Fig. 21, may be labeled ( K , F 2 , F, MF1, and the corresponding dissociation

FIG.21. Vector model for H2+. The strong magnetic electron nuclear interaction couples the total proton angular spin momentum Z and the electron spin angular momentum S to form the resultant F2 which, in turn, by a much weaker magnetic interaction, couples with the rotational angular momentum K to give the total angular momentum F; from Dehmelt and Jefferts (1962).

rates may be expressed as the quantum mechanical averages of sin’ (K, E) with E taken parallel to the z axis A look at Table VI, which gives R values obtained in this fashion, shows that, nevertheless, appreciable structure remains in the dissociation rates. Intense rf transitions between any two sublevels s, t will equalize the populations in these levels on the average and cause the sum of the initial populations a, + a, to decay with the average dissociation rate (R, + R,)/2while without rf the populations in each of the two levels would decay at their respective (different) rates. In this way, the residual ion number remaining after a fixed interaction period is made to reflect the action of the rf transitions (Dehmelt and Jefferts, 1962) quite analogous to the method described in Section 4.1.

145

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

TABLE VI PHOTODISSOCIATION RATESR OF hfs SUBLEVELS OF THE THREE LOWEST ROTATIONAL STATES MOLECULAR IONFOR ELECTRIC LIGHTVECTOR PARALLEL TO STATIC MAGNETIC

OF THE H2

+

FIELD".^

K

2

F2

F

gF

IMI

R

100a4

-215

312 112

0.650 1.350

7.43 0.45

+215

512 312 1/2

0.500 1.100 1.400

13.50 1.23 0.37

+546

+22/45

312 112

1.200 0.800

0.82 4.08

+544

+ 1019

112

1.000

1.83

+378

+2/5

512 312 1/2

0.750 1.050 1.200

4.98 1.50 0.82

-898

-219

312 112

0.750 1.250

4.98 0.67

-973

-219

1/2

1.000

1.83

+2

1/2

1.000

1.83

WF(Mc/sec)

112

--

0

Hfs energies W , ,g factors gF, and residual population numbers a4 after irradiation for four dissociation time constants are listed also. Initial populations a,, have been taken equal to unity. This table is from Dehmelt and Jefferts (1962).

An important difference is that the polarized light alone only causes an align-

ment of the sample with equal populations in levels with f MF. The basic apparatus used in the first experiments is shown in Fig. 22. Operating parameters of the trap and detection electronics are similar to those used in the ,He+ experiments. As a measure of the alignment building up during the interaction cycle it was, however, most advantageous to count the H + ions formed during a short period at the end of it. These early experiments established the workability of the new technique by the observation of the strong magnetic resonance at lgFl N 2/5 shown in Fig. 23, which is due to a number of hfs states; see Table VI. Magnetic resonances for all states listed in Table VI, except for the unobservable ones with F = .), were later investigated by Richardson et al. (1968). Analysis of the g,-factor ratios obtained in this work

H . G. Dehmelt

146

Elliptical mirror Mercury arc lamp

Brewster's ongle polorizer

ode surfaces of tube rotational symmetry

Electron gun structu Cathode and heater Glass-to-metal sea

Steady magnetic field H, directed into page

Direction of r f mag field

FIG.22. Diagram of trap and optics. Basic arrangement with which first experiments on alignment and magnetic resonance of H,+ were performed. The trap electrodes were formed by metallizing appropriately shaped regions of the Pyrex interaction tube; from Jefferts (1 962).

67647Hz

61069 HZ

FIG. 23. Unresolved magnetic resonance signal of the ( K F 2 F ) states (1, 3/2, 5/2), (2, 1/2, 3/2), (2, 1/2, 5/2), all having lgFI 2 2/5, of the H2+ ion in a field HO u 115 mG.A point-by-point digital analyzer display obtained by Richardson (unpublished) in 11 hours is shown.

147

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

for the (K, F,, FI states ( 1 , 1/2, 3/21, ( 1 , 312, 3/21, and (1, 312, 5/21 yielded the first value for the electron spin-rotational coupling constant in H,’, d = +32.2 f 5

MHz

since the corresponding interaction causes appreciable mixing of the first two states with F = 3. Somewhat later the much more accurate, but only, absolute values

I~(u

=5

f 1)1 = 30,239 f 1 kHz,

Id(u = 6 f 1)1= 28,092 f 1 kHz

were measured (Jefferts, 1968), u referring to the vibrational quantum numbers, by direct observation of transitions in the 70-MHz region between the levels F = 3 and F = 3 of the K = 2, I = 0 state., In this work the greatly refined apparatus shown in Fig. 24 was used. Theoretical estimates of the coupling constant ranged from d = - 150 MHz (Dalgarno et al., 1960) to d = + 122 MHz (Burke, 1960). blarlzed Light beom%

E

1

U FIG.24. Sophisticated apparatus used in experiments on the spin-rotational fine structure of H,+. Note selective ejection and counting by electron multiplier of H+ and H2+; from Jefferts (1968).

Most of the hfs spectrum for the k = 1 state, which extends into the 20 cm region, has now been observed; Jefferts (to be published).

H. G. Dehmelt

148 5.2

"

BOLOMETRIC " DETECTION OF THE e-CYCLOTRON RESONANCE

Even in a perfect vacuum, one may expect that the energy absorbed by an electron gas from the rf field when the cyclotron resonance is excited will quickly lead to a detectable. increase of its translational temperature via e-e collisions. Formula (2.40) should be valid for electrons in a magnetic field as long as the cyclotron radius is appreciably larger than the average distance between electrons. Monitoring the electron temperature by the techniques described in Section 3.3.3 now provides a convenient " bolometric " method (Dehmelt and Walls, 1968) for the detection of the cyclotron resonance of the electrons. The sensitivity of this method may be judged by a look at Fig. 25,

-

Frequency

-

Frequency

FIG.25. (a) Temperature increase of the electron gas as the exciting frequency is swept through the cyclotron resonance. The sidebands separated by multiples of w,,, , are caused by the magnetron motion through the inhomogeneous microwave field. The temperature scale is correct to approximately 50°K.The slight slope of the base line is due to losses from the initially injected electron sample during the duration of the sweep; which was approximately 4 min. (b) The cyclotron line with (1) and without (2) parametric postmultiplication of the electron-gas temperature. Off-resonance points of (1) and (2) refer to room temperature. The data were obtained with the apparatus shown in Fig. 9 containing about los electrons. From Dehmelt and Walls (1968).

which demonstrates that cylclotron resonance excitation resulting in a temperature increase of only a few degrees Kelvin are easily detectable. The technique may even be capable of detecting the spin resonance of the electrons provided a sufficient relaxation linkage between the cyclotron motion and the spins can be established. Also, in other ion gases favorable relaxation conditions may make rotational, electronic, and hfs spectra observable. It is also possible to employ cyclotron heating of such intensity that the electrons are boiled out of the trap and to detect the resonance by counting the reduced

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

149

electron number (see Fig. 19). Here relativjstic shifts can be kept small by using very shallow traps (Dehmelt, 1961) but may become appreciable in traps several volts deep (Graeff et al., 1968).

6. Spectroscopic Line-Shifts and -Broadening The general situation is in many respects similar to that encountered in the hydrogen maser (Kleppner et al., 1962). However, specific effects arise from the periodic character of the ion motion in the trap and the inevitable electric trapping fields. The discussion of most of these effects may be kept very brief since so far they have been too small for experimental observation. Numerical examples of the more important ones have already been given in Section 4.1.3.

EFFECTS 6.1 DOPPLER Dicke (1953) has shown that a large fraction of the energy radiated from an oscillator, whose motion may be arbitrarily fast, appears in the form of a spectral line of natural width and unshifted frequency as soon as the motion is restricted to a region of dimensions 544.The symbol 1 here refers to the wavelength(s) of the radiation field (free space, cavity) with which the oscilIator is interacting. Similar considerations underly the Mossbauer effect. A simple way of understanding the above result is available by considering an oscillator which is executing an oscillatory motion around the maximum of a standing-wave pattern of a radiation field from which it is absorbing energy. If the above condition holds the moving oscillator will see a radiation which is weakly amplitude modulated at twice the motional frequency and therefore contains, besides the strong, unshifted carrier, weak side bands at twice the oscillatory motional frequency and its harmonics. Consequently, as long as the rf field is essentially constant over the ion sample, the dominant, unshifted carrier will cause an easily identifiable signal. In the presence of nonvanishing gradients, side bands due to z and r motions and also the Larmor precession of the ion orbits in the magnetic field will be expected and have been observed (Schuessler et al., 1969). Unlike the first-order Doppler shift the relativistic second-order Doppler shift does not vanish. The shift is negative, and it holds with formulas (2.18) and (2.19) and the assumption 3kTi 5 e D introduced in Section 2.5 16v/vl = KE/(rnc2) = 3kTi/(mcZ)c eD/(mcZ). (6.1)

In the ,He+ hfs experiment 6v from this cause has been estimated as - 4 Hz, and it is the most important shift. If the ion orbits with their spread in kinetic energies were stable for indefinite periods, one would expect that this shift would be accompanied by a broadening of comparable magnitude. Various

150

H . G. Dehmelt

randomizing effects which are fast compared to the spin-exchange time T, should, however, greatly reduce the broadening through averaging of the kinetic energy. Formula (6.1) clearly indicates the advantage to be gained from the use of low temperatures and heavy ions. 6.2 ELECTRIC FIELDEFFECTS The electric fields used in the trapping of the ions will, of course, always cause a Stark effect and limit the usefulness of the method to ions not exhibiting strong first- or second-order Stark shifts. This is not too serious as by far the most ions only show small second-order effects. Most favorable in this respect appears to be the situation found for the hfs spectra of single-electron atoms in S states. Fortson et al. (1964) have measured for the 0-0 transition of the hydrogen grounds state hfs

6 v / v = -3.4 x ~ O - ~ ~ E [E]~ =, V/cm, (6.2) while for the analogous transition in such a highly polarizable atom as Cs Haun and Zacharias (1957) found only

6v/v = -2.5 x 10-16E2, [El = V/cm. (6.3) A theoretical formula which agrees with the measurements on H and which may be generalized to hydrogenic systems has been given by Schwartz (1959), 6 v / v = -(14.91 ao4)/(Z6e2)E2 [cgs], (6.4) showing a strong inverse dependence on the ionic charge. For the 3He+ hfs this shift is negligible. The electric fields in the trap are, by virtue of the ion motion, also the source of small magnetic fields of a magnitude which, in cgs units, is reduced by the factor v/c. In past experiments this quotient did not In electric fields of the order 1 [cgs] or 300V/cm thiscorresponds exceed to motional magnetic fields of order lo-' G. This is not a negligible value when systems with g factors as large as 2.8 MHz/G are involved. Analogous magnetic ac fields with frequencies equal to the various combinations of multiples of O,,W,,and R might become especially annoying if they coincide with an internal ion frequency. For low energy ions concentrated in the fieldfree center of the trap all trapping field effects are, of course, correspondingly reduced.

FIELDEFFECTS 6.3 MAGNETIC The rf spectroscopy experiments of the University of Washington group have been troubled by problems due to 60 Hz magnetic hum fields of 2.0.5 mG since it was impractical to provide better shielding in these early experiments. Because of the resulting frequency modulation the field-sensitive a, c,

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

151

d, e, andftransitions of 3He+ showed a large number of side bands spaced at 60 Hz and spread over a region =2 kHz wide, the swing of the frequency modulation. Other small-field fluctuations sufficed to wash this pattern out, making it impossible to resolve the individual sidebands. For the chosen value v, = 10 MHz this corresponds to a relative broadening of these lines, 6v,/ v, N 2 x The displacement of the 0-0 transition, 26 N 23 kHz, depends quadratically on v,. This transition would therefore be expected to show a corresponding modulation swing of 4 x or 9 Hz; however, since this value is much less than 60 Hz the sidebands will be weak, and no broadening of the carrier will result from this cause. Because of the quadratic dependence of 6 on the magnetic field, ac fields may, however, cause noticeable dc shifts. In this connection the circular magnetic rf fields associated with the displacement currents from the electric trapping fields at R may become important. In the experiments referred to these were less than 2 mG which under the worst possible circumstances H , = 0, would have led to an average value A6 5 Hz. These rf fields may also cause internal ion transitions when resonant with them. In contrast to hum fields, static inhomogeneities of the magnetic field are much less serious since the ion motion tends to average them out. However, Majorana flops occurring in inhomogeneous magnetic fields when one of the motional frequencies coincides with an internal ion frequency may cause lifetime broadening of the states involved. These flops, which are simply due to the fact that the ion in its own reference frame may now see magnetic rf fields at all of the motional frequencies and various combinations thereof, may even be turned to good use if it is desirable to establish relaxation channels. The usefulness of this procedure may beextended by applying inhomogeneous magnetic rf fields making it possible to induce the flops at various sum and difference frequencies (cf. Section 4.2.2). 6.4 COLLISION EFFECTS All collisions with other atomic systems are potentially capable of shifting and broadening the rf resonance lines of the hyperfine spectrum of the ions under study. The broadening may either be caused by the modification of the hfs interaction during the collision or by collision-induced relaxation effects shortening the lifetime of states involved. We now will briefly discuss in order of decreasing importance some such collision processes, beginning with two resonant ones. Charge exchange with the parent atom He+ + H e + H e + H e +

(6.5)

may terminate the existence of a given hfs state by exchanging the nucleus of a ,He+ ion for a ,He nucleus in another spin state and thereby cause lifetime broadening. It is the process which makes it more difficult to orient ,He+ ions

H , G. Dehmelt

152

than 4He+ ions. To the extent that one may assume the same nucleus will not be involved in two consecutive such collisions without having its spin state thoroughly disoriented in the meantime, no shift will result. The velocity dependence of the cross section is shown in Fig. 10. For low velocities the collision frequency tends to a finite value. The spin-exchange process He+?

+ Csl + H e t l +

Cst

(6.6) is also resonant. It causes lifetime broadening through termination of the electron spin state of the ion (Wittke and Dicke, 1956). Spin-exchange shifts may be estimated as follows. Assuming unpolarized Cs atoms and ,He+ ions, the collisions result in the formation of short-lived complexes of which threefourths are in the triplet state and one-fourth in the singlet state. The hfs associated with the ,He nucleus in the triplet molecular complex is essentially the same as in the free ion while this hfs vanishes for the singlet state. Consequently, the ,He+ ions moving through a Cs beam of particle density will have their hfs turned off for a small fraction, of the time. For f one takes 4/3xr&, the fractional spin-exchange volume, where rSEis one-fourth of defined by nr& = ZQ, . The fractional spin-exchange shift of the hfs separation Av is then approximately given by the quantity$ The above effects may also be caused by other paramagnetic atoms and molecules present in the residual gas such as 0 2 .Spin exchange between ions will vanish as the energy is reduced because of the ion-ion repulsion. With respect to the use of inert buffer gases, cf. Section 3.3.2, collisions of the type

-

(6.7) are also of interest. Experimental evidence shows that the spin states of S-state paramagnetic ions are only moderately more sensitive to such collisions than paramagnetic atoms (Ackermann et al., 1967). Hg+T+He+Hgt&+He

7. Conclusion In the preceding chapters, we have seen that it is possible to carry out highresolution spectroscopy of ions by storage-collision techniques. The resolution achieved for the hydrogen atom by very specialized techniques has so far not been attained in ion experiments. However, it appears that high-resolution spectroscopy of stored ions as a field is only in its infancy. The technique is potentially applicable to all, ground state or metastable, atomic and molecular ions, singlyor multiply charged, and not restricted to S states. For the purpose of orientation, hfs or fs state selection of the stored ions, other reactions than those discussed here are likely to be used in the future. These may include optical pumping of the ions, ion-molecule reactions at low energies, charge exchange including charge exchange of ions with isotopic atoms or molecules

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

153

at low energies, superelastic collisions, electron impact ionization, and electron impact dissociation of molecules and molecular ions. No ultrahigh resolution technique for neutral atoms of comparable generality seems to be available at present. The techniques described may also be useful in collision studies at low and ultralow energies involving polarized ions.

ACKNOWLEDGMENT The author is indebted to his colleagues and collaborators Dr. E. N. Fortson, Dr. H. A. Schuessler,Dr. C. B. Richardson, Dr. T. S. Stein, and Dr. G. H. McCall and to his graduate students S. C. Menasian, D. A. Church, F. L. Walls, and K. Zieher for discussions and for reading the manuscript or parts of it and offering suggestions for its improvement. The manuscript was typed by Miss Karen Haskell.

ERRATA FOR PART I Formulas (2.43) and (2.44) given in Part I of this article should read

Tc = (nmaxhYc*N 3-”l’(~/qmsx)Tc**

(2.43)

On pages 63, 65, and 66 read F??< b, V, = (16r/3)rq3,and q/qms.= 150, respectively. On page 72, read T,= 40 rnsec and t, =0.6 msec instead of T,= 200 msec and t, = 3 msec. Also on page 72, read: Wuerker, R. F., Goldenberg, H. M., and Langmuir, R. V. (1959). J. Appl. Phys. 30,342 and 441.

REFERENCES Ackermann, H., Z u Putlitz, G., and E. W. Weber (1967). Phys. Letters MA, 567. Bloom, A. L. (1962). Appl. Opt. 1, 61. Burke, B. F. (1960). Astrophys. J . 132,514. Church, D., and Dehrnelt, H. G. (1969). J. Appl. Phys. 40. Dalgarno, A,, Patterson, T. N. L., and Sommerville, W. B. (1960). Proc. Roy. SOC.(London) A259, 100. Dawson, P. H., and Whetten, N. R. (1968). J. Vacuum Sci. Techn. 5, 1, 11. Dehrnelt, H. G. (1961). Private communication. Dehmelt, H. G. (1968). Proc. Intern. Symp. Phys. One- and Two-Electron Atoms, Munich, September, 1968.

154

H. G. Dehmelt

Dehmelt, H. G.,and Jefferts, K. B. (1962). Phys. Rev. 125, 1318. Dehmelt, H. G., and Major, F. G. (1962). Phys. Reu. Letters 8, 213. Dehmelt, H. G., and Walls, F. L. (1968). Phys. Rev. Letters 21, 127, Dicke, R. H. (1953). Phys. Rev. 89, 472. Fischer, E. (1959). Z. Physik 156, 1. Fortson, E. N., Kleppner, D., and N. F. Ramsey (1964). Phys. Rev. Letters 13, 22. Fortson, E. N., Major, F. G., and Dehmelt, H. G. (1966). Phys. Rev. Letters 16, 221. Graeff, G., Klempt, -, and Werth, G. (1969). Z. Physik 222,201. Graeff, G . , Major, F. G., Roeder, R. W., and Werth, G. (1968). Phys. Rev. Letters 21,340. Haun, R. D., Jr., and Zacharias, J. R. (1957). Phys. Rev. 107, 107. Huggett, G. R., and Menasian, S . (1965). Private communication. Jefferts, K. B. (1962). Thesis, Univ. of Washington, Univ. Microfilms, Ann Arbor, Michigan. Jefferts, K. B. (1968). Phys. Rev. Letters 20, 39. Jefferts, K. B. (to be published). Kleppner, D., Goldenberg, H. M., Ramsey, N. F., (1962). Phys. Reo. 126, 603. Landau, L. D., and Lifshitz, E. M. (1960). “Mechanics.” Pergamon Press, Oxford. Major, F. G . (1962). Thesis, Univ. of Washington, Univ. Microfilms, Ann Arbor, Michigan. Major, F. G., and Dehmelt, H. G. (1968). Phys. Reu. 170,91. Menasian, S . (1968). Private communication. Mulliken, R. S. (1939). J. Chem. Phys. 7 , 20. Novick, R., and Commins, E. D. (1958). Phys. Rev. 111, 822. Paul, W., Osberghaus, O., and Fischer, E. (1958). Forschungsber. Wirtsch. Verkehrsministeriums Nordrhein-Westfalen No. 41 5 . Purcell, E.M., and Field, G . B. (1956). Asfrophys.J. 124, 542. Rapp, D., and Francis, W. E. (1962). J. Chem. Phys. 37, 2631. Rettinghaus, G. (1967). Z. Angew. Phys. 22,321. Richardson, C. B., Jefferts, K. B., and Dehmelt, H. G . (1968). Phys. Rev. 165, 80. Salmona, A. (1965). Compt. Rend. 260,2434. Schuessler, H. A. (1968). Bull. Am. Phys. SOC.13, 1674. Schuessler, H. A., Fortson, E. N., and Dehmelt, H. G. (1969). Phys. Rev. (to be published). Schwartz, C. (1959). Ann. Phys. 6, 156. Shockley, W. (1938). J. Appl. Phys. 9, 63.’. Sirkis, M. D., and Holonyak, N. H. (1966). Am. J . Phys. 34,943. Sternheim, M. M. (1963). Phys. Rev. 130,211. Wittke, 5. P., and Dicke, R. H. (1956). Phys. Rev. 103, 620. Wuerker, R. F., Shelton, H., and Langmuir, R. V. (1959). J . Appl. Phys. 30, 342. Zwanziger, D. E. (1961). Phys. Rev. 121, 1128.

THE SPECTRA OF I I MOLECULAR SOLIDS 0. SCHNEPP Department of Chemistry University of Southern California Los Angeles. California

I . Lattice Vibrational Spectra ........................................

A . Introduction ................................................ B. General Theory .............................................. C. Intermolecular Potentials ...................................... D . Complete Brillouin Zone Treatments ............................ E. Inorganic Molecular Solids .................................... F. Organic Solids ................................................ G . Intensities of Lattice Vibration Spectra .......................... I1 Intramolecular Vibrational Spectra ................................ A . Introduction ................................................ B. General Theory .............................................. C. Dipole-Dipole Treatments .................................... D . Atom-Atom Interaction Potential Treatments .................... E Applications ................................................ F Intensities of Intramolecular Vibrational Spectra .................. G Induced Spectra .............................................. 111. Spectra of Solid Hydrogen ........................................ A Introduction ................................................ B . Pure Rotational Spectra of Solid Hydrogen ...................... C Vibration-Rotation Spectra of Solid Hydrogen .................. D . Spectra of Cubic Hydrogen .................................... References .....................................................

.

. . . . .

155 155 156 160 160 164 170 174 176 176 177 179 181 182 186 187 187 187 188 191 196 197

.

I Lattice Vibrational Spectra A . INTRODUCTION

Lattice vibrations of molecular solids involve motions of the molecules as a whole. Such vibrations can for many purposes be divided into two types. translational motions in which the centers of mass of the molecules are displaced from their equilibrium positions and librations for which the centers of mass are fixed whereas the molecules perform quasi-rotational motions. The importance of such lattice motions in molecular solids was recognized 155

156

0. Schnepp

early with respect to the x-ray determination of crystal structures (Cruikshank, 1958). More recently it has become clear that the investigation of the lattice dynamics of such solids provides an important source for information concerning intermolecular potentials. The first attempt to calculate librational frequencies from intermolecular potentials in the form of atom-atom interaction terms was made by Dows (1962b) for the ethylene crystal. Some attempts were made to deduce the frequencies of lattice vibrations indirectly from x-ray investigations (Cruikshank, 1958) and from the infrared spectra of molecular solids in the region of intramolecular vibrations where lattice modes have been observed in combinations with the intramolecular vibrational frequencies (Dows, 1959). Direct observations of lattice vibrations was made possible by the development of good Raman techniques which permitted the investigation of small frequency displacements. Kastler and Rousset (1941) investigated naphthalene solid, Fruhling (1950) investigated solid benzene, and the Raman frequency of hexamine (hexamethylene tetramine) was reported by Couture-Mathieu et al. (1951). Recent developments in the Raman techniques by the application of laser excitation has made further progress possible (see Sections I.E, 1.F). In many cases only part of the lattice vibrations can be observed in the Raman spectrum. Therefore, the significant progress made in far-infrared instrumentation has been a further stimulus to this field. During the past five years, commercial spectrometers and interferometers for far-infrared spectroscopy have become available. Anderson et al. (1964) were the first to apply interferometry to the problems of lattice vibrations. These authors investigated the hydrogen and deuterium halides; Walmsley and Anderson (1964) investigated the halogen crystals; and Anderson and Walmsley (1964) reported the far-infrared absorption spectrum of C 0 2 and that of ammonia and hydrogen sulfide (Anderson and Walmsley, 1965). This pioneering work was expanded in the following years by other groups. No comprehensive review on the subject of lattice vibrations of molecular solids is available at this time although it has been briefly treated in reviews dealing with the infrared spectra of molecular crystals (Dows, 1963,1965,1966). On the other hand, a number of good reviews on the lattice motions of ionic and metallic crystals are available (Mitra and Gielisse, 1964; Martin, 1965). The classical source for lattice dynamics is the book by Born and Huang (1 954). Another standard reference is the book by Ziman (1960).

B. GENERALTHEORY To illustrate the treatment, we shall describe the work of Walmsley and Pople (1964) on the intermolecular vibrations of solid C 0 2 in some detail. The displacement of theqth molecule in thepth cell is denoted by r,(p/q) where

THE SPECTRA OF MOLECULAR SOLIDS

157

ct can assume the three designations x,y , z. The spatial direction of the linear molecule is denoted by the unit vector

AucP/q) = A,OcP/q) + Aa(P/q)

(1)

where Aao is the equilibrium value and &(p/q) is the displacement coordinate. Each molecule has only two rotational degrees of freedom, and therefore a redundancy is included if a here is allowed to assume three designations x, y , and z as above. The redundancy condition in this case is obtained in the form ofEq. (21,

which simply signifies that the unit vector has retained its length and only changed its orientation. Next, symmetry coordinates of the crystal space group are formed and this is achieved by a transformation of the molecular displacement coordinates to linear combinations which belong to irreducible representations of the translation group. These symmetry coordinates are of the form r:(q) = N-'"

A:(q) = N-l"

C expC2nik P 1expC2nik

R(p)]ra(p/q) R@)]la(p/q).

(3)

P

Here N is the number of unit cells in the crystal, k is a wave vector, and R@) is the position vector of the pth cell. The use of such coordinates will assure the factorization of the energy problem or the problem of the normal vibration frequencies of the system into problems concerning a particular point in the Brillouin zone designated by the wave vector k. This problem can then be formulated generally in terms of k and solved for any point in the zone. Since, however, the transition probability from the ground state to a single phonon state (excitation of one quantum of lattice vibration) will vanish both for the Raman effect as well as for infrared absorption except for k = 0, Walmsley and Pople treated only this point of the zone usually designated by r. In principle, the vibrational degree of freedom excited will have k equal to the wave vector of the radiation interacting with the solid. However, this wave vector will be very close to zero since the wavelengths of light involved are very long compared with the lattice spacing. For k = 0 further simplification of the problem is possible by making use of the symmetry at this point, which is the full factor group symmetry (Koster, 1957). For solid CO, this factor group is T,,,and therefore symmetry coordinates can be formed which belong to irreducible representations of the factor group by forming the proper linear combinations of k = 0 functions of

0.Schnepp

158

the type given in Eq. (3). The resulting symmetry coordinates are of the following form: (i = 1, . . . , 12) Si = T/,aqr:(q)

2

a.9

where r;(q) and AaO(q) are given by substituting k = 0 in Eq. (3) rao(q) = N-1’2 C r a W q ) , P

naO(q)

= N-”2

C Aa@/q>.

(5)

P

The crystal structure of C 0 2 , as already mentioned, belongs to space group Th6.The molecular centers are located at face-centered cubic (fcc) sites and the molecular axes are oriented parallel to the body diagonals of the cube. There are four such directions and accordingly there are four physically equivalent moelcules in the primitive unit cell. We therefore expect 4 x 3 translational degrees of freedom for every point k in the Brillouin zone. At the center of the zone, k = 0, the three acoustic frequencies are zero and represent the translational motion of the crystal as a whole. Thus we expect nine translational optical modes. Straightforward group theoretical analysis gives the result that this 9 x 9 reducible representation contains the following irreducible representations of the factor group T,,: A,

+ E, + 2T,.

Since the translations x, y , z belong to the representation T,. we expect two absorption bands in the far infrared. To solve the frequency problem, the potential energy in the harmonic approximation (i.e., for small displacements) is expanded up to second-order terms in the displacements. For k = 0 this is best done in terms of those linear combinations of the molecular displacements which form the symmetry coordinates Si given in (Eq. 4). The potential energy then has the following form : 12

where the force constants Fij are the second derivatives of the crystal potential with respect to the symmetry displacements. The frequencies are then given by the roots of the secular equation of dimension 12 IFij - M o 2 hijl = 0

(8)

where M is the mass of the molecule. Since the potential energy has been written in terms of symmetry coordinates, this determinant will break up into smaller blocks, each for an irreducible representation of the factor group. As usual, for degenerate representations, there will be an identical block for

THE SPECTRA OF MOLECULAR SOLIDS

159

each line of the representation. As already pointed out three of the roots of the twelve-dimensional problem will be zero and these can be identified in advance and separated from the problem in order to facilitate its solution. The librational lattice motions are treated in a parallel manner. For a centrosymmetric molecular site, as is the case in the CO, lattice, the symmetry coordinates formed from the translational displacements (4) belong to u representations, whereas the symmetry coordinates formed from the librationa1 displacements are even and belong to g representations. This can be understood if it is considered that a rotational displacement keeping the center of mass of the molecule fixed preserves the center of inversion, whereas the displacement of the center of mass bf the molecule in a translational displacement destroys that center of inversion. As a consequence of this differentiation between the two distinct types of lattice motions the translations and librations are accurately separable at k = 0 for this case. As formulated by Walmsley and Pople the quasirotational displacement coordinates are twelve in number, three components for each molecule in the unit cell. However, there are only two rotational degrees of freedom for a linear molecule and as a result four redundant coordinates have been introduced. It is therefore necessary to modify the procedure in order to take account of this fact. We therefore conclude that the librational motions span a reducible representation of order 8, and this reduces into irreducible representations of the factor group T,, as follows :

EB-t2T,. (9) Accordingly, we expect three Raman lines in the low frequency region. As already stated, the problem is similar to that of the translational motions except for the replacement of the mass of the molecule by the moment of inertia I in Eq. (8) and the problem of redundancies. It should be pointed out, however, that it is not necessary to introduce these redundancies as formulated by Walmsley and Pople. An alternate scheme can be used whereby two angular displacement coordinates are assigned to each molecule and the problem is then straightforward and does not involve any redundancy (Kuan and Schnepp, 1968). This is of particular advantage if the lattice dynamical problem is to be solved through the Brillouin zone. It is appropriate to digress here from the contents of the paper by Walmsley and Pople to point out what steps have to be taken in order to obtain solutions for points k # 0. In this case it is not as advantageous to use symmetry coordinates since the applicable symmetry is greatly reduced from that of the factor group symmetry. For a general point, there is, in fact, no more than trivial syrnmetry C, apart from the translational symmetry of the lattice. We therefore expand the potential energy in displacement coordinates of type (3) and calculate the appropriate force constants on this basis. It must

I60

0.Schnepp

also be remembered that for k # 0 the translational motions are no longer separable from the librations and therefore the problem has to be solved in full. If redundancies are not included, this implies that a secular problem of order 20 must be solved for every point of the Brillouin zone (every value of k). Three of those solutions will be those of the acoustic branches and the remaining belong to the optical branches. C. INTERMOLECULAR POTENTIALS It is now clear that the lattice dynamical problem can be solved once a potential for the crystal is formulated. It has generally been assumed that the potential of the crystal can be expressed as the sum of pair potentials. There has been some discussion of this assumption in the literature, but it is probably valid for molecular solids (Williams et al., 1967; Dymond and Alder, 1968). Two general types of pair potentials have been used in this field. The first, applied by Walmsley and Pople (1964) to the calculation of the k = 0 lattice frequencies of C 0 2 , consists of a central forces term in the form of a 6-12 Lennard-Jones potential and an angle-dependent term for which a quadrupole-quadrupole potential is used. The second type of intermolecular pair potential is expressed as a sum of atom-atom interaction terms between neighboring molecules. This latter type was developed for organic molecules in which case it seemed originally that the hydrogen-hydrogen repulsions were the dominant interactions. The first application of such a potential to the calculation of librational lattice frequencies is due to Dows (1962b), who used an exponential repulsion term in accordance with the approximate quantum mechanical calculation by deBoer (1942). Kitaigorodskii (1966) derived parameters for atom-atom potenials from crystal structures of aromatic systems. Such a potential has the form

- AIR6 + B exp( - CR)

(10)

where R is the interatomic distance of the interacting molecules. Coulson and Haigh (1963) listed parameters for hydrogen-hydrogen potentials and Williams (1967) determined potential parameters for nonbonded H-H, C-H, and C-C interactions from a large number of crystal properties, including structures, elastic constants, and sublimation energies of nine aromatic hydrocarbons. Potential constants determined by other workers in connection with specific applications will be discussed in the following sections. D. COMPLETE BRILLOUIN ZONETREATMENTS Cochran and Pawley (1964) discussed the theory of diffuse scattering of x rays by a molecular crystal, hexamethylene tetramine. In the course of this

THE SPECTRA OF MOLECULAR SOLIDS

161

work they studied the lattice dynamics of this solid, which is body-centered cubic (bcc) with one molecule per primitive unit cell. Both translational and librational degrees of freedom were considered and the lattice dynamics problem solved throughout the zone. Dispersion curves were computed for some symmetry directions and the density of state curve was computed. They parametrized the lattice dynamics problem and used four parameters for nearest-neighbor interactions and four more for next-to-nearest-neighbor terms. However, only three independent observables were available to evaluate these parameters, i.e., two independent elastic constants and the Raman frequency which had been observed at 40 cm-’ (Couture-Mathieu et al., 1951; Cheutin and Mathieu, 1956); this frequency represents the k = O optical libration of the molecule. The study is of great fundamental interest inasmuch as it is the first such investigation and demonstrates the mixing of translational and librational degrees of freedom throughout the zone. This work was followed up by Pawley (1967), who studied the lattice dynamics of naphthalene and anthracene. This author used a specific intermolecular potential which was expressed in terms of sums over atom-atom interaction terms of the form of Eq. (10). The parameters used were those given by Kitaigorodskii (1966). The lattices of naphthalene and anthracene are more complex inasmuch as they are of lower symmetry (space group P2,/u, C,”,), i.e., they are monoclinic crystals with two molecules per unit cell. Since the molecules are not linear, there are three rotational degrees of freedom for each and as a result we have six rotational and six translational degrees of freedom that must be considered at each point in the zone. Contacts up to 5.5 A were included in the calculation, but it was shown that only a 1 % change in the resulting frequencies occurred if the lattice sums were cut off at 5 A distance. Dispersion curves were given only for the [OIO] symmetry direction since this is the only symmetry axis in the crystal, and as a result, for any point along this direction in the Brillouin zone, the modes can be classified as symmetric or antisymmetric with respect to the two-fold rotation (symmetry species A and B, respectively).The results at k = 0 can be compared to known Raman frequencies. In the paper these values are compared to the results of Wilkinson (1 966) and the comparison was judged to be reasonable. No attempt was made in this calculation to adjust the parameters of the potential to give better agreement with measured observables. The results of this calculation and the experimental results of Wilkinson quoted here are included in Table I, which lists the naphthalene lattice vibrations as observed and calculated by different authors. Note that librations are of symmetry species A , and B, and are Raman active. Since the molecules are located at centrosymmetric sites, librations and rotations are separable at k = 0. Translations are of species A , and B, and are infrared active by contrast. The far-infrared data were not available at the time. Two sets of calculated results of Pawley

0.Schnepp

162

TABLE I LAITICEVIBRATIONS OF SOLIDNAPHTHALENE Observed Species

Clh A,

B,

Raman

IR

KandR"

Wb (300°K)

I' (77°K)

127 76 54 109 74 46

146 90 71 124 86 59

120 88 69 141 81 56

A.

B. a

Handsd (300°K)

98 53 66

Calculated Handsd (300°K) 115 85 58 82 70 42 97 54 74

119 86 62 93 77 47 104 58 76

PC (300°K) 134 94 61 126 70 77

139 92 62 129 77 49 88 47 58

Kastler and Rousset (1941). Wilkinson (1966) (from Pawley, 1967). Ito ef al. (1967). Harada and Shimanouchi (1966). Pawley (1967).

are given in Table I. The first set was calculated by assuming rotation of the molecules to be about the principal axes of the molecules whereas the second set is a more correct set, and it was found eventually that in some cases the actual normal mode represents libration of the molecule about axes considerably different from those of the molecule. In other words, the normal modes represent combinations of molecular motions and these are not bound to the molecular frame. The author also pointed out that the calculation shows the symmetric mode in each pair but one of the symmetric and anti-symmetric librational modes ( A g ,B,) to be of higher frequency whereas previously the opposite was assumed (Kastler and Rousset, 1941). Pawley also points out that his calculation neglected interactions between lattice modes and internal molecular vibrations, some of which occur at frequencies as low as 200 cm-', and therefore some serious errors may be expected due to this approximation. The density-of-states curve peaks for naphthalene at about 40, 55, and 80 cm-'. Also elastic constants that had been measured by ultrasonic methods were calculated and found to give a reasonable fit to the experimental data. Reference to Table I indicates that the infrared-active modes of naphthalene as calculated by Pawley are similarly in agreement with the observed values

THE SPECTRA OF MOLECULAR SOLIDS

163

of Harada and Shimanouchi (1966) within the same limits of error of about 10 cm-' as the Raman-active frequencies. It should also be pointed out that the density-of-states curves are probably of importance in determining the combination frequencies of lattice vibrations with internal molecular vibrations in the infrared and with electronic-vibrational spectra in the ultraviolet. For such combinations, k of the lattice vibration need not be zero but only the total lattice vector need give that value (i.e., the sum of the lattice vibration wave vector and that of the other combining degree of freedom). Unfortunately, the limits of error of the calculation are at this time such as not to allow clear differentiation between the k = 0 frequencies and the frequencies at which the density-of-state curve peaks. Most recently, Ron and Schnepp (1968) have undertaken a calculation of the lattice vibrations throughout the Brillouin zone for a-N,. The work was undertaken in an effort to account for large differences in line widths observed in the far-infrared lattice vibration spectrum of this solid. Two types of potential were used in this work. First, a central forces 6-12 Lennard-Jones potential was used. The crystal structure is very similar to that of solid COz already described, i.e., the molecular centers are located at fcc sites with molecular axes along the body diagonals of the cube. There are four molecules per unit cell. The accurate crystal structure was reported to be somewhat different from this structure but the deviations were neglected in this work. When the central force model as described was used, the problem was similar to that of solid argon, which has recently been treated in full (Grindlay and Howard, 1965), except that the frequencies are distributed differently throughout a zone of different structure. The argon crystal is face-centered cubic, but its primitive cell contains only one atom. As a result only acoustic branches occur. The difference between the two cases can best be visualized by considering the dispersion curves in the [lo01 direction. The length of the primitive unit cell is, in the case of argon, half the cubic cell parameter, whereas it is the whole cubic cell parameter in the case of nitrogen. As a consequence, the Brillouin zone is larger in the case of argon; and if we want to modify results of the argon problem to fit the case of nitrogen, we have to terminate the zone halfway and fold the dispersion curves backwards such that the points at the edge of the zone for argon fall at k = 0 for a-N, . The density-of-state curve is unaffected. The same problem was also treated by using an atom-atom potential with inverse 6-12 power interaction terms. The parameters for this potential were those determined in the course of a calculation for k = 0 (Kuan and Schnepp, 1968). The results of this work will be discussed further in the next section. The group theoretical analysis of lattice vibrations in molecular crystals throughout the Brillouin zone has recently been discussed by Chen and Dvorak (1968). They applied their discussion based on the theory of space

164

0. Schnepp

groups to hexamine (hexamethylene tetramine) which was treated by Cochran and Pawley (1964) as discussed above. Chen and Dvorak list symmetry points and symmetry directions in the Brillouin zone and discuss the point groups appropriate to these regions. They show that irreducible representations, the degeneracies, and the types of motion determined by the degree of mixing between translational and librational displacements. The second example discussed by Chen and Dvorak is adamantane, which also has one molecule per primitive unit cell and has an fcc structure. As a third example, these authors chose NH4CI.

E. INORGANIC MOLECULAR SOLIDS 1. Carbon Dioxide As already described, Walmsley and Pop.; (1964) treated solid CO,. They determined the normal frequencies of the translational and librational modes for k = 0. The intermolecular pair potential used has already been referred to and the molecular quadrupole moment is known. They obtained very good agreement between calculated and observed translational modes, but there were considerable discrepancies for the librational frequencies. Table I1 summarizes the frequencies of solid CO, . The T, modes were observed in the far-infrared by Anderson and Walmsley (1964) and by Ron and Schnepp (1967). The librational frequencies have been observed in the Raman spectrum by Ito (1968) and by Cahill et al. (1967a).

2. Solid a-Nitrogen

The far-infrared spectrum of crystalline nitrogen below the phase transition at 36°K (a-phase) has been reported by Anderson and Leroi (1966) and by Ron and Schnepp (1967). The latter authors also performed calculations similar to those by Walmsley and Pople for C 0 2 , using Lennard-Jones potential force constants available in the literature and the experimentally measured quadrupole moment of the molecule. The results are compared to the experimental frequencies in Table 111. The Raman frequencies have been reported by Cahill et al. (1967b) and have been remeasured by Brit et al. (1969). Both these groups agreed inasmuch as they found only two lines, whereas three Raman-active modes are expected. Again, agreement between calculated and observed values was very good for the translational modes but poor for the librations. Relative intensity measurements of the Raman lines coupled with calculations of these intensities indicate that there is accidental coincidence of two of the librational frequencies (Brit et al., 1969). Recently, calculations have been carried out using atom-atom potentials

TABLE I1 LATTICE VIBRATIONS OF SOLID Translational modes

Librational modes

Calculated

Observed -~

Symmetry

W and P'

A and W* Rand s'

Tu (Qz)

74 77 94 113

68 114

E" A.

T.

(Qi)

Intensity ratio; I J I z (R and S ) :

coz

5.4

Observed

Calculated

Symmetry

Wand P

E,

35 48 88

T, T,

I' (Dg) (195°K) 65 83 (1 12)

Lh (170°K)

70 85

124

L (90°K)

I (77'K)

I (4.2"K)

74 91.5 131

72 92 136

72 92 136

-l

2i

; 4

P

s

2.5 P m

Frequencies given in reciprocal centimeters. From Walmsley and Pople (1964), uiz = 4d(o/R)" - (o/Wl E = 3.18

x

ergs,

+Q-Q (T

= 3.72

Walmsley and Pople (1964). Anderson and Walmsley (1964). Ron and Schnepp (1967). Ito (1968). Dows (1959). Cahill et a/.(1967a).

x

cm.

0.Schnepp

166

TABLE 111 LATTICEVIBRATIONS OF %LID

a-NZ n*b

Translational modes Calculatedb

Observed

Symmetry

R and Sc K and Sd

A and Le R and S’

Tu (Qz) Tu Q(d

49 71

48.8 70

Intensity ratio, IJIZ : a

Librational modes Calculatedb

Observed

c, L, LO Symmetry

K and S

BRS’

Eo

27.8 38.4 70.2

-

T, To

33

37

1 f0.2

2.3

Frequencies given in reciprocal centimeters. /LIZ = ~E[(u/R)”- (u/RYI

E = 1.18 X

+Q -Q

ergs, Ron and Schnepp (1967). Kuan and Schnepp (1968). a Anderson and Leroi (1966). Brit et af. (1969). Cahill et ol. (1 967b).

u = 3.71 x

cm.

’

of the 6-12 type (Kuan and Schnepp, 1968). The results are summarized in Table IV, in which they are compared to the new assignments. Recently, St. Louis and Schnepp (1969) found that the two far-infrared absorption bands of u-N, in the far-infrared have very different line widths. The lower frequency line at 48.8 cm-’ has a line width smaller than 0.3 cm-’, whereas the higher frequency line at 70 cm-’ has a half-intensity width of 6 cm-’ (Table IV). This observation was made for a sample condensed in the liquid state in a closed cell which was then further cooled to form solid phase p and then further cooled through the transition temperature to give samples of phase c1 at 15°K. This observation motivated the lattice dynamics calculation referred to in the previous section (Ron and Schnepp, 1968). The aim was to investigate phonon densities of states which would be available for the relaxation of these two optical phonons into two other phonons subject to conservation of total lattice vector k = 0 and to conservation of energy. Translational lattice motions only were considered in this work. The ratio of the appropriate two-phonon state densities was found to be of the order of 20 indicating that the difference in line width of these two optical modes can probably be accounted for in terms of the relaxation of the optical

THE SPECTRA OF MOLECULAR SOLIDS

167

TABLE IV

LATTICE VIBRATIONSOF SOLIDa-Nz'Pb ~ _ _ _ _ _

~~

Translational modes

Symmetry ~~

Librational modes

Calculated

Observed

Calculated

K and S'

S and Sd

Symmetry

K and S

48.6

EP

35.3

Observed

c,L, L8 BRS'

~

Tu (Qz) Tu (Qd

72.5

48.8 (0.3) 70 (6)

Intensity ratio Zl/12(infrared):

1.05

1 f 0.2

I,

33 (1.5)

I = 3.6

To

TP

Intensity ratio (Raman):

35.4 44.8

31 (1.5) I= 1

2.3

3.6

Frequencies given in reciprocal centimeters (with half-intensity widths in parentheses). From Kuan and Schnepp (1968), 4

u12

= 4EC 1(u/&)12- ( O / W I a= 1

ergs, u = 3.387 x Kuan and Schnepp (1968). St. Louis and Schnepp (1969). Cahill et al. (1967b). Brit et al. (1969). E

J

= 0.415 x

cm.

mode into two phonons, as has been assumed for other systems (Kleinman, 1960). Further work is necessary before more definitive conclusions can be drawn.

3. Hydrogen Hardy et al. (1968) have recently discovered far-infrared absorptions due to optical phonons in solid hydrogen and solid deuterium in the low-temperatureordered state below the transition point at about 1.5"K. Below this temperature the crystal structure is face-centered cubic but neutron diffraction studies (Mucker et al., 1966) have shown that the crystal structure is lower than face-centered cubic. Theoretical work (Homma et al., 1967; Raich and Etters, 1968; Ueyama and Matsubara, 1967) predicted that the low temperature structure would involve an ordering of the rotational axis of quantization. The structure would thus be analogous to that of a-N2 except that here the quantization axis for the angular momentum would be [l 111 instead of this

0. Schnepp

168

being the direction of a fixed molecular axis. As a result, the J = 1 rotational state, which is the lowest state for orthohydrogen or for paradeuteriurn, would be split in the axial field of this site, M = 0 being lower relative to M = k 1, and this splitting was predicted to be of the order of 10 cm-'. Thus at temperatures corresponding to energies appreciably lower than this order of magnitude, a rotational ordering would occur since all molecules would then be in the M = 0 substate. Since now four sublattices physically distinct from one another occur, optical phonons are expected. These have, indeed, been found for enriched orthohydrogen and paradeuterium. Three absorptions have been recorded in each case, the lowest being very sharp, i.e., half-intensity width of less than 1 cm-' at 62.2 and 57.4 cm-' for hydrogen and deuterium, respectively. The second absorption line at 80 and 74.5 cm-', respectively, is broader and has a half-intensity width of about 4 cm-' ; and the highest energy line at 93 and 85 cm-'. respectively, has the greatest half-intensity width, which is about 10 cm-'. As discussed in the preceding section, or-N, and CO, have two lattice bands as expected from group theoretical treatment. Solid hydrogen is, however, basically different from all other solids discussed here. If a pair potential derived for gas phase interactions is employed, negative force constants are obtained for the lattice; and as a consequence, imaginary frequencies are found (deWette and Nijboer, 1965; Van Kranendonk and Sears, 1966). It must, therefore, be concluded that the harmonic approximation breaks down for these solids and this is correlated with the fact that the zero-point motions are large in amplitude. It turns out that the zero-point energy is approximately half of the heat of sublimation per molecule. Good agreement was obtained by Hardy et a/., between measured frequencies and those calculated on the basis of a modified potential fitted to measured parameters of the hydrogen solid. It should be noted that more fundamental treatments of this type of solid have recently been described (Nosanow, 1966; Fredkin and Werthamer, 1965) which formulate the treatment of " self-consistent phonons ". However, this treatment has not as yet been applied to solid hydrogen. 4. Miscellaneous Solids

Both Anderson and Leroi (1966) and Ron and Schnepp (1967) have measured the far-infrared absorption spectrum of CO. However, the disorder prevailing in this solid precluded accurate theoretical treatment although Ron and Schnepp carried out frequency calculations. Anderson et a/.,(1964) have reported the far-infrared spectra of hydrogen and deuterium chlorides and bromides at 77°K in the spectral region 20-400 cm - The absorptions were assigned as lattice vibrations and translational and librational modes were distinguished on the basis of frequency changes on

'.

THE SPECTRA OF MOLECULAR SOLIDS

169

deuteration. It was shown by these authors that their results are consistent with a crystal structure containing four molecules per unit cell in a planar array. Some intermolecular force constants were derived from the experimental results. Recently, Arnold and Heastie (1967) have investigated the farinfrared absorption of these solids in different phases, two phases for HCl and three for DBr. The absorptions due to the librational modes broaden faster with rise in temperature than the translational bands. Leech and Peachey (1968) have investigated the frequency distribution of phonons in the low temperature phase of solid HCl. Results of calculations have been reported for two possible space groups. Force constants were obtained by fitting the parameters of an atom-atom potential function to obtain agreement with observed infrared absorption frequencies. The infrared spectrum of crystalline H F and DF have been described by Kittelberger and Hornig (1967). A lattice band was observed near 200 cm-' for both H F and D F and the Raman spectra were found to contain one other lattice mode each. Raman and infrared lines are coincident and this observation has important implications for structural considerations. Cahill et al. (1967b) have reported the low frequency Raman spectrum of N,O at 90 and 170°K. The far-infrared spectra of crystalline acetylenes C,H, and C2D2 have been reported by Anderson and Smith (1966). Two bands were observed in each case and these were assigned as translational lattice modes on the basis of the frequency change on deuteration. Deductions were made relevant to the crystal structure. The results were found to be consistent with a D,,,factor group with two molecules per unit cell on sites of symmetry C Z hThis . work was done at 77°K or below the phase transition at 133°K. Low frequency Raman results were also quoted in this work. The translational lattice modes of the solid halogens Cl, , Br, , and I, have been observed in the far infrared by Walmsley and Anderson (1964). Two low frequency bands were observed in accordance with theoretical predictions on the basis of the known structure. Weak absorption bands, that were assigned to the intramolecular stretching frequencies, were also observed in this work. Reasons for the infrared activity of these modes are discussed. Anderson and Walmsley (1965) described the far-infrared spectra of ammonia, H,S, and their deuterated analogs. Giguere and Chapados (1966) reported the far-infrared spectra of solid H,0, and D,02. All the lattice vibrations predicted by group theoretical analysis were observed. A number of papers have appeared in recent years which treat the optical spectra of ices. Bertie and Whalley (1964a,b) discussed the infrared spectra of two ices. The translational lattice vibrations of orientationally disordered crystals have been discussed by Whalley and Bertie (1967) and the same authors (Bertie and Whalley, 1967) have applied this theory to the infrared spectrum of ices Ih and Ic. The frequency spectrum of hexagonal H 2 0 ice as

170

0. Schnepp

investigated by inelastic neutron scattering has been reported by Prask et al. (1968). F. ORGANICSOLIDS 1. Benzene

Information concerning benzene lattice vibrations known as of this time is summarized in Table V. The low frequency Raman spectra that were reported by Fruhling (1950) were recently reinvestigated at lower temperatures by Ito and Shigeoka (1966a). The far-infrared absorption spectrum of films condensed from the vapor on a cold surface has been recently reported by Harada and Shimanouchi (1967). Harada and Shimanouchi (1966, 1967) also carried out calculations on the basis of intermolecular pair potentials in the form of atom-atom interaction terms of the form of Eq. (10). They first took into account five H-H interaction pairs but found that the calculated frequencies were much lower than those observed. They thereupon included C-H interactions and found that the repulsive exponential terms both for the H-H and C-H interactions were decisive. Five pairs of H-H and ten pairs of C-H interactions were used. The exponential parameters [C in Eq. (lo)] were taken from the work of deBoer (1942) for H-H and from Abe et al. (1966) for C-H. Then the preexponential constants [B in Eq. (lo)] were fitted to the experimental observables, i.e., the Raman and infrared data. In this way good agreement was obtained, particularly for the Raman-active frequencies. The potentials so determined were further tested by application to H-H interactions for intramolecular vibrations in the gas phase. Good consistency was found for CH4 and C6H6. Similarly the C-H potentials were tested by applying them to intramolecular F-matrices where these appear as off-diagonal terms. Good agreement was found for CH,OH, CH,CN, C2H4, C2H6, and C,H6. 2. Naphthalene The lattice dynamics of naphthalene as already described has been discussed by Pawley (1967) and the available information concerning lattice vibrations is compiled in Table I. The early low frequency single-crystal Raman work by Kastler and Rousset (1941) has recently been supplemented by the work of Wilkinson (1966) and the work of Ito et a/. (1967). The farinfrared spectrum of naphthalene has been reported by Harada and Shimanouchi (1966). These authors also performed calculations at k = 0 and found again that they had to use H-C interaction terms in addition to H-H interaction terms. Two sets of calculated results of Harada and Shimanouchi

171

THE SPECTRA OF MOLECULAR SOLIDS

TABLE V

LATTICE VIBRATIONS OF SOLID BENZENE Raman-active modes

Species

0bserved

F" D2h

(-3°C)

Calculated

I and Sb (138°K)

H and Sc (-3°C)

H and Sc

H and Sc

H-H

H-H

H-H C-H

79 57 28

96 71 34

102

120 70 33

121 83 45

131 99 65

76 69 56

91 84 67

106 1 02 82

100

116 86 76

128 96 81

79 57 126

-

126 90

73 60

-

(138°K)

(138°K)

80 55

Infrared-active modes

Species

Observed

Dih

H and Sc (1 40°K)

Calculated

(-3°C)

H and Sc (138°K)

H and Sc (138°K)

58 49 26

68 56 35

86 66 54

85 70

58 34

68 44

79 66

94

57 31

67 38

86 55

94

44 30

56 35

87 49

H and S'

A.

Fruhling (1 950). Ito and Shigeoka (1966a). Harada and Shimanouchi (1966,1967).

0. Schnepp

172

(1966) are given in Table I. The first set was obtained by considering H-H interaction terms only, whereas the second set included C-H interaction terms. The two columns of calculated results by Pawley (1 967) in Table I represent two states of sophistication where in the first stage the librations of the molecules were assumed to occur close to the molecular axes, whereas the second set was obtained by determining the best frequencies for the crystal. In this latter calculation, as already discussed, Pawley found that some of the librations occur about axes which are as much as 30" inclined to the molecular principal axes.

3. Anthracene Pawley (1 967) has discussed the lattice dynamics of anthracene on the basis of Kitaigorodskii (1966) potentials throughout the Brillouin zone. Ito et al. (1967) have tabulated the Raman spectra of anthracene in the low frequency range as obtained by laser excitation. Colombo and Mathieu (1960) have described the optical properties and Raman spectra of solid anthracene. Pawley (1967) quotes Raman data by Wilkinson (1966). Bree and Kydd (1968) have measured the infrared spectrum out to 50 cm-'. 4. Ethylene

Table VI summarizes the information concerning solid ethylene. The even TABLE VI OF SOLD ETHYLENE" LATTICE VIBRATIONS

Species

DZL B3P

B2#

B1 0 B3u B2U

Observed

Calculated

B and Hb [IR(vib)]

B and Rc (far IR)

14 11

-

-

-

-

-

B and Rc Dd

I

90 40

-

-

-

-

75

40 -

-

12 (1.5)

110 (4)

-

100

-

61

-

I1

13

93 34

a Frequencies given in reciprocal centimeters (with half-intensity widths in parentheses). Brecher and Halford (1961 )-infrared combinations. Brit and Ron (1969tfar infrared. Dows (1962b).

THE SPECTRA OF MOLECULAR SOLIDS

173

vibrations have been observed only in combination with intramolecular vibrations by Brecher and Halford (1961). No Raman data are available. The far-infrared spectrum was recently observed by Brit and Ron (1968). Calculations were performed earlier by Dows (1962b) on the basis of atomatom interactions but only for the even vibrations. Brit and Ron (1969) have carried out two sets of calculations. one making use of a Lennard-Jones center-center potential (column I in Table VI) and a second calculation using exponential repulsion terms for the H-H interaction (column IL). Although three infrared-active vibrations are predicted only two have been observed. Brit and Ron used mostly samples prepared by spraying a vapor on a cold surface. However, in one experiment they used a sample prepared as liquid in a closed cell and then cooled to solidification. They observed that the line width was appreciably decreased by this sample preparation, and these line widths are given in parentheses in Table VI. This observation of line-width dependence on sample preparation is in accordance with the observations of St. Louis and Schnepp (1969) for a-nitrogen.

5 . Polyethylene The lattice motions in crystalline polyethylene have been discussed by Tasumi and Shimanouchi (1965). These authors calculated the translational lattice modes and compared their calculation with the observation of one far-infrared absorption at 73 cm-' as reported by McKnight and Moeller (1964), Bertie and Whalley (1 964a,b) Frenzel and Butler (1964), and Genzel (1964). The calculated frequency was 76 cm-'. The second expected infraredactive mode has not been found until recently when Dean and Martin (1967) reported a weak absorption line in a sample cooled to 2°K at 109 cm-'. These authors also reported a shift of the previously observed line to 79.5 cm-' at 2°K.The calculated value for the higher frequency mode by Tasumi and Shimanouchi was 105 cm-'. In view of the difference in lattice dimensions expected for the different temperatures, the agreement between calculated and observed frequencies is most satisfactory. Dean and Martin (1967) also reported a weak line at 110 cm-' for paraffin samples at 2°K. Tasumi and Krimm (1967) have discussed the effects of cell dimension changes with temperature on the frequencies of lattice motions. They found that it was possible to account for the observed frequency changes in the lower frequency line on the basis of such dimensional variation. 6 . Miscellaneous Solids

Ito (1964) observed the lattice vibrations of solid methyl iodide by Raman scattering. All six frequencies were observed thus confirming the low symmetry of the crystal structure.

174

0. Schnepp

Ito and Shigoeka (1966b) have investigated the lattice vibrations of solid pyrazine at four different temperatures and have compared these observations with calculations performed on the basis of correlations with the work of Harada and Shimanouchi (1966). Of particular interest is the N-H force constant in this solid since it represents a hydrogen bond interaction. It was found that this force constant is about one-tenth of that of the 0-H hydrogen bond force constant in crystalline formic acid. Very good agreement was achieved between calculated and observed frequencies. Ito et al. (1967) have also tabulated the low frequency Raman spectra of p-dichlorobenzene and p-dibromobenzene. Colombo (1967) has reported the low frequency Raman spectrum of crystalline p-toluidine. Brot et al. (1968) have studied the far-infrared spectra of liquid and crystal tertiary butyl chloride. This study is of interest in connection with rotation in the solid phases and their correlation with the motions in the liquid phase. The lattice vibrations of thiourea and the deuterated compound have been studied by Takahashi and Schrader (1 967). Calculations of these frequencies were performed and were found to be in good agreement with observed frequencies in the Raman and far-infrared spectra. The far-infrared spectra and structure of crystalline hydrazine have been discussed by Baglin et al. (1967). It is found that the crystal structure can be characterized on the basis of these studies. Shimanouchi and Harada (1964) have applied their computational methods for the vibrational degrees of freedom of molecular solids to cyanuric acid, uracil, and diketopiperazine. They also measured the farinfrared spectra between 300 and 90 cm-'. These authors characterized the hydrogen bond force constants in these solids. Temperature dependences of line widths in Raman spectra have been studied by Colombo and Moreau (1967). Such a study for cc-hydroxynaphthalene crystal has been performed by Korshunov et al. (1967).

G . INTENSITIES OF LATTICE VIBRATION SPECTRA Poll and Van Kranendonk (1963) treated the infrared absorption intensities of lattice branches accompanying induced infrared absorption in solid hydrogen. These treatments, as described in Section 111. C.2,, were based on molecular quadrupole-induced mechanisms. Schnepp (1 967) has formulated the theory for the far-infrared absorption intensities of the pure lattice vibrations in molecular solids. For crystals made up of nonpolar molecules, a molecular quadrupole-induced mechanism is again formulated. Thus, the absorption intensity is proportional to the square of the molecular quadrupole moment and the square of the molecular polarizability, and inversely proportional to the tenth power of the lattice constant. The calculation is carried out in full for the space group Pa3 (T:) with four molecules per unit cell suit-

THE SPECTRA OF MOLECULAR SOLIDS

175

able for CO, and approximately applicable to a-N, . Ron and Schnepp (1967) extended this theory to the accurate structure of a-N,, which is P213 (T4). Schnepp (1967) also discussed the intrinsic transition moment which occurs if the molecules making up the crystal have a dipole moment. In this case librational modes have an intrinsic moment which can be calculated. He also considered the derivative of the polarizability with respect to a lattice motion in connection with developing the theory of the quadrupole-induced intensity. This formulation is directly applicable to the calculation of Raman intensities of lattice motions as has also been pointed out previously by Van Kranendonk (1960) in connection with his interpretation of the spectra of solid hydrogen. Ron and Schnepp (1967) found that the intensity theory gave good order of magnitude descriptions of the observations for EN,, C O , , and CO. St. Louis and Schnepp (1969) have measured the absolute intensities of the farinfrared absorption lines of a-N,. They find that the measured intensities are lower than those calculated by a factor of at least 3 and possibly 4. The uncertainty arises from the observation that the intensity is strongly temperature dependent, increasing with lowering of temperature and falling to zero as the temperature approaches the transition temperature between u- and fl-N, at 36°K. The discrepancy between calculated and measured intensities is probably due to errors involved in the multipole expansion of the charge, i.e., in the representation of the molecule as a point quadrupole. David and Person (1968) have investigated the intensities of the farinfrared lattice vibration lines of the halogen crystal. These authors found a very good correlation between the measured absolute intensities and the molecular properties, quadrupole moments, and polarizabilities, in accordance with the theory of Schnepp (1967). Friedrich (1967) has discussed the absolute infrared intensity of the librational mode of crystalline OCS. Since this molecule has a dipole moment the librational intensities can be calculated as discussed above. Without considering the induced intensity, the agreement between calculated and measured intensities here is excellent. Takahashi and Schrader (1967) have discussed the intensities of the lattice vibrations of thiourea. These authors also limited their discussion to the intensity arising from the libration of the polar molecules. No absolute intensity measurements were carried out, however, The assumption made by both Friedrich and Takahashi and Schrader to the effect that the intrinsic intensity due to the dipole moment of the molecule dominates and far exceeds the induced intensity is borne out by the results of Ron and Schnepp (1967) for CO. Although the dipole moment of CO is very small (0.1 D), the intrinsic transition moment is already twice the expected quadrupole-induced moment. This theoretical value is now believed to be high on the basis of the results of St. Louis and Schnepp (1969) for a-N, . Brit and Ron (1969) have discussed the infrared absorption intensities of the translational lattice mode lines of ethylene.

0. Schnepp

176

They found that the measured intensities agreed qualitatively with those calculated from the quadrupole-induced theory of Schnepp (1967). Hardy et al., (1968) have very recently observed the far-infrared-active optical phonon spectrum of solid hydrogen and deuterium in the ordered state, i.e., below the phase transition at 1S"K. The structure of this low temperature phase is believed to be analogous to that of cc-N,. Therefore, the theory for induced absorption intensity of Schnepp (1967) is expected to be applicable. It was indeed found in this case that agreement between experimental and calculated intensities is within a factor of 2 for solid hydrogen, with the experimental intensity being lower, and the agreement was excellent for solid D2 , The superior agreement of experimental intensities with the quadrupoleinduced mechanism theory is to be correlated with the fact that the anisotropic part of the intermolecular potential of hydrogen molecules is very adequately described by a quadrupole-quadrupole interaction term (Van Kranendonk, 1960), whereas such a potential is not applicable to nitrogen (Ron and Schnepp, 1967; Kuan and Schnepp, 1968).

11. Intramolecular Vibrational Spectra

A. INTRODUCTION A number of reviews have been written on this subject during the last decade. The references are as follows: Dows (1966), Dows (1963, 1965), Mitra and Gielisse (1964), Vedder and Hornig (1961). It is largely the object of the present section to bring these reviews up to date. The near-infrared spectrum of solid hydrogen is a topic of such importance and depth that it will be treated in a separate section (Section 111). The original incentive to the development of the field of infrared spectroscopy of molecular crystals was twofold. On one hand it seemed promising to study the absorption spectra of single-oriented crystals in polarized light in order to obtain information helpful for the assignment of intramolecular vibrations. Since the molecules in the crystal are rigidly fixed in space, it was thought that it should be possible to relate the results of polarized light investigations to the symmetry of the vibration observed (Pimentel et al., 1955). It became clear, however, that a good theory of the molecular vibrations on the solid was indispensible, and this was indeed available at the time (Halford, 1946; Hornig, 1948; Winston and Halford, 1949). Dows (1966) has sketched the background theory briefly but with great clarity.

THE SPECTRA OF MOLECULAR SOLIDS

177

B. GENERAL THEORY The ground state wave function for the crystal is given by the product of ground state vibrational molecular functions:

Here q designates the site in the unit cell and p numbers the unit cells. An excited state function for the crystal might be written as a product representing all molecules but one in the ground state and the r, s molecule in an excited state i,

This function describes a localized excitation, localized on the molecule r, s, and does not possess the translational symmetry of the crystal. The correct crystal function belonging to representation k of the infinite translation group is a Bloch-type function

c

. 1 Qr’ = - exp(ik R,)Y;,, . J

N

S

If the crystal structure contains a primitive unit cell with more than one physically distinguishable molecule, i.e., if r as used in Eqs. (12) and (13) has more than one value, then further linear combinations between functions of the type given in Eq. (13) have to be formed in order to diagonalize the crystal problem. A set of solutions are found by diagonalizing the matrix of the crystal problem. A set of solutions are found by diagonalizing the matrix of the crystal problem for every value of k in the first Brillouin zone. It is easily shown that only the states of k = 0 can interact with light, and this selection rule is based on the fact that the wavelength is very much longer than the unit cell dimension. The potential Vis written as a sum whose first term is the sum of the intramolecular potentials and is called Vo and the additional term represents a sum of two-molecule interactions

It is justified in all cases discussed in the literature to treat the sum over two-molecule interaction terms as a perturbation since for molecular crystals the interaction energy is small compared to molecular energies. The perturbed energies obtained by first-order perturbation theory are give by the roots of the secular determinant

IH’ + E(5 - 5,Jl = 0

(15)

0. Schnepp

178

where X is the matrix of the perturbed roots Izi = 4n2cZvi2and

H$ = +ia

1’V,,,;,,,

dr.

u,u

Here + i a is one of the linear combinations of terms (13) which now belongs to one of the irrreducible representations of the factor group of the crystal. In this case, the determinantal Eq. (15) factors into blocks according to the irreducible representations. The first-order solution for the frequency of the ccth component of the multiplet associated with the ith nondegenerate vibration of the free molecule is given by = hcvio

+ D’ + M i a

with

~

‘

9

”

=

c

B a r B a u /(+:,s

+:,u>*

v r , s ; u , v <+y,s

+t,u>

d7.

(19)

u,u

Here, the Barare the coefficients in the symmetry combination 01 of functions where obviously the index r refers to a particular site in the unit cell or, in other words, to a sublattice consisting of translationally equivalent molecules. The term D’is an energy shift term and gives the frequency shift of the vibrational level in the crystal as compared to that in the free moelcule. Equation (18) shows that this is a term analogous to a Coulomb term. On the other hand, the term M i . agiven by Eq. (19) respresents a splitting term and has a different value for each k = 0 component in the crystal exciton band. If there are Z molecules in the primitive unit cell, then there will be Z different k = 0 states. Each such state is characterized by a different symmetry will have a different combination of functions (13) and for each such state Mi.“ value depending on the coefficients Barwhich appear in the symmetry function. The selection rules are then found by considering the final crystal states of a given symmetry. These states are of course classified according to the factor group of the crystal. This group consists of operations as in a point group except that it is coupled with translations and as a result may contain screw axes instead of pure rotation axes and glide planes instead of reflection planes. General methods of determining selection rules in crystals and the formulation of crystal functions have recently been summarized by Kopelman (1976a). This author defines an “interchange symmetry” which is made up of the factor group operations interchanging equivalent crystal sites. Kopelman thus formalized and described methods which have been used by workers in the field. Crystal symmetry coordinates and crystal symmetry have recently been discussed by Oehler and Giinthard (1968). Qri

THE SPECTRA OF MOLECULAR SOLIDS

179

C. DIPOLE-DIPOLE TREATMENTS The vibrational exciton theory was based on the electronic exciton theory of molecular solids (McClure, 1959). In this case, the first attempts at quantitative calculations were made for an intermolecular potential which was expanded in a multipole series retaining the first term, the dipole-dipole term. Thus the transition dipole moment which could be obtained from absorption intensities was used to carry out these calculations. Similarly, Hexter (1962, 1963) attempted to correlate the absorption intensities with the energy differences between the various components CI of crystal absorption lines. Hexter had varying success in these correlations and it seemed that the dipole-dipole interaction is only important in the case of very intense infrared absorption bands. Fox and Hexter (1964) formulated the exciton theory for a cubic symmetry crystal using the dipole-dipole term as interaction potential. For high symmetries the crystal is optically isotropic and only states belonging to the triply degenerate representation of the translations x, y, z are infrared active. Three cases are discussed according to the number of crystal lines to be expected corresponding to a given molecular vibrational exciton. The first case is the so-called one-T case. The space group of solid carbon dioxide is T : and that of carbon monoxide is similar except that here the molecule has no center of symmetry and as a result also the crystal space group is of lower order, namely, it is T4. In both cases there are four molecules per primitive unit cell which have different axis directions in space. For a nondegenerate molecular vibration four k = 0 states are obtained and the reducible representation of these can be reduced by group theoretical techniques to give A,, + T,, or in the case of CO, A T. Only the triply degenerate representation T,, or T is infrared active, as already pointed out. As a result only one line is predicted for the crystal, and this is indeed found experimentally for CO (Ewing and Pimentel, 1961) and for the antisymmetric stretching vibration v 3 of C 0 2 . The energy difference between the two states A and T can be calculated but cannot be observed. Other examples of the one-T case are v3 and v1 of solid N,O, which is isomorphous with solid CO and belongs to the space group T4 with four molecules per unit cell, and v 3 of solid SiF,, which is body-centered cubic (bcc), space group Td3.As already pointed out no splitting is expected from the crystal field theory. However, it is important to note that a crystallite shape dependence, that may produce further splitting is predicted by Fox and Hexter (1964). This shape splitting or shape dependence of the energies is due to the fact that a dipole interaction is assumed as the dominant perturbation mechanism and the resulting dipole-dipole lattice sum does not converge. As a result the energy is a function of the surface shape of the crystallite.

+

180

0. Schnepp

Maki (1961) and Ewing (1962) have indeed observed a shoulder on the high frequency side of the crystal absorption of a-CO. If this structure is indeed due to internally ordered crystals, it must be attributed to the shape splitting and thus confirms this theoretical prediction. No indication of splitting has been observed for solid NzO. The antisymmetric stretching frequency v3 of CO, has been observed by Osberg and Hornig (1952). These authors report a weak satellite but assign this to a reflection phenomenon. The second case considered by Fox and Hexter (1964) is the two-T case encountered when the doubly degenerate vibration v, of COz is treated. Because of the two fold degeneracy of the molecular excited state, eight k = 0 states occur in the solid and these reduce to give the representations E. 2Tu.Here two absorptions are predicted and an energy difference can be observed. The energy difference between the two-T states is given by

+

+?/,

p 2h, a - 3 .

(20)

Here p is the transition dipole moment, a is the cubic lattice constant; and h, is a lattice sum which here has a value of 4.333. The intensities of the two-T components are given by 1, = +(2 T

J3,

(21)

and the intensity ratio between these bands is given by Eq. (22)

I-/Z+ = 1.47.

(22)

Solid COz does, indeed, have two well-resolved peaks in the region of v, with a frequency difference of 5 to 6 cm-' and a relative intensity of 1.37 to 1.49. If the sample is prepared by fast deposition from the vapor on a cool surface, then three additional weak absorptions have been observed within a few reciprocal centimeters of the strong absorption doublet. Fox and Hexter (1964) attempted to fit all these components to the theory which takes into account the dependence of the spectrum on crystallite shape. It was found, however, that not all components could be fitted with respect to energy and intensity distribution. For the two intense components the energy splitting is about 100 % too high as calculated from this dipole interaction theory. On the other hand the intensity ratio is in agreement with the theory within experimental error. The spectrum of the chlorate ion C103- has been reported by Hollenberg and Dows (1960). Small particles that were suspended in a mull were used in the experiment. For v3 a splitting of 17 cm-' was observed, and this is again SO-lOO% from the calculated result for a flat slab-shaped crystallite but fits for a needle shape. For solid N,O only one absorption has been reported for v z , but this spectrum has not been studied at high enough resolution to observe the predicted splitting.

THE SPECTRA OF MOLECULAR SOLIDS

181

The third case considered by Fox and Hexter is the three-T case. This is encountered in phase I1 of solid methane. In this case, as the descriptive term states, three infrared-active components are predicted. There have been some reports of observation of three components but no splitting is predicted in the dipole-dipole approximation for this solid. Also the reported splitting magnitudes are unreasonable for a shape splitting. As a result it is concluded that the splittings observed are due to forces different from electrostatic dipole-dipole interactions. It is concluded that for intense infrared transitions the splitting energies observed are to a large degree accounted for by the dipole interaction approximation. However, this interaction term is far from generally satisfactory. This conclusion is in agreement with previous observations and in particular the same order of magnitude for dipole-allowed and for molecular dipoleforbidden transitions in the crystal (Dows, 1966). The most cited example is that of solid benzene where El, and E,, vibrations exhibit very closely similar splitting energies in the neighborhood of 3 cm-', whereas the former is dipole-allowed in the molecule, and the latter is dipole-forbidden, and therefore the dipole interaction theory would clearly predict decisive differences (Dows, 1966). D. ATOM-ATOM INTERACTION POTENTIAL TREATMENTS Another type of intermolecular interaction potential is expressed in terms of a sum over nonbonded pairs of atoms of neighboring molecules in the crystal. This formulation implies that an intermolecular pair potential can be expressed as a sum of atom-atom interaction terms. DeBoer (1942) showed that the repulsive overlap energy of two hydrogen molecules can be expressed as a sum of four atom-atom exponential terms. This was the result of a very approximate quantum mechanical calculation. Dows (1962) applied this type of potential to solid ethylene. A number of atom-atom interaction potentials have been proposed in the course of recent years, all of which are of the form - AIR6

+ B exp( - CR).

(23)

Kitaigorodskii (1966) reviewed earlier work along these lines. This author proposed sets of constants A , B, and C for Eq. (23) for C-C, C-H, and H-H interaction terms which were derived from the crystal structures of hydrocarbons. Pawley (1967) used the potentials proposed by Kitaigorodskii to calculate the complete lattice dynamics of naphthalene and anthracene. This work is discussed further in Section I concerned with lattice vibrations. Williams (1967) determined potentials of the type of Eq. (23) from the crystal structures and properties of nine aromatic hydrocarbon solids. He determined the C-C repulsive exponential coefficient from a calculation based on the

182

0.Schnepp

interplanar spacing and compressibility of graphite. He also adopted the H-H repulsive exponential coefficient given by deBoer (1942). Altogether 77 properties of nine aromatic hydrocarbon solids were used, including the crystal structures, elastic constants, and energies of sublimation.

E. APPLICATIONS 1. Single-Crystal Studies of Condensed Gases

Halford and his group made a large number of contributions to the study and understanding of infrared spectra of molecular crystals. This group developed a technique for growing single crystals of such substances as ethylene (Brecher and Halford, 1961), cyclopropane (Brecher et al., 1961), acetylenes (Freund and Halford, 1965a), and diborane (Freund and Halford, 1965b). The work on ethylene was particularly profitable inasmuch as it has proved to be difficult to determine the full crystal structure by x-ray techniques.

2. Ethylene The x-ray investigation of solid ethylene determined the position of the carbon atoms only and this skeleton forms a crystal of space group Dii-Pnnm, However Brecher and Halford (1961) were able to decide on the basis of this x-ray structure and the infrared absorption of single crystals in polarized light that the molecular site was centrosymmetric and the splittings observed determined unambiguously that the structure had to be of symmetry C,, . The crystal structure was then unambiguously determined as Cz,-P2,/n. Dows (1962a) also investigated solid ethylene at high resolution and was able to observe the crystal splittings better although he used polycrystalline samples and therefore could not use polarized light techniques. The splittings which he observed were consistent with the space group assignment of Brecher and Halford. Dows (1962b) used the H-H repulsion potential of deBoer to calculate force constants and vibrational frequency splittings by the perturbation methods outlined. He obtained reasonable agreement between calculated and observed values and concluded that the H-H repulsion parameters derived from the experimental splitting energies were consistent both with the potential proposed by deBoer and that of Amdur et al. (1961), which was of the form I/ = 1.44/r6." where the energy as given in electron volts and r is in angstroms.

3. Polyetliylene Tasumi and Shimanouchi (1965) have treated the splitting observed in the infrared absorption spectra of crystalline polyethylene and normal paraffins.

THE SPECTRA OF MOLECULAR SOLIDS

183

These solids are orthorhombic and contain essentially infinitely long polymethylene chains; and the factor group of these solids is D Z h .A UreyBradley force field was used which was based on previous molecular normal coordinate analyses. As intermolecular perturbation potential only short range H-H interactions were included in the calculation. In the infrared spectra doublets are observed for the CH, scissors and rocking vibrations due to intermolecular interaction (Krimm et al., 1956). These splittings are about 10 cm-'. Four H-H distances of two CH, groups were included in the calculation. The force constants were adjusted to fit the observed splittings in the infrared and to fit the Raman data (Brown, 1963). These force constants were compared to force constants calculated from exponential-6 potentials of the type given in Eq. (23) as tabulated in the literature (Coulson and Haigh, 1963) and also to the potential proposed by deBoer (1942). Reasonable agreements were obtained with some of these potentials, the latter being among the successful ones. Tasumi and Krimm (1967) studied the relative importance of transition dipole coupling and intermolecular H-H interactions. They found that the former is of minor importance in determining the crystal field splittings of vibrational lines.

4. Benzene The splittings of the vibrational absorptions of crystalline benzene and naphthalene have been studied by Harada and Shimanouchi (1966) and very recently by Rich and Dows (1968). The infrared absorption spectrum of single crystals of benzene were investigated in polarized light by Zwerdling and Halford (1955). These authors did not know the orientation of their crystal independently but determined the developed plane and the axes in that plane by invoking oriented gas intensity rules (see below). Dows (1966) has described the high resolution spectra of polycrystalline samples which exhibit very distinct and well-resolved splitting patterns. The A,. fundamental which is nondegenerate is expected to give rise to three absorption lines in the infrared and this number of lines is indeed observed. Zwerdling and Halford were able to distinguish to some extent between the different components by the use of polarized light but their sample was too thick to give them clear resolution. The in-plane E l , vibration v,,, which occurs near 1030 cm-' and vl,, which is of symmetry E,, , are seen to exhibit six components each, in accordance with predictions for a degenerate molecular mode. The former vibration is infrared active in the molecule, whereas the latter is forbidden, but in the crystal all u vibrations are allowed although they are considerably weaker. However, as already referred to previously, the splitting is of the same order of magnitude for both these bands, namely, in the neighborhood of 10 cm-' for the whole pattern. Kopelman (1967b) has recently discussed

184

0. Schnepp

Zwerdling and Halford's assignments and interpretation. Harada and Shimanouchi calculated the splitting energies of the benzene vibrations in the solid in order to account for the observations of Hollenberg and Dows (1962). At the time of this work, the crystal structure had only been described at - 3"C, whereas the experimental observations of the splitting were made at 85°K and 155°K. Three sets of atom-atom intermolecular potentials were used. The potential constants were fitted to the vibrational absorption splittings as well as to the known lattice vibrations. The latter have been discussed in Section I. F of this review. It was found that the force constants obtained here fit very well on the curve of the second derivative of deBoer's exponential repulsive H-H potential. Rich and Dows (1968) included twenty H-H contacts, comprising four contacts at each of five different distances between 2.4 and 2.7 A. They used the potential V(ergs) = 1.2 x lo-'' exp( - 3.54R) which is deBoer's potential. These authors get very good agreement for some of the modes, particularly the out-of-plane A,, mode, but the inplane vibrations were somewhat less successful. They also calculated the splitting expected for the totally symmetric breathing mode at 992 cm-' which is Raman active. Four components are predicted all of which would be observable. These components are predicted to range over a frequency of 1.2 cm-'. Raman spectra have been reported by Ito (1965), by Gee and Robinson (1967), and by Ito and Shigeoka (1966a,b). However, no splitting was observed although the investigations included low temperature work down to 2°K (Gee and Robinson, 1967). Similar results were obtained for the inplane degenerate Eze mode at 606 cm-' for which eight components are expected to cover a range of 8.6 cm-'. Gee and Robinson indeed, have observed four lines spread over 6 cm-' in this region. No definitive assignments, however, have been possible to date. Ito and Shigeoka (1966a,b) have reported observing crystal field splittings for v , (e,,) and vg (e,,) in the Raman spectra of benzene and perdeuterobenzene solids.

5 . Naphthalene Harada and Shimanouchi (1966) also considered solid naphthalene but confined their work to the calculation of lattice vibrations in view of ambiguities they felt existed concerning the intramolecular force field in their previous work. Rich and Dows (1968) made an extensive investigation of the splittings of molecular vibrations in solid naphthalene. Four sets of potentials were used, namely those due to Kitaigorodskii (1966), to Williams (1966) (two sets), and to deBoer (1942). They included 156 H-H pairs, 348 H-C pairs, and 192 C-C pairs. These contacts included all nonbonded interactions up to 5A. The authors conclude that the C-C interactions are unimportant but that the

THE SPECTRA OF MOLECULAR SOLIDS

185

C-H interactions must be considered. This conclusion is in agreement with the work of Craig et al. (1965) and Harada and Shimanouchi (1966). Agreement was obtained in the general trends of the splitting energies. Large calculated splittings corresponded to large observed splittings of the order of 10 cm-' and small splittings of the order of 3 cm-' corresponded to small observed splittings but the quantitative agreement is only moderate. As already described, the information concerning g-mode splittings is limited. However, it seems that the factor group splittings for such modes are generally small (Gee and Hanson, 1967). According to the calculations some splittings of g modes are larger than 1 cm-', but generally the predicted values are indeed smaller than those calculated for u modes. The larger splittings observed for out-of-plane vibrations are also reproduced by the calculations. Very significantly, the sign of the splittings, i.e., the polarization of, for example, the higher frequency component was correctly predicted in almost all cases. The spectrum of naphthalene crystal was reinvestigated by Rich and Dows in order to improve on previous work. The best quantitative fit was obtained by using the potential constants of Williams (1966). It is therefore concluded that potentials which are useful in accounting for properties such as elastic constants, coefficients of expansion, and heat of sublimation can also account for the splittings of the vibrational infrared bands. Ito et al. (1967) have recently observed polarized Raman spectra of single crystals of naphthalene, anthracene, p-dichlorbenzene, and p-dibromobenzene using argon laser excitation. They have assigned the Raman-active molecular and lattice vibrations of these solids and have generally found their results to be in agreement with previous assignments. 6. Miscellaneous Studies. Vibrational Assignments The investigation of the spectra of molecular crystals has for some years been a valuable aid in the assignment of molecular vibrations, as already pointed out. Some recent applications of this method will be listed here. Bree and Kydd (1 968) have measured the infrared spectrum of anthracene and have assigned all bands. Loisel (1968a, 1968b) has investigated naphthalene and quinoline. LeRoy and Jouve (1967) and LeRoy and Thourenot (1967) have reported on solid N,O and cyclopentene; 1,3,4-oxadiazole has been investigated by Christensen et al. (1968), urea by Nefeolov and Failkovskaya (1966), pyrimidine by Sobrana et al. (1966), and glycine derivatives by Novak and Cotrait (1966). Dows (1965) has completed the assignment of cylohexana fundamentals on the basis of a solid state investigation. Kyogoku et al. (1967) have investigated the infrared spectra of single crystals of l-methyl thymine, 9-methyl adenine, and their one-to-one complex.

186

0. Schnepp

7. Miscellaneous Studies. Crystal Structures

In cases in which x-ray investigations are incomplete, infrared and Raman spectra of solids have been helpfufto characterize crystal structures. Marzocchi et al., (1966) have investigated the structure and phase transition in CH21, and Kartha (1967) has assigned structures for CH,CI, , CH,Br2 , and CH212. Dows (1965) has partially characterized the low temperature crystal structure of cyclohexane and Marsault and Dumas (1967) have observed changes in the absorption spectrum of 1,4-dioxan. Brunel and Peyron (1967a,b) have obtained interesting results which bear on the crystal structures of HCl and HBr. Information on the crystal structure of ethylene oxide has been deduced by LeBrumant and Maillard (1967), of SO2 by Gerding and Ypenburg (1967), of cyclopentane by LeRoy (1967) of thiophene, furan, and pyrrole by Loisel and Lorenzelli (1967), and of 9,lO-anthraquinone by Pecile and Lunelli (1968). The absorption spectrum of solid HF has been reported by Kittelberger and Hornig (1967), and structural implications of their observations have been discussed by them also. The Raman spectra of solid CH, and CD, were reported by Anderson and Savoie (1956). These authors observed changes in the vibrational spectrum over the temperature range 10-77"K7 which includes all three solid phases of CD, and phases I and 11 of CH, .

F. INTENSITIESOF INTRAMOLECULAR VIBRATIONAL SPECTRA The intensities of the infrared absorptions of molecular crystals have been the subject of some attention. Dows (1966) has discussed this question and has evaluated the validity of the " oriented gas " model. This model assumes that the transition moments of the molecules are unaffected on condensation and therefore the dipole moments of the transition in the condensed phase can be obtained as a vector sum of the oriented molecular transition moments. In some cases, however, large quantitative changes have been observed on condensation and therefore the oriented gas model cannot be accepted as valid. In benzene, for example, one band is weakened by a factor of 3, on condensation, whereas another band is doubled. The out-of-plane A , , band remains practically unchanged on the other hand. The change of intensity is particularly severe for hydrogen-bonded solids and for HCI and solid state intensity is six times that of the free molecule. To date, it has not been possible to account for these considerable intensity changes on crystallization, theoretically, and the problem is still unresolved at this time. On the other hand, Person and co-workers (Cook et al., 1967) have found a number of instances in which the absorption intensity did not change greatly on solidification.

THE SPECTRA OF MOLECULAR SOLIDS

187

Recently, an investigation of the vibrational infrared spectrum of solid naphthalene has been reported by the technique of attenuated total reflection (Yamada and Suzuki, 1967). Single crystals and polarized light were used. This seems to be the first report of the application of this technique to solids, but it has been very successfully employed in the investigation of the absorption intensities of liquids (Crawford et al. 1966).

G. INDUCED SPECTRA Intermolecular forces can induce absorption in the crystal where the molecular vibration is inactive in the vapor. Many examples are known from the study of polyatomic molecules in the crystal state. For example, all u modes of benzene become weakly infrared active by this mechanism (Harada and Shimanouchi, 1966; Rich and Dows, 1968; Zwerdling and Halford, 1956). A number of examples of induced infrared absorptions are now known for solids of homopolar diatomic molecules. The well-studied case of H, is discussed in detail in the next section of this review. In this case the fundamental molecular vibration becomes active because the molecular site is not centrosymmetric in the crystal of hexagonal close-packed structure (Van Kranendonk, 1960; Van Kranendonk and Karl, 1968). The fundamental absorption of O2 in the infrared spectra of u- and 8-oxygen has been reported by Cairns and Pimental (1965). These authors conclude that the absorption appears in violation of selection rules since the 8 phase was known to have the 0, molecular sites centrosymmetric. Since the absorption intensities were similar for the two phases, Cairns and Pimentel concluded that the molecular sites are also centrosymmetric in the u phase. This has since been confirmed by the work of Barrett et al., (1967). The weak transition moment is ascribed to the perturbation by lattice imperfections. Walmsley and Anderson (1964) have reported weak absorptions assigned to the fundamental molecular vibrations in crystals of chlorine, bromine, and iodine. Again, the molecular sites are centrosymmetric and static crystal imperfections or lattice vibration coupling must be responsible for the nonvanishing transition moments. 111. Spectra of Solid Hydrogen A. INTRODUCTION Because of its importance, a separate section is devoted here to solid hydrogen, which has proved to be spectroscopically of great interest. This solid is unique inasmuch as the molecules rotate virtually freely as is evidenced by the fact that the rotational constants remain virtually unchanged on condensation. Rotation in the solid perseveres to very low temperatures because the

0. Schnepp

188

spacing of rotational levels is large (of the order of 350 cm-') compared with intermolecular binding energies. However, intermolecular interaction causes the delocalization of rotational excitation and this gives rise to a rotational exciton band of width 20 cm-'. As a result structure is observed in the spectrum. The solid parahydrogen spectra have been intensively investigated experimentally by Welsh and his group and theoretically by Van Kranendonk and co-workers. No comprehensive review of this beautiful work has been available. The early work has been reviewed by Vedder and Hornig (1961) and, while the present review was in preparation, an updated summary of the theory of the spectra of solid parahydrogen was published (Van Kranendonk and Karl, 1968). This publication covers the frequencies but does not discuss intensities or coupling of the molecular motions with phonons. B. PUREROTATIONAL SPECTRA OF SOLIDHYDROGEN

I . General Theory The theory of the rotational states of solid hydrogen has been described by Van Kranendonk (1959, 1960) and Van Kranendonk and Karl (1968). For a given molecular rotational state J , the 2 N ( 2 J 1) crystal states are described by the Bloch-type functions

+

Jlm'(k)

= N-"'

C exp(ik .Ri) [Om(Ri)f O,,,(Ri + T)]. i

+

(24)

Here m denotes one of the (25 1) substates of the rotational state J and k is the wave vector denoting the irreducible representation of the translation group; R i is the position vector of one of the two molecules in the unit cell i of the hexagonal close-packed crystal and R i + T is the position vector of the second molecule in the same unit cell characterized by Ri. There are N unit cells in the crystal. Since the wavelength of light is large compared to the unit cell dimensions, it can be shown that only the states of k = 0 will appear in the spectrum. Furthermore, since the crystal has a center of symmetry midway between the two molecules of the unit cell, only the even states to an interchange of these two molecules will be Raman active whereas the odd states will be infrared active. Quadrupole-quadrupole interaction between the molecules in the lattice causes the broadening of the molecular J = 2 state into a band which is calculated to have a width of approximately 30 cm-'. However, the experimentally observable k = 0 levels are predicted to form a triplet of levels representing transitions from J = 0 to the states m = 0, + 1 , + 2 of J=2.

THE SPECTRA OF MOLECULAR SOLIDS

189

2. Raman Spectra The pure rotational Raman spectra of solid hydrogen and deuterium were described by Bhatnagar et a/. (1962). For solid parahydrogen the spectrum contains three sharp lines of half-width 0.3 cm-' at 351.84, 353.85, and 355.83 cm-'. The spacings are nearly equal and amount to 2 cm-'. The'Raman spectrum of solid orthodeuterium also contains three narrow lines that are partially resolved and have frequencies 176.8, 179.4, and 182.0 cm-'. In normal hydrogen (25% para and 75% ortho), two asymmetric and broad bands are observed which peak at 353.5 and 586.8 cm-'. Clearly, the higher frequency band is due to ortho-H, . The Raman spectrum of solid HD in this region consists of one broad band centered at 268.7 cm-'. All these features, except the higher frequency band of normal H2,are ascribed to the transition S,(O) (J = 0 to J = u in the ground vibrational state u = 0) of the parahydrogen molecule and the higher frequency band of normal H , represents the transition So(l), ( J = 1 to J = 3, u = 0) of the orthohydrogen molecule. The three components of So(l) observed in the Raman spectrum of solid parahydrogen have been assigned to the states +m+(0) with the different absolute m values 0, f 1 , f 2 (Van Kranendonk, 1959,1960; Van Kranendonk and Karl, 1968). The spacing between these levels calculated on the basis of quadrupolar forces is about 10 % larger than the observed spacing of 2 cm-' The triplet pattern is centered about the level m = k 2 . It is interesting to note that the triplet spacing as experimentally observed is not accurately symmetric. Van Kranendonk considered a general crystal field perturbation of arbitrary intermolecular forces in an ideal lattice and found that even then the spacings are expected to be equal. The deviation of 0.03 cm-' is then attributed to lattice imperfections or lattice vibrations.

3. Infrared Spectra The So(0)line has been measured by Kiss (1959) in the infrared absorption of solid parahydrogen. The measured frequency is 355.6 0.3 cm-'. Van Kranendonk (1960) has given the theory of this absorption. It is interpreted as occurring due to the quadrupole-induced transition moment. The induced dipole moment is assumed to be additive, i.e., p=Cpi i

(25)

where i designates one of the molecules in the crystal and the sum extends over the whole crystal. Investigation of the transition moment. (+o

I clk I \Ir,*(k)>

(26)

190

0.Schnepp

immediately gives the result that transitions are infrared active only to states of k = 0 and those that are odd to the interchange of two molecules in the unit cell, in other words, states of the type $,,,-(O). The symbol pk designates a component of the dipole moment p, Furthermore, quadrupole-induced transitions will only occur from the ground state Jr, to the excited states of in = 2 2 . These transitions are also shown to be polarized perpendicular to the hexagonal axis. The calculated frequency is 355.7 cm-', which is in very satisfactory agreement with the experimental value. This frequency is also in satisfactory agreement relative to the frequency of the Raman lines. The integrated absorption intensity has also been caclulated by Van Kranendonk (1960) on the basis of the quadrupolar induction mechanism. The calculated integrated absorption intensity is 0.77 x sec-' cm3, whereas the experimentally measured intensity is 0.52 x 10-'3sec-'cm3 (Kiss, 1959). The discrepancy of 30 % is ascribed to the sharpness of the absorption line which precluded satisfactory spectroscopic resolution, and thus presumably caused an error in the intensity measurement. It is of interest to note that the lattice sum that enters into the calculations of the integrated absorption intensity would vanish if the crystal structure were face-centered cubic. This property arises from the fact that in this structure the molecular sites would be centrosymmetric, whereas for the hexagonal close-packed (hcp) structure the center of symmetry lies midway between the two molecules i n the unit cell, as already pointed out. Therefore, the observation of the S,(O) line strongly supports the hcp crystal structure (Van Kranendonk and Gush, 1962). Kiss (1959) also measured the infrared absorption due to the excitation of two rotational excitons, i.e., a line assigned to S,(O) S,,(O). The measured sec-' cm3, or of very similar integrated absorption intensity is 0.6 x order as the single-rotational exciton transition. Van Kranendonk (1960) has calculated the quadrupole-induced moment for this transition and has obtained sec-' cm3. The discrepancy is ascribed the theoretical value of 0.49 x to limited experimental accuracy due to the overlap of the phonon branch, which accompanies this transition. It turns out that the coefficients in the expansion of pk are small as compared to those which are responsible for the single-exciton transition moment. However, the lattice sum which occurs in the calculation of the single phonon transition is small due to cancellations caused by symmetry. This lattice sum squared occurs in the expression for the integrated intensity. On the other hand, the expression for the integrated intensity of the double-rotational exciton absorption contains a lattice sum which is the sum of square terms and therefore no cancellations occur. As a result the final values for the two cases are very similar. The pure rotational infrared spectrum of solid hydrogen is not available in the published literature. Van Kranendonk (1960) refers to measurements of frequency and intensity by Kiss (1959) which are contained in a thesis. From

+

191

THE SPECTRA OF MOLECULAR SOLIDS

the theoretical paper of Van Kranendonk (1960) and from a latter paper by Poll and Van Kranendonk (1962), it appears that both single- and doublerotational exciton absorption lines are accompanied by phonon branches which occur due to the coupling of the rotational motion with the lattice vibrations of the solid. Phonon branches will be discussed in connection with the infrared absorption of solid hydrogen in the region of the molecular vibrational fundamental band.

C. VIBRATION-ROTATION SPECTRA OF SOLIDHYDROGEN I . Raman Spectra

Bhatnagar et al. (1962) observed very sharp lines (half-width 0.1 cm-') in the Raman spectrum of solid parahydrogen at 4149.81 and 4485.85 cm-'. These lines were assigned to the Q,(O) and S,(O) transitions, respectively. In solid normal hydrogen also the Q,(l) line has been recorded. The corresponding lines have also been observed in the Raman spectrum of normal deuterium, orthodeuterium, and HD. Van Kranendonk (1959, 1960) and Van Kranendonk and Karl (1968) have developed the theory of the vibrational exciton band and of the vibrationrotation levels of solid parahydrogen. The width of the virbrational band of the Y = 1 molecular level was found to be 2.4 cm-'. The broadening of the molecular level into a crystal band is due to dependence of the intermolecular dispersion forces on the internuclear distance in the molecule. Again, only the k = 0 levels in the band will be active. Also here, analogous to the situation in the rotation band, two such levels exist, one which is symmetric and the other antisymmetric with respect to an interchange of two molecules in the same unit cell. Their energies are given in terms of a parameter E' which is called the vibrational coupling constant. Relative to the band origin the energies of these two states are E+(O)= -68' and E-(O) = 0. The reference energy (E' = 0) is skifted from the gas phase due to isotropic intermolecular forces. This shift has been determined empirically from the S,(O) line of the Raman spectrum. Again, only the state symmetric to interchange of molecules is Raman active. Using the empirical values of the reference energy as described above, the theoretical frequency value corresponds to the observed frequency for the Q,+(O) line of 4149.81 cm-' with the vibrational coupling constant E' = 0.40 cm- The calculated value is 0.50 cm- and this agreement indicates that the main contribution to the vibrational coupling is indeed due to dispersion forces. In accounting for the S,(O) line which appears as a single sharp line in parahydrogen, Van Kranendonk (1959, 1960) considers the coupling between vibration and rotation in the solid. As has already been described, exciton bands are formed which have their origin in the molecular rotational levels (width 20 cm-' for J = 2) and vibrational levels (width 2.4

'.

'

192

0 . Schnepp

cm-’ for u = 1). In addition, there exist 5N degenerate “impurity levels.” These represent the u = 1, J = 2 excitations on the same molecule and these levels lie below the rotational band. These states are localized and do not broaden to form a band. The band states are Raman inactive in the approximation that the polarizability is independent of intermolecular interaction. The inactivity is due to the fact that the band states represent vibrational and rotational excitations on different molecules as opposed to the bound localized states. The bound states are Raman active and give rise to the observed single narrow S,(O) lines. The frequency of the crystal S,(O) line is lowered relative to that of the free molecule by two terms. The first represents a shift due to isotropic forces in terms of a parameter which is empirically determined from the measured frequency. The second term is due to quadrupolar forces and depends on the quadrupole coupling which can be evaluated theoretically and the rotation-vibration interaction constant which is known from the gas phase spectrum. This part of the theory is elaborated upon by Van Kranendonk and Karl (1968). The experimental solid state frequency is 4485.8 cm-’, or it is shifted to lower frequency by 11.7 cm-’. The S,(O) line is also split similar to the S,(O) into three components by the anisotropic forces of the crystal field. By using the splitting parameter determined from the observed splitting of the S,(O) line, the spacing between the lines is calculated to be 0.35 cm-’. It is seen that the splitting here is an order of magnitude smaller than that for the S,(O) lines, as and a result the components have not been resolved by the experiment. Soots et ul. (1965) have studied the Raman spectrum at a series of orthoparahydrogen ratios. The Ql(0) line was found to split into a number of closely spaced lines for ortho concentrations below 0.20. These lines are interpreted as due to localized states with different neighbor interactions; the concentration is too low for coupling between orthomolecules and exciton band formation. On the other hand, a single maximum was observed for higher concentrations of o-H, , which is characteristic of the vibrational exciton of J = 1 species. Rosevear et al. (1967) have found an anomaly in the intensity ratio of the Q,(l) and Q,(O) vibrational lines in the Raman spectrum of solid H, and D, . The anomaly has been found to persist in the liquid state. 2. Infrared Spectra

The infrared vibrational fundamental band of solid hydrogen has been described in detail by Gush et al. (1960). At low resolution the band consists of three absorption bands, each 3040 cm-’ wide and increasing in intensity with frequency. Each of these bands is accompanied by a broad band adjacent to it and extending to higher frequency. The three bands have been assigned as related to the Q, S(O), and S(1)

THE SPECTRA OF MOLECULAR SOLIDS

193

transitions of the molecule. The broad features to higher frequencies of each of these bands have been interpreted as phonon branches. The Q phonon branch is by far the most intense and is much more intense than the narrower Q zero-phonon band. On the other hand, the S phonon branches are much less prominent relative to the respective zero-phonon bands. In addition, two weak features have been observed to higher frequency and these have been assigned as S(0) + S(l) and 2S(O).These are double transitions. The spectrum, as described above, is characteristic of normal hydrogen. Under high resolution, each of the narrower bands can be resolved into a number of components, and these high resolution spectra have been studied as a function of orthopara concentration. The Q band consists of three components, Q(l), Q(O), and a third line of higher frequency. With decreasing ortho composition, the Q(0) line increasingly dominates the Q(1) line as expected, but in addition the total intensity decreases and the whole Q zero-phonon branch disappears in the limit of pure parahydrogen. However, before this happens the three features which compose this branch are narrow and very well resolved at low ortho concentration (2) %. Van Kranendonk (1960) has developed the theory for the infraredfundamental induced band of solid hydrogen. The infrared-active k = 0 state of the vibrational exciton band is that which is antisymmetric to interchange of the two molecules in the unit cell. The pure vibrational transition to this level is designated Q,-(0), and it is predicted to be at 4152.2 cm-' as compared to the observed frequency of 4153.0 cm-'. The intensity is expected to depend on the overlap induction term, but the corresponding lattice sum vanishes for the hcp structure. As a result, the zero-phonon Q branch is infrared inactive in pure parahydrogen. The sharp line corresponding to this transition in solid hydrogen containing 2% orthohydrogen is due to the presence of the orthohydrogen impurity (Sears and Van Kranendonk, 1964). As already described, for such a solid three sharp features are observed. The Two at lower frequency are assigned to Ql(l) and Q,(O) and they are due to the presence of single-orthohydrogen molecules in an environment of parahydrogen. Since the Q,( 1) line is characteristic of the impurity orthomolecules, it remains sharp since the exictation is localized. On the other hand, the Q,(O) transition has a width of about 3 cm-' because of the vibrational exciton band width. The J = 1 level of the orthohydrogen molecules is threefold degenerate and in a perfect hcp lattice this degeneracy is not removed. However, removal of the degeneracy is expected to arise from the large amplitude zero-point vibration of the lattice, which exists in hydrogen due to the low mass relative to the intermolecular forces. This problem has been investigated by Van Kranendonk and Sears (1966). The resulting expected splitting has, however, not been observed. The third Q branch feature at 4155.0 cm-' is ascribed by Sears and Van Kranendonk (1964) to pairs of orthohydrogen

0. Schnepp

194

molecules and the transition is to the u = 1 vibrational exciton band of the parahydrogen crystal accompanied by an orientational transition in the pair of orthomolecules on neighboring sites. The orientational states of the pair of orthomolecules are 9 in number and their degeneracy is expected to be removed by the quadrupole-quadrupole interaction. No structure however, has been seen due to this removal of degeneracy. The integrated intensities of the Q,(O) and Q,( 1) lines have been calculated and agreement with experiment is satisfactory. The S(0) branch of solid normal hydrogen has been observed to consist of three features at high resolution. These are assigned S,(O), Q,(l) + S,(O), and Q,(O) + So(0). In pure parahydrogen the S,(O)line is extremely sharp. The Q,(O) S,(O), on the other hand, is about 20 cm-' wide and has a number of peaks (Gush et al. 1960). Van Kranendonk (1960) has interpreted this spectrum in terms of his theory of rotational and vibrational excitons. The rotation-vibration exciton band is, as already described, about 20 cm- in width. The band states give rise to the transition designated Q,(O) + S,(O) to indicate that the vibrational and rotational excitations are never located on the same molecule and the selection rule k = 0 requires only the sum of the wave vectors of the rotational band and of the vibrational band states to vanish. For this reason the total exciton bandwidth is observed. On the other hand, the transition designated S,(O) is to a bound state in which the rotational excitation J = 2 is localized on the vibrating molecule in the state u = 1. This localized state lies below the exciton band, and the transition to it is therefore a sharp line. Now in the infrared, only the substates m = 1 are active similar to the situation in the pure rotational So(0) transition. In the vibrational Raman spectrum all substates of J = 2 are active, but as already described the splitting is very small. Therefore, it is predicted that the Raman and infrared lines coincide, and this is indeed found experimentally. At very low orthohydrogen concentrations, as mentioned above, the 20-cm- wide Q,(O) + So(0)rotation-vibration band transition exhibits a number of peaks. Gush and Van Kranendonk (1962) have investigated this structure. These authors have considered the partial removal of the 15-fold degeneracy of a pair of molecules, one parahydrogen ( u = 1, J = 2) and the second orthohydrogen ( u = 0, J = 1). This degeneracy is removed by the intermolecular interaction due to the quadrupole-quadrupole terms. It was found that the overall structure of the observed band could be reproduced on the basis of this theory, and further improvement was obtained when the quadrupolehexadecapole term was included in the calculation. Van Kranendonk ( 1 960) calculated the quadrupole induced transition intensity for the S,(O) and Ql(0) + S,(O) transitions. The theory is similar to that developed by the same author for the pure rotational infrared transitions. Again, the integrated absorption intensity of the localized state S,(O) is

+

'

THE SPECTRA OF MOLECULAR SOLIDS

195

obtained in terms of a lattice sum where cancellation due to crystal symmetry occurs. The intensity of the Q,(O) + So(0)transition to the rotation and vibration exciton band states is found to depend on the sum of square terms in the corresponding lattice sum, and as a result no symmetry cancellations occur. sec-' cm3, The total absorption intensity is predicted to be 0.48 x which is in excellent agreement with the experimental value of 0.45 x lo-''. Also, the theoretical ratio of the intensities of these two transitions agree very well with the observed value. As already described above, very intense phonon branches have been observed in the infrared spectrum in the region of the vibrational fundamental (Gush et al., 1960). The Q phonon branch is 200 cm-' wide, and its intensity overshadows that of the zero-phonon branch. In addition, the Q phonon branch remains intense in pure parahydrogen whereas the Q zero-phonon branch disappears. On the other hand, the Raman spectrum does not contain the phonon branches since the polarizability is not affected by molecular displacements. Poll and Van Kranendonk (1962) have described the theory of the phonon branches. The induced transition dipole moments are strongly dependent on intermolecular separation, and therefore the vibration-rotation transitions couple the lattice and give the phonon branches their intensity. On the other hand, it is found that the sharp internal lines are not broadened by lattice vibrations and that their intensities are not appreciably affected. In the theory, coupling between the internal molecular rotational states and the lattice modes are neglected and a product function is used. The integrated absorption intensity is formulated as the square of induced moments which are obtained by summing over the contributions by the molecules in the crystal. For single transitions such as S,(O), Q,(O), cross terms are thus obtained of the type p i p j where i and j refer to different molecules, whereas for double transitions such as Q,(O) + So(0),the integrated intensity is obtained as a sum of square terms. The double sums contained in these expressions are contracted by summing over the complete set of lattice vibration states n, and as a result second-order terms are obtained whose expectation values have to be calculated only over the lattice vibrations of the ground state. The moments are expanded in molecular displacements and finally the total intensity is obtained in terms of the expectation value of the square of molecular displacements in the zero-lattice vibration states. The expectation values are then equated to average square displacements and these quantities are evaluated from both the Einstein and the Debye models for lattice vibrations. It is found that the Ql(0) phonon branch intensity is entirely due to overlap distortion terms. Agreement is obtained within 20%, the Debye model giving a somewhat better fit than the Einstein model. The S(0) phonon branch intensity is found to be due to quadrupolar forces. Again, agreement within 20% is obtained for the lattice vibration models used.

196

0. Schnepp

Crane and Gush (1966) have studied the infrared absorption spectrum of solid D, and of solid HD. In the latter case, now J = 1 species persists to low temperature, and therefore the spectrum is analogous to that of pure parahydrogen. Van Kranendonk and Karl (1968) conclude that agreement between experimental and theoretical values for a wide range of observables justifies the conclusion that the intermolecular potential assumed is well based. This potential consists of a 6-12 Lennard-Jones term and an angle-dependent term, the quadrupole-quadrupole interaction.

D. SPECTRA OF CUBICHYDROGEN Clouter and Gush (1965) first discovered changes in the infrared vibrationrotation spectrum of normal hydrogen occurring at the temperature at which a &type specific heat anomaly had been known to exist, i.e., near 1.5"K. These authors found that the S,(O)and S,(1) lines disappear completely as the sample is cooled through this temperature. As has been described, Van Kranendonk (1960) showed that these lines appear in the infrared absorption due to quadrupolar intermolecular interaction. This mechanism is able to induce a transition moment only as long as the molecular sites are not centrosymmetric, as in the hcp structure (Van Kranendonk and Gush, 1962). Therefore, the disappearance of these absorption lines in Clouter and Gush's work proves that a sturctural change occurs in the solid at the transition temperature. Furthermore, the lower temperature phase must have centrosymmetric molecular sites. Accompanying the disappearance of the S, lines, changes occur in the phonon branches, and an intensity decrease of about 25 % is observed for the zero-phonon Q band. Both of these changes support the above conclusions. The structure of the low-temperature phase of solid n-hydrogen has been established to be face-centered cubic (Schuch and Mills, 1966; Schuch et al., 1968; Mucker et al. 1966). It is also known that the phase transition temperature is very sensitive to orthopara composition and occurs only in samples containing at least 60 % orthohydrogen ( J = 1 species). The disappearance of the So(l) line in the pure rotational infrared spectrum of paraenriched samples of deuterium has also been observed at the phase transition temperature by Hardy et al. (1968). The far-infrared absorption spectra (50-100 cm- ') of enriched orthohydrogen and paradeuterium due to optical phonons has been discussed in Section I. E.3 of this review dealing with lattice vibrations. Briefly, the observation of optical phonon absorptions in the low temperature cubic phase (Hardy et al., 1968) proves unambiguously that the crystal is composed of a number of physically distinct sublattices. The discovery of this spectrum

THE SPECTRA OF MOLECULAR SOLIDS

197

thus confirms theoretical predictions (Homma et al. 1967; Raich and Etters, 1968; Raich and James, 1966) according to which ordering of the rotation in the J = 1 state is expected in the cubic phase. The four fcc sites have axial symmetry at each of the sites. The axial symmetry causes the splitting of the J = 1 state into substates m = 0 and m = f I, the former being lower in energy by about 10-15 cm-’. Therefore, at the temperature of the phase transition, orientational ordering of the molecular rotation takes place. Mucker et al. (1966) reported evidence for a Pa3 crystal structure from a neutron diffraction study of solid paraenriched deuterium.

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0.Schnepp

Novak, A., and Cotrait, M. (1966). Ann. Chim. (France) 1, 263. Oehler, O., and Giinthard, Hs.H. (1968). J. Chem. Phys. 48,2032. Osberg, W. E., and Hornig, D. F. (1952). J. Chem. Phys. 20, 1345. Pawley, G. S . (1967). Phys. Status Solidi 20, 347. Pecile, C., and Lunelli, B. (1968). J. Chem. Phys. 48, 1336. Pimentel, G. C., McClellan, A. L., Person, W. B., and Schnepp, 0. (1955). J. Chem. Phys. 23. 234. Poll, J. D., and Van Kranendonk, J. (1962). Can. J. Phys. 40, 163. h a s k , H., Boutin, H., and Yip, S. (1968). J. Chem. Phys. 48, 3367. Raich, J. C., and Etters, R. 0. (1968). Phys. Reu. 168,425. Raich, J. C., and James, H. M. (1966). Phys. Rev. Lett. 16, 173. Rich, N., and Dows, D. A. (1968). Thesis, Univ. of Southern California. Ron, A., and Schnepp, 0. (1967). J. Chem. Phys. 46, 3991. Ron, A., and Schnepp. 0. (1968), unpublished. Rosevear, A. H. M., Whiting, G., and Allin, E. J. (1967). Can. J . Phys. 45, 3589. Schnepp, 0. (1967). J. Chem. Phys. 46, 3983. Schuch, A. F., and Mills, R. L. (1966). Phys. Rev. Letters 16, 616. Schuch, A. F., Mills, R. L., and Depatie, D. A. (1968). Phys. Rev. 165, 1032. Sears, V. F., and Van Kranendonk, J. (1964). Can. J. Phys. 42, 980. Shimanouchi, T., and Harada, I. (1964). J. Chem. Phys. 41,2651. Sobrana, G., Adembri, G., and Califano, S., (1966). Spectrochim Acta 22, 1831. Soots, V., Allin, E. J., and Welsh, H. L. (1965). Can. J. Phys. 43, 1985. St. Louis, R. V., and Schnepp, 0. (1969). J. Chem. Phys., in press. Takahashi, H., and Schrader, B. (1967). J. Chem. Phys. 47, 3842. Tasumi, M., and Knmm, S . (1967). J. Chem. Phys. 46,755. Tasumi, M., and Shimanouchi, T. (1965). J. Chem. Phys. 43, 1245. Ueyama, H., and Matsubara, T. (1967). Prog. Thoert. Phys. (Kyoto) 38, 784. Van Kranendonk, J. (1959). Physica 25, 1080. Van Kranendonk, J. (1960). Can. J. Phys. 38,240. Van Kranendonk, J., and Gush, H. P. (1962). Phys. Letters 1,22. Van Kranendonk, J., and Karl, 0.(1968). Rev. Mod. Phys. 40, 531. Van Kranendonk, J., and Sears, V. F. (1966). Can. J. Phys. 44, 313. Vedder, W., and Hornig, D. F. (1961). Advan. Spectry. 2, 189. Walmsley, S. H., and Anderson, A. (1964). Mol. Phys. 7, 411. Walmsley, S. H., and Pople ,J. A. (1964). Mol. Phys. 8, 345. Whalley, E., and Bertie, J. E. (1967). J. Chem. Phys. 47, 1264. Wilkinson, G. (1966). unpublished; quoted by Pawley (1967). Williams, D. E. (1966). J. Chern. Phys. 45, 3770. Williams, D. E. (1967). J . Chem. Phys. 47, 4680. Williams, D. R., Schaad, L. J., and Murrell, J. N. (1967). J. Chem. Phys. 47, 12. Winston, H., and Halford, R. S . (1949). J. Chem. Phys. 17, 607. Yamada, H., and Suzuki, K. (1967). Specfrochim. Acta 23A, 1735. Ziman, J. M. (1960). “Electrons and Phonons.” Oxford Univ. Press (Clarendon), London and New York. Zwerdling, S., and Halford, R. S . (1955). J. Chem. Phys. 23,2221.

THE MEANING OF COLLISION BROADENING OF SPECTRAL LINES : THE CLASSICAL-OSCILLATOR ANALOG* A . BEN-REU VEN The Weizrnann Institute of Science. Rehooot. Israel

I . 1[ntroduction ....................................................... 201 204 I1 m e Fourier-Transform Method ...................................... 4. Radiation Damping ............................................. 204 B Time Correlations and the Spectral Shape Function .................205 C. The Undamped Oscillator ....................................... 208 210 111. [mpact Damping .................................................. A Complete Interruptions ........................................ 210 B. An Interruption Model .......................................... 211 C. Sense-Reverting Collisions ...................................... 212 D Application to Rigid Linear Rotors ............................... 215 IV Complex Oscillators ................................................ 217 A Line Overlapping ............................................... 217 B Collision and Doppler Broadening ................................ 220 V Statistical Broadening .............................................. 221 A . Adiabatic Perturbations .......................................... 221 B. Adiabatic against Impact Pressure Shifts ........................... 224 IC The Method of Moments ........................................ 225 VI Resonance Broadening.............................................. 228 A Resonance Broadening as a Many-Body Effect ...................... 228 B Holtsmark's Model ............................................. 229 IC A Quantum-Mechanical Justification .............................. 231 D The Finite Size of Molecules ..................................... 232 References ........................................................ 234

.

.

.

.

. .

. . .

.

. . . .

.

I Introduction The theory of collision broadening of spectral lines has undergone several metamorphoses since its early beginning in the days of Michelson and *This research has been supported in part by the Air Force Cambridge Research Laboratories through the European Office of Aerospace Research. OAR. United States Air Force. under contract AF 61(052).838 . 201

202

A . Ben-Rewen

Lorentz. The simple model of Lorentz, of an ensemble of classical harmonic oscillators (representing the spectral line) interrupted by randomly occurring collisions, gave way to increasingly more sophisticated theories, using a variety of quantum-mechanical methods.' Most notable advances in the quantum-mechanical theory were the introduction of scattering matrix techniques (Anderson, 1949 ; Baranger, 1958 ; Fano, 1963), thermodynamics of irreversible processes (Fano, 1963), and, most recently, many-body formalism (Ross, 1966; Bezzerides, 1967). A major advance in the theory has been the introduction by Fano (1963) of the Liouville space formalism. In this formalism, the basis for describing the effect of collisions is shifted from energy eigenvalues and eigenfunctions to resonance frequencies and optical transition amplitudes. In this manner, it is duly recognized that the wave functions of the upper and lower levels of an optical transition should not be treated as being perturbed independently, but it is rather their coherent product that matters. A line in the spectrum, associated with the optical transition i - f , is represented in Liouville space by a unit vector l i ) ( f l (which is a Schrodinger operator formed by the product of a bra and a ket vector). The strength and position of the line are closely related to an amplitude (il p a If) (where p a is a component of the dipole moment of the molecular system) and an angular frequency m i / . This description immediately raises association with the classical theory of complex harmonic oscillators. The complex oscillator is represented by a set of normal-mode unit vectors nk, amplitudes Qk, and oscillation frequencies mk . Furthermore, the equations of motion of the Liouville space vectors and of the classical normal-mode coordinates are formally similar. So, the Liouville space formalism is, in a sense, a natural extension of the classical-oscillator models of early collision-broadening theories. This formal similarity should not be pushed too far. It is not correct, for example, to view the oscillator theory as a classical approximation from which the quantum-mechanical theory is obtained by the process of quantization. The quantum operator li)(fl is restricted to a space of two levels only, whereas the field operators obtained by quantization of the harmonic oscillator act in the infinite Hilbert space of the quantum oscillator. The molecular system is not a quantum oscillator. But it is formally similar, in a sense, to a classical oscillator. This analogy may be helpful in providing models with which various collision-broadening effects can be given a simple interpretation, or by which various results can be derived in an easily conceivable manner. The fruitfulness of this approach in stimulating a better underAn exhaustive review of the theoretical work on collision broadening, from the early works of Michelson and Lorentz to the mid-fifties, can be found in the monograph of Breene (1961).

THE CLASSICAL-OSCILLATOR ANALOG

203

standing and creating new concepts may be demonstrated in connection with four apparently unrelated recent developments in the theory. The first occasion to which we refer is the solution to the problem of the collapse of microwave resonance spectra into spectra of a nonresonant type at pressures where damping rates become comparable to the resonance frequencies (Ben-Reuven, 1965b, 1966). In this demonstration of the advantages of the Liouville space formalism, the collapse of the resonance mode is fully accounted for by introducing an additional cross-relaxation parameter 5. The same line shape expression could have been obtained, however, by a simple purely classical model in which 5 represents the rate of a particular type of collision that inverts the motion of the harmonic oscillator. Another occasion is connected with a recent treatment of the effect of collisions on the Doppler broadening of spectral lines (Rautian, 1967; Rautian and Sobel’man, 1967). A simple classical model, treating the various Doppler-shifted frequencies as a band of distinct normal modes, leads to the same integral equations which are the basis of the treatment of Rautian and Sobel’man. The third occasion was brought forward by the realization of the important contribution of the reorientation of molecules by collisions to the broadening and shifting of spectral lines. Although this had already been pointed out by Spitzer (1940), proper consideration of this effect has been made only in recent treatments (Ben-Reuven, 1965a; Gordon, 1966a). This could hardly have happened if classical models were consulted more frequently. Indeed, Gordon (1966a) has shown that cross sections for broadening and shifting of lines due to rotational phase shifts and reorientations can be computed using a completely classical model. One has to assume that the molecule is represented by a classical rotor with a fixed unperturbed rotational frequency equal to the resonance frequency of the spectral line. Given a potential function for the interaction with perturbing molecules, the phase shifts and reorientations of the rotor can be calculated by integrating along the classical collision trajectories. The fourth occasion is connected with the treatment of resonance broadening by Reck et al. (1965). A conjecture concerning the square root density dependence of the line width due to resonance transfer of excitation between similar molecules, obtained under certain conditions, reaffirms in fact the conclusion reached by Holtsmark (1925), using a simple classical model of coupled oscillators. A resort to classical models shows that, at the wings of the line, resonance broadening is after all a binary collision effect, linearly dependent on the density. It seems desirable, therefore, to regain the line of thought of Lorentz and his contemporaries, and rewrite classical-theoretical models for the various types of damping phenomena (such as impact, statistical, or resonance

A . Ben-Reuven

204

broadening) starting from, as much as possible, a common point of view. In doing so, we can exploit the insight provided by quantum theory and the vast experience gained in line shape studies and avoid certain wrong conceptions which have persisted in earlier works.

II. The Fourier-Transform Method A. RADIATION DAMPING

The emission of radiation near a characteristic resonance frequency is associated in the classical theory with an electrically charged oscillator of mass m,charge e, and harmonic angular frequency oo. Let the oscillator be stretched to the amplitude Qo at the time t = 0 and then be left to oscillate freely. Its displacement, when not disturbed, should vary in time according to

where 9(t) =

1 0

when t > O when t < O

is the “step” function. As an accelerated charge, however, it must emit radiation at the rate (cf. Heitler, 1954) G ( t ) = 3eZ(1’/c3.

(3)

Its motion cannot therefore be strictly harmonic ; energy will be radiated at the expense of diminishing the amplitude. As a result, the spectral distribution of the emitted light will have significant components spread continuously over a finite range of frequencies, of the order of magnitude of the rate of dissipation of the oscillator energy, instead of the single component at o = a,,. The oscillator, so to speak, undergoes natural or radiation damping. Provided the damping is weak enough, its effect is introduced by simply adding into the right-hand side of Eq. (1) a factor exp(-Etj, where E is the rate of radiation damping. I n what follows we shall be concerned with another source of damping, namely, collisions of the radiating oscillator with other particles. Collision damping is usually much stronger than radiation damping, and the latter can be neglected in dealing with the motion of the oscillator under the effect of collisions. The presence of natural damping is, nevertheless, essential for understanding the meaning of relaxation of systems of finite size. In relaxation theories it is customary to consider a microscopically small part of the system -one molecule, say-whose periodic oscillations are perturbed by an

THE CLASSICAL-OSCILLATOR ANALOG

205

infinite number of other particles, acting as a thermal reservoir. An aperiodic motion, such as an exponentially damped oscillation, then results. In finite conservative systems, however, multiperiodic motions may still exist with discrete resonance frequencies (compared to the continuous spectrum of a damped oscillator). In the classical theory, these periodic motions are associated with the discrete normal modes of the complex system. A discrete spectrum would mean that (though the chance may be exceedingly small, owing to the complexity of the system) the excitation may periodically return to the same molecule if it cannot eventually escape by such means as spontaneous emission. A closed finite system cannot strictly act as a thermal bath; it must be coupled to the "outer world" to form one. Radiation damping provides for such a link (though it is not the only conceivable one; the system may also interact directly with the surrounding room by collisions of the gas molecules with the walls of the container). The interaction with the radiation field, mathematically expressed by the addition of the damping parameter E , secures a truly aperiodic motion. Periodic motions of the macroscopic system, superimposed on the oscillations of its microscopic part, will then be completely smeared out by natural broadening (or by other means of relaxation of the macroscopic system), to appear as a single smooth line in the spectrum.

B. TIMECORRELATIONS AND THE SPECTRAL SHAPEFUNCTION Light is emitted in ordinary experiments by a macroscopically large number of oscillators, the motion of which is incoherent. We would, therefore, expect the power spectrum per oscillator to be simply related to the Fourier transform of an ensemble average of Q(r), but in such a manner that the incoherence of phase (and the randomness of orientation) of the various oscillators will not make it vanish. The most obvious suggestion is to try and relate the power spectrum to the autocorrelation of eQ(r) in time,

where the brackets stand for the ensemble average. Consider Q ( t ) as an unspecified function of the time and its Fourier transform Q(o).In general, Q(o)=

1; Q(t)e-'"' dt m

and co

Q(t)= (27c)-' -m

Q(co)e'O' do.

206

A . Ben-Reuven

SOY m

Q(t)= -(2n)-1/

oZQ(o)eiat do

(7)

-W

and

[(l(t)lZ= (4~')~'

IIwa*(o')Q ( w ) o Z o f 2exp[i(o - w')t] do dw'. (8)

By transforming back, we have m

[(l(t)I2

Q(f')~zd2

= (4n2)-' lTJTQ(t") -m

- oft"- (o- o')t])do do'dt' dt"

x exp( - i[wf'

= (4n2)-' I f l / Q ( f " ) Q(t"

+ z)w2ofZ

-m

x exp[- i(o - o')(t"- t ) - i o z ] do d w ' d t " d z ,

where z = t'

(9)

- t".

At this point, the ensemble average is introduced. The correlation (Q(t") Q(t" + z)) will then appear in the calculation in the time interval z, given the initial time t". By the ergodic hypothesis of statistical mechanics, averaging over an ensemble is equivalent to averaging over all initial states. The brackets are therefore independent of t", which appears now only in the exponent in (9). Given any well-behaved function f(w),

jm f ( o fexp[i(o ) - o')?"] dt" do'

1-

-m

W

= 2n

f(w') 6 ( o - 0') do'= 2 n f ( o ) , (10)

m

where 6 is the Dirac delta function. Therefore, performing these integrations, the mean power emitted is ( G ( t ) ) = (2e2/3c3)(2n)-'

jm

fm

-m

-

w4(Q(0) Q(z))e-'"'

-m

d o d?. (1 1)

Notice that ( G ( t ) ) is no longer time dependent. It follows that m

( G ( t ) ) = (2/3c3)

0

m4F(o) d o =

/

m

0

G(o)do,

(12)

where F ( o ) = ( e 2 / n )Re( f m ( Q ( 0 ) Q(z))e-'"' d r ] -W

(13)

207

THE CLASSICAL-OSCILLATOR ANALOG

is the so-called spectral shape function, and (14)

C(O) = (2/3c3)04F(w)

is the power emitted in a unit interval of o,as may be seen by considering an ensemble of harmonic oscillators. The latter conjecture is true if the ensemble averaging is made over randomly distributed initial times and phases of the excitation. The absorption coefficient K of a gas of n, similar oscillators per unit volume is related to G(w) by Kirchhoff's law,

K ( 4 = n, G(o)/cU(o), (15) provided the radiation is sufficiently weak so as not to cause appreciable deviation from thermal equilibrium. The term U(w) is the energy density (per unit volume) of blackbody radiation in a unit range of w. In classical physics, (16)

U(W)= w ' k T / n ' ~ ~ ,

where kT is the classical value of the mean energy of each normal mode of the radiation field at the temperature T. If kT is not much larger than hw, kT should be replaced, according to Planck's law, by ho(e"w'kT- I)-', giving K(u) = n , ( 2 ~ ' ~ / 3 h ~ ) ( dI)F(o). ~'~~ (17) In dealing with harmonic oscillations, it is sometimes more convenient to use complex quantities, the time variation of which is described by exp( f.ioot) instead of sine or cosine functions; i.e.,

+

X ( t ) = Q ( t ) i(mwo)-'P(t), Y(t) = Q(t) - i(mwo)-'P(t) = X*(t),

(18)

where Q and P are, respectively, the canonical coordinates and momenta, The use of the ergodic hypothesis enables us to choose the initial phase of the oscillation at will. We choose, therefore, as initial conditions those appropriate to the example of Eq. (I), namely lim X(t) = t-ro

(OQo

when t > 0 , when t c0.

Then, written in terms of X and Y,

F(o)= X F C ( 4

+ Fc( - 41'

where F,(w) = n-'Re(~om8c(r)e-i"' dr

dr).

A . Ben-Reuven

208

The second term in (20) arises from the similar contribution of Y(z), noting that E(o) = X*(-o). (22) F(o)is thus an even function of w . Though only positive values of w are of physical significance, it is formally possible to associate with a term in the “ physical ” region of positive o,around a resonance frequency wo ,a similar term in the “unphysical” region around - a o , with the latter, however, extending a “ tail ” to positive frequencies. As will be illustrated later, each line is composed of “ positive” and “ negative” resonance terms. It follows from (21) that, no matter how X(t) varies with time,

lOmF(o)do

m

=4

F,(o) do = +e’(QoZ) = +@( +0)

=A,

-m

(23)

where by + O we mean the limit as t approaches zero from the positive side. Equation (23) is sometimes referred to as the “ law of spectroscopic stability.”

C. THEUNDAMPED OSCILLATOR Consider, again, the undamped oscillator of Eq. (l), Q(t) = Re{X(t)} = Re{O(t)Qo exp(ioof ) } . The phase-space variables X(t) and Y(t) obey first order differential equations,

(d/dt - iwo)X(t) = d(t)Qo,

(d/dt + io,)Y(t) = S(t)Qo ,

(24)

where the singular source at t = 0 results from the step function O(t). A Fourier transformation of Eqs. (24) gives

+

i(w - wo)%(o)= Qo , i(o oo)P(o)= Qo . (25) The solution of these equations is obtained by a method known from the theory of Green’s functions. To cope with the divergences at w = w o , formed in solving for %(o)and fr(w), we add to wo either + ie or - ie, where E is a positive increment which is ultimately made to vanish. The general solution is an arbitrary linear combination of the two corresponding solutions, I.e.,

where a, + a- = I . The correct combination is determined by the boundary conditions (19), in the following manner: Consider the transformation back to X(t),

209

THE CLASSICAL-OSCILLATOR ANALOG

performed by contour integration in the complex w plane. In case (a) t > 0, the integration path can be closed by a semicircular arc in the upper half-plane (Im w > 0), the contribution of which will vanish as the radius of the arc is made infinitely large. Nonvanishing contributions to the integral in (27) come only from poles encircled by the path, namely at w = coo + ie. Therefore X(t) = a+Qoexp[(iw, - ~ ) t ]

for t > 0.

(28)

In case (b), t < 0, the integration path should be closed in the lower halfplane, encircling the pole at w = wo - ie, with the result

+~ ) t ]

for t < 0.

(29) The minus sign results from traversing the path clockwise. The boundary conditions (19) force us to choose X(t) = - a - Q o exp[(iwo

a - =o. with poles at w Following a similar solution for %?(o), a, = I ,

Q(o)= +[8(0)+ f’(w)]

+

-

(30) = -coo

= frQo{[i(w wo - i ~ ) ] - ’ [i(w

f ie, we get

+ wo - zk)]-’}. (31)

Since, in the absence of collisions, all the oscillators of the ensemble behave in the same manner, F(w) =

(t)(

(w

E

+ + (w + + 6(w + oo)},

- wo)2

= A{6(w - 0 0 )

E2

E

wo)2

+ E2

1

(32)

as E -+ 0, where A = $e2( Q o z ) . The spectrum of the undamped oscillator thus consists of an infinitely sharp line at w = wo . The “negative resonance” term 6(w wo) should be retained, however, since in reality no oscillation will go on undamped. The total power emitted is given by

+

(C) = ImG(w) d o = (w~/3c3)e2Q02. 0

(33)

Notice that E has the right sign necessary to keep F(o)positive. But the sign of E is determined solely by the boundary conditions (19) which are an expression of the condition of causality: Q(t) has a nonvanishing value only after the initial displacement Qo has taken place. It can be shown that similar arguments hold also in the quantum-mechanical case. Thus, e.g., causality turns out to be the reason why, as a linear response to a radiation field, a system in thermal equilibrium always dissipates (absorbs) energy. The same solution would have resulted, with the same sign of E, if radiation damping were retained in the form of a factor exp(-et) added to Q(t).The only difference is that E is then a finite physically meaningful quantity.

A . Ben-Reuven

210

III. Impact Damping A. COMPLETE INTERRUPTIONS

The idea of a sudden interruption of the oscillation by a strong collision originates from the works of Michelson and Lorentz in the earliest days of pressure-broadening theory. Suppose the oscillator is enclosed in a gas of N identical particles, contained in a box of volume V . The gas is supposed to be dilute so that, .to a good approximation, the various particles move in a random, uncorrelated manner, and the time during which each of them interacts with the oscillator is negligibly short compared to the mean time between successive collisions. Let the oscillator be perturbed strongly at each collision so that every phase and orientation are equally probable after the collision. In the process of averaging over the outcoming phase or orientation, this amounts to cutting the amplitude to zero. It is still necessary to average over the random temporal distribution of collisions, which occur at a mean rate y. Let At be a time interval short enough to incorporate, at most, only one collision with the oscillator, yet sufficiently long compared to the duration of the collision. The probability for a collision to occur in At is therefore y At. Suppose the collisions are independent of each other, so that the effect of each collision does not depend on the past history of collisions. In calculating the mean change of O,(t) in the interval t to t + At, we may therefore average first over the collisions in At, for a given X ( t ) , independently of the averaging of X ( t ) itself. Since a single collision changes X by A X = -X,

(34)

the mean change of X in At is Aco,,X(t) = - v X ( t ) At. (35) In a “coarse grain” sense, considering the collisions infinitely short, At may be treated as a differential. The equation of motion of the undamped oscillators, Eq. (24), should then be modified to (d/dt + y - iwo)X(t)= B(t)Qo. (36) Inserting the solution of (36) in (21), the Lorentz line shape

F,(o) = n - ’ e 2 ( Q o 2 )

Y ( w - wo)2

+y2

(37)

is obtained. The full expression for F(w) which follows from (20) and (23), (38) is also known as the Van Vleck and Weisskopf (1945) line shape.

21 1

THE CLASSICAL-OSCILLATOR ANALOG

B. AN INTERRUPTION MODEL

The oscillation need not be completely interrupted by collisions in order to damp the radiation. Other ways by which a collision may affect the oscillator are (a) changing its amplitude, (b) shifting its phase, and (c) reorienting it in space. Explicit reference is made here to linearly polarized oscillations only. Other types of polarization, such as circular, may be expressed as linear out-of-phase combinations of the three components of linear polarization. Let a collision of type s change the amplitude of the oscillation by a factor a,, shift its phase by q,, and reorient it from X to R,X, where R, is an orthogonal rotation matrix (or a complex unitary matrix of unit modulus, if nonlinear polarizations are considered). A collision of that type changes x by AX = (a,eiqsRs- l)X (39) where a,, qs are assumed to be independent of X. If the rate of collisions of type s is v, , then Atoll X( t ) = - v,( 1 - a, e'"R,)X( t) At . (40)

c S

Included in the summation over s is an averaging over an isotropic distribution of the orientation of the collision trajectories in space. The matrix R, then becomes invariant, or scalar, and may be replaced by cos O,, where 8, is the angle by which X is rotated. Consequently, Aco,lX(Q= -(y - id)X(r)Ar,

where y =

c v,(l - a, cos

(41)

9, cos e,),

(42)

S

and 6=

1v,a, sin q, cos 8,.

(43)

S

The resulting line shape is Lorentzian with a shifted peak, F,(o) = n-'eZ(Qo2)

Y

(w - wo - S ) 2

+

72'

(44)

The combined effect of phase shifts and amplitude variations was first calculated by Lindholm (1 942). The classical reorientation effect was not treated properly until the recent work of Gordon (1966a). This reorientation effect, well known to be the major cause of relaxation in nonresonant (or dielectric) relaxation, was unjustifiably neglected for a long time in the study of optical spectra [in spite of Spitzer's (1940) observation of its importance].

212

A . Ben-Reucen

The parameters that distinguish a collision of type s, and the corresponding mean rate v,, have yet to be specified. Whenever the inner structure of the perturbing particles is immaterial, the collision trajectory is completely specified, apart from its orientation, by the relative velocity u and the impact parameter b. The rate of collisions in the range b to b bd and u to u + do, isotropically averaged over orientations, is

+

v, = nb 2nb db vg(u) do,

(45)

where nb is the number density of perturbing particles. Here g(o) is the Maxwellian distribution of velocities at the temperature T, g(u) = 4 n ( r n , / 2 ~ k T ) ~exp( / ~ v-~m , v2/2kT),

(46)

where m, is the collision reduced mass. Equation (44) represents the spectrum of a damped oscillation if y is positive. From (42) it follows that y is positive if c(, < 1 for all collisions. If values of x , greater than unity are allowed, the positive-definite nature is not ascertained, i n general. Therefore, the above model does not guarantee damping. This might be guaranteed by some further specifications of the nature of the collisions and the causes of amplitude variations. This question does not arise in the quantum-mechanical impact theory (cf. Anderson, 1949), where owing to the unitarity of the scattering matrix the oscillations are always damped.

C. SENSE-REVERTING COLLISIONS Damping is frequently associated, by the suggestion of Lorentz (1915 ) , with the existence, on a phenomenological level, of a velocity-dependent friction force. Yet, it was Lorentz himself, in his famous Note 57, who pointed at the difference between the effects of interruptions by collisions and those of the friction force. The resulting line shapes, in the two cases, are different and become similar only when y < oo. Since at the time of Lorentz radio and microwave spectroscopy did not yet exist, and for all practical purposes the two line shapes seemed to coincide, little attention was paid to the difference. It is nonetheless remarkable that this distinction has managed to avoid recognition even after more than two decades of microwave spectroscopy, and so many years after spectra had been measured with y comparable to w o . This failure of recognition led to the use of the wrong line shape expression for the interpretation of the data, and consequently to the discovery of some “shifts” of an exceptional behavior which could not be explained by an impact theory. The most outstanding observations of this kind were made with the inversion spectrum of ammonia in the microwave region (Bleaney and Loubser, 1950; Birnbaum and Maryott, 1953).

213

THE CLASSICAL-OSCILLATORANALOG

A quantum-mechanical derivation of an appropriate expression for microwave line shapes, in the impact approximation, was presented recently (Ben-Reuven, 1956b, 1966). An observation was made there that this line shape can also be obtained classically, if two distinct processes are considered as jointly affecting the oscillator. One process is the interruption by collisions of the type discussed in Section II1,B. The other process is an interruption by collisions that revert the sense of the oscillation, having the same effect as a velocity-dependent friction force. These collisions leave the diplacement of the oscillator unaltered while reverting its momentum, i.e., AQ

= 0,

AP = -2P.

(47)

As before, we average over collisions in a time interval At, AcollQ = 0 ,

AcollP= -2cP At

(48)

c

where is the mean rate of momentum-reverting collisions. This amounts, in the " coarse grain " sense, to adding - 2cP to the time derivative of P. But this is precisely the form in which velocity-dependent friction is phenomenologically introduced into the equations of motion of the oscillator. Let us proceed now to express (47) in terms of the complex variables X and Y . Using ( 1 8), we may write AX

= i(moo)-'AP =

-2i(moo)-'P = - ( X - Y)

(49)

and AY = -i(m~o)-'P = -(Y

- X).

(50)

That is, a sense-reverting collision interchanges X and Y. The resulting equations of motion are (d/dt

+ 5 - ioo)X - cY = h(t)Qo,

ro--?+

(d/dt

+ 5 + iwo)Y - (X = S(t)Qo. (51)

Their Fourier transform is, in matrix notation, i(w

+it)+ i)

(*(coJ=

(:I)*

(52)

The solution follows immediately, by inverting the matrix in Eq. (52):

ro+ 7 )+

+ 2ior

oo2- o2

(53)

Thus, i(o

+ oo)+ 26

X(0) = wo2 - w 2 + 2iwc Q o .

(54)

A . Ben-Reuven

214 from which follow

and

When

C Q oo,and Iw - oolQ oo, as the case is in optical spectra, we get e2( Q o 2 ) F,(w) % 71

(0

c - coo)’

(57)

+ 5’’

which is similar to (44),with C instead of y, and hence the source of confusion between the interruption and the friction models. Equation ( 5 5 ) vanishes whenever oo= 0. But spectra of zero resonance frequency are known to exist; they are caused by reorientation by collisions of otherwise stationary (nonoscillating) dipoles. This process (Debye, 1929), known by various names as nonresonant, orientational, dielectric, or Debye relaxation, is certainly an interruption-type process.’ Since the two models were taken to be equivalent, it led to rejecting the friction model in favor of the interruption model. Having now recognized that the two models represent two distinct processes, it seems most natural to incorporate both of them in the equations of motion. Letting y’ be the rate of interruption-type collisions [Eq. (42)], we then get [d/dt ( 7‘ - i ( ~ 0 S)]X - CY = 6(t)Qo, [d/dt y y l i(w0 6)IY - yx = 6(t)Qo. (58)

+ + + + +

+ +

Again, using Fourier transforms,

(W)= ( K W

-

””;

8) + Y i ( 0

+ wo-[+ 6) + y

)-l(:;),

(59)

where y = y‘

+ y.

(60)

A straightforward calculation yields X(W)

=

i(w

+

+ uo+ 6) + y + C

(o 6)’ - w z + y z

- yz + 2iwy Qo

For recent theoretical treatments of nonresonant absorption and Debye relaxation in gases see Ben-Reuven (1966), Birnbaum (1966),and Gordon (1966b).

THE CLASSICAL-OSCILLATOR ANALOG

215

Hence

(62)

and A 2(Y - C)w2 + 2(Y - C)C(oo + 4z+ Y 2 - C21 F(o) =n [(oo 6 ) 2 - w2 y z - 5 2 3 2 402y2

+

+

+

(63)

is the resulting shape function (Ben-Reuven, 1966). The Van Vleck and Weisskopf shape function (38), with y now representing the full relaxation rate y‘ + C, is approached as a limiting case of (63) when 4 wo 6. In the case wo 6 = 0, however, Eq. (63) attains the Debye shape for nonresonant relaxation,

+

+

A 2y’ F ( o ) =n o z + y’2 ~

where only y’ participates. The quantum-mechanical equivalent of y’ and C has been discussed by Ben-Reuven (1966). The functions X and Y are the classical counterparts of the operators li)(fl and If)(il, respectively.Therefore, theequivalent to collisions which transform X into Y are collisions which cause transitions from i to f and fromfto i ; namely, inelastic transitions between the two levels of the observed line. D. APPLICATION TO RIGIDLINEAR ROTORS

We have so far avoided the question of how to calculate q, ,9,, or a, which appear in the interruption model of Section II1,B. One usually uses quantum theory for the description of internal states of molecules in such calculations. The treatment of the collision problem, having to deal with the many channels of molecular collisions, is very cumbersome, in spite of the many simplifying assumptions ordinarily introduced. It is therefore gratifying to know that, at least in some cases, various line shape parameters may be calculated by integrating the classical equations of motion of the colliding molecules treated as classical particles (Gordon, 1966a,c). Consider, for example, linear molecules as rigid rotors. A linear rotor with a moment of interia Z has the quantized resonance frequencies

o,=hz-’j

o’= 1,2, ...),

corresponding to the transitions between the rotational levels j - 1 and j . We can look upon this rotor as an “angular harmonic oscillator” with the

21 6

A . Ben-Reuven

fundamental frequency h l - ' and its harmonics. Consider as an initial state a particular value of j , so that w j is the classical angular frequency of rotation, and hj is the magnitude of the associated angular momentum, which has a fixed orientation in space. As demonstrated by Gordon (1966a), a rotational phase shift q and an angle describing the reorientation of the angular momentum cector in space may be obtained from the classical equations of motion, given an angle-dependent interaction potential. Gordon, for example, uses dispersion forces for HC1-noble gas collisions, including the loaded-sphere correction (Herman, 1963, 1966). It may be necessary to introduce a few quantum-mechanical modifications, utilizing the correspondence principle. Thus, e.g., the effect of inelastic collisions between rotational states may be introduced by a relation between the quantum transition probability and the classically calculated change in the magnitude of the angular momentum as a result of the collision. In vibration-rotation spectra, it is also necessary to introduce a vibrational phase shift, due to anharmonicities. This may be accomplished by calculating the vibrational perturbation quantum mechanically, and integrating over the time along the classical trajectory (Gordon, 1966a). An interesting analogy to the case of sense-reverting collisions, discussed in the previous section, is encountered in an application of the linear rotor model to the problem of line overlapping in vibration-rotation spectra (Gordon, 1966~).A vibration with an angular frequency w, modulated by a rotational frequency w j gives rise to the doublet of combination frequencies 0, & w j I An inversion of the orientation of the angular momentum by a collision amounts to a change of sign of w j , therefore transferring from one member of the doublet to the other. This is similar to the effect of sensereverting collisions on the w,, " modes " of the classical oscillator. More generally, a collision alters the orientation of j by a certain angle. The final orientation may be looked upon as a superposition of the initial state and the state of inverted momentum, to a degree determined by the reorientation angle. Therefore, reorienting collisions are partially effective as " sense-reverting" collisions. Their outcome is the merging of the doublet into a single line at o,as the rate of collisions is increased. This is essentially Gordon's explanation of the collapse of the P and R branches into a Q branch observed in the vibration bands of compressed CO gas by Vu et al. (1963). A question may be posed regarding the foregoing interpretation. In the previous section the sense-reverting collision model has been treated as a mathematical analog of the collapse of resonance lines into a nonresonant spectrum. The physical processes involved are, however, quite different. In the classical analog, the relevant collisions are elastic ones, in which only the sign of P is inverted. Contrarily, in the quantum-mechanical theory, inelastic collisions connecting the initial and final levels of the line are responsible for

*

THE CLASSICAL-OSCILLATOR ANALOG

217

the collapse. Similarly, in vibration-rotation spectra, inelastic transitions between rotational levels are responsible for the mixing of the P and R branches. Gordon, nevertheless, considers the elastic reorienting collisions as actually causing the mixing of the P and R branches in a model which represents an approximation to the real physical system. This apparent contradiction may possibly be settled by a closer look at the nature of the classical approximation. One should remember that the classical oscillator is an analog of the optical transition between two levels, and collisions affect the two levels simultaneously. Consider, for example, the doublet R(J - 1) and P(J). In order to transfer from the one line to the other, it is necessary to have inelastic rotational transitions in both lower and upper vibrational states. But the energy given to the molecule in the lower state is just compensated by energy taken in the upper state. Since, in the classical limit, there is no way of telling in which state the molecule is, no energy is transferred to the molecule on the average. The manner in which the classical expression results from the quantum formulas and, particularly, how reorienting collisions emerge as the dominant process, still need a rigorous demonstration

IV. Complex Oscillators A. LINEOVERLAPPING

So far we have considered only three-dimensional single-mode harmonic oscillators. Let us now suppose that the oscillator is a complex system, con. sisting of s normal modes with angular frequencies ol, o2, ..., u s Collisions may couple various modes, exciting one mode at the expense of others, in addition to damping individual modes. We then have cross relaxation between overlapping lines. In addition to averaging over the effects of collisions in the ensemble, we shall from now on also average over all possible initial configurations Qo = (Qoi, Qoz

3

-

4

,QoA

(65)

using a distribution function

The distribution of initial configurations is assumed to be independent of the positions and the momenta of the perturbing particles. It is therefore possible, as before, to average first over all collisions, given a fixed initial configuration, and then use (66) to average over the configurations of the oscillating system.

218

A . Ben-Reuuen

Suppose a collision changes the 3s-dimensional vector A’, which describes the configurations of the oscillator at t, into X’. The transformation is described by X’ = S X where S is a complex matrix in our 3s-dimensional space. As in the impact model of Section III,B we assume that S is independent of X itself. It may be simplified by rotational invariance if collisions are isotropically distributed. Then the three Cartesian components will transform independently by the same (now s-dimensional) matrix S. We may, finally, add collisions of the sense-reverting type, but then S will be a 2s-dimensional matrix acting on 2, = ( X d , * *

f

9

x,,,

y,1, *

-

* 9

Yo,>,

(67)

where a = x, y, or z. Let w(a) be the mean rate of collisions of the type n, represented by the matrix S(a). The mean rate of change of 2, by collisions is then

The 2s equations of motion of the components of 2, are, in matrix notation, [dldt

- i(R, + A,)l2,

(69)

= a(t)Qoa,

where Qo, is now Q o ~= ( Q o a i ,

* * * 9

Qoas 9 Q0.1,

. * QoA

and R, is the real diagonal matrix whose eigenvalues are -0,s.

W,1,...,~,s,-Wo1,...,

Here, A, is the complex matrix Ac = i

1w(a)[I e

- S(a)]

Im du 2nbug(u)[I

= in, Jomdb

0

- S(b, u ) ] ,

(70)

where I is the unit matrix. The last equality in (70) holds if S does not depend on the internal state of the perturbing particle. The element i, j of Ac signifies an excitation of thejth mode by a collision, the oscillator having an excited ith mode before the collision. If these transfers of excitation occur with welldefined phase shifts, S will be a complex matrix. Equation (69) is an eigenvalue problem in which we may consider R, as an “unperturbed” Hermitian matrix and A, as a non-Hermitian “ perturbation.” The eigenvalues of R, + Ac are complex; their real parts correspond to shifted resonance frequencies, and their imaginary parts to damping rates (or line widths). The spectral shape function will not be, however, a sum of Lorentzian terms since the transformation that diagonalizes Ro + Ac is not unitary, and Q& will be transformed into a complex quantity.

THE CLASSICAL-OSCILLATOR ANALOG

219

If the elements of A, are small in comparison to the separations between the resonance frequencies w o j , then Ac may be treated by perturbation techniques. It is convenient to split A, into a sum of two matrices, one Hermitian and one anti-Hermitian, Ac = Ac

+ ir,.

(71)

The off-diagonal elements of ir,, treated as a small perturbation, have a peculiar property : instead of repelling two eigenvalues of the unperturbed matrix Roaway from each other, as the off-diagonal elements of an Hermitian perturbation would do, they attract them toward each other, causing two distinct resonance lines to collapse into each other. We had an example of this effect in Section II1,D [Eq. (59)], in which momentum-reflecting collisions, represented by c, act on the 2 x 2 matrix of eigenvalues ooand -coo as an off-diagonal anti-Hermitian perturbation. The result was a "shift" of the two eigenvalues toward each other (or of the resonance frequency toward zero) when 5 is increased by raising the gas pressure, ending up with the nonresonant Debye shape. In general, processes that cause the mixing of any closely spaced resonance modes w1 and w2 do not contribute to the line width in the limit where the cross-relaxation rates are much larger than Iwl - w21 and the two lines completely merge. If, in addition, the two modes have equal amplitudes, we get in this limit a single line with position and width given by

-

00

+ s + ir = +(a,+

02)

+ i c v(0) a

x (1 - ",(a)

+ Sz(41 - 3CSld4 + S21(4119

(72)

where

c v ( M - S 1 2 ( 4 + SZl(d3, U

which is essentially the cross-relaxation rate, is subtracted from the total width. Such behavior is typically observed in the exchange narrowing of nuclear magnetic resonance lines (Abragam, 1961). More generally, a band of N closely spaced resonance frequencies oOf (f= 1, . . . , N), with corresponding amplitudes X a J , will merge by cross relaxation into a single line which, to a first approximation, has the mean position and width

where Q,, is an N-dimensional vector.

220

A . Ben-Reuuen

B. COLLISION AND DOPPLER BROADENING The translational motion of the " oscillator" itself, as a particle of finite mass, has been so far disregarded. Owing to the Doppler effect, similar oscillators moving at different velocities to or from the observer will be seen radiating at different frequencies; their displacement from the resonance frequency at rest is proportional to the longitudinal component of the velocity, u,,

- (un/c))*

(74) A Maxwellian distribution of velocities will therefore show up as a Gaussian distribution of frequencies in the spectrum (neglecting damping). An ensemble of such oscillators may be treated as an ensemble of oscillators of varying resonance frequencies; to different values of un correspond different normal modes with the appropriate value of w D .The Doppler-broadened line may be considered as a continuous band of resonance lines. If only the internal state of the oscillator is affected by collisions, then the internal and translational motions may be treated independently and the resulting shape of an isolated line is a convolution of a Gaussian and a Lorentzian shape. Collisions, however, change u,,. This effect may result in the narrowing of the line width with increasing pressure at sufficiently low pressure, where Doppler broadening still dominates over collision broadening (Galatry, 1961, and references therein). Considering the Doppler-broadened line as a continuous spectrum of eigenvalues, the effect of impacts can be expressed by an equation similar to (69), but with oD(un)

=

(75) being a kernel of an integral over un' (appearing in the equation for the normal mode corresponding to un) instead of a matrix. The result is an integral equation, = Ac(un 9 on')

i[a - ~O(vn)]z,(w;

vn)

-i

I

OD

-m

dun' &(On

u n ' ) z a ( ~ ; )0 ;

= Qou(vn)-

(76)

This equation is essentially equivalent to the one used by Rautian and Sobel'man (1967) in their recent treatment of combined Doppler and impact broadening. Collisions represented by the kernel A, couple the " normal modes," as in the discrete case. By making them collapse into each other, the Doppler-broadened line is narrowed. This behavior is reminiscent of motional narrowing in nuclear magnetic resonance (Abragam, 1961). There the spin-spin coupling takes the place of Doppler broadening in our case, and the temporal variation of the spin-spin interaction is the equivalent of changing the velocity by collisions.

22 1

THE CLASSICAL-OSCILLATOR ANALOG

Let S(un,un’; a) be the velocity-dependent collision matrix, and gn(un)the Gaussian distribution of radiating molecules. Then, with &(On

3

un’)

=i

C [d(un - On’) - S(vn

on’

;a)Iv(c),

d

(77)

we get by application of (73) (assuming the amplitudes are velocity independent), in the limit of high relaxation rate, the impact-broadening result 8,

+ 6 + il = oo+ i C u(a)[1

- ~(a)],

(78)

0

where

represents the effects of collisions averaged over velocities of the radiating molecules.

V. Statistical Broadening A. ADIABATIC PERTURBATIONS Consider, again, the single-mode oscillator, contained with the gas of N identical particles in a large box of volume V. Let us assume now that the gas perturbs the oscillator in an adiabatic manner, that is, the oscillator’s angular frequency at any moment t is given by mo‘W = 0 0

+ Ao(q(09 P(t)),

(80)

where the the perturbation Am depends on the position of the gas in phase space, 4 = (q1,qZ . * * 2 q N ) ,

p = (Pl, P Z

9

3

*

--

Y

PN),

(81)

at that instant. Given the initial configuration q(O), p(O), the displacement of the oscillator at t > 0 will be

X(t) = e(t)Qoexp

(82)

In calculating the correlation function, Eq. (82) has to be averaged with a distribution function f ( q , p) d3Nqd3Np,with q = q(O), p = p(0). For simplicity’s sake, we shall assume that the motion of the various particles is uncorrelated, and that their perturbations are additive, i.e., N

222

A . Ben-Reuven

and

N

c

Pi).

P) =i = l

(84)

The superscripts ( I ) denote single-perturber functions. The single-perturber distribution functionf'" approaches the Maxwell-Boltzmann distribution as lqil increases (the oscillators being at the origin). If the volume Vis sufficiently large, thenf") must be inversely proportional to it in order to remain properly normalized : lim f("(q, p) d3q d3p = V-'g((pl)4np2 d p d3q. Iql-rm

(85)

With the brackets denoting the averaging over the perturbing gas (an averaging over a distribution of initial values of Q may be postponed till the end of the calculation), and using Eqs. (83) and (84), (exp i

1;Am(?') d t )

= (exp i

j: Am(')(?') d t ) N

= (1 - V - ' ( V ( 1 - exp i = exp( - n( V ( 1 - exp i

j: Am(')(t) dt')))n"

1:

Am(')(t) d t ) ) }

where n = N / V is the number density. The last equality in (86) was obtained by applying the binomial theorem in the limit V 4 00 (keeping n finite). The remaining average is independent of the volume, owing to (85), if Am decreases rapidly enough as 191 increases. Equation (86) is still a complicated function of t , and further approximations may be required in order to obtain a simple expression for F(o).The so-called statistical (or static) approximation is contained in the assumption that the motion of the particles is so slow that in a typical time interval zero to t, the configuration of the particles alters only insignificantly. In that case

1;

Am(?') dt'

t

Am(q, p )

(87)

where q = q(O), and p = p(0). Equation (86) attains a particularly simple form if it is furthermore assumed that the perturbation is sufficiently weak to allow expanding the exponential within the brackets in a power series of t. Then, to second order terms, (exp i

1,Am(tf)d t )

= exp{ -n( V [ - i t A d 1 )

+ ft2(Ao(1')2])}.

(88)

THE CLASSICAL-OSCILLATOR ANALOG

223

The spectral shape function obtained by Fourier transformation is Gaussian, i.e., Fc(w)= (In 2/n)”’y-’ e’(Qo2) exp{ -(In 2 ) ( 0 - coo - ~ 5 ) ~ / y ’ > , (89)

with a shift of the peak

6 = n ( V Ad’)),

(90)

and half-width at half-intensity y

=

((2 In 2)n( V(Aw(1))2)}1’’.

(91)

The approximation of weak interactions is a rather poor one since the power series appears in the exponent. To be of any use, it must be valid for time intervals of magnitude comparable to y - l . But then the second order term in (88) is of the order of magnitude of unity, and it is very unlikely that all higher-order terms are much smaller than unity. One usually must resort to better approximations, such as those made by Margenau (1951) (see also Breene, 1961). It has been argued by Margenau and Jacobson (1963) that, given any line shape, as long as it is sharp and symmetrical around its peak, Eq. (90) may still be valid if the perturbation is adiabatic. Suppose the line is sufficiently sharp so that its width is considerably smaller than wo . We may then neglect the term F,(-w) as well as the asymmetry caused by the factors preceding F(o)in (14) or (17). If Fc(w)is symmetrical around the peak then the position of the peak will be given by the mean value of the frequency, defined as Q

W=

j-

m

oFc(w)d ~ / ! : ~ F , ( w ) dw.

(92)

But by the rules of Fourier transformation, it follows from Eq. (82) that (5 =

-iC(4d0@,(01,= +ol@(+O>

+ n(

= o0

V Aw(’)(p, q))

= WO

=0

0

+ 6,

+ ( A N q , PI) (93)

where by t = + O we mean that the zero limit is approached from the positive side. Equation (90) has been used as the starting point for several attempts to calculate infrared line shifts (Buckingham, 1962; Bridge and Buckingham, 1963; Margenau and Jacobson, 1963). One should be cautious, however, when that depends on the state of the oscillator using a distribution function f(’) itself. The calculations may then lead to different values for the shift of corresponding lines in absorption and emission, in plain defiance of Kirchhoff’s law-and of experiment (Jaffe et at., 1963).

224

A . Ben-Reuven

B. ADIABATIC AGAINST IMPACTPRESSURE SHIFTS The consequences of the adiabatic assumption do not involve, in any explicit manner, the duration of the collision though the validity of the assumption may depend on how fast the collision occurs. It is therefore possible to envisage a case in which the conditions allow both the impact and the adiabatic assumptions to hold. The perturber particle may adiabatically change the state of the radiating oscillator, finally leaving it in the same state (same amplitude and oscillation frequency in the classical sense ; same energy eigenstate in the quanta1 sense) as before the collision, with only a shift of its phase. But this change may take place in a relatively short time, compared to the mean time between collisions during which the oscillation continues uninterrupted. If so, we face a difficulty, for the adiabatic assumption leads to Eq. (90), where the line shift depends linearly on the perturbation of the frequency, whereas the impact approximation requires that the line shift depend linearly on the sine of the phase shift. The two results agree only when the phase shifts are much smaller than unity and sin 9 x q . It could be argued that the adiabatic assumption, in the strict sense of Eq. (82), may not be used at all. So, for example, even though an adiabatic collision leaves the oscillator’s amplitude and frequency ultimately unaltered, it may change its orientation. Suppose, nevertheless, that the adiabatic assumption is still valid. Let us review, then, what the respective results of the impact and adiabatic theories, Eqs. (43) and (93), do imply. The impact approximation tells us that the shape function of a single isolated line is Lorentzian around the resonance frequency. The exponentially decaying correlation function is correct only for time intervals appreciably longer than a typical duration of a collision At so that the transient effects of the collision may be neglected. Therefore, the Lorentz shape is correct only in a finite range of frequencies Am 6 l/At. Equation (43) tells us then where the peak of the Lorentz curve is (provided, of course, 6 < Am). Equation (93) provides, in contrast, a measure of the mean frequency, in the calculation of which we have to integrate over the whole line [Eq. (92)]. The two results coincide only if Fc(o) is symmetric around E and its first moment converges rapidly enough when the limits of integration are extended to infinity. But the Lorentz function, which is asymptotically proportional to offers a very poor convergence. Indeed, only the principal value of E , N

P[E] = Nlim +m

I-N

oF,(w) d o / / I m F c ( o )do,

(94)

is well defined. The slightest asymmetric deviation from the Lorentz shape out in the wings may place the mean frequency, calculated by Eq. (92), off the peak. Since the line is Lorentzian only in a restricted range, and practically

THE CLASSICAL-OSCILLATOR ANALOG

225

nothing is known about the shape further in the wings, it may well be that the mean frequency does not represent the position of the peak at all. Reliable measurements of line wings are rarely carried out beyond several line widths away from the peak. The position of the peak is usually measured with comparatively much higher precision. Therefore by " shift," we generally mean either the position of the peak or any related parameter on which F,(w) depends in a simple preestablished manner. With Lorentz shapes, we can measure the shift 6, but we can hardly expect to have a reliable measurement of the mean frequency.

C. THEMETHODOF MOMENTS In order to judge whether or not the line is Lorentzian and whether the position of the peak can be represented by the mean value 0, we resort to Van Vleck's method of moments (Abragam, 1961). The kth moment of the line is defined by

100

Mk

=

m

F,(w)(w - 0)& do

IL

F,(o) do.

(95)

In a Lorentzian line, all integrals with k > 1 diverge. However, we can use 1 = a, so the approximation of truncating the line at the frequencies Iw - 0 that F,(o) vanishes further in the wings. The moments of the line are then finite, and given by k-1

k-3

+

* *

-1

(k even),

(96)

( k odd). In the limit a/y -+ co, we have

W(W2 = b/6)(a/Y)

+

00.

(97)

Therefore, if the calculated moments of a line diverge very rapidly with k, we can quite confidently expect it to be Lorentzian at least in the wings. A typical example of better converging moments is provided by the Gaussian line that we encountered in Section V,A. For a Gaussian given by (89), we have (Abragam, 1961) Ak(k - l)(k

- 3)

1

Mk=(O

(k even) ( k odd),

where A

= (2 In 2 ) - ' I 2 y .

(99)

A . Ben-Reuuen

226

In this case, M 2 is simply A2 and ( 100)

M4/(M2)2= 3.

Let us calculate now the moments of a particular example of adiabatic perturbation. By the rules of Fourier transformation we have, in general, M k = [(- i d/dt)ke-i"'@,(r)],,+ o / @ , ( + ~ ) .

(101)

Applying (101) to the adiabatic approximation (80) through (84), we get Mk

=

(CwL3'(q,p ) - G I k ) ,

( 102)

where (83) is used in averaging over the perturbers. The single-perturber function Am(')can be renormalized by subtracting from it the infinitely small quantity (0- o 0 ) N - ' so that Mk

=

([2

i= 1

Am'l)(qi? pi)]').

(103)

This renormalization amounts to choosing W as the origin on the frequency scale. Equation (103) may be written (in the limit of large N ) as a polynomial in the density n,

The leading term at low densities is the binary collision term

More generally,

where {v,} denotes any set of nonnegative integers obeying the two conditions m

mv, = k ,

m

vm

=j ,

(107)

and the weight of each set is determined by the coefficients Dk,j({V,})= k ! [ n ( m ! ) V m V , ! ] - l . rn

(108)

As a result of our renormalization, the first moment vanishes and, therefore, all terms with v1 > 0 vanish. Thus, for example, M 2 = n ( V(Ao'")2)

( 109)

THE CLASSICAL-OSCILLATOR ANALOG

227

and

M4 = n( V(ACO(’))~) + 3n2( V ( A W ( ~ ) ) ~ ) ~ .

(1 10)

The renormalization correction is infinitely small whenever m > 1, and therefore we can insert back in (109) and (1 10) the orginal A d ’ ) . Consider now the example of the van der Waals gas, with a billiard-ball distribution in which f‘” equals its asymptotic value (85) whenever IqI is larger than the molecular diameter r o , and vanishes at smaller separations ( V is then thefree volume of the gas). The perturbation is of the inverse power type,

A d ’ ) = C(r0/r),F,

(1 11)

where r = 191, the strength of the perturbation at r = r,, is given by C,and F is a geometrical factor (of unit order of magnitude) which contains all angular coordinates. In ordinary dispersion forces, p = 6. The leading term of the kth moment is now

N ( ( A w ) ~ )= (2/rr(k - 1))Ck4,(k)a.

(112)

a= +nro3n

( 1 13)

Here,

is the ratio of the unpenetrable volume around a molecule to the specific volume per molecule in the gas, and

The second term in (110) is of the order of u2 and is therefore negligible in dilute gases, where u < 1.

(1 15)

Equation ( I 12) is similar in form [aside from the numerical factor $,(k)] to the leading term in (96) for even k . Therefore in the extreme case (1 15), the behavior is quite similar to that of a truncated Lorentzian shape, C being usually several orders of magnitude larger than the line width. The odd moments do not vanish, in general, and therefore 0 does not represent the position of the peak. The actual values of the position and width depend on the properties of the line wings, to which the truncation is only a crude approximation. One would expect appreciable deviations from a Lorentz shape at IW - 0 1 2 T-’, where T is a measure of the duration of a collision. Since T is inversely proportional to the relative velocity of the colliding particles, results are generally velocity dependent, in contradiction to ordinary statistical-broadening theories.

228

A . Ben-Reuven

VI. Resonance Broadening A. RESONANCE BROADENING AS A MANY-BODY EFFECT

’

When two similar polar molecules encounter each other, one at the lower level of a resonance line and the other at the upper, they may exchange their excitation states by means of dipole-dipole interaction. The excitation is transferred with well-defined phase and polarization. Therefore, the radiation field cannot distinguish which particle is excited (provided the wavelength is much longer than a typical intermolecular distance). Matters are further complicated by the long-range nature of the dipole-dipole interaction. For example, attempts to calculate pressure shifts using first order perturbation may lead to diverging integrals (Mizushima, 1961). Resonance exchange, so it seems, is therefore a many-body effect. Wherever it predominates, one should study carefully whether it is allowable to analyze it in terms of binary collisions only, as is the case in impact broadening. A binary collision approximation, with the resulting linear dependence of the relaxation parameters on the molecular density, is undoubtedly valid if the coherent resonance transfer is destroyed. As was pointed out by Mead (1966), this is possible if the Doppler broadening exceeds the rate of resonance transfer. Owing to the Doppler shift, the colliding molecules are then in different modes of excitation, in the sense discussed in Section IV,B, and they cannot be in a state of exact resonance with each other. The impact (binary collision) approximation may also be valid if the damping by means of other interactions is stronger than resonance transfer. Then the interaction should only be modified by a symmetrization of the wave function of the binary pair to account for the similarity of the two resonating molecules (Di Giacomo, 1964).3 There still remains the case of pure resonance broadening. This problem has been studied recently by Reck et al. (1965). They show that under certain conditions the line width may effectively be of the order of y x (p2/fi)(n/ro3)1/2

where p is the dipole matrix element for the resonance transition, n is the number of oscillators per unit volume, and ro is a “minimum approach” distance for a colliding pair. Replacing p with the classical “oscillator strength,” one obtains, up to a numerical coefficient, the result which has been derived by Holtsmark (1925) using a very simple classical model. In the following section, a somewhat modified version of Holtsmark’s The resulting additional terms may also be obtained from the many-body theory (Ross, 1966) by reckoning the first-order diagrams in the expansion in powers of the density.

THE CLASSICAL-OSCILLATOR ANALOG

229

theory will be presented, followed by a quantum-mechanical justification in Section VI,C. The outcome of this theory is that (1 16) is obtained by taking the square root of the second moment of the resonance-broadened line. As we saw in Section V,C, this is correct only if the moments do not diverge rapidly (that is, if M4 is not much larger than M 2 2 ) .A simple scrutiny of the higher moments in Section VI,D will tell us that this is not so in dilute gases, even in the case of pure resonance broadening. As in the case of nonresonant broadening discussed in Section V,C, the leading term in each even moment is a binary collision term. B. HOLTSMARK’S MODEL

Consider a static configuration of N similar oscillator particle^,^ couplied by dipole-dipole forces. The classical Hamiltonian of such a system is H

=

1PA212m+ A

A

+moo2 QA2

+ 11+e2

QA *

A#B

KAB * QB

.

(1 17)

Here KAB is the traceless symmetric tensor (irreducible tensor of the second rank; Fano and Racah, 1959), (118)

KAB= r d ( 1 - 3rAB rAB

where r A B is the radius vector from particle A to B, and I is the unit tensor. Hamilton’s equations of motion are

the prime signifying a summation over B # A . Using the transformation (18) to the complex variables X and Y, we get

Here we have a secular problem in 6N dimensions. The 3N eigenvalues w, and the 3N eigenvalues -wo of the uncoupled equations are perturbed by the dipole-dipole coupling. Assuming the mean value of the coupling is much smaller than w, (a narrow distribution of eigenvalues around o,),we can neglect the coupling between the X’s and the Y’s, thus reducing the problem to a 3N-dimensional one, XA =

- iw, XA - ie2(2rno,)-

1’

KAB

B

XB

.

(121)

The motion of the particles is neglected both by Holtsmark (1925) and by Reck et al. (1965).

230

A. Ben-Reuven

The spectral shape function is given by a generalization of the singleparticle expression (21),

Fc(w) = (e2/n) Re(X*(O).[i(o

- R - i~)]-'.X(o)),

(122)

where

X

=W x l , ~ j m * * *

Y

XzN)

is a 3N-dimensional vector, and

R = woZ + Ke2/2moo,

(124)

with Z being the unit matrix, and K the interaction matrix in 3N dimensions. The averaging over the ensemble is carried out over all spatial configurations and initial states of the system. Let us assume that (a) all particles are homogeneously distributed in space and that (b) all oscillators are excited with equal amplitudes and polarizations but with random phases. Assumption (a) implies that (besides a possible cutoff at interparticle contact distances) the molecules are distributed independently of each other, as in Section V,C. This assumption is further discussed in Section V1,D. Assumption (b), about the incoherent excitation of the molecules, is implicit also in the work of Reck et al. (1965) (their resolvent operator is composed only of diagrams that end up with the same molecule they begin with). By this random phase assumption, we exclude coherent excitations, such as Dicke's (1 954) superradiant state. We are justified in making this assumption if the total integrated intensity is proportional to the gas density. The mean resonance frequency 0 and the second moment can be calculated from (122) by the method of moments. The mean value of K vanishes, either because of the random phase assumption or because of the tensor character of KAB. The mean frequency is thus "W

J-m

The second moment is

W

Following the two assumptions made above, we get M z = (e4/12mZw,2N)(KZ),

( 127)

THE CLASSICAL-OSCILLATOR ANALOG

23 1

where

Let us introduce now, as in Section V,C, the single-particle homogeneous distribution with a cutoff,

Then, in the limit N M2 =

+ oc), with

12m2w,

n = N / V constant, we get

SK(r) : K(r) d3r =

2ane4

3m2oO2ro3 '

(130)

The root-mean-square spread of frequencies is then

This is, up to a numerical factor, Holtsmark's (1925) result (the difference in the factor is apparently due to Holtsmark's assumption of parallel dipoles). Here, A is simply related to the line width only if M4 is not much larger than A4. Before checking with the higher moments, however, let us discuss first the quantum-mechanical justification of Holtsmark's model.

C. A QUANTUM-MECHANICAL JUSTIFICATION Consider the particular case of the two-level system, in which all particles in equilibrium are at the ground state5 and the radiation excites one of them to level i. This is an approximate description of, for example, the S-P transition in alkali atoms. For simplicity, we assume that, in accordance with this example, state f is nondegenerate, while i is triply degenerate. We shall also assume that the static approximation is valid, and calculate matrix elements of the interaction with fixed intermolecular coordinates. The ground state of the system is, according to our assumptions,

Is>

IfAfB

*.

*f~>.

(132)

The state where particle A is excited with polarization a is lead

l L f B

. . .fN>,

(133)

etc. The dipole-dipole interaction between A and B, V A B ,can only couple leaA)with I e b B ) , where b may be different from a since VA, is a tensor force. Thus, VAB I g> = O* < f i b 1 v(rAB)l ia f>lebB>, vAB I eaA> = b

232

A . Ben-Reuven

The dipole moment component pa, can be expanded in terms of the complete set of basic vectors in Liouville space, pa, =

C ( j IpaAI k) Ij)(kI . ik

(135)

Let us consider that component which is relevant to the resonance excitation of molecule A , namely, (ia

lpal .f>leaA>(gl

xaA*

(136)

The equation of motion of this operator is

(d/dt)X,, = - ih-"H, X,,],

(137)

where

is the Hamiltonian of the system, including the internal degrees of freedom ( H A ) of the particles and their interaction. The part of H which gives a nonvanishing contribution to (137) is H A + CB'V A B .Using (134) and (136), we get d

- x,, dt

= -ih-'[H,,

X,,] - i h - ' c ' [VAB,XaA] B

= -ioifXoA-

ih-'cc'(fibIV(rAB)Ii , f ) X b B . b

B

(139)

The similarity to the classical model equations (121) becomes evident on making the replacement ~ ( i Ipal , f)l2h-'

-,e2(2rnw0)-'

(140)

in the matrix element of the dipole-dipole interaction. It is possible to give a more general proof for multilevel particles in thermal equilibrium by using the density matrix formalism. Both resonant and nonresonant parts of the interaction will then appear in the analysis, making it possible to judge their relative importance. This, however, reaches beyond the limited scope of this article. D. THEFINITE SIZE OF MOLECULES

Consider the kth moment, calculated with (122) under the two assumptions made in Section V1,B. In this moment will appear a summation over products of the type KAB KBC

'.' K Z A

THE CLASSICAL-OSCILLATOR ANALOG

233

with k - 1 intermediate steps. It is possible that the same particle will be involved in several of these steps. We may then expand M k, as we did in the nonresonant case in Section V,C, as a power series of the density. The power of the density is determined by the number of distinct intermediate particles. Take, for example, the fourth moment. To M4 contribute terms with one distinct intermediate particle (binary collision terms), symbolically denoted as

A

-B -A - B - A,

terms with two intermediate particles,

A-B-C-B-A

A-B-A-C-A,

or

and terms with three distinct intermediate particles,

A - B - C - D - A. The contribution of the binary collision terms to M4 is

c

M,(1)= 3(e2/2mWo)4~

riiz d3r~,(Fi~)

=+ ( e 2 / 2 m ~ o ) 4 a ( ~ ~ , ) ,

(141)

where a is defined in (113), and is a shorthand equation for the geometrical factor in (118). Substituting

C = e2(2m00ro3)-’

(143)

for the strength of the interaction at the cutoff, we encounter a situation similar to the one discussed in Section V,C, with p = 3. Terms with two intermediate particles are proportional to a’, and so on. These terms are negligible in comparison to (141) in the dilute gas limit a < 1. Binary collision terms, proportional to a, are the leading terms in all even moments k 2 2, the kth moment being of the order of magnitude of Cka.The line will closely resemble a Lorentzian line, at least in the wings, if C is much larger than the line width. The magnitude of C can be obtained by substituting (140) in (143). In typical atomic or molecular spectra of dilute gases, C is indeed much larger than the line width. We can therefore use the impact approximation [with Di Giacomo’s (1964) modification] to calculate the line width parameter. One should be aware, however, that the diverging behavior of the moments is an indication of a Lorentz-like shape only in the wings of the line, for which we may fit the impact-approximation line width parameter. It tells us nothing about the center of the line, which hardly affects the moments. There the many-body effects discussed by Reck et al. may become prominent.

234

A . Ben-Reuuen

A novel feature is encountered in the odd moments of (122). In all moment calculations using a random phase assumption, we start from and end up with the same molecule A. The radius vectors in each term then form a closed polygon and are therefore linearly dependent. As a result, the odd product of F A B factors will not generally vanish in the averaging over an isotropic distribution, even though F A B is an irreducible tensor of the second rank. The leading term in each odd moment k 2 3 is, however, a tertiary collision term, proportional to u2, since a polygon with an odd number of edges must have an odd number of vertices. The line shape is therefore slightly asymmetrical, with the peak shifted with respect to the mean frequency ooby an amount proportional to the square of the density. We should notice in passing the effect of using more rigorous distribution functions in the ensemble average. Expanding the distribution function as a power series of the interaction [the zero order term being the independent particle distribution (129)], we get extra terms, multiplied by factors of the type (C/kT)m, in the calculation of the kth moment. The number of these terms is greatly reduced by the renormalization of the distribution function. In hot enough gases, where C is not considerably greater than kT, the leading term in the even moments will still be linear in the density, and our previous conjectures concerning the density dependence will remain essentially unaltered. The binary collision approximation may break down near the line center, as has been shown by Reck e l al. (1965). It should be commented, however, that in their semiqualitative arguments concerning the dipolar interaction, the finite size of the molecules has not been taken into account. As a result the effect of the dipolar interaction may be somewhat overestimated. In conclusion, it seems that atomic and molecular gases at ordinary pressures and temperatures may usually be treated with the binary collision impact approximation. Only under restricted conditions, provided resonance dipole interaction dominates, will the line center be modified by many-body effects. ACKNOWLEDGMENTS The author is grateful to R. Zwanzig for the suggestion to study the classical analog. Thanks are also due to H. Friedmann for many illuminating discussions, and to G. Bimbaum, R. G. Gordon, and C. A. Mead for their helpful comments.

REFERENCES Abragam, A. (1961). “The Principles of Nuclear Magnetism.” Oxford Univ. Press, London and New York. Anderson, P. W. (1949). P h p . Rev. 76, 647.

THE CLASSICAL-OSCILLATOR ANALOG

235

Baranger, M. (1958).Phys. Rev. 111,481,494;112,855. Ben-Reuven, A. (1965a).J. Chem. Phys. 42,2037. Ben-Reuven, A. (1965b).Phys. Rev. Letters 14,349. Ben-Reuven, A. (1966).Phys. Rev. 145,7. Bezzerides, B. (1967).J. Quant. Spectry. Radiative Transfer 7,353;Phys. Rev. 159, 3. Birnbaum, G. (1966).Phys. Rev. 150,101. Birnbaum, G., and Maryott, A. A. (1953).Phys. Rev. 92,270. Bleaney, B.,and Loubser, J. H. N. (1950).Proc. Phys. SOC.(London) A63,483. Breene, R. G. (1961). “The Shift and Shape of Spectral Lines.” Pergamon Press, Oxford. Bridge, N. J., and Buckingham, A. D. (1963).Trans. Farahy SOC.59, 1497. Buckingham, A. D.(1962). Trans. Faraday SOC.58,449. Debye, P. (1929).“ Polar Molecules.” Dover, New York. Dicke, R. H. (1954).Phys. Rev. 93,99. Di Giacomo, A. (1964).Nuovo Cimento 34,473. Fano, U. (1963).Phys. Rev. 131,259. Fano, U.,and Racah, G. (1959).“Irreducible Tensorial Sets.” Academic Press, New York. Galatry, L. (1961).Phys. Rev. 122, 1218. Gordon, R. G. (1966a).J. Chem. Phys. 44,3083. Gordon, R. G.(1966b).J. Chem. Phys. 45, 1635. Gordon, R. G. (1966~). J. Chem. Phys. 45, 1649. Heitler, 0.(1954).“The Quantum Theory of Radiation,” 3rd ed. Oxford Univ. Press, London and New York. Herman, R. M. (1963). Phys. Rev. 132,262. Herman, R. M. (1966).J. Chem. Phys. 44,1346. Holtsmark, J. (1925).Z. Physik 34,722. Jaffe, J. H., Friedmann, H. Hirshfeld, M. A., and Ben-Reuven, A. (1963). J. Chem. Phys. 39, 1447. Lindholm, E. (1942). Thesis, Univ. of Stockholm, Uppsala. Lorentz, H. A. (1915). “The Theory of Electrons,” 2nd ed. Dover, New York. Margenau, H. (1951).Phys. Rev. 82, 156. Margenau, H., and Jacobson, H. C. (1963).J. Quant. Spectry. Radiative Transfer 3, 35. Mead, C.A. (1966). Unpublished preprint. Mizushima, M. (1961). Intern. Con$ Spectral Line Shapes Mol. Interactions, Rehovoth (Unpublished). Rautian, S. G. (1967). Soviet Phys. JETP. (English Transl.) 24,788. Rautian, S . G.,and Sobel’man, I. I, (1967). Soviet Phys. Usp. (English Transl.) 9,701. Reck, G. P., Takebe, H., and Mead, C. A. (1965).Phys. Reo. 137,A683. Ross, D.W. (1966). Ann. Phys. 36,458. Spitzer, L. (1940).Phys. Rev. 58,348. Van Vleck, J. H.,and Weisskopf, V. (1945). Rev. Mod. Phys. 17,227. Vu, H.,Atwood, M. R.,and Vodar, B. (1963).J. Chem. Phys. 38,2671.

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THE CALCULATION OF A TOMIC TRANSITION PROBABILITIES R . J . S . CROSSLEY Department of Mathematics. University of York York. England

I . Introduction ..................................................... A . Limitations ................................................... B. Experiment ...................................................

237 238 240 ............................................ 243 243 I1 General Formulas for the Dipole Approximation ..................... 111. Approximate Wave Functions : General Considerations ................248 Configuration Interaction .......................................... 254 255 IV. Criteria for Calculation ........................................... Test of Criteria on Two-Electron Systems............................ 256 257 V . Variational Wave Functions ....................................... A The Hartree-Fock Method ..................................... 258 264 B Other Variational Wave Functions .............................. VI . Semiempirical Methods ........................................... 267 A Methods for Simple Transitions ................................. 267 B. Transitions Involving Equivalent Electrons ........................ 271 273 VII. Perturbation Treatments ........................................... A The Z-Expansion Method ...................................... 273 B. Other Perturbation Methods .................................... 278 VIII Sum Rules. Bounds. and Variational Principles ...................... 279 A. Sum Rules .................................................... 279 B. Bounds and Variational Principles ............................... 280 IX Summary ........................................................ 281 References ....................................................... 288

.

. . . .

.

.

.

I Introduction Spectroscopic data for atoms and atomic ions fall into that category of fundamental scientific material for which justification of measurement or calculation is scarcely necessary. The measurement of the wavelengths of spectral lines and the subsequent deduction of the energy levels is usually not a difficult matter [see the excellent review of EdlCn (1964) and the atomic energy level tables of Moore (1949)l . The study of the intensities of these spectral lines is however. both experimentally and theoretically. a much more formidable problem . Many techniques exist both for measuring and for 237

238

R . J. S. Crossley

calculating line strengths, or equivalently, oscillator strengths or transition probabilities, so that a literature search for the strength of a particular line will often reveal a number of results by different methods. These results will, however, frequently differ from each other even by orders of magnitude. There is therefore a prime need for a critical review and assessment of atomic transition probability data with the object of establishing, so far as practicable, recommended values for each transition together with estimates of error limits. This need is now all the more pressing in view of current research, especially in plasma physics and astrophysics, which requires data for a very wide range of atoms and ions. To a great extent it is being met by the work of the United States National Bureau of Standards (NBS) through the National Standard Reference Data System (NSRDS). Already a critical data compilation of atomic transition probabilities for atoms hydrogen to neon and their ions has been published (Wiese el al. 1966), and a second volume is in preparation (sodium to calcium). In addition Glennon and Wiese (1966, 1968) have produced a bibliography of atomic transition probabilities which is now complete to December 1967. The existence of these compilations fortunately relieves the present author of any obligations to completeness so far as bibliography and tables are concerned. Accordingly the emphasis in this review will be on the description of approximate methods of calculation of atomic transition probabilities in current use, and on the estimation and improvement of their accuracy. A few tables of results are included merely to illustrate arguments made in the text. Since some of these tables are referred to in a number of different contexts, the tables are collected at the end of the article. References cited are intended to be representative of the principal lines of inquiry, rather than comprehensive. Our objectives are thus analogous to those of Foster (1964) in his excellent review of the corresponding experimental procedures. In the remainder of this section we outline certain limitations of the present review and give references to cover matters which have been excluded (Section 1,A); we then make some comments on experimental work (Section 1,B); finally we very briefly discuss applications (Section 1,C). A. LIMITATIONS We will concentrate entirely on the calculation of allowed (electric dipole) transitions between atomic states described by the Russell-Saunders coupling scheme (L-S coupling; see, e.g., Condon and Shortley, 1935). In practice, this is not as serious a limitation as it may seem as, principally due to the work of Garstang, calculations for intermediate coupling and for forbidden transitions have been very thoroughly reviewed. Furthermore, intermediate coupling calculations make use of transition probabilities calculated in a

ATOMC TRANSITION PROBABILITIES

239

definite coupling scheme, usually L-S coupling, the transformation to intermediate coupling being carried out by a standard method which depends on the energy levels of the system (Condon and Shortley, 1935). Thus our considerations will be directly applicable to intermediate coupling calculations. For a review of recent calculations in intermediate coupling see Layzer and Garstang (1968).' Forbidden transitions have been extensively reviewed by Garstang (1962a); more recent work is covered by Layzer and Garstang (1968). We will not consider here the matrix elements involved in these calculations; however, many of our observations about wave functions apply also to forbidden transitions. One point of some interest here is the relative magnitudes of contributions to a line strength from the various electric and magnetic multipole effects (Garstang, 1962b, 1967; Dugan, 1965; Mizushima, 1966). For allowed transitions in L-S coupling, however, it is clear that, except possibly in the most pathological cases, contributions other than electric dipole may be safely ignored (Condon and Shortley, 1935). In almost all calculations except some for helium-like systems (two electrons) determinations of the wave functions are carried out in some form of the central field approximation in which the wave function is expressed as a linear combination of products of one-electron spin orbitals; each spin orbital is a product of space and spin parts, and the space part (the orbital) is in turn a product of a radial function and an angular function which is of spherical harmonic form. Consequently, the transition integral (see Section 11) may be expressed as the product of an integral involving the radial parts only (the radial transition integral) and factors resulting from integration over the angular and spin parts (Condon and Shortley, 1935). The angular and spin factors are readily evaluated for simple cases; see the useful tables of Allen (1963) who gives further references. These calculations have been extended to more complicated configurations by the methods of Racah (de-Shalit and Talmi, 1963), and recently extensive tables have been given by Shore and Menzel(1968). Further results are given by Crossley and Parkinson (1969). In view of the standard nature of the method and this extensive coverage in the literature, we will confine our attention here to the calculation of the radial transition integral. Coupling schemes of relevance to atomic spectra are discussed by Cowan and Andrew (1965). Methods of calculation of transition probabilities in the various coupling schemes are thoroughly described in the book by Levinson and Nikitin (1962) [see also the recent paper by Warner (1968)l. These calculations are usually carried out by Racah methods which start from This review of atomic transition probability calculations came to my attention when the present work was almost completed. I am indebted to Professor Layzer for sending it in advance of publication.

240

R . J . S. Crossley

transition matrix elements in L-S coupling and recouple by means of a transformation which again depends only on the angular and spin parts of the one-electron spin orbitals. Accepting the central field apprcximation, this is a well defined process and the only remaining factors of interest are the radial transition integrals discussed in this article. We further confine ourselves to transitions between bound states of atomic systems, excluding transitions to autoionizing levels which have recently been well described in the review of bound-free transitions (photoionization) by Stewart (1967) [see also Peach (1967b,c)]. Peach (1965, 1967a,c) discusses free-free transitions. Two-photon transitions have been discussed by Levinson and Nikitin (1962), Dalgarno (l966), and Dalgarno and Victor (1 966). Previous reviews of calculations of allowed transition probabilities have been given by Garstang (1955), Trefftz (1958), and Griem (1964), and more briefly by Nicholls and Stewart (1962) and Wiese et al. (1966). A much more comprehensive review, including some previously unpublished material which will be referred to below, has been given by Layzer and Garstang (1968). Finally, an unpublished review by Varsavsky (1958) has been of value in preparing the present work.

B. EXPERIMENT As mentioned above, the experimental measurement of atomic transition probabilities has been thoroughly reviewed by Foster (1964). However, since the accuracy of a calculation is frequently assessed by the extent to which it agrees or fails to agree with experimental results, it is necessary for the theoretician to have a working knowledge of the accuracy which may be expected from the various experimental techniques. As these have changed and developed considerably since Foster’s review, it seems worthwhile to give here a short list of references to current experimental work. For a complete bibliography we refer to Glennon and Wiese (1966, 1968). Important additional references are the forthcoming reviews by Wiese (1968a,b) of lifetimes and arcs, respectively, and the report of the Tucson conference on beam-foil spectroscopy (Bashkin, 1968) which also covers other techniques and applications. Cunningham and Link (1967) have commented briefly on the available methods, drawing a distinction between those methods which measure emission and absorption intensity and require knowledge of vapor density for the deduction of transition probabilities, and those methods, usually measuring lifetime, which are independent of vapor density. The measurement of vapor density almost always causes an uncertainty of at least 10% in absolute transition probabilities. The necessity of making assumptions concerning local thermodynamic equilibrium and radiation trapping (Olsen

ATOMIC TRANSITION PROBABILITIES

24 1

1963; Dickermann and Deuel, 1964; Griem, 1964) often increases this error considerably. These methods may be much more reliable, however, in determining relative transition probabilities, which then may be put on an absolute scale by means of a single measurement (or calculation) by some more reliable absolute method. For some transitions, e.g., between highly excited levels, this may be the most satisfactory procedure. Emission methods have been reviewed by Olsen (1963) and some recent references are Popenoe and Shumaker (1969, Malone and Corcoran (1966), Woodgate (1966), Shumaker and Popenoe (1967), Corliss and Shumaker (1967), Frish (1967), and Vaughan (1968). Some comment is perhaps necessary on the very extensive tables of oscillator strengths of Corliss and Bozman (1962). These results are of no great accuracy, the authors themselves suggesting errors of 40-50 % in relative values within a single spectrum, and of 70-100% in absolute values. In addition, there appear to be errors in the second section of iron-group elements due to the ionization potentials used, and also due to energy calibration below 2500 A (Morozova and Startsev, 1965; Warner and Cowley, 1967). New emission measurements have been reported for iron by Corliss and Warner (1966) [see also Corliss and Tech (1967a,b)], and for singly ionized elements of the iron group by Warner (1967). Corliss (1967) has given a recalibration of intensity for the ultraviolet region. Measurements in absorption may be made either using a line source (line absorption) or a continuum source (total absorption); a very full description of both methods is given by Mitchell and Zemansky (1934). Foster has described the atomic beam experiments of R. B. King and colleagues, who have since introduced more sophisticated techniques (Lawrence et al., 1965) and estimate their errors as 10-20%. Their relative errors should of course be much better than this. Other total absorption measurements have been made by Kozlov et al. (1966) and Bieniewski et al. (1968). Chashchina and Schreider (1967) have used the line absorption method, and Lvov (1965) has ingeniously combined both methods to eliminate some of the experimental errors (even so he suggests only 20 % accuracy). The Rozhdestvenskii hook method has undergone some development since Foster’s review; more recent descriptions have been given by Penkin (1964) and Marlow (1967). In the former, Penkin gives revised results of earlier measurements on the basis of the vapor pressure data of Nesmeyanov (1961). A more significant advance, however, is the combining of the method with total absorption measurements by which it is possible to avoid vapor density determination altogether. This method must be counted as one of the most accurate experimental methods for transition probabilities. Among recent papers are Slavenas (1966) (hook method) and Penkin and Shabanova (1967) (hook-absorption method). Other methods involving dispersion effects have recently been suggested. Buckingham (1962) has discussed the Kerr effect

242

R. J. S. Crossley

(electrooptical birefringence) and calculations by Boyle and Coulson ( I 966a,b) and by Bogaard et a/. (1967) suggest this would provide a viable technique for oscillator strength determination. The last reference also discusses the analogous magnetic effect (Cotton-Mouton effect). Measurements of lifetimes proceed by exciting the atoms of interest either optically or by electron impact and studying the subsequent decay by one of a variety of techniques. In favorable circumstances an accuracy for the lifetime of better than 10% is frequently possible. In the simplest cases in which deexcitation is possible only to one level, the transition probability is just the inverse of the lifetime; in other cases it may be necessary to use relative transition probabilities (branching ratios) from other sources. Sometimes it is possible to give a bound. Errors may arise from, e.g., neglect of forbidden transitions or of cascade processes from higher levels. The latter difficulty may be reduced by carefully controlled excitation processes. Optical excitation thus has an obvious advantage, but recently controlled slow electron beams have been developed [e.g., Karstensen (19691. Optical excitation techniques have been reviewed by Budick ( I 967) who discusses magnetic double resonance and level-crossing experiments, including the Hanle effect (zero-field level crossing). A simple description of the Hanle effect has been given by de Zafra and Kirk (1967). Electron excitation methods have been reviewed by Series (1959). Some recent references are: Savage and Lawrence (1966) (electron excitation, phase shift); Cunningham and Link (1967) (optical excitation, phase shift); Lurio (1965), Barrat et a/. (1966), Gallagher (1967), and Budick (1968) (optical excitation, Hanle); Klose (1968) and references cited there (electron excitation, delayed coincidence) ; and Altman and Chaika (1965) (optical excitation, double resonance). Investigations using shock tubes have been carried out by Garton et at. (1964), Shackleford (1965), Brown (1966), Coates and Gaydon (1966), Byard (1967), Huber and Tobey (1967), and Roberts and Eckerle (1967); however conditions in the shock d o not appear to be sufficiently understood for this method to be reliable. A discussion of quantitative spectroscopic studies with shock tubes has been given by Wurster (1963). An important new method, especially valuable for the determination of lifetimes of excited states of ions, is the beam-foil method of Bashkin and colleagues, recently reviewed by Bickel (1967a). The excited ions are produced by firing a beam of atoms through a thin foil which thus acts as a very well-defined ion source. Lifetimes are determined by studying the subsequent decay. Recent references are Kohl (1967), Heroux (1967), and Bickel (1967b, 1968). Heroux suggests errors in lifetime of 5-15%; this is perhaps a little optimistic. The most recent work is described in the forthcoming volumes edited by Bashkin (1968). Occasionally, helpful information may be obtained from astrophysical observations; see, e.g., Underhill (1968).

ATOMIC TRANSITION PROBABILITIES

243

C. APPLICATIONS The renewal of interest in accurate determinations of transition probabilities is due very largely to astrophysical applications concerning stellar atmospheres and in particular the chromosphere and corona of the Sun. Here highly ionized atoms play an important r61e. The subject has been reviewed in the volumes edited by Hubenet (1966) and Perek (1967); recent developments affecting the solar corona have been described by Wimel-Pecker (1967). In the laboratory accurate transition probabilities are of value in plasma diagnostics (Griem, 1964, Hinnov, 1966), studies of line broadening (Griem, 1964, Vaughan, 1968), populations in a discharge (Parkes et a/., 1967), optical excitation processes (Kraulinya et al., 1966; Frisch and Klucharyov, 1967), and in investigations of conditions in shock tubes (Coates and Gaydon, 1966). Because of difficulties of measurement and approximations in theory, knowledge of transition probabilities to within 10% would usually be satisfactory in these applications; however, very few transition probabilities indeed are known with this degree of certainty (Wiese et al., 1966). We will therefore adopt 10% as a target of accuracy in the discussion of methods of calculation which follows. 11. General Formulas for the Dipole Approximation

In the following we use atomic units, the unit of energy being twice the ionization potential of the hydrogen atom in its ground state, that is, two Rydbergs. A complete list of these units is given by Shore and Menzel (1968, pp. 1 13-1 20). Following Condon and Shortley (1935, p. 97), the atomic levelcorresponding to a given value of the total angular momentum quantum number 5 is, in the absence of external fields, (25 + 1)-fold degenerate; that is, it consists of (25 + I ) states each specified by a different value of M , the quantum number of the z component of total angular momentum. A transition between leoels gives rise to a line. The radiation from a transition between states is called a component of the line. Consider the transition from an upper (initial) state a to a lower (final) state 6 of an atomic system. Then the emitted irttensity may be written I(u + 6) = N, Eab&(a + 6) (1) where Eab = E, - Eb

(2)

and E,, E,, are the energies of states a, 6; N ( a ) is the number of atoms in state a and &(a + 6) is the spontaneous transition probability for the transition

244

R. J. S. Crossley

(the number of transitions per second). The term Z(a + b) is thus expressed in atomic units of energy per second. From the quantum theory of radiation (see, e.g., Heitler 1954), it follows that the intensity may be expressed in terms of the component strength S(a + b) defined by

that is, it is the square of the dipole-length matrix element. ri denotes the position vector of the ith electron; i is summed over the N electrons of the atom. Then d ( a + b) = 4 a 3 E z b s(a + b )

(4)

where u is the fine structure constant (in atomic units CCL= 1 where c is the velocity of light). The lgetime z(a + b) of the state a against the decay a -,b is z(a + 6) = {&(a + b)}-I,

(5)

and so the lifetime z, of state a (against all decays) is given by 7,

=

{F

d ( a + b)]-'

(6)

where the summation is over all states b lower than a. The emission oscillator strength fob is defined by fob

=

-3Eab

S(a -P b).

(7)

We now consider transitions between leuels A , B of total angular momenta JA, J,. The intensity of the emitted line is obtained by summing (1) over initial and final states

where we assume natural excitation in which each state of the initial level has the same population N , . However, the Einstein spontaneous transition probability d ( A + B ) satisfies, analogously to (l), Z(A + B ) = NA E A B d ( A 4B )

(9)

where, because of the degeneracy of the states of the levels A , B, EAB=

Eab,

(10)

and where NA is the number of atoms in level A . Now assuming natural excitation, N A = gA Na

(1 1)

245

ATOMIC TRANSITION PROBABILITIES

where gA = (25, and using (4)

+ 1) is the weight of level A . Hence comparing (8) and (9)

where the line strength S(A -+ B) is given by S(A +B ) =

1S(a

+

b).

(13)

a. b

The (line) absorption oscillator strength f B Ais defined in terms of the line strength by gBfBA = 3 E A S S(A

B,

(14)

and is related to the (line) emission oscillator strength f A B[cf. Eq. (7)] by gBfBA = -gAfAB

(15)

The phrase " oscillator strength " will imply absorption unless otherwise stated. Because of the symmetry of (1 5) (the sign is unimportant), tables often list gf values. Here g is the weight of the initial level, whether for emission or absorption. In L-S coupling further degeneracy occurs in theory within a term characterized by the total orbital and spin angular momentum quantum numbers L and S between the levels characterized by different values of J . Equations (12)-(15) hold good in this case if A , B are interpreted as referring to terms, provided the summations are extended over all levels of each term. The weight g for the term ( L , S ) is then given by g

= (2L

+ 1)(2S + 1).

(16)

Equation (13) now gives the multiplet strength and (1 5 ) the (multiplet) gf value. In practice the levels within a term are separated by the spin orbit interaction. Provided this splitting is small compared with the separation of the terms A , B, no great loss of accuracy is incurred by using the average energy difference of states (that is, the weighted energy difference of levels) for the excitation energy EAB . For other definitions and numerical relations, we refer to the literature [e.g., Allen (1963)l. We now concentrate on the matrix element which appears in the component strength S(a -+ b) of Eq. (3). Standard quantum mechanics leads to three different forms of this matrix element: the dipole-length form

R . J . S. Crossley

246 the dipole-velocity form

where the change of sign follows from the anti-Hermitian nature of V ; and the dipole-acceleration form

.

The summations are over the labels i = 1, . . , N of the electrons of the system. When la) and Ib) are eigenfunctions with eigenvalues E,, Eb of the same Hamiltonian H , where ff=-+xAi2+V (20) i

and Visa potential energy, the expressions (17)-(19) are equal to each other. However, it is nor necessary that this H is the exact Hamiltonian of any problem. Thus in certain cases Hartree calculations will yield equal matrix elements (1 7)-( 19), but the corresponding Hartree-Fock calculations will not (Ivanova, 1963); this is so if the Hartree calculation is made for a single active electron in the field of a frozen core, and exchange is present in the HartreeFock calculation; see also LaPaglia (1967). If we further assume that H is the Hamiltonian of an N-electron atomic system, then the potential energy V is given by 1/= - Z x r ; ' + x r , j * (21) i< j

i

where 2 is the nuclear charge, ri the distance from the nucleus to the ith electron, and r i j the separation of the ith and jth electrons. In this case, the dipole acceleration form may be written

Z E ~ ; ' ( U ~C r i r i 3 1 b ) = Z E ; ~ ( ~L Ir i r r 3 l u ) . i

i

(22)

The proof uses the fact that

( V , + V2)f(r12) = 0

(23)

where f i s any differentiable function of rl . The equations shown in (17)-(19) and (22) hold under very general circumstances. The case of the dipole-velocity form has been considered by Green and Weber (1950). A sufficient condition is Hala) = Ea'la)

and

HbIb) = Eb'Ib)

(24)

for some (not necessarily the same) Hamiltonians H a , Hb with eigenvalues Ea', Eb' not necessarily equal to E,, E b .Proofs that Eqs. (17)-(19) are equal

247

ATOMIC TRANSITION PROBABILITIES

to each other, assuming the physical Hamiltonian and exact eigenfunctions, have been given by, e.g., Bates (1961), who used Green's second theorem, and by Bethe and Salpeter (1957, p. 251) by considering commutators of operators. This latter method has been generalized and extended by Chen (1964) who employed the off-diagonal hypervirial theorem,

( 4 CH, w1 Ib)

= Eub

( 4 WIb)

(25)

where Wis an arbitrary time-independent operator. Assuming la) and Ib) are eigenfunctions of H with eigenvalues E,, Eb , and recalling Eq. (2), Eq. (25) is easily proved. Substitution for W of, in turn, r i , [ H , r i ] , and [H, [H, essentially reproduces the proof of the equality of (17), (18), and (19) given by Bethe and Salpeter. We now see very clearly, however, that the process may be continued apparently without limit. One more step leads to a fourth form of the dipole matrix element

xi rill

xi

xi

(the circumflex as usual indicating the unit vector) in which the operator is again anti-Hermitian. Chen has shown, however, that because of the requirement that [ H , W ] l b ) be normalizable, formula (26) is not applicable to S states. Clearly the procedure may be continued to yield further formulas, but with increasing restrictions in their application. Very recently, Chong (1968) has used Eckart-type wave functions in calculations of transition probabilities for helium-like ions using all four matrix elements (17), ( I 8), (22), and (26). Some of his results are reproduced here in Table 11. The striking feature is that, whereas the dipole-length and dipole-velocity results agree with each other, and the dipole-acceleration and Eq. (26) results agree with each other (in both cases the agreement improves with excitation of the states involved in the transition), these two pairs of results fail to agree, and in the case of helium fail by a factor of 2. Closer inspection shows that to a remarkably high degree of accuracy, the ratio of acceleration to velocity results is as 2 : Z - I . We may analyze these results in terms of the conditions for equality between the various matrix elements given above. The near equality of length and velocity results suggests that the Eckart wave functions are close to being eigenfunctions of some common Hamiltonian H' [see the discussion following Eq. (20)] which will include an expression V' for the potential energy which approximates the true electron interaction term I/rl2 by a sum of one-electron terms symmetric in rl and r 2 . However in calculating the dipole-acceleration form by Eq. (22), it is assumed that H' is the exact Hamiltonian [Eqs. (20) and (21)]. The much more stringent assumption made at this stage is the qualitative reason for the lack of agreement between the velocity and acceleration forms. We may

248

R . J. S. Crossley

quantify this argument by considering the more consistent method of calculating the acceleration form by Eq. (19) in which, for the potential energy V, we take the approximate one-electron potential V' described above. The problem now is to find an expression for V'. In the cases of interest here the electron involved in the transition is highly excited and so moves in a potential of roughly (2 - l ) r - This consideration suffices to show that acceleration matrix elements calculated by Eq. (22) will be reduced in the ratio ( Z - 1) :Z , precisely in accordance with Chong's results. This argument is similar to that given by Stewart (1967) for photoionization. The crux of this matter seems to be Eq. (23). No symmetric one-electron potential approximating t;; will be annihilated by (V, + V,). Thus we may conclude that the acceleration matrix element in the form (22) should not be used with noncorrelated wave functions. In the discussions which follow, we will concentrate on the length and velocity forms.

',

III. Approximate Wave Functions: General Considerations In nonrelativistic quantum mechanics the calculation of the dipole matrix element can be carried out exactly only for the one-electron problem (Bethe and Salpeter, 1957, Sect. 63; Green et al., 1957). For any more complex atom, it will be necessary to employ approximate wave functions to describe the upper and lower states of the transition. In general these will not be eigenfunctions of the same Hamiltonian and consequently the different forms of the matrix element (17)-(19) will lead to different results. The problem now is to decide which forms of the matrix element are appropriate to the various possible types of approximate wave functions. In a very qualitative way, we may distinguish between the forms (17), (18), and (19) by observing that in the dipole-length form the contribution to the matrix element from regions of large r is important, whereas in the acceleration form it is the region near the nucleus which is important. The velocity form lies between these two cases. This is strikingly illustrated by considering the Is - 2p transition of the hydrogen atom. Figure 1 [from Ehrenson and Phillipson (1961)] shows the radial factors of the integrands of the length, velocity, and acceleration matrix elements plotted against r ; the vertical scale is chosen so as to normalize each radial integral. It will be observed that the acceleration form in particular concentrates on a narrow region of r. This leads us to consider regions in which approximate wave functions are welldetermined. In view of the very qualitative nature of this argument and the importance which is attached to it, it seems worthwhile to refer briefly to the very extensive literature on the calculation of the bound-free absorption coefficient for

249

ATOMIC TRANSITION PROBABILITIES

the hydrogen negative ion. This involves calculating the transition matrix element between the ground state of H- and the singly ionized system. The argument suggested above was used by Massey and Bates (1940) to justify the use of a plane-wave continuum function in the dipole-length matrix element. Subsequent calculations of the absorption coefficient were reviewed by Chandrasekhar (1944), who pointed out that an increase in complexity of the Hylleraas-type variational wave function corresponding to a very small change in energy gave a very large change in the absorption coefficient. This he attributed to the variational wave functions being poorly determined by the energy criterion at large r. In a subsequent paper (Chandrasekhar, 1945), ~

~

f'\

0.4-

I

;

I I

I

\

\\

\

\

\

\

FIG.1 . Ratio I of the radial dipole integrand to the radial dipole integral against radius r for the Is-2p transition of hydrogen: -, dipole length; - -, dipole velocity; -, dipole acceleration. [From Ehrenson and Phillipson (1961).]

--

calculations were carried out using a 6- and an 11-term Hylleraas trial function and all three forms of the matrix element. The velocity form was shown to be much more stable under variations of the wave function than either of the other forms. This was attributed to the energy criterion leading to good determination of the wave functions in the regions (medium values of r ) of importance in the velocity matrix element. Subsequent calculations using 20and 70-term wave functions by Geltman (1962) have confirmed this result. Furthermore Geltman shows that the velocity form gives excellent agreement with the experimental results of Smith and Burch (1959). For a further discussion see Stewart (1967, p. 45).

2 50

R.J. S.Crossley

It could be argued that H- constitutes a particularly sensitive case; however, precisely the same problem arises in the photoionization calculation for He (Stewart, 1963, p. 346). We must expect the sort of problem discussed here to be even more important for systems with more than two electrons since it is usually necessary to employ approximate wave functions of types much less sophisticated than the wave functions available for two-electron problems. On the other hand, for bound-bound transitions the simple arguments concerning dependence of the different forms of the matrix element on different regions of space do not necessarily follow, since the importance of a region heavily weighted by one bound-state function may be reduced by the second function taking very small values. Some insight into the accuracy of different types of wave functions may be obtained by evaluating expectation values of various operators for twoelectron systems with such wave functions and comparing results with those obtained using sophisticated wave functions (Stewart, 1963) of the Hylleraas, Kinoshita, or Pekeris type. Such calculations were carried out by Cohen (1961), who showed that the Hartree-Fock wave function for helium of Green et al. (1954) leads to expectation values of the operators rl , r ; , and r;' which agree within 0.5 % with values derived from the wave function of Pekeris (1959). Operators which weight regions nearer to or further from the nucleus give less satisfactory agreement. More recently, Chen and Dalgarno (1965) carried out similar calculations using the simple screened hydrogenic product function for helium, with and without correlation and in both the closed- and open-shell approximations, for the operators rln, n taking integer values from - 1 to +9. Again, good agreement is obtained with the more sophisticated calculation for n = - 1, but the discrepancy increases to as much as 100% at n = 9. These results reinforce the argument that a variational wave function is only reliable in the energy-important region and so only good for evaluation of quantities roughly dependent on (we note that normalization rio is always given exactly). This in ensures that the expectation value of turn reinforces the argument that the dipole-velocity matrix element is to be preferred when a variational wave function is used. This may also be argued in another way: the virial theorem, which is always satisfied by a Hartree-Fock wave function and, obviously, by any wave function which includes a scaling parameter, ensures that a good total energy estimate implies a good kinetic energy estimate; that is, an energy-optimized wave function will give a good result for ( p 2 ) , the expectation value of the square of the momentum. Hence we may expect it to give reasonable values of ( p ) and so reasonable values of dipole-velocity matrix elements. A step in the direction of putting these qualitative arguments on a mathematical basis may be made as follows. Assume that the available approximate

xiEir;'

ATOMIC TRANSITION PROBABILITIES

251

wave function la’) may be related to the unknown exact wave function la) by la’) = la)

+ pa)

(27)

where 16a) is small. Then the dipole-length matrix element i

1

I

where for simplicity, we assume that the wave function J b )is known exactly. Now the dipole-velocity formulation gives

The first terms of the right-hand sides of (28) and (29) are equal to the exact value. The errors due to use of the approximate wave function are given by the second terms. In the dipole-velocity form, this error is inversely proportional to the excitation energy. This suggests that the velocity form will be less sensitive to errors in the wave function for large excitation energies. A more sophisticated version of this argument has been given by Dalgarno and Lewis (1956) who expanded 16a) in terms of the complete set of eigenfunctions of the exact Hamiltonian and so concluded that if the excitation energy is larger than that associated with any other possible transition, then the velocity form is to be preferred. Calculations of the photoionization of helium support this hypothesis (Dalgarno and Kingston, 1958). From Eq. (19) follows an obvious corollary that the dipole-acceleration form is likely to be especially sensitive to the approximate wave function for small excitation energies, and reliable for very large excitation energies. This is borne out by the satisfactory results which Bagus (1964) obtained with the acceleration form for X-ray transitions. There is, however, a serious difficulty in estimating the error in a calculation of the transition integral (17), or any of the equivalent integrals, due to the fact that contributions to the integral may cancel. Pfennig et al. (1965) give an example of complete cancellation in Be. A very simple measure of this cancellation, suggested by Bates (1947), is obtained by calculating, in addition to the radial transition integral R=

JO

P,,,,,rPnIdr

(30)

(for simplicity we assume we need only consider radial integrals over a single electron), the integral m

X

=

0

IP,.,. rPnlldr,

(31)

252

R.J. S. Crossley

where P,,,and P,,,,, denote the initial and final radial wave functions with normalization integral

Jbm(P,,J2dr = 1. Then

0 < lRl/X< 1

(33)

the value 1 signifying no cancellation and the value 0 signifying complete cancellation. This simple test is useful for picking out transitions with heavy cancellation which in consequence cannot be calculated reliably. A more sophisticated test of cancellation has been suggested by Layzer (1961). Using the same notation, we define Rn1by

and Rn.l.,correspondingly. Then, by Schwarz's inequality,

o G I R I / ( R , , , R , , , ~ ,G) ~1~ ~

(35)

where complete cancellation in R of course gives the lower limit 0, while = P,,,,,. The expression (35) the upper limit can only be attained when P,,, thus gives an indication of the similarity of P,,,and P,,.,.. Calculations (Layzer and Garstang, 1968) suggest that a value greater than 0.7 usually ensures a reasonable estimate of the transition probability. When the length form is used in the central field approximation there are certain transitions for which no cancellation occurs since neither radial function contains a node. These are transitions of the type n, I + n + 1, 1 + 1 with I = n - 1. The above considerations suggest that these nodeless transitions may be particularly amenable to calculation. Cancellation always occurs when the velocity form (18) is used because the derivative of the wave function is involved (c.f. Fig. 2). This suggests that the length form is to be preferred for weak transitions. Another problem arises over the excitation energies required for (18) and (19) and for converting dipole matrix elements into oscillator strengths, transition probabilities, and emission intensities (see Section I). Green and Weber (1950) and Green et al. (1951) discussed these problems and concluded that calculated energies should be used in preference to experimental energies. However, their evidence is very slight and, although there is a possible qualitative argument in favor of using the calculated excitation energy with the dipole-velocity matrix element on the grounds that the two errors involved might cancel, in the absence of any sound theoretical argument

253

ATOMIC TRANSITION PROBABILITIES

I.o

2.0

3.0

4.0

5.0

r(ad

FIG.2. Transition integrands for a 2s22pz-2s2p3 transition in carbon calculated with HartreeFock radial functions: -, transition integrands; - -, 2p function; - - -,2s function. (From Weiss, 1967c.)

it seems preferable to use the most accurate excitation energy available. It is the nonrelativistic energy which is required so the calculated value should be used for one- and two-electron problems and the experimental value in all other cases, except possibly when the relativistic error is expected to be very large as might be the case for an X-ray transition in a very heavy atom. These conclusions are in broad agreement with those of Bagus (1964). A possible source of error in this procedure, however, arises from the fact that identification of spectral lines is sometimes incorrect. A well-known example of this arises even in a two-electron spectrum-that of Li', where Pekeris (1962) showed by calculation that the ls2s('S) - 1s2p('P0) line is at 9584 A and not at 8517 A as identified by Series and Willis (1958). The recent literature of spectral analyses is covered by the bibliography of Moore (1968). In practice, calculations are frequently made using both length and velocity matrix elements, agreement being interpreted as a favorable sign although it is clear that this is not a guarantee of accuracy. Often the arithmetic mean of the two results is taken, and it is interesting to note that there is in fact some justification for doing this. Seaton (1951) has suggested the geometric mean should be taken, pointing out that this enables the f value to be expressed independent of the energy [Eqs. (14), (17), and ( I 8)] ;furthermore, this choice enables the Thomas-Reiche-Kuhn sum rule (see Section VII1,A) to be applied

254

R.J. S. Crossley

to calculations with approximate wave functions, provided only that they form a complete set. Hansen (1967) has also advocated the geometric mean; he demonstrates that under certain conditions of configuration interaction (the case of Be discussed in the next section is an example) the geometric mean is affected by the perturbation only in second order. Since it is rare for the length and velocity results to differ by a factor of more than 2 [although Garstang and Hill (1966) give an example in Ca" with a factor of 1001, it follows that usually the geometric mean will differ by only a very few percent from the arithmetic mean.

CONFIGURATION INTERACTION The simplest picture of a complex atom is one in which each electron moves individually in the field of the nucleus screened by the other electrons. Each electron is described by a scaled hydrogenic function and the electron wave function of the whole atom is the Slater determinant of these one-electron functions [see, e.g., Slater (1 960)]. This model admits the familiar configuration description, e.g., the ground state of Be ls22s2('S).Slater developed this kind of model by using the variational principle and allowing for more general radial dependence of the one-electron functions (spin orbitals). He was able to show that certain ratios of term intervals could be calculated independent of these radial functions. This is the crux of the central field model. Condon and Shortley (1935) compare many of these term-interval ratios with experiment; they show that although agreement is sometimes good, it is often not. Much improved agreement may be obtained by allowing for configuration interaction, either within a variational treatment (especially the Hartree-Fock method; see Section V,A) or within a perturbation treatment (Layzer, 1959; see Section VII). The wave function will now become some linear combination of single configuration wave functions of the same total space and spin symmetry. The important perturbers have been generally supposed to be terms of energy close to the term of interest. However, Layzer (1959) has argued that configurations within the same complex (that is, with the same symmetry and set of principal quantum numbers) are important (see Section VII). In the following, we take particular interest in three examples of configuration interaction-the simplest Be('S)

lsz22s2 and

ls22p2

and the two classic examples of Condon and Shortley (1935, pp. 366369) [but note that the work on Mg by Bacher (1933) was in error and later corrected (Bacher, 1939) with much improved agreement]

ATOMIC TRANSITION PROBABILITIES

255

Mg ('D) 3s3d and 3p2 and A1 ('0) 3s23d and 3s3p2. In the case of Mg, there is a third configuration 3d2in the same complex (that is, hydrogenically degenerate with 3s3d and 3p2), and in the case of A1 several further configurations (Godfredsen, 1966). To illustrate the effectof coniiguration interaction on transition probability matrix elements consider the resonance transition of Be : ~ S ~ ~ S ~ ( ~ S ) ls22s2p('Po). In the single configuration model the calculations reduce essentially to the matrix element (24 r I2p>. Adopting a two-configuration description of the ground state a(ls22s2)+ fl(ls'2p2), where a and fl are constants (mixing coefficients),the matrix element becomes (a & fl)(2sl r 12~). According to the relative sign of a and j3 and the f sign, the transition may be strengthened or weakened, but to discover which requires a calculation. Because of the anti-Hermitian nature of V [Eq. (lS)], however, it follows, provided experimental energies are used, that if the length matrix element is strengthened, the velocity matrix element is weakened, and vice versa (LaPaglia and Sinanoglu, 1966; Hansen, 1967). Thus lack of agreement between length and velocity results may well indicate a configuration interaction effect, and taking the arithmetic or geometric mean would qualitatively correct for this. This is well illustrated by the results for Be given in Table I11 which suggest that configuration interaction weakens the 2s'('S)-2s2p('P0) transition; correspondingly it would strengthen the 2p2('S)-2s2p( 'Po) transition. In more complicated cases, however, this simple analysis may not apply. Cases also arise in which we can predict a definite weakening due to configuration interaction. Consider in A1 the transition 3sz3d('D)-3s24f (2Fo). If we describe the ground state by two configurations, a(3s23d) + fl(3s3p2),and notice that the transition 3~3p'(~D)-3~~4f(~F~) is forbidden (two-electron jump) we see that the effect of the configuration interaction is to reduce the transition matrix element by a factor a, and so the strength by a factor a'. Since this particular configuration interaction is very strong, it is possible that the oscillator strength may be halved by this effect. This example also shows how an apparently forbidden transition may become allowed through configuration interaction. The 3s3p2-3sz4f transition has borrowed some of the strength of the 3s23d-3s24f transition. IV. Criteria for Calculation

It may be helpful at this point to summarize our conclusions so far. We advocate the length matrix element when the transition energy is small, and

256

R . J. S. Crossley

especially for transitions without change of principal quantum number. The velocity form should be used for higher energies. In intermediate cases the geometric mean is the best compromise. The experimental energy should be used in all cases except when there is doubt over the spectral identifications. Transitions free of cancellation (nodeless transitions) should be especially reliable. Anomalously weak lines are probably cases of severe cancellation and so are unreliable. Configuration interaction can have very large effects and therefore must be allowed for. TESTOF CRITERIA ON TWO-ELECTRON SYSTEMS Because of the practicability of introducing coordinate systems which include the electron separation r1 explicitly, the solution of the Schrodinger equation for two-electron ions has been developed to the stage where greater accuracy is achieved by calculation than by experiment [for a review see Stewart (1963)l. Transition probabilities calculated from these wave functions are of very high accuracy and consequently are of importance to us here in testing some of the criteria of the previous section and later in providing standards for comparison with less sophisticated calculations. The most accurate calculations, using 220-term wave functions for the ls2('S) and ls2s('*'S)-ldp and ls3p('! ' P o ) transitions in helium with length, velocity, and acceleration matrix elements, are those of Schiff and Pekeris (1964). Their final results (which use their calculated energies) appear in Table I ; for their detailed results discussed here, we refer to the original paper. The wave functions of the S states were optimized under a constraint to ensure the correct asymptotic behavior, while the P states were determined by the energy criterion alone. Additional results were presented for the IsZ('S)-ls2p and 1s3p('P0) transitions with both functions optimized under the asymptotic criterion. This variety of results allows interesting comparisons of the convergence and accuracy of the different methods of calculation. Taking first the calculations in which only the S-state function is asymptotically correct, the velocity form gives excellent convergence with expansion length in all cases, the greatest variation being 0.0008 in the f value (1.3%). The length form does almost as well and improves with decreasing excitation energy. The acceleration form gives adequate results for large excitation energy, but fails badly in other cases [e.g., for 2s('S)-2p('Po) there is a variation with expansion length of 50 73.When both wave functions are asymptotic, the convergence of the velocity form worsens while that of the length form remains about the same (it must be borne in mind that the asymptotic constraint leads to a poorer overall representation of the eigenfunction for a given expansion length). All this evidence is in line with the qualitative criteria discussed above.

ATOMIC TRANSITION PROBABILITIES

257

More recently, Weiss (1967b) has presented calculations of oscillator strengths for the helium isoelectronic sequence using trial expansions of Hylleraas type (Stewart, 1963) with about 52 terms. He gives results for the and ls3d(" 3D)-ls2p,3p,4p(1*3P0)in heliumtransitions ls2('S), ls2s,3s('. 3S), like ions He-Ne" with both length and velocity forms. The transitions in He which were also calculated by Schiff and Pekeris (1964) are given in Table I. All the results agree to better than 2.5%, which suggests that the remainder of Weiss' calculations for which correlation will be less important should also be of this accuracy. It is noteworthy that if we apply the simple criterion of using the length form for transitions without change of principal quantum number and the velocity form in other cases to Weiss' results, we get the best agreement with Pekeris in four of the six transitions; of the remaining two cases, Pekeris' result for the l ~ ~ ( ~ S ) - l s 3 p transition (~P~) has not converged very well (this may be a case of cancellation). Recent work on correlated wave functions of He has concentrated on the introduction of powers of ( r I 2+ r22)1/2and of logarithmic terms [log(r, + r2) and log(r12 + r2')] in the wave function; see Frankowski and Pekeris (1966) (ground state) and Ermolaev and Sochilin (1968) (ls2s'S) for history and references. No transition probability calculations have been done using these wave functions. The use of Hylleraas-type wave functions for larger atoms is very difficult. The most recent calculation for the ground state of Li is that of Larsson (1968) who gives references to earlier work. The ground state of Be has been investigated by Szasz and Byrne (1967) with correlation only between equivalent electrons. It would be of some interest to calculate transition probabilities using these functions for the lower states and cruder wave functions for the upper states. V. Variational Wave Functions One of the earliest attempts at solving the Schrodinger equation for a complex atom was made by Hartree (1927), and it is interesting that the methods he employed predated wave mechanics. He conceived of each electron moving in the field of the other electrons and the nucleus and tried to set up a differential equation for the motion of each electron (single-particle model). Each electronic orbital was to be determined at large distances by assuming the electron to move in some averaged field (Coulomb method) and at small distances by estimating orbitals for each electron, computing the effect on a given electron of the other electrons thus obtaining a revised orbital for the given electron, and iterating this process to self-consistency ; this is the self-consistent field method (SCF). The two parts of the solution

258

R. J . S. Crossley

were then to be fitted at some intermediate distance. Slater (1930) and Fock (1930) showed that the coupled differential equations of the self-consistent field method could be obtained on the assumption of the single-particle model from the variational principle

that the trial energy E T (the expectation value of the true Hamiltonian H with respect to a normalizable trial function I,!J~) must exceed the lowest eigenvalue E of H . This result still holds when the further physical requirement of antisymmetry is imposed on the wave functions of the SCF and variational approaches (Fock, 1930), which thus become Slater determinants. These simple ideas underlie the developments of the next few sections. A. THEHARTREE-FOCK METHOD

The Hartree-Fock method is well known as the most successful method for determining wave functions of complex atoms, but because a number of different approximations are used, we will state here what we mean by “the Hartree-Fock method ” (HF): a single configuration wave function of correct symmetry (i.e., an eigenfunction of L2, S2, L,, and S,) consisting of one or more Slater determinants of single-particle spin orbitals, each orbital being a product of a radial function and a spherical harmonic, equivalent electrons having the same radial function. The spherical harmonic angular dependence of the orbitals indicates that this is a central field model. Physically, this means that the influence on a given electron of the other electrons is averaged over all of their possible positions (in fact except for ‘ S states the spherical harmonic requirement is more restrictive than this). This specification is sufficient for lowest states of a given symmetry; other excited states will be discussed below (Section V,A,4). It will be observed that this is a very restricted version of the general approach of the previous section; the restrictions are, of course, introduced to facilitate solution of the Hartree-Fock equations. Variants of the Hartree-Fock method either impose further restrictions which further simplify solution, or lift some of the restrictions; reviews have been given by Hartree (1957), Lowdin (1959), Nesbet (1965), Freeman and Watson (1965), Jucys ( 1967),2 and Sharma (1968a). Unfortunately, different authors are inclined to use the same terminology for different techniques which can lead to confusion.

* The name of A. P. Jucys, through double transliteration, appears in the literature variously as Iutsis, Jutsis, Yutsis.

ATOMIC TRANSITION PROBABILITIES

259

The usual procedure (Hartree, 1957) is to employ the variational principle to derive the coupled integrodifferential Hartree-Fock equations. These are then solved by computer, programs being available both for the numerical method (Hartree, 1957; Froese, 1963, 1965c; Dow and Knox, 1966; Mayers and O’Brien, 1968) and for the analytic method (Roothaan and Bagus, 1963; Malli, 1966; Synek et al. 1966; and Pfennig et al., 1965). The latter method, first suggested by Coulson (1938), employs a set of basis functions with variable parameters for each radial function. It is often referred to as the HartreeFock-Roothaan method. The numerical method of solution has the advantage that it avoids the arbitrariness of the choice of basis functions in the analytical method which might affect the convergence of an expectation value. A very good example of this is provided by the hyperfine structure calculations of Larsson (1968), which, despite a very accurate Li wave function of the Hylleraas type, show very poor convergence. However, transition probabilities do not appear to be very sensitive to details of the wave functions employed (Pfennig et al., 1965) except that with the analytic method, care must be taken with the asymptotic form of the functions. Kelly (1964a) suggests an error as large as 10 % could arise here. Approximate methods, such as the “ configuration average” method of Slater (1960) and Herman and Skillmann (1963) employed by Kelly (1964b), are unnecessary and should be avoided since they ignore the details of the configuration structure in which we are interested. Furthermore, Berrondo and Goscinski (1969) have recently shown that wave functions calculated by methods of the Hartree-Fock-Slater (HFS) type (for reviews see Cowan et al., 1966 and Siegbahn et al., 1967) do not satisfy the virial theorem [HF wave functions satisfy the virial theorem since they are derived from the variational principle of Eq. (36)]. Nevertheless, many calculations of transition probabilities have been made by these methods (Zare, 1967; Cowan, 1966, 1967, 1968; Fawcett et al., 1968). Calculations using the Hartree-Fock method described here have been made by very many authors. We refer to the bibliography of Glennon and Wiese (1966, 1968) for references. 1. Conjiguration Interaction in the Hartree-Fock Method It is possible to allow for configuration interaction in the Hartree-Fock method, but to do this consistently requires an iteration process for the mixing coefficients (Hartree et al., 1939; Hartree, 1957) which makes the calculations very lengthy. However, programs have recently been developed to do this both numerically (Froese and Underhill, 1966; Froese, 1967a) and analytically (Hinze and Roothaan, 1967). Both methods yield energy levels satisfying Slater interval ratios in very good agreement with experiment

260

R . J . S. Crossley

(Hinze and Roothaan, 1967; Bagus and Moser, 1968). The latter authors show, using the numerical program of Froese, that the (3P-'D) : ('D-'S) ratio in C agrees with experiment to 1 % , compared with the single-configuration result of 21%. However the result depends on a particular choice of configurations; further light is cast on this by the pseudonatural orbital calculations of Weiss (1967~)(see Section V,A). Calculations of transition probabilities by this method (often called superposition of configurationsSOC) have been made by Froese and Underhill (1966), Froese (1967b), and Froese Fischer (1968a,b). Some problems which arise over orthogonality of the orbitals will be mentioned below. A simpler, but less exact, method of introducing configuration interaction is to treat each configuration separately and determine the mixing by diagonalizing the energy matrix-this may be called the configuration interaction Hartree-Fock method (CIHF). Transition probabilities have been calculated in this way by, among others, Biermann and Trefftz (1949), Trefftz (1949, 1950, 1951), Steele and Trefftz (1966), and Froese (1964, 1965a,b). These calculations, and also energy calculations, cast light on the configurations which should be included in the calculations. They reveal the importance of configurations in the same complex. Recently, Silverman and Brigman (1967) have argued that the complex has a significance in variational calculations as fundamental as it does in the Z-expansion method [Layzer (1959) and Section VII] ; calculations thus support this argument. The next most important configurations appear to be those in which one valence electron has its principal quantum number increased by one, but the effect of these configurations is small except for neutral and near-neutral systems. Tables 111-VI illustrate the effect of configuration interaction on oscillator strengths. It will be seen that changes of a factor of 2 are common. Thus if any accuracy at all is required it is essential that appropriate configuration interaction is allowed for in the calculations.

2. Other Extensions of the Hartree-Fock Method Current interest in energy calculations with the Hartree-Fock method lies in the relaxation of the restriction that each equivalent electron has the same radial function (variously called the unrestricted, extended, or openshell method). The subject has been reviewed by Freeman and Watson (1965). Some more recent references are Bagus and Liu (l966), Lefebvre and Smeyers (l967), Jucys (1967), and Jucys et al. (1968). The onlycalculations of transition probabilities by these methods appear to be those for He discussed in Section V,A,5. Herman and Skillmann (1963) discuss spin orbit and relativistic corrections in the Hartree-Fock scheme. It is also possible to carry out calculations

ATOMIC TRANSITION PROBABILITIES

26 1

directly in a relativistic scheme (Hartree, 1957; Mayers and O’Brien, 1968). Other generalizations of the Hartree-Fock method are described by Hartree (1946). None of these methods seems to have been used for transition probability calculations. 3. Restrictions of the Hartree-Fock Method

An expression for the off-diagonal matrix element of an operator between single Slater determinant wave functions A and A’ has been given by Lowdin (1959, without assuming any spin orbitals orthogonal. For the one-electron dipole-length operator of Eq. (17), this expression is

where summation is over the spin orbitals k of Slater determinant A and similarly k’ of A ’ ; D:;, is the cofactor of the element ( k l k ’ ) of the determinant of orthogonality integrals between the spin orbitals of A and A’. In the Hartree-Fock method orthogonality is assured except between spin orbitals which differ only in their principal quantum numbers, not only in a calculation on a single state but also between two different states. In the former case, complete orthonormality is obtained by applying constraints to the variational procedure (Hartree, 1957; Sharma, 1968a), but it is not possible to obtain complete orthonormality between spin orbitals of different states. This leads to contributions of different kinds to the transition moment (37). It is simplest to consider an example. Consider the He transition 1 ~ 2 s ( ~ S ) - l s ’ 2 p ( ~(the P ~ )prime serves merely to distinguish the two 1s spin orbitals). Equation (37) will reduce to ct(

1s I ls‘)(2sl r ) 2 p )

+ p(2s I Is’)(

Is1 r 12p)

(38)

where a, p are constants including the angular integrals, and the remaining integrals in Eq. (38) are now just radial integrals. In general, we expect (1s I 1s’) x 1;

(2s I 1s’)

%

0.

(39)

Similar considerations apply to the operators of Eqs. (18) and (19).3 The electron involved in the transition 2s-2p is called the active electron; if the relations (39) are assumed to be equalities, then Eq. (38) reduces to 4 2 4 r 12p) (active electron approximation). This has been used by Veselov For similar reasons Hartree-Fock calculations may yield nonzero probabilities for two-electron jumps; e.g., a transition 2s2p-3s3d has a matrix element proportional to (2sl3s)(2pl r 13d).

R . J . S. Crossley

(1949), Bagus (1964), and Synek et al. (1967). In simple cases the error is small, but in more complicated configurations (last reference cited), it can exceed 10 % . Synek et al. (1967) also consider assuming orthogonality except between valence electrons. Other approximations arise in the calculations of the Hartree-Fock functions themselves; for instance, the valence orbitals might be determined in the field of a fixed core [frozen core; Kelly, (1964a)l. Errors of 510% can arise from this procedure. Bagus (1964) considers a method (frozen orbital) in which all the orbitals are drawn from a single calculation on a system with one more electron. The active electron approximation is of course fully consistent with this method. The errors can be large. Attempts have been made to improve the accuracy of some of these methods by including a core polarization potential ar - 4 in the equation of the active electron (Hartree, 1957,p. 162;Bersuker, 1958;Mayers and O'Brien, 1966), the constant CI being chosen to yield the best fit to the observed energy levels. Weiss (1967a)has suggested that a polarization potential would improve a full Hartree-Fock calculation by reintroducing distortions of the core eliminated by the averaging process of the Hartree-Fock method (see Section V,A); this is in accord with the 2-expansion analysis of Cohen and Dalgarno (1966) (see Section VII). Some of the approximate methods discussed in this section effectively assume that orbitals may be transferred from one state of a system to another. Roothaan and Bagus (1963) point out the danger of this; the ls22s2p configuration of Be gives rise to two terms, ' P o and ' P o . The peak of the 2p orbital of the singlet state occurs at twice the radius of that of the triplet state. Even greater changes can occur with 3d orbitals (Webster, 1968); however, the problem seems less serious in configurations with a number of equivalent electrons. The purpose of the approximations described in this section (Hartree, 1957) was, of course, to simplify calculations done by hand. With a computer they seem pointless, besides being, as has been shown, inaccurate.

4. Orthogonality Problems in Excited States

A number oE problems arise in setting up Hartree-Fock equations for states which are not the lowest states of their symmetry. Sharma and Coulson (1962)pointed out that if the 1s and 2s orbitals of He ls2s('S) are constrained to be orthogonal, an inconsistency arises [see also Coulson ( 1 965)]; they recommended use of nonorthogonal Hartree-Fock equations (Fock, 1930j. Again, it would appear to be consistent with the variational procedure to insist that the ls2s('S) state be orthogonal to the lsz('S) state. Such a con-

ATOMIC TRANSITION PROBABILITIES

263

dition would obviously severely complicate calculation, since the 1s orbitals of the two states will not be the same. These problems have recently had considerable attention (Cohen and Kelly, 1965; Froese, 1966, 1967c,d; Sharma, 1968b,c), but no very clear prescription seems to have emerged. What is worse, different methods lead to considerable differences in the oscillator strength. Some comparisons are given in Table VI for the astrophysically interesting ion Si +. Roothaan and Kelly (1963) have argued that using the variational principle one should orthogonalize to exact lower state wave functions. Since these are not known they recommend imposing no orthogonality restrictions and hoping for approximate orthogonality to the known approximate wave functions. This is usually satisfactory for simple configurations. Perkins (1966), however, has suggested that in certain cases the Hylleraas-UndheimMacdonald theorem (see Section V,B) will apply to Hartree-Fock wave functions with consequent automatic orthogonality; calculations do not appear to bear this out. Finally, we may observe that the Hartree-Fock method (Section V) was originally founded on the uufbau principle and was independent of the variational principle. Thus, it is fully consistent with the Hartree-Fock method to ignore orthogonality to lower states (Hartree, 1957, p. 37). We will no longer ensure upper bounds to the energies of excited states, but this is not our prime consideration. What we need is good excited state wave functions. 5. Comparison of Results for Helium

The existence of the calculations of Schiff and Pekeris (1964) and Weiss (1967b) for helium described in Section IV allows a direct assessment of the accuracy of approximate calculations for this atom. Extensive Hartree-Fock calculations have been carried out by Vizbaraite et al. (1956) who simplified the method for the more highly excited states, and by Trefftz et al. (1957) who employed both correlated and uncorrelated wave functions in both the closedand open-shell approximations with both length and velocity formulas. Calculations in both open- and closed-shell approximations, but using only the length formula, have been made by Froese (1967c,d); she also presents results for the isoelectronic ions Li+ and BeZ+. Cohen and Kelly (1967) use the frozen-core approximation to calculate transition probabilities between excited states of He, Li’, Be”, B3+, a nd C4+. Selected results are given in Tables I and 11. Comparison with the effectively exact results (Table I) is interesting; taking the ordinary Hartree-Fock calculations first, for the six transitions where comparison is possible the error ranges up to 13+% with the length form and up to 28% with the velocity form. Introduction of a

264

R . J . S. Crossley

correlating factor (Trefftz et a]., 1957) brings about a considerable improvement, the largest errors now being 84% (length) and 8 % (velocity). Correlated open-shell calculations (only two comparisons available) reduce the largest errors to 1 % (length) and 2 % (velocity). However for these two transitions the correlated closed-shell calculations lead to greatest errors of only 5 % (length) and 3%%(velocity), and the open-shell method without correlation is not very successful. This suggests that the bulk of the error in the HartreeFock calculation can be avoided by inclusion of a simple correlation term in the closed-shell calculation. As might be expected, this correlation correction is particularly important for transitions from the ground state, which, as the results show, are not amenable to calculation by the ordinary Hartree-Fock method alone. Trefftz et al. (1957) recommend the use of the dipole-velocity formula with calculated energies; as discussed above, there is no sound reason for using calculated energies. It is interesting to note that, apart from the resonance transition, which is so strongly affected by correlation in the Isz shell, very good results may be obtained from the ordinary Hartree-Fock calculations for helium by application of the qualitative arguments advanced above. If we exclude the ls2('S)-ls3p('P0) and 1~2s(~S)-ls3p(~PO) transitions where the matrix elements are very small and we accordingly expect cancellation difficulties, and use the length formula for transitions where there is no change of principal quantum number and so small excitation energy and the velocity formula in other cases, then the errors in the three transitions remaining are ls2s('S)-ls2p('P0)

(length)

3.5 %

1s2s('S)-ls3p('P0)

(velocity)

1.9 %

1 ~ 2 s ( ~ S ) - l s 2 p ( ~ P ~(length) )

3.3 % .

These results are very satisfactory. For the isoelectronic ions, the general agreement (Table II), even for resonance transitions, is very much better. In conclusion, it appears that transition probabilities can be calculated by the Hartree-Fock method to an accuracy of better than 10 % in reasonably simple configurations, provided configuration interaction is taken into account and suitable transitions are selected and treated according to the criteria suggested above.

B. OTHERVARIATIONAL WAVEFUNCTIONS Simple analytical trial functions for the variational method are described by Slater (1960, Vol. I, p. 348). Those of Morse et al. (1935) are the most complex [corrections and extensions of the original work have since been

ATOMIC TRANSITION PROBABILITIES

265

published: Tubis (1956), Morse and Yilmaz (1956); and see Slater (1960 loc. cit.)]; a simple scheme is suggested by Slater (1960, Vol. I, p. 368). Energies obtained with single configuration functions of this kind are inferior to Hartree-Fock energies, and there is no reason to suppose that transition probabilities calculated with such functions will be other than correspondingly inaccurate. Nevertheless, many calculations of this type have been carried out by Russian workers; for example, Vainshtein and Yavorskii (1952) used Slater-type functions, and Bolotin and Jucys (1953) used Morse-type functions with configuration interaction. A brief review of work up to 1956 is given by Vainshtein (1961); a more recent paper discussing the convergence of the multi-configurational approximation is that of Jucys et al. (1962). Vetchinkin (1963a) discusses the choice of approximate wave functions with particular reference to transition probability calculations. These calculations, while showing the importance of allowing for configuration interaction, especially within the complex as predicted by Silverman and Brigman (1967), lead to unsatisfactory results. The obvious development is to include much larger numbers of configurations in the interaction, that is, to use a variational trial function consisting of a linear combination of Slater determinants each built up from, for example, Slater orbitals. Alternatively, one might take a few determinants and allow the orbitals much more complicated forms, e.g., expansions in Laguerre polynomials (Hylleraas, 1937; Stewart, 1963). In both cases the physical concept of the configuration is more or less abandoned (cf. the discussion of Green et al., 1965). Calculations of this sort for the ground state and a few excited states of two-electron ions have been reviewed by Stewart (1963), and Weiss (1963) has calculated the 2s-2p transition of the Li sequence. More recently, extensive calculations for the ground and excited states of helium have been carried out using Slater orbitals by Green et al., (1965), and the results have been used in the calculation of helium oscillator strengths (Green et al., 1966a). The results of this work, some of which are included in Table I, are most encouraging. First, the method gives better energies than Pekeris for the ls3s('S) and 1 ~ 6 s ( ~states S ) upwards (showing, as is to be expected, that electron correlation is more important in the triplet states than in the singlet states) even though only eight configurations are used for the 1 ~ 6 s ( ~wave S ) function. In 100 cases out of 203, the length and velocity forms lead to results agreeing within 1 %. For the lsz('S)-lsnp('Po) transitions, Green et al. (1966b) give additional results using somewhat different wave functions. Green et al. prefer the 1966a results (which are in fact more recent) since they give better agreement between length and velocity forms, but we observe that the 1966b work, in fact, gives better agreement for n = 7, 8. In this way, agreement between length and velocity for the whole series (n = 2-8) can be obtained to 13 % . Agreement with Pekeris (Table I) is to 5 % (six transitions, length and velocity).

266

R. J. S. Crossley

The usual criterion of taking the length form for no change of principal quantum number reduces this to 33 %; omitting the single transition Is2s(’S)ls3p(’Po) which appears to be a cancellation case (or using the length form for this transition; as illustrated by Fig. 2, the length matrix element is less affected by cancellation than the velocity) gives agreement to 12 % . These results are extremely satisfactory, and all the more so because this essentially rather simple single-particle approximation configuration interaction method may be extended to more complex systems without the difficulties that arise in methods described above. There are no rI2coordinates (which make integration difficult) and no orthogonality problems since in a straight variational method the Hylleraas-Undheim-Macdonald theorem (Hylleraas and Undheim, 1930; Macdonald, 1933) ensures the orthogonality of excited states provided the variational functions are drawn from some complete set (Green et al., 1965). Weiss (1 967c,d) has carried out calculations of transition probabilities in C, C’, Mg, Al’, and Siz+ by a variant of this method which has very interesting features. He writes the wave function Y in the form

where @ is the Hartree-Fock-Roothaan wave function. Correlation is then allowed for by the Slater determinants miwhich are the (properly symmetrized) @ with one or two outer orbitals replaced by orbitals representing excited electrons (virtual orbitals) constructed from the Slater orbitals of the Roothaan set, possibly extended. The number of added configurations is minimized by making use of the pseudonatural orbital technique of Edmiston and Krauss (1966) applied to a single pair of orbitals chosen for maximum effect. Up to fifty configurations were employed. Even so energy convergence is rather slow. Convergence is also slow for the transition probability calculations, and the agreement between length and velocity forms is often rather poor. Both convergence and agreement are better for the two valence electron Mg series results than for the four and three valence electron C and C+ results; both improve with increasing ionization. Comparison with experiment is possible in only a very few cases. Weiss suggests a 25 % accuracy, but using the criteria advanced in this article an accuracy of l0-15% seems possible, although the evidence is very slight. Further reliable experimental data would be very valuable here, and it would be interesting to see the results of still more extensive calculations allowing, for instance, greater flexibility of the inner electrons. Finally we mention some recent variational calculations on small atoms, mostly for ground states: He: Auffray and Percus (1962); He and Li (ls22s and ls22p): Brown and Fontana (1966); Be: Kotchoubey and Thomas (1966),

ATOMIC TRANSITION PROBABILITIES

267

Bessis et al. (1967), Bunge (1968); B: Brigman and Silverman (1966), Schaefer and Harris (1968). The Be and B results show, once again, the importance of configuration interaction within the complex. Schwartz (1963) has considered the problem of estimating the convergence of a variational calculation. Goodisman (1966) (see also Vetchinkin, 1963b) has tried minimizing the width (H’>-(H)’ in place of the usual variational method. This should make the local energy +-‘HI+! as constant as possible, leading to wave functions more suited to the calculation of properties; however, Goodisman’s results are disappointing. 1. Screening Methods

Variational trial functions usually include a factor exp{ -n-’Z‘r}, where Z’ is a scaling parameter. We may interpret Z’ physically as the effective nuclear charge; Z ’ = ( Z - s) where s is the screening parameter, measuring the shielding effect of the other electrons on the nuclear field, In calculations on complex atoms s is usually, and unphysically, taken to be the same for each electron. Layzer (1959, 1967) has presented a more sophisticated screening theory in which s depends on the quantum numbers n, 1 of each electron. However, this method has not yet been used for the calculation of transition probabilities. Simple screened hydrogenic functions have been used by Varsavsky (1958, 1961) to calculate transition probabilities for a wide range of ions of astrophysical interest. His results, however, are in poor agreement with other methods (Bagus, 1964), and it seems probable that his work contains numerical errors. An improved treatment incorporating some of the features of Layzer’s screening theory has been given by Naqvi (1964). Comparison with experiment suggests that his results are not satisfactory either.

VI. Semiempirical Methods A. METHODS FOR

SIMPLE

TRANSITIONS

In Section V, we outlined the approach to atomic structure calculations of Hartree (1927), and pursued in Sections V,A,1-5, the development of the self-consistent field method. In this section, we take up Hartree’s proposal of finding an asymptotic solution of the equation for the active electron, assuming it to move in the field of the nucleus and some averaged electron distribution. The method is thus appropriate to systems with one active electron separated from and moving in the field of a core, e.g., the alkalis, or

R . J. S. Crossley

268

singly excited states of more complex atoms. These one-electron problems are discussed in Section V1,A. Extensions of the method to handle more complex cases involving electrons equivalent to the active electron are discussed in Section V1,B. As pointed out by Chandrasekhar (1944), the dipole-length matrix element will be appropriate to calculation of transition probabilities between such states. This approach will prove useful for those cases for which it can be shown that the region of the active electron wave function strongly affected by the core makes a negligible contribution to the transition integral. 1. The Bates-Damgaard Method

The simplest approximation we can make to describe the field of the core is that of placing the whole core charge at the nucleus, giving a Coulomb potential ( Z - N + 1)r-' for an N-electron atom with nuclear charge 2. This is a central field model, and handling the angular parts of the wave function in the usual way leads to the equation 1d2R Z-N+1 2 dr2 -Ir

--

(

I(1+1) 2r2

for the radial function r - ' R of the active electron with azimuthal quantum number 1. Bound-state solutions of Eq. (40) are given by the Whittaker functions in terms of the energy E ; the early mathematical development was by Waller (1927) and Hartree (1927). Trumpy ( 1930) proceeded semiempirically, giving - E the value of the experimental ionization potential. He employed numerical integration to obtain R, and so (see also his review: Trumpy, 1931)fvalues for transitions in the principal series of Li and Na. Hylleraas (1945) discussed solutions of Eq. (40) for S states (Le., I = 0), expressing R as a contour integral in terms of the effective principal quantum number n* defined by 1 Z-N+12

E = - - 2(

n*

),

the experimental ionization potential being again used for - E. General formulas for S - P transition probabilities were obtained on the assumption that for P states n* took integral values, that is, ignoring the quantum defect p = (n - n*) for these states. Bates and Damgaard (1949) used the same principles but a different mathematical method involving series expansions of the Whittaker functions to obtain their well-known expression for the radial transition integral in terms of two functions which they tabulate. Many authors have given f

ATOMIC TRANSITION PROBABILITIES

269

values calculated from these tables (Goldberg ef al., 1960; Allen, 1960, 1963; Houziaux and Sadoine, 1961); a particularly extensive tabulation has been given by Griem (1964). Results from different authors sometimes vary slightly due to the use of differing ionization potential data or to different cutoff procedures in the summing of the series contributions. Bates and Damgaard have given a very careful discussion of the accuracy of the method; comparison with experiment suggests that their claims are overcautious. An accuracy of 10 % seems possible for the simple transitions discussed here which fall within the range of the tables given by Bates and Damgaard. Less accuracy must be expected outside this range for reasons stated by Bates and Damgaard. There is, however, one difficulty which arises over normalization. The normalization integral for the Whittaker function corresponding to integral principal quantum numbers was evaluated by Waller (1927). For nonintegral effective principal quantum numbers Hartree (1927) was only able to derive a recurrence relation which was consistent with Waller’s formula in the integral case. Hartree and Bates and Damgaard therefore assumed that Waller’s formula also held in the nonintegral case and checked the accuracy of this assumption by numerical examples. The problem was investigated using the quantum defect method (see next section), by Ham (1955) and Seaton ( 195q4, who showed that this assumption was incorrect, an additional factor [(n*) being necessary, where

where the quantum defect p is regarded as a continuous function of effective principal quantum number v. Thus the Bates-Damgaard formulation is satisfactory provided p is a slowly varying function of v at v = n*. This is generally so except when n* is small. Seaton (1958) gives an example of this, and a simple means of calculating the correction approximately from experimental data [an exact calculation would require knowledge of all the eigenvalues of Eq. (40)]. Armstrong (Armstrong and Purdum, 1966) has considered the effect on normalization of allowing the core potential to be energy dependent, as would be necessary, for example, for transitions from states involving equivalent electrons. Recently, Bates and Fink (1 968) have suggested a new normalization factor for cases where n* < I + 1, obtained by suitably truncating the asymptotic expansion of the normalization integral. Calculations by Armstrong and Purdum (1966) suggest that the error from using the Bates-Damgaard normalization does not exceed about 2 % when n* > I + 1. An alternative Normalization for S states has been considered by Foldy (1958).

270

R. J. S. Crossley

procedure, which has been investigated by Layzer (Layzer, 1967; Layzer and Garstang, 1968) would be to choose the normalization factor so that some atomic property, most obviously ( r ) , is given correctly. The drawback of this approach is that the advantage of the Bates-Damgaard method is lost if it is necessary to use some more sophisticated wave function to calculate ( r ) . Bates-Damgaard results for resonance transitions in alkalis and alkali-like ions are compared with other calculations and with experiment in Table IV.

2. The Quantum Defect Method The philosophy of the quantum defect method is to look for quantities, such as the quantum defect p, which vary slowly from state to state of a quantum mechanical system, to formulate quantum mechanical problems in terms of these quantities, and so to develop a method for the prediction of the behavior of (usually) higher states from the behavior of lower states found by experiment or calculation. For the single active electron case, the development of the theory is due to Ham (1955) and Seaton (1958). The applications are usually to continuum states. Burgess and Seaton (1958, 1960), in a treatment of photoionization, have given a formula for bound-bound transition probabilities obtained by a method similar to that of Bates and Damgaard. Their results are in good agreement with Bates-Damgaard provided the effective principal quantum numbers of the states involved in the transition differ by at least 1.5. Compared with Bates-Damgaard, Burgess and Seaton claim similar accuracy and a wider validity. Their method seems only to have been used in the context of investigations of the validity of Coulomb methods in complex systems; these, and developments of the quantum defect method itself, will be discussed in Section V1,B. 3. Other Methods A number of other methods suitable for one-electron spectra may be mentioned. Ionescu-Pallas (1966) has presented an explicit formula for atomic lifetimes. His approach is that of Bates-Damgaard, but with the simplest asymptotic form for the wave functions. A rather naive treatment leads, surprisingly, to very good results. Stone (1 962) has calculated oscillator strengths for cesium by numerical integration of Eq. (40) with a potential chosen to reproduce the lowest energy levels (he obtains a good fit to forty levels) and a spin orbit term. Two other methods are based on Eq. (40), but with different forms for the core potential : the Wentzel-Brillouin-Kramers method and the ThomasFermi method. The former is fully described by Condon and Shortley (1935,

ATOMIC TRANSITION PROBABILITIES

27 1

pp. 341-344 and 147-148) [the only more recent work appears to be that of Minnhagen (1949)l; the latter, however, has been the subject of recent interest. Stewart and Rotenberg (1965) treated the active electron as moving in the field of the appropriate ion described by the Thomas-Fermi model (Condon and Shortley, 1935) scaled by a factor chosen so that the experimental ionization potential of the active electron is obtained. Lawrence (1967) has developed a similar method, except that he uses the ThomasFermi potential given by Latter (1955). These methods have wider validity than the Bates-Damgaard method. Petrashen' and Abarenkov (1954) used the Hartree-Fock method to obtain the core potential and obtained wave functions for the active electron by numerical integration, starting with the asymptotic form at large r derived from the experimental ionization potential. Thus their method has much in common with the original proposal of Hartree (1927). Although the method is rather laborious, it has been used by Anderson et al. (1956) and Anderson and Zilitis (1964). Because of the more accurate description of the core potential, these methods should be more accurate than Bates-Damgaard. In practice the improvement seems to be marginal. The extensive tabulation of the BatesDamgaard method ensures that it retains its usefulness. The other methods are, however, useful for those cases which fall outside the Bates-Damgaard range of validity. The accuracy of all these methods is good (10%) for transitions of the one-electron kind described here, except possibly in cases exhibiting strong cancellation. Some comparative results for alkalis and alkali-like ions are given in Table IV. B. TRANSITIONS INVOLVING EQUIVALENT ELECTRONS Attempts have been made to extend most of the methods described in the previous sections to transitions in which the active electron is excited from or to a configuration containing equivalent electrons, for example, the transition Be: 2s2('S)-2s2p(lPo). It can hardly be expected that methods designed to treat problems with one active electron well separated from a core should be successful in such cases. Nevertheless, these methods often give satisfactory results at least for the simpler cases such as the above example. It is worth asking why this is. Norman( 1965)and Gruzdev( 196h,1967a,b)haveinvestigated this problem in relation to the behavior of the quantum defect. Gruzdev argues that reliable results may be obtained when the quantum defect is a very slowly varying function of energy. This is an indication that configuration interaction is unimportant (Condon and Shortley, 1935, pp. 367-369). Numerous results for such cases have been given by Gruzdev and Prokof'ev (1966) using both the Bates-Damgaard and Burgess-Seaton methods.

272

R. J. S. Crossley

In cases in which configuration interaction is important, the quantum defect p will vary strongly with principal quantum number. Equation (42) then suggests that the normalization of the Coulomb wave functions will be strongly affected. This is consistent with discussions given by Bates and Damgaard (1949) and Armstrong and Purdum (1966). On the other hand, it appears that the use of the experimental energy often approximately corrects for the configuration interaction effect. Norman (1965) has proposed a way of allowing for configuration interaction from a study of the quantum defect curves; alternative procedures have been suggested by Garstang (1962~) and Garstang and Shamey (1967). Some results appear in Table VI. Applications of one-electron methods to complex systems without regard to these finer points have been made by many authors. Bates and Damgaard (1949) discuss the choice of ionization potential, but all authors seem to use the experimental value without regard to the change of energy of the equivalent electrons. Anderson et al. (1 967a,b) extend the Petrashen'-Abarenkov method (they use a scaled Hartree-Fock core potential in this work) to transitions in Mg and Ca, Ionescu-Pallas (1966) extends his method to two valence electron cases, and Stewart and Rotenberg (1965) apply their method to a variety of complex systems. 1. The Many-Channel Quantum Defect Method The quantum defect method discussed in Section VI,A,2 has been extended to many-channel cases by Seaton (1966). This theory may be used to calculate fvalues with allowance for configuration interaction. Moores (1966) presents results for the principal series of Ca [4s2('S)-4snp('Po)] using the wave function of Chisholm and Opik (1964) for the ground state, which was obtained from a treatment of the two valence electrons moving under an effective potential, allowing for configuration interaction with 4p2('S) and 3d2('S). The upper state is treated by the quantum defect method, allowing for interaction with 3d4p('P0) and 4p5d('P0). Satisfactory results are obtained only for the lowest levels (n = 4, 5, 6). It seems that further allowance for configuration interaction would be necessary to improve the results for higher n. It may well be that Ca is especially awkward because of the empty 3d subshell under the partly-filled N shell. From the standpoint of the complex, this would suggest a large number of interacting configurations. The same reason may underlie the difficulties found with calculations on Ca' (Douglas and Garstang, 1962). 2. Helliwell's Method

Helliwell(l964) has devised a method for systems with two valence electrons which is a mixture of the Bates-Damgaard and Hartree-Fock methods. The

ATOMIC TRANSITION PROBABILITIES

273

Hartree-Fock equations are solved for the outer electrons moving in the asymptotic field of the core. An inner boundary condition for the wave functions is obtained by assuming that the outermost node of the radial functions occurs at the same radius as in the singly ionized system (e.g.. the outer node of the 3s radial function of Mg 3s’ occurs at the same radius as the corresponding node of the 3s function of Mg’ 3s). Numerical evidence is given that this is a reasonable assumption. The position of the node is determined from a Bates-Damgaard calculation using the experimental ionization potential. Results of this “nodal boundary condition” method are given for Iv3S1,3P0transitions; they are not very good. This is presumably because the only allowance made for configuration interaction is through the use of the experimental energy. VII. Perturbation Treatments

A. THEZ-EXPANSION METHOD The Z-expansion approach to atomic structure rests upon a simple device introduced by Hylleraas (1 930) for helium-like systems (Bethe and Salpeter, 1957, pp. 151-3). If we rewrite the N-electron atom Hamiltonian of Eqs. (20) and (21) in terms of a unit of length of Z au and a unit of energy of 2’ au (thus different units for each N-electron ion), we obtain H = { -4

C vi2i

r,:’} i

+ z-’{i< C rill}. j

(43)

The Z-expansion method divides this Hamiltonian as shown by the brackets into a one-electron zero order Hamiltonian Ho and a two-electron perturbation Hl with the natural expansion parameter 2- Rayleigh-Schrodinger perturbation theory then leads to expansions for the energy and wave function of the form (in the original atomic units)

’.

E = EoZz+ E I Z + Ez + E3Z-’

II/ = ljoz3”+ ljlzl/’+ I j 2 2 - 1 / 2

+ e m -

+ .. * *

(44) (45)

Ho is a sum of hydrogenic Hamiltonians, so is a linear combination of Slater determinants of hydrogenic orbitals. El may be calculated analytically by standard degenerate perturbation theory, the degeneracy in Hobeing that of the complex. Eo depends only on the principal quantum numbers, so every configuration with the given set of principal quantum numbers and symmetry must be included. The extension of the theory to N-electron systems is due to Layzer (1959), Linderberg and Shull(1960), and Crossley and Coulson (1963). Very extensive

R. J. S. Crossley

274

calculations of El and the mixing coefficients of Ic/o have been made by Godfredsen (1966). Results even to this simple order are of great value in the analysis of spectra of highly ionized atoms (EdlCn, 1964). Calculations to higher order are much more difficult. For two-electron systems calculations have been made to high order using the Hylleraas variation-perturbation method (Scherr et al., 1966). E2 for the ground state of the Li sequence has been calculated by Layzer et al. (1964, 1968) using results for the He sequence. Similar calculations by Chisholm and Dalgarno (1966) introduce a sophisticated method which has also been applied to the Li sequence 1 ~ ~ 2 p (state ~ P (Chisholm ~) et al., 1968). They show that Ic/l, which satisfies the equation (Ho - Eo)Ic/, + (HI - ~,)Ic/, = 0,

(46)

may be expressed in terms of the of two-electron problems. Seung and Wilson (1967) have thus calculated E3 for the Li sequence ground state, using the two-electron first order wave functions of Scherr et al. (1966). The Hartree-Fock model may be treated in the same way, using the same zero order Hamiltonian (for a review see Sharma, 1968a), yielding an energy expansion EHF= E , Z 2

+ EYFZ + E y + E y Z - ' + . * .

(47)

in which in general only the zero order energy contribution has the exact value of Eq. (44).Then the correlation energy (Lowdin, 1959) is given by

E - EHF= ( E l - EYF)Z+ ( E 2 - EyF)+ *

*

.

(48)

The correctness of the Z dependence of this expression is borne out by the calculations of Clementi and Veillard (1966). However, the leading term may be eliminated by introducing a particular configuration interaction Hartree-Fock method; we simply adopt the zero order mixing required in the exact problem. Because of the simplified nature of the Hartree-Fock model (single-particle functions with prescribed angular dependence), the first order wave functions may be found by solving an equation similiar to (46) numerically (Froese, 1957), variationally (Dalgarno, 1960), by Laguerre expansion [Sharma and Wilson (1968) and references cited there], or in closed form (Cohen and Dalgarno, 1961; Cohen, 1963; Schwartz, 1959), thus allowing direct (but difficult) calculations to third order in energy. Froese (1958) has obtained numerical solutions of the equations for Ic/!F. Open-shell HartreeFock treatments have also been considered; for a review see Sharma (1968a). Expectation values of one-electron operators may also be evaluated by the 2-expansion method; the subject has been reviewed by Hirschfelder et al. (1964). Briefly, it is clear that expectation values may be evaluated to the order to which the wave function is known. Adopting the Hartree-Fock

ATOMIC TRANSITION PROBABILITIES

275

model with zero order mixing, properties may be obtained correct to first order. Difficulties encountered in using the I,$?“ may be circumvented by means of Dalgarno’s “interchange theorem ” (Dalgarno and Stewart, 1957; Hirschfelder et al., 1964). This method requires the solution of a differential equation often much simpler than Eq. (46). For transition probabilities we require off-diagonal matrix elements. The theory just described has been extended to this case by Cohen and Dalgarno (1964) and Borowitz and Vassell (1964). In principle any of the matrix elements (17, (18), (19) might be expanded, but Stewart (1968) has shown that the length form is the most appropriate. This yields the expansion

Using the exact energy expansion of Eq. (45), it follows from Eqs. (3), (13), and (14) that the oscillator strength has the expansion

f = b,

+ b , Z - ’ + bzZ-’ +

* a *

(50)

as first pointed out by Weiss (1963). Where the excitation energy is known, however, it is better to calculate the f value from the 2 expansion of the transition matrix element (49) using this information in Eq. (14). The quantity a, is the matrix element between hydrogenic functions [formulass and tables aregiven by Betheand Salpeter (1957, pp. 262-264)] and al may be determined by means of the interchange theorem. It is also possible to carry out expansions in a screening approximation in powers of (2 - s), where s is a screening constant. This makes better physical sense, and s may be chosen so that the first order contribution vanishes (Dalgarno and Stewart, 1960). This leads to a final expression for the matrix element

ICi riJI,$B)I

= I a d z - s)-’I

(51)

where s = a, a, which includes higher order contributions. For energy this approximation corresponds to a simple variational treatment (Froman and Hall, 1961). For other properties although the choice of s has some variational significance, a formal justification is lacking, despite investigations in terms of hypervirial theorems (Robinson, 1965; Sanders and Hirschfelder, 1965; Hirschfelder and Sanders, 1965; Bangudu and Robinson, 1965), the first order density matrix (Hall et al., 1965) and the Hellmann-Feynman theorem (Jones, 1965), but the evidence generally justifies this choice. Calculations of the transition matrix element have been performed in this way for the He sequence [Cohenand Dalgarno (1964): 1s2s(1~3s)-ls2p(1~3P0); One side of Eq. (63.5) of Bethe and Salpeter (1957) should be multiplied by -1, both in their phase scheme and in the phase scheme of Condon and Shortley (1935).

276

R . J. S. Crossley

Dalgarno and Parkinson (1968): 1~3s('*~S)-"3p(''~P~); Ali and Crossley (1969) : 1s3p(' *3P0)-1s3d(lV3D)] ; for all 1s22sa2pb-ls22sa-'2pb+' transitions (Cohen and Dalgarno, 1964); for the 3~(~S)-3p('P~) and 3p('Po)-3d('D) transitions of the Li sequence (Ali and Crossley, 1968) and of the Na sequence (Crossley and Dalgarno, 1965); for all M-shell transitions of the Mg sequence (Crossley and Dalgarno, 1965); and for certain transitions of the A1 sequence (Crossley and Parkinson, 1969). Froese (1965a) has performed calculations to second order, using numerical solutions of the first and second order HartreeFock equations, for all 1 ~ ~ 2 ~ ~ 2 p ~ - 1 ~ transitions, ~ 2 ~ " - ~and 2 pfor ~ ~M-shell ' transitions in the Na and Mg sequences. She introduces a Z-dependent screening parameters = so s,Z-', so that her matrix element takes the form

+

It is also possible, but considerably more difficult, to use the exact 2 expansion rather than the Hartree-Fock scheme. Calculations have been made for the He sequence ls2s(' 93s)-ls2p('93P0)(Cohen and Dalgarno, (Dalgarno and Parkinson, 1968), and for 1966), for ls3~('~~S)-ls3p('~~P~) 1s3p(' ,3P0)-ls3d(' (Ali and Crossley, 1969). For transitions involving change of principal quantum number, the differential equation that arises in the interchange theorem becomes difficult to solve in closed form. However, series expansion solutions have been used by Sando and Epstein (1965), who calculated the He sequence 1s' and 1s2s('S)-ls2p('Po) transitions as a (Z- s) expansion with the screening s arbitrary, and extensive calculations for ls2-lsnp('Po) have been made by Dalgarno and Parkinson (1967), n being treated as a parameter. The great value of the 2-expansion method is that a single calculation gives results for a whole isoelectronic sequence. It is to be expected from the formulation that the accuracy will increase with increasing ionization. The interesting problem is the accuracy at the neutral end of the sequence. Many of the authors mentioned above compare their results with those of other methods; some comparisons are given here for the He sequence (Table 11) and for FeI4+ (Mg sequence, Table V). Taking first transitions without change of principal quantum number in the Hartree-Fock approximation, Cohen and Dalgarno (1964) and Froese (1965a) showed that agreement with ordinary Hartree-Fock calculations is excellent for L-shell transitions with two or more degrees of ionization, both without and with configuration interaction. The error reaches only 5 % for neutral and singly ionized atoms. Results for M-shell transitions are not quite so successful, but good with about four degrees of ionization (Crossley and Dalgarno, 1965; Froese, 1965a). Mg sequence transitions have been the subject of calculations by Zare (1 967), who used Hartree-Fock-Slater functions with configuration 13D)

277

ATOMIC TRANSITION PROBABILITIES

interaction, and Weiss (1967d), who used his pseudonatural orbital method (Section V,B) ; two of Weiss' comparisons with 2-expansion results are reproduced here in Fig. 3. Two points may be noted here. First, it is sometimes the case that the radial transition integral (R, I r I R B ) of the active electron passes through zero for some value of Z along the isoelectronic sequence. The screening approximation of Eq. ( 5 1) cannot represent this and should therefore be abandoned in such cases (which we observe are once again cases of

t

0

0.02

0.04

0.06

0.08

I/Z

FIG.3. Comparison of results of the pseudonatural orbital method (PSNO) with the 2-expansion method for transitions in the magnesium isoelectronic sequence. Upper pair of curves: 3s3p('P0 j 3 s 3 d P D ) ; lower pair: 3s2('S)-3s3p('P0): -X , PSNO (results for Fel4+ from Froese, 1964); - - - 0,Z expansion (Crossley and Dalgarno, 1965). [From Weiss, 1967d.l

cancellation). Another problem may occur in cases of strong configuration interaction (Crossley, 1969); states are labeled by the single configuration with the laGest mixing coefficient, so zero order mixing may indicate a different state from mixing calculated by other means. Calculations in the exact 2-expansion scheme naturally give better results, but except where correlation is important the improvement is marginal and scarcely justifies the increased complexity of the calculations (Dalgarno and Parkinson, 1968; Ali and Crossley, 1969). Less satisfactory results are obtained for transitions involving a change of

218

R. J. S. Crossley

principal quantum number (Dalgarno and Parkinson, 1967); the screening approximation is probably less accurate here. Layzer and Garstang (1968) suggest, on the basis of calculations by Froese, that the matrix element expansion [Eq. (49)] may not converge in some cases; however, only asymptotic convergence is required. The success of the method and the regularity of the results suggest that the very simple procedure might be employed of fitting Eq. (49) or (50) to calculated or observed results (Wiese and Weiss, 1968). This has been done by Cohen (1967a) for f values [Eq. (50)] of the He and Li sequences. For no change of principal quantum number b, = 0 and 6 , takes the hydrogenic value; bz may be estimated from the calculations of Weiss (1963, 1967b) for Ne" and Nee+. With change of principal quantum number, bo takes the hydrogenic value and b, may be estimated from the same sources. In this way, a single very accurate calculation for an ion may provide accurate data for a whole isoelectronic sequence, except for the first few members where the accuracy may fall off. Some of Cohen's results are included in Table 11. B. OTHERPERTURBATION METHODS Recently, many forms of perturbation theory have been applied to atomic calculations with the object of calculating correlation energy. Claverie et al. (1967) have briefly surveyed and related the possible methods (RayleighSchrodinger, Brillouin-Wigner, Brueckner-Goldstone, and Bethe-Goldstone) and reviews of the various approaches by Sinanoglu, Lowdin, Brueckner, Kelly, and Nesbet appear in the volume edited by Lefebvre and Moser (1969). Very little work indeed has been done on transition probabilities. Possibly the most natural approach is to apply Rayleigh-Schrodinger perturbation theory with Hartree-Fock as the zero order problem. This has been done by Weiss and Martin (1963), by Sinanoglu, as mentioned above, and by Byron and Joachain (1966, 1967), who use the Hylleraas variation perturbation method. The only transition probability calculation is that of LaPaglia and Sinanoglu (1966) for the Be sequence resonance transition; their results appear in Table 111. This calculation, however, amounts to no more than an allowance for configuration interaction in the ground state between 2s' and 2p2('S). Kelly (1964) has applied Brueckner-Goldstone theory and obtained oscillator strengths for Be: 2sz('S)-2snp('P0) transitions (n = 2-8). His method, which is also based on a Hartree-Fock zero order problem, is extremely complex. Altick and Glassgold (1964) have used the random phase approximation (Pines, 1955; Valatin, 1961) to correct a zero order Hartree problem and obtain oscillator strengths for 'S - ' P o transitions in Be, Mg, Ca, and Sr. Their results are not very successful.

ATOMIC TRANSITION PROBABILITIES

279

VIII. Sum Rules, Bounds, and Variational Principles A. SUM RULES Oscillator strengths satisfy sum rules of the form

where summation is over all possible states A of the system (except B), including the continuum. Setting k = 0 gives the well-known ThomasReiche-Kuhn sum rule S(0) = N

(54)

where N is the number of electrons in the system. Other values of k give quantities which may be determined by experiment; a complete list has been given by Stewart (1967), and a general derivation by Jackiw (1967). A review of applications in two-electron systems has been given by Stewart (1963); Dalgarno and Davison (1966) have described applications to the calculation of long-range forces. Knowledge off values can thus be used to determine properties; two recent references are Bell and Dalgarno (1965) and Bell and Kingston (1967). The reverse procedure has sometimes been used to estimate f values. For example, the Thomas-Reiche-Kuhn sum rule has been used to put experimental relative f values on an absolute scale [e.g., Allen (1960)l. A common situation is that f values are known for the lowest lines of a series; then the higher lines may be estimated by fitting sum rules to experimental or calculated data. Bell and Kingston (1967) worked in this way for He, assuming thatf values along the Is2-lsnp(’Po) series obeyed the formula

f.= A n - 3 + Bn-4 + Cn-’.

(55)

The n - 3 term describes the hydrogenic behavior and has often been used in the study of series regularities o f f values (Penkin and Shabanova, 1965; Kobzev et af., 1966; Wiese and Weiss, 1968). The continuum contributions may be obtained from the photoionization literature with the help of the reviews of Samson (1966) and Stewart (1967) and the bibliography of Kieffer (1968). Satisfaction of a sum rule is of course no guarantee of the accuracy of individual f values; Green et af. (1951) provide an example. Sometimes, however, it is possible to obtain bounds on f values (Dalgarno and Kingston, 1959) or on other properties (Cohen, 1967b).

280

R . J . S. Crossley

B. BOUNDS AND VARIATIONAL PRINCIPLES The most unsatisfactory feature of the methods for calculating transition probabilities described so far in this article is that in no case are we able to do better than guess the accuracy of the results. It is well known that in energy calculations it is possible to calculate rigorous upper and lower bounds to the energy (even though the lower bounds may not be very accurate). It would be a very great advance if we could give similar bounds to calculated transition probabilities. Such bounds may be derived from variational principles, and it is important to distinguish between rigorous (strong) and effective (weak) bounds. Weak bounds may be obtained in a situation in which the variational principle ensures zero first order contribution and a second order contribution of definite sign, the assumption being made that higher-order contributions are negligible. The simplest procedure is essentially a generalization of the use of the interchange theorem and the screening approximation in Z-expansion theory (Section VI1,A). An approximate wave function is taken to satisfy an effective Hamiltonian; the first order correction of Rayleigh-Schrodinger perturbation theory is then evaluated by means of the interchange theorem. Expectation values have been improved in this way by Delves (1963a), Chen and Dalgarno (1969, Somorjai (l966), and Tuan et al. (1966). Delves (l963a) goes one stage further by constructing variational principles for the expectation value t o j r s t order; that is, he makes the second order contribution zero. He also gives a formulation for off-diagonal matrix elements (Delves, 1963b), but it is complicated to use. He reports no calculations. Buymistrov (1963) has generalized Delves’ treatment to apply to forbidden transitions, and by using bounds developed by Braun and Rebane (1967), Dmitriev and Yuriev (1967) have formulated bounds for forbidden transition intensities. Delves’ method has also been discussed by Aranoff and Percus (1968) who give weak bounds for expectation values, but not for off-diagonal matrix elements. The Rebane bounds have recently been discussed by Epstein ( 1 968). The only actual calculations of bounds for transition probabilities appear to be those of Jennings and Wilson (1967). Their method is an extension of the procedure of Bazley and Fox (1966) and yields weak bounds which are however rather impressive (see Table I). Weinhold (1968a,b) has obtained strong lower bounds to expectation values and, by an extension of his method (Weinhold, 1968c), upper and lower bounds to sum rules. His method permits bounds to be determined given values of other sum rules. Knowledge of individual oscillator strengths improves the bounds [see also Gordon (1968)l.

ATOMIC TRANSITION PROBABILITIES

28 1

IX. Summary With few exceptions, the calculation of atomic transition probabilities to an accuracy of 10% is a very difficult task. The exceptions are one-electron systems, for which exact calculations are possible (Section III), and twoelectron systems, for which the use of coordinates involving the interelectron separation explicitly is practicable (Section IV and Tables I and 11). For more complex systems variational wave functions have the widest applicability (Section V), but it is essential to make proper allowance for configuration interaction both in the Hartree-Fock method (Section V,A,1) and in other variational schemes (Section V,B and Tables I11 and V). Difficulties may arise in establishing a consistent procedure with the HartreeFock method in cases involving excited states not the lowest of their symmetry, and in cases of strong configuration interaction (Sections V,A,I and V,A,5, and Table VI). Transitions involving a single active electron moving in and separated from a core, in particular alkali-like ions, may be treated by a Coulomb method with generally good accuracy (Section VI) ; the various methods yield similar results (Table IV). Transitions which do not involve change of principal quantum number are amenable to treatment by the Z-expansion method (Section VIl); good accuracy is obtainable for systems of at least a few degrees of ionization (Tables I1 and V, and Fig. 3). For the general case it seems probable that at present the best results are to be achieved by the intelligent combination of results from different methods, both experimental and theoretical. Thus accurate relative values from, e.g., an emission experiment might be put on an absolute scale by the use of a single result for some suitably chosen transition obtained from either experiment or calculation. Reference has been made to the recent experimental literature (Section I,B), and criteria have been suggested for obtaining the greatest accuracy in calculations (Sections I11 and IV). Very little discussion has been presented for systems involving more than about 20 electrons; this is because very little data is available. It does appear, however, that considerable difficulty would be encountered in carrying out accurate calculations for systems involving numbers of equivalent electrons with more than one open shell. Further work is needed here.

ACKNOWLEDGMENTS This review is based on an earlier unpublished report prepared by the author for the G.C.A. Corp. in connection with contract NAS9-5589 with NASA, and on further work carried out at Harvard College Observatory with the support of NSF. Part of the revision

R . J. S. Crossley was carried out during a stay with the Quantum Chemistry Group of the University of Uppsala, Sweden, made possible by a generous study visit grant from the Royal Society. I take this opportunity to thank Professor Lowdin, Dr. Calais, and Dr. Goscinski for their very warm hospitality. I am indebted to many authors who have allowed me to use unpublished material: Professor D. R. Bates, Professor A. Dalgarno, Professor R. H. Garstang, Professor D. Layzer, Dr. A. L. Stewart, Dr. B. C. Webster, Dr. F. Weinhold, and Dr. A. W. Weiss. Professor S. Ehrenson and Dr. Weiss have kindly allowed me to reproduce figures from their own work. Dr. D. W. 0.Heddle and Dr. Weiss read the manuscript and offered much appreciated criticisms. Miss Elizabeth Taylor and Mrs. Sinikka Wyatt translated articles from German and Norwegian, respectively.

TABLE I gf VALUES FOR HELIUM*

I ~ Z ( ~ S ) - ~ ~ ( 1sZ(’S)-3p(’P0) ~PO) 1

Pekeris’ Error boundsb(%)

0.0736 0.0735 7.4

0.2759 0.2754

0.2761 0.2759

0.0734 0.0730 0.0729 0.0730 0.0706 0.0763

HF

0.2583 0.2835

HF Corr. HF UHF Corr. UHF

0.2583 0.2557’ 0.2883 0.3141

0.2604’ 0.2503 0.2436 0.2695

0.1162 0.259

0.3059

Cohen‘ Bates-Damgaardh

V

0.2762 0.2762 1.14

UHF

Froese‘

1

( 0 ) 0.2762 (0 0.2762 1.07

Weiss’ Greend

Trefftz’

V

HF

0.0734 0.0731 8.5

0.0764 0.0831

0.0652 0.0714

0.0334 0.0713

0.0857

2 ~ ( 1 ~ 2 ~ ( 1 2~(159-3~(1~0) ~0) z r ( 3 ~ ) - 2 ~ ( 3 ~ 02) 4 3 ~ ) - 3 ~ ( 3 ~ 0 ) 1

V

1

I

V

0.3764

0.3764

0.1514

0.1514

2.4

40

10.2

45

1.6173

1

V

0.1935

0.1935

0.3764 0.3774 0.1478 0.1506 0.3773 0.3950 0.1513 0.1540

1.6173 1.6203 0.1923 1.6194 1.6461 0.1932

0.1902 0.2004

0.3901

1.6599

0.1704

0.1596

1.6173

V

0.3903

0.3395

0.1597 0.2100

0.1543 0.1615

1.6731

1.8330 0.1707 0.2394

0.1509 0.1710

0.3932 0.390

0.3461

0.1616 0.160

0.1555

1.6648 1.614

1.9218

0.1496

* Experimental energies are used in all results except Pekeris. I, length; v, velocity.

0.1693 0.189

Schiff and Pekeris (1964): Method (D): no asymptotic restriction on P-state wave functions. Method (C):P-state wave functions constrained to give correct asymptotic form. * Jemings and Wilson (1967): Bounds apply to Pekeris’ results by method (D). Green et al. (1966a): Configuration interaction. ‘Weiss (1967b): Hylleraas expansion method. ‘Froese (1966, 1967d): HF: Hartree-Fock (numerical) allowing for nonorthogonality between Ins> and In’s). UHF: open-shell treatment of ground state lsls‘ (IS). Trefftz et al. (1957): HF: Hartree-Fock (numerical). Corr.: correlation is introduced by a factor (1 cull) in the wave function which no longer satisfies the virial theorem. UHF: ground state treated as l~ls’(~S). Results marked 1 corrected by Trefftz (private communication). Cohen and Kelly (1967): Hartree-Fock (numerical) with “frozen core” 1s orbital. Ground state treated as lsls’(’S) with Is frozen. Hence the unsatisfactory results. Bates and Damgaard (1949).

’

+

TABLE I1 gf VALUESFOR SELECTED MULTIPLETS OF HELIUM-LIKE IONS*

1s2( 'S)-l s2p(1PO)

Weiss"

I v

Froese (1 966)b Froese ( I 967)' Dalgarno and Parkinsond Dalgarno and Parkinson' Cohen* Bates and Damgaard'

I I

I I I

I

0.276 0.276 0.283 0.258 0.373 0.390

0.457 0.457 0.464 0.451

0.259

0.431

0.510

0.526

0.552 0.552 0.559 0.552 0.584 0.598 0.559

0.609 0.609

0.647 0.647

0.631 0.643 0.614

0.663 0.674 0.650

is2s(3~)-1s2~(3~0) Weiss"

I 0

Froese (1967)' Cohen and Kellyh

I I u

Cohen and Dalgarno (1966)' Cohen and Dalgarno (1964)' Cohenf Bates and Damgaard'

I I I I

1.617 1.620 1.660 1.665 1.992 1.475 1.552

0.924 0.926 0.939 0.943 1.188 0.893 0.918

1.614

0.918

0.639 0.641 0.647 0.650 0.849 0.628 0.640 0.627

0.488 0.488

0.394 0.395

0.494 0.660 0.483 0.490 0.483

0.398 0.539 0.391 0.396 0.393

is2P(3~0)-1s 3 d ( 3 ~ ) Weiss"

I 0

Chongk

I v U

Cohen and Kellyh

f I

v

Cohen' Bates and Damgaardg

* Experimental energies are

I I

5.476 5.509 6.016 6.773 24.071 19.024 5.773 5.497 5.580

5.619 5.637 6.186 6.794 13.514 10.773 5.759 5.611

5.751 5.764 6.449 6.915 11.048 9.159 5.835 5.742 5.724

5.844 5.854 6.641 7.013 9.998 8.553 5.900 5.833 5.832

5.909 5.918

5.956 5.903 5.904

used in all cases except Chongk. I, length; v , velocity;

a, acceleration; f, fourth matrix element of Chen

(1964)[Eq. 261.

Weiss (1967b): Hylleraas-type wave functions. Froese (1966): Hartree-Fock (numerical) with open-shell treatment of ground state. Froese (1 967): Hartree-Fock (numerical). Dalgarno and Parkinson (1967):Exact Z expansion. Dalgarno and Parkinson (1967): Z expansion of Hartree-Fock. Cohen (1967a):Z expansion fitted to results of Weiss" for Nee+ Bates and Damgaard (1949). * Cohen and Kelly (1967): Hartree-Fock frozen-core approximation. ' Cohen and Dalgarno (1966):Exact Z expansion. J Cohen and Dalgarno ( 1964): Z expansion of Hartree-Fock. Chong (1968): Eckart-type wave functions. f

284

TABLE 111 gf VALUES FOR THE RESONANCE LINElS22s2('StlS22S2p('p0)OF BERYLLIUM-LIKE IONS*

Ion : Configurations in ground state: Bolotin and Jucys" Weissb

S2

1 I V

Froese' Froesed Trefftz'

I

I I V

Sinanoglul

Be

B+ s2,P2

2.09 1.81 0.95 1.97 2.00 1.79' 0.96'

1.23 1.24 1.15

I V

HansenO Cohen and Dalgarnoh Bates and Damgaard'

I I

1.31' 1.70

+

s2

s=,p2

s2

1.55 1.49 0.71 1.54

0.96

1.18 1.11 0.50

s2, p2

0.54

1.31

0.91 1.37

1.59

S2

0 4

s2,P2

0.77 1.04 1.08

S2

+

s2, P2

0.74

0.93 0.89 0.39

0.59

0.77 0.74 0.32 0.79

0.48

0.71 0.67 0.73 0.75 0.74 0.72

0.89 0.43

0.57 0.52 0.59 0.60 0.59

0.74

0.49 0.43 0.49 0.50 0.49 0.50

1.16

1.11

1.16 1.20 1.18' I .33' 1.25' 1.15

N'+

C2

0.62 0.85

0.59

0.88

0.36 0.52 0.72 0.74

* The results show the very marked effect of interaction between the ls22r2and ls22p2 configurations in the ground state. Experimental energies are used in all cases. I, length; v , velocity. Bolotin and Jucys (1953): Variational treatment with functions of Morse et al. (1935). 'Weiss: Unpublished Hartree-Fock calculations (see Wiese et al. 1966). Froese (1967d): Numerical Hartree-Fock. Froese (1966): Numerical Hartree-Fock with open-shell treatment of the ground state. Trefftz: The single configuration results are from Pfennig er al. (1965) and the two-configuration results from Steele and Trefftz (1966); analytical HartreeFock method. LaPaglia and Sinanoglu (1966): Perturbation theory applied to the single configuration results of Pfennig ef al. (1965) to allow for the twoconfiguration interaction. Hansen (1967): Geometric mean of length and velocity results of Pfennig er al. (1965) (single configuration) and of LaPaglia and Sinanoglu (1966) (two-configuration). * Cohen and Dalgarno (1964): Z expansion method. Bates and Damgaard (1949): Results of Griem (1964). The results of Pfennig et af. (1965) for Be were corrected by Steele and Trefftz (1966). The results of LaPaglia and SinanogIu (1966) and of Hansen (1967) which were thus invalidated have been recalculated by the present author.

,.

' '

'

286

R . J . S. Crossley TABLE IV gf VALUES OF THE RESONANCE LINEns(zSl,z)-np(2P!/2)OF THE ALKALIS AND ALKALI-LIKE IONS**'f

Experiment

Hartree-Fock

Atomn Hk

Mr

Other

11:47d]

Li

2

0.96

Na

3 1.53

1.30

K Rb Cs Fr

4 1.37 5 1.60 6 1.31 7

1.31 1.32

131'

n

Be+ Mg+ Ca+ Sr+ Ba+ Ra+

2 3 4 1.66 5 1.52 6 1.32 7

Hk

1

[(pol.)

v

1.4Ie 1.18'

Coulomb approx. u

1.W 1.03b

1.02" 1.05"

1.64'

BD

AZ

SR

1.00

0.99

0.97

1.62'

1.25

1.30

1.30

1.43"

1.31 1.35 1.50

1.39 1.45

1.40

1.51

1.56

Stone

1.63

No data available

Experiment Ion

1

CI

Hanle

Hartree-Fock

I 0.68b

v

CI

I(pol.) u (pol.)

0.73b

1.56'

1.24'

2.13k

1.25k

u

0.67b 0.69' 1.23"

1.32 1.42 1.48

I

Coulomb approx.

1.44'

1.36'

BD 0.67 1.19 1.33 1.47 1.62

SR Zwaan

1.25 1.46

No data available

* I, length; u, velocity; (pol.), with polarization term.

t Hk:

Hook method (Penkin, 1964). Mr : Lifetime by magnetorotation (Stephenson, 1951). Hanle: Lifetime by Hanle effect (Gallagher, 1967). CI: Configuration Interaction.

BD: Bates and Damgaard (1949). AZ: Anderson and Zilitis (1964). SR: Stewart and Rotenberg(l965). Stone: Stone (1962). Zwann: WKB method (Zwann, 1929).

Weiss (1963): Similar results have been obtained by Ivanova and Ivanova (1964). Weiss (1963). Brehm et al. (1961) and Hulpke er al. (1964): Lifetime by phase-shift method with optical excitation. Karstensen (1965): Lifetime by phase-shift method with slow electron excitation. Chapman et al. (I 966). Bersuker (1958). Weiss (1967a). * Biermann and Liibeck (1948). ' Weiss (1967a): Similar results have been obtained by Douglas and Garstang (1962). 'Trefftz and Biermann (1952): Similar results have been obtained by Douglas and Garstang (1962). Garstang and Hill (1966).

287

ATOMIC TRANSITION PROBABILITIES

TABLE V ABSOLUTE MULTIPLET STRENGTHS FOR ('D)-('Po) TRANSITIONS IN FeI4+ Single configuration

Configuration interaction

('D)-('Po)

HF"

Z b

HF"

Zb

3d2-3p3d 3~3d-3~3d 3~3d-3~3~ 3p2-3p3d 3p2-3s 3p

0.537 0.470 0.768 0.026 1.564

0.536 0.468 0.731 0.025 1.532

0.659 0.689 1.438 0.003 0.575

0.61 1 0.628 1.449 0.006 0.599

" HF: Hartree-Fock calculations of Froese (1964) using the dipolelength formula. Z: Z-expansion calculations of Crossley and Dalgarno (1965). Further results and comparisons are given in this reference. TABLE VI

VALUES OF SELECTED MULTIPLETS IN Si+ **t ~~

Exptl.

CI

PSNO

SOC

3sz3p(zPo)-3s23d(ZD) 3.43" 7.19 3 ~ ~ 3 p ( ~ P ~ ) - 3 ~ 3 p<0.025" ~ ( ~ D ) 0.039 3~~3d(~D)-3~~4p(~P~) 0311 0.44 3~3p~(~D)-3s~4p(~P~)

6.22 0.103 0.438 0.406

0.361

4 17b 3~~3d(~D)-3~~4f(~FO)

5.99

9.31

9.54

3s3p2(2D)-3s24f(ZFO)

0.92

1.43

10: 1 5cb]

Pol.

GSd

GS'

10.33 11.42 0.029 0.017 0.208 0.171

9.90

GSf BD

7.88 0.166 0.123

1.82

2.34

3.06

3.85

3.16

4.88

4.94

6.28

9.68

9.12

* The multiplets selected illustrate the difficulty caused by the strong configuration interaction between the 3s23dand 3 ~ 3 p ~ ( terms. ~D) t Exptl: " Savage and Lawrence (1966): Lifetime by phase shift with electron excitation. Hey (1959): Emission. Woodgate (1966): Emission. PSNO : Pseudonatural orbital method (Weiss, unpublished; quoted from Froese Fischer, 1968a). SOC : Hartree-Fock superposition of configurations method (Froese Fischer, 1968a). CI : Hartree-Fock configuration interaction method (Froese and Underhill, 1966). Pol. : Hartree-Fock single configuration with polarization (Biermann and Liibeck, 1948). GS: Garstang and Shamey (1967); these authors use three methods: Hartree-Fock radial integrals; SIater parameter fitting of energies. Hartree-Fock radial integrals; center-of-gravity fitting of energies. Bates-Darngaard radial integrals; center-of-gravity fitting of energies. BD : Bates and Damgaard (1949).

288

R . J. S. Crossley

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ATOMIC TRANSITION PROBABILITIES

29 1

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TABLES OF ONE- AND TWOPARTICLE COEFFICIENTS OF FRACTIONAL PA RENTAGE FOR CONFIGURATIONS sAsIy4 C. D. H . CHISHOLM Department of Chemistry, The University, Shefield, Yorkshire

A . DALGARNO Harvard College Observaiory and Smithsonian Astrophysical Observatory Cambridge, Massachusetts

and F. R. INNES Air Force Cambridge Research Laboratories, Bedford, Massachusetts

I. Introduction

.....................................................

11. The Calculation of cfp.. ........................................... 111. Applications of Two-Particle cfp ...................................

IV. Description of Tables.. ............................................ References .......................................................

.297 .301 .308 .309 ,334

I. Introduction Single-particle coefficients of fractional parentage (cfp) of the kind worked out by Racah (1943) have effected a great simplification in calculations of matrix elements of one- and two-particle operators for both discrete and continuum states. The most complete tabulations of single-particle cfp are those of Nielson and Koster (1963) and Karaziya et al. (1967). Two-particle coefficients of fractional parentage are similarly useful particularly in studies of two-particle interactions [cf. Racah (1943), Elliott Cf. Trees (1951), Horie (1953), Innes and Ufford (1958), Rohrlich (1959), Slater (1960), Levinson and Nikitin (1965), Judd (1963), Mapleton (1963, 1968), Fano (1965), Wybourne (1965), Kaxaziya et al. (1967), Smith et al. (1967), Smith and Morgan (1968), Shore and MenzeI(1968), Rudge (1968), and Henry et al. (1969). 297

C . D . H . Chisholm, A . Dalgarno, and F. R . Innes

298

et al. (1953), Levinson (1957), French (1966), Armstrong, (1968)]. Most of the earlier work was restricted to cfp within one set of equivalent orbitals. Many-particle cfp spanning more than one equivalence set assume a special utility when atomic eigenfunctions are introduced in which correlation or configuration mixing effects are included. The general two-particle cfp are the subject of this review. The earliest application of fractional parentage concepts is contained in a paper by Bacher and Goudsmit (1934) and the term " fractional parentage" was introduced by Menzel and Goldberg (1936). Parentage expansions achieve antisymmetry by sums over intermediate terms rather than by permutations of coordinates or labels. Write the wave function, assumed antisymmetric, of an N-electron atomic system in a state specified by quantum numbers r as (rI N ) . A parentage expansion expresses (r IN) as a vector-coupled sum over intermediate states of products in some prescribed coupling arrangement, ((rlI N,)(T, I N2) * * (r, I NM)),of antisymmetric wave functions ( r iI N J , each of which contains N i( < N ) particles such that

-

M

1N i= N.

i= 1

If the symmetry division is twofold so that the products contain only pairs, (r, 1 M ) ( T , 1 N - M ) , the coefficients of the products in the expansion of (r I N ) are M-particle coeficients of fractional parentage (r,,T2 I }r). Most applications of cfp have assumed the wave functions (r,I N i ) to be constructed from single-particle orbitals (y I 1), characterized by the singleparticle central field quantum numbers y = (nlmlm,), and much of the early theoretical development was restricted to systems involving equivalent orbitals (nZ)q. Coefficients which work entirely within sets of equivalent orbitals have a number of special properties. We shall call such cfp substantive. We shall call those cfp which involve more than one set of equivalent orbitals adjective. A standard notation for substantive cfp was introduced by Racah (1943). In the forms (P1rl, lq2r21}rq1+~~r) or (/q1+q2r(11q2r2, Pr,) the director

I

I} or { 1 leads from lower to higher symmetry. The wave function

be called the postparent. Of the wave functions (lq2r2 lq2) and (lqlrll q l ) , one is the preparent and the other is the coparent. In the evaluation of the matrix element of an operator, it is convenient to identify as coparent that wave function upon which the operator acts. Ordinarily the coparent contains fewer orbitals than the preparent. For the present treatment of cfp, we shall locate the first one or two particles in the coparent, ordered as in the postparent.

(lq1+q2r qlq2)may

TABLES OF CFP

299

We use a comma to indicate the division of a wave function into two parts, each of which is separately antisymmetrized. Then the original single-particle substantive form of Racah (1943) is

The coparent is the last particle. But we may write alternatively

(w)= C (f, fn-T1l}Pr)(/, fn--'rlr) rl

in which the coparent is t h e j r s t particle, the two cfp differing merely by a recoupling phase and a transposition phase (Racah, 1943). We choose to locate the coparent first in the cfp and to place the preparent immediately behind the director. A more general form, suitable also for adjective cfp, was used by Jahn (1954), Levinson (1957), Innes and Ufford (1958), and Kamuntavichyus (1967). The addition of momenta can be carried out in a variety of sequences or orderings. Innes and Ufford (1958) represented the construction of a wave function from z momenta, proceeding by successive vector addition from left to right, in the lineal coupling format

The parameter T ispecifies the coupling scheme of the set 4' and also the principal quantum number ni . If the sets are placed in normal order (Condon and Shortley, 1935) so that in a lineal coupling format the group nil? stands to the right of the group n i l f i when nj > ni or if nj = ni when fj > f i , the coupling arrangement is conventional. A cfp is similarly conventional if the coupling arrangements in its parents are conventional and if the particles are placed in numerical order in the wave functions. All entries in the tables are conventional in this sense, though the recouplings within the cfp are often unconventional. Unconventional cfp differ from conventional cfp by a transposition phase and a recoupling. Adjective cfp can be partitioned. The first part is a product of substantive cfp, each of which detaches one or more orbitals from a set of equivalent orbitals. The second part is the unitary recoupling transformation coefficient that relocates and recouples the detached orbitals and it is expressible in terms of 3nj coefficients together with various weights and phases. The remaining part consists of a transposition phase, found by counting the intervals through which the orbitals have been moved and a combination of multinomial coefficients appropriate to the symmetry divisions which have occurred.

300

C.D . H. Chisholm, A . Dalgarno, and F. R. Innes

Consider the cfp which is obtained in the SL coupling scheme by detaching an s and a p orbital from the wave function (sa,P”JpqK

(spi,(sA- l ~ i l r ) ~ ‘ plq~- ” ( } ( ~ a ~ s ” ) ~ p q ~ ) or ( I a’bJ’cK”l}abJcK).

The coparent is spl. The two substantive cfp are @,a’ I }a) and (p,c‘ I }c). The unitary recoupling coefficient is

(spZ(a’b)(J’C)K”KI (sa’ )(ab)J(pc’)cK) and apart from weights and phases it is a product of a 6j and a 9j coefficient (cf. Table I). It contains five component momenta, six intermediate momenta, and a resultant momentum, making twelve parameters in all. The coefficient is accordingly a (factorizable) 12j coefficient. Substantive cfp may be studied through their properties under conjugation, correspondence, and reciprocity, through Redmond’s formula (Redmond, 1954), and generally through group-theoretical methods. The study of adjective cfp involves in addition certain aspects of the theory of angular momentum that relate to the 3nj coefficients. In addition to their unitary property, 3nj coefficients have associative or group properties which embrace various composition formulas (cf. Yutsis et al., 1962). The composition formulas can be cast into the form of relationships between adjective cfp, the adjective character consisting essentially of a recoupling of component momenta. The original composition formula, due to Racah (1943), refers to a twoparticle substantive cfp:

(in-2rl, i2r21}znr) =C([rl-~[r~])l’~ rl

(i2r2(1z, i)(in-2rl, z1}ln-T) x (Piri;r1r2) where, in the SL coupling scheme, [r]= ( 2 s + 1)(2L + l ) , S and L x

being, respectively, the total spin and orbital angular momentum quantum numbers contained in the specification r, where (T,lTE; r1T2)represents the product of two W coefficients (Sl&5*; S1S2)(L11LI; L1L2),and where the summation is over all allowed intermediate terms. We have modified the notation for Racah’s W coefficient W(abcd;ef)-a 6 j coefficient-by omitting the designating letter W, as we shall do throughout. Because simple relationships exist between many cfp, tables of all oneparticle or two-particle cfp would contain a substantial element of redundant

TABLES OF CFP

30 1

information. Call the set of cfp appropriate to a specific configuration and coupling arrangement ( S ’ S ’ ~ ) ~a ~bloc. We regard as redundant any cfp in one bloc that, apart from recoupling phases, a transposition phase or an overall normalization factor, has occurred in another bloc. Redundancies arise in the cfp appropriate to the configuration sAsfppqfrom simple interchange of s and s’, from replacing s’ by s or s by s’ and from the simple addition or subtraction of s2 ‘ S or s” ‘ S . For the tables, we have adopted the phase scheme of Racah for substantive cfp (Racah, 1943). The other phases are standard (Condon and Shortley, 1935; Edmonds, 1957; Messiah, 1962; Shore and Menzel, 1968).

II. The Calculation of cfp We shall restrict our description of the methods of calculation of cfp to in the those needed for three equivalence sets (and in particular to sAs‘”pP4) conventional coupling arrangements. The separated orbitals that constitute the coparent will be brought out to the left and they will be attached to the first one and two particles. Numerical order is used for all the particles. The recoupling coefficients that occur in the second part of a cfp may be written in terms of recoupling formats described by Yutsis et al. (1962). The first of these formats is the lineal and conventional one, A , . For three momenta it has the form ((. .) .), and for four the form .) .) .). The second recoupling format, A , , for four and five momenta is represented by, respectively, ((* .)(**)) and ((* .)((. .) *)),and the third, A , , for five momenta by (((- *)(..)) No other format is required for conventional cfp in three equivalence sets. If we denote the number of component momenta in a recoupling coefficient by an external subscript, the recoupling coefficient for the separation of one momentum from two equivalence sets has the structure ( A , I A,J3 and the recoupling coefficient for the separation of one momentum from each set the structure ( A , I A1)4. If three equivalence sets are affected in the recoupling, the transformation coefficients have the structure ( A , I Ao)4 or ( A , I A1)4 for the separation of one momentum and ( A , or ( A , l A 2 ) 5 for the is at separation of two momenta from different sets. The format ( A , I most a 6j coefficient, (A, I A1)4 at most a 9 j coefficient, and ( A , I A1)4 a product of two 6j coefficients (Yutsis et ul., 1962). The recoupling coefficients for the given formats and for given orderings of the component momenta are listed in Table I for all types of cfp ( a - A p q ) or (ub - Apq) in which one component, a, or two, ub, are detached from (sAsSIp)p4.In no case do we require to divide a single equivalence set into more than two parts. Table I also displays the structures of the recoupling coefficients in terms (((a

a).

302

C. D. H. Chisholm, A. Dalgarno, and F. R. Innes

of 3nj coefficients. They are appropriate generally to any three equivalence sets. None of the recoupling coefficients involves a nonfactorizable 12j coefficient ; nonfactorizable I2j coefficients may occur when unconventional coupling arrangements are employed. The momenta specified in the recoupling coefficients consist of the component momenta only. If any one among the intermediate or resultant momenta which also occur in the coefficients happen to vanish, one or both 3nj coefficients degenerate into 3(n - 1)j coefficients. General formulas for the required recoupling coefficients in terms of 6j and 9 j coefficients are given below ; the intermediate and final momenta are

TABLES OF CFF'

303

((12345)A1 I (13245)A2),: ((12>434>(b5)egI (13)c(24)4f51g) = (4b5)eg I (ab)(f5>g>((12>a(34)bfI (13)@4)df) =

[::;I

~ C ~ l ~ ~ l ~ ~ l C ~ l C ~3 l4~ fb l ~ ' ~ 2 ~ ~ ~ 9 ~ ~ f *

We specialize now to the SL coupling scheme and we introduce the notation that a single letter refers to a pair of spin and orbital angular momenta. We use capital letters for intermediate and resultant momenta and lower case for component momenta. Recoupling coefficients and 3nj coefficients, written with momentum pairs as entries, imply the product of two such coefficients, one containing the spin angular momenta and the other the orbital angular momenta (cf. Innes and Ufford, 1958). Let j, k ,and I represent momenta (of a single orbital or of several equivalent orbitals) belonging, respectively, to the three equivalence sets. For the cfp with j, I, jk, or j l as coparents, we require (j,a'bJ'cK'K I (ja')abJcK) = ([J'][K'][a][J1)'/2(jJ'Kc; JK')(ja'Jb;U J ' )

( I , abJc'K'K I abJ(1c')cK) = ( [ K ' ] [ C ] ) ' ' ~ ( ~ C Kc'K). 'J; If j or I contains 0, the corresponding part of the coefficient is unity. For the separation of two momenta, we require (jkZ,a'b'J"cK"K 1 (ju')a(kb')bJcK) = ([z][J"][a][b][K"][J)1~2(IJnKc; JK")

x

[:::I a'

b'

J" ;

(jlZ,a'bJ'c'K"K I (ju')abJ(lc')cK)

1';

I I

= ( [ ~ [ J ' ] [ ~ ] [ J ] [ K " ] C C ] ) ' / ~c'

K"]

x (ja'Jb;U J ' ) .

In these formulas the primes and double primes indicate removal of one and two momenta from the momenta found initially in the postparent.

C. D . H.Chisholm, A. Dalgarno, and F. R.Innes

304

If I

= ‘Sin the first coefficient, (jk0, a’b‘J”cK I (ju‘)a(kb‘)bJcK)= ([a][b]/[ j])’/2(Jb‘aj;a’b)

and if I

= ‘ S in the second (jl0, a‘bJ’c’K”K 1 (ja’)abJ(lc’)cK)

The parameters can be zero in the spin or orbital part but not in both, so that only one of two coefficients paired together is simplified. The third part of the cfp consists of an overall normalization and a transposition phase, each division of the wave functions introducing its own normalization and transposition phase. For a twofold symmetry division and two eqiiivalence sets, two very simple cfp are (C’111,

1;11221r2IlP11@I2 IP2)

and

x

Wl

I9

11) w 2 2

,Id

In the more general cases, the normalization will contain three multinomial coefficients, and the recoupling will be more elaborate. Table I1 lists the normalizations and transposition phases for one- and two-particle cfp involving three equivalence sets. The nontrivial adjective cfp were calculated using the following formulas: (s, sa - ‘ a ’s”bJ’p4cK’I}saus’”bJp4cK)

([U][J’][J][K’])’’~ x (sa’Jb; aJ’) x (sJ’Kc; J K ’ )

-

) A+P+4

112

( p , pq~1c’(}p4~)([K’][~])1~2 x ( p K ’ c J ; Kc’)

305

TABLES OF CFP

x ( [ K ” ] [ C ] ) ’ ~ ~ ( Z K Kc”) ’’~J; (ss’l, s”“’s’’-l

b ’J ”p 4cK”l}s’us’”bJp4cK)

x (ZJ“Kc;JK“) x

( s p l , sA- ‘a’s’’bJ’pp- ‘c‘K‘‘l}s’us‘”bJp4cK)

X

“

:I.

([J’][U][J][C][Z][K”])’’~(SU’J~; aJ’) x p J’ c’ Z K“ K

Composition formulas provide a useful check on the direct procedure of calculating adjective cfp from substantive cfp. The composition formula for substantive cfp

(z2rl,in-2r,j}l”r) = 1 ([r1][r1]p2 r2

x

(z2rl{1i, I)(/, in-2r21}~n-1r1)

in;

x ( I , ~-~r~(}i~r)(r~ rlrl),

demonstrates a process which is useful also for adjective cfp. Normalization and transposition phase changes must be effected. In the case of two inequivalent detached orbitals (ll‘)I = 2-{(l,

Z’)I - (-)t+t’-’(r,/ ) I } ,

but ( P ) I = (1, Z)Z. Accordingly, a factor of 2ll2 must be inserted on the right-hand side of the composition formulas for two particles. Then for cfp in which two particles

306

C. D.H. Chisholm, A. Dalgarno, and F. R. Innes

jk orj1 are detached the composition formulas are ( j k l , a’b’J”cK’’(}abJcK)= 2l”

C ([I][K’])’’2

J ’ , K’

x ( j k K K ” ;lK’)(j, a‘bJ’cK’l}abJcK)

x ( k , a’b’J”cK”l}a’bJ’cK’)

and ( j l l , a’bJ”’K”(}abJcK)= 2lI2

1 ([l][K’])’’2 K’

x (j1KK”;l K ’ ) ( j ,a’bJ’cK’1)abJcK) x (1, a’bJ’c’K’’l}a’bJ’cK’).

There remains to describe the calculation of the substantive cfp. It is useful to refine the specification of a term SL by the addition of the seniority number u (Racah, 1943) or, alternatively, a quasi-spin quantum number Q (Judd, 1967). The wave functions that can be constructed from the equivalence array pq may be divided into six strings labeled by uSL = UI or by QSL = QI. In the seniority scheme, there are two strings with seniority number 3 inpq, one containing the single wave function 34Sand the other the single wave There are two strings with seniority number 2, and they contain function 32D. pairs of wave functions labeled by 23Pand 21 D (found in p 2 and p“). In the single string with seniority number of unity, there are three wave functions lzP(found in p , p 3 , and p 5 ) , and in the single string with a seniority number of zero there are four wave functions ,‘S (found in p o , p2,p4, and p 6 ) . As orbitals are added to an equivalence set, the new terms that occur are those for which u = q. The calculation of single-particle (substantive) cfp within an array uses the properties of conjugation, correspondence, and reciprocity together with a procedure of building up from smaller to larger values of q. The buildup procedure is needed when new terms enter into the calculation. The buildup procedure is carried out by a recursive formula derived by Redmond (1954)

= ~ ( u Iu’I’) ,

- ~([I][I’])’’~ 1( l ~ z ’ lz*I’) ; V’I’

The particular application to cfp containing new terms has been described by Innes (1967). It is conventional to base the buildup procedure on the specification that ( l o , I I }I) and (I, 1I }I2) are positive (and equal to unity).

307

TABLES OF CFF'

Conjugation refers to the symmetry about the half-filled shell (Racah, 1943)

(I"

-n

- 1Z',11 } I" -"])a= ( - ) I + x [(n

-*([If] / + l)/(rn - n)I1/'(Inl, ZI}Zn+'1')9, in = 41 + 2,

I'

where the subscripts 9 and W distinguish between different phase schemes. In the formula and in Racah's seniority scheme, Y refers to a phase scheme for wave functions belonging to equivalence sets l 4 for which q < [fl and W to another phase scheme for wave functions belonging to l 4 for which q > [I] (Condon and Shortley, 1935; Racah, 1942, 1943). The Y and W schemes differ by a phase (-)"/' if u is even and ( -)("-')I2 if u is odd. The phase modification ensures that the various possible conjugate pairs of wave functions (r 11') and (r I l m - q can ) be combined into a unique wave function (0's I 1"). Correspondence similarly leaves term designations unaltered

Reciprocity connects cfp in which the term designations of the preparent and the postparent are interchanged

(I"-'u

+ IZ', 11}2"01) = (-)I-I'+*([Z']/[rp x

[Y

"1

- u)(n 1/2(1"uZ,lI}P+ ' u + 11'). - n - o)n

in

+

Correspondence and reciprocity take a simple form, when expressed in quasi-spin (Judd, 1967). Similar formulas may be written down for two-particle substantive cfp. However, the two-particle substantive cfp in the tables were actually calculated using the composition formula (Racah, 1943), and a limited check was made by the use of conjugation, correspondence, and reciprocity relationships. The calculations were performed with the coparent taken last, as done originally by Racah (1943). The coparent was then shifted into the first position by the use of the interchange formulas (1,') , I"-1Zy}Pz) = (-)I+I1-*(1n-'Z1,

(z& z)rz,P - ~ Z ~ ~ }=I(-)r2-r(zn-2z2, ~Z)

l(n)l}zv)

308

C. D . H . Chisholm, A . Dalgarno, and F. R . Innes

The phase in the first interchange formula consists of the recoupling phase ( -)' +I1 - I and the transposition phase ( - ) , I 1 , and the phase in the second interchange formula consists of the recoupling phase ( - ) r 2 + r 2 - 1 and the transposition phase ( -)4r2, but both I, and 41' are even.

111. Applications of Two-Particle cfp Two-particle cfp are useful in the evaluation of matrix elements of two-particle operators between wave functions constructed from singleparticle orbitals and in the evaluation of the matrix elements of one- and two-particle operators between wave functions which include the effects of two-particle correlations. If G is a two-particle scalar operator, its diagonal matrix element (PI1 G I P I ) may be written as a sum of two-particle matrix elements (/'I, I G I 1, I,) according to the formula (Racah, 1943) (l"1 (GIP I ) = +n(n - 1)

1

11. I 1

x (PI{I I" - 2 1 1, P I , ) ( I" - 2 1 1, 121, I} I"1) x (121, IGI 12Zz).

A more general relationship in which an adjective cfp appears is that for the off diagonal matrix element ( / " II G 1 l n - ' I ' l ' l )

x

(PI,, In-2Z1[}P1)(lI'1,,

l"-2zll}l"-~~'I'1)

x (121, (GI11'12).

If the wave functions describe an independent-particle model, the sums over the intermediate terms can be carried out and only substantive cfp are required, but if Il'I, represents a correlated pair, the sum over I, becomes a weighted set of two-particle matrix elements and adjective cfp are needed. Two-particle cfp are necessary when the wave functions are more accurate than independent-particle wave functions and two-particle cfp can play an important simplifying role in many-electron theories of correlation. Indeed the earliest many-electron theory of correlation (Bacher and Goudsmit, 1934) is also the earliest use of fractional parentage and the algebraic coefficients occurring in the Bacher-Goudsmit theory are related to one and two-particle cfp (Meshkov, 1953;Trees, 1954). A similar development by Sinanoglu (1961) also contains algebraic coefficients which are simply expressed in terms of cfp (Chisholm and Dalgarno, 1966). Many-electron theories show that the contribution to the second-order energy from double excitations is a

TABLES OF CFP

309

weighted sum of pair-energies. The use of a parentage expansion shows that the weights are the squares of two-particle cfp (Chisholm and Dalgarno, 1966). Two-particle adjective cfp have also been used in a simple case to calculate off-diagonal matrix elements of the single-particle electric dipole transition operator and diagonal matrix elements of the two-particle mass polarization operator (Dalgarno and Parkinson, 1968).

IV. Description of Tables Following the example of Nielson and Koster (1963), the cfp are written as quotients of surds and integers. Parentheses are used to indicate square roots, and integers are separated by a period. Thus

a(b)”2/cd(ef)”2= a(b)/c.d(e. f ).

A few trivial cfp are listed in Table IV. Otherwise, the cfp are listed in the order (s - A w l , (P - k ) (ss‘ , - k ) (SP , - b q ) , ( p 2 - Apq), appearing successively as Tables VI-XVIII. Both within a particular table and from one table to the next, the bloc designation (Apq) increases in odometer fashion. Furthermore, the postparent is increased first, the preparent second, and the coparent last, following Hund’s rules order. The number of cfp in thz bloc is given in square brackets next to the bloc designation. Adjacent to it stands the overall normalization and transposition phase appropriate to the bloc. The individual cfp are specified by the momenta I of the coparent, the intermediate momenta J* (if necessary) and the resultant momenta K* of the preparent, and the intermediate momenta J and resultant momenta K of the postparent. The momenta I, J * , K*, J, and K appear in the first five columns of Tables VI- XVIII. Frequently, momentum values that are uniquely specified by the other momenta or by the structures of the parents are omitted from the tables, and a blank space implies a repetition of the symbol above. Thus in Table XVI for ( p z ; Ooq), the row 3P 2P 4 s

+1

in the bloc (003) reads in full I 3P

K” K 2P 4s=

+1

and refers to the cfp ( P 2 3 p ,p 2 p I )P3 4s); and in Table XIV for (sp; 1 lq), the row

3P

. * *

2 s 3s 4 P

-1/(3)

310

C. D. H. Chisholm, A . Dalgarno, and F. R. Innes

in the bloc ( I 11) reads in full I 3P

J’ 2s

K”

2s

J 3s

K

4 P = -1/(3)

and refers to the cfp

((V)3p,(S’pO) ’s I >(ss’>(3sP> 4p). Table I11 is a guide to the location in Tables IV and VI-XVIII of a particular cfp. It also contains directions for the calculation of the redundant cfp that are not given explicitly in the tables. The redundancies are the interchange of s and s‘, the replacement of s by s‘, and the addition of s’ ‘s or s” ‘ S . Three examples provide a sufficient demonstration of the use of Table 111. (i) The cfp (sp, s”pq-’ (}ss”pq) is obtained by multiplying the cfp (sp, p 4 - l I}spq), found in Table XIII, by a factor {(q + l)q/ x (q -I- 3)(q 2,} l’’. (ii) The cfp (s’sSL, ss‘pq J}s2s”pq)is obtained by multiplying the cfp (ss’SL, ss‘pq I }s2s”pq),found in Table XII, by a phase factor ( -)’. (iii) The cfp (s’p, ss’J’pq-‘ 1 }ss”pq) is obtained by multiplying (sp, ss‘J‘p4-’ I >s’s’p~), found in Table XV, by a phase factor ( - ) J ’ .

+

Table V contains a list of the six cfp that vanish because of triangle rules upon component momenta and not upon the intermediate and final momenta. The cfp in Tables VI-XVIII are identified by intermediate and final momenta. Within a bloc, it is not necessary to specify component momenta when no more than one equivalence set is a p set and the others are s sets.

TABLES OF CFP

31 1

TABLE I OF SOME RECOUPLING COEFFICIENTS FOR ELEMENTS CONVENTIONAL cfp

Types

Recoupling coefficients'

Structure in 3nj

((s - Am):

In general Ifh=l (p" In general (ss' - Am):=

In general Ifh=l -h w ) : In general Ifh=l Ifq=n I f h = l and q=n

((SP"

a Yutsis et a[. (1962), Chap. V. A o , A , , and Az indicate the coupling formats (. .(((. .) . ) . ) .), ((. .)(...((. .) .). .)), and .) (* *))(. .((. .) -). .)), respectively. The couple (A, p) is unaffected here. If q = n, the coefficient is just a 3j (a triangular delta function). If both h and p = 1, the coefficient is again a 3j only.

.

.

.

..

.

(((a

312

C. D.H. Chisholm, A . Dalgarno, and F. R . Innes

TABLE I1 NORMALIZATIONS AND TRANSPOSITION PHASES Type

Value

TABLES OF CFP

313

TABLE 111

FINDING INDICATOR FOR CONVENTIONAL ONE-AND TWO-PARTICLE cfp IN s, s', ORBITALS. TYPESs, s', p, ss', s's, sp, s'p, s2, sS2,and p2

AND p

Types

A P L q

Posit ion

S

1 0 0 - 6 1 1 0-6 1 2 0-6

Table IV Table VI Add s2 to ( 1 , O ) . Factor: (q l)1/2/(q 3)'12 From (1, 1) 'S. Factor: l/z. Table VII Add s2 to (2,O). Factor: (q 2)'I2/(q 4)'"

2 0 0 - 6 2 1 0-6 2 2 0-6 s' (e.g., s-s'

on s-, as

noted)

P

noted)

0-6

0-6

1 1 2 2

0

0

0

0

1 2

1 1

0 1

1

2

1-6 1-6 1-6

2

1

1-6

2 2

1 2

1-6

0-6

- 6 - 6 - 6 1-6 1-6 1-6

1-6

1 1 0 - 6

SSI

s's (e.g., s-s'

1

2 1 2 0 1 0 2 0

0 0

on ss' as

+

+

Table XI1

1 1

2

2 1 0-6 2 2 0 - 6

+

Add s2 to (0, 0); or, from (1,l) 'S. Factors those of (0,2) Add s2 to (0,l). Factor that of (1,2) E.g., add s2 to (0,2). Factor: (q 2)'I2/(q 4)'12

2

0-6 0-6

+

+

2 2

1

+

Table VIII Table IX (sws') Add s2 to (0,O);or, from (1, 1) ' S . Factors: q1I2/(q 2)'12; dB Table IX Table X Add s2 to (1,O). Factor: (q l)'I2/(q 3)'/'

Table IV Table XI

0-6 0-6

+

Table IV From s - (2,O). Phase exp., 0 From s - (1, .)1 Phase exp., J From s - (2, 1). Phase exp., J' From s -(I, 2). Phase exp., 0 From s - (2,2). Phase exp., 0

1 2 0 - 6 1

+

s-s'

+

on (1, 2). Phase exp., Z

From ss'- (1, .)1 Phase exp., Z From ss' - (2,1). Phase exp., 0 From ss' - (1, 2). Phase exp., 0 From ss' - (2,2). Phase exp., Z

314

C. D . H. Chisholm, A . Dalgarno, and F. R . Innes

Table 111 (continued)

sp (cf. s)

s‘p (cf. s’; or, e.g., s c t s ’

on sp as noted). (These procedures are also available for cfp of type ST.)

PZ (cf. P)

1 1 1

0 1 2

1-6 1-6 1-6

2 2 2

0

2

1-6 1-6 1-6

0 0 1 1 2 2

1 2 1 2 1 2

1-6 1-6 1-6 1-6 1-6 1-6

1

Table XI11 Table XIV Add s2 to (1,O). Factor: [(q l)~I”~/[(q +3)(q 211”* From (1, 1) IS.Factor: d2 Table XV Add sz to (2, 0). Factor: [(q 2)(q 1)1’/2/[(q 4)(q 3 W 2

+

+

0 2 0 - 6 1 2 0 - 6 2 2 0 - 6

Table JV Table IV Table IV

1 2

2-6 2-6 2-6

Table XVI Table XVII (s-s‘) Add s2 to (0.0); or, from (1, 1) ‘S. Factors: [q(q - l)]’/*/[q 2)

+

x(9+

1 1 1

0 1 2

2-6 2-6 2-6

2

0

2-6

2 2

1 2

2-6 2-6

+

From sp - (1,O). Phase exp., 0 From sp -(2,0). Phase exp., 0 From sp - (1, 1). Phase exp., J From sp -(2, 1). Phase exp., J‘ From sp -(l, 2). Phase exp., 0 From sp - (2, 2). Phase exp., 0 Table TV Table IV Table IV

0

+

+

2 0 0-6 2 1 0-6 2 2 0 - 6

0 0 0

+

1)11/z:42

Table XVII Table XVIII Add s2 to (1, 0). Factor: [(q l)q11’2/[(q 3)(q 2)11/2 Add sz to (0,O);or, from (1, 1) IS. Factors those of (0, 2) Add s2 to (0, 1). Factor that of (1, 2) E.g., add s2 to (0, 2). Factor: [(q 2)(q 1)11/2/[(4+ 4)(q 311’/’

+

+

+

+

+

+

TABLES OF CFP

315

TABLE IV

SOME TRIVIAL cfp

TABLE V SIXVANISHING cfp OF TYPE(pa;AM) p21

ab

3P ID 1D 3P 1D 1D

s-

2s

S-

2s 2s

S-

ss'

ss' ss'

J* = J

3s 3s 3s

~"(p')

1D 3P 1D

1D 3P 1D

K* = K" 20 2P 20 30 3P 30

ab SS-

sss' ss' ss'

. I

2s 2s 2s 3s

3s 3s

4~') 1D 1D 3P 1D 1D 3P

K 28 20 2P 30 30 3P

C.D.H. Chisholm, A . Dalgarno, and F.R. Innes

316

TABLE VI (s; 1 l d 110 -

[21

2s 2s 111 -

[5]

2s

112

[lo]

2s 2s

2s

2s 2s 2s 2s 113 2s 2s 2s

2s

2s

3s

2s

IS

[I61

3s 3s

3s 3s 3s

2P +P

1s

2P 2s 2D

3s

2P 2s

3s 1s

IS

5P 3D 3P

3s 1D 1P 1s

5s

3P 5s 3s 5s 3s

3s 3s 3s 3s

IS

3P

IS 3s

2P

1P 3P 1P 3s [lo]

4P

2s

4P 2P 4P 2P 2s 2D 2P 2s

2D

2s 2s 2s 2s 115 -

[5]

1s

3s

2s

+1/(2.3)

2s 2s

2s 2s

3s 35 3s

5P 3D 3P

1s

3s

3s

1s

ID

3s IS

1P IS

+1/(7)

3P 3P

3s 3s

4P 2P

IP

6s 4D 4P 4s

2D

ID

114

+1/(5)

3D

3D

2s

+I12

35

ID

4P 2P

IS

4P 2D 4P

3D

2s

+1/(3) 3P 3P 1P 3P 1P

2s

2s

+1/(2)

3P 1P

116 2s 2s

[2]

IS

+1/2(2) 2s

3s

IS

3s IS

TABLES OF CFP

TABLE VII (s; 214)

210 -

2s

3s

2s

2s

1s 2s 211

-

[51

2s

3s

2s

+1/(2) 2s

3P

2$

3s IS 3s

2P

2s

1P

214

POI

2s

3s 3s

2s

212 -

[lo]

2s

3s

5P

3s

3P

+(2)/(5) 2s

3s IS

3D

3s

3P

2s

2D

-

1P

3s

3s

IS

1s

[I61

2s

3s 3s

2s

2s

3s 35

2s

3s 3s IS

215 2S

2s 5s

ID

2P

2s

1P

+(2)/(7) 2s

4P

3s

3D

2s

2D

IS

ID 3P 2s

2P

3s

1P 3s

2s

IS

1s

3s

2s

[5]

+I12

3s

4P

3s

2P

2s

3P

2s

1P

1s

4s

2s 4D 2D

25

3D

4P

2s

3P

1s

2P

2s

2s

2s

1s 2s

2D

5P 3P

3s

2P

+1/(3) 65

3s

1s

2s

3s

213

2s

ID

1s

2s

2s

4s

IS

4P

1s

2s

IS 3s 3s IS 3s IS

4P 2P

1s

2s

3s

3s

2P

1s

216

[2]

!S

3s

3s

1s

1s

+(2)/3 2s

2s

C.

318

D.H . Chisholm, A . Dalgarno, and F. R. Innes

TABLE VIII (Pi W)

001

(I]

OM

ti

2P

2P

t

I

[6]

2P

ti

4s

3P

2D

002

[3]

2P

2P

003 2P 2P

[6]

K

3P 3P 3P

2D

ID

t (3)/2

2P

2P 2P

1s

+

2p

+ (3)/(5)

005 4s 2D

ID 2P

2P t 1

+I + I

- 1/(2)

[3]

2P

- 1 /(2)

ID IS

ID

- (5)/3(2)

006

1s

t (2)/3

2P

-

1/2 1

t1

3P

+ 1/(2) 2P

- (5)/2(3) - 1 /2

2P ti

- 1/(3)

+ 1/(3) t 1/(3.5)

[I1 2P

IS

t 1

TABLES OF CFP

101

[21

2P 2P

102

[TI

104

-1/(2) 2s 2s

3P 1P

2P

2P

-2/(5)

5s 3D 3P

-(2) / (3 )

2P 2P

3P 3P 1P

4P 2D

2P

3P 1P 3P

2P

2P

319

3s 3D 3P

2P 2P

2P

(151

2P

3D 3P 3s 1D 1P 3P 1P

2s

-(3) / 2 4P 4P 2D 2P 4P 2D

5s

2P

3D

105 3P

2P

2P

2s 4P

2P

2P 2D

1D

2P

2P 2D

1P

2s

[TI

2P

2s

-(5)/(2.3) 4P

3P

2D 2P

2P

2P

2D

ID 1P

1P

103

4P

3s

2P

2s 2D

1P

2P 2s

2P

3P 1P

2s

320

C. D. H. Chisholm, A . Dalgarno, and F. R. Innes

TABLE X (P;llq) 111 -

r31

2P

+i/w

2P

3s

3s

4P

2P

3s

3s

2P

1s

1s

2P

1 I2

[Ill

ZP

3s

4P

3s

3s

4P 2P

3s

3s

3D

35

1s

3P 1P 3P

1s

3s

1D 30

35

IS

3P 3s 1P 3P

1s

+1/(2)

zp 2P

3s

4P

3s

5P 3D

3P

1s ZP

1s

2P

3s

4P 2P

3s

3s

2P 2P

1s

IS

3s

2P 2P

3s

ID 1P

ZP

IS

2P

IS

1s

113 -

~ 5 1 +(3)/(5)

2P

35

5P

2P

3s

2P

2P

3s

3P

3s

114

[251

2P

35

+(2)/(3)

6s

3s

5P

3s

3D

35

3P

4D 4P

ZP

3s

4s 4D

6S

4P

5P

4D

2D ZP 4D

3s

3D 3P 5P

3s

4P

3s

3D 3P 3s 5P

2P

3s

4P 4s 2D 2P

3s

4s

2s

IS

3P IS

2s

1P

3s 3s

1s

ZP

ID

ZP 1s

2D

4s ZD

1s

TABLES OF CFP

TABLE X (continued)

2P

3s

2P

IS

2P

35

2P

IS

I15 -

1113

2P

3s

2P

3s

IS

116 -

131

2P

3s

2P

IS

2P 4P 2P 2D 2P 2D 2P 2s 2P

3s

3s

IS

ID

3s

1P

1s

IS

+(5)/(7) 5P 3D 3P 3s 3D 3P 3s IP 3P ID IS

3s

4P

3s

2P

1s

+(3)/2 4P 2P 2P

3s

3s

1s

1s

32 1

C. D. H . Chisholm, A . Dalgarno, and F.R. Znnes

322

120 -

3s

3s

1s

3s 3s

121 -

3s

IP 5s 3s

3s

1s

3s 3s

1P 3P

IS

1P

1101

2s

1P

+1/(5) 4P

3s

2s

4P

1s

t 1 /2(3)

3s

3P

2s

1P

t 1 /(2.5)

IS

1P [lo]

-

4P

2s

4P

t (5)/3(2.7)

3s 3s

2P 2D

2s

2D

t 1/(2.7)

2s

2P

t 2/30)

2s

2s

t 1/(2.7)

t 1/2(7)

2s

2P

3s

2P

3s IS 3s

2P

4P

2s

IS 2s

125

[5]

3s t(2)/(3.5) 2s

5s

3s 3s

IS

IS 3s 3D

25

3D

126 3s

IS ID

- 1/(2.3.7) - 1/3(2.7) - 1/(2.3.7) - 1/(2.3.7)

t1/(2.7) 3P

2s

3P

1P 3P 1P

2s

1P

IS 5s

- 1/(2.3.7) - (2)/3(7)

2s

IS

[16]

1 /(2.3.5)

3s IS

3s

2s

- 1 /(2.3.5)

t(2)/(3.7)

4P

3s

- 1 /2(3.5) - 1/(2.3.5)

t 1 /(2.5)

3s

3s

3s

ID

2D

3s 3s

2s

2s

2s

IS

- 1/(2.3.5)

- 1/(2.3.5)

IS

IS

t 1 /(3.5)

3D ID

2P 2D

3s 3s

3s

3P

IS

124

123 -

2s

1s

3s

122 -

3P

IS

1s

[2]

- 1/2(2.7) -

1/2(2.7)

t (3)/2(2.7)

- 1/2(2.7)

t1/3(2) 2s

2s

2s

t 1/2(2.3)

- 1/2.3(2)

TABLES OF CFP

220 -

(21

IS

3s

3s

IS

IS 221 -

-(2)/(3)

[31

-(2)/(5)

3s

3s

4P

3s

3s

2P

IS

1s

222

-

181

3s

3s

5P

3s

3s

3P

IS

IS

3s

3s

1P

3s

3s

3D

IS

-2/(3.5)

IS

IS

ID

3s

3s

3s

IS

1s

IS

223 -

[lo]

3P

6s 4s

3s

3s

3s

IS 3s 3s

3s

2s

3s

4D

/3 t

IS

ID

IS

1s

-2/(3.7)

3s

IS

3s

3s

IS 3s 3s

1s

3s 3s

1s

IS

3s

3s

5P

3s

3s

3P

45

4P 2P

1s

2P

- 1/(3.7) - (2)/(3.7)

t 1/(3.7)

-

/(3.5) /(3.5)

IS

1s

3s

3s

IP

3s

35

3D

IS

1s

1D

3s

3s

3s

1s

IS

IS

225

[3]

3s

3s

4P

3s

3s

2P

1s

IS

226

IS IS

t 1/(3.7)

2D

1/(3.7)

2P /(2.5)

IS

323

2D

3s 1s

[2]

s

3P

S

ID

IS

1s

IS

2P

-1/3

- 1/3(2) t 1/2.3

- I /2.3

-2/3(5)

3s

3s

IS

IS

IS

1s

- 1/(3.5) - 1/3(5)

C. D . H . Chisholm, A . Dalgarno, and F. R. Innes

324

101

[2]

104 -

t1

[IS]

t(2)/(5)

3P

3P

3P

4s

1P

1P

1P 3P

2D

3P 3P

ZP 2D

1P 3P

2P

1P 3P

45

103

[IS]

+1/(2)

3P 3P

3P 3P

1P 3P 3P

3P

5s

3D

3P 1P 3P 1P 3P 1P

105

ID 3P

3P

3s

1P 3P 1P

3P

ID 1P

106

3P 1P 1P

3P ID

IS

[7]

3P IP

2s

+1 /( 3 )

ID

3P

1D

2P

1P 3P

3P 3P 1P 1P

IS

2P

2P

3P

1D

2D

ZD

3P

IP 3P 3P

4P

3P

IS 3P 1D

1P

1s

[21

+(2)/(7) 2P

2s

325

TABLES OF CFP

~

111 -

[51

2s

3P 3P 1P 3P 1P 112 -

-1/(3) 3s 3s

2s

4P 2P

1s

[I91

3P 3P 1P

-1/(3) !4P

3P

3P

1P

3P 1P

3P

3P 3P 1P 3P 3P

1P 3P 1P 3P

3s 3s

5P 3D

3s

3P

1s

3s

3s

IS

ID

3P

3s

1P

1P 3P 1P

1s

1s

3P

1P

1P

3P 1P

113 3P 3P 1P

[43]

2P 4P

3P 1P

3s

4P

4s

3P 3P 3P 3P 1P

2D 2P 2s 4P

3s

3P

2P 4P

1s

3P 3P

2P 4P

3s

3P 1P 3P 1P 3P 3P

2D

2D

2P 4P

1s

2D

1P

1P

3P 1P 3P

2D

3P

3P 1P

1P 3P

3P

-(3)/(2.5) 4P

3s

6S

4P

3s

4D

3P 1P

2P

3P 3P 1P 3P 1P

4P

3P 1P

2s

3P 3P 1P

4P

3s

20 2P

20

1s

2P

C.D.H. Chisholm, A . Dalgarno, and F. R. Innes

326

TABLE XIV (continued)

-

t 1/(2.3.5)

3P

1 /4(5)

3P

1P

+ (3)/4(5)

1P

3P

-

1/2(5)

3P

t 1/(2.3.5)

3P

ID

-

3P

I P

3P

3P

3P

2P 2s

- 1/2(3.5)

1P 3P

4P

3P

2P

3s

2s

- 1/(5)

1P

114 3P

1/(3.5)

-2/(3.5) 5s

3s

5P

3P

3D

3P

3P

3P

3s

3P

3D

1P 3P

3P

3s

3D

3s

IS

1D

5s 3D

3s

3P

3P

3P

1P

1D

- 1 /2(5)

1 /(3.5)

1P

1P

t 1/2(3.5)

t 1/2.3(5)

3P

3D

3P

3P

+ 1/2(5)

3P

3s

+ 2/3(5)

+

1P

ID

+

1P

1P

+ 1 /2(5)

3P

3P

1P

1P

- 1/(2.5)

1/(2.3.5)

t 1/2.3

0

1P

115

t 1 /3(2)

3P

0

1P

3P

[I91

4P

3P

2D

t 1/2(5)

3P

2P

3P

2s

ID

t 1 /3(2)

3P

4P

t 1/(2.3.5)

3P

2D

+ 1/3(2)

1P

+

3P

-

3P

1P

3P

5s

3P

3D

1s

1/3(2.5)

1 /3(2)

t ll2.3

1P

3s

IS

1P

1s

- 112.3 - 1 12(3.5) 112(3)

- 1 /(5) - 1/(3.5)

3s

4P

-

(5)/3(7)

t 1 /(3.7)

t 1/(2.3.5)

1P

+ 1/2(5)

-(5)/(3.7)

3s

1P

- 1 /(3.5) - (3)/2(5)

113

-

t 1/2(3.5)

3P

1P

1/2(3.5)

(2)/(3.5) t 1 /(3.5)

+ +

1P

3P

3P

-

3D

ID

1P

3s

3P

3P

3P

- 1/2.3

1/2(5) 1/(2.3.5)

3P

3P

t 1/3(5)

1P

+

- 1/2(3.5) - 1 /2(5)

1P

1 /2(3.5)

- 1/3(2.5)

1P

[43]

1P

3P

+ 3s

2P

- (5)/3(7) - I /3(7) 3s

2P

1 /3(7)

- (2)/3(7)

t (5)/2.3(7)

-

(5)/2(3.7)

+

1 /2(3.7)

- 5/2.3(7)

TABLES OF CFP

TABLE XIV (continued) 25

3P 1P 3P

4P

3P 1P

2D

3P

2P

1s

1P 3P

ZS

1P

116 3P

[51

-(3)/(2.7) 3P

3s

35

1s

IS

IP 3P

1P

3P

3P

1P

1P

327

C . D . H. Chisholm, A . Dalgarno, and F. R. Innes

328

213

211 3P 1P

IS

3P 3P

3s

1P

IS

2s

1P

t 1/(2.3)

3P

3s

5P

t 1/2(3)

1P 3P

3s

3P

3P

IS

3P

3s

5P

3P

3s

3D

3s

3P

- l/2(3) - 1 /2 - 1/2(3)

3P

3P 1P 3s

2P

3P

IS

3P

3s

4P

3P

3s

2P

2D

3P

3s

4P

3P

3s

2P

2s

2P

IS

3P

3s

4P

3P

3s

2P

1P

1s

3P

-

1P

1/(2.3.5)

2s

2s

IS ID

3P

1s

t 1/(5)

3P

3s

1P

t 1/2(2.5)

3P

3s

5P

t (3)/2(2.5

3P

3s

3D

3s

3P

(3)/2(2.5

1P

t 1/2(2.5)

3P

- 143.5)

1P

-

3P

1/2(2.5)

IS 3s

3s

1P

1/(5)

3P

IS

ID

t 1/2(2.5)

3P

3s

1P

t (3)/2(2.5

3P

IS

1s

3P

3s

5P

3P

3s

3P

+

- (3)/2(2.5

t 1/2(2.5)

2s

1P

t 1/2(2.5)

- (3)/2(2.5

1P

3P

t 1/(2.5)

t (5)/2(2.3 3 P

1P

1P

1P

-

IS

1P

3P

t 1/(2.3)

- ll(2.5) 25

1P 3P

2s

1P

212 -

3P

2s

2s

329

TABLES OF CFP

TABLE XV (continued)

1P 3P 1P 3P 3P 3P 3P

1P

3P 3P 3P 3P 3P 3P 3P 1P 3P 1P 3P

IS 3s

1P

3s 3s

3D

1s IS

2s

ID

3P

ID 1P

1P 3P 3P

3s 3s

3D

3s

3P

3P

1s

3P 1P 1P 1P

3s

3s

3P

IS 3s IS

ID

1P

214

[431

3P 3P 1P

3.5

6s

35

4D

3P

3s

4P

3s

4s

2s

1P

3s 3s

2s

3s

4D 2s 4P

3s

ZD

ZD

1s 3s

2P

1s

1P

3P

3s

1s

.3P

3s

4D 4P

3P 3P 3P 1P 3P 1P 3P 1P

3s

4s

+2(2)/(3.7) 2s

4P

IS

3s

2P

IS

IP

iP 3P 1P 3P 1P 3P

1s 3s

ZD

t (5)/3(2.3.7

IS 3s

2D

IS 3s

3P

IS

1P 3P 1P

3s

2P

2s

ZS

2P

C. D . H . Chisholm, A . Dalgarno, and F. R. Innes

330

TABLE XV (continued) ~

3P

3s

4P

3P

3s

2P

1P 3P

IS

2s

2s

t 1 /(2.3.7)

[I91 3s

5P

3P

35

3D

3s

3P

0 02 -

3s

3s

ID 1P

1s

1s

3s

3D

t I /(2.3.7)

3P

3s

3P

3P 3P

IS 3s

3s

t (5)/(2.3.7)

1P

IS

ID

t (5)/2(3.7)

1P 1P

3s

1P

IS

IS

t16

[5]

t 1/2(7)

3P

3s

4P

3s

2P

t 1/2(2.7)

3P 1P

t 1/2(3.7)

3P

IS

t 1/2(2.3.7)

1P

0

- 1/2(7)

1s

1P 3P 1P

3P

3s

3P

t (5)/2(2.3.7)

IP 1P 3P

2s

IS

3P

t(5)/(2.7)

3P

3P 3P

- 1 /(2.7)

t 1 /(2.7)

1P

3P

t 2/(3.7)

2s

1P

- (5)/2(2.3.7) - 1/2(2.3.7) - 1/2(2.3.7) - (5)/2(2. 7 ) + (3)/2(7)

- (3)/2(2.7) - 1/2(2.7) - (5)/2(2.3.7) + (3)/2(2.7) - 1/2(2.3.7)

+l/(3)

2s

2s

t ll(2.3)

+ 1/4(3) t I/4

- I/4

+ 1/4(3)

ID

ID

K

IS

003 -

ID

IS

3P

3P

3P

ID

ID

3P

IS

IS

1D 3P

005 -

ID

3P

4s

1s

3P

2D

(61

+I

ID 004 -

3P

3P

1D

ID IS

IS 006

-

3P

3P

3P

3P

ID

1D

3P

1s

1s

2P

[31

+I

33 1

TABLES OF CFP

1 02

[41

+11(3)

+ 1/(3) + 1 /(3)

3P

2s

4P

ID

2s

2D

3P

2s 2s

2P

- 1/(3)

2s

+ 1/(3)

1s

103 3P

5s

+

1/(2)

3P ID 3P 3P

3P

3D

+

1/(2.3)

- 1/2

+ 1/2(3)

1P

ID

- 1/(2.3) - (5)/2.3

1s

t 113

3P

3P

3P 3P

1P 3P

3s

3P 3P ID

1P 3P

ID

1P 3P

1P

1P

- 1/2(3) - 1/(2.3) +

1/(3)

+

112

- 1/2 - 1/2

- (5)/2.3

+

1s

1/3

104

+ 1 /(2.3)

3P ID 1s

3P 3P 3P 3P

2D 2P 2s 4P

ID

2D

IS

2s

3P

4P

3P 3P ID

2D 2P

2P

1s

3P

3P ID

2P

3P ID

ID 3P 1s

105 3P 3P ID

IS

3P 3P

106

- 1 /(5)

- 1/(2.3.5)

1P

3P

3P 3P

+

+ (7)/(2.3.!

3s ID 1P 3D

IP

JP

- 1/(2.3) - 1/(2.3.5) - 1 /(2.3) 1 /(2.3.5)

3P

3P ID

ID ID

- 1/(2.3.5)

2D

3P 3P

3s ID 1P

1s

-

[41

+(5)/(7)

3P ID

4P

3P

2P

1s

2s

2D

25

C.D.H. Chisholm, A . Dalgarno, and F. R. Innes

332

TABLE XVIII

(P’; 119)

[El

1s

t1/(2.3)

3P

3s

3s

5P

3P

iD 3P 3P

3s

3s

3D

ID

3s

3s

3P

1s

IS

3s

3s

ID

IS 3s

IS

ID

3P

3s

1P

1s

1s

IS

1s

3s

3s

3P

4?

3s

114

[411

3P

3s

5P

3s

3P

3s

4P

3s

4D

3P

3s

ID

5P

3P

ID

ID

3s

1s

3P

1P

3s

3P

4P

3P 3s

4s

2P

3P

1s

3P

3s

3P

IS 4P

3s

2D

2P IS

1s

3P ID

5P

4P 2P

3s

3s

2P

3P

-

1 /3(5)

-

1 /3(2)

1/(3.5) 1/3(5) 1 /3(5)

3D

t 1/3

3P

3P

t 1/(2.3.5)

-

1s 3P

3s

3P

1P

1s

3P

1s

3P

ID

3P

IS

113 1/3(5)

t 1/3(5)

+ (2)/3(5) 1s

t (2)/(3.5)

-

ID 3s

1/3(5) 1/3

3P

3P

ID 3P

3s

-

t (7)/3(5)

ID

ID 3P

3D

3D

3P

3P

3s

1s

2P

1/3(5) 113

t 1 /(2.3.5)

3P

3P

1/3

-

3P

4P

+ 1/(2.5)

3D

3P

3s

2/(3.5)

3P

6s

4P

5P

-

3s

3s

3s

-

4P

3P

-

IS 1251

2P

t 1/(2.3.5)

ID

3s

ID

2s

+(2)/(5)

3P

3P

(3)/2(5) 1/20) t 1/(3.5)

2P

fi

t(3)/(2.5)

-

1s

1s 3P

IS

t 1 /(3.5)

1s

1/3 1/3(5)

- l/3 - 1/3(5)

TABLES OF CFP

333

TABLE XVIII (continued)

3P

3s

5P

3s

3s

- 1/3

3P

3P

- l/3 - 1/(3.5)

1s

3s

t 2/3(5)

ID

3D

3P 3P ID

3P

3s

3P

2D

ID

+ (7)/3(5)

ID

1s

-

3P ID

3D 3s

IS

ID

ID

1P

35

1P

-

1/3(5) 1/3(5) 1/3 (2)/(3.5) 1./3(5)

ID

ID

- 113 - 1/3(5) - 1/(5) - 1/3

1s

1s

+ 2/3(5)

1s

[25]

3P

3s

3P

3P

1s

1s

t(2.5)/(3.7)

6s

3s

4P

-

3P

2P

ID

1s 116 -

[El

3P

3s

3P

+(3.5)/2(7) 5P

ID

3D

3P

3P

IS

3s

3s

3s

1s

1s

IP 1s

3P

+ (2)/3(7)

3P

3P

4s

-

ID

ID

3P

2D

t (2.5)/3(3.7)

1s

1s

2/3(3.7)

1s

ID

-

1s

4s 2D

3P

- (5)/3(2.7) (5)/3(2.7)

2s 1s

3P

4P

ID

2P

IS

1/(2.7) +'5/3(2.3.7)

- (5)/(2.3.7)

3s

2P

3P

4D

ID 3P

4D

3P

3P

4

3s

1/(5)

3P

1s

3P

3P

- 1/3(5)

3P

3P

2s

-

3P

1s

ID

2P

3P

4P 4s

1P 1s

3P

334

C.D . H . Chisholm, A . Dalgarno, and F. R . Innes REFERENCES

Armstrong, L., Jr. (1968). Phys. Rev. 172,18. Bacher, R. F., and Goudsmit, S. (1934). Phys. Rev. 46,948. Chisholm, C. D. H., and Dalgnrno, A. (1966).Proc. Roy. SOC.(London) ,4290,264. Condon, E. U., and Shortley, G. H. (1935). “The Theory of Atomic Spectra.” Cambridge University Press, London and New York. Dalgarno, A., and Parkinson, E. M. (1968).Phys. Reo. 176,73. Edmonds, A. R. (1957). “Angular Momentum in Quantum Mechanics.” Princeton Univ. Press, Princeton, New Jersey. Elliott, J. P., Hope, J., and Jahn, H. A. (1953). Phil. Trans. Roy. SOC.(London) A246,241. Fano, U. (1965).Phys. Reu. 140, A67. French, J. B. (1966). in “ Many-Body Description of Nuclear Structure and Reactions” (C. Bloch, ed.), (Proc. Infern. School Phys.). Academic Press, New York. Henry, R. W. J., Burke, P. G., and Sinfailam, A.-L. (1969).Phys. Rev. 178,218. Horie, H. (1953).Progr. Theoref.Phys. (Kyoto) 10,296. Innes, F. R. (1967).J. Mufh. Phys. 8,816. Innes, F. R., and Ufford, C. W. (1958).Phys. Reo. 111, 194. Jahn, H. A. (1954).Phys. Reo. 96,989. Judd, B. R. (1963). “Operator Techniques in Atomic Spectroscopy.” McGraw-Hill, New York. Judd, B. R. (1967). “Second Quantization and Atomic Spectroscopy.” Johns Hopkins Press, Baltimore, Maryland. Kamuntavichyus, G. P. (1967).Lifoo. Fir. Sb. 7,No. 3, 553. Karaziya, R. I., Vizbaraite, Ya. I., Rudzikas, Z. B., and Yutsis, A. P. (1967). “Tables for calculating Matrix Elements of Operators for Atomic Quantities” (in Russian). Computation Centre, The Academy of Sciences, Moscow. Levinson, I. B. (1957).Tr. AN Lifoo. SSR B4,17. Levinson, I. B., and Nikitin, A. A. (1965). “Handbook for Theoretical Computation of Line Intensities in Atomic Spectra” (translated from Russian). Daniel Davey, New York. Mapleton, R. A. (1963).Phys. Rev. 130, 1829. Mapleton, R. A. (1968).J. Phys. B, (Proc. Phys. Soe.) 1,850. Menzel, D. H., and Goldberg, L. (1936).Asfrophys. J . 84,1. Meshkov, S. (1953). Phys. Reo. 91,871. Messiah, A. (1962).“Quantum Mechanics,” Vol. 11. Wiley, New York. Nielson, C. W., and Koster, G. F. (1963). “Spectroscopic Coefficients for the p”, d”, and f” Configurations.” M.I.T. Press, Cambridge, Massachusetts. Racah, G. (1942).Phys. Reo. 62,438. Racah, G . (1943).Phys. Reo. 63,367. Redmond, P.J. (1954).Proc. Roy. SOC.(London) A222, 84. Rohrlich, F. (1959).Astrophys. J. 129,441,449. Rudge, M. R. H. (1968). Reu. Mod. Phys. 40,564. Shore, B. W., and Menzel, D. H. (1968). “Principles of Atomic Spectra.” Wiley, New York. Sinanoglu, 0. (1961).Phys. Reo. 122,493. Slater, J. C. (1960).“Quantum Theory of Atomic Structure.” McGraw-Hill, New York. Smith, K. L., Henry, R. J. W., and Burke, P. G. (1967).Phys. Rev. 157,51. Smith, K. L., and Morgan, L. A. (1968).Phys. Reu. 165,110.

TABLES OF CFP

335

Trees, R. E. (1951). Phys. Rev. 82, 683. Trees, R. E. (1954). J. Res. Natl. Bur. Std. (US.)53, 35. Wybourne, B. G. (1965). “Spectroscopic Properties of Rare Earths.” Wiley (Interscience), New York. Yutsis, A. P., Levinson, I. B., and Vanagas, V. V. (1962). “The Theory of Angular Momentum” (translated from Russian). Israel Program for Scientific Translations, Jerusalem.

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RELATIVISTIC Z-DEPENDENT CORRECTIONS TO ATOMIC ENERGY LEVELS HOLLY THOMIS DOYLE Harvard College Observatory Cambridge, Massachusetts

I. Introduction ....................................................... 337 11. The Relativistic Z-Dependent Theory ............................... .342 A. The Hydrogenic Problem.. ..................................... .344 B. The Many-Electron Problem. ................................... .347 C. Screening Methods .............................................. 350 111. Irreducible Tensor Expansions of the Electrostatic and Breit Interaction Operators, ....................................................... ,352 IV. Antisymmetrization. .............................................. ,357 V. Reduction of Matrix Elements to Matrix Elements between One- and Two-Electron States. ........................................... 363 VI. Results, Comparisons, and Conclusions .............................. ,369 A. Discussion of the Calculations and Numerical Results .............. ,369 B. Comparisons with Experiment. .................................. .406 C. Concluding Remarks ........................................... .410 References ....................................................... .412

I. Introduction A nonrelativistic perturbation calculation of atomic energy levels with a sum of single-electron hydrogenic Hamiltonians as unperturbed Hamiltonian and the electrostatic interaction of the electrons as perturbation operator results in an expansion of the nonrelativistic energy in inverse powers of the nuclear charge Z m

EN, = Z2

1En, Z-"

n=O

(1)

This method of calculation has the advantage of determining a given energy level simultaneously for all the members of an isoelectronic sequence. Layzer and Bahcall (1962) extended the Z-' expansion formalism to include relativistic corrections. The relativistic energy E = E N R + ER has a double power-series expansion in 2-' and E = a2Z2,

c c En, Z-"ern, m

E =Z2

m

n=O m=O

337

338

Holly Thomis Doyle

where 01 is the fine structure constant and the coefficients En, are functions of EZ. Layzer and Bahcall approximated the total relativistic Hamiltonian by H = Ho V B, where the unperturbed Hamiltonian Ho is a sum of hydrogenic Dirac Hamiltonians of the electrons moving in the Coulomb field of the nuclear charge Z, and the perturbation V + B is the sum of the electrostatic and Breit interactions between electrons. Applying degenerate first-order perturbation theory to a subset of hydrogenically degenerate states of the same parity which Layzer (1959) called a complex, Layzer and Bahcall showed that inclusion of the mixing between the finite, generally small, number of configurations in the complex results in energy calculations that correctly predict the leading two terms of the 2-' expansion, that is, all terms of the form Z 2 ~ and " ZE"for m 2 0. As in the nonrelativistic theory, matrix elements of H between states belonging to different complexes were shown to be of order Z-2 relative to the leading terms. However, unlike the nonrelativistic theory, which is complete to all orders of perturbation theory, the relativistic theory is incomplete since H does not include the Lamb shift and other interactions of relative order Z-' with respect to the leading terms. Consequently the perturbation V + B may not be used to find second-order relativistic corrections. Also unlike the nonrelativistic theory, it is not possible to factor the nuclear charge Z out of the matrix diagonalization, and the relativistic energy eigenvalues have constant coefficients in the power series expansion (2) only for the special case of a single state in the complex with a given angular momentum J. Nevertheless, it is convenient to refer to calculations as " complete through En,,,"when they are correct to order n in Z-' and m in E , even if no constant coefficients Enmexist. The importance of the relativistic contribution is illustrated in Table I for the ionization potential of the ground state ls22s 2S of the lithium sequence. The Z-' expansion of this ionization potential is the difference of the Z-' expansion of the total energies of the ground states of the lithium-like and helium-like ions. For ions of the lithium sequence theoretical values are available for the nonrelativistic energy expansion coefficients E,, , El, (Linderberg and Shull, 1960), E,, (Chisholm and Dalgarno, 1966), and E,, (Seung and Wilson, 1967). For the helium sequence ions in the ground state Midtdal (1965) has calculated the nonrelativistic energy coefficients En, , 0 < n < 21. Layzer and Bahcall(l962) have calculated E,, and E l , for the ground states of both the lithium and helium sequence. In Table I

+ +

AE=AENR+AER+ AEp+AE,,

where the nonrelativistic energy is AENR= 0.1252, - 0.397805212

+ 0.25048256 + 0.031742322-',

TABLE I

NONRELATIVXSTIC AND RELATIVISTIC CONTRIBUTIONS TO THE LITHIUM SEQUENCE IONIZATION POTENTIAL Z

AENR

AER

3 4 5 6 7 8 9 10 11 12 13 14 15

0.1926477 0.6671973 1.3928048 2.3689417 3.5953808 5.0720087 6.7987626 8.7756047 11.0025109 13.4794652 16.2064566 19.1834771 22.4105205

-0.00011927 -0.00014959 -0.00013214 +0.00039375 t0.00133872 0.00306337 0.0058781 0.0101433 0.0162693 0.0247163 0.0359946 0.0506645 0.0693352

AEP -0.0000009

-0.0000019 -0.0000027 -0.0000034 -0.0000038 -0.0000040

-0.0000041 -0.0000044 -0.0000043

-0.0000046 -0.0000045 -0.0000047 -0.0000046

A EL

AE

-0.0000091 -0.oooO258 -0.oooO573 -0.0001092 -0.0001870 -0.0002964 -0.0004429 -0.0006317 -0.0008673 -0.0011540 -0.0014952 -0.001894 -0.002352

0.1925184 0.6670199 1.3926128 2.3692229 3.5965287 5.0747716 6.8041937 8.8779387 11.0179086 13.5030230 16.2409515 19.232243 22.477499

EXP

0.19815770" 0.66928833b 1.393994 2.370209' 3.5975376 5.0759294 6.805793 -

11.01985 13.50442 16.243 19.2348 22.4771

2RM 219457.28" 219461.26b 219463.76 219464.60' 219466.02" 219467.10' 219468.32 219468.64 219469.38' 219469.661 219470.16= 219470.32 219470.76

EXP- AENR

EXP- AE

0.00510 0.002091 0.001189 0.001267 0.002156 0.003921 0.00703

0.005631 0.002268 0.001381 0.00986 0.001009 0.001 138 0.001599

0.01733 0.0250 0.0367 0.0513 0.0665

0.00194 0.00140 0.0023 0.00252

-

-

--0.O004

~

'Johansson (1959). Johansson (1962). Bockasten (1956). 'Hallin (1966). Eriksson and Isberg (1963a,b). Risberg (1955,1956). w w

\o

340

Holly Thomis Doyle

and the relativistic energy is

AER = 0.0390625~r~Z~ - 0.200141901~Z~. The mass polarization correction AEp has been calculated by Prasad and Stewart (1966) for 3 < Z < 8. Their results fit the approximate formula AEp = p/M( -0.0202 + 0.038), where p is the reduced mass of the electron and M the nuclear mass, and we have used this formula for higher Z . For the Lamb shift AEL we have used a formula for the lowest-order Lamb shift of a hydrogenic 2s electron given by Erickson and Yennie (1969,

where m is the mass of the electron. The table is in atomic units. The experimental energies are taken from Moore (1949) unless otherwise noted. The elementary mass correction was made by converting from inverse centimeters to atomic units using the appropriate Rydberg constant 2R,. It can be seen from Table I that, while the relativistic corrections make little difference for low Z,for Z > 1 1 they reduce the discrepancy between the predicted and experimental energies by a factor of ten or better. The nonrelativistic coefficients El have been calculated by Layzer (1 959) and by Linderberg and Shull (1960) for the ground terms in even and odd complexes with up to ten electrons (lP, 29, 1 < p < 2, 0 < q < 8, and by Godfredsen (1963, 1966) for low-lying terms of even and odd complexes (12, 2', 3') and (12, ,'2 318-') with 1 < r < 6. Theoretical relativistic corrections through El 1, i.e., through terms of order u2Z3 in the Z-'expansion scheme, have been calculated exactly for the ground state of the He isoelectronic sequence by Dalgarno and Stewart (1960b) and Layzer and Bahcall (1962). The latter also calculated Ell for ls2s lS3Sand 183s lS3Sin the He sequence for ls22s' S and ls23s2Sin the Li sequence as linear combinations of the two-electron coefficients. The term Ell has also been determined using an interchange theorem for ls2s lS3S by Stewart (1966)and for ls22s2Sby Cohen and Dalgarno (1963).Dalgarno and Parkinson (1968),in a general discussion of the calculation' of properties of many-electron systems from those of two-electron systems, have shown explicitly that Ell for the ls22s ' S state may be evaluated as a linear combination of the two-electron coefficients. These relativistic calculations were for two- and three-electron complexes with only one state of a given angular momentum J (requiring no matrix diagonalization) and with only s electrons (for which certain terms in the relativistic operator I/+ B are identically zero). We have extended these calculations of El, to include 2p ( j = 3) and 2p ( j = 4) electrons for atomic systems with up to ten electrons in the first two shells and up to four

RELATIVISTIC 2-DEPENDENT CORRECTIONS

341

states in a complex with a given J. We again note that in many cases there is no single 2-independent value of El, for a particular energy level. In these cases the calculations are complete through Ell in the sense that all terms of order 1 in Z and in E are included in the perturbation calculation, the distinctive dependence on 2 and E being lost in the separate matrix diagonalization for each 2. In addition to the theoretical calculations of the relativistic Z-'-expansion coefficients, there have been reported a semiempirical study of the Z-lexpansion coefficients and a number of variational calculations of relativistic corrections. Sherr et al. (1962), using the known coefficients Eoo and Elo,fitted the available experimental data obtaining semiempirical coefficients for the various powers of 2 for the ionization potentials of the ground states of ions with up to ten electrons, 1 < Z < 20. Accurate variational calculations for two-electron ions include the ground state energies of He-like ions, 1 < 2 6 10 (Pekeris, 1958), the 1x2s '.'S states H-, He and Li' (Pekeris, 1962a, b), the fine structure of the ls2p 'P state of He (Schwarz, 1964), the lsnp 1,3Pstates, n = 2-4, of He, and the ls2p 'P state of Li' (Schiff et al., 1965a, b). A three-electron calculation has been done by Bell and Stewart (1964), who evaluated the fine structure of the ls'np *Pstates, n = 2-4 and 3 < 2 < 9, of the lithium sequenceusingrelatively simple wave functions of the open-shell type. It is not necessary to go to second-order perturbation theory to improve on the predictions of the first-order nonrelativistic theory. Another possibility is to choose zero-order wave functions that allow somewhat for the interaction of the electrons, such as scaled or screened hydrogenic wave functions. The problem of systematically choosing the " best " screening parameters has been discussed by Layzer (1959). He used the variation principles to derive a coupled set of simultaneous algebraic equations for screening parameters that minimize the nonrelativistic energy. Layzer et al. (1967) have used this method to calculate nonrelativistic screening parameters. We have employed these nonrelativistic screening parameters in a calculation of expectation values of the relativistic operators using screened wave functions for a few cases in order to illustrate the effect of screening. In using nonrelativistic screening parameters to calculate atomic properties other than the nonrelativistic energy we should note that Dalgarno and Stewart (1960a) showed that the screening parameters which make the firstorder energy correction vanish are not necessarily the best screening parameters for other atomic properties. They found that high accuracy is obtained by choosing screening parameters that make the given atomic property stationary with respect to first-order variations, a procedure that results in different screening parameters for different atomic properties.

342

Holly Thomis Doyle

A simple nonrelativistic screening procedure that can provide improved energy predictions is the method of “ simple screening constants,” described by Layzer (1959). These are obtained from an unscreened first-order calculation by neglecting external screening and choosing screening parameters so that the two leading nonrelativistic terms in the Z - ’ expansion of the energy are correctly predicted by a single screened zero-order expression. Since a zero-order screened expression of the form E,,,(Z - a)’ contains terms of order Z d 2relative to the leading term, an estimate of E,, is included in the screened expression and significant improvement of the energy prediction results in many cases. An extension (see Section 11) of this method of simple screening constants to the relativistic corrections, determining a relativistic yields results which simple screening constant from the known value of Ell, include an estimate of and are quite accurate in some cases. The determination of this relativistic simple screening parameter has particular significance for the classic problem of extrapolating relativistic term splittings for highly ionized atoms which are unobserved in the laboratory. In the limit of infinite Z the first two terms in the Z - ’ expansion determine the screening parameter in a representation of the level splitting by a screened Sommerfeld formula. Exact knowledge of E , , is therefore equivalent to knowing the screening parameter at infinite 2. This formally reduces the classic problem of extrapolation along the isoelectric sequence to an interpolation problem for these cases, as discussed in Section VI, C . In Section 11, we review the relativistic 2- ‘-expansion theory of Layzer and Bahcall. We briefly discuss the extension of Layzer’s elementary screening theory to relativistic corrections. In Section 111, we derive the expansion in irreducible tensors of the electrostatic and Breit operators in two forms, one suitable for calculations in L-S coupling, and the other for j-j coupling. Sections IV and V concern the evaluation of matrix elements of symmetric sums of two-electron operators connecting many-electron states. Such matrix elements are expressed as linear combinations of the matrix elements between two-electron states. The matrix elements are further reduced to linear combinations of products of radial integrals and single-electron matrix elements. Finally, Section V l reports some new calculations based on the formulas developed in the preceding sections and presents a discussion of the results and comparisons with previous calculations and experiment.

II. The Relativistic 2-Dependent Theory The total Hamiltonian of a many-electron atom may be written (3) H = Hatom + H r a d + Hint where Hatomis the relativistic Hamiltonian of the atom in the absence of a

RELATIVISTIC Z-DEPENDENT CORRECTIONS

343

radiation field, including the instantaneous Coulomb interaction of the charged particles. The term Hladis the Hamiltonian of a free radiation field and Hi,, is the interaction Hamiltonian of the weak coupling between particles and field. We can further divide Ha,,, into

v = Ce2/rij i<j

(5)

is the electrostatic interaction between the electrons, and H,, is a sum of single-electron Hamiltonians N

Ho= C H i , i= 1

and H i is the Dirac Hamiltonian for a single electron in the field of a nucleus of charge Z H i = mc2pi + cai pi - Ze'lr, ,

(7)

where /? and the vector operator Q are 4 x 4 matrix operators acting on the spin variables alone. These four operators /?,a', a', u3 anticommute and their squares are equal to one: (a')' = p' = 1

. .

a l u ~

=

--Cciai

ui/?= -pai,

i#j, i,j = 1, 2, 3.

We use the representation

where a is the 2 x 2 Pauli spin matrix operator whose components also anticommute and have squares equal to one. aid = 6"

Our representation for a is

and

the unit matrix.

+ icijkak,

i, j = 1,2, 3.

344

Holly Thomis Doyle

We take the radiation field to be in the vacuum state. In this case Hrad does not contribute to the energy. Here, Hintis treated as a small perturbation with Hat,,+ Hradas the unperturbed Hamiltonian. The first-order energy contribution of Hintis zero since Hintcreates or destroys a photon. It takes (Hint)’,that is, both creation and destruction of a photon, to get back to the original state; thus the first nonzero energy shift arises from the secondorder contributions. These can be divided into two types: first, emission and reabsorption by the same electron (the Lamb shift); and second, emission by one electron followed by a second electron (the Breit shift). The larger of these is the Breit shift. Breit (1929, 1930, 1932) approximated this second-order energy shift as the expectation value of an operator B=

C - e2/2rij(ai* aj + (ai .Pij)(aj Pij))

i<j

(9)

between zero-order eigenfunctions of Hat,, . A. THEHYDROGENIC PROBLEM

(h

We first consider the single-electron problem. If we choose natural units = m = c = 1) then ct = e’/hc = e2 z 1/137. Then Hibecomes

H i = pi + ai pi - Ze2/r,. (10) If we further change the unit of length to ao/Z = h2/me2Z= l/eZZ,then ri = ri’@ = r,’/e’Z (1 1) p i = pi’e2Z (12) so that Hi = pi + e’Za, p i - (e2Z)’/ri, where we have dropped the primes. Now (e’Z)’ = a’Z’ = E , and we see that Z appears in H i only in the combination e’Z = E“’, H i = pi + &‘‘’ai p i - e / r i , (13) and therefore solutions Y of Hi” = EY

-

depend on Z only through E . The solutions Y are well known analytically as functions of E . They are 4-component vectors which can be written as column vectors with two spinor components

In this case, the eigenvalue equation (14) becomes, writing out the matrices

345

RELATIVISTIC 2-DEPENDENT CORRECTIONS

for pi and ~

l

and ~ , dropping the i's since we are considering only one electron,

or, writing this out in two equations, we have Y+

+ E ~ / ~*p)Y ( o

- (E/T)Y+ = EY+

(15a)

and -Y-

+ &'/'(a

*p)Y+- ( c / r ) Y - = EY-.

(1 5b)

For a derivation of the exact solutions to these equations, see Bethe and Salpeter (1957). Simultaneous solutions of these two equations are of two types. In the first case Y - is of order e l l 2 compared to Y'. We see that the two terms involving E in the first equation are small compared to the other two and hence approximate solutions are E z 1 = mc2, the mass energy. In the second case, the opposite condition holds, Y + is of order compared to Y-. Now the E terms in the second equation are small and approximate eigenvalues are E z - 1 = -mc2. These negative energy solutions are the positrons, and the positive energy solutions are the electron solutions. The solutions are known analytic functions of E , and in our calculations we used the following E expansion of the positive energy electron solutions. Y+ Y = (y-)=

(

+ E Y l + + .. . + &Yl+ ...)

YO+

E1/2(Yo-

1 *

Again we note that the Yo+, Y1+, . . . , Yo-, etc., are independent of Z, the entire Z dependence of Y being through the explicit E'S. We note that this is also an expansion in v2/c2since, in our units, e2Z _ -- - Nmu2 = (e2Z)2 = E. c2 mc2 - ( a o / z ) Consequently, Y is the nonrelativistic wave function. We are now in a position to show that the matrix elements of order c1l2 = u/c. From Eqs. (8) and (16), we have v2

0'

(;:iai;::)

ct

are of

+ (Y-lOlY"+) = (YO+ + EYl+ + . . . lOlE'/2(Y'b- + &Y;- + . . .)) + (E'/2(Yo- + + . ..)lOlY'b+ + &Y';++ ...) = E'/2((Yo+IulY;-) + (Yo-lalY;+)) + higher order terms.

(17)

:(I=;(

): ;I);

= (Y+lOIYf->

EY1-

This is of order

since Yo* and o have no E dependence.

346

Holly Thomis Doyle

There are several operators which commute with H i . They are K = f l ( 1 + G * L), with eigenvalues K ; J 2 = K 2 - 4, whose eigenvalues j ( j +_ 1) are not independent of those of K, and are given by j = J K J- f ; J,, with eigenvalues m ; and the parity operator PR, with eigenvalues k 1. A convenient representation for Hiis a set of simultaneous eigenkets of H i , K and J2,and J , , which we can write InKm),

Each spinor component, '4' or Y - , is an eigenfunction of S2 and L2,and thus we can use the somewhat more familiar quantum numbers s, I, j , and m to describe

which has an E expansion

+ + ... E ~ / ~ ( R ~+(R~Z+I + ) ,) , & +. R&+), R&+),E

(rlnicrn) =

The eigenvalues I + and by 1: 1+ =

IKI,

(19b)

I- of the upper and lower spinor components differ II+

=

IKI

- 1,

if if

K

is positive, is negative.

K = ( K J- 1, I - = JKJ, Note that n, I + , j , and m are equivalent information to n, K , and m and we

can write

a notation which is somewhat more familiar since nl-, j are the quantum numbers of the state in the nonrelativistic limit. In our calculation there are four types of electrons, described in the

lnl+

i) notation by

RELATIVISTIC 2-DEPENDENT CORRECTIONS

347

for which K = - I , K = - 1, K = + 1, and K = - 2 , respectively. Note that inj-j coupling 2pIl2and 2p,/, are not equivalent electrons. The eigenvalues of the ket Intim) depend on n and IKI, and are given by the Sommerfeld formula

which has an E expansion

& = 1 - 2n 2 (1 + a l &+ . .. + a k & k + .. .),

(22)

say, where the 1 is the mass energy (rnc’) and the terms in brackets are the relativistic hydrogenic energy in atomic units, excluding the mass energy. Note that these terms reduce to the usual nonrelativistic hydrogenic energy in the limit E + 0.

B. THEMANY-ELECTRON PROBLEM The unperturbed Hamiltonian for many electrons, Eq. (6), is N

H, = C H i , i= 1

(23)

which determines the motions of N electrons in a nuclear field with no interactions between the electrons or with the radiation field. Eigenkets are constructed from products

of single-electron hydrogenic kets by standard jj-coupling methods. First equivalent electrons are coupled together and antisymmetrized. Then successive groups are coupled, Is to 2s, the result to 2 p I l 2 ,etc., and finally the whole result is antisymmetrized. This procedure is discussed in some detail in Section V. Now since we considered only K- and L-shell electrons, a sufficient set of quantum numbers is just the number of each of the four types of equivalent

Holly Thomis Doyle

348

electrons, the angular momentum of each of the four groups, and the way in which the angular momenta are coupled to make a total angular momentum. Thus the eigenkets of Ho upon which we based our calculations may be written

This ket has q1 1s electrons coupled so their resultant angular momentum is J1,q2 2s electrons coupled to give angular momentum J 2 , and J1 is coupled to J , to give angular momentum 1,. The q3 equivalent 2pl12electrons are coupled to give J , , J3 is coupled to f, to give I , , and finally 1, is coupled to the resultant angular momentum J4 of the 2p3/2 group to give 14, the total angular momentum of the ket. The antisymmetrization operator d completes the antisymmetrization with respect to exchange of inequivalent electrons. The energy of this state is just the sum of the one-electron energies, Eq. (22) $Qniji+l/2 i=l

=q1811

-k

( q 2 f q3I821

+ q4&22*

(25)

We recall here the definition of a complex as the set of states belonging to principal quantum numbers (ni)and a definite parity p. A subcomplex is the subset of these states belonging to quantum numbers(n,j,)p. The configuration is another subset, corresponding to (nili), and the intersection of configuration and subcomplex, (ni l i j i ) ,is also important in our discussion of screening. Another notation for the complex, (lp243'.. .), with a suffix zero for odd parity, is similar to the notation of Eq. (24). Now we are ready to turn on the perturbations. We recall that the Breit operator is an approximation to the second-order energy shift arising from Hint,and is correct only for calculations to first order of Z-'. Although the expression for the Breit shift was derived from a perturbation development which used Ha,,, = Ho + V as the unperturbed Hamiltonian, Layzer and Bahcall (1962) showed that the contribution of B to the energy shift can equally well be evaluated in a representation based on eigenkets of Ho as the unperturbed Hamiltonian and with V + B as the perturbing operator, that is, to the same, first-order, accuracy in Z-'. We thus use the representation of Eq. (24), with zero-order energies given by Eqs. (21), (22), and (25), to calculate the first-order energy of V + B, Eqs. (5) and (7),

v + B = i< Cj (e2/rij)(l- *(u,

uj

+ (ui

Pij)(u,

* fij))).

(26)

Since we changed coordinates, using ao/Z as the unit of length, for Ho ,

RELATIVISTIC 2-DEPENDENT CORRECTIONS

349

Eqs. (11) and (12), we must do the same for V + Bywhich becomes, again suppressing the primes, I/

+ B = x(e4Z/rij)(l - +(ai- aj + (ai 3ij)(aj Pij))) = 6Z-l (l/rij)(l - +(ai a j + (ai Pij)(aj Pij))). *

i<j

i<j

(27)

From Eqs. (6),(131, and (27), we may now write the total Hamiltonian in the form H = H , + V + B = Ho + Z - l H , , (28) where H, and H , depend on Z only through E , and Z-' is a natural parameter for a perturbation expansion. The first-order energies are the eigenvalues of the matrix H ( n i ) pof Ho + V + B evaluated between all states in , matrix the complex (ni)p. This matrix breaks up into blocks H ( n r ) p Jsince elements connecting states of different J vanish. The matrix of the unperturbed Hamiltonian, H O ( n i ) p is J , diagonal and its elements have an E expansion which is a summation of E expansions of Sommerfeld terms, Eq. (22). A typical element looks like N

+ E(hO0+ chol + E2h02+ * - * ) ,

(29)

where

and so forth. Matrix elements of V + B between states of the form (19b) have an E expansion EZ-l(hlo + Eh,, + 2 h 1 2 + (31) a * * ) .

The leading coefficient, h , , , is the nonrelativistic matrix element of V evaluated between the leading, i.e., nonrelativistic, terms in the E expansion of the kets. The term B does not contribute to h,, since, as we saw from Eq. (17), the matrix elements of ct are of order so that those of (ai ,aj + (ai,rij)(aj,rij)) are of order E . Thus the leading contribution of the matrix element of B is of order E'Z-', and h , , consists of this term plus the second term in the E expansion of the matrix element of V. For cases with a single element in a block H(ni)pJ (one state in the complex with the given J ) , the energy is just given by the matrix elements themselves, and the coefficients h,, are the Z-independent coefficients in the double power series expansion of the energy, Eq. (2), (32) If there is more than one state in the complex with the given J, however, En, = hn,

*

350

Holly Thomis Doyle

Eq. (32) no longer holds. Since 2 enters each matrix element in several different powers it cannot be scaled out of the matrix, and thus a separate diagonalization must be carried out for each Z . The diagonalization leads to a single energy in which the energy contributions from the various orders are inseparably combined. We report in this review calculations of matrix elements through hll, and hence energy coefficients up to El, for the single-element cases. The diagonaliization of matrices with more than one element leads to first-order energies complete through Ell and gives the mixing coefficients for the zero-order states. In our calculations we evaluated the perturbation operators V and B between relativistic wave functions. The calculations could alternatively have been done to the same accuracy using two-component wave functions and either a Pauli approximation to the operators (see Bethe and Salpeter, 1957, p. 178) or by applying a Foldy-Wouthuysen transformation to the Hamiltonian to eliminate the “odd” terms [terms with only nondiagonal matrix elements such as a, Eq. (8), which mix the large and small components of the wave functions (Foldy and Wouthuysen, 1950; Chraplyvy, 1953a,b)]. It is these terms which cause both upper and lower components of Y to occur in each of Eqs. (15a) and (15b). The Foldy-Wouthuysen transformation removes this coupling and splits Eqs. (15a) and (15b) into separate equations for Y + and Y -. Either of these procedures applied to V yields extra terms in addition to e 2 / r i j ,a spin-orbit coupling term and a term due to the spread in charge of the electron (for a discussion see Itoh, 1965; Armstrong, 1966, 1968). The reduced form of B includes spin-spin, orbit-orbit, and spinother-orbit terms. In evaluating the Pauli approximation to the Breit operator nonrelativistic wave functions may be used [R;, in Eq. (19b)l. However, this is not accurate enough for a Pauli approximation to V to the same order. Bahcall (1960) showed that the first two terms in the E expansion of R: [Eq. (19b)l must be retained if all terms of order E’Z-’ are to be included.

xi<

C. SCREENING METHODS As mentioned in Section I, we have followed Layzer (1959) in considering two types of screening. In the variational screening theory each orbital is assigned a screening parameter derived from a variation principle. The programs we have written allow for the introduction of screening parameters into the calculation of the radial integrals. We have taken values of the variational screening parameters computed by Layzer et al. (1967) and screened the relativistic wave functions to calculate energies for a number of examples.

RELATIVISTIC %DEPENDENT CORRECTIONS

351

The elementary screening method does not involve any new calculation of the radial integrals using screened wave functions. Instead the energy is given by a simple screened formula in which the screening parameter is determined from an unscreened first-order calculation. On the basis of Moseley's law, Layzer (1959) suggested representing nonrelativistic term energies by an expression which is the eigenvalue of a nonrelativistic screened hydrogenic Hamiltonian : E& = -

m

qm(Z- o,J2/2nm2.

(33)

On comparing this with the nonrelativistic Z- '-expansion, Eq. (l), clearly

This definition of a screening constant can be extended to the relativistic corrections by representing the terms of order one in E in a form suggested by the success of the empirical regular doublet law,

and identifying with the terms of order one in similar to Eq. (34), R

Ell = 2 x q m

nm

E

= a2Z2, we find a result

(i-i).

From calculations for successive groups of equivalent electrons we can establish the omR.In practice, the number of cases for which this analysis is possible is small, namely, those states for which a constant Z-independent Ell exists. We note that the idea of two different screening constants for each group of equivalent electrons, one for the nonrelativistic and the other for the relativistic interactions, is in harmony with the findings of Dalgarno and Stewart (1960a) and Cohen and Dalgarno (1961a) that better results are obtained from the screening approximation when different screening parameters are chosen for different operators. They generalized the above procedure to the case of any atomic property for which the first two terms in a Z - '-expansion are known ( L ) = AZ"

+ Bzn-'+ O(zn-2).

(37)

This may be replaced by a single screened expression of lowest order in Z-', thereby yielding an estimate of O(Zn-2) terms. The screening parameter

352

Holly Thomis Doyle

is chosen so that the first-order correction is zero, thus making the property stationary with respect to first-order variations in a:

( L S )= A(Z - a)”

(38)

a = - B/nA

(39)

with

yields approximate Z ” - 2terms of the form

In conclusion, we remember from the discussion of the derivation of the Breit interaction that the present relativistic theory is known to be incomplete at order EZO,or a2Z2since interactions such as those giving rise to the Lamb shift have been neglected. Thus we must be skeptical of an attempt to predict explicitly neglected quantities by an extrapolation of those included. Despite the last of sound theoretical justification, however, the method of elementary screening constants gives surprisingly good agreement with experiment in several cases, as discussed in Sections VI,B and C.

HI. Irreducible Tensor Expansions of the Electrostatic and Breit Interaction Operators It is convenient to evaluate the matrix of the perturbing operators in a representation based on eigenkets of angular momentum operators. This is because the algebraic techniques developed by Racah (1942a,b, 1943,1949) for eigenkets of angular momentum operators are available. In using Racah methods to compute matrix elements between two such kets, we must express the operator as a series of irreducible tensor operators referred to the same origin as the angular momentum operators. If the irreducible tensor operators are products of one-electron operators, then the whole matrix element can be written as a linear combination of products of radial integrals times reduced matrix elements of one-electron operators between one-electron states times recoupling coefficients. The recoupling coefficients can all be found using the Racah algebra. The expansion of the electrostatic interaction operator is r;; = r;’

1 pkpk(cosoI2) 1 (- ~ > ~ [ k ] ” ’ p ~ ( ~ , ~ ~ , k ) ~ , k

= r;’

k

353

RELATIVISTIC Z-DEPENDENT CORRECTIONS

where

+

if rl > r 2 , [ k ] = 2k 1, if rl < r 2 , p = r < / r , .

The (C,")i = (4n/[k])'/2 Ykq(6i, 4i) are constants times spherical harmonics, and ( C k D k ' ) QisK the tensor product of rank K of two tensors of rank k and k', respectively. The notation follows that of Edmonds (1957). We require an analogous expansion for B, that is, an expansion of the form B = m r 1 , r2)(TlkTzk)O,

where thef's are functions of the radial variables only and T: is an irreducible tensor of rank k with respect to rotations of both the spin and orbital spaces of the ith electron. This form of expansion is particularly well suited for calculations in j-j coupling. We will also find an expansion suited for calculations in L-S coupling: =c f k l ,

k l . kl', k 2 ' . K((s:1sk,2)K(o:1'022)

k ' K O 3

where the (S:LSi2)K is a tensor product of spin operators for both electrons and (O::O,ki)Kis a tensor product of orbital operators. Talmi (1953) derived the tensor expansion of the tensor force ( ( v 2 > 2 ( r 1 2 r 1 2 ) 2 ) 0 v(r12) r:2

for a general V(r12)in both L-S and j - j coupled form. The term B is partly of this form (with (r replaced by a) and the expansion we obtain here is a special case of Talmi's more general expression. The Breit interaction B1 between two electrons labeled 1 and 2, respectively, is [Eq. (911 B12 =

-(e2/2NGYa1

a2)

+ rT2Ya1 * r12Ma2 - r 1 2 ) ) .

Writing these in tensor notation, and recoupling, we have

(al - a 2 )= -JSo and

(a, Thus

-

r12)(a2 * r12) =

-(1/J3)(a,a,)Or:, t- J3((ala2)2(r12r12)2)0

354

Holly Thomis Doyle

Innes and Ufford (1958) give the expansion (written here in slightly altered notation).

+ ((k + 2)[k + 2])"'(1

- p z) Pk (C,kC";'2)z r>

1 r> k = O

1 1 2

-2(o

0 0) (C1kC2)2 ,

)

(43)

where

We substitute this result as well as the expansion of l / r l z , Eq. (41), into the Breit interaction, Eq. (42), obtaining

B12= - (ez/2)[ -(4/J3)(alaz)0r;1

(- l)k[k] llzpk(ClkCzk)o k

This form would be convenient if we were computing matrix elements between kets which were eigenfunctions of L2, L , , S2, and S, . But for calculations in j-j coupling, it is better to have an expansion in which the spin and orbital operators for each electron are multiplied together before they are multiplied by those of any other electron. This requires a recoupling of into ( ( C ( ~ C ~ C5')j)O. ~ ) ~ ( C First ( ~ consider the each product ((alaz)L(ClkC:')L)o L = 0 case. Then,

355

RELATIVISTIC &DEPENDENT CORRECTIONS

0

1 [k + 111/’ ( ( q ~ , ~’(a2 ) ~ cZk)’+ + +-a [kl”2 ‘)O.

Next consider L = 2, k

(45)

= k’:

(ii

+ [2]1/2[k + 1I1I2 k k+l

x ((a1C1 1

Finally, for L = 2, k‘ = k

k:

1) (46)

(a2 C2k)k+1)0.

+ 2, we find 1

((ala2)2(C,kCk=2)2)0= C [2]u]

((a, CCk)’(a, Ck,‘2)i)o

j

= [2]1’2[k

+ 1I1l2( k i 2

i

k+l

k+2 k+l 0

x ((a, C
)

.

(47)

We substitute Eqs. (45)-(47) into B12 as it appears in Eq. (44)and collect terms. The coefficient of ((alClk)k-1(a2C2k)k-1)o is

eL 2

= - - (1 - 6 k O ) ( - l ) k f 1 2 [ k ] ' i 2 p k / r > .

(49)

The coefficient of ((alC,k)k"(a2 C 2 k ) k + 1 )is0 1 [k

+ l]l/2

(-4)

+ [2]'"[k + 11112( k

( - UkCkl

k k

1/2 k

P /r>

l)Js[kl((!l

; ;)

_ - - e2 2 ( - 1 ) ~ + ' -

2 [k

+ 1]1/2( k + 2)pk/r,.

(50)

The last coefficient is that of ( ( a , C < k ) k + l ( aC, k + 2 ) k + 1which ) o is

-

7

[2]'/'[k

+ 1]1/2

(ki2

+

e2 --( - l ) k [ ( k l)(k

2

k+l

+ 23 + 2)31/2([k][k Ck 11 ) +

(1 - p 2 ) p k / r > .

(51)

357

RELATIVISTIC Z-DEPENDENT CORRECTIONS

Combining these four expressions, we obtain B12in the desired j-j coupled form :

+ [ kk++ 2

11112

+ 21)1'2P"(l - p2)((a, + (- ljk((k + l)(k + 2) CklCk Ck + 11

c,k)k t 1

This form of the expansion was used in the calculations reported in Section VI.

IV. Antisymmetrization This section reviews the derivation of an explicit form for the antisymmetrization operator that may be used to reduce the expression for a matrix element of the general symmetric sum of two-electron operators between antisymmetrized states. The antisymmetrization operator is well known (see, for example, Horie 1964; Layzer et al., 1964; Fano, 1965; and Shore, 1965). We follow Layzer ef al. (1964) in applying it to partially antisymmetrized wave functions and in the derivation of the corresponding normalization. The calculations reported here used the relativistic forms of the operators and wave functions which made it convenient to usej-j coupled antisymmetric kets. Both the perturbations V and B are symmetric two-electron scalar operators of the form

T=

1tij,

(53)

i<j

where k

tij

k O

(54)

= lfk(ri 9 rj)(?i Y j k

and y: is a single-electron irreducible tensor operator of rank k. For V, yik = C,: and for B, yik = (miC;')'l [Eqs. (41) and (52)]. The j-j coupled eigenfunction of Ho , Eq. (24),

I ) = I(.

* *

((al)"Jl(a2>"J2Y2 . .

f

(a,>Jr)~,>,

(55)

Holly Thomis Doyle

358

denotes a state in which q, equivalent electrons having quantum numbers a, (in our case a,,,= n, l,+jm)are coupled in an antisymmetric fashion to a total momentum of J,. Then J1 and J, are coupled to give angular momentum I , , and I, is coupled to J,,,+' to give angular momentum for m > 1. Such a state is already antisymmetric with respect to permutations of equivalent electrons. If Sqmis the symmetric group of permutations of the qm electrons of type m, then G = S,, x * . x Sqr, the direct product of r such groups, is the group of all permutations P such that 6 , P I ) = I ), where 6u is the sign of P.However, I ) is not antisymmetric with respect to permutations of inequivalent electrons. We wish to find the operator d which will complete the antisymmetrization. The total number of electrons is Q = 4,. The symmetric group S, of all permutations on Q objects contains G as a subgroup. If P is any permutation in S, then 6,P6,.P' has the same effect on I ) for any P' in G. That is, the left coset of G in S, which contains P has a unique effect on I ) (up to sign). The left cosets of G are a nonoverlapping set whose elements exhaust S,. Each coset has the same number of elements as every other. Hence there are q = Q ! / ( q l ! * * . q r ! ) of them. We choose a representative member from each one and apply

xm

where the sum is over representative members P belonging to R . Note that if P , P' are representatives of different cosets, then P I ) is orthogonal to P'I ), i.e., ( IP-'P'I ) = 0. This is because P-'P' either changes the quantum numbers of at least one electron or else it is a member of G. In the first case ( IP-'P'I ) = 0, and in the second case P and P' are in the same coset, since P-'P' E G implies there exists P in G such that P-'P' = P", or P' = PP" so that P and P' are in the same left coset. Next we show that 6,P( ) is antisymmetric. To see this we apply any P in S, to it. We have a, P' 6, PI ) = 16,6, P'PI ).

zpER 1

PER

PER

But the 6,. 6, P'P are another set of representatives of the cosets; hence their sum has the same effect on I ) as the original set. That is a, 6 , P'PI ) = 6 , PI ).

c

c

PER

PER

This shows that z p e R 6 , P is some constant multiple of d , the operator we are looking for. To determine the constant, consider 1 = ( Id+dl) = Iconst12q( Id1 ) = Iconst12q.

RELATIVISTIC 2-DEPENDENT CORRECTIONS

Thus, we may take the constant = 1/& = [(ql ! expression for the antisymmetrization operator is

d = [(ql! *.. qr!)/Q!]”’

* *

359

qr!)/Q!]’/2, and the final

C 6,P. PER

(57)

Note that the representatives P and the qm’s will in general be different for different states. Suppose we wish to evaluate the matrix element of an operator T that is a symmetric sum of two-electron operators ti,, Eq. (54), between two antisymmetric states IY) = d ( Y ) / (*** ((uI)Q’Jl(u,yu,)z,

-

*

(u,)?J,)z,)

(58)

and IY’) = d(Y’)I(** * ((u1)~~’51’(u2)~~’J2’)zz’ * * ( u r ) q y ; ) .

The matrix element is (Y ldt(Y)Td(Y’)lY’)

The operator T commutes with all permutations. Also Pt = P-’ is a permutation with the same sign as P. Hence

where the last sum is over P” = PtP‘, another set of representatives of the left cosets of G(Y’). Therefore the matrix element in Eq. (59) reduces to

Consider a single term in this sum p = (Y I ti,6,<

P I Y’).

Assume we have numbered the electrons so that as many electrons as possible match, where we mean that the ith electrons match if they have the same quantum numbers in 1”) as in IY’). Actually, numbers can be assigned only to groups of equivalent electrons, not to the electrons themselves; e.g., the first three electrons could be 2p1,,, the fourth and fifth could be 3s, etc. Then if more than two electrons are unmatched, the matrix element will vanish. If P’ = 1, only the ith andjth pairs can be unmatched in p if the result is not to be zero. Any other permutation in d must change the quantum numbers of at least two electrons. If it changes exactly two, the matrix element will vanish unless those two are the ith a n d j t h ; that is, they must be the pair

Holly Thomis Doyle

360

affected by the operator t i j . In this case P' belongs to the same coset as P i , , the transposition of the ith and jth electrons. If P' changes the quantum numbers of more than two electrons, p again vanishes. We have established that once the electrons have been put in matching order, the states 1") and 1"') must satisfy one of the following three conditions if the matrix element M , M = (YItij

c LjPPP'pP'),

(61)

P' E R'

is to be nonzero: (1) qm = q m ' ,for all m. In this case the two states are in the same configuration. Either: (a) The ith andjth electrons are equivalent, in which case the coset of P i j is not represented in d ( Y ' ) . In fact, every permutation in the sum (except the unit operator) changes the quantum numbers of at least two electrons (transposition cosets being the only ones that change only two), Hence

M = (Y I tij I Y').

(62)

Or: (b) The ith andjth electrons are inequivalent. Then P i j is in the sum and every other permutation in XI("') except 1 changes the quantum numbers of other electrons and M reduces to M

=

(Y I f i j ( l - P i j ) I Y').

(63)

Summing over these possibilities to obtain the matrix element of the whole operator T, Eq. (60), yields (Y Id+TXIIY ') =

i

(!f)(y"iljy')

m=l

i, i E mth group

where the coefficients

1

+

m
qmqn(yltijly'),

(64)

i E mth group j E nth group

(";.) and qmqnare the numbers of identical matrix

elements. If i, i refer to equivalent electrons and j , j refer to equivalent electrons, then (Y I t i j I Y') = (Y 1 t i j I Y')

(65)

since both 1") and IY') are antisymmetric under permutations interchanging i with i a n d j with j . The number of ways i, i can be chosen so that they are both in the mth group is

(";.I .

The number of ways i, j can be chosen so that

i is in the mth and j in the nth group is qmqn.

36 1

RELATIVISTIC Z-DEPENDENT CORRECTIONS

(2) A single electron is unmatched. Suppose this electron is in the mth group of equivalent electrons of lY) and the m’th of (Y’). Then (Y I t i j P 1”’) = 0 unless either the ith or theJth electron is in the mth group for IT). Suppose it is the ith. There are then three possibilities: (a) Thejth electron belongs to neither the mth nor the m‘th group. Then 1 and - P i j contribute and no other permutation does. We obtain

I

M = (Y I tij(l

- P i j ) Y’).

(66)

(b) The jth electron belongs to the mth group for both lY) and IY’). Then P i j is in d ( Y ’ ) ,but not in d ( Y ) .We have

M = (Y I tij(l - P i j ) I Y’)) = (Y I tij - Pijtij I Y’) = 2(Y

(67)

I tij 1 Y’)

since P i j commutes with tij and (Y I ( - P i j ) = (Y 1. (c) Thejth electron belongs to the m’th group for both P i j is not in d ( Y ’ )and

1”)

and IY’). Then

M = (Y I tij I Y’).

(68)

Summing over these possibilities reduces the matrix element of T to (Y I d+( Y) T d ( Y ’ )IY’ )

Ic .-

= (q/q’>1/2

q m q i i i ( Y l r i j ( 1 - Pij>lY‘>

j E mth gro,up iii#m,m

1

+ 2(y)
j

C

E mth

(4m4~,)1/2{qiii
group M#m,m‘

+ qm’<~l~ijlY’) + qm,(yltijl‘Y’>,

(69)

where i is the number of the unmatched electron and it belongs to the mth Note that if we had begun by moving group for IY) and the m‘th for 1”’) . at(”’) to the left instead of moving d t ( Y )to the right, the primed q’s and m’s would have been unprimed, and vice versa, leaving the result unchanged, as it should. (3) Two electrons cannot be matched. Let them belong to groups m, and m2 for IY), and to m,’ and m,’ for IT’). Then t i j contributes to the

Holly Thomis Doyle

362

matrix element only if i and j refer to the unmatched electrons. If this is the case, there are four possibilities: (a) The unmatched electrons are inequivalent for both 1”) and (Y’), i.e., m, # m, and m,‘ # m2‘. Then Pij is in both d ( Y ) and d ( Y ’ ) ,so that

M = (Y I t i j ( l - P i j ) I Y’). There are qm,qm2ways of choosing i, j in this case. The matrix element of T, from the above and Eq. (60), is (Y Id+( Y ) T d (Y ’)IY ’)

M = (qki*qkz,q m i qrnz)i’2<~Itij(1 - Pij)I‘Y‘>* =(q/q’)’/2qm,

qm2

(70)

(b) The ith and jth electrons are equivalent for I”), but not for IY’), i.e., m, = m2 and m,’# m,‘. Then P i j is in at’(”’), but - P i j I Y) = 1”). Since t i j commutes with P,, ,we have

M = (Y I t,(l - P i j ) I Y’) = 2(Y I t j j I Y’). In this case there are

p:I)

ways of choosing i, j so that the ith andjth

electrons are both in the m,th group of IY). Therefore, (Ylsdt(Y)Td(’u’)lY’)= (q/q’)”2 (q;i)M = (qk,, qk2*qm,(qrn,

- l))1’2<~ItijI~’)*

(71)

(c) m, # m2 but m,’= m,’. Then Pij is not in d ( Y ’ )so that

M = (Y I t i j I Y’). There are qmlqm2ways of choosing i and j . Thus (YIdt(WTdW’)IY’) = (q/4f)”24m1 qrnz(WI tijI‘Y’> = (qA,,(qk1* - 1)qml qm~)’/2<’YItijly’),(72)

which is symmetric with case (b), as it should be. (d) m, = m, and m,’= m,‘. Then Pi, is not in either d ( Y )or d ( Y ’ ) ,and

There are

el)

M = (Y I t i j I Y’).

ways of selecting i and j in this case.

RELATIVISTIC Z-DEPENDENT CORRECTIONS

363

To summarize the results of this section the problem of finding matrix elements of symmetric two-electron operators between antisymmetric states has been reduced to that of calculating direct and exchange elements of certain two-electron operators between states which are antisymmetric only among equivalent electrons.

V. Reduction of Matrix Elements to Matrix Elements between One-and Two-Electron States Both the electrostatic interaction operator V and the Breit operator B are symmetric sums of two-electron operators of the general form [see Eqs. (41) and (52)] k k’ k”

tij = f(ri 9 rj) (Yi Y j 1

9

(74)

where 7: is an irreducible tensor operator of rank k acting on the spin and orbital variables of the ith electron. Matrix elements of such operators can be reduced to linear combinations of matrix elements between two-electron kets and still further to linear combinations of products of radial integrals and matrix elements of single-electron operators between single-electron states. The development here is based on that of Innes and Ufford (1958) which has been further detailed by Kelly (1959), but is extended to include the case when the kets are not in the same configuration. Another approach to this problem that simplifies the handling of recoupling coefficients has been developed by Fano et al. (1963) and Fano (1965). The matrix elements to be evaluated are (Y I t i j 1’4”) and often the exchange element (Y I t i j P i jI Y‘), where r i j is given by Eq. (74) and (Y) and 1”’) are kets of the form

ly) = I(. .. ((al)41Jl(a2)4zJ2)~2 * * (arlqrJrVr), where a,,, specifies the quantum numbers n,, I,’, andj,,, . The notation is the have been matched same as in Eq. (24). Suppose the electrons of I”), 1”’) as well as possible (as described in Section V). Then we have assigned qn, electron labels to the mth group, for m = 1 . .. r. Since tij operates on the ith andjth electrons only, we first recouple the electrons in 1”) and IY’) so that the ith and jth are coupled to each other and then to the rest. If the ith andjth electrons are equivalent, both in the mth group, say, then

Holly Thomis Doyle

364

we may uncouple them from the group using two-electron fractional parentage coefficients,

The groups may then be recoupled so that the pair is to the right of the ( m + 1)th group as follows: 1'4') =

C ( U ~ ~ - ~ JU,~JI)J,)~U~,'J~ ,,,, ... Im-l(~4,m-2Jm, u,~J)J,Z,

I,)

J, Jm

A succession of recouplings, each moving am2Jmone group further to the right, results in

x

I(* * ((a!'J1 * * * ~ i ~ - * J , ,a4,",Jm+ , ) f ~ ,)I,,+ 9

* * *

1,amZJ)I,), (76)

where the Coef is the product of the fractional parentage coefficient, the 6-j symbols, and a phase factor. Similarly, if i a n d j are inequivalent, say they belong to the mth and m'th groups, we uncouple each of them and move them right, finally coupling them together as the last two electrons. This gives, assuming m c m',

RELATIVISTIC Z-DEPENDENT CORRECTIONS

365

where the Coef is a product of two fractional parentage coefficients (&- ‘J, a, I)J,),

(U‘Zm-

‘J,.

a,. I)J, ),

and 6-j symbols with a phase factor. Writing both 1”) and 1”)’ in this form, we can reduce the evaluation of the direct matrix element to a sum of matrix elements between two electron states. The reduced matrix element is

1

( Y ~ ~ t i j= ~~Y Coef*(Y) f) (@J,

- - - I r ( q aj)JIrIItijll

a’‘Z1‘~~’ 1 * * *

Z,‘(uifujf)J’Irf) Coef(Y’)

r

x

n

arprp,((ai

p=m

aj)JIItijII(ai’ai’>J’>.

Carrying out the sum over all quantum numbers except J and J’, we have

1F ( y , y’,i, j , J, J‘) ((aiaj)JIIfijII(ai’ai’)J’>.

. I J’ ,

(78)

Thus the many-electron problem reduces to a linear combination of twoelectron problems. The problem can be reduced still further by taking into account the explicit form of the operator, Eq. (74). It is a product of a radial function and two single-electron operators. Recall that ((alaz)JI and I(a’la’z)J’)are vector coupled products of single electron kets which may be written in split notation, as in Eq. (16):

Also tij

=f(ri

k

3

r j ) (Yi Y j

k K

1

*

Each y k in t i j is a 4 x 4 matrix operator on the angular and spin variables. In split notation

where each ( Y ‘ ) ~ .,s, t = &, is a 2 x 2 spin matrix.

Holly Thomis Doyle

366

For example, one piece of the Breit interaction, from Eq. (52), is t i j = (-ilk+ '2pk/r,

Here f(ri, rj) is (-

k+2

((aiCik)k+l(aj k )k + l )0 .

+ 1I1l2 l ) k + 1 2 ( p k / r , ) ( k+ 2 ) / [ k + 1]'12),

cj

[k

( Y : ) ~ - ~= (a,c:))k+', In this case, then

( Y ? ) ~=~( Y : ) ~ ~= 0, and ( Y : ) ~ - ~= (ajC;)k+l,for s = &. t i j = (-

1)'+'2(pk/r,)

k +2 0

Performing the matrix multiplication, we find, for the direct matrix element,

In the example where t i j is given by Eq.(74), only terms connecting a large component with a small component contribute, so that there are only four terms in the sum instead of sixteen:

RELATIVISTIC 2-DEPENDENT CORRECTIONS

367

In the particular example we have been considering, this is

x R ~ ~ ~ ( r l ) R ~ ~ ! ( ~dzr ,) dr, ~12~,z

(84) Note that in the last two equations J appears only in [ J ] and in the 9-j symbol and that both of these are independent of s, t . Thus substitution of Eq. (83) into Eq. (78) gives x

(UISII

(aCk)k+ Ila;-”(az‘ll(aCk)k+ Ila;-‘).

F(Y, Y’, i , j , J , J’)([J][J‘][K])1/2

x

2= f ~ R ~ ( r i ) R ~ ; ( fr(jr)i , r j ) R ~ ; , ( r i ) R ~ , , ( r j )dri r i 2drr~j

ss’tr’

) ‘ ; .1

(85) and the summation over J , J ‘ may be carried out for any operator of the . resulting coefficient depends on K , k, and k‘. form f( r i , r j ) ( ~ i k y i k ’ ) ~The This suggests that we may introduce formally a single-electron unit operator defined by x (aisII (Vi”),,,

Ila:s’>(ai’ll

tt*

( a IIUklla’> = 1,

(86)

which differs from the definition of the unit operator dr)of Racah (1942b) and the definition of U k k of Innes and Ufford (1958) in that it does not vanish for inequivalent electrons. Then the matrix elements of the tensor product of two such unit operators is just

(87) a sum of products of fractional parentage coefficients, 6- and 9-j symbols, and a phase factor. The matrix elements of t i j may then be written in the transparent form

Holly Thomis Doyle

368

('yll

tij

ll'y')

= ('Yll ( U , k q Y II'Y')

x

c

ss'rr'=

x

f

J R z ( r i ) R c ( r j ) f ( r ir,)R:;,(ri)Ri,,(rj)rizrjz , dr, dr,

(Q~SII(~:)ss,

IIay' > *

IIaY'>tr*

(88)

We now turn to the exchange matrix elements. We can proceed in the same way to the expression in Eq. (78) where instead of t i j we have t i j P i j as operator. Applying P , to the right-hand side and expanding, we have ((aiaj)JII tijpij Il(ai'aj'>J'> - ( -)ji'+J.j'-J'((u, Uj)JII

- (-

1;

t , j [l(Ui'Ui')J')

~>""~'""[J][J'][~])'~~

jj'

j;

:) J'

Thus the exchange matrix element reduces very similarly to the direct, except that it requires the exchange matrix elements of the product of unit operators. These could be evaluated by the same method as in the direct case; however it is convenient to express them in terms of the direct matrix elements. This method is similar to that of Racah (1942b). Substitution of the identity

RELATlVlSTIC 2-DEPENDENT CORRECTIONS

369

(93) Thus if the direct matrix elements (Y II(VikUi”’)’IIY’)are known, the exchange elements can be found as an appropriate linear combination of them. When K = 0, as it does for V and B, then k = k’ and I, = I, so that Eq. (93) reduces to (YII (UikU:)OPij II‘Y’)

VI. Results, Comparisons, and Conclusions A. DISCUSSION OF THE CALCULATIONS AND NUMERICAL RESULTS We have written a computer program that uses the formulas developed in Sections 11-V to compute energies of order Z2,€Z2,Z and EZ for atoms with up to ten electrons in the first two shells. With this program, we computed unscreened energy levels including first-order nonrelativistic and

Holly Thomis Doyle

370

TABLE I1 STATES WHOSEENERGIES WERECALCULATED"

Complex

I I J

jj-Coupled states

I

LS-Coupled states

Two electrons

(1 220) (1'2')

0

ls2 0

ls2 'So

0

ls2s 0

ls2s 'So

I

ls2s I

-

ls2p 3P0 ls2p 'PI Is2p 3P1

2s2 'So

2p2 3P0 2p2 'So

1

2p2 3P1

2

2p2 3P2 2p2 D 2

It

2s2p 3P0

(1022)"

2

2s2p 3P2 Three electrons

lS22P1/2t

ls22p 2P1/2

The boxes contain states grouped by complex (nr)pand total angular momentum J. Subrnatrices (H+V + B ) c , , , , ~were computed between all states in a box and diagonalized to give energies complete through El 1. (I

RELATIVISTIC 2-DEPENDENT CORRECTIONS

TABLE I1 (confinued) Complex

(1222)

( 1222)"

(1223)

1 1 J

j-coupled states

LS-Coupled states

371

372

Holly Thomis Doyle TABLE I1 (continued) Complex

IJI

jj-Coupled states

I

Six electrons

(1225)o

(1226)

(1326)o

LS-Coupled states

RELATIVISTIC 2-DEPENDENT CORRECTIONS

373

TABLE I1 (conrinued) Complex

I 1 J

I

jj-Coupled states

LS-Coupled states

Nine electrons (1227)

I

ItI

ls22s2(2P1/2(2pJ/2)4~)t

1-1

I

is22s22p5 2pIl2

1

I

Ten electrons

I

I

relativistic corrections for low-lying levels of atoms with two to ten electrons in the K and L shells. These energies are the eigenvalues of H = Ho+ V + B evaluated between degenerate unperturbed states, i.e., between states belonging to the same complex. This matrix, H(nf)p, breaks up into submatrices H(nf)pJ belonging to different values of the total angular momentum J, since matrix elements connecting states of different J vanish. The states for which we computed energies and eigenfunctions are listed, by complex, in Table 11. Although the j-j coupled form was used in the computations, eigenkets are closer to L-S than j-j coupled form for low 2, so after diagonalization we have denoted energies by their L-S labels. Experimentally states are labeled in L-S notation, and both L-S andj-j forms are listed in Table 11. The boxes in Table I1 contain sets of states degenerate in zero order belonging to the same complex and the same J, that is, states which are mixed by the diagonalization. have the form of power series in Z , Eqs. (29) and (31), Matrices H(n,)pJ

+ + h,,z +

(H(~, )~~>~/ a = ' hooz2 h,,a2z4

+ h,, a4z6+ + ... + h2, + -.,

hIla223

(95)

where the coefficients are constants, the first row is given by the Sommerfeld formula, hl is the nonrelativistic electrostatic interaction term, and hl is the relativistic part of V and B. The power series expansions of this type for the matrices of all two-electron complexes and most three-to-ten-electron complexes in the K and L shells are listed in Table 111. If only one state in a complex belongs to a given total angular momentum J , the entry in Table I11 is an expansion for the energy itself. For complexes in which several states belong to the same angular momentum J, Z cannot be factored out of the diagonalization process because matrix elements involve

,

w

4 P

TABLE 111

POWERSERIESEXPANSIONS OF THE MATRICES H(n,)PJ 1s’ 0 (1Sb)o (1Sb)l (1s2p1/2)0

.OZ’ -0.6252’ -0.6252’ -0.6252’

+0.6252 0.2318242 +0.1879292 +0.2257282

-1

+

-0.25a’Z4 -0.1640625a’Z4 -0.1640625a’Z4 -0.1640625a’Z4

+0.480140a’Z3

+0.1694682a’Z3

+0.0769352a’Z3 +0.219768a’Z3

Y

-0.6252’ 0.2371082 -0.1 640625a2Z4 +0.132621a2Z3

+

3-

2

-0.0160943Z

E’

+0.0257505a2Z3

eb ip“

-0.6252’ +0.248488Z -0.1328125a’Z4 +0.0532108a’Z3 (lS2p,/z)2

-0.6252’

+0.2257282

-0.1328125a’Z4

+0.0406387a’Z3

TABLE 111 (continued)

I

(2s’)o

(2P1/2)20

-0.252’ 0.1503912 -0.078125a’Z4 +0.0608867aZZ3

+

-0.02929692

(2P3/2)’ 0 -0.04143202

-0.00863 166a’Z3 -0.252’ +0.1816412 -0.078125a’Z4 0.W20556a’Z3

-0.0225343a’Z3

0.02485922

+0.00572270a’Z”

+

~

-0.252’ +0.1992192 -0.015625a’Z4 +0.0419483a2Z3 -0.252’ W P l /’P -0.252’ (b2Pwz)o (2P11~2p3/~)1 -0.252’

+

0.13281252 +0.13281252 0.1640625Z

+

-0.078125a’Z4 -0.046875a2Z4 -0.046875azZ4

+0.06573374a2Z” +0.03554706a22’ 0.06136694aZZ3

+

TABLE 111 (continued)

-0.252' 0.15234382 -0.0781 25a2Z4 +0.06280762a2Z3

+

-0.02762142

-0.00805105a2Z3

-0.252' +0.1718740Z -0.046875a2Z4 +0.05245237a2Z3

Y

P

E'

-0.252' +0.1781250Z -0.046875a2Z4 +0.04637502a2Z3

+0.009943692

+0.002687615a2Z3 -0.25Z' +0.17109402 -0.015625a2Z4 0.02228190a2Z3

+

TABLE 111 (continued) Three Electrons - 1.1252’ - 1.1252’

-1.1252’

+ 1.0228051672 + 1.09352612

+1.09352612

-0.2890625a2Z4 -0.2890625a2Z4 -0.2578125azZ4

+0.6802815a2Z3 +0.7889552a2Z3 +0.5708461 a2Z3

Four Electrons IS%*

lS’2S’ 0

0

- 1 .25Z2

+1.5710012 -0.328125a2Z4 +0.9413102a2Z3

-0.02929692 -0.00863166a2Z3 -

1 .25Z2 1.7436932 -0.328 1 25a2Z4 +1.1898263a2Z3

+

-0.041432032 -0.02253434a2Z3 +0.024859222

+0.005722700azZ3 - 1.1252’

+ 1.7612712

-0.265625a2Z4

+0.7035009a2Z3 1S2(2p&J~/z)1 - 1.252’

+1.726114782

-0.296875dZ4

+0.9410286a2Z3

W

u 00

TABLE III (continued) ls2(~1,22P3/2)2 - 1 .25Z2

+1.7401772 -0.296875a2Z4 +0.9260366a2Z3

+0.0099436892 +0.00268762a2Z3 -

1~~(2~2p,~,)O - 1.252’ - 1.252’ lS2(2S2p31z)2

+ 1.624143852 +1.624143852

-0.328125a2Z4 -0.296875a2Z4

1S2(2S2P1/2)1 1s2(2r2p1/2)1

1 .252’ +1.7331462 -0.265625a2Z4 +O.6838344a2Z3

+ 1.O548308a2Z3 +0.8O65350a2Z3

lsz(2S2P3/z)l

- 1.252’

+ 1.6436752 -0.328125a’Z4 + 1.051905a’Z3

-0.027621362

-0.00805105a’Z - 1 .252’

+ 1.6632062 -0.296875a’Z’ +O.82344O3a22”

TABLE III (continued)

1~'2.?2~~/2

-1.375Z2 2.3344492 -0.3671875a2Z4 1.377204a2Z3

+

+

-0.041432032 -O.02253434a2Z3 - 1.375Z2 +2.5755002 -0.3046875a2Z4 1.1 1 6310azZ3

1s2(2Pi/z(2P3/z)'o)

+

1~'((2~2p1/2)12p312)

-1.3752' 2.4231392 -0.3359375~~2~ 1.224533a2Z2

+

+

+0.023754372

+0.006713141a'Z" -1.3752' +2.4063422 -0.3046875a2Z4 +0.9592968a2Z3

w

00

0

TABLE 111 (continued)

1s22.T22p3 /2

- 1 .375z2

+2.3344492

-0.3359375a’Z‘ 1.1 1579O’Z’

+

-0.02929692

-0.029296872

0.0

+0.00863 1M2Z3

+0.01 59342u2Z3

- 1.3752’

+2.5579222 +O.O1111738Z -0.3359375~~2~ +0.00300484a2Z3 1 .384527u2Z3

+

t0.017578122 f0.004046560~~2~

- 1 .37sz2 -0.01 11 17382 +2.5368282 -0.3046875~~2~ -O.00300484a2Z3 1.107888u2Z3

+

- 1.37522

+2.5579222 -0.2734375a2Z4 +O.8289379u2Z3

TABLE III (continued)

- 1.375Z2 +2.4364202 -0.3671875a2Z4 1 .517046a2Z3

+

-0.03382912

-0.00986049a2Z3 - 1.3752’ +2.4481392 -0.3359375a’Z‘ 1.2563Oa2Z3

+

+0.02485922 +0.00572270a2Z3

+0.023920802 +0.00697242a2Z3 - 1 .375Z2 +2.4539982 -O.3046875a2Z4 +0.9874159a2Z3

1s2&2P1/2)0 2P3/2 -1.3752’ +2.4129822 -0.3359375a2Z4 1.239421a2Z3

+

+0.025719032 +0.000926317a’Z” - 1.3752’ +2.4387642 -0.3359375~’Z~ 1.244469a2Z3

+

+0.78611772

+0.002124746a2Z3 -0.027739872 -0.00821467~~2~

-1.375Z’

+2.4551702 -0.3046875~~2~ +0.9804MZ3

TABLE 111 (continued)

l ~ ~ 2 S ~ ( 2 p ~ ~ ~ 2p 1.5Zz ~~~)2 +3.2760222 -0.375a2Z4 1.598059a2Z3

+

+0.00268761 a2Z3 -1.5Z2 +3.2689912 -0.40625a2Z4 +1.312551azZ3

-0.01593419a2Z3

-0.029296872

0.0

0.00863166a2Z3

-

-1.5Z2 +3.5432452 -0.34375a2Z4 1.601509aZZ3

+

+O.OO9943689Z $0.00268761 a2Z3 -1.5Z2 $3.5502762 -0.3125a2Z4 +1.288122azZ3

RELATIVISTIC 2-DEPENDENT CORRECTIONS

383

384

8

Holly Thomis Doyle

TABLE III (conrinued) Seven Electrons

ls2~22P,/2(2P3/2)20 4 - 1 .625Z2

+4.406267Z -0.3828125a2Z4 + 1.872105a’Z’

-0.04143203Z -0.02253434a2Z3 ~~

~

~

-1.6252’ +4.713724Z -0.3u)3125azZ4 1.513127a2Z3

+

8 ls2b22p1~2(2p3,~)’2

- 1.625Z2

t4.385173222

-0.3828125a2Z4

+1.844943a’Z3

w

m v,

TABLE III (continued)

- 1.6252’ +4.3675952 -0.3828125a2Z4 1 .863683cr2Z3

+

-0.011 117382

+0.0

+O.O1111738Z

-0.00300484a2Z3 - 1 .625Z2 4.3886892 -0.3515625~~2~ 1.541428a2Z3

+

+

+0.00300484cr2Z3 -0.029296872

+0.00863166a2Z3 - 1 .625Z2 +4.713724Z -0.3515625a2Z4 1.850607a2Z3

+

+0.0175781Z +0.00404656a2Z3 -0.029296872 +0.01593419a223 - 1.625Z2 +4.3886882 -0.4140625a2Z4 +2.183627a2Z3

TABLE I11 (continued)

- 1.752’

+0.024859222

+5.6970582 -0.421 875a2Z4 2.504049a2Z3

+

-0.04143203Z

+0.00572270a’Z3

-0.02253434a’Z

- 1.752’

+5.6794802 -0.359375a2Z4 + 1.792586a’Z3

-0.029296872 - 0.00863166a2Z3

-1.752’ +6.0552972 - 0.359375a2Z4 +2.121986a2Z3 1S22S2(2p1/2(2p3/2)3 3)l

- 1.752’

1Sz2S’2p:/2(2p3/z)’ 2

+5.661902482

-0.3906250

- 1.75Z2 5.6689332 -0.421 875a2Z4 2.484382a2Z3

+

+

’4

+2.142683azZ3

+0.009943692 +0.00268761 a2Z3

- 1.752’ 5.6759652 -0.390625a2Z4 2.127691a2Z3

+

+

E

.i

w 00 m

TABLE 111 (continued) ~~

1s2((~2p1~2)O(2p3/2)4 0)O

- 1 .75Z2

+5.834185912

1s'((B2Pi

-0.359375a2Z4

+ 1 .946549u2Z3

(2P3/d4 011

12) 1

- 1.752' +5.853717Z -0.359375a2Z4 1 .943623u2Z3

+

-0.02762142 -0.00805 1

0 5'2' ~

- 1 .75Z2 +5.873248Z -O.39O625u'Z4 +2.314052U'Z3

lS2(2s2P:/z(2P3/z)3#)2

- 1.752'

+5.834185912

-0.390625~'Z~

+2.297147u'Z3

TABLE 111 (confinued)

Nine Electrons ls”2s2(2P1/2)’(2P3,2~3t t ls2~’(2P~/z(2P3/z)4O)

-1.8752’ - 1.8752’

1~’(2s1/~(2p~/~)’(2p3/2)*0)

- 1.8752’

+

7.134334872 +7.134334872 +7.337868302

+

-0.42968751~~2~ 2.817253a’Z” -0.3984375a2Z4 +2.436468a2Z3 -0.3984375a2Z4 +2.616752a2Z3

Ten Electrons

Is’2s”(2P1/2)’(2P3/2)4 0

-2.02’

+8.770829762

-O.4375a2Z4

+3.172405a’Z”

Holly Thomis Doyle

390

terms multiplied by several different powers of 2.The matrices in Table 111 must then be diagonalized separately for every Z. Hence, the resultant energy cannot be written as a simple power series in Z with constant coefficients. We have computed the matrix of Eq. (95) to terms of order a2Z3 for the low-lying levels of the complexes listed in Table I1 and its eigenvalues for 2 = N up to 25 or 30. The results are too lengthy to be included here.' Instead we present a sample of the results for four cases. Table IV contains the theoretical energies complete through Ell for the levels ls22F22p 'Pill and ls22p3 'PlI2 which mix with each other, for ls22s22p2P31zwhich mixes with three higher levels (whose energies are not included here), and one unmixed level ls22p3 'D,,,whose energy is a simple power series. TABLE IV TOTAL ENERGIES COMPLETE THROUGH El 1 FOR SEVERAL

5

-22.74046

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-35.54439 -51.10415 -69.42235 -90.50205 - 114.34679 - 140.96058 - 170.34787 -202.51361 -237.46321 -275.20254 -315.73794 -359.07623 -405.22469 -454.19106 - 505.98357 -560.61091 -618.08224 - 678.40720 -741.59589

-21.46556 -34.01360 -49.31656 -67.31662 - 88.19635 -111.77871 -138.12708 -167,24519 -199.13718 -233.80159 -271.26134 - 311.50373 -354.54048 -400.37766 -449.02176 -500.47964 -554.75857 -611.86618 -671.81050 -734.59996

BORON SEQUENCE

-22.74113 - 35.54520 -51.10489 -69.42261 -90.50125 - 114.34407 - 140.95479 - 170.33756 -202.49694 -237.43794 -275.1 6599 -315.68694 -359.00707 -405.13312 -454.07221 - 505.83194 - 560.42030 -617.84572 -678.1 1709 -741.24369

LEVELS

-21.60586 -34.18206 -49.51322 -67.60151 -88.44951 -112.06019 - 138.43688 - 167.58334 - 197.07746 -234.20246 -271.68454 -31 1.95525 - 355.02026 -400.88566 -449.55790 - 501.04386 -555.35076 -612.48625 - 672.45836 -735.27548

We have differenced the total energies for a number of cases to predict ionization potentials and line energies for comparison with experimental differences along isoelectronic sequences. To illustrate the contribution of successive higher-order terms these energy differences are calculated in three levels of approximation and compared with experiment (Moore, 1949, unless otherwise noted) in Figs. 1-13. The first approximation, AENR, is the nonTables like Table IV for all the levels calculated are available at Harvard College Observatory.

391

RELATIVISTIC Z-DEPENDENT CORRECTIONS

1

1

I

I

I

I

I

I

0.090

0

0.085

16~

A

0

A

A

0.080

2

0.075

I

I

D

0.07C He I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

9

10

FIG. 1 . The unscreened term difference (ls2p 3P0- Is2 'So) of the He sequence. 0 Exp - A E N I I0 , Exp - AE,.,, , A Exp - AE; 0 A same as 0 0 A but variationally screened; Exp - AE'"; vertical error bar shows the size of the indicated fraction of Exp.

+

392

Holly Thomis Doyle

I

1

1

1

l

a

0.065

a 0.060

0.055

a a

4

A 8

4

A

8

0.050

8

'P-

A

8

$9

0.045

's

I

2

He

I 8

I

I

I

I

I

I

I

I

3

4

5

6

7

8

9

10

FIG.2. The variationally screened term difference (ls2p 3P0 - Is* 'So) of the He seA same as 0 0 A but quence. 0 Exp - AENR, Exp - AElomm,A Exp - AE; 0 variationally screened; Exp - AE'*; vertical error bar shows the size of the indicated fraction of Exp.

n

+

393

RELATIVISTIC Z-DEPENDENT CORRECTIONS

0.00100

0.0005-

0

Y,

A

*

*

*

0

-0.0010-

-0.0015

A

!

-0.0005-

f

0 A

A

+l

A

-

A B

-

-0.0020

A

-0.0025 0

-0.0030-0.0035

B

9-3

P i

52-

-

5I

2

I

3

A He

I

4

I

5

I

6

I

I

1

I

7

8

9

10

394

Holly Thomis Doyle

1

I

1

I

1

1

I

0.030

0

0.020

0

0.010 0 0

A

d

0

A A

0 El

-0.01c

A

I

I

3

I

I

4

5

I

6

A

El

He 2

I' -4

I

I

I

7

8

9

I

I

1 0 1 1

I

I

1213

FIG.4. The unscreened term difference (ls2p lP1- Is2 'So) of the He sequence. 0 Exp - A&,, 0 J W - AEmmm, A Exp - AE; 0 A same as 0 0 A but variationally screened; Exp - AEas;vertical error bar shows the size of the indicated fraction of Exp.

+

395

RELATIVISTIC2-DEPENDENT CORRECTIONS

-0.100

-

-0.110

--

II

I I

I

I

I

D 0

-0.120

--

0

0

*lo-*

b B

--

-0.130

B

00 A

A'

AA

A

. -0.140-

B

-0.1 50 -0'150

' A A

T

-.

7

2S 2S

I4 6

k

3 4 5 5 6 7 8 9 10 11 I;

13 14 lk

FIG.5. The unscreened term difference (2p 2P1/2 - 2s 2Sl/z)of the Li sequence. 0 Exp - AENR,0Exp - AE,,mm,A Exp - AE; 0 A same as 0 0 A but variationally screened; Exp - AE"; vertical error bar shows the side of the indicated fraction of Exp.

+

396

Holly Thomis Doyle

-0.03C

I

I

-0.04C

-0.05C

2

a

-0.060

-0.07C

-0.08C

I

I

3 4 5

6

I

1

1

7 8 9 10 11 12 13 14 15

FIG. 6. The variationally screened term difference ( 2 p - 2s zS1/2) of the Li sequence. 0 Exp - A E N R ,0Exp - AE,.,, , A Exp - AE; 0 A same as 0 0 A but variationally screened; Exp - AE"; vertical error bar shows the size of the indicated fraction of Exp.

+

397

RELATIVISTIC Z-DEPENDENT CORRECTIONS

0.05-

0

0.04-

0

0.030

0.02-

0.01

-

-0.01

-

0 0

0

0

-0.02-

0

2

P I

-0.03-

0

%-

Li 0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

FIG. 7. The unscreened relativistic splitting (ls22p 2P3,2 - 2P1,2) of the Li sequence. 0 Exp - A E N ~0 , Exp - AE,,,, , A Exp - AE; 0 A same as 0 0 A but variationally screened; Exp - AE"; vertical error bar shows the size of the indicated fraction of Exp.

+

398

Holly Thomis Doyle

I

1

I

I

1

-

0.002

A

A

0.001

-

A

t lo-*

A

o.Ooo1-

-

0-

-0.0001

-&*, 1

+

. m u

A

f : *. 8 8

8

:mi

-

-0.001

2P'

FIG. 8. The variationally screened and simply screened relativistic splitting (ls22p 2P3,2- 2Pl,2) of the Li sequence. 0 Exp - AENm,0Exp - AE,,,, , A Exp - AE; 0 A same as 0 0 A but variationally screened; Exp - AEbs;+, Bell and Stewart (1964). vertical error bar shows the size of the indicated fraction of Exp.

+

RELATIVISTIC 2-DEPENDENT CORRECTIONS

0.010-

-

,

I

,

1

,

I

I

1

1

1

-

0.005

399

1

A

-

A A

ro A

-

A

A

A

A

00

"f

'r

t

0

tnl-

[rl

0

itti

-0.005

-

-

-0.010

-0.015

1

-

0

'P3Pr 'S-

4

I

5

Be

'

6

'

7

'

8

I

9

I

I

I

I

I

I

10

11

12

13

14

15

FIG.9. The unscreened, variationally screened, and simply screened relativistic splitting (1s22F2p3P2- 3P0)of the Be sequence. 0 Exp - AENn, Exp - AE,,,, , A Exp - AE; 0 A same as 0 0 A but variationally screened; Exp - AE"; vertical error bar

shows the size of the indicated fraction of Exp.

+

400

Holly Thomis Doyle

1

I

I

I

I

1

-0.030-

-

-0.040-

0

-

-

-0.050

0

-0.060 -

-

0

0

-0.070-

Q

-0.080-

8

2

8

'

' A

a/k10-2

I

-

0 0

-

0

-0.090-

0

-

-0.100 0

-0.1 10 -

-0.120

-

IS-,

I

6

I

7

0

c

D&

-

I

I

I

1

8

9

10

II

I

12

I

I

I

I

13

14

15

16

FIG.10. The unscreened term difference (2p' ' D l - 2p2 'Po)of the C sequence. 0 Exp A same as 0 0 A but variationally - AENR,0Exp - AE,,,,, A Exp - AE; 0 screened; Exp - A P ; vertical error bar shows the size of the indicated fraction of Exp.

+

401

RELATIVISTIC 2-DEPENDENT CORRECTIONS

I

A

I

1

A

2S-

2P=?= F

A

A 0

A A

a

-.

0.001

0: -0.001

.

A

A

.. ..:.. 8

A

A

4

I

R

I.[

I

-

FIG.11. The unscreened, variationally screened, and simply screened relativisticsplitting (ls22s22p52P1,2 - 2P3,z) of the F sequence. 0 Exp - AENR, 0Exp - AE,.,,, A Exp - AE; 0 A same as 0 0 A but variationally screened; + Exp AE"; vertical error bar shows the size of the indicated fraction of Exp.

Holly Thomis Doyle

402

0

0

FIG. 12. The unscreened term difference @2p6 'SI12 - 2p' 2P112) of the F sequence. 0 Exp - A&R, 0Exp - AE,,,, , A Exp - AE; 0 A same as 0 0 A but variationally screened; Exp - AEss; vertical error bar shows the size of the indicated fraction of Exp.

+

RELATIVISTICZ-DEPENDENT CORRECTIONS

403

FIG.13. The variationally screened term difference (2r2p6 *SI,Z - Zp5 zP,,z) of the F sequence. 0 Exp - AENR,0Exp - AE,,,, , A Exp - AE; 0 A same as 0 0 A but variationally screened; + Exp - AESa;vertical error bar shows the size of the indicated fraction of Exp.

404

Holly Thomis Doyle

relativistic approximation, that is, including contributions from h,,Z2 + h l o Z only. These results are identical with those of Layzer (1959) and Linderberg and Shull(l960). We note that they can be represented by a twoterm series with constant coefficients, €,,Z2 + E l , Z , even in cases requiring matrix diagonalization, since h,, for each complex is a multiple of the unit matrix and 2 can be factored out of h l o Z . Apart from relativistic terms, the difference between experimental and theoretical energies should approach constancy, i.e., E 2 , , as Z increases. The second approximation, AESomm,includes nonrelativistic and Sommerfeld terms. As can be seen clearly in the figures, the effect of these terms in most cases is to overcorrect for the discrepancy between the nonrelativistic predicted and observed energy differences. The third approximation AE includes the relativistic contribution of the electrostatic and Breit interactions to first order; i.e., it contains the firstorder unscreened energy complete through terms of order EZ= u 2 Z 3 .

Screened Calculations Two methods of improving the unscreened calculations by screening were suggested in Section I1,C. The first of these uses variational screening parameters in the radial wave functions. Insofar as screening the nucleus approximates the electronic electrostatic interactions, we expect the screened unperturbed Hamiltonian to be closer than the unscreened to the total Hamiltonian. Hence the unperturbed screened energies should be closer to the final energies and the matrix elements of the screened perturbing operator should be smaller in absolute value than the unscreened matrix elements were. Finally, we expect an improved energy prediction. The nonrelativistic screened energy is just N

ELR= -

C(Z - ~ , ( Z ) ) ~ / 2 n ~ , i=1

where si(Z) is the variationally determined screening parameter. The s(2) have a Z - l expansion of the form (Layzer, 1959) s(2) = so

+ Sl/Z +

* * *

.

(97)

Therefore, Eq. (96) includes an estimate of both €, and E 3 , . This estimate of E2, is incomplete, but comparison with experiment shows that the estimate of E30 can be useful for low 2. For states which are the only one in a complex belonging to a particular total angular momentum, the L-S andj-j coupled states are identical. Under these circumstances it was possible to insert the nonrelativistic variational

RELATIVISTIC Z-DEPENDENT CORRECTIONS

405

screening parameters into the program to obtain variationally screened energies and we computed screened energies for several such states. The cases for which we calculated variationally screened energies are the same cases for which the unscreened energies are expressed as a power series in Z-’ and E in Table 111. Unfortunately, no such power series exists for the screened energies. In the variational screening theory each orbital has its own screening parameter. Where Zr appears in wave functions of the unscreened theory ( Z - s,(Z))r appears in the screened wave functions. In generai (except for such cases as 1s’) unperturbed kets are products of orbitals involving different (2 - s,). Hence, Z cannot be scaled out of the radial integrals which arise in the course of evaluating V + B. Thus, V + B had to be evaluated separately for each 2 and the results cannot be represented as a power series in 2-’ with constant coefficients. To facilitate comparisons, we calculated energy differences for screened states in three levels of approximation, just as we did for the unscreened energy differences. The first, nonrelativistic, approximation is given by

AEiR = A E ~+)AEQ

(98)

with N

Eag) =

1 - (1/2ni2)(2 -

SJ’

i=l

(99)

and

which vanishes for the optimal set of variational screening parameters (si). The screening parameters were available to only four or five places (Layzer et al., 1967). The resulting values of €&(;), while small, were not quite zero, probably due to the lack of more places in si. We therefore included E#;) in ELR. The next approximation AEiomm, which includes the Sommerfeld terms as well as the nonrelativistic contributions, is given by N

AE”,,,, = AEhR

+ A 1 - (1/2ni2)(2 - s ~ ) ’ [ u , E+~ a , ~ ~ ’ ] , i=1

(101)

where E , = a2(Z - s,)’ and a,, a2 are the Sommerfeld expansion coefficients as in Eq. (22). Note that this approximation includes an estimate of AEll and AE21and higher-order terms. Finally, the first-order screened calculation, complete through El includes the relativistic contribution of V and B and the part of the nuclear interaction which is not included in the screened unperturbed Hamiltonian :

406

Holly Thomis Doyle

We illustrate the effect of variational screening in Figs. 2, 3 , 6 , 8,9, 11, and 13 where the difference between the experimental and theoretical screened energy differences is plotted. The second method of screening discussed in Section II,C was the method of elementary screening constants. We have used this method only for 2 , ,ls22s2p 3 ~ 0 , particularly simple cases, namely, 1s’ ‘s, 1s22p ’ P ~ 3,1 ~ and ls22s2 2P112,3 1 2 , all cases of relativistic splittings for which expansions with constant coefficients were available. For the relativistic splittings the set of screening parameters cri are the same for both states, and the nonrelativistic energy difference AEgR is zero. From Eqs. (33) and (35), the energy difference is

The two states under consideration differ only in that the outermost electron belongs to different IC, say u1 and ic2 . Then the energy difference in Eq. (103) reduces to

In all four relativistic splittings we calculated, the outermost electrons are a 2pIl2 and a 2p312,u1 = 1 and K~ = 2, so that

AEss = a2(Z -

(105)

where OR = AJq1.

For purposes of comparison Table V lists the four types of screening parameters discussed here and in Section V1,C.

B. COMPARISONS WITH EXPERIMENT For relativistic splittings the nonrelativistic energy differences are exactly zero ; hence the relativistic energy prsdiction can be compared directly with experiment and the error determined precisely. The predicted unscreened ionization potentials and term differences, however, do not include the large nonrelativistic constant contribution E2,, (or any higher-order terms). The

TABLE V COhPARISON OF THE DIFFERENT TYPES OF SCREENING PARAMETERS FOR THE

Z -N

+1

ls2p Variational Empirical

ls22p Variational Empirid"

6 7 8 9 10 11 12 13 14 15

0.91 11 0.8368 0.8021 0.7828 0.7706 0.7623 0.7562 0.7516 0.7479 0.7450 0.7426 0.7406 0.7389 0.7374 0.7361

1.9548 1.8738 1.8168 1.7791 1.7529 1.7339 1.7195 1.7082 1.6992 1.6918 1.6856 1.6803 1.6759 1.6720 1.6686

a3

0.7178

1.6176

1

2 3 4

5

Simple screening constants Edlen (1964).

ONR

0.90291

OR 1.43304

ON' 1.8741

2.020 1.937 1.892 1.856 1.844 1.822 1.811 -

2p312 - 2p1/2SPL.Il7ING IN FOURCONFIGURATIONS ls22r2p Variational Empirical 2.2423 2.1698 2.1211 2.0873 2.0627 2.0441 2.0296 2.0180 2.0085 2.0006 1.9939 1.9882 1.9832 1.9789 1.9751

2.303 2.191 2.152 2.119 2.101 2.086 -

2.065 -

1.9145

s

1.74486

ON' 2.4054

is22~2~5 Variational Empirical 3.8644 3.8204 3.7885 3.7640 3.7445 3.7285 3.7151 3.7037 3.6939 3.6853 3.6778 3.6711 3.6651 3.6597 3.6548

3.234 3.203 3.183 3.169 3.157 3.142 3.137 3.128 3.120 -

3.1 11 3.107 -

3.5575 OR 1.98637

dM 4.4600

OR

3.04628

408

Holly Thomis Doyle

screened predictions include an estimate of only part of E,, . Therefore, in roughly estimating the accuracy of the energy differences from an examination of the figures, we considered only departures from constancy (without making explicit estimates of E20 or &,). That is, we describe the accuracy as the ratio of the departure from constancy to the absolute value of theexperimental energy difference. In this spirit, therefore, we find that for screened and unscreened energy predictions of ionization potentials, the departure from constancy generally amounts to around one part in 10’ or lo4 of the absolute value of the ionization potential. For term differences the departure from constancy is about one part in 102-103,with the screened values usually showing some improvement over the unscreened. Layzer (1 959) differenced experimental term differences along an isoelectronic sequence in order to extract El, for comparison with his theoretical calculations. In certain cases, this procedure did not approach the constant nonrelativistic theoretical prediction with increasing Z . Without exception, these were cases of differences in which the terms belonged to different configurations. Layzer suggested that this was the effect of the different relativistic contributions to terms in different configurations. We found this suggestion confirmed by our work in the following sense. In Layzer’s cases of poor agreement (terms in different configurations), we found that the relativistic corrections significantly improved the energy predictions, i.e., improved constancy along the isoelectronic sequence. For term differences between terms in the same configuration, the relativistic corrections in many cases did very little if anything to improve constancy along the isoelectronic sequence. These remarks apply to term differences (screening doublets), of course, and not to relativistic splittings (regular doublets), for which both levels are in the same configuration, but for which the energy separation is entirely relativistic. The accuracy of the predictions of relativistic splittings is poorer than that of ionization potentials and term differences, and in rough proportion to the absolute values of these energy differences. This suggests that the calculations have a certain intrinsic inaccuracy which is approximately the same in energy units for all levels, and consequently introduces the largest uncertainty into the smallest quantities being predicted. The size of the difference between experimental and predicted relativistic splittings generally ranges from a few parts in 10 to a few parts in 100, and in this case we find a strong systematic difference between screened and unscreened predictions. In every case the screened prediction is superior to the unscreened, and the discrepancy averages almost a factor of ten smaller in the screened case. We note that all of the above comments applied to the best unscreened and

RELATIVISTIC 2-DEPENDENT CORRECTIONS

409

screened estimates, i.e., complete through Ell. We now draw some distinctions between the three levels of approximation : nonrelativistic, nonrelativistic plus Sommerfeld terms, and complete through Ell. We have seen how screening picks up higher-order nonrelativistic terms, such as part of Ez0 and E 3 0 , which generally improves agreement with experiment for very low 2. But for large Z , and for predictions of unobserved highly ionized atoms, the screening calculations add little or nothing to the nonrelativistic calculations. In fact the unscreened and screened nonrelativistic energies look equally poor and asymptotically parallel for increasing Z . However, for relativistic calculations screening methods can be useful in one way or another in all types of energy differences. This may seem to contradict our earlier comment that the accuracy of screened and unscreened results are essentially similar for ionization potentials. This latter is the case, but the value of screening for ionization potentials lies in the observation that the improvement made by including E l , is systematically less in the screened calculations than in the unscreened (see Figs. 5 and 6 for a particularly clear example). That is, E,,,, - Es is generally much smaller than E,,,, - E. Since E' and E are not much different for ionization potentials, we conclude that E,,,, sometimes contains a reasonable estimate of Ell, and for much less work. For the term differences screening again improves the nonrelativistic contribution, especially for low Z , thus improving the overall constancy in most cases. And, in spite of the nonrelativistic origin of the variational screening parameters, the addition of screening improves energy predictions for relativistic splittings in all cases. Generally, the screened E,",,, - Es is still much smaller than the unscreened E,,,, - E for both term differences and relativistic splittings so that there is still value in a screened Sommerfeld calculation, but now the screened Ell, which produces the exact Ell plus estimates of higher-order terms, is a great improvement over both the screened E,,,, and the unscreened total prediction E. In the preceding remarks, screening refers to variational screening. In commenting above that the screened Sommerfeld terms give ionization potentials which include some estimate of E l , for less work than a full calculation of El,, we assumed that the variational screening parameters are already available. With this in mind, we come to what may be the most surprising conclusion about the screening calculations, namely, that in every case where we applied the method of elementary screening constants this simple screening method led to results which were as good or better than all other methods. This is surprising when we notice that it is a result of using only the information in the unscreened relativistic calculations of Ell. For example, in the nineelectron case of the fine structure of the ls22s22p5 'P term, Fig. 11, the

410

Holly Thomis Doyle

simple screening method predicts the splitting at the highest observed point, Z = 20, within 0.0007 a.u., nearly fifty times smaller than the same information in the unscreened relativistic prediction, which is off by 0.033 a.u. The simple screening expression reproduces the El term exactly, and its improved energy prediction results from its apparently very reasonable estimate of E,, based simply on the ratio of Ell to Eol, Eq. (39). Not only is it easy to determine, once El is known, compared to a variational screening calculation, but it results in a single screening constant for all Z. This has a striking application to extrapolations along isoelectronic sequences discussed below. The relativistic splitting of the ls22p 'P term was also computed (through Z = 9) by Bell and Stewart (1964) who used variational wave functions of the open shell type. We plotted their results in Fig. 8. Although their wave functions were slightly more complicated than ours, since they use two different screening parameters for the two Is electrons, their results seem to be slightly less good than the variationally screened result. While the elementary screening approximation at Z = 9 is apparently no better than the variational for large Z (assuming Bell and Stewart's values continue much the way our variationally screened approximation does) it is as good as either of these more complicated calculations.

C. CONCLUDING REMARKS In conclusion, we make the following four observations : First, the calculations of Ell have been extended to include 2p1,, and 2p,,, electrons and are now available for all states with ten or fewer electrons in the first two shells. Second, the straight unscreened calculation of El leads, in virtually every case we have considered, to a clear improvement in the comparisons with experimental data. Out of forty-one energy difference predictions (Doyle, 1968) only three showed no improvement, and in many cases the deviation from constancy for the largest experimentally observed Z improved by more than an order of magnitude. These comments are equally applicable to the three types of energy differences-ionization potentials, term differences, and relativistic splittings-though the percentage accuracy achieved diminishes as the energy difference decreases. We conclude that knowledge of Ell has improved the theoretical predictions of atomic energy levels for a variety of cases in the first row of the periodic table and should be extended to larger numbers of electrons. Third, we suggest that when these calculations are extended, it may be worthwhile to apply one or more of the methods of screening, which has proven itself a useful tool in improving energy predictions beyond that possible with just the exactly known terms in the Z-' expansion. We note in this connection the remarkable accuracy of the method of elementary

41 1

RELATIVISTIC 2-DEPENDENT CORRECTIONS

screening constants. This screening method contains only the information from the unscreened first-order calculation of El, and Ell, yet in every case it produced energy predictions which are as accurate or more accurate than all other methods including variational screening calculations of El itself. Fourth, and finally, we pointed out in the introduction that the knowledge of El, allowed a determination of the value of the screening parameter in the Sommerfeld formula for the relativistic splitting for infinite Z. The extension of exact Ell values to the 2p orbitals enabled us to determine that screening parameter for some extremely well-studied and important splittings in the Li, Be, and F isoelectronic sequences. Edlen (1964) has summarized the experimental study of these 2p splittings along isoelectronic sequences in a graph showing the variation of azp (which is determined from the “ spin-orbit ” integral cZp) plotted against N / ( Z - a). On the basis of smooth fits through these points, he extrapolated values for these splittings, and the corresponding screening parameters, that had not been observed in the laboratory. We have plotted the screening parameters for infinite Z, determined from our values of El 1, on Edlen’s graph at N / ( Z - a) = 0. The result is shown in Fig. 14, where we see that these points agree very well with smooth extrapolations of the corresponding experimental values.

1.8 -

2.0’

c L i I

GI 2.4 -

2.2

-.=N

tj,

2.6 2.8

-

3.0

-

3.2

-

3.4 I 2.4

FI

*

z=w I

2.2

I

2.0

I

1.8

I

1.6

I

1.4

4

I

1

1.2 1.0 N/(Z-u)

1

0.0

I

0.6

I

0.4

1

0.2

I

I

0.0

FIG.14. Screening Parameters for the “spin-orbit’’ interaction of 2p electrons along isoelectronic sequences. N is the total number of electrons. The points at Z = a, are theoretical. All others are experimental.

412

Holly Thomis Doyle

As we mentioned earlier, the above result changes the classic extrapolation problem for highly ionized unobserved ions in these cases to a problem of interpolation along isoelectronic sequences.

ACKNOWLEDGMENTS We wish to thank Professor David Layzer for his continuing interest in this work and Professor A. Dalgamo for reading the manuscript and making several helpful suggestions. This work was supported in part by a grant from the National Science Foundation.

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This Page Intentionally Left Blank

Author Index Numbers in italics refer to the pages on which the complete references are listed.

A Abarenkov, I. V., 271,294 Abe, A., 170,197 Aberth, W., 83, 85,86, 87, 107 Ablow, C. M., 58,60,107 Abragam, A., 219,220,225,234 Ackermann, H., 152,153 Adembri, G.,185,200 Albritton, D. L.,3,42,46,55, 56 Alder, B. J., 160, 198 Ali, M. A., 276,277, 288 Allen, C.W., 239, 245,269,279,288 Aller, L.H., 269, 286,289,291 Allin, E. J., 188, 191, 192, 194, 195, 197, 198,200 Altick, P. L., 278, 288 Altman, E. L., 242,288 Amdur, I., 182, 197 Amme, R.C.,65, 108 Anderson, A., 156, 164, 165, 166, 168, 169, 186,187,197,200 Anderson, E. M., 271, 272,286,288 Anderson, P. W., 202, 212,234 Andrew, K. L., 239,290 Aranoff, S., 280,288 Armstrong, B. H., 269,272,288 Armstrong, L.,Jr., 298,334,350,412 Arnold, G.M., 169, 197 Atwood, M. R.,216,235 Auffray, J. P., 266,288

B

Bacher, R. F., 254,288,298,308,334 Baglin, F. G.,174, 197 Bagus, P. S., 251, 253, 259, 260, 262, 267, 288,294 Bahcall, J., 337, 338, 340,348,413 Bahcall, J. N., 350,412 Bandzaitis, A. A., 265,292 Bangudu, E. A., 275,288 Baranger, M., 202,235 Barnes, A. H.,58,107 Barrat, J.-P., 242, 288 Barrett, C.S., 187, 197

415

Bashkin, S., 240,242,288 Bass, A. M.,, 4,55 Bates, D.R., 81, 82,86, 107,247,249, 251, 268, 269, 272, 283, 284, 285, 286, 287, 288,293 Bazley, N. W., 280,288 Beaty, E. C.,8,55 Bell, K. L.,279,288, 341, 410,412 Bell, R.J., 279, 288 Belyaev, V. A., 68,70,105,107 Ben-Reuven,A.,203,213,214,215,223,235 Bergmark, T.,259,295 Berrondo, M., 259,288 Berry, H. W., 65,107 Berry, R.S., 87, 88,89, 107, 108 Bersuker, I. B., 262, 286,288 Bertie, J. E., 169, 173,197,200 Bessis, G.,267, 288 Bethe, H. A., 247, 248, 213, 275, 288, 345, 350,412 Bezzerides, B., 202,235 Bhatnager, S. S., 188, 191, 197 Bickel, W. S.,242,288, 289 Bieniewski, T.M.,241, 289 Biermann, L., 260,286,287,289,295 Birnbaum, G., 212,214,235 Blanc, J., 182, 197 Bleaney, B., 212,235 Bloom, A. L., 134,153 Bockasten, K., 339.412 Bogaard, M. P.,242,289 Bohme, D.K.,48,49,50,55,56 Bolotin, A. B., 265, 285, 289 Born, M., 156,197 Borowitz, S., 275, 289 Boutin, H., 170,200 Boyd, A. H., 81,82, 107 Boyle, L. L.,242,289 Bozman, W. R.,241,290 Brackmann, R.T.,70,71, 107 Branscomb, L. M., 43, 55 BratoZ, S., 267,288 Braun, P. A., 280,289 Brecher, C., 172, 173, 182, 197

416

AUTHOR INDEX

Bree, A., 172, 185,197 Breene, R. G., 202,223,235 Brehm, B., 286,289 Breit, G., 344, 412 Brezhnev, B. G., 68,70, 105,107 Bridge, N. J., 223, 235 Brigman, G. H., 260,265,267,289,295 Brit, M., 164, 166, 167, 172, 173, 175, 197 Broida, H. P., 2,4, 5,55,56 Brot, C., 174, 197 Brouillard, F., 80,81, 82, 83, 107 Brown, R. G., 183,197 Brown, R. T., 266,289 Brown, W. A., 242,289 Browne, J. C., 52,55 Brunel, L.-C., 186, 197 Buckingham, A. D., 223,235,241,242,289 Budick, B., 242,289 Bunge, C. F., 267,289 Burch, D. S., 249, 295 Burgess, A., 270,289 Burke, P. G., 297,334 Burke, B. F., 147,153 Burt, J. A., 3,42, 52, 54, 55,56 Bush, S. F., 174, 197 Buska, 2.A., 271,288 Butler, J. P., 173, 198 Buymistrov, Y. M., 280, 289 Byard, P. L., 242,289 Byers Brown, W., 274,275, 292 Byme, J., 257, 295 Byron, F. W., 278,289 C Cahill, J. E., 164, 165, 166, 167, 169, 197 Cairns, B. R., 187, 197 Califano, S., 185, 186, 199, 200 Carlston, C. E., 62, 107 Chaika, M. P., 242,288 Chandler, C. D., 248,291 Chandrasekhar, S . , 249,268,289 Chantry, G. W., 174,197 Chapados, C., 169, 198 Chapman, R. D., 286,289 Chashchina, G. I., 241,289 Chen, J. C. Y.,247,250,280,284,289 Chen, S. H., 163,197 Cheutin, A., 161, 197 Chisholm, C. D. H., 272, 274, 289, 308, 309,334,338,412

Chong, D. P., 247,284,289 Chraplyvy, Z. V., 350,412 Christensen, D. H., 185, 198 Chupka, W. A., 76,107 Church, D., 120, 122,124, 153 Clarke, W. H., 286,289 Claverie, P., 278, 289 Clementi, E., 274, 289 Clouter, M., 196,198 Coates, P. B., 242, 243, 289 Cochran, W., 160, 164,198 Cohen, M., 250, 262, 263, 274, 275, 276, 278, 279, 283, 284, 285, 289, 340, 351, 412 Cojan, J.-L., 242,288 Colombo, L., 172, 174, 198 Commins, E. D., 139,154 Compton, D. M. J., 78,108 Condon, E. U., 238,239,243,254,270,271, 275,290,299, 301, 307, 334 Cook, C. F., 186,198 Cook, C. J., 58,60,83,85, 107 Corcoran, W. H., 241,293 Corliss. C. H., 241, 290 Corsiglia, L., 262,295 Cotrait, M., 185, 200 Coulson, C. A., 160, 183, 198, 242, 259, 262,273,289,290,295 Couture-Mathieu, L., 156, 161, 198 Cowan, R. D., 239,259,290 Cowley, C. R., 241,296 Craig, D. P., 185, 198 Cramer, W. H., 67, 107 Crane, A., 196, 198 Crawford, B. L., 187, I98 Cremer, J., 156, 161, 198 Crossley, R. J. S . , 239, 273, 276, 277, 287, 288,289 Cruikshank, D. W. J., 156, 198 Cunningham, P. J., 240,242,290 Czyak, S. J., 241,289 D Dalgarno, A., 52, 55, 70, 71, 107, 147, 153, 240, 250, 251, 262, 274, 275, 276, 277, 278, 279, 280,284, 285, 287, 288, 289, 290, 308, 309, 334, 338, 340, 341, 351, 412

Damgaard, A,, 268,272,283,284,285,286, 287,288

AUTHOR INDEX

417

Erastov, E. M.,68,70, 105, 107 David, J. C., 175, 198 Erickson, G. W., 339, 340,412 Davis, B., 58, 107 Erickson, R. A. 167, 196, 197, 199 Davison, W. D., 279,290 Eriksson, K. B. S., 412 Dawson, P. H., 124, 153 Ermolaev, A. M., 257,290 Dean, G. D., 173,198 deBoer, J., 160, 170, 181, 182, 183, 184,198 Etters, R. O., 167, 197, 200 Debye, P., 214, 235 Evenson, K. M., 5 , 56 Dehmelt, H. G., 111, 115, 116, 118, 120, Ewing, G. E., 179,180,198 122, 123, 124, 127, 128, 129, 130, 133, F 135, 136, 137, 138, 139, 142, 144, 145, 148,149,153,154 Fahlman, A., 259,295 Failkovskaya, 0. V., 185, 199 Delfosse, J. M., 80, 83, 107 Fano, U., 202,229,235,297,334, 357, 363, Delves, L. M., 280,290 412 Demtroder, W., 286,289 Farragher, A. L., 53, 54,56 Depatie, D. A., 196, 200 Dettmar, K.-H., 263, 264, 283,295 Fawcett, B. C., 259,290 Deuel, R. W., 241,290 Fehsenfeld, F. C., 2, 3, 5 , 6, 38,40, 41, 42, 43,46,47,48,49,50,52,55,56 deWette, F. W., 168, 198 Dicke, R. H., 149,152,154,230,235 Ferguson, E. E., 2, 3, 6, 38, 40, 41, 42, 43, 46,47,48,49, 50, 52, 55,55, 56 Dickermann, P. J., 241, 290 Di Giacomo, A., 228,233, 235 Field, G. B., 131, 154 Diner, S.,278, 289 Fink, W. H., 269,288 Firsov, 0. B., 67, 107 Dmitriev, Y.Y., 280, 290 Fischer, E., 125, 142,154 Douglas, A. S.,272,286,290 Dow, J. D., 259,290 Fite, W. L., 53, 54, 56, 70, 71, 76, 106, 107 Dows, D. A,, 156, 160, 165, 172, 173, 176, Flory, P. J., 170,197 181, 182, 183, 184, 185, 186, 187, 198, Fock, V., 258,262,290 199,200 Foldy, L. L., 269,290, 350,412 Fontana, P. R., 266, 289 Doyle, H. T., 410,412 Dugan, C. H., 239,290 Fortson,E. N., 111,115,116,118,130, 133, Dumas, G., 186, 199 135, 136, 137, 138, 139, 149, 154 Dunkin, D. B., 2, 47, 48, 49, 50, 55, 56 Foster, E. W., 238,240,290 Durig, J. R., 174, 197 Fouracre, R. A., 50, 56 Dushman, S., 18,24,56 Fox, D., 179, 180,198 Dvorak, V., 163, 197 Fox, D. W., 280,288 Dymond, J. H., 160,198 Francis, W. E., 67, 108, 125, 126, 154 Frankowski, K., 257,290 Fredkin, D. R., 168, 198 E Freeman, A. J., 258, 260, 291 Eckerle, K. L., 242,294 French, J. B., 298,334 Edltn, B., 237, 274, 290,407,411,412 Frenzel, A. O., 173, 198 Edmiston, C., 266, 290 F r e u d , I., 182, 198 Edmonds, A. R.. 301,334, 353,412 Friedman, L., 75, 108 Ehrenson, S.,248, 249,290 Friedmann, H., 223, 235 Ekin, J. W., 2, 56 Friedrich, H. B., 175, 198 Elias, L., 42, 56 Frisch, S. E., 243,291 Elliott, J. P., 297, 334 Frish, M.S., 241, 291 Entemann, E. A., 76, 100, 105, 108 Froman, A., 275,291 Epstein; S. T., 274, 275, 276, 280,290,292, Frwse, C., 259,260,263,274,276,277,283, 294,295 284,285,287,291,341,350,405,413

418

AUTHOR INDEX

Froese Fischer, C., 260, 287, 291 Fruhling, A., 156, 170, 171, 198 G Galatry, L., 220, 235 Gallagher, A., 242, 286, 291 Garstang, R. H., 239, 240, 252, 254, 270, 272,278, 286, 287,290,291, 293 Carton, W. R. S., 242, 291 Gaydon, A. G., 242,243,289 Gebbie, H. A,, 156, 168. 174, 197 Gee, A. R., 184, 185, 198 Geltman, S., 249, 291 Genzel, L., 173, 198 Gerding, H., 186, 198 Ghosh, S. N., 67, 107 Gielisse, P. J., 156, 176, 199 Giese, C. F., 74, 75, 107 Giguere, P. A., 169, 198 Gilmore, F., 100, 107 Gioumousis, G., 74, 75, 107, 108 Glassgold, A. E., 278, 288 Glennon, B. M., 238,240,243,259,285,291, 296 Godfredsen, E. A., 255, 274, 291, 340, 341, 350,405,412,413 Goldan, P. D., 3, 6.56 Goldberg, L., 269, 291, 298, 334 Goldenberg, H. M., 149, 154 Goldschmidt, Z., 363, 412 Goodisman, J., 267,291 Gordon,R. G.,203,211,214,215,216,235, 280,291 Goscinski, O., 259,288 Goudsrnit, S., 298,308,334 Graeff, G., 140, 141, 142, 149, 154 Green, L. C., 246, 248, 250, 252, 265, 266, 279,283,291 Griem, H. R., 240, 241, 243, 269, 285, 292 Grinberg, R. O., 271, 288 Grindlay, J., 163, 198 Grossgut, P., 262, 295 Gruzdev, P. F., 271,292 Giinthard, Hs. H., 179,200 Gush, H. P., 190, 192, 194, 195, 196, 198, 200

H Hagen, G., 101, 103, I08 Haigh, C. W., 160, 183, 198

Halford, R. S., 172, 173, 176, 182, 183, 187, 197, 198,200 Hall, G. G., 275,291, 292 Hall, L. C., 186, 198 Hallin R., 339,412,413 Ham, F. S., 269,270,292 Hamrin, K., 259, 295 Hansen, A. E., 254, 255, 285,292 Hanson, D. M., 185, 198 Harada, I., 162, 163, 170, 171, 174, 183, 184,185,187,198,200 Hardy, W. N., 167, 176, 196, 199 Hare, W. F. J., 192, 194, 195, 198 Harris, F. E., 267, 294 Harris, P. M., 167, 196, 197, 199 Hartree, D. R., 257,258,259,261,262, 263, 267, 268,269,271,292 Hartree, W., 259, 292 Hasted, J. B., 67, I08 Haugsjaa, P. O., 65, 108 Haun, R. D., Jr., 150, 154 Haurwitz, E., 264, 285, 294 Heastie, R., 169, 197 Hedman, J., 259,295 Heitler, O., 204, 235 Heitler, W., 244, 292 Helliwell, T. M., 272, 292 Henglein, A., 74, 108 Henry, R. W. J., 297, 334 Herman, F., 259, 260, 292 Herman, R. M., 216,235 Heroux, L., 242, 292 Hexter, R. M., 179, 180, 198, 199 Hey, P., 287, 292 Higuchi, S., 185. 199 Hill, S. J., 254, 286, 291 Hinnov, E., 243, 292 Hinze, J., 259, 260, 292 Hirschfelder, J. O., 274, 275, 280, 292, 294, 295 Hirshfeld, M. A,, 223, 235 Hollenberg, J. L., 184, 199 Holonyak, N. H., 112, 154 Holtsmark, J., 203, 228, 229, 231, 235 Homma, S., 167, 197, 199 Hope, J., 297, 334 HorAk, Z., 274, 293, 357, 413 Horie, H., 297, 334, 357, 413 Hornig, D. F., 169, 176, 180, 186, 188, 199, 200

419

AUTHOR INDEX

Houziaux, L., 269, 292 Howard, R., 163, 198 Huang, K., 156, 197 Hubenet, H., 243,292 Huber, M., 242,292 Huggett, G. R., 120, 154 Hughes, T. P., 412 Hulpke, E., 286, 292 Hummer, D. G., 70, 71, 107 Hylleraas, E. A., 265, 266, 268, 273, 292 I Innes, F. R., 274, 289, 297, 299, 303, 306, 334, 363, 367, 413 Ionescu-Pallas, N., 67, 108 Ionescu-Pallas, N. J., 270, 272, 292 Isberg, 1-1. B. S., 339, 412 Ishii, H., 68, 108 Ito, M., 162, 164, 165, 170, 171, 172, 173, 174,184,185,199 Itoh, T., 350, 413 Ivanova, A. N., 286,292 Ivanova, A. V., 246,286,292

J Jackiw, R., 279, 292 Jacobson, H. C., 223, 235 Jaffe, J. H., 223, 235 Jahn, H. A,, 297,299,334 James, H. M., 197, 200 Jarvis, S., 34, 56 Jefferts, K. B., 124, 142, 143, 144, 145, 146, 147, 154 Jennings, P., 280, 283, 292 Jernigan, R. L., 170, 197 Joachain, C. J., 278, 289 Jorgens, K., 263, 264, 283, 295 Johansson, G., 259, 295 Johansson, I., 339, 413 Johansson, L., 339, 4 / 3 Johnson, N. C., 265,266,283,29/ Jones, L. L., 275, 292 Jouve, P., 185, 199 Jucys, A. P., 2.58, 260, 263, 265, 285, 289, 292,296 Judd, B. R., 291, 306,307,334

K Kaminskas, V. A., 260, 292 Kaminsky, M., 78, 108

Kamuntavichyus, G. P., 299, 334 Kantserevichyus, A. I., 263, 296 Karaziya, R. I., 297, 334 Karl, G., 187, 188, 189, 191, 192, 196, 200 Karlsson, S.-E., 259, 295 Karstensen, F., 242, 286, 292 Kartha, V. B., 186, 199 Kasner, W. H., 103, 108 Kastler, A., 156, 162, 170, 199 Kaufman, F., 2, 56, 243, 294 Kaveckis, V. J., 260,292 Kelly, H. P., 278, 293 Kelly, P. S., 259, 262, 263, 283, 284, 289, 293,294,363,413 Keyser, L. F., 243, 294 Kieffer, L. J., 279, 293 King, R. B., 241,293 Kingston, A. E., 251, 279, 288, 290 Kirk, W., 242,296 Kiss, Z. J., 189, 190, 199 Kitaigorodskii, A., 160, 161, 172, 181, 184, I99 Kittelberger, J. S., 169, 186, 199 Kleinman, D. A., 167. 199 Klempt, 140,154 Kleppner, D., 138, 149, 150, 154 Klose, J. Z., 242, 293 Klucharyov, A. N., 243,291 Klump, K. N., 167, 176, 196, 199 Knight, R. E., 274, 294 Knox, R. S.,259, 290 Kobzev, G. A., 279, 293 Kohl, J. L., 242, 293 Kolchin, E. K., 265, 266, 283, 291 Kopelman, R., 179, 183, 199 Korshunov, A. V., 174, 199 Koster, G. F., 157, 199, 297, 309, 334 Kotchoubey, A., 266,293 Kozlov, M. G., 241, 293 Kraulinya, E. K., 243, 293 Krauss, M., 266,290 Krawitz, E., 252, 279, 29/ Krikorian, E., 182, 197 Krimm, S., 173, 183,199, 200 Krueger, T. K., 241, 289 Kuan,T., 159, 163, 166, 167, 176, 199 Kushnir, R. M., 67, I08 Kydd, R. A., 172, 185, 197 Kyogoku, Y.,185, 199

-.

420

AUTHOR INDEX

L Lafferty, J. M., 18, 24, 56 Landau, L. D., 110,154 Langrnuir, R. V., 125, 154 LaPaglia, S. R., 246, 255, 278, 285, 293 Larson, A. C., 259, 290 Larsson, S., 257, 259, 293 Lassier, B., 174, 197 Latter, R., 271, 293 Laufer, J. C., 164, 166, 197 Lawrence, G. M., 241, 242, 271, 287, 293, 294 Layzer, D., 239,240,252,254,260,267,270, 273, 274, 278, 294, 337, 338, 340, 341, 342, 348, 350, 351, 357, 404,405, 408, 413 LeBrumant, J., 186,299 Lecluse, Y., 242, 288 Leech, J. W., 169,199 Lefebvre, R., 260,278, 293 Leroi, G. E., 164, 165, 166, 168, 197 LeRoy, A., 185, 199 LeRoy, M. A., 186,199 Levinson, I. B., 239,240,293,297,298,299, 300, 301, 31 1,334,335 Lewis, J. T., 86, 107, 251,290 Lewis, M. N., 250,274,291 293,357,413 Lezdin, A. E., 243, 293 Liang, C. Y., 183,199 Liberman, D., 259,290 Lifshitz, E. M., 110, 154 Lifson, H., 341, 413 Lindberg, B., 259, 295 Linderberg, J., 273, 293, 338, 340, 404, 413 Lindgren, I., 259,295 Lindholm, E., 211, 235 Link, J. K., 240,241,242, 290, 293 Liu, B., 260, 288 Lowdin, P.-O., 258, 261, 274, 293 Loisel, J., 185, 186, 199 Longrnire, M. S., 182, 197 Lorents, D. C., 83, 85, 107 Lorentz, H. A., 212, 235 Lorenzelli, V., 186, 199 Loubser, J. H. N., 212,235 Lubeck, K., 286, 287,289 Lunelli, B., 186, 200 Lurio, A., 242, 293 Lvov, B. V., 241,293

M McClellan, A. L., 176,200 McClure, D. S., 179, 199 McClure, G. W., 70, 71, 108 McDaniel, E. W., 23,28,56 Macdonald, J. K. L., 266,293 McIllwain, J. F., 65, 108 Mackie, J. C., 89, 107 McKnight, R. V., 173, 199 MacNair, D., 6, 56 Magnuson, G. D., 62,107 Maier, 11, W. B., 74, 75, 107 Maillard, D., 186, 199 Major, F. G., 116, 124, 127, 128, 129, 136, 138, 140, 141, 142, 149, 154 Maki, A. G., 180,199 Malli, G. L., 259, 293 Malone, B. S., 241, 293 Malrieu, J. P., 278, 289 Mann, J. B., 259,290 Mapleton, R. A., 297, 334 Margenau, H., 223,235 Marino, L. M., 58, 108 Marlow, W. C., 241, 293 Marsault, J. P., 186, 199 Martin, D. H., 156, 173, 198, 199 Martin, J. B., 278, 296 Maryott, A. A., 212, 235 Marzocchi, M. P., 186, 199 Mason, E. A., 182,197 Mason, R., 185, 198 Massey, H. S. W., 249, 293 Mathieu, J. P., 156, 161, 172, 197, 198 Mathis, R. F., 86, I08 Matsen, F. A., 341, 413 Matsubara, T., 167,200 Mayers, D. F., 259, 261, 262, 293 Mead,C. A., 203,228,229,230,234,235 Menasian, S., 120, I54 Menzel, D. H.,239,243,295,297,298,301, 334 Meshkov, S., 308, 334 Messiah, A., 301, 334 Meyer, L., 187, 197 Midtdal, J., 338, 413 Miller, G. H., 65, 108 Miller, R. E., 164, 165, 197 Mills, R. L., 196, 200 Minnhagen, L., 271,293 Mitchell, A. C. G., 241, 293

42 1

AUTHOR INDEX

Mitra, S. S., 156, 176, 199 Mizushima, M., 228, 235, 239,293 Moeller, K. D., 173, 199 Moore, C. E., 237, 253,293,294, 340, 390, 413 Moores, D. L., 272, 294 Moreau, D., 174, 198 Morgan, L. A., 297, 334 Morozova, N. G., 241,294 Morse, P., 264,265, 285,294 Moruzzi, J. L., 2, 56 Moser, C. M., 260,278, 288.293 Mucker, K. F., 167, 196, 197, 199 Miiller, E. A., 269, 291 Mulder, M. M., 250,291 Mulliken, R. S., 143, 154 Murez, C., 267, 288 Murrell, J. N., 160, 200 Mutsuda, H., 197,199 Myers, B. F., 89, 96, 108

N Nakayama, K., 68, 108 Naqvi, A. M., 267,294 Nefeolov, A. V., 185, 199 Nesbet, R. K., 258,294 Nesmeyanov, An. N., 241,294 Neynaber, R. H., 58, 62, 64, 67, 72, 75, 89, 90,96, 108 Nicholls, R. W., 240,294 Nielsen, 0. F., 185, 198 Nielson, C. W., 297, 309, 334 Nielson, J. T., 185, 198 Nijboer, B. R. A., 168,198 Nikitin, A. A., 239, 240, 293, 297, 334 Nikonova, E. I., 241, 293 Nordberg, R., 259,295 Nordling, C., 259, 295 Norman, G. E., 271, 272,279,293,294 Nosanow, L. H., 168, 199 Novak, A., 185, 200 Novick, R., 139, 154 0 O’Brien, F., 259, 261, 262, 293 Oehier, O., 179, 200 Opik, U., 272,289 Ogryzlo, E. A., 42, 56 Okada, K., 197,199

Olsen, H. N., 240,241,294 Orr, B. J., 242, 289 Osberg, W. E., 180, 200 Osberghaus, O., 142,154,286,289 Oskam, H. J., 23,56

P Pack, J. L., 89, 108 Parkes, D. A., 243, 294 Parkinson, E. M., 239, 276, 277, 218, 284, 290, 309,334,340,412 Parkinson, W. H., 242,291 Patterson, P. L., 8,47, 55, 56 Patterson, T. N. L., 147, 153 Paul, E., 286,292 Paul, W., 142,154,286,292 Pauling, P., 185, 198 Pawley, G. S., 160, 161, 162, 164, 170, 172, 181,198,200 Peach, G., 240,294 Peachey, C. J., 169, 199 Peacock, N. J., 259, 290 Pecile, C., 186, 200 Peden, J. A., 53, 54,56 Pekeris, C. L., 250, 253, 256, 257, 263, 283, 290,294,341,413 Penkin, N. P., 241, 279, 286,294 Percus, J. K., 266, 280,288 Perek, L., 243,294 Perkins, J. F., 263, 294 Person, W.B., 175, 176,186, 198,200 Peterson, J. R., 83, 85, 86, 87, 107 Petrachen’, M. I., 271,294 Peyron, M., 186, 197 Pfennig, H., 251, 259,285,294 Phelps, A. V., 2,56, 89, 108 Phillipson, P. E., 248,249,290 Pimentel, G. C., 176, 179, 187,197, 198 Pines, D., 278,294 Poll, J. D., 174, 191, 195, 200 Polyukh, B. M., 67, 108 Popenoe, C. H., 241,294,295 Popescu-Iovitsu, I., 67, 108 Pople, J. A., 156, 160, 164, 165, 200 Potter, R. F., 67, 108 Poulet, H., 156, 161, 198 Prasad, S. S., 340, 413 Prask, H., 170,200 Prats, F., 363, 412 Prokof’ev, V. K., 271,292

422

AUTHOR INDEX

Purcell, E. M., 131, 154 Purdum, K. L., 269,272,288

R Rabinowitz, P., 341, 413 Racah, G., 229,235,297, 298,299 300,301, 306, 307, 308, 334, 352, 367, 368, 413 Raich, J. C., 167, 197, 200 Rainis, A. E., 259,295 Ramsey, N. F., 138, 149, 154 Rapp, D., 67, 108, 125, 126,154 Rautian, S. G., 203, 220,235 Rebane, T. K., 280,289 Reck, G. P., 203, 228, 229, 230, 234, 235 Redmond, P. J., 300,306,334 Rees,D. I., 275, 292 Reeves, E. M., 242,291 Refaey, K., 76, 107 Repka, R.,262,295 Rettinghaus, G., 116, 117, 154 Reuben, B. G., 75,108 Rich, N., 183, 184, 187, 200 Richardson, C. B., 145, 146, 154 Risberg, P., 339, 413 Roberts, J. R.,242, 294 Robinson, G. W., 184, 198 Robinson, P. D., 275,288,294 Roeder, R. W., 140, 141,142, 149, 154 Rohrlich, F., 297.334 Rol, P. K., 76, 100, 105, 108 Ron, A., 163, 164, 165, 166, 167, 168, 172, 173,175,176,197,200 Roothaan, C. C. J., 259,260,262,263,292, 294,295 Rosevear, A. H. M., 192,200 Ross, D. W., 202,228,235 Rostagni, A., 65,108 Rotenberg, M., 271, 272, 286, 295 Rothe, E. W., 58,62,64,67,68,75,90, 108 Rousset, A., 156, 162, 170, 199 Rozett, R. W., 105, 108 Rudge, M. R. H., 297,334 Rudzikas, Z. B., 297,334 Rusetskii, Y. S., 174, 199 Rush, P. P., 248, 291 Russell, M. E., 76, 107 Rutherford, J. A., 68, 78, 86, 106, 108

S Sadoine, M. P., 269,292

Salmona, A., 140,154 Salpeter, E. E., 247,248,273,275,288,345, 350,412 Samson, J. A. R.,279,294 Sanders, F. C., 274, 294 Sanders, W. A., 275, 292, 294 Sando, K. M., 276,294 Santry, D. P., 185,198 Saulgozha, A. K., 271,288 Savage, B. D., 242,287,294 Savoie, R.,186,197 Sayers, J., 2, 56 Schaad, L.J., 160,200 Schaefer, H. F., 267,294 Scherr, C. W., 274,294, 341,413 Schettino, V., 186, 199 Schiff, B., 256, 257, 263, 283,294, 341, 413 Schiff, H. I., 2, 3, 6,42, 54,56 Schliiter, A., 263, 264, 283, 295 Schmeltekopf, A. L., 2, 3, 6, 38,40,41,42, 43,46,47,49, 52,55,56 Schnepp, O., 159, 163, 164. 165, 166, 167 168, 173, 174, 175, 176, 196, 197, 199, 200 Schrader, B., 174 175,200 Schreider, E. Y., 241, 289 Schuch, A. F., 196,200 Schuessler, H. A., 111, 115, 116, 118, 129, 130, 133, 135, 136, 137, 138, 139, 149, 154 Schwartz. C., 138, 150. 154, 267, 274, 294, 341,413 Sears, V. F., 168, 193,200 Seaton, M. J., 253,269,270,272,289, 294, 295 Sena, L. A, 67, 108 Series, G. W., 242,253,295 Seryaltov, K. I., 279, 293 Seung, S., 274,295, 338,413 Shabanov, V. F., 174,199 Shabanova, L. N., 241,279,294 Shackleford, W.L., 242, 295 Shalit, A. de., 239, 295 Shamey, L. J., 272,287,291 Sharma, C. S., 258,261,262,263,274,295 Shelton, H., 125, 154 Sheridan, W. F., 67, 107 Shigeoka, T., 170, 171, 174, 184. 199 Shimanouchi, T., 162, 163, 170, 171, 173 174, 182,183,184,185, 187,198,200

423

AUTHOR INDEX

Shockley, W., 112, 154 Shore, B. W., 239, 243, 295, 297, 301, 334* 357,413 Shortley, G. H., 238,239,243,254,270,271, 275,290,299, 301, 307,334 Shull, H., 273, 293, 338, 340, 404, 413 Shumaker, J. B., 241,290,294,295 Siegbahn, K., 259,295 Silin, Yu. A., 243, 293 Silvera, I. F., 167, 176, 196, 199 Silverman, J. N., 260, 265, 267, 289, 295 Silverman, R. A., 341, 413 Sinanoglu, O., 255,278,285,293, 308,334 Sinfailam, A.-L., 297, 334 Sirkis, M. D., 112, 154 Skillmann, S., 259, 260, 292 Slater, J. C., 254, 258, 259, 264, 265, 295, 297,334 Slavenas, I.-Yu. Yu., 241, 295 Smeyers, Y. G., 260, 293 Smimov, B. M., 70,71, 108 Smith, A. C. H., 70, 71, 107 Smith, D., 2, 50, 56 Smith, K.L., 297, 334 Smith, M. W., 238,240,243,285,296 Smith, S. J., 249, 295 Smith, W. H., 169, 197 Snow, W. R., 71,107 Snyder, R., 341, 350,405, 413 Sobel’man, I. I., 203, 220, 235 Sobrana, G., 185,200 Sochilin, G. B., 257,290 Sommerville, W. B., 147, 153 Somorjai, R. L., 280,295 Soots, V., 192, 200 Sorokina, E. S., 272,288 Spitzer, L., 203,211, 235 Startsev, G. P., 241,293, 294 Stebbings, R. F., 68,70,71,107,108 Steele, R., 251, 259, 260, 285,294,295 Stephenson, G., 286,295 Stemheim, M. M., 140, 154 Stevenson, D. P., 74, 108 Stewart, A. L., 240,248,249,250,256,257, 265, 275, 279, 290, 294. 295, 340, 341, 351, 410, 412, 413 Stewart, J. C., 271,272,286, 295 St. Louis, R. V., 166, 167, 173, 175, 200 Stone, P. M., 270, 286, 295 Strotskite, T.D., 265, 292

Suterland, G . B. B. M., 183, 199 Suzuki, M., 162, 170, 172, 174, 185, 199 Suzuki, K., 187,200 Swirles, B., 259, 292 Synek, M., 259,262,295 Szasz, L., 257, 295

T Takahashi, H., 174, 175, 200 Takebe, H., 203,228,229,230,234,235 Talhouk, S., 167, 196, 197,199 Talmi, I., 239, 295, 353, 413 Tasumi, M., 173, 182, 183,200 Taylor, R. L., 89, 107 Tech, J. L., 241, 290 Thomas, L. H., 266,293 Thompson, D. P., 274,293,357,413 Thourenot, J.-C., 185, 199 Tobey, F. L., 242,292 Trees, R. E., 297, 308,335 Trefftz, E., 240, 251, 259, 260, 263, 264, 283,285,286,289,294,295 Treuil, K. L., 164, 165, 197 Trujillo, S. M., 58, 62, 64, 67, 72, 75, 89, 90,96,108 Trumpy, B., 268,295 Tsuboi, M., 185,199 Tuan, D. F.-T., 280,295 Tubis, A., 265, 295 Turner, B. R., 68, 78, 86, 106, 108

U Ueyama, H., 167,200 Ufford, C. W., 297,299,303,334,363, 367, 413 Underhill. A. B., 242, 259, 260, 287, 291, 295 Undheim, B., 266,292 Utterback, N. G., 65,108

V Vainshtein, L. A., 265, 295 Valatin, J. G., 278,295 Vanagas, V. V., 300, 301, 311,335 Van Kranendonk, J., 168,174,175,176,187, 188, 189, 190, 191, 192, 193, 194, 195, 196,198,200 Van Vleck, J. H., 210,235 Varsavsky, C.M., 240,267.296 Vassell, M. O., 275, 289

424

AUTHOR INDEX

Vaughan, J. M., 241, 243,296 Vedder, W., 176, 188,200 Veillard, A , , 274, 289 Veselov, M. G., 261, 296 Vetchinkin, S. I., 265, 267, 296 Victor. G. A., 240,290 Vizbaraite, Ya. I., 263, 265, 292, 296, 297, 334 Vodar, B., 216, 235 Vu, H., 216,235 W Waber, J., 259, 290 Waller, I., 268, 269, 296 Walls, F. L., 123, 148, 154 Walmsely, S. H., 156, 160, 164, 165, 168, 169,187,197,200 Warneck, P., 2,56 Warner, B., 239, 241, 290, 296 Wasserman, J., 187, 197 Watson, R. E., 258,260,291 Weber, E. W., 152, 153 Weber, N. E., 246,252,279,291 Webster, B. C., 262, 296 Webster, H. C., 58,108 Weinhold, F., 280,296 Weiss, A. W., 253, 257, 260, 262, 263, 265, 266, 275, 217, 278, 279, 283, 284, 286, 296 Weisskopf, V., 210, 235 Welsh, H. L., 188, 191, 192, 194, 197, 198, 200 Werth, G., 140, 141, 142, 149, 154 Wertharner, N. R., 168,198 Whalley, E., 169, 173, 197, 200 Whetten, N. R., 124,153 White, D., 167, 196, 197, 199 Whiting, G., 192,200 Wiener, J., 87, 88, 89, 108 Wiese, W. L., 238, 240, 243, 259, 278, 279, 285,291 Wilkinson, G., 161, 162, 170, 172,200

Williams, D. E., 160, 181, 184, 185, 200 Williams, D. R., 160, 200 Willis, K., 253, 295 Wilson, E. B., 274,280,283,292,295, 338, 413 Wilson, R. G., 274,295 Wirnel-Pecker, C., 243, 296 Winston, H., 176,200 Wittke, J. P., 152, 154 Wolf, F. H., 76, 108 Woll, J. W., 250, 291 Woodgate, B., 241, 287, 296 Wouthuysn, S. A., 350, 412 Wuerker, R. F., 125, 154 Wurster. W. H., 242, 296 Wybourne, B. G., 297,335

Y Yadav, H. N., 70, 71,107 Yamada, H., 187, 200 Yavorskii, B. M., 265, 295 Yennie, D. R., 340,412 Yilmaz, H., 265,294 Yip, S., 170, 200 Yokoyama, T., 162, 170, 172, 174, 185, 199 Young, A., 341, 350,405, 413 Young, L., 264,285,294 Ypenburg, J. W., 186,198 Yuriev, M. S., 280,290 Yutsis, A. P., 297, 300, 301, 311, 334, 335

Z Zacharias, J. R., 150, 154 Zafra, R. L. de, 242,296 Zare, R. N., 259, 276,296 Zeniansky, M. W., 241,293 Zilitis, V. A., 271, 272, 286, 288 Zirnan, J. M., 156, 200 Zu Putlitz, G., 152, 153 Zwaan, A., 286,296 Zwanziger, D. E., 140,154 Zwerdling, S., 183, 187, 200

Subject Index A Absolute cross section determinations in merging beams, 61, 65, 70, 75, 83, 98, 100 Absorption hook method, 241 methods for measurement of atomic transition probabilities, 241 oscillator strength, 245 total, 241 Acceleration matrix element, 248 Acetylene derivative solids, vibrational spectra of, 182 Acetylene solid, lattice vibrations of, 169 Adiabatic perturbations, 221 Alkalis, semi-empirical methods for solution of wave functions of, 267 Ammonia solid, lattice vibrations of, 169 Amplitude modulation, 21 1 Anomalously weak lines, 256 Anthracene solid infrared spectra of, 172 lattice dynamics of, 161 lattice vibrations of, 172 Raman spectra of, 172 vibrational spectra of, 185 9,lO-Anthraquinonesolid,vibrationalspectra of, 186 Antisymmetrization, 357-363 operator, 357, 359 Associative detachment, 50, 51 Atomic beam experiments, 241 Atomic energy levels of many electron systems double power series expansion of, 337, 348-350 energy level differences, 390-403 hydrogenic problem, 338, 344 many electron problem, 347 Atomic transition probabilities, 237 ff Aufbau principle, 263

B Bates-Damgaard method, 268 normalization, 269

Beam-foil method for lifetime measurement of excited states. 242 Beam-foil spectroscopy, 240 Benzene solid intermolecular potential for, 170 lattice vibrations of, 170 vibrational spectra of, 183-186 Beryllium, see also Isoelectronic sequences comparison of screening parameters, 407 extrapolation of screening parameter, 41 1 Binary collisions, 228, 233 Binary rate constants, 54 Bloc designation, 309 Boron, see also Isoelectronic sequences total energies, 390 Bound-free absorption coefficient for hydrogen negative ion, 248, 249 Bound-free transitions, 240 Bounds and variational principles (for calculating transition probabilities),280 Branching ratios in atomic transitions, 242 Breit interaction, 338, 344 irreducible tensor expansion of, 353-357 j-j coupled form, 357 L-S coupled form, 354 Pauli approximation to, 350 Z dependence, 349 Breit-Rabi diagram of 3He+, 130 Broadening of spectral lines collision, 201 Doppler, 203, 220, 228 impact, 220,224,228 resonance, 228 statistical, 221 Bromine solid lattice vibrations of, 169 vibrational spectra of, I87 Brueckner-Goldstone theory, 278 C

Carbon, see Isoelectronic sequences Carbon dioxide solid general theory of, 156-160 intermolecular potential for, 160 lattice vibrations of, 156-160, 164-165

425

426

SUBJECT INDEX

infrared intensities due to 175 vibrational spectra of, 179-1 80 Carbon monoxide solid infrared intensities of, 175 lattice vibrations of, 168 vibrational spectra of, 179, 180 Cascade processes in atomic transitions, 242 Central field approximation, 239 model, 254 cfp (coefficient of fractional parentage), see Fractional parentage CH2Br2solid, vibrational spectra of, 186 CH2Clz solid, vibrational spectra of, 186 CH212solid, vibrational spectra of, 186 Charge transfer, merging studies of, 62 ff cross sections, 65, 70, 83 excited states, 68 ion-ion reactions, 80 ion-molecule reactions, 62 neutral-neutral reactions, 89 noise problems, 65, 70 Chlorate ion in solids, vibrational spectra of, 180 Chlorine solid lattice vibrations of, 169 vibrational spectra of, 187 Coefficients, multinomial, 304 Coherent signals, 112 Collision(s) binary, 228,233 broadening, 201 effects, 151 electron+lectron, 148 heavy particle, 59 sense-reverting, 212, 216 Complex definition, 338 table of calculated complexes, 370-373 Component strength (of spectral lines), 244 Configuration(s) in atomic systems, interaction of, 260 superposition of, 269, 272 average, 259 interaction, 254,255, 259, 277 Conjugation about half-filled shell, 307 Cooling of trapped ions collisional, 120 evaporation, 120

general, 119 radiative, 120, 121 Correlated wave functions, 257 Correlation theories, 308 Cotton-Mouton effect, 242 Coulomb method for solving of Schrodinger equation, 257 Coupling, 299 ff conventional arrangement, 299 intermediate, 238, 239 lineal, 299 L-S, 300, 303 nonconventional arrangements, 302 schemes, 239 Cyclohexane solid, vibrational spectra of, 186 Cyclopentene solid, vibrational spectra of, 185

D Dalgarno interchange theorem, 275 Damping impact, 210 radiation, 204, 209 Debye relaxation, 214 Dephasing effects, 110 Detection circuits low noise, 116-119 Rettinghaus, 116 Diborane solid, vibrational spectra of, 182 1 ,4-Dioxan solid, vibrational spectra of, 186 Dipole acceleration, 246 length, 245 matrix element, 244, 247 velocity, 246 Dipole-dipole forces, 229, 231 Dispersion effects, use for absorption measurements, 241 Doppler broadening, 203, 220, 228 Doppler effects, 149 Doppler shift, 149

E Effective principal quantum number, 268 Electric dipole transitions, 238 Electric and magnetic multipoles, 239

SUBJECT INDEX

Electronxyclotron resonance, 148 Electron-ion collisions, theory for, 100 Electron-ion dissociative recombination, merging beams studies of, 101 ff Electron spin resonance of stored ions, 140 Electrostatic interaction irreducible Sensor expansion of, 552 Pauli approximation to, 350 Z dependence, 349 Emission methods for measurement of atomic transition probabilities, 240 ff Emission oscillator strength, 244 Energy levels of the Hz-H2+ systems, 143 Ethylene oxide solid, vibrational spectra of, 186 Ethylene solid infrared spectra of, 172-173, 175 intermolecular potentials for, 172-173 lattice vibrations of, 172-173 vibration spectra of, 181-182 Exchange narrowing, 219 Excitation energies, 252 Excited states in merging beam studies, 68, 76, 78, 85, 100 orthogonality problems in, 262 Expectation values of one-electron operators, 274 Experimental measurement of atomic transition probabilities, 240 Extrapolation along isoelectronicsequences, 342 reduction to interpolation, 41 1 4 1 2 F f values, 279 Fluorine, see also Isoelectronic sequences comparison of screening parameters, 407 extrapolation of screening parameter, 41 1 Foldy-Wouthuysen transformation, 350 Forbidden transitions, 238, 239, 255 Forces, dipole-dipole, 229, 231 Fractional parentage coefficients of, 297 ff many-particle, 298 redundant, 3 10 single-particle, 297 two-particle, 297, 298 Free-free transitions, 240 Furan solid, vibrational spectra of, 186

427

H Halogen solids, infrared intensities due to lattice vibrations of, 175 Hanle effect, 242 Harmonic oscillations, 207 Harmonic oscillator, 204, 217 Hartree-Fock method active electron approximation, 261 configuration interaction, 258 core polarization, 262 extension of open shell method, 260 for frozen orbitals, 262 orthogonality, 261 Hartree-Fock model, 274 HartrmFock-Roothaan method, 259 H2+ ion alignment of, 142 energy levels of, 143 magnetic resonance of, 146 photodissociation of, 142 ff vector model of, 144 ion, hyperfine structure of, 129, 139 Helium, see also Isoelectronic sequences afterglow, 7-9 comparison of screening parameters, 407 expansion coefficients, 338-340 isoelectronic sequence, 257 variational calculations of, 341 Hexamethylene tetramine solid, lattice dynamics of, 160-161 Hook method (for absorption rneasurements), 241 Hydrogen bromide solid lattice vibrations of, 168-169 vibrational spectra of, 186 Hydrogen chloride solid lattice vibrations of, 168-169 vibrational spectra of, 186 Hydrogen flouride solid lattice vibrations of, 169 vibrational spectra of, 186 Hydrogen maser, 149 Hydrogen peroxide solid, lattice vibrations of, 168 Hydrogen solid infrared intensities due to lattice vibrations, 176 lattice vibrations of, 167-168 phase change in, 196-197 rotational spectra of, 188-191

428

SUBJECT INDEX

vibrational spectra of, 187-197 vibration-rotation spectra of, 191-196 Hydrogen sulfide solid, lattice vibrations of, 169 Hylleraas-Undheim-Macdonald theorem, 263,266 Hylleraas variation-perturbation method, 274 Hyperfine structure of 3He+ion, 129 Hypervirial theorem, off diagonal, 247

I Ice, lattice vibrations of, 169-170 Impact broadening, 220, 224, 228 damping, 210 Incoherent signal, 123 Infrared spectra of anthracene solid, 172 of benzene solid, 170 of ethylene solid, 172-173 of naphthalene solid, 170-172 of polyethylene solid, 173 Intensities of spectral lines, 237 ff Interaction configuration, 254,259, 272, 277 cross sections, 65 dipole-dipole, 23 1 of ions with tuned circuit, 112-116 Interchange theorem, 280 in 2-expansion method, 276 Intermediate coupling, 238, 239 Intermolecular potentials of benzene, 170, 184 of ethylene, 172-173 of naphthalene, 170-172, 185 Iodine solid lattice vibrations of, 169 vibrational spectra of, 187 Ion cooling, 119 Ion-ion collisions, 116 interactions, 113 reactions, 80 Ion-molecule reactions (in merging beams), 62 Ion oscillations, excitation of, 109 Ion sampling, 10-14 Ion sources dc discharge, 6

electron impact ionization, 6, 36 microwave discharge, 5 secondary reactions, 37, 38 Ion temperature, 122 Ionization, merging beams studies of, 90 ff cross sections, 98 excited states, 100 Ionization potential, 390, 406, 408-410 for lithium sequence, 338-340 Isoelectronic sequences, 278, 337 ff beryllium, 377-378, 399, 41 1 boron, 379-381, 390 carbon, 382-384,400 extrapolation along, 342, 41 1 flourine, 389, 401403, 41 1 helium, 340, 374-376, 392-394 lithium, 338, 340, 377, 395-398, 41 1 neon, 389 nitrogen, 385-386 oxygen, 387-388

K Kerr effect, 241 Kirchoff’s law, 207

L Lamb shift, 338, 344 Latticq dynamics of ammonium chloride solid, 164 of anthracene solid, 161 of hexamethylene tetramine solid, 160161, 164 of molecular solids, 160-163 of naphthalene solid, 161-163 Lattice vibrations of t-butyl chloride solid, 174 of cyanuric acid solid, I74 of p-dibromobenzene solid, 174 of p-dichlorobenzene solid, 174 of diketopiperazine solid, 174 of hydrazine solid, 174 of a-hydroxy-naphthalene solid, 174 of methyl-iodide solid, 173 of molecular solids, general theory of, 155-160 of polyethylene solid, 173 of pyrazine solid, 174 of thiourea solid, 174 of p-toluidine solid, 174 of uracid solid, 174

429

SUBJECT INDEX

Level-crossing experiments, 242 Lifetime of excited states, 242, 244 Line-broadening, 149 Line components (spectral), 243 Line overlapping, 216, 217 Line shape Lorentz, 210 microwave, 213 Van Vleck and Weisskopf, 210 Line-shift, 149 Line strength, spectral, 238, 239, 245 Liouville space, 202, 203, 232 Lithium, see nlso Isoelectronic sequences comparison of screening parameters, 407 extrapolation of screening parameter, 41 1 ionization potential of, 338-340 variational calculations, 341 Lorentz line shape, 210,211,224, 227 Lorentzian shape, truncated, 227 Low-noise detection circuits, 1I6

M Magnetic double resonance (in spectrosCOPY),242 Magnetic resonance of .He+ and H2+, 127 Many-channel quantum defect method, 272 Many particle cfp, 298 Matrix elements, 245, 248 between antisymmetric states, 359-363 direct, 365-368 double power-series expansions of, 374389 exchange, 368-369 off diagonal, 308 of one-particle operators, 297 of two-particle operators, 297, 308 Merging beams, 57 ff cross section, 61 current studies, 105 deamplification factor for, 60 history, 58 inclined, 59 overlap integral for, 61 principles for heavy particle collisions, 59 theory for electron-ion collisions, 100 Methane solid, vibrational spectra of, 181, 186 Method of moments, 225, 232 Methyl iodide, lattice vibrations of, 173 Mode mixing, 219

Molecular solids, 155 ff, see also Lattice vibrations Moments, method of, 225, 232 Microwave line shapes, 213 Microwave spectroscopy, 212 Multinomial coefficients, 304 Mutual neutralization, merging beams studies of excited states, 83, 85

N N 2 0 solid, vibrational spectra of, 179, 180, 185 Naphthalene solid, 161-163 infrared spectrum of, 162-163 intermolecular potentials for, 170-1 72 lattice dynamics of, 161-163 lattice vibrations of, 170-172 Raman spectrum of, 162-163 vibrational spectra of, 184, 185, 187 Narrowing, motional, 220 Neon, see Isoelectronic sequences Neutral reactant production, 40-44 Nitrogen, see Isoelectronic sequences a-Nitrogen solid infrared intensities due to lattice vibrations of, 175 lattice dynamics of, 163 lattice vibrations of, 164-167, 175 Nitrous oxide solid, lattice vibrations of, 169 Nodal boundary condition method, 273 Nodeless transitions, 252 Noise temperature, 123 Nuclear charge expansion method, see Z - expansion

'

0

OCS solid, infrared intensities due to lattice vibrations of, 175 Off-diagonal hypervirial theorem, 247 Operators one-particle, 297 two-particle, 297 Orbitals equivalent, 298, 303 frozen, 262 inequivalent detached, 305 separated, 301 single particle, 298 virtual, 266

430

SUBJECT INDEX

Oscillations, harmonic, 207 Oscillator, harmonic, 204,217 Oscillator strength, 238 1,3,4-0xadiazole solid, vibrational spectra of, 185 Oxygen, see Isoelectronic sequences Oxygen solid, vibrational spectra of, 187

P

Pair-energies, 309 Paraffins, normal, solids, vibrational spectra of, 182 Parentage expansions, 298 Pauli approximation, 350 Penning trap, 119,123 Perturbations, adiabatic, 221,224 Phase shifts, 211 Phase transposition, 304,312 Phase transition of solid hydrogen, 196-197 Photodissociation, 142,143 Planck’s law, 207 Polyethylene solid infrared spectrum of, 173 lattice vibrations of, 173 vibrational spectra of, 182 Pressure shifts, 223,224 Pyrazine solid, lattice vibrations of, 174 Pyrimidine solid, vibrational spectra of, 185 Pyrrole solid, vibrational spectra of, 186

Q Quadrupole ion trap, 1 1 1 mass spectrometer, 13 Quantum defect, 27&272 Quantum numbers, 299,300 quasi-spin, 306,307

R Radial transition integral, cancellation in,

251

Radiation damping, 204,209 Radiation trapping, 240 Radiative cooling, 120 Radiative recombination, merging beams studies of, 87 Raman spectra of anthracene solid, 172 of benzene solid, 170

of naphthalene solid, 170-172 of pyrazine solid, 174 Random phase approximation in perturbation theory, 278 Rate constant calculation, 14-32 temperature variation of, 49,50 Rearrangement reactions, merging beams studies of absolute cross sections, 75,100 of excited states, 76,78 of ion-neutral collisions, 72,76 of neutral-neutral collisions, 89,100 Recoupling, 299,300 Relative magnitudes of contributions to line strength, 239 Relativistic term splitting, 342,406-411 accuracy of prediction, 408 comparison with experiment, 393, 397-

399,401

extrapolation of, 342,411 in lithium, 410 in regular doublets, 408 Relaxation, Debye, 214 Reorientation of molecules, 203, 21 1 , 216 Resonance broadening, 203,228 magnetic double, 242 Rettinghaus detection circuit, 116,117 rf spectrum of H2+, 142 Rotor, rigid linear, 215

S

Schrodinger equation, methods of solution for configuration average, 259 Coulomb, 257 Hartree-Fock, 258 self-consistent field, 257 Screened hydrogenic functions, 267 Screening, 341-342,35&352,406-41 I approximation, 275 calculations, 404-407 comparison of screening parameters, 407 comparison with experiment, 406-41 1 elementary, see Simple screening hydrogenic, 341 parameters, 276,341-342,350-352,404-

407,411

43 1

SUBJECT INDEX

single screening method, 342, 350-352, 406,409-41 1 variational, 341, 350, 404-406 Self-consistent field method for solving of Schrodinger equation, 257 SiF, solid, vibrational spectra of, 179 Signal-to-noiseratio, 123, 134 Single particle coefficients of fractional parentage, 297 model in variational wave functions, 257 operators, 297 orbitals, 297 Slater determinant, 254, 258 Slater-type wave functions, 265 SOz solid, vibrational spectra of, 186 Sommerfeld terms, 347, 349, 373, 404 approximate energies through, 404-405 formula, 347, 373, 41 1 Spectral shape function, 205, 207 Spectroscopy, microwave, 212 Spin-exchange collision, 131 Spin-orbital, 239 Spin-orbit interaction, 350, 41 1 Stark effect, 150 Statistical broadening, 221 Statistical weight of atomic states, 245 Stored ions, 109 ff charge exchange collisions, 125 collisions, 151 counting of, 124 detection, 124 electron spin resonance, 140 hyperfine structure, 129 magnetic field effect, 150 spectroscopy of, 109, 120 ff spin-exchange collisions, 125 spin exchange cross section, 125 with polarized atomic beams 124 Stark effect, I50 thermometry of, 120-122 Sum rules, 279 Superposition of configurations, 269

Van der Waals gas, 227 Van Vleck and Weisskopf line shape, 210 Vector model for H,+,144 Vibrational spectra of molecular solids, 176 ff general theory, 177-1 82 induced spectra, 187 intensities in, 186-187

T Term differences, 406,408-410 accuracy of prediction, 408 comparison with experiment, 391-392, 394-396,400,402403

Weak transitions, 252 Wave functions correlated, in two-electron systems, 257

Thermodynamic equilibrium in atomic states, 240 Thymine, I-methyl-,solid, vibrational spectra of, 185 Thiophene solid, vibrational spectra of, 186 Thiourea solid, infrared intensities due to lattice vibrations of, 175 Thomas-Fermi method, 270 Thomas-Reiche-Kuhn sum rule, 279 Three-body rate constants, 49 Transition($ bound-free, 240 electric dipole, 238 forbidden, 238, 255 free-free, 240 in helium, 256 integral, 239 probabilities, 240, 252 Transposition phase, 299 Trapped ions, see Stored ions Triangle rules, 310 Two-particle coefficientsof fractional parentage, 297, 298 Two-particle interactions, 297 Two-particle operators, 308 Two-photon transitions, 240 Tuned circuit noise, 1 15

U Urea solid, vibrational spectra of, 185

V

W

432 division of, 299 Wave function coparent, 298 postparent, 298 preparent, 298

SUBJECT INDEX

Wentzel-Brillouin-Kramersmethod, 270

Z Z-expansion method, 213 Z - expansion nonrelativistic, 337, 340-342 relativistic, 337, 342-350 semi-empirical, 341