THE FOUNDATIONS OF NEUTRON TRANSPORT THEORY RICHARD K. OSBORN College of Engineering, Unlvertlty of Michigan Ann Arbor, Michigan
SIDNEY YIP Massachusetts tn,tltute of Tecflnology Cambridge, M....chu.etts
Prepared under the auspice. of
the Division of Technical Information United States Atomic Energy Commission
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Preface There arc at least three reasons why the authors felt that a m ODO-graph such as this might prove useful. Fo r th e past fifteen yean or so there have appeared many texts and trea tises which have presented extensive studies at aU levels of sophistication of the solutions of the neutron t ransport eq uation . However, the: origins and limitations of this equatio n have: been given little or no attention. But the fission reactor teebD.ology [like the fusion technology and many other areas of modem engi neering) is demanding a deepening awareness of the subtle relationship between microsco pic cause and macroscopic effect. Thus we felt that an initiation of an exploration into the foundation of the neut ron transport equation was a needed co mplement to the examinelion of its solutions . The: subject matte r summarized in this monogra ph was initially generated in bits and pieces within the context of various courses offered to the nuclearengineering studentsat the University of Michigan.
Thus a second reason for the preparation of this materialin its present form was to provide an integrated treatment of an integral topic. For example. it is qu ite conventional to separate the discussion of the transport equation from the study of microscopic reacti on ra tes. Thi s is both natural and necessary from the pedagogical point of view. particularly at the introductory level. Nevertheless it seems important that at so me point the essential unity of these concepts be restored, and this unity manifests itself in the study of the origins of the transport equationnot its solutio ns. Thirdly. it is probably inevitable that the analytical tools available to the engineer at any given insta nt in time will eventually become inadequate to his tasks. Indeed this may be the case in th e reactor technology today with respect to th e matter of interp reting neutron 6uotuaticn measurements. Thus a potentially practical purpose may be served by this work in that it suggests a path way along which generalization of the usual description of the reactor may be sought dedu ctively rathe r than indu ctively. v
vi
r ue
fOUNDA TIONS Of N EUTRON TRA NSPORT T Il EOR.Y
This book is not intended to be a text book, nor is it a imed at partieular areas of specialization, It deals with a smal l, well-defined topic. which, however, has broad implications. It is thus anticipated that graduate students, teachers. and research workers in nuc lear engineering. ph ysics, and chemistry (many of the principles and techniques of an alysis ca rry over intact from a study of neu tron transport to the study of th e kinetic th eory of reacting gases) might find herein something of interest to them. We have used whatever math ematical tools and ph ysical noti ons we have found necessary or co nvenient-usual ly without providing any ba ckground information. Nevertheless we have attempted to present the argument in a sufficiently self-contained way that the bulk of t he discussion can be followed without too much reference to background material . No attempt has been made to compile a comprehensive bibliography. In fact the referencing is ad mittedly spotty, cas ual and enormously incomplete. However some care bas been taken to see to it tha t points of connection between the topic discussed here and related topics dis-cussed elsewhere arc referenced for the reader's general interest, Also some forethought was exercised to supply references which in themselves provide good bib liographies. The authors arc grateful to Professor George Summerfield for his careful reading of the manusaipt and his helpful comments and criticisms, to Professor Ziya Akcasu for his assistance with t he perturbation method used here fo r the calculation of nuclear reaction rates. and to Mr. Malcolm Fe rrier whose interest in and encouragement of this work was crucia l to its fruition . One of us (S.Y.) gratefully ack nowledges the University of Michiga n Institute of Science a nd Technology for a postdoctoral fellowship and the Michigan Memorial-Phoenix Laboratory for hospitality during the course of this work.
Table of Contents Lbt of Symbols
vii
I Jotnldodloa II A Tnosport Equtioa ill ~ Pbue S pace: . • • • • .
8
9 12
A. Some Basic Fonnalism • • . . • B. A Ki ndt<: Equ ation for F(X. K, t) C. The Streaming Term . . • D. The Collision Terms . • • • • • B. Etrect of an External Fiel d • . •
20 22
26
m Neutroo-DUdtar lateractioal: Mainly Nuclear Coasidendoal
29
A. Formal Development of the Transition Probability . 8 . Radiative Ca pture . . . • . . C. Elast ic Scattering . . . . . . . . . . . • . • . D. Fission and Inelastic Scatteri ng • • . . • • . . •
30
E. The Neutron BalanceEquation in Continuous Momentum Space IV NtIItroa-Dudear Interactioas: Medium Eft'ectI • A: Tran sport in an Ideal G as . Ra diative Capture • Elas tic Scattering •
8 . Transpo n in Crystals . Radia tive Ca pture: . Elastic Sca ttering • The Thermal Average
VSpcdalT
SO
SS
sa 62 6S 6S 69 7S 76 80
91 .
A. Neutron Thermodynamics . B. H igher Order Ne utro n De nsities-Particu1arly the Doublet De ...
sity. Iadex
40
9S 9S 104
.
. vii
125
,
List of Symbols In the followingsome of the more frequently used mathematical symbols are defined. Whenever possi ble the equation in which the symbol is first introdu ced is given and th e read er is referred to the text for dC finition . So me sym bols are used for more t han one qu antity, but their meanings sho uld be clear from the co ntext. 4
A A(K 'k' ; Kkl Q
a +(X , K, 5), a( X. K, $)
-
Atomic mass Scatt eri ng "frequency", Eq . 5.2 " Free-atom " scatteri ng length Creation.destructlonoperators for neutrons at (X, K, s) - Scatteri ng length s, see Eqs. 4.71,
4.72, and 4.76
B HI
W,(P', Pl, B:"(P", P, P')
E(X, X)
E. E.
s.. . ~ . E.
E:
s:• •
A - M,u I) C' - Fission freq uencies. Eqs.2. 80 and 5.43b
- Densityoperator.Eq.229;seealso
D
£.'"
- em+ M
I
Eq.4.45 - Density matrix, Eq.2.28 - Neutron energies. mlJ'12 or ft'K'(lm - Step function, Eq. 2.2 - Nuclear recoil energy, Eq.4.33 - Photon energy. 1101 - Relative energies. Eq .4 .4 - External energy of mass A nucleu s in exter nal state k - Internal energy or mass A nucleus in internal sta te ~ - Excitation energy or l'lLh level in mass (A. + I) co mpo und nucleus
- £:+1_ B .4+ 1 viii
ix
LIST OF SYMBOLS
I. F(X. K, t) or
r:,{X, K, t )
F,{X, K, t)
r,"(X, K, t)
- e:+
1
+ ". + 8 'HI
- Coarse-grained neutron singlet density, Eq.2.27 - Spin-dependent coarse-grained neutron density, Eq.3.6S - Coarse-grained photon density, Eq.3.67 - Coarse-grained singlet density for 4 panicles, Eq. S.44a
Fl(X. K. X'. K.', t) or f1-)(X, K. X'. ){', I)
rt(X. Ie, X', K', t)
f{X, K, t) I.(X, k, t)
- Coarse-grained neutron doublet density. Eq.2.70
- Coarse-grained doublet density for IS particles, Eq .S.44b - Coarse-grained cross density for neutrons and IS particles. Eq. S.44c - Neutron density, Eq.2.30
- Density of mass A nucleus.
G(r, t)
Eq.4.27 - Scattering frequency, Eq.2.80 - Detection frequency, Eq.S.4S - Resolvent operator. Eq.3.2 - See Eq.4.78
H.
- Hamiltonian of system which
H' A
- See Eq.2.42 - b(a, where It is Planck's constant
' (PO - P) ,v(p -+ P') G(z)
interacts with the neutrons
u»
- Modified Bessel function of order n and argument x
K
- Neutron wave vector. P IA; also IS
{X,}, { X }"
- A set of J neutron wave vectors - A let of J wave vectors which 000 tains (does not contain) the wave
particle wave vectors
{Krl,
k
«Xl;,)
vector K - Wavevectorand spinJabels of the J neutrons produced by fission
or
- Quantum label nuclear external state, Eq .3 .38
II
1111 IO UNDAT IONS Of N £UTkON Tk .... NS ..UIlI T1 1[Oll. V
~
k. L I .Y'( , ' M
-0 , )
m
m, N(X, X, ,), N '(X, X, s)
"In)
.... p p
- Wa",e vector of a nucleus, p!" - Boltzmann's co nstant - Cell length in coarse-grained con figuration space - Superscript or subsc ript denotes a pa rt icular nucleus - Fiss ion frequ ency, Eq.3.99 _. Mass of the nucl eus - Neutron mass - Mass of Lth nucl eon - Neut ron occ upation nu mber at (X, K. s) in the state III) , In' ) - Nu mbe r of mass A nuclei in the spatial cell centered at X - Nuclea r density - Neutron state. Eq.2. 18; system state, Eq. 3.35 - T otal number opera tor - See Eq.4.45 - Neutron momentum ; momentum of III part icle - Di stribution of external n uclear stales - Nucleoni c momentum conj uga te to - Neutron momentum t ransfer d ivided by II - T otal reac tion rate - Scattering reacti on rate, sec Eq.3.95 and 3.97 - Position vector of center-of-mass of hh nucleus - Red uced transition ma trix, Eq .3 .33 - Reduced reaction ma trix with neutron and photon nu mber dependence extracted, Eq.3.SI - Position of Lth nucl eon in hh nucleus
r:-
Q
..•.
R,(X, X) R,(X, K ), R,(X, K ' - K)
R,
'-'
LIST Of
Sy~nOI.S
.I.,i~ .
RJ X. K) _. Sec Eqs.3.42, 3.51. 3.52. and 3.94 K..... ..,«(KsL>. ' : ..«Ks},) - See Eqs.3.8I, 3.82, 3.96. and 3.98 R,(X. KJ. R,(X, X' ~ KJ - See Eqs.3.68 an d 3.69 R:'''.' -,.l&.. r:1r.,r ,.l& . , . - See Eqs .3 .87 and 3.88 l'I:..&'.... l ••• ~ r . ,&,.... .I'...I.. - Neutron spin orientation label s - Shift function, Eq. 3.49 s. U, - Reduced potent ial for clas tic potential sca tteri ng by hh nucleus, Eq.3.73 - See Eq.4.19 0, - Nuclear matrix element (or emisU:!o(U::..) sion (absorption) of a neutron by Ith nucleus - Nuclear matrix element for emissio n (absorption) of a photon by hh nucleus. Eq.3.5S R:;.•• Rl·.·.I·.U•• r:
..
v
V
v· V' T
oo(r ). t' .(r )
,.
- V" + V' N ucleer velocity
-
-
x,
-
z
-
Neutron-nuclear interaction Photon -n uclear inte raction Neutron velocity Potential (unctions for elastic p0tential scattering. Eq.3.71 Transition probability per unit tim e, Eq .2.S6 See Eq .3.39 See Eqs.2.6S. 2.67. 2.73. and 2.74 Position vector locating t he center of a spa tia l cell Equ ilibrium position vector of the /th nucleus in a crystal See Eq. 4.4S; see also Eq.S .23
Gruk Letters - Quantum la bel of nuclear internal
stale - (k. T)- 1 - Width fu nction, Eq.3.SO
xii
T H E FOU N D AT IO NS OF N EUT a O N TkA NSP OkT T Il EOkY
r .(z) ;IF.
Q
•;. Au . A. i I'
('.( X. K )
n n(E- E', O)
.IX, K, x) I w
- Partia l rudiaticn (ncutron) width for e th level - Width and shift function - E.. -Elf.' - Dirac delta - Kronecker delta - See Eq.4.45 - Scattering angle (labo ratory) - Photo n wave vector - Photon polarization ; also as effective range in neutron -nuclear intera ction - See Gj .3.72 - Reciprocal neut ron wave vector, K- ' - Reduced mass ; also as chemical potential, Eq.5 .6 - Neutron number o perator, Eq.2.J6 - Macroscopiccross section for capture , detection , fission. sca ttering and to tal reaction - Microscopic cross section - Energy and angle d ifferential scattering cross sectio n, Eq.4.36 - Microscopic cross sectio n for potential, resonant, and interference scatterings - Cell function, Eq.2. 1 - See Eq.4.25 - Oscillator frequency - See Eq.3.17
I
Introduction Theoretical studies of neutron d istri butions arc usually based upon the transport eq uat ion
(:, + • . V + VE,).lt:x••, /) . - S(x,', t) + I d'" '' l:b,•.}....(•. - ' }Ax, ",I}
(I.I)
Here I is a macroscopiccross section, :F is a scattering frequency, and S is a neutron source which mayor may not depend upon f. Th e distribution funet io nJtx. Y, I), is the neutron singlet density in phase space or the expected number of neutron s pe r unit phase vol ume to be fou nd at the phase poi nt Ix, y) at ti me t. The motivation for initiating a cri tical analysis of the neutron transport equa tion does not arise from the fai lure of th is eq ua tio n in p ractical a pp licatio ns. Q uite the contrary. in fact. the usefulness of Eq.I .l in an overwhelmingly large class of pr oblems of interest in reactor tech nology has certainly been established beyond doubt. The com plete acceptance of thi s equatio n, however , has resulted in an at mosphere in which little co nsiderat ion has been given t o th e exploration of its foundations." This is unfortunate in our opi nion for, as th e util ity of Eq. l. 1 increases, our need to und ersta nd its origin and validi ty is also enhanced. The conventional method of deri ving the ne utro n transport equation is based ma inly on plau sibility argu ments. " F rom a purely th eoretical po int of view it is unsatisfactory for at least two reasons. The approach is phenomenological in that the result is not derived from a more fundamental descri ption . Secondly, the vari ous cross sections arc introduced • A . ummary or the basic assumption. or eeetrcn lransport theory hat been l iven by Wianer. l
I
2
TH E f U UND AT IO NS Of' NEU T RON TRA NSPO RT TH I;OR Y
empirically, hence the t rea tment must rely upon other mo re q uan titative theor ies for their calc ulation. Consequently it seems desirable to turn to a microscopic th eory to see to what extent th is well-kno wn and highly usefu l description can be justified from a more fundamental start ing point. Because th e neutron transport equation ca n be considered as a special linear varia nt of th e Boltzmann equation, one may look for a parallel develo pment in the kineti c t heo ry of gases where the latter eq uat ion plays a similar ly dominant role. It is to be obse rved that the Bolt zma nn eq uation, which was aJsofirst obtained on an intuitive basi s. had been in usc some 32 yean before attempts at systematic derivations based on the Liouville equation were made. J Fo rthepurpose o f studyi ng dilute systems, neither the structure nor the ph ysical content of the original equation has been altered by th e later. mo re rigo rous investigations. On the other hand. th ese de rivations have co ntributed significantly to our detailed und erstanding of the validity of thi s famous equation . Moreover, the deductive approach has yielded a logical basis for developing tran sport theories for dense gases an d liqu ids, an area of research th at is cu rrently act ively being pu rsued . To a certai n extent, similar advances can be achieved by a systematic study of the basic equation in the theory of neutron transport. Even if we a re only able to du plicate the results o f the co nventional treat ment, the analytical app roac h sho uld yield a set of sufficient co nditions for the applicability of Eq. 1.1. Since the anal ysis would then begin at the level of "first principles", the description can be made essentially selfco nta ined in the sense that both the met hod used for deri ving the tran sport eq uation a nd tha t used for evaluating t he relevant cross sections ca n be developed from the same basis. Finally, this app roach sho uld also be suita ble for studying the higher-order neutron densities t hat would arise in problems concerni ng fluctuations and co rrelations. In the present wo rk. an attempt is made to realize these anticipations. It is well known that an axio matic bersis for t he theoretica l examinatio n of macroscopic systems is provided by the Lio uville equation . Although there is a deceptively classical appearance to Eq. I.I, it turns out to be pract ically necessary for us to start with the qu a ntum Liou ville equation . Thi s necessity manifests itself almost as soon as the problem is posed, and in several different ways. In the first place, such an app roach requires so me notion of a Hamilton ian for the system and, thou gh such a notion is at best fuu y an d incomplete in th e present
I NT RO D U CTI O N
3
problem , it is almos t unthinkable classically. That is to say. there does not seem to be any classical formalism suita ble for treating systems in which particles of a given kind are not conserved. On the other hand, quantum field t heory pr ovides a co nvenie nt and powerful meth od of describin g the destruction and creation of particl es as a result of interactions. There is a mo re fundamental rea son for resorting to quantum analysis in de veloping a more quantita tive and unified theory of neutron balance. This arises becau se neutron -nuclear interactions are truly quantum phenom ena ; in particular the existence of resonance implies that the discre tness of nuclear ene rgy states will have an explicit inlIucnce on the behavior of the neutron d istribut ions. Again it is difficu lt. and inappropriate in our opinion, to treat this aspect of the problem in classical terms, particula rly since quantum-mechanical calculations exist that allow us to take into account not only the energy states of the individual nucle us but also those o f the macroscopic medi um (e.g. phonon states in a crystal). Ano ther reason for the quantu m treatment lies in th e proper interpretatio n of an observable density in phase space. In the abse nce of furthe r q ualifying comments, the density described by Eq. 1.I is am biguous. As already ment ioned, /(x, Y, t) d Jx d Jr rep resents the expected number of neutrons to be fou nd in the phase volume d Jx d Jp about the phase point (x, Y) at time t.ln view of t he uncertainty principle this interpretation beco mes meani ngless, if the volume element d Jx d Jt' is tak en in a limiting sense, and if not in a limit ing sense then how? In a classical treatment thi s issue does not a rise, but o ne enco unters other difficulties in interpretati on." Rather than giving up the physical mean ing of the distribution function, we find it necessary, if only as a mailer of principle, to give a more concise definition of the particle density of interest. As we shall demo nstrate, it is possible in a quantum fonnulation to answer th is question operationally a nd unambiguously, t ho ugh not necessaril y uniquely. This follows because quantum field theory provides us qui te naturally with an operator representative for me number of particles of a given type: in a phase cell of sufficient volume. There are other peculiarly qu antum effects which in principle will modify the struct ure of th e transport equation in its final Corm. Wh ile • See Grad . reference 3, p.2l 8.
4
TH E FOUNDATIONS Of NEUTllON TRA NSPORT THEORY
these details are not expected to be of much practical significance . it _ seems appropriate that in a systematic study the ir existence should at Jeast be recognized. For example, the velocity distribution of fully thcrmalized neutrons, strictly speaking. should be of the Fermi-Dirac type rather than the well-known Maxwellian . Admittedly it is difficult to visualize situa tions in which the two distributions are distin guishable; nevertheless, this property of the neutron d istribution can be shown to be a consequence of the derived transport equation. In concluding these introductory remarks we emphasize that the primary aim of the following work is to provide a more logical foundation of neutron transport theory in its present state of development. Our efforts will be devoted almost exclusively to the ju stificati on of the transport eq uation within the framework of current analytical meth ods . lt will he seen that we are only partially successful in thi s attempt, because n num ber of approximati ons are required in th e derivation. and th ese will be merely sta ted but not analyzed and evaluated. Evidently if we arc to understand the quantitative valid ity of Eq.I.I, these ap-proximation s mu st be thoroughly studied. The approximations in question are exceed ingly difficult to clarify. and we do not pretend that we eith er understand all their implications or are even able to carry out the necessary an alysis required to attempt this understanding. That the theor etical defense of the mathematical description represented by Eq. I.I. which ma y appear quite self-evident on intuitive grounds. requires such involved considerations is an indication that detailed understanding of neutron transport at the microscopic level is still lacking. Consequently there is much to be said for th e attempt to delin eate the specific area s of difficulty in opera tional terms, even if the effort to evaluate the approximations has yet to be expended . In (his sense the present work. which consists mainly of putting known results into new perspe ctive, rep resent s only the first portion of the ultimate solu tion. It is clear that to co mplete this solution further investigations. highly specialized and even more involved , are required. Although no truly new results. except for the generalization to the st udy of higher-order den sities, are obtained, the present approach brings many aspects of the neutron pr oblem into contact with other microscopic transport th eories. Thi s identification, which has not been emphasized previously. is potenti ally quite useful since one can then apply to th e study of neutron transport the powerful methods of analysis extensively developed in recent studies of irreversible processes.
I NT ..ODU CTION
5
The work is presented in four parts. In Chapter II a kinetic equation having the same physical content as that of Eq. I.1 is derived in coarse-grained phase space. The coarse-graining procedure used allows US to define an appropriate neutron singlet density, which avoids the difficulty in interpretation mentioned earlier. The time evolution of this density is studied under the assumption that there exists a time scale long compared to neutron-nuclear interaction times but short compared to some average time between interactions." That an approximation of this type seems necessary is quite evident in view of the fact that the desired transport equation treats transport (free flight) and collision processes separa tely. In the present approach these two processes are mo reover treated with different approximatio ns. The con- . venrional description of transport , characterized by the term e . V/. is explicitly obtained and. as one may expect in a coarse-grained analysis, this is only an ap proximate result. To see this one merely has to note that particles of a given momentum require a volume for localization of least linear dimension greater than the associated de Broglie wavelengths. But the difference between the distribution function on opposite sides of this volume cannot generally be represented solely in terms of the first derivative. Collision effects are discussed only very schematically, primarily to exhibit the resulting equation in a form that is readily identifiable with Eq.l.1. In fact, when quantum effects and the discrete natu re of phase space are ignored, and when finite differences are interpreted as derivatives, the conventional neut ron transport equation emerges as the desired result of this study. Detailed considerations of binary neutro n-nuclear interact ions are discussed in Chap ters III and IV. A brief development of damping theory is first given. This theory enables us to calculate tran sition probabilities for direct and resonan ce reactions taking into account both nuclear and macroscopic medium effects. Since we do not intend to present a detailed theory of neutron-induced reactions, we consider only the most relevant reactions from a general standpoint" namely radiative capture, elastic and inelastic scattering. and fission. Moreover. our attention to details must necessarily decrease with increasing complexity of the interaction. As the discussion must touch upon nuclear • The exi5tence o r widely d ifferent time scales is an important assumption basic
to the study o r irrevers ible processes in p.teS .t low density.' For example. Mori aod eo-workers 4 have emphasized lhat this pmccdure is equivalent to Kirkwood'. toneql(
of tilTlMmoothing.'
6
TH [ rOUNDATIONS OF NEUTRON TRA NSPORT TH EORV
forces (which arc less than fully understood at the present time), OUT treatment must be in some respects implicit rather than explicit. However, for the purpose of obtaining useful estimates of reaction rates, a number of well-known results can be deduced with a minimum knowledge of details of nuclear forces. Appropriate formulas for the transition probabilities are derived in Chapter III. These formulas arc then examined for some of the salient features of the specifically nuclear aspects of these reactions. To complete the discussion of neutron-nuclear collisions, the influence of the macroscopic medium on the reactions is studied in Chapter IV. Here only radiative capture and elastic scattering (both potential and resonance) by nuclei in gases and crystals are considered. The emphasis is on idealized systems because exact ca1culations are then feasible. The results obtained also provide the usual starting point for deriving appr oximate but more useful expressions for these cross sections. Finally, in Chapter V two disparate and specialized aspects of neutron balance are discussed. The first has to do with the nature of the velocity distribution of thermal neutr ons, while the second relates to the study of higher-order densities. The discussion of thermodynamic distributions is conventional to the theory of gas dynamics, and it is included here because the results are frequently referred to in the neutron context. It is not a discussion of neutron thermaTization, but only of some of the concepts that underlie what might be called neutron thermod ynamics. The second topic, however. is unconventional in that it represents a systematic development of the theory of second- (and , by implicit extrapolation) higher-order distributi on functions in systems in which particles of various kinds are created and destroyed. These distribution functions are of direct interest in interpreting fluctuati on a nd correlation experiments on multiplying systems. Since the implications of the present theory are still currently being explored, we shall restrict our attention to the development of general equation s and only some brief remarks concerning applications. (, ~ . -,
1. Eugene P. Wigncr, Proceedings of the Eleventh Symposium in Applied Mathe matics, American Mathemat ical Society, Prov idence. R.I., 1961, p.89. 2. A.M.Wcinoog and E.P. Wigner. T1te PhYJicQ/ TMory 01 Ntutron CllDin h ac--
INTRODU CTION
1
University of C hi.:agu Press, Chicago, 111.• 1958; R.V .Meghrebiian and D.K.Holmes, Reactor AnQ/ysis, McGraw-Hili Book Company, Inc•• New York, 1%0; B.Da...ison , N~utron Transport 71rrory, Oxford Uni "'crsity Press, London, 1957. J. H.Grad, Handbuchder Physik,XII, 205(1958) . The literature on the Boltzmann equation is enormous. Orad's worle. is cited because it is one of the most com pn:bensiYC critica1ltudics of the Boltzmann equation from the point of view of classical mec:hanics . Here the important work of Born and G reen, Kirkwood, Boaoliubov, and others arc also mentioned. Interested reade r should also ICC E.O. D. Cohen in FundanwnlQ/Problenu in Stallnjcol Mechtufics, North-Holland, Amsterdam" 1962, and Moti er ai., reference 4. 4. H.Morl, I.Oppenheim and J.Rou in StudWI in Statistical M~dan/cs, NorthHolla nd.Ams1eJ'da m,I962, edited by J.deBoer a nd O .E. Uhle nbeck , Volume I. S. J.G.Kirkwood, J. CMm . Phys. 14 : 180 (1946); IS: 72 (1947). tOr!,
II
In
A Transport Equation Coarse-Grained Phase Space
It was po inted our in the introductory remarks t hat the conventional neutron density [(x. Y, t) cannot be interpreted as an observable quantu m-mechan ical en tity because of the uncerta int y principle. To avoid t his difficulty, we shall at the outset introdu ce a discrete phase space. Such a space ca n be generated by dividing t he cont inu um into cells and then representing all points in each cell by the coordinate of its cent er . Part icle de nsities a re the n defined in terms of t hese coarse-gra ined co ordinates. and no attempt is to be mad e to determi ne the locatio n of a part icu lar pa rticle within any given cell. In " multiplying a nd/or an absorbing systcm ,thc number of neutrons in a given region in phase space is consta ntly changing, not on ly due to the nnt urnl flow of these particles but also du e to fission and absorp tio n proc esses. T he creation and destru ction of neutro ns can be qui re co nvenient ly described in the form alism of second qu antiza tion by representing t he pa rticles by a two-component spinor field operat or 'PJ(x), where j = I or 2 is the spino r index. The field for ma lism plus a procedure for coarse-grai ning phase space ena bles us to obtai n a particu lar rep resentation of the number ope rator, the eigenvalue of which gives the nu mber of particles in a given region in phase space. In terms of the nu mbe r ope rator, a coarse-grained qu antu m-mechanical ana logue o f fi x. v, I) can be defined, This new neu tro n density will be the quantity for which a transport equa tion is deduced . a nd t hus it prov ides the hasis of the present investigatio n of neutro n transpo rt t heory.
,
9
A TR ANSP ORT EQ U ATIO N IN ' H AIB SPA C E
A. Some Basic Formalism We will first review some o f the fundamental concep ts and introd uce the notations that will allow us to define a coarse-grained particle density in t he next sectio n. To introduce the coarse-gralaed phase space in operational terms we divide the co nfiguration space into ide ntical cu bical cells with edge length L. Let an ar bitrary point in confi guration space be denoted as x and let the set of posi tion vectors {Xl denote the cell centers . The coarse-graining procedure now consists of introducing the cell funct ion, I q{X, K. x) - L- l 12 E(X, x) etll:·· (2.1) where £(l(. x) ~
IT• £(X" x, ) ,.,
E(Xh x.) = 1 ·
(2.2)
X, - LI2 <
when
=- 0
X,
< X, + Lf2
otherwise
The cell functi on q.(X, K, x) is seen to describe a plane wave which is nonvani shing only within the cell centered at X. In Eq.2.2 the step fun ction. E(X, x), is not defined at the end po ints. However, if it is represented by the integral ;(, + 1.11
£(X"
xJ -
f
"lx, -
y) dy
(2.3)
%,- I./J
it can readily be shown tha t
E(X" X , ± L/2) - j
aE -
ax, (X" x ,) -
6(x. - X,
(2.4)
+ L/2) - 6(x. - X, - L/2)
(2.5)
These relations desc ribe th e behavio r o f the cell fun ction at the boundaries, and will be used. in the description of particle transport. For mathematical convenience we wiU apply peri odic bo undary conditions at the interfaces and thereb y restrict the co mpo nents of the wave vector to take on discrete va lues, K, - 2nM.IL. where M , is any positive or negati ve integer or zero . Hence the decomposition of configuration space res ults in a transformation of the co ntinuous momentum space to a la ttice of discrete points . The coordinates of X and 11K are to be regarded as coarse-grained variables in our description of particle dens ities. The phase point
ex."K)
10
Til" rOUN DA TIO N S or N E U TR ON TR ... ~s r O RT nIHH!. "
is seen to represent a cubical region of volume 11-' in phase space. Any par ticle found in th is volume will be assigned the coordinates of thc pha se point. The uncertainty in position and momentum implied by this procedure is therefore consistent with the uncertainty principle. The eell functions, g{X, K, x), by virtue of the pro perties of the step function , pro vide an analytical means of dividing the configuration space into cells. They can be used to obtain an ope rator representation of the neutron fi eld in the coarse-grained coordinates. Since these functions fo rm an o rthonormal and complete set, i.e.
f d )x 'I,-(X, K, x) r (X', K', x) L q ' (X, K, x) q(X, K, x ')
x." the spino r field can be expa nded as
,,Ix) =
L
= <'J xx- b"".
(2.6)
~ .l(x - x')
(2.7)
a(X, K, , ) u'(' h 'X. K, xl
(2.8 )
x.It••
where the functionsuJ(s). S = ± I, are the components of unit vectors in spin space which may have the simple representations," u(l) -
(~)
u( - 1)
~ (~)
(2.9)
Furthermore, they have the properties that-
.-%L u;(s) u.(s) = lJJ.
(2.10)
u;(s) u;(s') -= 6...
(2.11)
I
where the superscript" +" denote s Hermitian conj ugate. Note that the index s labels the orienta tion of the neutron spin. The coefficient in the expansion of 'PAx) , a(X, K, s) "" J dJx 1j!·(X. K, x) u; (s) 'PAx)
(2.12)
is an operato r governed by the same commuta tion relat ions specified for the field ope rato r. For neutron s and oth er fermions the operators satisfy nnticomrnutution rules, ['PJ(X), 'P:(x')]+ ... (}J. o(x - x ' )
(2. 13)
['PAx), 'P.(x')]~ - ['P;(x), 1fI:(x')] + = 0 - We employ the convention in which repeated spinor indices are summed.
A YIlA NSP OIlT I:.QU AY IO N I N PHA S E S P ACE
II
Using Eq .2.12 we lind [a(X . K, s), a+(X', K', s')] • - dxx' JJlI[' d."
[a(X, K, s), a(X', K', s1 J. - [a'(X, K, s), a'(X', K', s')I , - 0 (2.14) For boson s Eq.2.13 and 2.14 would still apply ifeverywbe re the anticommutator (A, B] , is repl aced by the commutator [A, B]. The operato r a(X. K, s) and its Hermitian conjugate are th e co nventional fermion de struction and creation operators. Thi s is best illustrated by considering th e effect ofthese op erators when acting 0)1 a given state. Explicitly.let us consider an operator whose eigenvalue gives the total number of neutrons in a given state. This operator"
.Y - Id' x y; (x) . ,(x) -
....L '
a'(X. K, s) a(X, K,s)
(2. 15)
is the total number ope rator and is term-wise Hermitian . A representation can always be found in which the operato r
.
e,(X, K, ,) - ..(x, K, s) a(X, K, s)
(2.16)
is diagonal (bence ce" is diagon al since the e's at different points co m. mute. ) t.e . e,( X, K, s) In> ~ N(X, K, s) In> (2.17) where N(X, K. s) is the number of neutrons at phase point (X. AX) with spin orienta tion s. Since the neutron is a fermion, the occupation number N(X. K, s) can be only zero or unity. The operator e l(X. K. $) is seen to be the number operator at the indi cated phase point. In the above representation . the state In) (ignoring other kinds of particles in the system for the moment) specifies the distribution of neutrons in X-K-.J spa ce as as the total number of neutrons in the stale. Thus
wen
In> - IN(X" K"s,) N(X" K" s,) ... N(X,K,,) ... > (2.18) with
L
N(X,K, s) - N
(2.19)
11,11,.
where N is the co rrespo nding eigenvalue of the total number operator ...Y. It will be convenient to replace the ordered arguments X. K. and s • We use Dirac', notalion of bras and keu. J
12
T H E FO UNDATIONS OF NEUTa ON TaA NSPOaT T H £oa y
by an ordered set of subscripts with one-to-one correspondence. Eq.2.18 then becomes In) - IN .NJ
.. .
NA
(2. ISa)
... )
By using the appropriate commutation relations and starting with Eq.2.17 one can readily show that for fermions! QA
In> = DANA I N I N~ ... I - N A .•. )
at In)
~
0,( 1 - NJ IN,N, ... N, ... )
(2.20) (2.21)
The phase factor DA arises because the stales before and after the operation of Q A and 01' must be properly labeled. For bosons one finds 0A
In) - [NAJ I/J ININ J •.. NA
-
I .••)
0: In> =- [N.l + 1)'IJ I NI N~ ... N.l + I ...)
(2.22)
(2.23)
Of course, the occupation numbers for bosons can be any positive integer or zero.
B. A Kinetic Equation for F(X, K, I ) Having introduced the neutron number operator in coarse-grained phase space, we can now define a particle density, which has the same interpret ation as that purportedly ascribed to f(x, V,I) a nd which will be suitable for use in deriving an approximate tran sport equa tion for neutrons. Let the state of the system of interest at time t be denoted by YJ(t). The expected number of neutrons per unit cell volume at the phase point <x, AK) is therefore given by F(X, K, I) - L- ' ( '1'(/)1•• 0', K) 1'1'(1»
(2.24)
with • •(X, K) -
The expansion
L a+(X, K. ,) aO', K, s)
(2.25)
•
'1'(1) -
L C.(/) III) •
(2.26)
A TRANS POR T EQU AT ION IN PH AlB SP AC E
13
results in an other form of the expectation value
FlX, K, I)
- L- '
where
-
L D_ (I) (nl. ,(X, K) 1m>
- L- ' Tr D('h,(X, K)
(2.21)
D..(I) - C:(I) C.(I)
(2.28)
is the von Neuma nn density matrix;" which is the qu an tum-mechanical equivalent of the classical Gi bbs ensemble.w- s The time dependence of F(X, K. r) is expressed th rough the density matrix operator which satisfies th e qu antum Liouville equation,'
a:, _~ (D, H]
(2.29)
H being the Hamiltonian of the system. It is wort h noting that th e trace is invariant under un itary transforma t ions; hence, the representa tio n in which Eq. 2.27 is evaluated may be chosen for convenience. Unless specifically sta ted otherwise , we sha ll ca lculate aU matrix elements in the representation which d iagonalizes th e number operator. In the sense of Eq.2 18, D.J..t) is seen to have the interpretatio n as the probability that at time t the syste m is in the state In) in whic h the number of neutrons and the ir d istributions in X·K-s space are specified. The funct ion V~ K, t) represents the expected number of neutrons with momentum P - IiK a nd any spin orientatio n in the cell centered at X at time t. Since F is the expect a tion value of an operator whose eigenvalues are positive or zero , it is grea ter t han or equal to zero everywhere a nd hence is appropriate as a particl e distribution func tio n. As de fined , Fi s a de nsity only in confi gura tion space a nd not in mo mentum space ; moreover, unli ke the fun ction/. it is not a distri bu tion in continuo us co nfigur ation space. The present de rivation of the t ransport equation act ually requires this discrete domain ; however, since co nventional results are usually expressed in a continuous momentum space, we will ultimately, whenever warranted, sum F over a small elemental volume d "K according to
L FlX,K,t) It.'s..
-
(.!:...)' F(X,K,I)d'K 2n
- j(X, K, I) d' K - j(X, P, I) d ' P'
(2.30)
- 'Ibe interpretation that a pure quantum-mechanical state corresponds to a classical enxmble d in aareement with van Kampen ;_·
ICC
also Fano."
14
HI E FO UNl1 AHON S Of NEU T RON T kANS I' O k l' HIH )k Y
It isf (X. P, 1)that is to be identified as the ana logue o f the conventional neutron density. It is perhaps of some value to digress and indicate briefly how the pre sent approach is related to the phase- space distribution function empl oyed in some recent investigations of transport phenomena. h·"" Con sider a generalized phase-space distribution function
dJy~:(X
K(X, k, I) = f
-
~)ew(x
- ~ ,x + ~ ,t)~{x + ~) (2.3\ )
where {'fl(X)} is an orthonormal and complete set of space functions and /!(l ' is a redu ced density matrix given by e(I J(X, x ', t) = Tr 'P;(x') 'PAx) D(t)
(2.32)
Th e function g(x, k, r} has been studied by Mori" in deriving the Bloch eq uation," and by One;' in the coarse-grained formalism, in deriving the UehJing-Uhlenbeck equation.!" It provides a convenient means with which one ca n obtain either the fine-grained or the coarse-grained distribution function s. For jf one uses plane wave for 'Pl(X), the result is equival ent to the familiar Wigner distribution function, 'oll
g(x, L, t) =
f
d Jye- "'·' l?o{x -
~, x + 1. 2'
I)
(2.33)
If the cell function is used the result is
g(x,X,K,I)
=
f
d 3y tp*( X, K, X -
x.,(X, K, x+ ~)
~)l?(l)(x
-
~,x
+
;,t) x (2.34)
The coarse-grained distribution function is then obtained by integrating g(x. X, K, f), G(X, K, I)
~
f d'x g(x, X, K, I)
"" Jd
3
x d 3 x'Ip*(X, K, x) 'f(X. K, x') Tr V,)'-(x) V'J(x' ) D(t)
(2.35)
In view of the spino r field expan sion, Eq.2.8, the above expression for G(X, K, f) is seen to differ from Eq.2.27 only by a volume factor.
A TRA NS P O R T EQ U A":'IQN IN P HASE SPA Cto.
IS
We now consider the time dependence of F(X, K, f ). If the system Hamiltonian is assumed not to be an explicit function of time,· then a formal solution to the operat or equation, Eq.2.29, is D(t + 1') _
e - f
D(f) eltH 11I
(2.36)
The Hamiltonian can be written quit e generally as H "'" T
+ H. +
V + V"
(2.37)
where T is the kinetic energy of the neutron s, H. is th at part of the energy of the system independent of the presence or absence of neutrons.t V is the energy of neutron-nuclear and photon-nuclear interaction s and Y-- is the energy of the neutron-neutron interaction. In the following we shall ignore Y-, as its effects are truly negligible in the studies of neutron transport in macrosco pic media. In the qu antized field formalism the nonrelativ istic neutron kinetic energy is of the form! .
T
=
-
•'f
2m d )x " j (x) V2 ",,(x)
(2.38)
where m is the neutron mass. This ope rator can be expressed in terms of coarse-grained coord inates by means of the spino r field expans ion. We obtain (2.39) T - 8 + T + T" • Problems with tin»dependent Hamiltonians are also of general intere$t. 12 In tbe present cue it miaht be realized if the neulrons were exposed to a lirne--varyin. aravitational or inhomogeneous masnetic field. These effects, however, are not liledy 10 be . i. nilicant in neutron transport theory. t In passin.. we observe lhal if H. is made sufficiently inclusive and if an app ropriate selc:ction of operator representat ives of dynam ical vari abks to be measured is made in any liven case, tben Eq.2.29 alon, with the ac:neral form o(an obser\'ab~ expectation value.w(t) - Tr D D(t) , encompass Maxwell's equations-hence all of classical elc:ctricity and ma anetism; equations for radia nt energy (photon) tta nsferl ).,,-hence aU of the equal ions of reactor sbieldin. as well as the theory cf photon interactions with m61te1' ; equations (or neutraJl ' and charged ~ 12 PI kinetics ; equations of Newton-hence all of clas.sicalmechanics ; ere. This is merely an involved way ofsuaa-in.lhat. in our opinion, Eq .2.29 andw(t). when a ppropriately phrased, provKte a suitaNe starlin. point for invcstiptions in virtually aU 1:nDchc:s of science and CR.il'JCUina. • The ratio o(the neutron density 10 the nuclear density in a reactor is at most 10- 1 or less. If one assumes that tbe cross aection for (If, II) scatterin , is roughly tbe same as tha t for (", p). then the mean free palh for ncutron.ncutron intenction is of order 10' em or more.
16
T H E FOUNDATIONS Of NEUTRON TRA NSPORT T HEORY
where (2.400)
x (£( X' , x) K' . VE<X, x) - £(X, x) K • V£(X', x)]
T"
"" ..!i:..L a+(X. K, s) a(X ', K'. s) J dJx c·· · 2mLJ x- ..· ,
CX
·-
XI
(2.40b) )(
x:".
x IV£ (X, x] . V£(X', x)1
(2.40<)
The term & represents the sum o f neutro n kinetic energies at every phase point . T his term will serve the useful purpo se (If determining the ncutro n slates between whic h collision-induced transitions take place. The term I, as will be seen presently, describes the tra nsport of neutrons from cell to cell. Th e term T" represents an apparent infinite cont ribution to the Hamiltonian. It is surmised th at this term is act ually mea ningless and sho uld hencefo rth be ignored." The Hamilton ian now appea rs as
H - 1" + H '
(2.41)
H' ... I + H. + V
(2.42)
with T his par ticular decomposition of II is mad e becau se we a nticipate t hat H ' is impo rta nt in con nection with collision processes only. In general it is not true that V is concerned solely with the effect of collisions on the variation of F. If th e particles interact with "externa l" force fields, or with each other or o ther kinds of particles thr ough forces charactcr izcd by effective ranges subs tantially greater than L, then a po rtion o f V should be inco rporated into the description of "transport". 1 1 T he Jong·ran ge part of Y wiD then provide smoo thly varying forces leading • Ktc ping T" would mean considering the expectation value Tr Dfr '.eJ (d. Eq. 2.SOol). To the same order ofapproxi marioo made in lhesubsequent analysis of the effects of p3r1icle stTeaming (Section C) one can demonstrat e that the co mmu tator, [T', a+<x, X) o(X, XU, vanishes identically. Th us, for the purpose of the pr esent derivalion the term T" actually Jives no contribu tion in the description or tran sport.
A T IUN SPO ll T EQUATI ON IN PHASE SpACE
17
to curved trajectories for the particles between impulsive events. For the present discussion, however, we shall assume that Y represents only extremely short-range interactions. The operators T and H' therefore will give rise to tran sport and collision phenomena respectively. It will be desirable to treat the effects of transport and collisions separately. To do thi s we first write V(r) _ e-,·(T' +Wl/' _ e-I,T'I. e- I'WI' Jft)
(2,43)
where J(r) is a unitary operator to be determined by the equation, oj
- u(t) J
(2.44)
01
and the boundary conditio n J(O) _ J,[being the identity operato r. The function &1(1) can be represented in 3 var iety of ways. Two such examples are u(1 ) -
L [H'
"
and 0(/ ) -
j "
- V ·...(I ) e'":" H ' c-"r/l U'(I )]
• (it/"'t't · L • •__ 0 nI m .
1H" [1' H"] '
•
J
• •
(2.45)
(2.46)
• +. ,, 0
where
U'(t ) .. e-""'I'
(2.47)
and (A. BJ. den otes the nth order commutator of A and B, i.e. [A, BJ. - B
lA, BJ, - lA, BJ [A, BJ, - lA, lA, BJ)
(2.48)
etc. Making use of lhis operator decomposition in Eq.2.36, we find that
Eq.2.27 becomes £(1 + ,) . L - ' T/ •• U(,) D(/) U+(,)
_ L _ 3 Tr c" r /, el e-r» U'(T) J(T) D(I) J +(T) U' +(T)
(2.49)
where use is made of the cyclic invariance of the trace. The dependence of F and e. on X and K will not be explicitly indicated when' no risk of confusion is incurred. 2 ~Yjp
18
T ill' fOUN O AT IONS O f NEU T RO N TRAN S POR T THEO RY
T hus far we have proceeded formally with out considering the str ucture of t he equ ation we ult imately wish to obta in. The fact that th e neutron tra nspo rt equa tion, Eq. L t, is a first-order linear differentia l eq uatio n in time suggests that. to derive a similar equa tion for F, Eq .2 .49 sho uld be examined for some sma ll time interva l, 1'. O ne ca n anticipate th at there will be a ra nge of interva ls. say 1'1 < T < 1'1, in which it is meani ngful to decompose Eq. 2.49 into terms describing eithe r t ranspo rt or collision effects. The description of transport is expected to be valid so lon g as T is less than some upper limit T 1. whereas the descriptio n of co llisions is expected to be valid for T greater tha n somc lower limit 1' •• These limits arc rather ill-defined a t this poi nt, hut a q ua litative estimate for 1'1 is suggested by th e requ irement th at ((JJF/t:t 2 )/(N'/ ot)] ~ 1'; 1. and for 1'. one may take th e neutron-nu clear interaction times to be discussed later. Accordi ng to the above considera tions we will treat l' as a small but finite interva l. Then F(1 + 1')
=
j, [T' . e.I + L - ' T r {el+-
"
I' [T' , el. I} x ._1; (j,W m! L...
---
x V '{,) J(, ) D(t) J+(,) V '+(,)
~
L-' Tr
{e, + :
[T' ,
e, J}
V'(,) J(,) D(l ) J+(,) V '+(,) (2.50)
Fo r sufficiently sma ll l' such that all terms in the m sum can be neglected , tr an sport is described by the seco nd term in Eq.2.50. Since this term is already proport iona l to l' we will keep onl y the leading term in the transformation V'(, ) J(,) D(t) J+(,) V '+(,)
~
D(l)
+ 0(,)
(2.51)
For th e first term in Eq.2.50 we witt ignore the effect of the ope rato r J. Th is ap proximation is justified by the fact that J(,) = 1
+ 0(,')
(2.52)
In more ph ysical term s the neglect of Jhere implies that in trea ting collisions in a given cell th e effects du e to par ticles outs ide the cell are ignored . Eq . 2.50 now becomes F(t + r] '" L -' Tr
{e,V'(,) D(t) V '+(,) + ~ (T' , e ,J D( t) }
(2.5Oa)
The present approximations result in a co mplete separa tion of the effects of T' and H'. and hence will lead to a transport equation in which
A T RA NS P OR T EQU A.TlON I N P HASE S PA CE
19
term s affected by transport or collisio n pr ocesses ente r independently of each other. This does not mean, however, that the momentum and spatial dependence of the solutio n is decoupled. In the representat ion which diagonalizes (I I th e first term in Eq.2.5Oa may be arranged to give
Tr e. U'(,) D(I) U'+(, )
Z
L H(X, K) D..(I) IU~,(')I'
.'
(2.53)
where we ha ve ignored the off-diagonal elements of th e densi ty mat rix." Since U'Cr) is un itary ,
IU;.(,)I' • I - L ' IU;.(')I'
(2.54)
•
where th e pri me on the summation indicates th at terms for which n "'" n' are to be excluded. Then Eq.2. S3 becomes
Tr
e,
U'(,) D(I) U'+(,)
z L' F(I) +
L 0••(1) IU;·.(,)I'
IN'ex, K) - Hex, K)]
(2.55)
Here the occupation nu mbers N' and N denote the eigenval ues of {II in the states In' ) and In) respectively. Combining Eq. 2.SOa and 2.55 we have i !FCX. K, I + ,) - FCX. K, I)J , - ' - - Tr [r, e,ex, K)JD(I)
.L'
~
where
L -,
.'L
W• .(,) D..(I) (N'( X. K) - Hex, K )]
(2,56)
W...(T) - IU;·.Cr)ll/T
The first term will be replaced by the time derivative of F, although in a strict sense it should always be th ought of as a finite difference. However, even for T 1 ::=:- 10- ' sec., coarse-graining of the time domain is not likely to be significa nt in most investigations of physical syste ms. T he remainder of th is cha pter will be devot ed to a reduction of the remaining terms in Eq. 2.56. It will be shown that the second term provides the co nventional descripti on of neutron transport, whereas the terms on the right-hand side pro vide the description ofintcractions. With these reductions, Eq .2.56 will then bea r co nsidera ble similarity to Eq.1.1. • This is equivalent to the so-called Random a p r/()rl Phase Approximation
which has been studtcd only in lI'Cfy special c:ases.I ,
20
T H E; fO UN D A.Tl O N S OF NEUTRO N T R A.NSPO R T TH EO R Y
C. The S tre aming Term In o rder to exhibit in Eq .2.56 the transport term that appears in Eq . 1. 1 it is necessary to evaluate the co mmutator, IT'" ,(X. K)]
- ~1 L 2mL
x
(a'(X',K', x') a(X" ,K" , x')• • ,(X, K») ,
X·. ll· '. ..•
..··.6·
J d1x eb .(lt"-K·)[E(X" , x) K" . VE(X', x)
- E( X', x) X ' VE(X" , x)]
(2.57)
On e ca n readily show, using the commutation relations given in Eq. 2. 14, th:1t [a+(X ' , K ', x') a(X", K" , s'J, a+(X, K , x) a(X, K, x)] ;: (a +(X', K', s) a(X, K.. s) du " du." - a+(X, K, s) a(X " , K", s) 6xX' 6"d 6...
(2.58)
T hus Eq. 2.57 assumes the form,
(r, . ,(X, Kl] - D - fro where
(2.59)
'.'
!J = - 'L a+(X' , K', s)a(X, K,s)J d1x e'· ·(K-K·) x 2mL 1 x·...·.• x I£(X , x) K ' V£(X', x) - £(X', x) K ' , V£(X, x)]
(2,60)
With t he help of Eqs.2.4 and 2.5 we find !J ...
~ ",L )
r
a +(X' , K'.s)o(X. K.s)J d1x E(X, x)EtX' ,x) c'· ·''' -ln x
X ·. k ·• •
xi.{x{,xI - Xj + ~) - .(x,-x; - ~)] - X;('(XI- x, + ~) - '(XI- x, - ~)]} L (±)(X, + XJ) x ",
x ((a+(X,
+ L. K J) - a'(X, - L. Kj ») a(X" X,)}
12.61)
A TJU NSP OJt,T EQU AT IO N IN PHASE SP ACE
21
where th e upper or lower sign is used depending upon whether MJ - (L/ De) (XJ - Xl) is even o r odd respectively in the Xl sum. In writing t he argumen ts of the operators we have suppressed the spin labels as well as th ose co mpo nen ts of the tw o phase po ints (X' , AK') and (X, AK) which are the same. The contributions from the K ... K ' terms in D and !J+ are easily recognized. Tak ing only these terms into account the transport term in Eq.2.56 becomes i A - Tr rT", , ,(x, K)] D(r) - - K · VF(X,K. r) AU m .
(2.62)
The gradient ope rat ion act ually rep resents a finite difference in the sense that
c1flX, K .I) _ L-J x BK, x Tr L {[a+(X I + L , X,.XJ.K,s} - a+(X I
-
•
L , X"X" K, s»)a{X,K,s)
+ a+<X,K.s)[a(X. + L,XhX, .K. s) - a(X . - L, X ,• X" K, 3)]} x x D(t)/2L
(2.63)
Eq.2.62 is seen to be the co nventiona l strea ming term which rigorously describes the flow of nonrelativistic, massiv e particles in free spa ce in both classical and fine-grained quantum the ories. The contribution from X +- X' terms in () do not lead to any readily interpretable result. However, they appear to describe the correlation of neutrons with different momenta in adjacent cells. Such effects may be regarded as corrections to the streaming term due to space-momentum coupling. Because these terms do not a ppear in a fine-grained theery, it is reason abl e to conclude that the co upli ng is a direct consequence of coarse-gra ining. Indeed, as L becomes arbitrarily small, th e separations between 11K, the momentum po int und er considera tio n, and othe r poin ts in the momentum lattice approa ch infinity. One may then anticipa te the K' sum to collapse to only the K - K' term. It would be of interest to investigate the qu ant ita tive effects of these terms. We observe that a typical term in () is
Tr a+(X,
+ L, K;) o(X" K,) D(I)
= ( '1'(1)1 a+(X,
+ L, K;) o(X" K,) IY'(I)}
(2.64)
22
TIll: tOU NIlATlONS OF NE UTRON T RANSP ORT TII EOR Y
which has the a ppearance of a reduced density in coarse-grained phasespacl: coordimucs (see Eq ,2.32) . Similar quanti ties have OCCII cncountcrcd in recent studies of many-body problems such as grou nd state energies, the nat ure of elementa ry excita tions. and th erm od ynam ics. I ' Th us the formalism and the techniques developed for those approaches " to a sta tistical theory of interacting particles may well be applied in t he present context to th e und ersta nding of the K #- K' ter ms.
O. T he Collision Tenn. We have derived in Eq. 2.S6 a kinetic eq uation for the coarse-gra ined neutron density F( X, K, t). The effects of neut ron-nuclear inte ractio ns are described hy Ihe transition probability per unit time, IV. ,., With regard to th e neut ron tran sport eq uatio n the reacti ons of primary interest
JY;...
W::1 N(X.. KJ )
• All interactions ere t.reated u binary collisions.
(2.65)
A TRANSPORT EQUATI ON IN PHAS E SPACE
23
where WK J is the reduced transition probability for the capture of a neutron at phase point (X"irK),· Note that if initially there is no neutron at (X, K), the present interaction, would have zero contribution, The sum over n' in this case implies a sum over all states in which the total neutron number is one less than the number in the state In) . It is effectively a sum over all X, and KJ ; however, because of the factor tN'(X, K) - N(X, K») all terms would vanish unless X. "'" X and KJ = K, in which case the factor becomes - 1. Hence,
I, - L-' ED••W.·.IN'(X,K) - N(X,K»)
.'
- - w~F(X, K, I)
(2.66)
Consider next the scattering'[ ofa neutron in cell X, from initial wave vector KJ to final wave vector K j • Thi s process is equivalent to the absorption of a neutron at (X j , K J) and the creation of one at (X" K ,), so that
W:.• - w:.,••,N(X" K,) [I -
N(X~ K,)
J
(2.67)
where ~J"I, is the reduced transition probability for the scattering of a neutron in cell X. from K J to K l . • The factor [I - N(X K.)!2] is the number dependence associated with the creation process" and we have assumed that the neutron spin orientation is random, i.e.
(2.68)
N(X, K. s) - N(X. K)/2
with N(X, K) equal to zero, one or two. Thefaetor [H '(X, K) - N(X, K)]
can be either + 1 or - J in this case depending upon whether (X" KJ) = (X, K) or (X" K ,)
-
0;
(X, K). Thus we find
'
L'I. - ED•• W;.[N'(X. K) - N(X,K») - -
~ D. N(X. K) t w:..•{1 - N(X~ K,)
+E
---
•
J
D.{I - N(X.2 K)J E"J w:.,•• N(X. K,)
[(2.69)
• Spatial dependence of iii will be suppressed for simplicity.
t The distinction between elastic and inelastic scattering is not necessary here. In the next chapter, the two processes will have to be treated separately• • We isnore here any spin-dependent effects in the ICatterina.
24
T HE FOUND AT IO NS OF NEUTR O N T R ANS P O IlT THEORY
It is obvious that the loss term in I, represe nts t he scattering of a neutron o ut of the phase poi nt (X, IiK) white the ga in term represents the scattering of a ne utron into (X. ilK), Whenever ](J "" K, - K the net contribution vanishes as expec ted . In Eq. 2.69 we ha ve terms proportion al to t he expectation value of a product of two nu mber operators which is a highe r-order de nsity a nd can be de fined as
F,(X, K, X', K', t) - L- . Tr .,IX, K) . ,IX', K',) D(I)
(2,70)
It is conv emion al tocall H x, K., l) asingletdensity and F ,(X, K. X' K', I ) a doublet density. The appearance of the doublet is a conseq uence of the qu antum sta tist ics. and hence these ter ms can be expected to van ish in the classical limit. To show this we need to transform Eq.2.69 to continuous moment um space according to Eq .2.30. It is fou nd t hat
L
I , - ,' .(K) d ' K - J,IP) d' P
(2.7 1)
I .d ' K
and .!,(P ) -
- /(X. P. /) I d 'P, w,(P - P ,)
+ I d'P,jIX, P"I) ...(P, - P) (2.,W + -2 -
- -(2;111) 2- J
f' f'
d P,fl(X. P, X, Ph I ) )l',(P - P I) d PJI1(X, P, X, P" I ) )l'.(PJ - P)
( 2,72)
Obse rve that w.... x,. whe n we summed over the elementa l volume d J K" becomes a distribution,
L wk.. Ii ,
'"
w,(K - K,) d JK. _ w.(P _ P,) d JP,
(2,73)
.. . . .S' K.
Th us in the classical limit (II _ 0) o nly the first two terms in Eq.2.12 survive. l astly we co nsider the contributions ari sing from fission processes. Let a neutron be absorbed at (X" KJ ) and th e resulting reac tion produce J neutr ons with momentum distribution specified by a set o f wave vecto rs, (K,J.I' We shall neglect delayed effects so these neu trons are all emitted in the spa tial cell X, and within the time interval r l ' The t ran sition probability for this event can be written as W: . =- w~' _ I.,I.I N(X" KJ) G(X , {K,ll)
(2.74)
A TRA NSPORT EQ UATION IN PH ASE S P AC E
2S
where w~J_ t",I J is th e red uced fission transition probability, and
G(X" (K,h) _
n [I
_ H(X" K,)]
IXIII
(2.75)
2
is the degeneracy factor which co ntai ns a product of J factors according to the J wave vecto rs in {K,}J ' It has just been sho wn that these terms lead to a dependence on higher-order densities which vani shes as Ii .... O. Since we are prima rily interested in the fission co ntri butio ns in the classical limit we sha ll rep lace G by unity in the followi ng. Hence
L) 1, - -
+
L D••N(X, K) L W:_I~IJ ( I •
L D.. L •
-
J.cI"IJ
J. IK ,I.I
W~'_ IX"JN(X, KJ) (l -
,
"
f QKI-',J) .·1 J
s....) . Lt'lQi, IKoI.l .. 1
+ [D..N(X. K) L W:"' I~I.lL (t'l - I)QKI"M •
+
1. 1...1.1
r D.. •
L Nex, K
J)
~ 1 "'IJ
•• 1
W:,_I~I,
,
L t'lQillldJ
(2.76)
.· 1
"
where QiIKM is t he probability t hat of the J neutrons emitted with mom entum d ist ribution {K.l J th ere are exactly t'l neutrons with wave vector K. The co llision terms, Eqs . 2.66, 2.69, and 2.76, alo ng with the transpo rt term , Eq.2.62, ca n be entered into Eq . 2.S6. Keeping in mind th at time and spatial derivatives are act ua lly finite differences, we may exhibit the resulting eq uation , in the absence of quantum effects, as
(~at + ~m K· V,
+ W)F(X, K,t)
- L F(X, KJ, ' )[WiJ-K+ L ",~,4 IK"" t IlJ
J. CK,I.I
.·1
t'lQiIIlM ]
(2.77)
where
(2.78) Fo r this equation to be directly compara ble to th e co nventiona l neutron transpo rt equation, it is necessary that we transform to continuous
26
TIn
FO U NDATIO NS OF NEU T RON TRAN SPO RT T H EO R Y
moment um spac e a nd express the transition pro bab ilit ies as cro ss sect ions, The presen t tr an sport equation then become s
P.]
" + -I' , V + - I , (P) f{X. P, t) -r[ ct m m
~ f d >po !: I(X, P ', I) [EJP ') F {P'
- P) +
In
2~(I") L '~(P', P)] .I,.
(2,79) where ~
m_
.., - - w P
l.~(P'} ' (P' -
P) -
!!!.. IY.(P '
r
- P)
The frequency Jf. is introduced such that S:(P', P) d 'P is the probability tha t a fission induced by a neutron at P' will prod uceexactly J neuIron s, /\ of whieh have mo menta in d'P about P.
E. Effect of an External Field In closing th is chapte r we consider briefly the effect of a time-indopenden t external field. ~... (x),· The Ha milton ian H is now modified by the additio n of V.... (2.81) In the decomposition of H we shall grou p V... with T' so that the effect of the exte rnal field appears onJy in the commutator. [Yn t . ~, (X, K»). It will be co nvenient to evalua te the present commutato r in a manner somewhat different from the way in which the streaming term is derived, We note that an integral rep resentation o f the number operator e I is
. ,(X. K) -
I d' ''' d >x ~'(X, K. x) 9'<X, K, x') .:(x) .'(x1
• An uample or lP•••(z) is the gravitational field.
(2.82)
A TRA N SP OR T EQU ,\ T I ON I N PHAS E. SPAC E
27
Then from Eqs.2.81 and 2.82 [VUlt ell =
f d- x dJx' d Jx" 9'· (X. K, x) tp(X, K, x') f •.,(X" ) x x [.; (X" ) .,.(x'1, . ;(X) . ,(x')]
- f d'x d'x' .'(X, K, x) o(X, K, x') .;(x) .'(x1 x x [~.. ,(x1 - ~",(x)l
(2.83)
Because of th e cell functi ons the x and x' integrals onl y extend over the cell centered at X. If now f ... is a slowly-varying function over a distance of order L, it may be approxima tely represented as
~...(x)
se ~...(X)
+ (x,
- X,) (~... ) a XJ
(2.84) lItJ- ZJ
Entering this expression into Eq.2.83 we find
W.... . ,]
", L-'(O:", ) vXa
fd'Xd'x'E(X,x')E(X,x) x
"._Z.
x (x; - xJ 'P;(x) 'PAx') e- ill: ·b;·- :IIl
- ,.(-o~ax}... )
" J - XJ
o£>lX, K) .
(2.85)
aKJ
The effect of an external field on the tran sport equation, Eq.2.56, is thus described by the addition al term
__,_. Tr I
n
Vu"el (X. K)J D(t) '" ( O~'" ... ) vX}
" J" llJ
oF(X, K, t)
"'p
(2.86)
v J
which, like the strea ming term, is a familiar result in fine-grained theories. Note t hat in the present instance coarse -grained mome ntum is treated like a continuo us variable. This procedure is acceptable so long as KJL )0 2n. which then represents a lower limit in the choice of cell size. A corresponding upper limit is determined by the truncated series express ion of fOIl ' Eq.2.84.
I. S.Ono, Prog. 1'Mor. Phys. (Japan), 11: 113 (1954). 2. L.I. Schiff,Quantum MechOlliu, McGra w-Hill Book Co., Inc., New York,19S5. 3. P. A. M. Dirac, The Principles ol Qu(Ultunr Mech(Ul/cs, Clare ndon Press, Oxford, 1958.
28
Till: FO U NDA TIO NS OF N EU T RON TRANS PO RT T HEORY
4. J. von Neuroan n, MafMmotical FDUNlaJIoM 0/ QlUUlrum Mtt:Jumi~s. Pr inc:w:ton Uni\lersity Press, Princeton, N.J., 1 9~~. ~. R.C.Tolman, 7'M Prjnciplts 0/ Statistical Mu hamcs, O"ford University Press, Oxfo rd , 1948. 6. E. W.Gi bbs, £/~rn~ntary PriMiples in Stafisti~u/ M~chanicJ, Yale University Press, Ne w Haven, 1902. ba . H.G . van Kampen, Procerdi"lJ 0/ tM J"Urt/utional Syrnposillrn 0" Tra"spor' Pro~fjs~s tn Statistical Mu hanics, Inte rscience Publi shers , New Yor k, 19SK. 6b. U.fano. &I)s. Mod. Phys., 19 : 74 (1957). 1. H.M ori, Prog. Th~or. Phys. (Japan), 9 : 473 (19~3). 8. A.w . Scene, Phys. RnJ., 105: S46 (1951). 9. F. Bloc h, Z~ilJ./. PAys_, 52 : SS5 (1928); 59 : 208 (1930). 10. L A. Uehling and G .E. Uh1erJbeck, PAys. h r ., 43 : 5S2 ( 1933). II. E. P. Wigl1C'r, Phys. Rn., 40: 149 (1932) ; J. H.I ,..,ing and R. W.Zwanzig. J. CI~m. PA)'s., I': 1113 (1951). 12. R. K.Osborn, Phys. Rn., 130 : 2142 (1963). 13. R.K .Osborn a nd E. H.KleYans, ANI. Phys., 15: lOS (1961); E. H. Klevans, Thesi s. Un iversity of Mictlipn, An n Arbor, Michigan, 196214. S.Chand nlsckha r, RDdiIJIlN TrlUlS/rr, Dover Publ ications. Inc., New York ,
1960. IS. H. G o ldstein, FwJ~"tal Asp«lJ 0/ /U«tor S";~ldillg, Addison-Wesle y, Reading, Mau ., 1959. 16. Sec, fo r examp le, G .W.Uhknbeck an d G. W. Fo rd, uclur~s ill Stalisti~u/ M~rJl(IlIirs, Arn«K:an Mathematical Societ y, Providence, R.I., 196). n.lo\/ un Hove, PAysira,11 : 517 (1955); 1. Luninger :md W.Kohn, Phys_ R~r •• 109 : 189:! ( 1958). 18. T.O. Schultz, " Qua ntum F ield Theory and t he Many-Body Problem", Space TlXhno lo8)' Labo rat ory Report STL/TR.(,().()()()().GR-332, 1960; L.P.K3wnolf and G.Baym, QlIQ1Jtum Statist/co/ M~chanics, Benjamin, Ne w Yo rk, 1962: D.P incs, Th~ Many-Body Prob/~m, Benjami n, New York, 1962.
ill
Neutron-nuclear Interactions: Mainly Nuclear Considerations In this cha pter we will undertake an investigation into the effects of
neutron-nuclear collisions upon the balance relation (2 56). Many kinds of nuclear reaction s may be initiated by such collisio ns. However, we shalt concentrate our atten tio n on only a few of them. There are at least two rea son s for thi s restricti on . In th e first place there are only a few such reactions that can be dealt with at all adequately by the rath er elementary analytical techniq ues that we en visage here. In the second place our ma in emphasis is on an illustrative investigation of the basi s
of the theory of the distributi on of relatively low-energy neutrons. Consequently, fission. radiative capture, and elastic scattering are probably a n adequat e: sa mple of rep resentative a nd significant interactions. There are two types of effects that must be taken into account in th e description of a collisio n process-the spec ifica lly nuclear effects and the effects of th e macroscopi c medium. The former depend upon nu clear force s, while th e latter depend upon the non-nuclear interactions of the nu cleus with its surro undings in the syste m. Because 'o f their importance. it is essential that the present discussion of the tran sition probability pe r un it time. W•.•• be made sufficiently genera! to include both nu clear and med ium effects. The specifically nuclear effects can be t reated by means of the steadystate theory of nuclear reaction s as, sa y, prese nted by Blatt and kopf ' and reviewed by Lane. J However, it is no t clear that che mica l binding effects ca n be conveniently grafted onto this elegant and rigorous theory of binary nucl ear reactions. At the opposite extreme we ha ve straightforwa rd perturbation methods for a successive approximation eval uation of W•.•• Althou gh such an a pproach enables easy incorpo ration of the medi um effects. this extreme must also be avoided since it a ppears that only po tential scatt ering can be readily and usefully
weiss-
29
30
TH E FOUNDATIONS Of NEUTRO N TRANSPORT THEORY
tr eated in this manner. For indirect processes, many of which co nt ributc s ign ificantly to (2.56), conventional perturbation methods are therefore not adequate. As a compromise, we will follow an approach originally developed by Heitler ? in the stu dy of photon interaction s with matter. The theory, someti mes known as damping theory,3 -' is sufficiently elementary so th at both medium and nuclear effects can be considered and is, at the same time, sufficiently sophisticated to aUow a useful exposition of the essential features of both types of phenomena. From the pointofviewofthe neutron transport equation, some ofthe co llisions of importance are those that result in capture, elastic and inelastic scaittcnng, and fission . We do not regard as an essenti al part of our purpose the detailed investigations of the specifically nuclear effects of the se reactions since very complete and thorough discussions lire available in the literature." For this reason many aspects of the following calcula tions will not be explored as fully as possible. Moreover, not all the reaction s are treated with equal emphasis. It will be seen that considerable detail is presented in the study of rad iative capture and elasti c scattering. wherea s the discussion s of inelas tic scattering and fission pro cesses are brief and, at best, descriptive. Th is by no means is intended to imply the relative importance of the reactions in the transport equation in general, although there arc special cases in which the effects of a given react ion or reactions are suppressed. What we attempt here, in essence, is to illustrate another a pproach to nuclear reacti on theory that is capable of producing, at least qualitatively, the co nvention al results and also allows a systematic treatment of the external degrees of freedom of th e nucleus . t
A. Formal Development of the Transition Probability The task of evaluating the transition matrix W. '. is essentially th at of determining the off-diagonal matrix elements of the "temporal evolution " o perato r U '(T). Thus far. the representation in which the matrix clement s a re to be calculated is onl y required to diagonalize th e neutron • Sec reference I for a genera l drscusstcn of t heory of nuclear reaction s. For
those aspects or particula r interest in reactor physics sec Weinberg and Wigner.'
t The present approach has also been employed in recent studies of photon transport in dispcrsiv e1 media . a nd of line shape theory." ·
NUCLEAI. CO NSI DBI. AT I O NS
31
number operator; otherwise it is unspecified. With the diagonaliza tion of el' we o bserve that 8 , the kinetic energy of neutrons withi n cetls.elso becomes diagonal. To develop a general expression for U;.(r) some consideration must be given to other degrees of freed om of the system. For the moment they need ONY to be introduced formally. detailed discussion being necessary only when a specific reacti on is to be investigated. We tberefore further speci fy that the abo ve representation also diagonalize the operator H., i.e. (3. 1) (I + H;) In) - <, In) The states In) are assumed to form an ort honormal and complete set. Alt hough H. describes the entire system exclusive of neutrons, the only part of it that will req uire our subseq uent attention. in view of the reactions of interest" is that relevant to the description of nuc lei (and of photons in the case of resonance capture). The operator U'('r). as given by (2.47), has as its Lap lace transform G(z) _
I' d,
J,
~
V '(,) e".." _ (z + iH' )- '
(3.2)
where H' ... I + H. + V. The fo rm of th is opera tor is particularly suitable for developing approximations. As will be seen, the present calcula tion provides an approximate exp ression for the otr-diagonal matri x elements of G(z) . O nce G•.J.z) is known , the inversio n then gives
U;.J..T) "" - I . :2:rn
I'H.
dz G... (z) eUI•
(3.3)
J - l ..
From Eq. 3.2 we have the matr ix equation
(: + it.)G_ + i.E V_.G..... - 6_ ,"
(3.4)
It will be convenient to treat the diagonal and nondiagona l parts of G sepa rately. For th is purpose we int roduce an operator Q such that
G••• - G.. Q... G•.•. for n JI' 11' . The diagonal elements of G then satisfy the format relation G..(z) -
(z + i,. + '2 r.)-. .~
(3.5)
(3.6)
32
where
THE FOUNDATIONS Of NEUTRON TRANSPORT THEORY
•
- r . (z) "" V"" +
2
I:
(3.7)
V••' G. ,•. Q•.•
•.".
An essential step in the development is the determination of Q"" .. From Rq. 3.4 we obtain
L V••.•G•..•.. Q• ..•. .........
[, + i(,.+ V••l)G•• Q••• +iV".+i Q•• .
- iV••. - i
=
+i
.......L
,. '
~ O ( 3.8 )
V••.. G. ..." Q...,..
LV"".. G. ... .. Q. ... G"" Q•• . ......
(3.9)
A useful, approximate solut ion to this equati on can be obtained by expanding Q as a power series in V. and ignoring the dependence of G Oil V. This is readily accomplished by writing V_ AV (3.10)
and considering A as a bookkeeping parameter ultimately to be evaluatcd at the unit point. We find, to second o rder in A
L V.... V. ...' ..."....
.
Z
+ IE.·· + -
and
i. 2
y,,(z) ~ iV••
if!
2
(3 .11) i' . "
+ L IV,,:I'
.·".Z+IE•.
(3.12)
With r .(=) given by Eq.3.12, the diagonal clements of G arc now explicitly determined by Eq.3 .6. The higher-order terms which have been ignored can be investigated. But we anticipate that the predominant features of the reactions of interest are usefully described by the two terms in Eq. 3.11. To this order of approximation, the off-diagonal elements of U '(T) are given as
If"'·
U ;. ,(T) ~ - . 2,..-u x
dz A.(z) A.(z) en /It x
r -I ",
l"...- ......Lv••.. .. V••.•. A• ..(z)] ,
(3 .13)
NUCLEAR CONSlDEJl.ATIONS
where A.(z) -
[
z + I~. +
'2 /~ y.(Z)
I'
33
(3.14)
The quantity y.(z) is the width and shift function for the energy level corresponding to the nth eigenstate of the system. It will be shown that when evaluated at z :oa - ie. the real part gives the shift of the unperturbed energy level du e to interactions. while its imaginary part describe s th e width and hence the finite lifetime of the state In) . The only reason that we cannot neglect these quantities completely in the present problem is that many of the neutron-nuclear reactions we are concerned with proceed via excited sta tes which are known to be significantly broadened. At the same t ime. in the systems of predominant concern here. it is most probable that the interacting nuclei are ini tially in their internal ground states. If also we consider onl y time intervals (T) long compared to the lifetimes of the intermediate states.· the final sta tes can also be taken as ground stales so far as the specifically nuclear degrees of freed om are co ncerned. Hence for our purposes the only sta les who se widths and shifts will ha ve apprecia ble influence on the co llisions are the intermed iate states. We will accordingly ignore the width and shift function s in A.(z) and A• .(z) in Eq .3.14. thus
U~ •.(T) se -1.
2rI,
f....
dz 8.(z)
x [ -IV••• -
where
B•.(z) e·'" x
I _I.,
LV. .. v..... l ...(z)] ........ '
(3.15)
(3. 16) In Eq. 3.15 the two terms represent the effects of direct and two-stage (co mpound nu cleus) processes respecti vely. It is anticipated that the first tenn will suffice to describe po tential scattering. whereas the seco nd term will lead to a description of resonance rea ctions (including fission). For elasti c scatt ering. both terms mu st be co nsidered simultaneously thus enabling an examination of potential scattering. elastic reso nance sca tteri ng. and the interference between them. The evaluation of the tint int egral in Eq.3.IS is a simple matter. and • Th is provides . qualita tive lower lim il for T. ) 00D0nt/Y1 .,
34
TIll
~O U NDATlON~ OF NE UTRON TIlA NSPOIlT T H l U KY
we find (3. 17)
where 11..,. = r• • and 11m... -= ~ . ... pressed as a convolution!" I/( r ) = _
If'.. · I
2,; = h
J
1- ' ~
dz ii.(: )
The second integral ca n be ex-
1'. "
B•.(z) A•.-t: ) eU •A
dr·A• ..(r - T') [
dr" 8 e-(r - r " ) 8.l r " )
O . IK)
where (3. 19)
and A.(r ) is the inverse: Laplace transform of A.,(z). A d iscussion of the funct io n A.(r) has been given by Akcasu.s We shall ad opt here a !>lightly d ifferen t approach. Since
we then have 13.20)
with
I f.... d: - -
,\.(.,(r ) ... 2.'"ti
,_1.•_
14'. ··1: ) B..(: ) c""
- _'Dri1_f.+I'" d=f"dr' -
· f
A."(1) B.(z) e'« - "' /A
0 h
, _ I...
d,' - A ••,(r ) 8..(1 - , 0) • A
The funct ion 8.( r - r' ) is given by (3. 19) a nd vanishes fo r , ' > r, so .t.{.( r ) - e- ,.. ·/Af· d r ' ..4• •.(1) e -'..·/A
• A
13.22)
Next we would like to extend the upper limit of integration to infinity. Th is procedu re is justified if A•..(r ') is negligible in the region r ' > r, and such is t he case jf in .4•..(:) the quantity ).. ..(:) is essent ially independent of =a nd has an imagi nary panmuch larger than 1ft . A.. W\.·
35
NUC LE Aa. CONS I DEa.A TIONS
sha ll show in the followin g, 1m Cr...) is a measure of the reciprocal of t he lifetime of the interme diate state, In" ) . Hence, by writing
(3.23)
U:..'
it is implied th at one is only look ing a t tha t part of which describes the completed transition fro m initial sta te, In) , to fina l sta te, In' ). We now obtain . ) A- ( . ) If"'-"" }t{'1 h(T) - ie-I••,{, [A- . " ( - IE. . " - It. .. e (3.24) t. -
F• •
Fo r a fixed t . the mos t impo rtant contribution a rises when t • . - 1'. ,thus h(r) - i(~UI..,) - 1 A•..(-it.)( 1 - e'"-')e- '-' (3.25) Combining this resu lt wit h (3. 17) we find
y.... V. ... .
' . + (hI2) Y. ·{ (3.26) Our res ult sho ws that the fu nct ion r... is to be evalua ted at - it •. The behavior of this fun ction al on g the ima gina ry a xis is rea dily examined by co mpa ring the boundary values of r . :·(z) as the axis is appro ac hed from both sides of the complex plane. O ne finds
lim :<_0 +
j/r
2
r .(x
lim ifr r .( - x 2
+ iy)
_ is.(y)
+ ! 1 ~(y)
(3.27)
!2 1'.{Y)
(3.28)
+ iy ) .. is.(y) -
:<_0 _
2
• This is merely to say that we anticipate the conserva tion of cne ro. To see tha t ' ... 0, consider
h(T). Cor large T. is sharply peaked about W _ lI(T) -
' e- '-'(frftl_'r
l )(
x [A e"<- it . )( 1 - e' -") -
_'_0"
_ _ ... .i.... e - ...• [ so IhI.:
fir~
term doll1inah.'S,
i" ,... (; iJ::~')e'-" + O(w~.)]
.
IA _--< -iF)T _ i
(,OA.'.) + O(W _,)] ilt.
36
T Hf FO UND ATIONS 0' NEUTR ON TRANSPOR.T TH EORY
where .' .(Y) - V. -
r .(y) ~ x
L IV••·I' '" ---..!.-
.....
y
(3.29)
+ ' .'
..L. . IV. ·1' 6(y + , •.)
(3.30)
In obtaining these expressions fro m (3.12) we have made use of the relationlim _1_ . =- _ i!iI' ! + :z6(y ) .~O X+ IY y
(3.3 1)
where (P(l lx) is the principal value of l /x. Thus in crossing the imaginary axis the value of Y. _ changes by an amount 2r . ..(y) which van ishes everywhere except at y - - '.0. no '" n", Hence r . ..{z) has branch points at z - -it.... no 'It n". T he tran sition matrix is o btained once U;" is kno wn, l.e.
IV..• 2,h V••. -
L .·· .....·
t . .. -
\'(1-:zrw... C''',w..,)
V. " V. .... " ( ') t . + - Y. " - 1£. 2
(3.32)
Fo r sufficiently large T. i.e., cu r ~ I. the last factor in Eq.3.32 is a sharply peaked function about cu =- 0.1 a nd can be replaced by a delta function. The quanti ty cu...'( win always be substantially greater than un ity if the width of the intermediate state is small compared to its energy abov e grou nd sta te, i.e., if " cu.._ > Y._ If such is not the case. the notion of the intermediate state becomes fuzzy and so does the concept of the transition probability per unit time. We have obtained a useful, though approximate, expression fOI W.... It is co nvenient to exhib it t his general result as (3.33) R• .•
-
"
. -.
v•.•.. V. ...
v-
2, V =,,
-
L.
£ • •• -
'.
+ -"Y. _{ .
Th e transition probability is independent of .. See Ueitler, rd. 3. p.70-
t See S:hiff. rd C1'C'fK'C II, p. I98.
2
T
l
(3.34)
- It.)
as one might expect in
N U CL EA I. CONS IDEI.A TlONS
37
the present situation. To develop exp licit cross-section formulas for the vari ous reactions, it is now necessary t o consider in more detail the states (In ) } and the matrix elements of th e interaction , V. The eigenstates (In )} were introd uced in Eq.3.1 simply as a diagonalizing representation for the kinetic energy of free neu trons in cells and for the t otal energy of the " system" with which the neutrons interacted acco rding to a potential V. We shall regard the " system" as an assembly of electrons, photons, and nuclei of various kinds . In th e p resent study we will ignore th e interacti ons between neutron s and the electrons 11 on the ground th at th ey have littl e effect on neutron transport. * Consequentl y, electron ic coordi na tes ap pear only in H •. We will also ignore the ph oton-neutron co upling; however, photon coordinates will a ppear in both H. and V because it is convenient to incorporate the energy of free photons in the former and the interaction of photons with nuclei in the latter. It is necessa ry to take explicit account of the photons only for the descript ion of radiative ca pture ; for the othe r processes to be considered here the presence of pho tons has little influence on the cross sectio ns. Following the above remarks, we exhibit the eigenstate In) as a product of elgenstates a ppropriate to each kind of pa rticle,
In> - IN.. ... >IN.., . ) IN•.•.•>
(3.35)
The eigenstates for the neutron s an d t he labels that characterize the m were in troduced in Chapter II. There it was mentioned that a neutron state In) , denoted here as IH I ...,. ), is co mpletely specified by a set of occupati on numbers for all spin and momentum sta tes and cell labels. From Eq . 2.18,
IHI
•I •• )
= IN(XI ' K I , .II) N(X " K I • .I,,) .•• N(XJ' K J, sJ) ••.)
(3.36)
It is a ppropria te to trea t the photons also by the field formalism. Then the photon eigenstates will be specified by a set of occ upat ion nu mbe rs for all polarization and momentum states and cell la bels,
IHI
••.,, )
= IN(X, .
"I. AI) N(X ;, "I, AJ) ... N(XJo "J' AJ) •.. )
(3.37)
where: N(X, " . A) is the number of pb otons in cell X witb momentum I\s and polari zation .t. Since: photons are bosons, this number can be any positive integer or zero. • The inclusion of I'Ie\Itron-doctron interactions entails no difficulty in princ:ipk .
.11(
1111 1lll lNI> AIIlI N S HI' N U l l Il U N
I Il AN .~ I 'U M I
1 111 0""
Th e eigen"IOI h:" fur the entire co llection of interacti ng nuclei a nd elect rons arc less eas ily desc ribed and mo re cumbe rsome 10 deal with. In the Iir"t plucc.Hk c the neutrons a nd the photons the nuclei a rc not co nserved, \>0 that one is tempted toward a field fo rmalism for their description . But on the other hand. the nuclei may well be localized, as ato ms ho und in crystals. thus ma king the application of field theory aw kward if not obscure. If. in fact, the nuclei (atoms or molecu les) arc in gas p hase. then their treatment in ana logy to th at of the neu trons and photons wou ld he quire appropriate. However , for th e gene ral discussion (more a pplicable to solids and liqu ids) we will make lI SC (}feigenvecto rs whose com p onents themselves are many-parti cle con figuration"pace wave fun ctio ns describing definite numbers of nuclei of definite kinds. Differen t co mpo nents would then descri be different num bers of nuclei of definite kinds . These eigenvectors will be presumed to be ort hon or mal, and it will be furthe r presu med that V has so me non vani shing off-diagonal matrix element s with respect to these rcpresenrations. As a no tat ion we will write
to represent a nuclear state with N(A 1 , 1*. 1 ' k I) nuclei of kind A I with interna l a nd exte rnal states specified by labels l' 1 a nd k I respectively, etc . It is important to keep in mind that the co mponents of thesc vector s are not functions in occ upatio n num ber space, but rather in ord inary configuratio n and spin space. WC will t reat th c various interactions sepa ra tely. Following th e approach o utlined in Chapter II we decompose all interactions into classes acco rdin g to the relative number of particles of a given kind in the sta tes 111 ) and Ill') . This will be seen to be a natural way of chosjfying the different binary neutron-nu clear reactions. Scatteri ng reactions, both potential a nd resonance scattering, are cha racterized by the same tot al number o f neutrons in the final slate as in the initial sta le. This is tru e for both clastic a nd inelastic events. altho ugh inelastic scatt ering rea lly belo ngs to a subclass in which the Dumber of photons in the final sta te differ!> fro m th at in the init ial state. " If th e neutro n and th e photon (s) are emit ted separately in an inelastic scatt ering process, :!ouch an event will req uire a descri ption that allo ws at least t WO intermediate states. • We co ntinue to treat the nuclei in both inilial a nd final stales as in thei r internal ground sta les.
,
NU( ~ LI: AIl.
J9
t:UN S I IJI .k A l ItINlO
Since the present t reatment is restricted to only one inte rmed iate state, our d iscussion of scatte ring will initially be limited to elastic pr ocesses. Later, we will assume that the approximation in which t he co mpo und nucleus decays to ground state by a simultaneous em ission of neutron and photon is adequate for treating inelastic scattering. Rad iative capture react ion s, as well as all other neutron capture pr ocesses which are followed by a decay to ground, are distinguished by on e less neutron in the final sta le than in the initi al sta te. Fina lly fission is a reaction in which the neutron number in the final sta te may be increased by one or more with respect to that of the initial sta te. Thus in the following we shall co nsider radiative capt ure, scattering and fission reactions. Though these hardly exhaust all the inte resting possibilities, they are the main processes that significantly influence neutron transpo rt in many rea ctor situa tions. As an initial step in the red uction of collision terms in Eq. 2.S6 we rewrite Eq .2.S6 as " + -h K' - " ) FlX K t) ( -ul m oXJ " = - V- I
+ V- '
-. -. L
W:.•D...
+
L IN'(X, K, $) -
V- I
W IG ..... .'a£o".. ~
L.
N(X, K, $»)
n
_
V- I
W~.D_
-'"-.-. . ... v- W"D
(3.39)
where we decompose the n' sum for a given n into sums co rrespo nding to the different types of W•.•. The terms proportional 10 W;'~ are all th ose for which the final sta tes contain the same total number as the initial and for which N '(X, K, s) = N(X, K, s) + I. The y are therefore the scatt ering gain contribut ions to the bala nce relation in the binary collision approximati on . An alogously, t he terms co ntai ning W:.~ con stit ute the scatte ring loss co ntri butio n," The term s co ntaini ng are all ihose (except fission) for which the total neutron nu mber in the final slate is o ne less tha n in t he initial state and for which N '(X, K, s) - N(X, K, s) - l. These rep resent t he effect of neutron capture reactio ns. The co mpa nion terms rep resen ting neutron ga in by emission from excited nuclei have been neglected in writin g Eq.3.39. t Finally the
W:.
• The scattering gain a nd loss term s will constsr of beth elastic and inelastic conlributions. t This is not jusli6ed if, say, the cc ncentraucn of photo neutro ns in the syskm is appreeia ble.
40
THE FOUNDA T IONS O F N EUT RON T AA NSP O Il T TH EO RY
W:.
term s containing are to represent the fission contributio n in which an arbitrary increase in tbe number of neutrons is allowed. A nu mber of other binary interactions co uld be included in Eq. 3.39, however. they are of more special interest- and need not be conside red in a general discussion of collision effects in neutron transport. The following section s in this chapter will be devoted to a study of th e specifically nuclear aspects of the various transition probabilities indicated in Eq .3.39. When reduced, th e collision terms will have the same form as those discussed in the previous chapter, but in the present instance explicit expressions for the reduced transition probabilities will be-derived. lnthc next chapter the infhicncc of macroscop ic medium effects will be investigated in some detail.
B. Radiative Capturet T he rad iati ve capture reaction (n, y) is not the simplest reaction con sidered in the present work. It is generall y viewed as a two-stage process involving t he passage through an intermediate state. Consequently, a more complicated description is required than that for the d irect pro cess of clast ic pote ntial scattering. However, a genera l treatment of elastic scatt ering must also include considerations of resonant scattering. a process of the same order of complexity as radiative capture. Thu s we shall first examine the (n, y) reaction and will make use of certa in features of the resonan ce proce ss in genera l in later discussions of clastic scattering. The (II, r) reaction is schematica lly represented by (3.40)
where we assume that the neut ro n intera cts with the n ucleus to form a compound nucleus whieh then decays directly to its ground state via the emission of a photon, The transition pro bability W;. associated • For clIample, the (II, 211) react lc n in beryllium . t Olher capru rc reactions such as (II', p) and (Ill, a) will not be: considered here . Their con tributions to the transport equation call usually be ignored (see, for t'llomple, reference 6, p. 51).
The reader may see Dresner" for a thorough investigation of the effects of
resonance 3M
NUCLEAR CONSlD8aATIONS
with this process is given formally by Eqs. 3.33 and 3.34. The potential Y describes both neutron-nuclear and nuclei-electromagnetic interactions. These interactions are presumed to be separable in the sense that
v _v'+vlt
(3.41)
where Vit involves the specifically nuclear forces and is that part of Y that cau ses the transition from init ial to intermediate state (neutron absorp tio n), and Y' is th e electromagnetic part that causes the transition from intermediate to final state (ph ot on emission). A consequence of th is sepa ratio n is that V will have no nonvani shing matrix. elemen ts in which bolh the neut ron an d the photon numbers are cha nged, " The reduced tran sition matrix thus beco mes .
Je. ~
:a. I fr
~ ....~
~."
v~..: v~.. fr - ~. + '2 y. ·.( -
.
r
(3.42)
It.)
It is now appropriate to obtai n a mo re explicit expression for the width and shift fun cti on y. We recall from Eq.3.12 the exp ression -Ay• •"\, -
2
il
.)
-
v•..•.. + I
IV.··. I'
....-: »,
(3.43)
E.
The m sum is to be regarded as a summation over all possible sets of neutron and photon occupation numbers, and over all statts of the nuclei. This sum can be decomposed int o contributions a rising from those states like the initial state, those like the final state, th ose like the intermediate state, and all other sta les for which Y•.•• does not vani sh. The contributions from the last class of states are not considered here and will henceforth be neglected. t Note th at by two alike states we mean that the number of any given kind of particles is the same in both states, and nothing is to be inferred abo ut their respective momentum and spa tial distributions. The co ntribution to the m sum in Eq.3.43 from stales like the initia l • In trutinl inelW:M: sealterinl wewill find it ~ to viola te th iscondition. This is, howew:r. because: we insist on usinl Eq.3.34 to de5cribe wha t is essentially a thrcc:-stap: proceu. t An cumple of such . contri bution is that which dc:scribel lhe decay or the intermed iate state by proton or aJpha pardcle emission as in (.. , p) or (... «) roact ions.
42
1 11 1 I U U~ IJA l"l UN5 Of
N l U I Il U "" '1 1l ", """ I' U k l
1 1I1Uk \
state lila)' he written in the form
I Vr.··.l l.,.P [l - N(X , KJ, 5J)]
.
(3.44)
E: +E,. -E,., -£:
£:
where \\ 1,.' dc\ignatc as E .. a nd respecti vely the kinetic energy of u neu tron with mome ntum IiK and the "external" energy of a nucleus o f
_.
r,
r-
- ---r----- -
','
,...,.
- 1--- ,
"
"
t'
-f-
r:: ' •
,I
. <'
, '"
' -. '
i
JJ . lo ·
'"
I...
.
.,
'"
F ig.J. I. f.ncrgy 1e vcl d ~gr,lm, ror tbe rorm:u ion and ccc..yor a compound nucleus; (a) In it ial Male I,,). (b) Interme(liale sta te III") , (c) Final sta le [n·).
mu..s II in "external" state k. Unless stated otherwise, the la bel k: denotes the sta le of translational motions of t he nucleus. We have as..umcd here that the total energy of a nucleus, can be expressed as the sum of its " internal" energy E: an d E: . The binding energy, (m + AI .. - At... r} c", is represented by B"+I, where all reference 10 nuclear lI\;l SSCS is to gro und state rest masses. The vari ous energies thai will enter into the pr esent discussion a re illustrated in the energy diugram given in Fig.3. !. In Eq. 3.44 we have extracted from the o ff-
E:•.
43
NU(.' L L A R C U N SI D U I.A T I O N S
diagon al matrix elements their dependence upon occu pation number in much the same manner as in Section 0 of Chapter II. The relevant pa rt ofthe potential here is seen to be Vif since the terms that cont rib ute are those: describing the em ission of a neutron. To illu strate how the sum ove r neutron occ upation numbers is pe rformed, we expl icitly d is-play the dis tribution of the pa rticles among cells in a given m state (see Eqs. 3.35 and 3.36) as 1m) - IN~ ) I N~.t ) IN'(X.. X. , J',) ... N '(XJ, KJ.J'J) ...)
- INi:l)
I N~~k )
IN"(X ., XI. SI) ... I - N "(XJo XI. SI) ... ) , (3.440)
where we have writt en the ph oton a nd ne utro n occ upa tion numbers relative to the interm ed iate state, In" ) - INL..t.) IN~'_ ) INxI.) . The photon distribu tion is not cha nged since we a re co nsidering only V.., an d in this pa rticu lar case the neutron with spin J'I at the phase poi nt (XJo "K/) is being emitted . So far as the su m ove r neutron occu pati on number is co ncerned the m sum now becomes a sum over XI . KI and SJ. since any of the neutron s present in any m state can be emitted. Wecan immedia tely set X J - X becau se on ly the neutrons at X are of interest and emission is pr esumed to ta ke place at the point of interaction. Thus the depe ndence upon neutron occupation num ber for the emission process is simply (I - N(XJ • KJ , -"JH. We have also atte mpted to show explicitly in Eq.3.44 those degree s of freedo m which will influence the: matrix elements of Vif. Since in the sta te 1m) the nucleus is in its "i nte rnal" gro und state, the dependence upon AJ is suppressed. In a similar man ner we may displa y the co ntributio ns from states like the final sta le as E IVl ... ··.t .... [ I + N( XJ• a:/ ' AJB (3.45) .~~I
E: +
Ec
,lP +
B H
I
-
E:/
I
-
E.
J
where E. represents the energy of a photon with moment um Ale, and the contribut ions from sta tes like the intermediate sla te as
On accoun t of t heir denominators. the term s in Eq.3.46 make a negli-
44
TH E FOUNDATIONS OP N'BUTJl.ON TJl.ANS PORT T H EOR Y
gible contribution. They describe an increase in the width and shift function due to a scattering interaction (elasti c if E~+ I is equal to E:' :-') between the compound nucleus and the neutron field. We expect the effects of this type of collision broadening to be relatively small, and will ignore such term s in t he following. The contributions to the m sum in Eq.3 .43, to a good approximation, are then given by expressions in Eqs. 3.44 and 3.45. It will be convenient to rep lace the momentum sums by appropriate integral s. Fo r typical quantization cells with characteristic length L ~ 10 _4 em, the momentum uncertainty IJ K/K (or Lhe/x ) is of order 10- 3 for 10- 3 eV neut rons (or keY photons). Thus in all practical cases the distribution of points in momentum space ma y be treated as essentially continuous. The second term in Eq. 3.43 then becomes
(3.47)
or
, Th e integrals in Eq.3 .47 may appear to be singular; however, wc recall from Section A that r•..( - ;e.) is to be evaluated in the sense of a limit, i.c. lim r•..(x - it.) . ... 0 ·
By ap plying. Eq.3.31 we find that Eq.3.43 may be written as (3.48)
45
NU C LE AR CU NS I D ER ATI O NS
where
!
r•..•..
_ !!.
(.!:. ./§~ -) ' ."1 LfdD.,{1V• .•.•••,.,,1. x
2 2n
x [1 - N(X, K;rJ)] E~'2h·~ _I" H "- I"" + I AI r . 'r! x
.,.,LIda. (IV:··....
2
1td/ 1 [l
n('!:-)' 2n"c x
~ N(X ,,,, AJ)]E! }I".d:+I.r..... I_. :,H (3.50)
The functio n r is now expressed in terms of its real and imaginary parts. When entered into Eq.3.42, s....... gives rise to a displacement of the resonance line the width of which is determined by In our discussion we shall merely note th e existence of s.... .. and will not be concerned with its effects. On the other hand, the existence of F. ... .. is obviously crucial in the development of a theory of resonance reactions, and we will shortly ret urn to more discussion of this quant ity. Entering Eq.3 .48 into Eq.3 .42 and aga in extracting neutron and photon number depe ndence from the matrix elements, we can exhibit the capture reaction matrix as
r........
Rl ·.·..l·.A~
~
;os
N(X. K, s) (I 2Jr e-
r A·.·J.'.AKI - £. Ii •..•..
+ N(X, ,,', A')] r ....J.'. A~ V:·"J.·.A-.- V::·..·, u .
£...... . " +J'l··...
£,"+ 1
+ ~"
-
s.
r
(3.51)
-
z:. -'2I r......
(3.S2)
The energy difference £::1 - SA+I is denoted here as e:,: I*. Note that
we are labe ling the matrix elements again by state labels rather than by occupation numbers. This is because the sum over occupation numbers in the final state in Eq.3.39 actually reduces to a sum over sta tes as in
46
Till
I' OU N U A T IO NS OF NEUTRON TRAN SP ORT rU f.OR Y
the curlier case s. T he sum o ver occ upa tion numbers in the initia l state is tu be c arried ou t formally accord ing to the spec ific dependence in dicated in I:q.3.51Thuv fur we have not given any specific ccnsiderution to the matrix. clcmcms ~)r r" an d V/II. The discussion of V' ca n be made more qean utati..c and will be considered first . The po rtion of the interactio n bet ween nuclei and electro magnetic field which describes single ph oton cmi ....ion or absorptio n is I I
L .!!:.. A (r~) . p~
V' -
(3.53)
1.I.m,.C
where the I »pccitics the nuc leus and L specifics the nucleon. The rnomentum of the nucleon is denoted by p and t he vecto r field A rep resents the t ransverse radi al ion field quantized in .01 manner wholly similar to the qcunuzatio n of t he neutron field in Cha pter II. The matrix clements o f r" then beco me
-'····,.aj.(K·) · p~ lk ..
VL·j·.,·....
I\.. )
I,L m ..c
:to
L,
where
u :~ · f KA)
"" ( 0
IL.!!:.. L " ' •.
r
U~~ ..(K·.l·)
e-'··f a" (K)' pI.
(3.54)
I",")
(3.55)
The label 0 i.. UM.'d hI denote the grou nLl -internal" 1>tilt..: uf the nucleus. To arrive .u the above factorizati on we have introduced the cente r-ofmass position vecto r, R, and the relati ve displacement , p, so t ha t = R, • These coord inates, however, a rc not independen t. T he mom entum p:. is co njuga te to r~ , an d t here fore consists of co ntribution s From center-of-mass motions as well as from relative motion s. But bccau-c the nuclear momentum is very small compared to the nucleonic momen t um we have neglected t he former and set p~ :::: p'-. It is o nly in rhis approxi mate sense that we may isolat e the effects du e to external now depend s solely upon intcrnal mot ions medium . The fact or and describes thc respon se of th e nu cleon s to the photon field . Since a pa rticula r Fo urie r co mpo nent of the neut ron field is invo lved lko ' an d since th e ra nge of neutron -nucl ear in t he mat rix. eleme nt, forces is small co mpa red to the dime nsion of t he quantizati on cell, it is expected that the matrix clements describing neutron a bso rptio n deco mpose in a fashion similar to t he fact oriza tion of the pboton-emis-
r:'
r:·.
U::..
V:..•..
41
NU C L E AR CO N SI D ER A TI O N S
sion matrix elements in Eq . 3.S4. We shall th erefo re write
Vf.·. ·-, ik>-
.:t:
L, ( k "
le"(·II'1 k > U~!o(lu)
(3.56)
Both matrix ele ments of V' a nd y JII a re seen to co ntai n t he sum o ver nucle i. These sums, however, will no t appear in the ca lculation of the reaction matrix (3.52). This is because such a reaction ma trix is intended to describe the evolution of t he system from a state characterized by a certain number of neu trons, photons, nu clei of masses ..4 a nd (..4 + J) to a sta te characterized by o ne less neutron, o ne more photon, o ne less ma ss x nucleus, and o ne more ma ss (..4 + I) nu cleus . The nucleuswhich absorbs the neutron must he the sa me nucleus as th at wh ich e mits the ph ot on, thus elemen ts of t he reactio n matrix between specified in it ial a nd fina l sta tes will depend o nly upon th e properties ofa single nucle us. The red uced rea ct ion matrix fo r capture now beco mes a sum of ma trices each a ppro pria te to an ind ivid ual n ucleus. Fo r the nucleu s designated by the la bel I we have
r.t!•.~ -. iIl.'
2n = -
"
r l··.··
U:~ ..(" ,A')U:'~o(lCs) ( k' l e -
·lk..) ( k" l e(J(·a, Ik )
bo· ·..
l
-EK - E.A - f.r .. 2 •
(3.57) where we ha ve igno red t he dependence of the level width a nd level shi ft upo n th e exter nal deg rees of freed om of the nucleus, a nd where &. " _ £: ~ I - 8.4+ 1
+
(3.58)
.J. ..
is the energy of the e t h level in t he nucl eus of mass (A + I) as seen by a free neutron in the laborat ory. If we assume for illu strative purposes tha t th e nuclei in the syste m a re char acterized by well-sep arated energy levels.s then Eq.3.S1 red uces to a sum of a single-level reso nan ces
r;/. 'j'.n.,
~
2rr
L IUC:..(Il:'l') U:~.(K.s-W
h . ·'
' L
x
(k·I . -····· lk..) ( k" I . " ·' I k )
•.' ,
" ." +
Em k"
-
E
,,
-
E' l
r
. ' I' -"2 .'.
(3.59)
• In the conventional theory of reson ance' •• onc introduces a lc\'(';l-spacing D which represeou the 8 \,(,;r'd ge separa lion between neighbo ring resonance lewis. Va l ~ (If 0 range from several hundred KeV for light elements down to a few eV for A :t: 100, a nd will in general decrease with increasin g excita tion eneriY. Thus it is ~ni ogful to spea k of isola ltd resonance levels only if « D.
r.·.
48
T ill' fOUND AT IONS OF N EUTR O N TR ANS POR T TH EORY
U:."
U:.
g incorporate all the responses in T he matrix clements a nd the interim of the nucleus to the reaction, und nrc co mplicated qUOIn· tine s which ca nno t be discussed qu antitatively in the present developmcnt. For our purposes it is sufficient to replace them by more famili ar q ua ntities. We observe th at the level width given in Eq. 3.50 ca n be identified as a sum of partial widths appropriate to th e decay of co mpound nucleu s by either neutron or photon emissio n. Specifically the radi ati on widt h for the ","th level is
where use has been made of Eq.3.54. We will assume that we may ignore t he facto r II + N(X, K . AJ)]. If we further assume that th e difference in "external" ene rgies, E:/ J , is negligible co mpa red to the excitation ene rgy of the com po und nucleus, then the sum over k J may be pe rformed 10 give
E: -
~ l~!!) ~ 7l(~cY(E~
+ BA+l )l
~
f
dQ"
I U:~o("AW
(3.61)
Using similar arguments and approximations we find the first term in Eq .3 .S0 10 be given by
~ r:~> " ; ( ~ .J';)'..fE. ~
f
dO, .
IU.~!,(K:lI'
(3.62)
which ca n be identi fied as the neutron width. In th e sense of the above a pproxi mations and if IU Il'12 and l U ~'1 2 can be considered as co nsta nts the se results show th at the radiation width is essentially energy independent . whereas the neutron width is proportion al to the neutron speed.- Eqs.3.61 an d 3.62 are useful in th at they allo w us to write the • For the case oru 23 S secOleksa.16 Because or its dependence upon (B,(+ 1)2 the rad iation width can be expected 10 decrease as If increases. The energy dependence or the neutron width is in agreement with the conventional results' for neutrons or zero angular momentum and therefore implies that I U~1 1 2 can indeed be treated as a constant so lona u the neutron enerty is not so high that neutron s with higher angular momentum beain to interact appreciably.
49
NU CLEA R. CONSIDeR ATIONS
clements 'of the reac tio n matrix in terms of level widths, and in the present treatment the la tte r qu antit ies will be treated as empirical parameters. It is expected that 1U:' ~o l ' is qu ite insen sitive to the directions' of K, so that we have"
-'2- r:~' '" 2x' (!:.. .j2m)' .JE;; L• IU; !,(Ks)I' lnfl
(3.63)
The same may be said for th e dependence of 1U.~!ol', altho ugh , as we will show later, the assertion is not necessa ry in this case. Furthe r pr ogress from this po int, at least so fa r as the redu ction of Eq.3.S9 to useful forms is concerned, req uires specific ass umption s rega rding the macrosco pic sta te of the system. It will be necessary to know whether the extern al degrees of freedom of th e nucle i are those appropriate to a system in solid, liq uid, or gaseous state in order to compute the ind icated ma trix elements. These matters will be considered in the following chapte r. In co nc1ud ing th is section we sha ll examine some of the more general aspects of the collision terms in the balance relation which describe the effect of radiative capture processes. These terms now appear in Eq.3.39 as
V-'
L W;.D_(t)
- V-'
_ '.
.
L
N(X. K. s)[1 + N(X• • •• ")J x
"'...1....
x
r.~ J.,.I:~ • • (t) M..E:. +1 - R A + 1 + E•. -
'et -
E..) (3.64)
Evidently the n sum leads to fun ct ion als of various doublet densities. However, to avoi d explicit co nsideration of these higher-order densities, we shall libe rally (and for the moment uncritically) replace averages of funct ions by functions of averages .f Thus,
V- '
L W:,D..{,)
_ ',
'"
L
"".'1.",
F.(X, K. ')[1 + F,.(X, x'. ') J x
x r';!..a s,Du(t) 6(E:.+ I
-
BA+I
+ E..
-
Et -
E..)
(3.65) • Th is is equ ivaic:nt 10 l he assumptio n that neut ron em iuion o r absorption is essentially spher ically symmetric, a condition usually valid at k:ast for E.. s 100 KeV.· t Had we retained the doublet densit ies then Eq.3 .39, which 1II3y be regarded as an equatk»n for the singlet dens ity, would be incom pic:te for the determination of FCX, K.I). An equ.lIion fo r the doublet densi ty is tberefcre neoessary, and we will find that it contains the tlipld densities. Hence, a n infinite set o f coupled equ atiol\l i. renel1lted . ~
~ -.j YI ,
so
Til l: FO U NDATIONS OF NEUTR ON T R ANS PO RT T HE O RY
where F.(X, K, t) is the expected number of neutrons per unit volume at time t wit h spin of and mom cntum 11K at X, f~(X, " , t) is the expected numbe r of photon s at time t with pol ari zat ion Aand momentum h" at X, and D_l( t) is the probability of finding the target nuclcus in the state k at tim e t , Fo r most appli cations involving the neutron transport equation th e neutron spin orienta tion is not a varia ble of interest," so that th ere will be no loss of generality if we assume the spins are randomly distr ibuted , or (3.66) F.(X, K, I) - 1 F(X, K, , ) Now Lq. J. 65 becomes
v-' _L'. W;.• D••(,) '" F(X, K,I) U·..·A· L (I .
+ F•.(X, , ',I») x
(3.67) The ca ptu re co ntributio n is thus in a convention al form of a rea ct ion rat e times t he neut ron densit y. In the follo wing ch apter we shall show how this reaction rate can be redu ced to the more famili ar expressions for th e cross sect ion.
C. Elastic Scattering Fo r neutrons with energies below th e inelast ic scattering th reshold , ab out I MeV for light nucl ei down to :::::- 100 kcv for high A, the only proee~~ ava ilable for th eir energy moderat ion is clastic sca ttering. t Th e neutron energy distribution as de termined From the tra nsport eq uatio n ca n be qu ite sensitive to the energy-t ram-fer mechani sms un derlying this t ype of collision. T he fact that the neutron sc attering ~111 be significantly influenced by the atomic moti on s of the system not only introduces add itional co mplexities into the transport equation at low energies, but also suggests the use of neutrons as an effective probe for the study o f solids and liquids. These remarks will be elab orated in • A possible exception could be the case of neutron tra nsport in inhom ogeneou s. mngnetjc field. Admitt edly this is not a system of pract ical interest. t For a discussion o r the slowing down or neutrons by elasticcollisions see Mar shak 11 a nd Ferziger a nd ZweireJ.l'
NUC LE A R CONS1DEk AT lONS
51
greate r detail in th e next chapte r on the basis of the development presented in th is section. T hen: arc two types of clastic sca tteri ng processes which sho uld be distinguished at the outset since they will require somewhat different trea t ments . The first pr ocess is like radiative capt ure in that a co mpo und nucleus is formed, but rather than decayin g by the em ission of a phcton the compound nucleus decays to grou nd state by the emission of a neu tron. This reaction is known as elastic reso nan t scattering. The seco nd process is a direct reaction known as potential scattering. which can be considered as tal ing place in the immediate vicinity of the surface of the nuc leus so that there is effectively no penetration." In genera l, potential scattering dom inates in energy regio ns away fro m any resonance, whereas within th e vicinity of the reso na nce peak resonant scattering dominates . In regions where both kind s of scattering are of th e same stre ngth it is known that apprecia ble inte rfere nce can exist. which is generally destructive at t he low-energy side and co nstructive at the high-energy side. t We shall th erefore consider both processes at the same time in orde r to include such inte rference effects in the present ana lysis. T he reaction matrix describing the sca ttering interaction is aga in given by Eq. 3.34 where now only V " , the nuclear part of t he potential, need s to be considered. Here the class of initia l and final sta tes is tha t cha racterized by the conserva tion of neutrons. phot ons, and nuclei. There are, however, two sub-classes corresponding to the increase and dec rease respectively of a neutron at th e phase po int of interest. In the binary co llision they constit ute th e scattering ga in a nd loss to the ba lance relation as indicated in the qualita tive discussion given in Chapter II. For the treatment of both dire ct a nd resonance processes we ass ume that v.. . has nonva nishing matrix clements between initial and final states as well as between intermediate state and final or initial state. The rea ction matrix can be written in a fonn simila r to Eq .3.SI a nd 3.52, (3.68) Rf!..·•·.•Ks N(X. K, s)[1 - N(X. K' , s')J ' : ... • O . reference 1, p. 393; sec also remarks by Lane: and Thomas, reference 2, p.26 1. t A rather sltikina: example of th is phenomenon is the sulfur resonance: line at ::c: 100 KeV (also the silicon line at ::c: ISO KeV).I.
52
l il t fOUND AT IONS OF NEU T JlON TRAN S .'ORT T HEORY
V:·k·",k··."
- •..•.. L
,
vZ··.··.£.,. . - !.... J'•..•.. 2
(3.69)
These two expressio ns are appro priate to collisions resulting in "scatteri ng loss" . Corresponding expressions fo r "scattering gain" arc obtained by merely interchanging the sta te labels (k. K. s) a nd (k'. K', s'). T he various energies appearing in Eq.3.69 are the same as those introduced in the previous section. The matrix elements of VN may agai n be factored a s indicated in Eq.3.S6. and we obtain
I' ~." . , - .. ... -
L t.··.·,
U~~ "
uZ!.o
8 •..
I
+ £: : 1 -
.. •.
a, Ik ")
1:.; -
£K -
f J~
1
..
(3.70) where we have introdu ced S. .. acco rding to Eq.3.58. The direct matrix elements. u •• ca n be est imated in ter ms of II specific model ora neutron-nuclear interactio n. It is to be emphasized that this use of a model does not affect th e other matrix elements of VN, those describing resonance scattering. Th is will become evident in the fo llowing, for the parameters of the model arc to be determ ined acco rding to co mparisons with data fro m low-energy pote ntial scattering. Since these paramet ers are fitted to experiments in th e sense of certain calculational approximations, l.e., th e Born ap p roximation o r firstorde r pert urba tio n theory as presented here, it is not clear that such a model should he employed in general theor etical a nalyses whieh are not restr icted to these ap proxi mations. Conversely, it will be seen that the model employed in the context of first-order perturbation theory can be adjust ed to describe the experimental results exceedingly well. and i!s usc enables one to explore the specifically macroscopic medium ef· fccrs with reasona ble co nfidence. The model. as we shall cons true it, is intr oduced for mally as the po-
V:..·.·.
NUC L E AR C O NS I D E R A T I O NS
tential ,
v - LI , d'x [V; (x) .,(x) v~(I x
- R,I)
+ •• . ' J+(x) (JJ. ,.(x) v~( l " - R,l)]
(3.71)
Th e spi no r field operator "AX) has been discussed in the previou s chap-ter. The components of the vecto r a are the Paul i spin matrices II a nd con sequently (112 represent s the intr insic spin (in units of Ai) of the neut ron. Th e vector I, represent s the observed a ngular momentum of the hh nucleu s in its gro und sta te. Instead of the delta function (pseudopotenti al) introduced by Fe rm i,10.21. we shall depart slightly from convent ion and sugges t t he use of Yuka wa functions to represent the short-range potentials O. I. e - ·/Al v:
v:.;-
g: -,
T he reasons fo r our preference for t his po tential fu nct ion will be made more expli cit lat er ; for the mo me nt we simply rem ark that it provides a cross sect ion for neutron-nuclear potential scattering with a somewha t expa nded range of qualitative validity. A more useful expression of V is obtained by using the spin or field expansio n in Eq . 2.8. We find V -.
L
a+(X, K, s) <J(X, K ',s')
OK'...'
L U~ -Q, s,s') . - .,...
(3,73)
,
where
Q - K - K', U,(Q. s, 3') "" V-I Jd )R E(X. R + R,) e/O· a
(3,74) )(
)( (v~(R) 6". + I I' ut es) O'/ .U.("') v~(R) ]
(3.75)
Th e Hermitian character of this coarse-grained po tentia l can be read ily demonst ra ted . for
L a+(X. K·. s') a( X, K, s) L U,( -Q. s. " ) 0"." xxx·...· I and since S I and 0' a re H ermitian, and is real, we ha ve ,-' 1R E(X. R + R.)e -lQ·a, )( U'+(Q,S'•.f) "" V -I V+ -
(3,76)
v:
Id
)( (v~(R) 6... - U~ - Q.s,s')
+ I, . uU s') 0':' u~(s) va R)] (3.77)
• 11Ic Fermi Ipproximation has been m:cnlly studted by Plummer 2 and Summerfield.u .
54
Till" I HIlN f)AT ION S Of NEU TRON Tk A N St'O Il T T1 lfO IlY
Thus by in terch anging (Ks) a nd (K'. s') in Eq.3.76we obtain v.. = v, which is o f co urse a necessary property of the potential. Th e inclu sio n of spin-dependent inte raction s, as represen ted by the secon d term in (3.7 1) or (3.75), necessita tes a slight mod ification of ou r tr eat ment of the matrix elements. Thus far . th e la bels k an d 0." have been used to specify the "external" a nd "interna l" states of a nu cleus, and the possible existence of a nuclear spin has no t been considered . However. the presence of a spin-de pende nt ter m in Vii no w requires the introduction (If a label to specify the spin snucs ofnuclci in their ground M
vZ·/C ·.·.u:.
=- L ( k ' i UJ(Q, s, s')
,
eIQ • ll
lk )
(3.78)
undthc clement s ofthc red uced tran sition mat rix for clast ic scattering. wuh th is Il;trt icular choice o f a potentia l, arc
r: ~I" "'U'. = ~'t I~
\
..~..
8 •.. + E:.t t
( k ' le- IX· · · · lk") ( k" It!&· · , IA; ) -
E: -
E.. -
2
.!... ro"
(3.79)
2
Although still rath er forma l, this expression will prov ide a suitable sta rting po int for explicit co nsiderations of medi um effects and for the in troduction of uscful assumptions an d approximations. We shall close th i!' 'iCctiulI by noting that the clast ic scancnng contrib ution s to the co llision terms in Eq.3 .39 may be d isplayed in a form analogous to Eq.3 .67 V-I L (W:.": - W:.~L) D..{I) ~ .
~ [I -
x:2 F(X, K, I)] L F(X, K', I) L i2 '::.•..: x'
• ·•·••
x
x d(E: + Eli - E: - Ed D•...{t)
- F(X, K. t ) x 6(E:
L [ 1 - ~ F(X, K', t)] L ! ':"1('0 1(. x
1('
2
+ £ " - £..". - Ed Du (t )
.·.·u2
(3.80)
ss
NU CLE AR CCINSID ER AT I O NS
D. FI••lon and Inelastic Scattering The co ntributio ns to the ba lance relation (3.39) from fission an d inelastic scattering. like those from elastic scatteri ng. also can be expressed as gains an d losses. the fission loss term combining with the radiative capture loss to account for the removal of neutrons by abso rptio n. The processes of fission and inelastic scattering are mo re complicated in that they should be properly rega rded as mu ltiple-stage (at least three) react ions, for it is like ly in fission that the compoun d nucleus decays to two excited fragments which then subsequently decay 10 less excited fragments via neutron and /or photon emiss io n. In inelast ic scattering. t he compound nucleus emits a neutron. leaving an excited nucleus which will de-ex cite by photon emission after some time lag." Thus a reaction theory capable of describing three-stage events is necessa ry to treat these processes. The reaction matrix given in Eq.3.34 can be used to describe only first- and second-order p rocesses. so that higher-order te rms would have to be included in Eqs.3.11 and 3.12. The gene ra lizatio n of the formal ism in Sect ion A is a straigh tfo rwa rd mutter ; however.
R::••.•'" ({ Kr }J) .. H(X, K. s)
n· (I
- H(X. KJo s})]
\ I"' IJ
V:,.,.•..•_({ Ks},) V:_._.....
r" (IKs ) ) - 2n" t,. , .~ } ~ ~ n
• ..• ..
'r:...•c. ({Kr} J).
I •..
+ E-..." - E-I
-
E
I" -
' r ." 2
r
(3.81) (3.82)
-
In Eq.3 .81 the prod uct is such that whenever {K} . s}) "" (K. s) the faclor (I - H{X. K. s)] is to be replaced by unity. Th is thc n en sures that a neutron ca n be em itted ha ving the samc mom entum and spin as that of the neutron initially a bsorbed. The matrix element describes the format ion o f a co mpo und nucleus a nd has been encountered previou sly. Its depe ndence upon "external" degrees of freed om is given by Eq.3.56. The ot her matr ix element. ,,{{Kr }}). represents the "falsification" mention ed above since it describes the decay of th e compoun d
V:'.,.•..•
• hlClastic scatterings in seneral need not involve the format ion of . compound nucleus. A number of direct reactions of this type have been cited in Lane and lbomas, reference 2, p. 264. tbe direct scauerml by • rota tinl nonspberical nudeus has been investipled by Chase, et 1If.2.
S6
rill
F fllJN Il A TI O N S Of N E UTIlON T R A NSPORT T1U ()R Y
nucleus into two fission fragments in external sta tes k. and internal states 1\,. a nd J neutrons with momentum and spin distrib ution { /(sIJ . In this C;lSC an expression similar to Eq.3 .56 for the "external state" dependence is not to be expected since neither of the fission fragmen ts may be regarded as located at the centre-of-mass posit ion of the compou nd nucleus and one has no knowledge of which of J neutrons is emitted by a given fragment. The fission contributions to the balance relation can be decompo sed into gain and loss terms dependi ng upon whether or not the d istribution (Ks}J contains the momentum and spin of interest. In the sense of ap proximations inherent in Eq.3.80. they may be displayed as
v · , L IN'CX, K, s) - N(X, K, s») W:.•D..(I)
-.
• L L
1J~ , 0 1 c li',,~
f"(X, K', I)
Il" [I
-
III:I,,&,
""'"r'
)( D' -o (/ ) 1t(E.,...
+
/ )] ! ,::. ,.•-• .•, (( K },,) x
2
E ,;. - E,)
L L u::o,
- fl X. K. I)
.!:2 f"(X, K"
W[I- ~flX, K/, /)]~ ,::....., ({KI;.)
t ';I ~&'I C , ~a'
)( Du(l) ,)(.E :
+ E,. - E,)
(3.83)
where r =-~( { K }) -
L r=.~{{Ksb)
(3.84)
I'll
The distributions {KlJ " and {K} ;" denote those which conta in and do not contain the momentum K respectively, The energy E, may be obta ined as follows. The total energy before fission is (3.85)
and . according to the present approach. the tota l energy after fission is (3.86)
where M I is the sum of the rest masses of the two fission fragments. Eo, is their internal energy (subsequently lead s to emission of Y. P. a nd neutrons), E l • is their "external" (kinetic) energy and E ta'l" is the kinetic energy of the fissio n neutrons. Thus £, =
-- -
- - --
--
t o' -
-
(m
+ M A) c J
57
NUC L E A R CONSI DE RA TIO NS
The sum of the last t hree term s in Eq.3.86 gives the energy released by the fission process. For U 13 S th is' is about 200 MeV, roughly the same as the energy due to th e ma ss difference. Hence fr om the standpoint of energy conservation there is effectively no fission threshold." Such will no t be the case for inelastic scatte ring. We now consider the decay of the co mpound nucleus by neutron emission'; the residu al nucleus, being in an excited state, then decays to ground sta te by gam ma emission . To treat t his pr ocess wit h the forma lism employed throughout this cha pter, it will be necessary to assume that a potential exists th at has non vanishing ma trix elements between sta tes in which both neutron and photon numbers are changed . We sha ll denote this po tential as V' since it does not co nform to the sepa ration (3.41) into parts purely electromagnetic or nuclear. The relevant elemen ts of the transition matrix for inela stic scattering, in the present approx imati on , are therefor e
RrK·.·.1· ~K.
= N(X, K, s)[1 - N(X, K',s')] [I
r/r·K·. ·..·1·./rKo ~
"
+ N(X, K', .t')lr~K·.·"·l ·. /rK>
Vl·K·.·..·1·. ~... " VZ....../r1U
-2n L
Ii~ ... .. ". .,,
+
E'" ~,,
-
E't
-
.
(3.87)
,
(3.88)
E Il:- -')'." 2
In this case we ma y expect the matrix element s describing the decay of t he co mpo und nucleus to be approximatel y fact orable as in Eqs.3.54 and 3.56,
"(k'l e - m,......., Ik" ) U'0." (K's"It ,." ) (3•89) V .·11:·.·..·1· /r ..... -..... i.. •
I
'
Since the nucleu s emitting the photon and neutron has to be the one that captures the neutron, we can again introduce the reduced reacti on matrix app ropriate to a single nucleus,
L ~ " ."
u~
... uZ·'·o ( k' I e- iR ' ·(~·. "·1 Ik" ) ( k." 1efI'~' Ik) 8." + Et!" l - E: - Ell. -
1
-I r ." 2
(3.90) • See, however, Weinberg and Wigner, reference 6, p.tOS, for cases in which A is even.
58
Tl lf In U NDATION S OF NE UTRON TaA NSP OR.T TIl £.ORY
The contrib utio ns to the balance:relation due to inelastic scattering now become
_. ~' fl X, X, , )] "'I(~.u,F(X, X', I)[ ) + F~(X, It, I») x
:::: [I -
F(X, K, I)
L
III·I(·. ·...~ ·..
[I -
.!::2 F(X, K',')] [I
+ F,. (X, 0', 1)) x (3.9 1)
The 10la1 scattering effects in Eq.3 .39 are therefore given by the sum of E q~.3.liO and 3.91.
E. The Neutron Balance Equation in Continuous Momentum Space The neutron balance equation, as given in Eq.3.39 has been reduced by a systema tic study of the various collision term s. Expressions for the associated tran sition probabilities, although still rather formal, have been derived using Heitler's damping theory. Having determined the explicit dependence upon neutron occupa tion num ber of each process, we th us ob tain an equation describing the neutron density F(X, K, I). This equ ation can be written as
..
[~ + -m K, _
iJX }
~
]
+ R,(X, K) F(X, K, t )
L F(X, K' , I)[R.(X ; K'
- K) + R.(X; K' - K))
(3.92)
where we have introduced the following reaction rates, Rr-{x) "" R.(:c) + R.(x) + R, (x )
R..(X. K) =
L ·.(1 + F,d X, 11:' , I)l J '.~k·A" U" I.·.·A
(3.93) )C
,.
NUCLE Aa CONSIOEIt ATI ONS
E
RJX. X) -
A/I·Il.·..ob
[I -
~2 F(X. X'. I) D ..(I)J
X
)( ! {,:;,.,..lXcr «..E: + E.. - E:: - Ed +
..
L' {t + FA' (X, ,,',I)] ':~~·'r'A·..lEl
x t~(E:'
)(
+ Elf, ' + E. - E: - E.)}
(3.95)
[I- ~2 F(X. X" t ) D ..(I)] 1.1E .,., 1"Il" ' J/ It
R,(X. K) -
x
UI" ] ' ..1
X! '::.,.lLr ({ K} ;.) lJ(E:
RJX; X' -
K) -
[1 - ~ F(X, 2
E
- E,)
{,.r~.l·.·"
M..E: + E.. -
!
+
L [t + FlX. '1.1)] ':~.l'.'" )( " b{E:' + E... - Et - EI[ - E.}}
no [I
U.,. , 11[1 /1t
(3.96)
X,')J E..D..{I) x
x
x
R,(X; X' - X) -
+ E..
- ~2
F(X,X /
E:' - EI[')
(3.97)
I)J
D ., .{I) x
• ·.. {.tl/ ..
x
i ' ::.,."-s-,.({K}n) b{E: + E... - E I ) (3.98)
The express ions for th ese reaction rates have bee n discussed in the previous sections in th is chapter. Eq.3.92 is now seen to be identical in structure to the conventional neu tron transport EQ. 1. I ; however. the present equation has been derived on the bas is of a discrete phase space. It has been indicated earlier that the distribution of momentum poi nts is so dense compared to resolutions in any practical measu rement th at no appreciable error can result by expressing Eq. 3.92 as an equation in continuo us neutron momentum space. This is readily acco mplished by the use of Eq. 2.30 whereupon we find
[ :' + ,.. V +
vx,(X,')]J(X, v,l)
- J d'.' v'J\X, 0', I) IEJJ(. 0') 3"(0 ' -
0)
+ E,(X, 0')
2(0' - 0)]
(3.99)
60
THE f'O UN DA TION S O f NE UT RON T RANSPO R.T T II EO RY
where we have introd uced the neutron velocity as a variable" = "Kim, and have expressed the collision rates in terms of corresponding macroscopic cross sectio ns, I •. The two frequencies, IF and 2 in Eq.3.99, are defined as follows:
L
R.(X; K '
~
K)
~
R.(X. K') .F(K '
~
=
v' l.'.(X,,')9"(" -+ , ) d 1v
K) d' K
K ~d'K
L
(J.!OO)
R ,( X ; K' ~ K) ~ R,(X, K) -"'(K' ~ K) d' K
K . d 'K
= v'L'I'(X , v') 2 (, ' .... ,) d1v
(J.IOI)
As usual, .~( ,' _ , ) d 1v is to be interpreted as the probabi lity that a given neutron scattered with velocity,' will have its fi nal velocity in dlv about " whereas 2(, ' _ , ) d 1v represents the expected number of neutro ns emitted with velocity in d lv abo ut v, prov iding tha t a fission event has been initiated by a neutron with velocity ,'. For the reactions of interest the cross sections and 2 are independent of the direction of the incident neutron . Moreover, !F often- depends only upon the initial a nd final speeds and the scattering angle. o = cos- I(v · , 'fl'I I"!). Inserting these simplifications into Eq.3.99 and assuming tha t the discrete configuration space can be replaced by a continuum, we finally o btain the neutron tra nspor t equation in a form that is co nventionally employed in all investigations of neutron slowing do wn. d iffusion , and thermalization. Referenc e. I. J. M. BlaH a nd V.F.Wcisskopf, T1!rorrlicol Nudear Physics, Jo hn Wiley So ns Inc.• New York, 1952. 2. A. M. l.ane lind R.O.Thomas, Rev. Mod. Phys., 3 0 : 251 (1958). 3. w.Hcjucr, Thr Quanlllm Throry of &diation, Oxford Uni versity Press, New Yor k. 1954, th ird edi tion. 4. E.Arnnus and W.Hcitler, Pr(IC. Royal Soc ., 2201\ : 290 (1953). 5. R.C.O'R ourk e, " Da mping Theory", Nava l Research Laboratory Report 5315, 1959. 6. A. M . Weinberg a nd E. P. Wigner, The Physical Theory of Nelltron Chain Reactors, Unlvcrshy of Chicago Press , Chicago, UI., t958. • All exce ption might be the scattering frequency for lo w-energy ncutron s ill crys tals.
N UCL EAIl C O NS1D EIl A TJO NS
61
7. E. H.KJcvans, Thesis , University of Michipn, An n Arbor, M ichigan , 1962. 8. A.Z. Akcas u, ~is. Univcwty o f M ichipn, Ann Ar bor, M ichipn, 1963. 9. A.Z. Akcasu , Uni versity of Michi gan Technical Report 04836-I·T, Aprill96J. 10. R. V. Churchill, O~ralionaJ Math~malits, McG raw-Hill Book Company, Inc., New Vor k, 19S8. 11. L.I. SChiff, Quantum M echanics, McG raw-Hili Book Company, Inc., New Yo rk, 19S5, seco nd edition . 12. L.L. Foldy, Phys. Rev., 87 : 693 (l9S2); Rev. Mod. Phys., 30 : 471 (l9S8). 13. L. Dresner, " Resona nce Absorpt ions o f Neut ro ns in N uclear Reacto rs" , O RNL-2659 (1959); Resotlatlce .Absorpl iotl i" Nud~ar Reactors, PcrgamonPreu, New Yo rk, 1960. 14. L.No rdhc im, " Theory of Resona nce Abso rpt ion", GA -638 (I 9S9). 15. J.B.Sa mpson and J.Cherni ck, hog. iff Nucl~ar &ergy, Series I, %13(1958). 16. S.Okksa, Pr«ed ings 0/ Brt>Ok./rawn COll/~renu 0" kSOttOlfCe AbsorpliOtl in NJlCl~ar ReM/ orS, BNL-43 3. S9 (1958). 17. R. E. Ma rshak , Rn. Mod . Phys., 19 : 185 ( 1947). , 18. J. H. Fcrziger and P. F.Z..-eifd, The S lowillZ Dow" 0/ NtIIlrOIlS in Nuc/NJ' Reactors. Pergamon Press (to be pu blished in 19M). 19. D.J.Hughes a nd R.B. SChwaru. Ne/llro" Crou S«IiotU, BN L-325 (1958), second edit ion. 20. E. Fer mi, RJ«rN Sd elf1i{iN, 7 : Il (1936); English translatio n availa ble .. USA EC Re pl. NP·2JSS. 11. a .Breit, Phys. Rev., 71 : l IS ( 1947). 22. l .P.Plummer a nd a .C.Summerficid , Phys. k ll. 131, 1IS3 (1963). 23. G .C.Su mmerflcld, .Ann. PhYJ.,l6. 72 (1%4). 24. D. M.Chase, L. Wik lS, and A. R. Ed mo nds, Phys. /In., 110; 1080 (1958).
IV
Neutron-nuclear Interactions : Medium Effects In detailed investigations of neutron transport in macr oscopic systems, the usc of adequate cross sections in the tran sport equation is essential. And adequacy here requ ires that the cross sections no t only reflect the specifically nuclear processes under consideration, but also all relevant environmental effects. The environment can significa ntly influence the description of the cross section in at least two ways. The dynamics and symmetries of the system can either sepa ra tely o r simultancously mod ify an observed react ion rate. The ra tio of nuclear fora: ranges to characteristic internu clear dislances is of the order of lO- s or less. Thus it is a nticipated that a given neutron will interact with the nuclei in any medium one at a lime. Nevertheless the probability of a collision between a neutron and a nucleus will be a ffec ted (because of the requirements of energy and momentum conserva tion) by the character o f the sta tes available to the tar get nucleus in the system. In turn, the nature of these states is determined by the dynamics of the macroscopic system. Funhermore, system dyna mics modifies reaction ra tes in still another way, since they will depend upo n the relative probabilities of finding a target nucleus in pa rticular avai lable slates before a collision occurs. The effects o n reaction rates dependi ng upon the natu re o f Ihe uvailahlc states for the nuclei me often referred to as "bind ing effects", whereas those depending upon the probabilities o f occupancy of these states urc called "Doppler effects" . System symmetries, which for practical purposes may be regarded as distinct from system dynamics , can also playa role in determini ng reaction rat es for neutrons at sufficiently low energies that their de 62
MEDI UM EFFECTS
63
Broglie wave lengths ap proach or exceed internuclear spacings. i.e., energies of the order of tenths ofan electron vo lt or less. The most striking exa mple of symmetry effects on neutron cross sections is proba bly Bragg scattering in crystals. For the very low-energy ne utrons for which symmetry effects markedly influence reaction rates. dynamical. effects of both kinds (binding and Doppler) are generally expected to be sign ificant also. Since molecular dissociation and crystal dislocation potential energies are typically of th e order of a few electron volts, it is an ticipated that. at least in principle. there will be neutron reaction rates that are affected by bo th aspects of system dyna mics, but not by sym metries. Fi nally, for still highe r-energy neutrons. bind ing effects should decrease in impo rtance and only the Doppler effect should remain as an influence on cross sectio ns. The expressions for neutron-nuclear reaction rates that have been derived in th e previou s cha pter implicitly include all of these effects. In th is cha pte r we sha ll explicitly investi gate some aspects of them. The followi ng discussion is restricted to radiative capture and elastic scatt ering because, for simple systems. the calculatio n involved is straightforward and the results obtained are of considerable interest from the standpoint of reactor analysis.Because of the co mplexities of inelastic scattering and fission reactions an d of o ur intention to describe them only qualitatively, a qua ntitative investigatio n of mediu m effects in th ese pr ocesses does not seem feasible at th is poi nt. Furth ermore, it is un likely that inelastic scatte ring reaction s will be o bserva bly sensitive to medium properties due to the large neutron energy required. It is also unlikely that fission react ion s will be influenced by binding. altho ugh Doppler effects may be impo rtant. One neutron-nuclear reaction in which med ium effects are promine nt is elastic potential scatte ring at low ene rgies. With the advent of highflux 'reactors an d the development of high-resolutio n neutron spectremerry, it has become feasiblc to mea sure in co nside ra ble detail the ene rgy and .angular distributions of thc scattered neutron s. These investigations not only provide cross sectio n data for reac tor calculations• • For cnmpk. the temperature depen
thermalizalion and diffuaion of neutroN in the cncraY n:aion z below 1 cV are of interest.
64
l lll; FOUN D AT IO NS Of NE UTRON TRA NS P OR T T HE O RY
hut also const it ute a quantitative means of st udyi ng a to mic motions in so lids a nd liquid s." In the latt er cases the emphasis is o n (he prop er int erpret at ion of the measu rement s. a nd fo r th is purpose a real ist ic descriptio n of the sca tterin g syste m must exist. Th e theory of neu tron sca tte ring by crys ta ls nnd low-density gases, on the ba sis of availab le models ca pable of representing quite acc ura tely the mot ion s in act ua l systems, has been de veloped to the exte nt that qua ntitat ive under standing o f the vario us proces:-.I'S involved is po ssible. On the other hand , the corres po nd ing th eory fo r liq uids (a nd, in fact, thc theory of liquid sta te in general ) sutlers fro m the lac k of a systema tic a nd relia ble description of molcculur ructions. Th e complexities o f these mot ion s make it necessary (0 introd uce simplifying assumption s a nd specialized mod els to ca rry out a n unulysis. Since much of the theory of neutron sca tte r ing by liqui ds is sl ill und e r development, thi s aspect of the investigat ion of medium effects will not be conside red in the present wo rk, l-ro m the reaction -rate expressions alre ady derived it ca n be seen that the effects of the external degrees of freedom of the nucl ei a re partiall y spec ified by the matrix elements of exp (iq . r), where q is either a neut ron o r ph oton wave vector. The direct calculation of these matrix clem ent s necessaril y involves specifica tion of dyu nmicul a nd symmet ry properties of the medium. For the pu rpo se of illustration we sha ll considcr two sim ple systems. the ideal gas and the Einstein cr ystal. AI· th ou gh these a re rather idealized descr iptions of ac tua l syste ms, they a rc capable of providing useful cross sections for realist ic react or ca lculaticns. Th e discussion presented here is primaril y concerned with the transla tional mo tion s of the atoms. Thus the results o bta ined, st rictly spea king, a re applicable only to mon at omi c syste ms. For polya to mic mo lecules the sa me procedure ma y be used to treat the center(If-mass degrees of freedom, but in addition molecular rotat ion s and int ernuclear vibrations must also be taken in to account.
• Fo r un extens ive list of references as well as a num ber of importan t papers see prflcccdinlt~ of the "Sy mposium o n Inelastic Scattering o r Neutro ns in Solids a nd Uqui,b " held in Vienna, 1960. a nd the proceed ings of a simitar symposium held in Chalk River, and in Bombay, 1964.'
the
65
MEDIUM EfFE CTS
A . Transport In an Ideal Gas The simp lest dynamical system appropriate for the discussion of medium effects is one in which the atoms do not interact appreciably with each other. The recoil of t hese atoms in a collision with neutrons will be like that of free particles, so th at only two properties of the system can be expected to influence the cross sections : the particle ma ss (which influences the magnitude of the recoil) and the temperature (which characterizes the average energy of the atoms). The use of a free-particle desc ription makes the cross-section calculations quite easy, and the results are oflen useful in transport problems, since systems other than dil ute gases can also be treated in thi s manner whenever neutron energies are such that chemical binding can be ignored. Radiative Capture
The reaction rat e describing radiative capture in isolated levels has been de rived in the preceding chapter. From Eqs.3.94 and 3.59 we have R, ...
L [I + F.l' (X' 1", t)] 6(E:,+I
I.·.....·
-
+ 4 - E: - E,,) x
BA+1
~
x IU::(IC' ).' W x
. L"
1U:~(KsW ~ Du(t) x
(k'i e- I.·,II., Ik" ) ( k" l elK .II., Ik) 8.
+ E:':' - E: - E" -
1
(4.1)
.!-r • 2
For a system of noninteracting particles with no internal degrees of freedom the Hamiltonian is simply the su m of individual particIe kinetic energies. The kth eigenstate of the system is a product of individualparticle cell eigenstates, each characterized by a wave vector which is discrete in exactly the same sense as for the neutrons. The two matrix clements in the k " sum can now be written as
( k'i e-/O<··R, Ik" ) (k" l ct K•R, Ik) = 6,,(k; + 1"
-
kl') 6,,(k;' - K - k l ) (4.2)
where k " kj', and k; a re wave vectors characterizing the hh nucleus in initial (mass A), intermediate, and final (mass (.4 + I» states respectively. The symbol 6,,(x - x') denotes a Kronecker delta. Because of ,
ChbomIYlp
66
.111. rOUNDATIO,," S Of NEUTkON TkAr«SI'Okl' r H U l k Y
Eq . 4.2 the a bsolute sq ua re of the k" sum beco mes
wh ere
h,(k ' + ,,' - K - k) (8. - En)! + (/:12)1
14.3)
_~)' At
(4.4)
£u _ ~(IiK 2 m
tIIM,'(m + M) and we have suppressed the subscript I. As o ne ma y expect. in a collision in which th e nucleu s is in mo tio n the effectiv e neutron energy is th e rela tive energy E Ja . In the energy-conserving de lta fun ction in Rr the molecul ar energies (E~ II'J,;1 :'2Ml arc exceedingly sma ll compared to the bind ing energy B' II or the photo n energy E•.. If we assume that the differe nce E.~ · ' ca n be effectively igno red ," th e summa tion over k' stales can be perfo rmed immedia tely since then only the Kronecker delt a depends upu n k·. Th e e nergy of the ph ot on em itted d urin g th e ca pt ure process is usually many o rders of ma gnitude grea ter than the characteristi c therm al energy kiT , where k . is th e Bolt zmann consta nt a nd T th e eq uilibrium tem pera tu re. Therefore it is usually justified to neglect in Fq.4.1 the d i ~lrihu l ion F. ,(X , ,,'. t) compa red 10 unity. We nuw have
I' '""
=0
- e;
Rr
,.. :
I
N.. E. , - B'H I - E,) ~ 1V:.'1 1 ~ 1U:~ 11 x DuH) E u )' +
(I~/2)1
(4.5)
II I!> co nvenie nt to replace the k sum by an a pp ropriate integral. Th i-, accomplished by letting the system volu me become a rbitrar ily lar ge and observing t ha t (4.6) L Du(l ) - P(k) d'k i~
h " J,I;
In Eq.4.6 it is ofte n assumed t hat the system is in a th erm odynam ic sla te so tha t P is time-indepe nde nt..:rhe sum over phot on momentum ca n a lso be replaced by an integra l a nd in so d oing we may introd uce th e rad iation a nd neut ron pa rtia l wid ths as given in Eqs. 3.61 a nd 3.6J . Th e rat io (4.7)
• Kecpilli thi, dilTcrc nu: entail! no difficullY in principk. The made here: (or convc:nic:ntt in alcuJation.
~ ppro"jma l ioll
.
i..
61
ME.DIUM EPPl eT!
is seen to be essentially unity in view of the neglect of molecular energies. Aner some simplification the absorption rate becomes
R. - ~ L r " ' r'''fd'k
ltI · ·
2mKLJ
(I .
P(k)
E,uP + (r ./2P
(4.8)
The microscopic cross section a is related to the reaction rate R by dividing the latter by the incident neutron speed and the nuclear density. Since R~ represents the neu tron absorp tio n rat e by mass If nuclei located in the co nfiguration volume specified by X. the I sum in Eq .4.8 merely gives a fact or of N A(X), where N A(X) is the to tal number of mass A nuclei in th e cell. The nuclear de nsity in th is case is N A(X) L _ J so that
• .!.K) - [L'/N.(Xl) E,(X, K) _ nJ 2
1
E ~") ~·)fdlk •
•
•
P("')
'(I . - E U )l
+ (r .12)1
(4.9)
where E~ is the macroscopic capture cross section and J - 11K. For systems in a thermodynamic state we may use for P(k) the MaxwellBoltzmann distribution , and we then find that t1~ depend s parametrically upo n the medium.te mperature. Eq.4.9 therefore gives the fa miliar single-level reso na nce capture cross sectio n." The energy dependence of each term in Eq.4.9 gives the so-called resonance line shape. In the limit of zero temperature P(k) becomes .l(k), and
(4.10) desc ribes ra diative capture by a stationary ab sorber. Note howeve r, Eq.4.1 0 st ill con tains the effect of reco il of the co mpound nuc leus. Each line sha pe in th is case is called " natural" , the Lorentzian being characterized by a widt h r .. /2. At finite te mperatures, the integral (4.9) gives a weighted supe rpo sitio n of ma ny Lo rentzians so the resul ting line shape can be significantl y broadened, bu t with an accompanying dep ression of th e peak value. This effect is known as " Doppler broadening" an d is of co nsiderable importance in stud ies of reactor safety and con trol, t • The effect or thermal mot ion u po n r.uJiat ive ca pture or neutrons by p.s-phase nuclei was first co nsidered by Setbe an d Placzek.· t For a re... iew or Doppler effect in tberma l read.OB ICC Peara:.,1 The effect in rilSt J'CilCl0 tS has been discussed by Fcshbach et oJ.' and by Nk:holson. 1 Rcomtly the: problem or nonuniform temperature distribution MS been in\'Cltiptcd by
olhoen.·
68
TH E FO UND ATIONS 0' NEUTRON TR ANSPORT THE ORY
for it is well known that the broad ening of a resonance line can cause a significant increase in the effective absorption in a system.
The integral in Eq. 4.9 can be reduced to a form that is conventi onal in the investigatio n of Doppler effect in reactors." In terms of velocity var iables.
P(V)
a (a) "" nA! c- r C Nl r Cllf d' V -:_ -;:C;7--'--= = . < 2 ~ .. (8. E,)' + (FJ2)'
(4. 11)
where
E, = pv! /2
T, -T-V. T - lzK/m (4.12)
V - hk/M Since y is a fixed vector in the integration, the integral becomes
v,
E," + (FJ2,' oy changing the order of integration and performing the Vintegral, we obt ain (4. 13) (4. 14)
where we have introduced the variables
x. - 2(E - IJ/F. Y - 2(1. - E,)/F.
(4.IS)
J! - 4m Ek.TJM
t. -
FJ~
• Sec, for example, L W.Nordheim, "Resonance Absorption of Neutrons", Lectu res a l the Mackinac: Wand Conference on Neutron PhY1ic$. June, 1961. aV3ilab&e u a report of lbc Midlipn MemoriaJ Phoenix Project. Uni\'enil)' of Michigan.
69
MEDI UW EFFECTS
In arriving at th is result it has been assumed that p ::::: m and that in the exponentia l
The integral V' has been studied extensively" an d its values as a functio n of ~ and x a re tabulated.' It is somewhat interesting to note that at very high temperatures (~ sma ll) the contribution to the integral comes mainly from y ",. O. The resonance line shape is the n essentially gOY· erned by the Gaussian exp ( - xJe /4), the width of which. 2(4Ek.T/A)l fJ. is known as th e Doppler width. The parameter E/2 therefore is the ratio of na tural width to Doppler width.
Elastic Salllering From the preceding section it is observed that the external degrees of freedom of nuclei in gases in.fluence a given collision only kinematically. Because not all ato ms move with the sam e velocity, the cross section appears as an average over a distribution (usually thermodynamic) o f target velocities. The same remarks are also applicable to elastic scattering. and in the case of potential scattering the average is rather easily pe rformed. The reaction rate describing an elastic pr ocess in which th e neutr ons suf!er ~n energy change of Ec ' - E c and direction change of COS- I (K • K') is given by
o
::II
R. -
(-2nL)' -2I L o(e: - E: U '
where
e_
2n A
+ E, - E, .) p •.(t) L e (4.16) .
u'
[L (kl ' -"" " U,I k' ) ,
- L !U~(Ks) U:~(K's') (kl . -..·.. lk") ( k" I ...·•.. \k· ) ."
. .+ .. .. ~..
0.
-
r'
£.ot. -
EC '
-
'2I r•
r ]
(4.17) in which we have replaced D• .(t) by p .(t) as in Eq .4.6. In c:alculating the various matrix elements, we note from Eq.3.7S in the expression for U, that the integrand co ntains the step function
r n l'
70
FUU NI>A nON S OF NEU TRON TIlANSPORT T U(O R ¥
0:.
E( X, R + R,) as well as Because of the short ran ge of nuclear forces L p )'1 ' an d we may effectively write the step functio n as E(X, R) a nd oblain (4. 18)
where
V,
~ L _l
JdlR e- 'Q·· {6••,v~(R)
+ I"
u; (S)~J lUl(S' ) v~ ( R)J (4.1 9)
Again , t he subscript I appropriate to the nuclear momenta in the Kr onecker delta is u nderstood. For potent ials which depend o nly upon t he magnitude Ilf R (as assumed here) 0, is real. Th e matrix clements in H which describe resona nt scau cring are given by Eq.4 .2 with K' replacing x', so the k" s um ca n be treated as befo re. T he momentum-con serving Kroocck cr deltas appearing in bot h terms of 8 involve only the neutron an d the I nuclcus,- the sum is there fore inco herent a nd may he removed uUhide the square of the absolute value. This sum agai n givesa facto r of N A ' If we further ass ume that the reso na nces do not overlap, H '" 2't N A "",,(k _ k' + Q ) x
•
X
[V1 + L 1V:.1 1 lU:ol 1 •
-
2U(I. -
£" 'l')
(I . - R. ,,,,)l + ( r J 2)1
IV:' l
l
]
(4.20)
In writing the cross term s in Eq.4.20 which repr esent the inte rfere nce between potential and resonant scatte rings, it has been assumed that the neu tron emission a nd absorption matrix elements are at most only weakly dependent upon momentum a nd spin so that
Thi s approximation eliminates the explicit occurrence of real terms proportional to i. The particular model descri bing potential scattering used here has been introd uced with a spin-dependent term . Spin effects ca n aho be taken into account in the analysis of resona nt scatt ering. alth ough th is pa rticular aspect has not been emphasized. In the interest of illustrat ing the dynamical consequences of macroscopic medium Iccts we shall ignore the effects arising from neu tron-nuclea r spin
er-
•
Th i~ i~
ooly true fe r ideal gases in whic:h there is no intetpartic:le interaction .
iU DIU M
ee eec r s
71
coupling in our discussion. This neglect implies the foUowing :
..L lJl - 2Ul , '
~ 1U:.( KsW I U~(K's'W ~ (;~2l J.I~N'Y L UIU0.. . 12 u'
"l:
(4.21)
nA V( mL 2 11"'~1) )·
Making use of these results a nd inserting Eq. 4.20 into R.. we obtain
I -
)( VI +
L
1.2 ['h L') u:..]' · m
•
U(&• -
E• ..• )[~lr. L' · ..]) m
E,n·)J + (/ : /2)2
(&.
(4.22)
where , IE .... E. - E• ., and we further suppress the superscript A in the energy symbols. At this poi nt it becomes convenient to treat the neu tron mo men tum asa continuous va riable; then 61: becomes a Dirac delta, ' ,(k - k' + Q) -
( ~)' 6(k
- k'
+ Q)
(4.23)
Moreover, it is a lso ap propria te to treat the k an d k' sums as integrals. In the case of spinless nuclei the potent ial [j characterizing t he dire ct process may be written as
[j "'L - l grd lRe-iQ ·· -"I~ ~ ~X
(4.24)
mL'
R
with (A/h)'
nQ) - 2gm I
+ (AQ)'
(4.25)
Compa ring with Eq .3.7S we see that thi s expression is valid so lon g as A.. 1.. Le., range of interaction sma Uco mpared to the linear dimension of the spatial cell. Since Ais of order IO- u em, this co nditio n is always fulfilled for a ny reasonable cho ise of L. If, on the other han d. we had
72
THE FO UNDA TIONS O P N BU T RO N T R A NSPORT TH EO R Y
used the Fermi pseudopotential instead of the Yukawa potent ial we would have o btained %_
(m ~ M)a
(4.26)
where I is the bo und-atom scattering length and a is the conventional " free-atom " scatte ring length. The final form of the elast ic-scattering reaction rate may now be expressed as
R. = !:.. !d 'kd 'k't,(X.k,, ') .l(E. - E•. + JE).l(k - k' + Q ) ", 1 X [
I
.f-
L
l.
HH ~'"/2)' - X(8 • .- t:.: .~ .)( J.I ~'tl/2) ] (4. _ E..._.)I + (/ : /2) 2
X
(4.27)
Here we have denote d (N...(X)!U] P(t ', t ) by fiX , t ', I) wh ich, as an ana logue of the neutron density, represent s the a verage number o f nuclei of mass A having mo mentum n ' to be found at X at time t. For z > O. the last term in Eq.4.27 shows that the interference between poten tia! and reso na nt scatt erings is des tructive at the low-energy side and constructi ve at the high -energy side of the resonance line. The results in the present sectio n ca n be summarized in terms of an equat ion describing the transport of neutrons in mon atomic gases in which the d om inant neutron-nuclear interactions are radiative capture a nd clastic sca ttering,
--:. + I,K • V + hK E,(K)]fl X , K, I) [ vI III m
-!d'K'flX, K'. ')C:')E.(K') F(K'~ K)
(4.28)
where the macroscopi c cross sections are 1:.(K) - E'(10
+ 1:.(K),
l.' (X) _ .....:!.... c- ~tt) c 4K~ 7· E.(X)
=
F·l.)!
14.29)
f . . (X. t .l)
d Sk
(8.
h' !d' kd ' k'd'K'/.lX, k. 1).l(E. _ E•. + JE) mK x
~
('. -
(4.30)
Eu)~ + ( rJ2)1
E,,)' + W./2) '
x
(4.3 1)
73
MED I UM EFFE CTS
and the scattering frequency F{K' _ K) is given by
l.~(K')"(K' -
K)
~ ~fd'kd'k'fA(X.k,.t) 6(E. mK'
- E., + ~E) x
x 6(k - k'+ Q)x
x [ra +
E ~(1r:1f12)a ,
- rll. - E" 'A ') (J.r~IfI/2)] (I , - E• .•.)' + (rJ2) '
(4.32)
The cross sections, through their dependence upon the nuclear density, are ofcou rse also functions ofposition. The transport Eq.4.28 has been extensively employed 'i n reacto r analysis. Usually. additional simplifications are introd uced in atte mpts to treat a problem analytically. For example, in neutron therrnalization and diffusion investigations it is conventional to assume "I/o" absorptio n- and neglect the effects of resonant scatte ring. The po tential pan of the scatte ring cross section can be further reduced. Ignori ng the contributio ns from resonant and interference ef· Iects we have from Eq.4.31
EJK)
~
(::lc' fd'K' d' kJlX. k, ' )6 (E. + ~ Q . k - ~E)
(4.33)
where E.. - (AQ) a12M is the recoil energy and we have suppressed the nuclear mass designation in j(X. k. f). Upon the use of the integral tcpresentat ion of the delta function th is expression becomes E,(E) -
' f" (E')"'f f" d, '(
..L :brA
0
dE' E
dJ]'
) T e-II'Ci'.. -.d.¥) /-
_.
(4.34)
where (4.35)
In Eq.4.3S a Maxwellian distribu tion is agai n assumed in evaluating the integral. Th e nuclear den sity [N(x)ILJ] is denoted here simply as n. The scattering cross section is now expressed in tenns of the more convention al energy variable. From Eq.4.34 we can identify a microscopic cross section a(E - E . 9) which descn bes the scattering o f a neutron from energy E to E with a specified chan ge of directi on of motion,
EJE) - n f dE" lID' alE - E". 0)
(4.36)
• n.e invene speed dcpendcncc o( lha capture croa section (ollows (rom Eq.4.30 and 3.6) for neutron eneraic* far below rcaonanca.
74
1 111
I tl U N t l "Tl H NS til' NEU T RON T R " "l S I' O RT l llHIRY
where (4.37)
The van ..hlc (J denotes the angle between K an d K'. The integratio n o f "over encf};:Y· o r angles!" ca n be carried out. These calcu latio ns, being q uire complicated and not pa rticula rly illuminating, will not he discus!-I.."tl here. On the o ther ha nd, the high-e nergy limit is interesting and read ily available. For E ~ I eV the nuclei can be taken to be initially at rcst,J(X, k, I) ~ no\(k ) a nd we obta in instead of Eq .4.37 ~E -
E', O) - r 1
(EE')'"
II(E. - , IE )
(4.38)
The angula r integral of this result is readily performed,
I + 2M A'(E
-
E'llA'
when
«s s: E'
~
E;
(4.39)
o therwise, o(E - E') ~ 0,
with r- "" 2~m()..''')land (\ "'" ((M - m)/( M + m»)!. If the second factor is replaced by unity then 0(£ _ E') gives the scattering freq uency fam iliar in reactor theo ry and leads to a con stant total cross section a(E) - 4J1c 1 • t Th is i.. eq uivalent 10 assuming th at r is a constant, which is actu ally valid for energies up to abo ut lQ4 eV. Beyo nd this region expe rimenta l results show a grad ual decrease in 0'(£) that ca n be filled q ualitati vely by 311 expression of th e form (I + PE)" , II being an adj ustable pa rameter. From Eq .4 .39 we find that a(E) is in fact gi v en by th is form , 1
) -
4:tc I +2MA'( I - . )EW
• Th is is equivalent to the limit of zero temperature.
t The same result is obtained by using the Fermi pseudopotenua l instead of the Yul:m ;l functions to describe neutron-nuclear interactio n.
75
MED I UM EF FEC TS
H rcsonant scattering is ignored, the tra nsport Eq.4 .28, upon making use of the above results, becomes
[ : , + Ii} . V +
Dl.~(V)Jf(X, Y,I) all
f d"v' v' / (X, v', I) E(v' _
D,
0)
(4.40)
with 1: = na. This Integro-dlfferen tial eq uation has been the fundamental equati on in many investigations of neut ron transport. I I The energy-independent form of Eq.4.40, the "on e-speed" transport equation, provides a problem which can be treated with mathematical ri gor. I1 • I J The tint two angular moments of Eq.4.40 give the diffusion equation which constitutes the analytical basis for many of the present studies of nuclear reactors.
B. Transport In Crystals It has been sbown in the preceding section that, in the absence of chemical binding, nuclear recoil effects on the cross sections can sometimes be analyzed by a straightforward calculation. For systems in which interatomic forces cannot be ignored this effect is in general considerably more compl icated. However, in the case of strong chemical binding, it is again possible to discuss medium effects in analytical terms, for then one can reasonably represent the at omic motions as oscillations and make use of well-developed dynamical models in solidstate theory. I .. Aner eliminati ng the dependence upon electronic coordinates" one obtains in the harmonic approximation, a description of nuclear motions identical to that for a set of coupled oscillato rs, which can then be decoupled by a transfonnation to normal cocrdinates. However it is our purpose to illustrate the general featur es of medium effects on neutron interactio ns. Thus to avoid an involved discussion of crystal dynam ics we shal1 restrict our considerations to a system describable by a set of uncoupled oscillators. It will be assumed that each nucleus experiences identical interactions with ill surroundings so that it executes isotropic oscillations abou t an equilibrium position (a lattice site in the crystal) independent of all other nuclei. From this it follows that the fundamental vibrat ional frequencies can all be taken to be the same.
76
T H E F OU N DA.TlO NS O P N S U T It.ON TI.A. N SP ORT TH EO Il,Y
The present model, th e Einste in crystal, is admiUedl y a severe idea lization of act ual atomic motions in bound systems.· Nevertheless, the results derived on the basis ofsuch a description are meani ngful and , like t he ideal-gas cross sections, often useful for practical calcula tions, In general it ca n be expected t hat the model will p rovide an adequate descripti on of integral properties of the cross sections, but is not suitable for q uantitative analysis of differential cross-section measurements. One ca n, however, extend the fonowing result s to more elaborate crystal descriptions ; the requ ired mod ifications being mainl y refinements of the mod el and not changes in the method of calculation. Radio/in.' Captu re
A natu ral extension of the investigation of neut ro n capt ure by free atoms- is the co rresponding treatment for ato ms hound in a crystal. T his problem was fi rst co nsidered by Lamb" whose work is of mo re general signifi cance because the process of radiat ive captu re is closely related to o ther nuclear pr ocesses characterized by a point interac tio n, a f:1I;:( that has attracted attention onl y recently.' · Fo r example, one encounte rs the sa me mat rix clements of the for m ( k l cxp (iq , R) W> in the probl em of emission a nd abso rption of nuclear ga mma rays. t With respect to neu tron abso rptio n, Lamb sho wed that, if th e latt ice binding is sufficiently weak. fhe reson ance line shape is the same as that ofa free nucleus but at a n effective temperature co rresponding to the average energy per vibrational degree of freedom o f the nucleus including zero point vibration. We shall first obtain the general cross sectio n and then show how th is limit emerges , T he reaction rat e to be studied is th at given in Eq.4.1. Since the ene rgy cigcnsturcs Ik ) arc no longer momentum clgcnstetcs, the matrix. elements must be obtained by a different approa ch. We shall agai n ignore the external energies E: an d E:, +l in the della function" and also • In so me crystal! it is possible that the vibra tional mot ions ca n be a dequa tely described by an Einstein model , Such an example could be the hydrogen atom in
zirconium hydride. 16
t Inracr, Lamb's lheory provided the initialexplanation crrecomessgamma-ray transitions, a process now known as the MOssbauer effcct! 9 • Although this approximation is conventional, it is here (as in Section A) not necessary. For n am plc, the method or analys is used in the rollowing to obtain elastic.scauC'ring cross section ca n be: equally well a pplied here.
77
MED IUM EffECTS
neglect the photon distribu tion function compared to unity. The expre ssion for R c becomes
where ( k" l e" :II, lk )
I. -
1
E~ - E: + E:.-;1 - ~ r.
(4.42)
where ns a result of perfo rming the k ' summation the k " sum appears o utside the square of the mod ulus, The quantity except for a constant multiplicative facto r, is identical to the expressio n considered by Lam b. U To carry out the k " summatio n, we rearrange the resonance denominator by writing
<>.
(E - trv-» ... ; [
so that
ds e- lr -I f: M
( >_J ds ds' e- UU' )l'f1 e,c,-,·ur. - Ial ( >r ( >r
;0;
L p .(t) ( k l e IC' - " IH e - 1K • II , e-·u - n u el1l. ·lI, lk)
•
(4.43) (4.44)
For an Einstein crystal in thermal equilibrium it is a straightforward matter to evaluate ( >r. The calculatio n is d iscussed in detail at the end of this section and we quote here only the result,
<
}r
•L
.. -.
.,. e- DJl:1
J.(P K2) e- JtlZ -1r..(.-,,)]
(4.45)
where D - '1 coth Z, P "'" '1 cscb Z, '1 - II/2MflJ , Z - frflJ/2k. T. w is the oscillator frequency and J. the modified Bessel function . Inserting this result into Eq.4.43 and perfo rming the indicated integrals we obtain an expression fo r ~ . Upon introduction of the partial widths as before. the cap ture cross section is given by E«
1111(1)1 + (rJ2,)1
(4.46) The integer 11 denotes the number of phonons that are created or destroyed depe nd ing upon the sign of 11. The case of n - 0 gives a rea
7K
1 11 1 H IUNU,o, T IUNS 01' NI.'U T RON TJl. AN S 1'OIt T lllH l lt y
son ancc line centered a t &• . T his abse nce of reco il co rrespo nds tu the fact that t he neut ron mome ntum is absorbed by the crysta l as a whole. As in most cryslal models, t he mass of ou r system is assu med to be infinite. In practice, however , the re will always be a finite, th ou gh vanishing small, amoun t of recoil. In general the line sha pe of eae h resonance ca n he qui te com plicated and may eve n sho w fine struct ure indicat ive (If pho no n t ransitio ns.'? It is of some interest to investigate the im plications of Eq. 4.46 in the limits o f strong and weak bind ing. The co ndition of tightly bo und nuclei is simply expressed by taking the vibrational frequency to be a rbitrarily large. In this limit the cross section becomes (4.47)
where by virtue of the sma ll-argumen t repr esentation of the modi fied Bessel functio n. [,(x ) ~ -1 .. _0 2 n!
(X)'
we have ignored all but the II "" 0 term . As on e can expect, there is no tem peratu re or reeoi l effect for rigidly fixed absorbers so eac h rcsonancc is desc ribed by its " nat ural't llne shape. It is to be noted tha t this result is not equ ivalent to the zero-temperature limit because in th ai limit the zero-po int vib rat ion effect is still to be taken into accou nt. For the latt er case oAK) is given by Eq . 4.47 multiplied by the facto r cxp (-lJKl). T he form of the cross sectio n in Eq.4.46 is not co nven ient for exam ining the weak binding limit. For this pu rpose we return to a co nsidera tion of ( ). Upon the intro ductio n of a delta funct ion and its integral representation , Eq.4.42 becomes
<)
=
f'
do ,)(" - E A + _, • ~
~ _1- J~ ' dt 2:1
."
I'
~
IK R .
L Pl ( k l ell ll e- 1K • •
1
Ik" ) 1
u .. ~ (8. - E.. - e)1
+ (l~ 12) l
e - 1II1 Ik" ) x
Ik ) f 7: d!,l {8 _ E _ 'I'
dl
( kl e-
u ··
x ( J.. " l e " ·1t
. : ~J~ '
EA. ~I) L P
<. e -1 1o: R{'Je lK
•
J(
~- j;~l (!
e -/I{...· -£.o:I-I"l'V l
R)
'
(1' {2)'
+ •
(4.48)
79
M I ,IH UM I •• I T T S
where ( c · ·.. · II U ' c'.. ·· h
s
L p . ( kl e- ·..·•..· c'..·•
•
110 )
::II
e,O I
(4.49,
As in EqA.45 it can be shown tha t g( t) ... '/KJ[coth z(cos h",t - I) -
i sin hm']
(4.50 )
If now g(1) is expa nded in a powe r series an d term s up to seco nd order in t a re retained , (4 .51)
with
£-
It", coth z - It...( ( n).,.
+ iJ
(4. 52)
where ( lJ) r is the ave rage osc illato r level. It is seen tha t Erep resents the average energy of a vibrat iona l degree of freedom includi ng zero-point vibra tion . For th e truncated series to be valid it is necessary that the integra nd in EqA.48 make negligible co ntribution whenever Iioot ? I. If th is is to be the case, the sum rt2 + ateg(t) must be lar ge and pos itive; in other wo rds we require (4.53) which is the condition of weak binding. The approximate fo rm of g(1) given by EqA.5 1 allows us to exp ress the crystal cross section in terms of the x) integral introduced fo r the free-atom cross section." '" We observe that Eq.4.48 ca n now be writte n as
,{.e.
(4 .54 )
where d - .. 4£.£ and the ~ subscript is suppressed. Agai n ma king use of the delta functio n and its integral rep resenta tion, we have (4.55) With the help of this expression the J' integrati on in Eq .4 .S4 yields . (4.56)
80
TIII~ fOUN DA TI ONS Of NBUTIt O N Tk A NSP O ltT TIlr:O k Y
with! "" 1'jtJ a nd x = 2(£.. - I - £,JI1: The qu antity x is defined here somewha t differ ently from that given in EqA .I5; the difference is of o rder (mI M )2. Eq . 4.56 shows that the reso nance line sha pe in crystals fo r the case of weak binding is the sa me as that in gases, bu t by comparing the two A' s the crystal line sha pe is see n to co rrespond to a n effective tempera ture of Elk•. Thi s result was first obta ined by Lam b."?
Elastic Scattering An ela stic sca ue rlng is a n in teraction in wh ich the number of all kinds of panicles. tra nslational kin etic energy, and momentum a re co nserved. Ev idently. in t he pre viou s chap ter because of our preoccupation with the specifically nuclear aspec ts of neutron-nuclear reaction s, we treat ed this notion rather casua lly. There, we implied tha t a process was clastic irthe initia l an d final "inte rna l" nucl ea r sta tes were the sam e. Literally, such a n implication is never j ustified. Practi cally, it is j ust ified in the present discussion if t he target nu clei are aggrega ted in a n ideal, mona to mic gas, since we ha ve igno red all neutron -electron interactions a nd hen ce electr onic excita tion of a toms. But also practically, it is not justified he re if the nuclei experience appreciable bind ing as the y do in molecules, crystals, and liquids. In fact. in these latter instan ces, a n ela st ic co llision is o ne in which both the "interna l" and «external" ini tial a nd final nuclear sta les a re the same. We shall henceforth adhere to this more careful interp retatio n. The redu ction of R, as given by EqA. 16 and 4. 17 to give cross section s describing po tential, resonant, a nd interference sca tte r ings can be carr ied o ut witho ut approxima tio n. Th e resona nt cross section will be exa mi ned in the limit of sho rt lifetime of the compound nucleus, which , as will be seen, is Quite simila r to the above weak binding limit. various aspect s of the potenti al cross section a re of inte rest, a nd these will be d iscussed and used to predict the beh avior of the total c ross sec tio n. Th e reduction of R. invo lves the evaluation of matrix elements of the type given in EqA.44. We rewrite (4.17) as
8 8, ~
I
u-
~
•
I8, ,.,
V,V" (kl.-"" R" lk') (k' i ."' .R, Ik)
(4.57) (4.58)
81
M IO DI UM EFFECTS
(4.59)
9
..
_
t"' k.
V iti U" (Viti' V lfl ') . 0.
.0
O.
.0
f ds ds' el'(.Ja+U/l lr a) e -,. '(4. -U/Jlr. ) x
II ' .
where J. - I . - E. and in writing Eq.4.60 it is assumed that the re-
sonances do not overlap. Note that now there will be contri butio ns from terms with I " 1'. these terms lead to d iffraction effects which for crystals cannot be ignored . The calculation of 8, is tedious but proceeds in a com pletely strai ght forwa rd manner. We sha ll d isplay onl y the resul ts in the form of a differential cross section in final neutron energy and scatteri ng angle. Le.
oJ£, - E, O) - a:(E' - E, &) + o~£' _ E, 6) (4.61) where th e superscri pts denote the potential. int erference, and resonant co ntr ibutio ns, and where
x 1. ,(PQ · K')1.,(PQ · K) 1.,(PK· K·HI . - E - (n, + n. ) ~wJ [I. - E -
( II,
+ II.)
+ b(JE) e-JlI;I....!- L' ~ .b,-sr) L N..
II'
•
fiw]1
+ (r.J2)1
I ,,(PK ' K ') (I . - E - IIAw) e (I . - E - nfiw)J + (r.12)1
al }
(4.63) • 0000nIY1p
82
THE! FOUNDATIONS O F NE!UTR ON TRANSP OR T'THEORY
a'.(E'-
)(
£,8) - (:r ~ [l~"J
{e-IJfI;I+J: " I., ~.'< _ }.'''' h(AE
)( I. ,(PKJ) I.,(PK' J) )(
X
[Ii,.,
x
- [n, + "I + III + n,JAw» )(
I..( PK · K')e -·'I.] e- I.,••,II. )(
[I. - E - (n, + n, + " . )"0) ) (4'. - E - (Ill + n) + 111)11«1] {(I. - E (n, + n. + n,) "w]) + (r J 2)'})(
)( HI. - E' + O(AE)e-I,,«,_I- L' ~. .,-.,.I )( N ..
- (n) + II) + nl)
"wI'
+ (1'J2)'
+ (1'J2}' )
/I '
I••(PK· K ') I ••(PK· f( ') [(4'. - E' - 111"(") (I. - E' - IIlltU) ) + (1' 12)' ] c- (·,··· Ill
)( .~. (4. - E' - nlm,,») + (/ : /2)'1 [(8~ - E' -
"J;/II)' + (/~/2P]
(4.64)
The equilibrium position of the hh nucleus (atom) is denoted by X" In each cross section the direct terms (I - /') have been separated from the "interference" terms (I '!' the latter involve no energy transfer and therefore contribute only 10 elastic processes." A discussion of the potential cross section will be deferred until later. From Eqs.4.63 and 4.64 it can be observed that the in8uenc.es of chemical binding upon resonance phenomenon are quite complicated and that interpretationof these resultsappears feasibleonly in the limiting cases.t The dependence of c(, upon the scattering angle appears solely in the argument of the modified Bessel functions and in Q. In the event of 90° scattering Eq.4.64 is considerably simplifiedsince all the n,'s except Il, and n, are zero and the corresponding I.,·s are to be replaced by unity. The exponential exp [ -D(K' + X")), in the resonant scattering cross section is known as the Lamb-Mcssbauer factor. This factor provides an euenuation of any resonance process that is influenced by
n.
• This is. direct COI'*llIuencc of the assumpt ion of iDdcpcndcn t vibcalions. In realistic model which lrats the atomic motioM U a:M1plcd OIciIIatlom lbete willlbcn be bolla dutic and inelastic "interference" or di1I'rac:tion cffcctl. t See. for example. the discussion of elastic raonant ICattcrin. by Trammell.JO Analotovl apreulonl employin. more realistic: mode" or crya,ata have hoen obtained by E.Wiulel' (unpublished). • mot"C
83
MEDI UM EPfE Cts
temperature and lattice binding through the parameter D. On the other band. the corresponding expon ential in the potential scattering cross sectio n is exp 1- D(K - IC.')']. which is the fami liar Debye-waller factor . The attenuation of d irect processes is therefore sensitive. in addition. to th e angular correlation between initial and final neutron momenta : This comparison is inte resting since it tends to suggest that ift he lifetime of the compound nucleus is very short the attenuation fact or of a resonan ce p rocess can co nceivably be expressed in an angular dependent form. We shall now investigate Eq.4.64 in such a short lifetime limit. As in the case of the weak binding limit in radiative capture. a ..timedependent" representation of is more convenient for the present purposes. If we consider on ly the direct terms of Bq.4.64. we have
a:
".W -
E,O) -
E'
dr e - II • •
x _1- [
2n
T (E)'" ~ [lr"']'
_.
.
f-
x
d.rds' e - h H'U"J J
e- ""-"I C,,.- rl~1l
(4.65)
0
. where
p(ss't) =
~( K ' g(1)
+ K" g'(s - s' - t)
+ K . K'[g(s') + g'(s) - g(s' + t) - gO(s - 1m
(4.66)
r
with g(/) given by Eq.4.SO~ For very iarge the contribution to the s and s' integrals will come mai nly from s, s' ~ ~I . This sugests that the terms in p. which depend onl y upon s or s' may be represented by truncated power series. Retaining only the first two terms we obtain
. (ss't) ~ -DQ' + iA..,(K· K1 (s - s') + . '(ss't) Jl'(u'l) - ~ ej:.... [K'
2
+ K" ej:---.. l
-
K · K'(ej:-
(4.67)
- e' -
)]
(4.68) where t'lj: - coth Z
± I. In Eq.4.68 th e double lip. denotes a SUM of
tw o terms correspond ing to upper and lower siJlls respectively. The
term s neglected in the above approximation are of order (Aws)' and higher, so a condition for Eq.4.67 to be applicable can be stated as r )0 ACtI. Thus if the lattice binding is small compared to the resonance
84
THE FOUNDATIONS OF NEUTRON TRA NSPORT THEORY
width the attenuation factor in If. ISalso effectively given by the DebyeWaller factor." Th e resonant-scattering cross section which one obtains by using the approximate form of p. is very similar to Eq.4.64. We will not exhibit this result, but instead if we introduce a further approximation by writing c" 'I"'I) ~ 1 + p.'(ss't) (4.69) we would obtain a.(E'
-+
E, 8)
~
(;yJl
e-
{
IHP ~ [J~N)J x 6(JE)
(II'. -E'
+ C ±Aw -(i /2)r.) (II'. -E' + C + (i/2)r.) ' lAw
(II'.
E' +C
(i/2)r.)(II'.
.
II
,
E',+C ± Aw + (i/2)r.>JJ
(4.70) with C "" h7](X. • X') .The first term in Eq.4.70 represents the contribution from elastic scattering, and, in the remaining terms, upper and lower signs denote inelastic events in which the neutron loses or gains energy by an amount Aw (one-phonon processes). Higher-order inelasticity has been neglected by virtue of our expansion in Eq.4.69. t In the remaining part of this chapter we shall restrict our attention to potential scattering only .- The cross section given in Eq.4.62 is seen to - I(wc interpret tbe compound nucleuslifellme as theinteraction time, the above condition implies that the coUision time be short compared to the characteristic vibration period in lbe lattice. This conclusion is in aeneral agreement with Trammel1.10 t This iJ somewhat similar to the "time" expansion tint introduced by Wick .11 • For potential scattering of neutrons by crystals the reader should see the excc:Uent review by Kothari and Sinpi;l1 for the tifne.dcpendcnt repraentation approacb soc Sjolandcr;u a number of fundamental upcctl of the aeneral tbcory have been reviewed in detail by Yip, Osborn. and Kikuchi,"
MEDI UM EFFE CTS
contain %simply as a multipJicative factor. For a system with nuclear spin / it will be necessary to conside r both tcnnsin (3.7S). l f furthermore we co nsider the system as an isotopic mixture with random distribution, then by ca rrying out the appropriate spin and isotopic averagcsU. J6 we find th at the quantity %J multi plying the direct and interference terms sho uld be replaced by and respectively, where
Q:
Q:
a' _ ( 1 + I a J ) + ( 1 I , 21 +1 · 21 +1
Q~' /
, ( 1+ I, + 1 ,)
QJ -
Q.
21+1
21+1
(4.71)
Q-
>
and the symbol < here denotes isotopic average . The quantities Q.. and characterize tbe. interactions in which neutron and nuclear spin o rientations are paraUei and an tiparalleJ, and are defined by
Q_
Q.. "'"
%0
1
+ - Xl 2
1
Q- -
+
I
(4.72)
%0 - - 1 '1
2
where
.
f
:~ d)R17o.,(R)~-IQ··
%0.1 -
(4.73)
In the special case of neutron-proton scattering Q .. an d a; would correspon d jc the conventiona l tri plet and linglet scattering length s, although in the present treatment they are functions of the moment um transfer. With the above mod ification the differential cross section for potential scatt ering- can be given in more general fonn,
(.!-)'f'[
oJ..£' _ £ 0) _ _ 1_ . ' 'be N" E x
A ,,{Q . I) -
1/ .
x
[a: r A.~Q . I) + a: r' A ,,{ Q . I)l
(4.74)
r r, (kl ...·""" .-.. .... Ik)
(4.75)
I
where
dl e- LU 0
•
It is sometimes conventional to speak of o. - Henceforth we I Upprcu the I upencript p.
II'
in terms of its coherent and
86
I II I' FO U ND AT I O NS Of NEU TJlON T R A NS PO k T T H EO RY
incoherent parts. Thus if we introduce coherent and incoherent sca ttering lengths as (4.76) 0:'" .. a~ ar•• == a~ - a ~ the cross sectio n becomes .'.(E' .... £. 0) =
--!.(~)"' f" dl '2-"'I NII E'
e -U1" 1
x
0
x [a~
r AIl(Q , I ) + a:.. :E AIl.(Q, I)l l
(4.77)
II'
Fo r the Einstein crystal this is not a part icular ly co nvenient representation so we shall co ntin ue to discuss the contributions from direct and interference scatt erings sepa rately. Thus far it has been possible to calculate ;1(Q, t) directly because for the simple systems under co nsideration. the exact eigenstates lk) arc known . For more co mplicated dynam ic systems such as liquids, this a pproac h is still straightforward but no w the ca lculation depe nds upo n less satisfying models for explicit forms of the wave function . There exists, however, an alternative and eq uivalent proced ure for formulating the general scatte ring problem, In this approach the cross section is expressed in term s of a space- and time-dependent functi on which describe... t he dyna mical p roperties of the scatte ring system, U so that the ap pro ximat ion in describing a complicated system the n enters into the determ inati on of this fun ction . This fun ction is defined as
G(r, I)
~ (2n) -' f
d' Q e-'Q·'
..!.. L A n- (Q, I ) Nil'
(4.7H)
and similar quanti ties for the' - ' and' 01< " ter ms only arc deno ted as GJr, t) a nd GJ..r, I). The cross sect ion is then expres sed as 3 four dim ensional Fourier tr an sform .1( £ ' - £ ,0) = - I
2n
(E)' I> E· -
(O~MS'.C(Q, dE) +
a:.. SCII{Q. ,IE}] (4.79)
J
I,Sc. ,,(Q, l iE) ... dId" G(r, I) eMO · r- . lh/I '
(4.80)
and SIM is obta ined by replaci ng Gwith G•. The functi on S is ca lled the scat tering law, and is a quantity in term s of which the scatteri ng data ca n be analyzed and presented for use in the tran sport eq ua tion. n · J1.lI
87
MEDIUM EFFE CTS
The function ,G(r, I), was introduced by van Hcve'" asa natural timedependent generalization of the familiar pai r distribution function g(r) which describes the average density distribution as seen from a given particle in the system. 2 ' Aside from neutron scattering, G(r, /) is in fact a quantity of general interest in the statistical the or y of many -body systems.u.)O From the reality of S one has G'(r,l) - G( - r, -I)
(4.81)
The fact t hat G is in general complex implies that it ca nno t be interpreted as an observable. As suggested by van Hove, und er classical conditions or more specifically when R(t) commutes with R, G gives the probability that given a particle at the origin at I = 0 there will be a particle at rand t. A number of attempts have been made to develop a theory of slow-neut ron scattering by liquid s on the basis of such an interpretation. ) 1 _ • We now return to mor e detailed consideration of neutron scattering by an Einstein crystal. F rom Eq.4.74 we can write the cro ss section as (1,(£ ' _ E, O) = 6(Ll E) e-DO'[a: l o{PQ2) - a~ l
+
6(.1E)a~e-DO' ~ l ~ elQ 'Kr
(E)'"
+ -E .
a~
•
e- DO'•__... [ 6{LlE - nli(l) l i PQ 2) e- -Z •.,
(4.82)
The elastic contributions a re exhibited in two separate terms. The second term con tains the interference factor
where now / extends over all the scatterers in the spatial cell and N is their total number. For a cell of char acteri stic length L ~ 10- 4 em, N is of order 1012 so tha t this factor gives the well-kn own Bragg condition for elastic interference scattering in the usual way. As a result of the assumption of uncorrelated vibrations, diffraction effects are seen • For a discussion of the classical limit of the cross section sec Aamodt, Case, Rosenbaum and Zweifel, PhyJ. Rev., 116 : 1165(1962). A dbcuss ion o r the classical limit of G(r, t) has been liven by Rosenbaum.21
88
TJllo nlllNI)A.TlONS 01' N£UTRON TRA NSI'ORT HlI,ORY
to be purely clastic. This will not be the case if we employ a model that describe s the nuclear motions as coupled oscillations. Z4 Th e 1/ T- 0 term s in o, constitute the inela stic portion of the cross section and these give rise to a set of equally-spaced lines in the spectrum corresponding to different phonon excitations. This structure is in ma rked contrast to the smoo th distribution predi cted by the gas result in Section A. Since the vibrational states arc stationary in the harmoni c approximatio n (infinite phonon lifetime) all lines have zero width." It can be observed that so long as energy conservation is satisfied any inclastic proce ss may occur. At T ... 0 the neutron cannot gain any energy becau se exp ( - nZ ). interpretable as a measure of the probability of lind ing the osc illato r in the nth eigenstate. vanishes. The exponential factor exp ( _DQZ) in Eq.4.82 is the qu antum a nalogue of the Debye-Waller factor originally derived in X-ray diffraction to account for the effects o f thermal motions of the scatte ring system. II auc nuatcs a ll proces ses, particularly at high temperature or small Z ; the effect does not vani sh entirely at T = 0 because of zeropoint moti ons of the scatterer, For very small Z the asymptot ic form of the mod ified Bessel function
... become s applica ble, the exponential part of which then cancels the Debye-wuller factor . Obviously the same situation hold s for large Ql so we sec that interference effects will be negligible in the region o f high momentum transfer. t Since f1.( E' _ E. II ) is the differential cross section in energy and angl e. the total potential scattering cross section a.( E') is obtained upon inlcgm ti ng Eq.4.H2 o ver !) and E . Because a.(E') enters directly as ;. parameter in the tra nsport equation, it is of som c interest to examine its beha vior on the basis of Eq.4.82. The macroscopi c system und er considera tion is in general not a single crystal, so the crass sectio n sho uld be uvcrugcd over crystal orientations. Thi s aspect, howeve r. is not relcvun t to our discussion. and therefore we will ignore it along • For disc ussions o f finite phon on lifetime in neutron sca Uering sec Maradudin a nd Fc in. 3l and Akcasu.:n t In a generaltheory which admits elastic as well as inelastic interference scat!ering the present remark applies only to the elastic portion which . howe ver, usu;llly provide s the dominant diffraction effect.
MIWIU14 EFFECTS
89
with spin and isotope effects. At very low neutron energies (E ' ~ lO-) eV) the cross section predicts no a ppreciable elastic processes because /o(x) is essentially unity (a~ = a~ Z2) and the wavelength is sufficiently long that the Bragg interference condition cannot be satisfied at any scattering angle. Also in this region (E' < Am) the neutrons cannot lose energy, so the onl y permissible process is that by which the neutrons gain energy. The cross sectlon therefore varies as l/v, and generally increases with temperature. As the incident energy is raised. elastic processes begin to contribute. A significant increase occurs when the Bragg condition which allows the largest wavelength is just satisfied. At still higher energies the interference term begins to be attenuated by the Debye-Waller factor, and, while th e cross section will continue to exhibit sharp jumps as additio nal sets of crystal planes give rise to interference scattering, the over-all oscillatory behavior is damped. For sufficiently fast neutrons (E ' ~ I eV) the dominant process is inelastic scatt ering in which th e neutrons lose energy. Here each scatterer can be treated as a free particle so that the result in Section A is applicable. In fact, in the weak-binding limit one can show that :2
a,(E') - 4n[Mx /(M
+ m)]'
The above remarks are illustrated in Fig.4.1 which IS In general agreement with ob servations for such scatterers as graphite, beryllium, and lead.'· All the discussions in this chapter have been concerned with monatomic systems and hence the center-of-mass degrees of freedom of the nucleus. However, in polyatornic systems, the neutron can excite all the degrees of freedom of the molecule so that internal molecular degrees of freedom also have to be considered. The intermediate scattering function A(Q. t) can be written as a product of two functions, one depending on center-of-mass translations and the other on the internal molecular motions. If rotation-vibration coupling is ignored, A can be further decomposed so that the effects of translation, rotation and vibration may be considered separately. From the standpoint of analyzing a particular experiment, it is important to treat the rotations pr operly since th eir energies are of the same order as those of translations. The presence of rotational transitions can therefore complicate any interpretation of the scattering data with regard to intermolecular forces.
90
T ill
FOUN D A T ION S O f NE UT R O N TR A NS PO R T T U EO R Y
Th e method of ca lculation presented in thi s section can be used to treat the nor mal modes of internuclear vibrat ion . T he influence of molecular ro tations has been investigated mainl y in neutron scattering by gases.9 . U The cross section of a free rotator can be obta ined rigoro uslY,)"'·'7 but the application of the formalism is rather involved. JI On the other band , in systems where appreciable or ientation -dcpendcm Mr _1OOI Of'
IlO1tMHM t N' t
S, An t . "1;
. . _......
r lt n_ .. ' o ot , ltOU _S( croOlt
tTs
I(
I
'------,~------,---, 00'
INCIOENT
0'
NEUTAON [ NEAGY. ~
Fi(l..4.1. Qualitative behavi our of lolal po tential scattering cross section.
forces exist, rotation al moti ons will likely beco me hindered. An intercstin g example is water where experi men ts have revealed promi nent modes of hindered rotation. This type of motion is stil1 not co mpletely understood, a lthough att empts to descr ibe its cff.x. ts in neut ron SC4lt· teringn .• o arc probably sufficiently accurate for th ermalizaticn ca lcul.uions.
MEDI UM EFfECTS
9\
The Thermal Average In the pr eceding cross-section calculations it was necessary toevaluate averages of matrix elements of the form
s.; = L r, (nl e"NI. e- lIt . R, e- IIH1A e' K • R " In) •
(4.83)
for an Ein stein crystal.' In this notation p. is the probability that initially the crystal is in a state specified by the eigenstate In) . H is the crystal Hamiltonian, K is a momentum vector and R, is the instantaneous position of the hh nucleu s in the crystal. Ifequilibrium position Xl is introduced then (4.84) (4.85) where we have let R, "" x, operator
+ u,
and have introduced the Heisenberg (4.86)
Since the Hamiltonian consists of a sum of individual particle Hamiltonians, the only part of H that does not commute with UI is H,. In Eq.4.8S the symbol ( Q>r denotes an appropriate average of the expectation value of the operator Q. This quantity is often called the thermal average because the crystal is assumed to be initially in a thermodynamic state. Note that WII' is a function of t only if 1 = 1'; this is the case of direct scattering which will be considered first . According to the Einstein model, nuclear vibrations are isotropic. so each of three directions of motion can be treated independently of the others. The fact that each nuclear coordinate is an independent oscillator coordinate reduces the calculation to a one-dimensional problern, i.e. (4.87) WI·'
=
L p.. ( n..t c":··· U ) e-Uf... In..) ~
For a crystal in thermodynamic equilibrium we have
(4.88)
92
T H E FOUNDA TIO NS OF NEUT RON TRANSPORT T HEORY
where", is the cha racteristic vibrational freq ue ncy and Z = f1w/2k. T. The ther mal average Wi. can be rewritten up on the use of a n operato r identity (4.90) which a pplies whenever operators A and B commute with their commutator [A, BJ. In our case (u~(t). u,,] is just a c-numbcr so that (4.9 1)
Th is exp ression can be furt her simplified according to a corollary to Bloch' s theorem , (4.92) whe re x is a multiple, or some linear co mbinatio n, of commuting oscillat o r coordinates a nd thei r conjugate mo menta . Thus
Wr~
=
exp { -
~;
[( U; (I )>r
+
(U~)T -
2 ) U,(t ) U")Tl }
(4 .93)
To eva lua te the indicated thermal ave rages in Eq.4.93 it is co nvenient to rep lace particle d isplacements by "creation" a nd " destruct ion" operators simila r to those introduced in Chapter II. The new operators arc go verned by the co mmutation rule (4.94)
and have the properties that
a IIJ~) = a ~ III,, ) =
J".. ln"
- I)
.;;:-+i In" +
0(1) In,,) = I)
In te rms of these operators,
••(1)
~
J;;.. e-'.... ln.. -
a+(1) In,,) '""'
J
h
2Mw
[0: (1)
I)
.;;:-+i e'''' III" + + 0.(1)]
I) (4.95)
(4.96)
and similarly fo r u. The following thermal averages are then readily found,
(4.97)
MEDIUM EFFECTS
where
L n.P.
a
e- u: (1 - e- 2Z)- l
-
93 (4.98)
~
It is seen that in Wf. the only dependence upon" is in x.;. Thus the" product in Eq.4.81 leads to a dependence only upon Kl as one would expect, and we find
s; ... where
•L
e -D,l"'
• _ _ a>
I.(PK2)
e-II( Z- ...ll
(4.99)
D ='lcothZ
e csch Z
P
=
~
- M2Mw
and use has been made of the generating function of the modified Bessel function of the first kind, e C,/Ul,H {., =
L•
,·I.(y)
(4.100)
• • - a>
In a very similar manner the corresponding result for interference scattering (I '" /') is Eqs.4.99 and 4.101 have been used to write Eqs.4.45, 4.62, 4.63, and 4.64. Ref.rence. I. R.M.Pearce,l. Nllc!eor EMrgy, All: ISO (1961); R.B.Nicholson, APDA-139 (1960). 2. M.S.Nelkin and E.R.Cohcn, Progreu In Nuclear ENrgy, SeriesI, 3: 179 (19S9); D.E.Parks, Neel: Sci. and Eng., 9 : 430 (1961); sec also Proceedings or Brookhaven Conterence on Neutron 11lermalization, BNlr719 (1962). 3. lnewtlc Scaltering of Neutrons in Solids and UqlddJ, International Atomic Energy Agency, Vienna, 1961 , 1963 (two volumes), 1965 (two volumes). 4. H.A.Bethe and G.Placzek. Phys . Rn., 51 : 450 (1937). S. H.Feshbach, G.Goertzel and H.Yamanchi, Nue!. Sci. and Eng., 1: 4 (l9S6). 6. J.E.Olhoeft, University or Michigan Technical Report 0426I·3-F, July, 1962. 7. L.Dresner, Nue!. Sc/.1VId Eng., 1: 68 (I9S6); G.M.Roe, KAPL-I241 (19S4). 8. V.I.8ailor, BNL-2S7 (19SJ); M.E.Rose, W.Miranker. P.Leak. L.Rosenthal and J. K.Hendricbon, WAPD-SR·S06. (1954) (two volumes). 9. A.C.Zemach and R.J.Glauber, Phys. Refl.• 101: 118 (19~6). 10. E.P.Wianer and J.E. Wilkins, AECD-227S (1944). II. B.Davison, NtutTon Traruport T'htDrY, Oxford University Press, London,19S7.
94
lil t IIl U NI)ATlONS O F NEUTRON TR A NSPOkl TIt IOO IlY
12. K.M .Casc, F.dc: HolTmann, a nd G.Placzek, /n troJucticm to tile Theory o{ Nrutro" Diffusion, U.S. Government Printing Office, 19S3. 13. K.M . Casc, AM . Phys. (N .Y.) 9: I (1960). 14. M. Bow a nd K. H uang, D)'1ICUPIico/1M'ory o{ CryMa/ Lultiu I, Ollford Ur uvers ily Press, Lond on, 19S7. IS. II.IJorn and R. Oppenheimer, AIIII. Fhys., 14: 4S7 (1927). 16. A.W.MI;Reynold s, M. S. Nelk in, M.N.RQStn blulh, and W.L.Whiuc more, Fnx u dinr l o{ ' M ~a)fld U"i trd NariOlU /nternariotlCl! Con!erellCe i" tM Pe~e{ul UJ<s o{ Atomi c EMrr y, . 6: (I9S8 ). 17. W. E.Lamb, 1'11)'$. k " ., 55 : 190 (1939). 18. W.M. Visscher, Ann. Ph, s. N.Y., 9: 194 (1960); M.S .Nelkin . nd D.E.Parls: Ph)'s. k "., . 19: 1060 ( 1960); K. S. Sinpi and A. Sjola nder, Ph,s_ Ron., 120,
m
109) (1960). R .L.M tl~\baucr , Z . Physik , 151 : 124 (l9S8); Ntl'u rwis,enscha{iell, 45 : 538 ( 19Slll; /.. N'ltur/l"sch., 1"- : 211 (l9S9); see 111$0 H. F raucnfuldcr, n,r Mo,f.fb
19.
v Special Topics In this chapter we examine some aspects of two interesting but specialized and unrelated topics. The first has to do with what might be called neutron thermodynamics. i.e.• the origi n. natu re a nd applicability of a certain time-independent, velocity-space dist ribution for neutrons and atoms which is achieved in special circumstances. Th is topic is specialized only when viewed in the context of the reactor. But it will also be seen to be an important part of the general subject of gas thermodynamics. The second topic is also to be regarded as specialized only when considered as a part of reactor technology. As presented here it is the beginning of a study of higher-orde r particle d istributions in reacto rs- in particular of a few relevant doubl et densities. Such studies lead to a quantitative appreciation of the phenomena of correlations and fluctuations in the distributions of vari ous kinds o f particles. In th is eonnection it is of interes t to note tha t reactor-type systems are perhaps uniquely suited to an expe rimenta l investigation of these matters.
A . Neutron T hermodynamics Most attempts at an analytical study of neu tron distributions in reactors explicitl y div ide the energy range into at least two parts . in each of which the neutron dens ities are treated according to approximations peculiar to th e range. The lower energy pa rt of thi s subdivision is referred to as the therm al ran ge-its upper limit being some few times the (kT) of the atoms in the system . The reference to thermal, however, is presumably not solely based on its demarcation being roughly tied to the mean energy of the atoms in the reactor but also to the expectation that . at least in many insta nces. the neutrons themselves in this
"
96
T HE FO U NDATIONS OF N EU T R O N TRA NSP ORT TH EOR Y
energy ran ge will be in a quasi-thermodynamic state. In so me specific instances this expectation has been essenti ally verified expe rimentally," hut in most cases it is defended merely on speculativ e gro unds . As a part of a st udy of the fundamentals of neutron tran sport theory, it seems appropri ate to pr obe a little for th e limitations on what can or can no t be asserted in this matter. The initial approach to the subjcct will be in term s of a very specia l problem . Consider neutrons and a single kind of ato ms mixed homogeneously in gas phase. Assume . however, that these distributions arc not in a steady sta te. It is rea sonable to expect that t he mixture will indeed eventua lly achieve some sort of steady state, so the qu estion is : Wha t ca n be said o f it? This, of co urse, is a familiar pr oblem in the kinetic theor y of gases. For reasons that will become apparent later, it will be assumed that th e only interaction between neutrons and nuclei and bet ween atoms that needs to be conside red is potential scatte ring. Neutro n radiat ive ca pture processes co uld be included in the argument if the inverse gamma-neut ron reaction was also considered and if kinetic equa tions for ga mma rays were adjoined to the equations for the neutron s a nd the ato ms. However. such a system (an aJogous to a chemically reactive gas) would be o ne which in the equilibrium state would not only be cha racterized by a spec ific velocity distribution for the pa rt icles and y-rays but a lso by It specific rat io of particle densities. A study of th e kinetics of such a situatio n might be interesting, but it is difficult to regard it as relevant. Reas ons for not including resonant-elastic scattering are a bit mor e obscure and hence will not be discu ssed at th is poin t. t According to these remarks, the neutron balance relation as cbrai ned from Eq A .28 is
'!
~ I d'K' d'k' d ' k A(K 'k ' ; Kk) [g(K) g.(k)f(K') J.(k')
"
- g(K')g.(k ' ) f( K)f .(k»
(5, 1)
where
h' , A(Kk; K'k ') = _ X_ j}(E t + E~ - E t ,
m'
- -
-
~
-
Ed d(k + K - k' - K ')
A(K 'k '; Kk)
• S..x , for example, B.T .Taylor, AERE-N/R.lOOS (t9S2). Sec, bowever, the footnote for Eq. S.l8.
t
(5 .2)
97
SPE CIAL TOPICS
In Eq.5.1 we have retained the facton g(X) - I - 4J1 J /(X) and g..( k) "" I + (2J1-)l I ..(k). The former enters because neutrons are fermions and the latter because it has been assumed, for the sake of illustration, that the nuclei are spinless bcsons. Strictly spea king. neither of these factor s should be given much consideration because of the extreme unlikelihood of finding real systems degenerate with respect to either neutrons o r nuclei. Nevertheless, it is correct to keep them, and the keep ing occa sions no difficulty. Th e extent of their practical significan ce will be discussed later on. To pr oceed furt her, a balance relat ion for th e nuclear distribution fun ction is requ ired. This could be deduced from first p rinciples j ust as has been done for the neutrons ea rlier, but such a deri vat ion would be repet itious. Hence, we merely note that
01. _
J d ' k ' d 'k; d ' k , A,(k'k ; ; "
,) (g. (k) g. (k,)f. (k')I. (k; )
- g. (k')g.(k ;)I . (k) / . (k,) ]
+ Jd'K' d'k' d' K A(I('k ' ; Kk) (g(K) g. (k) j(K')/ . (k') - g(K ' ) g.(k') J(K )I. (k) ]
(5.3)
The first term on the right-hand side describes atomic collisions with A I being the scatteri ng frequency appropriate to elastic collisions betwe en neutral ato ms, while the second term represent s the effect on the at omic distribution due to neutron-nuclear collisions. Now note that a sufficient condition th at the neutron and nuclear dis tribution functi ons be independent of the time is the vanishing of the intcgrands in Eqs.S.t and 5.3, Le.,
1(1(') I.(k') I(K) I.(k) g(K') g.(k') - g(K) g. (k)
(5.4a)
I.(k ,) -I.(k') - -I.(k;) -- -g.I. (k) - -g. (k') g.(k ;) (k ) g. (k ,)
( 5Ab)
and
Since the pr imed and unprimed variabl es arc essentially the pre- a nd p o st-collision momentum variables for particles experiencing elastic collisions, it follows t hat the logarithms of the fact ors in Eqs.5.4a and 5.4b arc at most linear , scalar com binati ons of the collisional in•
Osbon/Ylp
r ill,
98
HI U N I>Al'IO NS O F N £UTRO N T II." NS I'U R T TII UIR \ "
variant s, i.c.•
In f(K)
g{K)
-= " + ~ . m~K
+ r E I:
15.5. )
. nd (5.5b)
whcrc e , {\ A. C, and r are six arbitrary co nstants. A little examination reveals that the arbitrariness in the co nsta nt vector C must be interpreted as II velocity shared by all of th e particles of both component s of the gas an d, as such, is ignorable in the present context. Afte r some rearran gement , one finds that Eqs.S.5a and 5.5b imply th at (5.60)
and (5.6b) In these latter expressions, P. fl. and flA are again a rbitrary con stan ts ; though the structure of these steady-state solutions to Eqs. 5.1 and 5.3 strongly suggests their interpretation as the therm od ynamic solutionsand hence the identificati on ofpas (k-n- ' and fl and flAas the chem ical pot entials for the neutrons and nuclei respectively. To reinforce this intcrprcuu ion , const ruct Sit) ~ - k. ! d 'K[f(K. I) Inf(K, I)
+ g(K, I) In g{K, I)]
- kB f d lk[f..(k, t) In!.t(k,t) - g A(k , t ) ln g A(k , t )]
(5.7)
T his function is studied because, when evaluated in the timc-Indcpcndent state correspo nding to the distributions (5.6a) and (S.6b), it is (to within a constant] the usually accepted expression for the entropy of
99
SPECIAL TOPI CS
latter distributions are to be interp reted as the thermodynamic dis-tributions of the gas mixture. Differentiating S we find that
dS _ - k.fd 'K af Inf _ k.fd'k af • ln f • dl
ill
01
(5,8)
where use has been made of t he fact that the to tal Dumber of particles of a given kind in the system is co nstant in time. Using Eqs.S.1 and S.3 10 elimina te the time deri vatives in Eq .S.8 and taking maximum advantage of the symmetries of the tran sition p roba bilities A(K'~' ; KIt) and A I (k 'k ~ ; kk ,). we find that
dS dl
-
~
-~ f x
d 'K'd'k 'd'Kd 'kg(K')g.(k')g(K) g.(k) A(K'k '; Kk)
In f( K)f. lk ) g( K') g.(k') [fIK' )f.(k') _ f (K)f.(k) ] g(K)g.(k)f(K')f.(k') g(K') g. (k') g(K)g. (k)
- J ~
d lk '
dlk ~ dlk d lk . g..(k')g..(k~)gA(k)gik l) x
x A I(k'k~ ; ll,) In fA( k)!A( k ,) gA(k ') g ..(k~ ) x g.(k) g.(k ,) f . (k') f . (k;) x [f.lk ')f.l k ;) _ f .lk)f.(k ,) ] " 0 g..(k ')g ..(k ~) g ..(k)gA( k,)
(5.9)
Th us the entropy function increases in time until the distributio n functions satisfy the conditions (5.4a) and (5.4b), at which time S beco mes maximum and statio na ry. In co nseque nce, we shall henceforth interpret the functions (5.68) a nd (5.6b) as the thermodynamic distributions for neutro ns a nd nuclei respectively in the " ideal gas" system. However. befo re these d istribut ions can be useful to us, somc estima te of the paramete rs I' a nd 1'.01 must be made. This is accomplished by the usual normalization requirement tha t the va rio us particle densities rep resent a definite Dumber of particles pe r em", Application of the requirement leads to th e observa tio n that the factors exp (PI') and exp (PPA) are exceedingly large, except for mos t unlikely co nditions of high density a nd/o r low temperature and/o r small ma ss particles. (Con ditions met, fo r exa mp le. by the gas-like conduction electrons in some
100
T HE FOUNDATIONS OF NIl UTa.O N Ta. ANSPORT T HEOR Y
met als at room temperature, by nearly zero- temperat ure gases -or liquid s of He4' and He', and by electron-proton gases in the co res of sta rs.) Consequently, for reasonable react or co nditio ns, we may approxi mate f( K) se e-,E"'/4n' r!~
and
f .. (k )
se e- '£I< /Stt'
eI"""
(5.10)
which, when pro perly normalized , are simply the usual Maxwell-Belt zmann distr ibution s for classica l gases . Wc shall regard them as so appro ximat ed for th e rest of the present d iscussion . It sho uld be reca lled that we have so regarded th e one for the neutro ns in th e pr eccding chapte rs also. The above discussion p rovides a fairly satisfying demonstrat io n o f th e pla usibi lity of the assert ion tha t the solutions (5.00) and (5.6b) (o r m ore pr acticall y (5.10» represent the thermod ynamic distribution s for the ne utron s a nd nuclei in gas pha se (assuming no sources or sinks and clastic scan cring o nly). However. it is a bit distu rbing that the demonstratio n was presented in so restrict ed a co ntext. Afte r all , most reactors so far have been cons tructed in the solid or liquid ph ase. Fu rther more most nuclcar environments in ter act with neutr ons in many othe r ways than clastic scattering. Man y of these interacti ons, such as rad iative capture. fissio n, a nd nucl ear inelastic scattering for example, arc true preventatives of the realization of the above thermod ynamic sta tes - at least under realist ic co nditio ns. But it is anticipate d -eand ha s bee n suggested expcrime nlally - tha t the above ther mal neu tron d istribution s will he reali zed in o ther stat es of matter than gas phase. T hus we present an argument IIr two more or less germane It! the point i ll a n c tlurt to reinforce that anticipation. First we not e that Eq .5.1 may be rewr itten as (bearing in mind the above assumption o f no ndegencrate syste ms)
.y - 1d' K' [flK') T (K'
.',
- K) - J( K) T (K - K'»)
(5. 1I)
where evidently .>'"(K' - K) .F (K - K ')
-I d ' k ' d'k A(K'k' ; Kk)J.(k') - I d'k' d' k A(K 'k' ; Kk) J.(k)
(5.120) (5,12b)
wnh the present phrasing of the eq uat ion for the neutron d istribu tion ,
SPECI AL TOPI CS
101
a sufficient condition for a steady state becomes
f(K1 ....(K· - K) - f(K) ....(K - K')
(5. 13)
The scattering kernel, !T, is essentially a momentum transfer cross section times the speed of the incid ent neutron . If we demand that this steady state be cha racterized by a MaxweUian neutron distribution, we find -after a few manipulations to extr act from :F the energy tr ansfer cross section - that
(5.14) That is. if th e stead y state is to be a thermal one for th e neutrons, then the effective energy transfer cross section (which of co urse is presumed to incorporate an ap propriate thermal distr ibution for th e scatterers) must satisfy a detail ed balance condition, Eq.5.l4.· It is notew orthy that the effective cross section for scatterers in the crystalline phase does indeed satisfy this condition as is evidenced in Eq .4.62. Thus it is suggested that the equilibrium distribution of neutrons in cryst als will also be Maxwellian. , In a seco nd attempt to give some force to this suggestion; we consider an H-theorem for the density matrix itself. Again. it is not so much a theorem as a plausibility argu ment. But when phrased in terms of the density matrix ra the r than the singlet densities it seems to represent a sign ificant generalizatio n of the abo ve discussion to arbitrary scatt ering systems. Recalling Eq . 2.56. we have
(5. 15) Agai n define a n entropy functi on by
S - - k . I D•• lnD••
•
(5.16)
If the tran sition proba bility. W, has certain symmetri es. it is easily demonstrated th at
dS ,, 0
dt
(5.17)
The necessary symmetry required of W in order that Eq.5.17 hold is • See th e d iscussion s o r Hurwitz. Nelkin and Hebetler, reference 2, Ap pendix A .
10 2
rill
.O UNUATI O NS OF NEU T Il ON TIlA NSPOIl T T HrORV
proba bly not known , but it is certainly sufficient thatWH
•
""
(S. IS)
W• .•
ACIU.lIly it is not difficult to show that Eq.5. 17 holds und er t he wea ker symmetry requirement"
L .,.'
W•.• D... =
L w... D•• .....
(S.1 9)
How ever, as we have seen, most of the useful representat ion s of Wfor the description of neut ron-nuclear reaction s in the energy range germane to therm ody nami c considerations actually sat isfy Eq . 5.18. Thus we will spe nd no effort here to explore the imp lication s of weaker requircm ems. Th e eq uality in Eq. 5. 17 o btains if a nd o nly if D•.•. "'" D•• for all Mates I" ) a nd Ill') fur whie h W••, does not itself vani sh . Recalli ng that W... is non zero on ly if E•. - E. , it seems evident that t he lime deriverive of the entropy will van ish whenever the density ope rato r ass umes the form o f a functional of the energy, H, i.e.,
D
~
D(H)
An argumen t sugges ting a ch oice of a particular funct iona l proceed s as follows, Co nsider a system co nsisting oftwo weakl y inte racti ng systems. The Hamilto nian will be of the form (S.20) • See Heiner, reference 3, Appendix 5. No te t hat from Eq.J.34 we have
R..._"' JV_."
L
V_..V _··.. I' +
A
iB. ·.
with A. ·· a nd B• . real, a nd y:,. "" Y.'. ( Y Hermitian). Thus W• . issymmetr ic if B _ 0 or Y. · Y. -•.. Y. ··. is real. In the case of either a d irect or a pure resonance event as in the cases of poten tial scallering and radiative ca pture, condition (5.18) holds to the order or the prcse:nt ca.kulations. HoweveT. when resonant sca uering is included the symmetry of the c;orrc$pondinl tra ns.ition ma tm depends upon propcttics of (he nuclear matrix elements, o«, which have not been discussed . By assuming and :' \<;0 no ovcrl::t pping reso nances at in Eq.4 .20. we have effectively asserted that Eq.S. IS is valid in this case as well. Note also that symmetry exists in the purely abs(.rbinl cnsc ont)' if the resonances do not overlap .
103
S PECI AL TOPICS
Suppose now that this system is left isolated for a sufficient time. A steady state will be reached which we anticipate will be the thermodynamic state. If the interaction between the systems is sufficiently weak, we further anticipate that the distributions of particles among the states cha racteristic of each separate system will be essentially determined only by the nature of that system-cexcept that each distribution willshare a parameter common to both. the temperatu re. That is, we expect that D(H'
+ H' + H" ) " D(H' + H') = D(H' ) D(H' )
(5.21)
A solution of the functional Eqs.5.20 (neglecting Hot.) and 5.21 is D - Z -I e-'f1
(5.22)
Z ... Tre- ,H
(5.23)
Tr D = I
(5.24)
where so that
Applying these arguments to a system consisting of a neutron gas interacting with nuclei. we have for the Hamiltonian
(5.25)
The thermal density matrix then becomes in this instance. D - 0-0-,
(5.26)
where exp [
A'K' a'(K. s) a(K. s) ] -n:.. -2m
(5.27.)
A'K' Tr exp [ - P L - a+(K ,s) a(K , s) ] , . 2m and /)' _
exp (- PH'] Tr exp (- PH' ]
(5.27b)
104
T U[ fO UND AT ION S O F Ni UT aON T R A NSPOR. T T HEOR Y
The neutron density corresponding to thi s density matrix is
ilK) = Tr (~ a'(K, s) a(K, S») D ~
Tr (~ a+(K, S) a(K, S») D"
(5.28)
A straightforwa rd calculation leads us again to the expression (5.6a). T hus by these a rguments also we find the conventional expression for a fermio n density in mo mentum space. However, in th is case the distribution of t he nuclei is not restricted to gas phase.
B. Higher·Order Neutron Densities-Particularly the Doublet Density To this point the discussion has been exclusively devoted to singlet densities (especially ne utron singlet densities) and approximate cq ualion s that describe the m. Actuall y this devot io n to singlet densit ies has been mo re appa rent than reaJ, since we have in fact slid over t he matter o f dealing with higher-order densities whenever co nfro nted with them. Many times above, we have casually replaced certa in avera ges of products by prod ucts of averages. Th us, wit hout explicit co mment, we have freq uently met , an d d isposed of, higher-order densities by approximating them by prod ucts of singlet densities. For deducing equatio ns 10 describe the singlet den sities, th ese approx imations a re expected 10 be justified in the context in which they ar e introd uced. Th at is, it is not anticipated that the interpretation of measur ements of quant ities determined primarily by mean values will be seriously falsified by ignoring fluctuations abo ut the mean . However, occas ionally expe rimen ts are designed fo r the explicit pu rpose of measuring-directly or indlrcctly-ethese fluctuations. It is perhaps obvious th at such observation s ca nnot he interpreted in terms of mean values (singlel densities) on ly. Th us, to developa framewo rk in which these obse rvat ions ca n be st udied a s well as in which h) investigate the impo rta nce o f the approximat ions referred Itl a bove, we tu m now to a brief examinatio n of higher-order stoc hastic q uantities, Actually we shall restrict our attention al most com pletely 10 second-order densities, although the generaliza tion neCC!o~l ry for the co nsideration of densities of a rbitra ry o rde r will be seen to be triv i;11 in principle but tedious in practice.
IDS
SPEC IA L TOPICS
We define a doublet density for neutrons by F';'(X, K, X', K', I) - L- .
.'L
Tr "(X, K, ,) a(X, K, s) .'(X', K', , ') a(X ', K', s') D(I)
- L-' Tr e,(X, K) ,, (X', K') D(I)
(S.29)
In the representation which diagonalizes the density operator with eigenvalue NO', K), we find F';~ X,
K, X', K', I) • L- .
L, N(X, K) N(X', K1 D..(I)
(S.3O)
In earlier sections we have consistently approximated averages of productslike the above by products of averages ;which, if done here,would lead to the statement that F';'(X, K, X', K',I) " F',"'(X, K,I) f1'>(X', K',I)
(S.3I)
As we have already mentioned, it is not anticipated that the error introduced by approximations like Eq. S.31 into the description s of the singlet densities themselves is important. However. the interpretation of experiments which, in one fashion or another, are designed to measure the difference between the left- and right-hand sides ofEq. 5.31 surely requiresa more elaboratetreatment of 1"1-'. Thus we shall briefly sketch the deduction of a transport equation for the doublet density defined in Eq.S.30. Recalling Eq.2.43 and the discussion leading up to it, we find that
a~·' _ L- 6 Tr ~
(T' , ele~ ] D(t)
" L [(e,e;).·.· - (e ,ea••] W•., D..(I) Using the relation
.'
[T'''?le'IJ - l? IT', l?i ] + [T',
ellei
(S.32)
(S.32a)
in connection with the arguments leading from Eq.2 .4S to Eq.2.49 facilitates the calculation of the first approximation to the transport terms, and we obtain U+ 11K} -[(it m
" L-'
- if- +s«; --
ax)
i'I ]
m ex;
L IN'(X, K) N '(X ', K')
.'
F"l '(X " K X ' , K ' , t )
- N(X. K) N(X', K')] W,., D.(I) (S.33)
106
' 1'111: I O U N DATl O NS OF NE U T R O N T RANS r URT TII EOR Y
As in Chapter II, the remainder of our tas k here is the red uction of the interaction terms o n the right-h and side of Eq. S.33to useful for m giving d ue regard to a ll nuclear and macroscopic med ium interactions which might significantly influence the distribution , ~. ). Also the details of cross-sectio n calculations proceed here essentially the sa me as in Chapters III and IV. Hence litt le of this detail need be recap itulated ; and, since o ur interest is in illustrati ng how a theory of tluctuations and co rrelati ons m:1 Ybe systematically co nstructed and no t in deriving working eq uations fu r the analyses o f realistic cases, we shall confine our attention 10 the relevant aspects of scattering. rad iative capt ure and I ission o nly. Then here, as in Eq . 3.39, we can write the right-han d side of Eq. 5.33 :I ~ the sum of three sets of term s, i.c.,
:
L~ '
UN'(X, K) N'(X', K') - N(X, K) N(X', K'») W;_. D~(t) ~-
+ L"
L IN'(X, K) N'(X' , J(')
- N(X, K) N(X ', K1J W: ; D_(t )
+ L~ .
L IN'(X, K) N '(X', K1
- N(X, K) N(X', K'»)
--
w; l D..{I) (5.34)
where W;.•. W;'•. and W:. are th ose elemen ts of the t ransition p robab ility requ ired for the d escription of ra diative capture, scatt ering a nd fission , respectively. Alt hou gh it is not the simplest interaction to deal wit h here, we shall first con sider the scattering term s. The poin t is that this interaction is sufficiently complicated to illustrate all the inte resting featu res of the influence of binary interactions on the time rate of change of the doublet density, and at the same time simple enough to be descri bed in so me detail. On the other ha nd, th e treat ment of the capt ure reactio ns here is almost an obvious a nd trivial generalizat ion of that required ea rlier in the discussion of the eq uat ion for the singlet density, whereas our discussion o f the fission co ntributions to Eq. 5.34 must necessarily be conli ned to results onl y, as the ir derivat ion is quite tedious tho ugh straightforward . To car ry o ut the sum over final states, it is co nvenient to d istinguish between the terms for which X + X' and those fo r which X = X'. We
107
SPEC I AL TOP ICS
thus write
I. - L-'
-
E [N'(X, K) N'(X', 1")
- N(X. 1') N(X'. 1")] W;.• D..(t)
- (I - 6" ,) L- · D N '(X,K) N'(X'. K? - N(X.K)N(X', K? JW;.•D..(t)
-' -'
+ 6" .L-· E [N '(X,K) N'(X', K ') - N(X,K) N(X', K')]W;.D••(t ) (5.35)
Ou r purposes here will be adequately served if we suppress i ll detail associ ated with cross-section calculat ions and trea t the distributi ons of ta rget nuclei as statisucally decoupled from the neu tron d istributions. It is to be emp hasized that the neutr on and nuclear d istributions will not be regarded as uncorrclated in all circumstances. In fact such a co rrelatio n is cruc ial to the interpretati on of certain fluctu ation experiments to be discussed late r. In view of these rema rks we find as in Eq.2.67 tha t
W:. -
w:.•••,N(X, K,) [I -
Nac; K,)]
(5,36)
where X designates the spa tial cell in which the scatte ring event takes place an d K: and K , represent the momentum o f the neutron before and after the collision. The meaning of the quantity ~._I:. is bes t described by the relations
r ~I_.'
Il,
-
!2..
irK: E/... K:) _ E/...P:) m m
(5.370)
• !2..l.'.(P:),(P: ... P 1)d )P,
(5.37b)
a nd
m
In Eqs.5.37a and 5.37 b E. is the usual ma croscopic cross sectio n and , is the scattering freque ncy. Entering Eq . 5.36 into Eq .S.35, the sum s over both final and initial states may be ca rried out quite straightforwar dly (bearing in mind. of course, that o nly binary collisions are to be co nsidered). If furthermore we pass to the con tinuum in mo men-
108
THE FO UNDA.TlONS OF NEUTRON TR ANSPOR T T IIEO RY
tum space. we find that J.
~ (~o)'
U
x { f"'(X 1 •
+ I(
f
d'P" : ' E.(P").F(P"
x
P " •X'• P ') " (:!no)' "'(X' P " " X' P ' • X• P)tf 2f )
d j p " p"
m
l.~(p") F(P"
{ J;.I(X , P, X'. P") -
_(hr.L )' !-m ~·.(P) 1(
~ P)
.... P') x
(C;"P f:- 1(X. P. X'. p ". X'. P')}]
x
'"'(X P X' P') - (:!no)' f'"'(X. P . X' . P' . X. 2) {f 1 . "
_(2.")' po L m
p")l
E,(P') x
x
{j''''(X 1 . P . X' , P ')
- (2)>0)' 2 f"J '(X . P , X' • P ' , X' •
+
(~~hr '}D' [J(P
- P') ; l:.(P)ftl(X ,.-)
- (2,',) ' "(P - P ')
~
E.(P)
f
f
d'P"ft'(X. P.
+ h(P - P') d'P " : ' E,(P ").F(P"
~
f
~',(P) 3"(P
_ P ') {I -
x, P ")3"(P ~ P")
P)f,'"'(X,P" )
" (2)>0)' b(P - P ') d' P" : ' E.(P ") .F(P" -
+ :.
p")l
(~o)' h(P -
P)f~"'(X.l·,X. P")
P ')! x
x {-f:"'(X. P) + (2d)) fr(X. P. X. P) + (2'tJi)3 rro: P, X. P')
109
SP ECIA l. TOPICS
""(X.P.X P ' • X• P ') _ (2)
p,} I
r &.(P1 JI'(p' - P) { + -;; 1 - (2)
"" (X•P•X P X P ') _ 2 JJ • •• + (2)
(2.d)J/t)~ p. x. P')
(2)
The first four terms in this expression are strict analogues of the corresponding scattering contributions to the balance relation for the singlet density as seen in Eq.S.1 for example. It is evident that the dependence of these terms upon the third-order densities is purely a quantum effect. and here also is of importance only for degenerate systems. The remaining terms. nonvanishing only at the points X -X' are classical and quantum contributions to a correlation effect resulting from the fact that at X = X' scattering can transfer neutrons between momentum cells within the doublet density. The factor (m AIL)' to which J. is proportional is common to all terms in the balancerelation and may be ignored throughout henceforth. In the classical limi t (limit as h _ 0) Eq ,5.38 becomes J. -
J
dJP" : . l.'.(P") '(P·' - P)/t\X. p ol. X'. P')
f
+ d'P" : ' &I.P") JI'(P" - 1")/ i"(X, P , X', P")
- .!.. &I.P)/i"(X, P, X ', P') - !:.- &.(P1 fl·'(X, p. X'. 1") m m +
6~:,
[ .l(P - P')
f
~
&I.P)fl"(X, P)
+
- .!.. &I.P) JI'(p m
P1 fl" (X.P) - r &.(r )F(p' - P)/ :·'(X. 1")] m . (5.39)
110
111 1 I O U N I>ATl O N S OF NEU T RO N TR ANSP O RT THEOR Y
The remainder of our considerations in this chapte r will be con fined to this classical limit. Our discussion of the " capture" contributions to Eq.5.34 can be brief. We find, in the continuum,
..
I, - t. .. L IN'IX, K) N'IX', K') - N(X , K) NIX', K')] W;' . D_I' )
!.. l.",( P)f~·)(X, P, X', P') -
.1, = -
n,
!:.. l.~.( P')f:-I(X, P , X', P ') m
p
,~ ... .
+ -r-r
(5.40)
The structure of this result is evidently the same as that seen in Eq.5.38 (o r 5.39). That is, the first two terms are a nalogous to the capture terms appearing in the singlet balance relation, whereas the last is inhomogeneous and implies a capture cont ributi on to correlation. These inhomogeneous terms a rc interpreted to imply correlation to the extent that their presence in the balance relation for the dou blet density prevents solutio ns of the form F," IX, P. X'. P')
E
F.· 'iX, P)f ;"'(X', P')
15.41)
The contribution of fissio ns to Eq.5.34 is ca lculated to be I, ~ I , " .I, "."
..L
IN'IX, K) N'(X', K') - NIX, K) N(X'. K'») W: . D,.iI )
_!... ~.:, ( P}J;·J( X" P, X', P ')
- !:-. l.·,( P')Jt'( X, P, X', P ' ) In
11/
~
J
d '/'" F,°'IX, p " , X', P') P" 2',( P" ) m
+ J"d ' pol n · '(X,
- j ;"'IX, P')
I~" X', Pol)
P" l.",(P")
~ B~(P", p ')
m
"
!:2~I P') ~> O:IP ', P) m J.
L > O: IP ", P) " Il.
III
S PE CIA L TOP ICS
f
+ d'P" It'(X, P ") 1: E,(P") ~> aB'..(P". P. P') m
,~
+ ol(P - P')fl" (X, P) 1: E,( P) m
(5.42)
In this equation, we have introduced so me frequency fu nctions re presenta tive of various aspects of the distribution of neu trons produced by fission. These are:" S: (P" , P) d ) P e The probabilily 01 a fission induced by a neutron et P" producing exactly J neu t ron s. a or which have momenta in d) P About P, (S.4Ja)
B:"cP" P, P') d 3 P d3 P' s The probability or a fission induced by a neutron at p" producing exactly J neutrone, e of which have momenta in d) P about P and 0 of which have roomenla in d 3 p' about P'. It is to be no ted thai B:o(P", p. P) I:; 6.... (P P') a:.(P", P). (S.43b) Evidently, for a given valu e of J, a and (f ta ke on aU integral valu es a nd zero subject to the condit ion that thei r su m not exceed J. Co nverting the left-hand side of Eq. S.34 to the continuum in mome nt um space, an d substituting Eqs.S.39. 5040 a nd 5.42 into the righthand side of Eq.5.34 provides us with a transport eq uation for the neutro n dou blet dens ity. One aspect of this equa tio n is notable a nd requires comment. Though the equation is inhomogeneo us (terms p roportiona l to the singlet density appear in it) it does not contain the usual inho moge neit ies proportional to the tr iplet densities (a t least not in th is classical limit). The absence of such term s is a direct consequence of neglecting neutron-neutron collisions and of treat ing the nu clear den sities as kn own and determined independently of the neutron distribution s. As we sha ll sec below, the latter of these simplificat ions will have to be d iscarded, a t least in pa rt , if certa in fluctu a tion experi ments a re to be u nderstood a nd a nalyzed . t A n example of problems in fluctu at ion a nal ysis might be t he moment a na lysis o f t he record of counts by a OF) neutron detect or. In this in• Sec also Eq.2.80 a nd Eq.3.JOI. Note that
r aS1
'.'
P) -
2'cP' ...
P)
t Fo r recent work on the lheory or neurrcn fluetualion in reaclOn see Pluta. J MaUhes,· an d Bell.' In all these inYCSlig.alions the theory W :l$ developed on the basis or anensemble probability for the reactor and the phenomenological derivalion of a n equation to describe it.
112
T H E FOUNDA TIONS Of NEUTRON T R A NSPO R T TH EOR Y
stance the particles detected a re the alpha particles produced in the B(n. He) Li reaction. The count record may be anal yzed in a variet y of ways. Perhaps th e mos t straightforward is to di vide the record into a large numhcr o f equal time intervals and reco rd the number of coun ts per interval. These numbers may then be evcruged , sq ua red and then averaged , etc. to obta in an y moment of'tbc a lpha particl e accumulat ion tha t is desired. T he same co unt record may then be redivided into time intervals of a different width and the moments reco mputed. Th is process is repeated un til the desired mom ent s ha ve been o btained as functions of the interval width . If, for example. the mean va lue o f the co un t rate changes with time, certa in ra ther obv ious refinements o f the ana lysis of the record must be introduced. The theor y 10 be ske tched here will deal with the first two moment s obtaine d as indicated . T he first th ing to be noted is that the actual observatio ns have noth ing directly to do with neut ron distributions at all. However, it will be seen tha t the seco nd mom ent of the alpha particle accumulatio n (which may in [ let be interp reted as the doublet den sity for the a lphas) is co upled to t he seco nd-order cross density for al phas a nd neu trons, Which. in turn. is co upled to the neutron doublet density for which an equa tion was deri ved above. The neutron d oublet and singlet densities arc co upled a lso to the doublet and singlet densities for the delayedneut ron precur sor s. We sha ll ignore the se latter co uplings (and hence delayed neutro ns) since they add great bulk to the an alysis bul nothing new in principle. Of co urse, the actu al interpretation o f the experiment requir es t heir conside ration. From these remarks , it is evident th at even the limited treatment envisaged here will require more equations than ha ve been derived so fur. Specifically we require ba la nce relations for the singlet and doublet dcnsh ics fo r the al pha pa rticl es and for the doublet alpha- neutron cross density. These densities are defined by F: -I(X, K . t) ~
F~ _ '(X ,
L -, T r a;(X, K) a,(X, K) D(I)
(5.440)
K, X', K', t ) ~
t.-. T r a:(X, K) a,(X, K) a:(X' , K') a, (X', K') D(/) (5A4b)
~
L - . Tr a:(X, K) ',(X, K) .:(X', K') .,(X', K')D(I) (5 .440)
113
SPE CIAL TOPI CS
where the (ce. n) subscripts and/or superscripts refer to quantities eppropriatc to alpha particles and neutrons respectively. Equations describing these densities in the classical limit and in the momentum continuum arc
+ P, -'-)f"'(X P (!.... vI M oXJ ' " =
+ (!.... vi =
f
f
d'P ' :
I)
E o(r ) , D(P' - P)f :"'(X, P', I)
J _0_ + Pj - '-)f'''(X P M (JXJ M aX; l ,
P X ' P' I ) .
•
,
d'P" : ' ED(P") .jOD(P" - P) f:d J(X"
f
+ dlP" : '
E ,,(P" ) .F D(P" -
f
(5.450)
r-, X . P" ,/)
nr;.. . '(X. P, X', P" ,/)
+ "(x - X'l,,{P - P') d'P" : ' E D(P") .!FD(P"-Plf:(X,P",I) (5.45b)
+ P (!... iJt M
J
.. f
- '-
c1Xl
+ Pj _ "_ + !::.. L·r{P' ») flt•• ,(x. P, X', P ',/) m I1Xj
m
d'P" : ' EJ.P" ) .:F(P" - P ')f:U)(X. p . X', p " , t )
+ f d ' P " P" I:y(P") f:
m
f
+ d 'P" : ' _ flu '
) "' 0
±A B~(P", P')f;"" (X, P, X" P"./)
. _ 0
I D(P") :F D(P" - P)f;-l(X'. r-. x, p OI, I)
!::.. E,J P' ) :FD( P'
_ P)ftc.l{X, P', /)
(S.4Se)
L' m
In these equations we have introduced the notation E ,,(P) for th e macroscopic cro ss section for the absorption of a neutron of momentum P by tbe boron in the detector, E,{P) - E,(P) + EtP) + E,{P) + E,,(P), and I/J(p _ P ') d ' P' for the probability that, if a neutron of momentum P is absorbed by a boron nucleus. an alpha particle of momentum P ' in d SP' will be pr oduced. The experiment with the alp ha-part icle count record referred to above consists of a measurement of the singlet and doublet density for the •
Ooltonl/Ylp
1 14
Till' fOUND 4 T ION S Of NEU TR O N TR4 NS P ORT T U IO OII.Y
alp ha part icles defined in Eqs .S.44a and 5.44b. T hus the int erpretation of that experiment in term s of system pa rame ters requires (a t least) the solution of the coupled system of Eqs. S.4Sa, S.4Sb, 5.4Se, S.34 (ta king inlO account Eqs .S.39. 5.40 an d 5.42), and the equat ion for the neu tron singlct density discussed in the previous chap ters. And even this formidahlc task is unrealist ic since it overloo ks th e influence of dela yed neutron s. The kinet ics of these " fluctuatio ns" a ppears to be sensit ive 10 these ne utro ns, a nd co nseq uently balance relat ions for the singlet and doublet densities for the de layed-neutron precu rsors mu st be considered alo ng with t hose just referr ed to. Ap proximate attemp ts to dcal realistically with these experi ments have been made. S T o facilita te a few final remarks regarding t he struct ure o f the cq uanons for the singlet and doublet neu tron densities. we recapitulate the m here. We display them this t ime as fun ction s o f velocity ra ther th an momentum and keep the explicit ind icat ion of the arguments of function s to a minim um . Correspo ndingly the eq uation for the singlet density reads
(5.46) and the equation for t he doublet density is
- J d ' .. .. G(V" ~
- V )! , (V " , V ') -
J d' V"
G(V" - V ') ! ,(V, V " )
" (X - X'I ('W - V' ) R. (V )!,(V) - G(V - V ')!,(V) - (;( V' _ V)! ,(V') + <>(V - V')j d ' V" R.( V" ) F(V" - V)! ,(V" )
+ J d'V" //( V"\V, V ' )! ,( V" )]
(5.47)
Here we have int rod uced the reactio n rates. R( V ) = V I( V ), and the frequencies (un no rma lized ); G(V" _ V) = R,( V") ,OW" _ V)
+ R.( V") ~> B:(V " , V) "
ilnd
//(V " IV, V' ) = R.( V" ) L naB~(V " \ V. V ')
,.
(5.48) (5.49)
11 5
S PECI .o\L T OPI C S
We have also ma de tbe identification (S.SO)
6(X - X ') = lJu '/U
where lJ(X - X') is to be inte rpre ted as a Dirac delta fun ction if we regard the domain {X} as co ntinuous. It is worth noting that
IG(V " - Vjd' V - R~V ")
and
I H (V "
+ R,(V")
( J )"
(S5l )
(5.52)
IV. V')d ' Vd ' V' - R,( V") ( J ') "
where, for exa mple, th e symbol (J )" represents the mean number of neutrons produced in a fission ind uced by a neut ron with speed V" . In ma ny practical applications. the depe ndence of ( J>" and ( J2 ) " upon the energy o f the fission-ind ucing neutro n is ignorable to a good approximation. O nly rudiment ary invest igat ions of the coupled systems of equation s like (5.46) and (5.47) for neutron distr ibu tions have been carried thr ough so far .'·ll Neverth eless preliminary study has suggested a number of interesting results which warrant some co mme nt here. However, these resul ts have been o bta ined in th e context of o ne-speed diffusion theory . Thus a co nsiderable amoun t of manipulation must be performed o n the above eq uations before they appear in a form suita ble to the present d iscussion. Most. if not all. of these manipulations are co nventional. and the refore they will be mere ly sketched with a minimum of detail. The redu ction to one-speed form is accomplished by integrating the equations for the singlet and doublet densities over all en ergies. To do this it is first necessary to tran sfo rm all functions from velocity spa ce with points labeled by the coordinates (v... v, . v.) to ene rgy-angle space with poi nts labeled hy (E, 0• •,,) or (E. 0 ) where 0 (0, '1') - y/v. Then it is necessary to decide how integrals ofproducts of the var ious func tional parameters with distri buti on functio ns are to be dealt with. This issue is subtle and com plex fro m the qualitative point of view and alm ost impossibly difficult fro m the qua ntitat ive point ofview. However. we shall treat itwith the utmost casual ness here, hoping th at the qualitative significance of co nclusions to be drawn subsequently will no t be serio uslyjeopard ized thereby. T o illustrate the po int and our treatment of it . co nsider
f"
£_ 0
dE v 2.\ v)[, (x. E. n , t) -
f"
dE R(v)[ ,(x. E. n, t)
0
2
, [ ,(x. n , t ) (5.53)
116
r o u N ll A TlO NS OF NEUTRON TRA NS POIl T HI.:OR Y
Til l
where
I,(x, 0, I) ea [ dEf,(X, E, 0 , I)
(5 ,54)
As definitio ns, these equations can hardly be q uarreled with, but as a step in the direction of reducing Eqs.5.46 and 5.47 to a more tractible form they arc purely formal. In fact, the averaged reaction rat c.v, is an unknown functio n ofx, O, and I in general. However, we shall treat it as a constant. T hen co nsider
J f dEllE· R(v)/l(x , E, 0 ; x', E ', D '; I ) e (lll(x , 0; x', 0 ' ; I) (5.55) i JdE dE' R(V')[l(X, E, 0 ; x', 0 ', E '; I) e (l' l l(x, n, x ', n ' ; 1) (5.56) and
n , x', 0 ' , I)
J; (x,
sa
II dE dE' f,(x , E, 0, x ', 0 ', E'; I)
(5.57)
Here we have introduced two new, unk nown d ifferent functions of the variable s x, O. x', n ', and I . But these too we will treat as constants. Furthermore we will rega rd them as the same constant. Still further we will con sider them to be the same as the correspondi ng reac tion rate, r, i.e., we set !.' = ft' = r, This sort of approximating will be assumed everywhere in the following d iscussion. With these remarks (among others unspok en) in mind, we find afte r integrati ng (5.46) and (5.47) over all energies iif, + " n - v! 1 + TTl, _ (J) rF f dD,! ,(n') ?I 4"
- ".r dU' p,(O' - O) f ,(O ') ~ S (J) r rU' vr, + t\O " vr. + 2rTl l -
j'J~ ,1,
411 'f'
-
(J ) "
4.>
... r,
f
fdU, I ,(O,O ') -
<.lU, p .(fi l
.....
+ x
dU I [l(O " n ')
Olf,(O, ,0')
1)
+ I ,(x', 0 ', /) S(x, n ,1)
.'l(x - x') ~O - n ')
1', 1,+ J
'.fd!J,p,(O, -
O ')/l(O, (
= II(x, n . n SiX', n ', t)
f
(5.58)
Six. n, I) + d(X - x') h(O - n') )(
dU,f,(O ,) [ ' , P,(O, - 0 ) + ( J ' )
;:Jl
117
SPECIAL TOPI CS
["p,(n' - n) + (J l:'] x')[,(n) ["p,(n - n ') + (Jl~" ]
-
(S.59)
In these equations we have introduced a source of neutrons th at is independent of the neu tron density, l.e., a source other than fission. This was accomplished by simply assuming that our system includes nuclei that decay spontaneously with the emission of a neutron. The inclusion of such a source is desirable and important since ultimately we wish to explo re fluctuat ions in both criti cal and subc ritica l systems. We hav e also introduced the symbo l P.(O _ 0 ') dD' to represent the probability that, given that a neutron going in the direction n is scattered, it is scattered to some dir ection fi' in dD', and v to represent the mean speed . Eq .5.58 is recognized as the conventional one-speed transport equation for the neutron singlet den sity. Eq.5.59 is the corresponding a na logue for th e neutron doublet den sity. It is important to note that the equation for the singlet is independent of the doublet and that the inhomogeneous terms in the equation for the doublet depend only on the singlet a nd the so urce. This linea rity of these equations is a direct consequence of ignoring the neutron-neutron interaction and of the ass umptio n that the atoms in our system a re in di stributions ind ependent of the presence of the neutron s. A diffu sion approximation to these equations is now obtained by sta ndard arguments. First, Eqs .5 .58 and 5.59 are integrated o ver all direction s,n; then they a re multiplied by n and again integrated over di recti ons of motion . Defining
/lJ(x, x', I)
J dD/I(x, n,/) es J dO n f r(x. n , I) es f dD dD' f J(X' n, x', n ' ; t)
"(x, x', l)
E
J df} dD'
(5.63)
" '(x, x', I)
E
JdDdD'
(S.64)
nl (x, I)
4»(x, t)
'5
(5.60)
(S.61) (5.62)
one o bta ins
iJII I
-
or
+ vV '+ + (r ..
- ( J) rF) n . = So
(S.6So)
.',+ ~ S ,
(S.6Sb)
-"- . + "f dfJ <1(<1 ' V)[, +',.+ 01
118
TH E FO UNDATIONS OF NEU T RON T R ANS P OR T THEOIlY
a nd
,..
+ II; SO + 6(x - x') [So +
n ,S~
f
+
,~,
f
«J-
I } l ) " n a]
(5.66a)
:, .. + v dOdO' a (a , V)!, + v dO dO' a (a' , V')!,
=
~ .'
'"
+ [2rT - ( J) r - pr. - ' .] $ .s~ + n ~S, + 3(x - x') [SI + ' ... -
( J ) T, +J
(S.66b)
+ l.JdD dD' 0 '(0· V) / l + vfdDd!.Y 0 '(0'· V')/l .. (2T T - ( J ) T, - pT. -
= n ,S;
+ 5+' + b(x
T.l $
- X')[S I
'
+ T A•
-
( J) T"ellJ
(S.66c)
At t his poi nt numerous ap proximations are made, The nonfissi on source is assumed to be spherically symmetric so t hat S , - o. Thc a mbiguo us tcrms arc approxima ted (albeit somewha t incon sistcn tly) accord ing to the assumption that the velocit y an isotropy of both densit ies is describable by a linear combinat ion of t he zeroeth a nd first spherical harmonics. The inconsistency . and its effects on the do ublet equations an d some of thei r sol utions, is discussed in detail in reference 9. It will no t be dealt with here since its removal grea tly complicates the formal appearance of the vario us doublet equa tions without significan tly altering their co nte nt so far as present illustr ative purposes are co ncerned . The above equatio ns th en beco me
+
++
So
(5.670)
. + -3 Vnl + 'u. - 0 4)1
(5.67b)
Onl
vI
v
vV .
tAlI I . ""
v
and
_.'11 ' + vV · 411 + vV' . $ ' + 24n 1
'" (5.683)
11 9
S PECIAL TOP ICS
-
,1
,"
+ (a: + '1'. ) II>
!.... a, 41"
+
-
~3 V'n, + (~ + ' 1'. ) CIt'
.S~
+
6(x - x') /X. (5.68b)
- So+'
+ 6(x - x') ~' (5.6&:)
Here we have int rod uced so me new notation. i.e.• (5.69)
a: ""'A - <J ) " r r . ..
and
fJ ..,
'f - P'. r~
+ «J-
(5.70)
1)2 ) ,,.
(5.7 1)
These syste ms of co upled equations can be redu ced to partial d ifferential eq ua tio ns fo r 1/ 1 a nd ", by st raightforward manipulat ions. The res ultant equations are of second order in the time for n , and third order in the time for " " ~ The time deriva tives o f higher order t han the first a re requ ired for the description of rela tively fas t tran sients. If attention is cente red in slo wly varying, or even tem porally asymptotic, solutions, t he time derivati ves in the cu rrent Eqs .S.67b, S.68b, and S.68c can be neglected. and o ne obtains (5.72)
and
an, at
v'
.,.,.....:._....,. (V2
+'u ) n l S~ + n~So +
+
3(a:
"'"
,
T
+
V'2) n 2
+ 2tvrJ
b(x - x') [So
V
3'rtt(.x + ' u ) 2<>v' 3'f.(,x + 'u)
+ {In,]
(S~VJn , + SoV' Jna b(x - x') V'n,
(5.73)
The arguments leading to Eq. 5.73 are a bit devious an d some explanalion is requi red . Accordi ng to Eq s.S.68b and S.68c, II> and «P' depend upon terms con ta ining Di rac de lta functions. This is all right as it sta nds but becomes a somewhat a mb iguo us ma tter when these relat ions are inserted in to Eq .S.68a. We have circumvented this ambiguity by replaci ng the delta function by a Gaussia n which is equivalent to a delta
120
rU E fO U NDATIONS Of NEUTRON TRANS P OkT THE OR Y
function in the limit o f van ishing width. But this replacement lead s to uncert a inty as ( 0 whether the singlet current in the term shou ld be evalu ated at the primed or the u nprimed. point. In the absence of clear cut evidence indica ting a choice in this matt er, we ha ve chose n to evalu ate these singlet currents as one-half of the sum o f the ir values at the two d ifferent points. Thus, for example, we have cmployetl the replace ment ",'I(x - x ·)
+.. .
t\
-=-:-,-=-
(5.14)
laking the limit D ..... 0 at a convenient point in the ana lysis. Eqs . 5.72 and 5. n co nstit ute the " diffusion de scription" of the neutron singlet a nd doublet densities. Before co nsidering a solution (0 these equation s in a n interesting "ped al case. it is useful to examine a general cha racteristic of the neuIron doublet density as described by Eq.5.41. For this purpose it is co nvenient to divplay Eqs.5.46 and 5.41 more co mpactly as
.,
cf, + Bj• ." S _.
(5.15)
nnd f:/~
"
+ ( B + 8 ')f,
=
Sf; + S11 + Il(x - x ') "'(' - v') S
+
~ x - X')
Jl .
(5.16)
Here aga in we have introduced the exte rnal source function, S(x , v, t ). The integration of Eqs.5,75 and 5.76 over energy yields Eqs. 5,58 and 5.59 includ ing so urce terms, The operator B we will refer to as t he Boltzma nn operator. Its definition is readily inferred from Eq.5,46. Similar ly t he oper ator r. appearing in on e o f the inhomogeneous term.. on the right -hand side of Eq.5 ,76, is obt ained by reference to Eqs, 5.47, 5.48, and 5.49. The prime on the Boltzma nn operato r in Eq. S.76 implies t hat it operates on the primed phase po int upon which f ,(x , v. x', y ',l) depends. For so me pu rposes it may be convenient to study the co rrela tion function (or gene ralized variance of the distribution) defined as Gl x, y, x'. Y',/ ) ". f ,ex, Y, x', ,.', t) - I .(x. Y, I) / I(x' , , ',I) (5.17)
121
SPEC IAL TOP ICS
Straightfo rward exploita tion of Eqs. 5.75 and 5.76 demo nstrates that the correlation function satisfies the equation
oG
-
a'
+ (B +
B')G = O(x: - x ')b(v - Y') s
+ I5(x -
x') rj.
(5.78)
The primary reason for reopening the discussion of the general Eqs.5.75 and 5.76, or 5,75 and 5,78, which we prefer in the present instance, is to extract a comment o n the so-called criticalstate ofa nuclear reactor. Thus we examine- in rat her formal terms-a solutio n to the source-free equations (5.79)
and
oG
-
a'
+
(B
+
B')G "'" O(x - X')rjl
(5.80)
Assume the existence of a co mplete, orthogonal (discrete and /or continuous) set of eigenfunctions and corresponding eigenvalues, which satisfy (5.81)
as well as appropriate no-reentrant-current boundary conditions over the surface of the reactor. Ofcourse. the existence of such a function set has not yet been established. Nevertheless we surmise that the pr incipa l inference to be drawn from the present argument has a wide validity, even though its logical defense is not immediately accessible. Thus, proceedi ng purely formally, we expand J ,(x , " I) -
L, Q,(/) .,(x, ,)
(5.82)
where the sum over the eigenlabels 'J" is a sum and /or an integration corresponding to their discrete and /or continuous distribution. We then find tha t daJ = - AJDJ
d,
(5.83)
so that (5.84)
and (5.85)
122
T Hr FOU N D AT IONS Of NEUTR O N TRA NS PO RT THE OR Y
We now assume that the eigenvalues can be ordered, e.g. (5.86)
and that ),0 is a discrete eigenvalue. Then the criticality co ndition is ),0 = 0, which implies (5.87) lim! . - 00(0) l'o(X, v) Now co nsider Eq.5.80, Expand G(x, v, x', v' , t)
=
Lp AJtO ) 'PAx, v) " t(X', v')
(5.88)
and h(x - x'» l j, < b(x - x') 0,(0) r~, ~ Th e coefficients Aj t then satisfy
r c" ~AX , y) ~.(x·, y') (5.89) "
dA Jt dt
( 5.90)
For j = k = 0 we have
A oo(t ) .... Aoo(O)
+ tCoo
(5.91)
and apparently G increases without limit as t _ 00 , unless Coo is zero whieh is generally not to be expected. Thus the correlation function and hence also the neutron doublet density acco rding to the p resent description -has no meaning in the critical state, if that sta te is realized acco rding to (5.87). T he significance and full range of validit y of this result is by no means understood at this point. Consequently we do not wish to d well upon it here nor to struggle much with its int erpretation. However, we do feci th at it is importa nt to point it out as a precautionary comme nt regardin g attempts to st udy neutron doublet den sities in critical systems. In par ticular , we o urselves are guided by it in restricting our consideration of solutions of Eq.5.73 to thos e app ropriate to subcritical systems only. Now returning to Eqs. 5.72 and 5.73, we seek their steady-state solutions in regions of large systems over which the neutron singlet density a nd so urce ruuy be regarded as space-indepe ndent. In such an instance, . (5.72) and (5.73) become (5.92) ". = sl&.
123
S PEC I AL TO PICS
and
= anlS
+ 6(x - x') [5 + pnd
(S.93)
where we have dropped the subscript on th e source symbol. Noti ng that "'Iru is generally very small compared to unity, and making use of (5,92), we may rewrite (5.93) as [- D(Vl
+ V' l) + 21.11.] n1
2n'
+ 6(x
"" _ ,
where we have defined
•
- x ') ('" + p) nl (5.94) (S.9S)
Because of the cha racter of the space dependence of the inhomogeneous terms in Eq. 5.94 it is evident that n1 depends only upo n R - x - x', Consequently (5.94) further simplifies to 1
[ -DV: + "'] n1(R) - _n , + 6(R) -'" +-P ", • 2
(5.96)
This equat ion is readily solved, yielding 1
'"
"l(R) "'"1 + n l -
where
,,1 _
+ P -e-'"-
2D
4:rR
",f D
(S.91) (S.98)
This is substantially the same solution th at was p resented in reference: 8. Defining a normalized correlation function as C(R) -
(S.99)
we have C(R) - • + P e' 2Dnl 4Jt:R
(S.IOO)
We also note that (5.98) may be written in terms of the diffusion length, L ... (1/3 I . E n ) " ' , and infinite medium multiplication constant, k - (j) EriE.. as I - k (S.lOI) L'
.' ---
124
1111, I'O U N D ATI O NS Of N EUT RON TR A NS POR T T lllOOM,Y
Thus we see that the co rrelation length is here given by ,,-I = L/ JI - k , Evidently the correlation extends over very large distances in large nearly critical systems for which ~ is small, as well as in nonmultiplying but weakly absorbing regions for which L becomes large. However, in the latter instance, oX + fJ becomes very small so that alth ough the correlation is widespread it is at the same time very weak. It is to be noted thai the normalized correlation varies inversely as the singlet density and hence becomes tess noticeable as this density increases. This is perhaps a n intuitively ant icipated observation. Reference. I. J. E. Mayer and M.G. Mayer, Slal istkal M~chQJIks, John Wiley & So ns, Inc., New Yo rk, 1940; L. D. La nda u an d E. M. Lifshitz., Stat istical Physics. Add isonWes ley. Reading, M a s~. , 1958. 2. H.Hu n.ill , M. S. Nclkin ,lnd G .J. Hebct ler, Nud . Set. and Eng., I : 280 (1956). J . w.Henjcr, The Quantum '1Mory a/ RadlaUon, O xford Universuy Press, New Yor k, 1954. third edition. 4. E.CG.Sliid..c1 bcrg, lie/v. Phys. Acto, 15: 511 (1952). 5. P. Plula, T hesis, Th e University o f Michigan. Ann A rbor. Michigan, 1962. 6. W. Matt hes, Nukfeonik. Ba nd 4, Heft 5: 213 (1962). 7. G . I. Dell. Ann. Phys., 21 : 243 (1 963). R. R. K.Oshu rn a nd S.Yip. Proceedings of the Conference o n Noise Analysis, U nivcrl ily of Florida, November, 1963. 9. R .K . O~tlllrn a nd M. Nmctson, J . Nlld. En~rg}', Purl Al B, 19; 619 (1%5) .
Index a lpha pa rt ic:le density, 112 binding effect, 62, 7S-90 Bohzma nn equation. 2 Boltzma nn operator, 120 bound.ry condition, 9 bou nd-atom scattering length, 72 Bragg sca tter ing, 87 cell fu nctio n for co arse gra ining, 9 c hemica l poten tial, 98 cohucnt scattering, 86 coll isional in.... ria nts. 97 compound nucleus. 40. 55 conti nuousmomcnl um 5pace, I), 24.58,
10'
entropy. 98 evolution. operator, 17, 31 external field, 26 Fermi-Dirac d istr ibut ion, 98 fission, 55, 110 free-atom scattering length . 72
~ncrilliud phase-space disl~ibution function, 1.( Ham iltonia n, 15. 102 H-theorem, 98. 101 inco heren t scattering. 86 inelutic scattering, 55, 80
correla t ions a nd ftuctuat ions, 6, 104 coreetenon length, 124 creation .nd des truction cperatc re, I I,
kinet ic enerS)'. IS
cross sect ions. see clastic: and inelutic: scattering, fission , a nd radiative ce peure
MvoweJJ.Boltzma nn d istribution, 100
.,
dampi nl theory. )() Debye-watler fact or. 8), 88 den sity mat rj x, 13, 101 derailed bala nce, 101 discre te pha se space, 9 Do ppler effect, 62- 75 Einstei n Cl)'s tal, 76 clastic: scatteri ng. SO, 69. 80, 106
lam boM6ssN uer factor. 82 Liouvi lle £(Iuat ion, 2, IJ
neutron balance, see transport eq ua tion neutron de nsity sina let, 12 doublet, 24 a lpha do ublet, 11) neut ro n pro ton sca tter ing, 85. 90 neulron spino r field, 10 neulron state, II, 31, 37 neutron width, 48 nuclear sta te, 37
'"
American Nucle.r Society
"d U.S. Atomic Energy Commission
_.....
MONOGRAPH SERIES ON NUCLEAR SCIENCE AND TECHNOLOGY
-""'.
ALLEN G.GRAV AlMl'lcan
.lOHN H.GRAHAM
SocI-trtor M. . .
AIMfkIlll Nuclear SocItoty
The Foundations of Neutron Transport Theory RICHARD K.DSBORN and SIDNEY VIP
Coolant Chemical Technology of Aqueous Heterogeneous Reactor Systems PAUL COHEN
Alkali Metal Handling and Systems Operating Techniques .I . W . MAUSTELLER. F. TEPPER and S. J . RODGERS
Ceramic Fuel Elements ROBERT •. HOLD IN
Non-Deltrudlve Fuel Auay WARREN J. McGONNA6LE
Fabrication of Refradory Metals .lAMES F.SCHUMAR and ROSS MAYFIELD
irradiation Behavior of Nuclear Fuels J. A . L. ROBERTSON
Fuel Elements In Operational Nuclear Power Reactors MASlSOUD T .!lIMNAD
Llquld .Metal Heat Transfer D .E.DWYER
Advanced Metalworking Processes f .E .BISHOP and " .L.DRRELL
Dispersion Type Fuel Elements A . N. HOLDfN
