Foundations of Neutron Transport Theory

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

GORDON AND BREACH, SCIENCE PUBLISHERS, INC . NEW YORK · LONDON· PARIS

Copyright

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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 = 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 - flightly 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 : - 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-'····,.aj.(K·) · p~ lk ..

VL·j·.,·....

I\.. )

I,L m ..c

:to

L, 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

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 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~ ~ ~Xr ( >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,



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'"(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)' 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)

... r,

f

fdU, I ,(O,O ') -