On pseudoreciprocity

On pseudoreciprocity

Richard Sanchez *, Simone Santandrea

Commissariat aÁ l'Energie Atomique, Direction des ReÂacteurs NucleÂaires, Service d'Etudes de ReÂacteurs

et de MatheÂmatiques AppliqueÂes, CEA de Saclay, France

Received 29 April 2000; received in revised form 9 May 2000; accepted 31 May 2000

Abstract

Under restrictive conditions on the functional dependence of the cross-sections, the energy-dependent Green's functions of the di�usion and the transport equations satisfy reciprocityrelations much like the familiar reciprocity relations, but without inversion of the particle

energy. These pseudoreciprocity relations, ®rst investigated by Modak and Sahni, are gen-eralized here to include, in particular, albedo boundary conditions as well as a mixture ofscattering isotopes. It is shown that pseudoreciprocity is the result of an underlying involution

in trajectory space di�erent from the mechanical involution that generates classical reciprocityrelations. # 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction

Reciprocity is a fundamental symmetry of the one-group linear transport anddi�usion equations. Classically the reciprocity property is proved by invoking thesymmetry relation between the Green's functions for the direct and the adjointequations (Bell and Glasstone, 1970; Case and Zweifel, 1967; Sanchez, 1998). Forthe di�usion equation this su�ces because the equation is self-adjoint, whereas forthe transport equation reciprocity holds only when the scattering is invariant underdirection inversion.Reciprocity is best expressed in terms of the Green's function. The latter is the

solution of the equation for a singular localized source. For the one-group di�usionequation the reciprocity relation for the Green's function can be written as

G y!x� � � G x! y� �; �1�

Annals of Nuclear Energy 28 (2001) 401±417

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0306-4549/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.

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* Corresponding author.

E-mail address: [email protected] (R. Sanchez).

where we denote by x the spatial location rx. Physically G y!x� � is the scalar ¯ux atx due to a unit isotropic source at y. For the transport equation this relation isslightly more complex because it involves inversion of the angular direction. With xindicating now the position rx;x� � in phase space, the reciprocity relation for theGreen's function of the one-group transport equation reads

G y!x� � � G Rx!Ry� �; �2�

where R is the operator that inverses directions, Rx � rx;ÿx� �. In the transportcase G y!x� � is the angular ¯ux produced by the elementary source � xÿ y� � �� rx ÿ ryÿ �

�2 x �y

ÿ �, where �2 is Placzek's delta function on the surface of the unit

sphere (Case et al. 1953),�

4�� 2 �0� �f 0� �d0 � f � �.For both, transport and di�usion, reciprocity can be extended in some special cases

to the energy variable. For instance, if the scattering kernel h x;E0 !E� � satis®es thedetailed balance relation

w E0� �h x;E0 !E� � � w E� �h x;E!E0� �;

then, the energy-dependent Green's function obeys the reciprocity relation (Bell andGlasstone, 1970; Sanchez, 1998)

w Ey

ÿ �G y;Ey!x;Ex

ÿ � � w Ex� �G x;Ex! y;Ey

ÿ �for diffusion;

w Ey

ÿ �G y;Ey!x;Ex

ÿ � � w Ex� �G Rx;Ex!Ry;Ey

ÿ �for transport:

�3�

Since a macroscopic quantity such as the Green's function is the statistical resultof microscopic events, i.e. particle motion and collisions, it is to be expected thatsymmetries at the microscopic level will induce a macroscopic symmetry. In thissense, reciprocity relations (1) and (2) can be viewed as global statements in the sensethat, at a deeper level, the Green's function is the compound result of the contribu-tions of all the trajectories that link the emission point y to the ®nal point x. Thisobservation leads us to write

G y!x� � ��

p !� �d!; �4�

where is the set of trajectories linking y to x and p !� � is the probability or weightof trajectory !. The set of trajectories can be further decomposed as � [n�0n,where n is the subset of trajectories with n collision events. By recognizing that forthe transport equation the angular direction of the particle may change as a result ofa collision we may indicate a trajectory as:

!n � y!x1; y1! . . .!xn; yn!x� �; !n 2 n; �5�

where xi and yi denote the positions at which the trajectory enters and leaves the i-thcollision, respectively. For the di�usion case we have simply xi � yi.

402 R. Sanchez, S. Santandrea /Annals of Nuclear Energy 28 (2001) 401±417

At the trajectory level reciprocity appears as a probability-preserving involutioniM: n!n; p !n� � � p i!n� �. For the di�usion case this involution acts as

iM!n � x! yn; xn! . . .! y1; x1! y� �; �6�

while for the transport case it involves inversion of directions

iM!n � Rx!Ryn;Rxn! . . .!Ry1;Rx1!Ry� �: �7�

That the reciprocity relation is a consequence of the much more elementary trajec-tory reciprocity is a fact that has been understood for a long time. For the transportequation this knowledge comes from the recognition that mechanical trajectories areinvariant under time reversal together with the assumption that collisions are invar-iant under direction inversion. The latter property holds, in particular, for isotropicmedia for which the scattering operator is invariant under rotations. The directconsequence of time homogeneity is that particle trajectories are invariant by timereversal because for a stationary force ®eld the equations of motion do not changewhen the sign of the time changes (Landau and Lifchitz, 1966): the particle canmove backwards along the trajectory with the opposite velocity at any given loca-tion. This property de®nes an involution in trajectory space iM : ! that sendseach trajectory ! � r t� �; v t� �; t 2 0;T� �� into its mechanical inverse, that is, into thetrajectory iM! � r� t� �; v� t� �; t 2 0;T� �� such that r� t� � � r Tÿ t� � and v� t� � � ÿv Tÿ t� �.Basically, it is this trajectory involution that is at the origin of the reciprocity of theGreen's function for the one-group transport equation. However, mechanicalreversibility does not hold in the presence of inelastic collisions or collisions with abackground medium that reaches thermal equilibrium independently (which is animplicit assumption of linearization) and, therefore, reversibility cannot be extended,in general, to the energy-dependent equation.The paradigm of mechanical reversibility has permeated the research in linear

transport theory and focused the research on reciprocity relations to those relationsinvolving the involution iM induced by mechanical reversibility. It is only recentlythat Modak and Sahni, inspired by numerical evidence from di�usion calculations,published two short papers on what they call `reciprocity-like' relations (Modak andSahni, 1996, 1997). These authors proved that, under restrictive conditions on thefunctional dependence of the cross sections, the energy-dependent Green's functionsof the di�usion and transport equations satisfy reciprocity relations, much like thefamiliar reciprocity relations, but without inversion of the particle energy. In ournotation these new reciprocity relations can be written as

G y;Es!x;Ef

ÿ � � G x;Es! y;Ef

ÿ �for diffusion;

G y;Es!x;Ef

ÿ � � G Rx;Es!Ry;Ef

ÿ �for transport;

�8�

to be compared to (3).From the point of view adopted in the present work these reciprocity relations,

that we prefer to call pseudoreciprocity relations, are based in an involution i of

R. Sanchez, S. Santandrea /Annals of Nuclear Energy 28 (2001) 401±417 403

trajectory space, !! i!, such that the particle moves on trajectory i! along theinverse geometrical path of trajectory ! but with di�erent energy at the same spatiallocation. Obviously this is not the mechanical involution iM induced by trajectoryinvariance under time reversal. In their work, Modak and Sahni proved pseudor-eciprocity for the multigroup di�usion equation in a homogeneous, non multi-plicative medium by assuming that the removal cross section was independent of theenergy (Modak and Sahni, 1996). They then extended this result to the energy-dependent transport equation in a heterogeneous medium by assuming further thatthe scattering kernel was the product of a function of the energy variable times afunction of the angular variables, with the latter satisfying reciprocity for directionreversal (Modak and Sahni, 1997). Both these results were obtained for problemswith vacuum boundary conditions.Under closer analysis it appears that the simplifying assumptions adopted by

Modak and Sahni amount to a particular factorization of the operators appearing inthe di�usion and the transport equation. This results in that the di�usion operator,ÿ r�Dr � �r, and the streaming operator, �r � �, are one-group like, while thecollision operator factorizes as a product of an operator acting on the angular vari-able (for the transport case) times an operator acting on the energy variable. It isthis property that results in the factorization not of the Green's function itself but ofeach of its components in a multiple collision expansion. The ®nal result is that eachone of the components of the Green's function is the product of the homologousone-group component times a function of the energy and, therefore, satis®es pseu-doreciprocity.As correctly pointed out by Modak and Sahni, the adjoint approach cannot be

applied to prove pseudoreciprocity because the latter involves energy exchangebetween the emission and observation points in phase space. Rather than using thelaborious method of their ®rst paper, in this work we have adopted and generalizedthe technique introduced in (Modak and Sahni, 1997) which is based on the familiarMonte Carlo approach to trajectory construction.In Section 2 of this paper we use a Neumann series technique to prove pseudor-

eciprocity for the linear transport equation. First we generalize the assumptionsadopted by Modak and Sahni to include an albedo condition and the dependence ofthe angular component of the collision kernel on the spatial variable. As previouslysurmised, it is proved that G n� � y;Es!x;Ef

ÿ �, the contribution to the Green's func-

tion from particles undergoing n collisions, equals the product of the correspondingone-group contribution, G n� � y!x� �, times a function of the initial and ®nal energies,Es and Ef. This ®nding has prompted us to consider the case of an isotopic mixtureof scatters with factorized kernels: by assuming that the energy operators commutewe were able to prove that the pseudoreciprocity relation holds also for this case.However, contrary to the single scatter case, for which there exists an underlyingtrajectory involution, for the multiple-scatter case we found a probability-preservingmapping of trajectory space that does not transforms single trajectories but rathersubsets of trajectories. Each one of these subsets comprises all the trajectories whichhave the same geometrical path and same initial and ®nal energies, regardless of thevalues of the intermediary energies between collisions. In the ®nal part of Section 2


we have generalized our results to the case of a factorized albedo. Section 3 isdevoted to the energy-dependent di�usion equation. In the ®rst part of this sectionwe construct a realistic boundary condition for which the one-group di�usionoperator is self-adjoint so that the associated Green's function obeys reciprocity. Wethen apply the Neumann series technique to prove pseudoreciprocity under theconditions that the di�usion coe�cient D and the total cross section � have the sameenergy behavior and that the scattering kernel factorizes. In the last part of thissection we choose, instead, to base our analysis on the use of a complete basis ofeigenfunctions for a laplacian-like operator. Pseudoreciprocity is then demonstratedunder the assumptions that D and � as well as the scattering kernel factorize as aproduct of a function of position times a function of energy, plus the condition thatthe scattering cross section is proportional to the total cross section. This resultcomprises, as a particular case, the result for a homogeneous medium with vacuumboundary conditions proved for the multigroup case in (Modak and Sahni, 1996). Asdiscussed inModak and Sahni, 1996, pseudoreciprocity relations can be put to work todemonstrate that the integral operator involved in the calculation of the eigenvalues ofthe transport equation is symmetric This point is considered in some detail in theAppendix.

2. Transport

This section deals with the transport equation. In the ®rst subsection we introduceour notation and discuss the familiar reciprocity property. Our aim is to analyze one-group reciprocity at the trajectory level. This result is then used in the second subsec-tion to demonstrate general results on pseudoreciprocity. Finally, these results areextended to a factorized albedo.

2.1. Reciprocity revisited

We consider the one-group transport equation in a domain D containing a nonmultiplicative medium and denote by x � rx;x� � a position in phase space. TheGreen's function obeys the equation

LÿH� �Gy � �y; x 2 XGy � �Gy; x 2 @ X

�: �9�

Here Gy x� � � G y!x� � is the angular ¯ux at x produced by an elementary sourceat y: �y x� � � � xÿ y� � � � rx ÿ ry

ÿ ��2 x �y

ÿ �, where �2 is Placzek's delta function on

the surface of the unit sphere,�

4�� 2 �0� �f 0� �d0 � f � �. The transport operatorcomprises the streaming operator L � �r � � and the collision operator Hf� � x� � ��Xh y!x� �f y� �dy. Notice that we are allowing for non local collisions. For localized

collisions, which is the usual assumption in particle transport, the kernel of the collisionoperator contains a delta function in space h y!x� �! � rx ÿ ry

ÿ �h y!x� �, and the

scattering operator becomes an integral operator over only the angular directions.


thus, localized collisions are characterized by the familiar collision operatorHf� � x� � � � �4��h rx;!x� �f rx;� �d. However, for easy of notation, we will con-sider the more general case of non localized collisions. All our results will apply, ofcourse, to the restricted case of localized collisions.Equation (9) is de®ned in phase space X � x; r 2 D; 2 4��

. The albedo opera-tor transforms functions de®ned on the exiting boundary of X; @�X � x; r 2f@D; 2 2�� g, on functions de®ned on the entering boundary @ X � x; r 2 @D;f 2 2�� g:

�f� � x� � ��@�X� y!x� �f y� �dby; x 2 @ X: �10�

Here dby � dSydy ny �y

�� is the volume element on the boundary. The albedo �represents either geometrical motions of the basic domain D or the e�ects of scatteringin the `external' medium surrounding the domain.Usually reciprocity relation (2) is proved invoking a global argument involving the

symmetry relation between the Green's function and its adjoint: G y!x� � �G� x! y� �, where the start indicates the adjoint. It can then be proved (Sanchez,1998) that (2) holds subject to the conditions:

h y!x� � � h Rx!Ry� �;� y!x� � � � Rx!Ry� �; �11�

where R is the operator that inverts the angular direction, Rx � rx;ÿx� �.However, for the transport equation the reciprocity relation is a direct con-

sequence of the reversibility of one-particle dynamic. This means that particle tra-jectories satisfy also the reciprocity principle. This is evident for the straight pathsbetween collision loci. Relations (11) extend reciprocity to collision events, eitherwithin the medium or when the particle attains the boundary. Because we will usetrajectory reciprocity to discuss pseudoreciprocity later on, we will reviewed thissubject here in some detail.Our approach is to introduce a multiple collision expansion for the Green's func-

tion, as obtained from the Newmann series for the inverse operator LÿH� �ÿ1:

G y!x� � �Xn�0

G n� � y!x� � �Xn�0

TH� �nT�y� �

x� �; �12�

under the condition THk k < 1. In this expression T � Lÿ1 is an integral operatorwith kernel the Green's function for uncollided particles t y!x� �. We note that thisGreen's function accounts for the action of the albedo on the entering boundary andobeys the reciprocity relation t y!x� � � t Rx!Ry� � (Sanchez, 1998).The expression for G n� � y!x� �, the contribution of G y!x� � of the particles hav-

ing experienced exactly n collisions, can be written as


G n� � y!x� � ��X2n

Yi�ni�1

t yiÿ1!xi� �h xi! yi� �" #

t yn!x� �dxndyn; �13�

where y0 � y; xn � x1; x2; . . . ; xn� � and dxn � dx1dx2 . . . dxn. By adopting the lan-guage of Monte Carlo, we note that the particle enters the i-th collision at xi andexits at yi. From the reciprocity properties for t y!x� � and h y!x� � it follows thatthe n-th component of the Green's function obeys also reciprocity:

G n� � y!x� � � G n� � Rx!Ry� �: �14�

However, a closer look to the proof shows that reciprocity extends to each of theelementary trajectories that contribute to the value of G n� � y!x� �. Denoting by pnthe integrand in (13):

pn !n� � � pn y!x1; y1! . . .!xn; yn!x� � �pn i!n� � � pn Rx!Ryn;Rxn! . . .!Ry1;Rx1!Ry� �: �15�

Then, identifying dxndyn with d!n and in view of (13) we can write

G y!x� � �Xn�0

�n

pn !n� �d!n

where !n is the trajectory de®ned in (5). This last formula is the practical realizationof the result that we advanced in (4). Note that the probability-preserving involution(7) is implicitly de®ned in relation (15).

2.2. Pseudoreciprocity

We prove here two results on pseudoreciprocity that generalize those in (Modakand Sahni, 1997). Following the work in this reference we assume that the total crosssection is independent of the energy,

� � �x; �16�

and that the kernel of the collision operator factorizes as follows:

h x0E0 !x;E� � � hx x0 !x� �hE E0 !E� �: �17�

This factorization implies that H � HxHE, where Hx and HE are integral opera-tors that act on the variables x � rx;x� � and E, respectively. Furthermore, thesetwo operators commute HxHE � HEHx and, since the operator Lx � �r � �x actsonly on the variable x, we also have the commutation THE � HET, where T is theinverse of Lx.


Consider now the energy dependent Green's function G y;Es!x;Ef

ÿ �solution of

the transport equation

Lx ÿH� �Gy;Es� �y�Es

; x;E� � 2 X� 0;1� �Gy;Es

� �Gy;Es; x;E� � 2 @ X� 0;1� �

�; �18�

where � � �x is a one-group albedo operator with kernel independent of the energyand �Es

E� � � � Es ÿ E� �.Following our earlier approach, the solution of this equation can be expressed in

the form of a Neumann series:

G y;Es!x;Ef

ÿ � �Xn50

G n� � y;Es!x;Ef

ÿ � �Xn50

TH� �nT�y�Es

� �x;Ef

ÿ �; �19�

where T accounts for the albedo �x. In view of the commutation properties betweenHE and the operators T and Hx we have:


ÿ � � THx� �nT�y� �

x� � HE� �n�Es

� �Ef

ÿ �� G n� � y!x� � HE� �n�Es

� �Ef

ÿ �;

where G n� � y!x� � is the n-th component of the one-group Green's function. Fromthis relation and from reciprocity (14) follows a pseudoreciprocity property forG n� � y;Es!x;Ef

ÿ �,


ÿ � � G n� � Rx;Es!Ry;Ef

ÿ �; �20�

and, consequently, for the energy-dependent Green's function:

G y;Es!x;Ef

ÿ � � G Rx;Es!Ry;Ef

ÿ �: �21�

This last formula generalizes the result obtained in (Modak and Sahni, 1997) byincluding a one-group albedo and the dependence of the angular scattering kernelhx x0 !x� � on the spacial variable r. As before, result (20) can be shown to derivefrom a ®ner-grade trajectory pseudoreciprocity:

pn�!n� � pn�y ÿ!Esx1; y1 ÿ!E1

. . . ÿ!Enÿ1xn; yn ÿ!

Ef

x� �pn�i!n� � pn�Rx ÿ!Es

Ryn;Rxn ÿ!E1. . . ÿ!Enÿ1

Ry1;Rx1 ÿ!Ef

Ry�:�22�

This formula proves that the general statement in (4) as well as the existence of theprobability-preserving involution i remain valid for the present energy-dependentcase. This involution, which is implicitly de®ned in (22), puts into equivalence twotrajectories, !n and i!n, that are geometrically identical and along which the particlesmove in opposite directions. However, the energy of the particles at a given position is


not the same for both trajectories so that, from the viewpoint of dynamical invarianceby time reversal, the two trajectories are not reciprocal one of each other. It isbecause of this fact that we have adopted the name of pseudoreciprocity.Next we consider the generalization of assumption (17) to an isotopic scattering

mixture:

h x0E 0 !x;E� � �Xk

hx;k x0 !x� �hE;k E0 !E� �: �23�

This formula implies that H �PkHx;kHE;k. We further assume that the operatorsHE;k commute among themselves

HE;kHE;l � HE;lHE;k; 8k; l: �24�

A non-trivial example of (23) in two energy groups is realized by the two-para-meter family of scattering matrices:

HE; �;�� a�b �� c

� ��25�

for ®xed values of a; b and c.For the present case we observe that the Neumann expansion in (19) is made up of

contributions in the form:

Gn� �k y;Es!x;Ef

ÿ � � Yi�ni�1

THx;ki

ÿ �T�y

" #x� �

Yi�ni�1

HE;ki

ÿ ��Es

" #Ef

ÿ �� G

n� �k y!x� �

Yi�ni�1

HE;ki

ÿ ��Es

" #Ef

ÿ �;

�26�

where we have introduced the multidimensional index k � k1; k2; . . . kn� � to indicatethe order in which the particle collides with the di�erent isotopes. This formula givesthe contribution to G n� � y;Es!x;Ef

ÿ �from all the trajectories for which the ®rst col-

lision is done with isotope k1, the second with isotope k2 and so on until the n-th col-lision. Also G

n� �k y!x� � is the contribution to the one-group Green's function from

this set of trajectories. Since one-group trajectories obey the reciprocity principle itfollows that pseudoreciprocity relations (20) and (21) apply also to the present case.Our change of variables transforms the set of trajectories

n Es!Ef

ÿ �� !n � y ÿ!Es

x1 k1� �y1 ÿ!E1. . . ÿ!Enÿ1

xn kn� �yn ÿ!Ef

x

� �; 8E1; . . . ;Enÿ1

� �

into the set


in Es!Ef

ÿ �� !n � Rx ÿ!Es

Ryn kn� �Rxn ÿ!E1. . . ÿ!Enÿ1

Ry1 k1� �Rx1 ÿ!Ef

Ry

� �;8E1; . . . ;Enÿ1

� �

while preserving the overall probability�n Es !Ef� �

p !� �d! ��in Es !Ef� �

p !� �d!: �27�

However, for the present case there is not a one-to-one mapping of trajectories thatpreserves trajectory probability and, consequently, pseudoreciprocity does not reachtrajectory level.The sets n Es!Ef

ÿ �and in Es!Ef

ÿ �contain all the trajectories that share the

same geometrical path, the same order of collisions with the di�erent isotopesmaking up the medium and the same initial and ®nal energies. The di�erence is thatthe geometrical path and the order of collisions for in Es!Ef

ÿ �are the opposite of

those for n Es!Ef

ÿ �. Note that the inversion of the order of collisions is necessary in

order to preserve the probability of the geometrical path. It is this inversion of scat-tering order that breaks down the probability-preserving involution of the single-scatter case. Fig. 1 shows a typical trajectory in n Es!Ef

ÿ �.

To further illustrate this point we consider the simple case of a two-group problemwith scatters of the type in Eq. (25). For a trajectory with two collision events !2 ��y ÿ!f x1�k1�y1 ÿ!th x2�k2�y2 ÿ!th x� we have

p !2� � � pg�1b �2 � �2c� � 6� p i!2� � � pg�2b �1 � �1c� �;

where pg � pg y!x1 k1� �y1!x2 k2� �y2!x� � is the probability of the geometrical

path and i!2 � �Rx ÿ!f

Ry2�k2�Rx2 ÿ!th Ry1�k1�Rx1 ÿ!th Ry�. Consider next the setof trajectories with the original particle in the fast group and the ®nal particle in the

Fig. 1. A two-collision trajectory in a medium containing an isotopic mixture of scatters. After its ®rst

collison the trajectory reaches the boundary of the domain and re-enters under the action of operator T.

The direct and reverse trajectories do not have the same probability.


thermal group passing from all the intermediary groups 2 f! th� � � !2; !~ 2f g,where !~ 2 � �y ÿ!

fx1�k1�y1 ÿ!

fx2�k2�y2 ÿ!th x�. Then

p 2 f! th� �� pg �1b �2 � �2c� � � �1�2b� � �p i2 f! th� �� pg �2b �1 � �1c� � � �2�1b� �:

Nevertheless, this property is so hard to believe that we have independentlychecked the pseudoreciprocity of the Green's function for this special case of twogroups and two isotopes with an ad-hoc computer program based on the use ofanalytical solutions in a heterogeneous slab geometry.

2.3. Factorized albedo

We end this section with a comment on the albedo. Until now we have assumed aone-group albedo with kernel independent of the energy. However, by using theNeumann expansion it is possible to extend our results to the case of a factorizedalbedo with kernel:

� x0;E0 !x;E� � �Xk

�x;k x0 !x� ��E;k E0 !E� � �28�

where, again, we take the �x;k to satisfy reciprocity, �x;k x0 !x� � � �x;k Rx ! Rx0� �.We also assume that the associated energy operations commute:

�E;k�E;l � �E;l�E;k; 8k; l:

Next, we choose to view the boundary condition as a boundary source and writethe transport equation as:

Lx ÿH� �Gy;Es� �b�Gy;Es

� �y�Es; x;E� � 2 X� 0;1� �

Gy;Es� 0; x;E� � 2 @ X� 0;1� �

�;

where we have maintained assumption (16). In this equation �b is a delta functionthat reduces integration in X to integration on @ X :�

X

�b x� �f x� �dx ��@ X

f x� �dbx:

Then the multiple scattering expansion for the Green's function becomes;

G y;Es!x;Ef

ÿ � �Xn�0


ÿ � �Xn�0

TH~� �n

T�y�Es

h ix;Ef

ÿ �; �29�


where H~ accounts for collisions in X as well as for ``collisions'' on the boundary@ X;H~ f � H� �b�� f, and T � Lÿ1 accounts for a non re-entering boundary condi-

tion. The Neumann series in (29) converges under the condition TH~ < 1.

Finally, we assume that the scattering operator factorizes as in (23) and extend thecommutation relations (24) to the albedo operators

HE;k�E;l � �E;lHE;k;8k; l:

Given these assumptions, it is clear that pseudoreciprocity results (20) and (21)hold again. For comparison we show in Fig. 2 the same trajectory of Fig. 1 but forthe case of a factorized albedo. The di�erence now is that each time that the particlereaches the boundary and re-enters the domain it may undergo a change of energy.Each one of these events are counted as one collision in the Neumann expansion.

3. Di�usion

Our ®rst approach to prove pseudoreciprocity relations for the energy-dependentdi�usion equation will parallel the multiple-collision argument used for the trans-port case. First, we will set up a one-group di�usion equation with appropriateboundary conditions for which the one-group Green's function is symmetric,

G y!x� � � G x! y� �; �30�

where now we use x to indicate the spatial position rx.

Fig. 2. A trajectory for a multigroup albedo. Here the operator T accounts only for straight motion

between collisions. Therefore, the albedo operator appears explicitly in the Neumann expansion and its

action is similar to that of H, except that usually involves a geometrical motion.


With the help of this reciprocity relation we will establish a ®rst pseudoreciprocityresult for the energy-dependent Green's function G y;Es!x;Ef

ÿ �under similar

assumptions to those introduced in the previous section. A di�erent approach, basedon the use of a complete set of eigenfunctions for the Laplacian, is applied in the lastsubsection to establish pseudoreciprocity under di�erent assumptions.

3.1. The one-group problem

We consider the one-group di�usion equation in a non multiplicative medium

LÿH� �Gy � �y; x 2 XC Gy � �C�Gy; x 2 @X

�:

where X is the geometrical domain D; @X its boundary and L � ÿr�Dr � � is thedi�usion operator. Green's function G y!x� � is the scalar ¯ux at x resulting from aunit singular source at y. As before, we adopt a general albedo operator of the form

�f� � x� � ��@X

� y!x� �f y� �dSy; x 2 @X:

The boundary operators C� are taken to be of the form C� � a� bn�r, where aand b are functions de®ned on @X and n is the outward unit vector at the surface.Under these conditions one can prove that reciprocity (30) holds if the operator A �Daÿ1 1ÿ �� ÿ1 1� �� b is self-adjoint with respect to the scalar producthf; gi � � @XdS fg� � x� �. In order to ensure that A is self-adjoint we will assume that � isself-adjoint and that functions a and b satisfy the relation Daÿ1 � cb, where c is anarbitrary constant. We note that this general boundary condition comprises thefamiliar boundary albedo condition, Jin x� � � �Jout� � x� �, where Jin and Jout are theentering and exiting currents, respectively, as well as the usual conditions of zero¯ux or zero gradient at the boundary, � � 0 and n�r� � 0. The current albedocondition is obtained for a � 1=4; b � D=2 and c � 8, while for the other twoconditions we set � � 0 and put b � 0 or a � 0 for the zero ¯ux and zero gradientconditions, respectively.If we de®ne a `trajectory' as in (5), then a Neumann-series argument shows that

reciprocity (15) holds for trajectory !n and its inverse as given in (6). This provesthat (4) holds also for the Green's function of the one-group di�usion equation.

3.2. Neumann series approach

In order to prove a ®rst pseudoreciprocity relation for the energy-dependentGreen's function we assume that the di�usion coe�cient and the total cross-sectionsshare the same energy dependence and write

L � w E� �Lx � w E� � ÿr�Dxr � �x� �;


where the lower index x indicates that the functions depend only on the spatialvariable. We also adopt assumption (17) for the scattering operator.Next, we write G y;Es!x;E� � � w E� �G y;Es!x;E� � so that the di�usion equa-

tion for the Green's function can be written as

�Lx ÿ H�Gy;Es� �y�Ens ; x;E� � 2 X� 0;1� �

C Gy;Es� �C�Gy;Es

; x;E� � 2 @X� 0;1� �

); �31�

where H � HxHE with HE a scattering operator acting on the energy variable withkernel hE E 0 !E� � � hE E 0 !E� �=w E 0� �.The usual Neumann expansion for this equation yields an expression like (19)

where now G! G;H! H and where T, the integral operator inverse of Lx, accountsfor the one-group albedo condition in (31). This proves that G�n� y;Es!x;Ef

ÿ �and,

therefore, G n� � y;Es!x;Ef

ÿ �and G y;Es!x;Ef

ÿ �satisfy pseudoreciprocity:


ÿ � � G n� � x;Es! y;Ef

ÿ �;

G y;Es!x;Ef

ÿ � � G x;Es! y;Ef

ÿ �:

Clearly, this demonstration can be generalized to assumption (23) with the com-mutation constraints in (24).

3.3. Eigenfunction approach

Consider the one-group eigenvalue problem:

ÿr�Dxr�x

�n � ln�n; x 2 X

Cÿ�n � �C��n; x 2 @X

):

Given su�cient regularity conditions for Dx and �x this equation has a countableset of eignfunctions �nf g with positive eignvalues. Moreover, the �nf g are orthogonalfor the scalar product:

f; g� � ��X

fg� � x� ��x x� �dx; �32�

and form a complete set of functions on the Sobolev space f r� �� of functions in D

such that�D

Dx rf� �2��x f 2� �

dr <1:

For the one-dimensional case this result is directly related to the classical Sturm-Liouville problem and has been abundantly discussed in the technical literature


(Iyanaga and Kawada, 1980). Here we will assume without further ado that it is alsovalid for the multidimensional case with a non multiplicative albedo, �

< 1. In thefollowing we will take the eigenfunctions normalized to 1.In order to exploit this result to prove pseudoreciprocity we assume the factorization

of the scattering kernel (17) together with:

D � DxDE; � � �x�E; �33�

where the lower index variable indicates that the function only depends on thatvariable. The resulting di�usion equation for the energy-dependent Green's functionreads:

�ÿr�DxDEr � �x�E ÿHxHE�Gy;Es� �y�Es

; x;E� � 2 X� 0;1� �C Gy;Es

� �C�Gy;Es; x;E� � 2 @X� 0;1� �

�: �34�

Under these assumptions, we look for a solution of the form:

G y;Es!x;Ef

ÿ � �Xn

an y;Es!Ef

ÿ ��n x� ��n y� �: �35�

Pseudoreciprocity for the Green's function will follow if the expansion coe�cientsan are independent of the variable y. To compute these coe�cients we replaceexpansion (35) into the di�usion equation for G y;Es!x;Ef

ÿ �, multiply the result-

ing equation with �n x� � and integrate over x. By using the orthonormality of the �nand by assuming that

Hx�n � cn�x�n: �36�

where cn is a constant, we obtain the following equation for an:

lnDE � �E ÿ cnHE� �an � �Es:

Because the ln are positive this equation has a solution under the constraint�ÿ1E cnHE

< 1. The equation also shows that an is independent of y and that,therefore, G y;Es!x;Ef

ÿ �satis®es pseudoreciprocity. For localized collisions the

operator Hx reduces to a multiplication by the di�usion cross section �sx so that (36)is equivalent to take �sx=�x constant.Di�usion Eq. (34) comprises as a particular case the di�usion equation in a bare

homogeneous medium. We conclude that our results generalize those obtained in(Modak and Sahni, 1996) for this particular case.

4. Conclusions

In this paper we have re-examined and generalized the pseudoreciprocity relationsderived by Modak and Sahni for the multigroup di�usion equation and for the


energy-dependent transport equation. Pseudoreciprocity relations can be establishedunder severe constraints on the functional dependence of the cross sections thatamount to the factorization of the di�erent operator kernels as products of func-tions of the space variable (and the angular variable for the transport case) timesfunctions of the energy variable. This factorization does not result on an equivalentfactorization for the energy-dependent Green's function but, rather, on the factor-ization of the multiple-scattering components of this function.It has also been shown that these new reciprocity relations relay on probability-pre-

serving transformations of the set of trajectories similar to the mechanical involutionresponsible for the familiar reciprocity relations, with the di�erence that for pseu-doreciprocity there is no inversion of the energies between the emission and the testlocations. From this point of view pseudoreciprocity and reciprocity relations arealike in that they are generated by probability-preserving transformations in trajectoryspace.Pseudoreciprocity relations can be used for formal veri®cations of di�usion and

transport numerical methods. But, our opinion is that their main interest resides inthat they reveal new facts about the intimate structure of the transport and di�usionoperators and, that for this single reason, they are of interest.This work is by no means complete. Our intention was to give a consistent view of

pseudoreciprocity and show its relationship to `normal' reciprocity. Surely theresults proved here can be generalized to other factorizations of the operators thatappear in the energy-dependent transport and di�usion equations. We hope that theapproach given here will stimulate further research in this area.

Acknowledgements

One of the authors (R.S.) would like to express his thanks to Tam P. for fruitfuldiscussions.

Appendix

Under limited constraints the eigenvalue problem for the energy-dependent di�u-sion or transport equations can be written as an equation involving an integraloperator. We prove here that pseudoreciprocity can be used to demonstrate that thisintegral operator is symmetric. For brevity we will consider only the transport case.We assume that ®ssion is isotropic and that there is at most a ®ssile isotope at any

spatial location or, equivalently, that all immixed ®ssile isotopes have the same ®s-sion spectrum �. We also assume that the production cross section factorizes as

v�f r;E� � � f r� �v�f r� �;

where f r� � is a positive function. Then the eigenvalue problem for the transportequation reads:


LÿH� � � 1

lP ; x 2 X

� � ; x 2 @ X

); �A1�

where P is the production operator

P � � r;E� � � 1

4�� E� �

�v�f r;E

0� � dE 0d 0 � � E� ��f r� �

pt r� �:

To obtain an eigenvalue equation for t r� � � ��f r� �p= 4��

h i �v�f E

0� � dE 0d 0 wecompute from Eq. (A1) by inversion of the transport operator LÿH. By realizingthat the inverse of this operator is an integral operator with kernel the Green'sfunction we write

x;Ef

ÿ � � 1

l

�X

��f

pryÿ �

t ryÿ �

dy

�� E� �G y;Es!x;Ef

ÿ �dEs:

Hence

t r� � � 1

l

�D

k r0 ! r� �t r0� �dr0 �A2�

with the kernel

k r0 ! r� � ��f r� �f r0� �p4�

�v�f E� �dE

�� E 0� �dE 0

�d

�G r0;0;E 0 ! r;;E� �d 0:

Therefore, if the cross-sections are such that pseudoreciprocity (8) applies, thenthe kernel of (A2) is symmetric, k r0 ! r� � � k r! r0� �, and we can conclude that allthe eigenvalues of (A1) are real.

References

Bell, G.I., Glasstone, S., 1970. Nuclear Reactor Theory. Van Nostrand.

Case, K.M., de Ho�mann, F., Placzek, G., 1953. Introduction to the Theory of Neutron Di�usion. Los

Alamos National Laboratory.

Case, K.M., Zweifel, P.F., 1967. Linear Transport Theory. Addison-Wesley.

Iyanaga, S., Kawada, Y. (Eds.), 1980. Encyclopedic Dictionary of Mathematics 309.B. MIT Press.

Landau, L., Lifchitz, E., 1966. MeÂ canique. Editions MIR.

Modak, R.S., Sahni, D.C., 1996. Some reciprocity-like relations in multi-group neutron di�usion and

transport theory over bare homogeneous regions. Annals of Nuclear Energy 23 (12), 965±995.

Modak, R.S., Sahni, D.C., 1997. On the reciprocity-like relations in linear neutron transport theory.

Annals of Nuclear Energy 24 (15), 1271±1275.

Sanchez, R., 1998. Duality, Green's functions and all that. Transport Theory and Statistical Physics 27 (5±7),

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