Annals of Nuclear Energy 28 (2001) 401±417 www.elsevier.com/locate/anucene
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 energydependent Green's functions of the diusion and the transport equations satisfy reciprocity relations much like the familiar reciprocity relations, but without inversion of the particle energy. These pseudoreciprocity relations, ®rst investigated by Modak and Sahni, are generalized here to include, in particular, albedo boundary conditions as well as a mixture of scattering isotopes. It is shown that pseudoreciprocity is the result of an underlying involution in trajectory space dierent from the mechanical involution that generates classical reciprocity relations. # 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Reciprocity is a fundamental symmetry of the one-group linear transport and diusion equations. Classically the reciprocity property is proved by invoking the symmetry relation between the Green's functions for the direct and the adjoint equations (Bell and Glasstone, 1970; Case and Zweifel, 1967; Sanchez, 1998). For the diusion equation this suces because the equation is self-adjoint, whereas for the transport equation reciprocity holds only when the scattering is invariant under direction 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 diusion equation the reciprocity relation for the Green's function can be written as G
y ! x G
x ! y; * Corresponding author. E-mail address:
[email protected] (R. Sanchez). 0306-4549/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-4549(00)00067-0
1
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where we denote by x the spatial location rx . Physically G
y ! x is the scalar ¯ux at x due to a unit isotropic source at y. For the transport equation this relation is slightly more complex because it involves inversion of the angular direction. With x indicating now the position
rx ; x in phase space, the reciprocity relation for the Green'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 ; ÿ x . In the transport case ÿ G
y!ÿx is the angular ¯ux produced by the elementary source
x ÿ y rx ÿ ry 2 x y , where 2 is Placzek's delta function on the surface of the unit sphere (Case et al. 1953),
4 2
0 f
0 d 0 f
. For both, transport and diusion, reciprocity can be extended in some special cases to the energy variable. For instance, if the scattering kernel h
x; E0 ! E satis®es the detailed balance relation w
E0 h
x; E0 ! E w
Eh
x; E ! E0 ; then, the energy-dependent Green's function obeys the reciprocity relation (Bell and Glasstone, 1970; Sanchez, 1998) ÿ ÿ ÿ w Ey G y; Ey ! x; Ex w
Ex G x; Ex ! y; Ey for diffusion; ÿ ÿ ÿ
3 w Ey G y; Ey ! x; Ex w
Ex G Rx; Ex ! Ry; Ey for transport: Since a macroscopic quantity such as the Green's function is the statistical result of microscopic events, i.e. particle motion and collisions, it is to be expected that symmetries at the microscopic level will induce a macroscopic symmetry. In this sense, reciprocity relations (1) and (2) can be viewed as global statements in the sense that, at a deeper level, the Green's function is the compound result of the contributions of all the trajectories that link the emission point y to the ®nal point x. This observation leads us to write
4 G
y ! x p
!d!;
where is the set of trajectories linking y to x and p
! is the probability or weight of trajectory !. The set of trajectories can be further decomposed as [n0 n , where n is the subset of trajectories with n collision events. By recognizing that for the transport equation the angular direction of the particle may change as a result of a 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-th collision, respectively. For the diusion case we have simply xi yi .
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At the trajectory level reciprocity appears as a probability-preserving involution iM: n ! n ; p
!n p
i!n . For the diusion 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 trajectory reciprocity is a fact that has been understood for a long time. For the transport equation this knowledge comes from the recognition that mechanical trajectories are invariant under time reversal together with the assumption that collisions are invariant under direction inversion. The latter property holds, in particular, for isotropic media for which the scattering operator is invariant under rotations. The direct consequence of time homogeneity is that particle trajectories are invariant by time reversal because for a stationary force ®eld the equations of motion do not change when the sign of the time changes (Landau and Lifchitz, 1966): the particle can move backwards along the trajectory with the opposite velocity at any given location. This property de®nes an involution in trajectory space iM : ! that sends each trajectory ! r
t; v
t; t 2
0; T into its mechanical inverse, that is, into the trajectory iM ! r
t; v
t; t 2
0; T such that r
t r
T ÿ t and v
t ÿv
T ÿ t. Basically, it is this trajectory involution that is at the origin of the reciprocity of the Green's function for the one-group transport equation. However, mechanical reversibility does not hold in the presence of inelastic collisions or collisions with a background medium that reaches thermal equilibrium independently (which is an implicit 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 relations involving the involution iM induced by mechanical reversibility. It is only recently that Modak and Sahni, inspired by numerical evidence from diusion calculations, published two short papers on what they call `reciprocity-like' relations (Modak and Sahni, 1996, 1997). These authors proved that, under restrictive conditions on the functional dependence of the cross sections, the energy-dependent Green's functions of the diusion and transport equations satisfy reciprocity relations, much like the familiar reciprocity relations, but without inversion of the particle energy. In our notation these new reciprocity relations can be written as ÿ ÿ G y; Es ! x; Ef G x; Es ! y; Ef for diffusion; ÿ ÿ
8 G y; Es ! x; Ef G Rx; Es ! Ry; Ef for transport; 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
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trajectory space, ! ! i!, such that the particle moves on trajectory i! along the inverse geometrical path of trajectory ! but with dierent energy at the same spatial location. Obviously this is not the mechanical involution iM induced by trajectory invariance under time reversal. In their work, Modak and Sahni proved pseudoreciprocity for the multigroup diusion equation in a homogeneous, non multiplicative medium by assuming that the removal cross section was independent of the energy (Modak and Sahni, 1996). They then extended this result to the energydependent transport equation in a heterogeneous medium by assuming further that the scattering kernel was the product of a function of the energy variable times a function of the angular variables, with the latter satisfying reciprocity for direction reversal (Modak and Sahni, 1997). Both these results were obtained for problems with 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 in the diusion and the transport equation. This results in that the diusion operator, ÿ rDr r , and the streaming operator, r , are one-group like, while the collision operator factorizes as a product of an operator acting on the angular variable (for the transport case) times an operator acting on the energy variable. It is this property that results in the factorization not of the Green's function itself but of each of its components in a multiple collision expansion. The ®nal result is that each one of the components of the Green's function is the product of the homologous one-group component times a function of the energy and, therefore, satis®es pseudoreciprocity. As correctly pointed out by Modak and Sahni, the adjoint approach cannot be applied to prove pseudoreciprocity because the latter involves energy exchange between the emission and observation points in phase space. Rather than using the laborious method of their ®rst paper, in this work we have adopted and generalized the technique introduced in (Modak and Sahni, 1997) which is based on the familiar Monte Carlo approach to trajectory construction. In Section 2 of this paper we use a Neumann series technique to prove pseudoreciprocity for the linear transport equation. First we generalize the assumptions adopted by Modak and Sahni to include an albedo condition and the dependence of the angular component of the collision kernel ÿ on the spatial variable. As previously surmised, it is proved that G
n y; Es ! x; Ef , the contribution to the Green's function from particles undergoing n collisions, equals the product of the corresponding one-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 mixture of scatters with factorized kernels: by assuming that the energy operators commute we 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 underlying trajectory involution, for the multiple-scatter case we found a probability-preserving mapping of trajectory space that does not transforms single trajectories but rather subsets of trajectories. Each one of these subsets comprises all the trajectories which have the same geometrical path and same initial and ®nal energies, regardless of the values of the intermediary energies between collisions. In the ®nal part of Section 2
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we have generalized our results to the case of a factorized albedo. Section 3 is devoted to the energy-dependent diusion equation. In the ®rst part of this section we construct a realistic boundary condition for which the one-group diusion operator is self-adjoint so that the associated Green's function obeys reciprocity. We then apply the Neumann series technique to prove pseudoreciprocity under the conditions that the diusion coecient D and the total cross section have the same energy behavior and that the scattering kernel factorizes. In the last part of this section we choose, instead, to base our analysis on the use of a complete basis of eigenfunctions for a laplacian-like operator. Pseudoreciprocity is then demonstrated under the assumptions that D and as well as the scattering kernel factorize as a product of a function of position times a function of energy, plus the condition that the scattering cross section is proportional to the total cross section. This result comprises, as a particular case, the result for a homogeneous medium with vacuum boundary conditions proved for the multigroup case in (Modak and Sahni, 1996). As discussed in Modak and Sahni, 1996, pseudoreciprocity relations can be put to work to demonstrate that the integral operator involved in the calculation of the eigenvalues of the transport equation is symmetric This point is considered in some detail in the Appendix. 2. Transport This section deals with the transport equation. In the ®rst subsection we introduce our notation and discuss the familiar reciprocity property. Our aim is to analyze onegroup reciprocity at the trajectory level. This result is then used in the second subsection to demonstrate general results on pseudoreciprocity. Finally, these results are extended to a factorized albedo. 2.1. Reciprocity revisited We consider the one-group transport equation in a domain D containing a non multiplicative medium and denote by x
rx ; x a position in phase space. The Green's function obeys the equation
L ÿ HGy y ; x 2 X :
9 Gy Gy ; x2@ X ¯ux at x produced by an elementary source Here Gy
x G
y ! xÿis the angular ÿ at y: y
x
x ÿ y rx ÿ ry 2 x y , where 2 is Placzek's delta function on the surface of the unit sphere,
4 2
0 f
0 d 0 f
. The transport operator comprises the streaming operator L r and the collision operator
Hf
x X h
y ! xf
ydy. Notice that we are allowing for non local collisions. For localized collisions, which is the usual assumption in particle transport, of the collision ÿ the kernel operator contains a delta function in space h
y ! x ! rx ÿ ry h
y ! x, and the scattering operator becomes an integral operator over only the angular directions.
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thus, localized collisions are characterized by the familiar collision operator
Hf
x
4 h
rx ; ! x f
rx ; d . However, for easy of notation, we will consider the more general case of non localized collisions. All our results will apply, of course, to the restricted case of localized collisions. Equation (9) is de®ned in phase space X x; r 2 D; 2
4 . The albedo operator transforms functions de®ned on the exiting boundary of X; @ X fx; r 2 @D; 2
2 g, on functions de®ned on the entering boundary @ X fx; r 2 @D;
2
2 g:
y ! xf
ydb y; x 2 @ X:
10
f
x @ X
Here db y dSy d y ny y is the volume element on the boundary. The albedo represents either geometrical motions of the basic domain D or the eects of scattering in 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 ; ÿ x . However, for the transport equation the reciprocity relation is a direct consequence of the reversibility of one-particle dynamic. This means that particle trajectories satisfy also the reciprocity principle. This is evident for the straight paths between collision loci. Relations (11) extend reciprocity to collision events, either within the medium or when the particle attains the boundary. Because we will use trajectory reciprocity to discuss pseudoreciprocity later on, we will reviewed this subject here in some detail. Our approach is to introduce a multiple collision expansion for the Green's function, as obtained from the Newmann series for the inverse operator
L ÿ Hÿ1 : G
y ! x
X n0
G
n
y ! x
X
THn Ty
x;
12
n0
under the condition kTHk < 1. In this expression T Lÿ1 is an integral operator with kernel the Green's function for uncollided particles t
y ! x. We note that this Green's function accounts for the action of the albedo on the entering boundary and obeys 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 having experienced exactly n collisions, can be written as
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n
"
G
y ! x
X2n
# in Y t
yiÿ1 ! xi h
xi ! yi t
yn ! xdxn dyn ;
407
13
i1
where y0 y; xn
x1 ; x2 ; . . . ; xn and dxn dx1 dx2 . . . dxn . By adopting the language of Monte Carlo, we note that the particle enters the i-th collision at xi and exits at yi . From the reciprocity properties for t
y ! x and h
y ! x it follows that the 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 the elementary trajectories that contribute to the value of G
n
y ! x. Denoting by pn the 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 dxn dyn with d!n and in view of (13) we can write X
pn
!n d!n G
y ! x n0 n
where !n is the trajectory de®ned in (5). This last formula is the practical realization of 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 (Modak and Sahni, 1997). Following the work in this reference we assume that the total cross section is independent of the energy, x ;
16
and that the kernel of the collision operator factorizes as follows: h
x0 E0 ! x; E hx
x0 ! xhE
E0 ! E:
17
This factorization implies that H Hx HE , where Hx and HE are integral operators that act on the variables x
rx ; x and E, respectively. Furthermore, these two operators commute Hx HE HE Hx and, since the operator Lx r x acts only on the variable x, we also have the commutation THE HE T, where T is the inverse of Lx .
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ÿ Consider now the energy dependent Green's function G y; Es ! x; Ef solution of the transport equation
Lx ÿ HGy ;Es y Es ;
x; E 2 X 0; 1 ;
18
x; E 2 @ X 0; 1 Gy ;Es Gy ;Es ; where x is a one-group albedo operator with kernel independent of the energy and Es
E
Es ÿ E. Following our earlier approach, the solution of this equation can be expressed in the form of a Neumann series: ÿ X
n ÿ X ÿ G y; Es ! x; Ef
THn Ty Es x; Ef ; G y; Es ! x; Ef n50
19
n50
where T accounts for the albedo x . In view of the commutation properties between HE and the operators T and Hx we have: ÿ ÿ G
n y; Es ! x; Ef
THx n Ty
x
HE n Es Ef ÿ G
n
y ! x
HE n Es Ef ; where G
n
y ! x is the n-th component of the one-group Green's function. From this ÿ relation and from reciprocity (14) follows a pseudoreciprocity property for G
n y; Es ! x; Ef , ÿ ÿ
20 G
n y; Es ! x; Ef G
n Rx; Es ! Ry; Ef ; and, consequently, for the energy-dependent Green's function: ÿ ÿ G y; Es ! x; Ef G Rx; Es ! Ry; Ef :
21
This last formula generalizes the result obtained in (Modak and Sahni, 1997) by including a one-group albedo and the dependence of the angular scattering kernel hx
x0 ! x on the spacial variable r. As before, result (20) can be shown to derive from a ®ner-grade trajectory pseudoreciprocity: Es
E1
Ef
Enÿ1
pn
!n pn
y ÿ! x1 ; y1 ÿ! . . . ÿ! xn ; yn ÿ! x Es
E1
Enÿ1
Ef
22
pn
i!n pn
Rx ÿ! Ryn ; Rxn ÿ! . . . ÿ! Ry1 ; Rx1 ÿ! Ry : This formula proves that the general statement in (4) as well as the existence of the probability-preserving involution i remain valid for the present energy-dependent case. This involution, which is implicitly de®ned in (22), puts into equivalence two trajectories, !n and i!n , that are geometrically identical and along which the particles move in opposite directions. However, the energy of the particles at a given position is
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409
not the same for both trajectories so that, from the viewpoint of dynamical invariance by time reversal, the two trajectories are not reciprocal one of each other. It is because of this fact that we have adopted the name of pseudoreciprocity. Next we consider the generalization of assumption (17) to an isotopic scattering mixture: X hx;k
x0 ! xhE;k
E0 ! E:
23 h
x0 E 0 ! x; E k
P This formula implies that H k Hx;k HE;k . We further assume that the operators HE;k commute among themselves HE;k HE;l HE;l HE;k ; 8k; l:
24
A non-trivial example of (23) in two energy groups is realized by the two-parameter family of scattering matrices: a
25 HE;
; b c 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: ÿ G
kn y; Es ! x; Ef
"
in ÿ Y i1
#
THx;ki Ty
x "
G
kn
y ! x
"
in ÿ Y i1
in ÿ Y i1
HE;ki Es
# ÿ HE;ki Es Ef
#
ÿ Ef ;
26
where we have introduced the multidimensional index k
k1 ; k2 ; . . . kn to indicate the order in which the particle collides ÿ with the dierent isotopes. This formula gives the contribution to G
n y; Es ! x; Ef from all the trajectories for which the ®rst collision is done with isotope k1 , the second with isotope k2 and so on until the n-th collision. Also G
kn
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 it follows that pseudoreciprocity relations (20) and (21) apply also to the present case. Our change of variables transforms the set of trajectories ÿ
n Es ! Ef Ef Es E1 Enÿ1 !n y ÿ! x1
k1 y1 ÿ! . . . ÿ! xn
kn yn ÿ! x ; 8E1 ; . . . ; Enÿ1 into the set
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ÿ i n Es ! Ef Ef Es E1 Enÿ1 !n Rx ÿ! Ryn
kn Rxn ÿ! . . . ÿ! Ry1
k1 Rx1 ÿ! Ry ; 8E1 ; . . . ; Enÿ1
while preserving the overall probability
p
!d! p
!d!:
n
Es !Ef i n
Es !Ef
27
However, for the present case there is not a one-to-one mapping of trajectories that preserves trajectory probability and, consequently, pseudoreciprocity does not reach trajectory level. ÿ ÿ The sets n Es ! Ef and i n Es ! Ef contain all the trajectories that share the same geometrical path, the same order of collisions with the dierent isotopes making up the medium and the same initial and ®nal energies. The ÿ dierence is that E ! E are the opposite of the geometrical path and the order of collisions for i
n s f ÿ those for n Es ! Ef . 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 scattering order that breaks down the probability-preserving involution of the singleÿ scatter case. Fig. 1 shows a typical trajectory in n Es ! Ef . To further illustrate this point we consider the simple case of a two-group problem with scatters of the type in Eq. (25). For a trajectory with two collision events !2 f
th
th
y ÿ! x1
k1 y1 ÿ! x2
k2 y2 ÿ! x we have p
!2 pg 1 b
2 2 c 6 p
i!2 pg 2 b
1 1 c; where pg pg
y ! x1
k1 y1 ! x2
k2 y2 ! x is the probability of the geometrical f
th
th
path and i!2
Rx ÿ! Ry2
k2 Rx2 ÿ! Ry1
k1 Rx1 ÿ! Ry. Consider next the set of 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.
R. Sanchez, S. Santandrea / Annals of Nuclear Energy 28 (2001) 401±417
411
thermal group passing from all the intermediary groups 2
f ! th f!2 ; !~ 2 g, f f th where !~ 2
y ÿ! x1
k1 y1 ÿ! x2
k2 y2 ÿ! x. Then p 2
f ! th pg 1 b
2 2 c 1 2 b pi 2
f ! th pg 2 b
1 1 c 2 1 b: Nevertheless, this property is so hard to believe that we have independently checked the pseudoreciprocity of the Green's function for this special case of two groups and two isotopes with an ad-hoc computer program based on the use of analytical 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 a one-group albedo with kernel independent of the energy. However, by using the Neumann expansion it is possible to extend our results to the case of a factorized albedo with kernel: X x;k
x0 ! x E;k
E0 ! E
28
x0 ; E0 ! x; E k
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 write the transport equation as:
Lx ÿ HGy;Es b Gy;Es y Es ; Gy;Es 0;
x; E 2 X 0; 1 ;
x; E 2 @ X 0; 1
where we have maintained assumption (16). In this equation b is a delta function that reduces integration in X to integration on @ X :
b
xf
xdx f
xdb x: X
@ X
Then the multiple scattering expansion for the Green's function becomes; n iÿ ÿ X
n ÿ Xh ~ Ty Es x; Ef ; G y; Es ! x; Ef TH G y; Es ! x; Ef n0
n0
29
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R. Sanchez, S. Santandrea / Annals of Nuclear Energy 28 (2001) 401±417
~ accounts for collisions in X as well as for ``collisions'' on the boundary where H ~ f
H b f, and T Lÿ1 accounts for a non re-entering boundary condi@ X; H
~ 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 the commutation relations (24) to the albedo operators HE;k E;l E;l HE;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 for the case of a factorized albedo. The dierence now is that each time that the particle reaches 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. Diusion Our ®rst approach to prove pseudoreciprocity relations for the energy-dependent diusion equation will parallel the multiple-collision argument used for the transport case. First, we will set up a one-group diusion equation with appropriate boundary 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.
R. Sanchez, S. Santandrea / Annals of Nuclear Energy 28 (2001) 401±417
413
With the help of this reciprocity relation we will establish a ®rst pseudoreciprocity ÿ result for the energy-dependent Green's function G y; Es ! x; Ef under similar assumptions to those introduced in the previous section. A dierent approach, based on the use of a complete set of eigenfunctions for the Laplacian, is applied in the last subsection to establish pseudoreciprocity under dierent assumptions. 3.1. The one-group problem We consider the one-group diusion equation in a non multiplicative medium
L ÿ HGy y ; x 2 X : C Gy C Gy ; x 2 @X where X is the geometrical domain D; @X its boundary and L ÿrDr is the diusion operator. Green's function G
y ! x is the scalar ¯ux at x resulting from a unit singular source at y. As before, we adopt a general albedo operator of the form
y ! xf
ydSy ; x 2 @X:
f
x @X
The boundary operators C are taken to be of the form C a bnr, where a and 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 ÿ1 Daÿ1
1 ÿ
1 b is self-adjoint with respect to the scalar product hf; gi @X dS
fg
x. In order to ensure that A is self-adjoint we will assume that is self-adjoint and that functions a and b satisfy the relation Daÿ1 cb, where c is an arbitrary constant. We note that this general boundary condition comprises the familiar boundary albedo condition, Jin
x
Jout
x, where Jin and Jout are the entering and exiting currents, respectively, as well as the usual conditions of zero ¯ux or zero gradient at the boundary, 0 and nr 0. The current albedo condition is obtained for a 1=4; b D=2 and c 8, while for the other two conditions we set 0 and put b 0 or a 0 for the zero ¯ux and zero gradient conditions, 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 proves that (4) holds also for the Green's function of the one-group diusion equation. 3.2. Neumann series approach In order to prove a ®rst pseudoreciprocity relation for the energy-dependent Green's function we assume that the diusion coecient and the total cross-sections share the same energy dependence and write L w
ELx w
E
ÿrDx r x ;
414
R. Sanchez, S. Santandrea / Annals of Nuclear Energy 28 (2001) 401±417
where the lower index x indicates that the functions depend only on the spatial variable. We also adopt assumption (17) for the scattering operator. Next, we write G^
y; Es ! x; E w
EG
y; Es ! x; E so that the diusion equation for the Green's function can be written as ) ^ G^ y;Es y Ens ;
x; E 2 X 0; 1
Lx ÿ H ;
31 C G^ y;Es C G^ y;Es ;
x; E 2 @X 0; 1 ^ E with H ^ E a scattering operator acting on the energy variable with ^ Hx H where H 0 ^ kernel hE
E ! E hE
E 0 ! E =w
E 0 . The usual Neumann expansion for this equation yields an expression like (19) ^ and where T, the integral operator inverse of Lx , accounts where now G ! G^ ; H ! H ÿ for the one-group albedo condition in (31). This proves that G^
n y; Es ! x; Ef and, ÿ ÿ therefore, G
n y; Es ! x; Ef and G y; Es ! x; Ef satisfy pseudoreciprocity: ÿ ÿ G
n y; Es ! x; Ef G
n x; Es ! y; Ef ; ÿ ÿ G y; Es ! x; Ef G x; Es ! y; Ef : Clearly, this demonstration can be generalized to assumption (23) with the commutation constraints in (24). 3.3. Eigenfunction approach Consider the one-group eigenvalue problem: ÿrDx r n ln n ; x Cÿ n C n ;
x2X x 2 @X
) :
Given sucient regularity conditions for Dx and x this equation has a countable set of eignfunctions f n g with positive eignvalues. Moreover, the f n g are orthogonal for the scalar product:
32
f; g
fg
xx
xdx; X
and form a complete set of functions on the Sobolev space f
r of functions in D such that
Dx
rf 2 x f 2 dr < 1: D
For the one-dimensional case this result is directly related to the classical SturmLiouville problem and has been abundantly discussed in the technical literature
R. Sanchez, S. Santandrea / Annals of Nuclear Energy 28 (2001) 401±417
415
(Iyanaga and Kawada, 1980). Here we will assume without further ado
that
it is also valid for the multidimensional case with a non multiplicative albedo, < 1. In the following 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 Dx DE ; x E ;
33
where the lower index variable indicates that the function only depends on that variable. The resulting diusion equation for the energy-dependent Green's function reads:
ÿrDx DE r x E ÿ Hx HE Gy ;Es y Es ;
x; E 2 X 0; 1 :
34
x; E 2 @X 0; 1 C Gy;Es C Gy;Es ; Under these assumptions, we look for a solution of the form: ÿ X ÿ an y; Es ! Ef n
x n
y: G y; Es ! x; Ef
35
n
Pseudoreciprocity for the Green's function will follow if the expansion coecients these coecients we replace an are independent of the variable y. To compute ÿ expansion (35) into the diusion equation for G y; Es ! x; Ef , multiply the resulting equation with n
x and integrate over x. By using the orthonormality of the n and by assuming that Hx n cn x n :
36
where cn is a constant, we obtain the following equation for an :
ln DE E ÿ cn HE an Es :
Because
the ln are positive this equation has a solution under the constraint
ÿ1 cn HE < 1. The equation also shows that an is independent of y and that, E ÿ therefore, G y; Es ! x; Ef satis®es pseudoreciprocity. For localized collisions the operator Hx reduces to a multiplication by the diusion cross section sx so that (36) is equivalent to take sx =x constant. Diusion Eq. (34) comprises as a particular case the diusion 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 relations derived by Modak and Sahni for the multigroup diusion equation and for the
416
R. Sanchez, S. Santandrea / Annals of Nuclear Energy 28 (2001) 401±417
energy-dependent transport equation. Pseudoreciprocity relations can be established under severe constraints on the functional dependence of the cross sections that amount to the factorization of the dierent operator kernels as products of functions of the space variable (and the angular variable for the transport case) times functions of the energy variable. This factorization does not result on an equivalent factorization for the energy-dependent Green's function but, rather, on the factorization of the multiple-scattering components of this function. It has also been shown that these new reciprocity relations relay on probability-preserving transformations of the set of trajectories similar to the mechanical involution responsible for the familiar reciprocity relations, with the dierence that for pseudoreciprocity there is no inversion of the energies between the emission and the test locations. From this point of view pseudoreciprocity and reciprocity relations are alike in that they are generated by probability-preserving transformations in trajectory space. Pseudoreciprocity relations can be used for formal veri®cations of diusion and transport numerical methods. But, our opinion is that their main interest resides in that they reveal new facts about the intimate structure of the transport and diusion operators 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 the results proved here can be generalized to other factorizations of the operators that appear in the energy-dependent transport and diusion equations. We hope that the approach 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 fruitful discussions. Appendix Under limited constraints the eigenvalue problem for the energy-dependent diusion or transport equations can be written as an equation involving an integral operator. We prove here that pseudoreciprocity can be used to demonstrate that this integral 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 ®ssion spectrum . We also assume that the production cross section factorizes as vf
r; E f
rvf
r; where f
r is a positive function. Then the eigenvalue problem for the transport equation reads:
R. Sanchez, S. Santandrea / Annals of Nuclear Energy 28 (2001) 401±417
1
L ÿ H P ; l ;
417
)
x2X ; x2@ X
A1
where P is the production operator
p 1
E vf
r; E 0 dE 0 d 0
E f
rt
r:
P
r; E 4 hp i To obtain an eigenvalue equation for t
r f
r=
4 vf
E 0 dE 0 d 0 we compute from Eq. (A1) by inversion of the transport operator L ÿ H. By realizing that the inverse of this operator is an integral operator with kernel the Green's function we write
ÿ 1 pÿ ÿ ÿ f ry t ry dy
EG y; Es ! x; Ef dEs : x; Ef l X Hence 1 t
r l
D
k
r0 ! rt
r0 dr0
A2
with the kernel p
f
rf
r0 vf
EdE
E 0 dE 0 d G
r0 ; 0 ; E 0 ! r; ; Ed 0 : k
r ! r 4 0
Therefore, if the cross-sections are such that pseudoreciprocity (8) applies, then the kernel of (A2) is symmetric, k
r0 ! r k
r ! r0 , and we can conclude that all the eigenvalues of (A1) are real. References Bell, G.I., Glasstone, S., 1970. Nuclear Reactor Theory. Van Nostrand. Case, K.M., de Homann, F., Placzek, G., 1953. Introduction to the Theory of Neutron Diusion. 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. MeÂcanique. Editions MIR. Modak, R.S., Sahni, D.C., 1996. Some reciprocity-like relations in multi-group neutron diusion 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), 445±478.
