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Time-dependent pseudo-reciprocity relations in neutronics
Annals of Nuclear Energy 29 (2002) 67–74 www.elsevier.com/locate/anucene
Time-dependent pseudo-reciprocity relations in neutronics R.S. Modak *, D.C. Sahni Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai-400 085, India Received 22 August 2000; accepted 13 February 2001
Abstract Earlier, certain reciprocity-like relations have been shown to hold in some restricted steady state cases in neutron diffusion and transport theories. Here, the possibility of existence of similar relations in time-dependent situations is investigated. # 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The existence of reciprocity relations in the linear neutron transport theory and in its approximate form namely the diffusion theory is well-known (Bell and Glasstone, 1970). In general, these relations are concerned with flux distributions obtained from different source distributions. The reciprocity relations for a steady state problem in a non-multiplicative medium can be written as: transport theory: Gðr; ; E; r0 ; 0 ; E0 Þ ¼ Gþ ðr0 ; 0 ; E0 ; r; ; EÞ
ð1Þ
diffusion theory: Gðr; E; r0 ; E0 Þ ¼ Gþ ðr0 ; E0 ; r; EÞ
ð2Þ
The Green’s function Gðr; ; E; r0 ; 0 ; E0 Þ in Eq. (1) denotes the angular neutron flux with energy E in direction at position vector r due to an external source of * Corresponding author. E-mail address: [email protected] (R.S. Modak). 0306-4549/02/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-4549(01)00028-7
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unit strength located at r0 which emits neutrons with energy E0 in direction 0 . The right-hand side (RHS) of Eq. (1) denotes a similar quantity for the adjoint problem where the source and field parameters are interchanged. Similarly, the G in Eq. (2) refers to total flux resulting from isotropic point sources of unit strength. The symbol ( ; ) separates the field and source variables. The function G obeys appropriate boundary conditions on the surface of the region. The solution to adjoint problem Gþ obeys the adjoint boundary conditions. It has been shown (Modak and Sahni, 1996, 1997) that, in the energy-dependent case, under some restrictive conditions on the behaviour of cross-sections, certain so-called ‘‘reciprocity-like relations’’ are obeyed which are different from the above reciprocity relations. These reciprocity-like relations do not involve exchange of energy at the source and field points and the adjoint problem need not be invoked in these relations. Recently, these relations have been re-examined and further generalised by Sanchez and Santandrea (2000) who have called them ‘‘pseudo-reciprocity relations’’. This term will be used hereafter in the present note also. These pseudo-reciprocity relations have been studied in the steady state only and they can be written as: transport theory: Gðr; E; ; r0 ; E0 ; 0 Þ ¼ Gðr0 ; E; 0 ; r; E0 ; Þ
ð3Þ
diffusion theory: Gðr; E; r0 ; E0 Þ ¼ Gðr0 ; E; r; E0 Þ
ð4Þ
In the present note, the possibility of existence of similar pseudo-reciprocity relations in the time-dependent case is investigated. It is well-known (Bell and Glasstone, 1970) that, in the usual time-dependent reciprocity relation, the time as well as energy variables at the source and field point are interchanged and the relation involves adjoint function also. Here, we look for pseudo-reciprocity relations in which neither time nor energy variables at the source and field point are interchanged and the adjoint problem is also not invoked. Thus, we seek to obtain following time-dependent pseudo-reciprocity relations where t > t0: transport theory: Gðr; E; ; t; r0 ; E0 ; 0 ; t0 Þ ¼ G r0 ; E; 0; t; r; E0 ; ; t0 ð5Þ diffusion theory: Gðr; E; t; r0 ; E0 ; t0 Þ ¼ Gðr0 ; E; t; r; E0 ; t0 Þ
ð6Þ
It may be mentioned that, earlier certain reciprocity relations have been identified in the energy-dependent case (Kuscer, 1962, 1967; Kuscer and McCormik, 1966) which are applicable to thermal neutrons at a uniform temperature. These relations
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do not involve adjoint problem and in this sense they are similar to the pseudoreciprocity relations. However, these relations by Kuscer do involve an exchange of energy values at the source and field point and in this respect they are different from the pseudo-reciprocity relations discussed here. The plan of this paper is as follows. In Section 2, the possibility of existence of time-dependent pseudo-reciprocity relations given by Eq. (5) in the transport case is examined. It is shown that the arguments used to obtain pseudo-reciprocity in the steady state transport case cannot be used in the time-dependent case. Hence, timedependent pseudo-reciprocity between directional source and angular flux seems improbable. In Section 3, a simplified case in transport theory is considered. This case is ‘‘close’’ to diffusion theory in some sense. The problem domain is considered to be an infinite homogeneous medium. Then there are no internal or external boundaries where diffusion and transport theory treatments differ. Moreover, the external source of neutrons needed to develop the reciprocity arguments is taken to be isotropic. It is shown that a time-dependent pseudo-reciprocity relation exists between isotropic source and angular flux (and, hence, total flux also) in this case. This suggests that the time-dependent pseudo-reciprocity may hold in some cases in diffusion theory also. In Section 4, the diffusion theory case is considered. It is shown that the timedependent pseudo-reciprocity relations can hold for a spatially uniform medium with no further conditions on cross-sections provided all the group fluxes vanish at the same extrapolated boundary. Section 5 gives the conclusions. At the outset, the nature of time-dependence considered in all the cases in this paper may be explained. We consider cases in which the material cross-sections as well as region configurations do not change with time. Thus, the problem is homogeneous in time. We consider an external point source which emits a pulse of neutrons only at a certain time instant (say, t0) with certain energy. The resultant energy-dependent neutron flux at various locations is a function of time t (>t0). This flux, denoted by Green’s function G will be used to develop reciprocity arguments. As a consequence of time homogeneity, G would depend only on the difference (t–t0).
2. Transport theory case While looking for pseudo-reciprocity in time-dependent transport theory, it is instructive to look at the arguments which led to pseudo-reciprocity (Modak and Sahni, 1997) in the steady state transport case expressed in the form of Eq. (3). It was argued that the left-hand side (LHS) of Eq. (3) is made up of contributions from several possible neutron trajectories from source point ðr0 ; E0 ; 0 Þ to the field point ðr; E; Þ. Corresponding to each such trajectory, it is possible to think of a reverse trajectory from source point ðr; E0 ; Þ to field point ðr0 ; E; 0 Þ which has the same probability provided some restrictions are placed on cross-sections. The reverse trajectory contributes to the RHS of Eq. (3). As a result, the LHS and RHS of Eq. (3) were shown to be equal. These restrictions are:
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1. There exists a finite non-multiplicative medium with boundary conditions of zero incoming angular flux; 2. The macroscopic total cross-section ðrÞ does not vary with energy; 3. The differential scattering cross-section is separable in angular and energy part so that s ðr; 0 ! ; E0 ! EÞ ¼ s ðr; E0 ! EÞ:fð0 ! Þ; 4. It is assumed that fð0 ! Þ ¼ fð ! 0 Þ: Sanchez and Santandrea (2000) have shown the relations to be valid for more general case of a mixture of commuting separable kernels under many other boundary conditions and even for non-local collisions. It should be noted that, in general, a particle travels different distance with given energy in the direct and reverse trajectories. Thus, the time spent in the flight is different for the direct and reverse trajectory. Hence, the particles in direct and reverse trajectory arrive at the respective field points at different instants of time even if their time of emission is same. Hence, one does not obtain time-dependent pseudo-reciprocity given by Eq. (5). This is true even if one relaxes some of the conditions stated earlier or even for an infinite homogeneous medium. As noted above, the only reason for non-existence of time dependent pseudoreciprocity relations is that the time of arrival of the particles in direct and reverse trajectories are different. This will not matter, however, if we were considering the asymptotic flux distribution (as t tends to infinity) resulting from a source that is constant in time. This is consistent with the existence of time-independent pseudoreciprocity relations.
3. A simplified case in transport theory It will be shown in Section 4 that for the diffusion equation, time-dependent pseudo-reciprocity relations do hold. Since the diffusion and transport equations are closely related, we wish to see if there is a more restricted form of time-dependent pseudo-reciprocity for the transport theory. With this in mind, we consider an infinite homogeneous medium. We assume that the probability of scattering from 0 to depends only on cosine of the angle between the two directions. We also relax the conditions (2) and (3) mentioned in the earlier section. The external source is always assumed to be isotropic. Suppose an isotropic source of unit strength placed at r0 emits neutrons of energy E0 at the time instant t=t0 . Let Gðr; ; E; t; r0 ; E0 ; t0 Þ denote the resultant angular flux at r in direction with energy E at a later time t. The problem has spherical symmetry if the origin is taken at the source point and as a result of this there is only one spatial variable namely jrr0 j and one directional
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variable namely cosine of the angle between and the radius vector (Carlson and Lathrop, 1968). Hence, one can write: r r0 ; t t0 ; E; E0 Gðr; ; E; t; r0 ; E0 ; t0 Þ ¼ G jr r0 j; ð7Þ jr r0 j If we interchange r and r0 , the RHS of the above equation is unchanged if the sign of is also changed. Hence, making the same changes must leave the LHS also unchanged. Thus, we get: Gðr; ; E; t; r0 ; E0 ; t0 Þ ¼ Gðr0 ; ; E; t; r; E0 ; t0 Þ
ð8Þ
This is time-dependent pseudo-reciprocity between isotropic source and angular flux. By integrating the above equation over , one can get: Gðr; E; t; r0 ; E0 ; t0 Þ ¼ Gðr0 ; E; t; r; E0 ; t0 Þ
ð9Þ
The above equation is a statement of time-dependent pseudo-reciprocity between isotropic source and total flux. Since the solutions of the diffusion equation approximate transport equation solutions in optically thick, homogeneous, nearly isotropically scattering regions, it is not therefore surprising that time-dependent pseudo-reciprocity can hold for the diffusion equation. In fact, our considerations in this section apply even when the scattering is highly anisotropic.
4. The diffusion theory case The diffusion theoretical results correspond to the average neutron distribution resulting only from those neutrons which have suffered a very large number of collisions. Moreover, in diffusion theory, one considers isotropic source and total flux of neutrons rather than directional sources and angular fluxes. Hence, from the validity of time-dependent pseudo-reciprocity between isotropic sources and total fluxes given by Eq. (9), it was felt that time-dependent pseudo-reciprocity relations may exist in diffusion theory also. Here, it will be shown that the pseudo-reciprocity relations given by Eq. (6) are obeyed in diffusion case under following conditions: 1. there exists a single homogeneous material in the region of interest so that the various cross-sections are not a function of position. The cross-sections and diffusion coefficient can vary with energy; 2. the flux vanishes at the same extrapolated boundary for any energy. Consider a finite homogeneous medium in which the above two conditions are obeyed. Let DðEÞ; R ðEÞ and S ðE0 ! EÞ denote the energy dependent diffusion coefficient, macroscopic removal cross-section and scattering cross-section from E0 to E. In order to develop reciprocity arguments, we consider following two distinct cases:
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Case A: There is a unit isotopic source of neutron at r0 which emits neutrons of energy E0 at the time instant t0 . Case B: There is a unit isotopic source of neutron at r which emits neutrons of energy E0 at the time instant t0 . Let Gðr; E; t; r0 ; E0 ; t0 Þ denote the total flux at r with energy E at time t > t0 arising from source described in Case A. It obeys the following time-dependent diffusion equation: ð 1 @G 2 ¼ SR ðEÞ:G þ DðEÞ:r G þ SS ðE0 ! EÞGðr; E0 ; t; r0 ; E0 ; t0 ÞdE0 v @t þ ðr r0 ÞðE E0 Þðt t0 Þ
ð10Þ
For brevity, the arguments of G are suppressed in the above equation. Let us consider the eigenfunctions of the Helmholtz equation over the region under consideration r2 n ðrÞ þ B2n n ðrÞ ¼ 0
ð11Þ
under the condition that n vanishes at the extrapolated boundary of the region where the diffusion theory fluxes vanish. It is well-known that there exist an infinite number of orthonormal eigenfunctions n ðrÞ labelled by n=1, 2, 3, . . . and they form a complete set. The approach followed here is based on expansion of Green’s function in terms of n ðrÞ and is essentially the same as used earlier by Sanchez and Santandrea (2000) and in the unpublished thesis by Modak (1997). Hence, G in Eq. (10) can be written as: X Gðr; E; t; r0 ; E0 ; t0 Þ ¼ an ðE; t; r0 ; E0 ; t0 Þn ðrÞ ð12Þ n
Let us substitute Eq. (12) in Eq. (10), multiply both sides by m ðrÞ and integrate over r. Using the orthonormality of n ðrÞ, we get: @am ðE; t; r0 ; E0 ; t0 Þ ¼ v R ðEÞ þ DðEÞB2m am ðE; t; r0 ; E0 ; t0 Þ @t ð
ð13Þ
þ v am ðE0 ; t; r0 ; E0 ; t0 Þ S ðE0 ! EÞdE0 þ vðE E0 Þðt t0 Þm ðr0 Þ
It is seen from the above equation that the spatial dependence of the coefficient am ðE; t; r0 ; E0 ; t0 Þ comes only from the function m ðr0 Þ occurring as a multiplicative factor in the source term on RHS. Hence, it is clear that am ðE; t; r0 ; E0 ; t0 Þ must have m ðr0 Þ as a factor. Hence, one can write: an ðE; t; r0 ; E0 ; t0 Þ ¼ bn ðE; t; E0 ; t0 Þ:n ðr0 Þ
ð14Þ
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From Eqs. (12) and (14), one gets: Gðr; E; t; r0 ; E0 ; t0 Þ ¼
X bn ðE; t; E0 ; t0 Þn ðr0 Þn ðrÞ
ð15Þ
n
It is obvious that, if rand r0 are interchanged, the RHS of the above equation is unchanged. Hence, the LHS is also unchanged for this interchange. This completes the proof of the time dependent pseudo-reciprocity relation given by Eq. (6). It may be mentioned that the zero flux boundary condition obeyed by solutions of Helmholtz equation [Eq. (11)] can be replaced by more general boundary condition such as a mixed boundary condition where a linear combination of n ðrÞ and its normal derivative vanishes at the physical boundary. The eigensolutions of Helmholtz equation still form a complete orthonormal set. Hence, even if the neutron flux obeys these different types of boundary conditions, the time-dependent pseudoreciprocity relations can be shown to hold exactly as above provided the boundary condition is identical for all the energies.
5. Conclusions The pseudo-reciprocity relations represent general symmetry properties of the solutions of energy-dependent neutronics equations, although in a restricted class of problems. Hence, it is of interest to identify as wide a range of situations as possible in which they hold. Earlier, pseudo-reciprocity between directional sources and angular fluxes had been shown to hold in steady state problems in linear neutron transport theory under certain conditions. Here, an investigation has been made to derive similar pseudo-reciprocity relations for time-dependent cases. The arguments based on neutron trajectories which were used in steady state case are found to be not valid in time-dependent case. This is because the times to cover the so-called direct and reverse trajectory by a neutron are different. As a result, time-dependent pseudoreciprocity between directional source and angular flux seems improbable. The timedependent pseudo-reciprocity, however, could be shown in transport theory for cases involving isotropic sources and infinite homogeneous medium even if the constraints on total and differential cross-section are removed. These are actually the situations under which diffusion theory better approximates transport theory results, thus suggesting that the time-dependent pseudo-reciprocity may hold in diffusion theory. This has indeed been shown to be true for a finite homogeneous medium in which the boundary conditions on the flux are energy-independent.
Acknowledgements It is a pleasure to thank Dr. R. Sanchez, CEN, Saclay, France for many thoughtprovoking and illuminating suggestions.
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References Bell, G.I., Glasstone, S., 1970. Nuclear reactor theory. Van Nostrand Reinhold, New York. Modak, R.S., Sahni, D.C., 1996. Ann. Nucl. Energy 23, 955. Modak, R.S., Sahni, D.C., 1997. Ann. Nucl. Energy 24, 1271. Sanchez, R., Santandrea, S., 2000. Submitted to Ann. Nucl. Energy. Kuscer, I., 1962. J. Nucl. Energy A/B 16, 379. Kuscer, I., 1967. Kernenergie 10, 265. Kuscer, I., McCormick, N.J., 1966. Nucl. Sci. Eng. 26, 522. Carlson B.G., Lathrop K.D., (1968) In Greenspan H, Keller CN, Okrent D, editors. Computing methods in reactor physics, Gordon & Breach, New York (Chapter 3). Modak. R.S., 1997. Studies on higher mode solutions of K-eigenvalue problem in neutron diffusion and transport theories. PhD thesis, University of Mumbai, India.
Time-dependent pseudo-reciprocity relations in neutronics R.S. Modak *, D.C. Sahni Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai-400 085, India Received 22 August 2000; accepted 13 February 2001
Abstract Earlier, certain reciprocity-like relations have been shown to hold in some restricted steady state cases in neutron diffusion and transport theories. Here, the possibility of existence of similar relations in time-dependent situations is investigated. # 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The existence of reciprocity relations in the linear neutron transport theory and in its approximate form namely the diffusion theory is well-known (Bell and Glasstone, 1970). In general, these relations are concerned with flux distributions obtained from different source distributions. The reciprocity relations for a steady state problem in a non-multiplicative medium can be written as: transport theory: Gðr; ; E; r0 ; 0 ; E0 Þ ¼ Gþ ðr0 ; 0 ; E0 ; r; ; EÞ
ð1Þ
diffusion theory: Gðr; E; r0 ; E0 Þ ¼ Gþ ðr0 ; E0 ; r; EÞ
ð2Þ
The Green’s function Gðr; ; E; r0 ; 0 ; E0 Þ in Eq. (1) denotes the angular neutron flux with energy E in direction at position vector r due to an external source of * Corresponding author. E-mail address: [email protected] (R.S. Modak). 0306-4549/02/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-4549(01)00028-7
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unit strength located at r0 which emits neutrons with energy E0 in direction 0 . The right-hand side (RHS) of Eq. (1) denotes a similar quantity for the adjoint problem where the source and field parameters are interchanged. Similarly, the G in Eq. (2) refers to total flux resulting from isotropic point sources of unit strength. The symbol ( ; ) separates the field and source variables. The function G obeys appropriate boundary conditions on the surface of the region. The solution to adjoint problem Gþ obeys the adjoint boundary conditions. It has been shown (Modak and Sahni, 1996, 1997) that, in the energy-dependent case, under some restrictive conditions on the behaviour of cross-sections, certain so-called ‘‘reciprocity-like relations’’ are obeyed which are different from the above reciprocity relations. These reciprocity-like relations do not involve exchange of energy at the source and field points and the adjoint problem need not be invoked in these relations. Recently, these relations have been re-examined and further generalised by Sanchez and Santandrea (2000) who have called them ‘‘pseudo-reciprocity relations’’. This term will be used hereafter in the present note also. These pseudo-reciprocity relations have been studied in the steady state only and they can be written as: transport theory: Gðr; E; ; r0 ; E0 ; 0 Þ ¼ Gðr0 ; E; 0 ; r; E0 ; Þ
ð3Þ
diffusion theory: Gðr; E; r0 ; E0 Þ ¼ Gðr0 ; E; r; E0 Þ
ð4Þ
In the present note, the possibility of existence of similar pseudo-reciprocity relations in the time-dependent case is investigated. It is well-known (Bell and Glasstone, 1970) that, in the usual time-dependent reciprocity relation, the time as well as energy variables at the source and field point are interchanged and the relation involves adjoint function also. Here, we look for pseudo-reciprocity relations in which neither time nor energy variables at the source and field point are interchanged and the adjoint problem is also not invoked. Thus, we seek to obtain following time-dependent pseudo-reciprocity relations where t > t0: transport theory: Gðr; E; ; t; r0 ; E0 ; 0 ; t0 Þ ¼ G r0 ; E; 0; t; r; E0 ; ; t0 ð5Þ diffusion theory: Gðr; E; t; r0 ; E0 ; t0 Þ ¼ Gðr0 ; E; t; r; E0 ; t0 Þ
ð6Þ
It may be mentioned that, earlier certain reciprocity relations have been identified in the energy-dependent case (Kuscer, 1962, 1967; Kuscer and McCormik, 1966) which are applicable to thermal neutrons at a uniform temperature. These relations
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do not involve adjoint problem and in this sense they are similar to the pseudoreciprocity relations. However, these relations by Kuscer do involve an exchange of energy values at the source and field point and in this respect they are different from the pseudo-reciprocity relations discussed here. The plan of this paper is as follows. In Section 2, the possibility of existence of time-dependent pseudo-reciprocity relations given by Eq. (5) in the transport case is examined. It is shown that the arguments used to obtain pseudo-reciprocity in the steady state transport case cannot be used in the time-dependent case. Hence, timedependent pseudo-reciprocity between directional source and angular flux seems improbable. In Section 3, a simplified case in transport theory is considered. This case is ‘‘close’’ to diffusion theory in some sense. The problem domain is considered to be an infinite homogeneous medium. Then there are no internal or external boundaries where diffusion and transport theory treatments differ. Moreover, the external source of neutrons needed to develop the reciprocity arguments is taken to be isotropic. It is shown that a time-dependent pseudo-reciprocity relation exists between isotropic source and angular flux (and, hence, total flux also) in this case. This suggests that the time-dependent pseudo-reciprocity may hold in some cases in diffusion theory also. In Section 4, the diffusion theory case is considered. It is shown that the timedependent pseudo-reciprocity relations can hold for a spatially uniform medium with no further conditions on cross-sections provided all the group fluxes vanish at the same extrapolated boundary. Section 5 gives the conclusions. At the outset, the nature of time-dependence considered in all the cases in this paper may be explained. We consider cases in which the material cross-sections as well as region configurations do not change with time. Thus, the problem is homogeneous in time. We consider an external point source which emits a pulse of neutrons only at a certain time instant (say, t0) with certain energy. The resultant energy-dependent neutron flux at various locations is a function of time t (>t0). This flux, denoted by Green’s function G will be used to develop reciprocity arguments. As a consequence of time homogeneity, G would depend only on the difference (t–t0).
2. Transport theory case While looking for pseudo-reciprocity in time-dependent transport theory, it is instructive to look at the arguments which led to pseudo-reciprocity (Modak and Sahni, 1997) in the steady state transport case expressed in the form of Eq. (3). It was argued that the left-hand side (LHS) of Eq. (3) is made up of contributions from several possible neutron trajectories from source point ðr0 ; E0 ; 0 Þ to the field point ðr; E; Þ. Corresponding to each such trajectory, it is possible to think of a reverse trajectory from source point ðr; E0 ; Þ to field point ðr0 ; E; 0 Þ which has the same probability provided some restrictions are placed on cross-sections. The reverse trajectory contributes to the RHS of Eq. (3). As a result, the LHS and RHS of Eq. (3) were shown to be equal. These restrictions are:
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1. There exists a finite non-multiplicative medium with boundary conditions of zero incoming angular flux; 2. The macroscopic total cross-section ðrÞ does not vary with energy; 3. The differential scattering cross-section is separable in angular and energy part so that s ðr; 0 ! ; E0 ! EÞ ¼ s ðr; E0 ! EÞ:fð0 ! Þ; 4. It is assumed that fð0 ! Þ ¼ fð ! 0 Þ: Sanchez and Santandrea (2000) have shown the relations to be valid for more general case of a mixture of commuting separable kernels under many other boundary conditions and even for non-local collisions. It should be noted that, in general, a particle travels different distance with given energy in the direct and reverse trajectories. Thus, the time spent in the flight is different for the direct and reverse trajectory. Hence, the particles in direct and reverse trajectory arrive at the respective field points at different instants of time even if their time of emission is same. Hence, one does not obtain time-dependent pseudo-reciprocity given by Eq. (5). This is true even if one relaxes some of the conditions stated earlier or even for an infinite homogeneous medium. As noted above, the only reason for non-existence of time dependent pseudoreciprocity relations is that the time of arrival of the particles in direct and reverse trajectories are different. This will not matter, however, if we were considering the asymptotic flux distribution (as t tends to infinity) resulting from a source that is constant in time. This is consistent with the existence of time-independent pseudoreciprocity relations.
3. A simplified case in transport theory It will be shown in Section 4 that for the diffusion equation, time-dependent pseudo-reciprocity relations do hold. Since the diffusion and transport equations are closely related, we wish to see if there is a more restricted form of time-dependent pseudo-reciprocity for the transport theory. With this in mind, we consider an infinite homogeneous medium. We assume that the probability of scattering from 0 to depends only on cosine of the angle between the two directions. We also relax the conditions (2) and (3) mentioned in the earlier section. The external source is always assumed to be isotropic. Suppose an isotropic source of unit strength placed at r0 emits neutrons of energy E0 at the time instant t=t0 . Let Gðr; ; E; t; r0 ; E0 ; t0 Þ denote the resultant angular flux at r in direction with energy E at a later time t. The problem has spherical symmetry if the origin is taken at the source point and as a result of this there is only one spatial variable namely jrr0 j and one directional
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variable namely cosine of the angle between and the radius vector (Carlson and Lathrop, 1968). Hence, one can write: r r0 ; t t0 ; E; E0 Gðr; ; E; t; r0 ; E0 ; t0 Þ ¼ G jr r0 j; ð7Þ jr r0 j If we interchange r and r0 , the RHS of the above equation is unchanged if the sign of is also changed. Hence, making the same changes must leave the LHS also unchanged. Thus, we get: Gðr; ; E; t; r0 ; E0 ; t0 Þ ¼ Gðr0 ; ; E; t; r; E0 ; t0 Þ
ð8Þ
This is time-dependent pseudo-reciprocity between isotropic source and angular flux. By integrating the above equation over , one can get: Gðr; E; t; r0 ; E0 ; t0 Þ ¼ Gðr0 ; E; t; r; E0 ; t0 Þ
ð9Þ
The above equation is a statement of time-dependent pseudo-reciprocity between isotropic source and total flux. Since the solutions of the diffusion equation approximate transport equation solutions in optically thick, homogeneous, nearly isotropically scattering regions, it is not therefore surprising that time-dependent pseudo-reciprocity can hold for the diffusion equation. In fact, our considerations in this section apply even when the scattering is highly anisotropic.
4. The diffusion theory case The diffusion theoretical results correspond to the average neutron distribution resulting only from those neutrons which have suffered a very large number of collisions. Moreover, in diffusion theory, one considers isotropic source and total flux of neutrons rather than directional sources and angular fluxes. Hence, from the validity of time-dependent pseudo-reciprocity between isotropic sources and total fluxes given by Eq. (9), it was felt that time-dependent pseudo-reciprocity relations may exist in diffusion theory also. Here, it will be shown that the pseudo-reciprocity relations given by Eq. (6) are obeyed in diffusion case under following conditions: 1. there exists a single homogeneous material in the region of interest so that the various cross-sections are not a function of position. The cross-sections and diffusion coefficient can vary with energy; 2. the flux vanishes at the same extrapolated boundary for any energy. Consider a finite homogeneous medium in which the above two conditions are obeyed. Let DðEÞ; R ðEÞ and S ðE0 ! EÞ denote the energy dependent diffusion coefficient, macroscopic removal cross-section and scattering cross-section from E0 to E. In order to develop reciprocity arguments, we consider following two distinct cases:
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Case A: There is a unit isotopic source of neutron at r0 which emits neutrons of energy E0 at the time instant t0 . Case B: There is a unit isotopic source of neutron at r which emits neutrons of energy E0 at the time instant t0 . Let Gðr; E; t; r0 ; E0 ; t0 Þ denote the total flux at r with energy E at time t > t0 arising from source described in Case A. It obeys the following time-dependent diffusion equation: ð 1 @G 2 ¼ SR ðEÞ:G þ DðEÞ:r G þ SS ðE0 ! EÞGðr; E0 ; t; r0 ; E0 ; t0 ÞdE0 v @t þ ðr r0 ÞðE E0 Þðt t0 Þ
ð10Þ
For brevity, the arguments of G are suppressed in the above equation. Let us consider the eigenfunctions of the Helmholtz equation over the region under consideration r2 n ðrÞ þ B2n n ðrÞ ¼ 0
ð11Þ
under the condition that n vanishes at the extrapolated boundary of the region where the diffusion theory fluxes vanish. It is well-known that there exist an infinite number of orthonormal eigenfunctions n ðrÞ labelled by n=1, 2, 3, . . . and they form a complete set. The approach followed here is based on expansion of Green’s function in terms of n ðrÞ and is essentially the same as used earlier by Sanchez and Santandrea (2000) and in the unpublished thesis by Modak (1997). Hence, G in Eq. (10) can be written as: X Gðr; E; t; r0 ; E0 ; t0 Þ ¼ an ðE; t; r0 ; E0 ; t0 Þn ðrÞ ð12Þ n
Let us substitute Eq. (12) in Eq. (10), multiply both sides by m ðrÞ and integrate over r. Using the orthonormality of n ðrÞ, we get: @am ðE; t; r0 ; E0 ; t0 Þ ¼ v R ðEÞ þ DðEÞB2m am ðE; t; r0 ; E0 ; t0 Þ @t ð
ð13Þ
þ v am ðE0 ; t; r0 ; E0 ; t0 Þ S ðE0 ! EÞdE0 þ vðE E0 Þðt t0 Þm ðr0 Þ
It is seen from the above equation that the spatial dependence of the coefficient am ðE; t; r0 ; E0 ; t0 Þ comes only from the function m ðr0 Þ occurring as a multiplicative factor in the source term on RHS. Hence, it is clear that am ðE; t; r0 ; E0 ; t0 Þ must have m ðr0 Þ as a factor. Hence, one can write: an ðE; t; r0 ; E0 ; t0 Þ ¼ bn ðE; t; E0 ; t0 Þ:n ðr0 Þ
ð14Þ
R.S. Modak, D.C. Sahni / Annals of Nuclear Energy 29 (2002) 67–74
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From Eqs. (12) and (14), one gets: Gðr; E; t; r0 ; E0 ; t0 Þ ¼
X bn ðE; t; E0 ; t0 Þn ðr0 Þn ðrÞ
ð15Þ
n
It is obvious that, if rand r0 are interchanged, the RHS of the above equation is unchanged. Hence, the LHS is also unchanged for this interchange. This completes the proof of the time dependent pseudo-reciprocity relation given by Eq. (6). It may be mentioned that the zero flux boundary condition obeyed by solutions of Helmholtz equation [Eq. (11)] can be replaced by more general boundary condition such as a mixed boundary condition where a linear combination of n ðrÞ and its normal derivative vanishes at the physical boundary. The eigensolutions of Helmholtz equation still form a complete orthonormal set. Hence, even if the neutron flux obeys these different types of boundary conditions, the time-dependent pseudoreciprocity relations can be shown to hold exactly as above provided the boundary condition is identical for all the energies.
5. Conclusions The pseudo-reciprocity relations represent general symmetry properties of the solutions of energy-dependent neutronics equations, although in a restricted class of problems. Hence, it is of interest to identify as wide a range of situations as possible in which they hold. Earlier, pseudo-reciprocity between directional sources and angular fluxes had been shown to hold in steady state problems in linear neutron transport theory under certain conditions. Here, an investigation has been made to derive similar pseudo-reciprocity relations for time-dependent cases. The arguments based on neutron trajectories which were used in steady state case are found to be not valid in time-dependent case. This is because the times to cover the so-called direct and reverse trajectory by a neutron are different. As a result, time-dependent pseudoreciprocity between directional source and angular flux seems improbable. The timedependent pseudo-reciprocity, however, could be shown in transport theory for cases involving isotropic sources and infinite homogeneous medium even if the constraints on total and differential cross-section are removed. These are actually the situations under which diffusion theory better approximates transport theory results, thus suggesting that the time-dependent pseudo-reciprocity may hold in diffusion theory. This has indeed been shown to be true for a finite homogeneous medium in which the boundary conditions on the flux are energy-independent.
Acknowledgements It is a pleasure to thank Dr. R. Sanchez, CEN, Saclay, France for many thoughtprovoking and illuminating suggestions.
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