On the singular structure of the uncollided and first-collided components of the Green's function

On the singular structure of the uncollided and®rst-collided components of the Green's function

R. Sanchez*Commissariat aÁ l'Energie Atomique, Direction des ReÂacteurs NucleÂaires,

Services d'Etudes de ReÂacteurs et de MatheÂmatiques AppliqueÂes, CEA de Saclay,

91191 Gif-sur-Yvette cedex, France

Received 17 September 1999; accepted 15 October 1999

Abstract

Explicit expressions are given for the uncollided and ®rst-collided ¯uxes originated from asingular source in a ®nite heterogeneous medium with vacuum boundary conditions. The

analysis of the singular behavior of the ®rst-collided ¯ux shows a known discontinuity inangle, but also a discontinuous limit for angular directions parallel or antiparallel to that ofthe original source that can be important for strongly forward scattering. # 2000 ElsevierScience Ltd. All rights reserved.

1. Introduction

The numerical solution of the transport equation is often based on an analyticaldevelopment of the angular ¯ux in terms of well-behaved functions (Bell, 1970). Forproblems with singular sources, the correct application of such methods requires aseparate and explicit evaluation of the singular components of the angular ¯ux,previous to expansion over regular base functions (Siewert, 1985). Such is the case,in particular, for the so-called searchlight problem where a monodirectional beam ofparticles impinges in a medium (see Fig. 1 for noti®cation). For this problem notonly the uncollided ¯ux but also the ®rst-collided ¯ux are singular. Indeed, if theincident beam enters the domain at r0 with direction, X 0 then it follows that theuncollided ¯ux �0��r;X � consists of a ¯ow of particles that streams forward indirection X 0 along the forward trajectory tF�r0;X 0� � r � r0 � R0X 0;R050f g.Therefore this ¯ux is a distribution with ®nite support the subset X�0� � r;X 0�f ; r 2tF�r0;X 0�g of phase space X � x � �r;X�f ; r 2 1;X 2 �4��g. Furthermore, it is

Annals of Nuclear Energy 27 (2000) 1167±1186

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easily surmised that the ®rst-collided ¯ux �1��r;X� is also singular. To simplify theargument, I will consider a homogeneous medium with isotropic scattering and dis-regard the change of energy with collisions. Note ®rst that the ®rst-collided sourceconsists of particles that stream isotropically out of the forward trajectorytF�r0;X 0�. Therefore the locus of the ®rst-collided ¯ux is the subset X�1� �x � �r;X �; r � r0 � R0X 0�f RX ;R0;R50;X 2 �4��g. If r is a point outside the

forward trajectory tF�r0;X 0�, then �1��r;X � is zero except for those angular direc-tions such that the back trajectory tB�r;X� � r0 � rÿ RX ;R50f g intersectstF�r0;X 0�. Since this angular subdomain is of measured zero, one may safely con-clude that at a point outside the forward trajectory the ®rst-collided ¯ux has a sin-gular behavior in angle. However, it is clear that for points that lie on the forwardtrajectory �1��r;X� is non zero for all angular directions. Nevertheless, even alongthese particular locations the ¯ux shows an abnormal behavior in angle at directionsparallel and antiparallel to the incident direction. Indeed, while the ®rst-collided ¯uxoriginates from a single collision center at all the other points in phase space, on thesubset �x � �r;X �; r 2 tF�r0;X 0�;X � �X 0� the ®rst-collided ¯ux results frommultiple, in®nite collisions over the forward trajectory. This singularity is best illu-strated by considering a pair �r;X 0� and taking the limit as rmoves continuously ontothe forward trajectory. The ¯uxes along the di�erent spatial locations are all zero untilthe point reaches tF�r0;X 0� where the ¯ux has a non zero value.The explicit forms for the uncollided and ®rst-collided angular ¯uxes for the search-

light problem were published in a paper by Siewert (1985). These formulas wereobtained via a Fourier transform technique and a complete discussion of all thesingularities of the ®rst-collided ¯ux was not given. At the time, other people,including myself, were asked to provide separate expressions in order to check theresults. Again, in a very recent work (Sanchez andMcCormick, in preparation), I havebeen involved with the calculation of a second collided ¯ux originating from a singularsource and had to reconsider the entire matter again. The purpose of the present work is

Fig. 1. Geometrical notation for the searchlight problem. At point r1 the ®rst-collided ¯ux is zero except

for directions X for which the back trajectory tF�r0;X0� intersects the forward trajectory. The ®rst col-

lided ¯ux is never zero at a point, like r2, on the forward trajectory. The ®rst-collided ¯ux is produced by a

single collision locus except for points on the forward trajectory in directions X � �X0, where the ¯ux is

the sum of contributions from collisions over a continuous range.

1168 R. Sanchez / Annals of Nuclear Energy 27 (2000) 1167±1186

to give a full derivation and detailed analysis of the structure of the uncollided and ®rst-collided ¯uxes produced by a localized source in a ®nite, heterogeneous domain. Asecondary aim of this paper is pedagogical. Since some of the technical details on thehandling of distributions are di�cult to dig up from textbooks and most of us areunfamiliar with the subject, I give a detailed derivation of my results as well as discusssome of the pathologies that one may ®nd when working with very singular objects.Both, direct calculation and Fourier transform techniques can be invoked to

obtain explicit expressions for the uncollided and ®rst-collided ¯uxes. In myapproach I have favored the direct evaluation, but in Appendix D I give a ¯avor ofthe Fourier technique by showing some of the details of the derivation that I believewas behind the results given by Siewert (1985). The framework is set up in Section 2,where basic formulas are obtained from the familiar multiple collision ¯ux expansion.Since the di�culties associated with the mathematical treatment arise from the spatialsingularity of the source and not from its particular angular structure; I have decidedto consider the case of a localized source of the form

S�r;X � � �3�r�g�X �; �1�

where g�X � is an unspeci®ed function that may also be a delta function in angle.Expressions for the uncollided and ®rst-collided ¯uxes are derived in Section 3.

The problem evoked earlier for the behavior of �1� in directions �X 0 for thesearchlight case, appears also for the source in (1) for directions parallel or anti-parallel to the streaming of uncollided particles from the initial source at the originof coordinates. As shown in Section 3, this problem has physical implications onlywhen the scattering cross section has singularities for fully forward and/or backwardscattering. The results in Section 3 are specialized to the searchlight problem in thefollowing section, where expressions similar to those derived by Siewert (1985) areobtained. Finally, in Section 5, I illustrate the problems that can appear when usingdistributions in curvilinear coordinates. These type of problems have their origin inthe singularities of the coordinate mapping in these geometries.Appendix A contains a short compilation of basic expressions for the three-

dimensional Dirac's delta in di�erent system of coordinates. The derivation of thetransformations between geometrically related Dirac's deltas, which are used inSection 3, is given in Appendix B. A con®rmation of the well-founded of thesetransformations is given in Appendix C, where the analysis of the delta-scatteringcase is carried out. All the results are easily extended to include dependence onenergy. The reader will ®nd in Appendix E a sample of results for the energy-dependent case.

2. Multiple collision expansion

I consider the solution of the stationary, one-group linear transport equation

�X �r � �� H � S; inX � 0; on @ X

�

R. Sanchez / Annals of Nuclear Energy 27 (2000) 1167±1186 1169

by a multiple collision expansion (Duderstadt and Martin, 1979). In this equation��r� is the macroscopic cross section, and H is the collision operator

�H ��r;X � ��4��

dX 0�s�r;X �X 0� �r;X 0�;

with kernel the transfer cross-section �s�r;X �X 0�. The equation is de®ned on thephase space X � r 2 D;X 2 �4��f g and it has non reentrant boundary conditionson @ X � r 2 @D;X �n� < 0f g, where n��r� is the outward normal at the domainsurface @D.In a multiple collision expansion, the solution is sought under the form of a sum,

�r;X � �Xn50

�n��r;X �;

where the nth term corresponds to particles that have undergone exactly n scattersevents. At any step of the expansion, the new ¯ux is computed in terms of the scat-tering source produced by the previous ¯ux as

�n��r;X � ��D

dr0��4��

dX 0S�n��r0;X 0�Gunc�r0;X 0 ! r;X �; �2�

where Gunc is the medium Green's function for collisionless particles and

S�n��r;X � � S�r;X �; n � 0��4��dX

0�s�r;X 0 �X � nÿ1��r;X 0�; n > 0

��3�

is the collision source from the previous ¯ux. With non re-entrant boundary condi-tions the Green's function is given by the expression (Case and Zweiful, 1967):

Gunc�r0;X 0 ! r;X � � �2�X 0 �XR� eÿ��r0;r�

R2�2�X �XR�; �4�

where �2�X �XR� is Placzek's delta (Case et al., 1953) over the unit sphere, ��r0; r� isthe optical path between the two points, R � rÿ r0 and XR � R=R. To be able touse Eq. (4) for non convex domains we adopt the convention that the optical dis-tance between two points, r; r0 2 D, is in®nity whenever the straight path linking thepoints crosses the boundary of the domain. Replacing the expression for the Green'sfunction in Eq. (2) and using the �2�X 0 �XR� to eliminate the integration overX 0 gives:

�n��r;X � ��D

dr0S�n��r;X � eÿ��r0;r�

R2�2�X �XR�: �5�


This result can still be simpli®ed by using spherical coordinates with origin r andby introducing the change of variables r0 ! �R;XR�; with r0 � rÿ RXR one hasdr0 � R2dRdXR and, after having used the �2�X �XR� to get rid of the integrationover XR, one obtains:

�n��r;X � ��10

dRS�n��r0;X �eÿ��r0;r��D�r0�; �6�

where

r0 � rÿ RX ; �7�

and where �D�r� is the characteristic function of domain D.In the next section, these formulas will be applied to the evaluation of the uncollided

and ®rst-collided ¯uxes produced from the localized source in Eq. (1).

3. Uncollided and ®rst-collided ¯uxes

For simplicity I will assume that the source intersects the support of the geome-trical domain D, 0 2 D� , and that this domain is starred, i.e. if a point r belongs to Dthen all points in the segment from the origin to r are also in D.The uncollided ¯ux can be evaluated by using source (1) into Eq. (6). The result is

simpli®ed by expressing the delta function in spherical-coordinates (see Appendix A):

�0��r;X � � g�X ��10

dR�3�r0�eÿ��r0;r��D�r0�

� g�X ��10

dR��Rÿ r� �2�X �er�r2

eÿ��0;r� � g�X � �2�X �er�r2

eÿ��0;r�: �8�

Observe that one could have obtained this result directly by using (5) instead of (6).Eq. (6) can also be utilized to calculate the ®rst-collided ¯ux, but ®rst one needs to

determine the ®rst-collided source. Inserting (8) into (3) results in

S�1��r;X � � g�er��s�r;X �er� eÿ��0;r�

r2

and replacing this source in (6) yields

�1��r;X � ��10

dRg�er0 ��s�r0;X �er0 � eÿ��0;r0�ÿ��r0;r�

r02�D�r0�: �9�

In this expression r0 and er0 � r0=r0 are functions of R via Eq. (7). For a better cal-culation of the collision kernel, it is expedient to introduce the change of coordinates


R ! �0 � X�er0 , where one recognizes � � X �e as a local angular component inspherical coordinates.The relations that help to calculate the change of variables can be obtained from

the projection of Eq. (7) onto X and then onto the vector perpendicular to X

passing through the center of coordinates (see Fig. 2):

�0r0 � �rÿ R;��1ÿ �02

pr0 �

��1ÿ �2

pr;

(

from which one obtains

R � r��ÿ �0��1ÿ �2

1ÿ �02

s: �10�

Notice that the above change of variables is not possible when �02 � 1 or,equivalently, when �2 � 1. This situation arises for er0 � �X , for which (7) yields

er0 � �X ! er � sg�R� r0�Xr � R� r0j j ; er � �X ! er0 � sg��rÿ R�X

r0 � �rÿ Rj j :

��11�

where sg�x� denotes the sign of x. When the behavior of the di�erential cross-sec-tions is regular around X 0 �X � �1, the singularity does not have any incidence inthe angular ¯ux since it produces local discontinuities of zero weight. This is not thecase when the di�erential cross-section is unbounded at one of at both of thesepoints. In the following, I will consider a singular behavior of the form:

�s�r;X 0 �X � � �s;reg�r;X 0 �X � � �s;��r��2�X 0 �X � � �s;ÿ�r��2�ÿX 0 �X �: �12�

Fig. 2. Geometrical notation for change of variables R ! �0 � X �er0 in Eq. (9). The change is possible

except when the back trajectory passes through the origin of coordinates, the locus of the singular source.

In that case er0 remains constant over continuous parts of the back trajectory.


Here `reg' indicates a regular behavior around X 0 �X � �1, and the other twocomponents account for singular behavior at X 0 �X � �1. For the general collisionkernel in Eq. (12), the ®rst collision ¯ux can be decomposed into three terms corre-sponding to the three scattering components:

�1��r;X � � �1�reg�r;X � � �1�� r;X � � �1�ÿ �r;X �: �13�

Since the scattering cross-section �s;reg does not have localized singularities atX 0 �X � �1, the regular contribution to the ®rst-collided ¯ux can be obtained byintroducing the change of variable (10) in Eq. (9):

�1�reg�r;X � �1

r��1ÿ �2

p ��ÿ1

d�0��1ÿ �02

p g�er0 ��s;reg�r0; �0�eÿ��0;r0�ÿ��r0;r��D�r0�: �14�

Further, the apparent singularity in the integrand can be eliminated with thechange of variables �0 � cos �0 to obtain

�1�reg�r;X � �1

r��1ÿ �2

p ��

d�0g�er0 ��s;reg�r0; cos �0�eÿ��0;r0�ÿ��r0;r��D�r0�; �15�

where

r0 � r�er ÿ sin��0 ÿ ��sin �0

X �; r0 � rsin �

sin �0:

Note that for a homogeneous material we have

��0; r0� � ��r0; r� � ��r0 � R� � �r cos � � sin � tan �0=2� �: �16�

Next, I consider the singular contributions to the ®rst-collided ¯ux (see Fig. 3).These are given by the expression

�1�� r;X � � g��X ��10

dR�s;��r0�eÿ��0;r0�ÿ��r0;r� �2��X �er0 �

r02�D�r0�:

These ¯uxes can be easily calculated using transformations (A12) and (A13). Aftersome algebra one obtains

�1�� r;X � � g�X � eÿ��0;r�

r2�2�X �er�

�r0

dr0�s;��r0er� �17a�

and �1�ÿ �r;X � � g�ÿX � e

ÿ��0;r�

r2��2�ÿX �er�

�rmax�er�

r

dr0�s;ÿ�r0er�eÿ2��r0er;rer�

� �2�X�er��rmax�ÿer�

0

dr0�s;ÿ�ÿr0er�eÿ2��0;ÿr0e��:�17b�


In these expressions rmax�X� is the distance from the origin to the boundary of thedomain in direction X. In the case of a homogeneous material, one can carry out theintegrals and the above formulas become

�1�� r;X � � �s;�g�X �eÿ��0;r�

r�2�X �er�

and

�1�ÿ �r;X � � �s;ÿ�ÿX �eÿ��0;r�

2�r2�2�ÿX �er��1ÿ eÿ2��rmax�er�ÿr��

��2�X �er��1ÿ eÿ2�rmax�ÿer��:

4. The searchlight problem

I now consider the calculation of the ®rst-collide ¯ux for the case of a beam ofparticles impinging on a ®nite homogeneous slab in 04z4a with isotropic scatter-ing. Now the source is also singular in angle so that g�X� � �2�X�X 0�. For theuncollided ¯ux, Eq. (8) gives

�0��r;X � � �2�X �X 0� �2�X �er�r2

eÿ�r:

In cylindrical coordinates with origin the point at which the beam enters the slabone has, according to the formulas in Appendix A,

Fig. 3. Geometrical notation for the singular case with the angular direction parallel or antiparallel to

unit vector er. In the parallel case there are two types of continuous contributions, those with R 2 �0; r�that come from forward (+) scattering, �coll � 1, and those with R > r that arise from backward (ÿ)scattering, �coll � ÿ1. In the antiparallel case all contributions are from backward scattering.


�0��r;X � � ��ÿ �0��ÿ�0�H�z��ÿ z

�

��1ÿ �2

p��ÿ '�

�0�eÿ�r; �18�

where H is Heaviside's step function, X � ��;��, r � ��; '; z� and the fact that�0 > 0 has been taken into account. Note that the �0 in the denominator is missingin Siewert (1985).The ®rst collided ¯ux can be obtained writing formula (15) in cylindrical coordi-

nates. However, it is much expedient to start afresh from Eq. (9), which for thepresent case reads

�1��r;X � � �s4�

�10

dR�2�X 0 �er0 �

r02eÿ��r

0�R�H�z0�H�aÿ z0�:

Using now transformation (A9) gives

�1��r;X � � �s4��0

�10

dR�2 �0 ÿ z0

�0X 0?

� �eÿ��r

0�R�H�z0�H�aÿ z0�: �19�

The two-dimensional delta function can be manipulated by grouping the terms in Ras follows

�2 �0 ÿ z0

�0X 0?

� �� 2�� ÿ RDe�� ÿ RD��'� ÿ��

��;

with

�� ÿ �z=�0�X 0?De� � X? ÿ ��=�0�X 0?

;

��20�

where e� � ��;�� is a unit vector on the xy plane.Next, by taking advantage of the delta in �� one can eliminate the integration in

R. Using �� ÿ RD� � ��=Dÿ R�=D it is found that

�1��r;X � �H zÿ ��D

� ��H aÿ z� ��

D

� � �s4��0

eÿ��z��0ÿ��=D ��0

��'� ÿ��D

;�21�

where I have used r0 � z0=�0 with z0 � zÿ �R. When specialized to the exiting ¯uxesat z � 0 and z � a, this equation yields the same results obtained in Siewert (1985).Note however that the notation is slightly di�erent. From the de®nitions in (20) onehas


�2� � �2 � z21ÿ �2

0

�20

ÿ 2�z

�0

��1ÿ �2

0

qcos�'ÿ�0�

cos '� � e� �� cos 'ÿ �z=�0�

��1ÿ �2

0

qcos �0

��

;

8>>>><>>>>:and

D2 � 1ÿ �2 � �2 1ÿ �20

�20

ÿ 2�

�0

��1ÿ �2��1ÿ �2

0�q

cos��ÿ�0�

cos� � e� �e� ��1ÿ �2�

pcos �ÿ ��=�0�

��1ÿ �2

0

qcos �0

D

:

8>>><>>>:Furthermore, for very forward scattering it may be necessary to include a singular

¯ux component of the type given in Eq. (17a).I end this section with a brief discussion of Eq. (21). First note that the use of the

new radial coordinates �� ; '�� is equivalent to adopt a variable system ofcoordinates with center C the projection on the xy plane of the intersection of theforward trajectory with the horizontal plane that contains point r (see Fig. 4). Inthese coordinates

r � R0X 0 � RX � �� R0 � �

�0R

� �X 0 ) �� R X ÿ �

�0X 0

� �:

The new radial vector �� vanishes when r lies on the forward trajectory, r � rX 0.But the presence of �� in the denominator of expression (21) is only the result of thelocal singularity of the cylindrical coordinates and does not create a physical diver-gence. On the other hand, D vanishes only when X � �X 0 and this invalidatesresult (21) since the �� ÿ RD� does not depend anymore on R and cannot be usedto eliminate the integration along R. The conclusion is that Eq. (21) is not valid in

Fig. 4. Geometrical notation for the searchlight problem showing the radial vector ��. The position of the

moving center of coordinates is noted by C.


this case. Physically one can argue that the ®rst-collided ¯ux is zero if r does not lieon the forward trajectory and X � �X 0. However, it is clear that this is not thecase when r lies on the forward trajectory. But in this last case expression (21) isuseless and, if necessary, one would have to resort to a direct estimation.

5. The vanishing delta function

A particular problem with curvilinear coordinates results from the singularities ofthe coordinate representation. In spherical coordinates the coordinate mapping issingular at the origin and for cylindrical coordinates it is singular on the z axis.Because of these singularities one has to be cautious when manipulating delta func-tions in curvilinear coordinates. As an example I analyze here the calculation of the®rst-collided ¯ux for a singular source of the form

S�r;X � � �3�r��2�X �ez�:

For simplicity, I will consider only the case of a homogeneous, in®nite medium.The uncollided ¯ux is given by Eq. (8) with the replacement g�X � ! �2�X �ez�.One can write the result in cylindrical coordinates by using formula (A10):

�0��r;X � � �2�X �ez�H�z��eÿ�z: �22a�

Next one computes the collided source and replaces the result in Eq. (6) to obtainan expression for the ®rst-collided ¯ux:

�1��r;X � � �s��10

dRH�z0��0�eÿ��z0�R�: �23a�

This expression is readily evaluated by observing that

��0� � ��ÿ RX?� � ��ÿ R��1ÿ �2

p��'ÿ��

�;

and by using the delta in � to eliminate the integration in R. The ®nal result is:

�1��r;X � � H zÿ ��1ÿ �2

p !�s�� 'ÿ��

��1ÿ �2

p eÿ� z��

��1ÿ�1��

p� �: �24�

But, assume that I had replaced �� in Eq. (22a) with ��=�2�� 2�=�,which is to the usual expression in cylindrical coordinates. Then

�0��r;X � � �2�X �ez�H�z� ��2��

eÿ�z; �22b�

so that now Eq. (23a) reads


�1��r;X � � �s��

�10

dRH�z0��02�eÿ��z0�R�; �23b�

where I have used ��2� � ��=�2��. Let see what happens if we try to use the deltafunction on �0 to eliminate the integration on R. For this we use formula (A6) buttake into account that the function has now two roots so that

��02� � 1

��1ÿ �2

p ��Rÿ R�� ÿ ��Rÿ Rÿ�2i sin�'ÿ�� ; �25�

where we have used �0 � �ÿ RX? to compute �02. The two roots in (25) havecomplex values except for ' � �:

R� � ��1ÿ �2

p e�i�'ÿ��; �26�

where i � ��ÿ1p.

Replacing these results in (23b) gives

�1��r;X � � �s��1ÿ �2

p �10

dRH�z0�eÿ��z0�R� ��Rÿ R�� ÿ ��Rÿ Rÿ�2�i sin�'ÿ�� : �27�

This expression shows that only for ' � � does the argument of the delta functionbecome real and thus allows to eliminate the integration in R. Since the support ofthe distribution �1��r;X� in the �';�� domain reduces to the diagonal, one maysuspect that the behavior of �1� has to be of the form f��; '; z; ��'ÿ��, so theonly thing that is necessary is to estimate f��; '; z; ��. However for ' � � both thenumerator and the denominator in (27) vanish and the entire expression becomesapparently meaningless.I insist that the problem has to do with the singularity of the coordinate repre-

sentation. To circumvent this di�culty one can reconsider Eq. (23b) under the lightof transformation (A14). The result is

�1��r;X � � �s�� 'ÿ��

�10

dRH�z0��ÿ R��1ÿ �2

p�eÿ��z0�R�;

which gives the correct expression in (24).But when working without care in curvilinear coordinates one often arrives at a

standstill with expressions like in (27):�10

dRf�R� ��Rÿ �ÿ i�� ÿ ��Rÿ �� i��2�i�

�28�

In this case, instead of going back and reconsider the change of variable, it is moreexpedient to adopt the following replacement


��Rÿ �ÿ i�� ÿ ��Rÿ �� i��2�i�

! ��Rÿ ��; �29�

which gives the correct result.

Acknowledgements

This work was supported in part by the US O�ce of Naval Research GrantN00014-96-1-0006 under a grant to N.J. McCormick.

Appendix A: The three-dimensional delta function in curvilinear coordinates

Consider the ordinary Dirac three-dimensional distribution:

�3�rÿ r0� � ��xÿ x0��yÿ y0��zÿ z0�: �A1�

In spherical coordinates, r � �r; �; '�, this distribution becomes:

�3�rÿ r0� � ��rÿ r0� �2�er �er0 �r2

; �A2�

where er � r=r is the unit vector in direction r, and where �2�X �X 0� is Placzek'sdelta function on the surface of the unit sphere (Case et al., 1953). By using sphericalcoordinates on the unit sphere, X � ��;��, one can write

�2�X �X 0� � ��ÿ �0��ÿ�0� � ��1ÿX �X 0�2�

: �A3�

Finally, in cylindrical coordinates, r � ��; '; z�, one has�3�rÿ r0� � ��zÿ z0��2��ÿ �0�; �A4�

where � � rÿ zez and

�2��ÿ �0� � ��ÿ �0�

��'ÿ '0�: �A5�

Also, when changing coordinates one may use the following formula (Schwartz,1951; Gel'fand and Shilov, 1968):

��f�x�� xÿ xroot�@xf�x��

x�xroot

; �A6�


where xroot is the single root of f�x� � 0. When this equation has several roots, a sumin the number of roots has to be used (Schwartz, 1951).

Appendix B: Some transformations between related distributions

In this appendix I establish some general relations for the distribution�2�X 0 �er0 �=r02 for r0 in the back trajectory of �r;X�:

r0 � rÿ RX ;R 2 �0;1�: �A7�

The technique consists of completing the distribution to the ordinary three-dimensional delta function. Consider, for instance, the following transformation:

�2�X 0 �er0 �r02

��10

dR0��r0 ÿ R0� �2�X 0 �er0 �r02

��10

dR0�3�r0 ÿ R0X 0�: �A8�

Next one writes the delta function in cylindrical coordinates and uses the delta inz0 to eliminate the integration in R0:

�2�X 0 �er0 �r02

��10

dR0��R0�0 ÿ z0��2��0 ÿ R0X 0?�

� 1

�0j jHz0

�0

� ��2 �

0 ÿ z0

�0X 0?

� �: �A9�

In this expression H represents Heaviside's step function, �0 � ez �X 0, and X 0? �X 0 ÿ �0ez is the component of X 0 on the xy plane. In particular, for R � 0 one has

�2�X 0 �er�r2

� 1

�0j jHz

�0

� ��2 �ÿ z

�0X 0?

� �;

and for X 0 � ez one gets the formula

�2�ez �er�r2

� H�z��2�� H�z��x��y�: �A10�

Next, I examine the particular case when X 0 is parallel or antiparallel to X . Thetechnique is as before, except that now I put together the two components along X .For example:


�2�X �er0 �r02

��10

dR0��r0 ÿ R0� �2�X �er0 �r02

��10

dR0�3�r0 ÿ R0X �

��10

dR0�3�rÿ �R� R0�X �: �A11�

By writing the delta function in spherical coordinates and using the ��rÿ �R� R0��to eliminate the integration on R0 one gets:

�2�X �er0 �r02

��10

dR0��rÿ �R� R0�� 2�X �er�r2

� H�rÿ R� �2�X �er�r2

: �A12�

Finally, for the case with X 0 � ÿX , operating as for Eq. (A11) gives

�2�ÿX�er0 �r02

��10

dR0��r0 ÿ R0� �2�ÿX�er0 �r02

��10

dR0�3�r0 � R0X �

��10

dR0�3�rÿ �Rÿ R0�X �:

For this case, one has to consider the two possibilities R > R0 and R < R0:

�2�ÿX�er0 �r02

��10

dR0 ��rÿ R� R0� �2�X�er�r2

� ��r� Rÿ R0� �2�ÿX�er�r2

� �� H�Rÿ r� �2�X�er�

r2� �2�ÿX�er�

r2:

�A13�

Similar expressions can be derived starting with �2�X�er�=r2. The resulting for-mulas are:

�2�X�er�r2

� �2�X�er0 �r02

�H�Rÿ r0� �2�ÿX�er0 �r02

and

�2�ÿX�er�r2

� H�r0 ÿ R� �2�ÿX�er0 �r02

:

An analogous technique can be applied to the delta in cylindrical coordinates. Let�0 � �ÿ RX? with R 2 �0;1�, and consider the following transformation:

��0��0��2�0

d'0��'0� ��0�

�0��2�0

d'0�2��0� ��2�0

d'0��ÿ R

��1ÿ �2

p��'ÿ��

�

� 2��ÿ R

��1ÿ �2

p��'ÿ��

�:

�A14�


Appendix C: The delta-scattering case

The simplest collision operator one may think of corresponds to a singular scat-tering kernel of the form:

�s�X�X 0� � �s�2�X�X 0�; �A15�

where, for simplicity, we have considered a homogeneous medium. The associatedcollision operator becomes

�H ��r;X � � �s �r;X �:

For this case one can obtain the total ¯ux by just replacing � with � ÿ �s in theexpression for the uncollided ¯ux in (7):

�r;X � � g�X � eÿ��ÿ�s�r

r2�2�X�er� �

Xn50

��sr�nn!

�0��r;X �; �A16�

where �0��r;X � is the uncollided ¯ux:

�0��r;X � � g�X � eÿ�r

r2�2�X�er�: �A17�

Eq. (A16) suggests a multiple scattering solution of the form

�n��r;X � � ��sr�n

n! �0��r;X �: �A18�

We can therefore use Eq. (6) to check this solution. For the delta scattering in(A15) S�n�1��r;X� � �s �n��r;X� and (6) gives:

�n�1��r;X � ��10

dRS�n�1��r0;X �eÿ�R

� ��s�n�1

n!g�X �

�10

dRr0neÿ��r0�R� �2�X�er0 �

r02: �A19�

Finally, we use transformation (A12) and, after recognizing that r0 � rÿ R, onegets

�n�1��r;X � � ��s�n�1

n!g�X �eÿ�r �2�X�er�

r2

�r0

dR�rÿ R�n

� ��sr�n�1

�n� 1�! �0��r;X �; �A20�

which con®rms the earlier assumption (A18).


Appendix D: An example of the Fourier transform technique

In this appendix, I revisit the searchlight problem to sketch the Fourier-analysistreatment. The starting point is the transport equation in a homogeneous slab writ-ten in cylindrical coordinates:

��@z �X?�r? � �� n� � S�n� 04z4a �n� � �n0 bd z � 0; � > 0 and z � a; � < 0

;

��A21�

where I have already introduced a multiple collision expansion. The boundary con-dition corresponds to a one particle impinging on the left side of the slab withdirection X 0:

bd��; zbd;X � ��2��2�X�X 0�

�0zbd � 0; � > 0;

0 zbd � a; � < 0;

;

8><>: �A22�

so that S�0� � 0.In the Fourier transform approach one introduces a pair of direct and inverse

Fourier transforms on the dual variables � $ !,

f�!� � F�f� � �1d�eÿi!��f��;f�� Fÿ1�f� � 1

�2��2�1

d!e�i!��f�!�:

8<: �A23�

Then, applying the direct Fourier transform to Eq. (A21) yields a one-dimensionaltransport equation for the transformed ¯ux �!; z;X�:

��@z � i!�X? � �� n� � S�n� 04z4a;

�n� � �n0 bd z � 0; � > 0 and z � a; � < 0

(�A24�

This equation can be integrated to obtain

�n��!; z;X � � eÿ��X?�Rbd�n0 bd�!; zbd;X � ��Rbd

0

dReÿ��X?�RS�n��!; z0;X �;�A25�

where H stands for Heaviside's step function, z0 � zÿ R�, ��X?� � � � i!�X?and

Rbd

z

�� > 0;

zÿ a

�� < 0:

8><>:


By applying Eq. (A25) to the evaluation of the uncollided ¯ux we readily obtain

�0��!; z;X � � eÿ��X0?�

z

�0�2�X �X 0�

�0: �A26�

The inverse Fourier transform can be carried out straightforwardly by noticingthat

F��2��ÿ �0�� eÿi!��0 : �A27�

With the help of this formula we obtain:

�0��; z;X � � eÿ� z

�0

�2��ÿ z

�X 0?��2�X�X 0��0

; �A28�

which is the same result as in (18).Next, (A26) is used to determine the ®rst-collided source S�1� and an expression

for the ®rst-collided ¯ux is then obtained by inserting S�1� in (A25):

�1��!; z;X � � �s4��0

�Rbd

0

dReÿ��X?�Rÿ��X0?�

z0

�0 : �A29�

At this point one has two choices. One choice is to go back to � space to obtain

�1��; z;X � � �s4��0

�Rbd

0

dReÿ��R� z0

�0��2��ÿ RX? ÿ z0

�0X 0?�

which is Eq. (19), so we can proceed like we did before.The other choice is to integrate Eq. (A29) to obtain the analytical expression

�1��!; z;X � � �s4��o�

eÿz��X01 �

�0

��X 0?��0

ÿ ��X?��

�1ÿ eÿ�Rbd�

��X?��ÿ ��X 0?�

�0�� A30�

and then transform back this expression to � space.

Appendix E: The energy-dependent case

The entire analysis for the one-group problem can be easily carried out in theenergy-dependent framework. This is due to the fact that the multiple collisionexpansion consists of a series of linear motions at constant energies intertwined withcollision events where the particle changes of energy. It su�ces, thus, to observe thatnow the collision operator comprises an integration over the energy


�H ��r;X;E� ��4��

dX 0�10

dE0�s�r;X�X 0;E0 ! E� �r;X 0;E0�;

but that Eq. (1) remains essentially the same

�n��r;X;E� ��D

dr0��4��

dX 0S�n��r0;X 0;E�Gunc�r0;X 0 ! r;X ;E�;

except for the fact that the Green's function has the energy as a parameter:

Gunc�r0;X 0 ! r;X ;E� � �2�X 0 �XR� eÿ��r0;r;E�

R2�2�X�XR�:

Therefore, Eq. (6) has to be replaced by

�n��r;X;E� ��10

dRS�n��r0;X;E�eÿ��r0;r;E��D�r0�: �A31�

Notice that this expressions is very similar to (6) except that the source and theoptical distance are evaluated at energy E. Since the source is produced by collisionsat a di�erent energy one expects to have optical paths at di�erent energies in the®nal ¯ux expressions. For instance, for a monoenergetic source of the form

S�r;X;E� � �3�r�g�X ��Eÿ E0�; �A32�

the angular ¯ux �n��r;X ;E� will contain a product of n� 1 exponential attenuationfactors for energies E0;E1; . . . ;En � Ef g corresponding to the initial motion of theparticle at energy E0 to reach the ®rst collision site, the nÿ 1 paths between colli-sions, and the streaming at the ®nal energy E from the last collision site to point r.The uncollided ¯ux produced by the source in (A32) is readily found to be

�0��r;X;E� � g�X ��Eÿ E0� �2�X�er�r2

eÿ��0;r;E0�: �A33�

Computing the corresponding source and replacing it in (A31) gives a generalexpression for the ®st-collided ¯ux

�1��r;X;E� ��10

dRg�er0 ��s�r0;X �er0 ;E0 ! E� eÿ��0;r0;E0�ÿ��r0;r;E�

r02�D�r0�:

�A34�

Next, by assuming a scattering kernel of the form of Eq. (13) but including energydependence, results again in three ¯ux components. The regular component is simi-lar to that of Eq. (14):


�1�reg�r;X;E� �1

r��1ÿ �2

p ��ÿ1

d�0��1ÿ �02

p g�er0 �

�s;reg�r0; �0;E0 ! E�eÿ��0;r0;E0�ÿ��r0;r;E��D�r0�;�A35�

whereas the expressions for the singular components are slightly more involved dueto the fact that the two particle tracks are done at di�erent energies. The result is

�1�� r;X;E� � g�X � �2�X�er�r2

�r0

dr0�s;��r0er;E0 ! E�eÿ��0;r0er;E0�ÿ��r0er;rer;E�;

�A36a�and

�1�ÿ �r;X � �g�ÿX �

r2��2�ÿX�er�

�rmax�er�

r

dr0�s;ÿ�r0er;E0 ! E�eÿ��0;r0er;E0�ÿ��r0er;rer;E�

� �2�X�er��rmax�ÿer�

0

dr0�s;ÿ�ÿr0er;E0 ! E�eÿ��0;ÿr0er;E0�ÿ��ÿr0er;rer;E��:�A36b�

For a homogeneous medium, the expression for the optical paths in Eq. (A35)simpli®es to

��0; r0;E0� � ��r0; r;E� � ��E0�r0 � ��E�R � r��E0� cos � � ��E� sin � tan�0

2�

to be contrasted with Eq. (16). Note also that physically one would expect a deltabehavior on the energy dependence of the singular scattering kernels �s;��r;E0 ! E�;for instance, for elastic scattering one has �s;��r;E0 ! E� � �s;��r��E0 ÿ E�.

References

Bell C.I., Glasstone S., 1970. Van Nostrand, New York.

Case, K.M., Zweiful, P.F., 1967. Linear Transport Theory. Addison-Wesley, Reading MA.

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

Government Printing O�ce, Washington, DC.

Duderstadt J.J., Martin W.R., 1979. John Wiley, New York.

Gel'fand, I.M., Shilov, G.E., 1968. Generalized Functions. Academic Press, New York.

Sanchez R., McCormick N.J., Propagation of a searchlight beam through strongly forward scattering

media. In preparation.

Schwartz, L., 1951. TheÂ orie des Distributions. Hermann, Paris.

Siewert, C.E., 1985. On the singular component of the solution to the searchlight problem in radiative

transfer. JQRST 33 (6), 551±554.


On the singular structure of the uncollided and first-collided components of the Green's function

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Transcript of On the singular structure of the uncollided and first-collided components of the Green's function