The Moments Method for the one-group transport and diffusion equations

The Moments Method for the one-grouptransport and diffusion equations

Richard Sanchez*

Commissariat a l’Energie Atomique, Direction de l’Energie Nucleaire,

Service d’Etudes de Reacteurs et de Modelisation Avancee, CEA de Saclay,

DM2S/SERMA, 91191 Gif-sur-Yvette cedex, France

Received 12 March 2003; accepted 1 June 2003

Abstract

We analyze the Moments Method for the one-group transport equation in an infinite, iso-

tropic and homogeneous medium and generalize some of the results found in the literature.We derive general recursion relations for global moments of the form of a product of apolynomial in (x,y,z) of order n times a spherical harmonic Ykl (�) and of the form r n times aLegendre polynomial of order k of �.r/r. We determine which moments can be obtained by a

finite recursion and obtain the maximum order of the eigenvalue of the scattering operatorthat enters the recursion relation, defining thus an equivalence class between scatteringoperators. We have also investigated and derived general recursion relations between one-

dimensional transport like equations for transverse moments of the form �n(e�.�/�)k, where �

is the radius vector in the xy plane and e� is the unit vector in the direction of the projection of� onto the xy plane. In particular, we investigate the expression for the mean distance to

absorption <r2> . The results are extended to the case of the one-group diffusion equationand an expression for <r2> is also derived for the particular case of a medium in thermalequilibrium.

# 2003 Elsevier Ltd. All rights reserved.

1. Introduction

The difficulty of obtaining an analytical solution of the transport equation ledearly researchers to devise means for the direct calculation of the spatial and angularmoments of the angular flux. These moments are often more easily accessible and

Annals of Nuclear Energy 30 (2003) 1645–1663

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give macroscopic information about the solution of the transport equation. In hisclassical analysis of slowing down, Marshak used Fourier transform techniques toderive an explicit formula for the slowing down length (Marshak, 1948), a quantitythat depends on the 0th and the second spatial moments of the scalar flux producedby a localized source.

In the early 1950s this approach was formalized as the Moments Method by Lewisand Fano (Lewis, 1950) who explicitly calculated global space and angle moments ofthe radiation distribution in an infinite, isotropic and homogeneous medium. Theseauthors considered a continuous slowing down model of transport that, in ournotation, can be written as the initial value problem:

@s þ� � r þ Sð Þ ¼ cSH ;

0; r;�ð Þ ¼ � rð Þ�2 � � ezð Þ;

limr ! 1 s; r;�ð Þ ¼ 0;

s > 0

s ¼ 0

s < 0

ð1Þ

where the variable s accounts for slowing down in energy, (s,r,�) is the radiationfield, � is the total cross section, c is the number of secondaries per collision and His the scattering operator

H ð Þ s; r;�ð Þ ¼

ðd�0P s;�0

��ð Þ s; r;�0ð Þ ð2Þ

with scattering kernel normalized to 1
ðd�P s;�0
��ð Þ ¼ 1:

In their work, Lewis and Fano projected Eq. (1) over a weight w r;�ð Þ ¼ p rð ÞYkl �ð Þ,where Ykl(�) is a spherical harmonic, and explicitly computed longitudinal, p(r)=zn,and transverse, p(r)=xn, moments of the radiation field:

� �

wsð Þ ¼

ðdrd�w r;�ð Þ s; r;�ð Þ:

The Moments Method has been applied to the calculation of transverse andlongitudinal moments of the stationary, one-group transport equation. Larsen andco-workers (Borgers and Larsen, 1995; Larsen, 1997; Franke, 1999) have applied theMoments Method to derive a closed system of lower dimensional equations for theexact radial moments of the radiation produced by the spreading of a beam infunction of the depth. Moments that can then be calculated without the need tosolve the initial equation. Larsen (1996) has applied the method to the time andenergy-dependent transport equation to compute the mean-squared displacement ofneutrons as a function of the initial energy and of the elapsed time and final energy,and Brantley and Larsen (1998) have used the Moments Method to derive physicallymeaningful identities for the ‘amplitude,’ the ‘center of mass’ and the ‘radius ofgyration’ in one-group diffusion and transport theory in an infinite slab geometry.More recently, Leakeas and Larsen (2001) have demonstrated an equivalencebetween space and angle moments with the eigenvalues of the scattering operator.

1646 R. Sanchez / Annals of Nuclear Energy 30 (2003) 1645–1663

In most of previous work relations between different types of moments have beenestablished only for a few first moments. In this work we proceed to a systematicinvestigation of the Moments Method in the stationary one-group transport theoryin an infinite, isotropic medium. We review different problems that have been pre-viously treated in the literature by the Moments Method and systematically con-struct general families of weight functions and derive general recursions for all themoments. For each family of weights we establish the conditions under which amoment can be calculated from a finite recursion and give the order kmax of themaximum eigenvalue of the scattering operator entering the recursion. Followingthe work of Leakeas and Larsen (2001), the value of kmax can be used to define anequivalence class between scattering operators: two scattering operators give thesame moment if they have the same kmax first eigenvalues. First we have investigatedgeneral factorized moments and then we have studied the moments generated byfunctions that are rotationally invariant. For completeness, we have also analyzedthe case of partial and transverse moments. In the former, the weights depend onlyon a few variables while in the latter the projection is done over a part of phasespace. For all cases we have derived recursion relations between the moments andgave rules for the dependence of the moments on the eigenvalues of the scatteringoperator. One of the motivations for this work was to find an explanation for thewell-known fact that the one-group transport migration area, that is, the meansquare distance from emission to absorption <r2> , only depends on the value ofthe mean scattering cosine ¼

Ðd��0

��P �0��ð Þ.

In this work we consider the stationary, one-group transport equation in an infi-nite, isotropic, homogeneous medium:

� � r þ�ð Þ ¼ c�H þ S;

limr !1

r;�ð Þ ¼ 0;ð3Þ

where (r,�) is the angular flux, H the scattering operator in Eq. (2) but without thes dependence, and the source S(r,�) is assumed to vanish for r!1 fast enough toensure an exponential decay of the flux as r!1.

The essence of the Moments Method consists on multiplying transport Eq. (3) bya function w(r,�) and integrating over phase space to obtain

� � �

w¼ H � �

wþ Sh iwþ

� ��:rw

; ð4Þ

where

� �

w¼

ðdrd�w r;�ð Þ r;�ð Þ

is the moment of with weight function w.In order to obtain meaningful expressions, in practice one chooses factorised

weights of the form w r;�ð Þ ¼ p rð Þq �ð Þ. In Section 2 we analyze the case when p rð Þ ¼

pn x; y; zð Þ is a polynomial of order n in the space variables and q �ð Þ ¼ Akl �ð Þ is areal spherical harmonic. We prove that a moment of order (n, kl) with n>0 can be

R. Sanchez / Annals of Nuclear Energy 30 (2003) 1645–1663 1647

written in terms of moments of order (0, k0l0) with k04k+n. Therefore, the (n, kl)moment depends at most on the first k+n eigenvalues of the scattering operator.The next section deals with weight functions that are invariant under rotations. Inthis case w r;�ð Þ ¼ rnPk � � r=rð Þ, where Pk is the Legendre polynomial of order k.We show that only some of the moments give rise to finite recursion relations, whilethe remaining moments generate infinite recursions and depend on all the eigen-values of the scattering operator. Only moments with n and k of the same parity andn>k give rise to a finite recursion and depend solely on the moment of order (0, 0).Moreover, the highest k order needed for the recursion is (n+k)/2. This is the case,in particular for the migration area <r2> . At the end of this section we investigatethe technique of transverse moments that was used in (Franke and Larsen, 1999).Contrarily to the approach previously analyzed, here the transport equation is pro-jected over only some of the variables, generating simplified transport equationsover the remaining variables for the moments. The case of the one-group diffusionequation is considered in Section 4, while conclusions are given in Section 5. Thereal spherical harmonics and an analysis of the scattering operator are given inAppendix A. In Appendix B we give an independent derivation for the migrationarea for the one-group transport equation.

2. Moments based on polynomials in the space variables

In practice one chooses factorized weights of the form w r;�ð Þ ¼ p rð Þq �ð Þ. A goodchoice for the spatial factors is to take p(r)=pn(r) to be a polynomial of order n in(x, y, z). Furthermore, in order to simplify the moment of the scattering term onemay take the angular factor to be a spherical harmonic, q �ð Þ ¼ Akl �ð Þ (Lewis,1950), so that

w r;�ð Þ ¼ pn rð ÞAkl �ð Þ: ð5Þ

The set of these weights form a complete basis for functions of (r, �).With these weights Eq. (4) yields:

�ak n;kl ¼ Sn;kl þ

ðdrd� r;�ð ÞAkl r;�ð Þ� � rpnðrÞ; ð6Þ

where fk and �ak=�(1�cfk) are the eigenvalue and the effective absorption for thespherical harmonics of order k, and where we have introduced the moments

n;kl ¼

ðdrd�pn rð ÞAkl �ð Þ r;�ð Þ ¼

ðdrpn rð Þ kl rð Þ:

We notice now that rpn rð Þ is a polynomial of order n�1 and that �Akl �ð Þ can bewritten as a linear combination of spherical harmonics of order k+1 and k�1. Thisentails that Eq. (6) can be utilized to iteratively calculate n,kl for n>0 in terms ofsource moments and of the 0,k0l0 with k04k+n, while the moments of order n=0depend directly on the source, 0;kl ¼ S0;kl=�a. Because the only coefficients that


enter these expressions are the �ak one concludes that for fixed �, c and source dis-tribution S(r,�), two scattering operators H1 and H2 that have the same eigenvaluesup to order K (note that the first eigenvalue is always 1) will produce the same fluxmoments:

n;kl

� �H1¼ n;kl

� �H2; kþ n4K : ð7Þ

However, it is well known (see Appendix B) that in an infinite, homogeneousmedium the mean square distance from emission to absorption depends only on thefirst two eigenvalues,

r2� �

¼2

�a�a1¼

2

�2 1� cð Þ 1� cð Þ; ð8Þ

which means that this moment of order n=2 and k=0 is the same for two scatteringoperators that have equal eigenvalues up to order K=1: f0=1 and f1 ¼ . On theother hand, Eq. (7) predicts the conservation of a moment of order n=2 and k=0only when the two scattering operators are identical to order K=2. This lack ofprecision of (7) is due to the fact that this relation has been established without usingneither the specific form of the polynomial pn(r) nor the detailed coefficients enteringin the expression for �Akl �ð Þ in terms of spherical harmonics. To illustrate thispoint we follow Larsen (1996) and analyze here the case of the calculation of <r2> .

For an arbitrary source distribution the expression for <r2> is:

r2� �

¼

Ðdr0d�0S r0;�0ð Þ�a

Ðdrd� r � r0j j2G r0;�0 ! r;�ð ÞÐ

dr0d�0S r0;�0ð Þ

where G r0;�0 ! r;�ð Þ is the Green function solution of the transport equationwith source � r � r0ð Þ�2 � ��0ð Þ. The integration over r and � can be done by trans-lating and rotating the coordinate axes so that the origin is at r0 and the z axis isparallel to �0. In these conditions the Green’s function does not depend on thevariables r0 and �0 and the above expression simplifies to:

r2� �

¼ �a

ðdrd�r2G r;�ð Þ;

where G(r, �) is the Green function solution of:

�:r þ�ð ÞG ¼ c�HGþ � rð Þ�2 � � ezð Þ

limr !1

G r;�ð Þ ¼ 0:ð9Þ

We now integrate this equation over phase space, first directly and then aftermultiplying by r2. The first integration gives the conservation relation�a

Ðdrd�G r;�ð Þ ¼ 1 while the second results in

r2� �

¼

ðdrd�G r;�ð Þ� � rr2 ¼ 2

ðdrd�G r;�ð Þ� � r:


alculate the integral on the right hand side we multiply Eq. (9) by �.r and
To cintegrate over phase space. The result is:
�a1

ðdrd�� rG r;�ð Þ ¼

ðdrd�G r;�ð Þ� � r � � rð Þ ¼

ðdrd�G r;�ð Þ ¼

1

�a;

where we have used the fact that rr requals the identity tensor I . From the last twoequations we obtain the value of <r2> given in (8). In the next section we generalizethis analysis to the general case of rn.

3. Weight functions that are invariant under rotations

Regarding the choice of the weight functions it seems appropriate to exploit thesymmetries of the problem. In an infinite and homogenous medium the transportoperator is invariant under arbitrary translations and rotations. Invariance undertranslations gives weights independent of the spatial variable and then the bestchoice is to use the spherical harmonics. These weights are those defined in (5) withn=0 and have been studied in the previous section. We consider here weights thatare invariant under arbitrary rotations. The general form of a function of (r, �) thatis invariant under rotations is w r;�ð Þ ¼ w r;� � erð Þ, where er=r/r is the unit vectorin the r direction. In the following we shall use factorized weight functions of theform:

w r;�ð Þ ¼ rnPk � � erð Þ ð10Þ

where Pk is the Legendre polynomial of order k. These functions form a completebase for an arbitrary function of r;� � erð Þ.

We consider first the integration of the scattering term. Noticing that Pk � � erð Þ

can be written as

Pk � � erð Þ ¼1

2kþ 1

Xl

Akl �ð ÞAkl erð Þ

we obtain
ðd�Pk � � erð ÞP �0
��ð Þ ¼ fkPk �0 � erð Þ ð11Þ

Ð Ð
and this entails d�Pk � � erð Þ H ð Þ r;�ð Þ ¼ fk d�Pk � � erð Þ r;�ð Þ so the scatter-ing term diagonalizes under projection. Hence, use of weights (10) in Eq. (4) gives:
�ak n;k ¼ Sn;k þ

ðdrd� r;�ð Þ� � r rnPk � � erð Þ½ �

with the moments

n;k ¼

ðdrd�rnPk � � erð Þ r;�ð Þ:


� � r

xt we consider the leakage term. Taking into account that rer ¼ I � ererð Þ=r
Newe can write:
� � r rnPk � � erð Þ½ � ¼ rn�1 nPk � � erð Þ� � er þ P0k � � erð Þ 1� � � erð Þ

2� �

;

where P0k xð Þ ¼ @xPk xð Þ. After use of the recursion relation between Legendre poly-

nomials the above expression reduces to

rnPk � � erð Þ½ � ¼rn�1

2kþ 1kþ 1ð Þ n� kð ÞPkþ1 � � erð Þ þ k nþ kþ 1ð ÞPk�1 � � erð Þ½ �:

Finally, replacing this result in the leakage term results in the recursion relation:

�ak n;k ¼ Sn;k þkþ 1ð Þ n� kð Þ

2kþ 1 n�1;kþ1 þ

k nþ kþ 1ð Þ

2kþ 1 n�1;k�1: ð12Þ

Next, we analyze the recursion relation (12). As shown in Fig. 1, the recursionfrom n,k to moments of smaller n index spreads out in (n,k) space: moment n,k

depends on two moments of index n�1, n�1,k+1 and n�1,k�1, then on three momentsof index n�2, n�2,k�2, n�2,k and n�2,k+2, and so on. Hence, n,k depends at moston n+1 moments of index n=0, 0,k�n, 0,k�n+2, . . ., 0,k+n. However, the onlymoment at which the recursion ends is 00. Notice also that the factor k in the sec-ond contribution on the right hand side cancels all moments with negative k index,while the factor (n�k) in the first contribution implies that all the moments on thediagonal, n,n, are recursively related to only the moment 00. Therefore, as shownin Fig. 1, only those moments that by recursion may reach the diagonal n=k will bethose for which the recursion ends with the moment 00. For one of such moments, n,k, we must have n+k even and n>k. Moreover, the highest eigenvalue needed inthe recursion will correspond to the value of the k index at the diagonal, (n+k)/2.One can conclude that if two scattering operators H1 and H2 have the same eigen-values up to order K, then

n;k

� �H1¼ n;k

� �H2; n5 k nþ kð Þ even; nþ k4 2K; ð13Þ

whereas moments with indexes (n,k) that do not comply with the above conditionsgenerate an infinite recursion and are equal only if H1�H2. In particular, relation(13) implies that the moment 20 is the same if the two scattering operators haveidentical eigenvalues up to order K=1. This proves again that <r2> depends onlyon the first two eigenvalues of the scattering operator.

3.1. Partial moments

The precedent technique does not allow to compute quantities such as <x2> , themean square distance to the x axis from emission to absorption. For this we could


turn to the general weights in (5) and to the corresponding recursion relation (6).Instead we take stock of the work we have done and use the following weights:

w r;�ð Þ ¼ r � eð ÞnPk � � eð Þ; ð14Þ

where e is a fixed unit vector and re ¼ r � e is the coordinate of r along the axis withdirection e. With the help of Eq. (11) we obtain the recursion relation

Fig. 1. Graphical analysis of recurrence relation (12). (a) Moment nk depends on two moments of

order n�1, n�1,k+1 and n�1,k�1, on three moments of order n�2, etc. Moment n�l,k+m, with l+m

even and mj j4 l, is affected by the product, wlm, of weights linking it to moment

n� l�mð Þ=2;k� l�mð Þ=2 : wlm ¼ n� kð Þðn� k� 2Þ . . . n� k� l�mþ 2ð Þ. (b) Recursions originating at points

with n5k reach the diagonal if n+k is even. Because diagonal moments are related by recursion to only

the moment 0,0, all moments with n5k and n+k even depend only on 0,0.


�ak n;k ¼ Sn;k þn

2kþ 1kþ 1ð Þ n�1;k�1 þ k n�1;k�1

� ð15Þ

for the moments

n;k ¼

ðdrd� r � eð Þ

nPk � � eð Þ r;�ð Þ:

Recursion (15) stops at n=0 where 0;k ¼ S0;k=�ak. Therefore any moment n,k

with n>0 can be computed from the moments 0,k0 with k04k+n. For instance:

�a 20 ¼ S20 þ2

�a1S11 þ

1

3

2S02

�a2þS00

�a

� � �:

For a unit point source, S r;�ð Þ ¼ � rð Þf �ð Þ withÐd�f �ð Þ ¼ 1, this formula gives

the value of x2e

� �¼ �a 20, the average value of the square distance to absorption

along direction e:

x2e

� �¼

1

3r2� �

þ 2�a

�a2S02

� ;

where <r2> is the mean square distance to absorption given in (8) andS02 ¼

Ðd�f �ð ÞP2 � � eð Þ. Note that the sum over three orthogonal directions i=1, 2, 3

of P2 � � eið Þ equals zero, so the last result is another way to compute <r2> .For an isotropic source, f �ð Þ ¼ 1= 4�ð Þ, we have S02=0 and therefore

x2e

� �¼ 1=3ð Þ r2

� �, a result that could have been obtained using the symmetry of the

problem. For a source in �2 � � ezð Þ we obtain:

x2e

� �¼

1

3r2� �

1þ 2�a

�a2P2 e � ezð Þ

� �:

This formula gives, in particular, z2� �

¼ 1=3ð Þ r2� �

1þ 2�a=�a2ð Þ, x2� �

¼ y2� �

¼

1=3ð Þ r2� �

1��a=�a2ð Þ and �2� �

¼ x2 þ y2� �

¼ 2 x2� �

, results that can be related to thoseobtained in (Lewis, 1950) as a function of the slowing down variable s.

3.2. Transverse moments

We end this section by investigating the case of transverse projections (Borgers,1995; Franke, 1999). We choose weights invariant by rotations around the z axis:

�n e� � e�� k

; ð16Þ

where � is the position vector on the xy plane, e�=�/� and e� is the unit vector inthe direction of the projection of � onto the xy plane, and integrate over the vari-ables (�, �) with � the angle between e� and the x axis. Following the work of


Franke (1999), our aim here is to obtain one-dimensional like transport equationsfor the moments

n;k z; ð Þ ¼

ðd�d��n e� � e�

� �k r;�ð Þ:

Consider the scattering term. With the expression for the scattering kernel in (23)and e� � e� ¼ cos �� ð Þ we have to calculate:

Ikl ¼

ð2�0

d�cosk �� ð Þcosl �� 0ð Þ:

This integral vanishes except if k+l is even and k5l:

Ikl ¼�

2k�1

kk� lð Þ=2

� cosl �0 � ð Þ; kþ l even; k5 l :

Then, since cosl �0 � ð Þ can be expressed as a sum of cosm �0 � ð Þ ¼ e�0 � e�� m

form+l even and 04m4l, we see that after projection over �n e� � e�

� �kthe scattering

term can be written under the form

H ð Þn;k z; ð Þ ¼1

2

Xl4 k=2

l¼0

Xm4 k=2

m¼l

�klm Hk�2l n;k�2m

� �z; ð Þ;

where

Hl ð Þ z; ð Þ ¼1

2

Xk5 l

k� lð Þ!

kþ lð Þ!Plk ð Þ

ð1�1

d0Plk

0ð Þ z; 0ð Þ;

and

�klm ¼1

22m

kl

� 1; m ¼ l;

�ð Þlþmk� 2l

m� l

k� l�m� 1m� l� 1

� ; m > l:

8><>:

Next, by taking into account that re� ¼ e e =�, the leakage term can be written as

� � r �n e� � e�� kh i

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2

p�n�1 n� kð Þ e� � e�

� �kþ1þk e� � e�� k�1

h i:

Finally, with the help of the previous results we can write the projection of trans-port Eq. (3) onto the weights (16) as the system of one-dimensional transportequations:


@z þ�ð Þ n;k ¼ Sn;k þHk n;k þXl4 k=2

l¼0

Xm4 k=2

m¼max l;1ð Þ

�klmHk�2l n;k�2m

þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2

pn� kð Þ n�1;kþ1 þ k n�1;k�1

� :

As before, the only point at which the recursion ends is n=0 and k=0. Then, theanalysis carried out for the weights (10) applies also to the present case, and thismeans that the one-dimensional transport equations for the moments with n+keven and n5k can be recursively solved. We give here the first six equations:

@z þ�ð Þ 00 ¼ S00 þH0 00;

@z þ�ð Þ 11 ¼ S11 þH1 11 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2

p 00;

@z þ�ð Þ 20 ¼ S20 þH0 20 þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2

p 11;


p 11 þ

1

2H0 �H2ð Þ 20;

@z þ�ð Þ 31 ¼ S31 þH1 31 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2

p2 22 þ 20ð Þ;


p 31:

The first three equations are those derived in (Franke, 1999).

4. Moments for the diffusion equation

The Moments Method can be straightforwardly applied to the one-group diffu-sion equation in an infinite, isotropic and homogeneous medium:

�Dr � r þ�að ÞF ¼ Q rð Þ;

limr !1

F rð Þ ¼ 0:

By projecting this equation onto weights of the form pn(r), where pn(r) is a poly-nomial of degree n in (x, y, z), we obtain:

�aFn ¼ Qn þD

ðdrF rð Þr � rpn rð Þ: ð17Þ

Since r � rpn is a polynomial of degree n�2 we get a recursion relation that gives themoment Fn ¼

Ðdrpn rð ÞF rð Þ in terms of source moments and of �0 or �1, according


to the parity, even or odd, of n, while the two first moments can be obtained directlyfrom the corresponding source moments, Fn ¼ Qn=�a for n=0, 1.

In the particular case when pn(r) is a polynomial of only one of the variables, forexample pn rð Þ ¼ xn, Eq. (17) gives the result obtained by Brantley and Larsen(1998):

�aFn ¼ Qn þDn n� 1ð ÞFn�2 ð18Þ

for the moments

Fn ¼

ðdrxnF rð Þ:

In the general case of a source depending on the three variables, one has to assumethat the source vanishes fast enough as y ! 1 or z ! 1 so that the moments ofthe flux are finite. If the source is independent of the variables y and z, then theprecedent equation can be obtained by integrating the diffusion equation in a infinitecylinder parallel to the x axis and of cross sectional area 1, and by using the fact thatthe current in directions perpendicular to the x axis is zero. Notice also that for aone-dimensional slab with a source Q rð Þ ¼ � xð Þ, Eq. (18) predicts the well-knownvalue x2

� �¼ 2D=�a.

We now examine the case with weight functions of the form rn. Accounting for thefact that r � rrn ¼ n nþ 1ð Þrn�2 one obtains:

�aFn ¼ Qn þDn nþ 1ð ÞFn�2 ð19Þ

with

Fn ¼

ðdrrnF rð Þ:

Thus, again the moment �n can be obtained iteratively from one of the first twomoments but the recursion ends only at n=0, not at n=1. For the case withQ(r)=d(r) and n=2 this recursion gives

r2� �

¼6D

�a: ð20Þ

With D=1/(3�) we have from the last equation r2� �

¼ 2= ��að Þ. However, if thediffusion coefficient accounts for the transport correction, D ¼ 1= 3�a1ð Þ, then diffu-sion theory predicts the same mean square distance to absorption than transporttheory. Also, a comparison between Eq. (12) for n=k=0 and Eq. (19) for n=0shows that diffusion theory preserves the integral of the flux. We note that muchmore general results, including energy and time dependence, have been establishedby Larsen (1996).


Following the terminology of Brantley and Larsen we can use Eq. (19) to calculatethe amplitude and the squares of the center of mass and of the radius of gyration.The results are:

F0 ¼Q0

�a;

rh iF¼ rh iQþ2D

�a

F�1

F0;

r2� �

F¼ r2� �

Qþ

6D

�a;

ð ð
where rnh if¼ drrnf rð Þ= drf rð Þ. This shows that, as asserted by Brantley and Larsen,
the amplitude and the squared radius of gyration are ‘universal’ properties of diffu-

sion. However, this is not the case for the squared center of mass.There are no easy extensions of result (20) to the continuous energy problem.

However, in the case of a local Maxwellian equilibrium it is possible to obtain arelation of the type in (20). To demonstrate this we will use the detailed principle bal-ance, according to which for a Maxwellian equilibrium in a non absorbing medium:

�s E0ð ÞP E0 ! Eð ÞM E0ð Þ ¼ �s Eð ÞP E ! E0ð ÞM Eð Þ;

where �s ¼ c� and M(E) represents the Maxwellian equilibrium at the mediumtemperature. This result may be used in systems with small absorption near thermalequilibrium. Then, for a factorized source distribution of the formQ r;Eð Þ ¼ q rð ÞM Eð Þ, use of the above formula proves that the solution of the diffu-sion equation in an infinite medium

�Dr � r þ�ð ÞF ¼ c�HF þQ r;Eð Þ;

limr !1

F r;Eð Þ ¼ 0;

where H represents here the diffusion scattering operator, is of the form F r;Eð Þ ¼

f rð ÞM Eð Þ with f rð Þ solution of the one-group diffusion equation:

�Dr � r þ�að Þf ¼ q rð Þ;

limr !1

f rð Þ ¼ 0:

In these conditions we obtain:

r2� �

¼

ðdEM Eð Þ r2 Eð Þ

� �¼

ðdEM Eð Þ

6D Eð Þ

�a Eð Þ;

where we have assumed a normalized MaxwellianÐdEM Eð Þ ¼ 1.


5. Conclusions

The Moments Method allows one to derive relations between global moments ofthe solution of the transport equation and the eigenvalues of the scattering operatorand, also, to obtain low-dimensional transport like equations for transversemoments. The basic approach is to multiply the transport equation by a weight andintegrate over all or some of the variables in phase space.

In this work we have considered the one-group transport and diffusion equationsand derive general recurrence relations between global moments. For weights thatare a product of a polynomial of order n in the space variables times a sphericalharmonic Akl (�) (Lewis, 1950), or only a polynomial for the case of the diffusionequation (Brantley and Larsen 1998), we found that all the moments with n>0 canbe written in terms of moments with n=0 that depend only on source moments. Forthe transport equation one can calculate an upper bound kmax=n+k for the max-imum order of the eigenvalue that enters the recursion relation, allowing thus todetermine in which conditions two scattering operators will have identical momentsup to some order.

However, as the example of the calculation of the mean square distance toabsorption <r2> (a moment of order n=2) shows, the upper bound kmax gives anoverestimation of the actual maximum order of the eigenvalue required to compute<r2> . This fact has motivated us to analyze weights that are invariant under arbi-trary rotations. Weights that can be written as a product of rn times a Legendrepolynomial of order k of � � r=r. We have derived a general recursion relation forthese weights and found that the only point where the recursion stops is (n=0, k=0).We have proved that the only moments with recursions ending at (n=0, k=0) arethose with n and k of the same parity and n5k, and we have proved that themaximum order of the eigenvalue needed in the recursion for one of thesemoments is kmax=(n+k)/2. In the case of the diffusion equation we have useddetailed balance to extend the result for <r2> to the case of a medium at thermalequilibrium.

Next, we have turned our attention to the derivation of partial transversemoments, as done in the work by Franke and Larsen, and we have used weightfunctions that are invariant under rotations around the z axis. Such weights areof the form �n e� � �=�

� �k, where � is the radius vector in the xy plane and e� is

the unit vector in the direction of the projection of � onto the xy plane. Wehave established a general recursion for these moments and found again thatonly the moments with n and k of the same parity and n5k give rise to a finiterecursion that ends at (n=0, k=0). All these transverse moments can be calcu-lated by recursively solving a system of one-dimensional transport like equa-tions.

It is clear from our derivations and that of previous authors that the MomentsMethod requires the use of a family of weights that is closed under the action of theoperators appearing in the equation. For instance, in the case of the one-grouptransport equation the family of weights has to obey the following replicationproperties:


� � rwk r;�ð Þ ¼Pk0�k0wk0 r;�ð Þ;

Ðd�P �0 ��ð Þwk r;�ð Þ ¼

Pk0�k0wk0 r;�

0ð Þ:

The interest of the Moments Method is that it gives an insight into global prop-erties of the solutions of the transport and diffusion equations, as well as a way toderive lower dimensional transport or diffusion like equations that allows one toobtain the behavior of some of the transverse moments of the solutions. By far themore interesting insight has come from the work of Leakeas and Larsen whoestablished a relationship between the eigenvalues of the scattering operator and theglobal moments of the solution of the transport equation.

Acknowledgements

The author thanks Ed Larsen, Mike Williams and Paul Reuss for their advice inthe earlier stages of this work. The calculation of <r2> in Appendix B has beenderived from a problem proposed for the examination of the Diplome d’EtudesApprofondies of Reactor Physics at the University of Paris XI.

Appendix A. Spherical harmonics and the scattering operator

In this work we use real spherical harmonics Akl �ð Þ; k5 0; lj j4 k �

defined asfollows:

Akl �ð Þ ¼ffiffiffiffiffiffi�kl

pPlk ð Þcosl’ 04 l4 k;

Ak;�l �ð Þ ¼ffiffiffiffiffiffi�kl

pPlk ð Þsinl’ 0 < l4 k;

where the

�kl ¼ 2� �l0ð Þ 2kþ 1ð Þk� lð Þ!

kþ lð Þ!

are normalization constants and

Plk ð Þ ¼ 1� 2

� �l=2@lPk ð Þ

is the Legendre function with Pk() the Legendre polynomial of order k.The spherical harmonics Akl(�) are a complete set of orthonormal functions in

L2(S), the space of square summable functions over the surface of the sphere S. Thereal spherical harmonics satisfy the following relations:


�kk0�ll0 ¼1

4�

ðd�Akl �ð ÞAk0l0 �ð Þ;

�2 �0��ð Þ ¼

1

4�

Xk;l

Akl �0

ð ÞAkl �ð Þ;

Pk �0��ð Þ ¼

1

2kþ 1

Xl

Akl �0

ð ÞAkl �ð Þ: ð21Þ

Any square summable function of � can be developed over the spherical harmo-nics. For instance,

r;�ð Þ ¼1

4�

Xk;l

Akl �ð Þ�kl rð Þ;

where the angular flux moments are:

�kl rð Þ ¼

ðd�Akl �ð Þ r;�ð Þ:

A1. Treatment of the scattering operator

In an isotropic media the scattering operator H is invariant under rotations andtherefore commutes with arbitrary rotations. Hence, the spherical harmonics form acomplete set of eigenfunctions of H. To obtain the eigenvalues of H it is convenientto expand P � ��0

ð Þ over Legendre polynomials:

P � ��0ð Þ ¼

1

4�

Xk5 0

2kþ 1ð ÞfkPk �0��ð Þ; ð22Þ

where the expansion coefficients fk are the average values of Pk � ��0ð Þ for the den-

sity of probability P � ��0ð Þ:

fk ¼ Pkh i ¼

ðd�Pk �0

��ð ÞP �0��ð Þ:

In particular f0 ¼ 1 f1 ¼ and f2 ¼ 1=2ð Þ 3 2 � 1� �

, where ¼ �0�� is the

cosine of scattering.Next, with the help of (22) and (21) we can evaluate the action of H on a function

of the angular direction. For example:

H ð Þ r;�ð Þ ¼1

4�

Xk;l

fkAkl �ð Þ�kl rð Þ:


e case where the function itself is a spherical harmonic we find:
In th
HAklð Þ �ð Þ ¼ fkAkl �ð Þ

which proves that the Akl(�) are eigenfunctions of the scattering operator witheigenvalue fk.

Finally, from the expressions for the spherical harmonics one finds the well-knownexpansion:

P �0��ð Þ ¼

1

4�

Xk5 0

2kþ 1ð ÞfkXl¼k

l¼0

2� �10ð Þk� lð Þ!

kþ lð Þ!Plk ð ÞPl

k 0ð Þcosl �� 0ð Þ�: ð23Þ

Appendix B. Direct calculation of <r2>

We consider one-group transport in an infinite, homogenous and isotropic med-ium with a unit source located at the origin of coordinates and emitting particles indirection �0. To compute \left hr2i we use a Neumann expansion:

r2� �

¼Xn5 0

ðPn rð Þr2dr;

wherePn(r) is the density of probability for all trajectories undergoing exactly n scatteringevents and ending with an absorption at r. The end position of such trajectories can bewritten as r ¼

Pi¼ni¼0li�i, where li is the distance travelled between the i-th and the

(i+1)-th collision sites, or, for i=0, the distance between the emission point and thefirst collision site, and �i is the angular direction after the ith collision. We cantherefore write: !

Pn rð Þ ¼ 1� cð ÞcnðYi¼n

i¼0

P lið Þdli½ �Yi¼n

i¼1

P �i�1 ��ið Þd�i½ �� r �Xj¼n

j¼0

lj�j ;

where the integration is over dl0. . .dlnd�1. . .d�n, P(l)=�exp(��l) is the density ofprobability for a length l between two collisions (or from emission to first collision)and P �0

��ð Þ is the density of probability for a particle entering a scattering indirection �0 to exit it with angle �.

With the help of the expression for Pn(r) we can write

r2� �

¼ 1� cð ÞXn5 0

cn < r2 >n; ð24Þ

where now

r2� �

n¼

ðYi¼n

i¼0

P lið Þdli½ �Yi¼n

i¼1

P �i�1 ��ið Þd�i½ �Xj¼n

j¼0

lj�j

!2


e mean square distance to absorption after exactly n scattering events.
is thBy expliciting the terms in the square of the sum,
Xj¼n

j¼0

lj�j

!2

¼Xj¼n

j¼0

l2j þ 2Xj¼n�1

j¼0

Xk¼n

k>j

ljlk�j ��k;

and by carrying out the integrations over the li and using the normalization condi-tion

Ðd�P �0

��ð Þ ¼ 1 we obtain:

r2� �

n¼

2

�2nþ 1þ

Xj¼n

j¼0

Xk¼n

k>j

ðYi¼n

i¼1

P �i�1 ��ið Þd�i½ ��j ��k

( ):

The remaining integrations over the �i are computed in decreasing order in i. Theintegrations for i>k give 1, the integration over �k yields �k�1, the next integra-tion gives 2�k�2 and so on until we reach the integration over �j that, because ofthe scalar product �j

.�j=1, gives 1, and so do the remaining integrations for i< j.Therefore, the multiple integral in the proceeding formula amounts to k�j and wecan write:

r2� �

n¼

2

�21þ

1

1� n� 2

1� n

1�

� � �: ð25Þ

For isotropic scattering, ¼ 0, collisions are uncorrelated and the mean-squaredistance after n+1 collisions is n+1 times the mean square distance for a singlecollision, r2

� �n¼ nþ 1ð Þ 2=�2

� �. On the other hand, for a completely correlated scat-

tering law of the type P �0��ð Þ ¼ �2 �0

��ð Þ, with ¼ 1, the mean square distanceincreases quadratically with the number of collisions, r2

� �n¼ nþ 1ð Þ nþ 2ð Þ=�2.

Finally, using this result in (24) and with the help of the formula �nnxn ¼ x@x�nx

n

we obtain:

r2� �

¼2

�2 1� cð Þ 1� cð Þ:

For isotropic scattering this formula gives r2� �

¼ 2= �2 1� cð Þ�

, whereas for ‘delta’scattering we have 2= � 1� cð Þ½ �

2. The latter value can be directly calculated bynoticing that the particles follow straight trajectories and that the density of prob-ability for absorption at a distance x, Pa xð Þ ¼

Paexp �

Pax

� �, predicts

r2� �

¼Ð10 dxPa xð Þx2 ¼ 1=�2

a.

References

Borgers, C., Larsen, E.W., 1995. The transversely integrated scalar flux of a narrowly focused particle

beam, SIAM. J. Appl. Math. 55, 1.


Brantley, P.S., Larsen, E.W., 1998. The center of mass and radius of gyration of neutron distributions.

Trans. Am. Nucl. Soc. 78, 114.

Franke, B.C., Larsen, E.W., 1999. 1-D calculations of 3-D coupled electron-photon beams. In: ANS Int.

Top. Mtg. on Mathematics and Computations, Reactor Physics and Environmental Analysis in

Nuclear Applications. M & C ‘99, Vol. 1, Madrid, Spain, pp. 446–455.

Larsen, E.W., 1996. How rapidly and how far do neutrons migrate? Ann. Nucl. Energy 23, 341.

Larsen, E.W., 1997. The amplitude and radius of a radiation beam. Trans. Th. Statist. Phys. 26, 533–554.

Leakeas, C.L., Larsen, E.W., 2001. Generalized Fokker-Planck approximations of particle transport with

highly forward-peaked scattering. Nucl. Sci. Eng. 137, 236–250.

Lewis, H.W., 1950. Multiple scattering in an infinite medium. Phys. Rev. 78, 526.

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The Moments Method for the one-group transport and diffusion equations

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