Milne problem for linearly anisotropic scattering and a specularly reflecting boundary

Annals of Nuclear Energy 27 (2000) 1483±1504 www.elsevier.com/locate/anucene

Milne problem for linearly anisotropic scattering and a specularly re¯ecting boundary M.A. Atalay * Institute for Nuclear Energy, Istanbul Technical University, Maslak, Istanbul, 80626, Turkey Received 30 July 1999; received in revised form 1 September 1999; accepted 2 February 2000

Abstract The Milne problem is investigated subject to re¯ecting boundary conditions. The original version of the problem with vacuum boundary condition is generalized assigning, to the surface x ˆ 0, a re¯ection coecient R…04R41†. Linearly anisotropic case is studied using singular eigenfunction method. The cases of non-multiplying and multiplying medium are covered. The separate treatment of non-absorbing medium is also included. The singular eigenfunction method yields good accuracy with an optimized ®rst order approach. Solution of the Milne problem is formulated in terms of characteristic parameters such as extrapolated endpoint, emergent angular distribution and total and asymptotic neutron densities. Numerical results for the analytically evaluated parameters are then presented. # 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction The Milne problem is one of the classical problems of radiative transfer and neutron transport. It involves a source far from the boundary, and the medium, as considered generally, is surrounded by perfectly absorbing medium at x ˆ 0. With no incoming particles at the boundary, the solution of this problem reveals many results which are quite important for neutron transport. The Milne problem with vacuum boundary condition is formulated and solved by Placzek and Seidel (1947) and Noble (1958) using Wiener±Hopf method. Variational approach was employed by Marshak (1947) and LeCaine (1947). The singular eigenfunction method of Case is also used quite e€ectively to obtain solutions. The formulation of the problem and * Fax: +90-0212-2853884. 0306-4549/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0306-4549(00)00011-6

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solution for isotropic and linearly anisotropic cases are given by Case and Zweifel (1967), McCormick (1964) and McCormick and KusÏ eÏer (1965). If vacuum is replaced with a re¯ecting medium, this version of the problem also yields signi®cant results as pointed out in Razi et al. (1991). The Milne problem with re¯ecting boundary conditions has been examined by Williams (1975) using Wiener± Hopf method and by Razi et al. (1991) using variational methods. The former work treats specular and di€usive re¯ecting boundary. The latter study considers only purely specular re¯ecting boundary as the present article does. The considered case in these works is restricted to medium in which only scattering collisions …c ˆ 1† are permitted. Furthermore only isotropic scattering case has been treated in all related works. In this paper, we apply singular eigenfunction method to Milne problem to obtain analytic solutions for linearly anisotropic scattering case that isotropic scattering results follow from this as a special case. Also we extend the Milne problem with re¯ecting boundary condition formulation to non-multiplying …c < 1† and multiplying …c > 1† medium cases. In terms of the singular eigenfunction methodolgy, c ˆ 1 condition exhibits a special case due to in®nite roots of the discrete modes and treated separetely in this paper. Numerical results for extrapolated endpoints, emergent angular distribution and asymptotic and total neutron densities are given. However, c 6ˆ 1 cases and anisotropy (restricted to linear) are attempted ®rst time by this study. Therefore, comparison of the results, with those of the literature is made in terms of availability of the results. 2. Linearly anisotropic scattering We begin by considering linearly isotropic scattering case. The one-speed, transport equation in the plane geometry for linearly anisotropic scattering is given in the form (Case and Zweifel, 1967) 

@ …x; † c ‡ …x; † ˆ @x 2

…1 ÿ1

d0 …x; 0 †…1 ‡ 3f1 0 †

…1†

where f1 is the mean cosine of the scattering angle in a collision. If c41 ‡ 1=f1 condition is satis®ed, Eq. (1) will have one-pair of discrete modes as in isotropic case (Mika, 1961). Thus, the general solution of Eq. (1) …1 dA…† v …x; † …2† …x; † ˆ a0‡ 0‡ …x; † ‡ a0ÿ 0ÿ …x; † ‡ ÿ1

where the discrete and continuum modes are given by 0 …† ˆ  …† ˆ

c0 d…0 † 2 0  

c d…† P ‡ l…†… ÿ † 2 ÿ

…3† …4†

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and l…† ˆ d…2 †…1 ÿ c tanhÿ1 † ÿ 3f1 …1 ÿ c†2 2

…5†

d…ab† ˆ 1 ‡ 3f1 …1 ÿ c†ab

…6†

The Milne problem involves a source far from the boundary at x ˆ 0 such that the neutron distribution approach to one of the discrete modes for large x …x; † !

x!1

0ÿ …x; †

…7†

Hence the Milne problem with purely specularly re¯ecting boundary is formulated subject to boundary condition at x ˆ 0 …0; † ˆ R …0; ÿ†

>0

…8†

where R is the re¯ection coecient …04R41†: In detail, R, here is considered to be specular re¯ection coecient charecterizing that the re¯ected particle direction depends upon its incident direction. In other words, perfect re¯ection takes place in the tangent plane at the surface point under consideration. On the other hand, if the particle is re¯ected from the surface in such a random way that all trace of their past history is lost, this di€erent mechanism of re¯ection is called di€use re¯ection that we neglect entirely in this study (Williams, 1971). The solution to the Milne problem can be taken linear combination of fundamental solution for half-range plus 0ÿ …x; †: Thus …1 …x; † ˆ 0ÿ …x; † ‡ a0‡ 0‡ …x; † ‡ dA…† v …x; † …9† 0

The solution in terms of angular distribution will be complete once the expansion coecients a0‡ and A…v† are determined. To use the boundary condition at half space surface, we ®rst write down Eq. (9) in the form …0; † ˆ

0ÿ …0; †

‡ a0‡

0‡ …0; †

‡

…1 0

dA…†

v …0; †

…10†

We use this in Eq. (8), and make use of the identities 0 …† ˆ 0 …ÿ† and  …† ˆ  …ÿ† to get a singular integral equation ‰1 ÿ Ra0‡ Š0ÿ …† ‡ ‰a0‡ ÿ RŠ0‡ …† ‡

…1 0

dA…†‰v …† ÿ Rÿv …†Š ˆ 0

…11†

…12†

To evaluate unknown expansion coecients, we need the following biorthogonality relations

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…1 0

…1 0

…1 0

…1 0

d0‡ …†‰0‡ …† ‡ Bcv0 =2Š …†…0 ÿ † ˆ ÿ…

d0ÿ …†‰0‡ …† ‡ Bcv0 =2Š …†…0 ÿ † ˆ …

c0 2 † X…0 †d…20 † 2

c0 2 d…20 †d…ÿ0 † † X…ÿ0 † 2 d…0 †

d …†‰0‡ …† ‡ Bcv0 =2Š …†…0 ÿ † ˆ 0

dÿ …†‰0‡ …† ‡ Bcv0 =2Š …†…0 ÿ † ˆ

…13†

…14†

…15† c2 0  d…20 †d…ÿ† X…ÿ† 2 d…0 †

…16†

These relations are given by McCormick and KusÏ cÏer (1965). We still need other relations noted by Atalay (1997)   h c i c2  0  d…0 † 1

…† ˆ X…0 † ÿ d0‡ …†  …† ‡ 2 4 0 ÿ   0

…17†

  h c i c2  0  d…ÿ0 † 1 ‡

…† ˆ X…ÿ0 † d0ÿ …†  …† ‡ 2 4 0 ‡   0

…18†

…1

…1

…1

h c i N…† …† … ÿ 0 † d0 …†  …† ‡

…† ˆ 2  0

…19†

  h 0 c i c2 0 1 0 d…ÿ † X…ÿ † 0 ‡

…† ˆ dÿ0 …†  …† ‡ 4 2  ‡  0

…20†

…1

where Bˆ

3f1 …1 ÿ c†…0 ÿ † d…0 †

…† ˆ

c 1 2 2 2 X…ÿ†…0 ÿ  †…1 ÿ c†…1 ÿ cf1 †

 ˆ …1† = …0†

…n†

ˆ

…1 0

dn …†

…21†

…22†

…23† …24†

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

…1

1487

…†d…2 † ÿz 0   … c 1 c2 2 2 2 dg1 …c; † d2 …2 †…1‡ † ‡ 3f …1ÿc†  d…ÿ † ln… ÿ z† ˆ exp‰ÿ 1 1 ÿ 2 2 0 …25†

X…z† ˆ

d

 N…† (  2 )ÿ1   cd…2 † ÿ1 2 2 2 2 ˆ d… †…1 ÿ c tanh † ÿ 3f1 …1 ÿ c†  ‡ 2

g1 …c; † ˆ

…v† ˆ d…2 †…1 ÿ

c  ‡ 1 ln † ÿ 3f1 …1 ÿ c†2 2 2 ÿ1

…26†

…27†

We now multiply Eq. (12) by ‰0‡ …† ‡ Bcv0 =2Š …†…0 ÿ †; and then ‰ …† ‡ cv=2vŠ …† and integrate over  from 0 to 1 to obtain ‰a0‡ ÿ RŠ0 X…0 †d…0 † ÿ ‰1 ÿ Ra0‡ Š0 X…ÿ0 †d…ÿ0 † …1 ‡ R dA…†X…ÿ†d…ÿ† ˆ 0

…28†

(   …c=2†2 d…0 † 1 ÿ ‰a0‡ ÿ RŠ0 X…0 † A…† ˆ ÿ N…† …† 0 ÿ     d…ÿ0 † 1 ‡ ‡ ‰1 ÿ Ra0‡ Š0 X…ÿ0 † 0 ‡    ) …1 d…ÿ† 1 ‡ ÿ R dA…†X…ÿ† ‡  0

…29†

0

and

Thus, we have reduced a singular integral equation given by Eq. (12) to Fredholm integral equations given by Eqs. (28) and (29). The treatment here is similar to the critical slab problem given by Mitsis (1963: isotropic scattering with vacuum boundary condition) and Atalay (1996, 1997: isotropic and linearly anisotropic

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scattering with re¯ecting boundary conditions). Using half-space orthogonality relations does not lead to a closed form solution but rather we require an iterative approach. For the ®rst order approximation, we neglect the integral term in Eq. (29). Order of the error involved in this truncation is discussed in Section 4. Thus we slightly rearrange the rest of terms in Eq. (29) and write the ®rst unknown expansion coecient in the form (   …c=2†2 0 X…0 †d…0 †  0 ÿ  ÿ ‰a0‡ ÿ RŠ A…† ˆ ÿ 0 ÿ   d…0 † N…† …† …30†  ) 0 X…ÿ0 †d…ÿ0 †  0 ‡  ‡ ‡ ‰1 ÿ Ra0‡ Š 0 ‡   d…ÿ0 † We use Eq. (30) in Eq. (28) and the de®nitions given by Eqs. (22) and (26), and also de®ne …1 g1 …c; † 2 X …ÿ†d…ÿ† …31† Kj ˆ c…1 ÿ c†…1 ÿ cf1 † dj 2 0 and obtain the expansion coecient a0‡ in the form    X…ÿ0 † d…ÿ0 † d…20 †K1 ÿ …0  ‡ 20 †K0 1 ÿ R K2 ‡ d…ÿ0 † X…0 † d…0 †

a0‡

…32†

   d…20 †K1 ‡ …0  ÿ 20 †K0 ‡R 1 ÿ R K2 ‡ d…0 † ˆ    X…ÿ0 † d…ÿ0 † d…20 †K1 ÿ …0  ‡ 20 †K0 1 ÿ R K2 ‡ R X…0 † d…0 † d…ÿ0 † 

  d…20 †K1 ‡ …0  ÿ 20 †K0 ‡ 1 ÿ R K2 ‡ d…0 † Thus Eqs. (9), (30) and (32) yield angular distribution for linearly anisotropic scattering. 2.1. Extrapolated endpoint The general solution for the Milne problem is given by Eq. (9). Here the discrete modes alone gives asymptotic density distribution as …x; †

ˆ

0ÿ …x; †

‡ a0‡

0‡ …x; †

…33†

The total asymptotic density distribution is obtained integrating Eq. (33) over all directions

as …x† ˆ

…1 ÿ1

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d

…34†

as …x; †

At the point in which as …x†vanishes gives the extrapolated endpoint z0 . Thus a0‡ ˆ ÿe2z0 =0

…35†

or z0 ˆ

0 0 ln‰ÿa0‡ Š ˆ ln‰a0‡ ‡ iŠ 2 2

…36†

2.2. Emergent angular distribution In order to evaluate the emergent angular distribution, we use Eq. (30) in Eq. (10), and manipulate the resulting equation to get 0 …0; †

ˆ 0ÿ …† ‡ a0‡ 0‡ …† … c0 1 …cv=2†2 d…† d ‰ ÿ 0 ‡ d…0 †Š ÿ ‰a0‡ ÿ RŠX…0 † 2 0 …†N…†…0 ÿ †… ÿ † … c0 1 …cv=2†2 d…† d ‰d…ÿ0 † ‡ 0 ‡ Š;  < 0 ÿ ‰1ÿRa0‡ ŠX…ÿ0 † 2 0 …†N…†…0 ‡ †… ÿ † …37† The integral expressions in Eq. (37) can be evaluated using complex contour integration. Using the details provided in the Appendix, we can show that the following identities are valid … c0 1 …cv=2†2 d…† d ‰ ÿ 0 ‡ d…0 †Š 2 0 …†N…†…0 ÿ †… ÿ †     c0 d…0 † 1 1 c0 c0 1 ÿ d…0 † ‡ ‡  ÿ  ;  < 0 ˆ ‡ 2 …0 ÿ † X…0 † X…† 2 2 X…†

…38†

… c0 1 …cv=2†2 d…† d ‰d…ÿ0 † ‡ 0 ‡ Š; 2 0 …†N…†…0 ‡ †… ÿ †     c0 d…ÿ0 † 1 1 c0 c0 1 ÿ d…ÿ0 † ÿ ‡  ÿ  ;  < 0 ˆ ÿ 2 …0 ‡ † X…ÿ0 † X…† 2 2 X…†

…39†

Using Eqs. (38) and (39) in Eq. (37), we obtain the emergent angular distribution after some algebra

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0 …0; †

ˆ

    X…0 † …0 ÿ † … ÿ  † ÿ …0 ÿ † 1 ‡ X…† 0‡ …† 1ÿ R ‡ ‰a0‡ ÿ RŠ X…† d…0 ; † d…0 ; †      X…ÿ0 † …0 ‡ † …ÿ † ‡…0 ‡ † 1 ‡ 1ÿ X…† 0ÿ …† ‡ Ra0‡ ‡‰1ÿRa0‡ Š X…† d…ÿ0 ; † d…ÿ0 ; †

<0 …40†

where a0‡ is given by Eq. (32). 2.3. Neutron density In Section 2.1. we de®ned total scalar asymptotic density distribution. On the other hand, the total scalar density distribution …x† ˆ

…1 ÿ1

d …x; †

…41†

and the deviation between total scalar density distribution and total asymptotic scalar density distribution is given by the relation …x† ˆ as …x† ÿ h…x†

…42†

where …1 h…x† ˆ ÿ dA…†eÿx= 0

…43†

To compute …x†, the appropriate expression is given by …x† ˆ ex=0 ‡ a0‡ eÿx=0

…1 1 …cv=2†2 ‰ ÿ 0 ‡ d…0 †Šeÿx= ‡ ‰a0‡ ÿ RŠ0 X…0 † d

…†N…†… ÿ 0 †  0 …1 1 …cv=2†2 ‰0 ‡  ‡ d…ÿ0 †Šex= ÿ ‰1 ÿ Ra0‡ Š0 X…ÿ0 † d 

…†N…†… ‡ 0 † 0

…44†

3. The nonabsorbing medium In singular eigenfunction treatment, when c ˆ 1, the eigenvalues of the discrete modes coalesce at the in®nity (McCormick and KusÏ cÏer, 1965; Case and Zweifel, 1967). Hence this case requires a special treatment. On the other hand, as stated in the introduction section, the Milne problem with gray boundary condition

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1491

treatment in the literature is restricted to this case. Therefore, a comparison of this article results with those of the related works is made possible considering the nonabsorbing medium. We take linearly anisotropic scattering case. The transport equation is 

@ …x; † 1 ‡ …x; † ˆ @x 2

…1 ÿ1

d0 …x; 0 †…1 ‡ 3f1 0 †

…45†

Because of the degeneracy in the discrete modes, a solution is sought in the form …x; † ˆ a1 …† ‡ a2 …†x

…46†

Substituting Eq. (46) into Eq. (45) and using appropriate normalization conditions, the required discrete modes can be found in the form 1 2

…47†

1 …x; †

ˆ

2 …x; †

3 ˆ ‰…1 ÿ f1 †x ÿ Š 2

…48†

Hence the general solution for the Milne problem is given by considering …x; † ! 2 …x; † x!1 …1 …49† …x; † ˆ 2 …x; † ‡ a0‡ 1 …x; † ‡ dA…†v …†eÿx= 0

Using the boundary condition given by Eq. (8), we obtain 3 1 …1 ‡ R† ÿ a0‡ …1 ÿ R† ÿ 2 2

…1 0

dA…†‰v …† ÿ Rÿv …†Š ˆ 0

…50†

We need again to determine unknown expansion coecients. The appropriate orthogonality conditions are given by McCormick and KusÏ cÏer (1965) …1 0

…1 0

…1 0

d …† …† ˆ 0

dÿ …† …† ˆ

…51† 32 4 …†

d …†0 …† …† ˆ

…†N…† … ÿ 0 † 

…52†

…53†

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…1 0

…1 0

d …†0 …†ÿ …† ˆ

d …† …† ˆ ÿ

32 0 …ÿ† 4 …†

…54†

 2

…55†

Multiplying Eq. (50) by …†and …†0 …† in turn, and integrating over  from 0 to 1, and using Eqs. (51) through (55), we get 3 1 3R …1 ‡ R† …1† ÿ a0‡ …1 ÿ R† …0† ‡ 2 2 4 and A…† ˆ ÿ

…1 0

d

2 A…† ˆ0

…†

  …1 3  2  …ÿ† …1 ‡ R† ÿ R dA…† 4 …†N…†

…† 0

…56†

…57†

As usual, we omit integral term in Eq. (57) and set up the ®rst order expansion coecient for A…† A…† ˆ ÿ

3 2 …1 ‡ R† 4 …†N…†

…58†

We note that Eqs. (51) to (55) are exactly the same form in the isotropic scattering case. This can be seen considering lim d…ab† ˆ 1

…59†

c !1

At the same time, we note that lim 20 …1 ÿ c†…1 ÿ cf1 † ˆ

c !1

lim …† ˆ

c !1

3 2X…ÿ†

lim …0† ˆ 1

c !1

1 3

…60†

…61† …62†

The rest of the required equations can be written down from isotropic counterpart by setting c to unity. We now substitute Eq. (58) into Eq. (56) and use Eqs. (61) and (26) and also de®ne

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

Kˆ

…1 0

dX2 …ÿ†g 1 …†

1493

…63†

where g 1 …† ˆ g1 …1; †: We then obtain a0‡ in the form a0‡ ˆ

  …1 ‡ R† RK 3 …1† ÿ …1 ÿ R† 2

…64†

3.1. Extrapolated endpoint Since we have for asymptotic angular distribution as …x; †

ˆ

2 …x; †

‡ a0‡

1 …x; †

…65†

for extrapolated endpoint, we consider …1 ÿ1

d

as …z0 ; †

ˆ0

…66†

Using Eqs. (47) and (48) in Eq. (66), we obtain z0 ˆ ÿ

a0‡ 3…1 ÿ f1 †

…67†

3.2. Emergent angular distribution We consider Eq. (49) for x ˆ 0 and  < 0, and use Eq. (58) to get …0; † ˆ ÿ

3 1 3…1 ‡ R† ‡ a0‡ ÿ 2 2 4

…1 0

d

v …† ;

…†N…†=

<0

…68†

Here, we use the following identity evaluated with complex contour integration   v …† 1 ˆ ÿ2 ‡  ÿ  ; c ˆ 1;  < 0 d

…†N…†= X…† 0

…1

…69†

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We then obtain the emergent angular distribution in the form   3 1 3…1 ‡ R† 1 ‡ a0‡ ‡ ‡  ÿ  ; …0; † ˆ ÿ 2 2 2 X…†

<0

…70†

3.3. Neutron density The total scalar density distribution in this case can be written as …x; † ˆ 3…1 ÿ f1 †x ‡ a0‡ ÿ

3…1 ‡ R† 4

…1 0

d

2 eÿx=

…†N…†

…71†

or for numerical evaluation, this can be put into the form …x; † ˆ 3…1 ÿ f1 †x ‡ a0‡ ÿ

…1 ‡ R† 2

…1 0

dg 1 …†X…ÿ†eÿx=

…72†

4. Numerical results and discussion In this section, numerical results and discussion of this study will be presented, based on the analytic solution of the Milne problem with re¯ecting boundary conditions. The solution of this problem follows a close route the critical slab problem with the same type of boundary conditions (Atalay, 1996, 1997). As stated by Kaper et al. (1974) for the slab criticality problem with vacuum boundary conditions, the singular eigenfunction method yields results in benchmark quality. However, in the solution Fredholm integral equation, an iteration is required until the convergence is attained. On the contrary, here A…† is simply truncated without further iteration to obtain a solution by the ®rst order approach. Therefore, we investigated the error involved in this approximation. For this purpose, we have numerically solved integral equations given by Eqs. (29) and (57) and estimated the error for the neglected terms for all values of c and R. For c 6ˆ 1 case, the ratio of the neglected term to the retained terms is always considerably less than 1% which is very small indeed. This includes for both real and complex A…†, namely the deviation in its real and imaginary part for complex A…†. On the other hand for c ˆ 1 case, the deviation is about 1% or somewhat larger if  is close to zero. However, this is due to fact that for this value of , Eq. (57) has endpoint singularity, which is believed to aggravate the error than actual order while we have used relatively simple solution method not considering this situation in detail. Thus, we conclude that the ®rst order approximation will be sucient for the required optimum accuracy. In this paper, we tabulated results. In Tables 1±4, the results for extrapolated endpoints are given depending upon f1 …f1 ˆ 0; 0:1; 0:2; 0:3†. Obviously, f1 ˆ 0 case represents isotropic case, while the tabulated results are outcome of linearly

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

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Table 1 The extrapolated endpoint for various values of c and R in isotropic scattering …f1 ˆ 0† c\R

0.0

0.1

0.2

0.4

0.5

0.6

0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 1.0 1.01 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

8.53829 3.92391 2.49473 1.8249 1.44085 1.19226 1.01806 0.889055 0.789569 0.717624 0.710447 0.703413 0.645971 0.592392 0.547144 0.50841 0.474869 0.445536 0.419659 0.396659 0.376078 0.357551

2.27753 1.4662 1.16917 0.98923 0.874223 0.863245 0.85263 0.76845 0.693779 0.633123 0.582702 0.540031 0.503394 0.47156 0.44362 0.418884 0.396822

1.68306 1.2701 1.07081 1.05321 1.03658 0.910202 0.805719 0.72496 0.660143 0.606701 0.561731 0.52328 0.489969 0.460799 0.435018

2.78782 1.67349 1.61893 1.57094 1.26283 1.05948 0.921642 0.819773 0.740425 0.676383 0.623343 0.57854 0.540101 0.5067

2.17925 2.06902 1.97776 1.47725 1.19813 1.02309 0.899207 0.805374 0.731112 0.670497 0.619868 0.576819 0.539685

3.00337 2.74207 2.54916 1.71588 1.3409 1.12389 0.976564 0.867823 0.783279 0.715165 0.658838 0.61132 0.570594

6.09488 4.6736 2.24083 1.62414 1.31586 1.12077 0.982753 0.878486 0.796216 0.729259 0.673478 0.626155

0.9

12.78803 6.59641 2.50649 1.75762 1.40409 1.18623 1.03456 0.921221 0.832498 0.760724 0.701216 0.650928

0.99

133.19375 8.78419 2.7349 1.87021 1.47808 1.24101 1.07786 0.956921 0.862801 0.787005 0.724388 0.671626

Table 2 The extrapolated endpoint for various values of c and R in anistropic scattering …f1 ˆ 0:1† c\R

0.0

0.1

0.2

0.4

0.5

0.6

0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 1.0 1.01 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

7.01416 3.48694 2.36767 1.81642 1.48338 1.25788 1.0939 0.96873 0.869759 0.796739 0.789385 0.782166 0.722727 0.666498 0.618395 0.576755 0.540344 0.508227 0.47968 0.454136 0.431142 0.410332

2.66305 1.60988 1.28605 1.0936 0.970887 0.959161 0.947821 0.85771 0.777403 0.711805 0.656969 0.610309 0.570039 0.53488 0.503884 0.476329 0.451659

1.90875 1.41353 1.18974 1.17023 1.15183 1.0128 0.898513 0.810301 0.739439 0.680896 0.631508 0.589159 0.552367 0.520057 0.491422

3.46547 1.86287 1.79881 1.74285 1.39277 1.1678 1.01702 0.906103 0.819843 0.750223 0.692511 0.643696 0.601745 0.565227

2.43135 2.29891 2.19073 1.61954 1.31203 1.12166 0.98763 0.886308 0.80614 0.740659 0.685901 0.639269 0.598978

3.36816 3.04675 2.81596 1.86824 1.45869 1.22455 1.06634 0.949762 0.859136 0.786066 0.725567 0.674451 0.630567

6.77209 5.09019 2.40395 1.74541 1.41839 1.21189 1.06585 0.955457 0.868252 0.797169 0.737849 0.687431

0.9

14.2089 7.07602 2.67096 1.87934 1.50696 1.27772 1.11809 0.998676 0.905072 0.829225 0.766227 0.712886

0.99

147.993 9.27806 2.89957 1.99211 1.5812 1.33282 1.16178 1.03484 0.935892 0.856071 0.790005 0.734229

1496

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

Table 3 The extrapolated endpoint for various values of c and R in anistropic scattering …f1 ˆ 0:2† c\R

0.0

0.1

0.2

0.4

0.5

0.6

0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 1.0 1.01 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

6.17714 3.29781 2.35014 1.86301 1.55852 1.34683 1.18954 1.06727 0.969016 0.895513 0.888058 0.88073 0.820012 0.761894 0.71159 0.667561 0.628654 0.593987 0.562873 0.53477 0.509241 0.485932

1.80801 1.43474 1.22391 1.09165 1.07906 1.07756 1.06737 0.884309 0.813869 0.754751 0.7042 0.660332 0.621805 0.58763 0.557058 0.529511

2.23437 1.59592 1.33845 1.31651 1.29588 1.14155 1.01625 0.920086 0.842965 0.779201 0.725287 0.678907 0.638465 0.602777 0.571009

5.38068 2.1007 2.02366 1.95699 1.5535 1.30316 1.13798 1.01734 0.923825 0.848411 0.785855 0.732856 0.687202 0.647349

2.74969 2.58627 2.47969 1.79414 1.45344 1.24601 1.10109 0.991908 0.905626 0.835132 0.776111 0.725754 0.682141

3.83499 3.42759 3.14539 2.05382 1.6042 1.35108 1.18122 1.05645 0.959562 0.881431 0.816676 0.761876 0.714733

7.6186 5.64877 2.60079 1.89451 1.54686 1.32822 1.17388 1.05728 0.965136 0.889961 0.827142 0.773662

0.9

15.985 7.64635 2.86917 2.0289 1.63583 1.39451 1.22666 1.10115 1.00273 0.9229 0.856514 0.80022

0.99

166.492 9.86295 3.09798 2.14187 1.71037 1.45001 1.27088 1.13795 1.03429 0.950592 0.881238 0.822603

Table 4 The extrapolated endpoint for various values of c and R in anistropic scattering …f1 ˆ 0:3† c\R

0.0

0.1

0.2

0.4

0.5

0.6

0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 1.0 1.01 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

5.83801 3.29973 2.42529 1.96351 1.67067 1.46528 1.31175 1.19181 1.09501 1.02231 1.01492 1.00766 0.947338 0.889371 0.838976 0.794653 0.755276 0.719976 0.688075 0.659026 0.632385 0.607783

2.10014 1.63032 1.39124 1.24679 1.23321 1.22011 1.11694 1.02599 0.952025 0.890184 0.837388 0.791563 0.751247 0.715367 0.683109 0.653843

2.77686 1.83613 1.52974 1.50458 1.48103 1.30801 1.17091 1.06731 0.98502 0.917397 0.860422 0.811487 0.768801 0.731081 0.697375

2.40836 2.31276 2.25342 1.75789 1.47799 1.2977 1.1679 1.06823 0.988387 0.922469 0.866799 0.818939 0.777192

3.16444 2.955741 2.82016 2.01424 1.63487 1.40937 1.254 1.13806 1.04706 0.97311 0.911442 0.858993 0.813673

4.45436 3.91725 3.56262 2.28592 1.78997 1.51672 1.33565 1.20381 1.10212 1.02058 0.953307 0.896611 0.848012

8.70697 6.21286 2.8447 2.08397 1.71454 1.48427 1.32288 1.2017 1.10649 1.02925 0.965059 0.910723

0.9

18.2686 8.33929 3.11446 2.21885 1.80395 1.55111 1.37638 1.24651 1.14526 1.06363 0.99616 0.939325

0.99

190.277 10.5708 3.34349 2.33203 1.87883 1.60713 1.42131 1.28422 1.17796 1.0927 1.02252 0.963623

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

1497

anisotropic scattering formulation. In each table, for some c values, no extrapolated endpoint value is reported, because, the total asymptotic neutron density will not vanish for x < 0: In other words, Eq. (34) has no real roots, if the re¯ection is present. In Tables 5 and 6, we tabulated emergent angular distribution …0; ÿ† for the cosine of scattering angles 0.0 and 0.2 in negative direction. In Table 7 the

Table 5 The emergent angular distribution, …0; † for various values of c and R and depending on variation of  in isotropic scattering …f1 ˆ 0† c\R

ÿ

0.0

0.1

0.2

0.4

0.6

0.8

0.9

0.99

0.1

0.1 0.3 0.5 0.7 0.9

0.05425 0.07048 0.09924 0.16603 0.49945

0.05880 0.07433 0.10258 0.16898 0.50209

0.06339 0.07821 0.10593 0.17194 0.50473

0.07263 0.08601 0.11269 0.17789 0.51006

0.08199 0.09389 0.11951 0.1839 0.51543

0.09144 0.10185 0.12639 0.18996 0.52085

0.09621 0.10586 0.12985 0.19301 0.52358

0.10053 0.10949 0.13298 0.19577 0.52604

0.2

0.1 0.3 0.5 0.7 0.9

0.10565 0.13883 0.19674 0.33054 0.99682

0.11478 0.14656 0.20343 0.33644 1.0021

0.12403 0.15436 0.21019 0.3424 1.0074

0.14285 0.17021 0.2239 0.35448 1.0182

0.16211 0.1864 0.23788 0.36679 1.0292

0.18184 0.20292 0.25213 0.37932 1.0404

0.19187 0.2113 0.25935 0.38568 1.0461

0.201 0.21893 0.26592 0.39144 1.0512

0.4

0.1 0.3 0.5 0.7 0.9

0.19772 0.26565 0.37932 0.63224 1.7597

0.21599 0.28114 0.39275 0.64411 1.7703

0.23476 0.29699 0.40649 0.65623 1.7812

0.27382 0.32986 0.43492 0.68129 1.8036

0.31496 0.36431 0.46465 0.70746 1.827

0.3583 0.40044 0.49576 0.7348 1.8514

0.38082 0.41916 0.51185 0.74893 1.864

0.40159 0.43639 0.52664 0.76192 1.8756

0.6

0.1 0.3 0.5 0.7 0.9

0.26713 0.36264 0.50751 0.78618 1.6039

0.29422 0.38587 0.52784 0.80426 1.6202

0.32258 0.41011 0.54903 0.82308 1.6371

0.38344 0.46186 0.59413 0.86309 1.6731

0.4503 0.51836 0.64325 0.90656 1.7121

0.52387 0.58019 0.69684 0.95393 1.7546

0.56343 0.61331 0.7255 0.97923 1.7772

0.60073 0.64447 0.75243 1.003 1.7985

0.8

0.1 0.3 0.5 0.7 0.9

0.28702 0.38916 0.5322 0.77612 1.3256

0.31804 0.41611 0.55601 0.79744 1.3449

0.35097 0.4446 0.58114 0.81992 1.3653

0.42319 0.50675 0.63581 0.86874 1.4094

0.50503 0.57673 0.69716 0.92344 1.4587

0.5982 0.65592 0.76641 0.98506 1.5143

0.64968 0.69952 0.80445 1.0189 1.5447

0.69915 0.7413 0.84088 1.0512 1.5739

0.99

0.1 0.3 0.5 0.7 0.9

0.11002 0.1451 0.17858 0.21209 0.24628

0.12905 0.16377 0.19678 0.22981 0.26352

0.15185 0.18604 0.21845 0.25088 0.284

0.21397 0.24641 0.27704 0.30773 0.33921

0.31481 0.34385 0.37134 0.39911 0.42784

0.50486 0.52675 0.548 0.57007 0.59353

0.67984 0.69481 0.71016 0.72691 0.74546

0.94886 0.95291 0.9591 0.9676 0.97857

1.0

0.1 0.3 0.5 0.7 0.9

1.08024 1.42247 1.74312 2.05592 2.36476

1.29589 1.64234 1.96506 2.27915 2.58887

1.56729 1.91796 2.24275 2.55811 2.86871

2.38879 2.74791 3.07682 3.39475 3.70712

4.04642 4.41399 4.74703 5.06752 5.38165

9.04855 9.42456 9.76173 10.0848 10.4007

19.0747 19.455 19.7942 20.1185 20.4353

199.671 200.055 200.396 200.722 201.039

1498

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

Table 6 The emergent angular distribution, …0; ÿ† for various values of c and R and depending on variation of  in anisotropic scattering (f1 ˆ 0:2) c\R ÿ 

0.0

0.1

0.2

0.4

0.6

0.8

0.9

0.99

0.1

0.1 0.3 0.5 0.7 0.9

0.05763 0.08239 0.12655 0.22932 0.74271

0.06194 0.08562 0.12899 0.23115 0.74407

0.06627 0.08886 0.13143 0.23299 0.74543

0.07498 0.09538 0.13635 0.23669 0.74817

0.08376 0.10195 0.14131 0.24042 0.75093

0.09262 0.10856 0.1463 0.24418 0.75371

0.09708 0.11189 0.14881 0.24606 0.75511

0.10111 0.1149 0.15108 0.24777 0.75637

0.2

0.1 0.3 0.5 0.7 0.9

0.11244 0.16066 0.24574 0.44263 1.4156

0.12113 0.16727 0.25082 0.44655 1.4186

0.1299 0.17393 0.25594 0.4505 1.4216

0.14768 0.18742 0.26632 0.45849 1.4277

0.16578 0.20113 0.27685 0.46661 1.434

0.18422 0.21506 0.28755 0.47487 1.4403

0.19356 0.22211 0.29297 0.47904 1.4435

0.20204 0.22851 0.29787 0.48283 1.4464

0.4

0.1 0.3 0.5 0.7 0.9

0.21065 0.2995 0.44807 0.76841 2.0192

0.22828 0.31333 0.45908 0.77726 2.0263

0.2463 0.32743 0.47031 0.78628 2.0336

0.28353 0.35646 0.4934 0.80482 2.0485

0.32238 0.38666 0.51738 0.82408 2.0641

0.36292 0.41805 0.54228 0.84407 2.0802

0.38384 0.43422 0.5551 0.85436 2.0885

0.40305 0.44904 0.56684 0.86378 2.0961

0.6

0.1 0.3 0.5 0.7 0.9

0.28485 0.39871 0.56817 0.87481 1.6378

0.31148 0.42038 0.58607 0.88975 1.6503

0.33918 0.44286 0.60462 0.90523 1.6633

0.398 0.49037 0.64376 0.93786 1.6907

0.46173 0.54159 0.68586 0.97293 1.7202

0.53085 0.59687 0.73122 1.0107 1.7519

0.56761 0.62617 0.75523 1.0306 1.7686

0.60202 0.65354 0.77764 1.0493 1.7843

0.8

0.1 0.3 0.5 0.7 0.9

0.31012 0.4229 0.56226 0.75716 1.0654

0.3445 0.45245 0.5879 0.77958 1.0851

0.38138 0.48404 0.61526 0.80347 1.1061

0.46366 0.55414 0.67583 0.8563 1.1525

0.55934 0.55414 0.67583 0.8563 1.1525

0.67153 0.72958 0.82679 0.98765 1.2677

0.73506 0.78287 0.87251 1.0274 1.3025

0.79717 0.83485 0.91707 1.0661 1.3364

0.99

0.1 0.3 0.5 0.7 0.9

0.12138 0.16015 0.19708 0.23393 0.27134

0.14203 0.18031 0.21664 0.25287 0.28967

0.16665 0.20423 0.2398 0.27527 0.31133

0.23305 0.26845 0.30181 0.33514 0.36916

0.33886 0.37017 0.39977 0.42958 0.46028

0.53188 0.55499 0.57739 0.60059 0.62513

0.70264 0.71813 0.73401 0.75129 0.77034

0.95298 0.95706 0.96328 0.97181 0.98278

1.0

0.1 0.3 0.5 0.7 0.9

1.08024 1.42247 1.74312 2.05592 2.36476

1.29589 1.64234 1.96506 2.27915 2.58887

1.56729 1.91796 2.24275 2.55811 2.86871

2.38879 2.74791 3.07682 3.39475 3.70712

4.04642 4.41399 4.74703 5.06752 5.38165

9.04855 9.42456 9.76173 10.0848 10.4007

19.0747 19.455 19.7942 20.1185 20.4353

199.671 200.055 200.396 200.722 201.039

asymptotic neutron density, as …x†, and in Table 8 the deviation between total and asymptotic neutron density, h…x† at x ˆ 0 are given only isotropic scattering. The total neutron density …x† at x ˆ 0 is evaluated using these tables and due to Eq. (42). In Fig. 1, we sketched the variation of asymptotic neutron density near half space surface for various values of R in nonabsorbing medium. As seen clearly in Eq. (72), this variation is linear. Fig. 2, on the other hand, represents the corresponding variation of the deviation from asymptotic density function, h…x†:

Table 7 The asymptotic neutron density, as …x† at x ˆ 0 for various values of c and R in isotropic scattering …f1 ˆ 0† 0.0

0.1

0.2

0.4

0.5

0.6

0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 1.0

1.0 0.99961 0.99310 0.97260 0.93666 0.88508 0.81496 0.71725 0.56381 0.21933 2.13134

1.1 1.09961 1.09312 1.07259 1.03636 0.98396 0.91195 0.81009 0.64638 0.26039 2.58974

1.2 1.19962 1.19328 1.17313 1.13737 1.0852 1.0127 0.90849 0.73676 0.30889 3.15962

1.4 1.39967 1.39403 1.37597 1.34352 1.29534 1.22675 1.12467 0.94658 0.43864 4.85679

1.5 1.4997 1.49463 1.4783 1.44879 1.40457 1.34079 1.24404 1.06968 0.52853 6.20706

1.6 1.59974 1.59539 1.58129 1.55562 1.5168 1.46006 1.37221 1.20817 0.64521 8.22622

1.8 1.79985 1.79737 1.78924 1.77424 1.7511 1.71628 1.65989 1.54552 1.02901 18.2846

0.9

0.99

1.9 1.89992 1.8986 1.89426 1.88619 1.8736 1.85437 1.82244 1.75446 1.37985 38.3641

1.99 1.98999 1.98985 1.98939 1.98853 1.98718 1.98508 1.9815 1.97352 1.91732 399.581

Table 8 The deviation from asymptotic neutron density, h…x† at x ˆ 0 for various values of c and R in isotropic scattering …f1 ˆ 0† 0.0

0.1

0.2

0.4

0.5

0.6

0.8

0.9

0.99

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 1.0

0.03551 0.07252 0.10575 0.12820 0.13995 0.14321 0.13924 0.12745 0.10321 0.041 0.39929

0.03518 0.07193 0.10504 0.12763 0.13977 0.14365 0.1405 0.12972 0.10656 0.04381 0.43922

0.03414 0.06988 0.1022 0.12446 0.13674 0.14118 0.13895 0.12949 0.10803 0.0462 0.47915

0.02999 0.06137 0.09003 0.11016 0.12186 0.12702 0.12673 0.12054 0.10423 0.04921 0.5590

0.02674 0.05489 0.08066 0.09893 0.10983 0.11507 0.11565 0.11125 0.09822 0.04942 0.59893

0.02283 0.04693 0.06906 0.08492 0.09462 0.09966 0.10095 0.0983 0.0888 0.04827 0.63886

0.01286 0.02649 0.03911 0.04835 0.05428 0.05782 0.05957 0.05961 0.05685 0.0385 0.71872

0.0068 0.01401 0.02072 0.02567 0.02894 0.03101 0.03224 0.03277 0.0323 0.02581 0.75865

0.00071 0.00147 0.00218 0.0027 0.00306 0.0033 0.00346 0.00357 0.00364 0.00359 0.79458

1499

c\R

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

c\R

1500

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

Fig. 1. Asymptotic ¯ux variation near half space surface for c ˆ 1.

Fig. 2. The variation of the deviation from asymptotic ¯ux near half space surface for c ˆ 1.

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

1501

Table 9 Comparison of results of nonabsorbing medium (c ˆ 1) and isotropic scattering cases with literaturea,b R

0.0

0.1

0.2

0.4

0.5

0.6

0.8

…z0 †…I† …z0 †…II† …z0 †…III† …z0 †…IV†

0.7104 0.71045 0.71045 0.710447

0.8585 0.86322 0.85859 0.863245

1.0437 1.05321 1.04378 1.05321

1.5993

2.7104

6.0436

1.59933 1.61893

2.0437 2.0674 2.04378 2.06902

2.71045 2.74207

6.04378 6.09488

0.1 0.3 0.5 0.7 0.9

1.08024 1.42246 1.74311 2.05592 2.36474

1.30250 1.64475 1.96535 2.27815 2.58698

1.58032 1.92239 2.24317 2.55597 2.86480

2.41360 2.75585 3.07645 3.38926 3.69808

4.08017 4.42242 4.74302 5.05583 5.36483

9.08023 9.42248 9.74308 10.0559 10.3647

0.1 0.3 0.5 0.7 0.9

1.08024 1.42247 1.74312 2.05592 2.36476

1.29589 1.64234 1.96506 2.27915 2.58887

1.56729 1.91796 2.24275 2.55811 2.86871

2.38879 2.74791 3.07682 3.39475 3.70712

4.04642 4.41399 4.74703 5.06752 5.38165

9.04855 9.42456 9.76173 10.0848 10.4007

1.795 1.7323 1.73205

2.217 2.1768 2.15052

2.7323 2.68047

4.3990 4.29779

7.7323 7.58736

17.732 17.566

… †…III† c ÿ

… †…IV† ÿ

…†…II† …†…III† …†…IV† d

3.05038 3.41372 3.7447 4.06391 4.37716 5.683 5.61713

0.9

0.99

12.7104 12.754 12.7104 12.78803

133.194

19.0747 19.455 19.7942 20.1185 20.4353

199.671 200.055 200.396 200.722 201.039

37.590

397.57

37.605

398.786

132.75

a

z0 , the extrapolation endpoint; ( ), the emergent angular distribution …0; ÿ† and …†, the neutron density at x ˆ 0, …0†. b (I) Williams (1975); (II) Razi et al. (1991); (III) Abdel Krim and Degheidy (1998); (IV) present study. c After abandoning normalization of results. d The total neutron density is calculated by using Tables 7 and 8 according to Eq. (42).

In all these tables, c ˆ 1 values are given according to the calculations of Section 3. In Table 9, as restricted to only nonabsorbing medium (c ˆ 1) and isotropic scattering, we compared our results with those of related works of the literature. For extrapolated endpoint, comparison is made with Williams (1975), Razi et al. (1991) and Abdel Krim and Degheidy (1998). Our results here especially agree well with those of Razi et al. (1991). The emergent angular distribution results are compared with the study of Abdel Krim and Degheidy (1998). Abondoning normalization in this study for comparison is required as indicated in this table, and the maximum deviation, between the results is about 0.1%. Finally, our results, for neutron density at x ˆ 0 are compared with Razi et al. (1991) and Abdel Krim and Degheidy (1998). For this purpose, our results for total neutron density have been reevaluated using Eq. (42) and asymptotic neutron density and the deviation between asymtotic and total neutron densities given in Table 7 and 8. Here reasonable agreement between results are also observed. 5. Conclusion The solution of the Milne problem with re¯ecting boundary conditions is attempted by a di€erent method for a larger spectrum of di€erent cases as compared

1502

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

to existed works in this subject. Thus, this paper made following contributions. As mentioned in the introduction section, the singular eigenfunction method is used ®rst time in this problem. Also, other workers studied only nonabsorbing medium …c ˆ 1† and isotropic scattering. We extended this to both non-multiplying …c < 1† and multiplying …c > 1† medium cases. Furthermore, we have treated linearly anisotropic scattering case in this study. The results presented in this paper, have comparable accuracy with other works which have used di€erent methods. This is due to inherent accuracy of the singular eigenfunction method that produced quite accurate results even by using the ®rst order approximation. Appendix A In order to derive the identity given by Eq.(38), we ®rst write down this in the form … c0 1 …cv=2†2 d…† d ‰ ÿ 0 ‡ d…0 †Š 2 0 …†N…†…0 ÿ †… ÿ † ˆ

c0 d…0 † 2 0 ÿ 

‡

c0 2

…1 0

d

…1 0

d

…cv=2†2 d…2 † c0 d…0 † ‡

…†N…†…0 ÿ † 2 0 ÿ 

…1 0

d

…cv=2†2 d…2 †

…†N…†… ÿ †

…A1†

…cv=2†2  ‰3f1 …1 ÿ c†… ÿ † ÿ d…†Š

…†N…†… ÿ †

Since 1=X…z† is analytic function on the ®nite plane except the cut from 0 to 1 in the complex plane 1 1 ˆ X…z† 2i

‡

dz0 0 0 C1 ‡C2 …z ÿ z†X…z †

…A2†

1=X…z† as C2 ! 1 behaves like z ÿ  (McCormick, 1964). Hence 1 2i

‡

dz0 1 ˆ ‡ …z ÿ † 0 0 X…z† C1 …z ÿ z†X…z †

…A3†

where the second term inside the parentheses is signed as positive because, while C2 is counterclockwise, the contour around the cut, C1 is clockwise. Since on the cut by de®nition   ‡ … 1 dz0 1 1 0 1 1 1 dz ˆ ÿ …A4† 2i C1 …z0 ÿ z†X…z0 † 2i 0 X‡ …z0 † Xÿ …z0 † …z0 ÿ z†

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

1503

where X …z† are the limiting values of the X…z† function in both sides of the cut from 0 to 1 in complex z-plane. We consider 1 1 X‡ …z† ÿ Xÿ …z† ÿ ˆ ÿ X‡ …z† Xÿ …z† X‡ …z†Xÿ …z†

…z† ˆ

…A5†

Now using de®nition (Case and Zweifel, 1967)

…z† ˆ

cz X‡ …z† cz Xÿ …z† ˆ 2 ‡ …z† 2 ÿ …z†

…A6†

or

2 …z† ˆ

c2 z2 X‡ …z†Xÿ …z† 4 ‡ …z†ÿ …z†

…A7†

where  …z† are the limiting values of …z† which is given by Eq. (27) on the cut. Using the relation ‡ …z†ÿ …z† ˆ N…z†=z

…A8†

and

…z†d…z2 † ˆ

 1  ‡ X …z† ÿ Xÿ …z† 2i

…A9†

we obtain

…z† ˆ ÿ2i

…cz=2†2 zd…z2 †

…z†N…z†

Using Eqs.(A5)±(A10), we ®nd …1 ‡ 1 dz0 …c=2†2 d…2 † ˆ ÿ d 2i C1 …z0 ÿ z†X…z0 † … ÿ z† …†N…† 0 Then by Eq. (A3) we obtain ®nally …1 …c=2†2 d…2 † 1 ˆÿ ÿ …z ÿ † d … ÿ z† …†N…† X…z† 0

…A10†

…A11†

…A12†

Using this result, we can evaluate the ®rst two integrals of the right hand side of Eq. (A1). On the other hand, the third integral has been given by McCormick (1964). …1 0

d

…cv=2†2  1 ‡ … ÿ † ‰3f1 …1 ÿ c†… ÿ † ÿ d…†Š ˆ

…†N…†… ÿ † X…†

…A13†

1504

M.A. Atalay / Annals of Nuclear Energy 27 (2000) 1483±1504

Thus, Eq. (38) is proved using Eqs. (A12) and (A13). Eq. (39) is also veri®ed by the same treatment. References Abdel Krim, M.S., Degheidy, A.R., 1998. Ann. Nucl. Energy 25 (4±5), 317. Atalay, M.A., 1996. Ann. Nucl. Energy 23 (3), 183. Atalay, M.A., 1997. Prog. Nucl. Energy 31 (3), 229. Case, K.M., Zweifel, P.F., 1967. Linear Transport Theory. Addison-Wesley, Reading, MA. Kaper, H.G., Lindemann, A.J., Leaf, G.K., 1974. Nucl. Sci. Eng. 54, 94. LeCaine, J., 1947. Physical Review 72 (7), 564. Marshak, R.E., 1947. Physical Review 71 (10), 688. McCormick, N.J., 1964. One-Speed Neutron Transport Problems in Plane Geometry. PhD thesis, The University of Michigan. McCormick, N.J., KusÏ cÏer, I., 1965. J. Math. Phys. 6, 1939. Mika, J.R., 1961. Nucl. Sci. Eng. 11, 415. Mitsis, G.J., 1963. Nucl. Sci. Eng. 17, 55. Noble, B., 1958. The Wiener±Hopf Technique. Pergamon Press, Oxford. Placzek, G., Seidel, W., 1947. Physical Review 72 (7), 50. Razi, N.K., El-Shahat, A., Mork, K.J., 1991. Physical Review A 44 (2), 994. Williams, M.M.R., 1971. Mathematical Methods in Particle Transport Theory. Butterworths, London. Williams, M.M.R., 1975. Atomkernergie 25 (1), 19.