transport theory of neutron wave propagation by eigenfunction expansion method

This article was downloaded by: [Stony Brook University]On: 26 October 2014, At: 08:04Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Journal of Nuclear Science andTechnologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tnst20

Transport Theory of Neutron WavePropagation by Eigenfunction ExpansionMethodTomejiro YAMAGISHI aa Department of Nuclear Engineering, Faculty of Engineering , OsakaUniversity , Suita-shi, OsakaPublished online: 15 Mar 2012.

To cite this article: Tomejiro YAMAGISHI (1972) Transport Theory of Neutron Wave Propagation byEigenfunction Expansion Method, Journal of Nuclear Science and Technology, 9:7, 420-429

To link to this article: http://dx.doi.org/10.1080/18811248.1972.9734871

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/tnst20

http://dx.doi.org/10.1080/18811248.1972.9734871

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

Journal of NUCLEAR SCIENCE and TECHNOLOGY, 9 (71, p. 420-429 (July 1972).

Transport Theory of Neutron Wave Propagation

by Eigenfunction Expansion Method

Tomejiro YAMAGEHI*

Received January 5 , 1972

Revised March 29, 1972

An eigenfunction expansion method is developed for the neutron wave propagation theory with a cut off cross section. The full and half range orthogonality relations for the eigenfunctions are derived, which are respectively applied to obtain an infinite space solution and the space eigenvalue formula.

The formula is tried on some actual evaluations, and found to reduce considerably the laborious procedure required hitherto. The existence of marked interference phenomena between the discrete and the pseudo modes is demonstrated by numerical calculations performed on the solutions for graphite and beryllium.

I. INTRODUCTION The transport theory of neutron wave propaga-

tion has in recent years been investigated by many a~thors""'~). They have mainly examined the existence and behavior of the fundamental space eigenvalue (complex relaxation length), or else have derived the formal solutions as a mathematical problem in various plane geometries. These efforts have however proved unsuccessful in interpreting the interference phenomenon observed experimentally by Utsuro et uZ . '~ ' in a graphite cube.

In polycrystalline materials such as graphite or beryllium, the penetration of the cold neutrons significantly affects the amplitude of the collided neutrons and this can be expected to be the cause of the interference phenomenon. This phenomenon was analyzed by Utsuro et u Z . ( ~ ) , Nishina et aZ.@) and Takaha~hi '~ ' on the basis of the energy de- pendent diffusion theory.

In the transport theory, the effect of neutron penetration is involved in the continuum contribution. Therefore, without estimating both the discrete mode and this contribution, i .e. the complete solution, it is impossible to explain to interference phenomenon. Little is, however, known about the behavior of the continuum contribution.

In this paper, we intend to examine the behavior of this contribution and then proceed to the details of the interference phenomenon, by nu- merically calculating the solution in an infinite plane geometry. To derive the complete solution,

we shall first develop an eigenfunction expansion method for the neutron wave transport theory with a cut-off cross section.

The complete set of eigenfunctions for the neutron wave transport equation has been derived by Kaper et u Z . ( ~ ) and Duderstadt"'. The present author(lO' and Klinc et ul.(") have, independently of each other, established the half range orthogonality relations for these eigenfunctions. Such relations are useful for the half space and finite slab

W e shall extend the orthogonality relations to the case of the total macroscopic cross section Z(v) containing a discontinuity, such as the Bragg cut-off in polycrystalline material. We shall hence derive the space eigenvalue formula, which will be found to reduce the laborious procedure of numerical experimentation hitherto required for the evaluation of the discrete space eigenvalue U O ( O ) .

The full range orthogonality will be applied to the derivation of the infinite space solution. From the numerical results of the solution, a marked interference phenomenon will be noted between the discrete mode and the cold continuum contribution (pseudo mode).

For simplicity, the isotropic simple scattering kernel

Z(v', v) =&(v)Kz(v') will be employed where Kl(v) and Kz(v) are

* Department of Nuclear Engineering, Faculty of Engineering, Osaka University, Suita-shi, Osaka.

- 38 -

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4

Vol. 9, No. 7 (July 1972) 421

chosen to let B(v’, v) satisfy the detailed balance condition and give the correct total scattering cross section(13’. This approximation may be rough in the energy exchange mechanism between neutrons and scatterers for polycrystalline media. In our present approach, however, the transport effect of the discontinuous mean free path can be precisely taken into account.

II. EIGENFUNCTIONS AND

area G = { [ lo< lpl 9 1 , V > V B } (thermal continuum), and G’=LUG. Here vt and V B are the thermal and the Bragg cut-off velocity respectively, Zo, 81 and Za are constant and assumed to be vBEl>vf80. The subsets L* and G* are defined in the same manner as before. The structure of the spectral plane is presented in Fig. 1. If V B = 0 (continuous cross section), the continuum L vanishes and the boundary of G+ i .e. 8G+ is prolonged from fo to the origin along a circle.

DISPERSION FUNCTION 0

In a plane geometry, we shall deal with the neutron wave transport equation -2

{.(v)+p&]f(.., P, v > 0) - = p i KI(v) Idfit{ dv’Kz(v’)f(s, p’, v’, o)

(1) 2 - 0 -6

which is the case of transverse buckling B=O in the equation treated previously(5’. Here a(.) = B(v) + iw/v , and other notations are conventional.

We introduce a new complex variable and function

( 2 )

Fig. 1 Structure of half range spectral plane for graphite

For the discontinuous cross section the integral in Eq. ( 4) must formally be written u, = E + io = Pula@)>

WG f)=+)f(s, P> v, o)/K(l(v). ( 3 ) j j , i ~ ( ~ ) o = j / ~ j o ( ~ ) o + jj, d2%j1(6)0,

( 6 ) in which the integral of the 6-variable over L corresponds to that of the v-variable over (0, v B ) ,

hence using relation (2) , we have

Then Eq. (1 ) can be written

(l+%&)@(x, 6) =+jj,$“Sj(u,)@(.., %),

( 4) where d2c=d[dli), and j (6) is defined by

(5)

while G’ is a two-dimensional domain in the %-

under the relation (2) . For convenience, we define the subsets G+’ corresponding to p 2 0 ; then

= j d%{ dvKl(v)Kz(v)o, plane defined by the set { c l O < 1 ~ 1 91, O<v<co} I, Q(t)

where Q(6) is the set of 71’s given by {vI%EL}. In the present case it becomes V<V<Z)B, where

with respect to the origin in the %-plane, and

If the discontinuity of the scattering cross sec- C’=G+’UG-’.

tion E,(v) is assumed in the simplified form

and the total cross section Z(v) is the sum of 8, and l/o-absorption, i.e., E(v) = Es(v)+B,vt/v. Then the set G’ is split into disjoint sets: line L= {[lo< 1fi1 9 1 , O < V < ~ B } (cold continuum) and

( 7 )

which is real, even, continuous decreasing from j o ( 0 ) to j o ( C B ) = 0, and satisfies a Holder condition on L(l4’, where [ B = ~ / c x ( ~ B ) . For [EG, jl(6) is given by Eq. ( 5 ) for ~ > v B .

In what follows, the integration over G will be regarded in the sense of Eq. ( 6 ) , and

j o ( 0 for ji(%) for ~ E G . j ( c ) = [

- 39 -

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4

422 J. Nucl. Sci. Technol.,

We assume the separation of variables in the form

a(.., 6) = d u , C) exp (-&), then Eq. (4) can be reduced to an eigenvalue problem :

When normalized for the eigenfunction

( 9 )

the general solution of Eq. ( 8 ) becomes

where 2(u) is a complex function to be determined

Let us introduce a complex function by de- by Eq. ( 9 ) .

fining

This function is even, because j(C) is even from the definitions ( 5 ) and ( 7 ) . In the ensuing ana- lysis, it is important to examine the property of the dispersion function A(2).

On a set of functions g(C) absolutely integrable over the bounded domain G with a well behaved boundary aG, we define the operator

According to the theory of the generalized analytic fun~tion"~', TGg(z) as a function of z is analytic outside G, exists almost everywhere in G and vanishes at infinity. It has a generalized derivative with respect to z, and

If f ( z ) = &g(z) is absolutely integrable over G, then

g ( z ) = Y ( 4 + T c f ( 4 , (13) where Y(z) is a function holomorphic inside G. With the operator T c and the definition of the integral over G, A(z) can be rewritten

Then it is analytic outside GI, continuous in G,

tends to A(m)=l-- d2Cj(I) at infinity, and 2 G '

satisfies

because the second term in the RHS (right-hand side} of Eq. (14) is analytic for ZEG. Across the cut L, A(z) is discontinuous and from the Plemeji formulas(16', we have

Ti . 2 A'(v)=A~(v)+-vJo(v) for u E L , (16)

where A+(A-) is the limit of A(z) with respect to 2 from above (below) the cut L, and

for U E L , (17)

where P represents the Cauchy principal value

integration. To determine the eigenfunction q(u, t) we

distinguish three cases : (1) For u$G' : introducing Eq. (10) into Eq.

( 9 ) and using the function A(z), a pair of discrete eigenvalues i u o are determined by A(fuo) = 0. The corresponding eigenfunctions then become

s

(2) For U E L and (3) for U E G : with use made of Eqs. (lo), (11) and ( 9 ) , A(C) is similarly determined :

(19) A(C)/jl(C) for C=G

2(6)=lAp(C)/jo(g) for CEL. The values of A(z) along aG+ and L+ are

shown in Fig. 2 for graphite. The point 61 determined by ImA(C1) = 0 is important for determining the existence and for estimating of the discrete eigenvalue. From the argument principle, if ReA(Cl)<O, a pair ~ U O exists, while for Red([,) > O the pair vanishes, and ReA(l;l)=O is then the critical condition for the existence of DO. In what follows, we shall treat the range of o where uo exists, and assume that the image of A(v) around L+ does not enclose the origin in the A- plane.

If we choose the principal branch such that I arg A I < R , In A(z) is discontinuous at the point

51 and the gap is - 2 ~ i in the positive direction

- 40 -

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4

Vol. 9, No. 7 (July 1972) 423

-I 1 The values of A+(z) are almost superimposed on that of k ( z ) .

Fig. 2 Boundary values of A(z ) in the A-plane

on aG+. This gap is preserved when the contour aG+ is squeezed continuously into the interior of G+, provided A(z)+O. In other words, the index I , of A(z) along aG+ is -1 so long as uo exists outside G+, and I , is invariant for the continuous variation of the closed contour 7 under the above condition, where the index is defined by the in-

teger I,=- d arg A, and r is the continuous

image of a closed contour in the z-plane. The index of A(z) along aG+ equals the sum

of the individual indexes of the zero point of A(.) inside aG+ (Ref. (I f l , p. 53), and the individual index may be either 1 or -1 depending on the sign of the Jacobian“’) at that point, i.e. J = IazA I ’- (a,A(220, where J is the Jacobian in the trans- formation z+A(z) (the case of the zero point with J = O is excluded). Then A(z) has at least one zero point inside aG+. In the range of o interesting us, it was confirmed from numerical calculation that there is only one zero point u1 in G+. It has also been found that A(z) is doubly transformed in some regions near aG in G, i .e. the mapping is two-to-one in these regions.

As the frequency o is increased, uo approaches G along a trajectory and at the critical frequency oc it attaines the continuum G. Above oc the total index of A(.) around aG becomes zero. From numerical experiments, the doubly mapped region in the A-plane may contain the origin for a limited range of o just above wc. Thus it is expected that uo still exists in G for a while as the “normal- zero”c1*) whose index is 1, until finally the zero points uo and UI in G+ completely vanish.

277 ‘ S I

III. FULL-RANGE ORTHOGONALITY AND INFINITE SPACE SOLUTION

We look for the orthogonality relations of the eigenfunctions over G’, and apply these to the deviation of the solution in an infinite system. For U, U’E L, the multiplication of p(u, f) by P(u’, f) must be performed in the sense of the direct product of the generalized function with the support L. As in the stationary case (where L lies on the real axis), we have

(20)

For v, U‘ EG, the ordinary partial fraction de- composition can be applied, and the term (742)’ in Eq. (20) disappears. For the cases of U = *DO, IJ’ E G and u=uO, U’ = -u, the relation (20) also. holds, but the &function term vanishes.

Making use of the identity (20) and the relation

p f j ( l l f P ( U , C ) = u f l ( a ) ,

SjG,d’fj(f)fP(U, f)Co(U’, I) = W ) W - U ‘ ) ,

which holds also for U = & U O , we obtain the fol- lowing full-range orthogonality relations

(21) ^ ^

Making use of the above orthogonality, it is easy to show that the eigenfunctions {p&, q(u, c ) } constitute the complete set over the full-

If a plane source is placed at z=O, the solution can be expanded in terms of the eigenfunctions under the boundary conditions O( -)_a, f) = 0 :

range GtCl2)(18).

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4


W e shall express these terms by f d ( z ) , f c ( z ) and f t ( z ) respectively. These terms may also be con- sidered projections of f(z, a) to the relevant spectrum in the spectral plane.

+{{,., d2uA(u)du , 6)e-"" for zso* (25)

From the source condition at z=O, we have

where S(C) represents a source, and is related to an actual source SO(,V, v) by S(') = [SO(Y, v)/fG(v)lg.

The expansion coefficients a+ and A ( v ) can easily be obtained by using the orthogonality relations (21)-(23), and are found to be

a+= b,/N+, (274 A (u) = W ) / N ( u ) , (27b)

where

b(v)=j jc,d25j(e:~(i)~(u, 5) for UEG',

(28) and b, are defined by Eq. (28) for u= ?DO.

Introducing these coefficients into Eq. (25), we have an intermediate solution @(z, 5) and from the relation ( 3 ) the final solution f ( s , p, v, a). In- tegration of f, thus obtained, by p(-1,1) and by v(0, 03) with a weight w(,u, v) yield the angle- velocity averaged quantity

where j , ( ' )= j (~ ) [w/Kz]~ . From Eqs. (25) and (27) this average can be written

and d, are defined by Eq. (30) for u = i u o . The first term on the RHS of Eq. (29)

represents the discrete mode wave and is usually called the asymptotic term, the second term represents the cold continuum contribution, and the third the thermal continuum contribution.

IV. HALF RANGE ORTHOGONALITY AND SPACE EIGENVALUE FORMULA

W e proceed to develop the space eigenvalue formula in the stationary case(14) in order to apply it to the present problem. This formula is based on the property of the half-range characteristic function X(z), which is derived as a weight function in the half range orthogonality. This orthogonality itself is vitally important for deriving the exact solution of half space problemsc18' and for obtaining an approximate solution for a finite slab problem'11'. We shall therefore, first derive the orthogonality relations for the eigenfunctions {q+(f), p(u, c ) } over the half-range G+'. The results are:

where A+(u)A-(u)/jo(u) for UEL+

for UEG+. n+(u)=uH(u) .

(33)

H(u)=A(uo-~))X(u) /A(u) (34)

The weight function H(u) is given by

in the whole complex plane, where A is a constant, and

for zGG+. For ZEG, X ( z ) is given by the product of the RHS of the above equation and A(z). The derivation of the H and X-functions from Eqs. (31) and (32) will be given in APPENDIX.

In addition to the above orthogonality, the integrals

- 42 -

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4

Vol. 9, No. 7 (July 1972) 425

are useful for half space problems'1s'.

through the convenient relation We shall consider another application of H(z )

A(Z)H(Z)H( -2) = 1, (35) under the condition H(O)=l. This relation may be proved by Liouville's theoremc1" and equally well by the residue theorem(l8'.

At infinity, Eq. (35) leads to H ( m ) = l/l'A(m) and then A is found to be K?@j. With the X-function, the relation (35) is rewritten in the form

~

X(z)X( -2) = A ( Z ) / A ( ~ ) ( ~ O ~ - Z ~ ) , (36)

which yields the space eigenvalue formulas in the same manner as in the stationary case(18). A direct consequence is

uo2= l/A(m)XZ(O). The functions A(z) and X ( z ) can be expanded at infinity

where

Introducing these into Eq. (36) and taking the limit z+m after rearrangement, we obtain uo2= a/A(m) +2b2-b12, or by further trivial manipula- tion,

with which the discrete space eigenvalue u ~ ( o ) can be estimated making use of the values A(z) along aG+' obtained in the course of the calculation for the solution (29), thus dispensing with the laborious numerical experiments required hitherto. Equation (37) also can be regarded as an alterna- tive expression of the dispersion law.

V. NUMERICAL RESULTS In the numerical evaluations, we have assumed

the simple scattering kernel employed previousIy'ls', i.e. Kl(v) =PBsvm(v) and &(v) =Zs(v) where

P- - dvZsvm (v) and m (v) = v2 exp { - (v/vt)'} . In this case, for small Za and o, the uo-formula (37) may be approximated by'"'

l-I uo2= a/A(m), (38)

where

a=-! R dv-m(v). Zs2v 3 0 a"v)

For the computation of A(.) along the boundary aG+, the expression

was adopted, and integration was carried out by Gaussian quadrature formula with 64-point mesh over (0, 5vt). When the integrand had a loga- rithmic singularity at v* for z ~ a G + , the domain was separated into (0, v*) and (v*, 5 4 , and in- tegrated by the same method for each interval. For the evaluation of A*@), i.e. Eq. (16), the variable u on L was expressed by u= l / a (V) (0 <V<VB) and the integration about the I-variable was transformed to that about the v variable :

The numerical results of uo(o) with use made of the formula (37) are presented in Figs. 3 and 4 for graphite and beryllium respectively, where they are compared with the results obtained by the conformal mapping method.

For the evaluation of f ( s , a), i.e. Eq. (29), an isotropic Maxwellian source and l/v-neutron counter were assumed : So(v)=m(v) and w(p, v) =l/v. Then S(c)=[l/vBS]c and j ,=j([)[ l / v Z S ] ~ , hence from Eqs. (28) and (30) it follows that b(u)=d(u) for UEG' and u = - ~ u o . This weight can be written in actual physical quantities as in the expressions of A(z) and A,(z). The

- 43

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4


.05 I / wi'""/sec;

R e % I c m " 1

0 cQ5 01 15

-: Eq. (37), --: Eq. (38), 0: Conformal mapping method, A: Warner et al."'

Fig. 3 Inverse space eigenvalues u0-I us. frequency-for graphite

integrals f c ( x ) and ft(x) were also transformed into integrals about real variables by similar procedure in the actual numerical evaluations.

The thermal continuum contribution ft(-.) affects the total amplitude 1 f ( x , o) I only near the source(zo), and for large x (>20 cm) it was ignored. The numerical results of I f i x , o) 1 , which are composed of f d ( x ) and fc(s) , are shown in Figs. 5 and 6 for graphite and beryllium respectively.

0 ( r o d / s e c ) \ 6 0 0 0

5 0 ' 00 x ( c m )

--: Ifd(Z)I, -'- : I f L r I , -: If(& W ) I

Fig. 5 Amplitude ~ I S . distance from source for graphite

.I8

.I 5

.I0

.05

-: Eq. (371, 0: Conformal mapping method, A: Wood (Sinclair phonon model, multigroup approximation) (19)

Fig. 4 Inverse eigenvalues u0-l us. frequency-for Be

In these results, as we gain distance from the source, the discrete modes Ifd(s)l become in- cressingly dominant until I f C ( x ) I becomes com- parable therewith; The dips seen in the curves are due to the interference between f d ( s ) and f c ( z ) . Beyond the dip, the cold continuum contributions become predominant and constitute the pseudo modes whose amplitudes decay roughly by

I , I 4 0 60 90

X (cm)

Fig. 6 Amplitudes I f ( r c , 0 ) 1 nus. distance from source-for Be

- 44 -

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4

Vol. 9, No. 7 (July 1972)

XO (cm-') XI (cm-') X. (cm-l) V B (cm/sec)

427

0.1285 x 10-l 0.385 0.736 0.320 x 0.123 x

0.2475 x 10-1

0.555 x lo5 0.976 x 105

exp(-Zmln x) as x+m. The total amplitude is presented against oscillation frequency in Fig. 7 where comparison is made with the experimental resultcz1'. The d e q dip due to the interference effect is located at x=lOOcm, which is larger than that observed experimentally. This discrep- ancy may be due mainly to the difference in the boundary conditions between experiment and theory.

l o z

c .- C a

P E c $

-a Y

8 10

E" a c .- - Q

The numerical calculations were performed on NEAC2200 Model 500 in the Computer Center of the Osaka University.

VI. CONCLUDING REMARKS

propagation theory for a discontinuous cross section. We can prove the completeness of the eigenfunctions for these cases by treating separately the continuum L and G. Over the area G(G+) and line L(L+) continuums the method of Kaper(12' and that of the author"s' can be applied respectively for the proof of completeness. The weight function H(z) in the half range orthogonality yields the function X ( z ) which plays an important role in the transport theory of a plane geometry, as in the stationary case(ls'. The space eigenvalue formula derived from the property of X ( z ) is found to be very useful in actual numerical calculations.

The argument principle determines only the number of zero points of A(z) , while the present formula determines even the coordinates of the zero point. Extension of the physical applications and mathematical generalization of the formula should be of interest.

From the numerical results obtained for the solution in an infinite system, the behavior of the cold continuum contributions were revealed for various values of oscillation frequency, and interference was observed between the discrete and pseudo modes.

The discrete mode is determined from the dispersion relation A(u0) = 0, while the continuum contribution depends upon various factors- the spectrum, strength and geometry of source, neutron counter, system type and system size. The temperature of the medium also affects Z(v) significantly below the Bragg cut-off velocity(22', and hence also the cold continuum contribution.

From the fact that the pseudo mode becomes dominant for sufficiently large distances from the source, the system size, i .e. the boundary conditions, becomes an important factor to be taken into account in polycrystalline media.

The half range orthogonality has the advantage of being straightforward in solving the Milne problem, and of permitting examination of the behavior of the neutron density near the free sur- face which depends upon the oscillation frequency. Our method can also be applied to the reduced three-dimensional transport equation. The numerical results for these problems are now in preparation.

We have derived the full and half range orthogonality of eigenfunctions in the neutron wave

- 45 -

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4


ACKNOWLEDGMENTS [ A P ~ T X ] The author’s thanks are due to Mr. A. Taka-

hashi for helpful discussion and encouragement, and to Prof. T. Sekiya for reading the manu- script.

Derivation of the H-function From Eq. (20), Eqs. (32) (for u+u’) and (31)’

can be reduced to equivalent but simplified form :

-REFERENCES-

KAPER, H.G., FERZIGER, J.H., LoYALKA,S.K.: IAEA Symp. Neutron Thermalization and Reactor Spectra, (1968), Vienna. DUDERSTDT, J.J. : Thesis California Inst. Tech- nol., (1968). WILLIAMS,M.M.R.: J. Nucl. Energy, 22, 153 (1968). WARNER, J.H., Jr., ERDMANN, R.C. : ibid., 23, 117 (1969); ibid., 23, 135 (1969). YAMAGISHI, T., TEZUKA, M., SEKIYA, T. : J. Nucl. Sci. Technol., 6(11), 611 (1969). UTSURO, M., SHIBATA, T. : ibid., 4(4), 205 (1967). UTSURO, M., INOUE, K., SHIBATA, T. : ibid., 5(6), 298 (1968). NISHINA,K., AKCASU,A.: Nucl. Sci. Eng., 39, 170 (1970). TAKAHASHI, A.: J . Nucl. Sci. Technol., 9(3), 172 (1972). YAMAGISHI, T. : Technol. Rep. Osaka Univ., 19, 301 (1969). KLINC, T., KUSCER, I. : Submitted for publication in ‘‘ Transport Theory and Statistical Physics ”. KAPER, H.G. : J. Math. Phys., 10, 286 (1969). CORNGOLD, N., MICHEL, P., WOLLMAN, W. : Nucl. Sci. Eng., 15, 13 (1963). YAMAGISHI, T. : J. Nucl. Sci. Technol., 8[8), 470 (1971). VEKUA, I.N. : “Generalized Analytic Functions ‘I,

(1962), Pergarmon Press. MUSKHELISHVILI, N.I. : “Singular Integral Equa- tions”, (1953), P. Nordhoff Ltd. Groningen, Netherlands. SUGIYAMA, S. : ‘I Non Linear Oscillations”, (in Japanese), (1965), Hirokawa Publ. Co., Tokyo. YAMAGISHI, T. : J. Nucl. Sci. Technol., 8[3), 153 (1971). WOOD, J.: J. Nucl. Energy, 23, 211 (1969). YAMAGISHI, T., SEKIYA, T. : Technol. Rep. Osaka Univ. 22, 1 (1972). TAKAHASHI, A,, SUMITA, K. : J. Nucl. Sci. Tech- nol., 5(3), 137 (1968). YAMAGISHI, T., DEGUCHI, I., SEKIYA, T. : ibid., 9[6), 375 (1972).

dzC5j(C)CH(C)P(u, 5) =AD for UEG+’ and U = U O . (-41)

The substitution of the explicit form of p(u, C) into Eq. (A l ) leads to

for UEG+’, (A2)

(A3)

T o solve these integral equations, we introduce a complex function defined by

+T~+[H(z)82&) 1 f A. (A4) This function is analytic outside G+’, tends to A at infinity and is discontinuous across the continuum L+. For UEL+, applying the Plemeji formulas, we have

from which M+(u)-M-(u)=aiujo(u)H(u)

for UEL+. (A6) Making use of Eqs. (16), (17), (A5), (A6) and the same procedure as in the stationary case(18), we find the relation equivalent to the orthogonality for UEL+:

M+(u)A-(u) = M-(u)A+(u) (A71 For UEG+, in view of Eq. (12), M ( z ) satisfies

With the relations (15), (19) and the definition of M(z) , Eq. (A2) can be rewritten in the form

M(u)=A(u)H(u) for UEG+. (A9) Then, Eqs. (A8) and (A9) yield

an a D a M o ~ ( ~ ) = a ~ ~ ( D ) for ~ E G + . ( ~ 1 0 )

-46-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4

Vol. 9, No. 7 (July 1972) 429

Equation (A3) directly reduces to

M(u0) = 0. ( A l l ) Equation (A7) is a homogeneous Hilbert problem and the solution is found by the method in Ref. (Is) (p. 88). On the other hand, Eq. (A10) is a generalized Riemann-Hilbert problem which can be solved with the aid of relation (13)“”. Then the solution which satisfies both Eqs. (A7) and (A10) is given by a product of these fundamental solutions and can be written in the form

M ( z ) = mo(z) exp I r (z) l , (A121 where Yo(z) is an arbitrary analytic function in G+ except VI, and

(A13) Since we are treating a case where the locus of A(z) around L+ in the A-plane does not enclose the origin, the integral over L+ in Eq. (A13) has no singularity. Then from the properties of M ( z ) the function TO(z) is determinedc1l) :

Yo(z) = A(uo-z ) / (u l -~ ) , and A is found to be H(co)A(w).

written(1z1 The integral over G+ in Eq. (A13) can be

InA(z)+ro(z) for ZEG+ ro(z) for . z E E ~ + ,

Tc+[a~Ind(z ) ]=

where ro(z)= --S 1 d f a 2ni aG+uc t - z ,

and G+ is cut out from f~ to ~1 along the discontinuity of InA, i.e. a curve C. Then we have(11)

which doesnot exhibit any singularity at 61 because the integral on the RHS of Eq. (A15) behaves as ln(f1-2) for z=f i (Ref. (16), p. 83). From the relations (A13), (A14) and (A15), the function M ( z ) can be expressed in the form

exp {r’(z)} for z$G+ A(z)exp {r’(z)} for ZEG+,

(-416)

M ( z ) = A--

where

For the sake of brevity, we shall represent the above integrals by

It can easily be shown that the function M(z) thus obtained by Eq. (A12) or (A16) indeed satisfies Eqs. (A7) and (A10) which are equivalent to the orthogonality relations.

Once M(z) is completely determined, then the weight function H(z ) will be given by Eqs. (A6) and (A9). By virtue of Eqs. (16) and (A7) for EL+, the weight is found to be

H(u)=M(u)/A(u) for VEG+’. From Eqs. (A8) and (A9), it follows that aDH=O for UEG+, i.e., H(u) is analytic in G+. From the property of M ( z ) and A(z), H(z) is analytic outside G-’ except at DI, where H(u) has a simple pole.

If we introduce the function X ( z ) by X ( Z ) = M ( z ) / A ( u o - ~ )

then it satisfies also Eqs. (A7) and (A10) and is non-vanishing in the whole plane except at infinity.

- 47 -

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

8:04

26

Oct

ober

201

4

transport theory of neutron wave propagation by eigenfunction expansion method

Documents