THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report HALF-SPACE GENERAL MULTI-GROUP TRANSPORT THEORY S. Pahor J. K. Shultis P. F. Zweifel ORA Project 01046 supported by: NATIONAL SCIENCE FOUNDATION GRANT NO. GK-1715 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR July 1968

HALF-SPACE GENERAL MULTI-GROUP TRANSPORT THEORY* S. Pahor, J. K. Shultist, and P. F. Zweifel Department of Nuclear Engineering The University of Michigan Ann Arbor *Supported by the National Science Foundation.'*On leave from the University of Ljublijana, Yugoslaviao tThis paper is based on part of a dissertation submitted by J. K. Shultis in partial fulfillment of the requirements for the PhoDo degree in Nuclear Science at; The University of Michigano 1

ABSTRACT A method for solving various half-space multi-group transport problems for the case of a general transfer matrix is explained.. The advantage of this method is that it readily yields numerical results. First, the nonlinear integral equation for the emergent distribution of the albedo problem is derived. Then, by using the full-range completness of the infinite medium eigenfunctions, the inside distribution is obtained. from the emergent distribution. Finally, the Milne problem and the half-space Green's function problem are solved in terms of the emergent distribution of the albedo problem and the infinite medium eigenfunctions. 2

I. INTRODUCTION A widely used. method, to treat the energy dependence of the transport equation is the multigroup approximation. However only recently have exact solutions to the multigroup transport equations been considered. By applying the singular eigenfunction approach of Case, [1] the solution of the infinite medium Green's function problem has been obtained for the two-group and the N-group cases3 Until now, the solution of half-space problems has been restricted to special caseso Several two-group problems have been investigated. In a paper on radiative transfer Siewert and Zweifel treated, the N-group case with the specific limitation that the determinants of the transfer matrix, C, and all its minors vanish.[6 Finally, for the case of symmetric transfer (of which the two group case is a special example [7]) Leonard and Ferziger showed how solutions to half-space problems can, in principle, be obtained by solving Fredholm equations.8 The purpose of this paper, therefore, is to consider half-space problems "of the N-group" isotropic transport equation with a completely arbitrary transfer matrix. However all of the above techniques depend critically upon a "half-range completeness" theorem whereby the solution of any half-space problem can be expanded uniquely in terms of only half of the infinite medium eigenfunctions of the transport equation. For the case of general transfer no such half-range theorem has been found and another approach must be used. Recently an approach which circumvernts this half-rarnge difficulty has been [91, [10] used by Pahor in the thermal neutron degenerate kernal case. It is this approach which will be used in this paper. First the emergent distribution for the problem is found. Then, once the angular flux is completely known at the 5

surface, the full-range completeness of the property N-group infinite medium eigenfunctions[3] can be used to obtain the complete solution inside the halfspace. The main problem then is the calculation of the emergent distributions. To this end, several different methods can be used.. Case has obtained a Fredholm equation for the emergent flux in terms of the infinite medium Green's function. Secondly, from an eigenfunction expansion of the problem, a Fredholm equation for [71, [1] the emergent d.istribution involving only eigenfunctions can be obtained. [ However, both'these methods are very difficult to evaluate numerically. A third approach, which readily yields numerical results for the emergent distribution, will be used in this paper. By applying the invariance principles of Ambarzumian and Chandrasekhar, [1 a nonlinear integral equation with a simple kernel can be easily derived for the general N-group albedo problem emergent distributions The plan of this paper is as follows: Section II reviews the known results and properties of the infinite medium eigenfunctions which will be needed later. The next section shows how the emergent distribution tuo the half-space albedo problem can be obtained in terms of'two fundamental matrix functions, J4a) and )IJ- e These functions satisfy a pair of coupled nonlinear integral equations which are quite amenable to solution by numerical means. In Section IV we demonstrate how the emergent distributions for the Milne problem and the halfspace Green's function can also be expressed in terms of these U and V matriceso Then by applying the full-range completeness theorme of the infinite medium eigenfunctions, the complete soluttions to these problems may be obtained. 4

II. EIGENFUNCTIONS OF THE MULTIGROUP TRANSPORT EQUATION The linear Boltzmann equation for N energy groups in plane geometry and with isotropic scattering and, fission can be written in the form x(x) + = C S dX' t(x,p1). (2.1) The vector 1(x, i) is an N-component vector, of which thei-th component, (i(x, ), is the angular flux for the i-thgroup. The components of the matrix Z are given by ai6ij, ai being the total interaction cross section for the ith group. Finally, the elements, cij, of the transfer matrix C describe the transfer of neutrons from the jth group to the i-thgroup. For an isotropically scattering and fissioning medium the c.i are given by ij 2 j-i i j. (2.2) S*1 S where aji, is the scattering cross section for the transfer of neutrons from the jth group to the ith group, a.f is the fission cross section for the jth group, 3 v. the number of fission neutrons produced by an incident jth group neutron, and J Xi is the fission spectrum fraction of the ith group. It is always possible to order the groups such that[7] c1 > a2 >.00 < ON' (2.3) and by dividing Eq. (2.1) by aN and measuring distance in units of the largest mean free path, a, one may set a = 1o[6] 9N N Using the analogy of the one-speed problem, [1] a set of eigenfunctions, Y(v,x,3) to Eqo (2.1) of the form 5

*(v,x,4) = eo-X/V(v,1) (2.4) is sought. Substituting this ansatz in Eq. (2.1), the following equation for the eigenvectors ~(v,) is obtained.: (I- ) - N(v,) = Cf' di 4(v,4'), (2.5) where E is the unit matrix. The explicit form of these eignevectors and their properties have been investigated, by several authors[2. 2[4'[8 ][ 5] In order to establish notation, the basic form and properties of these eigenfunctions will [33 be briefly quoted. We will use, with slight changes, the notation of Yoshimurao The eigenvectors can be written in the form (v,) = P F(v,4) b(v) + G(v,4) X(v), (2.6) where P denotes the Cauchy principle valueo The matrices F(v,p) and G(v,L) are defined as.F(.z.. =' j. (2-7) and [G(z,L)]i 6= 65(Cz-p)a.j, (2.8) A simultaneous equation for the unknown b(v), which satisfies b(v) =..! d,.' v,') = C a(v), (2.9) and the unknown vector \(v) is obtained by substituting Eq. (2.6) into Eq. (2.9); namely 6

Q(v)b(v) I= dCt G(v, M) X(v) (2.10) -1' \ where ~(z) = c - P f1 F(z,)d[. (2.11) 1.0 %-~ -1 i To solve for b(v) and X(v) the eigenvalue spectrum is divided into two regions. (a) Region I: v((-l,l) In this region there may exist an even number, say 2M, of discrete eigenvectors, which in component form are written as (Vos)] = o [bv )]i = 1-N, s = 1 2M, (2.12) where b(vos) is a well defined vectorio3] The discrete eigenvalues, v, s = 1 2M, are solutions of the dispersion relation det Q(v ) = o (2.13) It can be shown that if Vs is a solution of Eq. (2.13) then also -vos and v o (complex conjugate) are eigenvalues with b(v ) b= vo) = *(o) * (2.14) OS Os Os Os For a symmetric transfer matrix, the discrete eigenvalues, if they exist, are either real or imaginary. [7 For a general system, on the other hand, there does not appear to be any a priori reason the expect that the discrete eigenvalues are not complexo However it may be argued on physical grounds that a 7

subcritical medium must have a real dominant eigenvalue (defined as the eigenvalue with the largest real part). [12 (b) Region II: vc(-1,1) This region is divided into N subintervals, vj, j = 1 - N, such that for 1 1 vcvj, Cj- I < I ~ U-. For the jth sub-interval, there are (N-j+l) linearly c1 independent eigenvectors, m(v,), where ith component has the form v~j (v0)]i = P aV- [bT(v)]i + 3(aciv-)[ 1(v)]i m = j N, j = 1 N, (2.15) where P indicates the Cauchy principle value. The vectors bm(v) and Xm(v) are J J also defined by Yoshimura.[ The eigenvectors of both regions, J(v,.'l), depend parametrically upon C. t If we denote by b (ve.) the eigenfunctions with C replaced by C (tilde denoting the transpose), we see from Eq. (2.5) (v,9, = L(v,[,c ). (2.16) In passing, it should be noted that b(v) and K(v) are even functions of v and hence the eigenvectors have the property J(v,-) = l(-vn) s(2.17) From the eigenvalue equation (2.5), one finds that the eigenvectors are orthogonal in the following sense: 8

f1 di t 1 (vpi) (v,i) = 0 if v' f v. (2.18) Moreover, it is possible to choose particular linear combinations of eigenvectors for the independent eigenvectors of each subinterval, vj, such that all the "continuum" eigenvectors are mutually orthogonal,[l3 ice., -jj. j(+ 1 t +, S(,) = +Nm(v)((, v, V ~ V.. (2.19) Similarly for the "discrete" eigenvectors, we have f1 d. 4 (v+os,4) (-+~Vos',) = ~+N5 s, s s 1 M. (2.20) -1 The normalization functions NT(v) and N are given by Yoshimura.[31 The eigenvectors, t(v,p-), of Eq. (2.5) have the very useful property that they are "full-range complete.,[3]'[15] This property may be stated in terms of the following theorem~ Theorem. The set of functions _(v,4), v E[-1,1] or v = ~+vo, s = 1 M, is complete in the sense that an arbitrary vector function X(4) defined for 4 4[-1,1] can be expanded in the form M M *G0) = a(Vos) j(v0oSl) + sZ a((-v ) &(-vSP) N N + EZ dv ( E A.(v) Km(v,)), (2.21) j=l vj m=j J where a(vos), da(-vos), and AQ(v) are uniquely determined expansion coefficients. 9

III. EMERGENT DISTRIBUTION OF THE ALBEDO PROBLEM In many problems in half-space transport theory, only the angular flux at the surface of the medium is needed, To this end., the emergent distribution of the half-space albedo problem will be considered in this section. The emergent distribution of this particular problem turns out to be of fundamental importance in determining the emergent distributions of all other half-space problems. Consider an albedo problem for which the incident neutron beam belongs to the i-th energy group. The angular flux of this i-thalbedo problem", 1i(o,.;x,.), is the solution of Eq. (2.1) with the boundary conditions (i) f~(O ~;O,4) = ei (4-4), > 0, > 4, (3.1) (ii) lim (,Olo;x, ) = 0, (3.2) where _i is a vector all of whose components are zero except the ith one which is unity. The N distinct albedo problems (one for each group) can be handled collectively by introducing the "albedo matrix".(O,;,0;x,)) defined. as,(?~0 o; X ) = [1 (O,p~;xo), J (2(Oo;5X,), (O0;x,4o;0; )] (35) This matrix is the solution of the transport equation (4 I E +) (O,.0o;x,4) = C1 d. (0,,o;x,4), (5.4) with the boundary conditions (i) (O,lo;O, ) = E(-lo), p > 0, O, > 0, (3.5) (ii) xi v(0,o;X,p; ) = 0 (3o6) From the definition of the albedo matrix, the quantity Y(O,. 0;0, o0) gives the emergent distribu;tion rem(4_), pi > 0, for any arbitrary incident inc(p.), > 0 as 10

em(_ o o-) = ap.' d(0 Id';0, -) tinc(t,). (3 7) 0 - In fact the emergent distribution of any half-space problem can be expressed in terms of the matrix ~(O,k,;0;,-p).o > 0, i > O. There are several methods for obtaining this portion of the albedo matrix without first solving the complete albed.o problem.[7]'[9]'[11])[14] Here we will obtain from "the principle of,,[14] invariance"[ a nonlinear integral equation for this reflected flux. The principle of invariance states that the reflected flux from a halfspace is unchanged by the addition (or subtraction) of layers of arbitrary thickness to (or from) the mediumo[14] Thus if *(x,11) is the angular flux at a distance x inside the half-space, the outwardly moving flux can then be expressed from Eq. (3-7) in terms of the inward flux as I(x,-/L) = /l dp''(0,';0,-p) A(x,l)'), 4 > 0. (5.8) 0 in particular, Eqo (3.8) gives for the ith albedo problem fi(O,o;x,-4) -= d' T(O,';O,-4).i(Oo;x,~'), (3o9) Finally if we treat the N-albedo problems collectively, the above vector equation yields the following relation for the albedo matrix: V(0,o;X,_4) = f1 dp' 1(0,1';0,-) v(O0o;X,p'). (35.10) 0 ~' To obtain a nonlinear integral equation for Y(O0,i1;0,-4), first differentiate Eq. (3.10) with respect to x and set x = 0. Then using the transport Eq. (3.4) to evaluate the derivatives, and employing the boundary condition (3.5) one obtains 11

; s (^ o,4) + o o(4 ) Z [E +', S(,4)] C[E + 1')]. 0.014 1 V(5-,11) where the generalized S-matrix is defined as S(o40,) = (O,0;O0,-p; ), I > 0. (3.12) Since each bracket on the right hand side of Eqo (3,11) is a function of only one angular variable, this equation may be written as ZsS(4,O) +ksI S40,4) Z = U()C v(40), ('313) with UT() - E + s(,), (3 14) 1 dp' V([) =E +; - S(,). (3.15) Unfortunately Z and S(.,- ) do not commute, and to obtain an equation more amenable to numerical solution, the definition of a "matrix direct product" is introduced. If D is the direct product (denoted by *) of A and. B, i.e, D = A * B, then in component form we have [.i= [A]j [B]ij i,j = 1 ~N. (316) It should. be noted. that the direct product operator is neither associative nor distributive with the conventional matrix product. 12

To obtain a system of equations equivalent to Eq. (3.11) involving only U and V, integrate Eq. (3.11) first with respect to o'. Then use the definition of U(l.), Eq. (3.14) and the direct matrix product notation to obtain 3u(G) = E + [I ft di^' A^(i,')*[u^(l)CG)], (5.17) where the matrix A(J,L') is defined as [A(',' )]ij =i. +j 118) Similarly by integrating Eq. (3.11) with respect to g an expression for (4.) is obtained: V(.) = E + f d.' A(',)*[U')C v()] (5.19) Equations (5.17) and (5.19) are two simultaneous nonlinear integral equations for the U and V matrix functions. For N = 1, they correspond to Chandrasekhar's [14] one-speed nonlinear H-function equation with U - V - H. The equations for U and V, as they stand, can be solved numerically by the method of successive iterations.[12] However it is possible to cast the U and V equations into a different form which does not involve direct matrix products and whose iterative convergence is much better than that of Eqs. (3.17) and (3.19). In Eq. (3.19) it is not possible to factor the term V(g.) outside of the integral because of the direct product. However, it is possible to transform this matrix equation into a system of vector equations in which such a factorization can be accomplished. 13

Consider first the integrand in Eq. (3.19). Substituting explicitly for A(i,pl), and denoting summation by the repeated index notation (where a repeated lower case Greek subscript signifies summation from 1 to N), this integrand can be written in component form as Uiv(-[I')o~V j(M) [A (, ) U(' )C V( l)] (3.20) Ij Cai [I + rj.' Now define the matrix Vk(|) all of whose elements are zero except the kth column which equals the column of the V(4) matrix, i.e., [Vk(4)]ij = [V([)]ij jk (.21) With this notation the integrand (3.20) becomes D(,G') U(4') c V (4) (3.22) where the diagonal matrix D k(i,4') is defined as -k(' ij = i4+ Sij. (3.23) Using this notation, Eq. (3.19) can be written in the form N 1 V() = Z7 v.(i) = E + t fo d4' D (~~') U(l') C V (24) i=l _ Similarly the transpose of Eq. (3.17) may be written as N () i= El U(4) =.E + Jo d' Dp(,') (4') C u(), (3525) where 14

[Tk )]i, = ^ ^ji5 (5.26) [Uk(g)]ij - [U(g)]ji $jk These two new matrix equations for JU(G) and V(p), Eqs. (3.24) and (3.25) can be reduced to systems of N vector equations. This is possible because of the particularly simple forms of the matrices Vk(g) and. Uk(). If one defines the vectors vi(O) and ui(4) as Vii( uil() ()u vi(p) = V21i(I) and ui() = Ui2(l) i = 1 N, (5.27) VN (ij ui2(J then Eqs. (3524) and (3525) become Vi() = ei + 1 d2i' D i(4,)') UiC (4v ), i = 1 N., (3528) and i(4) =e+ 4 + dp' Di(4,4') V(G) C i(G), i = 1 N Oi 0 V_ ~~~~~~~(3.29) This system of vector equations is exactly equivalent to the U and V matrix which involved direct products (Eqs. (3517) and (3.19)). However now we can factor vi(g) or ui(.) from the integrands to obtain vi(4) = [E - od. Di(4,[) U(t) C] ei (3.o) and, ui(g) = [E- 11 d ~DDi(l9 ) Vg>') C]1 ei. (5.51) 15

It has been found. that this system of equations is also solved. readily by method. of successive iterations. However, the convergence rate is significantly [12] better than that of the iterative solution of Eqs. (3.17) and (3.19).2 Once the U and V matrices have been calculated, the generalized S(Go, ) function is readily obtained from Eq. (3513)o Thus the emergent distribution for the ith albedo problem is in view of Eqs. (351), (533), (357) and. (3.12) j(O0,;O,-) = s(%, ~)t, e>0. (>,32) ri(o'o;O,,) 14 (3 3) At the end. of this section it should. be mentioned, that in solving the albedo problem for a given transfer matrix 9, we have, in fact, also solved. the albedo problem for the transposed transfer matrix C. To show this, let us consider the soltuion jt(O, il;x, l) of the transport equation t 1 (7x E + i O) 41(0,4;x,4) = C l 4t(Oi1;xp ) di.' (3.33) which satisfies the boundary conditions (i) vt(O,1l'O,L) = ej 5(G1-4), l1 > 0, Y > 0, (3.34) (ii) lm *t(Oi;x x,) = 0. (35.5) To find. the relationship between the emergent distribution ji(Oo;,,-4) defined by Eqso (2.1), (351) and (3.2), and. t(o,1;O,-i) first multiply Eq. ~-I~j (2.1) from the left by t.(O, l;x,-ji); then multiply Eq. (3.33), with p replaced by -Cp, from the left by t-(O,toj;x, p)o Subtraction of these two results and integration over pI from -1 to 1, and over x from 0 to o, yields the identity 16

-1 0 f dlt [i Jf dx E [^t(~, li1;x,.-t) jr(~0p;x, i)] = 0. (5.56) Use of the boundary conditions Eqs. (31l), (3.2), (3-34) and (3355) and integration by parts of the above equation gives a generated reciprocity relation ht [o (O,41;0,-t )ei -41 qj Vi(O,0o;O,-4) * (3-57) As before we introduce an albedo matrix t(O; ) defined as tt t t J (0,pl;xp,) = [$2(0,l;xi ), 1(041;x,4).o.~N(0,|l;x,4)], (3.38) and. the generalized St (l, t) matrix S (41,) = t(041;0, -) ~ (3539) Then it follows from Eqs. (3~3), (3.12), (3537), (3538) and (3.39) that S( Jo,|jl) and S (Ll, o) are related by the equation S (410 = o) 5('O, 41) - (3540) Finally, by defining Ut(-) E+;l dHISttpV GI ) (3.41) and Vt(t) = E + f^, S^t(') (5,.2) 17

it follows that u ( p) = v ()M (3.43) and. vt ( ) = U3 () 5.44) IV. EMERGENT DISTRIBUTIONS OF OTHER HALF-SPACE PROBLEMS In this section it is shown that the emergent distributions for the generalized Milne problem and the half-space Greents function problem can be expressed. in terms of the generalized S-function or the U and V matrices of the previous sections. (a) The Generalized Milne Problem For every positive eigenvalue v(0Ol) or v = vos, s = 1 M, a Milne problem can be defined. Denoting its solution by tv(xp), it is defined as the solution of the transport equation, Eq. (2,1), with the following boundary conditions: (i) ~v((O, ) = 0, 4 > 0, (41) (ii) im (x) = (-v,4)eXv, v > 0, (4.2) where i(-vg,) may be any of the eigenvectors-regular or singular. The first step in obtaining the solution is to find the emergent distribution, v (O,-4), > Oo Consider a solution of the transport equation, jx,L), defined as (x,P) = v (x,P) +~ a(X,p) (4.3) 1.8

where a (x,, ) is an albedo problem solution with the boundary conditions (i) {a(OA) = ~(-v,), [ > 0, (4.4) (ii) m ta(x,) = 0 (45) Hence from Eq, (4,3), *(x,(i) must have the boundary conditions: (i),(0o,-) -= (-v4,),, > O (4.6) (ii) lim x ) - g-v,^e / (4.7) ilearly the unique solution for t(x,p) is -(x,p) = P(-vI)ex/v (4,8) Equations (4.3) and (4o8) then yield for the emergent Milne distribution v(o, ) - (,-) ~ t( ) - ta(o~-O), > 0 (4.9) The emergent albedo distribution, a (0,- ), can be expressed in terms of the S-function. From Eqs. (357)., (3512) and (4.4) (0,9-4) =- d4's(' t4)L(-vp,'); (4l10) and hence the emergent Milne distribution in terms of the S-function is (01 = v) 3 - 1-J d.'E',-vn) (4.11) In the same way we obtain the emergent distribution, < t(0O,-) for the transposed transfer matrix C 19

t(0,) = t(V) 1do' St(,) t(-v,) (4.12) Zt(vI) - 0 Once the S-function has been determined, these equations could be used to obtain numerical values for the emergent Milne distribution. However, in any computational scheme only the U and V-functions would be obtained, and thus this emergent distribution should be expressed in terms of these single variable functions. This reduction of Eq. (4.11) leads to a far simpler equation for numerical evaluation. From Eqs. (3.135), (3.23), (3.25) and (3.26) it can be shown that the S(MoC ) matrix may be written in the form (W'M) = MWM'U ([)C V(G'),D1(,G)), (4.13) where the double index notation is again used to denote summation. Recall also that the eigenvector, y(v,-l)'), v > 0, [l' > O, in view of Eqs. (2.6) and (2.9), can be expressed as 4(v,-.') = F(v,-'C^S a(v), v > 0, i'> O. (4.14) If the diagonal matrix Mk(v,P,M4o) is defined as ~Mk(V'~,IJ,) = Dk(lJ^,II' (v,1), (4.15) then the integrand of (4.11) is,1 ^ V N j2 v) (4.16) 20

This expression can be considerably simplified by considering the explicit form of Mk(v,l,)J-' Substitution of F and Dk from Eqs. (2.7) and (3.23) yields (in component form) (4.17) ^[k(V'~'~) )]ij (aiv+,)( iJ+ak) Eij The identity 1 1 1 F k 1 (~i4+a~k')(Gav+ i) a7kv-i a +a (4018) may be written as 1 1 P _k 1 P 1 (a i~k~- oif+')k' -Fi I')P- (4~19) (io+ak )(ovM) aki a kV-1 aGi4+aGk4 a ~ kvi -k v+1 (4i+ This result transforms Eq. (4o17) to I - _1 _ k 1 1 1 1~[M^,k(v,^ ) =^iJkvt ^ A l j+ - i j j airi (4.20) and since pb +d 1 d - b+d4-] Y (4.21) Eq. (4.20) yields Substitution of this result into (4.16) and use of (4.14) gives for the emergent distribution 21

V rlv —l~trl PV 1 K(O- ) = (V ) V- G(4)C diV(4 )I(V 1) + 4 1 )C dl'V(i')D (l,4')C a((v) * (4.23) This last term may be further simplified by considering the nonlinear integral equation for the U-function. The transpose of Eqo (3526) is u ) = E + () dl'v(i'))(,4'2), (4,24) or solely in terms of Uk(J) Egk(t)J Ek + Uk(41)C J d.V(p' ))k(,L'), (4025) where [Ek] ij i Hence the emergent distribution is,V(o,-4) = (V,) a- Pv- ()c1 C d'V(l')I(v,-') + TVr U ((Ti)SEC a() v) However, from Eq. (2.6) (v) - = V- E C a(v) + G(v,,)X(v) (4.27) Combining the last term in (4o26) w-ith jjv,[i) the emergent distribution simplifies to (4.28) t (0n-k) = i(7v', L(v) + ^" ()O4' )F(v,-' (). Finally writing this equation completely irn terms of the matrix JU(p), the emergent distribution of ~he generalized. Milne problem is given by the very simple equation 22

,(0o,-) -= vA(v (v) + PF(vt)U()h(v), v > 0, 4 > 0, (4~29) where the constant vector h(v) is h(v) = C - Jf d.4'V(4')F(v, -')C a(v). (430) (b) Half-Space Green's Function As a final example of the use of the generalized 5,-function technique, the half-space Green's function problem will be solvedo The half-space Green's function, with the source neutrons belonging solely to the ith energy group, gi(xo,o-0;xJ), is defined by the equation (~ F E + S)gi(Xo O;x ) = C dJ.'gi(Xoio;x,4') + &(j.-io)6(x-xo)ei, (4 7 ax Atci(Xo,~o;X,)7 x > 0, (4351) with the boundary conditions (i) gi(Xoo;0,i) =, O. > 0, (4~32) (ii) lim gx;x) 0 (4355) X40 g(Xo,%;x,~) = 0 o ( 405) The first step towards obtaining the solution, is to determine the emergent distribution, gi(x,po;,O-j.), > O0 Consider the Green's function to be composed of two parts: g j(x,,;X, ) + la(XJ,), x > 0, (434) problem solution ar (x,x)., satisfies the homogeneous transport equation with the boundary conditions 23

(i) a(0,) = -g~(xO,Yoi;O,;n), 0 > 0 (ii) lim a(xpa ) = 0 (4.35) Clearly gi(xo,[,o;xl,) defined by Eq. (4-34) satisfies Eq. (4o31) and has the required boundary conditions. The emergent distribution of the albedo solution, ta(O,-,), can be expressed in terms of its incident distribution from Eq. (357). Hence from Eqo (43.4) the emergent distribution gi(xo, o;0O,-s) is i.(x^o=;0,-) = g(xo',O0,-i) - 1 f1 di'S(', )g(Xo, 0;O,') * (4056) V. COMPLETE SOLUTIONS TO THE HALF-SPACE PROBLEMS Once the emergent distributions of half-space problems are known, the use of full-range completeness and orthogonality of the eigenvectors readily yield the coefficients of an eigenfunction expansion of the flux inside the medium. In the following the complet-e sol.ution to the half-space albed.o, Mllne and. Green's function problems will be obtained. (a) Albedo Problem: First, we seek the complete solution for the ith albedo problem in the foim M / M 1i(0~,o;X,I) = 0(v ( vVo,)e x/os + E (-vo )~-vos )ex/os S=l 0,OS ( sS=1 Os N ri N + Z f' AA(v) ~(m(v,)e X/V + A( —v)j m(-v p)ex/v je( l=l' e (5.1) rjlm=j where 24

nj -- and. 0o 0 In view of the full-range completeness of the eigenvectors ~j(~v,O), vC(0,1), v = v s, we can now determine the expansion coefficients in Eq. (5.1) so that the above equation with x = 0, equals the known surface distribution0 The expansion coefficients a(+voS) and A(~+v) are readily obtained by applying full-range orthogonality relations (Eqso (2o19) and (2.20)) and Eqs. (531) and. (3o32)o Explicitly =^('os) ~ Os (Cos4o r e (5.2) 5 "sL N~os and. A(+v) = + (~+v, )e ( ) (~v,-)S(4o,)e v > 0o (5-3) n Appendix A i is shown tha he funion satises certain elatn In Appendix A it is shown that the ~SS(Jop) function satisfies certain relationships with the eigenvectorso From Eqso (A-5) and (3-40) it is seen at once that a(-voS) and Am(-v), v > 0, are identically zeroi Thus we see that the boundary condition at infinity, Eqo (306), is satisfiedo Therefore, the expansion Eqo (5el) with a(-v o) and Am(-v) set equal to zero represents the complete solution of the problemo This in'turn implies also the half-range completeness of the eigenvectors,v,t), ve(0,l), v V o=, (b) Milne Problem Since half-space albed.o problem can always be expanded, in terms of only the decaying eigenfunctions9 Eqo (403) shows that the solution for the generalized Milne problem can be written as 25

t(Xi ) = (- e + a(vo)(vsl N fN + Zf Am (z )m(v' ( )V eXx/v (5 4) j=l - m=j J Setting x = 0 in this equation and. using Eqs. (4.1) and (4.29) one finds from the full-range orthogonality relations that 11 fl T.t c(Vos) = S- S dl l (voOS4)tv(~,-) - f d.it(vos,) ((v,,()x(v) + PF(v,n)U(I4)j(v)) (5.5) Ns 0 and A>(v') m=- f1 d5t~m(v', )(G(v,)vJv) + PF(vy)_U(4)h v)}. (5.6)' _N.(v,) 0 Often the Milne problem of most interest is the one related to the largest discrete eigenvalue, v o The asymptotic behavior of this particular problem for large x is as -x/v~ C(X,) = (-v,)eXi/V + o(v )O(v 11)e (5 7) A quantity of interest for this problem is the extrapolated end point, xo, defined such that as s 1 asl a s, (5-8) p (Xo) = div(xso, ) =a(v)e + av)(v)v)exO/V (08 -1 Solving for xo and substituting for a(v ) from Eq, (505), the extrapolated end point is XO = - 2^ tnN t d~in(v,)l(,-u),' (5,9) 26

or in terms of the TU ) matrix I~ 1 1 - a 2 xo = - 2 n no 1 d[ia (v~)C F2(v.4 )U()h(v o (5o10) (c) Green's Function From Eqo (4.34) we see that the complete solution of the ith half-space Green's function can be written as M -X/Vos gi(xo'o;x, =) = i(xoo;xg) + se a(v os)(vos,)e s1 O5 )o -,/ 0 Z d N X/V dv Am(v )Z j(v,[t e-/, (5.11) j=l jm knj A ^ J j1 ^=j Using full-range orthogonality relations and Eqs. (43.6) the expansion coefficients are readily found to be 11 t r4S(. 11 dvos0) = - N f d.t44 (vos4[) (XOGi - duxS(4t Gi o 0 3 (5.12) and Am Nv) f= -, A( = 1 v 1 di0^ I 2;] (5135) 27

APPENDIX A As previously mentioned, Eqso (3515) and (3.17) are the multigroup generalization of Chandrasekhar's one-speed. H-function nonlinear integral equation. Since [16]-[18] this one-speed equation does not have a unique solution, one suspects that the nonlinear integral equation for,J(-,GI) and therefore also for the equations for U(4) and, VJ() are not uniquely solubleo Proceeding as in the one[181 [19] speed. case or in the case of the degenerate kernel approximation it can indeed be shown, that for each discrete root, vos, Eqo (3.11) admits a "nonphysical" solution denoted by Sos(oI, ).o It is equal to Sos(o lo, = + 2 ) Jv (~ O,-4) (A.l) where (os) Os v ( ) o It is possible, however, to give a set of conditions which must'be satisfied by the physical solutions S(J%,t) or U(') and V(j). The eigenfunctions t(vl)e-x/v, Re(vw > 0, are solutions to the transport equation; and since they tend. to zero for large x, they are solutions to half-space albedo problems with incident distributions given by ~(v,), 4 > O0 Thus from Eqs. (3.7) and. (3.12), the S(o,~) function must satisfy -(v,-l) = f1 dlp j-))ll v,VL ), p. > 0, Re(v) > O (A.5) 28

Entegrating this condition over J. from 0 to 1 and using Eqs. (3.15) and (2,17), Ne obtain -1 o J d rvi4) = a(v) = d v,) (A.4) Similarly by consid.ering t (v,)ex/v, Re(v) > 0. as albedo problem solutions of the transport equation for a transposed transfer matrix C from Eqs. (Ao3) nd (2.17) one has the conditions on the St(o0,.4) matrix t =1 1 t t,(- ) = J1 =d4 (4/ r )(-v,-p'), Re(v} >0. (Ao5)!gain integrating over 4 from 0 to 1 and using Eqso (5340), (3.14) and (2.17), Cq. (Ao5) yields d (-v,) = (v) = d(-v ) Re(v) > 0o (A 6) Eqs. (Ao4) and (Ao6) for the discrete roots vos, s = 1 ~ M are 2M conditions which the physical J(gJ-) and jy(-) functions must satisfy. In one-speed case these equations becomes identical, and it has been proved that they are a sufficient condition to uniquely specify the real physical H-function given by the nonlinear integral equation (3-17)o [9][18] Also for the degenerate kernel approximation, Pahor, using a corresponding set of discrete eigenfunction conditions, proved. that, these conditions were sufficient for uniquely specifying his generalized S-functiono ] Although it has not been possible to show that the discrete root conditions for the general multigroup case are a set of sufficient conditions, it is felt that they are a severe restriction on the possible solutions of Eqso (3ol7) and (3519), and in all likelihood they are sufficiento 29

Therefore, in iterating Eqs. (3.17) and (3518), the conditions Eqs. (A.4) and (A.6) must be used as a check. At the same time, an estimate of the accuracy of the iterations can be obtained, from these conditions. 50

APPENDIX B NUMBERICAL EXAMPLE As an illustrative example, the emergent distribution for a 6-group halfspace Milne problem was calculatedo* A 2% enriched uranium medium was chosen and the cross sections were calculated from 6-group tables developed, by Hansen for fast assemblieso[]'2 Table I lists the problem parameters. For these cross sections, the largest (or dominant) eigenvalue (calculated from Eq. (2o13)) was found to be 12o379 cmo The U and V matrices were calculated from Eqs. (3.30) and (5531) and then the emergent distribution for the largest eigenvalue was found from Eq. (4.29). The results are shown on Figure 1. Finally from Eqo (5.40) the extrapolated endpoint for this particular problem was calculated to be 2.218 cmo *Information on the numerical techniques, computer programs and other examples can be found in Ref, [12]. 31

REFERENCES 1. K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, Mass. (1967). 2. C. E. Siewert and P. S. Shieh, J. Nucl. Eng. 21, 383, (1967). 3. T. Yoshimura (submitted to Nucl. Sci. Eng.). 4. R. Zelazny and A. Kuszell, Ann. Phys. (New York) 16, 81, (1961). 5. D. Metcalf (University of Michigan doctoral thesis), Ann Arbor, Michigan (1968). 6. C. E. Siewert and P. F. Zweifel, J. Math. Phys. 7, 2092, (1966). 7. S. Pahor and J. K. Shultis (submitted to J. Math Phys.). 8. A. Leonard and J. H. Ferziger, Nucl. Sci. Eng. 26, 181, (1966). 9. S. Pahor, Nucl. Sci. Eng. 26, 192,(1966). 10. S. Pahor, Nucl. Sci. Eng. 29, 248, (1967). 11. K. M. Case, Proc. Symp. on Transport Th., Am. Math. Soc. April (1967), to be published. 12. J. K. Shultis (University of Michigan Doctoral Thesis) Ann Arbor, Michigan (1968). 13. V. A. Ambarzumium, Astr. J. Sov. Union 19, 30, (1942). 14. S. Chandrasekhar, Radiative Transfer, Dover, New York (1960). 15. R. Zelazny and A. Kuszell, Proc. Seminar on the Physics of Fast and Intermediate Reactors, Vol, II, IAEA, Vienna (1962). 16. T. W. Mullikin, Astrophys. J. 139, 379 (1964). 17. T. W. Mullikin, Astrophys. J. 139, 1267 (1964). 32

REFERENCES (Concluded) 18. S. Pahor and I. Kuscer, Astrophys. J. 143, 888 (1966). 19. S. Pahor (submitted to Nucl. Sci. and Eng.). 20. Reactor Physics Constants, 2nd ed., Argonne National Laboratory Report, ANL 5800, (1963). 21. Hansen, G. E., "Properties of Elementary, Fast Critical Assemblies," Proc. Geneva Conf. (1958). 33

ACKNOWLEDGMENT One of us (S. P) wish also to express his gratitude to the members of the Department of Nuclear Engineering, The University of Michigan, for hospitality shown him and to the Institute of Science and Technology for financial support. 54

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FIGURE CAPTION Figure 1. Normalized emergent Milne distribution. 36

\ \a. \ I -o D (0 D O09 ~ 0 ~o\ \ \ UI o 0 I 0r bJ oj o N~ cp r o o NOInnBIW.LSIO IN39I3V13 3AILV133 57