THE U N I V E R S I T Y OF M I C H I G A N COLLEGE OF ENGINEERING Department; cf Nuclear Engineering Technical Report SOLUTION OF THE TWO-GROUP NEUTRON TRANSPORT EQUATION PART I D. R_ Metcalf P. F. Zwe.ifel ORA Project 01046 supported by: NATIONAL SCIENCE FONUWATION GRANT NO. GK-1713 WASHIN"GT)ON, D, C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1968

SOLUTION OF THE TWO-GROUP NEUTRON TRANSPORT EQUATION PART I* D. R Metcalft and P. F. Zweifel Department of Nuclear Engineering The University of Michigan Ann Arbor, Michigan Number of Pages: 30 Number of Tables: 1 Number of Figures: 0 *Based on a PhoD. thesis submitted by one of'the authors (DoRoM ),to The university of Michigan, Work supported in part by National Science Foundatio.r..o tAtomic Energy Commission Predoctoral Fellow (1964 6;7). Presernt adreiss. Department of Nuclear Engineering T. University of Virginia Charlott. sevile Virginia 1

Proposed Running Head: Two-Group Neutron Transport-I Mail Proofs to: Professor Dale R. Metcalf Department of Nuclear Engineering School of Engineering and Applied Science University of Virginia Charlottesville, Virginia 22901

ABSTRACT The two-group one-dimensional neutron transport equation with isotropic scattering is studied. No analytical solution is found, but the equations are cast in a form which is convenient for numerical computation. This computation involves the solution of two coupled singular integral equations. The explicit form of these equations is obtained for two half-space problems — the Milne problem and the constant isotropic source problem.

I. INTRODUCTION In recent years, a great deal of effort has gone into the solution of the one-speed neutron transport equation. Most of this work is reviewed in a recent book. Attempts to deal with the energy dependent transport equation have been less successful., although a certain amount of progress has been made. These attempts have more or less followed two lines, a multigroup approach and a continuous energy treatment. In the multigroup method, energy effects are treated in considerably grosser fashion than in the continuous energy scheme but in the latter spatial effects have been quite difficult to include in 2 much detail. However, in a recent paper, it was shown that the multigroup scheme is equivalent to a continuous energy scheme with a particular degenerate scattering kernel, A good review of the continuous schemes has been 3,4 given by Kuscer, to which we refer the reader for references. In the present paper we consider the multigroup approach, in particular the two-group case. As usual, we limit ourselves to a single space dimension and to isotropic scattering. Our method should be readily generalizable to mere than two groups and to anisotropic scattering. The two-group case has been studied previously by Zelazny and Kuszellui They succeeded in proving a "completeness theo-rem" which in effect tells us that the normal modes which we develop are adequate tt) solve all infinite and semi-infinite medium problems, We depend upon this theorem, but because the results are not presented:in a form convenient for numerical computation much of their analysis is repeated. Analytical solutions have been obtained in 6 two recent papers for a very special problem, but the restrictions upon the 4

parameters involved are invalid in the neutron transport case. (The work was applied to radiative transport.) Siewert and Shieh7 have carried out some preliminary work on the more general problem considered here, but their results can be applied directly only to infinite medium problems. Recently some 8 unpublished work along similar lines to that presented here has come to our attention. In Section II the eigenvalues and eigenfunctions for the two-group transport equation are discussed. Essentially these same eigenvalues and eigenfunctions were obtained by Zelazny and Kuszell 5 although, as in Ref. 6, we select certain convenient linear combinations of these eigenfunctions. In Section III we follow the Case approach in deriving a convenient pair of coupled equations for the expansion coefficients of an arbitrary functio n. Some important simplifications are made in these equations by the judicious use of a two-group X-function identity. These coupled equations are easily solved by an iteration technique using a computer program. (Nt analytical solution has been found and in fact it appears unlikely that an analytical solution exists ) The numerical. techniques and explicit results 9 are presented in a companion paper. The application to typical half-space problems is made in Secticon IV.:In particular, the solutions to the two-group Milne and constant s urce pro.blems are presented. II. EIGENVALUES AND EIGENFUNCTIONS The one-dimensional, two-group transport equation for isotropic scattering can be written as 5

I j(z, 0 + _ V(zlp) C f 1(ztll)d4 (1) Here *(zl~ /t1(ZI4\ /C11 C012 \42(Z4)) C21 C22) and ~ a. 5 C > 1 0 1 where Cij = oij/202 and 0 o1 /G2 1l(z)4) and 42(zpt) are the angular fluxes in grcups one and twc, respectively; distance is measured in units of mean free path of the second group, We have (without loss of generality) ordered the groups such that c2 < 01 where o1 and 02 are the macrcscopic total cross sectio. ns of the respective groups and o.js are the macroscopic scattering transfer cross sections which describe the scattering from group j to group io As usual, we assume solutions of the form r(z,ni) = e Z/ F('q,)o (2) This ansatz when substituted into Eq, (1), and after cancellation of the space dependence, yields A( ~F( 3'- C I F _(, 4)6

where CM -4 0 Ao (4) The eigenvalues (k is the eigenvalue) and the associated eigenfunctions of Eqo (3) are discussed in some detail in Ref. 6~ From the matrix A we note that three separate regions of the eigenvalue spectrum must be considered depending on the vanishing or nonvanishing of the respective terms in A. We list here the three regions of the eigenvalue spectrum with their associated eigenfunctions as obtained in Ref. 6. Region 1: qc[-l/a, 1/a] with degenerate continuum eigenvectors C11TIP + [l -21Cjj1T(Gn)16(7-ii) \ F1 (rai) = ( (a) C21P 2C and at1z) 2C1(29_(f )6(o ) I) Fl j (5b) C221P + [-2C22T(l)1b(1-k) Here T(q) = tanh-1 he Region 2: ~c[-l, -1/a] and [l1/, 1]: The continuum eigenfunction is F(2)(,p) / aq-lA + x (rl ) (l_

where t(~) 0 C22 - 2-C(1/oq) (7) and (%n) ~= 1 - 2nC227(r) - 2rC11.T(.l/,:) + 4Cr27(n)T(l/oq.) (8) We have introduced the abbreviation C det C The symbol P in the above equations indicates as usual that integrals invo!lving these eigenfunctions are to be interpreted in the sense of the Cauchy principal value. Region 3: ~j[-l1,1 Here we have the discrete eigenvectors F. (.L): /C12/ni/j-T 7 F:.G) F ()' (9) tJ-, Fi~ 2'() where t([i) is defined by Eq. (-) and the eigenvalue ji is the positive rcct cof the dispersion equations Q(z) - 1 - 2CLlzT(l/oz) - 2C22ZT(1/Z) + 4Cz T(l/z)T(l/oz) =. (1)) For the problems we have studied there are only tw) real roots to Eq. (1C) which we denote by q`1. In general, there are either two or four roots which are either real and/or pure imaginary, and thus occur in:+ pairs. These rcots have been studied in considerable detail by Baran and we reproduce in Table

1 his analysis of the eigenvalues. We note that Q(z) is analytic in the ccmplex plane cut along the real axis from -1 to 1. Introducing the notation i(iN) = 1, EcRegion i C,'i (11) we easily find 1 - 2C2211(pT) - 2C~1pj(oi)T1(p.) - 2Cjiir(l/G0)G2(4) + 4 + 4C2(T()l/1/)2() - 22CE () 2 2i [C22+Cl G1 ( )-2Cl( ol ) 1 () -2CpI'r (4)@1 () - 2C41 T(l/ct)G2(W1)]* (12) Here - represent the boundary values of Q(z) as the branch cut is approached from (bbelow) iLe., QI(4) = limit Q(Itic). It is convenient to use the following linear combination of Region 1 eigenvectcrs: 1 F(1) 1 F(1) (13a) 11 C1 - Explic itly, 1 ( 1 - ___________ 6((_t.) [cii-2an,~(n)1 Cl1 C1 2( k- ) CL 2C11 The following linear combination is a convenient eigenvectcr valid over the full range, 6C[-ll]: ~2(1r, t) = 2C1()1()F1) + [ 1 + F(2)(p, ~)eG2(n). (14a) 9

Again the explicit form is / C1E2P/(Co -1) ~2(T ) -j! |.(14b) Here we have defined g%() C22 - 2CqT(ao)91(%) 2Cr'(l//ao)G2(n) (15) and (l)'1 1 2C22r'(r) - 2C2C\T(o1)Ql/) - 2CllT(l/ogr)2(r) + 4CP2:(q) x T(l/O])G2(rq)+4Cr2q(r,)T(orl)Gl(3i) = + T2f2CG1(f) (16) 2 Thus, the eigenvectors consists of two continuum modes _1(n, i) [Eq. (13b)] c[-l1/, 1/o]; js(y,1p) [Eq. (14b)], qr[-1,l]; and the discrete modes Fi_+(L) [Eq, (9)]. A half-range completeness theorem for these modes has been proved in Ref, 5. In the next section we reduce the basic equation to cne convenient for numerical analysis, III. TWO-GROUP EQUATIONS The half-range completeness theorem states that an arbitrary (two ccmponent) function *( ) can be expanded in terms of half the eigenvectors if we consider only half of the range, eg., pg > 0O Specifically, _( ) = = X A i+ F+() + i) + + f 2(r)/2(r,)df, p > O. (17) 10

Typical proofs involve the solution of this equation for the unknown coefficients Ai+ and aj(r). This procedure also has the virtue of providing the required answer, i.e., the expansion coefficients. In the case considered here (as is stated in Section I) no analytical solution has been found. However, in Ref. 5 the existence of a unique solution has been proved. Thus, we develop a numerical solution to Eq. (17) by a method that parallels closely typical half-range completeness proofs. By appropriate use of half-space Xfunction identities, we obtain a form which is found to be most convenient for numerical analysis. We delete the discrete mode from Eq. (17) the discrete modes will be introduced later), substitute the explicit form of ~j(r,k) and ~2(N,4), Eqs. (13b) and (14b), into Eq. (17) and perform the integrations over the delta functions to obtain 1() = ( ) + C12FP t t2(C)di (18) oC1 1 1rT1 - and 2( ) P = C,/ - 2Cp r(t)d - 1 [C1-2 )al(p ( ( ) )1 (1) C11C1 0 C11 C12 + P fl T12(rq)g(q)dq + 1\()O2(p). (19) Making the change of variable p + po in Eq. (18) and solving for ai(p), we obtain,(II) = oC11Jii(up.) C11C12P /1 rP~(n)dri. (20) o rl-p 11

This equation will determine al(p) once CU2(p) is known. We now obtain the equation for C2(G) by inserting Eq. (20) into Eq. (19) to yield V2 (k) + Ca p f1 nl(a )d + a [C2-2c4-(i)]*jl(ac)((4) C12 0 C 2 CP fl/a I'd' Ip rf C2(r()d + [C11-2CI-T(0I)]- 1( ) fP J 2(()d o q - o q-~ o q-p + p f1 n2(r)g(qn)dn + (Ol)ac2(G). (21) o r-et10 Using the Poincare-Bertrand formula and the partial fraction decomposition _ rl_'.1 (p11 _ + (22) we perform an integration over dt' in the first term on the right-hand side of Eq. (21) to obtain CP,, d 1 2( T)jdq 1 q9Z2( ) dT Q (: 1I CP fl/ dr' p qnC,2(rTd]' = p zl nc2(rpdi L(i)CpQn (.- - 1) + @2(i)Cpin (1 L - i (_)Cln (n - 1) (23) - 2(N)Cn (1 - ) Ci2t2()() We can write Eq. (15) as g(r+) = C22 - C+n C(1 ) 1()n ( - 1) + C2()n (1 - (24) The insertion of Eqs. (23) and (24) into Eq. (21) yields (after some cancellation and rearrangement), the compact form 12

2 (25) Here we have defined xV( ) 2= V() + Co p _1 _(o_)d + C l(a)1l(4)[C1 -2CTr()] (26) and k(qrpi) ~= nn (- + l (27) We note that V'([') is a known function. The second term on the left-hand side of Eq. (25) is a nondegenerate Fredholm term. If this term were absent Eq. (25) could be solved directly by standard procedures. If it were degenerate, the method of Shure and Natelson could be used. However, neither of these conditions obtain and no closed form solution of Eq. (25) has been found. Thus, we shall describe an iterative procedure in Part II of this work9 similar to that used by Mitsis12 to solve the critical problem and by McCormick and Mendelscn in treating the slab albedo problem. We define,"(4) _= J(4) + C fSl 42(1 )k(1n)dn. (28) o l-s Next we introduce a function N(z) defined by N(z) 1 ~ l /_ 2(~)d,. (29) 2ii o T-z 13

If c62(q) of "class G" exists then N(z) has the following properties: l, N(z) is analytic in the complex plane cut along the real axis from 0 to lo 2. N(z) 1l/z as z + oo, 5. N"(Ct) 1 P f1 rIC62rk)d ~ 2I 2iti 0 F4 2 We note from Property 3 that a,( =) N= ()-N-(~) (30) Inserting Eqs. (28) and using Property 3 in Eq. (25) we find (after some rearrangement) that'"(~) = N*(G)*Q+(k) - Q-()N-((). (31) We recall that Q(z) is analytic in the complex plane cut from -1 to 1. From Property 1, N(z) is analytic in the complex plane cut from 0 to 1. This requires (as in one-speed theory) the introduction of a function X(z) such that IX+() Q+- 4 > > o. (32) X-(~) Q-(_) The X(z) function satisfying the necessary restrictions for the half-range and for one pair of discrete roots only is X(z) = 1 exp 1 Arg +()d (33) 1-z M; o p-z The ratio condition, Eq. (32), is inserted into Eq. (31) to yield r(r,"(J) = N7+()>x+(4 ) - N-(t)X-(H) (34) 14

where 7(4) = x+( (35) Assuming that the left-hand side of Eq. (34) is known, we can write the solution as N(z) - l l ( x. (36) 2jtiX(z) o [A- The Plemelj formulae10 give the boundary values of N(z) from Eq. (36). We insert these boundary values into Eqo (30) to obtain 1 1 1 P f 2 xi x+X ) X-( ) +l (1 + ) ()T(37) 2i VX+(O) X-( ) (However, this is not a solution because we note from Eq. (28) that the unknown a2(It) is still contained in *1"(i)t) We next define 1~-l() - 1 1 1i (38a) and Q(x2) X )= (>)+( )Q.( (38b) The last form of Eqs. (38) is derived from Eqs. (32) and (35). By defining the singular integral operator O([i) by O(~)q(~) = ~(~)p Sl y(i()>(k,)dci, +.2(ki)4(i) (39) o 15

we can rewrite Eq. (37) in the compact form c2(kt) = o(0 )( ). (40) Property 2 requires that N(z) 1/z as Izl oo. We note from Eq. (33) that X(z) - /z as z + o. Thus, from Eq. (36), we must require (~) +(). C f drr 2()k( ) = 0 (41) where we have used Eq. (28). For the case of one pair of discrete roots we have the discrete eigenfunctions available to satisfy Eq. (41). We make the replacement of _([) in Eq. (41) by (k) - A+ F1.(t). (42) We recall that *'(p) is defined by Eq. (26) and is a functional of the components of l(kt), i.e., 41(Pt) and 42(p-) We define 0+( k) as the corresponding functional of the components of F1+(~). Thus the replacement given by the expression (42) is equivalent to replacing \'(k) in Eq. (41) by *'(4i) - A+/+(4i) * (43) With this replacement made in Eq. (41) we then solve for A+ to yield bl?(U),'(~)d + C l Y(r)d4 ~ rlTA.:kt 4 ())dq(d.d (44) Likewise with (43) inserted into (28) and subsequently into Eq. (40) we obtain c,(p) = O(p)[ E'(p)-A,~,(p) ] * CO(>) /1 rg2(r,)k(n,~)d q (45) o rl-_ -

Finally, the replacement of i(a>o) by li(ao)-A+Fl+,l(k) in Eq. (20) yields a~(~) = oCll[1l(oa4)-A+Fl+,l(o)] - C11C12P f r )d. (46) An iteration procedure for solving Eqs. (44) and (45) for A+ and a~2(0) is discussed in Ref. 9. Providing this procedure converges, we can then insert A+ and a2(kt) into Eq. (46) and solve for al(i) completely determining all expansion coefficients. A number of important simplifications are now made for the terms containing ~+([) in Eqs. (44) and (45). First, from Eq. (26) we can write the explicit form of /+(p) as +([) = F1+,2(k) + C p fl rFl+,l(aCo) + a F1+,1(_O_)Q(0) [Ci —2CiT(C. (47) Next we insert the discrete modes as given by Eq. (9) into Eq. (47) and perform the integration to obtain ~+(II) l= __ [C22+Clll(1)+C~ il()~n (1l 1) + CkQ2(j)~n (1 - - 2CGQi(,),T(i) - Crln (1 + -(48) We define the function f() = (2-( = C22 + C19(kt) - 2C91( )[T(k) - CkG~(4t)~n (1 + _) + C.i~l(p)in _1 - CW~2()~In + 1 + C~@2(.)in -(1 -l). 17

=+(iz) =-Tl- [f(p)-Ck(rll,k)]. (50) The substitution of Eq. (50) into the integrand of the denominator of Eq. (44) yields 1 7(k)+(I)df = -rlX(kl) - C f )(/)rlk(Tl,>)dk (51) 0o o z1-P where we have used the X-function identity X(z) = f1 ((5)f(p)d. (52) 0 The proof of this identity parallels closely that for the one-group case. Next we consider the term 0(t)/+(4) in Eq. (45). With 0(j) given by Eq. (39) and /+([) by Eq. (50), we write 1 y( 0f(it')dK + 12([I)rif(GI) - CO(p) k k 1)] (53) The partial fraction decomposition inserted into the first term on the right-hand side of Eq. (53) yields.(p)P.I.1 Y(P' )rlfz-~(Z' )clp' n1 r( )f(4 )dU o ()K -I f (')f(,' )d' j. (54) o'rlz —'. The last term in brackets in Eq. (54) is -X(nz) and the first term can be written in terms of the boundary values of X(z). Explicitly,

p /1 y(~')f(~')d' = - X+(-I)+X-(0) = _ Y( [Q+(,) + 2-( )]. (55) 0 2 2i The last form in Eq. (55) was derived by using Eqs. (32) and (35). We insert Eq. (55) into Eq. (54) to obtain I()p f 7(,)1rf(k,)d? =.(L) L [X(r) - )(k) (Q+(I)+r ()]. (56) o (k —')( kj-' ) a1-4 2p We insert Eqs. (38), (56), and (49) into Eq. (53) to obtain (after some calcellation) O().( ) = (O)rX(-) _ CO() ( (57) Now Eqs. (51) and (57) can be substituted into Eqs. (44) and (45) to obtain a somewhat simpler form, o1()'()c + C 1 Y(k)d 1,(7)k(,. ) d A+ = (58a) Ca(p2) =- O(M)1,() + CO(() j1 pa(n)k(f,l)dn 0 -A+ ( )x() _ CO() (,1 (58b) rl__ -L l_- l (8b) We recall that the kernel k(rj,[i) is defined by Eq. (27). The operator O([1) is given by Eq. (39) and ~1(pi) by (38a). Equations (58a) and (58b) are the set which will be treated numerically,9 as simultaneous equations for the expansion coefficients A+ and aQ2( ). After A and a2(1i) are obtained from the numerical solution of Eqs. (58a) and (58b), GQi(~) is computed from 19

el( ) = Cll | l(a )- A+ C121 |- C11C12P f1 2( )d.~ (58c) This completes the reduction of the general two-group expansion to a form convenient for numerical analysis. In the next section, we shall find some specific forms for v'(k) in Eqs. (58a) and (58b) and for \l(ak) in Eq. (58c). We are then able to simplify the terms involving'(j) in Eqs. (58a) and (58b). IV. APPLICATION TO MILNE AND CONSTANT SOURCE PROBLEMS (a) Milne Problem We define the two-group Milne problem ina manner similar to the onegroup case. The solution must satisfy the conditions,(o,,) = 0, 1 > 0 (59a) and (59b) *(zM) ~ F_(M)e /b.) Z-Moo The solution which obeys (59b) is expanded in the two-group normal modes of the transport equation as _(z,9) = A_ F1+ A (+)ez/Tl + lA+ Fl+()e-Z/l + / _ l )ez/odq + l 2(n)~(,z/)e A dn. (60) 0 We use the boundary condition given by Eq. (59a) (normalize by setting A_ = 1) to obtain X (k) = - 1(P) = A+ F1+(j) + Al(~)( ) + f 2()2(,)d 0 0 (61) 20

The appropriate Vm(4) for the Milne problem [see Eq. (26)] is closely related to the j+(i) as defined by Eq. (47). In fact *m')= ~G O+ 1.(62) This means that [see Eq. (50)] ~m(t) = _.1 [f(~)-Ck(-l,#)j. (65) *' GO (63) Thus we have from Eqs. (51) and (57) the result that 11, y(~)xm(i)dit = ~-X(-~1) + C f.(l ),1k(-1,l )d (64) and 0( o)() =- _ ~(k)ix(-n_) + co(U) Lk(-g. (65) We insert Eqs- (64) and (65) into Eqs. (58a) and (58b) to yield -AX(-) + C l L + C 1 y()di 1 A -r1X(l) - C bl 7(r1k(flk)d (66) and C2(GO) = - 1,(k)1X(-n1) + COGO) 71k(-T, 1p)1 + CO(() f TP2(rq)k(,q4)dTj _ A+, 1) - c Ln( ) _ [(%) 1 (67) Since z(ai) = C12hl (68) a( rll-+) 21,

we have from Eq. (46) that c61() = -C11C12r1 + )- ClCl2P r(dr (69) Equations (66), (67), and (69) are the final reduced equations for the Milne problem which are solved by numerical methods. We note that Eqs. (66) and (67) are two coupled equations for the expansion coefficients A+ and c62(ji). The operator O(p) in Eq. (67) is a singular integral operator which requires special treatment for numerical analysis,9 otherwise the solution is quite straightforward. Finally with A+ and U2(pi) known, Eq. (69) provides the solution for a1i(). We can easily prove from the form of the operator 0(i) and Eq. (58b) that U2( ) + 0, [ = 1/a and i = 1. (70) This result is important because otherwise the angular flux would be singular at these values of ki. With the expansion known from the computer solution of Eqs. (66), (67), and (69), the two-group angular fluxes are calculated from Eq. (60). The eigenvectors given by Eqs. (13b), (14b), and (9) are substituted into Eq. (60) and the integration performed where possible to derive l(z = C22Ale Z1 + 1 + C12P 1 ~2( )e Z/ dl (71) where -1 K 4 < 1, 22

and 2(lt(l)e /1 + lt( ql)A+eZ/ll C p /o r(l(r)e lz/d l+~ T-~ Cll~C11C12 - i() [Cii-2CiT()]i()e-z/ + P 1 rg(r)c 2(n) e- Z/Td Ci2C 11 o + %(G)aa2()eZ/k [@1([)+e2(kl)], e [ -1,1. (72) The total flux and current for each group are derived by appropriate integrals over dkt and idk,, respectively. We shall use superscripts on the p's to indicate group number. Thus, P(1)(z) = 1 (z,ki)dt = 2C12lT(1/Tl/jl)[e Z/l1+A+e Z/1l] + f/ o 1 C11 o x UOl()e/ df ] - C12 f rX2(r])e- [g(k)-C22]dl (73a) P(l)(Z) = 1 ( = 2C12eZ/1[1-[l lT(1/ 1)] + 2C+121e/A+ x [oliT(l/cnq)-l] + 1 q /" rpq(r)e Z//dt - 2C12 T rjp2(W)e'/r'dr - CC 12 1 2(2() -z/e [g()-C22]d] (73b) p (2) (z)= f *2(Zo,)d = 2B1t(B1)T(l/fI )[e z1+A+e /1 2C x fl/ r( rl)r( ) e dld - 1 f1/E[C 1-2CnT(r) ]Cl() e-Z) d + 2 1 lg(l)aQ2(l)T()ez/ dR + l k(l)2()e d (73c) 23

p(2)(z) = 21 2(z,.)d = 2f1t(ri)e [1 ( + -l x A+[n1T(1/1l)-l1] -l(~) [T(2C)-l] e dn C11C21 1C1 2/71 1*2 1 C CT r/2(r)[Cll-2CllT(r)]e r dn + 2 lg(f)cC2(f) x [fT(r)-l]e/fl dB + I (r)2(h)e eZ/ d~. (73d) The extrapolation distance is given by either Eq. (75a) or (73c). Thus we wish to determine zo such that eZ l + A+e -Z l (74). The solution for zo from this equation gives the same result as in the one1 speed case, = - n ( A) I (75) Again we emphasize that for one pair of discrete roots the extrapolation distances are equal for each group. For four discrete roots we would calculate a different extrapolation distance for each group. Baran has studied this problem in detail. (b) Constant Isotropic Source Problem Assume an isotropic source So02 in Group 2. (The associated problem with a source in group one is virtually identical.) We now seek solutions to the inhomogeneous transport equation which vanish at +oa subject to the condition, _(o,) = O for o > O. (76) 24

The solution consists of a particular integral x (z,kt) plus a solution of the homogeneous equations,,h(Z,). The latter consists only of those modes which vanish as +*o, i.e., _h(z, ) A+e a 1 F+(+)j(>)ez d + 1 ()( ) _z (77) The particular integral can be found from the two-group transport equation [Eq. (1)] in the limit as zoo. In this case, _(z,i) approaches a constant value denoted for each group by 1S and *2S' With *r(z,k) constant, Eq. (1) reduces to a pair of simultaneous equations which are easily solved to yield V 2SC12 S (78) is (l-2C22)(ao-2C1~) C12C21 and S(a-2C1j) 2S =S(o-2C) = S2 (79) (l-2C22) (o-2C) -i4C12C21 where S2o 02 The complete solution is written as S1 + A Ae z 1/% ()( (zla) = (s) + A+e /l F +(h) 6+ /1 / F i +() e d+ C 22()~2(r, i )e dI2 (80) + 1 Y,(n,,(nc_, PzA,9 (AO

By setting z = 0 in Eq. (80) and applying Eq. (76), we obtain _S- ) = l/ Uj(f)BI(rI4 +)dl + l O62(7)J2(fl,)df + A+ 1() (81) We note from Eq. (81) that 4r((r2(i) = _ S2 (82) By inserting Eq. (82) into the appropriate terms of Eq. (26) and then performing the integration on the second term on the right-hand side, we find (F =+ - $2+ C ()n(1 1 + C@()n (1 - 2CwiT(O)Q1(O) + CiQo()I].* (83) The term in brackets in the second term on the right-hand side of Eq. (83) has terms similar to Eq, (49). In fact we can write f-. - 1 -) xVt) = - ~SJCS2 C + f([i) + Cn 1- C22. (84) We recall from Eqs. (57) that and are required in order to calculate the expansion coefficients A+ and 2(p). By the same method that was used in Section III to simplify O(4)/+(pt), we can easily prove 26

O( )(~) ( = O. (85) Also from Eq. (52) we have that lim zX(z) = 1 y() f()d. (86) But from Eq. (33) X(z) -1 z-*oo z therefore, from Eq. (86), f1 7y()f(i)d =- 1. Thus for constant source problems, we obtain O(L)S( = O(O) [w - S12C Q~n (1 + 15 (87a) and fl OTS )t ( )ds +; (1,) Lw _- S1C G~n (1 + 1 )] d (87b) o C12 C12 where = -S2 S1C + aS1C22 (88) C12 C12 and S1 and S2 are defined by Eqs. (78) and (79), respectively. By inserting Eqs. (87) into Eqs. (58), we obtain C=-S +f dliy( k) L cSgC Pin(l + i-) + C f1 7(p.)d-t fl Tp2(k)k( TjyL)dfl A+ = -Xq)- C 1 (rl-J1d (89) 27

and ~(kt) = o(,) - ( 1 + CO() C1 2.o -lk - A+_i CO(i) r rk(rt) (90) The final equation for aj(O) is obtained from Eq. (58c) where 1(vasj) = -S1 Explicitly, aj(O) = -aCs - CC121+ - 11C12p' 2(n)dn. (91) We note the similarity of Eqs. (89), (90), and (91) to the corresponding equations for the Milne problem. We refer the reader to the earlier comments on solution techniques for the Milne problem [see paragraph following Eq. (69)]. The angular fluxes for the constant source problem are given by Eqs. (71) and (72) where we replace the first term in the right-hand side by Si and S2, respectively~ The neutron current in each group is given by Eqs. (73b) and (73d) where we delete the first term (term with positive exponential) on the right-hand side. For the total flux we replace the first term on the righthand side (i.e., term with positive exponential) of Eqs. (73a) and (73c) by 2S1 and 2S2, respectively. 28

REFERENCES FOR PART I 1. K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley Publishing Company, Reading, Massachusetts (1967). 2. J. C. Stewart, I. Kuscer, and N. J. McCormick, Ann. Phys., 40, 321 (1966) 3. I. Kukc'er, "Advances in Neutron Thermalization Theory," paper given at I.A.E.A. Symposium on Neutron Thermalization and Reactor Spectra, Ann Arbor, Michigan (1967), to be published. 4. P. F. Zweifel and E. In5b'nu, Editors, Developments in Transport Thleory, Academic Press, London (1967). 5. R. Zelazny and A. Kuszell, Ann. Phys. 16, 81, (1961); also in Physics of Fast and Intermediate Reactors, I.A.E.A., Vienna (1962). 6. C. E. Siewert and P. F. Zweifel, Ann. Phys. 36, 61 (1966) and J. Math Phys. 7, 2092 (1966). 7. C. E. Siewert and P. S. Shieh, J. Nuc. Energy, 21, 383 (1967). 8. R. Zelazny, Private Communication (June, 1967). 9, D. R. Metcalf and P. F. Zweifel, to be published. 10. N. Muskhelishvili, Singular Integral Equations, Nordhoff, Groningen, Holland (1953). 11. F. C. Shure and M. Natelson, Ann. Phys. 26, 274 (1964). 12. G. J. Mitsis, Nucl. Sci. Eng. 17, 55 (1963). 13. N. J. McCormick and M. R. Mendelson, Nucl. Sci. Eng. 20, 462 (1964). 14. This definition is correct only for the two-discrete eigenvalue case. 29

TABLE 1 THE ZEROS OF ThE DISPERSION FUNCTION Conditions Roots %11 CTC2 < o/2 2 Real C o C11 + Ca2 > ci/2 2 Imaginary C11 + GC = a/2 2 Infinite C11 + UC22 2C < a//2 Real C < 0 I I C +~cC - 2C > a/2 2 Imaginary 11 22 C +uCT 2C = u/2 2 Infinite 11 22 C>O ~~~~~~~~~~~C11 + cC22 - 2C >ci/2 2 Infginite C + cC 2C < a/2 2 Real 11 22 a >o0 C11 ~ ciC 22 -2C > /2 2Imaginary C > 2CT(1/oJ) C + cyC 2C ai/2 2 Infinite 22 11 22 C < u/2 and C > 1/2 2 Real 11 - 22or and C > u/2 and C <1/2 2 Imaginery 11 - 22 j C11 +CTC22 - 2C < a/2 4 Real C>O C11 K o/2 and C22 < 1/2 C11 + 0C22 - 2C > u/2 1 2 Real and 2 Imaginary C < 2C(/ C + C22 - 2C = u/2 2 Real and 2 Infinite [22 _ _ _ _ _ _ _ _ _ _ 11 a C11 + 0C22 - 2C < cT/2 4 Imaginary 11 >cT/2 and C22 > 1/2 C11 + 0CY22 - 2C > cT/2 2 Real and 2 Imaginary _ __11+ C22 - 2C = a/2 2 Imaginary and 2 Infinit