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## Material Information- Title:
- Spatially dependent integral neutron transport theory for heterogeneous media using homogeneous Green's functions.
- Series Title:
- Spatially dependent integral neutron transport theory for heterogeneous media using homogeneous Green's functions.
- Creator:
- Church, John Phillips,
- Place of Publication:
- Gainesville FL
- Publisher:
- University of Florida
- Publication Date:
- 1963
## Subjects- Subjects / Keywords:
- Approximation ( jstor )
Boundary conditions ( jstor ) Geometric planes ( jstor ) Geometry ( jstor ) Greens function ( jstor ) Neutrons ( jstor ) Plane geometry ( jstor ) Point sources ( jstor ) Scalars ( jstor ) Speed ( jstor )
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- University of Florida
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- University of Florida
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- Copyright John Phillips Church. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 028393469 ( alephbibnum )
13311489 ( oclc )
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SPATIALLY DEPENDENT INTEGRAL NEUTRON TRANSPORT THEORY FOR HETEROGENEOUS MEDIA USING HOMOGENEOUS GREEN'S FUNCTIONS By JOHN PHILLIPS CHURCH A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1963 To my former teacher Dr. Dudley E. South ACKNOWLEDGMENTS The author wishes to express his appreciation to his chairman, Dr. G. R. Dalton, for his guidance and assistance throughout the course of this investigation. In addition, the author is indebted to the staff of the Computing Center of the University of Florida for their assistance, and to the Computing Center and the Department of Nuclear Engineering for their combined financial support of the computational work for this dissertation. The author has been ably assisted in the prepa- ration of this dissertation by his wife, Nancy, who typed the numerous drafts and the final manuscript; its final form owes much to her concern. It is difficult to express in words the author's appreciation of her faith in him. She has made the path to his goal less difficult. TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . LIST OF ILLUSTRATIONS . . . LIST OF TABLES . . . . Y TI D~ T I .N1 TR .W I A CHAPTER I CHAPTER II I1.1 11.2 CHAPTER III II1.1 111.2 111.3 CHAPTER IV IV.1 IV.2 CHAPTER V V.1 V.2 Page S. iii * vi S. viii 1 . . . . # . DEVELOPMENT OF THE HOMOGENEOUS MEDIUM GREEN'S FUNCTION METHOD . ANGULAR INTEGRATION . . Expansion of Angular Dependence in Spherical Harmonics and Zeroth Harmonic Approximation .. Advantages and Disadvantages of Homogeneous Green's Function Method SPEED DEPENDENCE . . . General Speed Dependent Equation . The Source Term .. . One-Speed Equation . . DIFFERENCE DENSITY METHOD . Development of Difference Density Equations . . . . Equivalence of Equations for N and D PLANE SLAB GEOMETRY . . One-Dimensional Problem . . Symmetry Considerations . . * J. S 4 S 15 S 15 S 26 28 S 28 S 30 S 34 S 36 * . . CHAPTER VI CHAPTER VII VII. 1 VII.2 VII.3 CHAPTER VIII VIII.1 VII.2 CHAPTER IX IX.1 IX. 2 IX.3 IX.4 NOMENCLATURE APPENDIX A APPENDIX B REFERENCES COMPARISON OF N AND D METHODS FOR SLAB GEOMETRY . . . THE FIRST-FLIGHT GREEN'S FUNCTION FOR A HOMOGENEOUS MEDIUM . . Differential Equation and Boundary Conditions . . . Analytic Form of G(v;x x') ... Monte Carlo Generation of G(v;xlx') SOLUTION OF THE TRANSPORT EQUATION . Spatial and Speed Dependence . Iteration Techniques .. . RESULTS AND CONCLUSIONS . . Comparison of the Homogeneous Green's Function Method with High Order P and Sn Methods . . . Monte Carlo Generation of the Green's Function . . . Iteration Convergence . . Generation of the Green's Function for the Multispeed Problem . . . . . . * DERIVATION OF Gf(v;xIx') . SAMPLE PROBLEMS . . 4 0 0 0 0 . . S 69 * 77 S 90 . 102 . 102 . 115 . 119 119 125 132 135 137 142 145 169 * 0 * 0 LIST OF ILLUSTRATIONS Figure Title Page 1 The Spatial Dependence of the h Functions of Equations (2), (7), and (8) . 13 2 Plane Geometry Transformation . 50 3 Symmetry of Neutron Flow'at the Boundaries of Unit Cells in An Infinitely Repeating Lattice . . . .. . 75 4 Slab Unit Cell with Mirror Boundaries at x-+b . . .. 85 5 Mirror Symmetry of First-Flight Green's Function About Plane at x-0 . . 97 6 Average Values of the Whole-Cell and Half-Cell Green's Functions for Problem 2, Appendix B . . 128 7 Average Value of the Whole-Cell and Half- Cell Green's Functions for Problem 4, Appendix B . . . . 129 8 Effect of Convergence Acceleration Techniques Applied to Problem 2 of Appendix B. Solution Iterated Until Residuals Were Equal to 10-5 of the Neutron Density at Each Spatial Point 133 9 Spatial Dependence of Scalar Neutron Density for Gold-Graphite Cell of Problem 1. 148 10 Spatial Dependence of Scalar Neutron Density for Fuel-Water Cell of Problem 2 .155 11 Advantage Factor for Fuel-Water Cell of Problem 2 by Various Approximation Methods 156 12 Advantage Factor Calculated by HGI Method vs. Number of Iterations for Fuel-Water Cell of Problem 2 . . 158 13 Advantage Factors for Problem 3 by Various Calculations . . . 161 14 Advantage Factor for Problem 4 by Various Calculations . . .. 165 15 Spatial Dependence of Scalar Neutron Density for Cases 1 and 2 of Problem 4 167 16 Spatial Dependence of Scalar Neutron Density for Cases 3 and 4 of Problem 4 168 vii Title Page Figure LIST OF TABLES Table Number Title Page 1 Relative Effort for Methods of Solution of End-Point and Midpoint Equations for a Total of M Spatial Intervals . 109 2 Breakdown of Computation Time for Problem 2, Appendix B .. .. 134 3 Parameters for Problem 1 . . 147 4 Parameters for Problem 2 . . 153 5 Parameters for Problem 3 .. . 159 6 Parameters for Problem 4 . . 163 viii INTRODUCTION Probably the most fundamental information needed to describe static behavior of a nuclear reactor is the speed and spatially dependent neutron collision rate, that is, Z(r,v)N(r,v)v. Such information is needed in order to predict long-term kinetic behavior of the reactor, burn-up, power level, etc. Thus.one needs to know the neutron density, N(r,v), over the entire speed spectrum and space of the reactor. Because of chemical binding and crystalline effects, the scattering kernels are considerably different for neutron speeds near thermal than for the higher speeds implied when one speaks of epithermal and fast speed ranges. In general, the absorption cross-sections of reactor fuels also have a different functional dependence on neutron speed at thermal speeds than at higher speeds. Finally, because of the tremendous range of speeds in reactor spectra, and because a critical reactor with no independent sources is an eigenvalue problem if the entire speed spectrum is considered, the problem of predicting the neutron density is usually divided into several speed ranges. The present paper deals exclusively with the prediction of the spatial and speed dependence of the steady-state thermal neutron density in heterogeneous media, although the basic method to be presented is equally valid for any speed range. There have, of course, been numerous approaches to various aspects of the problem which can be very loosely divided into two classes according to whether the integro-differential or the integral form of the transport equation was solved. The methods of solving the integro-differential equation differ to a large extent only in the choice of functions used to expand the angular dependence of the terms in the equation. The expansion functions include spherical harmonics (1-3), Tchebycheff polynomials (4-7), and first order polynomials, that is, trapezoids (8, 9). The methods of solving the integral equation have been a bit more varied, due perhaps to a more intuitive derivation of the equation to be solved. As Osborn (20) has pointed out, some approaches, particularly as applied to thin absorbers (10, 11), have separated the problem into two parts--first, a calculation of the absorption rate in the absorbing media in terms of the density at the surface of the absorber and second, a density depression problem of evaluating the density at the surface of the absorbing media in terms of the density in an all-moderating media. Other integral approaches (12-14), based on transport theory, depend on the fact that the absorption rate depends only upon the zeroth angular moment of the neutron density, the solution for which is independent of higher angular moments if scattering is isotropic. The present paper describes the latter type of approach. In brief, the method to be presented uses a kernel for a homogeneous finite medium to formulate an integral transport equation which predicts the speed and spatially dependent neutron density in a heterogeneous, finite unit cell. CHAPTER I DEVELOPMENT OF THE HOMOGENEOUS MEDIUM GREEN'S FUNCTION METHOD Assume that the neutron density in a general three-dimensional medium satisfies the time independent linear Boltzmann equation, and let N(r,vn)drdvd. denote the steady-state angular neutron distribution, that is, the number of neutrons in the volume element dr about r having speeds in dv about v, and whose directions of motion lie within the solid angle dn about n; N(r,vn) will subsequently be called the angular density. Define V(r,v) to be the probability per unit time that a neutron of speed v will suffer a scattering collision in dr about r with a nucleus which is in motion with respect to the laboratory system. Further, define y(r,v) to be the probability per unit time that a neutron of speed v will be absorbed in dr about r by a nucleus which is in motion with respect to the laboratory system. Finally, define P(r,v-,v4*-nn) dv dgto be the probability per unit time that a neutron of velocity v'r.' will suffer a scattering collision in 4r about r with a nucleus which is in motion with respect to the laboratory system such that the neutron emerges from the collision with speed in dv about v and direction of motion in dn about n. Then let S(r,vr)drdvdn denote the source term, that is, the number of neutrons per unit time being added to the volume element dr about r, having speeds in dv about v, and whose directions of motion lie within the solid angle dn about n. The transport equation which takes into account the relative velocity between the neutrons and collision nuclei which are in thermal motion can then be written as vn 7-N(r,vn + [V(r,v)+ A(r,v)] N(r,vr S(r,vrn) + fdv*d' P(r,vN(v,-. N(r,v'rD. (1) Equation (1) is an integro-differential equation; to make the problem of its solution determinate it is necessary to specify boundary conditions. The boundary conditions follow from the physical interpretation of N(r,vn). For example, at an interface between two media, any packet of neutrons characterized by the vectors r + Rn and vn will contain exactly the same number of neutrons when it enters one medium as when it left the other; in other words, N(r + Rn,vn) is a continuous function of R for r + Rn at the interface. For a complete discussion of this and other boundary conditions applied in neutron transport theory, the reader should consult Davison (2). Consider the space of interest to be divided into R regions, the medium of each region being assumed homogeneous (the derivation of Eq. (1) assumes that the medium is isotropic) so that the spatially dependent probabilities V, ), and P in Eq. (1) are constant with respect to r throughout each individual region. Define a unit function, h, such that h(p-s) 1 if p > s = 0 ,if p s. (2) For each interface between regions define an index k and a position vector T such that -1k_ e JA. Then h(rk-r) h(Ek-l-r) = 1, if Ek-r1 r Ek (3) = 0, if r e k-, or if r = rk where r is on the line between rk and :k-l. One can then use Eq. (3) to separate the r dependence of the probabilities V, Y, and P in Eq. (1) in a special way. In general, for any operator 0, one can write that R S(r,)- kl [h(Ek-r)-h(k-l-r)] (k,) (4) Use Eq. (4) to rewrite Eq. (1) in the form R yvt* N(r,vn) + [ [h(rkE-r)-h(rk_,-r)] S[v(k,v)+ (k,v)] N(r,vn) R 8(r,vr) + 1 [h(rk-r)-h(!k-1-)] dv'/ df' P(k,vI-v,n'-,O)N(r,v'n. ') (5) Choose a particular region, say k', and remove the term for the k' region from the summation on the LHS of Eq. (5), meanwhile placing all other terms of that summation on the RHS. Thus Eq. (5) becomes vrL N(r,v*) + [h(Ek,-r)-h(rk'_-r1) [V(k',v)+ /(k,v)] N(r,vn) R S(r,v) [(-r)-h( -r] k+k' [V(k,v)+ y(k,v)] N(r,vn) R + 1 [h(Ek-r)-h(k_-1r)] k-i dv' dn P(k,vi-v,n'~-N(r,v'n') (6) Now using the identity h(p-s) I h(s-p) (7) one can write h(!k,-r) h(Ek, -r) [1 h(r-k,)] h(k,-_l-r) -1 [h(r-r,.)+h(r,_-r)] (8) Substituting Eq. (8) into Eq. (6) and putting the [h(r-r,) + h(Ek'~,i-)] term on the RHS gives va*VN(r,vm) + [V(k',v)+'(k',v)] N(r,vn) SS(rvn) + [h(r-,)+h(r .,-r) [V(k',v)+ '(k',v)] N(r,vn) [ [h(rkr)-h( ~.-"r) [V(k,v)+ '(k,v)] N(r,vn) k+k' R + E [h(rki-r)-_h(!k-1L] (9) dv' dn' P(kv-'-,v,nt.) N(r,v'n') Define a function gk' (r,vLr' ,vtn.') by requir- ing that it satisfy the equation obtained by replacing the RHS of Eq. (9) by the Dirac delta function product 6 (r-r') d (v-v') & (n-n'), subject to the same boundary conditions as Eq. (9). Thus TVgk T (rn r'Iv' 1) + [v(k',v)+ W/(k',v)] g' (r,vnlr' ,v'i') g(r-r')&(v-v')s(n-n') (10) The function gk (r,vnCr',v'.n') has a physical significance; gk (r,vnr r',v'n') drdvdn is the number of neutrons in the volume element dr about r with speeds in dv about v and with directions of motion lying within the solid angle dn about n due to a unit point source at r' emitting one neutron per unit time in direction n' with speed v' in a first-flight medium which has the same properties as the medium in region k'. It is important to understand fully the term "first-flight medium" as used here. To be more explicit, a first-flight medium is a medium in which all collisions with nuclei result in removal of the neutron from the population. Thus the removal probability for the Green's function problem is equal to the total collision proba- bility in medium k' of the real problem, and the neutrons denoted by gkt(rvn l r*,v') are first-flight neutrons in the sense that they have suffered no collisions enroute from the source point to the field point. The term first-flight neutrons will be used throughout the following work. The subscript k' will be dropped from gk (r,vr'iE*,v't) in the following work, it being understood that the function depends on the properties of the particular region, k', chosen. The function g(rvnjr',v'n') can be used as a Green's function to convert Eq. (9) to the following integral equation (see Chapter 7 of reference (16)). N(r,vn) dr' dv' dn' g(r,vn r v'n) ,v'f) rt v' n' + [h(r'-Ek)+hc(Ek.,-l-') [V(k',v')+ 2'(k',v')]N(r',v'n') [h(!k-r' )-h(rkl-r')] [V(k' vt)+ Y'(k',v')] N(r' ,v') k+k' R + [h(Ek-r)-h(k-r1] dv"d P(kvyrv',n"-)N(r'vlg) . Since Eq. (10) describes the neutron density in a medium in which all collisions result in removal of the neutron from the population, the only way a neutron which is emitted at r' may arrive at r is by direct flights, that is, with no collisions enroute. It will be assumed and later justified that events at the bounda- ries do not change the speed of the neutron. Hence if a neutron of speed v' is emitted at r' and arrives at r, its speed must still be v'. From this discussion it can be concluded that the speed dependence of g(r,vfir',v'n') includes a delta function in speed so that g(r,vn r' ,v'') g(v;r,~ r' ,n.' )(v-v'). (12) The product solution, Eq. (12), is a result of the fact that since there can be no scattering in the medium, then there is no interdependence of nr*n' and v-v'. Note that g(v;r,- lr',') ) still has an implicit dependence on v due to the v dependence of V(k',v) and ((k',v) in Eq. (10). Substituting Eq. (12) into Eq. (11) and carry- ing out the integration over v' yields N(rvn) dr d g(v;r,nl r,f' )S(r,vn') +[h(r'-rk )+h(r, l-rt)] [V(k' v)+ Z(k',v) N(r',vn) Sh(Ek-r)-h(-r'i )][V(k,v)+ (k,v)] N(r',vn') k+k' R + [h(rk-r')-h(k1-r')] (13) k-1l --k -- -- dv" dn" P(k,v"--V ,n"-nr )N(r',v"n ) . The integration over r' may now be considered. The meaning of the functions [h(r'-rk,)+h(Ek1-r' ) and [h(rk,-r')-h(rk_'-r') may be easily deduced from Eqs. (2), (7) and (8) and is illustrated in Figure 1. The function [h(r'-!k,)+h( k,l-r')] has unit value every- where except in region k' where it has zero value, while [h(k-r')-h(Ekl-r')] has zero value everywhere except in region k where it has unit value. It can easily be seen that the r' integration of the second term on the RHS of Eq. (13) is an integration over all regions except 1 0 - 1 h(r-'k +-r' ) 0 ---- 1 h(r'-r) + h(rk--r') 0 - 1 h(r~kt-re )-h(rk' _1-r') 0 ------ Figure 1 The Spatial Dependence of the h Functions of Equations (2), (7), and (8) the k'th one while the r' integration of the remaining step-function terms is a sum over k of r' integration over each individual region k. Thus one can write Eq. (13) as N(r,,vn) dr' dn' g(v;r,, r',n')S(r',vn') rt n R + ki [v(k',v)+ (k',v)] V(k,v)+ -k,v)] dr' dn' g(v;r,nrr',nt)N(r',vn') r'in k r' R + Z dr' dnY g(v;r,n jr,nt) k-1 ~ - r'in k n' (14) dv" d"P(k,v"- v,n'L.' )N(r',v"r") . Note that the k-k' term in the first summation is automatically zero, so that no special notation is needed on the summation symbol. CHAPTER II ANGULAR INTEGRATION 11.1 Expansion of Angular Dependence in Spherical Harmonics and Zeroth Harmonic Approximation The angular integration of Eq. (14) may be performed as follows. Expand all the angular dependent terms of Eq. (14) in the complete orthonormal set of spherical harmonics as defined in Chapter 7 of reference (15). Thus oo n N(r,vn)- Nm(r v)y-(n) (15) n-0O m--n co n S(r,vn) E Sm(r,v)Ym(n) (16) n-O m--n n n oo a P(k,v'"Uv,n'!n) Pa(k,vv)yb()b*(). (17) a-0 b--a a- a The expansion for the Green's function is more complicated. Because of possible events at the bounda- ries enclosing the space of interest, g(v;r,nI r' ,') depends explicitly on both n and nr' and not just on r*n' as P(k,vn.v,n'n ) does. For the moment only the n.' de- pendence of g(v;r,n r',nh) is represented by an expansion in the complete orthonormal set of complex conjugate spherical harmonics. Thus g(v;r,n r'l,n') gs;rn r')Y (). (18) s-0 t--s Substitute Eqs. (15) through (18) into Eq. (14) to obtain N(r,v ndr' dn' gt (v;rnJf)Yt* (A) : m(r',v)Ym(n!) Ss,t nm r' n' + R v(k',v)+ (kt,v)0 [V(kv)+ Y(k,v) Sfdr' dn E gg(v;r, lr')Y t*(n) N(r (n' S s,t s n,- r'in k n' + rl drt dn' g (v;r,nr')Y (') r'in k n1' (19) ab a a n - Equation (19) may be simplified considerably by interchanging the order of integration and summations and by using the orthogonality relation of the spherical harmonics, namely, n)Yb n, a Smb (20) where Sn,a is the Kronecker delta function. After carrying out the angular integration, Eq. (19) becomes N(r,vn) dr' E ge(v;r,nlr')Sm(r',v) n,m rf + [V(k ,v)+ ](k ,v) [V(k,v)+ ?'(k,v)1 Sdr g gm(v;r,nr')Nm(r',v) n,m r'in k (21) + R dr' dv" gn(v;rn l')Pn(k,vZ-v)o(r',v") kl 1 f n,m r'in k Equation (21) is an expression for the total angular neutron density and is exact; no approximation has been made so far. Now, the relative isotropy or anisotropy of the n' dependence of g(v;r r',n') depends on the boundary conditions applied to Eq. (10) and also on the properties of region k'. Recalling the physical meaning of g(v;r,nlr ,n.'), one observes' that n' denotes the angular dependence of the unit point source in a medium in which all collisions result in removal of the neutron from the population. If the space of interest is of infinite extent, then g(v;r,n r',n') has a double delta function behavior in direction, namely S(-n*') -(-r')] , which merely describes the fact that since scatters are not considered in the calculation of the Green's function the direction of motion at r not only must be the same as the direction of motion when emitted at r', but also must be pointed along the vector connecting the field and source points. In this case, the complete set of coefficients g (v;r,,nir) would be required in Eq. (18) to satisfactorily describe g(v;r,n r',n'). The problem is entirely different, however, if one imposes perfectly reflecting boundaries around the space of interest including the source as would be done if one were calculating a unit cell in a repeating array. Collisions or events at the boundaries will reflect neutrons back into the space of interest with some direction nL. If the neutron then arrives at r it will have direction n, such that n + n' and n (r-r'), although there llbe a definite relation between although there will be a definite relation between , n'_, and (r-r*). Further, if the medium characterized by [V(k',v)+ ,(k',v)] has a mean free path that is long relative to the distance between boundaries, then the neutron may be reflected at more than one boundary before arriving in the element of volume dr about r. Now the integral of the angular density due to a unit point source per solid angle, integrated over all directions of emission n', is simply the angular density due to an isotropic point source of one neutron per unit solid angle or a total of 41T neutrons and is denoted by g(v;r,nar'). Thus g(v;r,nlr') g(v;r,nlr',n')dn'. (22) From the discussion above regarding reflections at the boundaries, it is obvious that in a finite space enclosed by reflecting boundaries an isotropic point source will produce a very nearly isotropic angular density g(v;r, i r'). In order to take advantage of the relative isotropy of g(v;r,n r'), one proceeds as follows. If one asks only for information about the number density N(r,v), the dependence of Eq. (21) may be integrated out to obtain N(r,v) dr' Z g(v;rlr )Sm rv) nm r' k V (k ,v)+ k',v) [V(k,v)+ /(k,v) dr' g(v;r r)lE (r',v) n,m r'in k (23) R m + dr' dv" n g(Vr'r)P (kv"-v)Nm(r',v") k-1 n,mn n - r'in k where N(r,v) -/N(r,v)dr (24) n Expanding the angular dependence of g(v;rl r') in spherical harmonics gives g(v;rn ') g(v;r r')Ym(n) (25) n,m The coefficients gn(v;rlr') in Eq. (23) are recognized as being equal to the coefficients in the angular expansion of g(v;r, nr'), a function which has been discussed above and which has been concluded to be nearly isotropic for a finite space enclosed by perfectly reflecting boundaries and containing a medium having a mean free path long relative to the distance between boundaries. Thus the coefficients gn(v;r r') form a rapidly decreasing sequence for increasing n. Further, since Eq. (1) includes a scattering term, the angular density N(r,vn) should be rather isotropic, even for rather strong absorbers. A unique advantage of the integral equation method over the integro-differential equation method thus becomes apparent; namely, the zeroth harmonic approximation disregards only products of small terms, that is, one assumes -1 -1 1 1 0 0 0 g0 N0 g1 N * g1 N1 1 1 Due to the orthogonality of the spherical harmonics, there are no cross products to consider such as g0 No or go N1 which might be of the same order of magnitude 0 0 as g0 NO' The situation with regard to the scattering term in Eq. (23) is even more favorable. One need not assume completely isotropic scattering, but merely make the less stringent assumption that the product of the zero order coefficients of the Green's function, the angular density, and the scattering kernel, is much greater than the product of higher order coefficients, that is, g1 PI Ni1 g0 P NO 9 10 P NO gl p N1 1 11 On the basis of the above discussion the zeroth harmonic approximation is made in Eq. (23) and only the n-0 terms are kept. Thus Eq. (23) becomes N(r,v) dr' g0(v;r r')So(r',v) r0 + [v,(k.v)+/(kv) [V~f,v)+(kv dr' g00V;r Ir)N (r',v) r'in k R O l o t + Z dr' dv" go(v;r r')P.(k,v.v)NO(r,v"). (26) +-1 kl d - r'in k The zero order coefficients may be put in a more recognizable form as follows. Substitute Eq. (15) into Eq. (24) to obtain N(r,v) Na(r,v) Ya(n)dn n,m Nm(r,v) Ym(c) -/f YO*() n,m -/41 N(r,v) 00* where use has been made of the fact that Y* (n) - YO n) 1/--1 Thus N0(rv) N(r,v) (27) Similarly, defining S(r,v) -s (r,vn)dc (28) rL and using Eq. (16), one obtains o 1 S(r,v) S(r,v) (29) 0 -- Define g(v;rlr) g(v;r,lr')dn (30) to be the number density of first-flight neutrons at r having speed v due to a point source at r' emitting 4TT neutrons isotropically with speed v in a homogeneous finite medium. Substitution of the angular expansion Eq. (25) into Eq. (30) then gives the result that g0(v;r ') 1 g(v;r r') (31) For the scattering kernel one has from Eq. (17) that P(k,v-.v, -)dn .- Z Pa(k,vi.v)Y b Y b d) b 0 abb* /yb ,d a,b P(kv"-.v) ,/4f YO* () PO(k,v!',v) (32) which is just the probability per unit time per unit speed v that a neutron having speed v" will suffer a scattering collision in region k and emerge from the collision with speed v. The zero subscript will be dropped and the function will subsequently be denoted simply by P(k,v!3.v). Finally, substituting Eqs. (27), (29), (31), and (32) into Eq. (26) gives N(r,v) dr' g(v;r r')S(r',v) r' R + 4 1 [(V(ktv)+ (k',v)] [V(k,v)+ Y(kv)] Sf dr/ g(v;rlr)N(r',v) r'in k I R k+ kl f dr' g(v;r r') dv"P(k,v'4v)N(r ,v"). r'in k (33) 11.2 Advantages and Disadvantages of Homogeneous Green's Function Method The preceding method of converting the integro- differential transport equation to an integral equation with a kernel defined by Eq. (10) has at least two advantages over the more standard methods of solution. First, as discussed above, the zeroth harmonic approxi- mation involves discarding only products of small terms as in Eqs. (22) and (23). For example, rather than neglect- ing the higher moments of the neutron density, as must be done when the integro-differential form of the transport equation is used, one need only neglect the products of the higher moments of the Green's function times the higher moments of the neutron density. A second advantage and a unique feature of the Green's function method described in this work is that the Green's function defined by Eq. (10) depends only on the dimensions and geometry of the outer boundaries of the real problem for which the neutron density is to be calculated, the total cross section of the medium in the k'th region of that problem, and the boundary conditions imposed on Eq. (10). Once the Green's function has been determined, it may be used to determine the neutron density in any heterogeneous medium, the only restrictions being that the outer boundaries of the neutron density problem must coincide with those of the Green's function problem, the boundary conditions must be the same for both g and N, and the total cross section of the k'-th region of the neutron density problem must be the same as that for the homogeneous medium of the Green's function problem. A disadvantage of the method is implied above; namely, if the outer boundaries and/or boundary conditions of the neutron density problem do not agree with those for which the Green's function has been determined, then a new Green's function must be determined. Another dis- advantage is that, as yet, the method has not been extended to higher order approximations such as keeping gm N P, g Pn N terms for n 0, although such an extension to higher moments would, in general, follow the work of Wilf (29), and would not alter the above mentioned advantages. CHAPTER III SPEED DEPENDENCE III.1 General Speed Dependent Equation Equation (33) is a general speed-dependent equation and is valid over the entire speed spectrum, the limits of the speed integration being from v" 0 to v" -oo. As indicated in the introduction, the present investigation will be confined to speeds in the thermal region. In practice, there exist no true sources for the thermal energy region other than slowing- down sources, and hence S(r',v) 0 in Eq. (33). The speed integration in Eq. (33) are then separated into two parts, one for 0 < v i v* and one for v =>v*. The speed v* represents the upper limit of the speed region of interest, and is characterized by the following assumptions (17): 1. For v = v*, the neutron density speed distri- bution is proportional to 1/v2 (that is, the neutron flux is proportional to 1/E). 2. The probability of a neutron being scattered from a speed below v* to a speed above v* is zero. 3. The motion of the collision atoms is ignored with respect to that of the neutrons with' speeds above v*. 4. The Maxwellian distribution for thermal neutrons or collision atoms has negligible magnitude for speeds above v*. 5. For neutron energies above v*, the energy transfer is computed assuming that the col- lision atoms are free from chemical bonds. This implies that v* should correspond to a neutron speed well above that required to excite vibrations in the collision molecules, so that the collision atoms would recoil freely at the instant of impact. 6. For neutron energies below v*, the energy transfer kernel P(k,vv-v) will depend on the particular scattering model chosen. A discussion of particular energy transfer kernels and their scattering models is given in references (17, 31). The S(r',v) term in Eq. (33) is then modified to include all neutrons scattered into a speed interval dv about v from speeds above v*. Thus Eq. (33) may be rewritten as N(r,v) drI g(v;r r')S(r',v) r'in k + 4 ( [V(kv)+ /'(k',v)] [V(kv)+ Y(kv)] dr' g(v;r r)N(r',v) r'in k V* + k dr' g(v;r r') dv" P(k,v"!.v)N(r',v") r'in k v"-0 (34) where co S(r',v) dv" P(k,v!.v)N(r',v") (35) v"-v* III.2 The Source Term Assumption (5) of III.1 implies that ordinary slowing down theory may be used to derive the scattering kernel for v v* (18, 19). Define m-1 --- (36) m+l Then ov is the minimum speed that a neutron of initial speed v can have after collision with a target nucleus of free atom mass m; the energy transfer probability, that is, the probability that a neutron of energy E' will after scattering have energy in the interval dE about E is (see (18)) T"(r,E..E)dE (dE for a2E' E E' -0 ,for E < a E' S0 for E E' (37) or, changing variables, 7l(r,vi.v)dv 7T(rE'-,E)dE dv rvdv so that 7rf(r,v.v) 7r(r,EL-.E) dE 1 dE mv SE' [la2(r)] dv l-o2(r)] (mv' 2v 2v ["I, vv for wav' v S v' S0 for v Oav' 0 for v >v' (38) Now, the differential scattering cross section, Lg(r,vi-av) dv, that is, the probability that a neutron of speed v' will be scattered at r into dv about v is equal to the product of the total scattering cross section at r for neutrons of speed v' times the proba- bility that the speed of the neutron after collision will be in dv about v. Thus Zg(r,v4..v)dv s(r,v')7r(r,v'.-v)dv. (39) Then the probability per unit time that a neutron of speed v* will suffer a scattering collision at r such that it emerges from the collision with speed in dv about v is simply P(r,v'.v)dv v'1%(r,v'Lv)dv (40) so that for v' > v* one has from Eqs. (38), (39) and (40) that P(r,v-.v) 2vv's(r,v) [1- (r ) v'2 2vZs(r,v') for Ov' s v s v' -[l-a(r)] v' 0, for v ctv' (41) 0, for v => v Assumption (1) of III.1 implies A(r) N(r,v') A for v' > v* (42) vt2 where A(r) is an arbitrary constant with respect to speed and is normalized to give a stationary distribution for N(r,v'). Substituting Eqs. (41) and (42) into Eq. (35) yields S(r',v) dv" 2vls(k,v") A(r') S1-a2(k) v" v"2 v"'v* [I I 2vs(k)A(r,) v/(k) -n [ldv" (43) [1-a2(k)] j v"3 d where Zs has been assumed to be independent of v in the speed range v* 5 v" v v/a(k). Carrying out the integration in Eq. (43) gives 8(r',v) V A(r') v2 (k) if ca(k)v* v s v. [1-o(k)] V*2 v2 0 if v < a(k)v* (44) where the constant A(r) determines the source strength; A(r) will be assumed to equal a spatially independent constant A, that is, one assumes a spatially flat slowing-down source distribution. As Honeck (22) has pointed out, A(r) could, if necessary, be approximated by the shape of the resonance group neutron density as predicted by diffusion theory and the use of suitable resonance group cross sections given in Weinberg and Wigner (19). II.3 One-Speed Equation To obtain the one-speed equation, one must assume separability of the spatial and speed dependence of the neutron density. Then integrating Eq. (33) over all speeds and using Maxwellian averaged cross sections, one has N(r) /dr'g(r r')S(r') r' + 1- [v(k')+ $(k')] -[V(k)+ Y(k)) dr' g(r r')N(r') r'in k + 1 V(-k) dr g(r r)N(r (45) r'in k where use has been made of the fact that Co P(k,vu.v)dv V(k,v") (46) v-0 which is just the total probability per unit time that a neutron in region k and having speed v" will suffer a scattering collision. In the one-speed case motion of the collision atoms may be ignored, so that in conventional notation Eq. (45) may be written as N(r) 1 dr' g(r r')S(r') r' + 1 (k')- (k)] dr' g(r|r')N(r') r'in k r'in k kj g(rfrl')N(r ) (47) Finally, the last two terms in Eq. (47) may be combined to obtain N(r) dr' g(r r')S(r') + k t(k')-a(k) dr' g(rr')N(r'). (48) r'in k CHAPTER IV DIFFERENCE DENSITY METHOD IV.1 Development of Difference Density Equations An alternative to the preceding method of describing the real neutron density, N, in a hetero- geneous medium is to define a difference density, D, as the deviation from a known density, N', for a similar or reference space. Although it may appear at the outset that such a method is analytically identical to that developed above for the real density N, nevertheless, a number of authors have found such a procedure to be advantageous (13) and, in fact, necessary (12) in order to obtain results for specific infinite geometry problems. It will be of considerable interest, therefore, to develop equations for the difference density, D, parallel to those above for the real density N. A direct comparison of the two methods will then be made in order to determine their relative advantages when applied to the specific problem to be considered in this paper. Thus the difference density D(r,vn) is defined such that D(r,vn) N(r,vri) N'(r,vn) (49) or N(r,vn) = D(r,vn) + N"(r,vn) (50) Using primed symbols to denote the reference space, one can write the transport equation for the assumed known angular density as v ~-IN'(r,v) + V'(r,v) + (rv) N'(r,Cvn) .L (51) S'(r,vn) + dv' dn P'(r,v-.v,n4 n)N' (r,v'n'). Substituting Eq. (50) into Eq. (1), subtracting Eq. (51) from the resulting equation, and grouping all of the N'(r,vn) terms on the RHS gives vnVD(r,vn) + [V(r,v)+ (r,v)] D(rvi) = S(r,vn) S'(r,vn) v(r_,v)+ (r,v)-[,v) (v)+ /(r,v)] N'(r,vr) + dv dn' P(r,v--v,n'-.*-0D(r,v'n') (52) + dv' d.n P(r,vv,nI)-P' (r_,vv,_)] ,n N(rv,',). As before, Eqs. (3), (4), and (8) are used to separate the spatial dependence of the terms in Eq. (52). Thus 38 vn-VD(r,vn) + [v(k',v)+ (k',v)] D(r,vn) - S(r,vn) S'(r,vn) + [h(r-rk,)+h(k'l-'r)] [V(k',v)+ K(k',v)] D(r,vn) rh(k-r)-h(rkl-r)] [V(k,v)+ K(k,v)] D(r,vn) k+k' - k 1 h(Ek-r)(k-1-^r)] [V(k.v)+ Y(k,v)] [V'(kv)+ 't(k,v)] N'(r,vr) + r h(rk-r)-h(kl,-r)] fdv dn' P(k,vi-*v ,nn)D(r,v'nW) k=1 f + h(k-r)-h('k-1-r) dv' di' [P(k,vI+v n!*n) (53) -P' (k,vv,L).)] N' (r,v'n'). The boundary conditions imposed on Eqs. (51) and (53) are the same as those for Eqs. (1) and (9). Thus the Green's function which satisfies Eq. (10) may be used to convert Eq. (53) into an integral equation as before. Then, using Eq. (12), one may perform the integration over v' to obtain D(r,vn.) dr' d' g(v;r,g r' n') S(r' ,v')-S' (r' ,vn ) + [h(r'- ,)+h(k, -rl') [v(k,v)+ ) k',v)] D(r',vn') R - [hr ktr')-h(h-1-r')1 [V(k,v)+ Y(k,v)]D(r',vn' ) k+k' R [h(Ek-r')-h(ik-lr')] [V(k,v)+ Y(kv) yv' (k,v)+ 1 (k,v)] N' (r' ,v') + kl[h(Ek-r')-h(rek_-r') dv" dr' P(k,v" vn'n*')D(r',v'tn") + k E [h(Ek-r' )-h(Ek-i-r') dvt" d [P(kv' mt-.') i (54) P' (k,v-vn'".)] N' (r' vn") . The r' integration for Eq. (54) is identical to that for Eq. (13) and the result is D(rv_) dr' dn' g(v;r,n r' ,n)[S(r',v f)-S'(r_',vn)] + V(k',v)+ (k',v)] [V(k,v)+ Yk,v) dr' da g(v;rn ir'L l )D(r' ,v') r'1in k 1' R + dr' dn' g(v;r,nlr'l,.') r'in k n dv" a" P(k,v" vn'1,-c n')D(r',v"n') - [V(k,v)+ Y(k,v)] [V'(k,v)+ (k,v) S dr' dn' g(v;rnl r',n')N'(r',vn!') rrin k ' R + dr, dn' g(v;r,nlr',n') dv"fdrl" P(kv4I.vI.nIn') r'in k T.' (55) P' (k,v"-.v,n "-.')] N' (r' ,vI'") . The angular integration presents no new problems and may be performed as in Chapter II. Making the zeroth harmonic approximation and asking only for the scalar neutron difference density, one obtains D(rv) d 1 dri g(v;rjr9)[ (r',v)-S'(r',v)] rt + ([V(k',v)+ (k',v)] 47r k-i [V(k,v)+ Y(k,v)]) dr' g(v;r r')D(r' ,v) r'in k R k dr' g(v;rlr') dv" P(k,vU*v)D(r',v") k-lr - r'in k R k-i + 1 471 4"/ Sdr' g(v;rr')N'(r',v) r'in k dr' g(v; rr') r'in k (56) *dv" P(k,vt v)-P' (k,v"-v) N' (r' ,v"). Writing Eq. (56) specifically for the thermal spectrum yields D(rv) dr' g(v;rr')[S(r'v)-S'(r',v)] r'in k R [V(k',v)+ (k',v)] [V(k,v)+ /(k,v)] dr' g(v;r r')D(r',v) r'in k S[vk,+ (k,v) )] - [V'(k,v)+ Y'I(kv)) 1 +1- I R + kw, v f V* 1 R 4+ F i dr' g(v;r r') dv" P(k,v"!v)D(r',v") r'in k v"'-0 -1 R i (k,v)+ (kv)] [V'(k,v)+ '(kv dr' g(v;r r')N'(r,v) r'in k R + k~ dr' g(v;rlr') r'in k (57) v* dv" [P(k,v,-.u.v)-P' (k,v"--.v)] N' (r' ,v") VV-O where, in region k, ro S(r',v) S'(r',v) dv" P(k,v-!.v)D(r',v") vl-v* 00 (58) + dv" [P(k,~,!v)-P'(k,v".v)] N'(r',v") v-v* and the scattering kernels for v > v* are determined from ordinary slowing down theory as in 111.2. Finally, the one speed version of Eq. (56) is D(r) dr' g(r r')S(')- ( r' 1 R + -- k + Rl 1 R +1 R 47( N-1 v[zt(k')- It(k)] dr' g(r r')D(r') r'in k v Zg(k) dr' g(r r')D(r') r'in k v [zt(k)- Z((k) drI g(r rf)N'(r') r'in k v[(k)- (k) dr' g(r r')N'(r') r'in k or if various terms are combined, D(r) /dr' g(r r')[S(r')-S'(r')] r' (59) + v k')- Zak)] dr' g(r r)(r r'in k S1 v[Za(k)- (k)] dr' g(rr')N'(r') (60) 47r T l k f i r'in k IV.2 Equivalence of Equations for N and D In order to establish the equivalence between Eqs. (34) and (57) and between Eqs. (48) and (60), the one-speed equations will be considered first. Subtracting Eq. (60) from Eq. (48) and using Eq. (50) gives N'(r) 1 dr' g(r r')S'(r') rt + k V[Zt(k*)- a(k)] dr' g(r )N(r') r'in k + 1 v[Za(k)- Z(k)] f drv g(rlr')N'(rt) k 47j r'in k _- fdr' g(r r')S'(r') 47r f rt + RF v[ l(k)- .(k)] drt g(rr)N(r'). (61) r'in k Comparing Eqs. (61) and (48), one observes that the requirement that Eqs. (60) and (48) be equivalent is simply that It(k') -= (k'), that is, the total cross section for the k'th region of the reference space be identical to the total cross section of the k'th region of the real space of interest. This simply says that the Green's functions in Eqs. (48) and (60) must be identical, that is, the Green's functions in Eqs, (48) and (60) must both satisfy Eq. (10) and its boundary conditions. But this is just the condition imposed in the original derivation, and hence Zt(k') is equal to Z (k'); equations (48) and (60) are therefore proved to be equivalent formulations of the same problem. Similarly, subtracting Eq. (57) from Eq. (34) and using Eq. (50) gives N'(r,v) = dr' g(v;rr')' (r ,) *E k-lf ~ # ~ r'in k 1 R 47T kl [V(k,v)+ a(k' ,v)] Iv(k,v)+ Y(k,v)] dr' g(v;r r')Nt(r',v) r'in k V* + k dr' g(v;r r') dv" P(k,v"..v)N (r' ,v") r'in k vIO 1 + k [V(k,v)+ ~(kv) [V'(k,v)+ f'(k,v)] r'in g(v;rlr)N'(r',v) r'in k Sdr' g(v;rlr') r'in k * dv" [P(k,v-t..v)-P' (k,v"!.v) N' (r' ,v") v" 0 1' R E-I r I r'in k dr' g(v;r r')8'(rt,v) + 1 [V(k'iv)+ K(k',v)] ['(k,v)+ (k.v)) dr' g(v;rjr')N'(r',v) r'in k v* S R + 1 dr' g(v;r r') dv" P'(k,,v'la)N'(r'v,v") r'in k V1=0 Comparing Eqs. (34) and (62) one again observes that, just as in the one-speed equation, the requirement that Eqs. (34) and (57) be equivalent is simply that [v(k',v)+ (k' ,v)] be equal to [V'(k',v)+ ('(k',v)], that is, the medium of the k'-th region of the reference space must be the same medium that is in the k'-th region of the real space of interest. As has been mentioned above, this condition was imposed in the original derivation so that the equivalence of Eqs. (34) and (57) is established. A discussion of the relative usefulness of the two formulations will be deferred until the general equations have been specialized to slab geometry. CHAPTER V PLANE SLAB GEOMETRY V.1 One-Dimensional Problem The equations of the previous chapters were written for a general three-dimensional geometry, and the Green's function g(v;r r') was interpreted as the number of first-flight neutrons per unit element of volume at r having speeds v due to an isotropic point source at r' emitting 47r neutrons per unit time having speeds v in a homogeneous medium. Before specializing Eqs. (34) and (57) to an infinite slab geometry it will be convenient to define a new Green's function G(v;r r'), such that G(v;r g(v;rr') (63) Thus G(v;r r') is the number density of first-flight neutrons at r having speed v due to a unit isotropic point source at r' emitting neutrons of speed v in a homogeneous medium. Rewriting Eq. (34) in terms of G(v;rlr') gives r'in k + 1 fV(k',v)+ /(k: ,v) -[V(kv)+ /(k,v)] */ dr' G(v;rir')N(r' ,v) r'in k v* (64) + 17 dr' G(v;r r') dv" P(k,vlvm)N(r',v") k-.f -I- f r'in k v"0O and similarly for Eq. (57). The transformation to plane geometry may be performed as follows. Referring to Figure 2, the element of volume dr' may be expressed as dr' 2 7Ta da dx' Define R r r' R r r'I (65) From Figure 2, and Eq. (65) one obtains R ( x-x'| 2 + a2)1/2 .x-_ In I r fixed field point r' variable source point Y cos G z / //'r O / X r x / L._ da dx' Figure 2 Plane Geometry Transformation so that for constant Ix-x'l dR = a da -Ix-x' a_2 dr . R 2 Hence dr' 2 7 R dR dx' _Hx-x'lX 2 d_ S2[Ix-x'I -dx' (66) where the i integration is from -1l to y-0, and the R integration is from R jx-x'I to R -oo . Observing that in plane geometry the r de- pendence of the functions N, N', D, S, and P is simply a spatial dependence on x, one may write Eq. (64) for plane geometry as R N(x,v) 1 dx' S(x',v) G(v;r r') 2 7r R dR x'in k R- x-x'I + 1 [v(k v)+ '(k',v)] (k,v)+ (kv)] Sdx' N(x',v) G(v;r r') 277R dR x'in k R-|x-x'l v* 00 + 1 dxt dv" P(k,vl-.v)N(x',v") G(v;r r') 27TR dR xin k v"-O R-Ix-x'i (67) Define G(v;xx') G(v;rr ') 2rrR dR (68) R- x-x'l The function G(v;x x') is the density of first-flight neutrons at x having speeds v due to a unit isotropic plane sourceat x' emitting one neutron per unit time with speed v in a homogeneous medium. This function is well known for an infinite medium (see Chapter 2 of Case, et al. (21)) and will be discussed extensively for a finite unit cell in Chapter VII. Using Eq. (68) to rewrite Eq. (67) gives R N(x,v) 4 /dx' G(v;x ,x)S(x,,v) x'in k + k1 [V(k',v)+ /(k',v)]-[V(kv)+ Y(k,v)] dx, G(v;x x')N(x',v) x'in k V* + dx' G(v;x x') x'in k v"0 Similarly, Eq. (57) for plane geometry becomes D(xv) / dx' G(v;x x')B(x'.v)-S'(x'.v)] x'in k + k1 [(k',v)+ (k',v)]-[V(kv)+ (k,v)] /dx' G(v;x x')D(x',v) x'in k + f dx' G(v;x x') x'in k v"-0 R -kx 3v" P(k,v-L.v)D(x',v") [V(k,v)+ Y(k,v)] [V (k,v)+ YO (k,v)] dx' G(v;x x')N'(x',v) x'in k (70) v* + dx' G(v;x x') dv" P(k,v'-wv)-P'(k,vtlv) N' (x',v"). x'in k v-0 dv" P(k,v"L.v)N(x',v") (69) V.2 Symmetry Considerations For subsequent work the space of interest will be assumed to be a plane geometry symmetrical about the plane x-0. In the summations over k as previously discussed, k-l denoted the left-most region in space. The regions will now be counted differently, letting k-l denote the region adjacent to, and to the right of, the x-0 plane. Thus a summation over k from k-l to k-R is replaced by a summation over k from k -R/2 to k R/2. Let R2 "- (71) denote the number of regions in the half-space. For a general function f(x) which is symmetric about x-0, that is, f(x) f(-x), one can then write that R E f(k)J dx' G(v;x x')N(x',v) k-l x'in k R2 R f(k) dx' G(v;xlx')N(x',v) k-R f(k) x'in k + ( f) dx' G(v;x x')N(x',v) k--R2 k- f(k) x'in k 0 b = dx' + dx f(x')G(v;xlx')N(x',v) (72) x'--b x'-O where b and -b denote the right and left boundaries of the space of interest. The notation used is defined such that cm c2 7C +f ... + + k (73) kl kn k-c1 k-en and (74) C2 cm c2 cm dx' + ... + dx' f(x') dx'f(x') + ...+ dx'f(x'). x'-c1 X' cn x'c 1 X'-cn In the first integral in Eq. (72) let x'--y, dx'=-dy. Observing that f(x)-f(-x) and N(x,v)-N(-x,v), one can write 0 0 dx'f(x')G(v;x x')N(x',v) dy f(-y)G(v;x -y)N(-y,v) x' -b y-b b /dy f(y)G(v;xl-y)N(y,v) (75) yZ0 Then let y-x' so that Eq. (75) becomes 0 b Sdx'f(x')G(v;xx')N(x',v) dx'f(x')G(v'x -x')N(x',v). x-b -0 (76) Substituting Eq. (76) into Eq. (72) yields R Z f(k) dx'G(v;x x')N(x',v) fk- x'in k b dx'f(x') [G(v;x x')+G(v;x -x')]N(x',V) x'-0 (77) f (k) dx' [G(v;x x')+G(v;x -x')]N(x',v) k-1n x'in k As noted above, G(v;x x') is the scalar density of first-flight neutrons at x due to an isotropic unit plane source at x'. If one defines H(v;x x') G(v;x x') + G(v;x -x') (78) to be the half-space Green's function, that is, the scalar density of first-flight neutrons at x due to isotropic unit plane sources at both x' and -x', then equation (69) may be rewritten for the half-space in the form N(x,v) dx' H(v;x x')S(x',v) x'in k + kV (V(k',v)+ Y(k',v) -fV(k,v)+ 7(k,v)1 f dx' H(v;x x')N(x',v) x in k v* (79) + dx H(v;xlx') dv" P(k,v'.v)N(x',v") x'in k v"-0 while Eq. (70) becomes D(x,v) R 2dx' H(v;x x') S(x',v)-S'(x',v)] x'in k + 2 ([V(k'.v)+ (k',v)]-[V(kv)+ (kv)] dx' H(v}x x')D(x',v) x'in k v* + T dx' H(v;xlx') dv" P(k,v!".v)D(x',v") x'in k v"-0 - ([V(k,v)+ /(k,v)]-[V'(k,v)+ Y'(k,v)] dx' H(v;xIx')N'(x',v) x'in k (80) v* + f dx' H(v;x x') dv" [P(k,v!.Lv)-P'(k,v!..v) N'(x'f,v") x'in k v"10 The one-speed equations may also be written for the one-dimensional half-space. Thus Eq. (48) becomes N(x) dx' H(x x,)S(x') x >0 + :2 v[Zt(k')- Ia(k)] fdx' H(x x')N(x') (81) x'in k while Eq. (60) becomes D(x) .dx H(x xI) [(x')-BS(xv) x') O 59 + E2 v[t(k')- Za(k)] dx' H(xx')D(x') x'in k i2 v (k)-a( j) dx' H(xJx')N'(x') (82) x'in k CHAPTER VI COMPARISON OF N AND D METHODS FOR SIAB GEOMETRY For the direct comparison of the formulation in terms of real density,N, with the formulation in terms of difference density, D, only the one-speed equations will be considered, namely Eqs. (81) and (82). Similar arguments would apply to the multispeed equations. Suppose that all R regions of the reference space contain the same medium and that the reference space itself constitutes one spatial period of an infinitely repeating lattice, that is, the reference space is a unit cell. Then the neutron density N'(x) in the reference space is that for an infinite homo- geneous medium and may be assumed spatially flat. Since the neutron density and medium are both spatially uniform in the reference cell, one may assume that the statement of neutron conservation holds not only over the entire reference space, but also over any arbitrarily small element of volume in that space. Thus at any point B'(x) S' v IN'. (83) Substituting Eq. (83) into Eq. (82) and cancelling terms where possible, one obtains D(x) -dx H(xlx')S(x') f',0 XI'-0 + a vt(k')- (k)] dx' H(xjx')D(x') x'in k v a(k)N' dx' H(x x') (84) x'in k Then using Eq. (50), one may combine terms to write D(x) dx, H(x x')S(x') X'-O + k v 2(k') f dx' H(x x')D(x') v Z.a(k) dx' H(x lx)N(x') (85) k-1 f x'in k Adding and subtracting the quantity v t(k') fdx' H(x x')N(x') k-1in k x'in k to Eq. (85) gives D(x) dx' H(x x)S(x') x 'O + 1k2 v[Zt(kt)- Za(k) f dx' H(x x')N(x') k-1 x'in k R2 I v lt(k') dx H(x x')N' k-in x'in k dxt H(x x')S(x') x*=0 + 2 v[t(k')- Za(k)] dx' H(x ')N(x') x'in k v I(k')N' dx' H(x x') (86) x '0 But from Eq. (10) v Zt(k')H(x x')dx' 1 (87) x 50 so that using Eqs. (50) and (87), one can reduce Eq. (86) directly to Eq. (81). The direct reduction of Eq. (82) to Eq. (81) having been established, the question remains whether the use of Eq. (82) has any advantages over the use of Eq. (81). It will be of particular interest to consider whether or not the complexity of Eq. (82) can be reduced by relating the real source S to the source 8' in the reference space. A few such special cases are considered below. In each case the assumption is made, as it was above, that all R regions of the reference space contain the same medium and that 8' and N' are spatially flat. Case 1 S(x) 8' Over All Space Equation (82) then becomes D(x) 2 v I(k')- a(k)] dx' H(xJx')D(x') x'in k va(k)- i(k)]N f dx' H(xxt) R- 2 tk'- a(k)] dxz H(x x')D(x') x'in k v Za(k)N' fdx H(x x') + vE(k' x'in k where Eqs. (83) and (87) have been used to obtain the last term of Eq. (88). Since S(x) = S' implies that S(x) is spatially independent, Eq. (87) may be used to show that the first term in Eq. (81) becomes dx' H(x xt')(xt) = 8 dx' H(x x') x'=O x '=O S' v St(k') (89) Thus Eq. (88) can be reduced directly to Eq. (81) in the same manner that Eq. (82) was reduced to Eq. (81), so that there is no advantage of the difference density formulation over that for the real density N. The reason for this is that, although it is given that S(x) S S', one must still calculate N' based on 8', and the N' terms then act as source terms in the differ- ence density equations. Case 2 S(x') 0 For Region k-1, S(x') 8' For Regions k 1 Equation (82) becomes D(x) dx' H(xIx')S'(x') x'in 1 + 2 k-i - 2 k-l v[ZCk)- (k)] dx' H(xIx')D(x') x'in k Sdx' H(x x')N'(x) :'in k To specialize further, let R2 2 and k' 2. Eq. (90) becomes D(x) S' dx' x'in 1 + v[Zt(2)-~a(1) H(x x') dx' H(x x')D(x') x'in 1 + v Z~(2) dx' H(x x')D(x') x'in 2 v[a(1)- a(1)] N' dx' H(xl x'). x'in 1 Using Eq. (83) one may write Eq. (91) as D(x) v[Zt(2), a(l)] idx' H(xI xtD(x') x'in 1 (90) Then (91) + v zS(2) dx' H(x Ix)D(x') x'in 2 a(l) N' dxv H(xlx') (92) x'in 1 or, again substituting Eq. (83) into the last term of Eq. (92), one finally obtains D(x) v[Zt(2)- Za()] f dx' H(xx')D(x') x'in 1 + v ZS(2) dx' H(x x')D(x') x'in 2 (1a(') 8' 7 a(2l) dx' H(x x') (93) x'in 1 For this special case, Eq. (81) becomes N(x) SI dx' H(xlx') x'in 2 + v[t(2)-Za(1)] /dx' H(xlx)N(xt) x'in 1 + v Zs(2) dx' H(xlx')N(x') (94) x'in 2 The obvious difference between Eqs. (93) and (94) is the source term. In a typical application of calculating the scalar neutron density in a space containing a gold foil in a graphite medium, the ratio Za(1)/Za(2) would become (1) Au 5.79 1 _7 1.94 x 104 a a a(2) ^C .000299 It can be seen that in a finite medium containing a highly absorbing region, even though it be of small spatial extent, the neutron density will not necessarily be a small deviation from a known density in a reference space with no absorber. In fact, the deviation in magnitude of the densities will be quite large, even though the spatial shape of the neutron density N may not be vastly different from the spatial shape of the known density N'. It seems obvious now that for problems involv- ing finite geometries the analytic method of solving for a difference density, D, offers no advantages over 68 the direct solution for the unknown density N; the difference density method may, indeed, be limited in usefulness to the solution of infinite geometry problems. CHAPTER VII THE FIRST-FLIGHT GREEN'S FUNCTION FOR A HOMOGENEOUS MEDIUM VII.1 Differential Equation and Boundary Conditions The first-flight Green's function 6k' (r,v r',v'n') for a general three-dimensional homo- geneous medium is defined such that it satisfies Eq. (10), namely va* Vgk' (r, vn r', v'n' ) + [V(k',v)+ (k' ,v)] gk'(rvnr' ,v'n') &(r-r )S(v-v )S(cn-n'); (95) further, gk (r,vnr',v'_1') is subject to the same boundary conditions that are applied to the angular density N(r,vn). Considering the RHS of Eq. (9) as a fictitious source Q(r,vn), one may rewrite Eq. (9) as vn.7VN(r,v) + [V(k',v)+ ((k',v)]N(r,v)) Q(r,vn) (96) Thus N(r,vn) represents the first-flight angular density in a medium as discussed in Chapter I, the removal pro- bability being equal to the total probability of col- lision in region k', with a source of neutrons Q(r,vn). If the homogeneous medium k' is of infinite extent, then the solution to the one-speed version of Eq. (95) is known to be (see Chapter 2, of Case, et al. (21)) t(k') -d gi (rlr S 2 n-r!' )'(n f-r') (97) where the superscript oo on g signifies that Eq. (97) is valid only for an infinite medium or for a finite medium enclosed by a non-re-entrant surface with vacuum boundary conditions, and r-r' denotes the unit vector (r-r')/jr-r' The extension to the speed dependent problem is obvious from the discussion of Chapter I and Eq. (12). Thus Zt(k',v) r-r' -v- S0, (r ', ,v'n') e (98) o(v-v')&(n-n')An ;Cc') . Integrating Eq. (98) over v' and n! gives t(k'v)lr-r' gg?(v;r,n|r') e t-----(k ,v ); (99) v r-r (99) integrating over n then yields k (v;r r') e -Ztk v) (-0oo v r-r 2 (100) where use has been made of Eq. (30). The simple form of Eq.' (100) is due to the fact that the medium k' is homogeneous so that the collision probability is not a function of the flight path of the neutron. As pointed out by Case, et al. (21), the case of a spatially variable collision proba- bility is a considerable complication because absolute coordinates instead of only relative coordinates between the field and source points enter the problem. It might be well to point out again that a unique advantage of the homogeneous Green's function method being described is that it completely avoids the complication of a spatially variable collision probability in the evalu- ation of the Green's function. From Eqs. (63) and (100) one can write the expression for G(v;r r') for an infinite medium. Thus (k';v _r-r'1 G- (v;rr') e(101) 47 v r-Ir' 2 From Eq. (68) one can then evaluate G(v;x x') for an infinite medium. Thus, using the transformations of V.1, one obtains ,3- ,,v Ir,-rIt(k' Gco (v;x x') 27r R dR 47T v r1-r' 2 R-Ix-x'I S e- t(k',v)R -1 dR (102) R-I x-x' The exponential integrals or En functions (21, 24) are defined such that En(x) e-xu u-n du (103a) 1 -i pn-2 e-X/P dp (103b) 00 Sxn-1 e-u u-n du (103c) u-x so that Eq. (102) may be written Goo (v;x x') L El [Zt(kv)Xx-x] (104) The Green's functions discussed above were, in general, for an infinite medium. The problem eventually to be considered, however, is that of a unit cell; that is, the space of interest represents one unit volume of an infinitely repeating set of unit volumes. The application of the unit cell approach is based on the fact that most heterogeneous nuclear reactors have fuel and moderator arranged in a repeating lattice structure. For calculational purposes a lattice is sub- divided into a number of identical unit cells (18, 19). The spatial and speed dependent neutron density, usually just the scalar density, is calculated for the unit cell and is subject to particular boundary conditions. Quanti- ties such as the thermal utilization are then calculated for the unit cell and are assumed to be valid throughout the lattice. The reactor is then treated as a homo- geneous reactor having the same thermal utilization as the unit cell, and the scalar neutron density is calcu- lated for the entire reactor. In most reactors the lattice is very large relative to the dimensions of the unit cell, so that it is not unreasonable to treat the unit cell as if it were in an infinite lattice. For a complete discussion of this assumption, see Chapters 7 and 18 of Weinberg and Wigner (19). If the assumption is then made that the lattice is infinitely repeating, then at steady- state the flow of neutrons in dn about direction ni across a point on the boundary between adjacent unit cells is equal to the flow in d_ about direction i where L2 is the mirror image of nl as illustrated in Figure 3. This is physically equivalent to the assumption that the cell boundaries are perfectly reflecting. Hence in the analysis that follows, neutrons which actually arrive at r in the unit cell by crossing the cell boundaries on first-flights can be considered to have originated within the unit cell at r' and suffered reflections at the cell boundaries before arriving at r. S2 180- - - T2 T 1 Figure 3 Symmetry of Neutron Flow at the Boundaries of Unit Cells in An problem in the following manner. Due to reflections at the cell boundaries, first-flight neutrons from an isotropic point source at r' can arrive at r by various paths, so that the total density at r is the sum over all possible paths i, of the densities of neutrons arriving at r by path i and having traveled a total path length Li. Alternatively, the neutrons at r can be con- sidered to have arrived by first-flights from i image sources at r located a straight-line distance L from soure i at located a straht-ine distance L from the field point r. The total angular density at r is then the sum over the i image sources of the contri- butions to the density at r from each image source at rt. (See Chapter 7 of Morse and Feshbach (16) for a discussion of the method of images.) Thus for a general homogeneous medium, one has from Eqs. (63) and (99) that t(k',v)L G(v;r,n r') e (t( rv)i) i 4 TTv2 -i (105) where Li is the total path length traveled by a neutron arriving at r along a first-flight path from an image source at ri. From the alternate point of view, Li is the path length traveled by a neutron along path i as it suffered reflections at the cell boundaries before arriving at r. Integration of Eq. (105) over n yields -" t(k',v)Li G(v;r r') 4 7tvvJLi (106) -O vLi2 (106) 4 7r vLi1 As one would expect, for an infinite medium or for a finite medium bounded bya non-re-entrant surface with vacuum boundary conditions Eq. (106) reduces to Eq. (101); for a finite medium bounded by a re-entrant surface with vacuum boundary conditions, Eq. (106) reduces to the infinite medium Green's function with spatially variable collision probability as given in Chapter 2 of Case, et al. (21), namely R (r sR/R, v) ds G(v;r r') e 2(107) 4 7 vR2 where Zt(s,v) 0 outside the bounding surface. VII.2 Analytic Form of G(v;xlx') The physical effect of introducing reflecting boundaries is to increase the density of neutrons at a point due to reflections at the boundaries which, in effect, send neutrons back into the finite unit cell. The total neutron density at r may be separated into two parts; namely, that due to first-flights from the source at r' within the unit cell and that due to neutrons that may be considered either as arriving from i image sources located at r' outside the unit cell or as having suffered one or more reflections at the boundaries after emission at r'. The first-flight angular density at r due to a unit isotropic point source at r' in a homogeneous medium is simply the infinite medium Green's function obtained from Eqs. (63) and (99), namely, 0,,nv) rr-,e oG (v;r,|r,) e- (a ^ '). 47T v r-r (108) Let Gfc(v;r,n r') denote the angular density at r of neutrons which do not arrive by first-flights from the unit isotropic point source at r' but instead arrive at r only after having suffered reflections at the mirror boundaries of the cell, or alternatively, arrive by first-flights from the i image sources located at ri. Then let Li,n denote the total path length traveled by a neutron along a path having a direction that is characterized by the unit vector ri-' pointing from the image source at r' to the j -ji field point r, the neutron arriving at the field point r after having suffered n reflections at the mirror boundaries of the cell. One can then write from Eq. (105) that Gfc(v;rnE I) z e- lt(k',v)Li,n _ n-l i 4 VLln 2 -- n (109) The double summation over n and i is, of course, equiva- lent to a single summation over all the image sources. The total Green's function for the unit cell is then G(v;r,n Pr') G (v;r,nlr') + Gfc(v'r,rl r') (110) Thus it is seen that Gfc represents a finite medium correction term to be added to GO. In V.1 the plane geometry equations were obtained by integrating the point source kernel over all the sources on the yz-plane at x'. For a unit cell with plane reflecting boundaries, a neutron which arrives at r after having suffered n reflections, and thus having traveled a path length Lin, can be con- sidered to have originated at a fictitious or image plane source located at a distance pLi,n from the field plane. The cell boundary at which the neutron made its last or n-th reflection lies between the field plane and the image source plane. The transformation from the point source Green's function to the plane source Green's function, that is Eq. (68), must be modified using the above method of images to locate image plane sources at distances pLi,n from the field plane. Referring to Figure 2, G(v;x,plx') is the number of neutrons with direction cosines between p and + du crossing a ring element of area 27 Tper unit time, the ring element being located in the plane at xt. If f is the azimuthal angle indicated in Figure 2, then dr du dP (111) Hence, using the transformations of V.1, Goo (v;x, Ix')dx dp dx' , 2 0 x d d G I fma0 f-2l1 (112) Since the sources are assumed to be symmetri- cal about the x-axis, G(v;r,flr') is a function only of r,r', and u. Thus the integration over c may be carried out in Eq. (112): 1 GC (v;x,p x') 47r2/ G(v;rl|r') X- 2 (113) P O =0 For slab geometry G(v;r,nlr') has a delta function behavior with respect to p' so that Eq. (108) becomes Gc (v;rn.r') - e- t(k',v) (x-x)/p| 4----x- ---------(114) 47yv(Ix-x'l /p)2 27r The p'-integration may then be carried out in Eq. (113). The result is Gc(v;x,pIx') - - (k',v) r-r 2v or using Eq. (2), G C(v;x,p x') - -Z'(k'.V) I (X-X'f)^l 2v II -Zt(k,v) (x-x')/P h(x-x') for f > 0 (115) h(x'-x) for < 0 2v I where the step function has been introduced to account for the fact that in an infinite first-flight medium, the flow of neutrons in the positive P direction is zero unless x is greater than x', while the flow of neutrons in the negative p direction is zero unless x is less than x'. For a more complete discussion see Chapters 2 and 4 of Case, et al. (21). In the same manner, one obtains the correspond- ing expression for the finite medium correction term. Thus Zt(k',v) (x-xi)/p GfC(v;x,pl x) e h(x-xi) for p 0 1 2v || (116) g(k v) I(x-x) I Ee -,- / h(xi-x) for p 0 i 2v i where the single summation represents a sum over all i image source planes located respectively at xi. The first two image source planes are illustrated in Figure 4. Substituting Eqs. (115) and (116) into Eq. (110), one can verify that each term in G(v;x,p x') satisfies the one-dimensional form of the source-free transport equation corresponding to Eq. (95), namely oG(v;x,p x') VU xy + v Zt(k',v) G(v;x,Ix') 0 (117) where v Zt(k',v) is equivalent to [V(k',v)+ (k' ,v) . Further, G(v;x,p x') satisfies the reciprocity theorem (21), that is, G(v;x,p x') G(v;x' x,-) (118) Finally, each term of Eqs. (115) and (116) satisfies the symmetry condition at the boundary x +b that the flow of neutrons in dp about p is equal to the flow in dp about -', that is, G(v;x,u x') x-l b G(v;x,- ix') .x-b (119) x-+b x-+b Since each term of the Green's function G(v;x,ujx') satisfies the original boundary conditions, it can be concluded that the function itself satisfies the boundary conditions. One can verify that, in agreement with sections 4 and 5 of Case, et al. (21), a source plane of strength qs at x' is equivalent to a discontinuity of the normal component of the angular current, where for an isotropic plane source, qs 1/2. The angular current is simply the number of neutrons crossing unit area perpendicular to the direction of flow per unit time and unit solid angle. In particular, for the infinite medium terms one can write vpG(v;xuIx') -x-x'+- vuG(v;xGiux') x -x'- q5s() " In an infinite medium the second term on the LHS vanishes for p >0 while from Eq. (115) the first term is equal to 1/2, that is, the source plane value. In the solution for the scalar neutron density in Eqs. (69) and (77) one needs the corresponding scalar Green's function G(v;x x') obtained by integrat- ing over all p the function G(v;x,p x'). Thus, 1 G(v;x x') G(v;x,p x')dp (120) P--1 Recall that G(v;x x') is the first-flight density of neutrons at x due to a unit isotropic plane source at x' in a homogeneous medium. Then, referring to Figure 4, one can see that for image sources to the right of x, that is, x => x, the neutrons reaching x must have a negative direction cosine u, while for image sources to the left of x, that is, xi <= x, the neutrons reaching x must have a positive direction cosine. Thus the contribution of the image sources is 0 1 Gfc(vI;x x) = Gfc(v;x,plx.) dp + Gfe(v;x,,Ix')di (121) 0- 1 0 where the first and second integrals represent the contributions of image sources placed respectively to the right and left of x. c7G x--2b-x' Image Source Plane at x--b-(b+x') Figure 4 x--b x-O 01800- -' 1,1 x-b x-2b-x' Image Source Plane at x-b+(b-x*) Slab Unit Cell with Mirror Boundaries at x-+b Letting Gtc denote one term of Eq. (116), one may write the first integral in Eq. (121) as 0 0 f fe(v e- t(k'tv) I(x-X)/IU Gi (v;x, |')du -d iI2vl I2 e- Zt(k',v)(x-xi)/l d (122) O-0 the last integral having been obtained from the preced- ing one by a successive substitution of variables, first letting y--u, and subsequently letting i-y. In general then, one can sum over all image sources and obtain the result that G (v;x x') f e t(k',v) (x-xi) d (123) Again, for a first-flight medium the range of integration over ) for the infinite medium contri- bution is from 0-O to P1. Thus one can write 1 G(v;xlx') f e- t(k',v) (x-x')/ d -O0 1 + -t(ck',v) (x-x') 4) i 2v (124) P-0 or from Eq. (103), G(v;x x') El ( t(k'v)x-x'j] (125) + E El[Zt(k',v) x-xil] ) To express G(v;xIx') as a function of the unit cell dimensions it will be instructive to return to the physical concept of reflection of neutrons in order to locate the image sources with respect to the cell boundaries. Using the notation for path length previously discussed with regard to Eq. (109), one can write the expression for GfC(v;x x') in Eq. (125) as Gc(v;x x') E E El[7t(k',v)ILi.nl] (126) n-l where PLi,n is, as before, the distance separating the field plane from an image source plane, the image plane corresponding to a neutron having made n re- flections at the mirror boundaries. In Eq. (126), the summation over i extends over all possible paths by which a neutron having its direction cosine equal to u can arrive at the field point after suffering n and only n reflections at the boundaries. In plane geometry, it is easily seen that for a first-flight medium there corresponds to any given p only two such paths for any n. Hence the summation in Eq. (126) runs from i-1 to i-2. Figure 4 illustrates the two possible paths, L1,1 and L2,1 corresponding to S= 0 and u < 0 respectively, by which a neutron can arrive at x after suffering only one reflection. The unit cell in Figure 4 has its midplane at x-O and the reflecting boundaries at x b. The cell width, that is, the distance between the two boundaries is 2b. The expressions for pLi,n can be obtained by simply adding the total path length a neutron travels while making n reflections. The expressions for pi,n depend on the orientation of the source plane and field plane x relative to each other and to the x-0 plane. It has been shown in V.2, however, that one need only consider field planes for positive x, because the solution of the half-cell is symmetrical with respect to x. The expressions for aLin are derived in Appendix A where it is shown that Eq. (125) may be written, for x > O, as G(v;xx') 1. E (k',v)Ix-x'] + 1 EI[Zt(k',v) 2nb-x'+(-1)nIxl 2v n-1 (127) + E[Zt(,v)12nb+x'-(-1)nx]) for x = 0. Note that Eq. (127) can be shown to be term by term equivalent to a more cumbersome series representation of G(v;x x') derived by Aswad (25). Finally, the half-cell Green's function defined by Eq. (78) becomes H(v;xlx') = E[l t(kO,v)|x-x'I] + E1[t(k',v)x+x'I 1 co + I E 1I[Zt(kv) 12"b'x-'l)n] n=1 + EL[Zt(k',v) 2nb.+x-(-1)nxi] + E,[Zt(k',v) 2nb+x'+(-l)nx] + El[Zt(k',v)12nb-x'-(-l)nxi] )(128) for x =0 and x' O 0 VII.3 Monte Carlo Generation of G(v;xlx') For complex finite geometries, the complete analytic form of the first-flight Green's function may be exceedingly difficult, if not impossible, to derive. In such cases, it may be advantageous and even necessary to represent the physical or mathematical system by a sampling operation satisfying the same probability laws. Such a process has come to be called a Monte Carlo method (26). Essentially, Monte Carlo methods are "paper experiments." The experiment consists of performing specific sampling operations many times, the sampling operations satisfying the criteria mentioned above. As in many other experimental procedures, the geometry of the problem does not greatly affect the difficulty of performing the Monte Carlo experiment. The con- struction of a function, say neutron density, by Monte Carlo is conceptually little more difficult for a square or hexagonal unit cell than it is for a slab or cylindrical unit cell. This is in striking contrast to the analytic description of the same function; there the solution for simple geometries does not, in general, give any indication of how to proceed to more complex geometries. Thus there is strong motivation to generate the plane geometry Green's function G(v;x x') by Monte Carlo. The transition to cylindrical or even more complex unit cell geometries would not be difficult; the general procedure, that is, the Monte Carlo sampling operations would be essentially the same. The Monte Carlo generation of G(v;xix'), that is, the first flight Green's function for a homogeneous finite medium in an infinite plane geometry with perfectly reflecting boundaries has been discussed extensively by Aswad (25) and will be outlined below. In plane geometry, a cluster of neutrons is considered to be emitted with unit statistical weight at the source plane x'. Its direction cosine, P, is chosen from a random distribution, the distribution being such as to represent an isotropic source, that is, a source which has equal probability for emission of neutrons in an element of solid angle dn about any direction n; thus in plane geometry the direction cosine P occurs with equal probability for values between -1 and +1. The random selection of P is done by Monte Carlo sampling. The projection of the path of the neutron cluster on the x axis is divided into closed intervals [Xm-l, xmJ [xm xm+1 ... and the statistical weight of the neutron cluster is computed at points Xm-l, xm, xm+l, *. The weight of the neutron cluster at some point Xm, denoted by W(xmjx'), is subject only to exponential attenuation along the path of travel, the mean free path between removal collisions being equal to the reciprocal of the total collision probability Zt(k',v). Thus W(xmIx') e- t ( ')/p (129) If xm+l > xm x', and if W(xmlx') and W(xim+xl') represent respectively the statistical weight of the neutron cluster at x, and xm+1, then the quantity [w(xmJx') W(xm1lx')] must represent the number of removal collisions per unit time that occurred in the closed interval [xm, xm+ . The neutron cluster continues along its original path until it arrives at a cell boundary where it is considered to be reflected and sent back across the cell with direction cosine p', where u*--u, that is, Ip' Iu." The number of collisions that occur in the interval [xm, Xm+1 as the neutron cluster passes through that interval is again recorded. The process is repeated as the neutron cluster suffers |

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angle indicated in Figure 2, then
dn. - dp d
Hence, using the transformations of V.l,
G00 (v;x,p xf)dx dp dx' Â«
2 0
- dx dp/ df
(111)
G(v;r,/i
r')(-27T)
x-x'
f
r
dp'
dx*.
I>*
(112)
Since the sources are assumed to be symmetriÂ¬
cal about the x-axis, G(v;r,nr') is a function only of
r,râ€™, and p. Thus the integration over |