A
accuracy of differencing vs. convergence, 76-77
Adams formula, 91 addition of matrices, 4
addition theorem for Legendre polynomials, 186, 325
adjoint
age-diffusion equations, 157-159 eigenvalues, 156-157
equations, 155-161 flux, 166-167 function, 155-161 Hermitian, 14 integral equation, 179 of a matrix, 14
multigroup equations, 166-167 operator, 155-161
slowing down density, 166 age-diffusion,
approximation, 151-155 equations, 151-155
matrix form of equations, 154 method of solution, 159-161 Aitkens δ2 method, 130 Allis, W P., 296
alternating direction implicit method 109-111, 131-134
stability of, 110-111 amplification matrix, 218-220
anisotropic scattering, 207, 266-267, 271 antisymmetric matrix, 13
area under curve, 53-55
associated Legendre polynomials, 325-326 attenuation in shields, 272-273
augmented matrix, 11-12 average
of Boltzmann equation, 298-299 cross-section, 162, 201
diffusion coefficient, 164 flux, 162, 201
logarithmic energy decrement, 153, 311-312
source density, 201-203
Β barn, 287
basis, 22 basis vectors
nonorthogonal coordinate system, 34-38 Bateman, H., 328
Bauer, W . F., 277 Berger, M. J., 232 bilinear form, 39
binary encounters, 289-290 binomial distribution, 252 biorthogonal
dual vector system, 38 functions, 155-157 vectors, 34-39 biorthogonality
of adjoint of integral equation, 179 condition, 36, 155-156
relationship, 34-36
Birkhoff, G., 146, 147, 178, 179 birth of a neutron, 261
fractional births, 262 block boundaries, 267-268
block tridiagonal matrix, 45-46, 81-82 Bode wig, E., 49
Boltzmann equation, 184, 200, 288-289 assumptions, 289-290
conservation of particles, 298
consistent with Newton's second law, 298
in cylindrical geometry, 297 derivation for neutrons, 284-288 derivation for photons, 291-293 energy conservation, 289-299 with isotropic scattering, 204 for neutrons, 288-289 for photons, 293
in plane geometry, 204, 297 in spherical geometry, 208, 291 boundary conditions
diffusion theory, 154 Dirichlet, 98
double PN method, 196 Marshak, 188-189 331
Neumann, 98
in numerical approximations, 93 PN method, 188-190
SN method, 211 boundary crossing
cylinder, 264 plane, 265 sphere, 264
ways of speeding up, 267 boundary crossing, checking, 276 boundary value problem
analytic solution of difference equation, 73-75
numerical solution, 96-97 types of, 98
C canonical form, 31 Carlson, B. G., 208, 231 Cartesian coordinates
use in Monte Carlo calculations, 263 Cashwell, E. D., 276
census, 240, 260 periods, 260, 273 time, 260
center-of-mass system, 301-302 definition, 306
central limit theorem, 249-250 Certaine, J., 232, 318, 323 Chapman, S., 296
characteristic equation, 23-24
characteristic roots of nonsymmetric ma
trix, 27-28 characteristics, 101 Clark, M., 178, 296 coefficient matrix, 11 cofactor, 7-8
collision transfer probability, 287, 318 column matrices, 2-3
commutative matrices, 5 eigenvectors of, 51 Compton process, 291-292 Compton relation, 225 Compton units, 292
congruence transformation, 19
congruential multiplicative method, 242- 243
conjugate gradient method, 136-139 conservation of energy, 302-303
conservation of linear momentum, 302 conservation of neutrons, 298
continuity condition, 188 convergence
and stability, 98-99 conditions for, 115
of difference solutions, 75-79, 98-99 of heat-flow difference equation, 78-79 of Laplace difference equation, 77-78 rate, 115, 120-122
test for, 113
versus accuracy of differencing, 76-77 Copson, Ε. T., 329
correction vector, 116-117 cosine of scattering angle, 304
cosines of a random direction, 277-278 counting, last collision method, 296 coupling between flux and slowing down
density, 153 Courant, R., 49, 147, 329 Cowling, T. G., 296 Cramer, H., 277 Cramer's rule, 8
criticality, 146, 190, 212-213 cross section
microscopic, 287 total macroscopic, 286 total reaction, 286 total scattering, 286
cumulative distribution function, 243, 245 Curtis, J. H., 277
D Davis, J., 231 Davison, B., 231 death
of a neutron, 260-261 weight, 271
decisions
boundary crossing, nuclear event, or census, 263
material with which neutron collides, 265 type of collision, 266
defect of basis, 22 definite sum, 58
delayed neutron precursors, 181 density of particles, 282-284
derivatives in terms of difference operators, 61, 82
determinants definition, 6-7
Laplace development, 7-8 products, 9
solution of linear equations, 8-9 theorems, 7
determinism, 274
diagonal dominance, 119, 122 diagonalization, 26, 28 diagonal matrix, 13 dietetics
control of weight, 271 difference
rule, 58
difference equations constant coefficients, 64
construction by physical means, 84 convergence of higher order, 77-79 formation of, 59-63
of heat-flow equation, 71-73 higher order, 61-62, 92-93 homogeneous, 64
inhomogeneous, 67-71 of Laplace equation, 74-75, 83 linear, 64
matrix form of, 79-82 non-uniqueness of, 59 order of, 63-64 repeated roots, 66-67 solutions of, 63-71 variable coefficient, 64 of wave equation, 83 difference operators, 55-59
backward, 56 central, 56 divided, 54
double integral in terms of, 95-96 expressions for second derivative, 59-62 forward, 55
inverse shift, 57 nth order, 56 shift, 56
differential distribution function, 243 selection from, 246
differential equations second order, 95-96
differential operator for the leakage, 168 Dirac delta function, 288, 292, 316 Dirac, P. A. M., 296
directional density, 282, 284, 293
directional flux expansion, 185 direction cosines, 266, 305
of isotropically scattered neutron, 246- 247
of random direction, 277-278 of random scattering, 247 referred to c-frame, 308 direction flux, 185, 282, 283, 284
direction of integration of equations, 212, 215, 220-221
Dirichlet boundary conditions, 98 discrete ordinate methods, 204-207 discrete SNy 213-216
discriminant, 97 displacement, 117
distance to nearest boundary cylinder, 264
plane, 265 sphere, 264
distribution functions, 243-249
distribution functions of more than one variable, 244
double integrals in terms of difference operators, 95-96
double Pjy approximation, 196-200 reduction to multigroup equations, 198-
199
second order, 234
Ε Edlund, M. C , 312
efficiency of rejection method, 246 Ehrlich, R., 178 ·
eigenvalues, 23-27 definition, 23
geometric interpretation, 34, 39 largest of matrix, 40
methods of determining, 140-146 multiplicity of, 23-24, 34 of real symmetric matrix, 24-26 eigenvectors, 23-27
expansions in terms of, 25 magnitude of, 24 orthogonality, 25
of real symmetric matrix, 23-27 scaling of, 24
eigenvector-eigenvalue equation, 23 elastic scattering, 196, 266-267, 286, 301
elliptic differential equations, 97 elliptic equations, 111-146 equivalent matrices, 18-19
error in numerical integration, 88-89, 91- 92, 94
error vector, 114-116 Evans, R. D., 296 Everett, C. J., 276 expansion
of arbitrary vector, 22 of directional flux, 185 of scattering function, 185 tests for adequacy, 230-231 expected value, 250, 273 explicit integration formulas, 91 extrapolated Liebmann method, 126
F factorial series, 57-59 Faddeeva, V. N., 49 Fano, U., 232, 329 fast fission factor, 261-267 Feshbach, H., 178, 231 fission, 266, 286
neutron spectral distribution, 310 source matrix, 171
sources, 157, 172 flux, 152, 212, 284 flux expansions, 197 flux vector, 170 forcing, 274 Forsythe, G. E., 146 Fort, T., 82 Fourier series, 105
fractional error, 252, 253, 255 fractional square error, 251 Frank, P., 329
Frankel, S. P., 147 Franklin, P., 312 Friedrichs, K., 147, 219
G Gantmacher, F. R., 49 Gauss quadrature, 205-207 Gauss reduction, 10-12, 111 Gaussian integration, 205-207
Gauss-Jordan reduction, 50 Gauss-Seidel method, 121 Gelbard, E., 194, 231
generalized boundary crossing routine, 279 geometric interpretation
eigenvalues and eigenvectors, 34, 39 matrix transformations, 31-34 Gerschgorin's theorem, 40, 119 Glasstone, S., 312
Goertzel, G., 277 Goldstein, H., 232 Gram determinant, 21-22 group constants, 201-204 group equations, 201-204
Η half-range polynomials, 197 Hamming, R. W., 82, 146
heat-flow difference equation, 71-73, 97 convergence of, 78-79
matrix form of, 79-80 solution of, 71-73 Hermitian adjoint, 14 Hubert, D., 49, 329
Hildebrand, F. B., 49, 82, 146, 147, 231 history, particle, 240
Hoel, P. C , 277
Holmes, D. K., 178, 231, 312 Householder, A. S., 49 Hurwitz, H., 178 Hyman, Μ. Α., 147
hyperbolic equations, 97, 99-106
I
implicit difference equation, 104 implicit formula, 91
importance
function, 155-161
physical interpretation, 159 importance sampling, 268-270 indefinite summation operator, 57 inhomogeneous difference equations, 67-71
method of undetermined coefficients, 68 method of variation of parameters, 68-71 inhomogeneous Dirichlet condition, 98 initial estimates, 175
initial value problems, 98 inner iteration, 176-177 instability, 98
integral conservation equation, 296 integral distribution function, 243-244 integral equation, 293-296, 299
inverse of a matrix, 14 inverse shift operator, 57
inversion of tridiagonal matrix, 46-49 irreducible, 40-41
isotropic scattering, 266 iterate, 117
iteration, 175-178 renormalization, 176
solution of multigroup equations, 175- 178
iteration matrix, 114, 118, 122-123, 129- 130
iteration methods, 111-146 nonstationary, 129-139 stationary, 114, 117-129
J Jacobi method, 121 joint c.d.f., 244, 245 joint p.d.f., 244 Jordan, C , 82
Jordan canonical form, 30-31, 115
Κ Kahn, H., 276, 277 Kalos, Μ. H., 277 Kaplan, S., 147
kinetics equations, 180-182 Kopal, Z., 146, 231 Kronecker delta function, 13
L
laboratory system, definition of, 301-302 Lagrange interpolation formulas, 88 Laguerre polynomials, 230, 327
definition, 327 derivative of, 327
orthogonality relations, 327 recurrence relations, 327 sum relation, 327 Lanczos,' C , 49, 146 Laplace development, 7-8
Laplace difference equation, 120-121, 124- 125, 126-128, 132-134
convergence of, 77-78 matrix form of, 80-82 solution of, 74-75 Laplace's equation, 97, 117
Laplacian in various geometries, 168 last collision, 296
Legendre polynomials, 184, 185 addition theorem, 325
associated, 325 definition, 325
generating function, 325 integral relations, 189 integrals involving, 326 orthogonality relation, 326 power series expansion, 325 recurrence relation, 326 lethargy, 304
Lewy, H., 147 Liebmarin method, 121 linear equations, 1-4
conditions for solutions, 12 solutions of, 7-12
linear independence tests for, 21-22 of vectors, 21-22 linear interpolation, 245
logarithm of a random number, 248-249, 263
lower triangular, 45 strictly, 45
Μ
macroscopic cross sections, 262, 286 Marchuk, G. I., 178, 194, 231 matrix
addition of, 4 amplification, 218-220 antisymmetric, 13 augmented, 11, 12
block tridiagonal, 45-46, 81-82 characteristic equation, 23-24 coefficient, 11
column, 2-3 commutative, 5 diagonal, 13 eigenvalues, 23-27 eigenvectors, 23-27 equality of, 4
equations, vector interpretation, 16 equivalent, 18-19
factorization, 46-49 functions of, 18
geometric interpretation of transforma
tions, 31-34 inverse, 14-15
inversion of tridiagonal, 46-49 irreducible, 40-41
iteration, 114, 118, 122-123, 129-130 lower triangular, 45
modal, 26, 29 multiplication, 2-5 negative definite, 31
noncommutativity of multiplication, 5 non-negative, 39-44
nonsingular, 8-9 nonsymmetric, 27-31 notation, 1-4 operations, 4 orthogonal, 15 permutation, 20 positive, 41 positive definite, 31 rank, 11-12 real symmetric, 24 reducible, 40 row, 2-3
secular equation, 23-24 submatrix, 6
supermatrix, 6 symmetric, 13 trace, 20 transpose, 13 tridiagonal, 45 upper triangular, 45 vector interpretation, 15 Mayne, A. J., 277
Meghreblian, R. V., 178, 231, 312 methods, 277
conjugate gradients, 136-139 separation of variables, 71, 74 simultaneous displacements, 117-121 steepest descent, 135-136
successive displacements, 121-125 undetermined coefficients, 68 variation of parameters, 68 Yvon, 196-200
metric tensor, 38
microscopic cross section, 287 Milne problem, 234-235 Milne-Thompson, L. M., 82 minor, 7
modal matrix, 20, 29 model problem, 117 moments method, 223-231
assumptions, 225
lack of truncation, 226-227
order of evaluation of coefficients, 227 outline, 223-224
reconstruction of flux, 229-230 moments method for neutrons, 315-323 Monte Carlo method
applications, 240-241
attenuation in shields, 272-273 boundary crossing, 264-265, 267-268 capture, 266
checking of, 276
distance to boundary, 264-265 for fast fission, 260-268 general rules of use, 275-276 outline, 239-241, 260-261 rejection method, 246 tallies
energy group, 266 spatial zone, 266 Morse, P. M., 178, 231 Moshman, J., 277
multigroup difference equations, 167-170 matrix form of, 170-175
multigroup equations, 161-165 multigroup transport methods, 200-204 multiplicity of roots, 23-24
Ν w, In event, 286
negative definite matrix, 31 net current, 284
net leakage, 285-286
Neumann boundary conditions, 98 neutron current, 215
Newton-Cotes integration formulas, 88
nonnegative matrices, 39-44 spectral radius of, 43-44 theorems about, 41-44
nonorthogonal coordinate system, 35- 39
nonorthogonal vectors
construction of dual system, 38 nonsingular matrices, 8-9 nonstationary methods, 129-139 nonsymmetric matrices, 27-31
eigenvalues of, 27-29 eigenvectors of, 27-31, 34-39 normal to surface, 33, 39 numerical integration, 87-90
by backward differences, 89 errors, 88-89
by forward differences, 88 starting, 93-94
by statistical procedures, 254-256, 268- 270
numerical solution of the multigroup equations, 175-178
Ο O'Brien, G. G., 147 operators
backward, 56, 59 central, 56, 60 forward, 55, 56, 60
interpretation, geometric, 60 inverse shift, 57
nth order, 56
properties of forward difference operator, 56, 58
relations among, 57 self-adjoint, 156 shift, 56 summation, 57 optical depth, 295
ordinary differential equations, 90-97 statistical solution, 256-259 orthogonal matrix, 15 orthogonal planes, 267 orthogonal transformation, 19 orthogonality relations, 73, 84, 197 orthogonalization of a set of vectors, 22-23,
137
outer iteration, 177-178
Ρ Pn approximation, 184-196 parabolic differential equations, 97 parabolic equations, 106-109
stability of, 107-109
parabolic interpolation methods, 277 partial difference equations, 71-75 partial differential equations, 97-99
elliptic, 97 hyperbolic, 97 parabolic, 97 types of, 97 Peaceman, D. W., 147
Peaceman and Rachford method, 109-111 Pearson, J., 231
permutation matrix, 20
Perron-Frobenius theorem, 42-44 placement of splitting planes, 272-273 positive definite matrix, 31
positive matrix, 41 power method, 140
principal axes of quadratic surface, 33, 39 probability distribution function, 243 probability of elastically scattering a
neutron, 309
probability of scattering, 292, 293
propogation constants in the P1 and P3
approximations, 233 propogation of errors
equations governing, 105 pseudorandom numbers, 242-243
Q quadratic form, 32
principal axes, 33 surface, 32, 135
tangent to, 33 quadrature formula, 88-90 quota sampling, 275
R Rachford, Η. H., Jr., 147 random distributions, 241-242 random number, 240-243 random walk method, 259
rank of matrix, 11-12 Rayleigh quotient, 144
reactor kinetics equations, 180-182 real symmetric matrix
diagonalization, 26, 32 eigenvalues of, 24 eigenvectors of, 25-27
interpretation, geometric, 31-34 recurrence relation
difference equations as, 65
for half-range Legendre polynomials, 197 for Legendre polynomials, 187
for scattering integrals in moments method, 320-321
reducible matrices, 40
reduction of n-th order differential equation, 94-95
rejection method, 246 relaxation, 117
renormalization, 144-145, 176, 194 repeated roots, 34
rescaling of sources, 194 residual
in conjugate gradient method, 138 interpretation of, 125
in simultaneous displacement method, 121
in successive displacement method, 125 vector, 116
reweighting, 270 Richardson method, 121 Richtmyer, R. D., 146, 219, 231 Robkin, Μ. Α., 178
Rose, D. J., 296 Rosenberg, R. L., 147
rotational symmetry of quadratic surface, 34
row matrix, 2-3
Runge-Kutta formulas, 93-94 Russian roulette, 270
S Sn method, 204, 208-213
for anisotropic scattering, 236-237 for cylindrical geometry, 237 sample mean, 253
scalar Helmholtz equation, 145-146 scaling, 144-145, 176, 194
scattering
anisotropic, 196 elastic, 196
function expansion, 185 inelastic, 196
integral
approximation by straight-line seg
ments, 318
in Boltzmann equation, 186
differences between neutrons and photons, 315
expansions in powers of atomic mass 318
in moments method, 317, 319 for neutrons, 317, 319 isotropic, 196
kinematical relations, 301-302 matrix, 171-172
neglect of energy-loss approximation, 317
probability of photon, 292 velocity relations, 304-308
scattering probability of a photon, 292 Schiff, D., 196, 231
Schiff, L. I., 296
Schmidt procedure, 22-23 secular equation, 23-24
selection from differential distributions, 246 selection from fission distribution, 262 selection of a neutron, 260
self-adjoint operators, 156 shift operator, 56
general expression for, 60 similarity transformation, 19, 24 Simpson's approximation, 89-90, 318 simultaneous displacements, 117-121
convergence of, 122-123 iteration matrix, 118
rate of convergence, 120-121, 123 residual, 121
test for convergence, 119 simultaneous equations, 9-12 singular matrices, 9
slowing down density, 153, 163 source function, 152, 212 special matrices, 12 spectral radius, 43
of matrix, 40
of non-negative matrix, 42-44 Spencer, L. V., 232, 329
Spencer-Fano method, 223 Spencer-Fano polynomials
adjoint polynomials, 328 definition, 327
derivatives of adjoint polynomials, 328 differential equations satisfied by, 328 integral relations, 328
relation to adjoint polynomials, 328 spherical harmonics approximation, 184
flux-current harmonic component rela
tions, 232
iteration procedure, 193
similarity to multigroup equations, 194- 196
in three dimensions, 232-233 splitting method, 270
splitting planes, 272 placement of 272-273 square matrix, 3, 44 stability, 218
condition for iterative method, 115- 116
condition for Pn method, 219-220 and convergence, 98-99
of difference approximation, 104 of iterations, 114
problem, 98 of Sn method, 223 standard deviation, 250
stationary iteration, 114, 117-129 statistical solution
to an ordinary differential equation, 256-259
Stein, P., 147 Stieljes matrix, 142 stochastic variable, 244, 245
streaming of neutrons in a vacuum, 293-296
strong perturbations, 274 subcritical assembly, 146, 178 submatrix, 6
successive displacements convergence of, 122-123 iteration matrix, 122 method of, 121-125 rate of convergence, 123 residual, 125
successive over-relaxation, 125-129 optimum factor, 129
summation
formulas, 57-59, 73, 82-84 operators, 57-59
rule, 58
supercritical assembly, 146, 178 supermatrix, 6
.S-wave scattering, 310 symmetric matrix, 13 systematic sampling, 275
Τ table look-up, 246 tallies, 274
energy, 266 zone, 266 Taylor series, 59
test for convergence, 113, 176 test for linear independence, 21-22 time-dependent transport methods, 216-
223
total multiplicity, 286-287
total number of mean free paths, 263 trace of a matrix, 20
trajectory method, 96 transfer integral
approximation of, 205 for multigroup equations, 203 transfer probability
for elastic isotropic scattering and fission, 308-312
for fission, 310 transformations
congruence, 19 orthogonal, 19 permutation, 19 similarity, 19 transport equation
assumptions, 289-290
one group for slab geometry, 204 for photons, 224
for spherical geometry, 291
transport term in Boltzmann equation, 290-291
in spherical geometry, 291 transpose of a matrix, 13 tridiagonal matrix, 45 Troost, M., 277
truncation error, 59-63, 89 as function of order, 62-63
in integration formula, 91 order of, 59-60
types of boundary conditions, 98
U unit matrix, 13 upper triangular, 45
strictly, 45
V variance, 250, 251, 267, 269
of binomial distribution, 253 variance reduction, 268-275
census periods, 273 expected values, 273 forcing, 274
importance sampling, 268-269 methods, 268
particle splitting, 270 quota sampling, 275 Russian roulette, 270 splitting planes, 272
weight standards, 271, 272, 274 when to use, 268, 275
Varga, R. S., 49, 131, 146, 147, 179 vector
interpretation of matrix equations, 16 method, 307
scalar product, 16 vectors, 15
linear independence of, 21-22 operations among, 15-18 orthogonalization, 22-23
velocity relations for nuclear events, 301- 312
vital statistics, 261 von Mises, R., 329
von Neumann method, 104-106 conservatism, 106
W Wachspress, E. L., 179 Wasow, W . R., 146 Watson, G. N., 328
wave equation, 83, 97, 99-104
solution of difference equation, 103-104 stability, 99, 104, 105-106
truncation error, 99 weight
control of, 271 standards, 271-272, 274 use of, 270
weighting coefficients, 228, 229 Weinberg, A. M., 178, 231 Whittaker, Ε. T., 328 Wick, G. C , 205, 231 Wigner, E. P., 178, 179, 231 Wilkins, J. E., 232
Wilson, Ε. B., 312
X, Y , Z Yvon, J. J., 231 Yvon method, 196-199
reduction to multigroup equations, 198- 199
zero matrix, 13 Ziering, S., 196, 231
