PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS

VOLUME XI

NUCLEAR REACTOR THEORY

AMERICAN MATHEMATICAL SOCIETY

190 HOPE STREET, PROVIDENCE, RHODE ISLAND

1961

http://dx.doi.org/10.1090/psapm/011

PROCEEDINGS OF THE ELEVENTH SYMPOSIUM IN APPLIED MATHEMATICS

OF THE AMERICAN MATHEMATICAL SOCIETY

Held at the Hotel New Yorker April 23-25, 1959

COSPONSORED BY THE OFFICE OF ORDNANCE RESEARCH

G a r r e t t Birkhoff a n d E u g e n e P . Wigne r

Library of Congress Catalog Card Number 50-1183

Prepared by the American Mathematical Society under Contract No. DA-19-020-ORD-4758 with the Ordnance Corps, U.S. Army.

Copyright © 1961 by the American Mathematical Society.

Printed in the United States of America. All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may not be reproduced in

any form without permission of the publishers.

CONTENTS

INTRODUCTION . . . . . . . . . . . v

Reactor Types . . . . . . . . . . 1 By ALVIN M. WEINBERG

Neutron Thermalization . . . . . . . . 20 By M. S. NELKIN

Deep Penetration of Radiation . . . . . . . 43 By U. FANO and M. J . BERGER

The Theory of Resonance Absorption . . . . . . 58 By L. W. NORDHEIM

Mathematical Problems of Nuclear Reactor Theory . . . 89 By EUGENE P. WIGNER

Diffusion Approximation to the Transport Equation . . . 1 0 5 By J . ERNEST WILKINS, Jr .

Positivity and Criticality . . . . . . . . 1 1 6 By GARRETT BIRKHOFF

Existence Theorems and Spectral Theory for the Multigroup Diffusion Model 127

By G. J . HABETLER and M. A. MARTINO

Transport Theory and Spectral Problems . . . . . 1 4 0 By G. MILTON W I N G

One-Dimensional Multigroup Calculations: Estimation of Group Constants . . . . . . . . . 151

By R. EHRLICH

Numerical Methods for Solving Multi-Dimensional Multigroup Diffusion Equations . . . . . . . . 1 6 4

By RICHARD S. VARGA

Monte Carlo Methods 190 By R. D. RICHTMYER

iii

iv CONTENTS

Transport Theory and Invariant Imbedding . . . . 206 By RICHARD BELLMAN and ROBERT KALABA

Numerical Solution of Neutron Transport Problems . . . 219 By BENGT CARLSON

Problems of Reactor Kinetics . . . . . . . 2 3 3 By HARRY SOODAK

Core Kinetics . . . . . . . . . . 2 5 6 By H. L. GARABEDIAN

Temperature Coefficients and Stability . . . . . . 2 8 9 By HARVEY BROOKS

System Kinetics . . . . . . . . . 309 By T. A. WELTON

Rearrangement Inequalities and Non-Linear Stability Criteria . 327 By W. K. ERGEN

EDITORS' PREFACE

The current era has been described as " the atomic age", and it seems probable that mankind will depend increasingly on nuclear energy during the next century. In the design of nuclear reactors, mathematical analysis already plays an important role.

Nevertheless, very few research mathematicians have so far devoted serious effort to the mathematical problems of nuclear reactor theory. The present volume is intended to increase the number of such mathematicians, by indicating the great variety of interesting mathematical problems encoun-tered in this fascinating field. As a by-product, it may help to put the design of future nuclear reactors on a more scientific basis.

The contributors to this volume having already done their part, we hope that both pure and applied mathematicians will accept the challenging invitation offered, thereby continuing the great tradition of Archimedes, Newton, Gauss, Fourier, Maxwell, Poineare, and many others. In this great tradition, each new major field of physical application has both suggested fundamental new mathematical concepts, and has owed much of its deeper development to the rigorous mathematical analysis of these concepts.

GARRETT BIRKHOFF

EUGENE P. WIGNER

INDEX

Accident studies, 254 Acoustical instabilities, 309, 319 Adjoint

eigenfunctions, 132-133 flux, 154, 261 operator, 130, 144

Adler, F. T., 58 Age theory, 107 Akcasu, Z., 252 Albedo, 69 Algebra, irreducible representation of,

99 Ambarzumian, 206 Amster, H. J., 37 Antithetic variates, 196 Asymptotic

flux, 102 flux shape, 282 method, semi , 49 shape, 278

Average, 194 chord length, 63 cosine of scattering angle, 109 lethargy loss per scattering collision,

109

Bade, W. G., 136 Banach lattice, 122-123 Bartels, W. J. C, 159 Bell, G. I., 78, 225 Bellman, R., 206 Berger, M. J., 18, 43 Bethe, H. A., 293 Binding, 35 Birkhoff, G., 5, 95, 99, 116,

166 Birkhoff-Varga theorem, 102 Blackness, 69 Blanket, 260 Bode, 291 Bohl, L. S., 160 Boiling, 295 du Bois, B., 24

Boltzmann equation, 6, 18, 22, 91, 93, 105-106 transport equations, 257

Born approximation, 28 Boundary conditions, 100, 107, 222

extrapolated, 165 Breit-Wigner formula, 58 Brockhouse, B. N., 21, 36 Brooks, H., 18, 91, 102, 156, 251, 254, 279,

289, 292, 310, 317, 328 Brown Recorder, 275 Buildup factor, 54 Burn out, 268

Calame, G. P., 156 Carlson, B., 18, 101, 151, 219 Carlvik, I., 76 Case, K. M., 159 Central-limit theorem, 195 Chandrasekhar, S., 206, 220, 226 Characteristic (See also Eigenvalue)

determinental equation, 281 value problem, 94 value problem, generalization of, 99

Chebyshev modified semi-iterative method,

179 polynomial, 155, 178, 180 semi-iterative method, 178

Chernick, J., 58, 70, 252 Chord length, average, 63 Coefficient

delayed, 297 delayed temperature, 296 Doppler temperature, 87 power, 289-290 prompt, 297-298 prompt positive, 298 steady state power, 290 temperature, 279, 289

Cohen, E. R., 271 Collatz, L., 170 Collision density, 60

reciprocity theorem for, 63

332 INDEX

Completeness, 95, 98, 271 Computing machines, 100 Conservation of neutrons, 23, 155, 219,

223-224 Consistently ordered, 173 Continuous spectrum, 95, 98, 145 Control rod, 267, 269

calibration, 267 Convergence, rate of, 171 Convergent, 171 Core, 260

kinetics, 256 Corngold, N., 41, 60 Cosine of scattering angle, 109 Courant, R., 153 Crank-Nicolson approximation, 185 Critical, 3, 117, 184

assembly, 191, 267 experiment, 100 length, 142, 206, 210, 215 radius, 14, 229 range of variables, 46 size, 14 zero power assembly, 271

Criticality, 116 constant, 3, 18 equation, 238

Cross section, 20-21, 23-24, 90-91, 158, 221

absorption, 58, 109 anisotropic, 91 differential energy-transfer, 29 effective absorption, 61 effective escape, 71 energy-transfer, 27, 39 for fission, 107 macroscopic, 105, 269 microscopic, 106 neutron scattering, 57 scattering, 58, 106 total, 106 transport, 109

Crystalline moderators, heavy, 31 Curlee, N. J., 244 Cyclic of index 2, 172-173

Dancoff, 74 Daneoff-Ginsburg correction, 76-77 Davison, B., 102, 114 Dayton, I. E., 86 De Broglie wavelengths, 28 Debye approximation, 33

distribution, 36

Delayed coefficient, 297 neutrons, 260, 311, 315-316 temperature coefficient, 296

Density, slowing down, 110, 272 Density function, moment, 216 Depletion codes, 161 Detailed balance condition, 22 Deutsch, R. W., 159 "Diamond" difference method, 222, 227 Difference equations, 222, 224, 257

finite, 155 Diffusion (See also Multigroup)

approximation, 219, 226 cooling, 25 constant, 257 equations, 257 theory, 17, 102, 151, 219-220, 223, 225-

226 Directions, 224 Disadvantage factors, 80 Discrete

directions, 221, 224 space matrix, 183

Distribution dominant (persistent), 116, 119 multivariate, 193 univariate, 193

Dominance ratio, 181 Dominant distribution, 116, 119 Doppler

broadening, 38, 59, 70, 80, 203 effect, 295-296 temperature coefficient, 87

Douglas-Rachford iterative method, 178 Dresner, L., 58, 69 Dynamic

behavior of nuclear reactors, 327 programming, 206

e-folding time, 259 EBR-1, 293, 298 (See Reactor) Edlund, M. C, 278 Effective delayed fraction, 240, 259 Ehrlich, R., 101, 105, 151, 164, 175 Eigenfunction, 23-24, 127, 129, 134, 143,

147 adjoint, 132-133 period, 271 principal, 130 quasi-positive, 133 reactivity, 271 spatial, 271

INDEX 333

Eigenvalue, 23-24, 130, 134, 137, 143, 147, 149, 154, 178, 180, 301

positive dominant, 132-133 positive, simple, 169 problems to initial-value problems, 206 simple, 119 time-independent problem, 133

Eigenvector, 169, 180 generalized, 138 positive, 119

Elimination of variable, 99 Elrod, H. G., 248 Energy

degradation, 43, 45 dependence, 213

Equivalence relations, 58, 72 Ergen, W. K., 252, 316, 327 Error, probable, 194 Escape probability, 63, 70-71, 79

rational approximation for, 71 Essentially positive, 135 (See also

Matrix) Euclidean group, 99 Excess prompt reactivity, 311 Excitation of modes, 241 Excursions, large, 299 Expectation, 194 Expected values, use of, 195 Experimental Boiling Water Reactor,

293 Exponential

distribution, 193 method, 226

Extrapolation distance, 16 linear, 102

Factoring, 174 Fano, XL, 18, 43 Fast effect, 102 Feedback, 250, 300, 306-307

function, 292-294, 298 non-linear, 294, 305

Feinberg, S. M., 15 Fermi

age-diffusion equation, 107, 109, 152 pseudo-potential approximation, 23

Feynman, R. P., 235 Fick's Law, 17 Field variable, 317 Finite difference equations, 155 Fission, iterated probability, 244 Fluctuations, 90, 235

Flux, 3, 44, 153, 207, 213, 261 adjoint, 154, 261 angular distribution, 17 asymptotic, 102 energy distributions, 7 neutron, 91 reflected and transmitted, 208, 211 spatial distributions, 12 time-dependent analyses of spatial

distributions, 277 total, 97 transform of, 46

Forsythe, 179 Fourier transform, inversion of, 48 Frankel, S. P., 172 Friedrichs, K., 153 Frobenius, G., 95, 120 (See Perron) Function fitting, 50 Fundamental mode, 302, 310

shape, 311

Garabedian, H. L., 18, 244, 256, 277-278 Gauss-Seidel method, 172 Gelbard, E., 155 General chain reactor equation, kernel

form, 1 Generation time, 5

prompt, 311 Geometry

cylindrical, 227 general, 218 plane, 227 spherical, 224, 227

Ginsburg, 74 (See Dancoff) Glauber, R. J., 29 Goertzel, G., 105, 196, 278 (See Selengut) Gourevich, I. I., 69 Graphite moderation, 34 Group, irreducible representation of,

99

Habetler, G. I., 5, 98, 120, 127, 166, 182 Hammersley, J. M., 196 Hardy, G. H., 327 Harmonics

method of, 278 spherical method, 219

Harris, T. E., 216 Helicity, 91 Hellstrand, E., 82 Helmholtz equation, 280

33 4 INDEX

Henry, A. F., 244 Hille-Yosida

theorem, 127, 146 theory, 136

Hinman, G. W., 58 de Hoffman, Frederic, 159, 235 Homogeneous

isotropic media, 44 (See Reactor)

Horowitz, 24 Howlett, J., 153 Hurwitz, H., Jr., 105, 153, 160, 248, 254,

292-293, 297 Hydrogenous moderators, 35

IM approximations, 67 Implicit alternating direction (IAD), 176-

177 Importance

sampling, 196 vector, 120

Indecomposable, 167 Infinite mass approximation, 68 Inhour equation, 259, 264 Inner iterations, 175, 186, 225-226 Inonu, 102 Instabilities

acoustical, 309, 319 thermal, 309

Interface conditions, 113 Interference scattering, 58 Invariance, 99 Invariant imbedding, 206 Irreducibility, 121 Irreducible, 167

linear manifolds, 99 representation of an algebra, 99 representation of a group, 99

Isotropic media, homogeneous, 44 Iterated fission probability, 244 Iterations (See Inner and Outer) Iterative method, Douglas-Rachford, 176,

178

Kahn, H., 195-196 Kalaba, R., 206 Kazarnovsky, M. V., 40 Kendall, D. G., 234 Kernel

migration, 116 positive, 5 slowing down, 272

Kinchin, D., 293 Kinetic

core, 25Q reactor, 18, 233 system, 279, 309 theory, 94

Kinetic equations, 290 generalized, 256, 259 primary, 257 solution of space-dependent, 271 space-independent, 265

King, J. S., 279 Kottwitz, D. A., 40 Krein, M. G., 127, 130 Krein-Rutman theorem, 127, 137

Lagrangian, 317 Large excursions, 299 Lehner, J., 121, 143 Length, critical, 142, 206, 210, 215 Lethargy, 8, 61, 66, 105, 152-153

groups, 164 loss per scattering collision, 109

Lewy, H., 153 Life history codes, 161 Line

absorption, 69 widths, 58

Linearity, 91 Linearized equations, 250 Linear system, three-term, 155 Linkage, 45 Lipkin, H. J., 252 Littlewood, J. E., 327

Jacobi matrix, 171, 173 method, 172

Jentzsch, 95 theorem of, 120, 127

Jorgens, K., 148 Judge, F. D., 160

Mandl, M. E., 153 Manifolds, irreducible linear, 99 Marshak, R. E., 105 Martino, M. A., 5, 98, 120, 127, 166 Matrix

diagonal, 128, 148, 169, 176, 185 discrete space, 183 (essentially) non-negative, 118

INDEX 335

Matrix (continued) (essentially) positive, 119, 135, 184-185 factorization, 158 irreducible, symmetric, and positive

definite, 168 Jacobi, 171, 173 non-negative, 118 Peaceman-Rachford, 177 positive, 119 primitive, 119 reducible, 119 reflection, 214 semi-irreducible, 119 statistical, 91 transmission, 214-215

Maxwellian distribution, 22, 26, 39 McGarvey, D., 210 Mean free path, 94 Metropolis, N. C, 190 Microscopic reversibility, principle of,

23 Migration kernel, 116 Minimax, 174 Mode, 300, 302-303

coupling, 303 excitation of, 241 persistent, 241 reactor, 236 (See Fundamental)

Moment, 46 conditions, 223 density function, 216 method, 49

Monte Carlo, 44, 63, 65, 75, 81, 190, 219

approach, 158 conditional, 196 differential effects in, 204 methods, 190 stochastic approach, 151

Morton, K. W., 196 Mullikin, T., 212 Multigroup, 220

diffusion model, 127 isotropic theory, 220, 224 one-dimensional calculations, 151 theory, 220-221

Multigroup diffusion equations, 105, 111, 127

multi-dimensional, 164 time-dependent, 137, 164 time-independent, 166

Multiple scattering, 206

Multiplication constant, 5, 238 factor, 286

Multiplicative process, 118, 121

NR approximation, 80 (See Resonance) n -velocity group

model, 148 theory, 149

Naimark, M. A., 128, 138 Nelkin, M. S., 11, 20, 107, 158 Neumann, H., 69 von Neumann, J., 190 Neutron

balance, 224 conservation, 23, 155, 219, 223-224 delayed, 260, 311, 315-316 distribution, 20 effect on the medium, 91 flux, 91 importance, 244 life cycle, 238 lifetime, 240 prompt, 272 scattering cross section, 57 slow scattering, 20 slow scattering from chemically

bound systems, 27 slow scattering properties of

water, 36 source, 272 temperature, 26, 295 thermalization, 20 velocity distribution in an infinite

homogeneous medium, 22 Neutron spectrum, 21

spatially dependent, 22 spatially independent, slow, 21

Neutron transport, 140, 219 in a one-dimensional rod, 140 in bounded geometries, 148 time-dependent in an infinite

slab, 142 Noderer, L. C, 278 Non-compact groups, 100 Non-linear, 295, 298

behavior, 308 equation, 91 feedback, 294, 305 oscillation, 298 problems, 252 stability, 327

336 INDEX

Non-linearity, 294 Nordheim, L. W., 11, 58, 94, 192, 204,

295 Normal

distribution function, 195 operators, 95, 99

Nyquist, 291, 303 diagram, 303 plot, 292

One parameter equation, 245 Operator, 144

adjoint, 130, 144 Optical reciprocity theorem, 117 Ordered, consistently, 173 Oscillation

non-linear, 298 small, 294 unstable, 297

Oscillator experiments, 304, 306 Outer iteration, 154, 175, 179, 181,

225 convergence of, 154

Parameter one equations, 245

Parks, D. E., 34 Peaeeman-Rachford

iterative method, 176, 178 matrix, 177 process, 177

Perron, O., 95, 120, 170 Perron-Frobenius theory, 166, 184 Pershagen, B., 76 Persistent distribution, 116, 119 Persisting distribution, 5 Perturbation, 239

theorem, 130 Pettus, W. G., 86 Phonon, 31

expansion, 33 Placzek, G., 4, 20, 159 Point spectrum, 145 Poison, 269 Polarization in collisions, 92 Polya, G., 327 Pomeranchouk, I. Y., 69 Poole, M. J., 37, 80 Porter-Thomas

distribution, 80 law, 203

Positive kernels, 5 uniformly, 124

Positivity, 116 Potter, R., 252 Power coefficients, 289-290 Precursor, 257 Programming, dynamic, 206 Prompt jump, 247

approximation, 248 Property (A), 172 Pseudo

-potential, 27 -potential approximation, 27 -random sequence, 190, 193

Quasi-primitive, uniformly, 124

Rachford, H. H., Jr. (See Douglas and Peaceman)

Radiation, deep penetration of, 43 Radiative transfer theory, 206 Radius

critical, 14, 229 spectral, 171

Randall, D., 244 Random

numbers, 193 walk, 206

Reactivity, 246 eigenfunctions, 271 excess prompt,* 311 generalized, 258 ramp change of, 265

Reactor aqueous homogeneous, 279, 309, 322 bare, 256 bare homogeneous, 116, 256 classification of, 8 dynamic behavior of nuclear, 327 dynamics, 289 EBR-1, 293 Experimental Boiling Water, 293 fast, 8 general chain equation—kernel

form, 1 general equation—Boltzmann

form, 6 heterogeneous, 15 homogeneous, 12 intermediate, 11 kinetics, 18, 233

INDEX 337

Reactor (continued) modes, 236 oscillator, 292-293 period, 265 reflected, 256 stable period, 259 statics, 289 subcritical, 279 thermal, 9 time-dependent problem for homo-

geneous, 133 transients, 300 types, 1

Rearrangement inequalities, 327 Reciprocity theorem for collision densi-

ties, 63 Reflector, 280 Relaxation factor, 173 Residual spectrum, 145 Resonance

capture, 191, 196 escape probability, 10-11, 192 narrow approximation (NR), 67 P, 80 scattering, 203 unresolved, 80

Resonance absorption, 58 for homogeneous mixtures, 60 in closely packed assemblies, 74 in heterogeneous systems, 62

Resonance integrals, 58, 79 for Th metal, 85 for Th02, 86 for U238 metal, 83 for U23802, 84

Richtmyer, R. D., 81, 101, 151, 190. (See M. G. Krein)

Roe, G. M., 160 Rosenberg, R. L., 172 Rothestein, W., 75 Routh, 297 Russian roulette and splitting, 196 Rutnam, M. A., 130

Sn approach, 222 difference equations, 221-224 difference method, 221 method, 18, 219-220, 226-227

Safanov, G., 105 St. John, D. S., 244 Sample

mean, 194

variance, 194 Sampling

importance, 196 model, 190 weighted, 195

Sampson, J. B., 70, 78 Scattering

angle, 109 cross sections, 58, 106 interference, 58 ker k (co, A; a/, A'), 44 multiple, 206

Schermer, R., 60 Schofield, P., 33 Schultz, M. A., 252 Seidel, 172 Selengut-Goertzel approximation, 152 Self

-adjoint, 144 -shielding, 82 -shielding factor, 159

Semi -asymptotic method, 49 -group, 146, 206

Serber, R., 235 Shape functions, 59 Sheldon, J. W., 182 Shielding calculations, 193 Shut-down, 268 Sigel, R., 293 Simmons, B. E., 279 Single-step method, 172 Size, critical, 14 Sjostrand, N. G., 279 Slowing down

density, 110, 272 equations, 65 integration of equations, 65 kernel, 272

Snow, R. N., 212 Soodak, H., 18, 91, 233, 256 Spectral

problem, 98, 140 radius, 171 theory for one-dimensional models, 138

Spectrum continuous, 95, 98, 145 neutron, 21 point, 145 residual, 145

Spherical harmonics expansion, 18 method, 219

338 INDEX

Spin, 91 Spinney, K. T., 70 Splitting, 196 Stability, 250, 279, 289, 291-293, 296, 315,

327 criteria, 313 non-linear, 327 small-oscillation, 292 sufficient condition for, 315

Stabilization time, 284 Stable, 298

inverse period, 247 "Star" method, 157 Stark R. H., 157 Static criticality factor, 238 Steady state, 221, 224-226

power coefficient, 290 Stein, P., 172 Stieltjes, T. J., 168 Stochastic

aspects, 233 initial-value problem, 191 process, 118 (See Monte Carlo)

Stokes relation, 210, 215 Straight ahead approximation, 48 Straus, 179 Subcritical, 3, 117, 184 Successive

line overrelaxation method (SLOR), 176 overrelaxation iterative method, 173 overrelaxation method (point) (SOR),

176-177, 179 Supercritical, 3, 117, 184 Supercriticality, 211 Surface

absorber, 103 scatterer, 103

Symmetry, spherical, 221, 224 System kinetics, 279, 309

Teller, E., 15 Temperature

coefficients, 279, 289 delayed coefficient, 296 profile, 279

Thermal expansion, 295 instabilities, 309 reactors, 9 utilization, 15

Thieberger, R., 252 Thomas, R. G. (See Porter)

Time dependence, 216 Time-dependent problem, one-dimen-

sional, 140 Total-step method, 172 Trammel, G. T., 319 Transfer function, 250, 291-293 Transference function, 93 Transformation properties, 99 Transient, 102, 224, 265

phenomena, 221 reactor, 300 solution, 265

Transport cross sections, 109 equation, 44, 90, 96, 105, 219-221, 223-

224, 226-228 in an infinite slab, 140 problem, 43 theory, 90, 92, 140, 206, 219-221 (See Neutron)

Tridiagonal, 174 Trotter, H. F., 196 Tukey, J. W., 196

Ulam, S., 190 Uniformly

positive, 124 quasi-primitive, 124

Ussachoff, L. N., 271

Van Hove, L., 20 Varga, R. S., 5, 95, 99, 101, 128, 151, 154,

164 (See also Birkhoff) Variable

critical range of, 46 elimination of, 99 field, 317

Variance-reducing techniques, 195 Variates, antithetic, 196 Variational

methods, 101 principles, 206

Velocity distribution following a fast neutron pulse, 22 in an infinite homogeneous medium, 22 in a reactor, 26 infinite medium, 27 spatially dependent, 39

Vernon, R., 58, 70 Volkoff, G., 4 Von Dardel, G., 26, 31

INDEX 339

Wachspress, E. L., 159, 182 Wallach, S., 248 Wanda Code, 284 Ward, A. G., 244 Water, slow neutron scattering properties

of, 36 Wave propagation, 206 Weight function, 50 Weinberg, A. M., 1, 95, 151-152, 161, 238,

242, 252 Welton, T. A., 18, 91, 252, 279, 309, 327

(See Breit) Wick, G. C, 47 Wick-Chandrasekhar method, 220, 226 Wick equation, 47 Width, practical, 70 Wielandt's inverse power method, 182 Wigner, E. P., 18, 67, 70-71, 89, 152, 192,

238, 242 Wigner-Wilkins equation, 158

Wikner, N. F., 34 Wilkins, J. E., Jr., 105, 151, 153, 248, 266 Wing, G. M., 5, 98, 121, 140

Xenon maximum, 268 transient, 268-269 weighting factor, 270

Yosida (See Hille) Young, D. M., Jr., 172 Yvon modification, 220

method, 226

Zemach, A. C, 29 Zirconium hydride, moderating properties

of, 37