Neutron Fluctuations

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ISBN: 9780080450643 | 0080450644
Cover: Hardcover
Copyright: 11/12/2007

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The transport of neutrons in a multiplying system is an area of branching processes with a clear formalism. This book presents an account of the mathematical tools used in describing branching processes, which are then used to derive a large number of properties of the neutron distribution in multiplying systems with or without an external source. In the second part of the book, the theory is applied to the description of the neutron fluctuations in nuclear reactor cores as well as in small samples of fissile material. The question of how to extract information about the system under study is discussed. In particular the measurement of the reactivity of subcritical cores, driven with various Poisson and non-Poisson (pulsed) sources, and the identification of fissile material samples, is illustrated. The book gives pragmatic information for those planning and executing and evaluating experiments on such systems. - Gives a complete treatise of the mathematics of branching particle processes, and in particular neutron fluctuations, in a self-contained manner; - The first monograph containing the theory and application of neutron fluctuations in low power ADS (spallation and pulsed sources); - Suitable as a tutorial and handbook/reference book for scientists and graduate students; - One of the authors is the founder of the mathematical theory of neutron fluctuations in zero power systems.

Preface	p. xi
Acknowledgement	p. xiii
List of most frequently used notations	p. xv
Physics of Branching Processes	p. 1
Basic Notions	p. 3
Definitions	p. 3
Equations for the Generating Functions	p. 4
Intuitive solution	p. 4
Solution according to Kolmogorov and Dmitriev	p. 7
Investigation of the Generating Function Equations	p. 11
Uniqueness of the solution, regular and irregular branching processes	p. 11
Moments: subcritical, critical and supercritical systems	p. 15
Semi-invariants	p. 17
Discrete Time Branching Processes	p. 20
Random Tree as a Branching Process	p. 23
Illustrative Examples	p. 25
Regular processes	p. 26
Explosive process	p. 31
Modelling of branching processes	p. 32
Generalisation of the Problem	p. 36
Joint Distribution of Particle Numbers at Different Time Instants	p. 36
Autocorrelation function of the particle number	p. 37
Branching Process with Two Particle Types	p. 39
Extinction and Survival Probability	p. 41
Asymptotic forms of the survival probability	p. 43
Special limit distribution theorems	p. 46
Injection of Particles	p. 55
Introduction	p. 55
Distribution of the Number of Particles	p. 57
Expectation, variance and correlation	p. 61
Limit Probabilities	p. 69
Subcritical process	p. 69
Critical process	p. 72
Supercritical process	p. 74
Probability of the Particle Number in a Nearly Critical System	p. 77
Preparations	p. 77
Equations of semi-invariants	p. 78
Determination of the approximate formula	p. 80
Special Probabilities	p. 82
Preliminaries	p. 82
The Probability of the Number of Absorptions	p. 82
Expectation of the number of absorptions	p. 86
Variance of the number of absorptions	p. 88
Correlation between the numbers of absorptions	p. 91
The probability of no absorption events occurring	p. 97
Probability of the Number of Detections	p. 102
One-point distribution of the number of detected particles	p. 103
Two-point distribution of the number of detected particles	p. 105
Probability of the Number of Renewals	p. 107
Expectation and variance of the number of renewals	p. 108
Correlation function of the number of renewals	p. 111
Probability of the Number of Multiplications	p. 113
Expectation and variance of the number of multiplications	p. 114
Correlation function of the number of multiplications	p. 116
Other Characteristic Probabilities	p. 119
Introduction	p. 119
Distribution Function of the Survival Time	p. 119
Number of Particles Produced by a Particle and Its Progeny	p. 121
Quadratic process	p. 123
Delayed Multiplication of Particles	p. 127
Expectations and their properties	p. 128
The covariance and its properties	p. 132
Properties of the variances	p. 135
Probability of extinction	p. 139
Process with Prompt and Delayed Born Particles	p. 141
Expectations	p. 143
Variances	p. 146
Branching Processes in a Randomly Varying Medium	p. 149
Characterisation of the Medium	p. 150
Description of the Process	p. 150
Backward equations	p. 151
Forward equations	p. 153
Factorial Moments, Variances	p. 154
The first factorial moments	p. 154
Properties	p. 155
Second factorial moments	p. 160
Variances	p. 161
Random Injection of the Particles	p. 165
Derivation of the forward equation	p. 165
Expectations, variances, covariances	p. 166
One-Dimensional Branching Process	p. 179
Cell Model	p. 179
Description of the model	p. 179
Generating function equations	p. 180
Investigation of the expectations	p. 182
Continuous model	p. 191
Generating function equations	p. 191
Investigation of the expectations	p. 192
Neutron Fluctuations	p. 203
Neutron Fluctuations in the Phase Space: The Pal-Bell Equation	p. 205
Definitions	p. 206
Derivation of the Equation	p. 207
The probability of no reaction	p. 208
Probabilities of the reactions	p. 208
Partial probabilities	p. 209
Generating function equation	p. 211
Distribution of neutron numbers in two disjoint phase domains	p. 212
Expectation, Variance and Covariance	p. 213
Expectation of the number of neutrons	p. 213
Variance of the number of neutrons	p. 214
Covariance between particle numbers	p. 215
Pal-Bell Equation in the Diffusion Approximation	p. 217
Derivation of the equation	p. 217
Expectation, variance and correlation	p. 220
Analysis of a one-dimensional system	p. 223
Reactivity Measurement Methods in Traditional Systems	p. 231
Preliminaries	p. 231
Feynman-Alpha by the Forward Approach	p. 234
First moments	p. 236
Second moments	p. 236
The variance to mean or Feynman-alpha formula	p. 238
Feynman-Alpha by the Backward Approach	p. 240
Preliminaries	p. 240
Relationship between the single-particle and source-induced distributions	p. 242
Calculation of the single-particle moments	p. 243
Calculation of the variance to mean	p. 247
Feynman-alpha formula with six delayed neutron groups	p. 248
Evaluation of the Feynman-Alpha Measurement	p. 250
The Rossi-Alpha Method	p. 253
Mogilner's Zero Probability Method	p. 257
Reactivity Measurements in Accelerator Driven Systems	p. 259
Steady Spallation Source	p. 260
Feynman-alpha with a steady spallation source	p. 260
Rossi-alpha with a steady spallation source	p. 263
Pulsed Poisson Source with Finite Pulse Width	p. 264
Source properties and pulsing methods	p. 265
Calculation of the factorial moments for arbitrary pulse shapes and pulsing methods	p. 268
General calculation of the variance to mean with arbitrary pulse shapes and pulsing methods	p. 269
Treatment of the pulse shapes and pulsing methods	p. 272
Rossi-alpha with pulsed Poisson source	p. 277
Pulsed Compound Poisson Source with Finite Width	p. 281
Periodic Instantaneous Pulses	p. 283
Feynman-alpha with deterministic pulsing	p. 283
Feynman-alpha with stochastic pulsing	p. 287
Rossi-alpha with stochastic pulsing	p. 290
Theory of Multiplicity in Nuclear Safeguards	p. 294
Neutron and Gamma Cascades	p. 295
Notations	p. 295
Basic Equations	p. 298
Neutron Distributions	p. 299
Factorial moments	p. 300
Number distribution of neutrons	p. 301
Statistics of emitted and detected neutrons	p. 303
Gamma Photon Distributions	p. 305
Factorial moments	p. 305
Number distribution of gamma photons	p. 306
The statistics of detected gamma photons	p. 307
Joint Moments	p. 309
Practical Applications: Outlook	p. 311
Appendices	p. 313
Elements of the Theory of Generating Functions	p. 315
Basic Properties	p. 315
Continuity theorems	p. 315
Generating function of the sum of discrete random variables	p. 320
On the Roots of Equation g(x) = x, o [less than or equal] x [less than or equal] 1	p. 321
A Useful Inequality	p. 322
Abel Theorem for Moments	p. 323
Series Expansion Theorem	p. 325
An Important Theorem	p. 327
Supplement to the Survival Probability	p. 330
Asymptotic Form of Survival Probability in Discrete Time Process	p. 330
The first step of the proof	p. 331
The second step of the proof	p. 331
Bibliography	p. 335
Index	p. 339
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