Neutron Fluctuations (eBook)

Front cover 1
Neutron Fluctuations 4
Copyright page 5
Contents 8
Preface 12
Acknowledgement 14
List of most frequently used notations 16
Part I. Physics of Branching Processes 20
Chapter 1. Basic Notions 22
1.1 Definitions 22
1.2 Equations for the Generating Functions 23
1.3 Investigation of the Generating Function Equations 30
1.4 Discrete Time Branching Processes 39
1.5 Random Tree as a Branching Process 42
1.6 Illustrative Examples 44
Chapter 2. Generalisation of the Problem 55
2.1 Joint Distribution of Particle Numbers at Different Time Instants 55
2.2 Branching Process with Two Particle Types 58
2.3 Extinction and Survival Probability 60
Chapter 3. Injection of Particles 74
3.1 Introduction 74
3.2 Distribution of the Number of Particles 76
3.3 Limit Probabilities 88
3.4 Probability of the Particle Number in a Nearly Critical System 96
Chapter 4. Special Probabilities 101
4.1 Preliminaries 101
4.2 The Probability of the Number of Absorptions 101
4.3 Probability of the Number of Detections 121
4.4 Probability of the Number of Renewals 126
4.5 Probability of the Number of Multiplications 132
Chapter 5. Other Characteristic Probabilities 138
5.1 Introduction 138
5.2 Distribution Function of the Survival Time 138
5.3 Number of Particles Produced by a Particle and Its Progeny 140
5.4 Delayed Multiplication of Particles 146
5.5 Process with Prompt and Delayed Born Particles 160
Chapter 6. Branching Processes in a Randomly Varying Medium 168
6.1 Characterisation of the Medium 169
6.2 Description of the Process 169
6.3 Factorial Moments, Variances 173
6.4 Random Injection of the Particles 184
Chapter 7. One-Dimensional Branching Process 198
7.1 Cell Model 198
7.2 Continuous model 210
Part II. Neutron Fluctuations 222
Chapter 8. Neutron Fluctuations in the Phase Space: The Pál–Bell Equation 224
8.1 Definitions 225
8.2 Derivation of the Equation 226
8.3 Expectation, Variance and Covariance 232
8.4 Pál–Bell Equation in the Diffusion Approximation 236
Chapter 9. Reactivity Measurement Methods in Traditional Systems 250
9.1 Preliminaries 250
9.2 Feynman-Alpha by the Forward Approach 253
9.3 Feynman-Alpha by the Backward Approach 259
9.4 Evaluation of the Feynman-Alpha Measurement 269
9.5 The Rossi-Alpha Method 272
9.6 Mogilner's Zero Probability Method 276
Chapter 10. Reactivity Measurements in Accelerator Driven Systems 278
10.1 Steady Spallation Source 279
10.2 Pulsed Poisson Source with Finite Pulse Width 283
10.3 Pulsed Compound Poisson Source with Finite Width 300
10.4 Periodic Instantaneous Pulses 302
Chapter 11. Theory of Multiplicity in Nuclear Safeguards 313
11.1 Neutron and Gamma Cascades 314
11.2 Basic Equations 317
11.3 Neutron Distributions 318
11.4 Gamma Photon Distributions 324
11.5 Joint Moments 328
11.6 Practical Applications: Outlook 330
Appendices 332
A. Elements of the Theory of Generating Functions 334
A.1 Basic Properties 334
A.2 On the Roots of Equation g(x)=x, 0 & #8804
A.3 A Useful Inequality 341
A.4 Abel Theorem for Moments 342
A.5 Series Expansion Theorem 344
A.6 An Important Theorem 346
B. Supplement to the Survival Probability 349
B.1 Asymptotic Form of Survival Probability in Discrete Time Process 349
Bibliography 354
Index 358
A 358
B 358
C 358
D 358
E 358
F 358
G 358
H 358
I 358
J 358
L 358
M 358
N 359
O 359
P 359
Q 359
R 359
S 359
T 359
V 359
Y 359
Z 359