Neutron fluctuations: a treatise on the physics of branching processes
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier
2008
|
| Edition: | Reprinted |
| Subjects: | |
| Online Access: | Publisher description Inhaltsverzeichnis |
| Item Description: | Includes bibliography (p. [335]-337) and index |
| Physical Description: | xvii, 340 p. ill., plate |
| ISBN: | 9780080450643 0080450644 |
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MARC
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| 100 | 1 | |a Pázsit, Imre |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Neutron fluctuations |b a treatise on the physics of branching processes |c Imre Pázsit ; Lénárd Pál |
| 250 | |a Reprinted | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2008 | |
| 300 | |a xvii, 340 p. |b ill., plate | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliography (p. [335]-337) and index | ||
| 650 | 4 | |a Branching processes | |
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Record in the Search Index
| _version_ | 1804137910223503360 |
|---|---|
| adam_text | Contents
Preface
x¡
Acknowledgement
x¡¡¡
List of most frequently used notations
xv
I. Physics of Branch ing Processes
ι
1. Basic Notions
3
1.1
Definitions
3
1.2
Equations for the Generating Functions
4
1.2.1
Intuitive solution
4
1.2.2
Solution according to Kolmogorov and Dmitriev
7
1.3
Investigation of the Generating Function Equations
11
1.3.1
Uniqueness of the solution, regular and irregular branching processes
11
1.3.2
Moments:
subcriticai,
critical and supercritical systems
15
1.3.3
Semi-invariants
17
1.4
Discrete Time Branching Processes
20
1.5
Random Tree as a Branching Process
23
1.6
Illustrative Examples
25
1.6.1
Regular processes
26
1.6.2
Explosive process
31
1.6.3
Modelling of branching processes
32
2.
Generalisation of the Problem
36
2.1
Joint Distribution of Particle Numbers at Different Time Instants
36
2.1.1
Autocorrelation function of the particle number
37
2.2
Branching Process with Two Particle Types
39
2.3
Extinction and Survival Probability
41
2.3.1
Asymptotic forms of the survival probability
43
2.3.2
Special limit distribution theorems
* 46
3.
Injection of Particles
55
3.1
Introduction
55
3.2
Distribution of the Number of Particles
57
3.2.1
Expectation, variance and correlation
61
3.3
Limit Probabilities
69
3.3.1
Subcriticai
process
69
3.3.2
Critical process
72
3.3.3
Supercritical process
74
3.4
Probability of the Particle Number in a Nearly Critical System
γγ
3.4.1
Preparations
77
vii
viii Contents
3.4.2
Equations
of semi-invariants
78
3.4.3
Determination of the approximate formula
80
4.
Special Probabilities
82
4.1
Preliminaries
82
4.2
The Probability of the Number of Absorptions
82
4.2.1
Expectation ofthenumberof absorptions
86
4.2.2
Variance of the number of absorptions
88
4.2.3
Correlation between the numbers of absorptions
91
4.2.4
The probability of no absorption events occurring
97
4.3
Probability of the Number of Detections
102
4.3.1
One-point distribution of the number of detected particles
103
4.3.2
Two-point distribution of the number of detected particles
105
4.4
Probability of the Number of Renewals
107
4.4.1
Expectation and variance of the number of renewals
108
4.4.2
Correlation function of the number of renewals
111
4.5
Probability ofthe Number of Multiplications
113
4.5.1
Expectation and variance ofthe number of multiplications
114
4.5.2
Correlation function ofthe number of multiplications
116
5.
Other Characteristic Probabilities
119
5.1
Introduction
119
5.2
Distribution Function ofthe Survival Time
119
5.3
Number of Particles Produced by a Particle and Its Progeny
121
5.3.1
Quadratic process
123
5.4
Delayed Multiplication of Particles
127
5.4.1
Expectations and their properties
128
5.4.2
The covariance and its properties
132
5.4.3
Properties ofthe variances
135
5.4.4
Probability of extinction
139
5.5
Process with Prompt and Delayed Born Particles
141
5.5.1
Expectations
143
5.5.2
Variances
146
6.
Branching Processes in a Randomly Varying Medium
149
6.1
Characterisation ofthe Medium
150
6.2
Description ofthe Process
150
6.2.1
Backward equations
151
6.2.2
Forward equations
153
6.3
Factorial Moments, Variances
154
6.3.1
The first factorial moments
154
6.3.2
Properties 155
6.3.3
Second factorial moments
160
6.3.4
Variances
іб!
6.4
Random Injection ofthe Particles
!б5
6.4.1
Derivation ofthe forward equation
165
6.4.2
Expectations, variances, covariances
!бб
Contents
¡x
7.
One-Dimensional Branching Process
179
7.1
Cell Model
179
7.1.1
Description of the model
179
7.1.2
Generating function equations
180
7.1.3
Investigation of the expectations
182
7.2
Continuous model
191
7.2.1
Generating function equations
191
7.2.2
Investigation of the expectations
192
II. Neutron Fluctuations
203
8.
Neutron Fluctuations in the Phase Space: The
Pál-Bell
Equation
205
8.1
Definitions
206
8.2
Derivation of the Equation
207
8.2.1
The probability of no reaction
208
8.2.2
Probabilities of the reactions
208
8.2.3
Partial probabilities
209
8.2.4
Generating function equation
211
8.2.5
Distribution of neutron numbers in two disjoint phase domains
212
8.3
Expectation, Variance and Covariance
213
8.3.1
Expectation of the number of neutrons
213
8.3.2
Variance of the number of neutrons
214
8.3.3
Covariance between particle numbers
215
8.4
Pál-Bell
Equation in the Diffusion Approximation
217
8.4.1
Derivation of the equation
217
8.4.2
Expectation, variance and correlation
220
8.4.3
Analysis of a one-dimensional system
223
9.
Reactivity Measurement Methods in Traditional Systems
231
9.1
Preliminaries
231
9.2
Feynman-Alpha by the Forward Approach
234
9.2.1
First moments
236
9.2.2
Second moments
236
9.2.3
The variance to mean or Feynman-alpha formula
238
9.3
Feynman-Alpha by the Backward Approach
240
9.3.1
Preliminaries
240
9.3.2
Relationship between the single-particle and source-induced distributions
242
9.3.3
Calculation of the single-particle moments
243
9.3.4
Calculation of the variance to mean
247
9.3.5
Feynman-alpha formula with six delayed neutron groups
248
9.4
Evaluation of the Feynman-Alpha Measurement
250
9.5
The Rossi-Alpha Method
253
9.6
Mogilners
Zero Probability Method
257
10.
Reactivity Measurements in Accelerator Driven Systems
259
10.1
Steady Spallation Source
260
10.1.1
Feynman-alpha with a steady spallation source
260
10.1.2
Rossi-alpha with a steady spallation source
263
x
Contents
1Ο.2
Pulsed
Poisson
Source with Finite Pulse Width
264
10.2.1
Source properties and pulsing methods
265
10.2.2
Calculation of the factorial moments for arbitrary pulse shapes and pulsing methods
268
10.2.3
General calculation of the variance to mean with arbitrary pulse shapes and
pulsing methods
269
10.2.4
Treatment of the pulse shapes and pulsing methods
272
10.2.5
Rossi-alpha with pulsed
Poisson
source
277
10.3
Pulsed Compound
Poisson
Source with Finite Width
281
10.4
Periodic Instantaneous Pulses
283
10.4.1
Feynman-alpha with deterministic pulsing
283
10.4.2
Feynman-alpha with stochastic pulsing
287
10.4.3
Rossi-alpha with stochastic pulsing
290
11.
Theory of Multiplicity in Nuclear Safeguards
294
11.1
Neutron and Gamma Cascades
295
11.1.1
Notations
295
11.2
Basic Equations
298
11.3
Neutron Distributions
299
11.3.1
Factorial moments
300
11.3.2
Number distribution of neutrons
301
11.3.3
Statistics of emitted and detected neutrons 3°3
11.4
Gamma Photon Distributions 3°5
11.4.1
Factorial moments
305
11.4.2
Number distribution of gamma photons
306
11.4.3
The statistics of detected gamma photons
307
11.5
Joint Moments
309
11.6
Practica
I Applications: Outlook
311
Appendices
313
A. Elements of the Theory of Generating Functions
315
Ал
Basic Properties
315
A.1.1 Continuity theorems
315
A.1.2 Generating function of the sum of discrete random variables
320
A.2 On the Roots of Equation g(x)=x,
ο <χ<ι
321
A.3 A Useful Inequality
322
A.4 Abel Theorem for Moments
323
A.5 Series Expansion Theorem
325
A.6 An Important Theorem
327
B. Supplement to the Survival Probability
330
B.i Asymptotic Form of Survival Probability in Discrete Time Process
330
B.1.1 The first step of the proof
331
B.1.2 The second step of the proof
331
Bibliography
335
Index
339
|
| adam_txt |
Contents
Preface
x¡
Acknowledgement
x¡¡¡
List of most frequently used notations
xv
I. Physics of Branch ing Processes
ι
1. Basic Notions
3
1.1
Definitions
3
1.2
Equations for the Generating Functions
4
1.2.1
Intuitive solution
4
1.2.2
Solution according to Kolmogorov and Dmitriev
7
1.3
Investigation of the Generating Function Equations
11
1.3.1
Uniqueness of the solution, regular and irregular branching processes
11
1.3.2
Moments:
subcriticai,
critical and supercritical systems
15
1.3.3
Semi-invariants
17
1.4
Discrete Time Branching Processes
20
1.5
Random Tree as a Branching Process
23
1.6
Illustrative Examples
25
1.6.1
Regular processes
26
1.6.2
Explosive process
31
1.6.3
Modelling of branching processes
32
2.
Generalisation of the Problem
36
2.1
Joint Distribution of Particle Numbers at Different Time Instants
36
2.1.1
Autocorrelation function of the particle number
37
2.2
Branching Process with Two Particle Types
39
2.3
Extinction and Survival Probability
41
2.3.1
Asymptotic forms of the survival probability
43
2.3.2
Special limit distribution theorems
* 46
3.
Injection of Particles
55
3.1
Introduction
55
3.2
Distribution of the Number of Particles
57
3.2.1
Expectation, variance and correlation
61
3.3
Limit Probabilities
69
3.3.1
Subcriticai
process
69
3.3.2
Critical process
72
3.3.3
Supercritical process
74
3.4
Probability of the Particle Number in a Nearly Critical System
γγ
3.4.1
Preparations
77
vii
viii Contents
3.4.2
Equations
of semi-invariants
78
3.4.3
Determination of the approximate formula
80
4.
Special Probabilities
82
4.1
Preliminaries
82
4.2
The Probability of the Number of Absorptions
82
4.2.1
Expectation ofthenumberof absorptions
86
4.2.2
Variance of the number of absorptions
88
4.2.3
Correlation between the numbers of absorptions
91
4.2.4
The probability of no absorption events occurring
97
4.3
Probability of the Number of Detections
102
4.3.1
One-point distribution of the number of detected particles
103
4.3.2
Two-point distribution of the number of detected particles
105
4.4
Probability of the Number of Renewals
107
4.4.1
Expectation and variance of the number of renewals
108
4.4.2
Correlation function of the number of renewals
111
4.5
Probability ofthe Number of Multiplications
113
4.5.1
Expectation and variance ofthe number of multiplications
114
4.5.2
Correlation function ofthe number of multiplications
116
5.
Other Characteristic Probabilities
119
5.1
Introduction
119
5.2
Distribution Function ofthe Survival Time
119
5.3
Number of Particles Produced by a Particle and Its Progeny
121
5.3.1
Quadratic process
123
5.4
Delayed Multiplication of Particles
127
5.4.1
Expectations and their properties
128
5.4.2
The covariance and its properties
132
5.4.3
Properties ofthe variances
135
5.4.4
Probability of extinction
139
5.5
Process with Prompt and Delayed Born Particles
141
5.5.1
Expectations
143
5.5.2
Variances
146
6.
Branching Processes in a Randomly Varying Medium
149
6.1
Characterisation ofthe Medium
150
6.2
Description ofthe Process
150
6.2.1
Backward equations
151
6.2.2
Forward equations
153
6.3
Factorial Moments, Variances
154
6.3.1
The first factorial moments
154
6.3.2
Properties 155
6.3.3
Second factorial moments
160
6.3.4
Variances
іб!
6.4
Random Injection ofthe Particles
!б5
6.4.1
Derivation ofthe forward equation
165
6.4.2
Expectations, variances, covariances
!бб
Contents
¡x
7.
One-Dimensional Branching Process
179
7.1
Cell Model
179
7.1.1
Description of the model
179
7.1.2
Generating function equations
180
7.1.3
Investigation of the expectations
182
7.2
Continuous model
191
7.2.1
Generating function equations
191
7.2.2
Investigation of the expectations
192
II. Neutron Fluctuations
203
8.
Neutron Fluctuations in the Phase Space: The
Pál-Bell
Equation
205
8.1
Definitions
206
8.2
Derivation of the Equation
207
8.2.1
The probability of no reaction
208
8.2.2
Probabilities of the reactions
208
8.2.3
Partial probabilities
209
8.2.4
Generating function equation
211
8.2.5
Distribution of neutron numbers in two disjoint phase domains
212
8.3
Expectation, Variance and Covariance
213
8.3.1
Expectation of the number of neutrons
213
8.3.2
Variance of the number of neutrons
214
8.3.3
Covariance between particle numbers
215
8.4
Pál-Bell
Equation in the Diffusion Approximation
217
8.4.1
Derivation of the equation
217
8.4.2
Expectation, variance and correlation
220
8.4.3
Analysis of a one-dimensional system
223
9.
Reactivity Measurement Methods in Traditional Systems
231
9.1
Preliminaries
231
9.2
Feynman-Alpha by the Forward Approach
234
9.2.1
First moments
236
9.2.2
Second moments
236
9.2.3
The variance to mean or Feynman-alpha formula
238
9.3
Feynman-Alpha by the Backward Approach
240
9.3.1
Preliminaries
240
9.3.2
Relationship between the single-particle and source-induced distributions
242
9.3.3
Calculation of the single-particle moments
243
9.3.4
Calculation of the variance to mean
247
9.3.5
Feynman-alpha formula with six delayed neutron groups
248
9.4
Evaluation of the Feynman-Alpha Measurement
250
9.5
The Rossi-Alpha Method
253
9.6
Mogilners
Zero Probability Method
257
10.
Reactivity Measurements in Accelerator Driven Systems
259
10.1
Steady Spallation Source
260
10.1.1
Feynman-alpha with a steady spallation source
260
10.1.2
Rossi-alpha with a steady spallation source
263
x
Contents
1Ο.2
Pulsed
Poisson
Source with Finite Pulse Width
264
10.2.1
Source properties and pulsing methods
265
10.2.2
Calculation of the factorial moments for arbitrary pulse shapes and pulsing methods
268
10.2.3
General calculation of the variance to mean with arbitrary pulse shapes and
pulsing methods
269
10.2.4
Treatment of the pulse shapes and pulsing methods
272
10.2.5
Rossi-alpha with pulsed
Poisson
source
277
10.3
Pulsed Compound
Poisson
Source with Finite Width
281
10.4
Periodic Instantaneous Pulses
283
10.4.1
Feynman-alpha with deterministic pulsing
283
10.4.2
Feynman-alpha with stochastic pulsing
287
10.4.3
Rossi-alpha with stochastic pulsing
290
11.
Theory of Multiplicity in Nuclear Safeguards
294
11.1
Neutron and Gamma Cascades
295
11.1.1
Notations
295
11.2
Basic Equations
298
11.3
Neutron Distributions
299
11.3.1
Factorial moments
300
11.3.2
Number distribution of neutrons
301
11.3.3
Statistics of emitted and detected neutrons 3°3
11.4
Gamma Photon Distributions 3°5
11.4.1
Factorial moments
305
11.4.2
Number distribution of gamma photons
306
11.4.3
The statistics of detected gamma photons
307
11.5
Joint Moments
309
11.6
Practica
I Applications: Outlook
311
Appendices
313
A. Elements of the Theory of Generating Functions
315
Ал
Basic Properties
315
A.1.1 Continuity theorems
315
A.1.2 Generating function of the sum of discrete random variables
320
A.2 On the Roots of Equation g(x)=x,
ο <χ<ι
321
A.3 A Useful Inequality
322
A.4 Abel Theorem for Moments
323
A.5 Series Expansion Theorem
325
A.6 An Important Theorem
327
B. Supplement to the Survival Probability
330
B.i Asymptotic Form of Survival Probability in Discrete Time Process
330
B.1.1 The first step of the proof
331
B.1.2 The second step of the proof
331
Bibliography
335
Index
339 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Pázsit, Imre Pál, Lénárd |
| author_facet | Pázsit, Imre Pál, Lénárd |
| author_role | aut aut |
| author_sort | Pázsit, Imre |
| author_variant | i p ip l p lp |
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| callnumber-subject | QC - Physics |
| classification_rvk | UG 3900 UO 6110 |
| ctrlnum | (OCoLC)254584660 (DE-599)BVBBV023483233 |
| dewey-full | 621.483101519234 621.48/3101519234 |
| dewey-hundreds | 600 - Technology (Applied sciences) |
| dewey-ones | 621 - Applied physics |
| dewey-raw | 621.483101519234 621.48/3101519234 |
| dewey-search | 621.483101519234 621.48/3101519234 |
| dewey-sort | 3621.483101519234 |
| dewey-tens | 620 - Engineering and allied operations |
| discipline | Physik Energietechnik |
| discipline_str_mv | Physik Energietechnik |
| edition | Reprinted |
| format | Book |
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| id | DE-604.BV023483233 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:38:45Z |
| indexdate | 2024-07-09T21:19:48Z |
| institution | BVB |
| isbn | 9780080450643 0080450644 |
| language | English |
| lccn | 2008295680 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016665323 |
| oclc_num | 254584660 |
| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | xvii, 340 p. ill., plate |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Elsevier |
| record_format | marc |
| spelling | Pázsit, Imre Verfasser aut Neutron fluctuations a treatise on the physics of branching processes Imre Pázsit ; Lénárd Pál Reprinted Amsterdam [u.a.] Elsevier 2008 xvii, 340 p. ill., plate txt rdacontent n rdamedia nc rdacarrier Includes bibliography (p. [335]-337) and index Branching processes Neutrons Fluktuation Physik (DE-588)4130671-5 gnd rswk-swf Neutron (DE-588)4041964-2 gnd rswk-swf Verzweigungsprozess (DE-588)4188184-9 gnd rswk-swf Neutron (DE-588)4041964-2 s Fluktuation Physik (DE-588)4130671-5 s Verzweigungsprozess (DE-588)4188184-9 s DE-604 Pál, Lénárd Verfasser aut http://www.loc.gov/catdir/enhancements/fy0833/2008295680-d.html Publisher description Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016665323&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pázsit, Imre Pál, Lénárd Neutron fluctuations a treatise on the physics of branching processes Branching processes Neutrons Fluktuation Physik (DE-588)4130671-5 gnd Neutron (DE-588)4041964-2 gnd Verzweigungsprozess (DE-588)4188184-9 gnd |
| subject_GND | (DE-588)4130671-5 (DE-588)4041964-2 (DE-588)4188184-9 |
| title | Neutron fluctuations a treatise on the physics of branching processes |
| title_auth | Neutron fluctuations a treatise on the physics of branching processes |
| title_exact_search | Neutron fluctuations a treatise on the physics of branching processes |
| title_exact_search_txtP | Neutron fluctuations a treatise on the physics of branching processes |
| title_full | Neutron fluctuations a treatise on the physics of branching processes Imre Pázsit ; Lénárd Pál |
| title_fullStr | Neutron fluctuations a treatise on the physics of branching processes Imre Pázsit ; Lénárd Pál |
| title_full_unstemmed | Neutron fluctuations a treatise on the physics of branching processes Imre Pázsit ; Lénárd Pál |
| title_short | Neutron fluctuations |
| title_sort | neutron fluctuations a treatise on the physics of branching processes |
| title_sub | a treatise on the physics of branching processes |
| topic | Branching processes Neutrons Fluktuation Physik (DE-588)4130671-5 gnd Neutron (DE-588)4041964-2 gnd Verzweigungsprozess (DE-588)4188184-9 gnd |
| topic_facet | Branching processes Neutrons Fluktuation Physik Neutron Verzweigungsprozess |
| url | http://www.loc.gov/catdir/enhancements/fy0833/2008295680-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016665323&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT pazsitimre neutronfluctuationsatreatiseonthephysicsofbranchingprocesses AT pallenard neutronfluctuationsatreatiseonthephysicsofbranchingprocesses |