Theoretical analysis of 2n accidentals rates - Revision history

Burgjeff: /* 1. Poissonian statistics */

2018-01-25T20:57:11Z

1. Poissonian statistics

Burgjeff: /* 1. Poissonian statistics */

2018-01-25T20:55:38Z

1. Poissonian statistics

Burgjeff: /* Reference of variables */

2018-01-25T20:54:11Z

Reference of variables

Burgjeff: /* Reference of variables */

2018-01-25T20:53:48Z

Reference of variables

Burgjeff: /* Reference of variables */

2018-01-25T20:53:30Z

Reference of variables

Burgjeff: /* Reference of variables */

2018-01-25T20:53:17Z

Reference of variables

Burgjeff: /* Same pulses */

2018-01-25T20:51:37Z

Same pulses

Burgjeff: /* Same pulses */

2018-01-25T20:51:15Z

Same pulses

Burgjeff: /* Same pulses */

2018-01-25T20:44:23Z

Same pulses

Burgjeff: /* Same pulses */

2018-01-25T20:42:54Z

Same pulses

@@ Line 69: / Line 69: @@
 The number of neutron-producing reactions occurring per pulse, <math>n</math>, is assumed to follow the Poissonian distribution.
 It can be seen from data, at least, that the number of photons per pulse follows the Poisson distribution (right). If ''m'' photons impinge on the target during a given pulse, each photon will have an independent probability, ''p'', of inducing a reaction. As a result, the distribution of the number of photo-induced reactions conforms to a binomial distribution with parameters ''m'' and ''p'', where ''m'' itself follows the Poisson distribution. A binomial distribution with the number of trials distributed according to the Poisson distribution, is also a Poisson distribution. Thus, the number of neutron producing reactions can safety assume to follow the Poisson distribution.
 ===2. Small detection probability ===

@@ Line 67: / Line 67: @@
 [[File:2NcorrPhotonPoissonCoincFit.png| thumb|right|Distribution of the number of photons measured in coincidence with a Poisson fit. ]]
-The number of neutron-producing reactions occurring per pulse, <math>n</math>, follows the Poissonian distribution.
+The number of neutron-producing reactions occurring per pulse, <math>n</math>, is assumed to follow the Poissonian distribution.
-It can be seen from data that the number of photons per pulse follows the Poisson distribution (right). If ''m'' photons impinge on the target during a given pulse, each photon will have an independent probability, ''p'', of inducing a reaction. As a result, the distribution of the number of photo-induced reactions conforms to a binomial distribution with parameters ''m'' and ''p'', where ''m'' itself follows the Poisson distribution. A binomial distribution with the number of trials distributed according to the Poisson distribution, is also a Poisson distribution. Thus, the number of neutron producing reactions can safety assume to follow the Poisson distribution.
+It can be seen from data, at least, that the number of photons per pulse follows the Poisson distribution (right). If ''m'' photons impinge on the target during a given pulse, each photon will have an independent probability, ''p'', of inducing a reaction. As a result, the distribution of the number of photo-induced reactions conforms to a binomial distribution with parameters ''m'' and ''p'', where ''m'' itself follows the Poisson distribution. A binomial distribution with the number of trials distributed according to the Poisson distribution, is also a Poisson distribution. Thus, the number of neutron producing reactions can safety assume to follow the Poisson distribution.
 ===2. Small detection probability ===

@@ Line 55: / Line 55: @@
 |<math> DP_{acc}</math> || represents the event that a two-neutron pair is detected when examining events across two separate pulses.
 |-
-| <math>E[X]</math> || Denotes the expectation value of some quantity X over large number of pulses.
+| <math>E[X]</math> || Denotes the expectation value of some quantity X over a large number of pulses.
 |}

@@ Line 55: / Line 55: @@
 |<math> DP_{acc}</math> || represents the event that a two-neutron pair is detected when examining events across two separate pulses.
 |-
-| E[X] || Denotes the expectation value of some quantity X over large number of pulses.
+| <math>E[X]</math> || Denotes the expectation value of some quantity X over large number of pulses.
 |}

@@ Line 56: / Line 56: @@
 |-
 | E[X] || Denotes the expectation value of some quantity X over large number of pulses.
+|}
 =Accidental estimation=

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 :where again, <math>n</math> is the number of neutron-producing reactions in a given pulse.
-The expectation of <math>m_{sp}</math> over a large number of pulses, is expressible in terms of <math>E[t_{sp}]</math> and <math>E[v]</math>, where the E denotes the expected value. This is possible because <math>m_{sp}</math> is a sum of products of independent random variables, and by the linearity of expectation, its expected value is equal to the sum of the product of the individual expectation values:
+The expectation of <math>m_{sp}</math> over a large number of pulses, is expressible in terms of <math>E[t_{sp}]</math> and <math>E[v]</math>, where E denotes the expected value. This is possible because <math>m_{sp}</math> is a sum of products of independent random variables, and by the linearity of expectation, its expected value is equal to the sum of the product of the individual expectation values:
 \begin{array}{lcl}

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 & = & \frac{1}{2} \lambda^2\bar{v}^2
 \end{array}
-where <math> \lambda</math> is the mean number of neutron producing reactions occurring per pulse. As argued in the section about the small detection probability approximation, the probability of detecting an accidental in some pulse, denoted by <math>P_{acc}</math>, is proportional to the number of accidentals available, times the two-neutron detection efficiency. To arrive at the final result, we take the expectation of <math>P_{acc}</math>:
+where <math> \lambda</math> is the mean number of neutron producing reactions occurring per pulse. As argued in the assumptions and approximations section, the probability of detecting an accidental in some pulse, denoted by <math>P_{acc}</math>, is proportional to the number of accidentals available, times the two-neutron detection efficiency. To arrive at the final result, we take the expectation of <math>P_{acc}</math>:
 \begin{array}{lcl}