Theoretical analysis of 2n accidentals rates

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Introduction

A given photon pulse may cause multiple neutron-producing reactions, ranging from zero to "infinity" reactions. The number of neutron-producing reactions in a pulse is hereafter denoted by [math]N[/math]. Being the number of neutron-producing reactions actually occurring per pulse, [math]N[/math] is assumed to follow the Poissonian distribution as a limiting case of the binomial distribution. Each neutron-producing interaction is said to produce [math]V_{i}[/math] correlated neutrons, where the random variable [math]V_{i}[/math] is the distribution of the number of neutrons produced in a single neutron-producing reaction. Each of the [math]V_i\text{'}s[/math] are independent and identically distributed random variables, so the purpose of the subscript is to distinguish between several distinct neutron-producing interactions which may occur in a single pulse.

The beam has a Bremsstrahlung end point of 10.5 MeV, which energetically allows for only two possible neutron-producing interactions, 1n-knochout and photofission. Thus, [math]V_{i}[/math] is equal to the photofission neutron multiplicity plus a contribution at [math]V_{i}=1[/math] from 1n-knockout events. The analysis that follows does not need to distinguish between 1n-knockout events and photofission events that emit a single neutron. In both cases, a single neutron is emitted and is uncorrelated with all and any other neutrons.

Variable reference

variable	Description
[math]n[/math]	A random variable for the number of neutron producing reactions occurring in a single pulse.
[math]v[/math]	A random variable for the number of correlated neutrons produced by a single neutron-producing reaction in a given pulse.
[math]v_i[/math]	A random variable with the same distribution as [math]v[/math]. The index only distinguishes between distinct and independent instances of [math]v[/math].
[math]\lambda[/math]	Poissonian mean for the number of neutron-producing interactions per pulse.

Section Title

Consider a pulse in which [math]n[/math] neutron-producing interactions occurred, where each individual reaction produces [math]v_i[/math] correlated neutrons, where [math]i[/math] ranges from 1 to [math]n[/math]. This event is denoted as, [math]p(v_1...v_n)[/math], and its probability by the expression, [math]p(v_1...v_n)[/math].

Accidentals

First, I seek to find the probability of detecting a specific pair of neutrons from this pulse. Let the pair be an accidental, meaning that the two detected neutrons were produced in separate interactions. The probability of detecting a specific neutron accidental pair, given that [math]v_1...v_n[/math] neutron producing interactions have occurred:

[math]p(d_id_j|,v_1...v_n)[/math]

[math]p(nv_1...v_n) = \frac{e^{-\lambda}\lambda^n}{n!}p(v_1)p(v_2)...p(v_n)[/math]