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_i[/math]	A random variable for the number of correlated neutrons produced by the [math]i\text{th}[/math] neutron-producing reaction in a given pulse. The index [math]i[/math], ranging from 1 to [math]N[/math], is used to distinguish between each of the distinct and independent reactions in a given pulse, thus all the [math]V_i'{s}[/math] are identical and independently distributed random variables

Probability of detecting an accidental from a given pulse

Consider a pulse in which [math]n[/math] neutron-producing interactions occurred, where each interaction produces [math]v_i[/math] correlated neutrons. Now I seek to find the probability of detecting one and only one specific pair of neutrons from this pulse. This pair is also an accidental, meaning that the two detected neutrons were produced in separate interactions.

[math]v_1v_2...v_n[/math]

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