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 exactly [math]v_i[/math] correlated neutrons, where [math]i[/math] ranges from 1 to [math]n[/math]. The probability of this event will be represented by the expression:

[math]p(nv_1...v_n)[/math].

Accidentals

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