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]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 occurred this pulse, is given by:

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

where,

[math]d_{i,j}[/math] is the probability of detecting both (and only both) of the neutrons of interest.

What the above statement is saying, is that the probability of detecting a particular pair of neutrons is independent of the how many other neutrons are produced during the same pulse. This is only an approximation, however, since each detector can register at most one hit per pulse. In other words, it is being assumed that neutrons do not "compete" against each other for a chance to be detected. The justification for this assumption is that each detector covers merely 0.5% of 4[math]\pi[/math] sr, and so the probability that two uncorrelated neutrons reach the same detector is on the order of 10E-6 times the number of neutrons emitted. I am making a point to address this because it accounts for the fact that the SP/DP accidental ratio is quite significantly greater than 1/2 for photons.

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