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, since 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].

• • introduce the result and a summary of what follows here**

Accidentals

First, I seek to find the probability of detecting a specific pair of neutrons from this pulse. Let the pair be an accidental so that they are uncorrelated. The probability of detecting a specific neutron accidental pair, given that [math]v_1...v_n[/math] neutron producing interactions occurred this pulse, is approximated 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, given that we know nothing else about neutrons except that they are uncorrelated.

What the above statement implies, is that the probability of detecting a particular pair of neutrons is independent of the how many other neutrons are produced during that same pulse. This is only an approximation, however, since each detector can register at most one hit per pulse, and so the effective efficiency of the entire array drops as the number of particles emitted increases. In other words, it is being assumed that neutrons do not "compete" against each other for a chance to be detected. 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]