Despite any researcher’s best efforts, a measured pulse height spectrum rarely con- sists of only the reaction of interest. Far more often theγ-ray spectrum resembles that of Figure4.10with contributions from various sources including environmental back- ground, beam-induced contaminant reactions, and, hopefully, the reaction of interest
itself. In order to determine the fractional contribution of each source of radiation to the experimental spectrum, each must be modeled and included in the spectral fit in the form of templates.
To this end, the extended binned likelihood function described by Barlow(1990) was adopted. This likelihood function essentially describes the probability,P, of ob- taining the data, D, from the fractions, F, of m number of templates across n bins. Using the notation ofDermignyet al.(2016), the expression for this function is given
byBarlow and Beeston(1993) as
P(D|F) = " n X i=1 Dilnfi−fi # + " n X i=1 m X j=1 ajilnAji−Aji # . (6.1)
For a single bin i in template j, aji is the observed number of events and Aji is the
predicted mean number of events. The total number of events contributed by all templates for a given bin is
fi = m X j=1 Adata Asim j FjAji (6.2)
where Adata and Asim
j are the total areas, within the fitted region, of the observed
spectrum and templatej, respectively.
Whereas the17O(p,γ)18F and22Ne(p,γ)23Na studies mentioned in the previous sec-
tion used likelihood maximization to estimate the template fractions, the template fractions in the 29Si(p,γ)30P analysis were found using the Bayesian approach pre-
sented by Dermigny et al. (2016). The primary advantage of embracing a Bayesian
strategy is the ability to derive probability density functions for the template fractions, which can then be used to calculate meaningful uncertainties as well as establish up- per limit values when necessary.
In this method, inferences are made using the multivariate joint posterior distribu- tion defined as
P(F|D) = RP (D|F)P (F)
FP (D|F)P (F)
(6.3)
where P(D|F) is the likelihood function given in Equation6.1 and P(F)is the joint prior probability function. For the latter term, a Jeffreys prior (Jeffreys,1946),
P (F) = m Y j=0 1 Fj , (6.4) was chosen.
The posterior distributions of individual source fractions were determined using this framework. First, the template fractionsFjwere sampled from the joint posterior
distribution (Equation 6.3) using a Metropolis-Hastings algorithm (Hastings, 1970). For the 29Si(p,γ)30P analysis presented in this dissertation, 80,000 iterations with a
20,000 sample burn-in period and a thinning interval of 5 were used to minimize autocorrelation in the combined posterior distribution (Hilbe et al., 2017). Template
fractions and uncertainties were then determined from the median and 68% highest density interval of the corresponding marginal posterior distributions, respectively. For template distributions with nontrivial density at zero probability, the fractional contribution is reported as an upper limit value defined by a 97.5% coverage inter- val. This analysis method is illustrated in Figure6.1which shows how the fractional contributions from each template combines to reproduce the observed data spectrum. Of course, the template fractions are not the desired final product of the29Si(p,γ)30P
data analysis. They can however be used to extract the sought-after information: the total number of reactions (NR) and primaryγ-ray branching ratios (Bj) of the reaction
of interest. To find these quantities, first one must use the sampled template frac- tions to find the number of events that a given template contributes to the measured
— Fit
— Data
29Si(p,γ)30P Templates
Template
Environmental Background
Beam-induced Contaminant Templates
12C(p,γ)13N 19F(p,αγ)16O
13C(p,γ)14N
Figure 6.1: Illustration of the fraction fitting data analysis method. An extended binned likelihood function is combined with a detailed GEANT4 model of the LENA detector system to simulate every source of radiation observed in the experimental spectrum. The product of these simulations is two groups of templates: 29Si(p,γ)30P
templates (shown in blue) and beam-induced contaminant reactions (green). Addi- tionally, environmental background, measured when no beam is present, is then con- verted to a template (grey). A Bayesian strategy is used to determine the fractional contribution of each template, and the fractional contributions combine to give an overall fit to the data spectrum. In the bottom right of this figure, the data spectrum (black) and the resulting fit (red) are displayed.
spectrum. This quantity is given by the expression Nj = Adata Asim j FjNjsim (6.5) whereNsim
j is the simulated number of events used to produce templatej. Typically,
only one template is generated for each source of contaminant radiation meaning that
Nj is equivalent to the total number of reactions from a particular source. This is not
the case for the reaction of interest because we are interested in its primary decay structure, and one template is produced for each purported primary branch as a re- sult. Consequently, the total number of reactions from the reaction of interest,NR, is
found by summing the number of reactions originating from all relevant templates,
NR = X j Nj = X j Adata Asim j FjNjsim. (6.6)
As will be shown later, the resonance strength can be derived from the total number of reactions, making this a particularly useful quantity to determine. Lastly, the branch- ing ratio for a given primary transition is then easily found from Equations6.5–6.6:
Bj = Nj
NR
. (6.7)
The analysis method presented in this chapter, unlike traditional analysis, does not require explicit corrections for various experimental factors such as detector ef- ficiencies and coincidence summing. As will be discussed in the next section, these effects are implictly included in the GEANT4-simulated templates instead.