In order to account for the data spectrum accurately, the source of every peak— resonance of interest, beam-induced contaminants, or environmental background— must be identified and then reproduced via simulation. First, recall the deexcita- tion of a compound nucleus via γ-ray emission during a nuclear reaction. Figure6.2
(Dermigny et al., 2016) helps illustrate how this can occur. For a theoretical reaction
A(p,γ)B, the deexcitation to the ground state, E0, begins at the compound level, Ex,
and can occur through a singleγ-ray or a complexγ-ray cascade. Transitions originat- ing from the compound level are called primary transitions. If the compound nucleus decays to the ground state via a singleγ-ray, the corresponding transition, Ex →E0,
is often referred to as the ground state transition. All other primary transitions must decay to a secondary state, e.g. E2, and emit aγ-ray with a characteristic energy given
by the energy difference between levels,Ex–E2 in this example. From here, the decay
continues to the ground state, either through one additional transition (E2 →E0) or a
γ-ray cascade (E2 →E1 followed byE1 →E0). Allγ-rays that originate from a state
other than the compound level are known as secondaryγ-rays.
Figure 6.2: The decay scheme of a theoretical reaction,A(p,γ)B, fromDermignyet al.
(2016). Primary transitions, originating from the compound state Ex, are shown as
To simulate the deexcitation of compound nuclei via γ-ray emission, a GEANT4 framework, incorporating the precise models of the LENAγγ-ray spectrometer (Sec- tion 4.3) and target station (Section 4.2) detailed earlier, was employed. For a simu- lated compound nucleus decay, allγ-rays emerge from the ion beamspot on the nu- clear target, and secondary decays are simulated by randomly sampling literature de- cay branchings. Interactions with the detection system and environment are tracked throughout the simulated event. To account for the number of counts one would see in the experimental spectrum, the energy deposition in the detection system is recorded for every event. This process is repeated many times to simulate a large number of de- cays, and the resulting simulated data set is processed as it would be in the laboratory. The ultimate result of this process is a template histogram, representing a particular contribution to the observed data, that is directly comparable to the experimentally measured pulse height spectrum.
For the analysis of the reaction of interest—or, more specifically, the resonance of interest—we seek primary branching ratios as well as the overall number of reac- tions. As a result, a template is generated for each potential primary branch of the reaction, each with an assumed branching ratio of 100%. This allows the fractional contribution from a given template to indicate the primary branching ratio as shown in Equation6.7.
Template generation for beam-induced reactions is achieved in a similar manner, but the focus is on quantifying the overall contribution each of these reactions makes to the spectrum. Thus, each contaminant reaction is usually accounted for with a sin- gle template derived from literature branching ratios because gaining an understand- ing of its decay structure is unnecessary. In the case of direct capture contaminant reactions such as12C(p,γ)13N, the level energy of the compound nucleus is adjusted
as required for the experimental proton bombarding energy.
to the experimental spectrum and must be included in the simulated spectra accord- ingly. However, the simulation of all sources of environmental background would be a laborious undertaking, and given that this type of background occurs regardless of whether an experiment is being conducted, the simpler approach was to measure the environmental background directly. Environmental background is comparatively weak and can change with meteorological conditions, so background measurements were usually taken on a daily basis after the29Si(p,γ)30P data acquisition had ended.
The resulting data was combined into one data set from which the required template was extracted.
6.2.1 Template Corrections
While simulated templates do an adequate job of reproducing the experimental spectra, there are a few instances in which corrections to the templates are required to account for physics not included in the GEANT4 simulation. These corrections include adjustments to the position of peaks, the overall width and shape of peaks, and the effect of angular correlations.
Peak Position Adjustment
Literature values are used to determine the energies at whichγ-ray peaks occur in the spectra, and while these energies are usually known to a high precision, they do not account for other particle kinematics. For example, it can be assumed thatγ-decay as a result of radiative capture reactions occurs on a short time scale such that the recoiling compound nucleus does not come to rest before γ-emission (Iliadis, 2015). The energy of the emittedγ-rays are fully Doppler shifted, but the GEANT4 simulation does not account for this effect. However, this usually results in an energy shift of only a few keV and can be easily adjusted without altering the bins surrounding the
affected peak.
Peak Width Adjustment
Template peak widths also require adjustment. These adjustments can require global or local adjustments to the templates.
First, it should be noted that the HPGe detector has a finite, energy-dependent energy resolution. It has a broadening effect on all peaks in a measured spectrum (for an example, see Figure4.10), so this effect must be reproduced in the simulated templates as well. To achieve this, each raw simulated spectrum is convolved with a Gaussian function of width σ(E). This width is found by measuring the FWHM of environmental and secondary transition full-energy peaks across a large energy range. Both of these types of peaks should be free of kinematic effects (e.g. Doppler shifts) that would impact peak widths. The energy resolution of the HPGe is roughly linear, soσfor a given peak is
σ(E) = FWHM 2√2 ln 2 ≈
FWHM
2.35 (6.8)
(Knoll, 2010). From the measured values ofσ for a variety of peaks, the linear σ(E) function can be found and the entirety of the simulated spectra broadened accord- ingly.
Two situations exist in which templates would require broadening local to particu- lar peaks. First, additional adjustment is occasionally required to account for Doppler broadening in full-energy primary transition peaks. This effect results in both broad- ening of the peak and the appearance of a tail on the low-energy side of the peak. To produce these effects in the required peaks, the γ-ray peak fitting algorithm devel- oped byMacMullin(2015), which allows for both peak broadening and a low-energy tail, was applied.
Additionally, the width of peaks produced by direct capture are often dependent on the target thickness which is not included in the GEANT4 simulation. Thus, any broadening of direct capture peaks seen in the data spectrum must be applied to the appropriate templates as well.
Angular Correlations
Sometimes, two radiations are angularly correlated because of the alignment of a particular nuclear level. This coupling can occur between two consecutive, outgoing γ-rays, but for the purposes of this work, the coupling of the incident radiation (in this case, proton bombardment) and primary transition γ-rays is of interest. In this scenario, the aligned nuclear level is populated unequally so that the subsequent de- cay of this level will result in an anisotropic γ-ray emission, or angular correlation. The angular distribution of theγ-ray emission is described by the angular correlation factor,
W(θ)≈1 +a2P2(cosθ) +a4P4(cosθ). (6.9)
In this expression, θ refers to the emission angle with respect to the incident beam direction in the center-of-mass frame, P2 and P4 are Legendre polynomials, and a2
and a4 are angular correlation coefficients of the second and fourth orders, respec-
tively. For29Si(p,γ)30P primary transitions where published angular correlation coef-
ficients were available, the primary transitionγ-ray distribution was simulated with a weighted probability for emittance given by
P(Ω)dΩ = W(θ)dΩ (6.10)