While some target properties can be estimated based on implantation parameters and the like, it is crucial to quantify key properties of nuclear targets and other ex- perimental parameters as the basis for an accurate nuclear study. To this end, target yield curves were employed to study the29Si target used in this dissertation. The aptly
named yield curve involves mapping the experimental “yield” of a nuclear reaction— the number of times a reaction’s characteristicγ-ray is observed versus the number of incident particles—over a range of bombarding energies.
The top panel of Figure5.4shows the cross section of an idealized resonance. The resonance energy is indicated by a dashed line through the center of the resonance
E Y σ Γ∼keV 50% ∆E 50% R→677 keV Elab r = 416 keV 29Si(p,γ)30P 0.0 0.5 1.0 1.5 2.0 Yield (counts/ µ C) 415 420 425 430
Bombarding Energy (keV)
Figure 5.4:Top:The cross section of an idealized resonance is shown here. The dashed line indicates the energy of the resonance. The width of the resonance,Γ, is measured at the FWHM of the peak. Bottom: This is the yield curve corresponding to the res- onance shown in the top panel. Using the dashed line from the top panel, it is clear that the inflection point on the front edge of the yield curve, corresponding to the 50% yield, occurs at the resonance energy. The FWHM, shown with arrows in this panel, gives the width of the target in energy units at the resonance energy. Right: A yield curve of the R→677 keV primary transition of theErlab= 416 keV standard resonance in29Si(p,γ)30P. The data points indicate the experimentally measured yield while the
red line indicates the fit to the data performed using the YCurveFitprogram. See
text for details.
peak, and the width of the resonance, Γ, is the full-width half-maximum (FWHM) of the peak as indicated by the dotted lines. For a target of width ∆E where ∆E > Γ, the resonant yield curve takes the form shown in the bottom panel of Figure 5.4. As indicated by the dashed line, the inflection point of the yield curve’s front edge corresponds to the resonance energy. The FWHM of the yield curve gives the target thickness in energy units at the resonance energy.
Many yield curve measurements were made over the course of the 29Si(p,γ)30P
study as these measurements can serve several purposes. For one, a properly under- stood yield curve can be used to derive a number of different target and experimental
properties. Yield curves can show an experimenter at what energy the maximum yield of a resonance occurs, and performing a measurement at this energy makes for an ef- ficient experiment. This use becomes even more important when the measurement of two closely spaced resonances is undertaken (see Section 7.2.1). Lastly and perhaps most importantly, yield curves are crucial for monitoring target stability because the maximum yield of a target yield curve decreases as a target degrades.
To measure the yield curve of the29Si target, the LENA I accelerator was employed
to generate protons in the range of 400–430 keV to find the experimental yield of the Elab
r = 416 keV standard resonance in 29Si(p,γ)30P (Table 7.3). The beam current was
maintained at≤ 2µA in order to minimize dead time. The intensity of the strongest primary transition, R →677 keV, was tracked over this energy range with the HPGe
detector operated in singles mode. The background-subtracted, dead time-corrected yield from one such measurement is shown by the data points in the right panel of Figure5.4. To monitor target stability, the yield of the29Si target was measured every
1.5–5 C accumulated on target.
While the yield is relatively simple to measure and the idealized yield curve is easy to interpret, it can be quite complicated to account for experimental factors (beam resolution and straggling, target heterogeneities, etc.) in order to extract the desired properties. For resonance yield curves specifically, several reasonable assumptions can be made to simplify matters (for a complete discussion, see Iliadis, 2015). The resulting expression for the experimental yield of a resonance is
Y(E0) = 1 εr Z E0 E0−∆E dE0 Z ∞ Ei=0 dEi Z Ei E=0 σ(E)g(E0−Ei)f(Ei−E, E0)dE (5.1)
where E0 is the mean bombarding energy, εr is the constant stopping power of the
target over the width of the resonance, ∆E is the target thickness, and σ(E) is the energy-dependent resonant cross section. The functiong(E0−Ei)gives the incident
Table 5.1. 29Si(p,γ)30PElab
r = 416 keV Yield Curve Analysis
Parameter Value
Maximum Yield 1.98±0.05 counts/µC
Yield Area 23.0±0.3 counts·keV/µC
Resonance Energy 416.61±0.09 keV
Target Thickness 11.39±0.26 keV
Beam Width 2.39±0.15 keV
Beam Straggling 2.69±0.31 keV
Note. — These are the results of a fit to a yield curve measured shortly before theElab
r = 221 keV measurement commenced (Section7.4).
beam width independent of the mean bombarding energy, whilef(Ei−E, E0)governs
the beam straggling in the target, which is independent of the incident energy.
A R program designed to fit experimental yield curves, YCurveFit, was writ-
ten by Richard Longland. This program uses Markov Chain Monte Carlo (MCMC) to fit the experimental yield, and from the fit, parameters in Equation 5.1 could be extracted. The red curve in the right panel of Figure 5.4 shows a fit found using
YCurveFit, and Table 5.1 presents the parameter values determined from this fit.
The yield curve shown in this figure was taken and analyzed shortly before theErlab= 221 keV study (Section7.4) began. Given the stability of the29Si target, all yield curves
taken during the course of the29Si(p,γ)30P study strongly resemble that of Figure5.4.
Between the resonance studies performed using the LENA I accelerator (Elab
r =
416 keV, 324 keV, and 314 keV) and theElab
r = 221 keV study with the LENA II accel-
erator, the LENA I bombarding energy was recalibrated for improved accuracy (for details, seeDermigny,2018). It should be noted that the yield curves (and correspond- ing bombarding energies) presented in Sections7.1–7.3were affected by the previous
LENA I energy calibration which resulted in an underestimate in the bombarding en- ergy. As this was only a calibration issue, there was no effect on the data presented in these sections, but the ostensible discrepancy could prove confusing to an uninformed reader.
CHAPTER 6: DATA ANALYSIS METHODS
For nuclear capture experiments, the ultimate desire is to further the understand- ing of both the decay dynamics and the frequency of the reaction. Experimentally, this translates to determining the primaryγ-ray branching ratios as well as the total num- ber of nuclear reactions originating from the reaction of interest. Traditional “peak- by-peak” analysis, involving the measurement of net intensities for all full-energy primary transition peaks with corrections applied for detector efficiency, has been successfully used in the past for relatively simple spectral analysis. Nevertheless, the efficacy of this method breaks down when any number of complications—coincidence summing, angular correlations, compound peaks, and the like—are encountered en masse.
Furthermore, the “peak-by-peak” analysis method is challenging to apply to γγ- coincidence spectroscopy used in the current29Si(p,γ)30P study. This data acquisition
method involves the roughly simultaneous detection ofγ-rays across multiple detec- tors. The observed data can then be refined using timing and energy gates to focus on the events of interest and reduce the impact of background sources (see Section4.3.4). As the result of this gating, the likelihood of observing an event of interest is signif- icantly improved, but the detection efficiency is now dependent on multiple factors in addition to the efficiency of the employed detectors. This added complexity means that it is substantially more difficult to correct full-energy primary peak intensities for detection efficiency.
(2016) was adopted instead of a traditional “peak-by-peak” method for the spec- tral analysis of the 29Si(p,γ)30P study. This method has been successfully applied to 17O(p,γ)18F and 22Ne(p,γ)23Na direct studies byBuckner et al. (2015) and Kelly et al.
(2017), respectively. In this technique, a binned likelihood function is used to produce Monte Carlo-simulated templatesextending across the entirety of the observed spec- trum. Singles orγγ-coincidence spectra can be easily produced with this method. A Bayesian approach is then applied to determine the fraction each template contributes to the observed spectrum. One can choose to fit only full-energy peaks, similar to the “peak-by-peak” approach, or the entirety of the experimental spectrum. Because the effects of complicating experimental factors (angular correlations, coincidence sum- ming, etc.) are implicitly included in the simulated templates, no corrections to the resulting fit are required, and the primaryγ-ray branching ratios and total number of reactions for the reaction of interest can be easily calculated from the template frac- tions. With these advantages in mind, a binned likelihood approach is clearly the best analysis method for the current work.
In this chapter, the binned likelihood approach and its application to the present
29Si(p,γ)30P study will be discussed. Section 6.1 will present the analysis method in
detail, followed by a discussion of simulated template generation (Section 6.2). Sec- tion 6.3 will discuss the method chosen to determine the goodness of fit. Lastly, a summary of the adopted analysis method will be presented in Section6.4.