Peak Fitting in Raw Spectra

A sample section of a raw spectrum for the detector at 0o_{is shown in Figure3.6. Similar}

results were produced for the 90o _{and 144}o _{detectors. The sample spectrum shows the}

200 4000 4500 5000 5500 6000 6500 7000 0 100 300 400 500 600 Counts

Gamma−Ray Energy (keV) 0

6793 keV 0

5183 keV 0

6176 keV

Figure 3.6: Sample raw spectrum at 0o_.

and 6793-keV bound states in 15_{O. The full-energy, first-escape, and second-escape} peaks are visible. These are the spectral lines that were used for the DSAM analysis.

After collating the spectra from all the individual short data runs into sum spectra for the three detectors, the first task in the data analysis was the precise determination of centroid positions for the spectral lines for the states of interest as well as for the spectral lines to be used in the energy calibration of the detectors. The analysis package GF3 was used for this purpose [Radford, 2000]. GF3 is part of RadWare, which is a software package for interactive graphical analysis of γ-ray coincidence data. GF3 models the response function of a HPGe γ-ray detector by using the convolution of a standard Gaussian plus a skewed Gaussian with a low-energy tail over a smoothed step function and a quadratic background. The functional form of the model, minus the step function and quadratic background, is shown in Equations 3.5 and 3.6. The

resulting shape of a theoretical response in shown in Figure3.7. y(x) =A exp x−c β erfc x−c √ 2σ + σ √ 2β (3.5) y(x) =Berfc x−c √ 2σ (3.6) x: Channel number

A, B: Peak height parameters

c: Centroid position

β: Decay constant for low-energy tail in skewed Gaussian

σ: Width parameter of standard Gaussian

Figure 3.7: The components of peaks as fitted by GF3 (reproduced from [Radford,

2000]).

function model is nonlinear in the adjustable parameters, and the number of parameters is moderately high, there is no way to be certain whether a given set of parameters constitutes a global minimum, and no procedure is guaranteed to find such a set [Press

et al.,1992;Spall, 2003]. Therefore, the approach to fitting the spectral peaks involved

a lot of painstaking trial and error in order to be reasonably confident that actual “best-fit” parameters were found.

Figures, 3.8,3.9,3.10,3.11, and3.13give examples of best fits toγ-ray peaks in the HPGe detector spectra. We used four sources with a total of seven points for energy calibration of the HPGe detectors: the 1461-keV and 2614-keV room background γ- rays from40_{K and} 208_{T l, the 1173-keV and 1332-keV lines from}60_{Co and the stopped} component of the 6130 keV γ-ray from the 19_{F(p, α}

2γ)16O reaction. The last source was needed to get calibration points in the vicinity of the Doppler-shifted secondary lines from 15_{O. Data were collected long enough to accumulate (0.8−6.0)·10}5 _net counts in the room background peaks, (1.5−2.5)·106 _{net counts in the} 60_{Co peaks,} and 1200 to 4500 net counts in the peaks from the 19F(p, α2γ)16O reaction. Figure

3.8 shows the fit to the 1173-keV γ-ray from 60Co in the 0o detector. This example is quite representative. The widths (FWHM) of the two60_{Copeaks in all three detectors} ranged from 2.4 to 2.6 keV. Fits to the 1461-keV and 2614-keV room background peaks and the 1173-keV and 1332-keV 60_{Co full-energy peaks in the 90}o _{and 144}o _detectors

were qualitatively similar. Theγ-rays from the19_{F(p, α}

2γ)16Oreaction presented a contrasting situation. Here, there was a strong difference in the line shapes between the 0o, 90o, and 144o detectors. Figures 3.9, 3.10, and 3.11 show full-energy peaks from this reaction at each detector angle. The escape peaks were also used in the calibration and show the same qualitative features. The most prominent feature of the three figures is the large Gaussian compo- nent of the line shape that arises when the recoiling16_{Onucleus from the}19_{F(p, α}

Figure 3.8: 1173-keVγ-ray from60_{Co in the 0}o _{detector, net area ≈2·10}6_,χ2 _{= 1.39.}

reaction emits a γ-ray after coming to rest in the target. The low energy tail in the 0o detector line shape and the high energy tail in the 144o detector line shape result fromα-particles from the 19F(p, α2γ)16O reaction in this energy region which have an angular distribution that is strongly peaked in the 0o direction relative to the incident proton beam as shown in Figure3.12[McLean et al.,1940]. In addition, theα-particles have a relatively high energy as compared to the incident protons, ≈ 340 keV [Tilley

et al., 1993, 2000]. The net effect was an angular distribution of recoiling 16_{O nuclei}

that was strongly peaked in the 180o _{direction relative to the incident proton beam,}

leading to the line shapes seen in the 0oand 144odetectors. The angular distribution of outgoing α-particles (and thus 16O recoil nuclei) was broad enough so that there were a significant number with an appreciable component of velocity in the±90o direction with respect to the incident beam, thereby giving rise to the “wings” on the main peak as seen in the 90o _{detector line shape.}

Fitting the peaks from the 19_{F(p, α}

Figure 3.9: 6130-keV γ-ray from16O in the 0o detector, net area≈4000, χ2 = 1.06.

Figure 3.11: 6130-keVγ-ray from16O in the 144o detector, net area≈4500,χ2 = 0.88.

lenge. Short of modeling the line shapes using a Monte Carlo simulation, there seemed to be little choice other than to treat the “wings” and “tails” of the line shapes as back- ground. Closed analytical expressions for these line shapes likely do not exist. This approach was of little concern for fitting the peaks in the 90o _{detector as the line shape}

is symmetric and different choices for the background parameters would only weakly affect the centroid determinations. The situation was different for the 0o _{and 144}o _de-

tectors. Here, different seemingly acceptable parameters for the background coupled with good fits to the main Gaussian peak could produce substantially different deter- minations for the centroid position. This problem exists for any fitting procedure of this type, however it could have been more serious for line shapes so different from the standard response model, such as those found in the 0o _{and 144}o _{detectors. This was}

reflected in the much larger, up to an order of magnitude, assignments of uncertainty to the centroid determinations for these unusual line shapes. However, the uncertainty in the centroid determination, as calculated by GF3, were overall quite small ≈ 0.1 channels at most. It turned out that the calibration residuals were far more important

Figure 3.12: Angular distribution in the CM system of α-particles from the 19_{F(p, α}

Figure 3.13: Doppler shifted 6793-keV γ-ray from 15_{O in the 0}o _{detector, net area}

≈1300, χ2 = 1.83, residuals are shown along the channel axis.

for the uncertainty in the calculatedγ-ray energies. The overall quality of the resulting calibrations will be discussed in Section 3.3.2.

Figure 3.13 shows the full-energy peak line shape and the best fit to the Doppler- shifted 6793-keV γ-ray from 15_{O in the 0}o _{detector. Both the full-energy and escape}

peaks were used in the analysis of Doppler-shifted γ-ray energies. We expected the escape peaks to exhibit the same Doppler shift since the shift in energy of the γ-rays emitted from the recoiling compound nucleus is the same irrespective of whether the full-energy, the full-energy minus 511 keV, or the full-energy minus 1022 keV is ulti- mately deposited in the detector. The escape and full-energy peak line shapes in the 90o

and 144o _{detectors look similar to that shown for 0}o_{. Data were collected long enough}

to accumulate 200 to 4000 net counts in the Doppler shifted full-energy and escape peaks (the lower figures for the escape peaks). Figure 3.13 clearly displays a classic Doppler-broadened line shape. Despite the existence of this broadening phenomenon,

0.0 0.5 1.0 1.5 2.0 2.5 0 5 10 15 20 Range Frequency

Figure 3.14: Distribution of χ2 _{statistic for all peak fits.}

counting statistics were not high enough to extract a lifetime from the line shape. As in the case of theγ-ray line shapes from the19_{F(p, α}

2γ)16Oreaction, the response func- tion model provided with GF3 was deemed flexible enough to fit these peaks. The fit shown in Figure3.13 is reasonable and theχ2 values for the other Doppler-broadened peaks ranged from 0.65 to 2.07 with the same distribution as shown in Figure 3.14.

Figure3.14shows the distribution of the “chi-squared” per degrees of freedom statis- tic for all of the peak fits. All fits shown in Figures 3.8, 3.9, 3.10, 3.11, and 3.13 have a reasonable value forχ2_{. However, Figure 3.14 shows that they were indeed the norm} and not best-case results. It can be seen that the most common value ofχ2 _{was indeed} in the vicinity of unity and rarely above 1.5. In addition, there were no instances of

● ● ● ● ● ● ● 1000 2000 3000 4000 5000 1000 2000 3000 4000 5000 6000 Channel Number γγ −ray Energy (keV)

Figure 3.15: 0o _{detector calibration curve (error bars smaller than points),χ}2 _{= 1.01.}