Analysis

In this section the final part of the analysis flow (Fig. 7.1) is presented: The fitting of the experimental spectrum to a spectrum in the MC set8. In a first step the observables are extracted from the spectra which is essentially a problem of determining the peak counts with high precision. In a second step the observables are compared, the best FCCD value is inferred and the uncertainties are propagated.

Data Selection

The analysis is performed on binned spectra in histograms from the MCA systems. The binning is roughly equal for all histograms with a bin width of ≈ 0.3 keV/bin. In some cases multiple measurements were taken for the same detector with the same source type. Especially in the beginning of the characterization campaign many different sources and combinations were tested to investigate systematics. For the final results of the 241Am and 133Ba methods only one measurement per method and detector is selected9. The selection is unbiased and prefers measurements that were taken under the same condition

8

In reality the observables from the MC spectra are interpolated.

9_{The combination of multiple measurements of the same kind was omitted due to simplicity and the}

(a)241Am ROI 1 (b) 133Ba ROI 1 (c)60Co ROI 1

(d)241Am ROI 2 (e)133Ba ROI 2 (f) 60Co ROI 2

Figure 7.7 Variation of peak regions for different FCCD values. Stronger peak count variations are observed for smaller γ-line energies. Simulated spectra are for GD91C.

(source geometry, distance, etc). In the case of 60Co there are additional systematics uncertainties which are uncorrelated between different measurements of the same detector (e.g. the activity of different 60_{Co source). Here, multiple measurements of the same}

detector are combined with a weighted average. The MC spectra are convolved with the energy resolution and binned and treated in exactly the same way as the experimental spectra10_.

Methods for Peak Count Determination

The peak counts are determined with two different methods: (1) A counting method with background estimation from side bands and (2) a fitting method. The counting method im- plies a linear background assumption. If this assumption is justified both methods should give the same result. The fitting methods may include a complex background p.d.f. and deviations from Gaussian shape for the signal p.d.f. which is used to investigate peal tails. The counting and fitting algorithms for each calibration source are described in detail in the appendix Sec. C.2. Here only the selected algorithms are briefly outlined and illus- trated. For 241Am and 133Ba the validity of a flat background assumption is limited due to the previously described peculiarities in the ROIs. Here the fitting method is chosen for the final analysis and the counting method used as a cross-check.

The two ROIs of241Am are illustrated inFig. 7.8for an experimental spectrum of GD91C and the MC spectrum with the best fitting FCCD. The fit function is constructed with Gaussian peaks and constant background section below and above the peaks. The back- ground sections are connected with a Gaussian commutative distribution function (c.d.f.) set to the same width as the peak. In the second peak window around 100 keV the branch- ing ratios of the two signal peaks and the background peak are tightly constrained. The measured spectra of all other detectors are shown inFig. C.10toC.13in the appendix.

10

7.1 Dead Layer and Active Volume Determination 93

(a)241Am measured (b)241Am simulated

Figure 7.8 Determination of peak counts in the two ROIs of241Am. The histogram contains data taken with GD91C (left) and the corresponding MC simulation (right). The fit function is shown separated into the signal (red) and background (blue) components. The legends show the counts in the respective peaks and the ratio defined as observable.

(a)133_{Ba measured (b)}133_{Ba simulated}

Figure 7.9 Determination of peak counts in the two ROIs of133_{Ba. The histogram contains data taken}

with GD91C (left) and the corresponding MC simulation (right). The fit function is shown separated into the signal (red) and background (blue) components. The legends show the counts in the respective peaks and the ratio defined as observable.

The two ROIs of133Ba are presented in Fig. 7.9. The signal peaks and background func- tions are constructed similar to those of 241Am. The double peak structure in the first peak window requires strong constrains on the branching ratio and energy resolution of the peaks. The ROIs for all other detectors are shown inFig. C.14toC.17in the appendix. The two ROIs for 60Co are shown in Fig. 7.10 for an experimental spectrum of GD91C. The 60Co peaks are single peaks with flat background regions below and above. Here the counting and fitting method yield practically identical results. The fitting method includes a low energy peak tail in the fit which is illustrated as the green function in Fig. 7.10b. A low energy tail could be a potential indication for charge collection deficiencies inside

(a)60Co 1173.2 keV (b) 60Co 1332.5 keV

Figure 7.10 Determination of the 60Co peak counts for as independent observables for 1173.2 keV (left) and 1332.5 keV (right). The histograms show the experimental spectrum of one of three 60_Co

measurements with GD91C. The legends show the peak count rate (red) and the event fraction in the low energy tail (green). The peak count rate is normalized to counts per second and Bq source activity.

the detector. The tail information is extracted for all detectors and the fitting method is chosen as reference for the60Co analysis.

The60_{Co peak counts are obtained by integrating the fit function without background. To}

estimate the peak asymmetry due to the tail, the peak counts are separated into a left side (Nleft) and right side (Nright) with respect to the mean value,. The asymmetry information

is converted into the fraction of events that are in the tail ftail:

ftail =

Nleft− Nright

Nleft+ Nright

. (7.6)

The presented choices of the fitting functions, their parameters, and their constrains, are the result of a long fine-tuning process. The challenge of the fitting methods is to construct an algorithm that can fit many experimental and MC spectra from different detectors and measurements without manual input. The number of individual fits sums up to 150 + 1 fits for each detector and method. The fit methods are constructed robust and occurrences of non-convergence are identified and flagged. Detailed tuning of individual spectral features have to be omitted in this automated approach. It was found that the selection choices of parameters do slightly influence the final results but do so for all detectors in a systematic way. The variations with different parameter choices are typically much smaller than the systematic uncertainty budget.

Fitting the FCCD After extracting the observable from the measured spectrum and from the set of MC spectra, both values are compared. The observable is plotted in Fig. 7.11afor the241Am method (O241Am Eq. 7.2) for GD91C. The 150 observable values

from the MC simulation are plotted in a histogram over the range of 0 to 1.5 mm FCCD. The histogram is interpolated with an empirical spline function such that the FCCD depen- dence of the observable in the MC is well described. The underlying physical dependence for the 241Am methods is an exponential function since the dominating effect is attenua- tion. The underlying physical dependence for60Co observables (O60Co 1/2 Eq. 7.4and7.5),

shown inFig. 7.11c and7.11d, is a cubic function since increasing the FCCD is decreasing the active volume. For the133Ba observable (O133Ba Eq. 7.3), shown in Fig. 7.11b, it is a

7.1 Dead Layer and Active Volume Determination 95

sloped surfaces and more complex volume description to the dependence. Especially for

241_{Am it was found that the simulation of a limited number of MC FCCD values interpo-}

lated with a simple exponential function introduces a bias, especially for conical diodes. This was often done in the past. The control of this effect, along with saving computing time, is a strong advantage of the DLPP approach developed for this work.

(a)241Am (b) 133Ba

(c) 60_{Co 1173.2 keV (d)}60_{Co 1332.5 keV}

Figure 7.11 Finding the FCCD with fitting the experimental observable to the set of MC observables for GD91C. The histogram entries are the MC observable values for each FCCD step. They are interpolated with a spline function (black). The MC uncertainties are shown in the band around this function (red). The horizontal line is the experimental observable value (black) with its statistical uncertainty (green). The vertical lines illustrate the best fit FCCD value (black), its statistical uncertaintiy (green), its systematic uncertaintiy (red) and their non-linear combination (blue). Note that lines may overlap in the plots.

The value of the measured observable is shown as a horizontal black line. Its 1 σ statistical uncertainty is plotted as green horizontal lines below and above this value. The intersec- tion of the experimental value with the MC curve is the best fit FCCD. The intersections of the experimental uncertainty values with the MC curve are the propagated statistical uncertainties on the FCCD. This uncertainty propagation naturally transforms symmetric uncertainties on the observable level into asymmetric uncertainties on the FCCD level. All uncertainties are treated asymmetrically in the following.

The statistical uncertainties in the MC simulation result in a band along the MC curve. This uncertainty is taken as the 1 σ uncertainty on each MC point. Due to the DLPP the

uncertainties of different points on the MC curve are correlated. The uncertainty of each individual FCCD value is taken as an approximation. All other systematic uncertainties are also applied to the MC curve. The individual systematic uncertainties are assumed to be uncorrelated with each other and added in quadrature. The total systematic uncer- tainty including the MC statistical component is plotted as the red band around the MC curve. A detailed description of the systematic uncertainties is given in the next section. The combined statistical and systematic uncertainty is determined from the intersection of the horizontal green lines with the red MC error band. The resulting total uncertainty on the FCCD is illustrated as blue vertical lines. Due to the non-linearity of the observable dependence, the statistical and systematic uncertainties on the FCCD cannot be simply distinguished. To separate the uncertainties, the following definition is used:

∆FCCDsyst= ∆FCCDtotal− ∆FCCDstat. (7.7)

As described above, ∆FCCDtotal is obtained with the propagation of statistical and sys-

tematic uncertainties (blue) whereas ∆FCCDstat is obtained with the propagation of only

the statistical uncertainty (green). Note that with this definition the simple sum of statis- tical and systematic uncertainty results in the total uncertainty.

The general sensitivity of the241Am, 133Ba and 60Co methods can be estimated with the relative change of the observable within a given range of FCCD values, e.g. for241_{Am in}

Fig. 7.11a the observable is≈ 880 at FCCD=0 mm and ≈ 290 at FCCD=1.5 mm. Hence the relative change is_{≈ 67 %. In contrast, the relative change of the observable is ≈ 40 %} for133_{Ba and only ≈ 27 % for}60_{Co within this FCCD range. A given uncertainty on the}

observable level will propagate stronger on the FCCD level for 60Co than for 241Am or

133_{Ba. The poor sensitivity to the FCCD for the}60_{Co methods has two reasons: (1)}60_Co

probes the volume of the detector and the dead volume is only a small fraction_{O(10 %) of} the total volume. (2) The energy depositions for 60Co FEP events is not distributed ho- mogeneously in the detector volume and is preferring the detector center over the detector surfaces (see argumentation around Fig. 7.3). Hence, the FEP counts in the 60_{Co γ-lines}

change only little with changing FCCD, making this method subject to strong influences from uncertainties in the peak counts.

For an unbiased comparison of experimental and MC spectra, the peak counts need to be determined in the same way and as precise as possible. This creates various difficulties: (1) The MC spectra include only the background created by the calibration source. The background in the experimental spectra includes additional background from natural ra- dioactivity. This issue was addressed with the fitting methods that only use a small section of the spectrum in which the absences of prominent background γ-lines is confirmed. (2) The peak shapes in the MC are exactly Gaussian with the width of the energy resolu- tion. Peaks in the experimental spectra may be slightly distorted due to DAQ effects or degraded energy reconstruction in small parts of the detector. This issue was investigated for241Am and133Ba including a tail to the fit function. No significant effect on the FCCD result was observed after reprocessing the entire analysis chain. The peak tails - although clearly visible in log scale - contain only an insignificant small number of events. An addi- tional argument is the presence of tails in all peaks which should cancel the effect to some extent in the ratio methods. For60Co the peak tails are included in the fit.

7.1 Dead Layer and Active Volume Determination 97

(3) The MC spectra include coincidences from γ-rays originating in the same nuclear decay. The experimental spectra additionally include random coincidences from γ-rays originating in separate decays. Random coincidences are clearly visible for the stronger241Am sources in e.g.Fig. 7.2aat 120 keV. Time and hardware constrains did not allow for a larger source distance. However, the probability for a random coincidence with a second γ-ray arriving at the same time is similar for any γ-ray energy; hence, it is canceled by the peak ratio. For the lower activity sources of60Co and133Ba no random coincidences are visible in the spectra.