Understanding the energy response function for each detector is crucial to the suc- cess of the Majorana Demonstrator. While a detector’s energy resolution must be known for 0νββ analysis, it is also crucial to understand a detector’s resolution as a function of energy in order to appropriately compare MC simulations to data (Chap- ter 5). A detector’s energy resolution can be well quantified if the detector’s response function is known, and as discussed in Chapter 3, a detector’s response function is best classified by simultaneously fitting several gamma peaks over a broad range of energy. The multi-peak fitting function is used to classify the response function for each working detector of the PC. The multi-peak fitting routine performs best with higher statistics and therefore is used on binned data taken with a 228Th line source. The routine is used on a total of five calibration sets in order to investigate how the peak fitting parameters change over time. The five calibration sets that are used in this analysis are referred to as calibration data sets: A, B, C, D and E. They are listed in chronological order and their relative timing with respect to one another can be found in Fig. 2.2. For each detector in each calibration data set, the multi-peak fitting routine is used to fit five gamma peaks, which are listed in Table 4.1. These five peaks are chosen because they have high relative intensities and are well separated (by at least 10σ) from other gamma peaks. For each peak, a fit range corresponding to (M ±10Σ) is used for the multi-peak fitting routine.
Table 4.1: The five gamma peaks fit for each detector in each calibration data set [NND].
Expected Pk Centroid (keV) Isotope Relative Intensity
277.371 (5) 208Tl 6.6% (3) †
300.087 (10) 212Pb 3.30% (4)
583.187 (2) 208Tl 85.0% (3) †
860.557 (4) 208Tl 12.5% (10) † 2614.511 (10) 208Tl 99.754% (4)†
† To compare 208Tl intensities to 212Pb intensity, multiply
the 208Tl intensities by 0.3594.
To fit multiple peaks over a range of energy, the parameters of the multiple peak- fitting function must be initialized. Because there are numerous parameters and the best fit is often at a very local minimum, it is crucial that the parameters be initialized with accuracy to aid the minimizing processor. Therefore the multi-peak fitting routine begins with two fitting algorithms to find the best values for parameter initialization. Upon successful completion of the first two algorithms, a good estimation for each pa- rameter of the multi-peak fitting function is found and used for parameter initialization in the third – and final – algorithm. The final algorithm ends with a “final fit” where no parameters are fixed and the minimization function and minimizer processor are optimized.
4.1.1 Fitting Algorithm I
The first algorithm finds approximate values for: µi,Ai,σ0, σ1 and σ2. The param-
eters µi and Ai are the Gaussian centroid and signal area of the ith peak, respectively.
The parameters σ0, σ1 and σ2 are three of the common parameters described in Sec-
tion 3.4. The algorithm consists of the following.
1. Find ymax, the maximum bin content of theith peak. Use the energy correspond-
2. Calculate the FWHM of each peak. Use the FWHM to find an approximate σi
and Ai.
• The variance of the Gaussian of the ith peak,σ
i, is approximated as: σi =
FWHMi
2.355 , (4.1)
where FWHMi is the FWHM of theith peak.
• The area of the ith peak, A
i, is approximated as: Ai =ymax
√
2π σi, (4.2)
where ymax is the maximum bin content of the ith peak.
3. Fit σ as a function of µ to find an approximate σ0, σ1 and σ2, where the fit function is: σ = q σ2 0 +σ21µ+σ22µ2 (4.3) 4.1.2 Fitting Algorithm II
The second algorithm finds the best initialization values for the parameters related to the tail in the multi-peak fitting function: bτ, mτ, bH and mH. In the second
algorithm, the multi-peak fitting function is used (Eqs. 3.9– 3.13, 3.16 – 3.18) but with the parametersbH and mH fixed in value. Based on experience,mH is typically on the
order of 10−6 and is therefore fixed to be 1.·10−6 in the fitting algorithm.
The parameter bH can vary greatly from detector to detector. The parameter bH
should never be less than zero as that would imply a negative tail component of the signal, and it should never be greater than one as that would imply a negative gaussian component of the signal. Since the realistic range of the parameter is limited, it is possible to explore the entire spectrum of possible bH values. To do this the parameter
is fixed at five different values: 0.0, 0.2, 0.4, 0.6 and 0.8. A new fit is performed for each different value of bH, where the parameters bH and mH are fixed and all other
parameters are allowed to vary. If a reasonable fit is obtained, the parameters resulting in that best fit are used as the initialization values for the parameters in the third and final algorithm.
It should be noted that for some detectors a reasonable fit can be found for more than one fixed bH value. In these cases the final algorithm is performed multiple times for
one detector, and as a result several “final fits” are attempted. However the parameter values that result from each attempted final fit are in agreement with one another and therefore only one unique best fit is found for each detector in a given data set.
4.1.3 Fitting Algorithm III
After the first two algorithms are performed and good parameter initialization val- ues are found, the final algorithm is performed. As discussed in Section 3.2, the final fit minimizes the log-likelihood function using the MINOS processor. It is often diffi- cult to obtain convergence with the log-likelihood function and the MINOS processor, and hence the final algorithm is implemented. The final algorithm, which is further explained in Section 3.2 is:
1. Chi-Squared minimization function; MIGRAD processor 2. Log-Likelihood minimization function; MIGRAD processor 3. Log-Likelihood minimization function; MINOS processor
The last step of the algorithm is the final fit. During this fit, no parameters of the fit function are fixed in value or limited in range. The one exception to this is when a detector’s best fit results in the parameter bH being consistent with zero (i.e. there is
no low-energy tail portion of the signal). In this case the final fit is performed withbτ, mτ, bH and mH fixed to be zero.