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Study of the parameters to be used in the analysis of the ²¹⁵Po decay in LaBr3 crystal and results of the analysis

by Silvia (May - June 2025, update October 2025)

1) DEFINITIONS OF THE ADC INTERVALS for the study of the ²¹⁵Po → ²¹¹Pb decay

According to our discussion on the data analysis for the measurement of the Po215 decay lifetime, I analyzed the first day of data (i.e. the first 2 files, 24.0 hours) to find the ADC intervals that give the best signal to noise ratio.

First I defined the cuts on the pulse shape parameters to be used with the LaBr3 data, that are different from those used in the GAGG analysis. I rejected events with baseline > -15126, abs(slope) > 1.2 e6 and delay < 0.8 e-7 or delay > 1.15 e-7. Then I select the alpha decay events requiring chi2amp/integral > 0.58.

I tried many combinations of ADC intervals for the selection of the pairs of events for the time-delay analysis. The ADC intervals can be contiguous or can also overlap to select the maximum number of decay events.

I construct an histogram with the time-delay delta-t distribution, with delta-t between -0.1 and 0.1 s, with a bin width=0.05 ms (4000 bins). I fit the histogram with an exponential function plus a flat background (for this test I use the Root fit option “LI”). I calculated the signal significance (sigma) as the ratio between the number Ns of decay events with time difference delta-t < 12 ms (integrating the best fit exponential function from 0 to 12 ms) and the square root of the average number of background events in an interval of 12 ms, sqrt(Nb).

The next table shows the best results (i.e. with the maximum sigma) obtained considering separately non-overlapping and overlapping ADC intervals. T1/2 is the half-life obtained by the fit.

ADC1 is the interval used to select the ²¹⁵Po decay events, ADC2 is the interval for the parent ²¹⁹Rn decay events. In case of overlapping ADC regions, if both events are inside the overlapping region, the pair is rejected, otherwise the same delta-t is counted twice, with opposite signs, forming a small excess with negative delta-t.

The overlapping intervals gives a number of sigmas slightly larger than the non-overlapping intervals.

Fig. 1a and 1b show the ADC alpha spectrum, where the intervals that give the best results are highlighted.

Fig.2 shows the obtained delta-t distribution in case of non-overlapping ADC regions:

2) SIMULATIONS

With these rates of decay events and background, one can make a simulation to study the dependence of the decay signal from different analysis parameters, as the bin width, the time range of the fit, and others.

For each parameter to be tested, we decided to do 10000 trials. Each trial consists in constructing a toy histogram containing the delta-t distribution of the exponential signal (with the “official” half-life T1/2 = 1.781 ms) plus a flat background, both corresponding to a run time of 22.5 days (i.e. the total run time minus one day), then doing the fit and comparing the resulting half-life with the official value.

In this simulation I use the number of events obtained above for the non-overlapping ADC regions, but I assume that the same conclusions can be valid also for the overlapping ADC region, since the difference is really small.

The total number of decay events in 24.0 h (integrating the exponential from 0 to infinity) is N = 305141, so multiplying this value by 22.5 one has the total number of decay events to simulate, Ntot = 6865673. Similarly, from the value of the background in 24.0 h, i.e. 143.0 events in a bin of width 0.05 ms, one can calculate the total background to simulate Btot = 143×22.5/0.00005=6.435 e7 s-1.

2a) Simulations: trying different bin widths of the histogram with time-delay distribution

I try to see the effect of the bin width on the result of the fit, changing the value of the bin width from 0.025 to 5.0 ms. For this test, I fix the time range of the delta-t histogram Tmin = -0.1 s and Tmax = 0.1 s.

For each bin width to be tested, I generate 10000 fake histograms containing an exponential distribution plus a flat background.

I fill each bin of the histogram with the number of events randomly generated from a Gauss distribution with mean value equal to the average number of events expected in that bin, both for the decay signal and for the background. Finally, for each of the 10000 fake histograms, I make a fit (with an exponential function plus a flat background) and I obtain a value of T1/2. The fit is done using the option “M”.

The option “M” improve fit results, by using the IMPROVE algorithm of TMinuit.

The averages of the 10000 values of T1/2 obtained by the fit and the corresponding rms of T1/2 distributions are shown in the following table, for each bin width:

One can see that for all the six tested bin widths, the T1/2 mean values are always consistent with the “true value” 1.781 ms within 0.00002 ms. The rms is around 0.00088-0.00090 ms (0.05% of the mean value), except for the last trial (T1/2 = 5 ms), where the rms increases to 0.00112 ms.

I think that with a 0.05-0.1 ms binning one can obtain a good result.

2b) Simulations: trying different fit options in Root

I try to change the fit method, considering the following Root options:

“M” Improve fit results, by using the IMPROVE algorithm of TMinuit.

“E” Perform better errors estimation using the Minos technique

“I” Use integral of function in bin instead of value at bin center

“L” Use log likelihood method (default is chi-square method). To be used when the histogram represents counts.

I tried these different combinations: M, MI, ME, L.

For each fit option, I construct 10000 fake histograms as before, fixing the bin width to 0.5 ms and Tmax = 0.1 s. The result does not change significantly, except for the option L, that gives an average value of T1/2 = 1.78201 ms, too different from the “true” value. The best result is obtained with the option “ME” that gives T1/2 = 1.78101 ms with rms = 0.000888 ms.

The option “I” makes the calculations much slower, so it's not worth using it, since the precision of the result does not increase.

In the following I will use the option “ME”.

2c) Simulations: trying different time limits of the histogram

I change the limits Tmin and Tmax of the histograms, using 2 different bin widths, 0.1 and 0.05 ms.

Here the results in case Tmin=-Tmax:

One can see that Tmax = 0.01 s gives a larger rms, otherwise all other values give a similar result, with the best one for Tmax=0.1 s.

As an example, Fig.3 shows one of the 10000 histograms with a bin width of 0.1 ms and Tmax=0.05 s.

Fig.4 is the distribution of the 10000 T1/2 values obtained with bin width 0.1 ms and Tmax=0.05 s.

I also try other time intervals, in order to exclude particular delta-t ranges that in the real data show problem.

The first problem is that a small excess for delta-t < 0 is always visible when the statistics is enough large, even in the case of non-overlapping ADC intervals, due to the pairs of events with energies falling in the “opposite” ADC interval.

The second problem is a lack of decay events with very small positive delta-t due to the data acquisition dead time (to be measured).

These “anomalies” are visible in Fig. 5a and 5b showing the real data (22.5 days) delta-t regions with a small excess for delta-t < 0 (left) and the lack of events in the first bin with delta-t > 0 (right).

To overcome these problems, first I try to see what happens by fitting only the positive part of the histogram (delta-t>0), and discarding the first bin, to avoid the dead time effect.

The fit of the only positive part gives a little worse result, as expected, in particular when the first bin is not used. In this case is better a smaller bin width and a larger Tmax.

So I fit the background separately in the negative part of the histogram (from delta-t -0.1 to -0.01 s to avoid the small negative exponential observed in the data) and then fit the positive part using the found background value.

The average background rate is (6.433 +- 0.003) 10⁷ ev/s.

The fit of the positive part of the histogram gives the following results (Chi2 is the average normalized Chi squared +- its rms):

The result is better, and if we must reject the first bin to avoid the dead time problem, the result improves using a small bin width and a larger Tmax.

With a bin width=0.05 ms and Tmax=0.1 s the rms is 0.00089 ms, corresponding to the 0.050% of the half-life value. Maybe we can improve a little the result excluding a smaller interval in delta-t (after the measurement of the dea time we can try).

3) COMPARING SIMULATIONS WITH REAL DATA

I analyzed all the data, except the first 2 files that has been used for the ADC intervals selection. The remaining data cover a time interval of 22.5 days.

In the following table the error on T1/2 is provided by the Root fit.

Then, according to simulations results, I fit the background separately in the negative part of the histogram (from -0.1 to -0.01 s) and then I fit the positive part using the found background value.

The average background rate is (6.471 +- 0.003) 10⁷ ev/s.

The results of the fit in the positive part of the distribution are:

The error on T1/2 found by the fit procedure agrees with the rms value obtained by simulations in the same conditions. The Chi square is consistent with the one obtained by simulations.

Since in Section 1 I found that the results obtained with one day of data by using overlapping ADC intervals are slightly better, now I analyze the data with that event selection.

Similarly to the above procedure, I calculate the background rate in the negative part of the histogram from -0.1 s to -0.01 s, and then I fit the positive part, obtaining this result:

In this case the average background rate is (6.987 +- 0.003) 10⁷ ev/s.

The result is slightly better than in case of non-overlapping ADC intervals. This is probably the best result that we can obtain.

4) STUDY OF THE STABILITY OF THE DECAY SIGNAL

To study the stability of the decay signal during the 22.5 days of measurement, I follow the same procedure adopted by Yinjie. I divide the total dataset (45 runs) in 9 periods of 2.5 days each, and for each period I do the same analysis described in the previous point.

The found 9 values of the half-life as a function of the period, and the distribution of these values are given in the next two figures.

The mean value of the 9 measurements is 1.7775 ms. The errors of the single measurements range between 0.00260 and 0.00261 ms. As Yinjie I found a large peak in the 4th interval, that however corresponds to 2.15 sigmas, so it is not statistically significant. The distribution of the 9 half-lifes has a standard deviation SD = 0.00224 ms, consistent with the value found by Yinjie. Hence the error on the mean value is SD/sqrt(9) = 0.00075 ms (0.042%). This is consistent with the statistical error found by fitting the global data in the previous analysis (0.049%). This means that the distribution of the 9 values is consistent with statistical fluctuations and it is not the result of some instability.

To further investigate the problem, I divided the data in 45 periods, each of 12h duration. The next two figures show the found 45 values of the half-life as a function of the period and the distribution of these values.

The mean value of the 45 measurements is 1.7776 ms. The errors of the single measurements range between 0.00576 and 0.00588 ms. The distribution of the 45 measurements has a standard deviation SD = 0.00517 ms. The error on the mean value is SD/sqrt(45)=0.00077 ms (0.043% of the signal), and also in this case is consistent with the statistical error found in the global analysis.

So, in my opinion, the decay signal does not show any anomalous fluctuations and can be considered stable in time.

To further demonstrate this, I compare this result with a simulation where the data are divided in N subsets.

I simulated the data taken in 22.5 days divided in N subsets, and for each subset I calculate the half-life. Then I calculate the mean value of the obtained N half-lifes and the standard deviation SD of the distribution of the N half-lifes. I have done 100 trials for each value of N and I calculated the average <SD> of the 100 values of the standard deviation.

I obtained:

N=1 <SD> = 0.00088 ms

N=9 <SD> = 0.00268 ms

N=45 <SD> = 0.00590 ms

So in simulations the rms of the half-life distribution increases in a way consistent with sqrt(N), as expected by a poissonian statistics, and our data are consistent with this behaviour.