Materiales y Métodos

Equation5.7shows that the gain is equal to the mean of the single-photoelectron peak. To find the single p.e. peak, several Gaussian distributions are fitted to the charge distribution of a readout channel. The Gaussian fit is centred on one bin of the histogram and covers ±30 % of the central value of the bin, either side of the bin. Each bin from 0 to 300 ADCs is covered in this way. Each mean of each fit is then added to another histogram, with the same binning as the charge distribution (figure 5.4(b)). If the bin with the highest number of mean values is more than double the next highest value, the centre of this bin is taken as the centre of a new Gaussian fit of the charge distribution spectrum. If it is less than double the channel is not fitted. The Gaussian distribution is fitted to ±30 % of this new centre value (figure 5.4(a)). This method achieves a constant value for the s.p.e. peak quickly, so can process many channels without intervention.

However, this method ignores the higher p.e. peaks and the pedestal by only fitting to ±30 %, so does not give any further information about the PMT.

5.3.3.1 Justification of Fit Method

Various binnings of the spectrum and ranges of the fit were considered. The fit was optimised to give results as close to the LI as possible for the FD data taken on the 25^th June 2007, although some optimisation was made for the ND on the data taken on the 6^thJune 2008. One date was chosen to optimise in order to get a start value as close as possible for the s.p.e. fit gain and the database gain. The full range of the spectrum could not be used to fit a Gaussian distribution to, as there are peaks associated with two and higher p.e. hits as well as the s.p.e. peak, which pull the mean fit higher. These higher peaks would vary at a different rate to the s.p.e. peak and thus introduce uncertainty to the drift calculation. To counter this only part of the spectrum was fitted. Spectra that have a higher s.p.e. peak

ADC

Figure 5.4: a) An example of a channel with its final Gaussian fit. The gain is 64.92. b) An example of the distribution of the mean values found by sweeping across the spectrum to find the s.p.e. peak. As the bin with highest frequency is at 64 ADC the final fit (that is shown in figure 5.4(a)) is then centred there on the spectrum and the fit values are found.

ADC

Figure 5.5: An example of a good ND channel: a) The fitted spectrum b) his-togram of means from sweeping the spectrum.

5.3. GAIN CALIBRATION WITH SINGLE-PHOTOELECTRONS 81 have a wider distribution than spectra that have a lower s.p.e. peak value. For this reason, a percentage of where the mean was thought to be was fitted rather than a fixed range. The figure of merit used to distinguish between the methods was to fit a Gaussian distribution to a histogram of (s.p.e.-LI)/(0.5(s.p.e.+LI) and then take the RMS, which is called spread. Table5.1 shows that 30 % coverage of the central bin value had least spread.

ADC

Figure 5.6: An example of a rejected ND channel: a) The fitted spectrum b) histogram of means from sweeping the spectrum.

To find the peak of the spectrum a sweep across the spectrum up to 300 adc, centring on all the bins fitting a Gaussian distribution, and the means of these distributions, were added to a histogram. The median and mode value of this histogram was used to find the centre of the main fit. Table 5.1 shows that the mode had less spread. The width of the s.p.e. peak should not be wider than the mean value of the s.p.e. peak. This made the spread value slightly worse, but made the fit more stable between runs. Table 5.1 shows that the finer the bins used to find the s.p.e. peak the lower the spread value, however, enough statistics need to be acquired with each PMT in each run to make a fit. Five adc counts per bin was chosen for the FD as in later runs not quite enough data was taken for fits to all channels. The final requirement was that the channel had to be good for all dates that were taken. This final requirement tightens the spread in both the ND and FD The fit that was found to work best in the FD was used for the ND.

However, some optimisation took place (table5.2). Ten adc counts per bin, rather than five adc counts per bin, was found to give a lower spread of values due to

Type Fixed Free Bias Spread Number

of fit in FD slope slope of entrees

Median R30 bin 5 0.999 1.156 0.024 0.0779 3600 Mode R30 bin 5 0.985 1.044 -0.000 0.0609 3600 Mode R20 bin 5 0.974 1.015 0.020 0.0801 3600 Mode R60 bin 5 0.979 1.030 -0.008 0.0634 3600 Mode R30 bin 5 fw 0.984 1.046 0.001 0.0626 3600 Mode R30 bin 2 fw 0.990 1.125 0.000 0.0614 3600 Mode R30 bin 10 fw 0.972 1.053 0.026 0.0774 3600 Mode R30 bin 5 fw height 0.983 1.046 0.001 0.0626 3600

Mode R30 1.100 0.002 0.0542 1501

bin 5 fw height all sets

Table 5.1: The slope of the profile of the 2D histogram between the s.p.e. gain and the gain found by the LI system in the FD for data taken on the 22/06/2007 (Figure 5.9(a)). The type of fit is described by whether the peak was found by using the median of fits in the first sweep or the mode, and how many adc counts were included in each bin. fw is the Gaussian distribution width has to be less than the found mean of the fitted Gaussian distribution and height is when the mode bin in the first sweep is 100 % more than the next highest bin. The second column shows the value the fit if the profile of the 2D histogram is forced through 0. The third column shows the value if the fit of the profile of the 2D histogram is allowed to float. The bias is the mean of the Gaussian distribution fitted to the distribution (s.p.e.-LI)/(0.5*(s.p.e.+LI)) and the fourth column shows the RMS of the Gaussian distribution (figure 5.9(c)). The lowest spread was chosen for the fit. However, due to limited statistics of later runs the 5 adc counts per bin was used.

5.3. GAIN CALIBRATION WITH SINGLE-PHOTOELECTRONS 83

Figure 5.7: 2D histogram comparing the s.p.e. gain to the LI system gain for the ND. a) Without the requirement that the mode has to be twice the value of the next highest bin. b) With the requirement the mode has to be twice the value the next highest bin. It can be seen that this extra cut cleans up many of the points where the s.p.e gain is low compared to that of the LI system.

fewer statistics in the ND run. Figure 5.7 shows that there are many PMTs that have low gain for the s.p.e. method. This only affected the ND as the PMTs have a continuos readout, and thus affected by where the trigger is set to separate hits.

A trigger level had to be set to distinguish between events which meant on some distributions the peak value was cut so the peak was not clear. To over come this an additional requirement that the bin with the highest number of mean values found had to be 100 % larger than the next highest bin. Figure 5.5 shows a ND channel that meet all requirements. Figure 5.6 show a spectrum that failed the requirement that the mode of the means be 100 % the next highest value. In the far detector this requirement had no effect.

Type Fixed Free Bias Spread Number of

of fit in ND slope slope entrees

Mode R30 bin 10 fw 0.811 0.861 0.082 0.2251 8012 Mode R30 bin 10 fw height 0.861 0.904 -0.106 0.1084 3341 Mode R30 bin 5 fw height 0.868 0.893 -0.070 0.1305 1454

Mode R30 0.854 0.886 -0.118 0.1021 1695

bin 10 fw height in both sets

Table 5.2: The slope of the profile of the 2D histogram between the s.p.e. gain and the gain found by the LI system in the ND for data taken on the 06/06/2008.

In the ND the mode and 30 % range was used from the FD and the width was limited to less than the mean value of the peak. Ten and five adc counts were included in each bin. Height is when the mode bin in the first sweep is 100 % more than the next highest bin. The second column shows the value of the fit of the profile of the 2D histogram when the fit is forced through 0. The third column shows the value of the fit of the profile of the 2D histogram when the fit is allowed to float. The Bias is the mean of the Gaussian distribution fitted to the distribution (s.p.e.-LI)/(0.5*(s.p.e.+LI)) and the fourth column shows the RMS of the Gaussian distribution (figure5.9(d)). The fit method with the lowest spread was chosen for the fit.

5.3. GAIN CALIBRATION WITH SINGLE-PHOTOELECTRONS 85 5.3.3.2 Other Fit Methods

Another fit was also considered that was used in [101]:

f (x) = N e^−λ 1

where xped is the pedestal mean, σped is the pedestal width, λ is the light level (mean number of photoelectrons), xpe is the mean of the s.p.e., σpe is the width of the s.p.e. peak width, df is the fraction of pulses that miss the first dynode and then strike the second dynode first and ds is the dynode scale (the amount of multiplication they miss). Figure 5.8 shows a result of the fit. The pedestal was fitted separately to help with fitting speed. This method has the advantage of tak-ing account of the Poisson nature of the s.p.e.. However, this method requires the pedestal to be present, which complicates the taking of the data. Also the function takes a lot of computer processing to come to a fit, which often depends on the start values. To decide which of the fits is correct requires human intervention.