Estimating the HV -> gain exponent for EEMC tower PMT's using slopes

These results are based on a set of runs taken near the very end of run 7, toward the end of a very stable fill.

• Runs 8171053 and 8171056 were taken with the then-current (Run 7 default) HV set, and together contain ~800k minbias events (run configuration was 2007EmcMbStatus).
• I then loaded a new set of HV's for all 720 endcap tower PMT's, and took ~1M events in run 8172001.
• Jason Webb and students at Valpo produced histograms and extracted slopes for each tower for each of the two data sets.

Our goal was to do a better job of matching the tower gains to their "ideal" values, and most of the HV changes were small - less than 10V - so this was not an optimum data set to determine the slope <--> HV relationship.  Nevertheless, the ratio of slopes for each tower matched quite well with the expected and desired change in gain. The new HV set had been calculated assuming an exponent of 9.3; more detailed fits suggest that 8.8 would be somewhat better.

The figure below shows the measured % change in slope vs the input % change in HV between the two data sets, along with curves showing the expected relation for three values of the exponent.

For the next figure, I calculate a value for the exponent (kappa) for each tower, according to

kappa = ln(slope1/slope2) / ln(HV1/HV2)

and then plot this versus the magnitude of the HV change. The scatter clearly decreases rapidly with increasing dHV, but values are centered around a kappa of just less than 9 for all HV changes.