Here is what I have so far for modifying the voltages on the small FMS cells to give a more uniform gain as a function of eta. In the interest of transparency, here is the procedure I currently follow:

1. Read the gain data from Jingguo's spreadsheet containing: 2009 voltages, 2009 gains and fits to the ADC vs. voltage calibration curves. The ADC vs. voltage curves that I reproduce from the fit parameters are uploaded as fmsAdcVsVoltage.pdf.
2. Scale the ADC vs. voltage curves by the 2009 gain at the 2009 voltage to get gain vs. voltage curves. These are uploaded as fmsGevVsVoltage.pdf.
3. Remove those cells whose ADC vs. voltage curves prove problematic (more details below).
4. Plot gain vs. the η of the physical centre of the cell (the η used for each cell can be seen in my previous blog post), then use a profile histogram to find the mean gain as a function of η (see figure 1).
5. Fit the large-η portion of this profile, where there is an approximate plateau in the gain vs. η. Use this as the constant gain to aim for in each band in 
   η.
6. Find the mean gain at each η from the profile histogram. I had been fitting the histogram and using the fit to evaluate the mean at each η, but didn't find the fitting to be very stable, so I use the histogram directly.
7. Scale the gain of each cell by the ratio between the desired mean gain and the mean gain at its η
8. Use the gain vs. voltage curves to find the voltage required to give this new gain.
9. Recalculate the new mean gain as a function of η to see how things have changed (see figure 2).
10. Print out a text file with the new gains and voltages in the same format as the input data, allowing iteration.

Figure 1: 2009 gain vs. cell η. The errors on the profile histogram are just supposed to be indicative of the spread, and aren't real statistical uncertainties.

Figure 2: Gain vs. cell η after a single iteration of the procedure.

To be clear, note that I don't individually scale the gain of each cell to the desired value, but scale to obtain a mean gain at the η of that cell equal to the desired value. Figure 2 shows the results after a single iteration. It looks to give a decent improvement except at small η (large θ). If the procedure seems sound, then what remains is to deal with certain problematic cells, which I will now describe in more detail.

Problematic cells

There are some cells that prove problematic, for the following reasons:

• Issue 1: There is no 2009 gain measurement (zero in the spreadsheet).
• Issue 2: There is no ADC vs. voltage calibration curve, for example detector 3, channel 102.

There are then two problems related to the following: sometimes the ADC vs. voltage curve "saturates" at 4048 (maximum value for 12 bits) at some voltage less than the maximum (FF in hexadecimal, 255 in decimal). An example is detector 3, channel 9 (see figure 3). Two possible things can then happen:

• Issue 3: The 2009 voltage measurement lies within the plateau, so the 2009 gain measurement and so does not even appear on the curve and I can't use it to extrapolate.
• Issue 4: The 2009 gain and voltage lie outside the plateau but the calculated new gain to extrapolate to is "above" the plateau, so a new voltage cannot be determined.

Figure 3: ADC vs. voltage of FMS subdetector 3 (south small cells) channel 9.

Because of the way I have written my code, I exclude cells with issues 1, 2 or 3 from the procedure, so there are all 476 small cells in the 'before' plots (figure 1), but only 339 in the 'after' plots (figure 2). Those cells experiencing issue 4 are included, but don't move (because I cannot determine a new gain/voltage) and can be seen as static points between the 2D plots. These are, I think, the main cause of the gains remaining low at small η in figure 2. While I know what the desired gain for these cells is, I don't know how to adjust the voltage to a new value to give that gain. Perhaps I am missing some piece of information or have misunderstood the information given to me.

I also haven't looked in great detail into how stable this procedure is under numerous iterations.

Code, plots and so forth can be found at /star/institutions/bnl/tpb/FMS/gains.

Questions, comments and concerns welcome.