[Halld-cal] Linearity requirements

[George Lolos]

Hi guys:

I have been thinking again about the perceived need for linearity  
measurements capability in the calibration system.  Let's look at the  
numbers a bit more carefully and, please, correct me if I am wrong:

Nocc = M x [1-exp(-PDE x Nph/M)]

Where:   Nocc is the number of pixels occupied by processing incident  
photons

                 M is the number of pixels in the array

                 Nph is the number of incident photons

The ratio, then, of Nocc/M is the degree of non-linearity.   For  
example, if M= 56,000 pixels, PDE is 0.20 and Nph ~ 50,000 per side  
per GeV, we get Nocc/M ~ 8.5%, assuming that ~20,000 photons go into  
only one array.  So, the number of ~10% non-linearity that Elton often  
quotes is in the right ball park.  So, the calculations are correct  
but is the input correct?

Let's start with the 56,000 ph/side/GeV.  This number comes out of  
Andrei's calculations based on the number of p.e.'s we get from the  
fibers using the Sr90 source and Irina's simulations of the energy  
deposition in the matrix.  However, these numbers were obtained with  
naked fibers, where the full effect of the outer cladding comes into  
play.  In the BCAL, the fibers trap the light in the region between  
the core and the first to second cladding volume, so the trapping  
efficiency will be reduced.  We will know the number of effective  
photons trapped in the BCAL when we get the cosmic ray measurements  
done.  In any case, the number will be less.

Now come the really significant assumptions:

Even at oblique angles (less than 20 degrees) is it reasonable to  
expect that all the photons on one side will go into a volume confined  
in two dimensions within 2 cm x 2 cm?  I have a difficult time  
accepting this and perhaps a real simulation of the shower generated  
by 1 GeV photon randomly distributed across the 2 cm width of a read- 
out cell will give us better answers.  The Moliere radius alone is  
over 3 cm so spill over will occur even if the photon was incident  
dead center on a read-out cell.  So, how many photons will be  
generated in a read-out cell as a fraction of the total energy  
deposited in the BCAL?  We need a better result to draw conclusions on  
this.

Even if we take the 56,000 photons per side per GeV, this number  
corresponds to the number of photons arriving at the interface of the  
BCAL with the light guide.  Here one has to start counting losses:   
losses crossing the glued interface from fibers to light guide.  Then,  
and this is a significant number, losses from end of light guide to  
the SiPM.  Unless I recall wrong, our measurements here between  
physical contact with Si grease between light guide and PMT, on one  
hand, and air gap on the other, the latter was around 50% of the  
former.  In other words, just by the air gap we lose about 50% of the  
light.   So, the realistic maximum number of photons reaching  
photosensors per side per GeV is much closer to 20,000 than to  
56,000.  If half of them are confined within the volume of one read- 
out cell (one SiPM), then Nocc/M  is ~ 3.5%.

Is such non-linearity an issue to try and monitor if it means more  
complicated calibration system?

George

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