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Re-calibrating UCAL to correct for the light absorbtion.

The algebra here provides technical background for the material discussed in  4.4.2. By $A$ and $C$ I denote the amplitude and calibration constant in case of no light absorbtion, and by $a$ and $c$ - the actual ones, respectively. If the light intensity in scintillator gets attenuated with distance $x$

$$a = A e^{-x/L},$$

and the scintillator length is $S$, then the amplitude observed in a single channel (averaged over all positions in $x$ and eliminating the pedestal) is not $A$ as it were in the absence of attenuation, but $a = AL/S(1-\exp(-S/L))$. Each UCAL tower is served by two PMT tubes, so we have $a_1$ and $a_2$; their gains are matched so that $<a_1> = <a_2>$. Calibrating a channel , I require that

$$( A_1 + A_2) L/S(1-\exp(-S/L)) c = E_0,$$ (98)

where $c$ is to be determined and $E_0$ is known. We will denote $(A_1 + A_2)C$ as $E_{cal}$ to indicate that this is the energy observable which we know how to calibrate.

The left and right PMT register the light which passes the distance $x$ and $S-x$, respectively (if the source is point-like). We use the product of the two so that our new observable is

$$E_{\times} = c \sqrt{A_1 e^{-\frac{x}{L}} A_2 e^{-\frac{S-x}{L}}} = c \sqrt{A_1 A_2} e^{-\frac{S}{2L}}$$ (99)

It is independent of $x$, but has yet to be calibrated, i.e. related to $E_{cal}$. Next we notice that $C=c L/S(1-\exp(-S/L))$ from  A.1, and that in the absence of absorption for the gain-matched tubes $\sqrt{A_1 A_2} = (A_1 + A_2)/2$, and therefore

$$E_{\times} = \frac{\exp(-S/2L)}{2L/S (1-\exp(-S/L))}(A_1+A_2)C = \frac{ <\sqrt{a_1 a_2}>}{<a_1+a_2>} (A_1+A_2) C,$$ (100)

where we recognized that the $S$ and $L$-dependent prefactor can be expressed in terms of the easily measurable averages. This means that calibration can be restored for the product with the help of the expression:

$$E_{\times} = \frac{<\sqrt{a_1 a_2}>}{<a_1+a_2>} E_{cal}$$ (101)

By itself, this result is natural and predictable. Its value lies in the fact that absorbtion length is now related to observables.