comments on timing averages

[Elton Smith <elton@jlab.org>]

I would like to comment on computing timing averages from various cells in
the Bcal. Please refer to GlueX-doc-804 (Analysis of the Bcal Hall-B Beam
Test), eqs 14 and 15.

Let me begin with the well-known relation between the time of impact time
t0 and impact point x (along the Bcal) and the measured right and left
times tL and tR:

tL = t0 + x/veff
tR = t0 + (L-x)/veff

veff is the effective velocity of the scintillation photons along the
Bcal, and L is the length of the Bcal. Then

x = (1/2)*(tL-tR)*veff
t0 = (1/2)*(tL+tR) - L/(2*veff).

Note that the decomposition is possible because of the linear relation
with x which is precisely matched (with opposite signs) between right and
left.  In particular, the equations for x and t0 are NO LONGER VALID if tL
and tR are weighted by their respective energies.

However, weighting different cells with their energies is perfectly
legitimate and appropriate in the following way:

Let deltat_cell = (1/2)(tL-tR), then

delta_x=veff* Sum_cells(deltat_cell * sqrt(EL*ER)/Sum_cells(sqrt(EL*ER))

Where we have assumed that deltat_cell is proportional to sig0/sqrt(Npe),
where sig0 is a constant (related to the single-photoelectron resolution).
The number of photoelectrons Npe is proportional to the energy deposited
in the module which is proportional to sqrt(EL*ER). This expression
averages the statistics from both sides, which happens when the two times
are averaged.

The above expressions can be taken over directly for estimating the
resolution on t0, since the (constant) expression L/(2*veff) does not
contribute to the jitter.



Elton Smith
Jefferson Lab MS 12H5
12000 Jefferson Ave
Suite # 16
Newport News, VA 23606
elton@jlab.org
(757) 269-7625
(757) 269-6331 fax