Angular Resolution

One of the goals of the analysis is the particle identification by cluster
recognition. In order to reconstruct the angle(s) of incidence I first
studied the angular resolution of our detector.

The idea is simple, looking for the mean of the shower distribution to
reconstruct the point of incidence and compare it with the thrown angle
in the simulation. To do so, I threw positrons uniformly all over the detector
array of last summer's run and looked for the crystal which had a contribution
to the total energy if the maximum was within the 'inner' crystals. Since
the shower develops in a cone touching some neighboring pyramids the portion
of energy contributed from a crystal was weighted by some exponent a
< 1 . So the formula to reconstruct the point of incidence becomes
where q is the fraction of energy E_{i}/E_{tot} . Now a
was to be optimized.

Therefor an angle g was defined describing the
deviation of the reconstructed position from the thrown position. This
angle is given by
and had to be minimized by varying a. Looking
for the mean of the distribution of g, a
was found to be 0.71 .

This gave an angular resolution of 3.7 degrees.

- A comparison of the positron and photon response at different energies can be found here.
- Due to the size of the crystal's front face with an opening angle of
approximately 12 degrees you do not reach an uniform distribution of the
reconstructed vectors (Fig. 1). This one could get by changing a
down to ca. 0.3 but - as one can see by looking at g
- is due to a smearing of the angular resolution (Fig. 2). The black dots
mark the theta vs. Phi position for the reconstruction while the yellow
dots represent the thrown angles.

This has to be compared with the 'starting'-value a=1 ; the number given in Ketevi's thesis was 5 degrees for the energy resolution. (I am still puzzled why this figure is approximately 2 times better than Ketevi obtained it. See page 158 of his thesis)

The main limit of obtaining a better angular resolution (besides the granularity) are the shower fluctuations which lead to a large uncertainty of the reconstructed position of incidence. This can be seen in the following picture where I threw positrons exclusively to a point 5 degrees in phi right of the center of the Pent. Without a weighting (or even more with a >1 ) the spot looks smaller, but is completely off, since the position of a crystal's center is overrated. Fig. 4 to 8 are showing the reconstructed spot size and differences between thrown and reconstructed angles at different weighting factors a.

a = 1 :

a = 3:

a = 0.71 :

a = 0.43 :

There is hardly a convenient way for a further improvement of the angular resolution. Only if one could retrieve the exact appearance of the shower and vary the weighting factor accordingly to the size/shape of the shower and the crystals which were hit, a result of better than 3 degrees could be expected.

Thomas Flügel 04-02-97