[Halld-cal] Curvature

[beattite]

Hello.

Last week, Mark asked for some sort of plot showing the effects of the 
shower curvature for why we might want to be doing a correction to the 
IU shower code, since it's been a while since Andrei's original work and 
we all want to be on the same page.

The attached picture is point r vs. point z for a simulated 5 GeV 
photon gun at 15 degrees.  The picture shows all reconstructed points in 
the run.  It gives the point z distribution for an incident photon at 15 
degrees (I chose such a large energy, 5 GeV, so that there are enough 
points in layer 4 to talk about).

If we expect the centroid of the point z distribution to lie along the 
trajectory of the photon, we would expect the centroid in the four 
layers to be r/tan(15):
   Layer 1 - 242 cm
   Layer 2 - 256 cm
   Layer 3 - 274 cm
   Layer 4 - 302 cm

What we actually see are centroids in the four layers of:
   Layer 1 - ~253 cm
   Layer 2 - ~257 cm
   Layer 3 - ~260 cm
   Layer 4 - ~265 cm

This is similar to what Andrei's plots from a lot time ago show, as in, 
the centroid in layer one is shifted in the positive z-direction from 
the photon trajectory while the centroids in layers three and four are 
shifted in the negative z-direction from the photon trajectory.  The 
centroid of layer two lines up nicely with the trajectory.

The IU code uses the Moliere radius around the trajectory to include 
points.  At low angles like 15 degrees, the spread of the cone in z is 
something like +\- 75 cm.  That is to say, the code is looking for 
points in the shower between z = 230 and z = 370 or so in layer four.  
While this does actually cover the shower including the curvature (in 
this case), it also has a huge acceptance in the positive z-direction 
that is unneeded and which possibly catches points which belong to a 
different shower.  So, the idea is to take the initial energy estimate 
from the IU shower along with the angle estimate of the IU shower (which 
is based off points in layers 1, 2, and 3, and so should be close to the 
angle of the actual incident particle), then use the curvature tables 
provided by Andrei which simulated in more detail the energy deposition 
of an incident photon in the BCAL at different angles and energies.  The 
tables give energy deposition centroids in each layer which we can then 
use to fill the showers with points rather than the Moliere radius that 
the IU code used.

The effect is less pronounced at, say, 50 degrees