Time of Flight Simulation

Incorrectly measured time of flight can result in tracks which are incorrectly assigned a large mass. In particular, any late tail in the resolution function will translate into a high mass tail. To study the time of flight resolution we have written a detailed simulation of the physical processes in the scintillator. This program uses the information from the GEANT hits banks described above to produce a simulated TDC and ADC value for each phototube. The energy loss, time and vertical position of all hits in a given counter are combined using geometry information (location and length of the counter, length of the light pipe) to simulate pulses from the photomultipliers at the top and bottom of each struck counter. A discriminator threshold is applied to get the time for each photomultiplier and the number of photoelectrons is added to get the pulse height. The GEANT hits file is then rewritten with the time and pulse height for each struck phototube included.

The timing is simulated by using each energy deposit in the scintillator to generate photons which are traced in time to the photocathode of the photomultiplier where photoelectrons are produced. Each photoelectron is added into an array which represents 50 psec time slices around the relevant time. After all energy deposits are processed, a transformation is applied to the array to represent dispersion in the photomultiplier and cable. The resulting array is then scanned to find the first crossing of the discriminator threshold. This value is then smeared randomly according to a gaussian to represent noise and other unaccounted effects. The final result is recorded as the TDC value for the photomultiplier. The number of photoelectrons is summed and recorded as the pulse height. We do not smear the pulse height since the width and shape of the pulse height distribution is expected to be totally dominated by Landau fluctuations which are included in the GEANT simulation.

The following processes are used in the simulation:

• A Poisson distribution for the number of photoelectrons. The mean is calculated by normalizing the energy deposit from GEANT to 85 photoelectrons for the average minimum ionizing charge 1 particle at 50cm from the end of the counter. An attenuation length of 110cm is used. This represents about one third the number of photoelectrons expected from a 1cm thick counter.
• Geometric dispersion (Fig. ). Photon paths are generated randomly in direction and the flight time to the photocathode is calculated and added to the time from GEANT.
• Photon production - a double exponential is used. The first exponential represents the primary (UV) scintillation in the polyvinyltoluene matrix which pumps the fluors. The fluors then decay with a second exponential time constant to give the visible light (see Fig. ). The time constants used are =0.5ns and =1.5ns.

    
  Figure: Geometric dispersion of light pulse in scintillation counters. The flight time to the photocathode for each photon is calculated for the time of flight simulation.
  

    
  Figure: Photon production in the scintillators is taken to be a double exponential.
  

  These and the attenuation length of 110cm are typical of fast plastic scintillators such as Bicron 418 or Pilot-U.

• Electronics - A double exponential pulse shaping is used to represent the dispersion in the phototube and cable.
• Discriminator - a fixed threshold is applied at about 10% of the average minimum ionizing pulse height.
• Everything else - the discriminator crossing time is randomly smeared according to a gaussian with width of about 220psec. (This is adjusted to give a resolution for the mean time in H3 of 200 psec, which is the design goal.)

Figure  shows the simulated pulses for four successive hits in H1.

  
Figure: Simulated pulse shapes for four successive hits in H1. The left and right plots are for the top and bottom phototubes.

Figure  shows a typical distribution of times for 200000 particles striking the center of one counter. The solid curve is the best fit Gaussian. The distribution clearly has a late tail.

  
Figure: Typical distribution of times for 200000 particles striking the center of one counter. The solid curve is the best fit Gaussian.