# Defining Occupancy

# General Occupancy

The occupancy measures the number of particles interactions per a detector cell per an event. For the CLAS12 drift chamber, there are 112 wires on each layer, with 12 layers within a region, giving 1344 cells. This can simply be defined as the "Unweighted Occupancy" for the CLAS12 DC and follows the equation:

*Unweighted*CLAS12 DC Occupancy

where

# CLAS12 DC Occupancy

The registering of a "hit" takes a finite time in which the detector and its associated electronics are not able to register an additional signal if it occurs. This time window is known as the "dead time" during which only limited events are registered. For Region 1:

Since the events are simulated outside the dead time constraints of the DC, we can factor in the number of event windows that occur by dividing the dead time window per region by the time that would have been required to produce the number of incident electrons given a known current.

When applying the Moller differential cross-section as a weight, this gives the CLAS12 occupancy as:

## Weighted Hits Occupancy

### Lab Cross-Section

Using the definition of the cross-section:

where the flux is defined as:

Making some assumptions that the flux can be taken over an same time range as the time found in the cross-section, which allows

For a LH2 target of length 5cm.

Additionally, for Moller Scattering, we can assume that almost 100% of the scattered electrons occur as events.

This allows us to rewrite the cross-section expression as,

we can define a differential scattering cross-section, , in the laboratory frame, where is an element of solid angle in this frame. Therefore, is the effective cross-sectional area in the laboratory frame for scattering into the range of scattering angles to and transverse angles to . Likewise, is the effective cross-sectional area in the CM frame for scattering into the range of scattering angles to and transverse angles to . By relativity, only the scattering angles are Lorentz contracted in the direction of motion, leaving the transverse angles invariant between frames of reference. Additionally, an effective cross-section with corresponding cross-sectional area is not changed when we transform between different inertial frames. The number of scattered particles and incident particles are invariant between frames, while the cross-sectional areas are also measuring the same relative space.

Since the expression for the differential cross-section for Moller Scattering is well known in the CM, we can solve for the minimum angle detected by the DC (.55 degrees in Theta) in the lab frame.

Here we will use a definite integral to find the cross-section in the lab. For a constant

this corresponds to a that will become smaller as increases. in both frames due to it being Lorentz invariant.