Monte Carlo Acceptance Simulations for the Prototype Active-Target Time-Projection Chamber

In a previous experiment (Fritsch et al., Phys. Rev. C $\textbf{93}$, 014321) the Prototype Active-Target Time-Projection Chamber (PAT-TPC) was used to investigate the $\alpha$-cluster structure of $^{14}$C by way of a 38~MeV secondary $^{10}$Be beam incident on a 90:10 He:CO$_2$ active target gas at the University of Notre Dame. The $^{10}$Be beam was produced by TwinSol and delivered to the PAT-TPC. In addition to measuring elastic and inelastic $^{10}\text{Be}+\alpha$ resonances, evidence of 3-body decays of $^{14}$C were observed in the data. Current work is being done to create a Monte Carlo simulation to calculate the detector acceptance for the 3-body decays at relevant reaction energies in order to produce normalized cross sections. Preliminary results will be presented.