  Magnetic bottle Time-of-Flight (ToF) spectrometers can measure the energy
spectra of all electrons emitted into a 2$\pi$ sr solid angle simultaneously,
greatly reducing data collection time. When the detection of the annihilation
gamma ($\gamma$) and the detection of the electron (e) are used as timing
signals for ToF spectrometers, the e-$\gamma$ time difference spectra
(e-$\gamma$ TDS) are reflective of the positron-induced electron energy
distributions provided the times between the impact of the positrons and the
emission of the annihilation gammas are short compared to the flight times of
the electrons. This is typically the case since positrons have short lifetime
in solids ($\sim$ 100 - 500 ps) compared to the flight times of the secondary
electrons ($10^2$ ns to $10^3$ ns). However, if the positron leaves the surface
as a positronium atom (a bound electron-positron state), the annihilation gamma
photons can be appreciably delayed due to the longer ortho-positronium (o-Ps)
lifetime. This can result in an e-$\gamma$ TDS having an exponential tail with
a decay constant related to the o-Ps lifetime. Here, we present an analysis of
the e-$\gamma$ TDS using a Monte Carlo model which estimates the spectral
contributions resulting from o-Ps annihilations. By removing the contributions
from the delayed gamma signal, the energy spectrum of Positron Impact-Induced
Secondary electrons (PIISE) can be isolated. Furthermore, our analysis allows
an estimation of the intensity of the exponential tail in the e-$\gamma$ TDS
providing a method to measure the fraction of positrons that form Ps at solid
surfaces without relying on assumed 100% Ps emitting surfaces for calibration.
