and of the presence of the boundary effects, at high temperatures ($T\geq T_{cr}\approx 0.1 $ MeV) we find a lower electron-positron flux as previously estimated. The numerical value of the critical temperature $T_{cr}$ depends on the surface potential of the star. We briefly consider the effect of the magnetic field on the pair creation process and show that the magnetic field can enhance drastically the pair creation rate.'
author:
- 'T. Harko$^1$ and K. S. Cheng$^2$'
title: 'Electron-positron pair production in the electrosphere of quark stars'
---

Introduction
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The Schwinger mechanism of pair production [@Sch51], first proposed to study the production of electron-positron pairs in a strong and uniform electric field, has been applied to many problems in contemporary physics. Strong electromagnetic fields lead to two physically important phenomena: pair production and vacuum polarization. A strong electric field makes the quantum electrodynamic vacuum (QED) unstable, and consequently it decays by emitting a significant number of boson or fermion pairs [@Gr85].

For a spin $1/2$ particle Schwinger’s predicted production rate per unit time and volume $w$ is given by [@Sch51; @So82] $$\label{prod}
w(E_0)=eE_0\int \frac{d^{2}k_{i}}{\left( 2\pi \right) ^{2}}\sum_{n=1}^{\infty }\frac{1}{n}\exp \left[ -\frac{\pi n\left( m_{e}^{2}+k_{i}^{2}\right) }{\left| eE_0\right| }\right] ,$$ where $m_{e}$ and $e$ 