Abstract: A new mechanism for free electron lasers, in which the surface potential of a ferroelectric single crystal may be in place of the wiggler magnetic field, was proposed by the author for the first time 1. Further studies have been made in this article. According to one-body z-coordinate equation of Lorentz equation, the research about simple pendulum equations and resonant behavior of electrons is carried out. The gain formula of the miniature free electron laser for the jth state of surface potential is given as follows: G _ j (t) = \frac 2 e ^ 4 F n _ j \lambda_ j\left(m c _ j 0 \Delta \omega_ j\right) ^ 3 C _ j ^ 4 f ^ 2 \left 1 - \exp \left(- \frac C _ j lW\right) \right ^ 2 \left 1 - \cos \left(\Delta \omega_ j t\right) - \frac12 \Delta \omega_ j t \sin \left(\Delta \omega_ j t\right) \rightThe simple pendulum equations are the same as given in literature 1 .The argument in literature 1 is substantiated.