This expression is similar to that obtained for the Schottky effect. The factor 2 in the exponent, that makes the barrier reduction in the Poole–Frenkel effect twice larger than that experienced in the Schottky effect, is due to the interaction of the thermally excited electron with the immobile positive charge of the ion acting as a trap center, rather than with its mobile image charge induced in the metal at the Schottky interface. Now if, without any applied electric field, the number of thermally ionized electrons is proportional to