Theory of Necessary Energy for Electron to Exit from conductor
A model of a conductor consisting of a positive charge that uniformly fills a certain volume and is absent outside it, and an electron gas whose total charge is equal to a positive charge are considered. The electron density outside the conductor should tend to zero, while remaining a continuous function of coordinates. It is assumed that the electron density is proportional to some continuous function, which depends on the coordinate perpendicular to the surface of the conductor. This function depends on the parameter, which is determined from the requirement of a minimum of additional energy arising as a result of the exit of a certain number of electrons outside the conductor. The inhomogeneous density of the negative charge does not compensate for the homogeneous density of the positive charge. As a result, an electrostatic potential is created inside the conductor. Above the Fermi level, an exponential potential well is formed. Zero of this potential well is located in the center of the conductor. Under the boundary condition that the eigenfunctions vanish at the conductor boundary, a discrete energy spectrum is obtained for an electron above the Fermi level. To go beyond the surface of a conductor, an electron from the Fermi level must receive energy no less than the lowest level of the energy spectrum of a potential well. This explains the laws of the photoelectric effect. The problem is considered for a spherical particle and a flat plate. The smoothing parameter and the minimum energy for the electron exit turn out to be different.
 Dubrovskyi, I (2018) Inhomogeneous density and energy spectrum of elementary excitations of electron liquid of conductors in magnetic field. BJMP,4(1), 313-322.
 Kratzer, A. und Franz, W (1960). Transzendente Funktione. Akademische Verlagsgesellschaft: Leipzig .
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