State of thermal equilibrium in an enclosure having tungsten walls and containing caesium vapor.—In a large heated enclosure having tungsten walls and containing Cs vapor in thermal equilibrium, there are ions, electrons and atoms, with concentrations given by $\frac{n_e{n}_p}{n_a}=K$ where the equilibrium constant $K$ can be calculated from the ionizing potential of Cs vapor. The interior of the enclosure constitutes a typical plasma with $n_e=n_p$. Near the walls are space charge sheaths in which particles of only one sign are usually present. The potential distribution of the weak fields that reach into the plasma are governed by the Debye-Hückel theory characterized by the Debye distance ${\lambda }_D$. The sheaths contain an excess (or deficiency) of ions and there is a corresponding sheath adsorption which is related to a negative surface tension (spreading force $F$) by Gibbs' adsorption equation.

Electrc image forces near plane metallic surfaces.—Still closer to the surface there is an image force sheath in which the distribution of ions and electrons is governed mainly by the image force $f={\left(\frac{e}{2x}\right)}^2$. Throughout both the space charge and the image sheaths, the concentration of electrons or ions (whichever is larger) is given by $n=\frac{\exp \left(\frac{x_s}{x}\right)}{8\pi x_s{(x+x_L)}^2}$ where $x_s$, the Schottky distance, is $\frac{e^2}{4kT}$ and $x_L$ is the $v$. Laue distance equal to ${\left(\frac{\mathrm{kT}}{2\pi e^2{n}_1}\right)}^{\frac{1}{2}}$, $n_1$ being the concentration at the boundary between the space charge sheath and the image sheath (corresponding to the saturation current).

Perturbation method for the study of image forces.—Within distances of a few Angstroms from the surface, the classical image force requires modification because the "effective reflecting plane" which determines the location of the image, changes its position as any given electron or ion approaches the surface. The image force for ions is thus greater than for electrons when these particles are at a given distance from the surface. A general method is devised for calculating the image force (a second approximation) acting on electrons very close to the surface. An approximate calculation of the electron distribution is made and then the perturbations produced by a given electron are used to determine the image force on that electron. In a modification of this method, the perturbation produced by a given electron in the distribution of neighboring electrons is considered to be characterized by a perturbation free path which is used to calculate the location of the reflecting plane and the resultant image force.

Application of methods to electrons with Fermi distribution.—These methods are used to calculate the image force on a given electron resulting from electrons near the surface of a metal (both inside and outside of it) which have a Fermi distribution consistent with the Poisson equation. A third approximation can then be made by considering that all of the electrons in the Fermi sheath are similarly acted on by an image force so that the sheath becomes much thinner than calculated by the second approximation. In Part II of this paper it will be shown that, because of the thinness of this sheath, it becomes possible to calculate the image force far more accurately by a new displacement method than by these two approximations. Similar image force considerations should govern all applications of the Fermi theory to concentrated electron atmospheres in which there are large concentration gradients.

• Received 29 December 1932

DOI:https://doi.org/10.1103/PhysRev.43.224