Electron Distribution in the Atoms of Crystals. Sodium Chloride and Lithium, Sodium and Calcium Fluorides

Determination of electron density by means of a Fourier analysis. The application of the correspondence principle by Epstein and Ehrenfest to Duane's quantum theory of diffraction leads to the conclusion that the electron density, $\rho \mathrm{}\left(\mathrm{xyz}\right)\mathrm{}$, at any point in the unit cell of a crystal may be represented by a Fourier's series the general term of which is $A_{n_1{n}_2{n}_3}sin(\frac{2\pi n_{1}x}{a_1}-{\delta }_{n_1})sin(\frac{2\pi n_{2}y}{a_2}-{\delta }_{n_2})sin(\frac{2\pi n_{3}z}{a_3}-{\delta }_{n_3})$ $A_{n_1{n}_2{n}_3}$ is proportional to the structure factor for x-ray reflection from the ($n_1{n}_2{n}_3$) plane, where $n_1$, $n_2$, and $n_3$ are the Miller indices multiplied by the order of reflection. Considerations of symmetry fix the values of the phase constants, and the assumption that the coefficients are all positive at the center of the heaviest atom in the unit cell fixes the algebraic signs. For crystals of the rock-salt or fluorite types the series becomes a simple cosine series in which the values of the structure factors previously determined by the author may be used as coefficients. If the atoms are assumed to possess spherical symmetry, the number of electrons in a spherical shell of radius $r$ and thickness $\mathrm{dr}$ is $Udr=4\mathrm{}\pi r^2\rho \mathrm{dr}$ and the total number of electrons in the atom is equal to the integral of $\mathrm{Udr}$. A. H. Compton has obtained the same expression for the electron density in a crystal, as well as a series expression for $\mathrm{Udr}$, on the basis of classical theory.