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, $\ensuremath{\rho}(\mathrm{xyz})$, at any point in the unit cell of a crystal may be represented by a Fourier's series the general term of which is ${A}_{{n}_{1}{n}_{2}{n}_{3}}sin(\frac{2\ensuremath{\pi}{n}_{1}x}{{a}_{1}}\ensuremath{-}{\ensuremath{\delta}}_{{n}_{1}})sin(\frac{2\ensuremath{\pi}{n}_{2}y}{{a}_{2}}\ensuremath{-}{\ensuremath{\delta}}_{{n}_{2}})sin(\frac{2\ensuremath{\pi}{n}_{3}z}{{a}_{3}}\ensuremath{-}{\ensuremath{\delta}}_{{n}_{3}})$ ${A}_{{n}_{1}{n}_{2}{n}_{3}}$ 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 $Udr=4\ensuremath{\pi}{r}^{2}\ensuremath{\rho}\mathrm{dr}$ 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.Results of the Fourier analysis. Application of this method of analysis to the calculated $F$ curve from a model sodium ion shows that the series converge rapidly when the $F$ values are uncorrected for the effect of thermal agitation, and that reliable results may be obtained after extrapolation of the experimental $F$ curves for light atoms to zero values of $F$. Curves are given which show the variation of electron density along the cube edges of the unit cells of NaCl, LiF, and NaF, and along the cube diagonal of Ca${\mathrm{F}}_{2}$. $U$ curves for the different atoms, showing the variation of $U$ with $r$, give the following information: (1) the points of the crystal lattice are occupied by ions (no a priori assumptions have been made concerning the amount of electricity associated with a lattice point); (2) the sum of the radii of any two ions in a crystal is approximately equal to the distance of closest approach as determined by ordinary crystal analysis; (3) the electron distributions in the ${\mathrm{Na}}^{+}$ of NaF and NaCl are markedly different, while the distributions in ${\mathrm{F}}^{\ensuremath{-}}$ from all three fluorides are practically identical; (4) there is evidence of the existence of electrons in shells which are in rough agreement with Stoner's scheme of electron distribution.