The Harmonic Analysis of Electron Orbits

Abstract

Harmonic analysis of penetrating electron orbits in the Bohr atom has been effected on the assumption that the outer segments of such orbits may be considered as parts of Keplerian ellipses and that the penetrating part of the orbit, which is traversed in a time short compared with the period of the Keplerian motion, may be represented arbitrarily as a continuation of the exterior motion. If the Fourier series is written in the form $x+iy=\Sigma (-\infty \mathrm{to} +\infty )C_{\tau }{e}^{2\pi i(\tau \omega +\sigma )t}$, the formula for the amplitudes is $C_{\tau }=\left(\frac{a}{2\pi }\right)\sqrt{{sin}^2\left(\frac{2\pi \sigma }{\omega }\right)+{\left(cos\right(\frac{2\pi \sigma }{\omega })-1)}^2} \Sigma \stackrel{\prime }{}(-\infty \mathrm{to} +\infty )b_m{J}_m\left(\rho \epsilon \right)\mathrm{with} b_m=\frac{(1+{\epsilon }^2)(\rho -m)+{\epsilon }^{\prime }}{{(\rho -m)}^2-1\mathrm{}}-\frac{\epsilon {\epsilon }^{\prime }+\frac{1}{2}\epsilon (\rho -m)}{{(\rho -m)}^4-4\mathrm{}}-\frac{\left(\frac{3}{2}\right)\epsilon }{\rho -m}$ where $a=\mathrm{major}\mathrm{axis}\mathrm{of}\mathrm{the}\mathrm{outer}\mathrm{segment}$; $\frac{\sigma }{\omega }=\mathrm{ratio}\mathrm{of}\mathrm{frequency}\mathrm{of}\mathrm{precession}\mathrm{to}\mathrm{the}\mathrm{frequency}\mathrm{of}\mathrm{Keplerian}\mathrm{motion}=2\mathrm{}\pi \mathrm{times}\mathrm{angular}\mathrm{separation}\mathrm{of}\mathrm{outer}\mathrm{segments}$; $\rho =\tau +\frac{\sigma }{\omega }$; $\epsilon$ is the eccentricity of outer segment; ${\epsilon }^{\prime }=\sqrt{1-{\epsilon }^2}$ The $J\text{'}\mathrm{s}$ are Bessel functions of the first kind. Tables are given of the values of $C_{\tau }$ for $\epsilon =.\mathrm{}3,.\mathrm{}6,.\mathrm{}866,1\mathrm{}$ and $\frac{\sigma }{\omega }=0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1\mathrm{}$. The error involved in the method may be large for high order harmonics or small values of $\epsilon$. In the case of some orbits of sodium ($3_1$, $3_2$, $4_2$, and $5_2$) the calculated values of the main coefficients agree fairly well with values obtained by Thomas from spectroscopic data by the method of Fues. Applications of this analysis to intensity relations in spectra will be made in a later paper.