Gaussian Band Shape Derived from Finite-Rate Oscillator Growth

Abstract

The accepted classical model for derivation of the frequency spectrum from the time-dependence of oscillator activity is considered. The exponential decay function giving rise to a Lorentzian band is modified to allow for a finite rate of growth of the oscillator, the resulting frequency spectrum is obtained, and the form normalized to unit band half-width is compared to the equivalent Lorentzian and Gaussian bands. It is shown that as the rate of oscillator growth decreases from infinity to one approximating the rate of decay, the resulting band contour changes from Lorentzian to near-Gaussian. At sufficiently fast growth-rates the band closely approximates a linear combination of Lorentzian and Gaussian.