CORRECTION OF ABSORPTION BANDS FOR EFFECT OF FINITE SPECTRAL SLIT WIDTH

Abstract

We consider an absorption band, not necessarily of Lorentzian shape, scanned over a finite region by a spectrometer whose slit width does not exceed the band width by a factor of more than a few-fold. The integral transformation which converts the true intensities incident upon the spectrometer, i $(\nu)$, into the apparent intensities transmitted by the spectrometer, $t=(\nu^{\prime})$ is given a discrete, approximate representation by the vector equation t=S i, in which the components of i and t are the values of the intensity functions at discrete equally spaced values along the frequency axis If S, the spectrometer matrix, is subjected to a physically reasonable limitation which renders it cyclic, it can be inverted exactly for a wide class of slit shapes, e.g. triangular, gaussian, trapezoidal. Thus in principle the tine spectrum can be calculated m terms of the apparent spectrum by $i = S^{-1}t$. In practice, however, t is not usually known with sufficient precision, since $S^{-1}$ magnifies the error typically by two orders of magnitude Instead of the exact inversion. Jin approximate, iterative procedure is employed, which will give i to 1\% precision for a satisfactorily wide range of conditions. Examples will be presented, both for a Lorentz true band shape and for certain experimentally observed bands, for $s/\Delta v_{t}$, as large as 3 (s = spectral-slit width, $\Delta v_{t}$ = true band width at ${^{1}/_{2}}$ maximum absorbance). The iterative calculations are easily programmed for an automatic digital computor.