Knowledge Bank

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 $i=S-1t$. 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 $s/\Delta vt$, as large as 3 (s = spectral-slit width, $\Delta vt$ = true band width at $\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}/2$ maximum absorbance). The iterative calculations are easily programmed for an automatic digital computor.