Knowledge Bank

It is customary to characterize the intensities of infrared absorption bands of condensed phase systems in terms of the peak absorbance intensity $(\ln (T0/T)\nu 0)$, the width at half maximal intensity $(\Delta \nu 1/2o)$ and the total band area $\int band\ln (T0/T)\nu d\nu$, where $\nu o$ is the wavenumber at the band maximum. These parameters are insufficient to define the profile of a symmetric band, and provide no information about the band asymmetry. The improved resolution of modern grating spectrophotometers make it desirable to characterize the band profile, and a method for its quantitative description has been developed, based on the evaluation of the truncated moments about the maximum ordinate. Such truncated moments are calculated on a linear scale of wavenumber in j-units where $j=(\nu -\nu 0)/1/2\Delta \nu 1/2o$. The $rth$ truncated moment is given by $$\mu_{r}(j)=\frac{\int^{+j}{-j} (\nu-\nu{0})^{2}\cdot\ln (\frac{T_{0}}{T}) (\nu -\nu_{0}) d\nu}{\int^{+j}{-j} \ln (\frac{T{0}}{T} (\nu - \nu_{0})\cdot d\nu}$$ and a program providing $\mu r(j)$ for $r=1,2,3$ and 4 at intervals of 0.4 j has been written for an I.B.M. 1620 computer. For symmetric bands the profile can be characterized by a plot of $\mu 2(j)/(1/2\Delta \nu 1/2)2$ against j. The nature of this relationship for the Gauss curve, the Lorentz curve, and selected absorption bands at different slit widths and time constants will be discussed. The band asymmetry can be defined by a plot of $\mu 3(j)/(1/2\Delta \nu 1/2)3$ against j. In principle the intrinsic asymmetry can be distinguished from instrumentally induced asymmetry by observing the effect of reversal of the scanning direction on the sign and magnitude of this relationship. Such truncated moments can aid in the evaluation of the comparative performance of infrared spectrophotometers and facilitate the adjustment of instruments for accurate data transfer. For this purpose it is desirable to know accurately the profiles of standard bands measured with known slit functions and known recorder time constant functions.