A Spectral Point-by-Point Catalog

The second approach makes no attempt to identify and characterize individual lines, but rather seeks to predict, on a point-by-point basis, the spectra as a function of temperature. Because of the exponentials in the Gaussian lineshape and in the lower state energy term, the absorbance as a function of frequency can be recast as

\begin{equation}A\left(\nu\right)=\frac{8\pi^3}{3ch}\sqrt{\frac{\ln(2)}{\pi}}\frac{nL}{Q}\frac{\nu_0}{\delta\nu_D}S_{ij}\mu^2 \left[1 - \exp{\left(-\frac{h\nu_0}{kT}\right)}\right]\exp{\left(-\frac{E_l}{kT}\right)}\exp{\left[-\ln(2)\left(\frac{\nu-\nu_0}{\delta\nu_D}\right)^2\right]}\end{equation}

with the Doppler width (HWHM) given by

\begin{equation}\delta\nu_D=\sqrt{\frac{2kN_aln(2)}{Mc^2}}\sqrt{T}\nu_0=W\sqrt{T}\nu_0\end{equation}

where $N_a$ is Avogodro’s number, and $M$ is the molecular mass. The absorbance normalized by the $nL/Q$ factor then becomes

\begin{equation}\frac{A(\nu)}{nL/Q}=\frac{8\pi^3}{3ch}\sqrt{\frac{\ln(2)}{\pi}}\frac{1}{W}\frac{\left[1-\exp{\left(-\frac{h\nu_0}{kT}\right)}\right]}{\sqrt{T}}S_{ij}\mu^2 \exp{\left(-\frac{E_l}{kT}\right)}\exp{\left[-\frac{\ln(2)}{W^2T}\left(1-\frac{\nu}{\nu_0}\right)^2\right]}\\=K\frac{\left[1-\exp{\left(-\frac{h\nu_0}{kT}\right)}\right]}{\sqrt{T}}\tilde{S}_{ij}\mu^2\exp{\left(-\frac{\tilde{E}(\nu)}{kT}\right)}\end{equation}

with

\begin{equation}\tilde{E}(\nu)=E_l+k\frac{\ln(2)}{W^2}\left(1-\frac{\nu}{\nu_0}\right)^2\end{equation}

According to equation (4) every frequency slice of the data (one for each temperature) can be represented by two parameters $\tilde{S}_{ij}\mu^2$ and $\hat{E}$. On line center, when $v = v_0$, $\hat{E}$ equals the lower state energy, while $\tilde{S}_{ij}\mu^2$ corresponds to the line strength. Off of line center, the meaning of $\tilde{S}_{ij}\mu^2$ and $\hat{E}$ are less physical, but equation (4) is still a valid fitting function for describing the spectral intensity for an unblended line.