Absorption Strengths

If the absorption coefficient of the THz power $P$ is defined by \begin{equation}\alpha = -\left(\frac{1}{P}\right)\left(\frac{\Delta P}{\Delta x}\right)\end{equation} in the microwave limit (where $h\nu << kT$), the peak absorption coefficient between two rotational levels $m$ and $n$ is \begin{equation}\alpha_{mn} = \frac{8\pi^2NF_m\nu^2}{3kTc\Delta\nu}\left|\mu_{mn}\right|^2\end{equation} where $NF_m$ is the number of molecules per unit volume in state $m$, $\nu$ is the rotational transition frequency $\Delta\nu$ the line width, and $\mu_{mn}$ is the dipole matrix element [1]. The transition moment $\mu_{mn}$ contains contributions from the components of the permanent dipole moment along each of the principal axes of the moment of inertia tensor, which determine the rotational selection rules.

For optimum sensitivity the gas pressure is adjusted in proportion to frequency so that the Doppler and pressure broadening contributions to the linewidth are equal and $N/\Delta\nu$ is independent of frequency. Because of degeneracy and rotational partition function effects, $F_m$ is often proportional to $\nu$ until declining Boltzmann population causes it to fall exponentially. These factors typically give rise to absorption coefficients that rise as $\nu^3$ to reach a maximum at some optimum frequency in the THz before declining exponentially. This effect can be seen in Figure: Rotational Interaction Strength above.