Collision-Induced Anisotropic Relaxation in Gases

An expression is derived for the cross sections, ${{\sigma }_j}^{\mathrm{}\left(x\right)\mathrm{}}$, for the collision-induced relaxation of the multipole moments of an ensemble of excited atoms in a state of total angular momentum $j$. Various general sum rules are obtained for the ${{\sigma }_j}^{\mathrm{}\left(x\right)\mathrm{}}$, as well as for the ${{\Lambda }_j}^{\mathrm{}\left(x\right)\mathrm{}}\equiv {{\sigma }_j}^{\mathrm{}\left(x\right)\mathrm{}}-{{\sigma }_j}^{\mathrm{}\left(0\right)\mathrm{}}$, the cross sections for reorienting collisions. For a $j=1\mathrm{}$ state, one obtains $\frac{{{\Lambda }_1}^{\mathrm{}\left(1\right)\mathrm{}}}{{{\Lambda }_1}^{\mathrm{}\left(2\right)\mathrm{}}}=\frac{5}{3}$. This result implies that the cross section for collision induced $\left|\Delta m\right|=2\mathrm{}$ transitions is always twice that for $\left|\Delta m\right|=1\mathrm{}$ transitions, independent of the assumed interaction potential. Absolute cross sections are obtained using the van der Waals collision model. Translational motion is treated in a classical linear path approximation. It is shown that the cross sections can be written in the form ${{\sigma }_j}^{\mathrm{}\left(x\right)\mathrm{}}=D{{\phi }_j}^{\mathrm{}\left(x\right)\mathrm{}}\frac{〈v^{\frac{3}{5}}〉}{〈v〉}$, where $v$ is the relative velocity between the colliding atoms, $D$ depends on the particular atoms and states considered, and ${{\phi }_j}^{\mathrm{}\left(x\right)\mathrm{}}$ is a geometric factor which is tabulated for $j$ values up to $j=8\mathrm{}$. The ${{\Lambda }_j}^{\mathrm{}\left(x\right)\mathrm{}}$ are related to the cross sections for transfers between magnetic sublevels. It is shown that in the present model a transfer from sublevel $\alpha \mathrm{to} -\alpha$ is forbidden for states with half-integer $j$ values.