Cusp Kernels for Velocity-Changing Collisions

We introduce an analytical kernel, the “cusp” kernel, to model the effects of velocity-changing collisions on optically pumped atoms in low-pressure buffer gases. Like the widely used Keilson-Storer kernel [J. Keilson and J. E. Storer, Q. Appl. Math. 10, 243 (1952)], cusp kernels are characterized by a single parameter and preserve a Maxwellian velocity distribution. Cusp kernels and their superpositions are more useful than Keilson-Storer kernels, because they are more similar to real kernels inferred from measurements or theory and are easier to invert to find steady-state velocity distributions.

Figure 1

A comparison of KS kernels (top) with various memory parameters $a$ and cusp kernels (bottom) with various sharpness parameters $s$. In contrast to KS kernels, cusp kernels display a sharp peak near the initial velocity $y=1$.

Figure 2

A comparison of a three-cusp fit and a measured collision kernel for Rb in He from Gibble and Gallagher (GG) [12] for the initial velocities $y=0.06$ (top) and $y=1.37$ (bottom). The GG kernel is represented by a model form with parameters from Table I of Ref. 12, normalized according to (2). The parameterized model form is estimated to agree with the measured kernel for Rb in He to within roughly 10% over the range of velocities shown [12]. The GG kernel and three-cusp kernel are solid lines, with the GG kernel (black) slightly sharper than the three-cusp kernel (red). The three-cusp kernel was fit to the GG kernel for $y=0.06$ (top), which gave the superposition parameters $[f_1,f_2,f_3]=[0.13,0.37,0.50]$ and $[s_1,s_2,s_3]=[7.8,27.2,500]$, as defined by (22). The individual cusps $f_j{C}_{s_j}(x,0.06)$ are shown by the dashed lines in the top panel. As the bottom panel shows, the same fit parameters give good agreement with measurements for $y=1.37$. See Fig. 10 of GG [12] for a comparison with KS fits to measurements.