Temperature-dependent planar potential for channeling

By effectively thickening the atomic planes, thermal agitation alters the potential between a plane and a channeled particle as evaluated by Lindhard in the "rigid" case. The influence of temperature on this "planar potential" has been calculated and the following universal expression derived: ${\mathcal{V}}_T\mathrm{}\left(x\right)=9.0\mathrm{}{Z}_1{Z}_2{e}^2{N}_{p}a{(u_T+0.55)}^{-\frac{1}{3}}{\left(1+\frac{0.17u_T+0.36\mathrm{}}{u_T+0.30\mathrm{}}{x}^{\{1.255-4.35\mathrm{}\times {10}^{-2\mathrm{}}{|u_T-1.77|}^3\}}\right)}^{-1\mathrm{}}$ $x$ is the distance to the geometrical plane, $u_T$ is the thermal root mean amplitude, both in units of Thomas-Fermi screening radius $a$, $Z_1$ and $Z_2$ are the atomic numbers of the target and the particle, and $N_p$ is the atomic density of the plane along which channeling is considered.