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: ${V}T(x)=9.0{Z}₁{Z}₂{e}²{N}ₚa{({u}T+0.55)}^{{-}1/3}{(1+{0.17{u}T+0.36}{{u}T+0.30}{x}^{{1.255{-}4.35×{}{10}^{{-}2}{|{u}T{-}1.77|}³}})}^{{-}1}$ $x$ 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.