Effective and Debye temperatures of alkali-metal atoms in graphite intercalation compounds

The vibrational energies of the alkali-metal atoms in graphite intercalation compounds have been analyzed to yield effective temperatures ${\mathrm{T}}_{\mathrm{\parallel }}$ and ${\mathrm{T}}_{\mathrm{\perp }}$ and the corresponding Debye temperatures ${\mathrm{\Theta }}_{\mathrm{\parallel }}$ and ${\mathrm{\Theta }}_{\mathrm{\perp }}$ parallel and perpendicular to the graphite planes. In the context, ${\mathrm{T}}_{\mathrm{\parallel }}$ and ${\mathrm{T}}_{\mathrm{\perp }}$ differ from the thermodynamic temperature as they include the kinetic energy of the zero-point vibrational motion of the alkali-metal atoms. It was found that the value of ${\mathrm{T}}_{\mathrm{\perp }}$ at 0 K is independent on the stage and that a universal linear relation of ${\mathrm{T}}_{\mathrm{\perp }}$ as a function of (${\mathrm{M}}_{\mathrm{X}}$d${)}^{\mathrm{-}1/2}$ exists (where ${\mathrm{M}}_{\mathrm{X}}$, the atomic mass of the alkali-metal atom and d, the distance between two graphene layers sandwiching an alkali-metal layer). This demonstrates that the effective force constant between the alkali-metal and the binding graphene layers, calculated per alkali-metal atom, is always the same for all compounds. By applying an “infinite-mass” correction to ${\mathrm{T}}_{\mathrm{\parallel }}$, a similar linear relation between the corrected values ${\mathrm{T}}_{\mathrm{\parallel }}^{\prime }$, ${\mathrm{\Theta }}_{\mathrm{\parallel }}^{\prime }$, and (${\mathrm{M}}_{\mathrm{X}}$d${)}^{\mathrm{-}1/2}$ has been found. These relations can be used for predicting ${\mathrm{T}}_{\mathrm{\perp }}$, ${\mathrm{\Theta }}_{\mathrm{\perp }}$, ${\mathrm{T}}_{\mathrm{\parallel }}^{\prime }$, and ${\mathrm{\Theta }}_{\mathrm{\parallel }}^{\prime }$ for cases for which no experimental data are reported.