Symmetry for lattice modes in ${\mathrm{C}}_{60}$ and alkali-metal-doped ${\mathrm{C}}_{60}$

We present group-theoretical analysis relevant to the interpretation of recent experimental observations of Raman-scattering spectroscopy in ${\mathrm{C}}_{60}$ and alkali-metal-doped ${\mathrm{C}}_{60}$. The group-theoretical analysis makes use of two space groups, both of which reduce the icosahedral point-group symmetry of the isolated ${\mathrm{C}}_{60}$ molecule to the cubic group ${\mathit{T}}_{\mathit{h}}$. The space group for the fcc arrangement of fullerene is ${\mathit{T}}_{\mathit{h}}^6$ and corresponds to the low-temperature solid phase of pristine ${\mathrm{C}}_{60}$. The bcc arrangement for solid ${\mathit{M}}_6$${\mathrm{C}}_{60}$, where M is an alkali metal, is consistent with the space group ${\mathit{T}}_{\mathit{h}}^5$ and is the symmetry considered for the fully doped alkali-metal material. The experimental Raman spectra show line splittings which are consistent with a symmetry lowering to ${\mathit{T}}_{\mathit{h}}$ symmetry for the fullerene molecule. However, the polarization selection rules observed experimentally indicate a further lowering of the symmetry, beyond cubic symmetry.