Modes of atomic vibration in the fourteen Bravais lattices

Summary

In this paper, the degeneracies and the directions of the normal modes of vibration in the fourteen Bravais lattices are worked out from symmetry considerations, making use of Sir C. V. Raman’s theory. Denoting the ratio of the displacements of adjacent atoms along the primitive translations by α, β, γ, these can have only the values + 1 or −1, so that the vibrations fall into eight types in the general case of no symmetry. In lattices possessing symmetry, however, some of the types could be brought over into others by the application of symmetry operations, and thus would be equivalent with the latter. Also, the directions of motion in any particular type of vibration may be equivalent. This is determined by selecting a group of operations which brings the atoms of the same phase to coincide for that type of vibration, and finding the directions of motion which satisfy the symmetry requirements of this group. In this way, both the directions of motion and their degeneracies are known. The number of distinct modes of vibration computed for the various lattices, are forΓ _tr 21;Γ _m 21,Γ _m′ 15;Γ ₀ 21,Γ ₀′ 15,Γ ₀″ andΓ ₀′″ 12;Γ _t 12,Γ _t ′ 8;Γ _rh 8,Γ _h 8;Γ _c 5,Γ _c ′ 4,Γ _c ″ 4. The modes are also described in as complete a manner as could be done by using considerations of symmetry alone.