Nanocrystalline nanowires: 2. Phonons

Nanocrystalline nanowires (NCNW) are fragments of bulk crystals that are infinite in only one direction. A construction is given for calculating eigenstates belonging to the symmetry labels (k,m) (wavevector and rotational quantum number). Vibrational harmonic eigenstates are worked out explicitly for a simple model, illustrating the general results: the LA mode has m = 0, while with sufficient rotational symmetry, the TA branch is doubly degenerate, has m = +/- 1, and has quadratic dispersion with k for k less than the reciprocal diameter of the NCNW. The twiston branch (a fourth Goldstone boson) is an acoustic m = 0 branch, additional to the LA and two TA branches.