Could you specify "cubic vs. hexagonal", please? Do you mean

-fcc vs. hcp

-Wurtzite vs. zinkblende

-other comparison?

And *which* phonon branch(es) are you interested in?

That is easy: First see problem 5 on page 449 of the book Solid State Physics, by Ashcroft & Mermin. Subsequently, replace the set of nearest-neighbour vectors of R=0 as given in Eq. (22.98) on page 449 by the relevant set of nearest-neighbour vectors as deduced from the expressions given in Eq. (4.9) on page 77 of the same book. In doing so, compare the expressions in the above-mentioned Eq. (22.98) with those in Eq. (4.8) on page 75. The sought-after dispersion relation is obtained by diagonalising the 3 x 3 dynamical matrix D that you thus obtain (see Eq. (22.60) on page 440).

Dear Fauth,

First of all

thanks for your quick response and interest.

I am talking about Wurtzite vs. zinkblende for longitudinal optical phonon branch (LO).

I am talking about the phonon dispersion relation ω(q)=√(A^2+B^2 √(1+cos(aq/2))) that is used to fit Raman spectra by spatial correlation model.

I am not an expert in Raman spectroscopy. Usually I guess you would consider phonons at q=0 only, unless you have very high density of states (e.g. a van Hove singularity at high symmetry points of the 1st BZ edge) giving rise to 2-phonon Raman losses. Is that correct?

(I ask this to understand why you are interested in dispersion formulae, which would probably not be necessary if you considered q=0 exclusively)

Dear Fauth, the zone center phonons are considered in the case of bulk only but for nano-dimension, not just q=0 phonons are considered rather a dispersion curve showing variation of phonon frequency with wavevector is required. The question is how the phonon dispersion curve will be modified for cubic and heaxagonal phases of a material?

Dear Nupur, that takes an interesting twist now, and I am not (yet) sure that it's this one that the thread opener intended to take (nanomaterials).

Wave vectors and phonon dispersion curves are concepts relating to (infinite & ideal) crystals, in principle. The smaller the objects the less you can equate their vibrational states to the bulk ones. Interesting question, to what extent you can model te Raman data with a bulk dispersion curve then. 

Wouldn't this mean that the selection rules for Raman processes somehow get "broken up" more quickly than the vibrational states losing their connection to the bulk ones? Any justification for this? (I'll be curious to see answers to that, but we shouldn't let go the original question out of focus. In principle, the first answer by B.F. tells us what to do.)

maybe a useful reference:

DOI:  10.1016/0022-3697(64)90127-1

Yet another remark: since the BZ's look quite different in the two cases, you might want to specify in which direction(s) you want to compare the dispersion relations [001] in the hexagonal phase will correspond to [111] in the cubic one etc. you cannot expect any of the dispersion relations to be isotropic, especially in the hexagonal case.

And maybe this can help you:

http://arxiv.org/ftp/arxiv/papers/1212/1212.1491.pdf

Edit: and maybe also this

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.59.5013

Dear Fauth, off course it's true that dispersion relation is not isotropic in case of hexagonal structure, but if there is no axis of preferential growth, then what could be the case?

Well you'd have to integrate over all directions or may hope that adequately averaging a finite number of high symmetry branches (e.g. the set Gamma-A, Gamma-K, Gamma-M, A-L, A-H, K-H, M-L) does the job for you. I don't know, however, how you'd have to deal with the fact that subsets of these branches have different (reciprocal) length in q-space.

Dear Behnam Farid, the problem given in Ashcroft & Mermin is about fcc monatomic Braivais lattice. The question is how we can obtain dispersion relation for WZ struture from ZB.