Hi Ali,  

Replace exact phonon dispersion relation and the density of modes

with the density of modes of an elastic continuum, but include a cut-off

frequency, the Debye frequency ωD.

Energy of system = Phonon Energy x Average number of phonons x number of modes.

For more ansewer detail check attached file.

Regards,

Dear colleague,

it is certainly possible to obtain a certain cut-off Debye frequency in the conventional way, as described within the pdf file attached by M. W. Azizi. But this quantity is of little use in practice. The essential point is the that Debye's obsolete theory is in fact NOT applicable to the overwhelming majority of physically realistic systems (like diamond, Si, Ge, III-V, II-VI materials, etc.). The reason is that the degree of phonon dispersion is as a rule significantly higher than admitted by the Debye model.

Consequently, it is an illusion to believe that it should be possible to precalculate heat capacities and other related thermodynamic quantities, by choosing simply a FIXED Debye temperature value (whichever this value would be), and using formulas of Debye type, that are hitherto (obviously "for simplicity") quoted in many solid state textbooks. In reality, the so-called Debye temperature is -per se - a strongly temperature dependent quantity, the detailed T-behavior of which, however, is often not known in due detail. Typical examples of carefully calculated TDetaD(T) dependences, e. g. for various II-VI materials, you can find in Sec. IV of my last paper (in AIP Advences 3, 0821208 (2013)), including several preceding papers (directly available via the Researchgate).

With best regards

R. Passler   

R. Pässler Sir can you tell me how the vibrational frequency of material can we get from phonon dispersion curve theoretically

Dear Attia Batool,

the basis for theoretical calculations of Debye temperatures and related quantities is given by material-specific phonon density of states spectra, whereas a mere knowledge of phonon dispersion curves, per se, is not sufficient for this purpose. The corresponding calculation procedure consists in two subsequent steps: (1) calculation of the temperature dependences of harmonic-lattice heat capacities and (2) transormation of these heat capacities into the respective Debye temperatures. From the latter follow readily the temperature dependences of other Debye-temperature-related quantities (energies or frequencies). The whole calculation procedure is described in detail in a separate pdf-file.

Roland Pässler