Consider a d-dimensional isotropic crystal with a simple cubic structure. Derive

Question

Consider a d-dimensional isotropic crystal with a simple cubic structure. Derive an expression for the low-frequency limit of the phonon density of state in the crystal. Sketch the DOS in the particular eases of d = 1, 2 and 3. Using this expression show that the heat capacity of the crystal at T rightarrow 0 turns to zero as Td. Derive an expression for the Debye frequency omega D of the d-dimensional crystal in terms of the velocity of sound and the atomic volume. Apply this expression to calculate omega D for the particular cases of d = 1, 2 and 3. Assume that the velocity of sound is independent of the direction or polarization. Note: The volume of d-dimensional space between two co-centric spheres of radii r and r+dr is alpha rd - 1 dr, where alpha is a geometric factor. Use specific values of alpha for the calculations in d = 1, 2 and 3. For the general dimensionality d, keep alpha as an unspecified parameter in all your expressions.

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