## Quadratic Debye

Posted by peeterjoot on January 19, 2014

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## Question: Quadratic Debye phonons (2013 midterm pr B2)

Assume a quadratic dispersion relation for the longitudinal and transverse modes

### Part a

Find the density of states.

### Part b

Find the Debye frequency.

### Part c

In terms of , and

find the specific heat for .

### Part d

Find the specific heat for .

## Answer

### Part a

Working straight from the definition

With and , this is

### Part b

The Debye frequency was given implicitly by

which gives

### Part c

Assuming a Bose distribution and ignoring the zero point energy, which has no temperature dependence, the specific heat, the temperature derivative of the energy density, is

### Part d

First note that the density of states can be written

for a specific heat of

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