# Heat capacity using Debye dispersion relation

• torch000
In summary, using the Debye dispersion approximation, the heat capacity of a harmonic, monatomic, 1D lattice can be calculated by adding together the heat capacities for two transverse modes and one longitudinal mode. The final solution, derived from Kittel, is given by C_{lattice} = \frac{\pi L k_{B}}{3h}\sqrt{\frac{M}{c_{p}}}KT(2\sqrt{5}+1) in the low temperature limit.
torch000

## Homework Statement

Using the Debye dispersion approximation, calculate the heat capacity of a harmonic, monatomic, 1D lattice. Next, find the temperature dependence in the low temperature limit. (Assume that the longitudinal mode has spring constant CL = C, and the two transverse modes both have spring constant CT = 0.2C. )

## Homework Equations

$C_{lattice}=k_{B}\sum$p ∫d$\omega$Dp($\omega$) $\frac{x^{2}e^{x}}{(e^{x}-1)^{2}}$ , p is over all of the modes
$d\omega$ $D_{p}( \omega)$ = $\frac{Lk_{B}T}{hv\pi}dx$
where $D_{p}( \omega)$ = density of states
$c_p$ = the spring constants for longitudinal and transverse

## The Attempt at a Solution

So I should end up with 3 heat capacities to add together, two transverse and one longitudinal.

$C_{lattice}=k_{B}\sum_p \int^{\inf}_{0} \frac{Lk_{B}T}{hv\pi}\frac{x^{2}e^{x}}{(e^{x}-1)^{2}} \omega D_{p}(\omega) = \sum_{p}\frac{Lk_{B}T}{hv\pi}\frac{\pi^{2}}{3}K\sqrt{\frac{M}{c_{p}}}$

$C_{lattice}$ = $\frac{\pi L k_{B}}{3h}\sqrt{\frac{M}{c_{p}}}KT(2\sqrt{5}+1)$
So this right here is my final solution, and I'm trying to see how far off base I am.

Last edited:
That was a mistake and should've been $(e^{x}-1)$, and the expression for $C_{lattice}$ is from Kittel if youre wondering where I got it.

## 1. What is the Debye dispersion relation?

The Debye dispersion relation is an equation that relates the heat capacity of a solid to its temperature. It takes into account the different vibrational modes of the atoms in a solid, and is based on the Debye model of solids.

## 2. How is heat capacity calculated using the Debye dispersion relation?

The Debye dispersion relation is used to calculate the heat capacity of a solid by integrating over all the vibrational modes of the atoms. This integration takes into account the temperature dependence of the vibrational frequencies and the number of atoms present in the solid.

## 3. What factors affect the heat capacity calculated using the Debye dispersion relation?

The heat capacity calculated using the Debye dispersion relation is affected by the number of atoms in the solid, the temperature, and the type of bonds present between the atoms. It also depends on the assumptions made in the Debye model, such as the solid being isotropic and composed of spherical atoms.

## 4. How accurate is the Debye dispersion relation in calculating heat capacity?

The Debye dispersion relation is a good approximation for calculating the heat capacity of most solids at low temperatures. However, it does not take into account certain effects such as anharmonicity and thermal expansion, which can affect the accuracy at higher temperatures.

## 5. Can the Debye dispersion relation be applied to all types of solids?

No, the Debye dispersion relation is most suitable for calculating the heat capacity of solids with simple crystal structures, such as metals and insulators. It may not be applicable to more complex solids, such as amorphous materials or those with strong covalent or ionic bonds.

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