hgandh
- 26
- 2
Ashcroft & Mermin Chapter 25, the Gruneisen Parameters are defined as:
$$\gamma_{ks}=-\frac V {\omega_{ks}} \frac {\partial {\omega_{ks}}} {\partial V}$$
where the normal mode frequencies are defined by the eigenvalue equation:
$$ M \omega^2 \epsilon = D(k) \epsilon $$
The volume of the crystal is defined by:
$$ V = V_o(1+\eta)^3$$ with $$\eta << 1$$
The effect of altering the volume is a change in the dynamical matrix that governs the eigenvalue equation. To linear order:
$$D(k) \to D(k) + \eta \delta D(k)$$
The normal mode frequencies change by:
$$\omega \to \omega + \eta \frac {\epsilon \delta D(k) \epsilon} {2M \omega} = \omega + \eta \delta \omega$$
The Gruneisen Parameter becomes:
$$\gamma = \frac {\epsilon \delta D(k) \epsilon} {6M \omega^2}$$
However, I am having trouble obtaining this result. After plugging in the expression for the new frequencies I have so far:
$$- \frac {1 + \eta} {3(\omega + \eta \delta \omega)} \delta \omega$$
I do not know how to proceed after this to get the required result.
$$\gamma_{ks}=-\frac V {\omega_{ks}} \frac {\partial {\omega_{ks}}} {\partial V}$$
where the normal mode frequencies are defined by the eigenvalue equation:
$$ M \omega^2 \epsilon = D(k) \epsilon $$
The volume of the crystal is defined by:
$$ V = V_o(1+\eta)^3$$ with $$\eta << 1$$
The effect of altering the volume is a change in the dynamical matrix that governs the eigenvalue equation. To linear order:
$$D(k) \to D(k) + \eta \delta D(k)$$
The normal mode frequencies change by:
$$\omega \to \omega + \eta \frac {\epsilon \delta D(k) \epsilon} {2M \omega} = \omega + \eta \delta \omega$$
The Gruneisen Parameter becomes:
$$\gamma = \frac {\epsilon \delta D(k) \epsilon} {6M \omega^2}$$
However, I am having trouble obtaining this result. After plugging in the expression for the new frequencies I have so far:
$$- \frac {1 + \eta} {3(\omega + \eta \delta \omega)} \delta \omega$$
I do not know how to proceed after this to get the required result.