Effect of Lattice Vibrations on Vacancy Formation.

[MathematicalPhysicist]

Homework Statement

2.10
To establish the effect qualitatively, consider the following crude model. Each atom vibrates as an independent three-dimensional Einstein oscillator of frequency $\omega_0$. Assume further that if a nearest-neighbour site is vacant, the frequencyof the mode corresponding to vibration in the direction of the vacancy changes from $\omega_0$ to $\omega$. Let $q$ be the number of nearest neighbours.

(a) Show that in this simple model, $$\Delta A = nqk_B T \ln(\frac{\sinh(\beta \hbar \omega/2)}{\sinh(\beta \hbar \omega_0/2)})$$

where $n$ is the total number of vacancies.

(b) Consider as an example a simple cubic lattice. Each mode then corresponds to the vibration of two springs. If one of them is cut, the simplest assumption one can make is: $$\omega = \omega_0/\sqrt{2}$$
Show that for high temperatures , $\beta \hbar \omega \ll 1$, $$e^{-\beta\Delta A/n}\approx 8$$
while for $\beta\hbar \omega \gg 1$, $$\Delta A \approx -3/2 n\hbar \omega_0 (2-2\sqrt{2}).$$

I am not sure I understand how did they solve question (b).

Here's the solution to question (b):

For $\beta \hbar \omega \ll 1$ we approximate $\sinh x \approx x$ and with $q=6$, the result follows immediately. Similarly, at low tempratures, $\beta \hbar \omega \gg 1$ we use $\sinh x \approx e^x/2$, and obtain the other limiting result.

Homework Equations

The Attempt at a Solution

For the first approximation I plugged everything to the identity in (a) and indeed got the approximation as it's written in the text, as for the second approximation I get:

$$\Delta A \approx 6nk_B T \ln(e^{\beta \hbar \omega_0/2(1/(\sqrt{2})-1) }) = \ldots =(-3/2) n\hbar \omega_0 (2-\sqrt{2})$$

Am I correct? Is there another mistake in a problem in this textbook of Bergersen's and Plischke's?

 

[TSny]

I agree with your result. It does appear that there is a (typographical?) error in the text solution.

 

[MathematicalPhysicist]

I agree with your result. It does appear that there is a (typographical?) error in the text solution.

Yes, it appears it got corrected in the third edition, I really should be using it now instead of the second edition.