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Homework Statement
I am not sure I understand how did they solve question (b).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##, $$\Delta A \approx -3/2 n\hbar \omega_0 (2-2\sqrt{2}).$$
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?