We start with | ||

Next we must do the differentiation,
i.e. form ¶:F/¶T |
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One can go straight ahead, of course.
But here comes a little helpful trick: Multiply skillfully by
and re-sort; you get T/T |
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Now we need to resort to approximations | ||

First we realize that whenever h ·w/2p << k,
thenT |
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This takes care of the first term. | ||

The second term needs a somewhat more
sophisticated approach. Substituting for xh ·
w/2p ·
, we can use a simple expansion formula, stop after the second
term and re-insert the result. This giveskT |
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That's as far as one can go. Now use
w' for the circle frequencies of the crystal
with a vacancy and form S_{F} = S' –
S |
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q.e.d. |
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Do the Math for the Formation Entropy

© H. Föll (Defects - Script)