Posted by Toby on January 27, 2014, 3:44 pm
Bit of a picky question - I'm going back over the derivation of Debye heat capacity in the book, and I think there should be an extra factor of N present. Here's my reasoning:
In section 2.2.2, Einstein's expression for the energy of a single particle in a 3D harmonic potential is modified to include a sum over the wave-vectors. After a little algebra the density of states g(\omega) is introduced as N\frac{9 \omega^2}{\omega_d^3} where the Debye frequency has a factor of n^{1/3} (number density) in it. This means that the density of states is proportional to N/n=V. This seems fine - we're still considering the energy of one particle, so it's good that the density of states for this one particle does not depend on N. In section 2.2.3 the cutoff frequency is defined as the frequency above which no sound modes exist.
"... with this frequency chosen such that there are exactly 3N sound wave modes in the system (three dimensions of motion times N particles).
3N = \int_0^\omega_{cutoff} d\omega g(\omega}. "
so there sum over all states (of one particle) is now equal to 3 times the number of particles? After this, an expression for the energy (still of one particle) is found to be proportional to N in the high temp limit. Have I missed something here?
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