Posted by Steve Simon on March 15, 2013, 11:16 am, in reply to "The Debye calculation"
The extra factor of three is out front. Each oscillator gets an energy hbar omega (n+1/2) where omega is the oscillator frequency. We just need to add up all the oscillation modes. For each wave vector k, in 3 dimensions there are three different oscillation modes  you can think of these as being the three directions of motion.. Or equivalently there is one longitudinal and two transverse modes at each k. 
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