# Phonon density of states in statistical thermodynamics

Phonon density of states (DoS), $$g_{(\omega_i)}$$, help understand the distribution of states across frequencies of vibration. $$\int g_{(\omega_i)}d\omega_i =$$ the number of states between $$\omega_i$$ and $$d\omega_i$$.

DoS values aren't integers, at least none in my calculations were. Wouldn't the possible number of states in any range of frequency be an integer, or is this some quantum/wave effect?

If we think about a crystal with $$N$$ atoms. Each atom has its own set of frequencies, so the equations above work for each atom. Codes like Phonopy output DoS as if it represented the frequencies of vibration of the whole crystal system of all $$N$$ atoms. The following equations are from Dr. Brent Fultz's review of the vibrational thermodynamics of materials. $$Z$$ is the canonical harmonic partition function.

$$$$Z_i = \frac{e^{-\beta \epsilon_i/2}}{1 - e^{-\beta \epsilon_i}}\tag{1}$$$$

$$$$Z_N = \prod_{i}^{3N}\frac{e^{-\beta \epsilon_i/2}}{1 - e^{-\beta \epsilon_i}}\tag{2}$$$$

The first equation is about each of the $$N$$ oscillators and the second about the whole system.

I'm confused with regards to the way lattice dynamics codes like Phonopy output DoS data. Can someone help me organize my thoughts here and put the right pieces in place?

• I just wanted to add that this is a really great, well-formulated question! Nov 10 '20 at 17:48

2. The density of states only makes sense as you take $$N\to \infty$$. For finite $$N$$, there is a finite/discrete number of phonon modes, and so the density of states would be a series of delta functions. As you take $$N\to \infty$$, the spacing between the modes goes to zero and you're left with a continuum and thus a continuous DOS.
• Understood. The cost of generating a new phonon mode is $\bar{h}\omega$, which isn't affected by phonon modes already present. Whereas, what you meant by saying "collective" was that all atoms at once vibrate because they all experience non-zero temperature, right? Nov 4 '20 at 17:41