Effect of the number of atoms in the basis to the heat capacity (component of the phonons) I am wondering what's the effect of the number of atoms in the basis onto the heat capacity (phonon part).
I've found this post: How the number of atoms in the basis affects the density of states? 
Here the answer says that the density of states is not affected due to the number of phonons in the basis. Can I conclude that there is no effect then onto the heat capacity, since U is given by $\int_{0}^{\omega_d}d\omega D(\omega)\cdot (\hbar \omega/(e^{\hbar \omega/\tau}-1))$ (and $C = \frac{dU}{dT}$)? Considering me there is nothing in this integral which is affected by the number of atoms in the basis?
 A: Atom lattice can be modeled as a phonon gas with constant volume. Then from gas laws and thermodynamics we know that constant volume gas has heat capacity :
$$ C_v = \frac {dQ_v}{dT} $$
Where $Q_v$ is a heat transferred to an object of volume $v$.
And specific heat capacity is :
$$ c_v = \frac {C_v}{n} = \frac 1n \cdot \frac {dQ_v}{dT} $$
where $n$ is amount of substance in that volume. So the answer is that heat capacity does depend on gas volume, because the bigger volume - more heat you need to transfer for raising volume's temperature by $1K$ degrees. However specific heat capacity is not dependant on volume, because it talks about heat transfer normalized by total substance amount, i.e. heat capacity for a unit substance amount.
EDIT
Probably at first glance relation of transferred heat to body volume and/or particle amount is not seen. Here's how to check that. Body absorbs heat transferred into internal energy and/or accomplishes some thermodynamic work (pushes hydraulic press wall, whatever) in case of gas.
Thus, this can be concluded as :
$$ Q =  \Delta U + W $$
Btw, I would not recommend skipping work part for a solid body in general, without thinking. Because solid bodies can also exchange heat for a work. For example, if you start heating a spring - it will begin to contract

In such case passed heat into a spring can be defined as :
$$ Q_{\textrm{spring}} = \Delta U + k\,\Delta x^2 $$
Further, body internal energy change fundamentally can be defined as :
$$ \Delta U = \Delta \sum_i N_i\,\epsilon_i $$
Where $N_i$ is number of particles in a microstate $i$ and $\epsilon_i$ is this microstate energy. Thus proved direct relationship of transferred heat to number of particles in the system.
A: Here's a thought experiment take on why there is no relation between the number of atoms in the basis and the heat capacity:


*

*The unit cell is not uniquely defined, and thus the basis is not uniquely defined. E.g., give me a unit cell, and I can combine two of those unit cells into a new unit cell with twice as many atoms in the basis. However, the material has not changed at all, so the heat capacity is exactly the same even tho the number of atoms in the basis has doubled.

*Now, you may say: this is just smoke and mirrors. I only want to work with a primitive unit cell. Fine, let's consider a 1D chain of identical atoms. Replace every thousandth atom with one of $1+\epsilon$ times the mass, where $\epsilon$ is a tiny amount. Now the primitive cell has a thousand atoms in it. Has the heat capacity changed? Not meaningfully. However, your unit cell is 1000 times bigger. So, the number of atoms in the unit cell is unrelated to the heat capacity.

