D-dimensional density of states and specific heats (phonons and electrons)

[tjny699]

Hi, I have a question about statistical mechanics.
How do you calculate the density of states for phonons and electrons in a d-dimensional system (at fixed chemical potential) and when the dispersion relation for the electrons is $$E(p)=A |p|^g$$ and for the phonons is $$w=v|p|$$

To get the specific heat one takes the temperature derivative of the energy, correct? How does all this change if you consider constant number of electrons rather than constant chemical potential?

I guess that's really the heart of my misunderstanding, how does the density of states and specific heat change when you consider constant electron number rather than constant chem. potential??

Thanks so much for any insights.

 

[kanato]

The density of states is

$$g(E) = \int d^dk \, \delta(E - E(k))$$

for a simple power law like that, it's fairly straightforward to compute the density of states.

I don't understand your second question. The heat capacity is defined as
$$C_V = \left( \frac{dE}{dT} \right)_{V,N}$$
which includes constant number of particles in the definition.

For your third question, the density of states of a non-interacting system is independent of electron number and chemical potential.

One thing to keep in mind is that for real materials, there is an extremely small change in the chemical potential from T = 0 up to the melting point of the material. So usually one does not need to worry about constant particle number vs. constant chemical potential.

 

[tjny699]

Hi kanato,

Thanks a lot for the reply.

I was under the impression that one could take the energy derivative with respect to temperate but also hold other thermodynamic quantities constant--the pressure for example:

$$C_P=(\frac{\partial E}{\partial T})_{P}$$

Is there a thermodynamic relation that related the chemical potential to this quantity?

 

[kanato]

Constant pressure heat capacity is actually the derivative of the enthalpy H = E + PV, not the energy:

$$C_P = \left( \frac{\partial H}{\partial T} \right)_{P,N}$$

Note that it is keeping particle number constant.

The basic relation for equilibrium thermodynamics is

$$T dS = dE + P dV - \mu dN$$

You can divide by any differential you want and take anything constant that you want. For example, dividing by dT taking P and N constant you get

$$T \left( \frac{\partial S}{\partial T} \right)_{P,N} = \left( \frac{\partial E}{\partial T} \right)_{P,N} + P\left( \frac{\partial V}{\partial T} \right)_{P,N}$$

The left hand side (if multiplied by dT) is the definition of the differential heat added in a reversible process, which is the definition of the heat capacity, constant pressure in this case. The right hand side is dH/dT at constant P. Note that the result would be different if the chemical potential is held constant. For one, the LHS would not have quite the same interpretation, and you end up with an extra term on the RHS involving dN/dT at constant P,mu.

It's not usual to need to consider the chemical potential when looking at heat capacities, so I am wondering what your motivation for this is?

 

[tjny699]

As for motivation: I'm a biochemistry student about to start work on a more physics intensive project and I got some old course notes form my friends to learn from. There is some discussion of the Debye theory of specific heats in arbitrary dimensions and I just want to make sure that I understand all of the details.

The notes contain some mysterious passages, like the question I originally posted.

I can get the phonon and electron specific heats in the low and high T limits. Then there is a question asking if these are ever comparable in magnitude. I think that I can answer this as well: in different temperature ranges you can obtain constraints on the dimension $$d$$ and exponent of the electron dispersion $$g$$ for which the two contributions have the same temperature dependence.

The next part of the notes asks to think about how all of this (DOS, specific heats, etc) changes when particle number is held constant instead of chemical potential.

There is also a thermodynamic relation that relates chemical potential to specific heat
$$C_p= -NT(\frac{\partial^2 \mu}{\partial T^2})_p$$
and
$$C_v= VT(\frac{\partial^2 P}{\partial T^2})_v-NT(\frac{\partial^2 \mu}{\partial T^2})_p$$

I'm not quite sure what that second derivative of chemical potential means and how to reproduce the results of the Debye theory for this case.

Thanks.