What I am trying to do is to compute the specific heat of a free electron gas in a conducting metal.

I am using Fermi-Dirac statistics as my framework to build off of. Importantly though, I want to understand what the specific heat does as a function of temperature, up to very high temperatures (millions of kelvin).

My governing equation is the definition of specific heat given in terms of the fermi-dirac distribution function $f$ and the density of states equation $g$

$$C_e(T_e)=\int_{-\infty}^\infty \frac{\partial f(\varepsilon,\mu,T_e)}{\partial T_e}g(\varepsilon)\varepsilon\ d\varepsilon,$$

The first consideration that I make is that the fermi level is a function of temperature, so I use the following conservation equation to solve for the fermi level $$N_e=\int_{-\infty}^\infty f(\varepsilon,\mu(T_e),T_e)g(\varepsilon)d\varepsilon.$$

where $N_e$ is the number of free electrons. For example in Copper it would be 1 electron/atom.

My question is around how to determine the number of free electrons in a metal as a function of temperature. All the analysis that I have seen just assumes a static amount of free electrons for a material, but we know as $T\to\infty$ the number of free electrons goes towards the atomic number of the metal.

My question is how can I solve for $N_e(T)$ if I know $N_e(T_0)$?

Could I use the same fermi-dirac distribution except this time set my electron density to be all of the electrons in the media, and solve for an energy level in which the high energy tail of the distribution at room temp equals the known number of free electrons. I could then watch how the population of electrons above and below that energy level evolve as a function of $T$?

Any guidance would be greatly appreciated.


1 Answer 1


While your definition of the heat capacity is correct, you are misinterpreting the second equation you write. The sum of the Fermi distribution multiplied by the density of states over all energy is equal to the total number of electrons not free electrons i.e. it says that the number of electrons doesn't change as you heat up the sample. It is necessary to calculate this as the temperature changes (at least for larger temperatures) because the position of the Fermi level will change, due to the uneven shape of the density of states.

Consequently, you don't need to know the number of free electrons to calculate the heat capacity using your equation (1).

  • $\begingroup$ Perhaps some of my confusion then comes from when I look at Fermi energys they seem to be calculated using the free electron density rather than the total electron density. One example hyperphysics.phy-astr.gsu.edu/hbase/Tables/fermi.html $\endgroup$
    – Andrew
    Jan 15, 2018 at 16:15
  • $\begingroup$ From my understanding the bookkeeping is done separately for electrons still bound to the nucleus and electrons in the conduction band which move freely through the medium. And the fermi liquid/gas approach is used for the conduction band electrons. $\endgroup$
    – Andrew
    Jan 15, 2018 at 16:22
  • $\begingroup$ The Fermi approach i.e. using the Fermi function, is done for valence and conduction electrons. You're right that they consider it in terms of free electrons, but they also note that they are using the density of conduction electron states, rather than the full density of states. In you consider all states, then the integral will be the total number of electrons. I admit I'm not sure why they are using only the conduction DOS, though. $\endgroup$
    – Nicholas
    Jan 16, 2018 at 8:27

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