Properties of Degenerate Electron Gas

[thepolishman]

TL;DR Summary

Is there a way to determine the properties of a degenerate electron gas?

Basically the thread title. For some background, I'm trying to model laser-material interactions, where I'm assuming that the laser is interacting with a free electron gas (copper). To model the interaction, I need to determine the properties of the electron gas, such as the heat capacity, effective number of free electrons, thermal conductivity, thermal coupling between the electrons and ions, etc.

Heat capacity is easy, and the analytical result is available in most textbooks over all ranges of temperatures (Thermal Physics by Kittel for example). However, looking at various papers, I can't find any equations that describe the other properties of a free electron gas. There are results for the low temperature (warm metal) and high temperature (plasma), but nothing in the intermediate range which is unfortunately the most relevant for my application.

One method that seems to be popular across various publications seems to be the Lee/More method ('An electron conductivity model for dense plasma', doi 10.1063/1.864744 for those with access). Reviewing the paper, it seems the method is simply an interpolation between the low and high temperature regime (I could be wrong though, I couldn't tell how the authors came up with the equations). Some examples from the paper include:

• τ_collision(degenerate)=τ_collision(classical)*[1+exp(-μ/kT)]*F_1/2(-μ/kT)
• k_e(degenerate)=k_e(classical)*20/9*F_4*[1-16*F_3^2/(15*F_4*F_2)]/[1+exp(-μ/kT)]/F_1/2^2

where F_j are Fermi-Dirac integrals of order j. In the low and high temperature limits, the above equations converge to the warm metal (highly degenerate) or plasma (classical) equations. However, as best as I can tell from reading the paper, in the intermediate temperature range the above equations are simply interpolations between the low and high temperature regimes. There is no way for me to really gauge how accurately these equations represent the intermediate temperature range.

So I guess that leads me to the equation I posed at the start. Is there any way for me to obtain the exact properties of a free electron gas in the intermediate temperature regime, either analytically or numerically? If not, why not?

 

[dRic2]

Summary:: Is there a way to determine the properties of a degenerate electron gas?

laser-material interactions,

What kind of interaction? Does it lead to a significant increase in the temperature of the sample?

At room temperature, the Fermi-Dirac distribution for electrons in metals is usually quite sharp ($T_{amb} \ll T_{F}$) and the system is very well approximated by the electron gas at zero temperature.

Anyway, for Copper $T_F = 80000K$ according to this https://scholar.harvard.edu/files/schwartz/files/13-metals.pdf; so I think you are safe to use the low-temperature limit.

 

[thepolishman]

Unfortunately, the temperatures that I'm interested are in the range of 10 eV or more (>~120,000 K), with the laser energy sufficient to cause material ablation and produce an expanding plasma plume. As a result, most low temperature approximations are no longer accurate. However, the temperature is not high enough that degeneracy effects can be ignored, and therefore classical (plasma) equations are also inaccurate.

To make my issue a little more clear, suppose I want to know the average velocities of electrons in a plasma following a Maxwell-Boltzmann distribution. To find the average velocity, I would use the equation given by: <v>=integral(v*f_MB(v), 0, Infinity) where f_MB(v) is the Maxwell-Boltzmann distribution function.

If I try the same method for a degenerate electron gas, the equation becomes <v>=integral(v*f_FD(E)D(E), 0, Infinity), where f_FD is the Fermi-Dirac distribution function and D(E) is the density of states for a free electron gas. At very low temperatures, I would find that the average velocity of the electrons is 0.75*v_F, or 75% of the Fermi velocity (given by v_F=sqrt(2*E_f/m_e)). This is the wrong result (or at least, not the one I'm looking for). While the average velocity of all the electrons (both degenerate and non-degenerate) is indeed 0.75*v_F, only non-degenerate electrons contribute to the properties of the material. The only electrons that are non-degenerate are those very close to the Fermi energy, with velocities given by v=v_F (not 0.75*v_F).

So I suppose my problem comes down to, how do I change my velocity integral (and other similar integrals), i.e. <v>=integral(v*f_FD(E)D(E), 0, Infinity), to only account for the properties of the non-degenerate or partially degenerate electrons.

 

[dRic2]

Unfortunately, the temperatures that I'm interested are in the range of 10 eV or more (>~120,000 K)

Sorry for the dumb question, but how can you have a solid at such high temperatures? I don't understand the setup of your problem.

Anyway, I also don't understand the way you want to calculate the thermal conductivity... why are you evaluating the average velocity? You said yourself that only electrons near the Fermi surface contribute to the thermal conductivity and indeed that is true, so your method is not correct.

A "simple" way to evaluate the thermal conductivity is to use the Boltzmann transport equation (https://en.wikipedia.org/wiki/Boltzmann_equation) and apply it to conduction electrons. If I remember correctly you can find a good explanation in Ziman's book "Principles of the theory of solids" (chapter 7 I think... I don't have this book at hand right now). After some cumbersome manipulations you should arrive at a formula more or less like this (from my notes):

where $\tau$ is the relaxation time (which you can also take as a constant), $v$ is the velocity and $f_0$ is the FD distribution. Now you see that $\frac {\partial f_0}{\partial E}$ is a very narrow gaussian-shaped function, centered around the Fermi level and indeed it refelts the property that only electrons near the Fermi level make a significant contribution.

 

[thepolishman]

Not really a solid, just a very dense electron gas. Using femtosecond laser ablation allows the material to reach several 10s of eVs with electron densities at or above the density of the solid. Since the pulse duration is on the order of ~100 fs, the material does not have sufficient time to expand while being heated by the laser pulse.

In any case, I think you gave me the answer I was looking for. From a glance, it seems like the classical 1/3*v^2*C_e*τ, where C_e is represented by the E^2/T*df/dE, all thrown into an integral. I'd definitely be interested in seeing the derivation. That said, is the Ziman book easy to understand? Other than Thermal Physics by Kittel, and a few publications, I don't have much knowledge on solid state physics. Let me know if you'd still recommend Ziman or if there's an easier book.

 

[dRic2]

From a glance, it seems like the classical 1/3*v^2*C_e*τ, where C_e is represented by the E^2/T*df/dE, all thrown into an integral.

nice intuition ;)

is the Ziman book easy to understand? Other than Thermal Physics by Kittel, and a few publications, I don't have much knowledge on solid state physics. Let me know if you'd still recommend Ziman or if there's an easier book.

Ok, so Ziman's book was my first and only book on solid-state physics. I also know Ashcroft and Mermin's book, but I used it more as a reference than as a textbook (and I don't think there is this derivation of the thermal conductivity but I may be wrong). Ziman's not easy, especially for a first introduction to the topic, but it is not exceedingly difficult either (the math, in particular, is not too complicated). Lots of things you may understand later if you keep studying solid-state physics, and I wouldn't bother too much. I really love the book, but I've seen Kittle's "Introduction to solid state physics" as the favored introductory textbook on the subject. I've never read it so I really don't know. I think you could try Ziman.