# Can we study degenerated electron gas on Earth?

The degenerate electron gas is supposed to be in heavy star's cores and white dwarfs, maybe few other places. D.E.G. has influence to a star dynamics, nuclear decays and many other things.

While trying to answer Is there something like "forced / induced electron-capture"? , I realized my own question - is there any possibility or a clever trick to study degenerate gas on Earth in laboratory?

Edit: I was recommended to specify more closely - so -I would like to see the effects of degenerate electron gas on nuclei, radioactivity. With filling the Fermi sea higher, one can not only change some halflives, but also some stable species can become radioactive.

And - I have no idea, how the dynamics of nucleus-nucleus collisions can change (if it changes). There should be some calculations of everything, but - is an experiment possible?

• The electrons in a common metal form a degenerate electron gas- there is a very well-defined Fermi surface. If this isn't what you had in mind you may need to refine your question... Apr 6, 2017 at 17:04

Consider the Fermi temperature of an ideal electron gas $$T_{F} = \frac{\pi^2 \hbar}{2 m_e k_B} \left( \frac{3 n}{\pi} \right)^{2/3}$$ Or $$T_F \sim \left(\frac{n}{10^{21} m^{-3}} \right)^{2/3} K$$ I.e. to make the Fermi temperature of the order of normal earthly temperatures of hundreds of Kelvin we need a density like $10^{24}$ electrons per meter cubed.

But this is not too difficult to reach, we know that one mole of matter, typically a small handful for normal solid elements, is $\sim 10^{23}$ atoms. We just need the electrons to be free, unconstrained within the atoms. We know of a case where electrons are almost freely flying around in a material, and that is metals. For instance in Magnesium, the density of conducting (free-ish) electrons is $\approx 8.6 \times 10^{28} m^{-3}$ so the electrons will be well degenerate even at room temperatures.

As a result, we can study the contribution of the degenerate electron gas to properties of metals. One historically very important result verified also experimentally is the fact that the heat capacity of most metals follows a law $$C_V = K_1 T + K_2 T^3$$ where $K_1$ corresponds to the contribution of the degenerate electrons and $K_2$ to the "normal" oscillations of the grid (phonons) in the Debye model.

Similarly, one finds "hard" contributions to compressibility from the degenerate pressure of the electron gas. The pressure of the free non-relativistic ideal electron gas is $$P = \frac{\pi^2 \hbar}{5 m_e} \left( \frac{3}{\pi} \right)^{2/3} n^{5/3}$$ The compressibility is conventionally expressed using the isothermal bulk modulus $K_T$ $$K_T \equiv n \frac{\partial P}{\partial n}|_{T=const}$$ which gives us an estimate on the degenerate contribution as $$K_T = \frac{\pi^2 \hbar}{3 m_e} \left( \frac{3}{\pi} \right)^{2/3} n^{5/3}$$ which is independent of temperature in the degenerate limit we are considering. This contribution to compressibility turns out to be in fact dominating to the compressibility of most metals.

You can contrast this with the compressibility of a classical ideal gas, where the same computation goes as $$P = n k_B T \to K_T = n k_B T$$ and the compressibility scales linearly with temperature.

A pedagogical introduction to the topic and how you get beyond the free, noninteracting electrons is given in Chapters 6,7 and 9 of Kittel's Introduction to solid state physics, and a more advanced treatment is in the first half of the canonical Solid state physics by Ashcroft and Mermin.

• Thanks, although it is not exactly what I expected, it is enlightening. Do you have some reference, where one can get some more complete picture? Like, relation of spatial and momentum picture, or something about 'hard' compressibility. ? Apr 10, 2017 at 7:39
• @jaromrax I made an edit to the answer which should be exhaustive enough. Of course, these are only quick estimates and to be able to compare with experiments you have to go at least to the approximation of an ideal electron gas in a periodic potential.
– Void
Apr 12, 2017 at 12:37
• Thanks for the effort. It is not the answer to my question, maybe there is no answer to it at all. I liked hearing about compressibility of metals, and reference to books. I appreciate it and assign the bounty. Apr 12, 2017 at 13:19
• @jaromrax Oh, I see, I had not noticed your edit. The point of induced electron capture in neutron stars is not really the degeneracy of electrons, but simply the heightened overlap of electrons and nuclei. In fact, the condition that the wave-packets of protons and electrons significantly overlap is the basis of the estimate of the neutron-star densities. Degeneracy just tells us that there will be a typical energy of the electrons essentially independent of the temperature.
– Void
Apr 12, 2017 at 21:22
• ... But this simply means that the conditions of heightened electron capture are essentially neutron-star densities $\sim 10^{17} kg/m^{-3}$. We will quite certainly never have that on Earth for macroscopic or even mesoscopic systems. I do not see any loopholes in this argument. Getting a little bit out of my depth here, I think the only way you could probably study the heightened electron capture by nuclei is in something like the planned Electron Ion Collider.
– Void
Apr 12, 2017 at 21:32

I'm not sure if this is what you're looking for, but people do directly study degenerate gases of Fermionic atoms, which using laser cooling and evaporative cooling to nanokelvin temperatures can be brought significantly below the Fermi temperature. At cold enough temperatures and under certain conditions, these gases behave universally, and are a reasonable model for a degenerate electron gas. See for example the work of Deborah Jin's group at the University of Colorado and that of Martin Zwierlein's group at MIT.

People do study the interactions of degenerate Fermi gases with impurity atoms (see the publications linked above and their references), however I'm not sure if there has been any focus on making direct connections to star dynamics or nuclear decays.

• Thank you, it is very interesting, a bit more about mutual relation of Fermions than an electron gas. I expected something like this, although it is not much comprehensible... I think I cannot spit the bounty, can I. Apr 11, 2017 at 9:10
• I have ment "split" the bounty. The papers, despite of being interesting, are more in the direction I expected, are too specific to understand from somebody from far. And they are lot about interactions of bose / fermi ultracold systems. Thanks. Apr 12, 2017 at 13:15