Neutron stars and condensed matter physics

In the wake of the remarkable results reported earlier this week regarding colliding neutron stars, I wanted to write just a little bit about how a condensed matter physics concept is relevant to these seemingly exotic systems.

When you learn high school chemistry, you learn about atomic orbitals, and you learn that electrons "fill up" those orbitals starting with the lowest energy (most deeply bound) states, two electrons of opposite spin per orbital.  (This is a shorthand way of talking about a more detailed picture, involving words like "linear combination of Slater determinants", but that's a detail in this discussion.)  The Pauli principle, the idea that (because electrons are fermions) all the electrons can't just fall down into the lowest energy level, leads to this.  In solid state systems we can apply the same ideas.  In a metal like gold or copper, the density of electrons is high enough that the highest kinetic energy electrons are moving around at ~ 0.5% of the speed of light (!).  

If you heat up the electrons in a metal, they get more spread out in energy, with some occupying higher energy levels and some lower energy levels being empty.   To decide whether the metal is really "hot" or "cold", you need a point of comparison, and the energy scale gives you that.  If most of the low energy levels are still filled, the metal is cold.  If the ratio of the thermal energy scale, \(k_{\mathrm{B}}T\) to the depth of the lowest energy levels (essentially the Fermi energy, \(E_{\mathrm{F}}\) is much less than one, then the electrons are said to be "degenerate".  In common metals, \(E_{\mathrm{F}}\) is several eV, corresponding to a temperature of tens of thousands of Kelvin.  That means that even near the melting point of copper, the electrons are effectively very cold.

Believe it or not, a neutron star is a similar system.  If you squeeze a bit more than one solar mass into a sphere 10 km across, the gravitational attraction is so strong that the electrons and protons in the matter are crushed together to form a degenerate ball of neutrons.  Amazingly, by our reasoning above, the neutrons are actually very very cold.  The Fermi energy for those neutrons corresponds to a temperature of nearly \(10^{12}\) K.  So, right up until they smashed into each other, those two neutron stars spotted by the LIGO observations were actually incredibly cold, condensed objects.   It's also worth noting that the properties of neutron stars are likely affected by another condensed matter phenomenon, superfluidity.   Just as electrons can pair up and condense into a superconducting state under some circumstances, it is thought that cold, degenerate neutrons can do the same thing, even when "cold" here might mean \(5 \times 10^{8}\) K.