## Friday, October 20, 2017

### 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.

DanM said...

Of course, with a Fermi temperature of 10^12 K for neutrons, you have a Fermi velocity which is about 40% of the speed of light. A non-relativistic treatment becomes questionable...

Douglas Natelson said...

So you do the relativistic version. No worries.

DanM said...

Yea but that's more work.

Anonymous said...

Hi Doug, so why can't we have much smaller clusters of neutrons? Say, 100 or 1000 neutrons? There is no electrostatic repulsion, and the strong force would hold them together?

Anonymous said...

Neutrons decay via weak interactions. In neutron stars the gravitational binding energy is strong enough to stabilise them, but not in smaller clusters, even many isotopes of atomic nucley. A typical example is the decay of carbon-14 into nitrogen-14.

Anonymous said...

That makes sense, thanks... so a small cluster is unstable, and a large one stable thanks to gravity. What is then the minimum number of neutrons needed for a ball of neutrons to be stable, in theory? In other words, can you imagine a body of neutrons significantly smaller than a neutron star, yet still stable?

Douglas Natelson said...

Thanks, Anon@12:34. I don’t know the minimum stable neutron cluster size. Electron degeneracy is basically what holds up white dwarf stars, meaning that gravity is insufficient to cause the p-e —> n plus electron neutrino inverse beta decay process until you get up to around 1.3 solar masses. I’m not sure what would happen, though, if you tried to chip off, e.g., an asteroid-sized blob of neutronium from a neutron star. It all comes down to the neutron fluid equation of state. Naively, the speed (via the Fermi energy) of the most energetic neutrons would exceed the gravitational escape velocity of the blob, and the blob would explosively evaporate since gravity would not be sufficient to bind it together.

Zach said...

I'm pretty sure that what would happen if you tried to chip off an asteroid-sized blob of neutronium from a neutron star is that you'd be ripped into a long, thin splatter of gore by tidal forces.

tonyon said...
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tonyon said...
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