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Astronomy and space

Astronomy and space

Pulsars, glitches and superfluids

01 Jan 1998

Astronomers have long been intrigued by occasional "glitches" in the rotation of pulsars.

These neutron stars usually rotate with such precision that they are known as the best timekeepers in the universe, but every so often their rotation rate suddenly increases. It is thought that these glitches are related to superfluidity inside the star, which allows the neutrons to flow without friction.

Now, a group of Italian scientists have studied the interactions between quantized vortices in the superfluid and the nuclei in the stars, which are widely thought to be responsible for the glitches (P Pizzochero et al. 1997 Phys. Rev. Lett. 79 3347). The work will have important implications for understanding the dynamics and thermal histories of neutron stars.

Neutrons become superfluid at temperatures equivalent to energies of a few MeV. All systems of interacting fermions (particles with half-integer spin) are expected to form a condensate of Cooper pairs at low enough temperatures – familiar examples are superfluidity in helium-3 and electron superconductivity. Neutron stars cool to below 1 MeV very soon after formation, and are therefore expected to have superfluid interiors.

Superfluids are characterized by a forbidden energy gap, which can be calculated using the Bardeen-Cooper-Schrieffer (BCS) equation. The pairing of protons and neutrons in the nuclei also leads to an ordered phase with an energy gap between paired (even) and odd nuclei. Indeed, the equation for the energy gap in nuclei is just a discrete version of the BCS equation. In nuclear matter, the attractive nuclear interactions lead to neutron superfluidity and proton superconductivity.

At the centre of a neutron star, where the density exceeds 2 ´ 1014 g cm-3, the neutrons form a homogeneous superfluid. But in the crust of the star, where the density is lower, this superfluid coexists with a crystalline lattice containing nuclei with comparable numbers of protons and neutrons. The BCS theory must then combine the discrete version for paired nuclei with the extended version for a superfluid.

Direct observational evidence for superfluidity in neutron stars comes from pulsar glitches. A superfluid can only rotate by forming quantized vortices, where the number of vortices per unit area is proportional to the rotation rate. Lines of vortices move outwards from the centre of the star, reducing the vortex density and hence the rotation rate of the superfluid. The coexistence of superfluid and lattice in the crust leads to very distinct rotational dynamics, since vortex lines become “pinned” to nuclei in the lattice. This means that it can take months or even years for the rotation rate to relax after a glitch. If the interior of the star were made up of normal matter, viscous processes would give rise to much shorter relaxation times.

Pulsar glitches are thought to be caused by a sudden release of pinned vortex lines, an idea proposed by Philip Anderson and Naoki Itoh in 1975. Vortex lines can also move continuously through the crust superfluid, either by thermal activation or by quantum tunnelling against pinning forces, a process known as vortex creep. Models indicate that both the unpinning that leads to glitches and the post-glitch relaxation due to vortex creep depend on the strengths of the pinning forces and the lag in rotation rate between the crust superfluid and the lattice.

In particular, a given pinning force can only sustain a certain amount of lag, beyond which the vortex line will become unpinned. Similarly, there is a steady-state lag, somewhat less than the critical lag, at which vortex lines will creep and allow the crust superfluid to relax with the normal matter in the crust. Angular momentum is therefore transferred from the superfluid to the normal matter, and the rate of energy dissipation is proportional to the difference in rotation speed between the two components. Limits on this rate of energy dissipation, obtained from observations of thermal luminosity from pulsars, constitute the most direct constraint on the pinning forces.

So what is the physical reason for pinning? A vortex line is a singularity in the phase of the wavefunction governing the particles in the superfluid. The velocity of superfluid circulating a quantized vortex line decreases as 1/r, where r is the distance from the vortex axis. The kinetic energy of each particle decreases as 1/r2, while the energy that each particle gains through being in the superfluid phase – the condensation energy – is given by ~D2/EF, where D is the superfluid energy gap and EF is the Fermi energy.

At some critical distance, called the coherence length, V, the loss in kinetic energy exceeds the condensation energy. At distances less than the coherence length, the vortex line consists of a cylindrical core surrounding the singularity, within which some particles are in the normal (non-superfluid) state. If the condensation energy varies on similar length scales to V, the vortex line will become pinned to certain preferred sites.

This is the situation in the crust of a neutron star, where the density and condensation energy of the superfluid have different values inside and outside the nuclei. The size of the nucleus is ~10 fm, comparable with the coherence length, while the lattice spacing is 30-50 fm. Pinning forces thus tend to pin the vortex lines to nuclei in the lattice.

The challenge is to calculate and compare the energies when a vortex line goes through a nucleus and when it does not. I made the first estimate of the pinning force in 1977 by simply comparing the condensation energies in these two cases. To estimate the condensation energy inside a nucleus, I combined the local density of superfluid neutrons and the value of D for a homogeneous neutron superfluid at that density.

In 1988, Richard Epstein and Gordon Baym considered how the superfluid density varied between the nucleus and the surrounding neutrons. They used the Ginzburg-Landau approximation, but this is only valid if the density varies over distances longer than the coherence length. However, the change in superfluid density between the inside and outside of a nucleus takes place over about 1 fm. Epstein and Baym therefore rescaled the coherence length by comparing their model’s results for finite nuclei with known experimental results. With this rescaling, sometimes by as much as a factor of 10, they obtained strong pinning energies of around 15 MeV. However, observations of the thermal luminosity of neutron stars give upper bounds on the critical lag between the lattice and the superfluid, ruling out such strong pinning.

Pizzochero and colleagues have instead used a local density approximation, which had previously been used to calculate the superfluid gap and specific heat in the crust of neutron stars. To calculate the pairing energies, they replaced the discrete bound-state energy levels in the BCS equation for nuclei in the crust with a continuous energy spectrum for a single particle. Since the gap and pinning energies mainly depend on the states near the Fermi level, this should be a good approximation to the pairing. The calculation also includes the kinetic energy contributions from the flow around the vortex line.

The results suggest that the pinning force between a nucleus and a vortex line is about 0.5 MeV fm-1, similar to my first rough estimate and about 10 times less than the results obtained by Epstein and Baym. These pinning forces are not strong enough to dislodge nuclei from their equilibrium sites in the lattice, so the vortex line probably only pins to nuclei that lie within its core. Estimates of the critical lag and rate of energy dissipation, assuming this “weak” pinning, agree with observations of thermal X-rays from neutron stars.

Understanding the pairing mechanism in the exotic setting of the neutron-star crust continues to intrigue theoretical physicists, and the microscopic model developed by Pizzochero and colleagues will be used to study other phenomena in neutron stars. It could help, for example, in studying the innermost crust layers, where the “nuclei” might be elongated into rod- or slab-like shapes. And by improving our knowledge of how pinning forces vary in the crust, the model could also provide clues to where vortices accumulate and how glitches are triggered.

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