Do the magnetic moments of neutrons mutually align to produce the magnetic field and emission beams of a pulsar and/or magnetar?

Question

In this graphic of a pulsar the emission beams are at the north and south poles of the neutron star's magnetic field.

I've read that the Earth's magnetic field is produced by a dynamo effect, whereby convection currents in the electrically conducting magma of the Earth's outer core create electric currents that produce a magnetic field.

But if most of the matter in neutron stars consists of superfluid electrically neutral neutrons, it seems to me that the dynamo theory would not work there, especially given that neutron stars which are pulsars have very strong magnetic fields.

When the core of a star's supernova explosion becomes a neutron star, it contains the original star's angular momentum in a much smaller object, so it spins very fast. The neutron star also contains the original star's magnetic field, so I assume its magnetic field likewise must be very strong. Would the magnetic field of the collapsing core of the original star cause the neutrons to align their magnetic moments? And is this the cause of a pulsar's strong magnetic field?

Answer 1

Some theories posit that a neutron star's magnetic field is residual, left over from the remnant's creation. Strong arguments are given for this in Flowers & Ruderman (1977)¹:

1. Dynamo mechanisms, which are responsible for the magnetic fields of many celestial bodies, cannot exist in mature neutron stars, because damping of fluid motion would have removed any significant movement of conductive fluid inside the remnant.
2. Permanent magnets cannot exist inside neutron stars, according to the present-day models (this, admittedly, is from 1977; since then, we have made more strides in understanding neutron stars).

Therefore, the magnetic field of a neutron star must be "fossilized". That's not to say these fields can't flair up again - in fact, arguments are presented in Price & Rosswog (2006) that during neutron star mergers shortly before a catastrophic event (e.g. a gamma-ray burst) the magnetic fields can be amplified considerably. However, the resulting merger may destroy the two bodies.

I should also mention a second major mechanism for magnetic field formation, discussed in (among others) (Spruit). In this mechanism, magnetic fields are generated through core collapse, through convection, field generation in "stable zones", and neutrino convection.

However, not everyone believes that magnetic fields cannot grow after neutron star formation. This overview discusses several models that say that thermal processes can occur within ~100,000 years of the neutron star's formation that can build up a magnetic field. There are two main models:

1. The battery model - originally proposed for "normal" stars like the Sun - states that different ionized components near the object's core behave differently due to different gravitational masses, with electrons wandering outwards a bit due to gravity and partial pressure. This acts like a battery, generating currents that in turn produce a magnetic field. However, matter in a neutron star is degenerate, and so a straightforward version of this mechanism is impossible. It is possible that temperature-dependent pressure could solve this, but the model is still not favored.

2. The thermoelectric mechanism solves the problem of degenerate components that arises from the battery model. It requires a non-zero vertical temperature gradient (which is present) and an existing "seed" magnetic field. The gradient brings "hotter" electrons up and "cooler" electrons down, which create a horizontal temperature gradient. This gradient requires pressure changes to go hand in hand with it, which thus brings about a thermoelectric field. The thermoelectric field helps the "seed" field grow.

   Here, the basic equation is $$\frac{\partial\vec{B}}{\partial t}=\overbrace{\vec{\nabla}\times\left(\vec{V}\times\vec{B}\right)}^{\text{Field convection term}}-\overbrace{\vec{\nabla}Q_0\times\vec{\nabla}T}^{\text{Battery term}}-\overbrace{\vec{\nabla}\times\left[\frac{\vec{\nabla}\times\vec{B}}{4\pi\sigma_0}\right]}^{\text{Ohmic decay term}}$$

There are various theories out there about why neutron stars have magnetic fields - some even say that neutron stars are giant magnets². The thing is, there is no consensus at the moment as to what the actual reason is. It doesn't seem like what you propose is correct, but we can't be sure because modeling convective processes under these conditions (as well as the interiors of neutron stars) isn't easy. For example, differential rotation needs to be accounted for. It appears that the answer to your question is "no", though.

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^{1 They also explain that at birth, the spin axis and magnetic dipole are nearly aligned, and that the dipole drifts over time, leading to a gap like the displayed in the picture linked to in the question.
2 The accompanying paper is Hansson & Ponga (2011).}

Answer 2

I've read that the Earth's magnetic field is produced by a dynamo effect, whereby convection currents in the electrically conducting magma of the Earth's outer core create electric currents that produce a magnetic field.

There's more than one way to skin Schrödinger's cat. See where we were talking about the Einstein-de Haas effect here. A bar magnet is like a solenoid, and both feature electrons going round and round. But the electron in itself "behaves like a tiny bar magnet". Or like a little compass needle. All the electrons in the iron core are spin-aligned like little compass needles all pointing North. Then when you change the direction of the current, they all spin-flip and point South.

But if most of the matter in neutron stars consists of superfluid electrically neutral neutrons, it seems to me that the dynamo theory would not work there

Sounds reasonable. A solenoid only has a magnetic field because the electrons go round and round whilst the metal ions don't. If the electrons were tied to the ions you wouldn't have an electric current, or a magnetic field. But does a neutron star actually consist of neutrons? Let's phone a friend: "Current models indicate that matter at the surface of a neutron star is composed of ordinary atomic nuclei crushed into a solid lattice with a sea of electrons flowing through the gaps between them. It is possible that the nuclei at the surface are iron, due to iron's high binding energy per nucleon". We don't actually know what a neutron star is really like.

When the core of a star's supernova explosion becomes a neutron star, it contains the original star's angular momentum in a much smaller object, so it spins very fast.

We are confident that neutron stars spin very fast. They were discovered by Jocelyn "no-Nobel" Bell Burnell.

The neutron star also contains the original star's magnetic field, so I assume its magnetic field likewise must be very strong.

But we don't actually know that it contains the original star's magnetic field. We have a law of conservation of angular momentum, but we don't have a law of conservation of magnetism.

Would the magnetic field of the collapsing core of the original star cause the neutrons to align their magnetic moments? And is this the cause of a pulsar's strong magnetic field?

Maybe. See this easy-reading NASA article about pulsars: "The 'pulses' of high-energy radiation we see from a pulsar are due to a misalignment of the neutron star's rotation axis and its magnetic axis". That suggests the intense magnetic field isn't due to the rapid rotation. And see this in HDE 22868's giant magnets reference: "Getting into the physics of Hansson and Ponga’s paper, they suggest that when a neutron star forms, neutron magnetic moments become aligned". Like little compass needles.