Master's Thesis

MASTER'S THESIS

Pulsars as "Neutromagnets" A Theory About the Magnetic Field of a Neutron Star

Anna Ponga 2014

Master of Science in Engineering Technology Space Engineering

Luleå University of Technology Institutionen för teknikvetenskap och matematik ABSTRACT

Pulsars are believed to be a kind of neutron stars. They rotate very fast; a period of a second is not uncommon. There are also millisecond-pulsars, as well as pulsars with a larger period of about eleven seconds, these are called magnetars. This thesis is about the magnetic field of these pulsars.

A normal pulsar has a magnetic field strength in the order of 1012 G (108 T), but there are special cases, the magnetars, that has a magnetic field strength of up to 1016 G (1012 T). The strong magnetic field affects the rotation period of these objects, hence the much longer period.

The model in question that is investigated in this thesis claims the neutron star, or pulsar, is a giant nucleus consisting of neutrons with all spins aligned. This is a possible source for the strong magnetic field, and it can also explain the stability of the field as well as why the magnetic field axis is not aligned with the rotational axis, which other models cannot explain in a satisfactory way.

With help of the Heisenberg Hamiltonian, which describes ferromagnetism, and the Argonne v14 potential, which is a nucleon-nucleon potential, it will be shown that this model is very possible.

SAMMANFATTNING

Pulsarer tros vara en sorts neutronstjärnor. De roterar väldigt fort, en period på en sekund är inte alls ovanlig. Det finns även ”millisekunds”-pulsarer, likaväl som det finns pulsarer som har en period på ungefär elva sekunder, dessa kallas magnetarer. Detta examensarbete handlar om magnetfältet hos dessa pulsarer.

En normal pulsar har ett magnetfält av storleksordningen av 1012 G (108 T), men det finns särskilda fall, de så kallade magnetarerna, som har en magnetfältstyrka på upp till 1016 G (1012 T). Det starkare magnetfältet påverkar dess rotationsperiod, därav den mycket högre rotationstiden.

Modellen som ska undersökas i detta examensarbete säger att neutronstjärnan, eller pulsaren, är en enorm kärna som består av neutroner med sitt spinn riktat åt samma håll. Detta är källan till det starka magnetfältet, och modellen kommer också förklara fältets stabilitet likaväl som varför magnetfältet inte behöver vara riktat åt samma håll som rotationsaxeln. Detta är något som andra modeller inte kan förklara på ett tillfredsställande sätt.

Med hjälp av Heisenberg Hamiltonianen, som beskriver ferromagnetismen, och

Argonne v14 potentialen, som är en nukleon-nukleon potential, kommer det att visas att denna modell kan stämma. .

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TABLE OF CONTENT

ABSTRACT ...... I SAMMANFATTNING ...... III TABLE OF CONTENT ...... V 1 INTRODUCTION ...... 1 1.1 Task ...... 1 1.2 Method ...... 2 2 STELLAR EVOLUTION ...... 3 3 NEUTRON STARS ...... 5 3.1 Discovery ...... 5 3.2 ”Orthodox” model of a neutron star ...... 7 3.2.1 The birth of neutron stars ...... 7 3.2.2 Structure of neutron stars ...... 8 3.2.3 Temperatures ...... 11 3.2.4 The origin of the magnetic field ...... 12 3.3 Pulsar specific properties ...... 13 3.3.1 Pulses ...... 13 3.3.2 Rotational period ...... 15 3.4 Magnetars ...... 16 4 MAGNETISM, THE HEISENBERG MODEL AND ARGONNE V14 POTENTIAL ...... 18 4.1 Magnetism ...... 18 4.2 The Heisenberg model...... 19

4.3 Argonne v14 potential ...... 23 5 A NEW MODEL FOR THE PULSAR MAGNETIC FIELD ...... 26 5.1 The Theory ...... 26 5.2 The physics behind ...... 27 5.2.1 The neutron ...... 27 5.2.2 Spin ...... 27 5.3 Calculations ...... 28 5.3.1 Number of neutrons ...... 28 5.3.2 The total magnetic field strength ...... 28 5.3.3 The interchange constant, J ...... 29 6 CONCLUSIONS AND DISCUSSION ...... 31 7 REFERENCES ...... 33 7.1 Books & articles ...... 33 7.2 Figures ...... 34 v

1 Introduction

There is an “orthodox” model for the magnetic field of neutron stars that for many years have been more or less accepted within the astronomical society (Manchester, Taylor, 1977). There are, however, a few problems with it; It does not explain: the strong magnetic fields satisfactorily; the stability of the magnetic field; how the magnetic field can be misaligned with the rotational axis. In this thesis a different model is presented and it will be shown that the strong magnetic fields of e.g. the magnetars, the stability and the misalignment can be understood with this model.

In chapter two the stellar evolution will be explained briefly. In the third chapter neutron stars and pulsars will be reviewed and the theories of today will be examined. After this, ferromagnetism and the Argonne v14 potential will be presented in the fourth chapter. In the fifth chapter a new theory for the magnetic fields of neutron stars and pulsars will be constructed and compared to known observational properties. And in chapter six the results and my own conclusions will be presented.

1.1 Task

The model that will be presented in this thesis suggests that the pulsar is a giant nucleus with all spins, and therefore all magnetic moments, directed in the same direction. This model would explain the strong and stable magnetic field of the pulsars which has not yet been done satisfactorily. The task at hand is to investigate whether or not this model can be valid or not. So the spin-dependent part of the binding energies will be examined as well as the Heisenberg Hamiltonian for ferromagnetism.

1.2 Method

Some simplifying assumptions will be made. These will be presented in chapter five. We will look at the Heisenberg Hamiltonian for ferromagnetism and the

Argonne v14 potential, which describes the nucleon-nucleon potential in a nucleus, and hence is a possible model for the nucleons in a neutron star.

2 Stellar evolution

Our universe is vast, and the number of stars one can see in the sky is uncountable. But where do these stars come from? There are something called nebulas, some of which can be seen with the naked eye, like the Orion nebula in Orion’s sword. There are different kinds of nebulas, emission, reflection and dark nebulas just to mention some. Stars are most often formed in the dark nebulas. The dark nebulae consist of dust grains and un-ionized atoms, there are also heavier atoms in the clouds, but dust grains and hydrogen are there in abundance. The reason why they are dark is because they are so opaque that they block any visible light coming from behind them and they are so cold that they do not emit any light themselves and hydrogen atoms may form molecules. Because of the vastness of the nebulas, several light-years across, many stars are formed simultaneously.

The star formation starts when the gravitational forces between the grains and molecules in the nebula overwhelm their kinetic energy. It starts to contract and since the gravitational forces are getting stronger the denser the cloud becomes, the more it contracts. It will continue to form a protostar, which will then become a main sequence star after several thousand years. A main sequence star is an “ordinary” star such as the Sun. If the mass is not sufficient to start hydrogen burning it will become a dark object or “failed star” such as a brown dwarf. And if the mass is too great they will rapidly evolve to super-luminous stars that will have an internal pressure that is greater than the gravitational pressure, and will expel its outer layers into space and disrupt the star.

Heavy stars will become main sequence stars faster than the light ones because they will come to start their hydrogen burning earlier. But they will also end their life as a main sequence star faster, since they will burn their fuel much faster. A main sequence star is a star like our own Sun; they are all burning hydrogen to create helium. When a main sequence star comes to this part of its life and it is running out of fuel, it will start to contract again, because the 3 radiation pressure will no longer be sufficient to balance out the forces of gravity. This collapse will continue until gravity is either balanced by quantum mechanical degeneracy pressure, in this case it will form a white dwarf, or it will continue to collapse until gravity is balanced by degenerate pressure between neutrons, and it will become a neutron star. If these forces still aren’t enough to stop the collapse, it will continue to collapse to form a black hole.

The Fermi pressure for degenerate matter is given by

휋2 푘푇 2 휋2 푘푇 2 휇 = 퐸퐹[1 − ( ) − ( ) + ⋯ ] (2.1) 12 퐸퐹 80 퐸퐹

Where T is the temperature, k is Boltzmann’s constant and EF, which is the Fermi energy, is given by

2/3 ℏ2 3휋2푁 퐸 = ( ) (2.2) 퐹 2푚 푉

N is the number of fermions in the system, V the volume, and m the mass of the fermion. This pressure and the centrifugal forces is what balances the gravitational forces so the collapsing star can reach a new stable state. The gravitational forces are given by

퐺푀 퐹̅ = − 푒̂ , (2.3) 푔 푟2 푟

Where G is the gravitational constant, M the mass of the star, r the stars radius and êr the unit-vector pointing out radially from the star.

3 Neutron stars

Neutron stars are the densest physical objects known to man. The only known object that theoretically has higher density is a black hole. In the neutron star the gravitational forces are balanced by the nuclear forces and baryonic Fermi pressure. In a white dwarf it is the degenerate pressure of electrons providing the Fermi pressure, while in neutron stars it is both degenerate electrons and degenerate neutrons, at different levels in the star. When the density reaches 4×1011 g/cm3 the Fermi pressure switches from consisting of degenerate electrons to neutrons. However, it is not until the density reaches 4×1012 g/cm3 that the pressure from the degenerate neutrons exceeds the pressure of the electrons.

Figure 1The distribution of neutron stars in the milky way .

3.1 Discovery

In 1934 Walter Baade and Fritz Zwicky suggested the existence of a neutron star-like object, with an extremely small radius and enormous density. During the coming 30 years other theories around neutron stars were stated, but it was not until 1967 that the first observation of a neutron star (a pulsar) was made, by the Ph.D student Jocelyn Bell and her thesis advisor Anthony Hewish. They had set up 2048 antennas over an area of four and a half acres. During this project where they were tuned in to 81.5 MHz. They were studying discrete radio scintillations from quasars passing through the solar wind. In July 1967 Bell found a bit of interference every 400 ft of her recorder. More careful 5 measurements showed that this interference occurred every 23 hours and 56 minutes. This meant that the source of these signals passed over the antenna array once every day. And it was determined that the source of these pulses was from an object outside the solar system. Doing more detailed measurements showed that these signals had a period of 1.337 s. For a while it was thought to be signals from another civilization, until a similar signal was found in another part of the sky.

Figure 2 Discovery of the first pulsar, PSR 1919+21

Signals from neutron stars had been recorded earlier, but they were not recognized. These signals had been filtered away as noise and interference. The intensity in the signals varies even though the period is constant, so this was a mistake easily done. This kind of neutron star that can be seen from earth is a specific kind of neutron star called a pulsar, which will be explained in a section 3.3. When the first neutron star had been detected, most of the radio observatories searched the sky for them, but the biggest effort was made by the large radio telescopes in Australia, USA and Puerto Rico. Today we know of several hundred pulsars spread over the sky, all of them in our own galaxy, their distribution can be seen in figure 1.

3.2 ”Orthodox” model of a neutron star

The most widely accepted model of a neutron star today can shortly be described, as a star with a solid crust with an underlying superfluid and possibly a solid core. In this section the life of a neutron star will be described as well as its structure, temperature and the origin of its magnetic field, as it is believed to arise in “the standard model”.

3.2.1 The birth of neutron stars

As mentioned earlier, neutron stars are formed during the collapse of a main sequence star, but they can also form from a white dwarf in a binary system that has accreted so much mass from its companion star that it has passed the Chandrasekhar limit. The Chandrasekhar limit is the upper limit for the mass a star can have that electron degeneracy pressure can support against gravitational collapse.

We will look at the case when a main-sequence star with mass M ≈ (10-25)Mʘ 30 (where Mʘ is the solar mass = 1.99 × 10 푘푔) dies. The collapse is a result of the star losing its hydrostatic equilibrium, the criterion for which is given by

휕 ln 푃 4 Γ = | | > , (3.1) 휕 ln 푉 3 if the star is non-relativistic. A star of this kind ends its life as follows. When the store of oxygen and other heavier elements are depleted, it will form an iron core which grows both in mass and temperature. After some ~107 years (Beskin al et, p64, 1993), the mass of the core will be ~10Mʘ, and the density will have reached 109 – 1010 g/cm3 at the core and its temperature will be at the critical 7 temperature of 107 K. When this happens the electrons will start to effect the equation of state due to the fact that they are becoming relativistic and are responsible for the pressure P. The photodisintegration reaction becomes energetically favored in the central regions, and matter neutronization is starting, this reaction looks as follows

56 4 훾 + 26퐹푒 ⇄ 13 2퐻푒 + 4푛 ⇄ 26푝 + 30푛. (3.2)

During the neutronization neutrinos leave the star and the electron density is decreased permanently, this is the cause of the loss of hydrodynamic equilibrium. Normally protons would be produced by the present neutrons, but due to the degenerate electrons there are no vacant states to occupy. The star will eventually reach a new stable state, which is in the form of a neutron star, (unless its mass is too great and it will form a black hole since the gravitational forces then cannot be balanced by the Fermi pressure). The creation of a neutron star from a white dwarf is similar, but it requires a binary system from which the white dwarf can accrete the mass necessary to evolve to a neutron star.

3.2.2 Structure of neutron stars

The outer crust of a neutron star consists of 56Fe nuclei and degenerate electrons if the temperature is below the critical temperature of 1010 K, and the nucleons have Fermi energy of 1 MeV. The outer crust is very rigid. One can see the structure of a neutron star in figure 3.

The inner crust has higher density which leads to a higher Fermi energy, around 25 MeV, and therefore the electrons can be captured by the nuclei. This layer consists mainly of neutron-rich nuclei and degenerate electrons. Beneath the inner layer there might be a shallow layer of normal neutrons.

Figure 3The structure of a neutron star. The top layer is a solid crystalline crust, underneath it there is a solid with free neutrons, next there is a layer of superfluid neutrons and then, possibly, a solid core.

When the density rises even more, the Fermi energy will rise, to approximately 100 MeV, and the neutrons will be released from the nuclei and form a sea of free neutrons. Here the neutrons are superfluid and the protons are super- conductive. Superfluidity means that there is no friction between the degenerate neutron matter, and superconductivity means that the protons can move around without being restricted by any electrical resistance. The electrons keep their natural behavior even at these pressures and densities.

If the neutron star has a radius large enough, there may be a possibility that it has a solid core. The density in the core would be about 1015 g/cm3. Here the Fermi energy would be so great that the neutrons could break down to quarks, so that i.e. muons and hyperons were created, or even a “quark-gluon plasma”, a state of liberated quarks and gluons. Muons are leptons, it is a “heavier electron”, whereas the hyperon is any baryon that consists of three quarks, where at least one is strange and none are charm or bottom quarks. Both have half-integer spin and thus are fermions.

Models of the neutron star like this is made from an integration of the general- relativistic equation of hydrostatic balance (Manchester, Taylor, p172, 1977)

푃(푟) 4휋푟3푃(푟) 퐺[휌(푟)+ ][푚(푟)+ ] 푑푃 푐2 푐2 − = 2퐺푚(푟) (3.3) 푑푟 푟2[1− ] 푟푐2

Where m(r) is the mass inside r, P(r) is the pressure and ρ(r) is the density at r. P(r) and ρ(r) are related by an equation of state, 푃 = 푃(휌).

푑푚(푟) = 4휋푟2휌(푟) 푚(0) = 0 (3.4) 푑푟

These equations, together with the equation for hydrostatic balance, give the criteria for hydrostatic equilibrium of a spherically symmetric cold star.

The Equation of state determines the upper mass limit. For a softer equation of state the upper mass limit of a neutron star would be 2-2.5Mʘ, this was determined by Pandharipande in 1976 (Pandaharipande, Wiringa, 1976). A softer equation of state means that there will be less pressure at a given density. A harder equation of state and with general relativistic effects allows a higher upper limit for the mass, this was shown by Rhodes and Ruffini already in 1974 (Rhoades, Ruffini, 1974). They came to the conclusion that the limit was around 3.2Mʘ. This together with observations of X-ray pulsars rules out the softer equations of state. There are numerous sets of equation of states for neutron stars with an upper mass limit ranging from 0.7 Mʘ to 3 Mʘ (Beskin, p71, 1993).

3.2.3 Temperatures

When a neutron star is formed it is very hot, T~1011 K, it will cool off very quickly until it has a temperature of around 1010 K. This happens through the so-called URCA process, where nucleons switch from being a proton to being a neutron and back again, as follows,

+ − 푛 → 푝 + 푒 + 휈푒̅ , + − 푝 + 푒 → 푛 + 휈푒.

The neutrinos and antineutrinos created in these processes escape the neutron star unhindered and the neutron star rapidly cools off. Due to this rapid cooling the solid crust and superfluid interior should form within an hour of its creation. During the next day or so the nucleons become degenerate and the URCA- process will cease. The neutron star will continue to cool down rather quickly for another 106 years. The first 103-104 years it will cool due to neutrino- antineutrino couple emission. After this it will start to emit X-ray photons and continue its cooling until it reaches a temperature around 104-105 K.

If one takes into account other forces acting on the star the “end” temperature will be higher. Even though the neutrons are superfluid other frictional interactions are causing a slowdown to these neutrons which results in thermal dissipation of rotational energy. With this included in the calculations T ~ 6×105 K. When this temperature is reached the star will continue to cool down in a much lower pace. At this temperature pulsars with low mass and large radii are visible in the optical spectra.

3.2.4 The origin of the magnetic field

As mentioned earlier the magnetic field of a pulsar is very strong, 1010-1012 Gauss (106-108 T) for most radio pulsars (Lyne, Graham-Smith, p17, 1998). The magnetic field is thought to be frozen into the star during formation, so the magnetic flux through the surface will be conserved. The flux is given by

Φ = 푩 × 푑푨, (3.5) ∫푆 where B is the magnetic field vector. If one ignores the geometry of the field, that means that the product of the star’s area and the magnetic field strength remains constant. With this assumption one can get the magnetic field strength from formula 3.6. Take some known properties from a white dwarf, BWD = 8 4 8 5×10 G (5×10 T), and a radius RWD = 2.9×10 cm. And a radius of RNS = 5.7×105 cm for the neutron star, you get

2 푅푊퐷 14 10 퐵푁푆 ≈ 퐵푊퐷 ( ) ≈ 1.3 × 10 퐺 (1.3 × 10 푇). (3.6) 푅푁푆

This shows that neutron stars can have very strong magnetic fields. The field strength chosen for the white dwarf in equation 3.6 is large compared to typical white dwarf field strength.

The original star probably had a magnetic field in the order of B ~102 G (10-2 T) and an average density of ~1g/cm3, and as the star contracts the magnetic field strength will increase proportional to of ~ρ-2/3, so at the final density of the pulsar magnetic field can reach the values that have been observed, ~1012 G (108 T), which is smaller than the field strength in the example above, equation 3.6.

3.3 Pulsar specific properties

Pulsars are a kind of neutron stars. It was in fact the pulsars that gave the “proof” as to why we believe that neutron stars exist. This is the only kind of neutron star one can observe directly and one can calculate its mass dynamically only if it is in a binary system, where it also may accrete material from its companion, which could be another pulsar, a white dwarf etc.

The naming of pulsar, like PSR 1919+21, tells what kind of object it is and where it is to be found. PSR stands for Pulsating Source of Radio, 1919 is the right ascension and 21 is the declination.

3.3.1 Pulses

There are three different ways one can obtain regular pulses: In a binary star system, pulsating stars, and rotating stars.

Since pulsars can have a period of just a couple of milliseconds the separation in a binary system would have to be very small, ~1.6×108 cm, for a period of 0.79 s, which is the average pulsar period. This distance is much smaller than the radius of the known white dwarf, Sirius B. So this rules out the binary system for being a source of these rapid pulses.

So we have pulsating stars and rotating stars left. An oscillating white dwarf has a period of 100 to1000 s, which is much too long for the rapid pulses of a pulsar. There is also a problem with oscillating neutron stars. Neutron stars are about 108 more dense than a white dwarf and the period of the oscillations are proportional to ρ-1/2. So a neutron star would oscillate with a period of approximately 10-4 s, which is much too fast for the slower pulsars that have a period of several seconds.

Left is the option of rotating stars. A rotating star, with the angular momentum as a neutron star has, would rotate with a fixed period with no or small 13 deviations. The angular velocity is limited by the gravitational forces to keep it together. If one assumes that it is a stable sphere, the maximum angular velocity could be found by putting the gravitational acceleration equal to the centripetal acceleration

푀 휔2 푅 = 퐺 (3.7) 푚푎푥 푅2

The minimum period is given by Pmin = 2π/ωmax, which gives

3 푃푚푖푛 = 2휋 √푅 /퐺푀. (3.8)

-4 So if one assumes a mass of 1.4 Mʘ, one gets a minimum period of 5×10 s.

The reason why there are pulses coming from these rotating star is because the magnetic axis is not aligned with the rotational axis, as shown in figure 4.

Figure 4. Misalignment of the magnetic axis and the rotational axis in a pulsar, which probably gives rise to the so-called beacon effect.

Most of the known pulsars emit radio pulses. There are also a few that emit gamma radiation and some that do indeed emit pulses that can be detected within the visible spectra. Examples of optical pulsars are the Crab and the Vela pulsars.

3.3.2 Rotational period

Figure 5. Distribution of rotational periods of known pulsars Pulsars have a rotational period from a couple of milliseconds to a couple of seconds, which is extremely fast compared to other objects of similar mass, like our own sun. The velocity at the equator of the pulsars and neutron stars must not surpass the speed of light. Calculations regarding the Crab pulsar, which has a period of 33 ms, shows that this particular pulsar cannot have a radius larger than 1700 km. The rotation period was one of the properties anticipated before they were observed As many neutron stars, and therefore pulsars, come from a core of a star, the conservation of angular momentum points towards a rapidly rotating neutron star/pulsar.

3.4 Magnetars

Magnetars was first discovered in March 1979 by the space-crafts Venera 11 and 12, roughly ten years after the discovery of the first pulsar. These space- crafts had been dropping off probes in the Venus atmosphere and were now circling the sun. They caught very strong signals in gamma rays, which for a short period of time made their radiation readings go from 100 to 200 000 readings per second. This signal was also picked up by nine other space-crafts in orbit.

This pulse consisted of so called “hard” gamma rays, that are highly energetic rays, and it lasted for only two tenths of a second. It was followed by “soft” gamma rays and x-rays, which dissipated over a couple of minutes. The upcoming four years there were another 16 bursts detected from the same direction, but none as strong as the original pulse.

In the mid-1980’s two more similar objects was discovered. At first these objects were catalogued as Gamma Ray Bursters (GRB). But their behavior was not consistent with that kind of object. So at a meeting in 1986 astronomers decided to name them Soft Gamma Repeaters (SGR). And it was only in 1998 that evidence of the existence of magnetars was presented, even though this theory was developed to solve another problem, it explained the SGR’s.

Magnetars are believed to be highly magnetized neutron stars, and their magnetic fields can have strengths up to 1015 G (1011 T), with an upper limit of 1017 G (1013 T). Compared to ordinary neutron stars which have a magnetic field strength of about 1012 Gauss, and a refrigerator magnet which have a strength of about 100 Gauss, this is a very strong magnetic field. If the magnetic field would become stronger than 1017 G it would theoretically dissipate because the fluids inside the star would mix.

Magnetars also have longer rotational periods than the more common radio pulsars have. It is due to its strong magnetic field, which acts as a break for the star remnant. Magnetars have a period of approximately 5-11 s (Kouveliotou et al, 2003), which is a lot longer than that of the radio pulsar. Because of its strong magnetic field it also has a short lifespan to being observed, it would only take it about 5 000 years to slow down to a rotational period of 8 s.

The reason for the bursts that have been observed from these stars is conjectured to be due to that the magnetic field changes shape and makes the crust of the neutron star move, which warms up its interior and makes the crust break. When the crust breaks a huge amount of magnetic energy is released and also soft gamma rays. The magnetic energy will then create a dense cloud of electrons and positrons, which will travel throughout the universe accompanied by the gamma rays.

4 Magnetism, the Heisenberg model and Argonne v14 potential

4.1 Magnetism

There are several different kinds of magnetisms. Ferromagnetism is probably the most well-known, which is also the kind we base this work on. There are two other kinds of magnetism that lies close to ferromagnetism, which is ferrimagnetism and antiferromagnetism. Antiferromagnetism is when there are equally many magnetic moments “up” as “down” and they are of equal strength. Ferrimagnetism is when there are equally many magnetic moments “up” as “down” but they are of different strengths. See figure 6 for closer explanation.

Figure 6 Magnetic moments in ferro-, ferri- and antiferromagnetsim.

Ferromagnetism is the case when all magnetic moments are oriented in the same way. The magnetic moment and the spin of a particle are always directed the same way.

The first models of ferromagnetism were derived simultaneously by Werner Heisenberg and P.A.M Dirac in 1926. Heisenberg’s model is the one used here. It will be examined more thoroughly in the next section. The Heisenberg model is difficult to solve exactly, and Hans Bethe was the first one to solve the one- dimensional case in the XXX model. That is the isotropic case which is consistent with a ferromagnetic environment. The Bethe ansatz was later used by Rodney Baxter, in 1971, to solve a more general model, the one-dimensional XYZ-model. The case of the one-dimensional XZY-model and any two- or three-dimensional case is very complicated.

4.2 The Heisenberg model

Let’s start out with a mathematical formulation of the Heisenberg Hamiltonian and then continue with a physical derivation (Ng, 1996).

Starting with the mathematical formulation of the model, using the same notation as (L.A. Takhtadzhan, L.D Faddev, 1979). A system consisting of N interacting half spin fermions, the Heisenberg Hamiltonian for a quantum mechanical system is given by

1 ℋ = − ∑푁 (퐽 휎푥휎푥 + 퐽 휎푦휎푦 + 퐽 휎푧휎푧 ). (4.1) 2 푛=1 푥 푛 푛+1 푦 푛 푛+1 푧 푛 푛+1

Jx, Jy and Jz are real constants. This Hamiltonian operates in a Hilbert state space, defined by

푁 2 ℌ푁 = ∏푛=1 ⨂픥푛 , 픥푛 = 퐂 . (4.2)

And the spin operators are given by

푗 푗 휎푛 = 퐼 ⊗ … ⊗ 휎 ⊗ … ⊗ 퐼 ( j = 1, 2, 3, 4; n=1,…,N ). (4.3)

The Pauli operators, which have the form

0 1 휎1 = 휎푥 = ( ), (4.4a) 1 0 0 −푖 휎2 = 휎푦 = ( ), (4.4b) 푖 0 1 0 휎3 = 휎푧 = ( ), (4.4c) 0 −1 휎4 = 퐼 (4.4d) in the orthonormal basis of C2;

1 0 푒 = [ ] , 푒 = [ ]. (4.5) + 0 − 1

j For σ n to act non-trivial only in the Hilbert state space we assume

푗 1 휎푁+1 = 휎1 (j = 1, 2, 3), (4.6) so the periodic boundary condition hold.

This model is known as the “XYZ-model”, which implies Jx ≠ Jy ≠ Jz. We also have the cases of the XXZ-model and the XXX-model, which has Jx = Jy ≠ Jz and Jx = Jy = Jz = J respectively. The case of the XXX-model with J>0 we have ferromagnetism, which is the case we will investigate further for our model. J is the exchange constant, and is a scalar, as mentioned above.

Next the physical derivation is provided. Starting with a two-electron system in a static Coulomb interaction potential. At position ℝ3electron j has a position as well as spin, so its wave function is the tensor product of the wave function 2 3 spaces, the spatial and spin, an element of 푯푗 ≅ 퐿 (ℝ )⨂픥푗. The total wave

20 function is then an element of the spatial and the spin component, 푯1⨂푯2 = 2 3 2 3 ℌ푠푝푎푡푖푎푙⨂ℌ푠푝푖푛, where ℌ푠푝푎푡푖푎푙 ≅ 퐿 (ℝ )⨂퐿 (ℝ ) and ℌ푠푝푖푛 ≅ 픥1⨂픥2.From this one can observe that the Hamiltonian describing the Coulombic interaction can be replaced by a spin Hamiltonian. The reasoning behind it will follow.

The coulomb interaction Hamiltonian is given by

2 2 ℏ 2 2 푒 ℋ푐표푢푙 = − (∇1 + ∇2) + , (4.7) 2푚 |풓ퟏ−푟2|

2 2 3 Where ∇ is the Laplacian operator on 퐿 (ℝ ), ℋ푐표푢푙 acts on 푯1⨂푯2 , and 푒2 is the electrostatic Coulumb repulsion potential. Here one can see that |풓ퟏ−푟2|

ℋ푐표푢푙 depends on r1 and r2, the spatial coordinates of the electrons only, and not on their spins, and therefore it acts on the spatial component, ℌ푠푝푎푡푖푎푙 , of

푯1⨂푯2.

According to the Pauli exclusion principle the total wave function describing this system must be antisymmetric. Since the total wave function must be an eigenvector of ℋ, one can see that it can be expressed as the tensor product

휓푠푝푎푡푖푎푙⨂휓푠푝푖푛of an eigenvector of ℋin ℌ푠푝푎푡푖푎푙 and a vector in ℌ푠푝푖푛. This indicates that one of 휓푠푝푎푡푖푎푙and 휓푠푝푖푛are antisymmetric while the other one is symmetric. The antisymmetric and symmetric spatial wave functions are given by

푎푛푡푖 휓푠푝푎푡푖푎푙 = 휓1(풓1)⨂휓2(풓2) − 휓2(풓ퟏ)⨂휓1(풓ퟐ) (4.8) And 푠푦푚 휓푠푝푎푡푖푎푙 = 휓1(풓1)⨂휓2(풓2) + 휓2(풓ퟏ)⨂휓1(풓ퟐ). (4.9)

2 3 Where 휓1 and 휓2 are wave functions in 퐿 (ℝ ) and they are determined by the time-independent Schrödinger equation and ℋ푐표푢푙.

The wave functions for the spin are given by

푠푦푚 휓푠푝푖푛 = | ↑↑⟩, | ↑↓⟩ + | ↓↑⟩, | ↓↓⟩ (4.10) and 푎푛푡푖 휓푠푝푖푛 = | ↑↓⟩ − | ↓↑⟩. (4.11)

These can be distinguished by their eigenvalues with respect to the spin operator 푥 푥 푦 푦 푧 푧 휎1 ∙ 휎2 = 휎1 휎2 + 휎1 휎2 + 휎1 휎2 ,

훼 where 휎푗 is the operator on 픥푗.

푎푛푡푖 푠푦푚 The spatial wave functions, 휓푠푝푎푡푖푎푙 and 휓푠푝푎푡푖푎푙, have different energies under the Coulomb Hamiltonian, ℋ푐표푢푙. Even if these energies have a spatial origin one can rewrite it as a spin-spin interaction. The spin Hamiltonian is then

3퐸푠푦푚+퐸푎푛푡푖 퐸푠푦푚−퐸푎푛푡푖 ℋ = + 휎 ∙ 휎 (4.12) 푠푝푖푛 4 4 1 2 and have the eigenvalue Eanti for the triplet symmetric spin state and Esym for the singlet antisymmetric spin state. This makes it formally equivalent to ℋ푐표푢푙. The constant term in the spin Hamiltonian can be ignored, since the energies are 퐽 relative, which leaves ℋ = − 휎 ∙ 휎 , where the exchange integral, J, is 푠푝푖푛 2 1 2 퐸푠푦푚−퐸푎푛푡푖 given by 퐽 = . The sign of J determines whether the system describe 2 an antiferromagnet (J<0), or a ferromagnet (J>0), that is, if aligned or antialigned spin are energetically preferred. By doing this the Hamiltonian have become much simpler.

In a solid which is magnetically ordered and that has no external magnetic field, the sum can be taken of all electron pairs to obtain the Heisenberg Hamiltonian:

1 ℋ = − ∑ 퐽 휎 ∙ 휎 . (4.13) 2 푖,푗 푖푗 푖 푗

The exchange integral falls off rapidly with distance for a solid with localized electrons, while it is the same for all nearest-neighbor electron pairs. This 퐽 indicates that the Hamiltonian can be written as ℋ = − ∑ 휎 ∙ 휎 , which 2 푛.푛. 푖 푗 sums up all nearest-neighbor pairs.

This Hamiltonian can be rewritten for the often used anisotropic case, which gives: 1 ℋ = − ∑ 퐽 휎푥휎푥 + 퐽 휎푦휎푦 + 퐽 휎푧휎푧. (4.14) 2 푛.푛. 푥 푖 푗 푦 푖 푗 푧 푖 푗

This is the generalized case for the XYZ-model, From this one can compose Heisenberg Hamiltonians for the XXZ-case, which is also known as the Ising- model. The case that is of interest here is the XXX-model, as mentioned before, as it describes a ferromagnet.

The Heisenberg Hamiltonian can be used in this case as it is valid for all ½-spin fermions, e.g. a neutron.

4.3 Argonne v14 potential

Here the Argonne v14 potential (Wiringa, Stoks, Schiavilla, 1995) will briefly be presented. It is a nucleon-nucleon potential that can be charge-independent when so is needed, as in the case of neutrons. First a many-body Hamiltonian will be given, then the operator form of the potential will be presented.

Figure 7. The Argonne v14 potential, where c is the central component, τ is the isospin component, σ shows the spin component and στ is the spin-isospin component of the potential.The Latter three are all attractive for small seperations, as for neutrons in a neutron star.

The many-body Hamiltonian that this is based upon is

2 ℏ 2 퐻 = ∑푖 − ∇푖 + ∑푖<푗 푣푖푗 + ∑푖<푗<푘 푉푖푗푘, (4.15) 2푀푖

By requiring that this Hamiltonian gives the correct trinucleon binding energy, it constrains the strength parameters of the NNN potential.

The operator form of the Argonne v14 potential is given by

푝 푣푖푗 = ∑푝=1,14 푣푝(푟푖푗)푂푖푗 (4.16)

p All the terms in O ij are charge-independent, which is the part that is of interest p here since neutrons are charge-free. O ij is given by

푝=1,14 푂푖푗 = 1, 흉풊 × 흉풋, 흈풊 × 흈풋, (흈풊 × 흈풋)(흉풊 × 흉풋), 푆푖푗, 푆푖푗(흉풊 × 흉풋), 푳 × 푺, 2 2 2 2 푳 × 푺(흉풊 × 흉풋), 퐿 , 퐿 (흉풊 × 흉풋), 퐿 (흈풊 × 흈풋), 퐿 (흈풊 × 흈풋)(흉풊 × 흉풋), 2 2 (푳 × 푺) , (푳 × 푺) (흉풊 × 흉풋) (4.17)

The spin-dependent parts of the potential are shown in figure 7. One can readily see how the potential depends on r, the separation of the nucleons.

5 A new model for the pulsar magnetic field

In this chapter a novel model of the magnetic field of pulsars will be introduced, a model that will automatically explain foremost three things: 1. the strong magnetic fields of pulsars; 2. why the fields are so stable; 3. why the magnetic axis does not have to be aligned with the rotational axis. At first the theory itself is presented, followed by some basics about the neutron and nuclear spin. Finally some calculations are made regarding the number of neutrons and the total magnetic field strength, including also the interchange constant J. Conclusions drawn from this will be stated and discussed in chapter 6.

5.1 The Theory

This theory investigates the possibilities of spin-polarized neutron stars, i.e. neutron stars being “cosmic permanent magnets”. The simple model of a pulsar as a rotating neutron star with a dipole magnetic field at an angle with respect to its orbital axis is probably basically correct, so it will be used. Some further simplifying assumptions will be made, they are listed below:

 The neutron star is composed solely out of neutrons. (Nearly true as- suming that quark stars do not exist. There are some observational indi- cations pointing to the conclusion that neutron stars indeed are composed out of normal nuclear matter.)

 The density is constant throughout the neutron star and roughly the same as the density of normal nuclear matter. (In reality, the density is a few times higher in the neutron star core, and much less in its thin crust.)

 The magnetic field is due to spin-alignment of the neutrons in the neutron star. This is motivated by the fact that aligned spins are energetically favored by the nuclear force. We thus assume that the neutron star is a “neutromagnetic” material (in direct analogy to ferromagnetic materi- 26

als). The orbital angular momentum does not contribute to the magnetic field as the neutrons are electrically neutral (no currents). The orienta- tion of the magnetic field in the star prior to, and during, the collapse will polarize the neutromagnet, and can thus readily explain why the magnetic field axis can differ from the rotational axis (this is difficult to explain in orthodox models where the magnetic field is supposed to be generated by rotating superconducting protons).

5.2 The physics behind

5.2.1 The neutron

The neutron is a baryon, and consists of three quarks (u,d,d). Since each quark has ½-spin, and the neutron consists of one up-quark and two down-quarks, the neutron has a total spin of ½. Just like the electron and proton it is a fermion, but it does not have a net electric charge, it is neutral. Due to its electrical neutrality it was originally surprising to find that it has a magnetic moment. Neutrons are usually found in the nuclei of atoms. Free neutrons are unstable and disintegrate through beta decay. They have an average life-time of approximately 15 minutes. Neutrons in a stable nucleus are however stable as the strong nuclear force stabilizes it (up to a point). The number of neutrons in the nuclei determines some of the characteristics of the element, it defines what isotope it is.

5.2.2 Spin

Spin is a nuclear property, a “state”, S, that is assigned. This state describes the intrinsic angular momentum of the nucleon. The spin is part of the total angular momentum I. The vector I is the sum of the orbital, L, and the intrinsic moment, S, which is given by

퐴 퐴 푰 = ∑푖=1(ℓ푖 + 풔푖) = 푳 + 푺 = ∑푖=1 풋푖. (5.1)

The quantum number has a connection to the vector I through

|푰| = √퐼(퐼 + 1)ℏ. (5.2)

The magnetic moment of a nucleon is always aligned with its spin.

5.3 Calculations

5.3.1 Number of neutrons

Here the number of neutrons in a neutron star will be estimated through their mass.

The most common mass of a neutron star is around 1.4Mʘ, where Mʘ= 30 −27 1.99×10 kg denotes the solar mass. A neutron has a mass of Mn=1.67×10 kg. The number of neutrons in a neutron star is then

30 1.4푀ʘ 1.4×1.99×10 57 푁 ≈ = −27 ≈ 10 . (5.3) 푀푛 1.67×10

5.3.2 The total magnetic field strength

The magnetic moment is always directed in the same direction as the spin of a nucleon. First the magnetic moment for a single neutron will be calculated and then the maximum magnetic moment for an entire neutron star, or pulsar, that the neutrons can give rise to, so it can be compared to the magnetic field strength of magnetars, the pulsars with the largest known magnetic field.

The magnetic moment for a neutron is given by

−26 휇푛 = 푠푔휇푁 = −9.6625 × 10 퐽/푇 (5.4)

-27 -8 Where μN is the nuclear magneton, μN = 5.05084×10 J/T = 3.15245×10 eV/T, g is the neutron spin factor, g = -3.8260837 +/- 0.0000018 and s is the spin of the particle, in this case s=1/2.

The total magnetic moment a neutron star could have is

57 31 푚 = 휇푛 × 10 = −9.6625 × 10 퐽/푇. (5.5)

Below follows the calculations of the total Neutron Star (N.S) magnetic field, using the equation for a far-field dipole moment (Griffiths, p246, 1999), we get,

푚휇0 2 퐵푁.푆 = 3 √1 + 3 sin 휆 (5.6) 4휋푅푁.푆

Where m is the magnetic moment, μ0 is the permeability of free space, RN.S = 10 km is the radius of the neutron star. If λ, which is the magnetic latitude, is set to 0°, the magnetic field strength for the magnetic pole will be calculated. According to this the highest value the magnetic field can have for any neutron 16 12 star is |BN.S |= 9.66×10 G (9.66×10 T), which is about a factor 100 larger than the observed magnetic field of a magnetar, which has a field strength 15 11 around BM = 10 G (10 T). At the equator, λ = 90°, the magnetic field for the neutron star would be twice as strong. This assumes that the neutron star is a dipole. Although the neutron star will not be perfectly polarized, and we have made many simplifying assumptions, there is still leeway to assume that the total magnetic field is due to spin-alignment i.e. “neutromagnitisation”

5.3.3 The interchange constant, J

Next the interchange constant, J, will be investigated a bit closer, as this decides whether or not one can view a neutron star as a ferromagnet.

The interchange constant is given by

2 (푔휇푛) 퐽(푚) = −2 3, (5.7) (푛푎0)

Where g is the neutron spin factor, μn the neutron magnetic moment, a0 the distance between the nucleons and n is an integer that describes how far apart the nucleons are in the order. n=1 is the case of the nearest-neighbour, n=2 is the next-nearest-neighbour, and so on.

To calculate the distance between the neutrons some approximations will be

4휋푟3 made. The volume of a neutron star with the radius 10 km is 푉 = = 4.2 × 3 1012푚3, which would give a cube of the same volume a side of 퐿 = 3√푉 = 1.6 × 104푚. If the neutrons were distributed equally over the cube one side

3 57 19 would have √10 = 10 neutrons. This would give a distance of 푎0 = 4.2 × 104 = 4.2 × 10−15푚 between the neutrons, if you take the size of the neutrons 1019 into account you would get a distance in the same range

4.2×104 푎 = − 1.65 × 10−15 = 2.55 × 10−15 푚. (5.8) 0 1019

Since this distance disregards both gravity and the strong interaction, the real separation should be even less, ≤ 1 fm. This distance gives the interchange constant J(1)= -3.6876×10-6. This indicates that aligned spin is favored according to the Heisenberg Hamiltonian. Since J(2) is a factor 1/64 of J(1), it is not needed to take this into consideration since its effect is small compared to n=1, which is the case of the nearest-neighbour.

6 Conclusions and discussion

Although this is a very oversimplified model of a neutron star, it can however simply explain a few unresolved properties about pulsars:

 The origin of the magnetic field, which in this model is simple and unavoidable.

 There is a natural maximum upper limit to the magnetic field, which coincides with the magnetic field strength of a magnetar (B≈1016 G)

 The exact period of the “lighthouse effect” requires a very stable magnetic field, which neutrons aligned in a normal “neutromagnet” automatically give rise to, in analogy to atoms in a normal ferromagnet.

As can be seen from the Heisenberg Hamiltonian an aligned spin configuration is favored, and according to the Argonne v14 Potential aligned spins favored when the nucleons have a distance less than ~0.6 fm. The assumptions made when presenting this theory, regarding the density of the neutron star, and the calculations made, tells us that the Argonne potential will not favor aligned spin in this case, but reality looks a bit different. Also the distance between the nucleons will be smaller the closer to the center of the star, as gravity also compresses matter towards the center, and there the Argonne potential will favor aligned spins. The interchange constant is also dependent of the distance between the nucleons, but as the sign will not change because of the nn-distance the Heisenberg Hamiltonian is not as sensitive to this assumption as the Argonne potential.

The simplification that the neutron star is a perfect dipole also adds to the inaccuracy of the calculations and results. However, if the calculations with the

Heisenberg Hamiltonian approximate reality the star would more or less be a dipole, as the naive light-house model of pulsars suggest.

The distance calculations where the volumes were remade to cubes made the calculations less accurate, but much easier to be carried out, so the inaccuracy it causes is small in comparison to the cumbersome calculations that would need to be done otherwise. And the distances are still in the same range as they would be if the “points in a sphere” calculations would be carried through.

These calculations and assumptions made make the theory possible. Further and more accurate calculations have to be carried out for a more exact result and to verify the conclusions made. Since a direct experiment of testing this theory is not possible today, studies of our most heavy nuclei must be done to see how common the occurrences of spin alignment are.

7 References

7.1 Books & articles

V.S Beskin, A.V Gurevich, Ya. N. Istomin, 1993, Physics of the Pulsar Magnetosphere, Cambridge University Press J.M Blatt, V.F. Weisskopf, 1979, Theoretical Nuclear Physics, Springer-Verlag New York inc. B.M Carroll, D.A Ostlie, 1996, An introduction to Modern Astrophysics IE, Addison-Wesley Publishing company, inc. R.C. Duncan, C Thompson, 1996, Magnetars. R.A Freedman, W.J Kaufmann III, 2002, Universe 6th Ed, W.H Freeman and Company D.J Griffiths, 1999, Introduction to Electrodynamics 3rd ed, Prentice Hall J.M Irvine, 1978, Neutron Stars, Clarendon Press K.S. Kane, 1988, Introductory Nuclear Physics, John Wiley & Sons, inc C. Kouveliotou, R.C. Duncan, C. Thompson, 2003, Magnetars, Scientific American, February 2003 A.G. Lyne, F Graham-Smith, 1998, Pulsar astronomy 2nd Ed, Cambridge University Press. R.N Manchester, J.H Taylor, 1977, Pulsars, W.H Freeman and Company D. C. Mattis, 2006, The Theory of Magnetism made simple, World Scientific Publishing Co. Pte. Ltd L.L. Ng, 1996, Heisenberg Model, Bethe Ansatz and Random Walks, Senior Honor Thesis, Harvard University V.R Pandharipande, R.B. Wiringa, 1976, A variational theory of nuclear matter, Nuclear Physics, Section A Volume 266, Issue 2, 9 August 1976, Pages 269-316. C.E Rhoades Jr., R Ruffini, 1977, Maximum mass of a neutron star, Phys. Rev. Lett. 32, 324–327 (1974), The American Physical Society. L.A. Takhtadzhan, L.D. Faddeev, 1979, The Quantum Method of the Inverse Problem and the Heisenberg XYZ Model, Russian Math. Surveys 34:5 (1979), 11-68

R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, 1995, Accurate nucleon-nucleon potential with charge-independence breaking, The American Physical Society, Volume 51, Number 1.

7.2 Figures

1. Taylor, Manchester and Lyne, Ap. J. suppl., 88, 529,1993 2. Andrew G. Lyne, Francis Graham-Smith, 1998, Pulsar astronomy 2nd Ed, Cambridge University Press 3. A.G. Lyne, F Graham-Smith, 1998, Pulsar astronomy 2nd Ed, Cambridge University Press. 4. http://www.alicesastroinfo.com/2007/07/ (2010-12-05) 5. Taylor, Manchester and Lyne, Ap. J. suppl., 88, 529,1993 6. Author; Jakob Schmid, Put together from several of his pictures found on wikipedia.org. 7. R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, 1995, Accurate nucleon-nucleon potential with charge-independence breaking, The American Physical Society, Volume 51, Number 1.