Fermions, Atoms and Neutron Stars

by Nicholas Mee on November 11, 2012

My article What on Earth is a Boson? has proved to be so popular that I have agreed to write a bit more about fermions and bosons. This article is about fermions, there will be a companion article about bosons very soon.

Stamp issued in Denmark to commemorate the 50th anniversary of the invention of the Bohr model of the atom by Danish physicist Niels Bohr.

The Mighty Atom

In 1913 Niels Bohr made one of the greatest breakthroughs in our understanding of the world around us when he constructed a model of an atom in which the electrons orbit the nucleus in a fixed set of energy levels. Bohr used his model to explain the lines in the spectrum of an atom, which had been used to deduce the chemistry of the stars as I described in an earlier article:  Twinkle Twinkle Little Star.

When an electron falls from one energy level to a lower energy level it emits a photon (particle of light) whose energy is equal to the difference in energy of the two energy levels. (This is what is represented by the equation shown on the stamp above.) As the electron can only exist in a discrete set of energy levels, the photons emitted in this way can only have a certain set of energies, each corresponding to a different colour. So the spectrum of any material formed from these atoms has of its own characteristic set of spectral lines – much like its very own barcode.

Bohr’s model was a great success and won him a Nobel Prize in 1922. However, Bohr felt that his model should be capable of more. If it really represented the structure of an atom, then it should be able to explain how atoms bond in chemical reactions and why each atom has its own particular chemical properties. Bohr made some progress along these lines, but eventually he had to admit defeat. He passed the problem over to a precocious physics prodigy – Wolfgang Pauli, an Austrian who had made a name for himself by writing a textbook on the new subject of relativity at the age of just 19.

The Exclusion Principle

The Austrian physicist Wolfgang Pauli who became known for his ascerbic wit and his harsh criticism of anyone whose thinking he considered sloppy. Rudolf Peierls wrote that a friend showed Pauli the paper of a young physicist which he suspected was not of great value but on which he wanted Pauli’s views. Pauli remarked sadly, “Not only is it not right, it’s not even wrong!”.

In 1925 Pauli came up with the fix that would enable Bohr’s model to explain ‘chemistry’.

The number of electrons in an atom is equal to the number of protons in the nucleus of the atom. This is because the charges on protons and electrons are equal in size, but the charge on a proton is positive and the charge on an electron is negative. With equal numbers of protons and electrons an atom is electrically neutral. Pauli realised that in Bohr’s original model we would expect that all the electrons in an atom would fall into the lowest energy level of the atom and just sit there. He decided that electrons must have a strange property that prevented this from happening. It has become known as the Pauli Exclusion Principle. And it implies that only one electron is allowed in any particular quantum state.

What this means is that in a hydrogen atom, which contains just one electron, this electron does indeed fall down into the lowest energy level. In helium, which contains two electrons, the second electron also falls to the lowest energy level, but it must be spinning in the opposite direction to the first electron. These two electrons have now filled the first energy level. In the the next type of atom lithium, there are three electrons – two go into the lowest energy level (with opposite spins), but the third can only fall as far as the second energy level. In this way the structure of atoms can be understood with the electrons filling the energy levels step by step and with no more than one electron occupying each possible state. Pauli realised that it is the outermost electrons – those in the highest partially filled shells, as they are known – that partake in chemical reactions, so Pauli was able to explain the chemical properties of an atom in terms of the number of electrons in the outer reaches of the atom.

This is discussed more fully in one of the early chapters of my book Higgs Force, so I won’t go any further into it here.

The Original Spin Doctor

In 1940 Pauli took a big step towards a theoretical understanding of this strange state of affairs by proving what is known as the Spin-Statistics theorem. Don’t be put off by the technical sounding name. What Pauli showed was that if relativity and quantum mechanics are combined into the theoretical framework that we know as quantum field theory, then particles are necessarily divided into two types that behave in very different ways and this classification depends solely on their spin.

Spin is the rate at which a particle rotates, as you would expect. It is an intrinsic property of a particle that cannot change without the particle changing into a different type of particle. Like many quantum properties spin comes in fixed lumps. In other words particles can only exist in states where they have multiples of a fundamental unit of spin. The smallest amount of spin that a particle can have (other than no spin) is half a unit of spin.

Bosons and Fermions

Particles with half a unit of spin are known as fermions. These particles include electrons, protons, neutrons, quarks and neutrinos. Each such particle shuns its colleagues, they must occupy different states, in other words they obey the exclusion principle. The term ‘fermion’ is derived from the name of the Italian physicist Enrico Fermi. (Although ‘paulions’ might have been more appropriate.)

Particles with a whole number of units of spin are bosons. For instance, the Higgs boson has zero spin, whereas photons and gluons have one unit of spin. These particles like to cluster in the same state. Bosons are named after the Indian physicist Satyendra Nath Bose.

Pauli’s analysis had shown that electrons obey the exclusion principle simply because of the rate at which they spin. Because of this behaviour fermions are the fundamental building blocks of matter. If electrons had no spin (like Higgs bosons) or one unit of spin (like photons) then they would all fall to the lowest energy level in an atom and there would be no chemistry, because it is the electrons in the outer unfilled shells that do all the chemical bonding. If electrons were bosons the universe would be a structureless mush.

The Nucleus

A similar situation prevails in the nucleus of an atom, where the neutrons and protons from which the nucleus is formed must each exist in its own distinct quantum state. The energy levels in the nucleus are not arranged in exactly the same way as the electron energy levels outside the nucleus, but the same general principles apply. Neutrons and protons fill up the energy levels step by step.

At an even deeper level neutrons and protons are each formed of three quarks. Indeed, most of the particles that are observed in particle accelerators such as the Large Hadron Collider are formed of quarks. These particles are collectively known as hadrons, hence the name of the Large Hadron Collider. Again, within all these particles the quarks obey the exclusion principle and this determines which particles exist and many of their physical properties.

Astronomical Exclusivity

The exclusion principle is extremely important in the physics of the very small, but quite remarkably it is also critically important in astronomy. In particular, the end point of a star’s life cycle.

Stars produce their tremendous energy output through nuclear fusion reactions, mostly by converting hydrogen nuclei, which consist of a single proton, into helium nuclei, which consist of two protons and two neutrons. The energy released in this way supports the star against the crush of its own gravity. But eventually the fuel runs out and the nuclear reactor in the star’s core is turned off. Once the energy supply is exhausted the star collapses under its own gravity. The final result depends on the mass of the star.

A star with the mass of the Sun will collapse until it forms an intensely hot and dense ball the size of the Earth that will eventually cool down by radiating its heat into the depths of space. These cosmic cinders are known as white dwarf stars. Remarkably, they are supported by nothing more than the exclusion principle. The electrons within the white dwarf must all exist in their own separate quantum state and this produces a huge resistance to their being squeezed any closer together. This is called ‘electron degeneracy pressure’.

Neutron Stars

In 1930 Subrahmanyan Chandrasekhar was awarded a graduate scholarship by the Indian government to continue his studies in physics at Trinity College, Cambridge. While on the voyage to England Chandrasekhar realised that there was a limit to the mass of a star that could be supported by electron degeneracy pressure. He calculated the maximum mass of a white dwarf star to be just under one and a half times the mass of the Sun. This is now known as the Chandrasekhar limit. Chandrasekhar was awarded the Nobel Prize in 1983 for his research into the physics and evolution of stars.

A representation of a neutron star showing the pulsar beams emanating from both magnetic poles. This is a frame from an animation on a multimedia CD-ROM called The Secret of the Universe that I wrote some years ago.

Stars with a mass greater than the Chandrasekhar limit have an even more dramatic fate. Beyond this limit, the collapse continues until the electrons and protons within the star undergo nuclear reactions in which an electron and a proton are converted into a neutron and a neutrino. A vast flux of  neutrinos then escapes into space and all that remains is a ball of neutrons with a radius of about 20 kilometres (12 miles). The entire star has been compressed to nuclear densities. It has, in effect, been converted into a gigantic atomic nucleus. This prodigiously dense stellar remnant is known as a neutron star.

Neutron stars are extremely weird objects. They rotate in a fraction of a second and have intense magnetic fields that generate beams of radiation that are fired into space from their two magnetic poles. Each time one of the poles points in our direction a pulse of radiation is detected. Astronomers call them pulsars.

Like white dwarfs, neutron stars are supported by the exclusion principle. But this time it is their component neutrons that resist being squeezed any closer together. Neutrons have almost two thousand times the mass of electrons and it is their greater mass that means that neutron degeneracy pressure is able to support matter at these incredible densities.

But there are limits to the mass that can be supported by neutron degeneracy pressure. It is thought that nothing can stop the relentless collapse of a star with more than about three times the mass of the Sun and that such a star must ultimately form a black hole.

Further Information

There is a lot more information related to this article in my book Higgs Force: Cosmic Symmetry Shattered.
http://quantumwavepublishing.com/higgs-force/

The Secret of the Universe
http://virtualimage.co.uk/html/the_secret_of_the_universe.html

 

{ 15 comments… read them below or add one }

Douglas November 11, 2012 at 5:55 pm

Thank you Nick, an excellent explanation.

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david stein November 14, 2012 at 7:46 am

Thanks. It is well written, but I will have to re-read it again to get full value. I will. David.

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Dr Tom Leigh November 14, 2012 at 8:07 am

Like many other non-mathematical people I have struggled to understand quantum physics. Your explanations open the doors of understanding like never before.

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Jack November 14, 2012 at 3:19 pm

“three times the mass of the Sun and that such a star must ultimately form a black hole. ”
The quote is from the above article. I am thinking that “mass” should possibly read “pressure”. This is from eso.org ,eso1034 “By comparison with these stars, they found that the star that became the magnetar must have been at least 40 times the mass of the Sun. This proves for the first time that magnetars can evolve from stars so massive we would normally expect them to form black holes. The previous assumption was that stars with initial masses between about 10 and 25 solar masses would form neutron stars and those above 25 solar masses would produce black holes.”. Basically they seem to say that the mass should be at least 25 times maybe 40. Or am I misreading something?

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Nick November 16, 2012 at 9:39 am

Thank you for pointing this out. I was in two minds about whether to qualify my statement when I was writing the article.
There is a good reason for the two quite different figures for the size of a star and its ultimate fate. When a very massive star has used up all its nuclear fuel its core collapses and it undergoes a supernova explosion. The amount of energy released by a supernova is enormous and it blasts the outer layers of the star into space. The classic example of this is the Crab Nebula which is formed of the outer layers of a star that was observed to undergo a supernova explosion almost one thousand years ago. Most of the material that formed the star is ejected into space in a supernova explosion, but the core of the star collapses to form a compact remnant which must be a white dwarf, a neutron star or a black hole. The masses that I was referring to in the article are the masses of these compact remnants not the mass of the original star. The Chandrasekhar limit on the mass of a white dwarf is well understood theoretically and backed up by observation of the masses of white dwarfs and neutron stars. Where the mass of a neutron star can be calculated, because it is in orbit around another star, it is always slightly more than the Chandrasekhar limit of around 1.4 times the mass of the Sun. The upper limit for the mass of a neutron star is not known with the same degree of accuracy, but it must be around 2 to 3 times the mass of the Sun.
What is much harder to calculate is what happens to a very massive star when it undergoes a supernova explosion. If a star that is 15 times the mass of the Sun undergoes a supernova explosion, it is very difficult to predict how much of its material will be blasted into space and how much will ultimately collapse to form the stellar remnant, so it is not certain whether it will form a neutron star or a black hole. This is why the estimates vary so much when astrophysicists try to predict how massive a star must be to form a black hole at the end of its life.

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John Riegel November 15, 2012 at 3:13 am

Thanks for the article, I am struggling with it, but am interested to the point where I will continue until I get a grasp.

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Mike November 15, 2012 at 9:28 am

Thank you so much.Now I understand so much more about the weird and wonderful world of quantum physics.I cant wait for the next instalment!

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kbhojapoojary November 15, 2012 at 1:53 pm

Thank you Lee, this article helped me to understand White Dwarf, Neutron Star and Black Hole and the Chandrasekhar Limit.

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umesh November 15, 2012 at 4:09 pm

nice article

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Rajendra November 16, 2012 at 4:04 pm

Great amount of knowledge in a nut-shell. Worth reading repeatedly.

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lopa November 18, 2012 at 9:09 am

Thanks for making it easy to understand.
Enriching with knowledge is highly appreciated.

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Srinath January 1, 2013 at 3:00 pm

Thanks again for another wonderful article !

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JimHenrie January 1, 2013 at 8:43 pm

Fascinating, and truly requires rereading for full digestion.

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Peter Wone September 19, 2013 at 2:30 am

Given that a neutron star is composed entirely of neutrons and therefore has no change, why does its spin induce a magnetic field?

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Nicholas Mee September 19, 2013 at 8:40 am

The composition of a neutron star is thought to be more complicated than the name would suggest. Moving inwards from the surface of the neutron star the pressure increases and this determines the type of material that is stable. The crust is thought to be made up of iron nuclei and electrons. Further inwards there is believed to be a layer composed of protons and electrons. Then the bulk of the material is composed of neutrons. The core may be composed of even more exotic material, such as heavier subatomic particles or even some sort of quark material, but no-one knows for sure.

The protons within a neutron star are expected to form superconducting currents and this might be the source of the intense magnetic field of the neutron star.

Although, as you say, neutrons are uncharged they still have a magnetic moment, which means that they carry a magnetic field. This is because a neutron is composed of three quarks – one up quark and two down quarks. These quarks are charged and as they spin within the neutron this produces a small magnetic field. So the magnetic moments of all the neutrons within the star will also contribute to the magnetic field of the neutron star.

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