Physics 6 Tutorial 15 - Spin

 Contents

This tutorial attempts to explain the concept of particle spin, which is not easy to understand.  Work through the tutorial carefully.

Many of the sources contradict each other and are hard to follow, which made the preparation of this tutorial rather difficult.

What is Particle Spin? (Extension for Pre-U students only)

Elementary particles such as electrons and quarks possess a property called spin, as do composite particles such as hadrons and atomic nuclei.  Spin is an intrinsic property, just like mass and charge.  Spin is a quantum property.  It has a value of half-integer or integer.  That means values of ½, or 0, or 1.  Other half integer or integer values are theoretically possible.

The spin also has a sign, indicating that is a vector quantity.  Spin can be +½ or -½, or +1 or -1.

The sign could be considered as up (positive) or down (negative), or clockwise (positive) or anti-clockwise (negative), depending on how you want to think about it.

The concept of spin is not easy, but I hope these notes will give you an idea of what it's about.

Stern Gerlach Experiment

Spin of electrons was first mooted as a result of the Stern-Gerlach Experiment.  You are NOT expected to know any details of the experiment at this level.  The experiment was set up like this:

Silver atoms from a furnace were injected into the apparatus.  The result was surprising.  The atoms split into two beams.  One half went upwards, while the other half went downwards.  No atoms went to the left or right.  If the apparatus was turned on its side, half of the atoms went to the right, and the other half went to the left.  None went up or down.

The key points are that:

• if there is a deflection due to the magnetic field, the atoms must be acting as little magnets;

• the silver atom is neutral, so that there is no way that the magnetic effect is due to a charge;

• silver is not a magnetic material.

Somehow the silver atoms were behaving like little magnet.

We can stop the downward beam by putting a screen in the way like this:

The we pass the upward beam into another pair of magnets that produce a non-uniform field.  We would expect the upward beam to be deflected upwards only.  Instead we get the beam split into two like this:

If we do this again and again, we see the same effect.

The neutral silver atom has a single outer shell electron.  It must be this electron that is responsible for the effect.  This was proved later by doing the same experiment with hydrogen atoms.  Follow this link if you want to know more.

How do we explain this?

We know that charged particles are deflected when they move through a magnetic field.  A stationary charged particle will not interact with a magnetic field at all.  We also know that moving charges generate a magnetic field.  The deflection is due to the interactions between the magnetic fields.

We can conclude that the silver atoms (and hydrogen atoms) in the experiment must have some kind of magnetic property.  In fact the atoms form a magnetic dipole, which has a north pole and a south pole.  The magnetic field is just like a bar magnet.  However the atoms are neutral.  Therefore a charge has got to be moving in a rotary motion to make the magnetic field.  (if it wasn't moving in a circle, the electron would separate from the atom.)  If something is moving in rotary motion it has to have angular momentum, just like an object moving in a straight line has linear momentum.

From rotational mechanics, we know the formula for angular momentum:

Angular momentum = moment of inertia × angular velocity

In code:

L = Iw

[L - angular momentum (kg m2 s-1); I - moment of inertia (kg m2); w - angular velocity (rad s-1)]

The units for L are kg m2 s-1.  Since the radian is a dimensionless unit, it gets left out.  This is consistent with the accepted quantum number of spin, which is derived from the Shortened Plank's Constant, ħ.  The term ħ (with the slash through it ("h-bar")) is the shortened Planck Constant, and is related to h , the Planck's Constant (6.63 × 10-34 J s) by:

We can, of course, give a value for ħ :

ħ = 1.06 × 10-34 J s

If we go back to base units:

1 J = 1 kg m2 s-2

Therefore if we multiply the energy by time, we get:

1 J s = 1 kg m2 s-2 × 1 s = kg m2 s-1

So we can say that the shortened Planck Constant is an angular momentum.

Two Possible Classical Models

Consider the silver atom:

Could the magnetic field be caused by the orbiting electron in the outer shell?  The first thing to say is that the orbit is not like a planet going around a star.  The orbit is all over the place and it's best modelled as a hollow sphere.  The moment of inertia is given by:

Silver is a heavy transition metal (with a relative atomic mass of 107).  Its diameter is 172 pm (1.72 × 10-10 m).  Therefore the radius is 8.60 × 10-11 m.  The mass of a single electron is 9.11 × 10-31 kg.

Electron spin is quantised.  It can only have two values:

+ħ/2  or -ħ/2

The value of the spin for an electron is:

5.28 × 10-35 J s

Let's work out the moment of inertia of the electron in its orbit:

I = 2/3 × 9.11 × 10-31 kg × (8.60 × 10-11 m)2 = 4.49 ×10-51 kg m2

If the angular momentum of the electron is 5.28 × 10-35 J s, we can work out the angular velocity:

w = L/I = 5.28 × 10-35 J s  ÷ 4.49 ×10-51 kg m2 = 1.18 × 1016 rad s-1

We can work out the linear speed of the electron using:

v = wr

v = 1.18 × 1016 rad s-1 × 8.60 × 10-11 m = 1.02 × 106 m s-1

However this does not explain how the electron paths in the magnetic field are only upwards and downwards.  The random movement of the electron as it goes around the orbit would mean that there could be any number of paths the silver atoms could take.  There is every possibility that the resulting little magnetic dipoles are perfectly horizontal, meaning that they would not be deflected at all.  This is not consistent with the real observation.

We can model the spin in a electron in a very simplistic way as the particle spinning on its axis.

In this case we can model the electrons as solid spheres.  Let's assume that the clockwise electron moves up, and the anti-clockwise goes down.  We would be right in thinking that not all the magnets are  pointing perfectly upwards or downwards.  The orientation is random.  It is tempting to think that the little magnets formed as a result of the magnets flipping upwards, or downwards, depending on their orientation.  However the electrons are spinning and act like gyroscopes.  Gyroscopes resist change in orientation due their angular momentum.  Let's look at this further:

The vector of the angular momentum is always perpendicular to the the torque acting on the spinning ball (which we are using to model the spinning electron).  The value of the torque remains constant, but the direction is constantly changing.  The gyroscope precesses in a circle that is on a horizontal axis.  The vertical component of the angular momentum vector points up (as in this case) or down.  The precession of the angular momentum vector produces the upwards component that results in an upwards magnetic field.  Or the downwards component makes a downwards magnetic field.

The moment of inertia is given by:

We can do similar calculations as before.  Electron spin is quantised.  It can only have two values:

+ħ/2  or -ħ/2

The value of the spin for an electron is:

5.28 × 10-35 J s

The classical electron radius is given as 2.82 × 10-15 m

Let's work out the moment of inertia of the electron in its orbit:

I = 2/5 × 9.11 × 10-31 kg × (2.82 × 10-15 m)2 = 2.90 ×10-60 kg m2

If the angular momentum of the electron is 5.28 × 10-35 J s, we can work out the angular velocity:

w = L/I = 5.28 × 10-35 J s  ÷ 2.90 ×10-60 kg m2 = 1.82 × 1025 rad s-1

We can work out the linear speed of the electron using:

v = wr

v = 1.82× 1025 rad s-1 × 2.82 × 10-15 m = 5.13 × 1010 m s-1

This means that the outer rim of the electron is travelling at a linear speed of 170 times the speed of light.  This is not possible.  Therefore this model has a fatal flaw.

We can't say that one electron is going clockwise or anticlockwise as such.  We cannot view the particle  like this, let alone decide what is the top or the bottom.  Electrons are not neat little blue marbles marked with e- ; they exist in a cloud of probability.  Also the model as shown can give other false results, for example, the size of an electron is bigger than an atom - clearly not true.

Quantum Model of Spin

The term spin has been carried over into the quantum nature of particles.  As said above, electrons are not neat little blue marbles.  They exist as little lumps of energy in a cloud of probability.  The closer we get to an electron, the lower the probability of catching it, so we don't know the precise nature of the little brute.   The precise nature of what gives an electron (or any other particle) the property of spin is not well understood.  However there are things we can say about it.  Spin has the following properties:

• Spin can have values that are integers (whole numbers) and half integers (an odd whole number divided by 2);

• The direction can change, but the particle cannot be made to spin faster or slower;

• Spin is described by a quantum number, s, which is defined by:

where n is any whole number.

Particles with spin have a value and a direction.  The SI units for spin are kg m2 s-1, but in particle physics the spin is given a dimensionless quantum number, with a positive or negative sign depending on the direction they are rotating in.  The way this is done is by dividing the angular momentum by the Shortened Plank's Constant, ħ.   As we saw above, it is related to h , the Planck's Constant (6.63 × 10-34 J s) by:

We can, of course, give a value for ħ :

ħ = 1.06 × 10-34 J s

Electron spin is quantised.  It can only have two values:

+ħ/2  or -ħ/2

We can divide these by ħ to write:

+1/2 or - 1/2

The value of the spin for an electron is:

5.28 × 10-35 J s

Rather than use the value of the spin, we use integer and half integer values.  We saw above that we can have spin values of 0, ½, 1, 1½, 2, etc.  There are relationships that show this, but you are not expected to know them at this level.  You can see them HERE.  You will study them at university.

At this point, it is sufficient to say that electrons have two states of spin 1/2.  They can be up (+) or down (-).

Electrons are elementary particles called fermions with a spin of ½.   Elementary particles with spins of 3/2, or 5/2 are not known to exist.

Two of the same fermions with a half-integer spin cannot occupy the same quantum state.   This is called the Pauli Exclusion Principle.  This means that if two electrons are in the same shell, their quantum numbers have to be different.  While charge and mass have to be the same, the spin numbers have to be different.

Spin is conserved in the same way as angular momentum is conserved.  This means that if the angular momentum is changed in a particle interaction, there has to be a corresponding angular momentum change elsewhere.  But the quantum number of spin appears not to have to be conserved in the same way as lepton number, charge, or baryon number.

Note:  the syllabus says that spin in antiparticles is opposite to the spin in particles.  There seems to be quite a lot of confusion among users or different forums about this.  Some users say that this is the case.  However other sources I have looked at say that the spin in particles and antiparticles is the same.

Spin and Particles

Fermions such as quarks and leptons have a spin of ½.   Elementary particles with spins of 3/2, or 5/2 are not known to exist.

Bosons have a spin of 1.  It is thought that the graviton has a spin of 2, while the Higgs Boson has a spin of 0.  Otherwise, bosons have a spin of 1.  Helium-4 can show similar properties to bosons.

Photons have spin of 0.

Quarks all have a spin of 1/2.  They are fermions.

Mesons are composite particles made of one quark and one antiquark.  Rather confusingly, they can have spins of -1, 0, and +1.  If the spin vectors are aligned upwards, the spin is +1.  If the spin vectors are aligned downwards, the spin is -1.  Of one spin vector is aligned upwards and the other is aligned downwards, the spin is 0.

Baryons are composite particles of three quarks (or three antiquarks if they are anti-baryons).  Each one of the quarks can have a spin of +1/2 or -1/2.  Therefore the four following spins are possible:

 Quark 1 Quark 2 Quark 3 Spin -½ -½ -½ -3/2 +½ -½ -½ -1/2 +½ +½ -½ +1/2 +½ +½ +½ +3/2

Although neutrons are by their nature uncharged, spin from the quarks allow neutrons to interact with magnetic fields.

Antiparticles have the same spin as particles.  Therefore an antibaryon will still have any one of these spin states.

Spin is an important part of quantum mechanics, a branch of physics that carries this description, "If you think you understand quantum mechanics, you don't."  You will have the chance to study it at university.