# Proton vs electron magnetic moments

For some reason I found myself wondering why we never talk about the contribution to the magnetic properties of materials due to the nuclear spins; we only ever worry about the electrons. After a short while I remembered that the magnetic moments of protons are vastly smaller than electrons (and that neutrons indeed also have a magnetic moment), and so I tried to remember why this was. Next I rediscovered that magnetic moments of particles have a mass dependence, i.e. they basically go as 1/m. This breaks a little bit for nucleons due to them being composite particles, but it seems good enough to get the intuition for the smallness of their moments.

So ok, there is some classical interpretation of this: If we imagine our particles as little spinning charged spheres, then sure if they have more mass then they require less angular velocity to possess a given amount of spin angular momentum, and since they spin slower it seems intuitive that they should produce smaller magnetic moments.

However since we are in the quantum world this picture makes no sense since angular velocity for point particles makes no sense, so it does not seem right to imagine that protons "spin slower" than electrons (although they are composite so perhaps it is slightly more valid: say we are talking about muons then).

So in what way should one think about the mass dependence of magnetic moments if the "rate of spinning" is silly? Perhaps if someone would remind me how we derive magnetic moments from QED or something that would also help.

## Answers and Replies

The Pauli equation, and hence the Bohr or nuclear magneton, can be derived from the Dirac equation.

If I recall correctly, the spin term is essentially a relativistic correction, so it is not surprising that it is proportional to 1/(mc^2).

http://en.wikipedia.org/wiki/Dirac_equation

The section "Comparison with Pauli theory" has a few words on this.

Bill_K
Science Advisor
[Why on Earth was this moved to General Physics??]

In the nonrelativistic limit, the Dirac equation reduces to the Pauli equation with the Hamiltonian

H = (1/2m)(σ·π)(σ·π) + eφ

where π is the canonical momentum, π = p - eA. Using the identity

(σ·A)(σ·B) ≡ (A·B) + iσ·(A x B)

and the fact that

π x π = ie x A = ie

where is the magnetic field, the Hamiltonian reduces to

H = (1/2m)(p - eA)2 - (e/2m)(σ·)

Thus the e/2m factor in front of the magnetic interaction term comes ultimately from the 1/2m factor in the kinetic energy, p2/2m.

edguy99
Gold Member
For some reason I found myself wondering why we never talk about the contribution to the magnetic properties of materials due to the nuclear spins; we only ever worry about the electrons. ....

So ok, there is some classical interpretation of this: If we imagine our particles as little spinning charged spheres, then sure if they have more mass then they require less angular velocity to possess a given amount of spin angular momentum, and since they spin slower it seems intuitive that they should produce smaller magnetic moments....

First on the proton/neutron spin, the field of NMR talks a lot about the spin of the proton. Also, in your classical interpretation, not only do you have to think about speed of the spin, but what does the object look like that is spinning. Compare the effort needed to flip a fast spinning bicyle wheel, to the effort needed to flip a small spinning ball of the same weight. Also, in the classical world, you may have a complex object where not all of the mass of the object needs to be "flipped" for the magnetic direction to change, imagine a heavy box containing a small spinning ball inside where the spinning ball is creating the magnetic field.

The idea that the electron (small mass) has a much larger magnetic moment then the proton or neutron (large mass) is pretty intriguing.

Thus the e/2m factor in front of the magnetic interaction term comes ultimately from the 1/2m factor in the kinetic energy, p2/2m.

Huh, well that makes sense I guess. I suppose ultimately that is its classical origin as well. Cheers.

First on the proton/neutron spin, the field of NMR talks a lot about the spin of the proton.

Indeed, however these are tiny effects and contribute basically nothing to the macroscopic magnetic properties of materials.

The idea that the electron (small mass) has a much larger magnetic moment then the proton or neutron (large mass) is pretty intriguing.

I agree. I am going to go and think about the Dirac equation and gyromagnetic ratios some more I think...