# What does spin degrees of freedom mean?

1. Aug 6, 2010

### inflector

What does "spin degrees of freedom" mean?

In the http://en.wikipedia.org/wiki/Spin_ice" [Broken], I find:
I've searched on the web for "spin degrees of freedom" and gather from the comment that this has something to do with magnetism since it appears that the sentence defines a magnet as a substance with spin degrees of freedom; but I can't find any definition for the term "spin degrees of freedom" itself.

Can anyone give me a fairly simple laymen's definition of what this means? Failing that, a technical definition along with the relevant topics I'd need to research in order to understand the definition would be a good second-choice answer.

Curtis

Last edited by a moderator: May 4, 2017
2. Aug 6, 2010

### alxm

Re: What does "spin degrees of freedom" mean?

Electrons are spin-1/2 particles, which means they can have two possible spin states (+1/2, -1/2, or 'up' and 'down'). If there's no magnetic field then the energetic state of the electron is independent of the spin state, so the Pauli principle (which says two particles cannot be in the exact same state at the same time) leads to you being able to have two electrons with opposite spin in any given energetic state.

Now, in most substances have a definite overall spin state (in most compounds it's zero, there's an equal number of 'up' and 'down' electrons). Since, due to the Pauli principle, if you have an 'up' and 'down' electron in a given energy state, and want to change the 'down' one to 'up', then it has to move to some other energy state that doesn't have an 'up' electron in it. Which usually means raising the energy, since the lowest energy states are always occupied.

So when you say that something has many spin degrees of freedom, you're saying that there are many electronic states (occupied and unoccupied) which have the same energy, or nearly the same energy, so that the electrons are free to occupy many different spin states (since changing the spin state won't increase the energy).

3. Aug 6, 2010

### peteratcam

Re: What does "spin degrees of freedom" mean?

From a theoretical perspective, spin ice is basically an Ising model, one of the most famous models of magnetism. The other famous model of magnetism is the Heisenberg model, and then all sorts of variations of it.

These models approximate solids which are insulators, so the electrons are not free to move around, but instead localized. So the position degrees of freedom of the electrons can be ignored, and all of the physics is in the spin degrees of freedom, which are fixed at the lattice sites.

When you think of magnet you might think of iron or other metallic magnets, but these are not normally what condensed matter physicists mean when talking about "a magnet". Iron is in fact what is called an itinerant magnet because the electrons are free to move about.

Frustrated magnetism is actually my field, so ask away.