How to model a permanent magnet?

[omoplata]

This is sort of a thought experiment. I want to find out if my approach to this is correct.

How can I analytically find the magnetic field from a permanent magnet?

My approach would be to consider the magnet to be a lattice, with each lattice point being a molecule with a magnetic dipole moment $$\vec{\mu}$$ (CAN typical permanent magnets be considered to be this kind of lattices?). Then I would calculate the magnetic field $$\vec{B_{i}}(\vec{r})$$ due a magnetic moment at a generic position vector $$\vec{r}$$. Spatial integration over the whole magnet would give me the total magnetic field $$\vec{B(\vec{r})}$$ at $$\vec{r}$$.

How do I calculate the magnetic dipole moment of each molecule?

If a typical permanent magnet indeed is a lattice, what makes all of them align in the same direction so all the magnetic dipole moments add up?

How does temperature affect this? Wouldn't the molecules oscillate as the temperature increases, and be less aligned in one direction as a whole?

How would the molecules affect each other? Would it be like the Ising model (I just learned about it), only more complex? Would they affect each other quantum mechanically, interfering with each others wavefunctions?

Do solid state physicists try to model magnets like this in real life?

Thanks

 

[JesseC]

How can I analytically find the magnetic field from a permanent magnet?

With some difficulty I imagine. I assume we're talking about ferromagnets as opposed to ferrimagnets? Ferromagnets are actually split into domains where most atomic magnetic moments are aligned. However separate domains may not be aligned ...

How do I calculate the magnetic dipole moment of each molecule?

Working out dipole moments of molecules is very difficult I think, whereas single atomic moments are easier to calculate with rough accuracy. It requires a knowledge of how the electrons in each atom are filling the outer orbital levels (Hunds rules), and then from there it is possible to estimate the magnetic moment of the atom/ion.

If a typical permanent magnet indeed is a lattice, what makes all of them align in the same direction so all the magnetic dipole moments add up?

The interaction between neighbouring magnetic dipoles is almost negligible in a ferromagnet. The dominant interaction is the exchange interaction.

How does temperature affect this? Wouldn't the molecules oscillate as the temperature increases, and be less aligned in one direction as a whole?

Yes, the degree of magnetisation is heavily dependant on temperature. Read about Curie's Law.

How would the molecules affect each other? Would it be like the Ising model (I just learned about it), only more complex? Would they affect each other quantum mechanically, interfering with each others wavefunctions?

Again: see exchange interaction. Also at the atomic distance scale everything is quantum mechanical!

 

[omoplata]

Thanks! I will look into the concepts you have mentioned.