Why classical mechanics is not able to explain the net magnetization in ferromagnets?

Question

Why classical mechanics is not able to explain the net magnetization in ferromagnets? why does exchange interaction can explain net magnetization in Ferromagnets, which is purely quantum mechanics??

Answer 1

Often, a statement like in the question is linked to the Bohr-van Leeuwen theorem. However, such a theorem does not say that classical electromagnetism can't explain any form of magnetism in materials. Should we need QED to explain magnets? What the theorem says is much more limited: classical statistical mechanics cannot explain magnetism if the electromagnetic field is introduced in the Hamiltonian via the minimal coupling hypothesis. It does not exclude the possibility of magnetism in classical systems via non-minimal coupling.

Actually, the correct statement is that the dominant interaction between local magnetic moments in a material is due to electrostatic effects induced by the antisymmetry of the electronic wavefunctions. However, this means that quantum mechanics is necessary for quantitatively modeling magnetic interactions in real materials. The step from interactions to ferromagnetism does not necessarily require Quantum Mechanics, as the well-known Classical Heisenberg model clearly shows.

Answer 2

Short answer follows from Bohr-Van Leeuwen theorem which showed that classical electromagnetism can't explain any form of magnetism in materials. Formal proof is long, but in short it reduces to the fact that averaged thermal magnetic moment of electron ensemble is zero, i.e. :

$$ \langle \mathbf {\mu } \rangle = {\frac {e}{2c}} \langle \mathbf {r} \times \mathbf {v} \rangle = 0$$

So magnetic properties of material can't follow from electron movement around nuclei. Instead they follows from the :

• spins of electrons, which has nothing to do with electron orbitals in an atoms and are quantized,- when projected along some axis takes only two possible values from $\pm \frac {\hbar}{2}$.
• Pauli exclusion principle which states that two or more fermions can't occupy same quantum state. This means that electron must "align" to neighboring electron spin, i.e. spin factors are not completely random in contrary to the classical magnetic moment of electron in electromagnetism.

Given that even in thermal equilibrium, spins can align so that total magnetic moment of material is not zero.

BTW, there are some semi-classical attempts to build a statistical mechanics explanation of ferromagnetism, such as Ising_model, but alas,- they are using quantized magnetic dipole moments of atoms as well, which take values of $-1,+1$. So it's not a "pure classical way", but more of perturbed theory, because no quantization exist in classical physics.