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When I have two magnets and try to rotate them so they are parallel, I feel a strong repulsive force which definitely tries to rotate them to an antiparallel notation.

However, when examining spin Hamiltonians, we see that in ferromagnetic cases parallel spins are actually energetically favored. I assume the confusion arises because a north/south pole analogy cannot be made using spins. But I'm wondering, what is the actual physical explanation of ferromagnetism?

Why are parallel spins energetically favored in ferromagnets?

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    $\begingroup$ It's a finely balanced tradeoff between the exchange and magnetic interactions. The fine balance is why ferromagnetism is a relatively rare phenomenon. I don't think there is a simple explanation for it. $\endgroup$ – John Rennie Oct 2 '17 at 15:43
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You are not wrong to think that, according to your classical intuition, magnetic dipoles should align in an antiferromagnetic fashion. It is true that the magnetic interaction between classical magnetic dipoles favors antiparallel alignment.

However, you are neglecting quantum mechanics. The quantum mechanical exchange interaction favors parallel alignment. In ferromagnetic materials, this exchange interaction overpowers the dipole-dipole interaction between spins, causing the spins to align in parallel.

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    $\begingroup$ So this 'favoring' is actually something we observed, but which we cannot explain? $\endgroup$ – JBrouwer Oct 6 '17 at 22:30
  • $\begingroup$ I tried to explain the exchange interaction below @MathematicalRain. There is no problem calculating this for an atom. $\endgroup$ – Pieter Oct 9 '17 at 16:05
  • $\begingroup$ No, @MathematicalRain. The 'favoring' is something that can be observed, and which we cannot explain using classical theory, but that we can explain using quantum theory. $\endgroup$ – Ian Oct 9 '17 at 18:58
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The exchange interaction that favors parallel spins is basically electrostatic. Consider Hund's rule for ions like Mn$^{2+}$ or Fe$^{3+}\!$. These have five electrons in their $3d$ orbitals. Because of the Pauli principle, electrons will be further away from each other in state where all spins are parallel with each other. This stabilizes the high-spin state.

Similar for the oxygen molecule - both $\pi^*$ electrons have their spins parallel. Compounds are more complicated. Manganese metal is an antiferromagnet. But the reason for that is still the Coulomb interaction between the electrons, not the magnetic dipole interaction.

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