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When talking about magnetism and the interaction of spins, the most basic model seems to be the Heisenberg model with an interaction of the form $ -J \vec{S}_1\cdot \vec{S}_2$. The origin of the contribution seems to be coming from the exchange interaction, where the Coulomb matrix element $\langle \phi_a^{(1)} \phi_b^{(2)} | V| \phi_a^{(2)} \phi_b^{(1)} \rangle = \int d^3 r \int d^3 r^\prime \varphi_a^\ast(\vec{r}) \varphi_b^\ast(\vec{r}^\prime) V(\vec{r}-\vec{r}^\prime) \varphi_b(\vec{r}) \varphi_a(\vec{r}^\prime) \delta_{s_1 s_2}$ gives a contribution only for parallel spins, introducing an energy shift between parallel and antiparallel spins.

My question is now if this is the only way that spins interact or at least why this is the only interaction mechanism I can find information about. Can spins also interact directly, for example since one spin induces a magnetic field that interacts with the other spins magnetic moment? Would this also lead to a contribution $\propto \vec{S}_1\cdot \vec{S}_2$? If so, how could one calculate the interaction strength? Is this just much smaller than the exchange energy and therefore neglected? Where can I find more about this in literature? Thank you in advance!

Stephphen
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1 Answers1

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Any relevant description of the magnetic properties of materials will give several interactions where only spins are involved. A very common and relevant process where most of them are involved is the creation of magnetic domains. The magnetic domains involve, the dipole-dipole interactions, the spin-orbit coupling and the exchange interaction. The process can be summarised as follow :

A material such as Iron or Cobalt with a significant coulomb repulsion $U \gg $ and localised orbitals fulfill the requirements to be ferromagnetic. The exchange coupling is positive $J>0$ for some spins, giving the Heisenberg model and a resulting magnetization.

However it is always unstable to align parallel dipoles. In this case, there is a dipole-dipole interaction ~$\propto \bf{S_1}.\bf{S_2}\frac{1}{r^3}$, which is weak compared to the exchange interaction ~$\propto \bf{S_1}.\bf{S_2} \frac{1}{r}$. This weak contribution becomes non négligeable with the size. A bulk ferromagnetic material has to decrease the dipole-dipole interaction, for this purpose domains are created separated by a domain wall. The wall itself has a cost because it flips the spins in a ferromagnetic material where they should be polarized. This wall will be very large without an anisotropy.

In some materials, the orbital angular momentum $\hat{L}$ is quenched because of the environnement and only the spin $s$ is a good quantum number. The remaining weak $\hat{L}$ is coupled to the spin and gives a lattice dependent spin-orbit coupling interaction $\Delta E_{SO} \propto \bf{L}.\bf{S}$, a self interaction with an angular dependency which reduces the wall width.

The final result is a bulk magnetic material with no effective magnetisation without a magnetic field encompassing several interactions.

M06-2x
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