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Can you give a basic explanation of what is crystal field anisotropy ?

What is the reason to arise ?

In spin ice it forces the dipoles to point in the local 111 direction.

For partially filled rare earth atoms hund rule requires S and L max. This leaves (2s +1)(2l+1) degeneracy which is partially lifted from the LS coupling. When inserted in anisotropic field inside crystal the expectation value , or mean value of L is 0 = 0 and the L is quenched leaving only S so is forced to point in some local direction, but this is if the field removes the deneracy ?

Is it this or it is much more complicated ?

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The $p$, $d$ and $f$ orbitals are only degenerate in a central potential. If you make the potential non-central, for example by putting the atom in an external magnetic field the energy levels are no longer degenerate. Putting the atom into a crystal is just another way of creating an non-central potential (though due to electostatic forces rather than magnetic). This is perhaps more clearly seen in the ligand field splitting you get when complexing your atom with suitable ligands.

I'm not sure if that's all you were asking or if there's something else. If I've missed anything perhaps you could edit your question accordingly.

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