Data schema for magnetic_state

[JosePizarro3]

@ndaelman-hu @JFRudzinski

This is part of my notes done in #12 . I moved them here for consistency.

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Short theory summary

Free atom magnetic_moment

The quantum numbers (L, S, J): depending on the relative size of the spin-orbit coupling, we need to use LS notation (small SOC) or JJ notation (large SOC). L, S are calculated from the individual electron ml and ms numbers as:

L = Σ_i ml_i
S = Σ_i ms_i
|L-S| <= J <= L+S

where i represents each electron in the OrbitalState(n, l, ml, ms).

To calculate L and S we need to know the OrbitalsState.occupation for each orbital, using Hund's rules:

1. Maximize S (first fill spin up 1/2, then spin down -1/2)
2. Maximize L (first fill larger ml and then smaller to negative)
3. Minimize SOC --> J=|L-S| if AtomState is less than half-full, J=L+S if AtomState is more than half-full.

For example, Mn+3 has a valence 3d4 AtomState:

The effective magnetic_moment is calculated using:

In this example, however, experimentally we find μ_exp=4.82 μ_B (instead of μ_eff=0).

The effect of the crystalline environment in the magnetic_moment

Due to the crystal environment, OrbitalState is feeling finite crystal_field / onsite_energy effects which depend on the symmetry of the crystal (e.g., perovskites break the d orbital symmetry into 3-fold degenerate t2g and 2-fold degenerate eg orbitals).

The Slater-Koster TB method is a good starting point to obtain onsite energies.

In the example for Mn+3, there is an interesting effect playing role: orbital quenching (important in d-orbital systems). If instead of L=2, we consider L=0, we get μ_eff=4.89 μ_B (close to the experimental value). This happens when the crystal field effect is larger than SOC, breaking Hund's rules.

Overall, we have 3 main energy scales which control the OrbitalsState which defines the AtomsState.magnetic_moment:

1. Crystal field
2. SOC
3. Hund's coupling (favors Hund's rule)
   
   

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Idea for the schema

We need to:

1. Check OrbitalsState.j_quantum_number and OrbitalsState.mj_quantum_number. Are these necessary?
2. Define AtomsState.L_total_quantum_number, AtomsState.S_total_quantum_number, AtomsState.J_total_quantum_number. Here we could think whether we want to keep a string using '(2S+1)LJ' notation.
3. Define AtomsState.magnetic_moment, AtomsState.spin_orbit_coupling (this is, the pre-factor on the L · S).
4. Define OrbitalsState.crystal_field (or OrbitalsState.onsite_energy). Related with Add onsite_energy for OrbitalsState #11
5. Is it possible to use Hund's rules + crystal_field + spin_orbit_coupling + OrbitalsState.occupation to obtain the quantum numbers L, S, J via normalization?
6. What about anisotropic spin Hamiltonians which go like H = D Sz^2 + E(Sx^2 - Sy^2)?
7. Via normalization using AtomsState.magnetic_moment, we can define ModelSystem.magnetic_state (or ModelSystem.magnetic_order): paramagnetic, ferromagnetic, antiferromagnetic (collinear or non-collinear, ferrimagnetic? altermagnetic?), spin glass.

We can also define ModelSystem.magnetization, which should be connected (perhaps, I am not 100% sure) with all the AtomsState[i].magnetic_moment. Finally, also add ModelSystem.magnetic_state_q_vector if possible.

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Note on where do we stop

I think we should not try to cover all potential cases in magnetism. For example, we can skip trying to characterize perfectly all the possible exchange interactions, and perhaps leave these for a near future.

[JosePizarro3]

@EBB2675

[EBB2675]

Thanks a lot! Let me make a draft ModelSystem.magnetization and ModelSystem.charge this week, that would go well for molecular systems, and we can discuss it there?