1. Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS Sub-atomic – pm-nm But with some surrounding environment Module 3 08/02/2001 Crystal fields

2. Intended Learning Outcomes (ILO) (for today’s module) 1.Explain why paramagnetism is T-dependent whereas diamagnetism is not 2.Estimate the value of the Curie constant for a given paramagnetic substance 3.Predict the ground state of ions by applying Hund’s rules 4.Explain the origin of the spin-orbit interaction, and describe its main effects 5.Compare Hund’s rule predictions with data on 4f and 3d elements 6.Describe how crystal fields arise 7.Explain phenomena such as crystal field splitting, Jahn-Teller distortions, low/high spin states

3. Flashback Einstein de Haas: -measure g-factor Diamagnetism: -small -T-independent -Orbital size Paramagnetism: -small -T-dependent ---> Curie law -Total angular momentum J

4. Van Vleck paramagnetism Another contribution to the paramagnetic susceptibility (there’s one more…mobile electrons – Pauli) If J=0, in principle there is no paramagnetic term. However, if we go second-order, and consider the possibility of excited states (off-diagonal matrix terms) with nonzero J, then we have: Which is positive (para), and T-independent. Why is it T-indepenent?? And why was the Langevin term T- dependent instead? John H. van Vleck, Nobel prize lecture

5. Lande’ g-value and effective moment J=1/2 J=3/2 J=5 Curie law:  =C C /T Estimate the Curie constant for a paramagnetic ionic salt with a=0.3 nm, J=S=3/2

6. Check: where are we? All atoms and ions are diamagnetic diamagnetism arises from a perturbation of the ground state diamagnetism is small and T-independent Whenever J differs from zero, we observe a paramagnetic response J can be either from OAM or from Spin or both paramagnetism is larger than diamagnetism but still small at room T The question now is: What gives angular momentum to an atom? Why are some atoms “more magnetic” than others? That’s what we focus on today.

7. The multi-electron atom and the Hund’s rules Find the electronic structure of Fe3+, Ni2+, Nd3+, Dy3+, and determine their spin configuration With many electrons, it gets messy. How do electrons “choose” which state to occupy? (1)Arrange the electronic wave function so as to maximize S. In this way, the Coulomb energy is minimized because of the Pauli exclusion principle, which prevents electrons with parallel spins being in the same place, and this reduces Coulomb repulsion. (2)The next step is to maximize L. This also minimizes the energy and can be understood by imagining that electrons in orbits rotating in the same direction can avoid each other more effectively. (3)Finally, the value of J is found using J=|L-S| if the shell is less than half-filled, J=L+S is the shell is more than half-filled, J=S (L=0) if the shell is exactly half-filled (obviously). This third rule arises from an attempt to minimize the spin-orbit energy. 2S+1 L J

8. Spin-orbit and the fine structure This is an opportunity to put QFT in action! Try to re-derive spin-orbit in a fully relativistic framework. For the atomic Hamiltonian we’ve considered so far, L and S were good quantum numbers. Problem is: they are not… For multi-electron atoms: Where the sign of Lambda depends on the shell occupancy.

9. Spin-orbit in the multi-electron atom For multi-electron atoms: Where the sign of the energy depends on the shell occupancy (see table). This justifies Hund’s third rule, whenever spin-orbit is a significant perturbation. If spin-orbit dominates (large atomic number, as it goes as Z 4 ), the L-S coupling scheme fails. Alternative: j-j coupling.

10. Composition of angular momentum Possibilities: J=L+S, L+S-1…|L-S| How many? Without spin-orbit, L and S are good quantum numbers (i.e. L and S are conserved), and J is not useful. With spin-orbit, L and S are not good quantum numbers (i.e. L and S are not conserved, although L 2, S 2 and J 2 are), and J becomes important. States are |L,S,J,M J >

11. Summary and example Fine structure of the Co 2+ ion: 3d 7 : S=3/2, L=3, J=9/2, g J =5/3

12. Data and comparison (4f and 3d) Hund’s rules seem to work well for 4f ions. Not so for many 3d ions. Why? How do we measure the effective moment?

13. Origin of crystal fields When an ion is part of a crystal, the surroundings (the crystal field) play a role in establishing the actual electronic structure (energy levels, degeneracy lifting, orbital “shapes” etc.). Not good any longer!

14. A new set of orbitals Octahedral Tetrahedral

15. Crystal field splitting; low/high spin states The crystal field results in a new set of orbitals where to distribute electrons. Occupancy, as usual, from the lowest to the highest energy. But, crystal field acts in competition with the remaining contributions to the Hamiltonian. This drives occupancy and may result in low-spin or high-spin states.

16. Orbital quenching Examine again the 3d ions. We notice a peculiar trend: the measured effective moment seems to be S-only. L is “quenched”. This is a consequence of the crystal field and its symmetry. Is real. No differential (momentum- related) operators. Hence, we need real eigenfunctions. Therefore, we need to combine m l states to yield real functions. This means, combining plus or minus m l, which gives zero net angular momentum. Examples

17. Jahn-Teller effect In some cases, it may be energetically favorable to shuffle things around than to squeeze electrons within degenerate levels.

18. Sneak peek 11 22 Interactions Ferromagnetism (Weiss)

19. Wrapping up Next lecture: Friday February 11, 8:15, KU Interactions (MB) Temperature dependencies Curie Law Van Vleck paramagnetism Hund’s rules Spin-orbit Crystal field Orbital quenching Jahn-Teller distortions