1. Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS Sub-atomic – pm-nm Module 2 04/02/2001 Isolated magnetic moments

2. Flashback What have we learned last time? You tell us…

3. Intended Learning Outcomes (ILO) (for today’s module) 1.Establish the link between g-factor and measured angle in the EdH experiment 2.Verify that magnetism in Fe is mostly from spin angular momentum 3.Describe what ”Diamagnetism” is 4.Estimate the diamagnetic response of He 5.Describe what ”Paramagnetism” is 6.Explain why some atoms are diamagnetic and why some are paramagnetic 7.List the three Hund’s rules and apply them to a few atoms

4. Einstein de Haas Consider the following scenario: A soft iron bar magnet placed in a torsion balance, initially demagnetized, saturates under the influence of the magnetic field generated by a coil. The angular momentum J associated with the acquired moment MV sets the magnet in motion. A laser beam is used to measure an equilibrium angle. Can you measure the g-factor from this experiment, and assess whether magnetism in Fe is from spin or oam? Parameters: R=4 cm, H=6 cm,  =10 -11 Nm  =7874 kg/m 3, M S =1707 A/m

5. Hamiltonian of an atom in a magnetic field Is the Hamiltonian gauge invariant? The Hamiltonian of an atom in a magnetic field has two main components: the “diamagnetic” term, and the “paramagnetic” term Spin (for each electron) OAM Initial Hamiltonian of the atom Hamiltonian in presence of a magnetic field Explicit version of the Hamiltonian

6. Magnetic susceptibility How strongly a material magnetizes in response to an external field Note: -Difference between intrinsic and experimental susceptibility (demag factor involved) -Other definitions of susceptibility: --molar (  V m ) --mass (  Focus on the initial magnetization

7. Diamagnetism Try this out with He. Does it match? All atoms and ions (except perhaps H+), have nonzero diamagnetic responses to an external field. It is, however, pretty small, and often masked by the paramagnetic response of atoms with a net moment (unpaired electrons) There’s another contribution: Landau diamagnetism (will be dealt with later)

8. Diamagnetic and paramagnetic susceptibilities

9. Paramagnetism – semiclassical If J is not zero, then you have paramagnetism Total angular momentum Energy of the moment at an angle What is the expected averaged moment along the field axis? Langevin function n is the volume density of moments

10. The J=1/2 case Estimate the paramagnetic susceptibility Several similarities Two spins, J=1/2, just two states (parallel or AP), to average statistically

11. Generic J and the Brillouin function

12. Lande’ g-value and effective moment J=1/2 J=3/2 J=5 Curie law:  =C C /T

13. 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

14. 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, and J=L+S is the shell is more than half-filled. This third rule arises from an attempt to minimize the spin- orbit energy. 2S+1 L J

15. Sneak peek

16. Wrapping up Next lecture: Tuesday February 8, 13:15, DTU (?) Crystal fields (MB) Einstein de Haas: how to distinguish L and S Diamagnetism: universal, temperature independent Paramagnetism: J.neq.0, temperature dependent Curie law and implications Lande’ g-factor Hund’s rules introduction