1. Magnetism and Magnetic Materials
DTU (10313) – 10 ECTS
KU – 7.5 ECTS
Sub-atomic – pm-nm
Mesoscale – nm-mm
Module 5
15/02/2011
Magnetic order
Macro – mm-mm

2. Intended Learning Outcomes (ILO)
(for today’s module)
List the various forms of magnetic order in magnetic materials
Calculate the room-T magnetization of a given ferromagnet
Relate exchange interactions with the ”molecular field” in Weiss models
Explain the peak in magnetic susceptibility at the Neel temperature in antiferromagnets

3. Flashback The spin Hamiltonian A new set of orbitals Superexchange
Crystal field splitting
The exchange integral

4. Ferromagnetism
In the Weiss model for ferromagnetism, exchange interactions are responsible for the huge “molecular field” that keeps moments aligned.
We define an effective field acting upon each spin due to exchange interactions
The Hamiltonian now looks just like the paramagnetic Hamiltonian, except there’s a field even with no applied field
We relate the molecular field with the “order parameter”, i.e. the magnetization

5. Review Brillouin paramagnetism
J=1/2
J=3/2
J=5

6. The spontaneous magnetization
By solving numerically the two equations, we determine the spontaneous magnetization (in zero applied field) at a given temperature
T>TC
T=TC
T<TC
Re-estimate the effective molecular field Bmf=lMS if TC is 1000 K and J=S=1/2.

7. The temperature dependence M(T)
The case of Nickel (S=1/2)
Near TC (mean-field critical exponent)
Estimate the room-T M/Ms of Fe (J=S=3/2, Tc=1043 K)
Low T (as required by thermodynamics)

8. Ferromagnet and applied field
T>TC
T=TC
T<TC
Increasing B

9. Origin of the molecular field
When L is involved (e.g. 4f ions), only a part of S contributes to the spin Hamiltonian: de Gennes factor
If we assume that exchange interactions are effective over z nearest-neighbours, we find: J L S
L+2S=J+S
(gJ-1)J
So that we reveal the proportionality between Tc and the exchange constant
This is valid when L is quenched (3d ions) and, therefore, J=S

10. Antiferromagnetism Staggered magnetization (order parameter)
Neglect those for now (but they are important for a realistic theory)
Staggered magnetization (order parameter)

11. The magnetic susceptibilities
Paramagnet
Ferromagnet
Antiferromagnet

12. AFM with a strong magnetic field

13. Types of antiferromagnetic order
Simple cubic
BCC

14. Ferrimagnetism and helical order
Ferrimagnets: important technologically for their non-metallic nature and flexible magnetic response

15. Sneak peek B M M
Shape effects and magnetic domains

16. Next lecture: Friday February 18, 8:15, KU room 411D
Wrapping up
Ferromagnetism
Spontaneous magnetization
Ferromagnetic-to-Paramagnetic transition at Tc
Antiferromagnetism
Susceptibilities and Curie-Weiss laws
Ferrimagnetism
Helical order
Next lecture: Friday February 18, 8:15, KU room 411D
Micromagnetics I (MB)