1. Magnetic and Nonlinear Dielectrics Properties

2. Magnetic Field Intensity = A/m

3. Measuring Magnetic Props.

4. Magnetic Field Intensity = A/m Magnetic Field In vacuum

5. In Presence of a Solid With material with Magnetization, M: Relative permeability In magnetism, the emphasis is on M and/or mag.

6. Most important Equation • Link between Macro and Micro: Saturation magnetization µion = net magnetic moment of ion or atom. A m2 V = volume of unit cell n = number of mag ions/atoms per unit cell Nmag = # of mag. Ions per m3

9. Orbital magnetic moment: Bohr magneton L = orbital angular momentum Q#

10. QMech solution • Spin Electron Moment Classic solution Spin quantum # = ± 1/2 Every unpaired electron = 1 µB

11. Moment of an atom In many systems, especially ceramic, L is quenched and:

12. For this class: Assume that orbital angular momentum is quenched, i.e. = 0 And every unpaired electron contributes 1 µB to total.

13. 2µB 5µB For this class you can assume that every unpaired electron contributes one Bohr magneton, µB to total.

14. Worked Example Example: What is Ms for Fe if µion for Fe = 2µB? Solution: Find radius and unit cell. Calculate unit cell vol. N mag = # of mag. atoms IN UNIT CELL and divide one by the other..

15. Para-, Ferro-, Ferri- and Anti-ferromagnetism

16. Unpaired electrons !!! No unpaired electrons no magnetism.

17. N = number of mag. atoms/m3 + µB 2 µB - µB For paramagnetic solids: B ≈ µoH is a very good approx. See worked example 15.1 Simplified More exact Factor of 3 comes from averaging.

18. Ferromagnetism Paramag. Ferromag Ferromag Curie-Weiss Law Above Curie temp. = Tc Below Tc spontaneous magnetization

19. Noting that Msat = Nµion and defining:  = mean field constant or coupling coefficient

20. More exact Factor of 3 comes from averaging over all directions. Only valid for T > TC.

21. As entropy disappears, then M = Msat = N µion where N is number of magnetic atoms/m3.

22. Antiferromagnetism and Ferrimagnetism Anti-ferrimagnetic Neel Temp. Ferrimagnetic

23. dx2- y2 Super- exchange leading to antiferromagnetism

24. Experimental Identification para 1/mag Antiferromag Ferromag or ferrimangetic 0 Temp. (K)

25. Problem Below Tc you have spontaneous magnetization…. In experiment, a virgin sample does NOT have a magnetization!

26. Domains and Domain Walls

28. Msat = N µion = N n µB where n is number of unpaired electrons/atom

29. Coercive Field, Hc Remnant magnetization, Br Coercive field Hard magnets: High Br and high Hc. Soft magnets Easy magnetization & demagnetization. Uses: permanent magnets motors, etc. Uses: transformer cores

30. Magnetic Ceramics Cubic Ferrites or Spinels Hexagonal Ferrites Garnets

31. Cubic Ferrites or Spinels Huge tonnage. Why?

32. O T c Oc 8 Fe3+ 8 Fe3+ 8 Fe2+ Cubic Ferrites or Spinels Normal spinels: 2+ in tetrahedral, T, sites 3+ in octahedral, Oc, sites. Inverse spinels: 1/2 of 3+ on O sites and 1/2 on T sites 2+ on Oc sites. Oc and T sites antiferromagnetically coupled, so they cancel each other out. Net = moment due to 2+ cations Examples: FeO.Fe2O3 or Fe3O4 - 24 cations/UC Sites, not oxygen Is this a normal or abinormal spinel???

33. Examples: FeO.Fe2O3 or Fe3O4 - 24 cations/UC O O 4µB per formula unit 8x4 µB per UC Measured = 4.1 8 Fe3+ 8 Fe2+ Examples: NiO.Fe2O3 - 24 cations/UC 2µB per formula unit 8x2 µB per UC Measured = 2.4 T T O O 8 Fe3+ 8 Fe3+ 8 Fe3+ 8 Ni2+

34. Examples: ZnO.Fe2O3 - 24 cations/UC O T 0 µB per formula unit Measured = 0 8 Fe3+ 8 Zn2+ O 8 Fe3+ Zn and Cd prefer Tetrahedral sites…. I.e. they form normal spinels.

35. Summary

36. Garnet Ferrites • When M goes to 0 is called compensation point.

37. Hexagonal Ferrites BaO6Fe2O3 Easy magnetization along c-axis.

38. Aligning in magnetic field. BH product enhanced

39. Ligand Field Theory

40. Hund’s Rules In isolated atoms or molecules, the electrons would rather NOT pair up… However, if penalty is too high, i.e. spacing bet levels is high, they will pair up….

41. Molecular Magnetism Fe2+ , d6 Low Spin Diamagnetic High Spin Paramagnetic You can thus get photoinduced magnetism ..

42. Applications • Transformer cores. Need to minimize losses. i) ferrites,

43. Para-, Piezo-and Ferroelectric Solids

44. Piezoelectricity Pyroelectricity - + T1 T2

45. Piezoelectricity: • Direct effect: Apply a stress leads to strain which leads to polarization. • Inverse effect: Apply E, get strain that depends on direction. • Not to be confused with electrostriction - which is change in shape upon application of E. • Pyroelectricity: • Has a spontaneous polarization in absence of E, for which P changes with ∆T • ∆P = l ∆T where  is pyroelectric coefficient. • Ferroelectricity: • Spontaneous polarization in absence of E, which may be switched in direction be application of E. • Characterized by hysteresis loops.

46. Do not confuse Electrostriction and Piezoelectricity Shrinkage Expansion

47. Arrows denote dipoles Centrosymmetric crystals cannot be piezo- or ferroelectric Non-centrosymmetric crystals can be piezo- but not ferroelectric. No permanent dipole Permanent dipoles = Ferroelectric

49. Paraelectric: already done… AKA dipolar polarization • Ferroelectric…. Already done… AKA ferromagnetism.

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