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Course outline: The outline can be downloaded here.

Class timings: 1:00 to 2:15 pm, Monday and Wednesday, Room: 301  SSE Complex.

Pre-mid term

sdTo access the video recording, click on the numbered links below.

Origin of Magnetic Moments (2 Lectures)

1: Isolated magnetic moments and paramagnetism

2: Forces and torques on magnetic moments

HW1: Isolated magnetic moments and paramagnetism (due date: Monday, 26 September, 1 pm), HW1 Solution

3: Symmetrization Postulate

  • Dipolar interaction, exchange interaction
  • Pauli’s equation, Pauli exclusion principle

4: The Helium Atom

  • Ground and Excited state

5: Molecular Hydrogen Ion

  • Heisenberg exchange Hamiltonian
  • Ferromagnetism, antiferromagnetism
  • Variational pricnciple

6: The Hydrogen Molecule

  • Bonding and antibonding molecular orbitals
  • Ionic bond, covalent bond
  • Exchange degeneracy in the molecule causes singlet-triplet energies to be different
  • London-Heitler approximation

7: Exchange Interaction

  • Direct exchange, Indirect exchange, Superexchange
  • Heisenberg exchange Hamiltonian
  • Example of finding energy splitting due to exchange Hamiltonian

8: Spin Orbit Interaction (SOI)

  • Semi-classical treatment
  • Quantum mechanical interpretation
  • Helium (He) atom,  Sodium (Na) doublet
  • Multielectron Atom, L-S and j-j coupling
  • Hund’s Rules:3d and 4f ions and their comparison
  • Effective magnetic moments of 3d and 4f ions

HW2: Exchange Interaction (due date: Monday, 10 October, 12 pm), HW2 solution

9: Crystal Field Interaction

  • Orbital quenching
  • 3d orbitals in Octahedral environment
  • Crystal field splitting

10: Crystal Field Interaction

  • t2g and eg Orbitals
  • 3d orbitals in tetrahedral environment
  • Jahn-Teller Effect
  • low spin and high spin configuration

11: Vibrating Sample Magnetometer

12: Diamagnetism of bound electrons and Pauli paramagnetism

  • Bound electrons in a solid
  • Canonical momentum
  • Diamagnetism, Paramagnetism
  • Van Vleck paramagnetism
  • Fermi Dirac distribution, Density of states
  • Spin split bands
  • Pauli paramagnetism

HW3: Crystal field interactions, diamagnetism, Pauli paramagnetism (due date: Monday, 31  October, 10 am). Assistive Mathematica file.

HW3 Solution: Assitive Mathematica File

13: Landau Levels

  • Spin-1/2 free electrons
  • Landau Gauge
  • Landau Quantization
  • Landau levels for electrons in a solid
  • Degeneracy of Landau levels
  • k-space in a magnetic field

14: Landau Levels

  • Density of states (DOS) in a magnetic field
  • Convolution of 1D and 2D DOS

15: Landau Diamagnetism

  • Van-Hove singularities
  • Integrated density of states
  • Change in Fermi level in a magnetic field
  • De Haas-Van Alphen oscillations
  • Magneto Oscillations (assistive files)
  • Comparison between Pauli and Landau paramagnetism

Midterm: Midterm Solution

16: Mean-Field theory

  • Magnetically ordered structures
  • Ferromagnetism, antiferromagnetism, helimagnetism, conical magnetism
  • Origin of mean-field theory
  • Curie-Weiss law, Curie temperature
  • Ferromagnetic (FM) ordering
  • Spontaneous magnetization

HW4: Weiss molecular field theory and Ferromagnetism (due date: Monday, 21  November, 1 pm), HW4: Solution

17: Ferromagnetism (FM)

  • Origin of Weiss molecular field
  • Susceptibility of FM beyond critical temperature

18: Antiferromagnetism (AF)

  • Antiferromagnetic ordering
  • Neel Temperature
  • Susceptibility of AF beyond Neel temperature
  • Susceptibility of AF below Neel temperature
  • Susceptibility parallel and perpendicular to B field
  • Canted Spins in antiferromagnets
  • Spin flip and Spin flop transitions (Assistive file)

19: Spin Waves

  • Magnetic excitations
  • Dispersion relation for Spin waves
  • Magnons
  • Temperature dependence of magnon numbers
  • Bloch Law, energy and heat capacity of magnons

HW5: Spin waves and magnons (due date: Monday, 28  November, 1 pm), HW5 solution

20: Quantum mechanical description of magnons

  • Second quantization formalism
  • Holstein-Primakoff transformations
  • Mapping of exchange hamiltonian under Holstein-Primakoff transformations
  • Tight Bonding Model
  • Diagonalization of hamiltonian by Fourier transform
  • Dispersion relation for magnons

HW6: Various aspects of quantum magnetism (due date: Monday, 12 December, 1 pm), HW6 solution

21: Landau theory of ferromagnetism

  • Symmetry breaking and order parameter
  • Power law for magnetization, phase transitions, critical exponents
  • Experimental methods to determine the Curie temperature (Arott plot)
  • Heat capacity of ferromagnet

22: Ising chains

  • Ising interaction
  • Role of entropy in ising chain
  • Heat capacity of finite ising chain
  • Tight binding inside a molecule (Hydrogen molecular ion)
  • Tight bonding hamiltonian
  • Bonding and antibonding molecular orbitals

23: Band magnetism

  • Tight binding chain
  • Hubbard model: interplay of kinetic energies and Coulombic repulsion

24: Stoner Model

  • Stoner criterion, Susceptibility in Stoner model
  • Qualitative analysis of ferromagnetism in 3d elements
  • Rigid band model
  • Weak and strong ferromagnets, Heusler alloys
  • Superexchange interaction in MnO
  • RKKY interaction