Lectures 3&4 (Exchange & Heisenberg Hamiltonian) Flashcards
How did Weiss attempt to explain FM?
1) Molecular / exchange field that aligns all m. moments (explains M(T))
2) Magnetic moments where M direction varies from one domain to the next (explains hysteresis loops)
What energy is from magnetic dipole interaction?
Magnetostatic
(energy between 2 magnetic moments)
What energy is from Coulomb interaction?
Electrostatic
Difference of r between parallel and antiparallel spins?
AP can occupy same orbital - r lower - higher energy (Coulomb)
P can’t occupy same orbital - r lower
Do P or AP spins have higher energy, following from Coulomb?
P lower energy than AP
In terms of spin, what do we want for FM?
Neighbouring spins ||
Hund’s first rule?
Maximise spin to obtain lowest energy
Will the overall 2-fermion wavefunction be symmetric?
No, antisymmetric
Which direction are spins in a symmetric spin wavefunction?
Symmetric
How does particle exchange manifest in the determinant?
Switching 2 columns (i.e. switching the positions of the 2 electrons)
In exchange, when will the wavefunction =0?
When any 2 rows/columns are equal
What is the symmetry of the overall wavefunction?
Antisymmetric
What does it mean physically, that a wavefunction is antisymmetric?
Is it not the same after exchange
What is the symmetric version of ø, before and after exchange?
Before:
ø = 1/√2 [ø_α(1)ø_β(2) + ø_α(2)ø_β(1)]
After:
ø = 1/√2 [ø_α(2)ø_β(1) + ø_α(1)ø_β(2)]
What is the antisymmetric version of ø, before and after exchange?
Before:
ø = 1/√2 [ø_α(1)ø_β(2) - ø_α(2)ø_β(1)]
After:
ø = 1/√2 [ø_α(2)ø_β(1) - ø_α(1)ø_β(2)]
What is the ø wavefunction?
Represent atomic orbitals i.e., positions of the electrons in space
Is the ø wavefunction discrete or continuous?
Continuous
Is the X wavefunction discrete or continuous?
Discrete
Why do we make spin point either up or down?
Can point anywhere - but we define a quantization axis (z-axis)
Measure projection of angle along z i.e., above horizontal = spin up
What is the singlet state?
Antisymmetric spin eigenstate
What spins does the antisymmetric spin eigenstate correspond to?
Antiparallel spins
When is the antisymmetric singlet state permitted?
When 2 electrons are in the same spatial state (same orbital)
How many electrons would doublets need?
3 electrons (for spin +/- 1/2)
What is the overall wavefunction for the singlet state?
1/√2 [ø_α(1)ø_β(2) + ø_α(2)ø_β(1)] * X_S
What is the overall wavefunction for the triplet state?
1/√2 [ø_α(1)ø_β(2) - ø_α(2)ø_β(1)] * X_T
What question do we ask to define the exchange integral?
How much energy do we save when doing from antiparallel to parallel spins?
How much energy do we save going from AP to P spins?
E_S - E_T / 2 = exchange integral
How can the exchange integral be defined?
E_S - E_T / 2
How is the energy from AP to P states defined (exchange energy)?
E = U_elec +/- J
For J itself +ve, what do +/- J correspond to?
+J = AP (singlet state)
-J = P (triplet state)
For J itself -ve, what do +/- J correspond to?
+J = P (triplet state)
-J = AP (singlet state)
What is the exchange integral J physically?
A correction to the classical Coulomb interaction which accounts for the PEP
When do spins of the same sign cost and save energy?
Cost energy when close, save it when further apart
In terms of J, when does AFM occur?
When J itself is -ve
How do electrons being on the same / neighbouring atoms affect J?
Same atom - J usually +ve (FM)
Neighbouring - molecular orbitals instead of atomic. Anti-bonding more energetically costly due to higher curvature so KE. Singlet favoured to J likely -ve
What are the types of molecular orbital?
Bonding / anti-bonding
Why is J likely negative if electrons are in neighbouring atoms?
Anti-bonding orbital (P spins) higher energy since spatial wavefunction has higher curvature and KE so singlet state (AP) is favoured
When might indirect exchange occur?
What does it mean, that the 2 electrons in exchange are tunneling?
The 2 electrons constantly change places
Can electrons in singlet / triplet states occupy the same space?
Singlet electrons can, triplet cannot since the wavefunction goes to 0 as r1 -> r2
What is the exchange interaction (in terms of swapping spins)?
When we swap 2 parallel spins, the sign of ø (and 𝚿) changes and the energy cost of electrostatic repulsion is minimised
Types of exchange?
- Intra-atomic - direct (Hund’s rules)
- Inter-atomic - direct (molecules), indirect (rare earths), superexchange
- s-d - direct, band mixing
- Itinerant - direct (Fe, Co, Ni)
What is superexchange?
Indirect exchange between non-neighbouring magnetic ions mediated by a non-magnetic ion in between
Is intra-atomic exchange direct? Is preferred state P or AP?
Direct, P
Is inter-atomic exchange direct? Is preferred state P or AP?
Molecules - direct, AP
Indirect - rare earth, P
Superexchange - indirect, AP
Is s-d interaction direct? Is preferred state P or AP?
Direct, P
Is itinerant exchange direct? Is preferred state P or AP?
Direct, P
