Lectures 3&4 (Exchange & Heisenberg Hamiltonian)

Question 1

How did Weiss attempt to explain FM?

Answer 1

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)

Question 2

What energy is from magnetic dipole interaction?

Answer 2

Magnetostatic

(energy between 2 magnetic moments)

Question 3

What energy is from Coulomb interaction?

Answer 3

Electrostatic

Question 4

Difference of r between parallel and antiparallel spins?

Answer 4

AP can occupy same orbital - r lower - higher energy (Coulomb)

P can’t occupy same orbital - r lower

Question 5

Do P or AP spins have higher energy, following from Coulomb?

Answer 5

P lower energy than AP

Question 6

In terms of spin, what do we want for FM?

Answer 6

Neighbouring spins ||

Question 7

Hund’s first rule?

Answer 7

Maximise spin to obtain lowest energy

Question 8

Will the overall 2-fermion wavefunction be symmetric?

Answer 8

No, antisymmetric

Question 9

Which direction are spins in a symmetric spin wavefunction?

Answer 9

Symmetric

Question 10

How does particle exchange manifest in the determinant?

Answer 10

Switching 2 columns (i.e. switching the positions of the 2 electrons)

Question 11

In exchange, when will the wavefunction =0?

Answer 11

When any 2 rows/columns are equal

Question 12

What is the symmetry of the overall wavefunction?

Answer 12

Antisymmetric

Question 13

What does it mean physically, that a wavefunction is antisymmetric?

Answer 13

Is it not the same after exchange

Question 14

What is the symmetric version of ø, before and after exchange?

Answer 14

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)]

Question 15

What is the antisymmetric version of ø, before and after exchange?

Answer 15

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)]

Question 16

What is the ø wavefunction?

Answer 16

Represent atomic orbitals i.e., positions of the electrons in space

Question 17

Is the ø wavefunction discrete or continuous?

Answer 17

Continuous

Question 18

Is the X wavefunction discrete or continuous?

Answer 18

Discrete

Question 19

Why do we make spin point either up or down?

Answer 19

Can point anywhere - but we define a quantization axis (z-axis)

Measure projection of angle along z i.e., above horizontal = spin up

Question 20

What is the singlet state?

Answer 20

Antisymmetric spin eigenstate

Question 21

What spins does the antisymmetric spin eigenstate correspond to?

Answer 21

Antiparallel spins

Question 22

When is the antisymmetric singlet state permitted?

Answer 22

When 2 electrons are in the same spatial state (same orbital)

Question 23

How many electrons would doublets need?

Answer 23

3 electrons (for spin +/- 1/2)

Question 24

What is the overall wavefunction for the singlet state?

Answer 24

1/√2 [ø_α(1)ø_β(2) + ø_α(2)ø_β(1)] * X_S

Question 25

What is the overall wavefunction for the triplet state?

Answer 25

1/√2 [ø_α(1)ø_β(2) - ø_α(2)ø_β(1)] * X_T

Question 26

What question do we ask to define the exchange integral?

Answer 26

How much energy do we save when doing from antiparallel to parallel spins?

Question 27

How much energy do we save going from AP to P spins?

Answer 27

E_S - E_T / 2 = exchange integral

Question 28

How can the exchange integral be defined?

Answer 28

E_S - E_T / 2

Question 29

How is the energy from AP to P states defined (exchange energy)?

Answer 29

E = U_elec +/- J

Question 30

For J itself +ve, what do +/- J correspond to?

Answer 30

+J = AP (singlet state) -J = P (triplet state)

Question 31

For J itself -ve, what do +/- J correspond to?

Answer 31

+J = P (triplet state) -J = AP (singlet state)

Question 32

What is the exchange integral J physically?

Answer 32

A correction to the classical Coulomb interaction which accounts for the PEP

Question 33

When do spins of the same sign cost and save energy?

Answer 33

Cost energy when close, save it when further apart

Question 34

In terms of J, when does AFM occur?

Answer 34

When J itself is -ve

Question 35

How do electrons being on the same / neighbouring atoms affect J?

Answer 35

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

Question 36

What are the types of molecular orbital?

Answer 36

Bonding / anti-bonding

Question 37

Why is J likely negative if electrons are in neighbouring atoms?

Answer 37

Anti-bonding orbital (P spins) higher energy since spatial wavefunction has higher curvature and KE so singlet state (AP) is favoured

Question 38

When might indirect exchange occur?

Question 39

What does it mean, that the 2 electrons in exchange are tunneling?

Answer 39

The 2 electrons constantly change places

Question 40

Can electrons in singlet / triplet states occupy the same space?

Answer 40

Singlet electrons can, triplet cannot since the wavefunction goes to 0 as r1 -> r2

Question 41

What is the exchange interaction (in terms of swapping spins)?

Answer 41

When we swap 2 parallel spins, the sign of ø (and 𝚿) changes and the energy cost of electrostatic repulsion is minimised

Question 42

Types of exchange?

Answer 42

* Intra-atomic - direct (Hund's rules) * Inter-atomic - direct (molecules), indirect (rare earths), superexchange * s-d - direct, band mixing * Itinerant - direct (Fe, Co, Ni)

Question 43

What is superexchange?

Answer 43

Indirect exchange between non-neighbouring magnetic ions mediated by a non-magnetic ion in between

Question 44

Is intra-atomic exchange direct? Is preferred state P or AP?

Answer 44

Direct, P

Question 45

Is inter-atomic exchange direct? Is preferred state P or AP?

Answer 45

Molecules - direct, AP Indirect - rare earth, P Superexchange - indirect, AP

Question 46

Is s-d interaction direct? Is preferred state P or AP?

Answer 46

Direct, P

Question 47

Is itinerant exchange direct? Is preferred state P or AP?

Answer 47

Direct, P