Lectures 9&10 (Magnons, Landau Theory & Anisotropy Intro)

Question 1

How can the dispersion relation for magnons be measured?

Answer 1

Inelastic neutron scattering

Question 2

How does inelastic neutron scattering work?

Answer 2

Neutron fired at sample, hits iron and spin waves excited

Neutron transmitted or reflected with lower energy (given up energy to create spin waves)

Higher k vector - higher energy spin waves excited

Question 3

What is each magnon equivalent to?

Answer 3

The reversal of a single-1/2 spin (but spread over the whole lattice)

Question 4

In terms of magnetisation, what can number of magnons be equated to?

Answer 4

∆M

Question 5

What is the Bloch law?

Answer 5

Far below Tc, that ∆M ~ T^3/2

Question 6

Briefly, describe how to derive the Bloch law

Answer 6

Need to find the number of magnons, which is the integral over the DOS * the occupation number

Use B-E stats since magnons are bosons

Use the dispersion relation to find simplify the density of states

Find ∆M/M0 = no. magnons / total no. of spins to get the Block law

Question 7

What are the experimental issues with the Bloch law?

Answer 7

In the lab, fairly easy to reach low temperatures in the magnon regime and measure M using magnetometer

But may not be possible to reach Tc and trace the full curve

Question 8

How can Bloch law be used to predict the Curie temperature?

Answer 8

Easy to reach low temps (magnon regime) and measure M using magnetometer

But may not be possible to reach Tc and trace the full curve

Fit Bloch law to see where M goes to 0

Question 9

Issues with using Bloch law to find Tc?

Answer 9

Only valid in low temperature regime (~0.5Tc)

But don’t know what Tc is, so don’t know where halfway is

More rapid dropoff than Bloch’s predicts

Question 10

What can we use to predict M(T) closer to Tc, instead of Bloch’s law?

Answer 10

Landau theory of phase transitions

Question 11

Is the Heisenberg Hamiltonian isotropic?

Answer 11

Yes

Question 12

How to derive the Langevin function?

Answer 12

For a FM particle in H, find the energy, find the expectation value of the component of µ parallel to H

Question 13

What does deriving the Langevin function from a FM particle in H show?

Answer 13

That our current definition of FM means that all particles are actually paramagnetic

Question 14

Is exchange alone enough to give FM?

Answer 14

No

Question 15

Without anisotropy, what does exchange do?

Answer 15

Couples all moments together into a giant spin that acts like a classical PM

Question 16

What can reduce thermal fluctuations?

Answer 16

Anisotropy term - holds moment in place at H=0

Question 17

What can easy and hard directions be thought of as?

Answer 17

Easy directions - energy wells

Hard directions - energy barriers

Question 18

When will a magnetic moment sit in an energy well?

Answer 18

As long as kT &laquo_space;barrier height