5.4 Magnetic Order

Question 1

What two conditons must be allowed in order for spontaneous magnetisation to occur?

Answer 1

• Spins on neighbouring atoms must be allowed to interact
• Must be below a critical temperature
• Allows for magnetisation in some materials even with no applied B field

Question 2

Why can interacting spins between neighbours cause spontaneous magnetisation?

Answer 2

Due to the exchange interaction between the spins

Question 3

Describe the pattern of the spins for a ferromagnet when T < T_c

Answer 3

All aligned with each other

• Thermal fluctuations are small compared to the alignment energy
• M is not 0

Question 4

Describe the pattern of the spins for an antiferromagnet when T < T_N

Answer 4

Each neighbour has the opposite spin to another neighbour

• Get same orientation along diagonals
• Magnetic ordering
• M = 0

Question 5

Describe the pattern of the spins for an ferrimanget when T < T_c

Answer 5

Like an antiferromagnet, except the down spins are smaller in size than the up spins

• 2 atoms in the basis anti align
• M is NOT 0

Question 6

What condition determines if we get ferromagnetism

Answer 6

If the exchange constant in the Hamiltonian, J_ij > 0

- Energy is lower if the spins align

Question 7

What condition determines if we get antiferromagnetism

Answer 7

If the exchange constant in the Hamiltonian, J_ij < 0

- Energy is lower if the spins anti-align

Question 8

How does considering the “nearest neighbour” model allow for simplification?

Answer 8

Sum over i,j and then extend this for the whole material

Question 9

What is an easy axis?

Answer 9

A direction in a crystal which the spins like to align

- Creates an asymmetic environment in the crystal

Question 10

What is a hard axis?

Answer 10

A direction in a crystal which the spins don’t like to align

Question 11

When do we get an easy/hard axis?

Answer 11

Easy - Kappa (anisotropy) is > 0 as the spins want to be parallel to Z
Hard- Kappa (anisotropy) is < 0 as the spins want to be perpendicular to Z

Question 12

Describe the Ising model

Answer 12

A special case where Kappa is large enough that spins only align along the Z axis
- Very simple model

Question 13

Why is the Ising model very simple, and what is it useful for?

Answer 13

There is only one degree of freedom

- Useful for looking at phase transitions from one magnetic state to another

Question 14

What is Weiss’ core assumption for his ferromagnetism model?

Answer 14

That spins align due to a molecular field that is proportional to the magnetism M

Question 15

What wil happen to the magnetic moments in a ferromagnet at absolute 0?

Answer 15

They will all align

Question 16

What happens to the magnetic moments as the temperature increases from absolute 0?

Answer 16

Thermal fluctuations start to have an increasing effect and will eventaully start to destroy the order

Question 17

In the spin 1/2 model, what are the solutions for M for low temperature?

Answer 17

3 solutions at 0 and ± the crossing point

- If M = 0, then any small fluctuations will move it to a non zero value. Not stable

Question 18

In the spin 1/2 model, what are the solutions for M for high temperature?

Answer 18

Only M = 0. No ordering at high temperatures

Question 19

In the spin 1/2 model, what are the solutions for M for Curie temperature?

Answer 19

Pass between two regimes. Have a similar sin(x) = x for small angles looking graph
- Valid in the paramagnetic (high temperature) state

Question 20

When is the Curie Weiss Law valid?

Answer 20

It is true just above T_c and in the paramagnetic state

Question 21

When do we use the spin J model?

Answer 21

When the spin is no longer 1/2

Question 22

Is the Curie Weiss law obeyed in the spin J model?

Answer 22

Yes, with a T_c that include the effects of J

Question 23

What is T_N?

Answer 23

The Neél temperature for antiferromagnetism

Above this, thermal effects dominate

Question 24

Describe how the Neél model for antiferromagnetism works

Answer 24

Split the crystal into two sub lattices

- One with diagonal spins up added to one with the diagonal spins down at B = 0

Question 25

How do the two sublattices interact for the Neél model?

Answer 25

Each sublattice experiences a molecular field proportional to the magnetism on the other sub lattice

Question 26

What happens to the magnetism at the Neél temperature for each sublattice?

Answer 26

It goes to 0

Question 27

How can we identify is a material is a magnet or not?

Answer 27

• Measure the magnetic susceptability in the paramagnetic state
• Fit it to χ ∝ 1 / (T - T_Weiss)

Question 28

Is the Weiss temperature is equal to 0, what type of magnet do we have?

Answer 28

A paramagnet

Question 29

Is the Weiss temperature is > 0, what type of magnet do we have?

Answer 29

Ferromagnet

Question 30

What is the critical temperature for a ferromagnet if the Weiss temp > 0?

Answer 30

T_c = T_Weiss

Question 31

Is the Weiss temperature is < 0, what type of magnet do we have?

Answer 31

Antiferromagnet

Question 32

What is the critical temperature for an antiferromagnet if the Weiss temp < 0?

Answer 32

T_c = - T_N

Question 33

What behaviour is observed at T = 0 and B is parallel to M?

Answer 33

Both sub lattices are saturated. The parallel components of both the magnetism and the magnetic susceptability are equal to 0

Question 34

What behaviour is observed at T = 0 and B is perpendicular to M?

Answer 34

The transverse/perp components of both B and M are not equal to 0
- The spins are tilted on both sub lattices

Question 35

What behaviour is observed at 0 < T < T_N and B is parallel to M_a?

Answer 35

M_a - sublattice with spins pointing upwards

• B enhances M_a, reduces M_b
• Strong effect on the susceptability

Question 36

What behaviour is observed at 0 < T < T_N and B is perpendicular to M?

Answer 36

The transverse/perp components of both B and M are not equal to 0
- The spins are tilted on both sub lattices

Question 37

What are M_a and M_b?

Answer 37

M_a is the sub lattice with spins up

M-B is the sub lattice with spins down

Question 38

How are the crysal atoms arranged in the rare earths?

Answer 38

In layers