5.4 Magnetic Order Flashcards
What two conditons must be allowed in order for spontaneous magnetisation to occur?
- 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
Why can interacting spins between neighbours cause spontaneous magnetisation?
Due to the exchange interaction between the spins
Describe the pattern of the spins for a ferromagnet when T < T_c
All aligned with each other
- Thermal fluctuations are small compared to the alignment energy
- M is not 0
Describe the pattern of the spins for an antiferromagnet when T < T_N
Each neighbour has the opposite spin to another neighbour
- Get same orientation along diagonals
- Magnetic ordering
- M = 0
Describe the pattern of the spins for an ferrimanget when T < T_c
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
What condition determines if we get ferromagnetism
If the exchange constant in the Hamiltonian, J_ij > 0
- Energy is lower if the spins align
What condition determines if we get antiferromagnetism
If the exchange constant in the Hamiltonian, J_ij < 0
- Energy is lower if the spins anti-align
How does considering the “nearest neighbour” model allow for simplification?
Sum over i,j and then extend this for the whole material
What is an easy axis?
A direction in a crystal which the spins like to align
- Creates an asymmetic environment in the crystal
What is a hard axis?
A direction in a crystal which the spins don’t like to align
When do we get an easy/hard axis?
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
Describe the Ising model
A special case where Kappa is large enough that spins only align along the Z axis
- Very simple model
Why is the Ising model very simple, and what is it useful for?
There is only one degree of freedom
- Useful for looking at phase transitions from one magnetic state to another
What is Weiss’ core assumption for his ferromagnetism model?
That spins align due to a molecular field that is proportional to the magnetism M
What wil happen to the magnetic moments in a ferromagnet at absolute 0?
They will all align
What happens to the magnetic moments as the temperature increases from absolute 0?
Thermal fluctuations start to have an increasing effect and will eventaully start to destroy the order
In the spin 1/2 model, what are the solutions for M for low temperature?
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
In the spin 1/2 model, what are the solutions for M for high temperature?
Only M = 0. No ordering at high temperatures
In the spin 1/2 model, what are the solutions for M for Curie temperature?
Pass between two regimes. Have a similar sin(x) = x for small angles looking graph
- Valid in the paramagnetic (high temperature) state
When is the Curie Weiss Law valid?
It is true just above T_c and in the paramagnetic state
When do we use the spin J model?
When the spin is no longer 1/2
Is the Curie Weiss law obeyed in the spin J model?
Yes, with a T_c that include the effects of J
What is T_N?
The Neél temperature for antiferromagnetism
Above this, thermal effects dominate
Describe how the Neél model for antiferromagnetism works
Split the crystal into two sub lattices
- One with diagonal spins up added to one with the diagonal spins down at B = 0
How do the two sublattices interact for the Neél model?
Each sublattice experiences a molecular field proportional to the magnetism on the other sub lattice
What happens to the magnetism at the Neél temperature for each sublattice?
It goes to 0
How can we identify is a material is a magnet or not?
- Measure the magnetic susceptability in the paramagnetic state
- Fit it to χ ∝ 1 / (T - T_Weiss)
Is the Weiss temperature is equal to 0, what type of magnet do we have?
A paramagnet
Is the Weiss temperature is > 0, what type of magnet do we have?
Ferromagnet
What is the critical temperature for a ferromagnet if the Weiss temp > 0?
T_c = T_Weiss
Is the Weiss temperature is < 0, what type of magnet do we have?
Antiferromagnet
What is the critical temperature for an antiferromagnet if the Weiss temp < 0?
T_c = - T_N
What behaviour is observed at T = 0 and B is parallel to M?
Both sub lattices are saturated. The parallel components of both the magnetism and the magnetic susceptability are equal to 0
What behaviour is observed at T = 0 and B is perpendicular to M?
The transverse/perp components of both B and M are not equal to 0
- The spins are tilted on both sub lattices
What behaviour is observed at 0 < T < T_N and B is parallel to M_a?
M_a - sublattice with spins pointing upwards
- B enhances M_a, reduces M_b
- Strong effect on the susceptability
What behaviour is observed at 0 < T < T_N and B is perpendicular to M?
The transverse/perp components of both B and M are not equal to 0
- The spins are tilted on both sub lattices
What are M_a and M_b?
M_a is the sub lattice with spins up
M-B is the sub lattice with spins down
How are the crysal atoms arranged in the rare earths?
In layers
