1.2 Landau Theory I Flashcards
What does Landau theory study?
The phase transitions of systems in thermodynamic equilibrium
What is the first postulate of Landau theory?
F(M) is the free energy that captures the state of the system
- M is the net magnetisation
What is the second postulate of Landau theory?
The system has time to reach equilibrium which is given by a minimum in F(M)
F’(M) = 0, F’‘(M) > 0
What do we assume about the form of F(M)?
It is a single law power expansion
F(M) = F(0) + F(1) M + F(2) M^2 etc.
Note the bracketted indices should be subscript
What are the three conditions we exert on F(M)?
- Insist M = 0 is an extremum, (F(1) is also 0)
- Insist on symmetry with M (equally likely to be +ve or -ve, so no odd powers of M F(1,3) = 0)
- Don’t care about F(0)
What happens if M=0 is not at an extremum?
We can translate it to an extremum where M’ = M + M(0)
- Translational invariance
After our three assumptions, what is the free energy of the system, and its derivative equal to, F(M), F’(M)?
F(M) = F(2) M^2 + F(4) M^4 F' = 2MF(2) + 4M^3 F(4)
How do we determine whether the free energy derivative is at a maximum or a minimum?
Depends on the paramaters F_k = F_k (T) where T is the only relevant parameter and close to some constant T_C
What do we determine the form for F(2) and F(4) is?
F(2) = α (T - T_C)
F(4) = β
Here alpha and beta are both positive constants
Describe the M-T graph for the solutions of M
Pitchfork bifurcation at T=Tc where you have a parabola shape for the solution M = +-sqrt ( α (T_C- T) /2β) and then a constant M = 0 solution which is a max pre T_C and a min post T_C
Describe the two F-T graphs we can sketch for different T when T > T_C and T < T_C
T > T_C: System is in a minimum state
General parabola centered about M = 0
T < T_C: System is in a maximum state:
Quartic cenetered with max at M = 0 then two minima at +- M(0)
