1. Introduction Landau Theory Many phase transitions exhibit similar behaviors: critical temperature, order parameter… Can one find a rather simple unifying theory that gives a general phenomenological overview of phase transitions ? Molecular field (Weiss ~1925): solve the Schrödinger equation for a one particle system but with an effective interaction potential : Several approaches : Microscopic model (Ising 1924): solve the Schrödinger equation for pseudo spins on a lattice with effective interaction Hamiltonian restricted to first neighbors

2. Introduction Landau Theory – Express a thermo dynamical potential as a function of the order parameter ( ), its conjugated external field (h) and temperature. Landau Theory : – Keep close to a stable state minimum of energy power series expansion, eg. like: – Find and discuss minima of versus temperature and external field. – Look at thermodynamics properties (latent heat, specific heat, susceptibility, etc.) in order to classify phase transitions

3. Broken symmetry Landau Theory a simple 1D mechanical illustration : d 0 x l let go with d > l o : equilibrium position (minimum energy) x = 0

4. Broken symmetry Landau Theory a simple 1D mechanical illustration : d 0 x l let go with d < l o : equilibrium position (minimum energy) x = x o 0 Order parameter @ critical value d c = l o spontaneous symmetry breaking dcdc d xoxo Only irreversible microscopic events will make the system settle at +xo or –xo when the system slowly exchanges energy with external world

5. Broken symmetry Landau Theory a simple 1D mechanical illustration : d 0 x l Taylor expansion of potential (elastic) energy

6. Broken symmetry Landau Theory a simple 1D mechanical illustration : d 0 x l Taylor expansion of potential (elastic) energy Change sign at d=d c !!! Does not change sign

7. h=0 Second Order Phase Transitions Landau Theory T >>T c T <<T c T =T c =0 stable above T c, unstable below T c

8. Second Order Phase Transitions Landau Theory Stationary solution : T T c T TcTc o

9. Second Order Phase Transitions Landau Theory Free energy : T T c T TcTc ( o ) - o Entropy : T TcTc S( o ) - S o No Latent Heat: T c S = 0

10. Second Order Phase Transitions Landau Theory Specific heat : T T c T TcTc c p - c o

11. Second Order Phase Transitions Landau Theory Susceptibility : T T c Curie law T TcTc

12. h Second Order Phase Transitions Landau Theory field hysteresis : T T c A 0 T T c A 0 h

13. Second Order Phase Transitions SUMMARY Landau Theory One critical temperature T c No discontinuity of,, S (no latent heat) at T c Jump of C p at T c c Divergence of and at T c Field hysteresis One critical temperature T c No discontinuity of,, S (no latent heat) at T c Jump of C p at T c c Divergence of and at T c Field hysteresis

14. First Order Phase Transitions: Landau Theory T > T 1 : o =0 stable T 1 > T > T o : o =0 stable o 0 metastable T o > T > T c : o =0 metastable o 0 stable T c > T : o 0 stable

15. T equ. ToTo TcTc T1T1 First Order Phase Transitions: Landau Theory T > T 1 : o =0 stable T 1 > T > T o : o =0 stable o 0 metastable T o > T > T c : o =0 metastable o 0 stable T c > T : o 0 stable Thermal hysteresis

16. First Order Phase Transitions: Landau Theory Steady state : T T c +

17. First Order Phase Transitions: Landau Theory Steady state : T = T o

18. First Order Phase Transitions: Landau Theory Entropy : T = T o A and 2 depend on T ! = 0

19. First Order Phase Transitions: Landau Theory Specific heat : T T 1 = 0 cpcp T1T1 coco

20. First Order Phase Transitions: Landau Theory Susceptibility : o = stable until T down to T o ToTo TcTc T1T1

21. First Order Phase Transitions SUMMARY Landau Theory Existence of metastable phases Temperature domain (T c T 1 ) for coexistence of high and low temperature phases at T o (T c < T o < T 1 ) both high and low teperature phases are stable Temperature hysteresis Discontinuity of,, S (latent heat), C p, at T c Existence of metastable phases Temperature domain (T c T 1 ) for coexistence of high and low temperature phases at T o (T c < T o < T 1 ) both high and low teperature phases are stable Temperature hysteresis Discontinuity of,, S (latent heat), C p, at T c

22. Tricritical point Landau Theory In the formalism of first order phase transitions, it can happen that B parameter changes sign under the effect of an external field. Then there is a point, which is called tricritical point, where B=0. The Landau expansion then takes the following form: Equilibrium conditions :

23. Landau Theory Potential : Tricritical point T>Tc: =0 T>Tc: 0 T TcTc

24. Landau Theory Entropy : A and 2 depend on T ! Tricritical point T>Tc: =0 T>Tc: 0 T TcTc S

25. Landau Theory Specific heat : Tricritical point T>Tc: =0 T>Tc: 0 T TcTc C p

26. Landau Theory Susceptibility : Tricritical point T>Tc: =0 T>Tc: 0 T TcTc T TcTc