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Introduction

Landau Theory. Several approaches :. Molecular field (Weiss ~1925): solve the Schr ö dinger equation for a one particle system but with an effective interaction potential :. Introduction. Many phase transitions exhibit similar behaviors: critical temperature, order parameter….

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Introduction

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  1. Landau Theory • Several approaches : Molecular field (Weiss ~1925): solve the Schrödinger equation for a one particle system but with an effective interaction potential : • Introduction • 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 ? Microscopic model (Ising 1924): solve the Schrödinger equation for “pseudo spins” on a lattice with effective interaction Hamiltonian restricted to first neighbors

  2. Landau Theory • Introduction • Landau Theory : • Express a thermo dynamical potential as a function of the order parameter (), its conjugated external field (h) and temperature. • 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. Landau Theory l d x 0 • Broken symmetry • a simple 1D mechanical illustration : • let go with d > lo: equilibrium position (minimum energy)  x = 0

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

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

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

  7. Landau Theory h=0   • Second Order Phase Transitions T >>Tc T =Tc T <<Tc =0 stable above Tc , unstable below Tc

  8. Landau Theory T  Tc T  Tc o T Tc • Second Order Phase Transitions • Stationary solution :

  9. Landau Theory (o) -  o T S(o) - S o Tc Tc T • Entropy : • Second Order Phase Transitions T  Tc • Free energy : T  Tc No Latent Heat: TcS = 0

  10. Landau Theory cp - co T Tc • Second Order Phase Transitions T  Tc • Specific heat : T  Tc

  11. Landau Theory -1 T  Tc T  Tc T Tc • Second Order Phase Transitions • Susceptibility : Curie law

  12. Landau Theory T  Tc A  0 h   T  Tc A 0 h • Second Order Phase Transitions • field hysteresis :

  13. Landau Theory • Second Order Phase Transitions SUMMARY • One critical temperature Tc • No discontinuity of , , S (no latent heat) at Tc • Jump of Cp at Tc • Divergence of  and  at Tc • Field hysteresis

  14. Landau Theory • First Order Phase Transitions: T > T1 : o=0 stable T1 > T > To: o=0 stable o0 metastable To > T > Tc: o=0 metastable o0 stable Tc > T : o 0 stable

  15. Landau Theory equ. T > T1 : o=0 stable T1 > T > To: o=0 stable o0 metastable To > T > Tc: o=0 metastable o0 stable T Tc To T1 Tc > T : o 0 stable • First Order Phase Transitions: Thermal hysteresis

  16. Landau Theory T  Tc + T  Tc • First Order Phase Transitions: • Steady state :

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

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

  19. Landau Theory = 0 cp co T1 • First Order Phase Transitions: • Specific heat : T  T1

  20. Landau Theory o= stable until T down to To -1 Tc To T1 • First Order Phase Transitions: • Susceptibility :

  21. Landau Theory • First Order Phase Transitions SUMMARY • Existence of metastable phases • Temperature domain (Tc T1) for coexistence of high and low temperature phases • at To (Tc< To < T1) both high and low teperature phases are stable • Temperature hysteresis • Discontinuity of , , S (latent heat), Cp,  at Tc

  22. Landau Theory • Tricritical point 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  Tc T • Tricritical point • Potential : T>Tc:=0 T>Tc:0

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

  25. Landau Theory Cp Tc T • Tricritical point • Specific heat : T>Tc:=0 T>Tc:0

  26. Landau Theory -1 T Tc  T Tc • Tricritical point • Susceptibility : T>Tc:=0 T>Tc:0

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