Download presentation

Presentation is loading. Please wait.

Published byLydia Knuckles Modified over 3 years ago

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 <

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

Similar presentations

OK

Kensuke Homma / Hiroshima Univ. from PHENIX collaboration

Kensuke Homma / Hiroshima Univ. from PHENIX collaboration

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on eddy current loss Ppt on diode family matters Ppt on sports day logo Ppt on national education day speech Ppt on db2 architecture Ppt on natural resources conservation Ppt on forward rate agreement calculator Ppt on quality education in india Marketing mix ppt on sony tv Ppt on success and failure articles