Chapter 4 Phase Transitions. 4.1 Phenomenology Basic ideas. Partition function?!?! Thermodynamic limit Statistical Mechanics 1 Week 4

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1 Chapter 4 Phase Transitions 4.1 Phenomenology Basic ideas Partition function?!?! Thermodynamic limit 4211 Statistical Mechanics 1 Week 4

2 4.1.2 Phase diagrams p S S+L S+G L S+G L+G G G T p solid triple point liquid gas critical point T V P V T system 4211 Statistical Mechanics 2 Week 4

3 B up down critical point phase boundary T magnetic system 4211 Statistical Mechanics 3 Week 4

4 4.1.3 Symmetry Crystal Ferromagnet Ferroelectric Superfluid Translational symmetry Rotational (time reversal) symmetry Inversion symmetry Gauge symmetry 4211 Statistical Mechanics 4 Week 4

5 4.1.4 Order of phase transitions G phase 1 kink in G at transition actual state: minimum G phase 2 1/T Fig. 4.5 Variation of Gibbs free energy for two phases 4211 Statistical Mechanics 5 Week 4

6 4.1.5 The order parameter System Order parameter Ferromagnet magnetisation M vector Ferroelectric polarisation along displacement axis P Scalar!!! Fluid density difference (n n c ) real scalar Superfluid 4 He ground state wavefunction Ψ 0 complex scalar Superconductor pair wavefunction Ψ s complex scalar Ising Ising magnetisation m real scalar Binary alloy species concentration x real scalar 4211 Statistical Mechanics 6 Week 4

7 In terms of the order parameter..... Order parameter in first and second order transitions 4211 Statistical Mechanics 7 Week 4

8 4.1.6 Conserved and non-conserved order parameters System Order parameter Section Ferromagnet magnetisation M non-cons. 4.3 Ferroelectric polarisation P non-cons. 4.6 Fluid density (n n c ) conserved 4.2 difference Superfluid 4 He ground state Ψ 0 non-cons. wavefunction Superconductor pair Ψ s non-cons. wavefunction Ising Ising m non-cons. 4.4 magnetisation Binary alloy species concentration x conserved Statistical Mechanics 8 Week 4

9 4.1.7 Critical exponents In terms of the reduced temperature T t = they are defined (using the ferromagnet variables for example) through T c T heat capacity C ~ c t α order parameter M ~ susceptibility χ ~ t t β γ equation of state at T M ~ B There are two more critical exponents, which are connected with the spatial variation of fluctuations in the order as the critical point is approached c 1 δ 4211 Statistical Mechanics 9 Week 4

10 The spatial correlation function for the order parameter is written as 0 ~ p M r M r e r l ( ) ( ) where M (in the ferromagnetic case) is the magnetisation per unit volume. From this we obtain two more exponents, ν and η. These describe the divergence in the correlation length l and the power law decay p that remains at t = 0, when l has diverged. The exponents are defined through correlation length l ~ t power law decay at T p= d 2 + η c ν where d is the dimensionality of the system Statistical Mechanics 10 Week 4

11 4.1.8 Scaling theory ( 2 ) γ = ν η Fisher law α 2β γ 2 γ νd + + = Rushbrooke law = β( δ 1) Widom law = 2 α Josephson law The experimental verification of these results is strong evidence in favour of the scaling hypothesis. Thus it would appear that the correlation length is the only length of importance in the vicinity of the critical point. A consequence of the hypothesis is that only two critical exponents need be calculated for a specific system. Note the Josephson law is the only one to make explicit mention of the spatial dimensionality d Statistical Mechanics 11 Week 4

12 4.2 First order transition an example Coexistence p critical point liquid liquid - gas coexistence gas V Coexistence region, showing p-v isotherms for liquid gas system 4211 Statistical Mechanics 12 Week 4

13 F F 1 F 0 F phase 1 (liquid) F 2 phase 2 (gas) v 1 v 0 v 2 v volume per particle Helmholtz free energy curve 4211 Statistical Mechanics 13 Week 4

14 Thus if a fraction α 1 of the particles is in regions of specific volume v 1 and a fraction α 2 = 1 α 1 in regions of specific volume v 2 then so that the fractions are given by vα + vα = v v α v, α v = = v v2 v1 v2 v1 Then the free energy of the inhomogeneous system will be given by F = α F + α F vf 2 1 vf 1 2 F1 F2 v 0 v2 v1 v2 v1 = Statistical Mechanics 14 Week 4

15 F fraction α 2 fraction α 1 phase 1 (liquid) this would be the single-phase state this is a point of lower F made up from fraction α1 of phase 1 and fraction α = 1 - α of phase phase 2 (gas) v 1 v 0 v 2 v volume per particle Double tangent construction for phase coexistence 4211 Statistical Mechanics 15 Week 4

16 df = SdT pdv so that p F = V ; T equality of pressure Statistical Mechanics 16 Week 4

17 4.2.2 Van der Waals fluid p 2 an p+ 2 ( V Nb) = NkT V liquid critical point superheated liquid forbidden region supercooled gas gas V Isotherms of the van der Waals equation 4211 Statistical Mechanics 17 Week 4

18 4.2.3 The Maxwell construction F b = F a + V b V a F V T dv V b = F a pdv. V a Integral independent of path: V b pdv = pdv. V a so that V b V b V a pdv = 0, pdv = 0. V a V b V a 4211 Statistical Mechanics 18 Week 4

19 4.2.4 The critical point p 2 p = 0 and = 0 2 V T V. T a 8a Vc = 3 Nb, pc =, kt 2 c = 27b 27b. -- Elegant derivation of Stanley Statistical Mechanics 19 Week 4

20 4.2.5 Corresponding states v = V, π = p, t = T V p T c c c then the van der Waals equation takes on the universal form: 3 1 8t π + v 2 = v 3 3. Critical compressibility ratio z c = p c V c / NkT c is predicted to have the universal value 3/8 = for all liquid gas systems. 4 He Ne A Kr Xe N 2 O 2 CO CH 4 p c V c /NkT c Statistical Mechanics 20 Week 4

21 TT / c gas liquid Ne A Kr Xe N 2 O 2 CO CH c Liquid - gas coexistence 4211 Statistical Mechanics 21 Week 4

22 Guggenheim s formula ρ l ρ c ρ c = 7 4 ( 1 T T c ) ρ c ρ g ρ c = 7 4 ( 1 T T c ) 1/3 1/ ( 1 T T c ) 3 4 ( 1 T T c ) Order parameter critical exponent β = 1/3 From van der Waals equation (Maxwell construction) ρ l ρ c ρ c = 2 ( 1 T T c ) ρ c ρ g ρ c = 2 ( 1 T T c ) 1/2 1/ ( 1 T T c ) ( 1 T T c ) + Order parameter critical exponent β = 1/ Statistical Mechanics 21a Week 4

23 4.2.7 Quantum mechanical effects TT / c 1.00 gas liquid He 4 He Ne A Kr Xe N 2 O 2 CO CH Liquid - gas coexistence including data for helium c 4211 Statistical Mechanics 22 Week 4

24 2π Λ =! mkt. Here m is the mass of the particles. This shows how Λ increases when m is small. At the critical temperature we can write Λ as Λ c = 3 4! 6π mε where ε is the energy parameter of the Lennard Jones potential, since 16 ktc = ε. 27 Then the ratio Λ c /σ compares the thermal debroglie wavelength with the interparticle spacing; σ is the length parameter of the Lennard Jones potential. We tabulate values of this ratio: 3 He 4 He Ne A Kr Xe N 2 O 2 Λ c /σ Statistical Mechanics 23 Week 4

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