Landau Theory of Phase Transition

5 Key excerpts on "Landau Theory of Phase Transition"

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  Statistical and Thermal Physics

  With Computer Applications, Second Edition

  • Harvey Gould, Jan Tobochnik(Authors)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  9 ........................................... Critical Phenomena: Landau Theory and the Renormalization Group Method W e discuss a phenomenological mean-field theory of phase transitions due to Landau and introduce the ideas of universality and scaling. The breakdown of mean-field theory close to the critical point leads us to introduce the renormalization group method, which has had a major impact on our understanding of phase transitions and quantum field theory. We introduce the renormalization group method in the context of percolation, a simple geometrical model that exhibits a continuous transition, and then apply renormalization group methods to the Ising model. 9.1 Landau Theory of Phase Transitions The qualitative features of mean-field theory can be summarized by a phenomeno-logical expression for the Gibbs function due to Landau. Although we introduce the Landau theory in the context of the Ising model, it can be applied to a wide variety of phase transitions, ranging from superconductors to liquid crystals and to discontinuous as well as to continuous phase transitions. One of the assumptions of the Landau theory is that a phase transition can be char-acterized by an order parameter , which for the Ising model is the magnetization m . The magnetization is chosen as the order parameter because it is zero for T > T c , nonzero for T ≤ T c , and its fluctuations (the susceptibility) diverge more strongly than any other quantity near the transition. 420 • CRITICAL PHENOMENA 0.03 0.00 –0.03 –0.5 0.0 0.5 g m b > 0 b = 0 b < 0 Figure 9.1. The dependence of the Landau form of the Gibbs function density g on the order parameter m for b = − 1, 0, and 1 with c = 16. The minima of g for b = − 1 are at m = ± 0.25. Because m is small near the critical point, it is reasonable to assume that the Gibbs function per unit volume g can be written as g ( T , m ) = a ( T ) + b ( T ) 2 m 2 + c ( T ) 4 m 4 − Hm .

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  Basic Statistical Physics

  • Nandita Rudra, P Rudra;;;(Authors)
  • 2009(Publication Date)
  • WSPC

    (Publisher)

  12.4 Landau’s Theory of Continuous Phase Transitions Landau classified phase transitions according to the behaviour of the order parameter η shown in Figures 12.4.1 & 12.4.2. In the case of discontin-T T c 0 Disorder e d Phase Ordered Phase Fig. 12.4.1 Variation of order parameter η as a function of temperature in the case of a discontinuous phase transition. uous phase transition the order parameter η changes discontinuously at the transition temperature T c , while for a continuous phase transition η changes continuously and vanishes at T c . In the case of liquid-gas transition the density ρ − ρ gas definitely satisfies this criterion as does magnetization M in the case of magnetic transition. η = 0 , in the higher symmetry disordered phase , (12.4.1) η = 0 , in the lower symmetry ordered phase . (12.4.2) 164 BASIC STATISTICAL PHYSICS T T c 0 0 Disordered Phase Ordered Phase η η Fig. 12.4.2 Variation of order Parameter η as a function of temperature in the case of a continuous phase transition. Order parameters for superconducting and Cu-Zn like order-disorder tran-sitions described in Section 12.3 also have this characteristics. It should also be noted that in a system there may be more than one phase transition point at different temperature like the case of α − Fe 2 O 3 which is paramagnetic above 950K and two antiferromagnetic ordering, one below 250K and the other in the range 250K–950K. Landau’s analysis is based on analytic expansion of the Free Energy F(p,T, η ) in powers of η : F ( p, T, η ) = F 0 ( p, T ) + α ( p, T ) η + A ( p, T ) η 2 + C ( p, T ) η 3 + B ( p, T ) η 4 . (12.4.3) This formula is valid only when there is no external field. In Landau’s theory of second order phase transition the expansion co-efficients α (p,T), A(p,T), C(p,T) and B(p,T) are analytic functions of the arguments. Lan-dau’s theory crucially depends on the symmetry argument that the two phases with η = 0 and η = 0 have different symmetries.

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  Phases of Matter and their Transitions

  Concepts and Principles for Chemists, Physicists, Engineers, and Materials Scientists

  • Gijsbertus de With(Author)
  • 2023(Publication Date)
  • Wiley-VCH

    (Publisher)

  The liquid–gas transition also belongs to this class (Problem 15.3). We limit ourselves entirely to this class, and some experimental results are given in Table 15.2. 15.2 Mean-Field Theory: Landau Theory In Chapter 12, we introduced Landau theory as a general thermodynamic theory of phase transformations and transitions. In this theory, a “generalized” Helmholtz energy F(;T,V ), dependent on temperature T and volume V , is minimized with respect to the internal vari- able (order parameter) for which we use here the symbol . We noted that in general  varies 462 15 Continuous Phase Transitions: Liquids ↔ Gases Table 15.2 Experimental values of critical exponents for various fluids. Compound       CO 2 0.10 0.321 1.24 4.85 0.57 ∼0 Xe 0.11 0.329 1.23 4.74 — 0.05 N 2 — 0.327 ± 0.002 1.23 ± 0.01 — Ne — 0.327 ± 0.002 1.25 ± 0.01 — Source: Adapted from Hocken and Moldover [5]. with position and we should use the energy density. Although we can use the energy itself for a uniform distribution of  as employed here, we will still use the energy density. For the liquid–gas transition, the order parameter is conveniently taken as the density difference Δ between liquid and gas. In general, we use ℒ () representing the grand potential density (Section 7.1) and assume that the Helmholtz part f () of ℒ () near T cri can be expanded in a power series in : ℒ () = f () − h = a 0 + a 1  + 1 / 2 a 2  2 − 1 / 3 a 3  3 + 1 / 4 a 4  4 + … − h The driving force h is the conjugate variable to the order parameter , that is h = f ()/. For a homogeneous fluid with order parameter  = Δ, the driving force h is the chemical potential . For continuous transitions, equilibrium requires that a 1 is absent and that a 4 > 0 for the system to be stable. If the order parameter  can have only the two values ±, as for the magnetization and polarization cases, the third-order terms must also be absent because ℒ () should equal ℒ (−).

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  Phase Transitions In Complex Fluids

  • Antonio Martins Figueiredo Neto, Pierre Toledano(Authors)
  • 1998(Publication Date)
  • World Scientific

    (Publisher)

  2 Landau theory for homogeneous systems 2.1 Magnetic materials. For this kind of materials there exists a temperature T c , known as Curie's temperature, such that: a) for T T c , M — 0 for H = 0 (they are in the paramegnetic state). The experimental data show, furthermore, that the magnetization M is a continuous function of the temperature. This phase transition can be easily described by means of a phenomeno-logical theory proposed long ago by Landau. The main ideas of this theory are the following. Let be F the thermodynamical potential, per unit volume, whose minimum corresponds to the stable state of the magnetic material under consideration. In the absence of external fields, F is expected to be a function of M, T and eventually of the pressure p: F = F(M : T,p). By taking into account that M is a continuous function of T and that M(T > T c ) = 0, for T near T c , M is very small (T < T c ). In this region F(M,T,p) can be approximated by means of a fourth order polynome of the kind F(M,T,p) = F(0,T,p)+R(T,p)M+±A(T,p)M 2 + S{T, P )M 3 + B{T, P )M (1) where and F(0,T,p) is the thermodynamical potential per unit volume, of the para-magnetic material. In the following F(Q,T,p) will be indicated by F 0 . Since in the bulk the states M and -M are equivalent, F(M,T,p) = F(-M,T,p). F(M,T,p) = F(0,T,p)+R(T,p)M+^A(T,p)M 2 + l -S(T,p)M* + B{T, P )M *■■--(£)«■ ^-(SL •*-K£L.-™-i(SL.. (2) (i) is(T,p)M 3 + iB(r,p)M 4 , 321 Consequently in (1) S(T,p) = R(T,p) = 0. Hence, for T « T c , the thermody-namical potential F for a magnetic material may be written as: F{M,T,p) = F 0 + ^A(T,p)M 2 + ^B{T,p)M 4 . (3) The thermodynamical coordinate M, corresponding to the stable state, minimizes F(M,T,p) given by (3). Simple calculations give dF — =A(T,p)M + B(T,p)M 3 = 0, (4) whose solutions are J f t -0 a n d M | = -| ^ .

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  Light Scattering Near Phase Transitions

  • H.Z. Cummins, A.P. Levanyuk(Authors)
  • 2012(Publication Date)
  • North Holland

    (Publisher)

  Along with the phenomenological approach accepted in the chapter, the quasi-microscopic approach which brings into use the renormalization group (71) so that 1/2 164 A.P. Levanyuk et al. has been actively developed. The quasi-microscopic approach has to do with the immediate vicinity of the phase stability loss points where thermal fluctuations of the order parameter become rather large, so that the Landau theory proves to be not applicable (this region of temperatures is usually called a critical or scaling region). As to systems containing defects this approach appears to be helpful only with additional assumptions of the weakness (extremely small value of η 0 ) of randomly distributed defects whose concentration satisfies the condition Nrl$>l. Using sophisticated methods of modern theory of phase transitions one has succeeded in an approximate calculation of the asympthotic temperature dependence of various physical quantities of a system with the defects within the scaling region (see, e.g., Ma (1976), Patashinsky and Pokrovsky (1982)). Such dependences are described by the so-called critical indices (critical exponents) which depend on the dimensionality of the system and its symmetry properties. Note, however, that for the majority of structural phase transitions the scaling region represents a very close vicinity of T c which has not been reliably observed in experiments up to now. For this reason one cannot speak of a comparison of the scaling-theory results with experimental data available. Owing to these circumstances we have not considered methods and results of the approach in the given chapter. There exists one more branch of the theory whose results are valid within the range of applicability of either the classical Landau Theory of Phase Transitions or the generalized Landau theory (Ginzburg and Sobyanin 1976) if a critical region is under consideration.

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