Ph 12c. Homework Assignment No. 7 Due: Thursday, 30 May 2013, 5 pm

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1 1 Ph 12c Homework Assignment No. 7 Due: Thursday, 30 May 2013, 5 pm 1. Scaling hypothesis from Landau theory In the Landau theory of second-order phase transitions, the Helmholtz free energy close to the critical point can be expressed as a function of the temperature τ and order parameter ξ in the form where A and B are positive constants and F(ɛ, ξ) = 1 2 Aɛξ Bξ4, (1) ɛ = τ τ c τ c (2) denotes the dimensionless deviation of the temperature from the critical temperature τ c. If λ is an applied external field that couples to the order parameter (for example, the applied magnetic field H), then we can perform a Legendre transform to obtain the Gibbs free energy G(ɛ, λ), a function of the temperature and external field: G(ɛ, λ) = [F(ɛ, ξ) λξ] stat wrt ξ. (3) That is, the Gibbs free energy is obtained by evaluating the expression in square brackets at the value of ξ where this expression is stationary with respect to ξ. We say that the Gibbs free energy G(ɛ, λ) satisfies the scaling hypothesis if, when ɛ and λ are small, G(Ω p ɛ, Ω q λ) = Ω G(ɛ, λ) ; (4) here Ω is a real scaling variable, and p, q are (not necessarily integer) exponents that encode the scaling properties of the system near the critical point. Show that the scaling hypothesis is satisfied in Landau theory for particular values of p and q, and find these values. (To perform the Legendre transform explicitly, you would need to solve a cubic equation. However, it is not necessary to perform the Legendre

2 2 transform explicitly to solve this problem. It is useful to notice that the formula for G(ɛ, λ) in eq.(3) would be unchanged if we rescaled ξ by a constant inside the square brackets, since we eliminate ξ anyway when we evaluate the quantity in square brackets at its stationary point.) 2. Critical exponents from the scaling hypothesis In many cases of interest, Landau theory fails, but the scaling hypothesis still holds with different values of p and q than the values predicted by Landau theory. Then various other critical exponents can be expressed in terms of p and q. For example, consider the isothermal susceptibility ( ) ξ χ τ =, (5) λ where the order parameter ξ is ξ = τ λ τ. (6) The critical exponent γ is defined by the behavior of χ τ for λ = 0 and ɛ close to zero: χ τ ɛ γ. (7) From χ τ = ( 2 G/ λ 2) τ and the scaling hypothesis, we find Ω 2q χ τ (Ω p ɛ) = Ω χ τ (ɛ). (8) Now we may choose Ω so that as ɛ approaches zero, Ω p ɛ is held fixed (hence Ω ɛ 1/p ); then We conclude that χ τ Ω 2q 1 ɛ (2q 1)/p. (9) γ = 2q 1 p. (10) Other critical exponents can be derived by similar reasoning. a) The exponent β describes how the order parameter ξ turns on for τ slightly less than τ c and λ = 0: Express β in terms of p and q. ξ ( ɛ) β. (11)

3 3 b) The exponent δ describes the relation between the order parameter ξ and the external field λ on the critical isotherm ɛ = 0: Express δ in terms of p and q. λ ξ δ. (12) c) The exponent α describes how the heat capacity C λ = τ( σ/ τ) λ diverges as ɛ approaches zero for λ = 0: Express α in terms of p and q. C λ ɛ α. (13) d) Check your answers for parts (a),(b),(c), and Problem 1 by verifying the Landau theory exponents: α = 0, β = 1/2, γ = 1, δ = 3. Because we have computed α, β, γ, δ in terms of p and q, we obtain two relations among the exponents. e) Express α in terms of β and δ. (This is called the Griffiths relation.) f) Express γ in terms of α and β. (This is called the Rushbrooke relation.) 3. Equation of state from the scaling hypothesis a) The equation of state is a relation among the variables ξ, λ and τ, where ξ is the order parameter ξ =. (14) λ The purpose of this problem is to show that the scaling hypothesis requires the equation of state to have a special form in the vicinity of the critical point ɛ = λ = 0. Using the scaling hypothesis and assuming ɛ > 0, show ξ as a function of ɛ and λ satisfies the relation ξ(ɛ, λ) ( = f λ/ɛ b), (15) ɛ a for some function f, and express the exponents a and b in terms of β and δ. (You are not being asked to determine the function f) Thus, if ξ ξ/ɛ a is plotted as a function of λ λ/ɛ b, data for different positive values of ɛ lie on the same curve. This prediction is well confirmed in various experimental studies of second-order phase transitions. τ

4 4 b) When an external field λ is applied, the field and order parameter are related by λ = F(ɛ, ξ). (16) ξ Using the Landau theory expression for F(ɛ, ξ) from Problem 1, show that the order parameter λ as a function of ɛ and ξ satisfies the relation λ(ɛ, ξ) ɛ b = h(ξ/ɛ a ); (17) find the values of a and b, and find the function h in terms of the parameters A and B in F(ɛ, ξ). As a check, compare your result with the values of a and b found in part (a), using the Landau theory exponents q and p found in Problem Landau theory of a tricritical point In the Landau theory of a ferromagnetic phase transition, the Helmholtz free energy F respects the symmetry m m, and can be expanded as a power series in the magnetization m: F(m, r, u) = rm 2 + um 4 + vm , (18) where we neglect the terms higher order in m. The Gibbs free energy is a function of r, u, and the external magnetic field h, defined as G(h, r, u) = [F(m, r, u) hm] stat wrt m. (19) That is, the Gibbs free energy is obtained by evaluating the expression in square brackets at the value of m where this expression is stationary with respect to m. A phase transition where h = r = u = 0 and v > 0 is called a tricritical point. The purpose of this problem is to calculate some of the critical exponents predicted by Landau theory for a tricritical point. a) The Gibbs free energy can be expressed in the form ( h G(h, r, u) = r a G u ) r b, r c, (20) for some function G. Find a, b, and c.

5 5 b) In terms of the Gibbs free energy G, the magnetization is m = For u = 0, the magnetization has the form for some function m. Find d and e.. (21) h r,u m = r d m(h/r e ) (22) c) For r = u = 0, the magnetization m scales as m h g. Assume that the function m(x) scales as m(x) x f as x, and find f and g by requiring that m(h, r, u) has a smooth limit (i.e., approaches a nonzero constant) as r 0 with u = 0 and h 0. d) Near the tricritical point the magnetization becomes very sensitive to small changes in h and u: ( ) m h Find j and k. u 1 h=u=0 r j ; ( ) m u h 1 h=u=0 r k. (23)

3. General properties of phase transitions and the Landau theory

3. General properties of phase transitions and the Landau theory In this Section we review the general properties and the terminology used to characterise phase transitions, which you will have already