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MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.8.334: Statistical Mechanics II Spring 2008 Test 1 Review Problems The test is ‘closed b ook,’ but if you wish you may bring a one-sided sheet of formulas. The intent o f this sheet is as a reminder of important formulas and definitions, and not as a compact transcription of the answers provided here. If this privilege is abused, it will be revoked for future tests. The test will be composed entirely from a subset of the following problems. Thus if you are familiar and comfortable with these problems, there will be no surprises! ******** 1. The binary alloy: A binary alloy (as in β brass) consists of NA atoms of type A, and NB atoms of type B. The a toms form a simple cubic lattice, each interacting only with its six nearest neighbors. Assume an attractive energy o f −J (J > 0) between like neighbors A − A and B − B, but a repulsive energy of +J for an A − B pair. (a) What is the minimum energy configuration, or the state of the system at zero temper-ature? (b) Estimat e the total interaction energy assuming that the atoms are randomly distributed among the N sites; i.e. each site is occupied independently with probabilities pA = NA/N and pB = NB/N. (c) Estimate the mixing entropy of the alloy with the same approximation. Assume NA, NB ≫ 1. (d) Using the above, obta in a free energy function F (x), where x = (NA−NB)/N. Expand F (x) to the fourth order i n x, and show that t he requirement of convexity of F breaks down below a critical temperature Tc. For the remainder of this problem use the expansion obtained in (d) in place of the full function F(x). (e) Sketch F(x) for T > Tc, T = Tc, and T < Tc. For T < Tc there is a range of compositions x < |xsp(T )| where F (x) is not convex and hence the composition is locally unstable. Find xsp(T ). (f) The alloy globally minimizes its free energy by separating into A rich and B rich phases of comp ositions ±xeq(T ), where xeq(T ) minimizes the function F (x). Find xeq(T ). (g) In the (T, x) plane sketch the phase separation boundary ±xeq(T ); and the so cal led spinodal line ±xsp(T ). (The spinodal li ne indicates onset of metastability and hysteresis effects.) ******** 1X X P P       2. The Ising model of magneti sm: The local environment of an electron in a crystal sometimes forces its spin to stay parallel or anti-parallel to a given lattice direction. As a model of magnetism in such materials we denote the direction of the spin by a single variable σi = ±1 (an Ising spin). The energy of a configuration {σi} of spins is then given by N 1 H = Jijσiσj − h σi ;2 i,j=1 i where h is a n external magnetic field, and Jij is the interaction energy between spins at sites i and j. (a) For N spins we make the drastic approximation that the interaction between all spins is the same, and Jij = −J/N (the eq uivalent neighbor model). Show that the energy can PNnow be written as E(M, h) = −N[Jm2/2 + hm], wit h a magnetization m = i=1 σi/N = M/N. (b) Show that the partition function Z(h, T ) = {σi}exp(−βH) can be re-written as Z = M exp[−βF (m, h)]; with F (m, h) easily cal culated by analogy to problem (1). For the remainder of the problem work only with F (m, h) expanded to 4th order in m. (c) By saddle point integration show that the actual free energy F (h, T ) = −kT ln Z(h, T ) is given by F (h, T ) = min[F (m, h)]m. When is the saddle point method valid? Note that F (m, h) is an analytic function but not convex for T < Tc, while the true free energy F (h, T ) is convex but becomes non-analytic due to the minimization. (d) For h = 0 find the critical temperature Tc below which spontaneous magnetization appears; and calculate the magnetization m(T ) in the low temperature phase. (e) Calculate the singular (non-analytic) behavior of the response functions ∂E  ∂m C = , and χ = . ∂T h=0 ∂h h=0 ******** 3. The lattice–gas model: Consider a gas of particles subject to a Hamiltonian N 2 XX p�i 1 H =+ V(�ri − �rj), in a volume V. 2m 2 i=1 i,j (a) Show that the grand partition function Ξ can be wri tten as   ∞  N Z N X 1 eβµ Y β X Ξ = d3�ri exp −V(�ri − �rj) . N! λ3 2 N=0 i=1 i,j 2R P (b) The volume V is now subdivided into N = V/a3 cells of volume a3, with the spacing a chosen small enough so that each cell α is either empty or occupied by one particle; i.e. the cell occupation number nα is restricted to 0 or 1 (α = 1, 2, · · · , N ). After approxima ting 3 PNthe int egra ls d3�r by sums aα=1, show that    nα N X eβµa3 β Xα Ξ ≈ exp − nαnβV(�rα − �rβ) . λ3 2 {nα=0,1} α,β=1 (c) By setting nα = (1 + σα)/2 and approximating the potential by V(�rα − �rβ) = −J/N , show that t his model is identical to the one studied in problem (2). What does this imply about the behavior of this imperfect gas? ******** 4. Surfactant condensation: N surfactant molecules are added to the surface of water over an area A. They are subject to a Hamiltonian N H = X p�i 2 +1 X V(�ri − �rj),2m 2 i=1 i,j where �ri and p�i are two dimensional vectors indicating the po sition and momentum of particle i. (a) Write down the expression for the parti tion function Z(N, T, A) in terms of integrals over �ri and p�i, and perform the integrals over the momenta. The inter–particle potenti a l V(�r) is infinite for separati o ns |�r | < a, and attractive for R ∞|�r | > a such that a 2πrdrV(r) = −u0. (b) Estimate the total non–excluded area avai lable in the positional phase space of the system of N particles. (c) Estimate the total potential energy of the


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