s i which can assume values s i = ±1. A 1-spin Ising model would have s i = 1, 0, 1, etc. We now restrict ourselves to the spin-1/2 model. Then, if there is also a magnetic field that couples to each spin, and assuming only nearest-neighbour interactions, we have the Hamiltonian H ({s k }) = J nn s i s j µ B H i s i. (3.51) nn stands for nearest neighbours. The number of nearest neighbours is called coordination number, z. In a two-dimensional square lattice z = 4. The coupling constant between spins is J, which here is assumed to be constant (i.e. not to depend on the location of the spins). In turn, we may have J > 0 or < 0. In Fig. 3.12 a possible configuration for a spin-1/2 Ising model on a square lattice is represented. Many variations of this model are possible. For example, the coupling constant J can be site-dependent. The Heisenberg model introduces a sophistication by way of a vector spin vector: H ({s k }) = J ij s i s j µ B H s i. (3.52) s i are freely-rotating vectors in the 3D space. One variation of this is the XY model, which is Heisenberg s model on the xy plane. The Gaussian, spherical, Potts and percolation models are also frequently used. The spin-1/2 Ising model with J > 0 is a toy model for i Figure 3.12: A possible spin conguration for an Ising model on a square lattice (in two dimensions). the ferromagnetic transition. The ferromagnetic transition occurs in ferromagnets, and is associated with spontaneous (i.e. at zero magnetic field) ferromagnetic order: above the critical temperature T c the magnetisation of the material is zero; below the critical temperature a spontaneous magnetisation arises: 0, T < T c, m(t) = 0, T T c. (3.53) 68
The transition is of second order at zero field, but it can also be reached at T < T c by varying the magnetic field from H < 0 to H > 0, and in this case it is of first order. Fig. 3.13 shows a schematic phase diagram in the T m and H T plane; note that the phase diagram is similar to that of the liquid-vapour phase transition, with H playing the role of pressure and the magnetisation per spin, m = M/N, m = N s i N = s i, (3.54) playing the role of density. The response function for a magnetic system is the magnetic susceptibility at zero field: ξ = m. (3.55) H H=0 As corresponds to a second-order transition, ξ [a second derivative of the free energy G(H, T)] diverges as T T c + at zero field. First derivatives, for example, the magnetisation M = ( G/ H) T, are finite and continuous at H = 0. Another quantity of interest is the correlation function, the analogue of the total correlation function h(r) for a fluid: Γ ij = Γ i j = s i s j s i s j = s i s j m 2. (3.56) This function remain short-ranged at H = 0 (it is an exponentially-decaying function of the separation i j ) except at T = T c, where it becomes a long-ranged function. It is easy to demonstrate that ξ = β N ( s i s j s i s j ) = β ij j ( s1 s j m 2) (3.57) [again this is analogous to the relation between κ T and the integral of h(r) for a fluid]. Figure 3.13: (Colour on the Web). Schematic phase diagrams in the m T and H T plane of the spin-1/2 Ising model. The shaded area represents the region of twophase coexistence. 69
3.4.1 Mean-field approximation for the 2D-Ising model: Weiss theory The Ising model in 2D is a cornerstone in the history of statistical mechanics as it is exactly solvable and contains a non-trivial phase transition (the ferromagnetic transition). This was shown by Onsager. The derivation of the exact solution is a bit cumbersome though. Therefore, we will only mention the exact solution without explicitely obtaining it. Here we resort to the mean-field approximation, which comes in various disguises. One of them is the Weiss or effective-field theory. The Weiss approximation consists of assuming that each spin (for instance the i-th spin) interacts not with the neighbouring spins in each system s conguration, {s j }, but with their averaged values, s j = m (Fig. 3.14). The energy of the i-th spin, ǫ i, is therefore written as ǫ i = J j i s i s j µ B Hs i = (Jzm + µ B H)s i µ B H eff s i. (3.58) The total energy is E = NJzm 2 /2 Nµ B Hm (the factor 1/2 accounts for the double counting of pair interactions). The underlying assumption is that the system is ideal (i.e. spins are non-interacting): every spin, with average magnetisation m, interacts with an external effective eld given by µ B H eff = Jzm + µ B H, equal to the real external field H plus an internal averaged field due to the neighbouring spins. But it is not an ideal approximation completely, since the effective field depends in turn on the average value of the spin, m: it is a self-consistent approximation, which expresses itself by means of the trascendental equation in m that we are about to derive now. Since spins are in principle ideal, the average value of s i can be obtained using its associated reduced Boltzmann factor exp (βǫ i ): m = s i e βǫ i s i =±1 e βǫ i s i =±1 = eβ(jzm+µ BH) e β(jzm+µ BH) e β(jzm+µ BH) + e β(jzm+µ BH) = tanh [β(jzm + µ BH)]. (3.59) This is a trascendental (self-consistent) equation in m [compare with Eqn. (1.83) where spin coupling was neglected]. Note that this equation does not explicitely depend on the dimensionality of the system but through the coordination number z. This is a typical feature of the mean-field approximation. 3.4.2 Mean-field approximation for the 2D-Ising model: Weiss theory Eqn. (3.59) has to be solved numerically. We consider the case H = 0. A simple means of obtaining the solution is based on inspection of Fig. 3.15. In the figure the straight line f(m) = m and the curve f(m) = tanh(βjzm) are plotted; the solution of Eqn. (3.59) is the point(s) where the two functions cross. It is immediately evident that, depending on the value of the temperature, two types of solution arise: 70
Figure 3.14: (Colour on the Web). A fluctuating spin s 0 embedded in a medium of spins with average value of their spin m. If the slope of the function f(m) = tanh (βjzm) at the origin is less than unity, then only one solution, m = 0, exists: The stable phase is the disordered phase (zero magnetisation). If the slope of the function f(m) = tanh (βjzm) at the origin is larger than unity, then two solutions, m = ±m 0 0, exist, corresponding to a degenerate ordered phase. The slope is: ( ) d Jz [1 dm [tanh (βjzm)] m=0 = tanh 2 (βjzm) ] kt = Jz m=0 kt 1. (3.60) The critical temperature T c, separating the two regimes of solutions, is therefore: kt c J = z. (3.61) As the number of neighbours increases, the critical temperature increases; this was to be expected, since more neighbours means stronger interactions and increasing ability of the system to counteract the disordering effect of temperature via fluctuations. For example, in the triangular lattice (2D), we have z = 6, and T c will be larger than that for the square lattice, with z = 4. For H 0 the solution of Eqn. (3.59) is as follows. We can write { [ ( f(m) = tanh [β(jzm + µ B H)H] = tanh βjz m µ BH Jz )]}, (3.62) so that the hyperbolic-tangent function in Fig. 3.15 is displaced to the left if H > 0 or to the right if H < 0, giving one solution with m > 0 and possibly two solutions with m < 0 in the first case and one solution with m < 0 and possibly two solutions with m > 0 in the second. This implies the equation of state H = H(m, T) shown in Fig. 3.16, which can in fact be obtained using Eqn. (3.59) and solving for H: βµ B H = arctanh m βjzm. (3.63) 71
Three temperatures are shown. The equation of state for T > T c is similar to that of a paramagnetic material. For T < T c a loop appears; this is similar to the van der Wals loop in the case of fluids. Here again we can interpret that the true phase transition proceeding at H = 0 between two states with magnetisation ±m(t) (T < T c ) is in fact given by the meanfield theory but a Maxwell construction must be applied (equality of free energies G 1 = G 2 in the two phases). In this case, due to the symmetry of the function H = H(m), the construction leads immediately to H = 0 for the location of the phase transition. There is a region of unstable states with H/ m < 0 (which implies a negative magnetic susceptibility, i.e. mechanical instability) and two regions of mechanically stable but thermodynamically metastable states. Figure 3.15: Graphical solution of Eqn. (3.59): points where the straight line f(m) = m (dashed line) and the curve f(m) = tanh (βjzm) (continuous lines) cross for three temperatures: kt/jz = 1/2 (T < T c, with solution at ±m), 1 (T = T c also with solution at m = 0) and 2 (T > T c, also with solution at m = 0). Solutions are indicated with filled circles. average value of their spin m. Based on all these considerations, we can represent the phase diagram in the T m plane, shown in Fig. 3.17. The dashed line is the line of spinodal points (where the magnetic susceptibility becomes infinite). The region inside the coexistence curve is the two-phase region, where the two phases coexist at composition given by the analogue of the lever rule. In this region we can distinguish between the region of spinodal decomposition inside the spinodal line, and the two regions of nucleation, between the coexistence and 72
spinodal lines (analogous regions exist for the liquid-vapour transition). These words refer to the different kinetic mechanisms by which the phase transition proceeds, a theme not covered by our equilibrium mean-field theory. In the nucleation region domains of the material with reversed magnetisation appear; they have to overcome an activation barrier but, once overcome, the domains grow without limit until all the material has reversed the magnetisation. In the spinodal-decomposition region there is no activation for the formation of domains and there is a fluctuation in the whole system, not just in local areas, that drives the transition. The prediction of meanfield theory is that there is a Figure 3.16: Equation of state H = H(m, T) for the spin1/2 Ising model, in meanfield approx- imation, for three temperatures: kt/jz = 2.2 (T < T c ), 1 (T = T c ) and 6 (T > T c ). phase transition at any dimensionality. This is wrong: there is no such transition in a 1D ferromagnet. To see this using a simple argument, let us consider the ground state (T = 0) of a row of spins (all pointing in one direction, say up ). The energy of this state is E 0 = NJ. We now consider a simple fluctuation from the ground state where half of the spins have flipped from up to down (thus creating a grain boundary). The energy is now E 1 = E 0 +2J and the excess energy per spin is e = (E 1 E 0 )/N = 2J/N. But for large N this is such a small energy that even for very low T these type of fluctuations, which destroy the ground state, will very likely happen: the ordered state is only possible at T = 0 or, if we want, the phase transition occurs at T = 0. In fact, including the entropy S = k log (N 1) [there are N 1 ways to locate the grain boundary], the associated free energy is G = 2J kt log (N 1), which is negative in the thermodynamic limit for all 73
temperatures. However, mean-field theory predicts a phase transition at kt c /J = z = 2! Incidentally, the same type of argument in 2D (i.e. the creation of grain boundaries) leads to a prediction of a finite critical temperature, actually quite close to the exact result. Let us now discuss the spin-1/2 Ising model with antiferromagnetic coupling. At H = 0, Figure 3.17: Phase diagram of the spin-1/2 Ising model, in meanfield approximation, in the temperature-magnetisation per spin plane, T m, at zero magnetic field. Temperature is measured in units of the critical temperature. The critical point is depicted by a circle. The shaded region is the twophase region, where the system is not thermodynamically stable and separates into two equivalent phases with opposite magnetisation. The dashed lines indicate the spinodal line where the magnetic susceptibility is infinity. the completely ordered phase on the square lattice consists of two interpenetrating square lattices, a and b, with opposite magnetisations m a = m b = 1. In the disordered phase m a = m b = 0 (Fig. 3.18). The condition m a + m b = 0 is in fact true for any T because of symmetry. Spins in a given sublattice do not in fact interact (for nearest-neighbour interactions) and are only subject to interactions with spins of the other sublattice. In mean-field approximation the Hamiltonian can be written H = J s i s j = n.n. i s (a) i H (a) + j s (b) j H (b), H (a) = Jzm a, H (b) = Jzm b. (3.64) Then: m a = s (a) i = s (a) i =±1 s (a) i =±1 s (a) i e βh(a) e βh(a) = tanh (βjzm b )tanh (βjzm a ), 74
m b = s (b) s (b) i = i =±1 s (b) i =±1 s (b) i e βh(b) e βh(b) = tanh (βjzm a ) tanh (βjzm b ). (3.65) Therefore, in mean-field approximation, each of the sublattice magnetisations, m a and m b, depend on T following the same law as in the ferromagnetic case. In particular, there is a phase transition between the disordered phase and the ordered phase (where the magnetisation of the sublattices are non-zero and with opposite signs) at a critical temperature kt c /J = z. This is ultimately based on the symmetry of H with respect to the two sublattices, a symmetry which is lost when H 0. In this case the transition temperature decreases until it becomes zero beyond some value of the magnetic field (Fig. 3.18). Figure 3.18: (Colour on the Web). On the left, phase diagram of the antiferromagnetic spin-1/2 Ising model, in the H T plane. The dashed line is a line of critical points (second-order phase transitions). On the right, depiction of the ordered and disordered phases, indicating the population of the two sublattices a and b. Let us calculate the critical exponents. The exponent β can be easily calculated from the trascendental equation (3.19). Setting H = 0 and recognising that close to the critical point m 1, we can expand the hyperbolic-tangent function: Then, since kt c = Jz: 1 T c T 1 3 m = tanh (βjzm) = βjzm 1 3 (βjzm)3 + O ( m 5). (3.66) ( ) 3 Tc m 2 m ± ( ) T 3/2 ( ) Tc T 1/2 3 (T c T) 1/2, (3.67) T Tc T so that β = 1/2, the same result obtained with the van der Waals equation for a fluid. The analogue of the compressibility for a fluid is here the magnetic susceptibility at zero field, ξ: ξ = m (T T H c ) γ, T > T c. (3.68) H=0 75
Differentiating (3.59) with respect to H: m H = { 1 tanh 2 [β (Jmz + µ B H)] } ( βjz m ) H + µ BH ( = (1 m 2 ) βjz m ) H + µ BH. (3.69) Solving for the derivative, and calculating at H = 0 (which means m = 0 for T > T c ): m = βµ B(1 m 2 ) H H=0 1 βjz(1 m 2 = βµ BT (T T ) c ) 1, (3.70) m=0 T T c which means γ = 1, again as in van der Wals theory. All other critical exponents can be seen to be equal to those derived from van der Waals theory (and in fact from any mean-field theory). The conclusion is that, at the level of mean-field theory, fluids and magnets are in the same universality class. 3.5 Improvement of Weiss theory: Bethe theory In Bethe theory correlations of two spins are taken into account. This is realised by considering fluctuations of one spin and of its z neighbours, i.e. a cluster of z + 1 spins rather than just one-spin clusters, as in Weiss theory. The z neighbouring spins are still coupled to their neighbours through the average magnetisation per spin m. For a 2D lattice in zero external field (H = 0) the energy associated to such a cluster is (see Fig. 3.19): z ǫ cluster = Js 0 s i (z 1)mJ Now the partition function of the cluster is Z cluster = e βj(s 0+(z 1)m)(s 1 +s 2 + +s z) s 0,s 1,,s z=±1 z s i. (3.71) = 2 z cosh z {βj[1 + (z 1)m]} + 2 z cosh z {βj[1 (z 1)m]}. (3.72) The average value of the central spin is s0 = 1 Z cluster [2 z cosh z {βj[1 + (z 1)m]} 2 z cosh z {βj[1 (z 1)m]}], (3.73) whereas the average value of one of its neighbours is s j = 1 Z cluster [ 2 sinh {βj[1 + (z 1)m]}2 z 1 cosh z 1 {βj[1 + (z 1)m]} 2 sinh {βj[1 (z 1)m]}2 z 1 cosh z 1 {βj[1 (z 1)m]} ]. (3.74) Since these two have to be the same for consistency, s 0 = s j, and we arrive at the equation e 2βJ(z 1)m = coshz 1 {βj[1 + (z 1)m]} cosh z 1 {βj[1 (z 1)m]} 76 (3.75)
Analysis of this equation can be done graphically, or else as follows: m = 0 is always a solution, so the two functions cross at this point. Since both functions are monotonically increasing but the exponential diverges whereas the ratio of hyperbolic cosines has a horizontal asymptote, they can only cross at an additional point (symmetric with respect to the origin in view of the symmetry m m of the equation) if the slope of the latter is larger or equal to the slope of the former, which is 2βJ(z 1); the condition is then: 2βJ(z 1) 2 tanhβj 2βJ(z 1) β c J = arctanh (z 1) 1. (3.76) For the square lattice (z = 4) this yields β c J = 0.3466 or kt c /J = 2.885..., to be compared with the exact result kt c /J = 2.269... and with the mean-field result ktc/j = 4. This is a considerable improvement over Weiss theory. However, it can be shown that the critical exponents derived from Bethe s theory have the classical (mean-field) values: Bethe s theory is a mean-field theory, more sophisticated than Weiss, but still mean field. Figure 3.19: (Colour on the Web). A fluctuating cluster of spins (in red) embedded in a medium of spins with average value of their spins m. 3.6 Mean-field approximation: Bragg-Williams theory The mean-field theory of Weiss can be obtained in a completely different way by writing an effective free energy that is minimised with respect to the magnetisation per spin m. This procedure is based on the result that the thermodynamic free energy is a minimum, at fixed values of the thermodynamic external parameters, with respect to the internal, microscopic variables. Here the internal variable is taken to be m. We construct the free energy, G(H, T; m) = E(H, T; m) TS(H, T; m) as follows. First, we have seen that the energy, in mean-field approximation, is E(H, T; m) = 1 2 NJzm2 Nµ B Hm. (3.77) 77
Now we write an entropy. Within mean-field approximation spins are uncorrelated, so that the entropy can be written as S = k log Ω, where Ω is the number of ways to impose a given value of the average magnetisation per spin, m, in a system of independent spins. If N + and N are the number of up and down spins, respectively, and if M = Nm is the magnetisation: Then: M = N + N, N = N + + N. (3.78) N + = 1 2 (N + M) = N 2 (1 + m), N = 1 2 (N M) = N (1 m), (3.79) 2 so that Ω = N! N +!N! = N! [ ] [ ]. (3.80) N N 2 (1 + m)! 2 (1 m)! After short manipulations, the entropy can be written as S Nk = 1 + m ( ) 1 + m log 1 m ( ) 1 m log. (3.81) 2 2 2 2 The free energy is then: G(H, T; m) = 1 2 NJzm2 Nµ B Hm [ ( ) 1 + m 1 + m + NkT log 2 2 1 m 2 ( 1 m log 2 )]. (3.82) The equilibrium state (value of m) of the system is obtained from: G = 0. (3.83) m Then: Rearranging the equation: eq. G m = NJzm Nµ BH + NkT 2 1 + m 1 m = e2β(jzm+µ BH) log ( ) 1 + m = 0. (3.84) 1 m m = tanh [β (Jzm + µ B H)], (3.85) which is the same trascendental equation for m as that obtained in the Weiss theory. 3.7 Ising model in 1D (Ising chain) In 1D the Ising model can be solved analytically. Let us deal with the case H = 0 first. The Hamiltonian is H = J N 1 78 s i s i+1, (3.86)
Figure 3.20: (Colour on the Web). Left: free boundary conditions (spins s 1 and s N only have one neighbour). Right: periodic boundary conditions (all spins have two neighbours). where we used free boundary conditions (i.e. the s 1 and s N spins only have one neighbour, s 2 and s N 1 respectively). The partition function is: Z N = s 1 =±1 s 2 =±1 s N 1 =±1 s N =±1 βj e N 1 s i s i+1. (3.87) The sum over s N can be done explicitely since s N only appears once in the sum: s N =±1 independent on the value of s N 1. Then we can write: e βjs N 1s N = 2 cosh βj, (3.88) Z N = (2 cosh βj)z N 1 Z N = (2 cosh βj) N 2 Z 2, (3.89) with Z 2 = e βjs 1s 2 = 4 cosh βj, (3.90) s 1 =±1 s 2 =±1 so finally Z N = 2(2 coshβj) N 1, (3.91) and the free energy is G = NkT log Z N = kt [log 2 + (N 1) log (2 coshβj)] = NkT log (2 cosh βj). (3.92) The last equality applies in the thermodynamic limit. In the case H 0 it is more useful to write the Hamiltonian using periodic boundary conditions: N H = J s i s i+1 1 N 2 µ BH (s i + s i+1 ). (3.93) 79
The partition function is then: Z N = s 1 =±1 s 2 =±1 = s 1 =±1 s 2 =±1 s N =±1 β e N N e β s N =±1 [ Js i s i+1 + 1 ] 2 µ BH(s i + s i+1 ) [ Js i s i+1 + 1 2 µ BH(s i + s i+1 )]. (3.94) At this point the use of transfer matrices facilitates manipulations. We define the matrix: ( ) ( P11 P P = 1, 1 e β(j+µ B H) e βj ), (3.95) P 1,1 P 1, 1 e βj e β(j µ BH) in terms of which the partition function can be written: Z N = { N } P si s i+1 = P s1 s 2 P s2 s 3 P sn s 1 s 1 =±1 s 2 =±1 s N =±1 s 1 =±1 s 2 =±1 s N =±1 = ( ) P N = Tr P N. (3.96) s 1 =±1 Now λ 1, λ 2 are the eigenvalues of P, then λ N 1, λn 2 are the eigenvalues of PN and The eigenvalues λ 1, λ 2 are: The free energy is λ 1,2 = e βj cosh (βµ B H) ± = e βj cosh (βµ B H) ± G = kt log ( λ N 1 + λ N 2 Z N = Tr P N = λ N 1 + λn 2. (3.97) e 2βJ cosh 2 (βµ B H) + 4 cosh βj sinh βj e 2βJ sinh 2 (βµ B H) + e 2βJ. (3.98) ) = kt N log λ 1 + log ( ) N λ2 1 + λ 1 NkT log λ 1 as N (3.99) since λ 1 > λ 2. The free energy is then: [ ] G = NkT log e βj cosh (βµ B H) + e 2βJ sinh 2 (βµ B H) + e 2βJ. (3.100) This expression reduces to Eqn. (3.92) at H = 0. The magnetisation is: m = s i = 1 N G H = sinh (βµ B H). (3.101) sinh 2 (βµ B H) + e 4βJ 80
From here we see that, at zero field, there can be no spontaneous magnetisation. The magnetic susceptibility is ξ = m β cosh (βµ B H) = = βe H H=0 (1 + e 4βJ sinh (βµ B H)) sinh 2βJ. (3.102) 2 (βµ B H) + e 4βJ ξ never diverges at finite T, but becomes arbitrarily large as T 0, indicating that we can assume a kind of phase transition to take place at this temperature (even though the ordered state can never be reached!). Next we calculate the correlation function: Γ j = s i s i+j s i s i+j = s i s i+j (3.103) (we used the fact that s i = 0 for all temperatures). First, we consider the more general function s i s i+j, and assume i and i + j to be far from both ends of the chain. Second, we take the ferromagnetic coupling constant of the pair l and l + 1 to be J l, a constant that will be set to J at the end of the calculations. The partition function will be We have s i s i+j l = 1 Z N = 1 Z N s 1 =±1 s 2 =±1 s 1 =±1 s 2 =±1 s N =±1 N 1 Z N = 2 (2 cosh βj l ). (3.104) s N =±1 l=1 β s i s i+j e N 1 J l s i s i+1 β (s i s i+1 )(s i+1 s i+2 ) (s i+2 s i+3 ) (s i+j 1 s i+j ) e The last equality follows from the property s 2 i = 1. Now consider the derivative: N 1 j Z N j β J l s i s i+1 = e J i J i+1 J i+j 1 J i J i+1 J i+j 1 s 1 =±1 s 2 =±1 s N =±1 = β j Therefore: s 1 =±1 s 2 =±1 s i s i+j l = s N =±1 ( 2 ) Z N β j β (s i s i+1 ) (s i+1 s i+2 ) (s i+2 s i+3 ) (s i+j 1 s i+j )e j Z N J i J i+1 J i+j 1 N 1 N 1 J l s i s i+1. J l s i s i+1. (3.105) (3.106) 2 Z N β j {cosh βj 1 cosh βj 2 (β sinh βj i ) (β sinh βj i+1 ) (β sinh βj i+j 1 ) β sinh βj N 1 } = tanhβj i tanhβj i+1 tanhβj i+j 1, (3.107) 81
where Eqn. (3.91) has been used in the last step. Now specifying at J l = J: The correlation function is s i s i+j = (tanh βj) j. (3.108) Γ j = s i s i+j s i s i+j = s i s i+j = (tanh βj) j. (3.109) Since tanh βj < 1, the correlation function is a decreasing function of distance j (as it should be). We define the correlation length, χ, as tanh βj = e 1/χ χ = [log (tanh βj)] 1, (3.110) so that the correlation function is an exponential: Γ j = e j/χ. (3.111) χ is a positive quantity, and quantifies the exponential decay of Γ. As T 0 +, we have tanh βj 1, and log (tanhβj) 0, so that χ, as would be expected if we were approaching a critical point. 3.8 The lattice gas and lattice-alloy models Here we consider some lattice models that can be formulated to treat a number of problems different from the ferromagnetic systems but that can be seen to map exactly onto a spin-1/2 Ising model. Binary alloy We begin with the binary alloy. Here we have two species A and B, and each lattice site can be occupied by either species. N A and N B are the (in principle fixed) number of particles of each species (no vacancies are allowed). Clearly mapping onto the Ising model can be made by assigning A species to up spin and B species to down spin (or the other way round). The Hamiltonian would be ǫ AA, s i = +1, s j = +1, H = n.n.ǫ si,s j s i s j, ǫ si,s j = ǫ AB, s i = +1, s j = 1 or s i = 1, s j = +1, (3.112) ǫ BB, s i = 1, s j = 1. In the Ising model we would have ǫ AA = ǫ BB = J and ǫ AB = J, so the binary-alloy model seems a bit more general. To see more clearly the possible configurations, we express the energy of the i-th configuration of the system, E i, as follows: E i = ǫ AA N (i) + ǫ AA ABN (i) + ǫ AB BBN (i), (3.113) BB where N kl are the number of kl pairs in the lattice, and kl are the energies associated with kl pairs. If z is the coordination number, it is easy to see that zn A = 2N (i) AA + N AB, (i) zn B = 2N (i) BB + N(i) AB (3.114) 82
by the following procedure: take A sites and draw a line to all its z nearest neighbours; then zn A lines will have been drawn, and there will be double lines between AA pairs and single lines between AB pairs (and no lines between BB pairs). Hence the first line of Eqns. (3.114) follows. Doing the same with B sites we have the second line. This means that only one of the three variables N kl are independent, and we choose N (i) AB as the independent one. Then: where E i = 1 2 ǫ AA ( ) zna N (i) AB + ǫabn (i) AB + 1 2 ǫ BB ( ) znb N (i) AB = z 2 ǫ AAN A + z 2 ǫ BBN B + ǫn (i) AB, (3.115) ǫ ǫ AB ǫ AA 2 ǫ BB 2 (3.116) is the mixing energy. The mixing energy gives the tendency for the system to keep different species apart (ǫ > 0), i.e. to macroscopically segregate into two phases (one rich in A s and the other in B s) or together (ǫ < 0), giving a uniform mixture. Note that the same construction can be made for the Ising model; in that case ǫ AA = ǫ BB = J, ǫ AB = J, and ǫ = 2J, so that the ferromagnetic Ising model gives phase separation into +1-rich and 1- rich domains. In the antiferromagnetic Ising model J < 0 and ǫ < 0 which corresponds to antiferromagnetic ordering; this case corresponds to a lattice of alternating A and B species in the binary-alloy case. Therefore, in the alloy language we may have: ǫ > 0: liquid binary mixtures where two phases, one rich in A and the other rich in B phase-separate below a critical (consolute) temperature T c. An experimental T x phase diagram of a mixture of sodium fluoride and aluminium fluoride is shown in Fig. 3.21(a). We define the composition or mole fraction x as x = N A N A + N B = N A N. (3.117) In the case of the figure, x corresponds to the fraction of the complex AlF 3 with respect to the total number. In the figure various transitions, involving liquid and solid phases, are visible, but here it is the one between two liquids, liq 1 and liq 2, that is of interest to us. ǫ < 0: solid binary mixtures where a lattice with compositional disorder (sites occupied with A or B more or less at random) becomes ordered (with sites of one sublattice mostly occupied by a given species in a regular arrangement) below some temperature; this is the so-called order-disorder transition, an example of which is the β-brass phase of CuZn, the ordered phase here being two interpenetrating simple-cubic lattices of Cu and Zn atoms, which are occupied at random above a given temperature, Fig. 3.21(b). In each of these two cases results from the Ising model can be used, since there is an exact mapping between the two models. In particular, the mean-field approximation of Weiss for a ferromagnet can be used to predict phase behaviour in binary fluid mixtures 83
Figure 3.21: (a) Phase diagram, in the T x plane, of the mixture NaF/AlF 3, with x being the mole fraction of the second. (b) Structure of β-brass, CuZn. or binary solid alloys. But let us rewrite the theory in terms of the new variable (number of AB pairs N AB or equivalently mole fraction x). Since particles are taken to be independent in mean-field theory, the number of N AB pairs can be obtained as follows: since zn/2 is the total number of bonds, the probability of having an AB pair in the i-th conguration is N (i) AB/(zN/2). We approximate this as N (i) AB 1 2 zn 2 ( NA N ) ( NB N ) = 2 N AN B N 2, (3.118) i.e. as the product of probabilities of having a site occupied by A and the other by B (in fact we know this is not quite correct because of correlations). Then: E i = z 2 ( ǫ AA N A + ǫ BB N B + 2ǫ ) N N AN B = Nz 2 [ǫ AAx + ǫ BB (1 x) + 2ǫx(1 x)]. (3.119) The entropy can be obtained from Eqn. (3.81) directly, taken into account the equivalence between magnetisation m and mole fraction x: m = N + N N N A N B N = 2N A N N = 2x 1. (3.120) Then, taking ǫ AA = ǫ BB for simplicity (which symmetrises the mixture), the grand potential is Ω NkT zǫ = x log x + (1 x) log (1 x) + kt x(1 x) µ A kt x µ B (1 x), (3.121) kt where an unimportant constant term zǫ AA /2kT has been omitted, and contributions from chemical potentials µ A and µ B (which play the role of the magnetic field in the Ising model), giving contributions µ A N A and µ B N B, added. Also, all energies can be measured in units of kt, so we can write ω = x log x + (1 x) log (1 x) + zǫ x(1 x) µ x, (3.122) 84
where ω = Ω NkT, ǫ = ǫ kt, and a constant term µ B /kt has been deleted. µ = µ A µ B, (3.123) kt Let us take ǫ > 0 (liquid binary mixture) and analyse the free energy with respect to x. The entropic term, called mixing entropy, S = Nk [x log x + (1x) log (1x)], (3.124) always favours mixing (disorder), having a maximum at x = 1/2 (complete mixing), and discouraging formation of pure phases (x = 0 or x = 1), for which entropy is minimum. The energetic term, E = NzkTǫ x(1 x), ǫ > 0, (3.125) have minima at x = 0 and x = 1, and a maximum at x = 1/2, favouring formation of pure phases (i.e. segregation into Arich and Brich phases). In Fig. 3.22 we plot ω(x) for µ = 0 (making the mixture fully symmetric with respect to x = 1/2) for various temperatures. For ǫ > ǫ c there is a region of instability; the critical point verifies ω x = 0, 2 ω x 2 = 0, log x 1 x + zǫ µ 2zǫ x = 0, 1 x + 1 1 x 2zǫ = 0, (3.126) with solution x c = 1/2 and ǫ c = 2/z or kt c /ǫ = z/2. For T < T c there are three solutions, one at x = 0 (unstable) and two symmetrically placed with respect to x c, at x = 1/2 ± x 0, corresponding to the two (stable) coexisting phases, one rich in A particles, the other rich in B particles. For ǫ < 0 the x variable is not a good one since we need something that measures the sublattice occupation. Once this is identified, one can follow an equivalent number of steps and treat the case of a solid binary alloy in the same vein. Lattice gas Consider now the lattice gas, a model meant to approximately represent gases and liquids. We introduce occupancy variables c i for each lattice site, so that c i = 0 if the lattice site is empty and c i = 1 if it is occupied. Exclusion effects typical of fluids at high density arise implicitely by not permitting more than one particle to occupy any given site. We consider an attractive interaction between nearest-neighbour particles, and write the Hamiltonian as H = ǫ c i c j µn = ǫ c i c j µ n.n. n.n. i c i. (3.127) To see that this model is isomorphic to the spin-1/2 Ising model, we introduce spin variables s i as s i = 2c i 1. (3.128) 85
Figure 3.22: Free energy ω as a function of composition x for reduced inverse temperature zǫ = 2.5, 2.0 and 1.5, which correspond, respectively, to temperatures below, equal to and above the critical temperature T c. The dashed line is the two-phase region of coexistence for the temperature. The Hamiltonian is then: H = ǫ 1 n.n. 2 (s i + 1) 1 2 (s j + 1) µ (s i + 1) 2 i = N ( ) zǫ 2 4 + µ ǫ i s j 4 n.n.s 1 ( ) zǫ 2 2 + µ i s i, (3.129) which corresponds to the Ising model with the identification J ǫ and µ B H (zǫ/2 + µ)/2. We can then say that there occurs a first-order liquid-vapour phase transition at a chemical potential µ = zǫ/2 at temperatures below kt c /ǫ = z/4, which is the critical temperature. 3.9 Landau theory As we have seen, in a second-order phase transition second derivatives of the free energy, such as the specific heat, the isothermal compressibility or the magnetic susceptibility, exhibit divergences at the critical point, while the first derivatives (entropy, magnetisation or density) remain continuous. From this point of view, all second-order phase transitions have something in common. In 1937 Landau extended this idea and formulated a theory which was meant to explain all second-order phase transitions from a common viewpoint. The Landau theory first identifies an order parameter, η, a parameter which distinguishes the high symmetry phase, where η = 0, from the low-symmetry phase, for which η 0. The order parameter may be a scalar, vector or tensor with respect to the symmetry 86
operations associated to the problem. For the moment we will assume it to be a scalar. Since the transition is continuous, the order parameter is continuous at the critical point (T = T c ). Then the dependences of the free energy are written as G = G(h, T; η) (h is an ordering field: a magnetic field in the case of spins, a chemical potential or difference in chemical potentials in a fluid, etc.) In the case of a magnetic field, the obvious order parameter is the magnetisation per spin, m. The second assumption of Landau s theory is that the free energy of the system can be expanded in powers of the order parameter. From this expansion, all critical properties of the system may be obtained. In the absence of an ordering field, h = 0, where the transition is to take place (external fields are usually coupled to the first power of the order parameter), Landau argued that only even powers of η can be included in the expansion if a symmetry (invariance of F) under the operation η η is assumed. Then we write: G(T; η) = A(T) + B(T)η 2 + C(T)η 4 + (h = 0). (3.130) This expansion should in principle be valid sufficiently close to the critical point, since in that region the order parameter is either zero (in the high-symmetry phase) or very small. For the sake of convenience, we express the temperature in terms of the scaled temperature Now we assume the validity of the expansions t = T T c. (3.131) T A(t) = A 0 + A 1 t + A 2 t 2 +, B(t) = B 0 + B 1 t + B 2 t 2 +, C(t) = C 0 + C 1 t + C 2 t 2 + (3.132) For the time being, we truncate the expansion (3.130) after the fourth power in η. This implies that the parameter C(t) > 0 for all t, so that the free energy is bounded from below (implying stability of the system). The equilibrium value of the order parameter is obtained by minimising the free energy with respect to η at a fixed value of the reduced temperature t: [ 2B(t)η + 4C(t)η 3 = 0 η = 0, η = ± B(t) ] 1/2. (3.133) 2C(t) Clearly, if B(t) > 0, the only possible solution is η = 0, i.e. the high-symmetry phase (stable for T T c ). In order for the low-symmetry phase to be stable, we must have B(t) < 0 for T < T c. The simplest possibility is to have B 0 = 0, C 0 > 0 so that close to the transition we can write G(T; η) = G 0 (t) + B(t)η 2 + C 0 η 4, B(t) = B 1 t, (3.134) 87