Lecture 15: Phase Transitions. Phase transitions

Transcription

1 Leture 15: Phase ransitions Continuous Phase transitions ims: Mean-field theory: Order parameter. Order-disorder transitions. Examples: β-brass (, Phase transitions Continuous phase transitions: our when the minimum in the thermodynami potential evolves smoothly into two equal minima. n example is seen in the model of phase separation, along the o-existene line (last leture. p solid liquid =0.6 =1. x vapour Inreasing order Low, ordered High, disordered / side: the phase transition as one moves aross the o-existene line (from liquid to vapour is fundamentally different. hat transition is known as 1 st order and there are minima in the potential throughout. In the transition the lowest minimum hanges from liquid to vapour (and vie-versa. May 05 Leture 15 1 May 05 Leture 15 1

2 Order-disorder disorder transitions Other examples (there are many: Isotropi nemati transition in liquid rystals: appearane of orientational order (liquid rystals have no long-range, positional order. Inreasing order Mean-field theory Order-disorder disorder transition in β-brass. a further example, whih we will follow in detail. rass is a 50:50, : alloy with a b... struture. t low temperatures, <460K, the and atoms for an ordered struture (eg. atoms in the body-entre sites in top diag. Isotropi liquid Nemati liquid rystal erromagneti - paramagneti transition: manifests itself as a spontaneous polarisation, in zero external field. Inreasing order Inreasing order Low, ordered High, disordered High-: paramagnet Low-: ferromagnet wo types of site all them: -sites and -sites. t high, equal probability for any site to be oupied by or. May 05 Leture 15 3 May 05 Leture 15 4

3 he order parameter Mean-field theory he mean-field theory ignores flutuations and spatial variation. hus a single parameter an be used to desribe the average state of the system. In the present ase, we look for a variable behaving like: 0 no order = Order parameter 1 full order with n as the number of atoms on sites n as the number of atoms on sites n + n = N. (N sites in all n order parameter with the desired property is n 1 1 ; n + = n = n = n + n hermodynami variables Internal energy, U, in the mean-field approx. ond energies - u ; - u zz ; - u z verage energy at eah site u = 1+ u u = site No. n n Prob. (1+/ (1-/ No. Prob. (1-/ (1+/ (( z (( uz (( 1+ (( 1 ( u + uzz ( u u u + ( u + u + u = onst. an ordering energy, per bond Note on signs: attrative interations orrespond to negative energies, u. he system will order if - bonds are preferred i.e. u z > u, u zz. In whih ase > 0. n n ( z zz z zz 4 May 05 Leture 15 5 May 05 Leture

4 Helmholtz funtion, o alulate, we need the entropy, S. Entropy, S. Number of arrangements of the atoms is N! g = n! n S = k ln = Nk! g k( N ln N n ln n n ln n [ ln ( 1+ ln( 1+ ( 1 ln( 1 ] ree energy, = U - S. = onst. N Nk and atoms [( 1+ ln( 1+ + ( 1 ln( 1 ln ] (Note: the form is similar to the phase separation example in the last leture. Equilibrium ours when is a minimum. Equilibrium value of the order parameter differentiating gives = N Nk[ ln( 1+ ln( 1 ] 1+ = N Nk ln 1 Equilibrium ours at the minimum = 0 fter a few lines of algebra we get tanh = = tanh k ( where beomes a ritial temperature at whih the transition ours (as we shall see. here is no simple, analyti solution to this non-linear equation. = k = 0 May 05 Leture 15 7 May 05 Leture

5 Determination of the order parameter = tanh ( he solution an be seen from a graphial onstrution: Plot = αx and = tanh(x, for different α=/ is varied. Intersetion gives the solutions ( ; = ( x = x tanh ransition in β-brass State of order: bove, the order parameter,, is zero and atoms have random lattie sites. elow, the order parameter inreases rapidly and approahes full order as =0.6 =1. x / Heat apaity One is known, other thermal properties an be alulated. e.g. thermal apaity C = U C / / May 05 Leture 15 9 May 05 Leture