Leture 13 Bragg-Williams Theory As noted in Chapter 11, an alternative mean-field approah is to derive a free energy, F, in terms of our order parameter,m, and then minimize F with respet to m. We begin with a slightly different definition for m, m = ( ) where = + is the total number of sites in the lattie. (13.1) The entropy assoiated with a given m is just the logarithm of the total number of onfigurations with a given and, S = k B ln = k B ln! (13.) (1+m)! (1m)! Applying Stirling s approximation, the above equation redues to, (1 + m) (1 + m) (1 + m) S = k B [ ln { ln (1 m) (1 m) (1 m) + ln }] (1 + m) (1 + m) (1 + m) = k B [ ln { ln + ln (1 + m) (1 m) (1 m) (1 m) (1 m) + ln + ln }] [ ] (1 + m) (1 + m) (1 m) (1 m) = k B ln ln and hene, s(m) S = k B [ln 1 (1 + m) ln(1 + m) 1 (1 m) ln(1 m) ] (13.3)
LECTURE 13. BRAGG-WILLIAMS THEORY 87 whih is just the entropy of mixing. What is the mean energy, Ē? To alulate this average, we have to evaluate the Hamiltonian reognizing that the spins interat with eah other. In the mean field spirit, as we saw for the Weiss Moleular field theory, we replae σ i in the Hamiltonian with its position independent average, m: Ē = J ij m = 1 Jqm (13.4) where, again, q = d is the lattie oordination number or the number of nearest neighbour sites for a d-dimensional hyperubi lattie. Let us now write down the free energy for this system, we have: f(t, m) F (T, m) = E T S = 1 Jqm + 1 k BT [(1 + m) ln(1 + m) + (1 m) ln(1 m)] (13.5) k B T ln Expansion of s(m) near T ow, in the viinity of T, m is small (as we have noted before), so we may expand s(m), and hene f(m), as follows, s(m) = s(0) + 1 s m + 1 4 s m 4 +... (13.6)! m 4! m 4 where, and, s m 4 s m 4 ( ) k B = (1 + m)(1 m) = k B 1 m + m = k B = k (1 + m) 3 (1 m) 3 B Therefore, the entropy is given as, s(m) = k B [ln 1 m 1 ] 1 m4 +... and the free energy beomes, f(m) = k B T ln + ( 1 Jqm + k B T 1 ) m + k B T 1 1 m4 or (13.7) f(m) k B = 1 (T T )m + T m4 1 T ln (13.8)
LECTURE 13. BRAGG-WILLIAMS THEORY 88 It is lear from the above equation that for T > T, the minimum in the free energy ours for m = 0. For T < T, where does the minimum in f(m) our? f(m)/k m = (T T )m + m = 3(T T ) T 4T m3 1 = 0 or m = ± 1/ T 3 T 1 (13.9) whih is the same result as derived for the Weiss Moleular Field theory! To expliitly see the behaviour of the free energy, as a funtion of the order parameter m, in the viinity of T, refer to the following figure, h = 0 F T>T T=T T<T m 13.1 Spins in an external field Let us now onsider our system of spins in the presene of an external magneti field, h, suh that the free energy beomes f f hm. Referring to Eq. 13.5 we have, f(t, m) = 1 Jqm hm + 1 k BT [(1 + m) ln(1 + m) + (1 m) ln(1 m)] k B T ln Minimizing the above expression with respet to m we have, f m = Jqm h + 1 1 + m kt ln = 0 1 m (13.10)
LECTURE 13. BRAGG-WILLIAMS THEORY 89 Solving for h we find, h = Jqm + 1 1 + m kt ln 1 m = Jqm + 1 [ ] kt m + m3 3 + m5 5 +..., m < 1 = Jqm + kt tanh 1m, or [ h m = tanh kt + T ] T m (13.11) whih again is the same result that we obtained from the Weiss Moleular Field Theory model. 13. Critial Behaviour of Mean Field Theories Using the expression for the average energy, Eq. 13.4 with the mean field result, Eq. 13.9, we have, Ē Jq T T 3 T (13.1) From this expression, we may alulate the heat apaity as follows, C h T T = E = 3Jq 3k T T T T (13.13) Sine m = 0 for T > T, this implies the average energy and, hene, the heat apaity are zero as we approah T from the positive side. The qualitative behaviour of the heat apaity atually demonstrates a usp disontinuity, as seen below. This behaviour is in ontrast to more exat theories and experiments whih demonstrate a power law dependene of the heat apaity, C h = B ± T T α (13.14)
LECTURE 13. BRAGG-WILLIAMS THEORY 90 C v disontinuity at T T T