Answer Happy

3.19. Starting with expression (13.3.8) for the partition function of a one-dimensional n-vector model, with Ji=n)', show that √ (4K²+1)+1 Lim n.N→RN 1 In Qu(nK) = V(4x² +1) - 1 - In [ ( 1}] 2 where K= BJ'. Note that, apart from a constant term, this result is exactly the same as for the spherical model; the difference arises from the fact that the present result is normalized to give ON = 1 when K=0. (Hint: Use the asymptotic formulae (for v > 1) r(v) (2./v)1/2(u/e)" and 1,(vz) = (2x v)-1/2 (22+1)-1/4" where n= V(+1) - In[{/(2+1)+1)/2].] 3.20. Show that the low-field susceptibility, xo, of the spherical model at T < Te is given by the asymptotic expression x0 = (Nu? /kgT) Nm3(T), where mo(T) is the spontaneous magnetization of the system; note that in the thermodynamic limit the reduced susceptibility, koTxo/Nu?, is infinite at all T < Tc. Compare to Problem 13.26.