For β-brass, let the sites occupied by Cu atoms at T = 0 be called the C0 sites and let th...

For β-brass, let the sites occupied by Cu atoms at T = 0 be called the C0 sites and let the T = 0 Zn sites be called Z0. At any temperature T, let r be the number of atoms in right positions (a Cu atom on a C0 site or a Zn on a Z0) and let w be the number of atoms on wrong positions (Cu on a Z0 site or Zn on C0). The long-range-order parameter σl is defined as σl  = (r − w)l (r + w). (a) What is σl at T = 0? (b) What is σl if all atoms are on wrong sites? Would this situation be highly ordered or highly disordered? (c) What is σl at Tλ, where the number of rightly and the number of wrongly located atoms are equal? (d) What is σl in each of the drawings in Fig. 7.12? Let np be the total number of nearest-neighbor pairs of atoms in β-brass and let np be the number of right nearest-neighbor pairs (Cu−Zn or Zn−Cu). The short-range-order parameter σs is defined as σs ≡ 2nrp/np − 1. (e) What is σs at T = 0? (f) What is σs in the T → ∞ limit of complete disorder? (g) What is σs in each of the drawings in Fig. 7.12? Note: Count only nearest-neighbor pairs; these are pairs with one atom at the center of a square and one at the corner of that square. (h) Sketch σl and σs versus T/Tλ using the results of this problem and the information in Sec. 7.5.