By Archer K.J., Dumur C.I., Joel S.E.

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Frustrated double chain (Fig. 1e) Magnetic susceptibility NA g 2 µ2B A χ = 8kT B A = U 2 exp(K 1 ) exp(K 2 ) + 2 1 − exp(2K 1 ) U + exp(3K 1 ) exp(−K 2 ) + exp(−K 1 ) exp(−K 2 ) − 2 exp(K 1 ) exp(−K 2 ) B = ER with E = exp(−2K 1 ) + cosh(K 2 ) + exp(−K 1 )R R = exp(2K 1 ) sinh2 (K 2 ) + 2 cosh(K 2 ) + 2 1/2 U = exp(K 1 ) cosh(K 2 ) + exp(−K 1 ) + exp(2K 1 ) sinh2 (K 2 ) + 2 cosh(K 2 ) + 2 1/2 Specific heat C p /R = (B/Z − A2 /Z 2 ) Z = exp(−K 1 ) + exp(K 1 ) cosh(K 2 ) + U 1/2 A = V + P/U 1/2 B = W + (U Q − P 2 )/U 3/2 U = exp(2K 2 ) sinh2 (K 1 ) + 2 cosh(K 1 ) + 2 V = −K 1 exp(−K 1 ) + K 1 exp(K 1 ) cosh(K 2 ) + K 2 exp(K 1 ) sinh(K 2 ) W = K 22 exp(K 2 ) cosh(K 1 ) + exp(−K 2 ) + K 1 K 2 exp(K 1 + K 2 ) + K 1 exp(K 2 ) sinh(K 1 ) P = K 2 exp(2K 2 ) sinh2 (K 1 ) + (1/2)K 1 exp(2K 2 ) sinh(2K 1 ) + K 1 sinh(K 1 ) Q = 2K 22 exp(K 2 ) sinh(K 1 ) + 2K 1 K 2 exp(2K 2 ) sinh(2K 1 ) + K 12 exp(2K 2 ) cosh(2K 1 ) + K 12 cosh(K 1 ) with K i = Ji /2kT ; i = 1, 2 This problem has been encountered in the Cu(II) compounds A3 Cu3 (PO4 )4 (A = CaII and SrII ) and Cu2 OSO4 .

3. An example which illustrates this case is provided by the series of solid state compounds formulated as Sr3 CuPt1−x Irx O6 which exhibit a chain structure formed by two alternating sites A and B (Fig. 6a). Site A is occupied Table 4. F/AF alternating chain (s = 1/2) A–H parameters for the rational expression of the susceptibility given as a function of polynomials in α, (α = J2 /|J1 |): X i (α) = x0 + x1 α + x2 α 2 + x3 α 3 . 02686769 S2i S2i−1 j ATr3 + BTr2 + C Tr + D ; + E Tr3 + F Tr2 + GTr + H NA g 2 µ2B kT ; χM = χr |J1 | 4|J1 | 10 1 One-dimensional Magnetism: An Overview of the Models Table 5.

The general approach of the mathematics, which is now briefly described, remains unchanged when dealing with more complex classical-spin systems. The partition function is first calculated for a finite length chain containing N spin vectors. We have: Z N (B) = d 0 d N exp(−β H ) (7) where β is Boltzmann’s factor 1/kT , and d i means integrating over all the directions (defined by the usual spherical angles θi and φi ) available to each vector ui . In order to perform the integrations, the argument of the exponential is written as a sum of terms, each one involving a pair of neighboring sites, say ui and ui+1 .