Theory of Alternating Antiferromagnetic Heisenberg Linear Chains

The exact eigenvalue spectrum and thermodynamic properties of the spin Hamiltonian $\mathcal{H}=2\mathrm{}\left|J\right|\Sigma \stackrel{\frac{N}{2}}{i=1\mathrm{}}({\mathrm{S}}_{2i}·{\mathrm{S}}_{2i-1}+a{\mathrm{S}}_{2i}·{\mathrm{S}}_{2i+1})$ are calculated for short chains of $N=4,6,8, \mathrm{and} 10$ spins, each of spin ½, for alternation parameter $a=0.2,0.4,0.6, \mathrm{and} 0.8$, with ${\mathrm{S}}_{N+1\mathrm{}}\equiv {\mathrm{S}}_1$. The behavior for $N=\infty$ is estimated by extrapolation. Comparison is made with the known results for $a=0\mathrm{}$ and 1. The ground-state energy, ground-state short-range order, energy gap, and triplet excitation spectrum are compared with various approximate theories. Zero-temperature infinite-chain magnetization curves are inferred from the finite-chain results. The energy, entropy, specific heat, and magnetic susceptibility for $N=10\mathrm{}$ are shown to approximate well the behavior for $N=\infty$ when $\frac{\mathrm{kT}}{\left|J\right|}>a$. The magnetic-susceptibility data on the free radical 2,2-bis ($p$-nitrophenyl)-1-picrylhydrazyl are shown to agree well with the theoretical results for an alternating chain with $a\approx 0.6$.