Theory of the Linear Heisenberg Antiferromagnet. Application to Paramagnetic Excitations in Organic Crystals

Abstract

We consider a linear array of tightly bound electrons that are exchange coupled to their neighbors. The Hamiltonian, in the Heisenberg model, is H = ∑ j=1 1 2 N {J(1+δ) S 2j · S 2j+1 +J(1−δ) S 2j · S 2j−1 − 1 2 J}, where δ is the alternation parameter and the exchange integral J is positive, corresponding to antiferromagnetic coupling. We investigate the solutions for the one‐dimensional antiferromagnet, for any |δ|≤1, by transforming the Hamiltonian first to Fermi creation and annihilation operators and then to pseudospin operators similar to the ones used by Anderson in connection with superconductivity. We thus take into account the coherence between antiparallel electrons in a self‐consistent manner. We exploit the similarity between the linear antiferromagnet and superconductivity theory to obtain the ground‐state energy and the excitation spectrum for arbitrary δ and temperature. We thus obtain a complete thermodynamic description. In particular, we calculate the paramagnetic susceptibility and compare it with experiment. At 0°K, the solutions we present agree with or improve upon previous calculations, for either the regular (δ=0) or the alternating (δ≠0) antiferromagnet, of the ground‐state energy, the excitation spectrum, and the short‐range order. Using the finite temperature extension of the theory, we find good agreement between the calculated and the experimental paramagnetic susceptibilities of organic crystals whose structures, so far as spin properties are concerned, may be approximated by either the regular or the alternating antiferromagnet. The temperature dependence of the excitation energies, a many‐body effect describing the decrease of coherence between antiparallel electrons with increasing temperature, agrees with the observed behavior of the singlet—triplet energy gap in paramagneticexciton systems. Thus, we obtain the extension of excitontheory to arbitrary temperature and alternation. The electron pairs forming triplet excitations are, for any alternation, increasingly separated at high temperatures, where, as expected, the electrons behave like a paramagnetic gas.