Theoretical and Mathematical Physics, 145(3): 1727–1735 (2005) A FEW EXACT SOLUTIONS FOR THE MODEL OF EQUAL SPIN–SPIN INTERACTIONS A. R. Kessel, ∗ R. R. Nigmatullin, † A. A. Khamzin, ‡ and N. A. Yakovleva § For the model of equal spin–spin interactions, we obtain exact Heisenberg expressions for the spin operator components and use them to find the relations that hold for spin operators appearing in the thermodynamic mean. We use the obtained exact relations for the means to describe the thermodynamics of the model under consideration. Keywords: spin–spin interactions, Heisenberg equations of motion, thermodynamics 1. Introduction The spin–spin interaction (SSI) has attracted the interest of researchers during several decades because it describes the fine properties of the dynamics and kinetics of electron and nuclear paramagnetic systems. The Hamiltonian of a (electron or nuclear) paramagnetic system is [1] H = H z + H d + H dd , H z = -  f h f S z f , H d = - 1 2  f,g A f,g S z f S z g , H dd = - 1 4  f,g B f,g (S - g S + f + S + g S - f ). (1) Here, S ± = S x ± iS y , A f,g = V f,g + J f,g , V f,g =  2 γ 2 R 3 f,g (1 - 3 cos θ f,g ), B f,g = -A f,g + J f,g , where S α f , α = x, y, z , is the α component of the spin S f located at the point f ,(R f,g ,θ f,g ,ϕ f,g ) are the spherical coordinates of the vector joining the locations of the spins f and g in the laboratory frame of reference with the z axis parallel to the constant magnetic field H 0 , γ is the gyromagnetic ratio, ω 0 = γH 0 is the Zeeman frequency, and J f,g is the exchange integral. In expression (1), we give only the secular part of the spin–spin and exchange interactions because this part commutes with the main Hamiltonian H z and the contribution of this part to the dynamics and kinetics of a spin system therefore considerably prevails over ∗ Deceased. † Kazan State University, Kazan, Russia, e-mail: nigmat@knet.ru. ‡ Kazan State Energy University, Kazan, Russia. § Kazan Physicotechnical Institute, Kazan Scientific Center, RAS, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 411–419, December, 2005. Original article submitted April 21, 2005. 0040-5779/05/1453-1727 c  2005 Springer Science+Business Media, Inc. 1727