We obtain the exact behavior of the cross-polarized cylindrical Einstein-Maxwell waves that generalizes the well-known Einstein-Rosen waves. In the presence of the second mode of polarization the outgoing waves interact with the incoming ones to exhibit an analogous effect of the Faraday rotation.

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The Schottky specific heats and magnetic susceptibilities of the Nd3+ in five doped garnets, GGG, LuAG, LuGG, YAG and YGG, have been calculated using effective spin Hamiltonian formalism in connection with the crystal field theory in a temperature interval of 30 to 500 K. Both heat capacities and susceptibilities are determined by two different methods. The first method is dealing with the diagonalization of the effective spin Hamiltonian and then obtaining the quantities in a very general way of quantum statistical mechanics. The second method is based on the cumulant expansion of partition function and utilizing the invariance properties of the trace of the Hamiltonian matrices. It is thought that the specific heat and the susceptibility calculations may help to refine the crystal field parameters of such crystals.

IntroductionHeisenberg models, one encounters great difficulties to solve them in the general framework of statistical and solid state physics. The critical structure of some systems is worth studying. These critical behaviours a r e well described by critical exponents /1/ near the critical point. In the following sections we write the eigenvalue equation of susceptibility and magnetization for a one-dimensional model obtained by the renormalization group approach and solve them in details. The nature of these equations, of course, depends on the mathematical models in hand as well as the assumed interaction po-

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