Study of Oscillations and Correlations in the Relaxation of a Model Spin System

A detailed investigation is made of the approach to equilibrium of system of spins with $I=\frac{1}{2}$. The spin-spin interaction ${\Sigma }_k{\Sigma }_{j<k}{B}_{\mathrm{jk}}{{\sigma }_x}^j{{\sigma }_x}^k$ with $B_{\mathrm{jk}}$ constant is treated exactly, while the additional interaction ${\Sigma }_k{\Sigma }_{j<k}{L}_{\mathrm{jk}}{{\sigma }_x}^j{{\sigma }_x}^k$ with $L_{\mathrm{jk}}$ depending on lattice vibrations is treated by means of the assumption of sufficiently short correlation times for the $L_{\mathrm{jk}}$ operators. All correlations between the $L_{\mathrm{jk}}$ are included. The effect of the correlations usually neglected is expressed in terms of a sum over states somewhat resembling an Ising-model partition function with the time replacing interaction strength. The oscillatory relaxation via the $B_{\mathrm{jk}}$ and the monotonic relaxation via the phonons compete with each other; interference effects between the two relaxation modes also occur; the origin and nature of the irreversibility are very different for the two relaxation modes.