Equations of Motion of Nuclear Magnetism

Abstract

A general formalism from a previous paper is applied to derive an exact equation of motion for the total magnetic moment M of a system containing a single species of nuclear spins in an arbitrarily time-dependent external magnetic field H. Then, the equation is simplified for $\mathrm{kT}$ large compared with $\ensuremath{\gamma}H$ and compared with microscopic anisotropic spin energies but not necessarily compared with microscopic exchange energies. Only this high-temperature approximation is needed for the equation to reduce to $\frac{d\mathbf{M}}{\mathrm{dt}}=\ensuremath{\gamma}\mathbf{M}\ifmmode\times\else\texttimes\fi{}\mathbf{H}\ensuremath{-}\ensuremath{\int}{0}^{t}\mathbf{K}(t,{t}^{\ensuremath{'}})\ifmmode\cdot\else\textperiodcentered\fi{}[\mathbf{M}({t}^{\ensuremath{'}})\ensuremath{-}{\ensuremath{\chi}}_{0}\mathbf{H}({t}^{\ensuremath{'}})]d{t}^{\ensuremath{'}},$ which is valid for systems with arbitrarily long correlations times. The system is assumed to be in equilibrium for $tl0$, but it may deviate arbitrarily far from equilibrium for $tg0$. The kernel K is a dyad that depends upon $T$ and upon H but not upon M. The equation is capable of describing nonexponential relaxation and therefore non-Lorentzian resonance line shapes. For arbitrarily large times $t$, it is valid for, but not restricted to, a many-spin system with strong dipole-dipole interactions between spins in a rigid lattice and with spin \ensuremath{\ge}\textonehalf{}. For a nonrigid lattice, no assumption is made that the lattice is or is not in equilibrium, that it is a thermal bath, or that the density matrix may be factored. Instead, in order to simplify the formal expression for K and thus the solution of the equation, slowly-varying-temperature, high-temperature, and high-frequency approximations are made so that $\mathbf{K}(t,{t}^{\ensuremath{'}})$ becomes a function of $t\ensuremath{-}{t}^{\ensuremath{'}}$ only. Finally, the approximation of neglecting the dependence of K on ${H}_{1}$ is made, so that K becomes diagonal and equal to a sum of terms each consisting of a simple oscillatory factor times a function that may be interpreted as the correlation function of a stationary random process. The corresponding correlation times need not be short, but for comparison with previous formalisms a short-correlation-time approximation is also made.