Nuclear Spin Diffusion Induced by Paramagnetic Impurities in Nonconducting Solids

It is shown that paramagnetic impurities can induce nuclear spin diffusion in nonconducting solids. The component of the impurity spin along the external magnetic field (assumed to be the $z$ axis), because of its interaction with the lattice, fluctuates. The resulting spectral intensity of the magnetic moment has components at all frequencies. The component at zero frequency creates a static magnetic field which is different at two neighboring nuclei, thereby splitting the levels $|\frac{1}{2},-\frac{1}{2}〉$ and $|-\frac{1}{2},\frac{1}{2}〉$, where the first and second quantum numbers refer, respectively, to the $z$ components of two neighboring spins. These are exact two-spin eigenfunctions if the interaction of the nuclear spins is neglected. When this is taken into account, the correct eigenfunctions to first order are ${\psi }_1=|\frac{1}{2},-\frac{1}{2}〉+\epsilon |-\frac{1}{2},\frac{1}{2}〉$ and ${\psi }_2=|-\frac{1}{2},\frac{1}{2}〉-\epsilon |\frac{1}{2},-\frac{1}{2}〉$, where $\epsilon$ is a small number. The Fourier component of the impurity spin at the frequency corresponding to the energy difference of ${\psi }_1$ and ${\psi }_2$ causes transitions between these states. This is a spin-diffusion process because $\epsilon$ is small. This means that Bloembergen's differential equation for nuclear spin-lattice relaxation in nonconducting solids must be generalized to include nuclear spin diffusion inside the critical radius.