General Theory of Cross Relaxation. I. Fundamental Considerations

A statistical analysis is made of spin transitions induced by dipole interactions which change the total magnetization while exactly conserving energy. The first-order effect of the dipole operator can be described by a function $\Phi \left(\omega \right)$, which is related to the level broadening observed in resonance lines. The second-order effect leads to a function $\chi \left(\omega \right)$ which represents the power spectrum of the dipole operator. The cross-relaxation probability $W_{\mathrm{CR}}\left(\omega \right)$ is given by the convolution of these two functions. $W_{\mathrm{CR}}$ is calculated explicitly in various approximations, without appeal to moments. For single-spin flips in magnetically dilute systems, the magnitude of $W_{\mathrm{CR}}$ depends linearly on the concentration $n$. There is a very sharp peak at $\omega =0\mathrm{}$ with a width proportional to the geometric mean of the resonance width and of the nearest-neighbor dipole energy.