Envelope Modulation in Spin-Echo Experiments

Expressions have been obtained for the envelope-modulation effect in spin-echo experiments of the two- and three-pulse type by partitioning the matrices which describe the evolution of the quantized system. The initial results are quite general and may be applied to a variety of systems. Simplified expressions are derived for the case of an electron spin transition split by small nuclear hyperfine interactions. The results are given in matrix product form. The problem of computing the envelope-modulation parameters in specific instances is discussed. Algebraic results are given for $S=\frac{1}{2}$, $I=\frac{1}{2}$ and $S=\frac{1}{2}$, $I=1\mathrm{}$.