Phase Memory in Electron Spin Echoes, Lattice Relaxation Effects in CaWO4: Er, Ce, Mn

The electron-spin-echo phase memory $T_M$ has been studied both experimentally and theoretically for the specific case in which it is limited by the lattice relaxation processes occurring in the sample. The relevant mechanism is as follows. Lattice relaxation of any spins, whether or not they belong to the species being observed, causes fluctuations in the local fields and so destroys the relations between precessional phases which lead to the generation of echoes. The effect of these fluctuations on the echo amplitude can be calculated by taking an ensemble average for the precessing spins and for all the environmental spins which give rise to significant time variations of the local fields in the sample. The problem reduces to that of finding a time and a space average. The space average has been obtained here by assuming a random distribution of spins in the paramagnetic sample, and by applying the statistical methods of Margenau. In order to obtain the time average, two models have been chosen to represent the time variation of the components ${\mu }_z$ for the relaxing spins. In one model, the ${\mu }_z$ are treated as Gaussian random variables (Gauss-Markoff model), and in the other the spins are assumed to make sudden jumps at random times between the "spin-up" and "spin-down" quantum states (sudden-jump model). Different forms of echo envelope are derived for the two models. Further differences in behavior will be observed, according to whether the sample is singly or doubly doped. If the sample contains only one spin species, $T_M$ becames shorter as the temperature is raised and as the lattice relaxation time $T_1$ is reduced. Initially, $T_M$ is limited by local field fluctuations and may be considerably shorter than $T_1$. Eventually, as the lattice relaxation of the precessing spins themselves becomes the dominant factor, $T_1$ and $T_M$ tend to the same value. If the sample contains two species $A$ and $B$, where $B$ relaxes more rapidly than $A$, then $T_M\mathrm{}\left(A\right)\mathrm{}$ and $T_M\mathrm{}\left(B\right)\mathrm{}$ both begin by shortening as $T_1\mathrm{}\left(B\right)\mathrm{}$ is reduced. For very small values of $T_1\mathrm{}\left(B\right)\mathrm{}$, however, $T_M\mathrm{}\left(A\right)\mathrm{}$ lengthens again. The rapidly fluctuating local fields due to the $B$-spins produce a diminishing effect on the $A$ spins, the phenomenon being analogous to motional narrowing. The form of the $A$-spin echo envelope in the limit of rapid $B$-spin relaxation does not depend on the model chosen to represent the time variation of ${\mu }_z$ during the relaxation of the $B$ spins.