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In transverse field (TF), as long as the random local magnetic fields (RLMF) due to nuclear dipole moments are small compared to the applied field Ho, their only effect will be to add or subtract their components parallel to Ho so that different muons precess at different frequencies and get out of phase with each other after a while. Such ``dephasing'' is technically not true relaxation, since the muon polarization can be fully recovered by a suitable 180o pulse of RF power; hence the terminology ``T2'' as opposed to ``T1'' and all that. If the RLMF change with time, whether due to nuclear spin fluctuations unrelated to the muon or to muon diffusion between sites with different static RLMF, then true relaxation can take place. Abragam provided a general relaxation function Gxx(t) from which the muon hop rate can be extracted given a knowledge of the static width of the RLMF distribution (assuming the latter has a gaussian shape). |
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In weak longitudinal field (LF) and (by extension) zero field (ZF) the situation is more complicated: all three components of the RLMF contribute to muon ``relaxation'' and the distinction between ``T1'' and ``T2'' becomes blurred. The static dipolar ZF relaxation function Gzz(t) worked out many years ago by Kubo and Toyabe can be ``dynamicized'' by several methods including that of Kehr, to produce dynamic functions that can be fitted to extract muon hop rates more precisely. |
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Analogous functions can be derived for lorentzian distributions of RLMF, which are more appropriate for dilute systems of strong moments. However, the lorentzian distribution is fundamentally non-physical, having an infinite second moment; in reality there is always a ``cutoff'' (maximum possible) local field so that fast enough hopping will cause the relaxation rate to decrease eventually. In a truly Lorentzian field distribution there is no such ``motional narrowing.'' |
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In the early 1980's Peter F. Meier and Moreno Celio showed that the ``static 1/3 tail'' of Gzz(t) for completely static muons in copper was in fact not static but exhibited small oscillations due to the fact that the muon forms a nearly closed spin system with its 6 nearest neighbour Cu nuclei and that system evolves quantum mechanically together. This effect was important because the resultant ``droop'' in the ``1/3 tail'' was being interpreted as evidence for slow muon hopping, the temperature dependence of which was crucial to the theory of quantum diffusion to which that work was contributing. |
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Shortly after that the error of ignoring quantum mechanics was dramatically illustrated by the first observation of ZF oscillations in LiF, a manifestation of the system of 3 spin-1/2 particles formed by the F- µ+ F- or FµF- or just FµF ion as the muon becomes strongly ``hydrogen bonded'' to two neighbouring fluorine ions. The time evolution of this spin system under symmetric dipole-dipole interactions is easily derived and matches the observed behaviour exactly (after an overall additional relaxation is imposed due to other nearby spins and/or other mechanisms. We would never be able to think about Gzz(t) the same way again! [Although some still try.] |
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It was quickly shown that the FµF ion forms not only in the ``rock salt'' alkali fluorides but in the ``fluorites'' (alkaline earth fluorides), in ZnF2, YF3 and in fact in all the insulating metal fluoride compounds. Moreover, the bonds between the muon and the fluorines are always almost exactly the same length, indicating that the formation of this ion takes energetic precedence over any considerations of lattice symmetry or structure. If you are doing an experiment on such a material, you may rest assured that the muon is in an FµF ion, whether the magnetic properties of the system allow you to observe it easily or not. |