Does Coherent Tunnelling of Positive Muons Cause Motional Narrowing?

Resonance lines broadened by dipolar interactions are a common phenomenon. The best studied cases are the nuclear magnetic resonance lines, for which Van Vleck's [ 1 ] classic analysis gives expressions for the moments of the shape. Recently work on muon spin resonance [2,3] stimulated questions of whether the linewidth depended on the number n of sites over which the muon moved coherently. Yaouanc [2] argued that there was no dependence on n, i.e. that the extension of the/a + wavefunction did not matter . This appears to be wrong, since it ignores an important difference in the nuclear magnetic and muon resonance experiments. Van Vleck's analysis is given most fully for like spins interacting. When one considers just a single spin A interacting with many spins B of different gfactor, it becomes clear that two types of ensemble average are involved, namely: (a) For a given site of spin A, there is a sampling of aH possible states of spins B; (b) For a fixed configuration of spins B, the spin A can be moved to moni tor all sites. If spin A lives long enough for the configurations of spin B to be sampled properly, these two averages should be identical. This is the case in the magnetic resonance of stable nuclei. However, for a muon in a typical solid, the muon lifetime (about 2/Is) is much less than typical nuclear magnetic reorientation times (normally ms). Thus each muon A samples a frozen nuclear spin configuration B, and this leads to two distinct types of sampling. In essence (ignoring limitations because only a few detectors are used), a muon spin rotation experiment monitors the spin reorientation angle and time delay between entry and decay for each muon. For present purposes one can regard the experiment as a measurement of the effective magnetic field (Happtie d + hA) in which muon A precesses. There are two distinct samplings of the internal field in the solid: (c) For a given nuclear spin configuration B, h A depends on whether the muon is localised at a single site or whether it tunnels over a group of N sites ( i ) , since the nuclear spin environment will differ among these sites. This is discussed quantitatively below. (d) Successive muons are injected at different times, and may end up in different parts of the target, so that the nuclear spin configuration B may change during an experiment involving many muons. This sampling parallels that for nuclear spins [(a) and (b) above]. Indeed, if the muon localises on a single site, the muon and nuclear cases are equivalent. Yaouanc argues that coherent tunnelling, i.e. the effect of (c) above, does not affect the observed spread in h A . We argue now this is wrong, and that tunnelling does reduce the observed width. The effective local field h A for muon A is the expectation value (q~A IhB(R)I gPA ) of a field h B (R) which varies in space but which is constant in time for a fixed nuclear spin configuration. For simplicity, we make two working assumptions. We write the muon wavefunction q~A in a tight-binding form, with i labelling the interstitial sites over which coherent tunnelling occurs: