In solid nitrogen muonium diffusion was studied using both longitudinal and transverse field muon spin rotation/relaxation techniques. From the relaxation rates of the muon polarization one can extract the muonium hop rate as a function of temperature. This is compared to the behavior predicted by theory, finding agreement at low temperatures only.
With a weak longitudinal magnetic field of a few Gauss applied to the sample, the muon and electron spins in muonium are strongly coupled by the hyperfine interaction. As the muonium atom diffuses among atoms of the host lattice, the nuclear hyperfine (nhf) interaction between the spins of the muonium electron and the randomly-oriented lattice nuclear moments undergoes fluctuations. The resulting time-dependent effective magnetic field felt by the muonium atom excites transitions among states of the muonium atom, and the original polarization of the muon spin is lost. Figure 6.18 shows the temperature and magnetic field dependence of the muon asymmetry measured in s-N_{2}in longitudinal field. To a good approximation, in weak magnetic fields the muon spin polarization relaxes with an exponential time dependence
P_{z}(t)=P_{z}(0) e^{-t/T1} | (11) |
(12) |
Figure 6.19 shows examples of muonium spin precession signals measured in a weak transverse field (wTF) of 5 G. In weak magnetic fields the two observable muonium frequencies and are very nearly equal (see Appendix A for definitions of these and the other muonium frequencies and )the splitting between these two lines is
(13) |
In transverse field the polarization relaxation function depends on the particle hop rate due to the motional narrowing of the precession frequency linewidth. Assuming that the local (effective nhf) field correlation function is described by
(14) |
(15) |
Figure 6.21 shows the muonium spin relaxation rates measured in s-N_{2}by TF and LF experiments. The qualitative temperature dependence of the relaxation rates and the hop rate is entirely consistent with the following interpretation. At low temperatures (T < 5K) muonium is nearly static; the TF experiment measured a T-independent relaxation rate as expected in the slow-hopping limit. Between about 9 K and 20 K the TF relaxation rate drops, as expected if the muonium diffuses faster as temperature rises. The LF relaxation rates in 4, 8 and 12 G reach their maxima, where
is satisfied, at progressively higher temperatures. Between about 11 K and 15 K the relaxation rate in LF becomes field-independent and approaches the TF relaxation rate, also indicating that the hop rate increases with temperature in this part of the data.Making use of Eqs. (6.13),(6.18) and (6.19) we can extract the muonium hop rate as a function of temperature, as shown in Fig. 6.22. The LF experiments measured a diffusion rate increasing as T^{6.7(1)} up to 15 K. The TF data extend this to higher temperatures where the correlation time is so short that the LF relaxation rate becomes nearly independent of magnetic field and approaches the TF relaxation rate. Between about 20 and 30 K the hop rate reaches its maximum; the TF relaxation rate become T-independent at about 0.4 .If the hop rate here is limited only by the coherent tunnelling bandwidth, we can estimate
It is also possible that the relaxation rate due to the motionally averaged nuclear hyperfine interaction with N_{2}moments is even smaller, but is overwhelmed by spin relaxation due to other causes, such as the muonium atoms diffusing to chemically active impurities or impurites with electronic moments such as O_{2} that immediately depolarize any muonium atom that strays near. Above about 50 K the TF relaxation rate increases as the hop rate drops with increasing temperature, a key characteristic of incoherent (2-phonon) quantum diffusion, although theory predicts a much weaker temperature dependence at such a large fraction of the Debye temperature.