3.1 Spin Exchange Relaxation of Muonium

In this section, we include a brief description of muonium (the bound hydrogenic atom $\mu^+e^-$) and its T₁ spin relaxation due to collisions with free electrons. A comparison with analogous nuclear spin relaxation is also made.

The hyperfine spin hamiltonian for an isolated isotropic Mu is:

where $\gamma_i$ are the gyromagnetic ratios, $A_\mu$is the Mu hyperfine parameter, ${\bf S}_i$ are the spins and ${\bf B}$ is the applied magnetic induction. This hamiltonian can be diagonalized analytically to give the field-dependent hyperfine energy levels which are plotted in a Breit-Rabi diagram, e.g. Fig. 3.10. The transition frequencies are conventionally[73] labelled $\nu_{ij} = (E_i-E_j)/h$, with E_i numbered according to Fig. 3.10. More detailed accounts of Mu (including anisotropic coupling) can be found elsewhere[73,92].

  

Figure 3.10: Breit-Rabi Diagram: The field dependence of the hyperfine energy levels (in Kelvin) of an isotropic muonium atom. The hyperfine coupling parameter here is the value for free Mu, A = 4.463302 GHz.

In metals, the predominant mechanism for T₁ relaxation of nuclear magnetization is via interaction with the conduction electrons within kT of the Fermi surface[93,94]. The interaction is usually modelled[94,95] by a direct hyperfine contact hamiltonian:

where the coupling A_n depends on the square modulus of the band electron wavefunction at the nucleus. The analogous coupling for a bare muon in conventional metals causes relaxation which is always too slow to observe on the muon timescale (i.e. $T_1 \gg 10 \mu s$) (see §3.2.4 of Cox[74]). However, T₁ can become short enough to observe in semi-metals such as graphite[96] and antimony[97] where the presence of a local electronic moment at the muon is permitted by weaker screening of the muon's coulomb potential. The actual coupling mechanism relevant for NMR T₁ in A₃C₆₀ has, in addition to (3.2), an important anisotropic contribution originating from the full electron-nucleus magnetic dipolar interaction, as discussed in detail elsewhere[34,98].

For endohedral muonium in A₃C₆₀, we are in the unusual situation of having a strongly bound paramagnetic muonium centre in a metallic environment. The interaction between a paramagnetic centre and the conduction electrons is more complicated than (3.2) because of the extra degrees of freedom of the bound electron. Nevertheless, we can model the interaction in a similar way: first, we neglect scattering into higher orbital states because such processes require orbital energies (10eV for vacuum muonium) which are not available at low temperature; second, we can neglect the direct muon-conduction electron coupling as mentioned above. The spin-independent Coulomb interaction together with the Pauli principle, can then be modeled by the simple spin-exchange hamiltonian:

where r is the separation of the scattering electron and the Mu atom, and J(r) is a short-range scattering potential[99]. The effect of this interaction is to randomly flip the muonium electron spin (Fig. 3.11), consequently producing a random modulation in the local field at the muon and causing the muon's spin to relax.

  

Figure 3.11: In each case a quasiparticle (QP) at the Fermi Surface with initial momentum ${\bf k}$ and energy e scatters magnetically from the local moment. a) Spin exchange of a muonium atom (with subsequent evolution due to the muon-electron hyperfine interaction $A_\mu$) vs. b) Electron-Nuclear spin-flip. In the spin exchange with the paramagnetic muonium, the electron Zeeman energies cancel; whereas in the latter, the nuclear and electron Zeeman energies do not.

The theory of such spin relaxation has been worked out in several contexts using various methods[100]. For the case of isotropic Mu hyperfine coupling $A_\mu$,the behaviour of the LF (T₁) muon spin-relaxation rate due to spin-exchange is divided into two regimes by a crossover when the rate of spin-exchange events ($\nu_{SE}$) equals the ``2-4'' muonium hyperfine frequency ($\nu_{24}$), which at high fields ($B \gg A_\mu / \gamma_e$) is approximately the electron Zeeman frequency. In the fast region ($\nu_{SE} \gg \nu_{24}$), the relaxation rate is approximately field independent, and in the slow regime the relaxation rate is governed by:

As Chow discovered[101], the relaxation rate in the case where the muonium hyperfine interaction is anisotropic can be dramatically different. There is a peak in T₁^-1(B) at a field determined by the hyperfine parameters which is the result of the geometry of the effective local field at the muon. While this peak has only been observed in doped crystalline semiconductors, it is expected to survive, in perhaps a very broadened form, an orientational powder average. From the high symmetry of the site at the centre of the C₆₀ cage, we expect Mu@C₆₀ have an essentially isotropic hyperfine interaction, but a priori one might expect that Mu inside the cage could bond to a single carbon, forming a highly anisotropic endohedral radical. Calculations[167] suggest that such a state is not stable. In pure and insulating alkali-doped C₆₀ phases[14,102], this conclusion is confirmed by observation of very narrow coherent spin precession lines from Mu@C₆₀. In contrast, the anisotropic exohedral radical in pure C₆₀ exhibits a much broader signal at low temperature (see Fig. 4.14). For nearly isotropic muonium with a large hyperfine interaction, the deviation from (3.4) will occur only at extremely high fields where the muon Zeeman interaction is comparable to the hyperfine interaction.

In analysis of the temperature dependence of the relaxation rate, it is of interest to consider the degree of inelasticity of the direct and spin-exchange scattering processes. In the case of the direct interaction (3.2), the nuclear spin and conduction electron spin flip-flop (Fig. 3.11), requiring an energy $\alpha_m=\vert(\mu_B - \mu_{nuc})B\vert$;whereas, in the muonium spin-exchange reaction (3.3), the electron Zeeman energies balance and the energy required is

independent of magnetic field. For the case of vacuum muonium $\alpha_m$ corresponds to a temperature of about 0.1K compared to an $\alpha_m(B=10T)$ of 13.4K for Korringa relaxation of nuclear spins.