Quantum effects in muon spin spectroscopy within the stochastic self-consistent harmonic approximation

Muon spin rotation experiments involve muons that experience zero-point vibration at their implantation sites. Quantum-mechanical calculations of the host material usually treat the muon as a point impurity, ignoring its zero-point vibrational energy that, however, plays a role in determining the stability of calculated implantation sites and estimating physical observables. As a first-order correction, the muon zero-point motion is usually described within the harmonic approximation, despite the anharmonicity of the crystal potential. Here we apply the stochastic self-consistent harmonic approximation, a quantum variational method devised to include anharmonic effects in total energy and vibrational frequency calculations, in order to overcome these limitations and provide an accurate ab initio description of the quantum nature of the muon. We applied this full quantum treatment to the calculation of the muon contact hyperfine field in textbook-case metallic systems, such as Fe, Ni, Co including MnSi and MnGe, improving agreement with experiments. Our results show that there are anharmonic contributions to the muon vibrational frequencies with the muon zero-point energies above 0.5 eV. Finally, in contrast to the harmonic approximation, we show that including quantum anharmonic fluctuations, the muon stabilizes at the octahedral site in bcc Fe.

Figure 1

(a) Evolution of the SSCHA muon frequency during the minimization steps. (b) Evolution of SSCHA muon energy as in Eq. (13) during the minimization steps. In both figures, the starting point for the minimization step number = 0 is that of the harmonic Hamiltonian except for the muon in octahedral site of Fe [Fe (oct)] when the starting potential is arbitrary.

Figure 2

100 random position generated using Eq. (B1) for the muon at the octahedral site in the Co-fcc unit cell. The equilibrium octahedral center is depicted by the pink sphere, while the small dark spheres represent the different random muon positions where the muon contact hyperfine field within point impurity treatment $B_c\left({\mathbf{r}}_{\mu }\right)$ was also calculated for the purpose of including the quantum effects of the muon.

Figure 3

Contact hyperfine field $B_c\left({\mathbf{r}}_{\mu }^{\text{eq}}\right)$ at the equilibrium muon implantation position ${\mathbf{r}}_{\mu }^{\text{eq}}$, the muon contact field averaged over the muon wave-function spread, $\langle B_c\rangle$, and experimentally observed values.

Figure 4

Evolution of the SSCHA muon frequency ($\tilde{\omega }$ in the upper panel) and those of Fe (nearly static low-frequency lines in the lower panel) during minimization for the muon in the tetrahedral site of bcc Fe. The figure depicts the expected anharmonicity effects on the SSCHA muon frequencies and nearly nonexistent anharmonicity effects on those of Fe, due to the large mass difference of the muon and Fe nuclei. The muon is $\approx 490$ times lighter.