Role of the transition state in muon implantation

In muon-spin-rotation experiments, positive muons are implanted in the material and come to rest in the unrelaxed host lattice. The formation of the final configuration requires a lattice relaxation which does not occur instantly. The present paper is concerned with the transition from the initial stopping state to the final muon configuration. We identify the often observed fast relaxing signal in muon experiments (e.g., in several oxides studied recently) with the transition state in this conversion process. This state is paramagnetic with a small hyperfine interaction (in the order of MHz) which fluctuates and averages to almost zero. Because of its apparent diamagnetic frequency behavior, the fast signal was in the past assigned to ${\mathrm{Mu}}^{+}$ or ${\mathrm{Mu}}^{-}$. We present evidence that this state is actually paramagnetic. The model presented in this paper is of importance for the interpretation of past and future $\mu \mathrm{SR}$ measurements.

Figure 1

Left: $\mu \mathrm{SR}$ signal of slightly Mg-doped zirconia ${\mathrm{ZrO}}_2$:Mg at 8 K (data from Ref. [6]). The signal (full data in the lower frame) is decomposed in a slowly relaxing component (solid line in the lower frame) and a fast relaxing component (upper part of the figure). Right: $\mu \mathrm{SR}$ spectra of ${\mathrm{Al}}_2{\mathrm{O}}_3$ at (a) 8(2) K and (b) 16(3) K (figure from Ref. [7], copied with permission). The solid lines are fit curves with a single exponential function but with different relaxations. The precession frequencies in all cases are close to the Larmor frequency of diamagnetic muon.

Figure 2

Final step in muon implantation: Neutral (${\mathrm{Mu}}^0$) and charged (${\mathrm{Mu}}^{+}$) muons come to rest at an interstitial site in the unrelaxed lattice (marked by $*$). ${\mathrm{Mu}}^{+}$ forms directly the final bound configuration, but ${\mathrm{Mu}}^{0*}$ may go through an intermediate (transition) state before the final configuration is formed.

Figure 3

${\mathrm{ZrO}}_2$:Mg at 8 K, decoupling curve in longitudinal magnetic field. The fraction of the non-relaxing signal (plus the slowly relaxing part which is not separated here) is represented. The fit with a dipolar-like hyperfine interaction (solid line) yields $D=-2.1\left(1\right)$ MHz.

Figure 4

Longitudinal-field relaxation rate of the fast relaxing signal in ${\mathrm{ZrO}}_2$:Mg at 8 K.

Figure 5

$\mu \mathrm{SR}$ frequency of ${\mathrm{ZrO}}_2$:Mg at $B=2$ mT as a function of temperature. The expected frequency from the calibration with an Ag sample is shown for comparison.

Figure 6

Schematic representation of the total energy for the conversion from atomlike interstitial to anion-bound muonium (adapted from Ref. [26]). The dashed line indicates the potential during the first steps of the lattice relaxation. The vertical lines (marked 1 and 2) indicate branching points (see text).

Figure 7

Recovery of the diamagnetic fraction as a function of temperature for ${\mathrm{ZrO}}_2$:Ca (from Ref. [6]).