Nuclear Magnetic Field in Solids Detected with Negative-Muon Spin Rotation and Relaxation

Using an intense negative muon (${\mu }^{-}$) source, we have studied the internal magnetic fields in a powder sample of magnesium hydride (${\mathrm{MgH}}_2$). By extracting the signal from the ${\mu }^{-}$ captured on Mg nuclei, we found that the negative muon spin rotation and relaxation (${\mu }^{-}\mathrm{SR}$) spectra clearly showed a Kubo-Toyabe-type relaxation, which indicates a random magnetic field at the Mg site. The field distribution width obtained is very consistent with the predicted value at the Mg site estimated by dipole field calculations, supporting our claim to have observed the nuclear magnetic fields of hydrogens in ${\mathrm{MgH}}_2$. As is the case with ${\mu }^{+}\mathrm{SR}$, ${\mu }^{-}\mathrm{SR}$ promises to soon be an indispensable tool for materials analyses.

Figure 1

(a) The crystal structure of tetragonal ${\mathrm{MgH}}_2$ drawn with vesta [13]. The lattice constants are $a=4.5180(6)\AA$ and $c=3.0211(4)\AA$ with space group $P{4}_2/mnm$ [14]. The time histogram of the TF-${\mu }^{-}\mathrm{SR}$ spectrum in ${\mathrm{MgH}}_2$ for the (b) forward counter and (c) backward counter, and (d) and (e) the reduced difference between the experimental data and fit result ($[N_{\exp }-N_{\mathrm{fit}}]/[N_{\exp }]$) as a function of the reduced experimental error ($\delta N_{\exp }/N_{\exp }$). In (b) and (c), red open circles represent the experimental data, green solid lines represent the fit result using Eq. (1), and blue solid lines represent the histograms of the five decay processes (see Table 1). In (d) and (e), since the fit was performed to minimize ${\chi }^2$, reduced ${\chi }^2(={\chi }^2/n_F)$ for the data is also shown, where $n_F$ is the number of degree of freedom.

Figure 2

(a) The TF-${\mu }^{-}\mathrm{SR}$ asymmetry spectra for ${\mathrm{MgH}}_2$ recorded with $H=50$ and 130 Oe. (b) The difference between the above two asymmetry spectra. In (a), the TF spectrum with 130 Oe is shifted upward by 0.05 for clarity of display. In (b), a solid line represents the fit using a combination of two cosine functions due to the TF oscillations with $H=50$ and 130 Oe.

Figure 3

The TF-, ZF-, and LF-${\mu }^{-}\mathrm{SR}$ asymmetry spectra for ${\mathrm{MgH}}_2$ recorded at room temperature. The TF spectrum was fitted by an exponentially relaxing cosine signal: $A_0{P}_{\mathrm{TF}}(t)=A_{\mathrm{TF}}\exp (-{\lambda }_{\mathrm{TF}}t)\cos ({\omega }_{\mathrm{TF}}t+\varphi )$ with $A_{\mathrm{TF}}=0.02177(14)$, $\varphi =-0.4(1.0)\mathrm{deg}$, and ${\lambda }_{\mathrm{TF}}=0.365(9)\mu {\mathrm{s}}^{-1}$, which corresponds to $1/T_2$ as well as $\mathrm{\Delta }[=0.545(9)\mu {\mathrm{s}}^{-1}]$. It is known that ${\lambda }_{\mathrm{TF}}<\mathrm{\Delta }$ [19], being consistent with the present result.

Figure 4

The TF-, ZF-, and LF-${\mu }^{-}\mathrm{SR}$ asymmetry spectra for ${\mathrm{MgF}}_2$ recorded at room temperature. The green bold line shows a best fit for the ZF and LF spectra using an exponential relaxation signal: $A_{0}P(t)=A_F\exp (-{\lambda }_{F}t)$ with ${\lambda }_F=0.77(5)\mu {\mathrm{s}}^{-1}$. Also, $A_{\mathrm{TF}}=0.0040(3)=A_F$, $\varphi =6(4)\mathrm{deg}$, and ${\lambda }_{\mathrm{TF}}=0.25(5)\mu {\mathrm{s}}^{-1}$ (see Table 1 and the caption of Fig. 3).