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Muons are particles with a spin of ½ that can be implanted into a wide range of condensed matter materials to act as a local probe of the surrounding atomic environment. Measurement of the muon’s precession and relaxation provides an insight into how it interacts with its local environment. From this, unique information is obtained about the static and dynamic properties of the material of interest. This has enabled muon spin spectroscopy, more commonly known as muon spin rotation/relaxation/resonance (μSR), to develop into a powerful tool to investigate material properties such as fundamental magnetism, superconductivity and functional materials. Alongside this, μSR may be used to study, for example, energy storage materials, ionic diffusion in potential batteries, the dynamics of soft matter, free radical chemistry, reaction kinetics, semiconductors, advanced manufacturing and cultural artefacts. This Primer is intended as an introductory article and introduces the μSR technique, the typical results obtained and some recent advances across various fields. Data reproducibility and limitations are also discussed, before highlighting promising future developments.

OGATA Koan (1810–63) was a physician and the director of Tekijuku, and he contributed to Western medicine in the late Edo period. Osaka University preserves two of his medicine chests. One of the chests, which was used in his last years (the second chest) contained 22 glass bottles and 6 wooden cylinders. These bottles and cylinders contained formulated medicines; however, about half cannot be opened because of the long-term storage. It is necessary to comprehend the physical property of both the containers and their contents for investigation of this adequate preservation method; however, destructive analysis is not allowed. To analyze the medicines sealed in the glass bottles, we focused on muonic X-ray analysis, which has high transmittance. First, we certified the analytical methods using a historical medicinal specimen preserved in Osaka University. Thereafter, we applied the method on the bottles stored in the second chest. X-ray fluorescence identified the glass of those bottles to be lead potash glass. Among these bottles, we chose the bottle with the label “甘,” which contains white powdered medication, for muonic X-ray analysis. We identified the contents of the medication in the glass to be Hg2Cl2. Through this study, we first applied muonic X-ray analysis on the medical inheritances and succeeded to detect the elements contained both in the container and in the contents of the sealed bottle. This would be a new method for nondestructive analysis of such cultural properties.

In order to study the effect of muonic-Li on the µ− SR spectrum for Li battery materials, the transverse field (TF) µ− SR spectra have been measured for natural Li metal and 6Li metal samples in the temperature range between 5 and 300 K. Here, TF denotes the magnetic field perpendicular to the initial µ− spin polarization. For both samples, the TF-µ− SR spectrum showed a clear oscillation without relaxation down to the lowest temperature measured: i.e., about 5 K. This clearly excludes the scenario that the muonic-Li starts to diffuse at a certain temperature between 5 and 300 K. Although the mechanism on such a zero TF relaxation rate is not fully understood, we could ignore the contribution from the muonic Li on the asymmetry spectrum for the Li-ion battery materials at temperatures above 5 K.

Hard carbon is the most common anode material for Na-ion battery. The structure of the hard carbon and the dynamics of Na-ion in hard carbon were studied with small- and wide-angle neutron scattering and muon spin relaxation technique. The neutron scattering revealed the increase of interlayer distance between graphenes and decrease of the size of nanopores with increasing sodium intercalation in hard carbon. The muon spin relaxation revealed that a systematic increase in the field fluctuation rate with increasing temperature evidenced a thermally activated sodium diffusion. Assuming the two-dimensional diffusion of Na-ion in the graphene layers, the self-diffusion coefficient of Na-ion was estimated to be 2.6 × 10−11 cm2/s at 310 K, with a thermal activation energy of 39(7) meV.

The internal magnetic field in a sodium battery compound, i.e., the paramagnet ${\mathrm{Na}}_{0.7}{\mathrm{CoO}}_{2}$, was investigated with negative muon spin rotation and relaxation (${\ensuremath{\mu}}^{\ensuremath{-}}\mathrm{SR}$), and the result was compared with the results previously obtained with ${\ensuremath{\mu}}^{+}\mathrm{SR}$. The majority of implanted ${\ensuremath{\mu}}^{\ensuremath{-}}$ is captured on an oxygen nucleus, while ${\ensuremath{\mu}}^{+}$ locates an interstitial site. Therefore, a ${\ensuremath{\mu}}^{\ifmmode\pm\else\textpm\fi{}}\mathrm{SR}$ work provides information on the internal magnetic field, which is formed by nuclear magnetic moments of $^{23}\mathrm{Na}$ and $^{59}\mathrm{Co}$, from the two different viewpoints. Besides a slight decrease in the field distribution width ($\mathrm{\ensuremath{\Delta}}$) around 300 K, the nuclear magnetic field detected with ${\ensuremath{\mu}}^{\ensuremath{-}}\mathrm{SR}$ was found to be almost static and temperature independent up to 400 K, even though Na ions are known to start to diffuse above 290 K based on ${\ensuremath{\mu}}^{+}\mathrm{SR}$, Na-NMR, neutron scattering, and electrochemical measurements. Such a discrepancy is caused by the fact that the Na contribution to $\mathrm{\ensuremath{\Delta}}$ is only about 3% at the O site whereas it is about 13% at the interstitial site, where the ${\ensuremath{\mu}}^{+}$ is presumably located.

Angle-resolved photoelectron spectroscopy has been used to follow the valence band and near Fermi edge electronic band structure in the quasi-1D compound NaV 2 O 4 . In this current study we have acquired the very first high-quality, high-resolution ARPES data from this material. Our data clearly reveal two distinct dispersive bands that cross the Fermi level at different k F . This is a clear signature that the electronic properties of this material is affected by the presence of a mixed valence state on the different vanadium chains and possibly also the low-temperature magnetic spin order.

In order to study a diffusive behavior of Li+ in Li intercalated graphites, we have measured muon spin relaxation (mu+SR) spectra for C6Li and C12Li synthesized with an electrochemical reaction bet ...

We have studied a mosaic of 1T-CrSe

We have studied a mosaic of 1T-CrSe$_2$ single crystals using $\beta$-detected nuclear magnetic resonance of $^{8}$Li from 4 to 300 K. We identify two broad resonances that show no evidence of quadrupolar splitting, indicating two magnetically distinct environments for the implanted ion. We observe stretched exponential spin lattice relaxation and a corresponding rate ($1/T_1$) that increases monotonically above 200 K, consistent with the onset of ionic diffusion. A pronounced maximum in $1/T_1$ is observed at the low temperature magnetic transition near 20 K. Between these limits, $1/T_1$ instead exhibits a broad minimum with a remarkable absence of strong features in the vicinity of structural and magnetic transitions between 150 and 200 K. Together, the results suggest $^{8}$Li$^{+}$ site occupation within the van der Waals gap between CrSe$_2$ trilayers. Possible origins of the two environments are discussed.

We applied a positive muon spin rotation and relaxation (μ+SR) technique to battery materials research by investigating Li diffusion in cathode and electrolyte materials. Recently, we have found th ...

The K2Cr8O16 compound belongs to a series of quasi-1D compounds with intriguing magnetic properties that are stabilized through a high-pressure synthesis technique. In this study, a muon spin rotation, relaxation and resonance (μ+SR) technique is used to investigate the pressure dependent magnetic properties up to 25 kbar. μ+SR allows for measurements in true zero applied field and hereby access the true intrinsic material properties. As a result, a refined temperature/pressure phase diagram is presented revealing a novel low temperature/high pressure (pC1 = 21 kbar) transition from a ferromagnetic insulating to a high-pressure antiferromagnetic insulator. Finally, the current study also indicates the possible presence of a quantum critical point at pC2 ~ 33 kbar where the magnetic order in K2Cr8O16 is expected to be fully suppressed even at T = 0 K.

Using an intense negative muon (μ^{-}) source, we have studied the internal magnetic fields in a powder sample of magnesium hydride (MgH_{2}). By extracting the signal from the μ^{-} captured on Mg nuclei, we found that the negative muon spin rotation and relaxation (μ^{-}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 MgH_{2}. As is the case with μ^{+}SR, μ^{-}SR promises to soon be an indispensable tool for materials analyses.

QC 20211215

The microscopic magnetic nature of the $A$-site ordered chromium perovskites ${\mathrm{CaCu}}_{3}{\mathrm{Cr}}_{4}{\mathrm{O}}_{12}$ and ${\mathrm{LaCu}}_{3}{\mathrm{Cr}}_{4}{\mathrm{O}}_{12}$ and their solid-solution system, ${\mathrm{Ca}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Cu}}_{3}{\mathrm{Cr}}_{4}{\mathrm{O}}_{12}$, with $x=0.2$, 0.4, and 0.8, has been studied with muon spin rotation and relaxation (${\ensuremath{\mu}}^{+}\mathrm{SR}$) measurements down to 2 K using a powder sample. For ${\mathrm{CaCu}}_{3}{\mathrm{Cr}}_{4}{\mathrm{O}}_{12}$, ${\ensuremath{\mu}}^{+}\mathrm{SR}$ revealed the formation of static antiferromagnetic (AF) order below 122 K ($={T}_{\mathrm{N}}$), although magnetization measurements showed a very small change at ${T}_{\mathrm{N}}$. Analyses of the internal magnetic field ${H}_{\mathrm{int}}$ at the muon sites, predicted with first-principles calculations, suggested $G$-type AF order as a ground state. For ${\mathrm{LaCu}}_{3}{\mathrm{Cr}}_{4}{\mathrm{O}}_{12}$ with ${T}_{\mathrm{N}}=225$ K, ${\ensuremath{\mu}}^{+}\mathrm{SR}$ also supported the presence of a $G$-type AF ordered state, which was recently proposed based on neutron diffraction measurements. However, the ordered Cr moments were found to change the direction at around 10 K. For ${\mathrm{Ca}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Cu}}_{3}{\mathrm{Cr}}_{4}{\mathrm{O}}_{12}$, both ${T}_{\mathrm{N}}$ and ${H}_{\mathrm{int}}$ at 2 K increase monotonically with $x$.

QC 20211215

We report a muon spin rotation and relaxation (μ+SR) study on Na0.7CoO2 powder samples, where the sodium (Na) has been intercalated via an electrochemical reaction inside a Na-ion battery. The zero field μ+SR measurement at T = 2 K shows a paramagnetic state for the as-grown sample whereas an antiferromagnetic (AF) ordered state is seen for the electrochemically cycled one. Furthermore, the temperature dependence of the muon-spin precession frequencies reveals a Néel transition temperature of TN = 22 K. The results demonstrate the importance of having high-quality homogenous samples, and put the existing NaxCoO2 magnetic phase diagram under debate. QC 20211124

QC 20211215

In this study, the magnetic ground state of the hollandite type material K2Cr8O16 was tuned by externally applied pressure and investigated using µ+SR method in Zero-field (ZF) and weak-transversefield (wTF) configurations. As a result, the obtained magnetic transition temperature for the measuredpressures differs notably from magnetization measurements. Moreover, both wTF and ZF data reveala transition between two different magnetically ordered states at low temperatures for higher pressures. Further theoretical and experimental studies are currently being planned in order to elucidatethe detailed nature of the magnetically ordered phase. QC 20211130

QC 20211124

In order to elucidate the dependence of the magnetic ground state on the Ca content (x) in Sr1-xCaxCo2P2 (0 ≤x ≤1, ThCr2Si2-type structure), we have performed muon spin rotation− and relaxation (μ+SR) experiments on Sr1-xCax Co2P2 powder samples mainly in a zero applied field. The end member compound, SrCo2P2, is found to be paramagnetic down to 19 mK. As x increases, such a paramagnetic ground state is observed down to 1.8K until x = 0.45. Then, as x increases further, a short-range antiferromagnetic (AF) ordered phase appears at low temperatures for 0.48 ≤ x ≤ 0.75, and finally, a long-range AF ordered phase is stabilized for x > 0.75. The internal magnetic field of the other end member compound, CaCo2P2, is well consistent with that of the A-type AF order state, which was proposed from neutron scattering experiments. The phase diagram determined with μ+SR is different from that proposed by macroscopic measurements. For an isostructural compound, LaCo2P2, static magnetic order is found to be formed below

Diffusion of ${\mathrm{Li}}^{+}$ in (111) oriented thin films of the spinels ${\mathrm{Li}}_{4}{\mathrm{Ti}}_{5}{\mathrm{O}}_{12}$ and ${\mathrm{LiTi}}_{2}{\mathrm{O}}_{4}$ has been studied with $^{8}\mathrm{Li}\phantom{\rule{4pt}{0ex}}\ensuremath{\beta}$-detected NMR in the temperature range between 5 and 310 K. In ${\mathrm{Li}}_{4}{\mathrm{Ti}}_{5}{\mathrm{O}}_{12}$, the spin-lattice relaxation rate ($1/{T}_{1}$) versus temperature shows a clear maximum around 100 K ($={T}_{\mathrm{max}}$) which we attribute to magnetic freezing of dilute ${\mathrm{Ti}}^{3+}$ local magnetic moments, consistent with the results of magnetization and muon spin relaxation (${\ensuremath{\mu}}^{+}\mathrm{SR}$) measurements. The decrease in $1/{T}_{1}$ with temperature above ${T}_{\mathrm{max}}$ indicates that ${\mathrm{Li}}^{+}$ starts to diffuse with a thermal activation energy (${E}_{a}$) of 0.11(1) eV. In ${\mathrm{LiTi}}_{2}{\mathrm{O}}_{4}$, on the contrary, as temperature increases from 200 K, $1/{T}_{1}$ increases monotonically up to 310 K. This suggests that Li also starts to diffuse above 200 K with ${E}_{a}=0.16(2)\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ in ${\mathrm{LiTi}}_{2}{\mathrm{O}}_{4}$. Comparison with conventional Li-NMR on ${\mathrm{Li}}_{4}{\mathrm{Ti}}_{5}{\mathrm{O}}_{12}$ implies that both $\ensuremath{\beta}$-NMR and ${\ensuremath{\mu}}^{+}\mathrm{SR}$ sense short-range Li motion, i.e., a jump diffusion of ${\mathrm{Li}}^{+}$ to the nearest neighboring sites.

In order to study a diffusive behavior of Li

The magnetic nature of a metallic two-dimensional triangular compound, ${\mathrm{CrSe}}_{2}$, has been investigated by muon spin rotation and relaxation ($\ensuremath{\mu}^{+}\mathrm{SR}$) measurements using both powder and single crystal samples. It is found that ${\mathrm{CrSe}}_{2}$ enters into a static antiferromagnetic (AF) ordered state below 157 K ($={T}_{\mathrm{N}}$). Furthermore, the AF state is slightly changed below around 20 K ($={T}_{\mathrm{N}2}$). Based on the analysis of the internal magnetic fields at the muon sites predicted with DFT calculations, collinear AF and helical ${120}^{\ensuremath{\circ}}$ AF are clearly eliminated for the ground state of ${\mathrm{CrSe}}_{2}$. The most probable one is an incommensurate spin density wave order.

Elementary excitations of the $S=\frac{1}{2}$ one-dimensional antiferromagnet ${\mathrm{KCuGaF}}_{6}$ were investigated by inelastic neutron scattering in zero and finite magnetic fields perpendicular to the $(1,1,0)$ plane combined with specific heat measurements. ${\mathrm{KCuGaF}}_{6}$ exhibits no long-range magnetic ordering down to 50 mK despite the large exchange interaction $J/{k}_{\mathrm{B}}=103$ K. At zero magnetic field, well-defined spinon excitations were observed. The energy of the des Cloizeaux and Pearson [J. des Cloizeaux and J. J. Pearson, Phys. Rev. 128, 2131 (1962)] mode of the spinon excitations is somewhat larger than that calculated with the above exchange constant. This discrepancy is mostly ascribed to the effective $XY$ anisotropy arising from the large Dzyaloshinsky-Moriya [T. Moriya, Phys. Rev. 120, 91 (1960)] interaction with an alternating $\mathbit{D}$ vector. ${\mathrm{KCuGaF}}_{6}$ in a magnetic field is represented by the quantum sine-Gordon model, for which low-energy elementary excitations are composed of solitons and antisolitons and their bound states called breathers. Unlike the theoretical prediction, it was found that the energy of a soliton is smaller than that of the first breather, although the energy of the first breather coincides with that observed in a previous electron spin resonance measurement.

Lithium diffusion in spinel ${\mathrm{Li}}_{4}{\mathrm{Ti}}_{5}{\mathrm{O}}_{12}$ and ${\mathrm{LiTi}}_{2}{\mathrm{O}}_{4}$ compounds for future battery applications has been studied with muon spin relaxation (${\ensuremath{\mu}}^{+}\mathrm{SR})$. Measurements were performed on both thin-film and powder samples in the temperature range between 25 and 500 K. For ${\mathrm{Li}}_{4}{\mathrm{Ti}}_{5}{\mathrm{O}}_{12}$ and above about $\ensuremath{\sim}200\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, the field distribution width ($\mathrm{\ensuremath{\Delta}})$ is found to decrease gradually, while the field fluctuation rate ($\ensuremath{\nu})$ increases exponentially with temperature. For ${\mathrm{LiTi}}_{2}{\mathrm{O}}_{4}$, on the contrary, the $\mathrm{\ensuremath{\Delta}}(T)$ curve shows a steplike decrease at $\ensuremath{\sim}350\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, around which the $\ensuremath{\nu}(T)$ curve exhibits a local maximum. These behaviors suggest that ${\mathrm{Li}}^{+}$ starts to diffuse above around 200 K for both spinels. Assuming a jump diffusion of ${\mathrm{Li}}^{+}$ at the tetrahedral $8a$ site to the vacant octahedral $16c$ site, diffusion coefficients of ${\mathrm{Li}}^{+}$ at 300 K in the film samples are estimated as $(3.2\ifmmode\pm\else\textpm\fi{}0.8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11} {\mathrm{cm}}^{2}$/s for ${\mathrm{Li}}_{4}{\mathrm{Ti}}_{5}{\mathrm{O}}_{12}$ and $(3.6\ifmmode\pm\else\textpm\fi{}1.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11} {\mathrm{cm}}^{2}$/s for ${\mathrm{LiTi}}_{2}{\mathrm{O}}_{4}$. Further, some small differences are found in both thermal activation energies and Li-ion diffusion coefficients between the powder and thin-film samples.

International audience; We have investigated the microscopic magnetic nature of a novel Na battery material, P2-Na0.5VO2, in which V ions form a two-dimensional triangular lattice, by means of muon-spin rotation and relaxation (μ+SR) measurements down to 50 mK. Although magnetization measurements indicated the presence of an antiferromagnetic transition at 13 K, the internal magnetic field due to the formation of magnetic order appears not at 13 K but at 2 K. Furthermore, the magnetic order is found to have a wide field distribution even at 50 mK. Such wide field distribution is reasonably explained by a combination of multiple muon sites and the formation of a long-range magnetic order, while the reliable spin structure is still unknown.

We have used the pulsed muon source at ISIS to study high-temperature Na-ion dynamics in the quasi-one-dimensional (Q1D) metallic antiferromagnet NaV2O4. By performing systematic zero-field and longitudinal-field measurements as a function of temperature we clearly distinguish that the hopping rate increases exponentially above Tdiff=250 K. The data is well fitted to an Arrhenius type equation typical for a diffusion process, showing that the Na-ions starts to be mobile above Tdiff . Such results makes this compound very interesting for the tuning of Q1D magnetism using atomic-scale ion-texturing through the periodic potential from ordered Na-vacancies. Further, it also opens the door to possible use of NaV2O4 and related compounds in energy related applications., Accepted for publication in Journal of Physics: Conference Series (2014)

In order to elucidate the magnetic ground state of Sr2VO4, we have measured muon spin rotation and relaxation (mu+SR) spectra of a powder sample in the temperature range between 1.8 and 140 K. As a result, we have clarified that the transition at 105 K is not magnetic but structural and/or electric in origin and found the appearance of static antiferromagnetic (AF) order below 8 K. Moreover, the distribution of the internal AF field was found to be very broad, even at the lowest temperature measured. These results are consistent with the formation of an orbital-stripe order with collinear AF order for the magnetic ground state of Sr2VO4.

In order to investigate the microscopic magnetic nature of a sodium phospho- olivine, we have measured μ+SR spectra for NaxFePO4 using a powder sample in the temperature range between 2 and 500 K. ZF-μ+ SR measurements below 200 K at TRIUMF showed the appearance of static magnetic order below TN ~ 61 K, where the susceptibility-vs.- temperature curve exhibits a sharp maximum due to an antiferromagnetic (AF) transition. The wide distribution found for the internal magnetic field (Hint) was explained by the formation of AF order with the spread in Hint due to random distribution of Na vacancies. At higher temperatures, the ZF-spectrum for Na0.7FePO4, obtained at J-PARC, is found to change from a low-T static behavior to a high-T dynamic behavior above ~ 300 K. This is consistent with the fact that Na+ ions are reversibly extracted from and intercalated into the NaxFePO4 lattice, while there is no NMR study on Na-diffusion in NaxFePO4 due to large Fe moments.

In order to study the phase diagram from a microscopic viewpoint, we have measured wTF- and ZF-$\mu^+$SR spectra for the Sr$_{1-x}$Ca$_x$Co$_2$P$_2$ powder samples with $x=0$, 0.2, 0.4, 0.5, 0.6, 0.8, and 1. Due to a characteristic time window and spatial resolution of $\mu^+$SR, the obtained phase diagram was found to be rather different from that determined by magnetization measurements. That is, as $x$ increases from 0, a Pauli-paramagnetic phase is observed even at the lowest $T$ measured (1.8~K) until $x=0.4$, then, a spin-glass like phase appears at $0.5\leq x\leq0.6$, and then, a phase with wide field distribution probably due to incommensurate AF order is detected for $x=0.8$, and finally, a commensurate $A$-type AF ordered phase (for $x=1$) is stabilized below $T_{\rm N}\sim80~$K. Such change is most likely reasonable and connected to the shrink of the $c$-axis length with $x$, which naturally enhances the magnetic interaction between the two adjacent Co planes., Comment: 5 pages, 5 figures, conference paper of muSR2014 (accepted for publication)

The magnetic properties of Cs${}_{2}$Cu${}_{3}$CeF${}_{12}$ were investigated through magnetization and specific heat measurements. Cs${}_{2}$Cu${}_{3}$CeF${}_{12}$ is composed of a buckled kagome lattice of Cu${}^{2+}$, which is stacked along the $b$ axis. The exchange network in the buckled kagome lattice is strongly anisotropic. Consequently, Cs${}_{2}$Cu${}_{3}$CeF${}_{12}$ can be divided into two subsystems: alternating Heisenberg chains with strong antiferromagnetic exchange interactions and dangling spins. The dangling spins couple with one another via effective exchange interactions, which are mediated by chain spins. The dangling spins are further divided into two subsystems, DS1 and DS2. The dangling spins in DS1 undergo three-dimensional ferromagnetic ordering at 3.14 K, while those in DS2 remain paramagnetic down to 0.35 K. The effective interaction between the DS1 spins is approximately expressed by the ferromagnetic $XXZ$ model with the $z$ direction parallel to the crystallographic $c$ axis. A magnetic phase diagram for $H\phantom{\rule{0.16em}{0ex}}\ensuremath{\parallel}\phantom{\rule{0.16em}{0ex}}c$ was obtained and was analyzed within the framework of the molecular field approximation. With increasing magnetic field, the dangling spins are polarized and the magnetization curve exhibits a wide plateau at one-third of the saturation magnetization.

Elementary excitations of the $S=\frac{1}{2}$ one-dimensional Heisenberg antiferromagnet ${\text{KCuGaF}}_{6}$ with exchange constant $J/{k}_{B}=103\text{ }\text{K}$ were investigated by high-frequency electron spin resonance (ESR) measurements combined with a pulsed high magnetic field. When an external magnetic field $H$ is applied in ${\text{KCuGaF}}_{6}$, a staggered magnetic field $h$ is induced perpendicular to $H$ owing to the staggered $\mathbit{g}$ tensor and the Dzyaloshinsky-Moriya (DM) interaction with an alternating $\mathbit{D}$ vector. Consequently, ${\text{KCuGaF}}_{6}$ in a magnetic field is represented by the quantum sine-Gordon (SG) model. We observed many resonance modes including a soliton resonance, breathers, interbreather transitions, and two-breather excitation. Their resonance conditions are beautifully described by the quantum SG field theory with one adjustable parameter ${c}_{s}=h/H$. To investigate the relationship between the Curie term due to the DM interaction and the proportionality coefficient ${c}_{s}$, magnetic susceptibility measurements were also performed varying the external field direction.

In order to clarify the reason why the hydrogen desorption temperature (Td) of MgH2 is lowered by milling, we have studied the change in a local nuclear magnetic field with temperature by means of μ+SR. We have found a very clear oscillation in the ZF-spectrum at 2 K for the "milled" and "milled with Nb2O5" samples, while such oscillation is weaker for the "as prepared" MgH2. It was also found that the oscillation signal is stable up to 250 K and is assigned mainly due to the formation of a H-μ-H system. At temperatures above ambient T, we also found that the ZF-μ+SR spectrum exhibits a static Kubo-Toyabe behavior due to the nuclear magnetic field of 1H. Furthermore, it was clarified that rapid H diffusion starts well below Td only in the milled samples, leading to the conclusion that the consequent enhanced diffusion rate in MgH2 is essential to accelerate the desorption reaction and to decrease Td.