# Local electronic structure of interstitial hydrogen in MgH2 inferred from
muon study

R. Kadono<EMAIL_ADDRESS>Muon Science Laboratory and Condensed Matter
Research Center, Institute of Materials Structure Science, High Energy
Accelerator Research Organization (KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan
M. Hiraishi  Institute of Quantum Beam Science, Graduate School of Science and
Engineering, Ibaraki University, Mito, Ibaraki 310-8512, Japan  H. Okabe  Muon
Science Laboratory and Condensed Matter Research Center, Institute of
Materials Structure Science, High Energy Accelerator Research Organization
(KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan  Institute for Materials
Research, Tohoku University, Sendai 980-8577, Japan  A. Koda  Muon Science
Laboratory and Condensed Matter Research Center, Institute of Materials
Structure Science, High Energy Accelerator Research Organization (KEK-IMSS),
Tsukuba, Ibaraki 305-0801, Japan  Department of Materials Structure Science,
The Graduate University for Advanced Studies (Sokendai), Tsukuba, Ibaraki
305-0801, Japan  T. U. Ito  Advanced Science Research Center, Japan Atomic
Energy Agency, Tokai, Ibaraki 319-1195, Japan

###### Abstract

Magnesium hydride has great potential as a solid hydrogen (H) storage material
because of its high H storage capacity of 7.6 wt%. However, its slow
hydrogenation and dehydrogenation kinetics and the high temperature of 300 ∘C
required for decomposition are major obstacles to small-scale applications
such as automobiles. The local electronic structure of interstitial H in MgH2
is an important fundamental knowledge in solving this problem, which has been
studied mainly based on density functional theory (DFT). However, few
experimental studies have been performed to assess the results of DFT
calculations. We have therefore introduced muon (Mu) as pseudo-H into MgH2 and
investigated the corresponding interstitial H states by analyzing their
electronic and dynamical properties in detail. As a result, we observed
multiple Mu states similar to those observed in wide-gap oxides, and found
that their electronic states can be attributed to relaxed-excited states
associated with donor/acceptor levels predicted by the recently proposed
“ambipolarity model”. This provides an indirect support for the DFT
calculations on which the model is based via the donor/acceptor levels. An
important implication of the muon results for improved hydrogen kinetics is
that dehydrogenation, serving as a reduction for hydrides, stabilizes the
interstitial H- state.

##  I Introduction

Hydrogen (H) storage materials are attracting attention for their importance
as a core energy carrier in hydrogen energy production and utilization systems
toward a decarbonized society.Dresselhaus and Thomas (2001); Schlapbach and
Züttel (2001) Therefore, there is a need for materials that can safely and
efficiently store H and easily control the absorption and release of H. In
particular, solid materials such as metal hydrides are characterized by high
safety and high volume and weight densities, and there are great expectations
for their practical use.Sakintuna, Lamari-Darkrim, and Hirscher (2007)

Among metal hydrides, magnesium hydrides have always attracted considerable
interest. Magnesium reacts reversibly with H to form hydride (MgH2), which has
great potential as a solid H storage material because of its high H storage
capacity of 7.6 wt%. However, its slow hydrogenation and dehydrogenation rates
and the high temperature of 300 ∘C required for decomposition are major
obstacles to small-scale applications such as automobiles.Sakintuna, Lamari-
Darkrim, and Hirscher (2007) Several experimental and theoretical studies have
already been published on H kinetics in MgH2 to solve the problem. However,
most of them have concentrated only on macroscopic analysis by thermodynamic
measurements of the effects of ball milling, heavy ion irradiation, or the
addition of trace metals or their oxides on the reaction kinetics. Huot  _et
al._ (1999); Oelerich, Klassen, and Bormann ; Fernández and Sánchez (2002);
Liang (2004); Conradi  _et al._ (2007); Du  _et al._ (2008); Evard, Gabis, and
Yartys (2010)

Understanding the atomic-level mechanisms of hydrogenation and dehydrogenation
processes is key to engineering solutions for accelerating these processes and
lowering the decomposition temperatures. In MgH2, the local electronic state
of isolated H at the interstitial site (Hi) is also one of the most
fundamental pieces of information in elucidating the rate-limiting processes
in H kinetics. Meanwhile, experimental means to study the microscopic states
of isolated H in trace amounts in solids are limited, and so far the ab initio
calculations based on density functional theory (DFT) have been the primary
tool for this purpose.Du  _et al._ (2008); Park, Janotti, and Van de Walle
(2009); Roy, Janotti, and Van de Walle (2013) However, there have been few
experimental studies to assess the theoretical predictions to date.

To overcome this situation, we have introduced muons as pseudo-H into MgH2,
and investigated their electronic states and dynamical properties in detail
using muon spin rotation ($\mu$SR). Because the muon mass is two orders of
magnitude (about 206 times) greater than the electron mass, the adiabatic
approximation is sufficient for understanding muon-electron interaction. In
fact, the difference in the Bohr radius between a muon binding a single
electron, known as muonium, and the corresponding H0 atom is only 0.43%,
implying that they have nearly the same electronic structure in the matter.
Hereafter, we will introduce the elemental symbol Mu (equivalent to H for
hydrogen) to denote the muon as pseudo-H, and refer to the Mu valence
electronic states as Mu+, Mu0, and Mu-. For deliberately avoiding distinction
between Mu+ and Mu-, both are referred to together as “diamagnetic Mu” and Mu0
is called “paramagnetic Mu”.

In comparing experimental and theoretical results, there is an important issue
to be pointed out at this stage. In conventional $\mu$SR experiments, muons
are implanted as a relatively high-energy ion beam (typical kinetic energy of
$\sim$4 MeV), which generates free carriers and excitons in insulator crystals
due to the associated electronic excitation towards the end of the radiation
track.Itoh (1997) These often propagate rapidly in the crystal, and there is
experimental evidence that Mu acts as a capture center to form relaxed-excited
states upon interaction with free carriers and/or excitons.Hiraishi  _et al._
(2022) Accordingly, these Mu states do not necessarily correspond to the
electronic state predicted from the thermodynamic charge conversion level
($E^{+/-}$) obtained by DFT calculations for thermal equilibrium conditions.
Moreover, Mu is often observed to be accompanied by fast spin relaxation,
which is due to spin/charge exchange interactions with excited carriers and/or
excitons. While these phenomena are not observed for H in thermal equilibrium
without such carriers, they can be regarded to simulate H in bulk excited
states induced by light or electric fields, and Mu is useful as a microscopic
probe for such states.

Another well-known evidence that Mu exhibits relaxed-excited states is that
two or more different electronic states of Mu are often observed
simultaneously in the same material. To understand this phenomenon, we
recently investigated the electronic states of Mu observed in various oxide
insulators and semiconductors, and compared them with the results of DFT
calculations in the literature. As a result, assuming that these relaxed-
excited states correspond to the electronic states associated with acceptor
($E^{0/-}$) and donor ($E^{+/0}$) levels predicted by DFT calculations (as
metastable states),Lichti, Chow, and Cox (2008) we found that the valence of
Mu can be predicted from the position of $E^{0/-}$ and/or $E^{+/0}$ in the
band structure.Hiraishi  _et al._ (2022) This allows us to describe the
previous experimental results in a unified manner, and we call it the
“ambipolarity model” because it is ascribed to the fact that the ambipolarity,
which is the true nature of H, is expressed through these relaxed-excited
states.

MgH2 is a typical ionic insulator with a band gap above 5 eV, and the
ambipolarity model combined with the results of DFT calculations for H
predicts that Mu simultaneously takes on both acceptor and donor-like
electronic states. The present experimental results are consistent with this
prediction, suggesting that the ambipolarity model holds semi-quantitatively
in MgH2. In the following, we will therefore organize the experimental results
in terms of this model and attempt to interpret them based on their
consistency with the predictions from the corresponding DFT calculations.

##  II Methodology

###  II.1 $\mu$SR Experiment

The sample used for $\mu$SR measurements was obtained from a commercial vendor
(Fujifilm Wako Chmicals Co.) as a powdered reagent sealed with inert gas.
Immediately after opening the container, approximately 2 g of the powder was
pressurized and shaped into a disk covered with aluminum foil, mounted on a
temperature-controlled cryostat, and then degassed into a vacuum. The $\mu$SR
experiments were performed using the ARTEMIS experimental apparatus installed
in the S1 area at MLF in J-PARC, and $\mu$SR spectrum [time-dependent positron
asymmetry, $A(t)$] was measured from room temperature to 5 K in zero magnetic
field (ZF) and under longitudinal magnetic fields (LF, 0.01–0.4 T).

The spin relaxation of the diamagnetic Mu is governed by a random local fields
at the Mu site ${\bm{H}}_{\rm n}$ from the nuclear magnetic moments, and the
magnitude of the relaxation rate is determined by the second moment of
${\bm{H}}_{\rm n}$,

$\Delta^{2}=\gamma_{\mu}^{2}\langle|{\bm{H}}_{\rm
n}|^{2}\rangle=\gamma_{\mu}^{2}\sum_{n}f_{n}\sum_{i}\sum_{\alpha,\beta}[\gamma_{n}(\hat{A}_{\rm
d}^{ni})^{\alpha\beta}\overline{{\bm{I}}}_{ni}]^{2},$ (1)

where $\gamma_{\mu}=2\pi\times 135.53$ MHz/T is the gyromagnetic ratio of Mu
spin, ${\bm{I}}_{ni}$ is the $n$th kind of nuclear spin at distance $r_{ni}$
($n=1,2$ for 1H, 25Mg) on the $i$th lattice point (with
$\gamma_{n}\overline{{\bm{I}}}_{ni}$ being the effective magnetic moment
considering the electric quadrupole interaction for ${\bm{I}}_{ni}\geq 1$),
$\gamma_{n}$ and $f_{n}$ are the gyromagnetic ratio and natural abundance (or
occupation) of the $n$th nuclear spin, $\hat{A}_{\rm d}^{ni}$ is the magnetic
dipole tensor

$(\hat{A}_{\rm
d}^{ni})^{\alpha\beta}=\frac{1}{r^{3}_{ni}}(\frac{3\alpha_{ni}\beta_{ni}}{r^{2}_{ni}}-\delta_{\alpha\beta})\>\>(\alpha,\beta=x,y,z),$
(2)

representing the hyperfine interaction between Mu and nuclear magnetic moments
at ${\bm{r}}_{ni}=(x_{ni},y_{ni},z_{ni})$ being the position vector of the
nucleus seen from Mu. The sum in Eq. (1) takes all $x,y,z$ for $\beta$, and
$x,y$ for the $\alpha$ components that are effective for longitudinal
relaxation when $\hat{z}$ is the LF direction; the $z$ component does not
contribute to the relaxation because it gives a magnetic field parallel to the
Mu spin.

When the coordination of the nuclear magnetic moment viewed from Mu is
isotropic and the number of coordination $N$ is sufficiently large ($N\gtrsim
4$), the time-dependent spin relaxation under zero external field is described
by the Gaussian Kubo-Toyabe (GKT) function

$G_{z}^{\rm
KT}(t;\Delta)=\frac{1}{3}+\frac{2}{3}(1-\Delta^{2}t^{2})e^{-\frac{1}{2}\Delta^{2}t^{2}}.$
(3)

In the case of fluctuating $\Delta$ over time (e.g., due to the self-diffusion
of Mu), $G_{z}^{\rm KT}(t;\Delta)$ is subject to the modulation and
adiabatically approaches exponential decay with increasing fluctuation rate
$\nu$, where the detailed lineshape of $G_{z}^{\rm KT}(t;B_{0},\Delta,\nu)$ as
a function of $\nu$ and LF ($=B_{0}$) are found elsewhere.Hayano  _et al._
(1979) For a small number of nuclei, the time evolution of the Mu spin
polarization exhibits considerable deviation form the GKT function. The exact
time dependence can be derived from the density matrix of the Mu-nucleus spin
system for the $N=1$ and 2 cases,Brewer  _et al._ (1986); Nishiyama,
Nishiyama, and Higemoto (2003); Wilkinson and Blundell (2020) and the latter
is introduced to describe H-Mu-H hydrogen bonding state.Kadono  _et al._
(2008); Umegaki  _et al._ (2014) Assuming a static collinear geometry with
$\mu^{+}$ at the center of the line joining two other nuclear spins ($I=1/2$),
the time evolution of muon polarization as a cubic average is calculated by
solving a simple three-spin model to yieldBrewer  _et al._ (1986)

$\displaystyle G_{z}^{\rm 3S}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{6}\\{3+\cos(\sqrt{3}\omega_{\rm
3S}t)+\alpha_{+}\cos(\beta_{+}\omega_{\rm 3S}t)$ (4)
$\displaystyle\>\>+\alpha_{-}\cos(\beta_{-}\omega_{\rm
3S}t)\\}e^{-\lambda_{\rm 3S}t},$

where $\alpha_{\pm}=1\pm 1/\sqrt{3}$, $\beta_{\pm}=(3\pm\sqrt{3})/2$,
$\omega_{\rm 3S}$ is the dipolar interaction frequency

$\omega_{\rm 3S}=\gamma_{\mu}\gamma_{1}/r_{\rm t}^{3},$ (5)

with $r_{\rm t}$ being the distance between $\mu^{+}$ and the proton nucleus,
$\gamma_{1}/2\pi=42.58$ MHz/T for 1H, and $\lambda_{\rm 3S}$ is the relaxation
rate describing the influence of local fields from outer-shell nuclear
magnetic momentsWilkinson and Blundell (2020) and spin/charge exchange
interaction.

In the case of fast fluctuation ($\nu>\Delta$), we have an approximated form
of the GKT function

$G_{z}^{\rm KT}(t;B_{0},\Delta,\nu)\simeq\exp(-\lambda t),$ (6)

where

$\lambda\simeq\frac{2\Delta^{2}\nu}{\omega^{2}+\nu^{2}},$ (7)

and $\omega=\gamma_{\mu}B_{0}$. Note that when the fluctuation rate satisfies
the condition $\nu\gg\omega$, Eq. (7) is least dependent on $B_{0}$.

The magnitude of $\Delta$ is sensitive to the size of the nearest-neighbor
nuclear magnetic moment $\gamma_{n}\overline{{\bm{I}}}_{ni}$ and the distance
$|{\bm{r}}_{ni}|$ from Mu, and the position occupied by Mu as pseudo-H can be
estimated by comparing the experimentally obtained $\Delta$ with the
calculated value at the candidate sites. We used the DipElec code Kojima  _et
al._ (2004) to calculate $\Delta$ for MgH2.

The $\mu$SR time spectrum shown by the paramagnetic Mu is complicated by the
hyperfine interaction between $\mu^{+}$ and the bound electron. In addition to
the limited time resolution of pulsed beam experiments, it is difficult to
directly observe the Mu spin precession motion associated with transitions
between hyperfine levels due to the dominance of fast spin/charge exchange
reactions in MgH2. Even in such a case, important information can be extracted
by examining the response of the time spectrum $G_{z}(t)$ to the LF in detail.

First, consider Mu0 under a magnetic field ($B_{0}$) applied parallel the
initial Mu polarization ($\parallel z$) without spin fluctuations (static). In
this case, we have

$G_{z}(t)=\frac{1}{2(1+x^{2})}\left[(1+2x^{2})+\cos(\omega_{c}\sqrt{1+x^{2}})t\right],$
(8)

where $x=B_{0}/B_{c}$ with $B_{c}=\hbar\omega_{c}/(\gamma_{\mu}-\gamma_{e})$
being the effective local field exerted on $\mu^{+}$, $\omega_{c}$ is the
hyperfine parameter (angular frequency for the isotropic Fermi contact term),
and $\gamma_{e}$ is the electron gyromagnetic ratio ($=2\pi\times 28024.2$
MHz/T).Patterson (1988) For the zero external field ($x=0$),

$G_{z}(t)=\frac{1}{2}(1+\cos\omega_{c}t).$ (9)

The implication is that a half of the ground state is not an eigenstate of the
hyperfine interaction, so the corresponding component exhibits oscillation
between the spin-singlet and triplet states with $\omega_{c}$. In a practical
material, $\omega_{c}$ is often smaller than that in a vacuum ($\omega_{\rm
vac}=2\pi\times 4463.3$ MHz) due to the effect of dielectric constant, etc.,
but it is often not visible in ordinary time-resolution measurements (i.e.,
the second term is zero when time-averaged). Thus, half of the implanted muons
effectively lose their polarization upon Mu0 formation (also called the
missing fraction), and only the triplet state [the first term of Eq. (8)]
becomes observable. The situation is similar for $B_{0}>0$, where the time-
averaged polarization at $t=0$ is

$G_{z}(t)\simeq\overline{G}_{z}(0)=\frac{1+2x^{2}}{2(1+x^{2})}.$ (10)

As $x$ is normalized by $\omega_{c}$, the field dependence of
$\overline{G}_{z}(0)$ varies with $\omega_{c}$. Moreover, Mu0 in MgH2 is
subjected to the nuclear hyperfine (NHF) interaction between the bound
electron and surrounding H atoms which have proton nuclear spins.Patterson
(1988) This further reduces $\overline{G}_{z}(0)$ to $1/2\times 1/3=1/6$,
reflecting the random orientation of the nuclear spins, and leads to a two-
step recovery of $\overline{G}_{z}(0)$ with respect to $x$.Beck, Meier, and
Schenck (1975) In any case, the magnitude of $\omega_{c}$ can be determined by
comparing the magnetic field dependence of experimentally obtained
$\overline{G}_{z}(0)$ with Eq. (10).

The effect of spin fluctuations on Mu0 under LF can be derived by
approximating the spin/charge exchange reaction with excited carriers by a
strong collision model. In particular, when the reaction rate $\nu$ is lower
than $\omega_{c}$, the time-dependent polarization is given by the following
equations,Kadono, Macrae, and Nagamine (2003)

$G_{z}(t)\simeq\overline{G}_{z}(0)\exp(-\lambda t),$ (11)
$\lambda\simeq\frac{\nu}{2(1+x^{2})}.$ (12)

When Mu0 exhibits self-diffusion, the fluctuation of the NHF interaction also
induces Mu spin relaxation. In this case, the magnetic field dependence of the
longitudinal relaxation rate is expected to be weaker than that in Eq. (12)
and to vary significantly with the jump frequency of Mu0.Kiefl  _et al._
(1989); Kadono  _et al._ (1990) However, the Mu0 relaxation rate observed in
MgH2 is so large ($>10^{1}$ MHz) even at LF = 0.4 T over the wide temperature
range (5–298 K) that such a possibility as the origin of longitudinal
relaxation seems unlikely. In the following, we assume that the primary cause
of the spin fluctuation for Mu0 is spin/charge exchange reactions with excited
carriers.

###  II.2 DFT Calculations

The donor/acceptor levels for H assessed in this work are those based on Ref.
[Roy, Janotti, and Van de Walle, 2013]. They performed DFT calculations with
screened hybrid functions of Heyd, Scuseria, and Ernzerhof (HSE) as
implemented in the Vienna Ab initio Simulation Package (VASP) code. The amount
of exact exchange mixed into the HSE generalized gradient functional is taken
to 33%, which reproduces the experimentally observed lower limit of the band
gap (5.2 eV). Defect calculations were performed for a supercell containing 72
atoms with $2\times 2\times 3$ iterations of the primitive cell. A mesh of
$2\times 2\times 2$ special ${\bm{k}}$ points and an energy cutoff value of
270 eV for the plane-wave basis set were used for the defective supercells.
For charged defects, the number of electrons in the supercell was varied
according to the format described in Ref. [Van de Walle and Neugebauer, 2004]
and the electrons were localized on the impurity.

We have also performed DFT calculations to investigate in detail the local
structure of H-related defects and to evaluate hyperfine parameters of Mu0 and
$\Delta$ of the diamagnetic Mu; the comparison of these with experimental
values allows us to identify the corresponding donor/acceptor states involving
Mu. The DFT calculations were within the generalized gradient approximation
(GGA) and the projector augmented wave method as implemented in the QUANTUM
ESPRESSO (QE) code.Giannozzi  _et al._ (2009) The defects were simulated using
48-atom host supercell with 2$\times$2$\times$2 iterations of the primitive
cell, and the Brillouin-zone integrations implemented by summations over a
Monkhorst-Pack $k$-point mesh of 3$\times$3$\times$5, with an energy cutoff of
100 Ry for the plane-wave basis set. The calculated lattice parameters
$c=0.301$ nm, $c/a=0.668$, and $u=0.304$ are in good agreement with
experimental values of 0.301 nm, 0.67, and 0.304.Bortz  _et al._ (1999) The
band structure calculated for special $k$-points is shown in Fig. 1, which is
in good agreement with the previous GGA calculations for the host.Park,
Janotti, and Van de Walle (2009) It is noteworthy that the band gap is
indirect (3.71 eV), and that the relatively steep dispersion near the
conduction band minimum (CBM) and valence band maximum (VBM) suggests high
mobility of excited carriers as well as a long lifetime.

Figure 1: Band structure of MgH2 calculated by QUANTUM ESPRESSO code (see
text). Filled circles indicate the conduction band minimum and valence band
maximum.

##  III Result

Figure 2 shows the observed ZF/LF-$\mu$SR time spectra at characteristic
temperatures. It can be seen in the spectra at 298 K that the signal from the
sample mainly consists of two components with nearly equal amplitudes, a
diamagnetic component roughly described by the Kubo-Toyabe function under zero
magnetic field,Hayano  _et al._ (1979) and a paramagnetic component whose
initial spin polarization is recovered by LF$\simeq 0.3$ T. On the other hand,
two more components occur at the lowest temperatures, suggesting that the
spectra consist of four components in total, as designated in Fig. 2a.

A closer look at the LF dependence of Fig. 2 implies that at fields as low as
10 mT, the initial asymmetry sharply decreases from 0.22 (=$A_{\rm tot}$) to
the value roughly corresponding to that of the diamagnetic component [$A(0)$
at ZF, $A_{\rm KT}\sim$0.09–0.14] within $\sim$0.5 $\mu$s at all temperatures.
This suggests that the amplitude corresponding to the spin-triplet state of
Mu0 [$\simeq(A_{\rm tot}-A_{\rm KT})/2$] is lost due to the fast spin/charge
exchange reaction with the charged carriers.Patterson (1988) In the spectrum
at 5 K, there is a non-relaxing asymptotic component whose amplitude recovers
with magnetic field at $t>0.5$ $\mu$s, suggesting a transition to the
diamagnetic state via an irreversible charge exchange reaction. Furthermore,
in the spectra at 56 K and 130 K, the final diamagnetic state shows a slow
longitudinal relaxation that is almost independent of LF, suggesting that fast
fluctuations of the internal magnetic field occur with increasing temperature
even in the final state.

Figure 2: $\mu$SR time spectra under zero and longitudinal magnetic fields in
MgH2 at typical temperatures. a) 5 K, b) 56 K, c) 130 K, d) 204 K, e) 298 K.
a) Right: time spectrum shows an initial paramagnetic Mu (Mup, whose asymmetry
is recovered by a longitudinal field) that exhibits conversion to a
diamagnetic Mu state (Mud). Another paramagnetic state (Mup2, showing fast
relaxation) and a diamagnetic Mu component showing the lineshape well
represented by the Kubo-Toyabe relaxation function (MuKT) can be seen. The
solid line shows the result of the global fit based on the model (see text).

The component showing slow relaxation in the ZF spectra can be attributed to
the diamagnetic Mu state(s), because the relaxation is completely quenched by
relatively small LF of 10 mT. It is described by the quasi-static Kubo-Toyabe
relaxation at 298 K, while it is composed of the Kubo-Toyabe component and
that showing relatively fast exponential relaxation at 5 K. Since previous
studies have interpreted the latter as part of the rotational signal
associated with the formation of the H-Mu-H complex,Umegaki  _et al._ (2014)
we also assume the similar situation in our sample. The detailed analysis on
the LF dependence of these spectra suggested contribution of another
paramagnetic species with a small amplitude at lower temperatures.

Figure 3: A schematic diagram of the phenomenological model involving Mup/Mud,
MuKT, Mup2, and Mu3S states and their spin/charge exchange reactions
represented by Eqs. (13)–(15). Here, the paramagnetic state are expressed as
para and the diamagnetic states as dia.

Based on the above observations, we assumed four components: a pair of Mu0
(labeled Mup) and a diamagnetic state (Mud) converted from Mu0, a diamagnetic
state showing Kubo-Toyabe type relaxation (MuKT), the secondary paramagnetic
state (Mup2), and the H-Mu-H complex (Mu3S, diamagnetic state). The time
spectrum was then analyzed by the least-squares method using the following
phenomenological model.

$\displaystyle A(t)$ $\displaystyle=$ $\displaystyle A_{\rm pd}G_{\rm
P}(t)+A_{\rm p2}G_{\rm p2}(t)$ (13) $\displaystyle+A_{\rm KT}G_{z}^{\rm
KT}(t)+A_{\rm 3S}G_{z}^{\rm 3S}(t),$ $\displaystyle G_{\rm pd}(t)$
$\displaystyle=$ $\displaystyle fe^{-(\lambda_{\rm
p}+\kappa)t}+(1-f)e^{-\lambda_{\rm d}t},$ (14) $\displaystyle\lambda_{\rm p}$
$\displaystyle=$ $\displaystyle\frac{\nu_{\rm p}}{2(1+x_{\rm
p}^{2})}\>,\>\>\>f=\frac{\lambda_{\rm p}}{\lambda_{\rm
p}+\kappa}\>,\>\>\>\lambda_{\rm d}\simeq\frac{2\Delta_{\rm d}^{2}\nu_{\rm
d}}{\omega^{2}+\nu_{\rm d}^{2}}\>,$ $\displaystyle G_{\rm p2}(t)$
$\displaystyle=$ $\displaystyle e^{-\lambda_{\rm p2}t},$ (15)
$\displaystyle\lambda_{\rm p2}$ $\displaystyle\simeq$
$\displaystyle\frac{\nu_{\rm p2}}{2(1+x_{\rm p2}^{2})}\>,$

where $A_{\rm pd}$ is the sum of partial asymmetry for Mup and Mud components,
$A_{\rm p2}$, $A_{\rm KT}$, and $A_{\rm 3S}$ are the partial asymmetry of the
respective Mup2, MuKT, and Mu3S ($A_{\rm tot}=A_{\rm pd}+A_{\rm p2}+A_{\rm
KT}+A_{\rm 3S}$), $f$ is the fractional yield of Mup, $\nu_{\rm p}$ is the
spin/charge exchange rate, $\kappa$ is the charge conversion reaction rate
from Mup to Mud,Kadono, Macrae, and Nagamine (2003) $x_{\rm
p}=B_{0}(\gamma_{\mu}-\gamma_{e})/\omega_{\rm p}$, $\omega_{\rm p}$ is the
hyperfine parameter (angular frequency) of Mup, $\Delta_{\rm d}$ is the
nuclear magnetic dipolar linewidth, $\nu_{\rm d}$ is the fluctuation rate of
$\Delta_{\rm d}$ including the effect of spin/charge exchange reaction. The
parameter $f$ is determined by the competition between $\kappa$ (varies with
temperature) and $\lambda_{\rm p}$, and it also depends on LF. For Mup2, the
relaxation due to the spin/charge exchange reaction ($\lambda_{\rm p2}$) is
considered as in the case of Mup. $\nu_{\rm p2}$ is the spin/charge exchange
rate for the Mup2 component, $x_{\rm
p2}=B_{0}(\gamma_{\mu}-\gamma_{e})/\omega_{\rm p2}$, $\omega_{\rm p2}$ is the
hyperfine parameter of Mup2. $G^{\rm KT}_{z}(t;\Delta_{\rm KT},\nu_{\rm KT})$
is the Kubo-Toyabe function for MuKT with $\Delta_{\rm KT}$ and $\nu_{\rm KT}$
being the linewidth and fluctuation frequency. $G_{z}^{\rm 3S}(t;\omega_{\rm
3S},\lambda_{\rm 3S})$ is the relaxation function for the H-Mu-H system [Eqs.
(4),(5)]. A schematic representation of the relationship between the four
states and the meaning of the parameters in the Eqs. (13)–(15) is shown in
Fig. 3.

Figure 4: Temperature dependence of the parameters obtained from the analysis
of ZF/LF-$\mu$SR time spectra: a) the initial asymmetry of each component, b)
the fractional yield of Mup, c) spin/charge fluctuation frequency of the Mup
component ($\nu_{\rm p}$) and conversion rate to Mud ($\kappa$), d), e)
nuclear dipolar linewidths of Mud and MuKT ($\Delta_{\rm d}$ and $\Delta_{\rm
KT}$) and their fluctuation frequencies ($\nu_{\rm d}$ and $\nu_{\rm KT}$), f)
hyperfine parameters of the Mup2 component ($\omega_{\rm p2}$) and its spin
fluctuation frequency ($\nu_{\rm p2}$), g) the precession frequency
($\omega_{\rm 3S}$) and relaxation rate ($\lambda_{\rm 3S}$) of the H-Mu-H
three-spin system.

In the curve-fitting analysis based on the least-squares minimization,
$\omega_{\rm p}$ showed strong correlation with $\nu_{\rm p}$ and $\kappa$,
and in the region where the latter is large (above $\sim$150 K, Fig. 4c), the
accuracy of $\omega_{\rm p}$ determination sharply dropped. This made it
difficult to estimate $\omega_{\rm p}$ only from the LF dependence of the time
spectra under such circumstances. Therefore, we performed analysis by fixing
$\omega_{\rm p}$ to the average value $\omega_{\rm p}=2\pi\times 721.1$ MHz
obtained in the preliminary analysis for temperatures below 50 K where $A_{\rm
tot}$ is nearly constant. The Mu3S signal was hard to be discerned from those
of other components above $\sim$150 K, and $A_{\rm 3S}$ was set to zero for
the analysis of data at higher temperatures. Meanwhile, since $\omega_{\rm
p2}$ showed a tendency of changing with temperature, we assumed that the
hyperfine interaction of the second paramagnetic component Mup2 could change
with temperature. As indicated by the solid line in Fig. 2, this
phenomenological model reasonably reproduces the experimental data including
the LF dependence at all temperatures. The temperature dependence of the
parameters obtained from the curve-fits is shown in Fig. 4.

The electronic state of Mu${}_{\rm p}^{0}$ is characterized by a relatively
large $\omega_{c}$ ($\simeq 16.2$% of $\omega_{\rm vac}$). As is evident in
Figs. 4a, b and c, the decrease in $A_{\rm pd}$ and total signal amplitude
$A_{\rm tot}$ seen over 50–150 K is in line with the decrease in conversion
rate $\kappa$. On the other hand, $\nu_{\rm p}$ appears to be almost constant
over this temperature range. Therefore, the decrease of $A_{\rm pd}$ and
$A_{\rm tot}$ suggests that the spin relaxation is induced by the fast
spin/charge exchange reaction ($\nu_{p}$) as the lifetime of the Mup state
($\kappa^{-1}$) becomes longer. The magnitude of $\Delta_{\rm d}$ provides a
clue to the local electronic structure of the final Mud state. As shown in
Fig. 4d, the average value of $\Delta_{\rm d}$ in the 50–150 K region is
2.71(2) MHz, which is an order of magnitude larger than that for the
interstitial site(s) (see below).

Regarding MuKT characterized by the quasi-static Kubo-Toyabe relaxation, the
magnitude of the linewidth $\Delta_{\rm KT}$ again provides a strong clue to
the microscopic local structure. As shown in Fig. 4e, $\Delta_{\rm KT}$ is
0.32–0.36 MHz in all temperature regions, while it is calculated to be greater
than 0.63 MHz for any interstitial site without lattice relaxation. Although
an exotic Mu state in which Mu is delocalized over two neighboring sites has
been proposed in previous studiesSugiyama  _et al._ (2019), another
possibility is favored from DFT calculations for H, as discussed below.

Although the accuracy of the deduced parameters shown in Fig. 4f seems limited
due to its small amplitude ($A_{\rm p2}/A_{\rm tot}\simeq$5–10%), we find for
the Mup2 state that the entire time spectrum can be consistently reproduced
when we assume that the relaxation is that exhibited by a paramagnetic state
of about $\omega_{\rm p2}\sim 2\pi\times 10^{2}$ MHz. The magnitude of
$\omega_{\rm p2}$ is about 2% of $\omega_{\rm vac}$ and significantly smaller
than $\omega_{\rm p}$. A closer look reveals that $\omega_{\rm p2}$ exhibits
three different electronic states around 50 K and 180 K, and the change in
$\nu_{\rm p2}$ also suggests that it has different electronic states at each
stage.

Finally, let us give a brief overview on the Mu3S state. The deduced H-Mu-H
signal exhibits relatively large relaxation rates at all temperatures,
$\lambda_{\rm 3S}\gg\omega_{\rm 3S}$, which makes it difficult to clearly
identify the oscillatory signal characteristic to the three-spin systems.
While the mean value of $\omega_{\rm 3S}/2\pi$ averaged over temperature can
be derived to be 0.41(2) MHz [corresponding H-Mu distance $r_{\rm t}=0.098(5)$
nm], the actual systematic uncertainty is presumed to be larger than that
determined statistically. The signal appears to marge with that of the Kubo-
Toyabe component above $\sim$150 K, implying the instability of the H-Mu-H
state and the occurrence of a thermal conversion (annealing) process to MuKT.

##  IV Discussion

###  IV.1 Electronic level and local defect structure of each Mu state

In order to compare these results with the predictions of ambipolarity model
plus DFT calculations, we extracted information related to the formation
energy of H-related defects from the literatureRoy, Janotti, and Van de Walle
(2013) and summarized it for Mu as shown in Fig. 5. The thermodynamic defect
formation energy $\Xi^{q}$ of Hq (with $q=0,\pm$ being the valence) in a
certain material is evaluated by DFT calculations, etc. using the following
formula,

$\displaystyle\Xi^{q}(E_{F})=E_{\rm tot}[{\rm H}^{q}]-E_{\rm tot}[{\rm
host}]+qE_{F}-\mu_{\mathrm{H}},$

where $E_{\rm tot}[{\rm H}^{q}]$ and $E_{\rm tot}[{\rm host}]$ are the total
energies calculated for the supercell with and without Hq between lattices,
respectively; $\mu_{\rm H}$ is the H chemical potential, and $E_{F}$ is the
Fermi energy. We assume that Mu’s experimental results correspond to
calculations in the limit where the added H is dilute, i.e., under H-poor
conditions in terms of chemical potential.

Figure 5: The formation energy ($\Xi^{q}$) of H-related defects vs the Fermi
level ($E_{F}$) in MgH2 obtained by DFT calculations. a) is for the case of
interstitial H atoms (H${}^{q}_{\rm i}$, $q=0,\pm$), b) is for the
interstitial molecules ([H${}_{2}]^{q}_{\rm i}$, $q=0,\pm,2\pm$). [HMu] and
[HMu]i respectively stand for the onsite and interstitial H-Mu molecules. The
paramagnetic state are expressed as “para” and the diamagnetic states as
“dia”. The donor/acceptor levels are determined as cross points between
$\Xi^{\pm}(E_{F})$ and $\Xi^{0}(E_{F})$ with their attributions to the
experimentally observed states indicated (see text). The dashed line shows the
intrinsic charge neutral level ($E_{F}^{\rm int}$) obtained from the DFT
calculation. (Adapted from Ref. [Roy, Janotti, and Van de Walle, 2013])
Schematic band diagrams for the electron energy associated with the
donor/acceptor levels (single-charge conversion levels only) in a) and b) are
shown in c) and d), respectively. The level indicated by a dashed line near
the CBM in d) is a shallow state suggested from DFT calculations. For the
corresponding local electronic structure, see Figs. 6,7.

Firstly, regarding the general trend of $\Xi^{q}$ obtained for interstitial H
(Hi) shown in Fig. 5a, the acceptor and donor levels ($E^{0/-}$, $E^{+/0}$)
are on the higher energy side than the double charge conversion level
($E^{+/-}$), both forming apparently deep ingap levels. This indicates that
the effective Coulomb repulsion energy between electrons in the same orbital
is negative (negative $U$) due to strong electron-lattice interactions. The
charged acceptor-like H (H${}_{\rm i}^{-}$, corresponding to Mu${}_{\rm
i}^{-}$) attracts Mg2+ while the electronic lattice interactions push out the
ligand H- atoms of the surrounding MgH6 by 0.052 nm.Park, Janotti, and Van de
Walle (2009) However, the associated defect level is expected to be close to
the VBM, comprising an effectively shallow level for electronic
excitations.Park, Janotti, and Van de Walle (2009) On the other hand, the
charged donor-like H (H${}_{\rm i}^{+}$, corresponding to Mu${}_{\rm i}^{+}$)
binds strongly to the ligand H, forming a [HMu] molecule corresponding to the
onsite H2 molecule.Park, Janotti, and Van de Walle (2009) The energetics for
the electronic parts are shown in Fig. 5c.

Figure 6: Local structure of interstitial H (Hi) in a) positive, b) neutral,
and c) hydride states which are substituted with Mu (red balls), viewed from
$b$ (left) and $c$ (right) axes, respectively.

Secondly, provided that Mu can form an interstitial molecule with a
preexisting Hi atom (denoted as [HMu]i to be discerned from the onsite [HMu]
molecule for Mu${}_{\rm i}^{+}$), the electronic states for the interstitial
H2 molecules ([H${}_{2}]_{\rm i}$) predicted by the DFT calculations can
provide the basis for discussing those of [HMu]i. As shown in Fig. 5b, the
[HMu]${}_{\rm i}^{0}$ state is stable over a wide $E_{F}$ region in the band
gap, showing lower formation energy than that for [HMu]${}_{\rm i}^{+}$ and
[HMu]${}_{\rm i}^{-}$ (i.e., positive-$U$), and can exist even in thermal
equilibrium. The $E^{0/-}$ and $E^{+/0}$ levels are closer to the band edge
than for the isolated Mu, and $E^{0/-}$ appears to overlap the conduction band
minimum (CBM). Regarding the local structure, the [HMu]${}_{\rm i}^{+}$
molecule is pulled to ligand H- to form a paramagnetic complex state (see
below). The double charge conversion levels for [HMu]${}_{\rm i}^{2+}$ and
[HMu]${}_{\rm i}^{2-}$ are closer to the charge neutral level ($E_{F}^{\rm
int}$) and form deeper levels. The [HMu]${}_{\rm i}^{2+}$ molecular states are
unstable, and both H and Mu are presumed to form onsite [H${}_{2}]^{0}$ and
[HMu]0 molecules with ligand H-, respectively. The energetics for the
electronic parts are shown in Fig. 5d.

In order to obtain further clues about the correspondence between these DFT
predictions and the experimental results for Mu, we have investigated the
local defect structure associated with these acceptor/donor levels in each of
the above two cases by QE calculations. The obtained local structures for Hi
and [H${}_{2}]_{\rm i}$ are shown in Figs. 6 and 7, respectively, where those
for Hi are in good agreement with earlier GGA results using VASP code.Park,
Janotti, and Van de Walle (2009) We found that the $E^{+/0}$ level associated
with Hi (corresponding to the onsite [HMu] molecule) and that of the
[H${}_{2}]_{\rm i}$ molecule (corresponding to the [HMu]i molecule) have a
finite unpaired electron density in the band gap (i.e., the paramagnetic
states). This is consistent with the experimental observation of two
paramagnetic states, Mup and Mup2. The hyperfine parameters calculated by
GIPAW module of the QE suite for these states are listed in Table 1. They
exhibit a common character that the Fermi contact interaction is predominant.
For the other diamagnetic states, the nuclear dipolar width $\Delta$ are
summarized in Table 2. Based on these DFT-QE calculations on H, we discuss the
correspondence between these levels and the experimentally observed Mu states.

| Magnetic dipolar term (MHz) | Principal axis
---|---|---
H${}_{\rm i}^{0}$ | $\omega_{xx}/2\pi$ | $-0.0295$ | $(0,0,1)$
$\omega_{yy}/2\pi$ | $-1.8829$ | $(0.5514,-0.8342,0)$
$\omega_{zz}/2\pi$ | $1.9125$ | $(-0.8342,-0.5514,0)$
| Fermi contact term (MHz) |
| $\omega_{c}/2\pi$ | $630.013$ | —-
| Magnetic dipolar term (MHz) | Principal axis
HI | $\omega_{xx}/2\pi$ | $-19.1146$ | $(0.4991,-0.8665,0)$
$\omega_{yy}/2\pi$ | $-19.5940$ | $(0,0,1)$
$\omega_{zz}/2\pi$ | $38.7086$ | $(-0.8665,-0.4991,0)$
| Fermi contact term (MHz) |
| $\omega_{c}/2\pi$ | $267.150$ | —-
| Magnetic dipolar term (MHz) | Principal axis
HII | $\omega_{xx}/2\pi$ | $-3.1894$ | $(0,0,1)$
$\omega_{yy}/2\pi$ | $-4.7335$ | $(0.31983-0.9475,0)$
$\omega_{zz}/2\pi$ | $1.9125$ | $(-0.9475-0.3198,0)$
| Fermi contact term (MHz) |
| $\omega_{c}/2\pi$ | $271.750$ | —-
Table 1: Calculated hyperfine parameters for the paramagnetic states: H${}^{0}_{\rm i}$ (corresponding to Fig. 6b) and those for each H in [H${}_{2}]_{\rm i}^{+}$ molecule (HI and HII in Fig. 7b). The values for Mu must be multiplied by 3.18. | Mui | [HMu]i
---|---|---
Electronic state | Mu${}_{\rm i}^{+}$ (= [HMu]0) | Mu${}_{\rm i}^{-}$ | [HMu]${}^{0,2\pm}_{\rm i}$
$\Delta$ (MHz) | 3.64 | 0.3685 | 3.2–3.9
Table 2: Calculated nuclear dipolar linewidth $\Delta$ for the diamagnetic Mu
states. Figure 7: Local structure of an interstitial hydrogen molecule in a)
[H${}_{2}]_{\rm i}^{0}$ and b) [H${}_{2}]_{\rm i}^{+}$ states with one H atom
substituted by Mu (red balls), viewed from $b$ (left) and $c$ (right) axes,
respectively. There are two inequivalent H denoted by labels I and II for b),
where the case with HI replaced with Mu is shown. The hyperfine parameters for
b) are distributed over four H atoms indicated by a dashed ellipse.

Most notably, $\Delta_{\rm KT}$ of the MuKT component (Fig. 4f) is in
excellent agreement with 0.3685 MHz expected for Mu${}_{\rm i}^{-}$ that
accompanies outward relaxation of the surrounding H-. Therefore, MuKT is
ascribed to the diamagnetic state associated with the $E^{0/-}$ level of
isolated Mui (i.e., Mu${}_{\rm KT}^{-}=$ Mu${}_{\rm i}^{-}$).

Although the experimentally estimated hyperfine parameters for the
paramagnetic states are considerably smaller than those in Table 1, the large-
small relationship $\omega_{\rm p}>\omega_{\rm p2}$ suggests that Mup
corresponds to the onsite [HMu] and Mup2 to [HMu]i. Then, from the viewpoint
of ambipolarity, the state paired with MuKT is Mup/Mud which is associated
with the $E^{+/0}$ level.

Regarding the Mup/Mud component, the magnitude of $\Delta_{\rm d}$ [$=2.71(2)$
MHz] for the Mud state is close to 3.64 MHz expected for the H-Mu molecule
([HMu]0, see Table 2). This is consistent with the speculation based on the
ambipolarity. More specifically, the experimental value can be explained when
the H-Mu distance is $r=0.083$ nm, about 11% larger than the p-p distance of
0.078 nm inferred from the QE calculation. The fluctuation frequency of
150–230 MHz indicated for Mud suggests that even after Mup destabilizes and
transitions to [HMu]0, a fast spin/charge exchange reaction persists with the
excitonic carriers. From the magnitudes of $A_{\rm pd}$ and $A_{\rm KT}$, we
conclude that the major part of the observed relaxed-excited Mu states can be
explained by pairs of Mu${}_{\rm i}^{+}$ (= [HMu]0) and Mu${}_{\rm i}^{-}$.

Since [HMu]${}^{q}_{\rm i}$ molecules are diamagnetic for $q=0$, $2\pm$ and
paramagnetic for $q=\pm$, Mup2 is speculated to be in the state associated
with the $E^{+/0}$ level in Fig. 5b (i.e., [HMu]${}_{\rm i}^{+}$). As shown in
Fig. 7b and Table 1, our QE calculations indicate that the local structure of
the [HMu]${}_{\rm i}^{+}$ state is complicated, resulting in the two
inequivalent H sites with different hyperfine parameters. This would suggest
that the local electronic structure is subject to complex modification upon
thermal excitation, which is in line with the complicated temperature
dependence of $\omega_{\rm p2}$ shown in Fig. 4f.

Coming to the diamagnetic [HMu]${}_{\rm i}^{0}$ state pairing with the
[HMu]${}_{\rm i}^{-}$ state, the ambipolarity model suggests that it
corresponds to the Mu3S state. Although the fast relaxation of the signal
makes it difficult to examine the local electronic structure in detail, it is
plausible that the Mu3S state consists of the [HMu]${}_{\rm i}^{0}$ state
bonded to a ligand H- via the hydrogen bonding. This is consistent with the
attribution of the Mu3S state to that associated with the $E^{-/0}$ level in
Fig. 5b. It is likely that the H-Mu-H signal reported earlier also stems from
the same complex state.

Figure 8: The electronic states of Mu estimated on the basis of the
ambipolarity and phenomenological model shown in Fig. 3. The possibility is
shown that [HMu]0 (Mu${}^{+}_{\rm d}$), Mu${}_{\rm i}^{-}$ (Mu${}_{\rm
KT}^{-}$), and [HMu]${}_{\rm i}^{0}$ (Mu${}_{\rm 3S}^{+}$) states may be
affected by spin fluctuations of the nearby H centers in the relaxed-excited
state (denoted as H∗), a factor not explicitly taken into account in the
analysis.

Applying these attributed electronic states of Mu to the phenomenological
model, Fig. 3 is updated to Fig. 8. Based on this, we consider a scenario that
explains the rough temperature variation of each state observed in Fig. 4.
$A_{\rm KT}$ shows a marked increase above 150 K, which is clearly correlated
with the increase in the charge transfer reaction rate $\kappa$ from Mup (=
[HMu]+) to Mud (= [HMu]0). This suggests that the transition from the [HMu]
molecular state to the Mu${}_{\rm i}^{-}$ state (via Mud) is accelerated with
increasing temperature. A similar transition from Mu3S to Mu${}_{\rm KT}^{-}$
is suggested to occur in parallel with that for the interstitial Mu.

Finally, regarding the initial electronic states of Mup/Mud-MuKT pair (i.e.,
the interstitial Mu), neutral and negative ionic states are predominant as a
whole. As shown in Fig. 5, the $E_{F}^{\rm int}$ level is close to the CBM,
and the formation energy at $E_{F}^{\rm int}$ for all electronic states is
neutral or negative ionic ground state. Therefore, the population of Mu
valence for the relaxed-excited states implies that the overall charge average
is close to that of thermal equilibrium. In other words, the reason why Mu
(and H) simultaneously assumes donor and acceptor states in the relaxed-
excited state is presumably to maintain the global charge neutrality of the
system consisting of the host and H.

In conclusion, assuming that the electronic state of Mu implanted into MgH2
takes the two pairs of defect centers (relaxed-excited states) associated with
the donor/acceptor levels for interstitial Mu and HMu molecule states
predicted from the DFT calculations, a scenario that coherently describe the
present experimental results can be constructed.

###  IV.2 Implications to the kinetics and diffusion of interstitial H

Now, let us discuss what we can learn for H kinetics from these results. It
has been reported that the addition of a few % of transition metals such as Ti
significantly improves the H kinetics.Liang  _et al._ (1999); Varin  _et al._
(2007); Amama  _et al._ (2011) DFT calculations suggest that, when a
transition metal exhibits negative $U$ character to accompany a double charge
conversion level $E^{+/-}_{\rm TM}$ in the host, the addition of the
transition metal is expected to shift the Fermi level by $\Delta
E=E^{+/-}_{\rm TM}-E_{F}^{\rm int}$.Roy, Janotti, and Van de Walle (2013) As
can be seen in Fig. 5a, for $\Delta E>0$ (as predicted for Ti), the
interstitial H is stable in the H${}_{\rm i}^{-}$ state, and this is assumed
to be the cause of the kinetic improvement by avoiding binding of Hi to the
ligand H-.

On the other hand, it is also reported that ball milling with a small amount
of Nb2O3 improves the kinetics.Hanada, Ichikawa, and Fujii (2007) From $\mu$SR
in the samples after milling with 5 wt% Nb2O3, a decrease of the
characteristic temperature $T_{\rm d}$ where diffusion motion of H and/or Mu
sets in (as inferred from from sharp decrease in $\Delta_{\rm KT}$) from 650 K
to about 450 K was observed.Umegaki  _et al._ (2014) Assuming that Nb2O3 is an
oxidation catalyst and that stabilization of the H${}_{\rm i}^{-}$ state is
the key to improving kinetics, it can be interpreted that Nb2O3 raises bulk
$E_{F}$ by causing dehydrogenation;

${\rm MgH}_{2}\rightarrow{\rm MgH}_{2-\delta}+\frac{\delta}{2}{\rm
H_{2}}+\delta e^{-}.$

The fact that a similar effect is obtained in the case of ball milling alone
suggests that the increased surface area also promotes dehydrogenation
reactions by atmospheric oxygen. It is paradoxical that dehydrogenation
corresponds to reduction for hydrides.

Meanwhile, it has also been reported that the formation of the Mu3S state
(corresponding to the H--H+-H- complex state) is promoted in those
samples.Umegaki  _et al._ (2014) Since such hydrogen bond formation is
expected to be rather unfavorable from a diffusion point of view, it is
assumed that the reason why H kinetics improves with Nb2O3 addition is because
the effect of stabilizing H${}_{\rm i}^{-}$ outweighs that of hydrogen bond
formation; the upward shift of $E_{F}$ will eventually lead to the
dissociation reaction,

${\rm H}^{-}\mathchar 45\relax{\rm H}_{\rm i}^{+}\mathchar 45\relax{\rm
H}_{\rm i}^{-}+2e^{-}\rightarrow{\rm H}^{-}+[{\rm H}_{2}]_{\rm
i}^{-}+e^{-}\rightarrow{\rm H}^{-}+2{\rm H}_{\rm i}^{-}.$

A similar situation is suggested from an earlier $\mu$SR study for the effect
of Ti-doping to NaAlH4.Kadono  _et al._ (2008); Hiraishi  _et al._

Regarding the Mu${}^{-}_{\rm KT}$ state corresponding to H${}_{\rm i}^{-}$,
the magnitude of $\Delta_{\rm KT}$ is found to vary little over the entire
temperature range, indicating that the Mu site remains unchanged over the time
range of $\sim$101 $\mu$s. The fluctuation frequency $\nu_{\rm KT}$ shows
nearly constant value of $\sim$0.1 MHz below 150 K, which is unlikely to be
attributed to thermally activated diffusive motion of Mu${}_{\rm KT}^{-}$. The
non-monotonic temperature dependence of $\nu_{\rm KT}$ above 150 K also
suggests that the cause is non-thermal. In addition, the paramagnetic Mu
components (Mup/Mud and Mup2) show fast spin and/or charge exchange reactions
with excited carriers even below 150 K. Considering that the frequency of
paramagnetic fluctuations ($>10^{2}$ MHz) are far from the sensitive range for
the diamagnetic Mu, these observations suggest that the fluctuations of
$\Delta_{\rm KT}$ is induced by the relaxation of proton spins of the nearby
H(’s) in the relaxed-excited states (denoted as H∗) generated upon muon
implantation (see Fig. 8).

Note that H∗ can be any of the observed four Mu states with H substituting for
Mu, which may be appropriately referred to Hp/Hd, HKT, Hp2, and H3S. The
proton spin relaxation rates $\lambda^{*}$ in these states can be evaluated by
reducing those for the diamagnetic Mu (inferred from Fig. 4) by a factor
$R^{2}\equiv[\gamma_{\rm p}/\gamma_{\rm\mu}]^{2}$ ($=0.0986$) for the same
fluctuation frequency as observed in Mu, e.g., $\lambda^{*}=R^{2}\lambda_{\rm
3S}\simeq 0.2$–0.5 MHz for the H3S state. This implies that further
experimental verification will be necessary to determine whether the
fluctuations of nuclear dipolar fields exerted from H around $\mu^{-}$Mg atoms
observed by negative muon $\mu$SRSugiyama  _et al._ (2018) can be attributed
to H diffusion alone, because the $\mu^{-}$Mg atoms can accompany nearby H∗’s
due to local electronic excitations which are much stronger than that in
positive muon implantation.Okumura  _et al._ (2021)

## Acknowledgement

We acknowledge helpful discussions with J. Sugiyama, and I. Umegaki. This work
was supported by the MEXT Elements Strategy Initiative to Form Core Research
Center for Electron Materials (Grant No. JPMXP0112101001), JSPS KAKENHI (Grant
Nos. 19K15033, 17H06153, 20K12484, 21H05102), Core-to-Core Program
(JPJSCCA20180006), and JST MIRAI Program (JPMJMI21E9).

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