Desorption reaction in MgH₂ studied with in situ μ⁺SR

Abstract

In order to study the mechanism determining the desorption temperature (T_d) of hydrogen storage materials, we have measured positive muon spin rotation and relaxation (μ⁺SR) in MgH₂ over a wide temperature range including its T_d. The pressure in the sample cell due to desorbed H₂ was measured in parallel with the μ⁺SR measurements under static conditions. Such in situ μ⁺SR measurements revealed that hydrogen starts to diffuse in MgH₂ well below T_d. This indicates the important role of hydrogen diffusion in accelerating the desorption reaction by removing the reaction product, i.e. H₂, from the reaction system.

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1 Introduction

Positive muon spin rotation and relaxation (μ⁺SR) techniques are widely used for studying local magnetic environments in solids^1,2 for both static and dynamic behaviors of the internal magnetic field(s) H_int at the muon site(s). However, due to limited counting rates for recording μ⁺SR spectra, past μ⁺SR measurements were usually performed under fixed conditions, i.e. at a constant temperature, pressure and magnetic field. Consequently, it was hitherto impractical to use μ⁺SR to study changes in H_int during chemical reactions of materials.

Recent developments in pulsed muon beams at ISIS in the UK³ and at J-PARC in Japan,⁴ together with the installation of multi-detector counting systems,^5–7 have increased the counting rate to above 100 M events per h. This means that one μ⁺SR spectrum with acceptable statistics is recorded within 5–30 minutes. Therefore, if we select a suitable material, μ⁺SR can provide information on the change in H_int during a fairly rapid reaction, in an in situ manner. Here we report our first in situ μ⁺SR study of magnesium hydride, MgH₂ (see Fig. 1), in order to reveal the change in H_int during its hydrogen desorption reaction at high temperatures.

Fig. 1 The crystal structure of MgH₂ in tetragonal symmetry with the space group P4₂/mnm together with the μ⁺ sites predicted by DFT calculations (see Section 3.1). The lattice parameters determined from synchrotron radiation powder diffraction data⁸ are a = 4.5198(7) Å, c = 3.0250(6) Å, and V = 61.80 ų. The pink lines represent the iso-surface of potential energy ϕ_E = 600 meV. A small black sphere represents the muon site, at which ϕ_E shows a local minimum.

An ideal hydrogen storage material is expected to absorb and desorb hydrogen gas reversibly without capacity fading, preferably at room temperature.⁹ Magnesium hydride (MgH₂) is known to be one of the promising hydrogen storage materials, because of its relatively low cost and high gravimetric and volumetric hydrogen densities.^10–13 However, its high enthalpy of decomposition leads to a high hydrogen desorption temperature (T_d) and slow reaction rate. Such disadvantages for applications are alleviated significantly by milling^14–16 and by adding catalytic agents, such as Nb₂O₅.^17–20 In fact, the measurements under dynamic conditions, i.e. with a temperature programmed desorption (TPD) and/or thermogravimetry (TG) technique, revealed that T_d = 710 K for bulk MgH₂,^21,22 while milling decreases T_d by about 100 K (ref. 21 and 22) and about another 100 K decrease can be achieved by milling together with 5 wt% of Nb₂O₅.^23–25 Such a large decrease in T_d, i.e. 200 K in total, and a simple crystal structure make MgH₂ a very attractive model material^11–13 to understand the way to control T_d in storage materials.

Based on macroscopic measurements and numerical calculations, the desorption reaction in MgH₂ is thought to consist of the following five steps under the assumption that a MgH₂ core is surrounded by a Mg shell in each particle:^12,26

(1) decomposition of MgH₂ at the interface between MgH₂ and Mg;

(2) diffusion of H across the Mg shell;

(3) penetration of H at the surface of the Mg shell;

(4) recombination of chemisorbed H and physisorption;

(5) desorption to the gas phase.

A past quasi-elastic neutron scattering experiment²⁷ revealed that the diffusion coefficient of hydrogen (D_H) in MgH₂ ranges below 2.5 × 10⁻⁹ cm² s⁻¹ even at 650 K, which is lower by at least three orders of magnitude than D_H in Mg (10⁻⁹ cm² s⁻¹ at 300 K).¹² Considering the rates of the above five reaction steps, the third and/or fourth steps are thought to be a rate-determining process for the overall desorption reaction,^12,19,26,28 although the nucleation of Mg and formation of the channels by Mg metal nuclei are proposed to be crucial for bulk MgH₂.²⁹ The hydrogen diffusion in MgH₂ is thus believed to be unessential for the desorption reaction in MgH₂. Nevertheless, a ¹H-NMR study on MgH₂,³⁰ which provides microscopic information of hydrogen dynamics, suggests a faster H motion even at 333 K in milled MgH₂ with Nb₂O₅ than in bulk MgH₂. Indeed, there are still large discrepancies between experimental results and calculated predictions for hydrogen diffusion in MgH₂.^31–36

We have therefore used μ⁺SR to study H_int in MgH₂ (where it arises mainly from the nuclear magnetic moment of ¹H), since the diffusion coefficients of Li⁺ and Na⁺ are estimated for many battery materials using the fluctuation rate of H_int detected with μ⁺SR.^37–46 Due to the fact that the natural abundance of ²⁵Mg with I = 5/2 is 10%, the μ⁺SR spectrum caused by the μ⁺s stopped in a Mg phase should be time independent, particularly above room temperature. Taking into account a very small concentration of diffusing H in Mg, H diffusion in Mg is invisible with μ⁺SR, but such behavior in MgH₂ should be visible. This is because H diffusion enhances the fluctuation rate of H_int in MgH₂ formed by ¹H. In fact, our previous μ⁺SR measurements on MgH₂ (ref. 47 and 48) showed a clear difference in H_int among the three MgH₂ samples: as-prepared MgH₂, milled MgH₂, and MgH₂ milled together with 5 wt% of Nb₂O₅. That is, although H_int is static up to T_d in the as-prepared sample, H_int of the two milled samples becomes dynamic below T_d.

Remembering that μ⁺ behaves as a light isotope of H⁺ in solids,^1,2 μ⁺ is expected to be mobile in any system where H is diffusing, regardless of the charge state of H. However, the differences between the diffusive behaviors observed by μ⁺SR in the three MgH₂ samples imply that μ⁺ diffusion is not equivalent to H diffusion in MgH₂. This naturally motivated us to study the hydrogen desorption reaction in MgH₂ with a more sophisticated μ⁺SR experiment, using a combination of an intense muon beam and simultaneous pressure monitoring—namely an in situ μ⁺SR technique. Here, we report the result of our first attempt to observe the diffusive behavior in MgH₂ with in situ μ⁺SR.

2 Experimental

2.1 Sample preparation

A powder sample of MgH₂ was purchased from Wako Pure Chemical Industries, Ltd., which will be referred to as ap-MgH₂ in the text. Then 3 g of ap-MgH₂ powder was milled with 300 g stainless steel balls in a stainless steel pot by using a planetary ball mill (Fritsch Japan Co., Ltd, P-5/4) for 24 hours; this sample will be referred to as m-MgH₂ in the text. Then a mixture of 2.85 g of ap-MgH₂ powder and 0.15 g (5 wt%) of Nb₂O₅ powder (Aldrich) was also milled for 24 hours in the same way as the above; this sample will be referred to as mwn-MgH₂ in the text.

2.2 μ⁺SR

In order to observe hydrogen dynamics, the μ⁺SR spectra were measured in a zero field (ZF), longitudinal field (LF), and transverse field (TF) on the EMU⁵ surface muon beam line at the ISIS facility of the Rutherford Appleton Laboratory in the U.K. Here TF (LF) means the magnetic field perpendicular (parallel) to the initial muon spin polarization. The statistics of each spectrum were 40 M events for ZF, 30 M events for LF, and 10 M events for TF with a counting rate of about 80 M events per h. Since the LF-spectra were measured at LF = 5 and 10 Oe, it took about 1.4 hours to obtain the data at one temperature point. Then, the sample was heated to the next temperature at a rate of 1–2 K min⁻¹. The temperature range for the measurements was maintained between 300 and 800 K using a furnace [Fig. 2(a)].

Fig. 2 (a) A schematic illustration of the experimental setup for in situ μ⁺SR measurements. Hydrogen gas in a sample cell is collected through a gas line equipped with a gas tank and a pressure gauge. (b) The sample cell made of stainless steel with a capillary to collect the liberated gas and gold O-rings for the sealing.

Each sample was sealed in a stainless steel cell (grade 316) with two gold O-rings [Fig. 2(b)] in a He-filled glove box. The window of the cell was also made of stainless steel foil with 50 μm thickness. Then the cell was mounted onto the furnace so that the direction of the muon beam is perpendicular to the window. As a result, the muons were implanted into the sample through the window. The released gas was collected through the aforementioned 1/16′′ capillary tube, also made from stainless steel, connecting the cell to a one liter tank [Fig. 2(a)]. Before the measurements, the cell, gas line, and tank were evacuated at room temperature, then, the exhaust valve was closed, and then, the sample was heated by the furnace. The total amount of the gas released was determined by measuring the pressure (p) in the gas tank with a Baratron pressure gauge. Prior to the in situ experiment at ISIS as explained above, μ⁺SR spectra were also measured at both TRIUMF and J-PARC.^47,48 The details of μ⁺SR techniques are explained elsewhere.^1,2 The obtained μ⁺SR data were analyzed with musrfit.⁴⁹

3 Results

3.1 Predicted muon sites

Knowledge of the site(s) for the implanted μ⁺s in the lattice is essential to understanding the μ⁺SR data. According to electrostatic potential (E) calculations using a first principles DFT+GGA method, there is likely one shallow E minimum for the implanted μ⁺ in the MgH₂ lattice (Fig. 1). However, the distribution of E in the lattice describes a “chimney-shaped” one-dimensional (1D) channel along the c-axis, connecting many local E minima between two neighboring hydrogens, as described in Section 4.1.

The zero-point energy (E_zero) of an H atom in solids is typically about 200 meV.⁵⁰ Since E_zero is approximately scaled by a factor of the inverse of the square root of the mass, E_zero for μ⁺ is expected to be about 600 meV. The “extended minimum” for E < 600 meV, within which the implanted μ⁺ will be located, is illustrated by the region within pink shaded cylinders in Fig. 1. This indicates that the μ⁺s will be in a delocalized state within the 1D channel, particularly at high temperatures. It should be noted that we ignored the local lattice distortion due to the implanted μ⁺ in the above calculations. Therefore, the observed μ⁺SR spectrum is probably either a superposition of multiple spectra from muons at different sites or the signature of a μ⁺ in a delocalized quantum state.

3.2 μ⁺SR spectra

The ZF-μ⁺SR spectrum exhibits a characteristic oscillation for the all three samples below 350 K [Fig. 3(a)], as already reported. This indicates the formation of a collinear three-spin-½ system (H-μ-H) and a two spin-½ system (H-μ), as is the case for many hydrogen storage materials.^51,52 The muon spin depolarization function in ZF caused by collinear H-μ-H [P^ZF_HμH(t)] is given by⁵³

Fig. 3 (a) The ZF-μ⁺SR spectrum of mwn-MgH₂ at 5 K; (b) the ZF- and LF-μ⁺SR spectra of mwn-MgH₂ at 480 K; and (c) the ZF-μ⁺SR spectra of mwn-MgH₂ at 480, 520, and 560 K. In (a), a solid line represents the best fit using a combination of one H-μ-H and eight H-μ signals. In (b) and (c), solid lines represent the best fit obtained using eqn (3). The fitted initial asymmetry is A₀ = 0.24 for (a) and 0.28 for (b) and (c); based on the measurement of a silver reference sample, this means that only a small fraction of the implanted μ⁺ form a hydrogen-like muonium (μ⁺e⁻) state in MgH₂ at 5 K.

On the other hand, P^ZF_Hμ(t) is given by⁵⁴

where ω_d = 2πf_d = (μ₀ħγ_μγ_H)/(4πr³), in which μ₀ is the permeability of vacuum, ħ is the reduced Planck constant, γ_μ is the gyromagnetic ratio of μ⁺, γ_H is the nuclear gyromagnetic ratio of ¹H, and r is the distance between μ and H in the H-μ-H and H-μ systems.

In fact, the ZF-μ⁺SR spectrum was well fitted by a combination of one P^ZF_HμH(t) and eight P^ZF_Hμ(t) signals, indicating the presence of multiple muon sites in the lattice. Since the fit shows that about 10% of the implanted μ⁺s form a H-μ-H state with r = 1.16 Å at 5 K, such muons sit at the middle of the two neighboring H⁻s. However, the distance between the two nearest neighbor H⁻s in the MgH₂ lattice is 3.27 Å.⁸ This means that μ⁺ attracts the two neighboring H⁻s, as in the case for alkali halides.⁵³ Considering the presence of the other eight muon sites evidenced by the formation of H-μ systems, the obtained μ⁺SR result is very consistent with the predictions from the first principles calculations.

At temperatures above 300 K, the ZF-μ⁺SR spectrum clearly shows a Kubo–Toyabe (KT) type relaxation [Fig. 3(b)] due to random nuclear magnetic fields. This means that both H-μ-H and H-μ are no longer stable at high temperatures, because of the thermal vibration of H, being consistent with the findings of ¹H-NMR.⁵⁵ Since the applied LF = 10 Oe significantly suppresses the relaxation by decoupling H_int,⁵⁶ the magnitude of the random nuclear magnetic fields at the muon sites is comparable to the applied LF. This is a typical value for nuclear magnetic fields in solids.^1,2,42

For both m-MgH₂ and mwn-MgH₂, as temperature increases from 300 K, the KT behavior changes from static to dynamic above around 450 K and finally the relaxation rate becomes very small compared with that at lower temperatures [Fig. 3(c)]. This indicates that the field fluctuation rate (ν) is very rapid compared with the static width (Δ) due to H_int, as in the case for Li- and Na-diffusion in battery materials.^37–46

In order to extract Δ and ν from the μ⁺SR data for the three samples, the ZF- and LF-μ⁺SR spectra were fitted by a combination of two dynamic Gaussian KT signals (G^DGKT)⁵⁷ and a time-independent background signal coming from the muons stopping in the sample cell:

A₀P(t) = A_KT1G^DGKT(Δ₁, v₁, t, H_LF) + A_KT2G^DGKT(Δ₂, v₂, t, H_LF) + A_BG, (3)

where A₀ is the initial (t = 0) asymmetry, A_KTi and A_BG are the asymmetries associated with the three signals (the first two from the μ⁺ stopped in the sample and the last from those stopped in the sample cell), Δ_i represents the static widths of the local field distributions at the μ⁺ sites, and v_i represents the fluctuation rates of the fields at those sites. For v = 0 and H_LF = 0 (i.e. ZF), G^DGKT (t, Δ, v, H_LF) becomes a static Gaussian Kubo–Toyabe function,⁵⁷

At first, we fitted all the ZF- and LF-spectra using a common A_BG over the whole temperature range and common, i.e. H_LF-independent, Δ_i and ν (v₁ = v₂) at each temperature using eqn (3). Then, since both A_KT1 and A_KT2 were found to be approximately temperature-independent, we finally used common A_KT1 and A_KT2 for fitting the ZF- and LF-spectra. Here, Δ corresponds to a spin–spin relaxation rate (1/T₂), while ν corresponds to a combination of the muon hop rate and the nuclear spin-lattice relaxation rate (1/T₁).

3.3 Nature of diffusion in the three samples

3.3.1 ap-MgH₂. Fig. 4(a) shows the temperature dependence of the pressure (p) in the cell. As temperature increases from 350 K, p suddenly starts to increase at around 670 K and levels off to a constant value at 700 K due to a hydrogen desorption reaction with a temperature width (δT_d) of 24.7 K. The desorption temperature for ap-MgH₂ is estimated to be T^mid_d = 677.7(5) K, at which p is 50% of the value at 700 K, i.e. p_max. We define δT_d as δT_d ≡ T(p = 0.9p_max) − T(p = 0.1p_max). The desorption reaction is thus confirmed to occur very suddenly, as already reported.^21,22

Fig. 4 The temperature dependences of the gas pressure and μ⁺SR parameters for ap-, m-, and mwn-MgH₂ samples: (a), (e), and (i) gas pressure in the cell; (b), (f), and (j) the field distribution static widths (Δ₁ and Δ₂); (c), (g), and (k) the field fluctuation rates (v₁ and v₂); and (d), (h), and (l) the asymmetries (A_KT1, A_KT2, and A_BG). Except for pressure, the values shown were obtained by fitting the set of ZF- and LF-μ⁺SR spectra with eqn (3). Dotted lines show T^mid_d determined from the p(T) curve.

Concerning the μ⁺SR parameters, both Δ₁ and Δ₂ decrease slightly with increasing temperature, but are roughly temperature independent up to T_d, below which Δ₁ of the dominant A_KT1 signal ranges between 0.33 and 0.31 μs⁻¹ (equivalent to 3.8–3.6 Oe) and Δ₂ of the smaller A_KT2 signal varies between 0.13 and 0.12 μs⁻¹ (equivalent to 1.5–1.3 Oe) [Fig. 4(b)]. Above T_d, Δ abruptly drops down to ∼0.07 μs⁻¹ (equivalent to 0.8 Oe), as the A_KT2 signal merges into the A_KT1 signal [Fig. 4(d)]. The presence of the two A_KT signals below T^mid_d suggests that about 30% of the implanted muons are located outside of the one-dimensional channel along the c-axis (Fig. 1), but still in the sample.

In addition, ν is very small and almost temperature independent up to around 670 K, but leaps up to ∼0.26 μs⁻¹ at 700 K and increases with further increasing temperature [Fig. 4(c)]. Based on the p(T) curve, MgH₂ fully decomposes into Mg and H₂ at 700 K. This indicates that the sudden decrease in Δ at T_d is most likely not caused by a motional narrowing behavior due to H and/or μ⁺ diffusion, but by a change in the phase from MgH₂ to Mg due to the hydrogen desorption reaction, as proposed by macroscopic measurements.^12,26

It should be noted that v/Δ₁ ∼ 1/9 at temperatures below T_d. Furthermore, the magnitude of Δ₁ is about 1/2 of that determined by dipole field calculations with Dipelec,⁵⁸i.e. Δ^calc = 0.63–0.66 μs⁻¹ (7.4–7.8 Oe) for an entirely static state. This suggests that μ⁺ in MgH₂ is in a delocalized state even at room temperature, as predicted from the first principles calculations (see Fig. 1). However, since both Δ and ν are almost independent of temperature, H is most likely not mobile but rather vibrating at its equilibrium position. We will discuss this later in Section 4.1.

3.3.2 m-MgH₂. The p(T) curve again shows a broad step-like transition at T^mid_d = 546.0(8) K with δT_d = 56.4 K [Fig. 4(e)], which is about twice as wide as δT_d for ap-MgH₂. This clearly shows that milling reduces T_d by about 145 K and makes the desorption reaction more moderate than that in ap-MgH₂.

The most significant difference between m-MgH₂ and ap-MgH₂ is found in the temperature dependence of Δ. That is, as seen in Fig. 4(f), although the magnitude of Δ₁ at 400 K is very close to that of Δ₁ for ap-MgH₂, Δ₂ is much larger than Δ₂ in ap-MgH₂. Since Δ₂ at 450 K is larger than Δ^calc, about 19% () of the implanted muons are thought to sit at fresh surfaces produced by milling, probably trapped by a dangling H⁻ bond. Furthermore, as temperature increases from 400 K, both Δ₁ and Δ₂ decrease monotonically up to around T^mid_d, and then decrease faster (larger |dΔ/dT|), and are likely to level off to a constant value at around T^mid_d.

By contrast, as temperature increases from 400 K, ν decreases slightly with temperature up to T^mid_d, and then increases rapidly with temperature [Fig. 4(g)]. The steady decrease of Δ₁ and Δ₂ below 500 K, at which p = 0, suggests a motional narrowing behavior due to H diffusion in m-MgH₂. This is because μ⁺ diffusion is temperature independent up to 700 K, based on the v(T) curve in ap-MgH₂. The decrease in ν also indicates that both species are diffusing below T^mid_d. However, above T^mid_d the phase change from MgH₂ to Mg is the predominant reaction, resulting in a decrease in Δ and a rapid increase in ν.

3.3.3 mwn-MgH₂. According to the p(T) curve in Fig. 4(i), H₂ starts to be released from mwn-MgH₂ at around 480 K, which stops at around 560 K, i.e. T^mid_d = 522(2) K with δT_d = 70.7 K. Furthermore, p increases linearly with temperature above 480 K and levels off above 560 K. This shows that milling with Nb₂O₅ reduces T_d further by about 20 K.

The temperature dependences of Δ₁, Δ₂, and ν are similar to those for m-MgH₂, particularly below 450 K, at which p = 0. Above 450 K, both Δ₁ and Δ₂ decrease with temperature more rapidly, but ν increases slowly compared with ν in m-MgH₂. Overall results reveal that the changes in Δ and ν below 450 K are caused by H diffusion in MgH₂, while those above 450 K are induced by a gradual phase change from MgH₂ to Mg due to a hydrogen desorption reaction.

The ratio between A_KT2 and (A_KT1 + A_KT2) is about 20%, which is comparable to that of m-MgH₂ (19%). Thus, the surface structure of mwn-MgH₂ is most likely the same to that of m-MgH₂.

3.3.4 Asymmetries for the three samples. Since milling reduces the size of each MgH₂ particle, the packing density and/or amount of the sample in the cell is naturally different among the three samples. This results in a different p_max for the three samples, but we could estimate the amount of each sample in the cell by comparing the p_max with that of ap-MgH₂ (), defining the normalized p_max as .

For samples exhibiting multiple asymmetries, each asymmetry is roughly proportional to the volume fraction of each phase. Since we used the same measurement setup for the three samples, the volume fraction (V_F) of the sample is given by

Therefore the normalized V_F, which is given by , also provides a rough estimate for the amount of each sample. In fact, as seen in Fig. 5, a linear relationship between Np_max and NV_F demonstrates the reliability of the above explanation.

Fig. 5 The relationship between the normalized volume fraction of the sample (NV_F) and the normalized maximum pressure (Np_max) for the three MgH₂ samples. Here, the volume fraction of the sample is defined as V_F = (A_KT1 +A_KT2)/(A_KT1 + A_KT1 + A_BG). For NV_F, each V_F is divided by V_F for ap-MgH₂. p_max is the pressure at the highest temperature measured [see Fig. 4(a), (e), and (i)], and, for Np_max, each p_max is divided by p_max for ap-MgH₂.

4 Discussion

4.1 Implanted muons in MgH₂

As predicted by first principles calculations, μ⁺ is most likely in a delocalized state in a 1D channel of MgH₂ even at room temperature [see Fig. 6(a)]. This has been confirmed by a very recent negative muon (μ⁻) spin rotation and relaxation experiment on ap-MgH₂.⁵⁹ That is, from the fixed position of μ⁻ captured on Mg, μ⁻SR shows that Δ = 0.520(7) μs⁻¹ [equivalent to 6.11(8) Oe], which is very close to the calculated value (Δ^calc = 0.5807 μs⁻¹) at the Mg site in MgH₂. By contrast, the present μ⁺SR shows that Δ ∼ 0.3 μs⁻¹ [equivalent to 2.55 Oe] for MgH₂ below 400 K, while the calculated Δ in the 1D channel ranges between 0.629 and 0.664 μs⁻¹ [equivalent to 7.39 and 7.80 Oe] [see Fig. 6(b)]. This clearly indicates that μ⁺ is not immobile, but delocalized along the 1D channel.

Fig. 6 The variation of (a) electrostatic potential (E) predicted by first principles calculations and (b) Δ estimated by dipole field calculations as a function of z along the line (0.5185, 0.9984, z) in MgH₂. In (a), the difference between the maximum E and minimum E is about 60 meV, suggesting that the μ⁺ is in a delocalized state, particularly at high temperatures. Since the zero-point energy of μ⁺ is expected to be about 600 meV (see in the text), the range of the vertical axis in (a) is between −50 and 600 meV.

Since ν for ap-MgH₂ is independent of temperature up to around 670 K, the delocalized state of μ⁺ is not altered with temperature in the temperature range between 300 and 760 K, probably due to a very small energy barrier in the 1D channel [see Fig. 6(a)]. Nevertheless, the Δ(T) curves show a motional narrowing behavior below 550 K for m- and 500 K for mwn-MgH₂ samples. This means that the delocalized μ⁺ feels a fluctuating H_int caused by H diffusion, which occurs even at temperatures for which p = 0. We wish to emphasize again that μ⁺feels a static and/or dynamic nuclear field formed by ¹H in MgH₂, but does not sense it in Mg due to dilute concentration of H. Consequently, μ⁺SR clearly detects H diffusion in MgH₂, where a desorption reaction does not occur, although it would be difficult to reliably estimate the diffusion coefficient of H under this circumstance, i.e. the observation of mobile H from a delocalized viewpoint.

4.2 Activation energy

When ν is due only to ion diffusion, the v(T) curve is well explained by a thermal activation process. However, the ν values in the three MgH₂ samples are not simply determined by a single diffusion process, as mentioned above. Hence, Fig. 7 shows instead the relationship between 1/Δ and inverse temperature. The activation energy (E_a) for 1/Δ₁ in m-MgH₂ and mwn-MgH₂ is estimated to be 0.045(2) and 0.040(2) eV, respectively, while E_a for 1/Δ₂ in m-MgH₂ and mwn-MgH₂ is 0.155(3) and 0.134(3) eV, respectively. This means that E_a values for m-MgH₂ are similar to those for mwn-MgH₂. These values are comparable to the diffusion activation energy for MgH₂ predicted by first principles calculations.^33,34 In contrast, past ¹H-NMR measurements on 1/T₁ in ap-MgH₂ revealed that E_a = 1.72 eV in the temperature range between 500 and 666 K,³⁰ but no data for m- and mwn-MgH₂ were reported.

Fig. 7 The relationship between Δ and pressure (p) for the three MgH₂ samples. Solid lines represent a fit obtained using the Arrhenius law Δ = Δ₀ exp(−E_a/k_BT) at temperatures below 500 K for m-MgH₂ and below 450 K for mwn-MgH₂. Here Δ₀ is the frequency factor, E_a is the activation energy, and k_B is Boltzmann constant.

Therefore, the milling is found to enhance H diffusion in MgH₂ through the formation of fresh surfaces. Such an enhancement leads to a decrease in T_d. However, since there are no clear differences in the μ⁺SR parameters below 500 K between m-MgH₂ and mwn-MgH₂, the role of Nb₂O₅ in reducing T_d is still not clarified. In order to further understand the dynamic behavior in MgH₂, we definitely need to separate the contribution of μ⁺ diffusion from the μ⁺SR data. For this purpose, we have initiated a negative muon spin rotation and relaxation experiment on MgH₂,⁵⁹ because μ⁻ is captured by Mg nuclei and thus μ⁻SR provides information on H_int from a fixed viewpoint.

5 Conclusions

We have carried out an “in situ” μ⁺SR experiment on MgH₂ to measure the pressure in the sample cell in parallel to the μ⁺SR spectrum under static conditions. The motivation behind this experiment is to clarify the role of H diffusion in solid MgH₂ in determining the hydrogen desorption temperature (T_d). This is, to our knowledge, the first in situ experiment to observe such a reaction in solids with μ⁺SR.

We have found that, as T_d is decreased by milling or milling with Nb₂O₅, the diffusive behavior of H in MgH₂ is observed well below T_d, where the Mg phase does not exist. This suggests that rapid H diffusion in MgH₂ is one of the key factors to reduce T_d, since the liberated H from the MgH₂ phase should be removed through the surface with H diffusion.

Finally, we wish to emphasize that in situ μ⁺SR experiments will provide clear insights into many reaction mechanisms through measurements of ion diffusion in solids and its thermal activation energy. This is because many reactions are often accompanied by mass transport in solids.

Author contribution statement

J.S. conceived the experiments. I.U. prepared and performed the experiments at ISIS together with J.S., H.N., Y.H., O.K.F., M. Månsson and the local ISIS support from S.P.C. J.S., H.N., and I.U. joined the μ⁺SR experiments at J-PARC and A.K. assisted them. J.S., H.N., T.N., E.J.A., and J.H.B. performed the μ⁺SR experiments at TRIUMF. M. Matsumoto and I.U. prepared the m- and mwn-MgH₂ samples. K.M. performed the first principles calculations. J.S. analyzed the data and created the first draft with I.U. J.H.B. and M. Månsson edited the draft, and all authors reviewed the manuscript in several steps.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We thank the staff of TRIUMF, J-PARC, and ISIS for help with the μ⁺SR experiments with the following proposal numbers: M1175, 2012A0089, 2013A0019, RB1510262, and RB1520259. We are particularly grateful to Chris Goodway, Paul McIntyre, Adam Sears, and Mark Kibble of the pressure and furnace group at ISIS, for their support that was essential for a successful experiment. J.S. specially thanks Dr Shinichi Towata of the Aichi Synchrotron Radiation Center for his encouragement to this work. M. Månsson and O.F.K. are partly supported by a Marie Skłdowska Curie Action, International Career Grant through the European Commission and Swedish Research Council (VR), Grant No. INCA-2014-6426 as well as a VR neutron project grant (BIFROST, Dnr. 2016-06955). Further support was also granted by the Carl Tryggers Foundation for Scientific Research (CTS-16:324 and CTS-17:325). This work was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, KAKENHI Grant no. JP23108003 and JP26108722 and Japan Society for the Promotion Science (JSPS) KAKENHI Grant no. JP26286084 and JP18H01863.

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