Observation of proton tunneling correlated with phonons and electrons in Pd

Abstract

Quantum-mechanical properties substantially affect the dynamics of light particles, including hydrogen. The temperature dependence of the tunneling rate provides a clue to understanding the quantum effects. Important physics behind quantum tunneling involves the correlation with surrounding environment, such as phonons and electrons. However, this correlation is not yet well understood due to the difficulty of directly probing hydrogen hopping. Taking advantage of nuclear reaction analysis and resistance measurements, we successfully observed hydrogen hopping from tetrahedral to octahedral sites in palladium. The hydrogen hopping is observed even at low temperatures owing to the quantum tunneling. The tunneling rate reveals a slightly positive and negative temperature dependence above and below 20 kelvin, respectively. The former is explained by the phonon effects. The latter is attributed to the nonadiabatic effects of conduction electrons, whose coupling constant K was derived to be 0.41 ± 0.03 from its temperature (T) dependence described as T^2K–1.

Temperature-dependent quantum tunneling was observed for hydrogen hopping from tetrahedral to octahedral sites in palladium.

INTRODUCTION

Hydrogen in materials finds various applications such as hydrogen storage and heterogeneous catalysis (1–3). Hydrogen diffusion is an elementary step for hydrogen storage and reactions and has long been studied to date. Hydrogen, as the lightest and smallest atom, manifests nuclear quantum effects including the zero-point vibration, discrete vibrational energy levels, and quantum tunneling (4). These quantum effects are believed to have a substantial impact on diffusion at low temperatures.

The interaction of hydrogen with surroundings such as phonons and electrons of host materials can be crucial for the hydrogen tunneling. It has been theoretically suggested that whereas phonon effects associated with lattice deformation bring about a positive temperature dependence in the tunneling rate, the effect of nonadiabatic electron-hole pair excitation due to the presence of the Fermi surface causes a slightly negative temperature dependence in metals (5–10). From an experimental viewpoint, however, such temperature-dependent tunneling was rarely observed except for some cases of H on metal surfaces (11–19). As for three-dimensional systems, the tunneling was only mimicked by muon, a light isotope with about one-ninth the mass of H, in the spin relaxation experiments in simple metals (20–23). Detailed experimental data of the temperature-dependent hydrogen hopping rate at low temperatures in materials are still lacking to comprehensively elucidate the quantum nature of hydrogen.

Absorbed hydrogen in metals occupies interstitial lattice locations. Figure 1A illustrates the face-centered cubic (fcc) lattice structure, where tetrahedral (T) and/or octahedral (O) sites are preferred by hydrogen. Hydrogen atoms are known to thermally diffuse between stable sites via a metastable site at elevated temperatures. At low temperature, the quantum nature appears, and the tunneling between them might also play a decisive role. Although it is recognized that the potential shape is crucial for the tunneling rate (24), the influence of the interaction with surroundings on the tunneling in an asymmetric potential between inequivalent sites such as the stable and metastable sites has hardly been considered even theoretically (25). In this regard, experimental identification of the hopping pathway is also important to address the quantum nature of hydrogen.

Fig. 1. Concept of probing hydrogen hopping.

(A) Schematic potential for hydrogen in palladium (Pd) and the corresponding crystal structure. Quantized energy levels of hydrogen are also illustrated by bars. Red and blue arrows represent the hopping pathway from O to T sites and from T to O sites, respectively, via the saddle point (SP). Wavy lines indicate potential tunneling processes. (B) Illustration of the experimental concept for analyses of structure and hopping of H by means of channeling ¹⁵N–nuclear reaction analysis (NRA), resistance measurements, and thermal desorption spectroscopy (TDS) combined with H ion implantation. imp. implantation; des. desorption.

Palladium (Pd) is a typical hydrogen-absorbing metal with an fcc structure. As schematically shown in Fig. 1A, the O site is the most stable, and the T site is metastable for hydrogen (26–29). Thermal diffusion of hydrogen (H) and deuterium (D) occurs with an activation barrier of 0.226 ± 0.006 and 0.206 ± 0.007 eV, respectively (30), where the rate-limiting step is the hopping from the O site to the saddle point (SP) between the O and T sites. Recent path-integral calculations suggested that the hopping is assisted by the tunneling around the barrier top at the SP below ~100 K (31, 32). The H hopping assisted by the tunneling was indeed experimentally verified around ~70 K in our previous study (33). The hopping from the stable O site to the metastable T site, however, inevitably involves the thermal excitation of hydrogen, which substantially contributes to the temperature dependence, rendering it difficult to examine the nature of tunneling. Although the hydrogen hopping from the metastable T site to the stable O site might directly reflect the tunneling characteristics without thermal excitation, it has not been reported so far.

Despite the increasing demand for identifying the hydrogen lattice location and measuring the hopping rate to elucidate the quantum properties, directly detecting hydrogen in materials is generally challenging due to its small scattering cross sections for electrons and x-rays. In the present study, we conduct nuclear reaction analysis (NRA) using the ¹H(¹⁵N,αγ)¹²C reaction combined with the ion channeling technique, called channeling ¹⁵N-NRA (34–38), where the lattice location of H can be identified on the basis of the incident beam angle dependence of the H-NRA yield. Furthermore, we use low-energy hydrogen ion implantation at low temperatures to place hydrogen atoms at metastable sites (39) and use resistance measurements to measure the hydrogen hopping.

We found that almost half of the H atoms implanted into Pd at low temperatures occupy the metastable T site and subsequently migrate to the stable O sites. The H and D hopping rates from the T to O sites were successfully measured from the resistance relaxation. It is revealed that the hopping occurs in a thermal manner above 70 K regardless of the isotopes. Below 70 K, whereas the D hopping is substantially suppressed, the H hopping still occurs with a nearly temperature-independent feature, confirming the quantum tunneling. The H hopping rate exhibits slightly positive and negative dependence on temperature above and below 20 K, respectively. These temperature-dependent hoppings demonstrate the phonon and electron effects on the tunneling process.

RESULTS

Synthesis of metastable PdH_x by hydrogen ion implantation at low temperature

A polycrystalline Pd thin film with a thickness of 10 nm was hydrogenated by hydrogen ion implantation with an incident energy of 500 eV at 7 K, where the ion dose was monitored by the sample current. A resistance (R) was measured during the implantation. An increase in the resistance was observed as shown in fig. S2. The atomic concentration of hydrogen (x) in PdH_x in accordance with the H implantation dosage was determined by NRA, and the implanted H was confirmed to be distributed in the entire film by the H depth profiles as shown in fig. S3. Figure 2 shows the resistance change (ΔR), its temperature (T) derivative (dR/dT), and simultaneously obtained thermal desorption spectra (TDS) for mass 2 in heating after the H implantation at 7 K. The resistance change during the subsequent cooling is also shown in the figure. As is apparent from the temperature derivatives in Fig. 2B, decreases in resistance were observed via two steps around 90 and 150 K with increasing temperature. Around 150 K, H₂ desorption was observed, and the increased resistance by the hydrogenation returned to that of Pd, showing that this resistance change is caused by the transformation from the hydride to metal phases. In contrast, the resistance decrease around 90 K was not accompanied by H₂ desorption. H₂ desorption peak around 50 K originates from the H₂ physisorbed on the manipulator surface. The resistance after the flashing at 130 K for the 1-mC H-implanted thin film is also shown in Fig. 2A, indicating that the resistance change around 90 K was irreversible. Figure S4 shows the resistances of the 6-mC H-implanted thin film during subsequent cooling from 89 and 116 K, respectively. An anomaly in the resistance around 50 K related to the ordering of H atoms in the O site, called the 50-K anomaly (40, 41), appeared after the 116-K flashing. These results indicate that the H implantation at low temperatures leads to the formation of a metastable state, which relaxes to the stable state with a decrease in the resistance.

Fig. 2. Simultaneous resistance and TDS measurements.

(A) Resistance changes during heating and cooling cycles after H implantation with 0.1 mC (green), 0.3 mC (blue), and 1.0 mC (black) at 7 K, respectively. The resistance change for the 1-mC H-implanted sample subsequently flashed to 130 K is also shown by the dotted line. The resistances are shown with respect to that (R₀ ~ 6 ohms) at 7 K without H. Res., resistance. (B) Temperature derivatives of the resistances. (C) Desorption spectrum of mass 2 for the 1.0-mC H-implanted sample (red). The data were taken after flashing the sample up to 70 K to reduce the background (B.G.) signal by the physisorbed H₂. Desorption spectrum without the H implantation is also shown (black) after scaling by 0.2.

Identification of hydrogen lattice location by channeling ¹⁵N-NRA

To elucidate the metastable structure fabricated at low temperature, we performed the channeling ¹⁵N-NRA. The experiment was conducted around the surface-normal direction for a Pd(100) single crystal by the ¹⁵N²⁺ ion beam with an incident energy of 6.42 MeV, which corresponds to the H probing depth of 9 nm from the surface. Rutherford backscattering spectrometry (RBS) was simultaneously performed by measuring the backscattered ¹⁵N ion yield. After the H implantation with an energy of 2.0 keV at an incidence angle of 45° at 50 K, PdH_~0.2 was formed at a depth of ~10 nm (fig. S5A), which mimics the hydrogenated Pd thin film with a thickness of 10 nm. Figure 3A shows the line scan profiles of the H-NRA and Pd-RBS yields simultaneously obtained around the <100> axis. A decrease in the H-NRA yield in accordance with the reduction of the Pd-RBS yield was observed. Figure 3B shows the line scan profiles obtained after a subsequent annealing above 80 K. The Pd-RBS dip structure remains almost unchanged, showing that the Pd lattice was not altered by the annealing. In contrast, the H-NRA yield at the <100> axis was reduced. Because the T site is located on the <100> axis channel but the O site is hidden behind the Pd atom as schematically shown in Fig. 3C (1 and 2), the decrease in the H-NRA yield intuitively indicates the migration of H from the T to O sites. Nuclear close-encounter probabilities (NEPs) calculated by Monte Carlo simulations using FLUX7 (42) are also shown in Fig. 3 (A and B), where the H site occupation of O and T sites is changed. NEPs for Pd and H can be compared with Pd-RBS and H-NRA, respectively. The NEP of H for structures in which H cooccupies both O and T sites, denoted as NEP(H), was given as a linear combination as NEP(H) = 2(1 – z)/(2 – z) NEP(H_O) + z/(2 – z) NEP(H_T), where NEP(H_O) and NEP(H_T) represent the NEPs for H at the O and T sites, respectively, and z denotes the T-site occupation ratio. The numbers of O and T sites per Pd atoms in the unit structure are one and two, respectively. The H vibration amplitudes were set to 0.27 and 0.18 Å for NEP(H_O) and NEP(H_T), respectively. Figure 3D shows the mean squared error (MSE) between the normalized H-NRA at 50 K and NEP(H) as a function of the T-site occupation ratio z. The MSE reached a minimum at z ~ 0.5, indicating that ~50 atm % of the implanted H atoms occupy the metastable T site, and the decrease in the NRA yield upon annealing revealed their migration to the stable O site. Therefore, the resistance decrease observed around 90 K in Fig. 3A is unambiguously attributed to the hydrogen hopping from the T to O sites.

Fig. 3. Identification of hydrogen lattice location by channeling ¹⁵N-NRA.

Line scan profiles of H-NRA and Pd-RBS obtained by ¹⁵N²⁺ beam with an energy of 6.42 MeV around the <100> axis for the Pd(100) single crystal along the dashed line in fig. S5B after (A) H implantation at 50 K and (B) subsequent heating to ~90 K. The error bars represent the statistical uncertainty. Dotted and solid curves represent NEPs calculated for Pd atoms [NEP(Pd)] and H atoms [NEP(H)], respectively. NEP(H)s consider the cooccupation of T and O sites by H with various T-site occupancies (labeled as T-site occupancy: 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0). This experiment was conducted once. Norm., Normalized. Schematic illustration of Pd(100) with H in (C) (1) T site and (2) O site. White and black spheres represent H and Pd atoms, respectively. (D) MSE as a function of the T-site occupation ratio between the line profiles of the normalized H-NRA at 50 K and NEP(H)s.

Measuring hydrogen hopping rate by resistance measurements

Figure 4 shows the time evolutions of the resistance of the Pd thin film after the H and D implantations of 1.0 mC at 7 K, which corresponds to the formation of PdH(D)_x with x ~ 0.2. Resistance decreases with time were observed above 70 K for both isotopes (Fig. 4B). Below 70 K, on the other hand, the resistance relaxation was only observed for the H-implanted sample (Fig. 4A). We fit the following formula R(t) to the time evolution of the resistance

$$R\left(t\right)=R_1{\mathrm{e}}^{-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\tau }$}\right.}+R_0$$

where R₀, R₁, and τ are constants. R₀ represents the resistance of Pd hydride with H atoms in the O site, including the residual resistance. In contrast, R₁ reflects the additional resistance contribution from H atoms temporarily occupying the metastable T site. Because the resistance decrease is due to the H(D) hopping, τ⁻¹ represents the H(D) hopping rate.

Fig. 4. Measurements of hydrogen hopping from T to O sites.

Resistance relaxation measurements at (A) low temperatures and (B) high temperatures for the 10-nm-thick Pd film after the H (red) and D (blue) implantation with 1.0 mC at 7 K. The resistances are shown in a normalized form with respect to the initial value, R_i. The labels denote the measurement temperatures for the isotopes. The dashed curves represent exponential fits.

Figure 5 shows the temperature dependence of τ⁻¹ obtained for H and D by repeating the resistance relaxation measurements at various temperatures. In each cycle, the sample was heated up to 380 K before the H or D implantation at 7 K, followed by the relaxation measurement. Because both R₀ and R₁ are associated with the metallic phase of Pd hydride, they tend to increase with temperature. However, as shown in Fig. 4A, the values of R₁ are substantially smaller at low temperatures. This can be attributed to local variations in energy levels of hydrogen in real systems. Consequently, hydrogen may not fully migrate to the O site via tunneling at low temperatures, resulting in a reduced R₁. In addition, an additional slow and nearly temperature-independent relaxation component appears to exist at low temperatures. According to the theory of tunneling in an asymmetric potential (25), the hopping rate tends to be temperature independent in a system where the energy level mismatch is large. Therefore, this component might also arise from the local variations. This slow relaxation is not caused by the temperature drift, as it was not observed for D. For the analysis at lower temperatures, we restricted the fitting range to the initial time domain to extract the temperature-dependent contribution. Above 70 K, the hopping rates of both H and D obey the Arrhenius law, showing that thermal processes are included in the hopping. Below ~70 K, in contrast, the D hopping was substantially suppressed, while the H hopping was still observed. This large isotope effect at low temperature is convincing evidence of quantum tunneling, which is consistent with quantum-mechanical calculations for H(D) atoms in Pd: In our previous study, it is revealed that there exists a quantum state extended by tunneling between the ground state in the T site (⁰T) and the second vibrational excited state in the O site (²O) for H, owing to the short hopping distance between the T and O sites and their energy level matching but not for D (33).

Fig. 5. Temperature dependence of hydrogen hopping rate.

(A) Temperature dependence of the hopping rate (τ⁻¹) derived from the resistance relaxation measurements for the 10-nm-thick Pd film after the implantation of H (red square) and D (blue cross). The error bars represent the SD. The enlarged views at (B) intermediate- and (C) low-temperature regions. The solid curve represents the fitting line described as T^2K–1. (D) Schematic energy diagram of H and D along the O-T and T-O pathways. Solid bars indicate the quantum energy levels of the isotopes.

We further examine the H hopping rate in the two temperature regions below 70 K, above and below 20 K. The H hopping slightly gets slower with decreasing temperature above 20 K, and, in turn, is enhanced below 20 K as shown in Fig. 5 (B and C). This different temperature dependence above and below 20 K can be understood in terms of the correlation with phonons and electrons, respectively. The Arrhenius plot between 20 and 70 K gave an activation energy of 3.1 ± 0.5 meV, which is in good agreement with the energy level difference of 1.8 meV calculated between the ⁰T and ²O states, indicating that the H tunneling is assisted by thermal phonon fluctuation to match the energy levels between those states. In contrast, the slight increase in the H hopping below 20 K is consistent with Kondo’s prediction based on the nonadiabatic effect of conduction electrons (8); the tunneling matrix is renormalized as T^2K, while an effective level broadening leads to T⁻¹. As a result, the temperature dependence is described as T^2K–1, where the proton-electron coupling constant K must be between 0 and 0.5 for a particle with one positive charge in fcc metals (9), and thus, 2K – 1 is predicted to have a negative value. A fit with a function of T^2K–1 shown by the solid curve in Fig. 5C gave a K value of 0.41 ± 0.03. The derived K value for H in Pd is larger than the Ks of 0.22 for muon in Cu, 0.32 for muon in Al, and 0.25 for H on Cu (16, 20–23). It is speculated that the large K is associated with the hopping pathway from the T to O sites, as theoretically analyzed by Regelmann: The K value approaches 0.5 when the potential is asymmetric between the initial and final states (25). The large K might be also associated with the electronic structure of Pd, characterized by electrons in the 4d orbitals with a large effective mass around the Fermi level. In previous studies, the K value was evaluated from the scattering phase shift based on the effective-medium theory, and it is shown that K approaches 0.5, as the averaged local electron density (n₀) around a hydrogen atom in metals is lowered (9, 43). For Pd, n₀ is pointed out to be relatively small due to its open lattice (44), which suggests a large K for H in Pd. Although further experiments in a variety of metals are needed to comprehensively clarify the nonadiabatic effects on hydrogen tunneling, the present results experimentally propose the importance of geometric and/or electronic structures for the hydrogen tunneling correlated with electrons.

Above 70 K in Fig. 5A, the activation energies were derived to be 64.1 ± 4.2 and 73.0 ± 3.9 meV from the slopes in the Arrhenius plots for H and D, respectively. According to the transition state theory, the activation energy reflects the energy level difference between the initial and transition states. These values are consistent with the calculated energy difference between the T site and the SP (31, 32), supporting the hydrogen hopping from the T to O sites. Figure 5D shows a schematic energy diagram for H and D at the O site, T site, and SP. The isotope dependence in the diffusion barrier comes from the zero-point vibrational energy (ZPE), which depends not only on the particle mass (m) but also on the potential curvature (k) as is described as $\text{ZPE}=\frac{1}{2}\hslash \sqrt{\frac{k}{m}}$ in a one-dimensional system. A smaller activation energy for H compared to D reveals that the contribution of the ZPE is larger at the initial state than that at the transition state, indicating that the initial site, i.e., the potential at the T site, is narrower than that of the SP. This isotope dependence is opposite to the trend of the macroscopic diffusion observed above 200 K in a previous study; the activation energies were experimentally derived to be 226 and 206 meV for H and D, respectively (30). Because the rate-limiting step of the macroscopic diffusion is the hopping from the O to T sites, the potential at the O site is shown to be wider than that of the SP. On the basis of these results, we can conclude that the potential for H is sharper in the order of the T site, the SP, and the O site, which is in good agreement with recent theoretical calculations with the ZPE corrections (31, 32). The quantum effects due to ZPE on the hydrogen hopping in thermal regimes depend on the hopping direction in an asymmetric potential. By analyzing the activation energy for each direction and its isotope effects, the characteristics of the potential landscape for hydrogen in metals can be experimentally identified.

DISCUSSION

Combination of metastable PdH_x formation by low-temperature hydrogen implantation, structure analysis by the channeling ¹⁵N-NRA, and probing the hydrogen hopping by electrical resistance measurements allowed for the determination of the temperature-dependent hydrogen hopping rate at low temperatures from the metastable T site to the stable O site in Pd. Whereas hydrogen thermally diffuses above 70 K, quantum tunneling is dominant below 70 K. Negatively and positively temperature-dependent hopping rates were observed for H below and above 20 K, respectively. They suggest the importance of the correlation of proton tunneling with electrons and phonons. The present study provides further insight into the quantum nature of hydrogen correlated with its surrounding environment in materials.

MATERIALS AND METHODS

Sample fabrication

A polycrystalline Pd thin film with a thickness of 10 nm was deposited on a glass substrate with a size of 8 mm by 8 mm by magnetron sputtering at room temperature. The polycrystalline growth was confirmed by the x-ray diffraction pattern as shown in fig. S1.

Resistance measurements and TDS

Electrical resistance is an exceptional probe for hydrogen-related dynamics such as absorption, desorption, and migration, owing to its high sensitivity to the electronic structures altered by hydrogen. In the present study, in situ measurements of in-plane electrical resistance are performed in an ultrahigh vacuum chamber with a base pressure below 10⁻⁷ Pa during hydrogen implantation at 7 K and TDS by heating the samples with a rate of +0.8 K/s. Au wires are fixed at the corners of the sample surface using In. Electrical resistance is measured using the four-probe method with a dc technique, applying a current of 1 mA using Keithley 2000 multimeter. The sample is cooled by a 4-K Gifford-McMahon refrigerator, with its cold end connected to the sample via a Cu rod. The sample stage is surrounded by an Al radiation shield maintained at ~45 K. The sample temperature is controlled between 7 and 380 K by a ceramic heater and is measured by a Si diode of Lake Shore DT670.

For relaxation measurements, we repeatedly performed H or D implantation, followed by resistance relaxation measurements. Each measurement cycle consisted of the following three steps: (i) hydrogen implantation at 7 K, (ii) setting the sample to a desired temperature and recording the time evolution of the resistance, and (iii) flashing the sample at 380 K. In step (ii), the radiation shield was closed, and the sample was heated at a rate of ~+2 K/s to the target temperature. Resistance relaxation measurements were conducted over a temperature range from 7 to 134 K in a nonsequential order. The flashing process in step (iii) ensured a consistent initial state before each measurement. According to transport of ions in matter simulations, vacancy formation of Pd due to H implantation at 500 eV is effectively negligible. For TDS, desorption of molecules with masses 2 and 4, corresponding to H₂ and D₂, respectively, was detected by a quadrupole mass spectrometer.

NRA and channeling ¹⁵N-NRA

NRA using the ¹H(¹⁵N,αγ)¹²C reaction was performed at Micro Analysis Laboratory, Tandem accelerator, The University of Tokyo (MALT). A monochromatized beam of ¹⁵N²⁺ ions accelerated to an energy of ~6.5 MeV by the 5-MV Van de Graaf tandem accelerator is delivered to samples in an ultrahigh vacuum chamber (base pressure ~ 10⁻⁶ Pa). The ion beam current and size on the sample were about 10 nA and 2 mm by 2 mm, respectively. The γ-ray with an energy of 4.43 MeV emitted from the ¹H(¹⁵N,αγ)¹²C reaction is measured with Bi₄Ge₃O₁₂ scintillators to detect H atoms. Depth profiles of H are obtained by scanning the incident ¹⁵N²⁺ ion energy by taking advantage of its narrow resonance energy width of 1.8 keV. This allows for a depth resolution of about 3 nm in Pd with a stopping power of 3.9 keV/nm. To determine the lattice location of H, the angular dependent H-NRA yield obtained by scanning the incident beam angle is examined, called channeling ¹⁵N-NRA. If H occupies the lattice site located on the channel under the channeling condition, then the H-NRA yield gets increased. In contrast, if the H is hidden behind the host lattice atoms, then the H-NRA yield is reduced. RBS is simultaneously conducted, where the ion signal backscattered from the sample is measured with a solid-state detector set at an angle of 135° from the ion beam. The ion channeling phenomenon is observed by reduction of the RBS yield. In this study, line scan profiles of Pd-RBS and H-NRA were obtained after a three-point moving average in the θ direction.

Beam trajectory simulation with FLUX7

The beam trajectories of the incident ions in crystals are calculated by the Monte Carlo methods using FLUX7 (42). The calculations are based on the binary collision of the incoming ions and target atoms using the Ziegler-Biersack-Littmark potential. The NEP of ¹⁵N²⁺ ions to H and Pd atoms can be quantitatively compared with the normalized H-NRA and Pd-RBS yields, respectively. The incident energy of ¹⁵N²⁺ ions was set to 6.5 MeV in the calculations. The beam trajectory of the incident ¹⁵N²⁺ ions is primarily determined by Pd atoms, regardless of the H location. Figure S7 shows simulated NEP patterns of Pd for the structures with and without H(D). No significant differences were obtained regardless of the presence of hydrogen, indicating that hydrogen has a minimal dechanneling effect under the present experimental conditions. Thus, NEP for Pd almost solely reflects the beam trajectory, whereas NEP for H depends on both H location and beam trajectory. In the present study, calculations were performed for Pd hydrides with a thickness of 100 nm, consisting of Pd atoms in the fcc lattice with a lattice constant of 3.89 Å and H atoms in the O or T site. The Pd vibration amplitude was assumed to be 0.11 Å, based on its value at 300 K, which was derived from the Debye temperature of Pd, 270 K. The divergence angle of the incident beam was optimized to 0.6° by comparing the Pd-RBS and NEP for Pd [NEP(Pd)] as shown in fig. S6. The H vibration amplitude of H at the O site was determined to be 0.27 Å by comparing the H-NRA data in Fig. 3B with NEP for H at the O site [NEP(H_O)]. This value is larger than the zero-point vibrational amplitude of 0.19 Å, calculated from the H vibrational energy at the O site, 60 meV, assuming a Gaussian distribution with a standard deviation of $\mathrm{\sigma }=\sqrt{\frac{\hslash }{2m\mathrm{\omega }}}$ , suggesting the influence of crystalline imperfections. The zero-point vibrational amplitude of H at the T site, calculated from the vibrational energy of 130 meV, is 0.13 Å, which is approximately two-thirds of that at the O site. Thus, NEP for H at the T site [NEP(H_T)] was calculated, assuming a vibration amplitude of 0.18 Å, corresponding to two-thirds of the experimentally obtained value at the O site. On the basis of these conditions, the T-site occupancy was evaluated. Line scan profiles of NEPs were obtained using the average values across the entire depth, followed by applying a three-point moving average in the θ direction.

Supplementary Materials

This PDF file includes:

Figs. S1 to S7

Table S1