Article

# First-principles study of the atomic volume of hydrogen in palladium

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## Abstract

The partial atomic volume of hydrogen, vH, is a fundamentally important thermodynamic parameter of interstitial metal hydrides in which dissociated H occupies interstices in the metal lattice. Such an important property should be able to be reliably calculated by a suitable theory or model in order to explain and understand its origin. In practice, vH is typically obtained by means of ab initio calculations founded on density functional theory (DFT), where the equilibrium lattice constant at zero temperature is found by minimising the Born-Oppenheimer energy. While the absolute lattice constants calculated in this way depend quite strongly on the DFT scheme employed, the present work showed that vH is rather robust against differing calculational approaches, thus making a meaningful comparison of theory and experiment possible. Comparing vH for PdnH (0 < n < 8) calculated with DFT and obtained from in-situ neutron diffraction measurements revealed a significant discrepancy when octahedral-only interstitial occupancy was assumed. Calculations for PdH with mixed octahedral and tetrahedral occupancy gave a value for vH in agreement with experiment assuming that PdH contains 15–20% tetrahedral H.

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... Even more fundamentally, it was reported recently that the atomic volume of H in Pd, as calculated employing six common "standard" DFT schemes, disagreed very significantly with experiment [28], suggesting that even the total Born-Oppenheimer energy is not reliable, in consequence of which the phonon properties also cannot be relied on. ...
... In the oct case the spread of Tc values between DFT schemes exceeded 2 to 1 for the same isotope: e.g. for PdH 45.5 K for PBE/PAW versus 19.9 K for PBEsol/PAW. In comparison, the values obtained by Errea et al. [16] after accounting for anharmonicity within one DFT scheme (LDA/USPP, predicting 5.0 K for PdH and 6.5 K for PdD) differ by a smaller factor from the This study has thus exposed a serious consequence of the basic problem of "standard" DFT applied to PdH uncovered by Setayandeh et al. [28], which is that not only are absolute lattice constants variably predicted between DFT schemes, the predicted atomic volume of H in Pd is in strong disagreement with experiment. Because phonon frequencies depend strongly on the absolute lattice constant, all predicted phonon-related properties must also vary depending on the particular DFT scheme employed. ...
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Realistic prediction of the superconducting transition temperature (Tc) for PdH is a long-standing challenge, because it depends on robust calculations of the electron and phonon band structures to obtain the electron-phonon scattering matrix element. To date, first-principles calculations employing density functional theory (DFT) have been based on selected exchange-correlation and core-electron approximations. Incorporating anharmonicity produced a more realistic value of Tc that nevertheless still disagreed strongly with experiment unless adjustable parameters were introduced. Here we consider how the value of Tc predicted using DFT in the harmonic approximation depends on the DFT scheme employed. The rationale for this work is that unless the calculation of Tc within the harmonic approximation is robust, albeit incorrect, there is not a solid foundation for incorporating anharmonicity meaningfully. Six combinations of exchange-correlation approximation (LDA, PBE, PBEsol) and core-electron approximation (PAW, USPP) were tested. Following a carefully systematic methodology, the calculated Tc was found to vary by a factor exceeding two across the tested DFT schemes. This suggests strongly that "standard" DFT, even including anharmonicity, is not reliable for PdH, implying that a higher-rung method will be needed to calculate a realistic lattice constant and phonon band structure, and so predict Tc accurately.
... Concerning the use of DFT in the context under consideration or similar contexts, we can add that the DFT studies focused on hydrogen absorption by metals are now numerous (briefly reviewed in [22]; for Pd and Pd hydride, see e.g. a recent review by Setayandeh et al. [28]). Referring to more recent studies, we may notice that Setayandeh et al. [29,30] have articulated the challenge of accurate description of electron and phonon band structures of the Pd hydrides including PdH and Pd 3 VacH 4 (the superabundant vacancy phase) [28]. Borgschulte et al. [31] have analyzed vibrational frequencies in the hydride (ZrV 2 H 2 ) where the distance between nearest-neighbour H atoms is appreciable shorter than in the conventional hydrides. ...
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At temperatures below 600 K, the isotherms of hydrogen absorption by Pd exhibit hysteresis loops related to the first-order phase transition or, more specifically, to separation of a diluted phase and hydride. According to the experiments, addition of even small amount of the second metal, e.g. Au or Ta, can appreciably suppress hysteresis. This interesting effect is important in various applications, e.g., in the context of fabrication of efficient hydrogen sensors. To clarify its physical background, we present statistical calculations of the hydrogen absorption isotherms for a series of binary alloys of Pd with Mg, Cu, Ag, Ta, Pt, or Au by using the values of the H-metal interaction provided by the density functional theory (DFT). Aiming at the situations with small amount (≤15%) of the second metal, the metal atoms in an alloy are considered to be located at random or with short-range correlations. In the random alloy approximation, appreciable suppression of hysteresis is predicted for all the additives under consideration except Cu. Concerning the correlations, we show that the tendency of metals to mixing (as, e.g., predicted for the Pd-Au or Pd-Ta alloy) is in favour of additional suppression of hysteresis whereas the tendency to segregation (as, e.g., predicted for the Pd-Ag alloy) makes the hysteresis loops wider. For Au and Ta, our findings are in good agreement with available experimental data.
... Interestingly, for chlorine adatoms when occupying the sulfur-poisoned surface, the local hydrogen adsorption gets stabilized. Setyandeh et al. 338 investigated the atomic volume of hydrogen in Pd using DFT. The first-principle's results were obtained in quite good agreement with the experimental outcomes in terms of partial atomic volume. ...
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The interaction potentials of the palladium and hydrogen sublattices at different hydrogen concentrations have been obtained in terms of the density functional theory and ab initio pseudopotentials. It has been shown that the anharmonicity of this interaction depends on the hydrogen concentration. The phonon spectrum of palladium hydride PdH has been calculated in the harmonic approximation and taking into account the anharmonic effects. The temperature-dependent effective potential technique accounting for the anharmonic effects of lattice vibrations has been described.
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The crystallographic properties of palladium at temperatures from absolute zero to the freezing point are assessed following a review of the literature published between 1901 to date. However values above 1100 K are considered to be highly tentative since they are based on only one set of measurements. Selected values of the thermal expansion coefficient and measurements of length change due to thermal expansion have been used to calculate the variation with temperature of the lattice parameter, interatomic distance, atomic and molar volumes and density. The data is presented in the form of Equations and in Tables whilst a comparison between selected and experimental values is shown in the Figures.
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Diffraction-based methods offer unique advantages for elucidating the pathways by which materials absorb and desorb hydrogen, especially when a phase change or the formation of new compounds is involved. In this case, the hydriding reaction may be followed via the changing crystallography of the phases involved in response to a change in temperature or hydrogen pressure. By using a fast diffractometer, the reaction kinetics may also be correlated to environmental conditions and the degree of completion of the reaction. In this paper we consider and model quantitatively the essential elements of a successful in-situ diffraction experiment with neutrons or X-rays under hydrogen pressures up to several kilobars: a gas manifold to accurately measure hydrogen uptake; a pressure cell designed for maximum detected intensity; means to exclude scattering arising in the cell as much as possible; methodology to correct for attenuation and subtract background intensity from the cell and environment.
Article
Ground-state properties of palladium-hydrogen systems are investigated in the framework of density functional theory in the local-density approximation. Norm-conserving ab initio pseudopotentials are used to describe the interactions between electrons and ion cores. The crystalline wave functions and the charge densities of the valence electrons are represented by a mixed basis containing plane waves and additionally localized d-like functions for palladium and s-like functions for hydrogen. Total energies are calculated for Pd-H systems with different hydrogen concentrations using PdnH supercells (n = 1,4, 8, 16, 32). We report on the cohesive properties as well as the diffusion potentials and vibrational energies for the interstitial hydrogen. Lattice relaxations are taken into account by calculating atomic forces according to the Hellmann-Feynman theorem and statistically relaxing the atomic positions.
Article
From the viewpoint of electron theory of cohesion, it is shown that anharmonic vibration of H(D) atom in PdHx(PdDx) alloy gives rise to an isotope effect such that φH ≡ mHω2H > φD ≡ mDω2D which is the basis to explain the inverse isotope effect in superconducting transition temperature of these systems in contradiction to the expectation from BCS theory.
Article
The question of whether in the two-phase region of a metal-hydrogen system the desorption branch of the hysteresis loop of an isotherm is nearer to equilibrium than the absorption branch (concept I) or the absorption and the desorption branches both deviate to a similar extent from equilibrium (concept II) is still an open question. After a review of the literature on this problem we try to solve it for the systems Pd-D2 and Pd-H2 as examples by applying a new method based on isothermal measurements of p(n) and χ(n) (p is the pressure of D2 or H2, n is the atomic ratio D:Pd or H:Pd and χ is the magnetic susceptibility) in the field about the critical point. The evaluation of the p(n) isotherms measured in the homogeneous solution phase above the critical temperature Tc yielded a value of the critical concentration nc = 0.257 ± 0.004, equal for Pd-D2 and Pd-H2. On the basis of this nc value the other critical data were redetermined; the values Tc = 556 ± 1 K and pc = 39 ± 0.5 bar obtained for Pd-D2 deviate markedly from those accepted so far.By means of the absorption and the desorption branches of the p(n) and the χ(n) isotherms measured across the two-phase region below Tc the boundaries of this region, i.e. the coexistence curve, could be determined. Different boundary lines for absorption and desorption were obtained in the Tvs. n diagram resulting in nc(abs) = 0.295 ± 0.005 and nc(des) = 0.255 ± 0.005, which were also equal for both isotopes. The coincidence of nc(des) with the nc value from the homogeneous region decides the question in favour of concept I. The shift of the coexistence curve obtained from the absorption measurements relative to that from desorption is interpreted as being a consequence of the constraining pressure on the more voluminous β phase. By means of this shift the hysteretic behaviour of the χ(n) isotherms also becomes understandable.
Article
We present results of our ab initio studies of electronic and dynamic properties of ideal palladium hydride PdH and its vacancy ordered defect phase Pd3VacH4 (“Vac” - vacancy on palladium site) with L12 crystal structure found experimentally and studied theoretically. Quantum and thermodynamic properties of these hydrides, such as phonon dispersion relations and the vacancy formation enthalpies have been studied. Dynamic stability of the defect phase Pd3VacH4 with respect to different site occupation of hydrogen atoms at the equilibrium state and under pressure was analyzed. It was shown that positions of hydrogen atoms in the defect phase strongly affect its stability and may be a reason for further phase transitions in the defect phase.
Article
The optical constants n and k were determined for some transition metals (Ti, V, Cr, Mn, Fe, Co, Ni, Pd) from reflection and transmission measurements on vacuum-evaporated polycrystalline thin films at room temperature, in the spectral range 0.5-6.5 eV. Three optical measurements were inverted to determine the film thickness d as well as n and k. The estimated error in d was ±2 Å and that in n, k was less than ±2% over most of the spectral range. Transmission measurements were made on films in the thickness range 200-500 Å. Many transition metals oxidize rapidly in the air and so measurements on those samples were performed in a nitrogen atmosphere. A detailed analysis of the effect of oxidation on the measured quantities indicates that it is small. The effects on the optical constants of the film thickness and the evaporation rate are discussed. Some recent theoretical calculations of the interband optical conductivity are compared with the results for V, Cr, and Ni. In addition, some other recent experiments are compared with our results.
Article
A detailed investigation of the electronic structure of palladium is presented in terms of two different band models: (1) ab initio calculations using the augmented-plane-wave method, and (2) calculations using the combined interpolation scheme augmented by inclusion of relativistic corrections. The width and position of the d-band complex are found to be particularly sensitive features of the electronic structure of palladium. A highly detailed density-of-states histogram, and estimates for the first and second derivatives of the density of states at the Fermi energy are derived. In addition, detailed comparisons are made with Fermi-surface-static susceptibility, and specific-heat experimental results. Estimates for the effects of manybody enhancements suggest that paramagnons raise the effective mass at the Fermi energy by only about 41%. Owing to the strong s-d hybridization in palladium, the Fermi surface is made up almost entirely of d-like states. Because the Fermi energy in palladium falls near the strongly spin-orbit split levels at X and L, spin quenching reduces the effective g factor at the Fermi energy from 2 to about 1.65. This increases an estimate of the effective Stoner-enhancement factor from 10 to about 15.
Article
The purpose of this paper is to explore the possibility of using augmented Slater-type orbitals (STO) as basis functions for electronic-structure calculations. STO's have a radial dependence given by rn-1exp(-ζr) and as a result have a number of important advantages. They are localized about sites and have the same asymptotic form as actual atomic orbitals. They are regular at the origin and possess analytic Fourier transforms. The Fourier transform can be manipulated to yield an addition theorem, that is, a reexpansion formula for an STO about another site which is similar to the one used for spherical Bessel functions. Augmenting the STO's with numerical solutions of the Schrödinger equation within touching spheres leads to a small secular matrix since the numerical functions are orthogonal to all the core states and the STO's are only used in the interstitial region. The method has been applied to copper, silver, and palladium using Chodorow-type potentials and accounting for all relativistic effects except spin-orbit coupling. The results on copper are in good agreement with previous calculations and with experiments. The results on Pd and Ag are in better agreement with photoemission experiments than fully self-consistent local-density calculations.
Article
We report a detailed augmented-plane-wave energy-band study and wave-function analysis of stoichiometric PdH which shows that, even though the Fermi surface of PdH is qualitatively similar to that of silver, the simple "proton model" is not valid. Instead, the screening of the proton in PdH is found to be larger than in an isolated H atom due, in part, to the formation of a H-Pd bonding band below the bottom of the d-band complex. This result, which is in qualitative agreement with Switendick's earlier calculation, is confirmed by ultraviolet photoemission experiments. A partial density-of-states (DOS) analysis in the energy range spanned by the six valence and conduction bands reveals the quantitative details of the bonding mechanism between the Pd and H constituents. At the Fermi energy, the high Pd d to H s DOS ratio ∼ 10.3 is found to be far higher than expected in silver, despite the fact that the Fermi-surface geometry is similar. The field-induced conduction-electron spin density at the proton site is evaluated from the wave functions at the Fermi energy. The calculated value of the spin-lattice relaxation rate arising from the contact term in the hyperfine interaction is found to be in good agreement with the experimental value of Wiley et al.
Article
Calculations of the electronic structure of transition-metal hydrides are applied to the cohesive energy of 3d and 4d monohydrides, and the single-particle lifetime of states in nonstoichiometric Cu and Pd hydrides. A simple formula is presented which delineates the principal contributions to the cohesive energy of the hydrides: (i) the formation of a metal-hydrogen bonding level derived of states of the pure metal band structure which have s symmetry about the site of the added proton, (ii) a slight increase in binding of the metal d bands due to the added attractive potential, and (iii) the addition of an extra electron to the metal electron sea. The calculations, corrected for Coulomb repulsion at the hydrogen sites, qualitatively reproduce the experimental trends of the heats of formation of the transition-metal hydrides. The single-particle lifetime calculations are in quantitative agreement with Dingle-temperature measurements and they correctly predict the existence of essentially undamped states on the hole sheets of the α-phase PdH Fermi surface.
Article
Absorption isotherms (25°) and thermodynamic parameters of absorption of deuterium by several platinum-palladium alloys have been determined. Relationships between the relative resistance of the alloys and their deuterium content have been established (25°). Results have been compared to the previously obtained data on the hydrogen-platinum-palladium system. The experimental differences between the heats of absorption of deuterium and hydrogen in palladium and several platinum-palladium alloys have been compared to predicted values for these differences. Lattice constants of the f.c.c. platinum-palladium alloys have been determined as a function of both the hydrogen and deuterium content of the alloys. Phase boundaries have been established using X-ray diffraction techniques.
Article
On the basis of the local density functional approximation the authors re-derive a local force theorem and, from this, a linearised expression for total energy differences. This can be used to decompose calculated heats of formation into angular momentum contributions constituting a basis for a bond analysis of heats of formation. They demonstrate this by applying the linearised theory to analyse results of augmented spherical wave calculations of the electronic structure and heats of formation of the transition-metal hydrides NiH, PdH, IrH, PtH and AuH.
Article
Hydrides of iron and iron-based alloys are thermodynamically stable only at hydrogen pressures in the gigapascal range and rapidly lose hydrogen under ambient conditions. At low temperatures, however, these hydrides can be retained in a metastable state at atmospheric pressure after being cooled under high pressure to liquid nitrogen temperature. This review will discuss the current state of studies on phase transformations in the Fe–H and related systems and also on the composition, crystal structure and physical properties of the hydrides, both under high hydrogen pressures and in the 'quenched' metastable state at ambient pressure. The studies at ambient pressure include magnetization measurements, x-ray and neutron diffraction, Mössbauer spectroscopy and inelastic neutron scattering. In the sections on Mössbauer and structural investigations of hydrides of Fe–Cr and Ni–Fe alloys new experimental results will be presented.
Article
Interstitial hydrogen contents and their associated volume increments have been determined for a variety of fcc metals and alloys. Using high pressure techniques, hydrogen contents approaching n = 1, where n = H-to-metal (atomic ratio), have been obtained. Despite electronic and initial volume differences amongst the fcc metallic matrices, all data fall onto a common relationship.
Article
An approach for electronic structure calculations is described that generalizes both the pseudopotential method and the linear augmented-plane-wave (LAPW) method in a natural way. The method allows high-quality first-principles molecular-dynamics calculations to be performed using the original fictitious Lagrangian approach of Car and Parrinello. Like the LAPW method it can be used to treat first-row and transition-metal elements with affordable effort and provides access to the full wave function. The augmentation procedure is generalized in that partial-wave expansions are not determined by the value and the derivative of the envelope function at some muffin-tin radius, but rather by the overlap with localized projector functions. The pseudopotential approach based on generalized separable pseudopotentials can be regained by a simple approximation.
Article
Previous calculations for Group IIIB transition metal and rare earth compounds with hydrogen showed that for the dihydrides a new band is added below the Fermi energy. This bond corresponds to the antibonding combination of the two hydrogen 1s-orbitals in the unit cell. The original transition metal s-band hybridizes with the bonding combination of the two hydrogens, the implications being that for compounds with only one hydrogen per unit cell no new band is formed and the additional electron goes into host metal states. This result has now been verified by calculations for YH and PdH. Previous conclusions about the relative stability of the trivalent trihydrides have been extended to Groups IVB, VB, and VIB dihydrides. The relative position of the added hydrogen bands below the Fermi energy determines the relative stability of a structure. This position depends primarily on interatomic distance. Thus the trihydride forms only for yttrium and the rare earths and not for scandium. The dihydride only forms for transition metals and rare earths for which the metal ion radius (which determines the metal spacing but not the structure) is greater than 1.25 Å. The monohydride forms only when the dihydride is relatively unstable and the density of states in the d-band is high, e.g. VH, NbH, TaH, NiH, and PdH. Quantitative substantiation of these ideas from band structure calculations will be presented.
Article
The limits nmax and nβmin of the two-phase region of PdHn can only roughly be estimated from the shape of the equilibrium isotherms pH2(n). Other methods applied so far do not yield more accurate results. More precise values can be obtained, however, from measurements of the magnetic susceptibility x as a function of the hydrogen content n at various temperatures.Such x(n) isotherms have been measured at temperatures between 20 and 300°C and H2 pressures up to 140 atm (0 · n · 0.8), using samples of Pd wire (1 mm) and Pd foil (33 μm). In the homogeneity range isotherms for adsorption and desorption were identical, in the two-phase region, however, hysteresis was always observed. Here the desorption curve was taken as the equilibrium isotherm, and was applied to determine the values of nmax and nβmin by extrapolation.Measurements on Pd black in the same region of pressure and temperature showed a number of peculiarities, for instance smaller values of susceptibility and smaller hysteresis loops as compared with bulk Pd. These can be attributed to the large specific surface area of Pd black as well as to its strongly distorted lattice structure.By means of the measurements on bulk Pd the position of the critical point of the palladium-hydrogen system could be redetermined with rather high precision: Tc = 291 ± 2°C; nc = 0.250 ± 0.005 mol H/mol Pd; Pc = 19.7 ± 0.2 atm H2.The measurements on Pd-black yielded within the limits of error the same values for the critical temperature and the critical pressure, whereas the value of the H/Pd ratio, properly corrected, was found to be a bit higher, namely nc = 0.260 ± 0.005.Die Grenzen nmax und nβmin des Zweiphasengebietes von PdHn können aus der Form der Gleichgewichtsisothermen pH2(n) nur grob abgeschtzt werden. Andere bisher angewandte Methoden liefern keine genaueren Ergebnisse. Przisere Werte können dagegen erhalten werden aus Messungen der magnetischen Suszeptibilitt x als Funktion des Wasserstoff-Gehaltes n bei verschiedenen Temperaturen.Solche x(n)-Isothermen sind im Temperaturbereich 20 bis 300°C und mit H2-Drücken bis zu 140 atm gemessen worden (0 · n · 0,8), und zwar an Proben aus Pd-Draht (1 mm) und Pd-Folie (33 μm). Im Homogenittsbereich waren die Isothermen für Absorption und Desorption identisch,im Zweiphasengebiet wurde dagegen stets Hysterese beobachtet. Hier wurde jeweils die Desorptionskurve als Gleichgewichtsisotherme angesehen und zur Bestimmung von nmax und nβmin durch Extrapolation herangezogen.Messungen an Pd-Mohr im gleichen Druck- und Temperaturbereich zeigten eine Reihe von Besonderheiten, z. B. kleinere Suszeptibilitt und schmalere Hysterese im Vergleich zum kompakten Pd. Diese können auf die große spezifische Oberflche des Pd-Mohrs zurückgeführt werden sowie auf seine stark gestörte Gitterstruktur.Die Lage des kritischen Punktes im Palladium-Wasserstoff-System konnte mit Hilfe der an kompaktem Pd durchgeführten Messungen mit verhltnismßig großer Genauigkeit neu bestimmt werden: Tc = 291 ± 2°C; nc = 0,250 ± 0,005 mol H/mol Pd; pc = 19,7 ± 0,2 atm H2.Die Messungen an Pd-Mohr lieferten innerhalb der Fehlerstreubreite dieselben Werte für die kritische Temperatur und den kritischen Druck, whrend für das H/Pd-Verhltnis unter Berücksichtigung erforderlicher Korrekturen ein etwas höherer Wert, nmlich nc = 0,260 ± 0,005 gefunden wurde.
Article
Most conventional work on metal hydrides has been restricted to compositions and temperatures at which they are stable under normal pressure. In this paper we describe some results of our recent high-pressure experiments which have been performed in order to extend our scope to the entire range of metal-hydrogen systems. Thus, general features of phase diagrams of binary metal-hydrogen systems, encompassing the whole composition range and temperatures up to the melting point of constituent phases, are established, and a general compression behavior of hydrogen atoms in metallic environment, including interstitial hydrogen and elemental metallic hydrogen, is inferred.