Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, … · 2019-12-19 · Hydrogen in...

Hydrogen in tungsten: Absorption, diffusion, vacancytrapping, and decohesion

Donald F. JohnsonDepartment of Chemistry, Princeton University, Princeton, New Jersey 08544

Emily A. Cartera)

Department of Mechanical and Aerospace Engineering, and Program in Applied andComputational Mathematics, Princeton University, Princeton, New Jersey 08544

(Received 14 August 2009; accepted 3 November 2009)

Understanding the interaction between atomic hydrogen and solid tungsten is importantfor the development of fusion reactors in which proposed tungsten walls would bebombarded with high energy particles including hydrogen isotopes. Here, we reportresults from periodic density-functional theory calculations for three crucial aspects ofthis interaction: surface-to-subsurface diffusion of H into W, trapping of H at vacancies,and H-enhanced decohesion, with a view to assess the likely extent of hydrogen isotopeincorporation into tungsten reactor walls. We find energy barriers of (at least) 2.08 eVand 1.77 eV for H uptake (inward diffusion) into W(001) and W(110) surfaces,respectively, along with very small barriers for the reverse process (outward diffusion).Although H dissolution in defect-free bulk W is predicted to be endothermic, vacanciesin bulk W are predicted to exothermically trap multiple H atoms. Furthermore, adsorbedhydrogen is predicted to greatly stabilize W surfaces such that decohesion (fracture)may result from high local H concentrations.

I. INTRODUCTION

One of the most crucial aspects to the development ofa viable fusion power reactor is the choice of materialsto act as plasma-facing components (PFCs). A largeamount of current research is aimed at characterizinghow various materials can withstand the high-energyparticle flux imposed on the first walls. For example,the divertor plate (in divertor-type reactors) may besubjected to particle fluxes up to 1024 m�2s�1 andheat fluxes up to 10 MW�m�2,1–3 while fluxes at otherplasma-facing walls can be orders of magnitude smaller.High energy particles can sputter surface atoms or pene-trate micrometers deep, leading to defect formation (e.g.,vacancies or interstitials) and transmutations (due toneutron bombardment). These processes will slowlyweaken and corrode the plasma-facing material (PFM).In addition, PFCs must retain structural integrity at hightemperatures (up to 900 �C). While both low-Z (e.g.,C, Be) and high-Z (e.g., W) materials have been consid-ered for use in PFCs, tungsten has emerged as one of themost promising materials for use in PFCs.2,4,5

Tungsten has several properties that make it well suitedfor use as a PFM. Current technology employs W as a

thin (1–3 mm) coating on graphite or carbon fiber com-posite tiles, but future PFCs might be composed simply ofbulk W.2,5 Tungsten retains strength at high temperaturesand can sufficiently conduct heat away from the surface,although underlying Cu-based components will likely actas future heat sinks.2 Ingress, transport, and retention ofradioactive tritium in PFMs is of great concern and theamount of hydrogen isotopes permanently retained intungsten is low compared to carbon-based materials.6

This is one of the main justifications for using W as aPFM. Another issue is that bombardment by high energyparticles causes sputtering of surface atoms and displace-ments of bulk atoms (vacancy/interstitial formation). Thesputtering rate of W is low, and W atoms promptly re-deposit when sputtered,4 resulting in decreased overallerosion. Several groups have investigated blistering onW surfaces caused by D,7–9 but the mechanisms leadingto blister formation are still debated. Cracking, blistering,and mechanical properties will be affected by hydrogenconcentration and mobility. In light of these concerns, it isnecessary to understand the thermodynamics and kineticsof hydrogen absorption into tungsten surfaces and thesubsequent behavior of hydrogen in bulk tungsten, includ-ing diffusion and trapping.

Tungsten surfaces have been well studied experimental-ly, since clean, low-Miller-index surfaces are easily ob-tainable. Special attention has been paid to the W(001)

a)Address all correspondence to this author.e-mail: [email protected]

DOI: 10.1557/JMR.2010.0036

J. Mater. Res., Vol. 25, No. 2, Feb 2010 © 2010 Materials Research Society 315

surface,10–15 which exhibits a (ffiffiffi2

p � ffiffiffi2

p) R45� reconstruc-

tion (see Fig. 1) below room temperature. At highertemperatures, a bulk-terminated c(1 � 1) structure isobserved,10 although ion-scattering experiments havesuggested a disordered surface.16 On the reconstructedsurface, the top layer of atoms displace 0.27 A in the[110] direction to form zigzag chains, which is thoughtto be caused by a strong-coupling charge density wave17

that stabilizes the nearly half-filled d-bands. Density-functional theory (DFT) employing the local-densityapproximation (LDA)18 and tight-binding theory19 cal-culations have confirmed this reconstruction. X-ray dif-fraction experiments20 and DFT calculations21 have alsoinvestigated interlayer relaxation of W(110). Additional-ly, tight-binding theory with a modified fourth momentapproximation was used to calculate relaxation and re-construction on seven different W surfaces.19

Numerous experiments have studied the adsorption of hy-drogen on tungsten surfaces: W(001),22–28 W(110),21,26–30

W(111),26,28,31 and W(211).32–35 Additionally, theoreticalinvestigations of hydrogen adsorption have focused on theW(001)36–39 and W(110)38–41 surfaces. Field-emission-fluctuation measurements29,30 and projector augmentedwave (PAW)-DFT-GGA (generalized gradient approxima-tion) calculations40 found that diffusion of H on theW(110)surface occurs between threefold sites with transition states(TSs) at bridge sites. However, H prefers lower coordina-tion bridge sites on the W(001) surface, as observed withlow-energy electron diffraction (LEED)22 and confirmedwith molecular dynamics simulations employing reactive

interatomic potentials.37 Recently, DFT-GGA calculationswere used to obtain potential energy surfaces for H2 onW(001) and W(110), to predict dissociative sticking prob-abilities.39 Molecular dynamics simulations predicted asticking probability <0.6 for low-energy (3–300 meV) hy-drogen atoms sticking to W(001).37 On W(110), surfacediffusion of H/D via tunneling plays an important rolebelow 130 K.42,43

Mass spectroscopy measurements detecting hydrogendegassing from bulk tungsten by Frauenfelder44 foundthat H sits at tetrahedral interstitial sites, characterizedby an endothermic dissolution energy (1.04 eV) and arelatively high activation energy (0.39 eV) for H diffu-sion [e.g., in another body-centered cubic (bcc) metal,Fe, the H diffusion activation energy is �0.1 eV45].Ultrasoft pseudopotential (USPP)-DFT-GGA calcula-tions recently confirmed the tetrahedral site preference46

and recent PAW-DFT-GGA calculations47 found a lowenergy barrier for diffusion between these sites using thedrag method (n.b., this is an unreliable algorithm forfinding transition states48). Ion-channeling experi-ments49 found that D also sits at tetrahedral sites in W.Recent experiments have focused on H, D, and T disso-lution and long term retention in tungsten, especiallytrapping behavior.50–55 Initial experiments using a In-111probe estimated that W vacancies can trap one or twoH atoms.55 Recent PAW-DFT-GGA predictions suggestvacancies can trap perhaps up to 10 H atoms simulta-neously, or up to eight atoms sequentially, with bindingenergies ranging from 0.4 to 1.2 eV,56 while self-trappingwas ruled out.47 Thermal desorption spectroscopy (TDS)experiments have been used to derive trapping energies ofaround 1.4 eV for H in bulk tungsten,53 while thermaldesorption of D in W found three traps with energies of1.03 eV, 1.34 eV, and 2.1 eV.52 The two lower valueswere attributed to trapping of the first (1.34) and second(1.03) D atoms at vacancies and the higher value wasassumed to be due to trapping near voids.In the current work, we investigate via Kohn-Sham

DFT calculations the thermodynamics and kinetics ofhydrogen absorption into tungsten and its subsequentbulk interactions. Specifically, we calculate surface ad-sorption and subsequent subsurface absorption of hydro-gen on W(110) and W(001), since these are low-energy,low-index surfaces likely to be present as a wall surfaceand have been studied experimentally. We identifysurface-to-subsurface diffusion pathways on these sur-faces and evaluate associated energy barriers and rateconstants. We analyze bulk diffusion, H-V binding,and energy barriers into and out of vacancy trappingsites, as well as overall diffusivities. We also estimatefree energies and equilibrium constants for H isotopeabsorption into defect-free and defective bulk W. Lastly,we analyze hydrogen adsorption at different cover-ages to predict (via a Born-Haber cycle57) the ease of

FIG. 1. Reconstructed W(001) surface with hydrogen (small spheres)

in short-bridge (SB) and long-bridge (LB) adsorption sites. Surface

layer tungsten atoms (large spheres) shift in the ½1�10� direction to

produce zigzag chains (dotted line) in the [110] direction. Second

and third layer atoms are dark gray. Schematics of surface diffusion

pathways are shown for SB-SB, SB-LB, and LB-LB. H is predicted to

diffuse along zigzag chains and to hop occasionally to adjacent chains

via LB intermediates; hopping from LB to LB sites is unlikely.

D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion

J. Mater. Res., Vol. 25, No. 2, Feb 2010316

hydrogen-enhanced decohesion at (110) and (001) planesin bulk W. We provide calculation details in Sec. II andthen present and discuss our results in Secs. III and IV.Finally, we summarize our work in Sec. V.

II. CALCULATION DETAILS

The Vienna ab-initio simulation package (VASP)58,59

was used to perform spin-polarized DFT calculations.We used the all-electron PAW method60,61 within thefrozen core approximation, and treated electron ex-change and correlation with the GGA.62 Spin-polarizedcalculations were performed due to the presence of anunpaired electron on H. Standard PAW ion-electronpotentials were used in which one and six electrons aretreated self-consistently for H and W, respectively.

The kinetic energy cutoff for the planewave basis setwas set to 500 eV; increasing the cutoff resulted invariations in the total energy of <1 meV/atom. We usedk-meshes of 8 � 8 � 8, 6 � 6 � 6, and 4 � 4 � 4 forcalculations of bulk 16-atom (2a0 � 2a0 � 2a0), 54-atom(3a0 � 3a0 � 3a0), and 128-atom (4a0 � 4a0 � 4a0)cells, respectively, where a0 is the lattice constant. 6 �6 � 1 and 8 � 6 � 1 k-meshes were used for the W(001)and W(110) surface calculations, respectively. As a re-sult, the k-point spacing for the surface cells was <0.028(2p/A), consistent with the bulk calculations. Thesek-meshes converged the total energy to <2 meV/atom.The Methfessel-Paxton (MP) smearing method63 wasused to integrate the Brillouin zone and account forpartial occupancies of the metal near the Fermi levelwith a smearing width of 0.1 eV, which keeps totalenergy errors below 1 meV. The MP method also allowsaccurate forces to be calculated for metals. Structureswere considered relaxed when all forces on the ionswere less than 0.025 eV/A. Finally, the energy of H2

(H) was obtained by placing the molecule (atom) in acubic box with sides of 12 A.

The ground state of tungsten is nonmagnetic witha bcc structure. The converged numerical parametersdiscussed above applied within PAW-DFT-GGA pro-duce a lattice constant a0 = 3.171 A and a bulk modulusB = 316 GPa. These are in good agreement with previousDFT-LDA calculations (a0 = 3.163 A64) and experiments(a0 = 3.16 A, B = 323.2 GPa65). Surface slab modelsconsisted of seven layers, with the top four layers allowedto relax and the bottom three layers held fixed to the bulkstructure to mimic a semi-infinite crystal. At least 10 A ofvacuum is added to separate slabs from periodic images.

As mentioned above, the W(001) surface undergoes areconstruction in which atoms shift slightly in the ½1�10�direction to form zigzag chains (Fig. 1). This is the low-temperature phase. Our calculations predict displace-ments of � 0.27 A with an energy gain of 0.14 J/m2,which agrees well with previous DFT calculations.18,39

The second layer atoms also shift 0.05 A, or �20% ofthe displacement of top layer atoms, which is also con-sistent with previous work.18 Experiments at high tem-peratures (>400 K) observe no surface reconstruction,so we also simulate this high-temperature phase. Onboth the reconstructed and unreconstructed surfaces, weplaced H in high and low symmetry sites to determineadsorption site preferences.

We calculate hydrogen adsorption energies, Eads, (andabsorption energies, Eabs, and dissolution energies, Es)according to:

DEads Hð Þ ¼ E WnHð Þ � E Wnð Þ �½E H2ð Þ ; ð1Þwhere WnH refers to the cell with an adsorbed(absorbed) H atom, and Wn is the energy of a W surfaceperiodic cell. However, when calculating dissolutionenergies, Es, Wn refers to a bulk periodic cell containingn W atoms (n = 16, 54, or 128) and WnH contains aninterstitial H atom. Total energies for D (or T) are iden-tical to H, for example, DEads(H) = DEads(D). However,frequencies (and consequently zero-point energy correc-tions and entropy factors) are affected by the mass dif-ference, as explained below.

With adsorption energies and clean W surface ener-gies, we can estimate the ideal fracture energy as afunction of H coverage via a Born-Haber cycle. Follow-ing the scheme of Jiang and Carter,57 we calculate thecoverage-dependent surface energy (half the ideal frac-ture energy), g(YH), according to:

gðYHÞ ¼ � EsðHÞ�YH=Aþ gð0Þþ EadsðYHÞ�YH=A ; ð2Þ

where YH is the fractional coverage of H, A is the areaof the surface periodic cell, g(0) is the surface energy ofthe clean surface, and Eads(YH) is the adsorption energyper H atom at coverage YH. Here, Es(H) is the dissolu-tion energy per hydrogen atom.

We define the hydrogen-vacancy (H-V) binding ener-gy, DEb(V+H), as the energy difference between an Hatom initially dissolved at a bulk interstitial site and anH atom then dissolved near a vacancy. The H-V bindingenergy, DEb(VHn+H), for sequential additional hydro-gen atoms is then calculated as:

DEb VHn þ Hð Þ ¼ EðW53VHnþ1Þ þ EðW54Þ� E W53VHnð Þ � EðW54HÞ ; ð3Þ

where E(W53VHn) refers to the total energy of nH atoms surrounding a vacancy in bulk W, E(W54) refersto the total energy of bulk W, and E(W54H) is the totalenergy of an H atom residing at tetrahedral interstitialsite of bulk W. In each case, the energy of bulk W refersto a 54-atom periodic cell of W, whereas the cases withvacancies are calculated with 53 W atoms in the periodic


J. Mater. Res., Vol. 25, No. 2, Feb 2010 317

cell, with the volume fixed at the equilibrium volume ofperfect bulk W. The binding energy for the first trappedH, DEb(V+H) is given simply by Eq. (3) with n set tozero. To compare to later experiments, we also considerthe trapping energy at a vacancy as the sum of theactivation energy for bulk diffusion, Ea, and the vacancybinding energy, DEb, as explained below.

The climbing-image nudged elastic band (CI-NEB)method was used to determine minimum energy path-ways (MEPs) for diffusion of H into surfaces andthrough bulk W.48 Atomic positions are linearly interpo-lated to set up initial guess images (interpolated struc-tures) along the pathway that are connected by artificialsprings to construct a “band.” The band is relaxedaccording to the CI-NEB algorithm to obtain the MEPand the TS. The energy difference between the minimumand the TS is Ea. As a post-relaxation analysis, themovement of the H atom (neglecting W atom displace-ments) is used to define the reaction progress, as plottedin Figs. 2–7. The absolute displacement of the H atombetween neighboring images is summed to give the cu-mulative distance of the nth image.

Vibrational frequencies are calculated at criticalpoints to confirm minima and characterize the nature ofeach TS; more than one imaginary frequency indicatesa higher order saddle point. Finite displacements of�0.015 A are made on the H atom to construct a numer-ical second derivative (Hessian) matrix from finite dif-ferences of the forces induced by these displacements.Tungsten atoms are kept fixed under the assumption thatlattice vibrations of the heavy metal atoms are sufficient-ly decoupled from the vibrations of the lighter H atom.Therefore, upon diagonalization of the Hessian weobtain three real frequencies at minima and two realfrequencies at TSs.

FIG. 2. Energy profile for hydrogen (small sphere) diffusion into

W(001) from the short-bridge surface site via Path I (see text). The

structure of the surface at critical points is displayed in the insets.


W(001) from the long-bridge surface site via Path II (see text). The

structure of the surface at critical points is displayed in the insets.


bulk-terminated W(001), Path III (see text). The structure of the sur-

face at several positions along the path is displayed in the insets.


W(110) from the threefold surface site. The structure of the surface

at critical points is displayed in the insets.



Zero-point energy (ZPE) corrections were calculated atcritical points from the vibrational frequencies: EZPE =½ Shnj, where nj are the real frequencies. For example,the vibrational frequency of the H2 molecule was cal-culated to be 4325 cm�1, which is close to the ex-perimental value 4301 cm�1,66 and corresponds toEZPE(H) = 134 meV. Frequencies for D and T are derivedby multiplying by the square root of the ratio ofmasses. Therefore, nD ¼ ffiffiffiffiffi

½p

nH and nT ¼ ffiffiffiffi1=3

pnH, which

results in: EZPEðDÞ ¼ffiffiffiffiffi½

pEZPEðHÞ and EZPEðTÞ ¼ffiffiffiffiffi

1=3p

EZPEðHÞ.Rate constants (k) for diffusion (bulk and surface-to-

subsurface) are calculated within the harmonic transitionstate theory (hTST) according to67

khTST ¼Q3N

i ninitialiQ3N�1i nsaddlei

e�ðEa=kBTÞ ; ð4Þ

where ninitial and nsaddle correspond to frequencies of theH atom at the initial minimum and saddle point, respec-tively. As mentioned above, only H is displaced in thefrequency calculations, so N = 1 in Eq. (4), giving rise tothree frequencies at the minimum and two frequencies atthe saddle point. ZPE-corrected Ea are used and entropiccontributions are neglected since they are small, asexplained below. The pre-exponential frequency factoris correspondingly adjusted for D and T isotopes, asdescribed above.

Bulk diffusivity is calculated from the empirical ex-pression for the Arrhenius diffusion constant:

D ¼ D0e�Ea=kBT ¼ n

6a2n0e�Ea=kBT ; ð5Þ

where n is the number of distinct escape pathways froma given minimum, a is the jump length, and n0 is the pre-exponential factor in Eq. (4).

Thermodynamic equilibrium constants, K, are calcu-lated from K ¼ e �DG=kBTð Þ where the Gibbs free energyis DG = DH – TDS. We approximate DH = DU since ourcalculations are at constant volume. We approximate theentropy changes as entirely vibrational entropy, Svib,which is expressed as68:

Svib ¼ kBX3Ni

ln 1� e�hni=kBT� �

þ hnikBT

� �e�hni=kBT � 1

� ��1

; ð6Þ

where ni are the frequencies. Since the H concentrationis small, configurational entropy changes are small andcan be safely neglected. Here, the initial state isH adsorbed on either surface and the final state is eitherH dissolved in bulk or H dissolved at a vacancy.

III. HYDROGEN ON TUNGSTEN SURFACES

A. Adsorption of hydrogen on W(001) and W(110)

On the W(001) surface, we simulate adsorption ofH on sites on the reconstructed surface (low-temperaturephase) and the bulk-terminated, or “unreconstructed”surface (high-temperature phase). Surface reconstructioncreates short-bridge (SB) sites and long-bridge (LB)sites with W-W distances of 2.82 A and 3.57 A, respec-tively. H is predicted to preferentially adsorb on theSB site on reconstructed W(001) (Fig. 1) with Eads =�0.92 eV/atom (�0.88 eV/atom with ZPE corrections)[Eq. (1)]. The SB site contracts further when H isadsorbed, decreasing to a W-W distance of 2.64 A.The LB site was also found to be a local minimum, with

FIG. 6. Energy profile for diffusion of H (small sphere) through bulk

W (gray) where the diffusion distance is the absolute displacement of

H. Only the six nearest atoms are displayed. At the transition state H

is coplanar with the three nearbyW atoms. Inclusion of ZPE-corrections

lowers the barrier to Ea = 0.38 eV.

FIG. 7. Energy profile for hydrogen (small sphere) diffusion from a

nearby tetrahedral site to an octahedral site at a vacancy in bulk

tungsten. Only atoms nearest the defect are shown.



Eads = �0.49 eV/atom (�0.46 eV/atom w/ZPE) and acontracted W-W distance of 3.21 A.

Adsorption on the bridge site of the bulk-terminatedsurface was found to be more exothermic, Eads =�1.20 eV/atom (�1.16 eV/atom w/ZPE), but theadsorbed H atom induces some surface reconstructionwhich could be responsible for the large energy gain.The W-W distance on the clean, unreconstructed surfaceis 3.171 A, but the W-W distance contracts to 2.61 Aacross the bridge site when H adsorbs, i.e., when the Watoms relax toward the adsorbed H.

Previous molecular dynamics (MD) simulations usinga reactive interatomic potential predicted weaker bind-ing: Eads = �0.38 eV/atom37 while DFT-GGA calcula-tions predicted a molecular adsorption energy of 1.9eV,41 equivalent to a dissociative adsorption energy perH atom of Eads = �0.95 eV/atom, which agrees wellwith our results.

To compare with experimental data, we assume thatthe activation energy for desorption is equal to theadsorption energy and we consider our ZPE-correcteddesorption energies per molecule: 1.76 eV and 2.32 eVfor the reconstructed and bulk-terminated surface, re-spectively. This assumption of equality between ad-sorption and desorption energies should be valid here,since H2 dissociation barriers on most transition metalsurfaces are small or zero.69 TDS experiments for highcoverages of hydrogen on W(001) observed a desorp-tion peak corresponding to 1.40 eV.26 Our values arehigher, but they reflect ¼ ML coverage where H-Hrepulsions are much smaller. At 1.0 ML coverage (oneH per surface W), we predict weaker binding due toincreased H-H repulsions: Eads = �0.68 eV/atom(�0.63 eV/atom w/ZPE) and Eads = �0.80 eV/atom(�0.75 eV/atom w/ZPE) for reconstructed and bulk-terminated W(001), respectively. The resulting molecu-lar desorption energy for the reconstructed surface(1.26 eV) agrees fairly well with the high coverage data,1.40 eV. However, since TDS experiments obtain thesedesorption peaks at �800 K, one would expect to seethe high-temperature phase, that is, the bulk-terminatedsurface. The predicted molecular desorption energy(1.50 eV) for the bulk-terminated surface is in slightlybetter agreement with the experimental data (1.40 eV).

The most favorable adsorption site on the W(110)surface is predicted to be the threefold hollow sitewith Eads = �0.75 eV/atom (�0.71 eV/atom w/ZPE).This is more exothermic than previous PAW-DFT-PBEcalculations which predicted an atomic adsorption en-ergy of �2.846 eV at y = 0.5 ML coverage,40 equiva-lent to a dissociative adsorption energy of Eads =�0.59 eV/atom. TDS exhibited desorption peaks cor-responding to activation energies of 1.17 eV/H2 and1.42 eV/H2.

26 Our calculated Eads per molecule withZPE corrections is �1.42 eV, in perfect agreement

with the second desorption peak. The peak correspond-ing to the smaller activation energy is likely due todesorption at higher coverages (again, lateral repul-sions between adsorbed H atoms will lower the de-sorption activation energy).

B. Surface diffusion of hydrogen

Surface diffusion of H on W(110) has been well stud-ied theoretically40 and experimentally30; we thereforefocus our attention on the W(001) reconstructed surface.Diffusion on the unreconstructed surface is difficult tomodel since the structure is metastable in our calcula-tions, which are performed at 0 K. We identify threepossible pathways to describe hops between sites: SB-SB, LB-LB, and SB-LB (Fig. 1). The energy barrier forSB to SB hopping is predicted to be 0.40 eV (0.35 eVw/ZPE). The energy barrier for an SB to LB hop is0.60 eV (0.53 eV w/ZPE) and the reverse process, LBto SB, has an energy barrier of 0.17 eV (0.11 eV w/ZPE)due to the thermodynamic stability difference of 0.43 eVbetween LB and SB sites. An NEB calculation of LB-LBdiffusion predicted a barrier of 0.44 eV, but the saddlepoint is second-order, suggesting direct LB-LB hops willnot occur. Overall, surface diffusion in the low coverageregime will consist of SB-SB hopping and occasionalSB-LB hops. As a result, H is expected to diffuse alongzigzag chains of surface W atoms in [110] directions(Fig. 1). A higher energy, two-step process, SB-LB-SB,allows an H atom to move to an adjacent zigzag chain.

C. Hydrogen absorption into W(001) surfaces

Three possible pathways were considered for surface-to-subsurface diffusion of H into W(001). Paths I and IIinvolve diffusion into the reconstructed surface from theSB and the LB surface sites (Figs. 2 and 3, respectively).Path III considers diffusion into the bulk-terminatedsurface from a bridge site (Fig. 4). Stable subsurfaceminima were identified and MEPs were calculated todetermine inward and outward energy barriers. In allthree pathways, diffusion is simulated for H penetratingpast the first and second layers.Stable subsurface sites for H were found to be tetrahe-

dral interstitials in all cases. H atoms placed in subsur-face sites between the surface layer and second layer ofreconstructed W(001) surface were unstable and rose tothe surface. However, a stable subsurface site was foundin the first subsurface layer of bulk-terminated W(001)(Fig. 4, top left inset). Energy differences (Eabs – Eads)and associated barriers with ZPE-corrected values inparentheses are listed in Table I. For the purposes ofcomparison, the energy differences listed in Table I arereflective of H absorbed to the same penetration depthinto the subsurface in all three pathways. For example,the “First step” refers to diffusion from the surface to the



subsurface site coplanar with second layer W atoms inall three (001) pathways.

Absorption sites in-plane with the second layer (sub1in Table I) were found to have Eabs = 1.07 eV, 0.74 eV,and 0.95 eV for sites along Path I, Path II, and Path III,respectively. Surface-to-subsurface diffusion along PathsI and III is energetically costly, with zero-point-corrected endothermicities > 2 eV. The endothermicityalong Path II is less only because the initial LB adsorp-tion site has higher energy than the other initial adsites.The absorption energies for H in the final sub-subsurfacesites (sub2, between the second and third layer) are stillendothermic: Eabs = 0.99 eV (1.12 eV w/ZPE), Eabs =0.87 eV (1.01 eV w/ZPE), and Eabs = 0.87 eV (1.00 eVw/ZPE) along Paths I, II, and III, respectively. Thesevalues are similar to the bulk dissolution energy, Es =0.96 eV (1.09 eV w/ZPE), suggesting that surface effectspersist only a few layers deep. Still, the 0.1 eV differ-ence between H absorbed under the SB site and underthe LB site indicates that the surface reconstruction has afairly large effect on the sub-subsurface sites, and indeedthe second layer W atoms shift parallel to the surfaceplane as mentioned earlier.

In the Path I scheme, H starts at the SB site and movesinto the surface by passing between the two surfaceW atoms forming the short bridge (see Fig. 2 insets).To accommodate the H, the two W atoms move apartslightly, from an initial separation of 2.64 A to a maxi-mal separation of 3.75 A when H is in-plane with thesurface layer atoms. By comparison, there is less dis-placement of surface W atoms in the Path II schemesince H is diffusing into an LB site where the W-Wdistance is large (3.21 A) initially.

Even though the SB site is the more stable adsorptionsite, the subsurface site under the LB site is the morestable absorption site. The second step of the process isexothermic for Path I, but endothermic for Path II(Table I). Still, the second subsurface site (between layertwo and three) along Path II is more stable than the

corresponding Path I site. An overall mechanism in-volving SB-LB equilibration (endothermicity DE =0.43 eV) followed by absorption into the subsurfacealong Path II (first step endothermicity, 1.23 eV, plussecond step barrier, 0.42 eV, see Table I) has a slightlysmaller total barrier (2.08 eV) than subsurface absorp-tion following Path I (2.13 eV). The overall barrier is theenergy difference between the initial state and the high-est energy TS. In Path II, the Eabs for H in the subsurfaceare both close to the bulk dissolution energy Es (0.96 eV)and the second step barrier (0.42 eV) is close to the bulkdiffusion barrier (see Sec. IV. A), which indicates thatonce H penetrates past the second layer, diffusion behav-ior is bulk-like.

Rate constants for H diffusion into the subsurfacewere calculated according to Eq. (4), where the activa-tion energy is the ZPE-corrected energy difference be-tween the initial state (surface adsorption) and highestenergy TS along the pathway. We predict k = 6.72 � 1012

exp(–2.17 eV/kBT) s–1, k = 5.24� 1012 exp(–1.72 eV/kBT)

s–1, and k = 7.38� 1012 exp(–2.28 eV/kBT) s–1 for Paths I,

II, and III, respectively. However, if we consider Path II toproceed starting from an initial SB adsorption site, thenthe rate constant becomes k = 5.66 � 1012 exp(–2.14 eV/kBT) s

–1. Equilibrium constants for H moving from a SBsite into the bulk are reported in Table II; we see thatH dissolution into perfect bulk W is thermodynamicallyvery unfavorable.

D. Hydrogen absorption into W(110)

Two stable interstitial sites were identified in the sub-surface layer of W(110). The surface-to-subsurface diffu-sion of H into W(110) is also endothermic (Fig. 5 andTable I). From the threefold surface site, H first movesto a subsurface site halfway between the surface layerand second layer (Fig. 5 insets), with Eabs = 0.93 eV[1.05 eV w/ZPE; Eq. (1)]. H then moves to a nearbysubsurface site closer to the second layer, with Eabs =1.11 eV (1.24 eV w/ZPE). At the first subsurface site,

TABLE I. Adsorption energy Eads, first subsurface absorption energy Eabs(sub1), second subsurface absorption energy Eabs(sub2), endothermi-

cities of the first step Eabs(sub1) � Eads(surf) and the second step Eabs(sub2) � Eabs(sub1), and associated energy barriers for surface-to-subsurface

diffusion of H into W (see also Figs. 2�5). All values reported in eV. Values in parentheses are ZPE corrected.

Surface Eads

Eabs

(sub1)

Eabs

(sub2)

First step Second step

Eabs

(sub1)�Eads

Inward

barrier

Outward

barrier

Eabs (sub2)

�Eabs (sub1)

Inward

barrier

Outward

barrier

W(001) Path I

(from SB)

�0.92

(�0.88)

1.07

(1.20)

0.99

(1.12)

1.99

(2.08)

2.00

(2.02)

0.01

(�0)

�0.08

(�0.09)

0.14

(0.08)

0.22

(0.17)

W(001) Path II

(from LB)

�0.49

(�0.46)

0.74

(0.85)

0.87

(1.01)

1.23

(1.31)

1.24

(1.29)

0.01

(�0)

0.13

(0.15)

0.42

(0.41)

0.29

(0.25)

W(001) Path III

(unreconstructed)

�1.20

(�1.16)

0.95

(1.09)

0.87

(1.00)

2.16

(2.25)

�2.16 �0 �0.08

(�0.09)

0.10

(0.03)

0.17

(0.12)

W(110) �0.75

(�0.71)

0.93

(1.05)

1.11

(1.24)

1.69

(1.76)

1.74

(1.77)

0.06

(0.01)

0.18

(0.19)

0.26

(0.20)

0.08

(0.01)



H is coordinated to two surface W and two subsurface Watoms, 1.01 A below the surface layer and 1.22 A abovethe second layer. At the second subsurface site, H is co-ordinated to one surface W and three subsurface W atomsand it is 1.56 A below the surface layer and 0.66 Aabove the second layer. Even at this shallow depth, Eabs isclose to Es, similar to the energetics of H in the (001)subsurface layers. The barrier for the first step is 1.74 eV(1.77 w/ZPE) and the second step barrier is 0.26 eV(0.20 eV w/ZPE) (Table I). Energy barriers for furtherpenetration intoW are expected to approach bulk diffusionbehavior discussed below.

As was done for W(001), we calculate the rateconstants [Eq. (4)] and equilibrium constants for absorp-tion into W(110). For the first step we predict k1 =9.07 � 1012 exp(–1.77 eV/kBT) s

–1. Combining the twosteps together results in a rate constant k12 = 1.07 � 1013

exp(–1.96 eV/kBT) s–1, which suggests H uptake into the

W(110) surface will be enhanced relative to the W(001)surface. Here, k12 = k1�k2/k–1, the product of the rateconstants of the two inward steps (k1 and k2) divided bythe rate constant for outward diffusion of the first step(k–1). Finally, overall equilibrium constants for H movingfrom the W(110) surface to bulk at several different tem-peratures are reported in Table II. Although again insig-nificant quantities of H are predicted to dissolve inperfect bulk W, the favorable thermodynamics of H dis-solution near vacancies increases H absorption by manyorders of magnitude, as we shall discuss further below.

IV. HYDROGEN IN BULK TUNGSTEN

A. Bulk diffusion

Dissolution energies were calculated [Eq. (1)] to beEs = 0.95 eV, 0.96 eV, and 0.87 eV for H in a tetrahedralinterstitial site in a 16-atom, 54-atom, and 128-atomperiodic cell, respectively. Frequency analysis con-firmed the tetrahedral site to be a minimum, in agree-ment with experiment,44,49 and recent USPP- andPAW-DFT-GGA calculations.46,47 Incorporating ZPE-corrections results in an Es = 1.09 eV for the 54-atomcell, in excellent agreement with Frauenfelder’s mea-

surement (1.04 eV).44 ZPE-corrected dissolution ener-gies for D and T are Es = 1.05 eV and Es = 1.04 eV,respectively. The octahedral interstitial site was deter-mined to be a higher order saddle point. The data thatfollow for bulk diffusion, vacancy formation, and H-Vtrapping were obtained using a 54-atom cell (3ao perside).Hydrogen diffusion through tungsten is modeled as

jumps between tetrahedral interstitial sites in the crystallattice. A diffusing H atom passes through an asymmet-ric triangle to reach a neighboring interstice and iscoplanar with these three atoms at the TS (Fig. 6). Wepredict an energy barrier for diffusion of Ea = 0.42 eV(0.38 eV w/ZPE) (Fig. 6). Recent PAW-DFT-GGA cal-culations predicted an energy barrier of 0.20 eV for thispathway,47 but we believe this result is questionablebecause the drag method was used, which is known tooften miss the actual saddle point.48 Moreover, no vibra-tion frequency analysis was reported in Liu et al.,47 sowe do not know whether they found an actual transitionstate to which a barrier can be associated.Following Wert and Zerner’s formulation of diffusion

through a bcc lattice70 and the harmonic transition statetheory,67 we calculate D0 = 8.93 � 10�7 m2/s fromEq. (5). This gives an overall diffusivity of D = 8.93 �10�7 exp(0.38 eV/kBT) m

2/s, which is close to Frauen-felder’s experimental value: D = 4.1 � 10�7 exp(0.391eV/kBT) m

2/s.71 This good agreement lends credence toour predictions for the kinetics of surface diffusion anddiffusion into the subsurface discussed above.Entropic barriers are usually negligible, especially

when the diffusing atom has little effect on the metallattice. Using Eq. (6) to calculate the entropy barrierresulted in entropy contributions, eDS

þ=kB , that range be-tween 0.93 and 1.25 for all isotopes and temperaturesconsidered. Since our entropy factors, which simplyscale the diffusion coefficients, are all �1, we neglectthem in our diffusivity calculations.Upon adjusting frequencies and ZPE corrections

to account for isotope mass, we obtain diffusivitiesfor D and T, respectively, as D = 6.32 � 10�7 exp(0.39eV/kBT) m

2/s andD = 5.16� 10�7 exp(0.40 eV/kBT) m2/s.

TABLE II. Equilibrium constants for H moving from an initial adsorbed state to a final dissolved state (bulk or vacancy). Gibbs free energies,

DG (eV), and equilibrium constants, K, are reported at 500 K, 900 K, and 1450 K.

W(001) (001) Surface ! bulk (001) Surface ! vacancy

T (K) DG K DG K500 2.02 4.45 � 10�21 0.693 1.04 � 10�7

900 2.01 2.16 � 10�13 0.773 1.35 � 10�5

1450 1.96 1.55 � 10�7 0.968 4.31 � 10�4

W(110) (110) Surface ! bulk (110) Surface ! vacancy

T (K) DG K DG K500 1.85 2.09 � 10�19 0.527 4.88 � 10�6

900 1.85 2.13 � 10�12 0.615 1.34 � 10�4

1450 1.82 4.79 � 10�7 0.828 1.33 � 10�3



Overall diffusivities for H, D, and T are listed in Table IIIfor several temperatures. The diffusivity increases bynearly three orders of magnitude over a 1000 K tempera-ture range for all three isotopes. The differences in diffu-sivity between the three isotopes are greatest at the lowesttemperature, as expected. However, they only differ atmost by roughly a factor of two.

B. Vacancy trapping

A vacancy was created by removing oneW atom from a54-atom cell and the vacancy defect formation energy wascalculated to be 3.44 eV according to the conventionalexpression of Gillan,72 which employs a periodic cell con-taining the vacancy with a volume scaled by (N – 1)/N,where N is the number of atoms. This differs only slightlyfrom the constant volume vacancy formation energy,which is predicted to be 3.32 eV, slightly larger thanrecent USPP- and PAW-DFT-GGA predictions.46,47 Thus,to mimic a macroscopic bulk sample with a small concen-tration of vacancies and hydrogen, we keep the cell latticeparameters fixed at perfect bulk W equilibrium valueswhen adding H atoms to the vacancy defect. We thereforecalculate the vacancy binding energies [Eq. (3)] usingconstant volume energies throughout (where the latticeparameters are those of perfect bulk W).

When trapped near a vacancy, H is predicted to sit atan octahedral-like site where it is coordinated to five Watoms, the sixth site being the vacancy (Fig. 7, rightinset). It is energetically favorable for an H atom to residenear a vacancy relative to a bulk tetrahedral site with abinding energy of Eb = �1.24 eV (�1.41 eV w/ZPE).Equilibrium constants for adsorbed H in equilibriumwith vacancy-trapped H (Table II) suggest the pres-ence of vacancies will dramatically increase overallH uptake.

We also characterized an MEP for H hopping into andout of a vacancy trap from a nearby interstitial site(Fig. 7). The dissolution energy at this initial tetrahedralsite (0.66 eV) is already lower than a normal bulk tetra-hedral site (0.96 eV). We predict that H diffuses betweenthree W atoms (similar to bulk diffusion) to reach theoctahedral site at the vacancy with a negligible barrierof 0.013 eV (�0 w/ZPE) and an overall exothermicity,DE = �0.94 eV (�1.10 eV w/ZPE), consistent with re-cent PAW-DFT-GGA predictions.56 Therefore, if an Hatom is close to a vacancy, it will be readily trapped.

As seen in other bcc metals, vacancies can trap multi-ple H atoms.73,74 Additional H atoms placed into octahe-dral sites surrounding the vacancy are predicted to bindexothermically to the vacancy (Table IV). In otherwords, it is energetically favorable for an H atom tomove from a normal bulk interstitial site to a vacancy,even if there is already an H atom near the vacancy. Thesix H-V binding energies listed are all negative, or exo-thermic, suggesting that a vacancy can trap up to at leastsix H atoms. Recent work predicted up to 10 H atomscould be trapped simultaneously at a single vacancy,ultimately forming H2.

56 Since it is much more energeti-cally favorable for H to sit near a vacancy rather than ata bulk interstitial site, H atoms can stabilize vacancies(and vice versa), an effect also seen in other metals.75–77

With just three atoms binding to a vacancy, the energygain (�3.60 eV) outweighs the energy cost to create avacancy (3.32 eV).

We can understand the trends in H-V binding energiesby considering H-H distances and subsequent repulsionsthat result. The second H placed into an octahedralsite opposite the first H, has a binding energy Eb =�1.24 eV, equal to the first binding energy. This config-uration is �0.05 eV lower energy than two H atoms inneighboring octahedral sites around the vacancy. Thedistance between octahedral sites surrounding a vacancywith no H atoms is 2.222 A (3.142 A) for neighboring(opposite) sites. This suggests some H-H repulsionoccurs when atoms are close, but there is little interac-tion when atoms sit far apart, explaining the identicalbinding energy for the first and second H atoms to reachthe vacancy. When three, four, five, or six atoms sur-round the vacancy, H-H repulsion causes some atoms tomove out of pure octahedral positions. For example, inthe stable five-atom configuration, four atoms (forminga square) move away from the fifth atom and toward theempty octahedral site. The four-atom configuration is asquare of H atoms—distorted slightly into a flattenedtetrahedron—with nearest-neighbor distances dH-H =2.07 A. The nearest-neighbor distance is an averageof 2.07 A for five H atoms and decreases to 1.95 A forsix H atoms. The crowding of these H atoms produces

TABLE III. Predicted diffusivity, D (m2/s), of H, D, and T through W

calculated from Eq. (5) at relevant temperatures.

Temperature Hydrogen Deuterium Tritium

500 1.38 � 10�10 7.32 � 10�11 5.27 � 10�11

900 6.81 � 10�9 4.11 � 10�9 3.13 � 10�9

1450 4.33 � 10�8 2.78 � 10�8 2.17 � 10�8

TABLE IV. Hydrogen-vacancy binding energies DEb [eV; Eq. (3)]

for increasing numbers of hydrogen atoms. Also given are the H-H

distances dH-H (A) for nearest neighbors (n) and atoms on opposite

sides of the vacancy (o). Up to six H atoms are predicted to exother-

mically bind to a vacancy. Values in parentheses are ZPE corrected.

Cluster Process DEb(VHn + H) dH-H

VH V + H �1.24 (�1.41) N/A

VH2 VH + H �1.24 (�1.40) 2.55 (o)

VH3 VH2 + H �1.12 (�1.14) 2.12 (n), 2.59 (o)

VH4 VH3 + H �1.00 (�1.12) 2.07 (n), 2.65 (o)

VH5 VH4 + H �0.94 (�0.91) 1.91 (n), 2.23 (n), 2.70 (o)

VH6 VH5 + H �0.54 (�0.79) 1.95 (n), 2.76 (o)



successively larger repulsions and hence successivelyweaker binding energies.

Previous thermal desorption studies assumed a maxi-mum of two H atoms (or D or T) being trapped pervacancy. Previous measurements found the trappingenergy for H in bulk W to be 1.40 eV.53 For D,two trapping energies were measured to be 1.03 eV and1.34 eV.52 These values reflect an overall energy barrierto move H from a trap to the bulk, which is estimatedfrom the sum of the vacancy binding energy andthe migration energy (activation energy for diffusion):Eb + Ea. Using our calculated values (with ZPE) of1.41 eV and 0.38 eV for these quantities, we predict anenergy barrier of 1.79 eV associated with the release ofone H atom from a vacancy trap. A trapping energy of1.40 eV would therefore be equivalent to a vacancy bind-ing energy of 1.02 eV, which could correspond with therelease of the third, fourth, or fifth H atom (Table IV).Thus, the measured trapping energy may be an averageof the three different values. Likewise, successive releaseof a sixth and fifth D atom from a vacancy corresponds totrapping energies of 1.10 eV and 1.32 eV, respectively,which agrees well with the previous measurement.52

C. Hydrogen-enhanced decohesion

We next consider the energetics involved in hydrogen-induced fracture of W between (001) or (110) planes,by construction of a Born-Haber cycle [see Eq. (2)].The exothermic adsorption of H atoms serves to stabi-lize the surface and decrease the fracture energy, asseen in Fig. 8. The ideal cleavage energy (2g) of Walong a (001) plane without reconstruction is 8.04 J/m2

(0.50 eV/A2). With reconstruction, 2g is reduced to 7.75J/m2 (0.48 eV/A2). At 0.50 ML of H, three differentsurface arrangements were considered in the recon-structed W(001) surface cell: two in a (

ffiffiffi2

p � 2ffiffiffi2

p)R45

cell and one in a (2�2) cell. A rotation was performed toaccount for zigzag chains in the [110] direction (Fig. 1)and k-point spacing for this cell was kept below 0.028(2p/A). A “ridged” arrangement in a p(

ffiffiffi2

p �2ffiffiffi2

p)R45

configuration, where all SB sites on one zigzag chaincontain an adsorbed H while the adjacent chain containszero adsorbed H atoms, resulted in the highest energy(Eads = �0.73 eV/atom). A staggered p(

ffiffiffi2

p �2ffiffiffi2

p)R45

configuration, where each zigzag chain contains parallelW-H-W bridges (i.e., alternating SB sites are occupied oneach zigzag chain) that are not parallel to W-H-Wbridges of adjacent chains, was more stable (Eads =�0.90 eV/atom). The lowest energy arrangement (Eads =�0.92 eV/atom) involves SB absorption in a c(2�2)configuration with all W-H-W bridges parallel on allchains, which maximizes the H-H distances.The embrittling effects of H in W are predicted to be

large compared to Fe and Al.57 In fact, the W(110)surface is predicted to cleave exothermically for highconcentrations of H. The origin of this effect is the largeEs for H in bulk W, which appears in Eq. (2). Highenergy H atoms incident on a W surface will penetrateand come to rest deep in the bulk. If the local concentra-tion of H atoms is high enough, the system can lower itstotal energy by cleaving along (110) planes, as predictedby the negative gY/g0 in Fig. 8. This may be an explana-tion for the surface-orientation dependence of blisteringobserved on W surfaces under exposure to deuteriumplasma beams.7 However, while exothermic cleavage isa driving factor, it is not the only factor to consider,since there is an energy cost associated with plasticdeformation that occurs with blistering.As noted previously,57 the Born-Haber cycle is a valid

means to calculate ideal fracture energies as a functionof local hydrogen concentration, if the dissolution ener-gy Es does not vary with H concentration and diffusionbarriers are small. The outward diffusion barriers (Table I)are small, but the diffusion barrier through bulk W is fairlylarge. Therefore, cleavage is only likely to occur whenH atoms are concentrated locally. Due to the favorabletrapping near vacancies, pockets of high H concentrationare likely to be present. Thus, under high neutron fluxesthat produce vacancies and high fluxes of H (or D/T)atoms, our calculations suggest vacancies can become nu-cleation sites for spontaneous crack growth and cleavage,possibly leading to blistering.

V. SUMMARY AND CONCLUSIONS

The kinetics of hydrogen absorption into tungsten,trapping near vacancies, and the effect hydrogen has on

FIG. 8. Ideal fracture energies as a function of hydrogen coverage in

(001) and (110) planes in bulk W [Eq. (2)], as compared to fracture

energies in Fe(110) and Al(111) (*Ref. 57). Values are normalized to

the ideal fracture energy of the metal in the absence of hydrogen. Due

to the high dissolution energy, cleavage along W(110) planes is ener-

getically favorable for high local H concentrations, that is, when

enough atoms are available to form 1.0 ML on both cleaved surfaces.



the cohesive properties of bulk tungsten have beenanalyzed with PAW-DFT-GGA calculations. Predictedadsorption energies are in good agreement with experi-ment, as are bulk diffusion constants. We also investigateddiffusion of H on the reconstructed W(001) surface, whichis predicted to occur along zigzag chains.

Absorption energies within the first three layers of Ware close to the highly endothermic bulk dissolution en-ergy. Energy barriers for surface-to-subsurface diffusionwere large and close to the overall endothermicity of theabsorption process. Consequently, the outward diffusionbarriers from the first subsurface sites are predicted to besmall, 0.06 eV. Overall, the energy barriers (Table I)and equilibrium constants (Table II) suggest H absorp-tion into the (110) surface will be more facile than the(001) surface. The diffusion kinetics of H from the firstsubsurface site to the second site was predicted to besimilar to the bulk diffusion kinetics. These short-rangesurface effects were also observed for H in Fe,78 anotherbcc transition metal.

Large energy barriers for absorption and an endother-mic dissolution energy suggest that the H concentrationin perfect bulk W will be low. This is also reflected in thelow equilibrium constants in Table II for adsorbed H inequilibrium with bulk dissolved H. However, H is pre-dicted to be trapped at vacancies and to greatly decreaseideal fracture energies. This is a combination of good andbad news as it relates to the efficacy of W PFCs. A highneutron flux is expected to result in increased defectformation, which could lead to a high concentration ofvacancies. Therefore, the amount of H retained in W willdepend on the radiation damage incurred.

Our results provide some insight into possible failuremechanisms encountered by excess H (or D or T) in W.H is predicted to be trapped at vacancies, and up to sixatoms can be trapped at a single vacancy. Also, ourcalculations indicate that H stabilizes surfaces and,therefore, is expected to stabilize voids. Due to the largeenergy difference between dissolved H and adsorbed H,we predict decohesion will be favored at a high enoughconcentration, such that one monolayer of H can form oneach cleaved surface. It might be possible to alleviatethese problems by keeping the W surface at a highenough temperature such that H has enough energy toescape traps and W vacancy-interstitial annihilation isenhanced.79 The temperature must still be kept lowenough to maintain overall bulk strength of tungsten. Amore detailed theoretical analysis of W void formationand fracture is warranted.

ACKNOWLEDGMENTS

We are grateful to Princeton University School ofEngineering and the Princeton Plasma Physics Labo-ratory (which is funded by the U.S. Department of

Energy) for funding and to NAVO DRSC for computertime. Figures 1–7 were created with the help of VESTAvisualization software.80

REFERENCES

1. R. Toschi, P. Barabaschi, D. Campbell, F. Elio, D. Maisonnier,

and D. Ward: How far is a fusion power reactor from an experi-

mental reactor. Fusion Eng. Des. 56–57, 163 (2001).

2. H. Bolt, V. Barabash, W. Krauss, J. Linke, R. Neu, S. Suzuki, and

N. Yoshida: Materials for the plasma-facing components of fu-

sion reactors. J. Nucl. Mater. 329–33, 66 (2004).

3. J.B. Condon and T. Schober: Hydrogen bubbles in metals.

J. Nucl. Mater. 207, 1 (1993).

4. D. Naujoks, K. Asmussen, M. Bessenrodt-Weberpals, S. Deschka,

R. Dux, W. Engelhardt, A.R. Field, G. Fussmann, J.C. Fuchs,

C. Garcıa-Rosales, S. Hirsch, P. Ignacz, G. Lieder, K.F. Mast,

R. Neu, R. Radtke, J. Roth, and U. Wenzel: Tungsten as target

material in fusion devices. Nucl. Fusion 36, 671 (1996).

5. M. Kaufmann and R. Neu: Tungsten as first wall material in

fusion devices. Fusion Eng. Des. 82, 521 (2007).

6. E. Serra, G. Benamati, and O.V. Ogorodnikova: Hydrogen iso-

topes transport parameters in fusion reactor materials. J. Nucl.Mater. 255, 105 (1998).

7. W.M. Shu, A. Kawasuso, Y. Miwa, E. Wakai, G.N. Luo, and

T. Yamanishi: Microstructure dependence of deuterium retention

and blistering in the near-surface region of tungsten exposed to

high flux deuterium plasmas of 38 eV at 315K. Phys. Scr. T. 128,96 (2007).

8. V.K. Alimov, J. Roth, and M. Mayer: Depth distribution of deu-

terium in single- and polycrystalline tungsten up to depths of

several micrometers. J. Nucl. Mater. 337, 619 (2005).

9. M. Poon, R.G. Macaulay-Newcombe, J.W. Davis, and A.A. Haasz:

Effects of background gas impurities during D+ irradiation on

D trapping in single crystal tungsten. J. Nucl. Mater. 337, 629(2005).

10. T.E. Felter, R.A. Barker, and P.J. Estrup: Phase-transition on

Mo(100) and W(100) surfaces. Phys. Rev. Lett. 38, 1138 (1977).

11. M.K. Debe and D.A. King: Clean thermally induced W[001]

(1 � 1)-](ffiffiffi2

p � ffiffiffi2

p) R 45� surface-structure transition and its

crystallography. Surf. Sci. 81, 193 (1979).

12. R.A. Barker and P.J. Estrup: Surface-structures and phase-dia-

gram for the H-W(001) chemisorption system. J. Chem. Phys.74, 1442 (1981).

13. D.A. King: Clean and adsorbate-induced surface phase-

transitions on W(100). Phys. Scr. T. 4, 34 (1983).

14. H. Landskron, N. Bickel, K. Heinz, G. Schmidtlein, and

K. Muller: LEED intensity analysis of the clean W(100) C(2x2)

surface reconstruction. J. Phys. Condens. Matter 1, 1 (1989).

15. M.S. Altman, P.J. Estrup, and I.K. Robinson: Multilayer recon-

struction of the W(001) surface. Phys. Rev. B 38, 5211 (1988).

16. I. Stensgaard, L.C. Feldman, and P.J. Silverman: Reconstruction

of the W(001) surface and its reordering by hydrogen adsorption,

studied by MEV ion scattering. Phys. Rev. Lett. 42, 247 (1979).

17. K.S. Shin, H.W. Kim, and J.W. Chung: Evidence for a driving

mechanism of the W(001) reconstruction. Surf. Sci. 385, L978(1997).

18. R. Yu, H. Krakauer, and D. Singh: Equilibrium geometry and

electronic-structure of the low-temperature W(001) surface. Phys.Rev. B 45, 8671 (1992).

19. W. Xu and J.B. Adams: Structure of 7 W-surfaces. Surf. Sci. 319,45 (1994).

20. H.L. Meyerheim, D. Sander, R. Popescu, P. Steadman, S. Ferrer,

and J. Kirschner: Interlayer relaxation of W(110) studied by sur-

face x-ray diffraction. Surf. Sci. 475, 103 (2001).



21. M. Arnold, G. Hupfauer, P. Bayer, L. Hammer, K. Heinz,

B. Kohler, and M. Scheffler: Hydrogen on W(110): An adsorp-

tion structure revisited. Surf. Sci. 382, 288 (1997).

22. P.J. Estrup and J. Anderson: Chemisorption of hydrogen on tung-

sten (100). J. Chem. Phys. 45, 2254 (1966).

23. R.A. Barker and P.J. Estrup: Hydrogen on tungsten (100) adsor-

bate induced surface reconstruction. Phys. Rev. Lett. 41, 1307(1978).

24. D.A. King and G. Thomas: Displacive surface phases formed

by hydrogen chemisorption on W(001). Surf. Sci. 92, 201

(1980).

25. A.H. Smith, R.A. Barker, and P.J. Estrup: Desorption of hydrogen

from tungsten (100). Surf. Sci. 136, 327 (1984).

26. P.W. Tamm and L.D. Schmidt: Binding states of hydrogen on

tungsten. J. Chem. Phys. 54, 4775 (1971).

27. E.W. Plummer and A.E. Bell: Field-emission energy-distributions

of hydrogen and deuterium on (100) and (110) planes of tungsten.

J. Vac. Sci. Technol. 9, 583 (1972).

28. J.B. Restorff and H.D. Drew: Surface reflectance spectroscopy of

hydrogen chemisorbed on W(100), W(110) and W(111). Surf.Sci. 88, 399 (1979).

29. R. Difoggio and R. Gomer: Diffusion of hydrogen and deu-

terium on the (110) plane of tungsten. Phys. Rev. B 25, 3490(1982).

30. S.C. Wang and R. Gomer: Diffusion of hydrogen, deuterium, and

tritium on the (110) plane of tungsten. J. Chem. Phys. 83, 4193(1985).

31. C. Dharmadhikari and R. Gomer: Diffusion of hydrogen and deu-

terium on the (111) plane of tungsten. Surf. Sci. 143, 223 (1984).32. O. Grizzi, M. Shi, H. Bu, J.W. Rabalais, R.R. Rye, and

P. Nordlander: Determination of the structure of hydrogen on a

W(211) surface. Phys. Rev. Lett. 63, 1408 (1989).

33. H.W. Fink and G. Ehrlich: Lattice steps and adatom binding on

W(211). Surf. Sci. 143, 125 (1984).

34. P.G. Flahive and W.R. Graham: Determination of single atom

surface site geometry on W(111), W(211) and W(321). Surf. Sci.91, 463 (1980).

35. R.R. Rye, B.D. Barford, and P.G. Cartier: Chemisorption of H2

on W(211). J. Chem. Phys. 59, 1693 (1973).

36. T.B. Grimley and M. Torrini: Interaction between two hydrogen

atoms adsorbed on (100) tungsten. J. Phys. C: Solid State Phys. 6,868 (1973).

37. K.O.E. Henriksson, K. Vortler, S. Dreissigacker, K. Nordlund,

and J. Keinonen: Sticking of atomic hydrogen on the tungsten

(001) surface. Surf. Sci. 600, 3167 (2006).

38. J. Zhang, Y.J. Yu, Z.X. Wang, W.N. Qin, Z.Y. Diao, and C. Hao:

Adsorption sites and states for H atom on W low-index surfaces.

Acta Chim. Sinica 65, 785 (2007).

39. H.F. Busnengo and A.E. Martinez: H2 chemisorption on W(100)

and W(110) surfaces. J. Phys. Chem. C 112, 5579 (2008).

40. A. Nojima and K. Yamashita: A theoretical study of hydrogen

adsorption and diffusion on a W(110) surface. Surf. Sci. 601,3003 (2007).

41. J.A. White, D.M. Bird, and M.C. Payne: Dissociation of H2 on

W(100). Phys. Rev. B 53, 1667 (1996).

42. R. Difoggio and R. Gomer: Tunneling of hydrogen in surface-

diffusion on the tungsten-(110) plane. Phys. Rev. Lett. 44, 1258(1980).

43. M. Kay, G.R. Darling, and S. Holloway: Comparing quantum and

classical dynamics: H2 dissociation on W(100). J. Chem. Phys.108, 4614 (1998).

44. R. Frauenfelder: Solution and diffusion of hydrogen in tungsten.

J. Vac. Sci. Technol. 6, 388 (1969).

45. Y. Hayashi and W.M. Shu: Iron (ruthenium and osmium)-hydro-

gen systems. Solid State Phenom. 73–75, 65 (2000).

46. K.O.E. Henriksson, K. Nordlund, A. Krasheninnikov, and

J. Keinonen: Difference in formation of hydrogen and helium

clusters in tungsten. Appl. Phys. Lett. 87, 3 (2005).

47. Y.L. Liu, Y. Zhang, G.N. Luo, and G.H. Lu: Structure, stability

and diffusion of hydrogen in tungsten: A first-principles study.

J. Nucl. Mater. 390–391, 1032 (2009).

48. H. Jonsson, G. Mills, and K.W. Jacobsen: Nudged elastic band

method for finding minimum energy paths of transitions, in Clas-sical and Quantum Dynamics in Condensed Phase Simulations,edited by B.J. Berne, G. Ciccotti, and D.F. Coker (World Scien-

tific, Singapore, 1998), p. 385.

49. S.T. Picraux and F.L. Vook: Deuterium lattice location in Cr and

W. Phys. Rev. Lett. 33, 1216 (1974).

50. R.A. Anderl, D.F. Holland, G.R. Longhurst, R.J. Pawelko,

C.L. Trybus, and C.H. Sellers: Deuterium transport and trap-

ping in polycrystalline tungsten. Fusion Technol. 21, 745 (1992).

51. K.L. Wilson, R. Bastasz, R.A. Causey, D.K. Brice, B.L. Doyle,

W.R. Wampler, W. Moller, B.M.U. Scherzer, and T. Tanabe:

Trapping, detrapping and release of implanted hydrogen isotopes.

Nucl. Fusion 1, 31 (1991).

52. M. Poon, A.A. Haasz, and J.W. Davis: Modelling deuterium

release during thermal desorption of D+-irradiated tungsten.

J. Nucl. Mater. 374, 390 (2008).

53. T.J. Venhaus and R.A. Causey: Analysis of thermal desorption

spectra to understand the migration of hydrogen in tungsten.

Fusion Technol. 39, 868 (2001).

54. A.S. Soltan, R. Vassen, and P. Jung: Migration and immobiliza-

tion of hydrogen and helium in gold and tungsten at low temper-

atures. J. Appl. Phys. 70, 793 (1991).

55. J.R. Fransens, M.S.A. Elkeriem, and F. Pleiter: Hydrogen vacancy

interaction in tungsten. J. Phys. Condens. Matter 3, 9871 (1991).56. Y.L. Liu, Y. Zhang, H.B. Zhou, G.H. Lu, F. Liu, and G.N. Luo:

Vacancy trapping mechanism for hydrogen bubble formation in

metal. Phys. Rev. B 79, 4 (2009).

57. D.E. Jiang and E.A. Carter: First principles assessment of ideal

fracture energies of materials with mobile impurities: Implications

for hydrogen embrittlement of metals. Acta Mater. 52, 4801 (2004).58. G. Kresse and J. Furthmuller: Efficient iterative schemes for ab

initio total-energy calculations using a plane-wave basis set.

Phys. Rev. B 54, 11169 (1996).

59. G. Kresse and J. Furthmuller: Efficiency of ab-initio total energy

calculations for metals and semiconductors using a plane-wave

basis set. Comput. Mater. Sci. 6, 15 (1996).

60. G. Kresse and D. Joubert: From ultrasoft pseudopotentials to the

projector augmented-wave method. Phys. Rev. B 59, 1758 (1999).

61. P.E. Blochl: Projector augmented-wave method. Phys. Rev. B 50,17953 (1994).

62. J.P. Perdew, K. Burke, and M. Ernzerhof: Generalized gradi-

ent approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).63. M. Methfessel and A.T. Paxton: High-precision sampling for

brillouin-zone integration in metals. Phys. Rev. B 40, 3616 (1989).64. K.W. Kwak, M.Y. Chou, and N. Troullier: First-principles study

of the H-induced reconstruction of W(110). Phys. Rev. B 53,13734 (1996).

65. C. Kittel: Introduction to Solid State Physics, 7th ed. (John Wiley

& Sons, Inc., New York, 2002).

66. K.P. Huber and G. Herzberg: Molecular Spectra and MolecularStructure IV: Constants of Diatomic Molecules (Van Norstrand

Reinhold Co., New York, 1979).

67. G.H. Vineyard: Frequency factors and isotope effects in solid

state rate processes. J. Phys. Chem. Solids 3, 121 (1957).

68. H.B. Huntington, G.A. Shirn, and E.S. Wajda: Calculation of the

entropies of lattice defects. Phys. Rev. 99, 1085 (1955).

69. J. Harris and S. Andersson: H2 dissociation at metal surfaces.

Phys. Rev. Lett. 55, 1583 (1985).



70. C. Wert and C. Zener: Interstitial atomic diffusion coefficients.

Phys. Rev. 76, 1169 (1949).

71. R. Frauenfelder: Solution and diffusion of hydrogen in tungsten.

J. Vacuum Sci. Technol. 6, 388 (1969).

72. A. Devita and M.J. Gillan: The ab initio calculation of defect

energetics in aluminum. J. Phys. Condens. Matter 3, 6225 (1991).73. Y. Tateyama and T. Ohno: Stability and clusterization of hydro-

gen vacancy complexes in alpha-Fe: An ab initio study. Phys.Rev. B 67, 174105 (2003).

74. A. Ramasubramaniam, M. Itakura, and E.A. Carter: Interatomic

potentials for hydrogen in alpha-iron based on density-functional

theory. Phys. Rev. B 79, 174101 (2009).

75. Y. Fukai: Superabundant vacancies formed in metal-hydrogen

alloys. Phys. Scr. T. 103, 11 (2003).

76. Y. Fukai and N. Okuma: Formation of superabundant vacancies

in Pd hydride under high hydrogen pressures. Phys. Rev. Lett. 73,1640 (1994).

77. G. Lu and E. Kaxiras: Hydrogen embrittlement of aluminum: The

crucial role of vacancies. Phys. Rev. Lett. 94, 4 (2005).

78. D.C. Sorescu: First-principles calculations of the adsorption and

diffusion of hydrogen on Fe(100) surface and in the bulk. Catal.Today 105, 44 (2005).

79. G.N. Luo, W.M. Shu, and M. Nishi: Influence of blistering on

deuterium retention in tungsten irradiated by high flux deuterium

10–100 eV plasmas. Fusion Eng. Des. 81, 957 (2006).

80. K. Momma and F. Izumi: VESTA: A three-dimensional visuali-

zation system for electronic and structural analysis. J. Appl. Crys-tallogr. 41, 653 (2008).



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