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Volume 83A, number 9 PHYSICS LETTERS 29 June 1981 ACTIVATION ENERGY IN ~ -PALLADIUM HYDRIDES M.A. KHAN, J. KHALIFEH and C. DEMANGEAT Laboratoire de Magnétisme et de Structure Electronique des Solides (LA au CNRS no. 306), Université Louis Pasteur, 67070 Strasbourg Cedex, France Received 23 February 1981 Revised manuscript received 13 April 1981 A tight-binding method is used to estimate the activation energy in a-palladium hydrides. We obtain an interstitial acti- vation energy of 0.26 eV for the indirect migration through the tetrahedral position. This is in fair agreement with experi- mental results. The data of 25 authors on the activation energy of method [6] seems to appear a convenient and reason- hydrogen at small concentration (a-phase) in palladium able approximation, so we have estimated [7] the have been compiled recently [I]. An estimation of this heats of formation of hydrogen at octahedral and tetra- activation energy can be made by using a phenomenol- hedral positions and shown that reasonable agreement ogical pair potential [21. The basic method used to cal- with experiment can be accounted for if the size effect culate the pair potentials involves (1) the selection of a [8] is taken into account. functional form (with a number of free parameters) The heats of formation of hydrogen atoms at octa- for the potential, and (ii) the variation of the param- hedral and tetrahedral interstitial sites and other less eters so that the various calculated quantities agree (in symmetrical positions in palladium metal were esti- a least-squares sense) with the corresponding experi- mated using a generalized tight-binding Slater—Koster mental data. Unfortunately, this type of calculation fit to the band structure calculated from first princi- does not take carefully into account the electronic ples for the host combined with one extra s orbital for structure of the alloy, each isolated impurity atom. The metallic spd orbitals In the case of hydrogen in aluminium [31 the acti- are !Rm)where R is a metallic site and m is the orbital vation energy has been estimated from the determina- symmetry. us) denotes the extra s orbital at the inter- tion of the heat of solution at different interstitial po- stitial site i and is the energy of this extra orbital. sitions, i.e. octahedral 0 and tetrahedral T sites and The perturbed hamiltonian, per spin, is given by: the midpoint between them. These three estimations treated within the spherical solid model lead to a good H=H 0 + is)E~(isJ + ~ Rd)vd(R)(RdI approximation of the interstitial activation energy. R,d This model in which the screening effects of the free- ms + (jRm)!3 . (isi + c.c.) , (1) electron-like conduction electrons are accounted for, R,m is appropriate for describing such simple metals as Al. The drawback of this method for transition metals is where H0 is the hamiltonian of the pure metal and c,c. that the effect of the more localized “d” electrons are denotes complex conjugate. I3~~ is the hopping inte- not incorporated reasonably in this way. A model [41 gral between IRm) and is). The diagonal disorder term has been proposed recently for the tight-binding elec- vd(R) in the d bands is introduced to be in agreement tronic structure of hydrogen, supposed at octahedral with Friedel’s rule: the number of external electrons position [5] in a palladium host. As the tight-binding brought by an interstitial impurity is equal to the total 0 031—9163/81/0000—0000/S 02.50 © North-Holland Publishing Company 457

Activation energy in α-palladium hydrides

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Volume 83A, number 9 PHYSICS LETTERS 29June 1981

ACTIVATION ENERGY IN ~-PALLADIUM HYDRIDES

M.A. KHAN, J.KHALIFEH and C. DEMANGEATLaboratoire deMagnétismeetdeStructure ElectroniquedesSolides(LA au CNRSno. 306), UniversitéLouisPasteur,67070StrasbourgCedex,France

Received23 February 1981Revised manuscript received 13 April 1981

A tight-binding method is used to estimatethe activation energy in a-palladium hydrides. We obtainan interstitial acti-vation energy of 0.26 eV for the indirect migration through the tetrahedral position. This is in fair agreementwith experi-mental results.

Thedataof 25 authorson the activationenergyof method[6] seemsto appeara convenientand reason-hydrogenat smallconcentration(a-phase)in palladium ableapproximation,so we haveestimated [7] thehavebeencompiledrecently [I]. An estimationof this heatsof formationof hydrogenat octahedralandtetra-activationenergycanbe madeby usinga phenomenol- hedralpositionsandshown that reasonableagreementogical pair potential [21.The basicmethodusedto cal- with experimentcanbe accountedfor if the size effectculatethe pair potentialsinvolves(1) the selectionof a [8] is takeninto account.functional form (with a numberof free parameters) The heatsof formationof hydrogenatomsat octa-for the potential,and (ii) the variationof theparam- hedraland tetrahedralinterstitial sitesandotherlessetersso that thevariouscalculatedquantitiesagree(in symmetricalpositionsin palladiummetal wereesti-a least-squaressense)with thecorrespondingexperi- matedusinga generalizedtight-binding Slater—Kostermentaldata. Unfortunately,this type of calculation fit to thebandstructurecalculatedfrom first princi-doesnot takecarefully into accountthe electronic plesfor the hostcombinedwith oneextras orbital forstructureof the alloy, eachisolatedimpurity atom.The metallicspdorbitals

In the caseof hydrogenin aluminium [31theacti- are !Rm)whereR is a metallicsite andm is theorbitalvation energyhasbeenestimatedfrom the determina- symmetry.us) denotesthe extras orbital at theinter-tion of theheat of solutionat different interstitialpo- stitial site i and is theenergyof this extraorbital.sitions,i.e. octahedral0 andtetrahedralT sitesand The perturbedhamiltonian,perspin, is givenby:themidpoint betweenthem.Thesethreeestimationstreatedwithin the sphericalsolid model lead to a good H=H0 + is)E~(isJ+ ~ Rd)vd(R)(RdIapproximationof the interstitial activationenergy. R,dThis model in which the screeningeffectsof thefree- ms

+ (jRm)!3 . (isi + c.c.) , (1)electron-likeconductionelectronsare accountedfor, R,m

is appropriatefor describingsuchsimplemetalsasAl.The drawbackof this methodfor transitionmetalsis whereH0 is thehamiltonianof the puremetal andc,c.that theeffect of the morelocalized “d” electronsare denotescomplexconjugate.I3~~is thehoppinginte-not incorporatedreasonablyin this way. A model [41 gral betweenIRm) and is). The diagonaldisordertermhasbeenproposedrecentlyfor the tight-bindingelec- vd(R) in the d bandsis introducedto be in agreementtronic structureof hydrogen,supposedat octahedral with Friedel’srule: the numberof externalelectronsposition [5] in a palladiumhost.As the tight-binding broughtby an interstitial impurity is equalto thetotal

0 031—9163/81/0000—0000/S02.50© North-HollandPublishingCompany 457

Volume83A, number9 PHYSICS LETTERS 29 June1981

numberof displacedstatesZ‘(EF) up to theFermi a

level.The heatof formation ~E’ per unit cell is defined

by the reactionof themetal(M) with hydrogengastoform thedilute alloy MH1 with hydrogenat the inter-stitial site i: x OT T IL

M(solid) + ~H2(gas) —~MH’(solid) — . (2)

Thereforethechangein energy is givenby:

~E~=E(MH~)—E(M)—~E(H2), (3)

whereE(MH’) andE(M) arethetotal energiesper unit x 00cell of thedilute alloy andthepuremetal,respectively.

Fig. I. Migration pathsof thehydrogenatombetweentheE(H2) is theHartree—Fockenergyfor the separation octahedralpositionsX and~: (a) indirect migrationthrough

of ahydrogenmoleculeinto its constituentelectrons thetetrahedralpositionwitha potentialbarrierof W1 (b)andprotons [9] . We shall approximatethequantity directmigrationbetweentheoctahedralpositions.Thenota-

E(MH’) — E(M) in eq. (3) by the differencein the tionsusedarethoseof table1.sumsof one-electronvalencebandenergies[7],

The following approximationshavebeenmadefor~E~5=E(MHi) —E(M) theestimationof

EF (i) The areapproximatedby the two-centrein-~f Z~(E)dE, (4) tegralsssa,spaand sduandrestrictedto first nearest= EF —

neighboursof hydrogen.

whereEF is theFermi levelof purepalladiumand (ii) Vd(R)is non zero only for Pd atoms,nearestZ~,(E)is thenumberof displacedstatesper spin a up neighboursto hydrogen.Moreovervd(R) is treatedinto energyE [4]: perturbationso thatthesecondtermof (5) is approxi-

matedby:

Z1(E)= _~ [arg~ — E°— ~°(E)) — ~ V~(R)flm(E),

R,m+ ~ arg(l — vd(R)G(E))]. (5) wheretheprime meansthat thesummationis restrict-RR

R,d ed to R first neighboursof hydrogenandnm(E)is the

zs2~°is essentiallyexpressedin termsof hopping inte- density of statesof d symmetry.grals: In all cases(excepttheoctahedralones)we have

~SG(E) = ~ (I3ms)

2GOrnmo(E) (6) founda hydrogenboundstate.WhenboundstatesareR,,n presentwe have to include in the local neutrality crite-

rion [4] the filling of this sboundstate,locatedatIn eq.(6) only intrasiteGreenfunctionsare retained, given by:

Friedel’srule requiresthat: 2N5(i,E~)= (8)

~ dL~(E)/dE~i(7)a A majordifferenceamongoctahedralsites(0), tetra-

Themostprobablestablepositionsof hydrogenin hedralsites(T), midpoints between0 andT (OT) andfcc palladiumareoctahedralat low temperatureand midpoints between0 and0 (00) (seetable 1) is thetetrahedralabove90 K [10]. This tetrahedralposition numberof first nearestneighbours.The valuesof theis usually associatedwith a point defectwhichcan be hoppingintegralsin thesedifferentpositionsresulta vacancy[101. Two pathsfor themigration havebeen from two oppositeeffects:investigated(fig. 1). (i) In theunrelaxedposition,the distancebetween

458

Volume83A, number9 PHYSICSLETTERS 29 June1981

TableILocationofthehydrogenatomin aPd host.ThedistancesbetweenthehydrogenandthenearestPd atomsaregivenin theunre-laxedposition.ThehoppingintegralsbetweenthehydrogenatomandthenearestPd atomaregiven in termsoføo,wherei~oisthehoppingintegralfor theoctahedralpositionanda is thelatticeconstant.

Locationof Nameof Number of DistanceH—Pd Valueofhydrogenatoms position Pd first neighbours thehoppingintegrals

X=0,0,0 0 6 a/2 130~a(1,1,l) T 4 a...J~/4 ~T=138130

(1, 1, 1) OT 3 a~./iT/8 130T= 1.48 Po(1, 1,0) 00 2 a~/~/4 13oo 1.8130

~ ~

thehydrogenatomandits first neighboursdecreases the00 andOT positionsdeducedfrom thevaluesob-if we go from an 0 to an 00 positionfollowing the tamedfor 0 andT positions.decreasingnumberof nearestneighbours,hencehop- The value of the~‘s for the tetrahedralpositionhasping integralsincrease, itself beenadjustedfrom theexperimentalsituation

(ii) On theotherhandthe relaxationhas theoppo- which statesthat the octahedralpositionis the mostsite effect andmorepreciselyisbiggerif thedistance stableone.For ~ 1.38~ we havefound thatL~.ETis smaller [11]. — L~i.Eç,,is 0.1 eV. Taking thevaluesobtainedfrom the

If the hoppingintegral~ for a distanceR0 is extrapolationscheme,i.e.~OT = 1.48i3~and~oo

known,thehopping integralj3 for a givendistanceR = 1.8~o (table 1), we havefound 0.29 eV for W2 andcanbe deducedfrom eithertheform of Ducastelle 0.26eV for W1 . This last valueof W1 comparesfavour-[12]: ably with theexperimentalcompilation[1]. The value

= ~0eP~O), (9) of W2 found for thedirect migration pathdoesnotdiffer appreciablyfrom the indirectmigration path

or the form of Heine [13] which has a R~~’1l) van- throughthe tetrahedralposition.ation (1 and1’ are orbital degeneracies).

A crucial point in the estimationof the jTs is then [1] J.Vdlkl andG.Alefeld, in: Hydrogenin metals,eds.G.thedeterminationof the displacementu(X) of the Alefeld andJ.Vdlkl (Springer,Berlin, 1978)p. 325.

nearestneighbourhostmetal atomat site ~. It is clear [2] M.I. BaskesandC.F. Melius, Phys.Rev.B20 (1979)3197.from lattice statics [14] that theatomic displacements [3] L.M. Kahn,F.PerrotandM. Rasolt,Phys.Rev.B21

u(X) can be deducedfrom the forcesF(X)throughthe (1980)5594.relation(a, y = x,y,z): [41M.A. Khan, J.C.ParlebasandC. Demangeat,Philos.

Mag.B42(1980)111.[51J.P.BugeatandE. Ligeon,Phys.Lett. 71A (1979)93.

Ua(X) = E G~~F7~), (10) [6] p.W.Bullet, Solid StatePhys.35 (1980)129.[7] M.A. Khan, J.C.ParlebasandC. Demangeat,J. Less

whereX andp arelattice sitesandGL is thestatic Common.Met. 77 (1981)1.Greenfunctionwhich is givenin termsof theforce [8] G.Moraitis andC. Demangeat,Inst.Phys.Conf. Sex.55

constantsof thealloy. EstimationsofF (j.i) for hy- (1981)583.drogenat an octahedralposition in a-palladiumhy- [9] C.D.GelattJr.,J.A.WeissandH. Ehrenreich,Solid

StateCommun.17 (1975)663.dridesareknown [15], but no estimationof GL in [10] J.P. Bugeat,D. Sc. Thesis,Universityof Grenoblethe sametype of approximationis actually available. (1979).

While it is almost impossibleto get a realisticesti- [11] N.A. JohnstonandC.A. Sholl, J.Phys.FlO (1980)mation of thedistancebetweenthehydrogenandits 2375.nearestneighbours,we cannotuseanyof thephenom- [121F. Ducastelle,D.Sc. Thesis,Universityof Paris(1972).[13] V. Heine,Solid StatePhys.35 (1980)1.enologicallawsdescribedpreviously.It is thenrather [141V.K. Tewary,Adv.Phys.22(1973)757.difficult to obtaina valueof the13’s for thedifferent [15] J. Khalifeh,G. MoraitisandC. Demangeat,J.dePhys.positionsso we havechosenalinearextrapolationfor submitted.

459