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

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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

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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.

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