Using Quantum Mechanics to Investigate the Location andDiusion of Hydrogen and its Isotopes in MetalsMatthewStephenDyerPeterhouseUniversityofCambridgeThisdissertationissubmittedforthedegreeofDoctorofPhilosophy.DeclarationThis dissertation is my own work and contains nothing which is the outcome of work doneincollaborationwithothers,exceptasspeciedinthetextandAcknowledgements.Thisdissertationdoesnotexceed60,000wordsinlength.UsingQuantumMechanicstoInvestigatetheLocationandDiusionofHydrogenanditsIsotopesinMetalsMatthewStephenDyerThe wave functions and energies of hydrogen and its isotopes, deuterium and tritium, werecalculatedinstoichiometric palladiumhydride (PdH), stoichiometric niobiumhydride(NbH), lithiumimide(Li2NH)andmodel potentials. Theywereusedtoinvestigatetheimportance in including quantum eects in determining the location of hydrogen in thesesystems.Therelativestabilityof thetwopotential sitesforhydrogeninpalladiumhydride, wasfound to have a large dependence on the zero point energy of the hydrogen atom. Hydro-geninlithiumimidewasfoundtobedelocalisedinsitesaroundthenitrogenatom. Thequantum tunnelling rate between these sites was calculated to be many times larger thantheclassicalrate.Anexpressionforthediusioncoecientintermsofthewavefunctionsandenergiesofasystem, derivedfromKubotheory, wasusedtocalculatethediusioncoecientsofhydrogenanditsisotopesinthesesystems. Thedierentprocesseswhichcontributetohydrogendiusionwerestudied. Itwasfoundthatitisnecessarytoincludequantumeectswhenconsideringthediusionofhydrogenthoughmaterials.Processesatenergieslowerthantheclassical barriertodiusionwerefoundtobeim-portantinallofthesystemsinvestigated. Inpalladiumhydride,theseprocessesusuallyinvolvedtunnellingfromtheoctahedraltothetetrahedralsite. Inniobiumhydridetun-nellingbetweenthegroundstates inneighbouringwellswas foundtocontribute stronglytodiusion.Calculationofthediusioncoecientinamodelsystemofaone-dimensionalpotentialcoupledtoaharmonicoscillator showedthat bothcoherent andincoherent tunnellingprocessescontributedtodiusion. Coherentprocessesoccurredwithoutachangeinintheharmonicoscillatorquantumstate,whereasincoherentprocessesinvolvedgainingorlosingphononsduringthetransition.Contents1 CalculatingDiusionCoecients 11.1 AnIntroductiontoDiusion. . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 EinsteinsDiusionCoecient. . . . . . . . . . . . . . . . . . . . . 21.1.2 TransitionStateTheory . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 QuantumDiusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.4 CouplingtoOtherNuclearDegreesofFreedom . . . . . . . . . . . 51.2 StatisticalMechanicsofNon-EquilibriumSystems . . . . . . . . . . . . . . 82 TheoreticalMethods 132.1 DensityFunctionalTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.1 FreeEnergyMolecularDynamics . . . . . . . . . . . . . . . . . . . 152.2 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 BlochTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 PalladiumHydride 223.1 PreliminaryStudies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1 BulkPalladium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 PalladiumHydride . . . . . . . . . . . . . . . . . . . . . . . . . . . 24iCONTENTS ii3.2 HydrogenEnergyLevels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 HydrogenDiusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.1 ExperimentalStudies. . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 SimpleEstimationofActivationEnergies. . . . . . . . . . . . . . . 333.3.3 KuboFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 OtherMaterials 494.1 NiobiumHydride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.1 HydrogenEnergyLevels . . . . . . . . . . . . . . . . . . . . . . . . 514.1.2 HydrogenDiusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2 LithiumImide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2.1 TheLocationoftheHydrogenAtoms . . . . . . . . . . . . . . . . . 634.2.2 Diusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 TransportinOneDimension 675.1 Amodelpotential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 ComputationalDetails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3 ResultsandDiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3.1 AnAsymmetricDoubleWellPotential . . . . . . . . . . . . . . . . 695.3.2 TransitionStateTheoryandArrheniusExpressions . . . . . . . . . 775.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81CONTENTS iii6 CouplingtoaHarmonicOscillator 826.1 TheBorn-HuangEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2 AnUncoupledBasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3 AModelPotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.3.1 ResultsandDiscussion . . . . . . . . . . . . . . . . . . . . . . . . . 916.4 ApproximationstoExactCouplingCalculations . . . . . . . . . . . . . . . 1036.4.1 Franck-CondonFactors. . . . . . . . . . . . . . . . . . . . . . . . . 1036.4.2 PerturbationTheory . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137 ConcludingRemarks 114Acknowledgements 117Bibliography 118Chapter1ClassicalandQuantumMechanicalApproachestoCalculatingDiusionCoecients1.1 AnIntroductiontoDiusionMuchof thecontentof thisthesisconcernsthephenomenonof diusion. Thissectiongivesabrief overviewof thesubject, bothfromaclassical andaquantummechanicalperspective. Hansen and McDonalds book, Theory of Simple Liquids[1] and Atkins book,Physical Chemistry[2] have been used for reference material throughout this introduction.Fromamacroscopicperspective,diusionisobservedasthetransportofmatteralongaconcentrationgradient. Fickslawstatesthattheuxofmatter,j (r, t),duetodiusionisproportional toconcentrationgradient, (r, t), where (r, t)istheparticledensityatpositionrandtimet.j (r, t) = D (r, t) (1.1)The constant of proportionality, D, is called the diusion constant or diusion coecient.It is a measure of the rate at which matter will ow along a given concentration gradient.Thecontinuityequationexpressestheconservationofmatterduringdiusion (r, t) +.j (r, t) = 0 (1.2)1CHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 2where =t. Itcanbecombinedwithequation1.1expressingFicksLaw, togivethediusionequation (r, t) = D2 (r, t) (1.3)1.1.1 EinsteinsDiusionCoecientEinstein[3] showed that,in a system of diusing independent classical particles,the solu-tionofthediusionequationgivesthefollowingrelationforthediusioncoecient, intermsofthemeansquaredisplacementoftheparticlesattime,t:D =limt16t|r(t) r (0)|2_(1.4)where representsastatistical average. Thisexpressionallowsthediusioncoecientto be calculated very simply from a classical molecular dynamics simulation in which thepositionsoftheparticles,r(t),arecalculatedateachtimestep.Inasystemwith nite volume equation1.4 willalways give adiusioncoecientof zeroif thelimitistakentoinnity, sincethemeansquaredisplacementcannotgrowlargerthanthesizeofthesystem. Ingeneral,however,theratio|r(t) r (0)|2_/6twillreachaplateauvaluebeforethesidesoftheboxesarereached. ItisthisplateauvaluewhichprovidesthedenitionofDforanitesystem.The displacement of a particle from its initial position r (0) can be expressed as an integralovertheparticlesvelocity,u(t).r(t) r (0) =_t0u(t

) dt

(1.5)Einsteinsexpressionforthediusioncoecientcanthenbeusedtogivearelationbe-tweenthediusioncoecientandthevelocityautocorrelationfunction.The classical time-correlation function between two variables, A(t) and B(t) is A(t)B(0).If AandBarethesamevariablethenthecorrespondingquantityisknownastheau-tocorrelationfunction A(t)A(0). Inanisotropicsystem, thevelocityautocorrelationfunctionis ux(t)ux (0) =13u(t).u(0).CHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 3Startingfromequation1.4andusingequation1.5:D = limt16t|r(t) r (0)|2_(1.6)= limt16t_t0dt

_t

0dt

u(t

) .u(t

) (1.7)Onchangingvariablest

,t

tot

,s = t

t

andintegratingbypartsthisgives:D =13_0dt u(t).u(0) (1.8)Thisexpressionisanexampleof agroupof relationsknownasGreen-Kuborelations.Theyare derivedbyconsideringthe statistical mechanics of non-equilibriumsystems(alsocalleductuation-dissipationtheory)andrelateamacroscopicquantitytothetimeintegral over a time-correlation function of microscopic quantities. They will be discussedinmoredetailinsection1.2.1.1.2 TransitionStateTheoryThediusionofparticlesinthecondensedphasecanoftenbesuccessfullymodelledasacollectionofparticlesjumpingfromonestablesitetoaneighbouringsiteviaanenergybarrier.Eyring[4]proposedthat,forasysteminwhichparticlesmovedbetweenequivalentsites,the diusion coecient, D, could be related to the rate constant for the activated process,k,andthedistancebetweenthesites,l,bytheexpressionD = l2k.Thisrelationshipcanbedemonstratedbyconsideringasimpleone-dimensional systemofidenticalsites,separatedbyadistancel. Therateofchangeofconcentrationatsiteiisgivenby i=

jjKjiiKij(1.9)wherethesumoverjisoverall othersitesandKjiistherateconstantfortheprocessof aparticlejumpingfromsitej tositei. Assumingthat jumps onlyoccur betweenneighbouringsitesandthattherateconstantsforthesejumpsisconstant, Kji=kforj= i 1andzerootherwise. Theexpressionfor ithenbecomes i= i1k 2ik + i+1k (1.10)CHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 4Thediusionequationgives i= D2ix2= Di12i + i+1l2(1.11)wherethesecondderivativeisfoundusingnitedierences. TheexpressionD=l2kisthenobtainedbycomparingequations1.10and1.11.Therateconstant,k,canbeestimatedusingtherelationproposedbyEyring[5]:k = kBThQQ0eEakBT(1.12)whereTisthetemperature,kBisBoltzmannsconstantandhisPlancksconstant. Q0isthepartitionfunctionwiththeparticleinthestablesite, whereasQisthepartitionfunction with the particle at the top of the barrier, neglecting the degree of freedom alongthe path that the particle is moving. is the transmission coecient and gives a measureof theprobabilityof theparticleremaininginthenewsiteratherthanre-crossingthebarrier.Assumingthatbarrierre-crossingsdonotoccurduringdiusionthisgivesthefollowingequationforthediusioncoecient:D =l2kBThQQ0eEakBT(1.13)This expression gives a semi-classical method to calculate the diusion coecient. It takesinto account the quantised energy levels of the diusing particles, but does not include anyeects due to quantum tunnelling or transitions between dierent quantum states duringdiusion. Itisassumedthatthemotionof theparticlecanbeaccuratelydescribedbyclassicalmotiononapotentialenergysurface.1.1.3 QuantumDiusionThe diusion of particles with small mass through a medium cannot always be describedby purely classical or semi-classical techniques. Quantum tunnelling at energies below theclassicalbarriertodiusioncancontributesignicantlytodiusion. Therearedierenttypesoftunnellingprocesseswhichcancontributetothediusionofquantumparticles.Tunnelling can occur between the ground states of two neighbouring wells. No activationCHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 5energy is required and the resulting diusion is largely temperature independent. Groundstatetunnellingoftenonlybecomesimportantatlowtemperatures.Diusioncanalsooccur duetotunnellingbetweentwoexcitedstates, or betweenanexcitedstate inone well andagroundstate inadierent well. These are activatedprocesses, but the activation energy may be well below that predicted by classical theories.Energy and momentum must be conserved in these transitions. Coherent tunnelling occurswhentheparticlestatesinthetwowellsarecoincidentinenergy. Ifthequantumstatesin the two wells are not coincident then the energy dierence can made up by transferringenergy to or from the lattice phonons in the system. This process is known as incoherenttunnelling. Theseprocessesaresummarisedingure1.1iiiiii(a)iviii(b)Figure1.1: Diagramsshowing(a)coherentand(b)incoherenttunnellingprocesses. Theredlinesrepresentparticleenergylevelsandthebluelineslatticephononenergylevels.Theguresshow(i)groundstatetunnelling, (ii)activatedcoherenttunnelling, (iii)theclassicalbarriertodiusionand(iv)activatedincoherenttunnellingAfullyquantummechanicalexpressionforthecalculationofthediusioncoecientcanbe derived by considering the statistical mechanics of systems which are not in equilibrium.1.1.4 CouplingtoOtherNuclearDegreesofFreedomThecalculationofdiusionorreactionratesthroughastaticone-dimensional potentialisoftennotsucienttocorrectlydescribethedynamicsofasystemwithmanynucleardegrees of freedom. Coupling between the motion of the diusing particle and the motionof itssurroundingscanhavealargeeectondiusionratescalculatedbothclassicallyandquantummechanically.IntheirbookChemical Dynamicsat LowTemperature, Benderskii et al[6] reviewsomeofthetechniquesusedtoincludethecouplingtoothernucleardegreesoffreedominanCHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 6approximateway.Itisnotusuallypossibletoincludeall of thenucleardegreesof freedomexplicitlyinacalculationofarateconstantordiusioncoecient, especiallyinthecondensedphase.Acalculationincludingthefull3Nnucleardegreesoffreedomislikelytobetoolargetobecomputationallyfeasibleforallbutthesmallestofmolecularreactions. Ingeneralitis necessary to divide the system into one or more reaction coordinates which are treatedexplicitly, andndsomewaytomodel theinteractionoftheremainingnucleardegreesoffreedom.Itissometimesasucientlygoodapproximationtotreatallmodesexceptthereactioncoordinate as a bath of oscillators which only act to modify the one-dimensional potentialthroughwhichtheparticlediusesorthereactionprogresses. Insuchanapproximationthebathof oscillatorsintroducesafrictional forceactingontheparticlemovingintheone-dimensional potential. The study of this system is known as the dissipative tunnellingproblemandisdescribedatlengthinCaldeiraandLeggettspaper[7].If the conditions are such that only the lowest eigenstates in the initial and nal positionsare occupied then the dissipative tunnelling problem can be reduced to that of a two-levelsystem[8]. Under the two-level system approximation the diusing particle is assumed totunnel instantaneouslywithrespecttolowfrequencyoscillationsinthebath, butwithatunnellingmatrixelementwhichismodiedbycouplingtothehighfrequencybathmodes.The dissipative tunnelling model is known to accurately represent the reduction in trans-portratesduetocouplingwithotherdegreesoffreedom. Inextremecasesthiscouplingcanfullylocaliseaparticleinonewell. However,itisalsothecasethatstrongcouplingwithdegreesof freedomoutsidethereactioncoordinatecanlowerthereactionbarrierorreduceitswidthandhencepromotetransitionsfromonewelltoaneighbouringwell.Couplingtothesemodesisnotwellrepresentedinthedissipativetunnellingmodelandifimportantmustbeincludedinadierentway.Inclusion of the promoting vibrational modes can be included by introducing an exponen-tial decayinthetunnellingmatrixelement, , whichdependsonthepromotingvibra-tionscoordinate, qp, andaconstant, , where1issmall comparedtothetunnellingdistance. = 0eqp(1.14)CHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 7The above approximation was used by Borgis etal[9][10] to include coupling eects whencalculatingprotonandhydrogentransfer rates insolution. Their methodallows thecalculationof hydrogentransfer rates inweakor moderatestrengthhydrogenbondedsystems across adoublewell potential, andincludes theeects of interactions withapolar solvent andthemolecular vibrations whichcausethebarrier touctuate. Thetransfer reaction rate is calculated by integrating over correlation functions available in amoleculardynamicssimulation.In the eld of hydrogen transfer in metals, Flynn and Stoneham[11] proposed a method forcalculatingdiusionratesofhydrogenin1970whichincludescouplingtophononmodesinthemetallattice.AfterusingtheBorn-Oppenheimerapproximationtoseparateouttheelectronicdegreesof freedom, the eigenstates of the nuclei are expressed as products of orthogonal hydrogenwavefunctions, whicharelocalisedat aparticular interstitial siteinthelattice, andalatticewavefunctiondescribingthemotionofthehostlattice.Expressionsforthejumpratebetweentwoneighbouringwellsarethenderivedstartingfrom the perturbation theory expression for the transition probability between two eigen-states. Theresultingexpressionsforthejumpratesatlowtemperaturesdependonthetunnelling matrix element between states in neighbouring interstitial sites and the energycostof distortingthehostlatticeduringthetransition. Someattemptwasalsomadetoincludetheeectsoflattice-activateddiusionbyconsideringthedependenceofthetunnellingmatrixelementonthelatticecoordinates.In2004SundellandWahnstrom[12]performedcalculationsontheniobiumhydrideandtantalumhydridesystems. ParametersforFlynnandStonehamsexpressionwerecalcu-latedbyconstructingathree-dimensionalpotentialenergysurfaceforhydrogeninthesesystems using density functional theory, and then nding the hydrogen ground state wave-functionsandenergiesonthissurface.Although Sundelland Wahnstroms results were infair agreement with experiment,theywereonlyabletocalculateratesoveraveryrestrictedtemperaturerange. TheiruseofFlynnandStonehamsexpressionrestrictsthevalidityof theircalculationstotemper-atureslowenoughtoassumethattheonlythehydrogengroundstatewavefunctionisoccupied,buthighenoughtotreatthelatticephononmodesclassically.In chapter 6 it is shown that the expression used in this thesis to calculate diusion coef-cientscanbeextendedtoincludetheeectsofcouplingtoafullyquantummechanicalCHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 8harmonic oscillator. In some cases it is not possible to separate outthe other nuclear de-grees of freedom on the grounds of having much slower or faster dynamics than hydrogen,anditishopedthatthismethodwill allowamoreaccuratecalculationof theeectofcouplingbetweenthemotionofhydrogenatomsandothernucleiinthesecases.1.2 Statistical Mechanics of Non-Equilibrium SystemsThetheoriesofclassicalstatistical-mechanicsofsystemsinequilibriumwereformulatedmanyyearsagobyscientistssuchasGibbsandBoltzmann.During the mid-twentieth century several studies were made into the statistical mechanicsofirreversibleprocesses,andparticularlyintotransportprocessessuchasdiusion,heattransfer, anduidow. It was shownthat transport properties, suchas conductivityandsusceptibility, canbecalculatedbyconsideringthelinearresponseofasystemtoaperturbationdueanexternalappliedeldorforce[13][14][15][16][17].Thetheoryresultsinexpressionswhichrelatemacroscopicallymeasurablequantitiestotheinnitetimeintegraloveranautocorrelationfunction.Kubo[14]beginsbyconsideringasystem,characterisedbyaHamiltonian,H,whichhasbeenperturbedawayfromequilibriumbyatime-dependentexternal force, F(t). Fromt = tot = 0thesystemisassumedtobeinequilibriumwithanequilibriumdensitymatrix0. Theexternalforceisappliedtothesystematt = 0.Itisassumedthattheperturbationissmall, astheforceisweak, andthatthechangeintheHamiltonianisgivenbyH

(t)= AF(t). Hethenlooksfortheresponsetothesystem in the linear approximation. This response, expressed in the change in the physicalquantityB,B(t),isthenexpressedintermsofthenaturalmotionofthesystem.Thedeviationofthedensitymatrixattimetawayfromitsequilibriumvalueis(t),suchthat(t)=0 + (t). Thetime-dependenceofthedensitymatrix, , isgivenbytheequationofmotion:d(t)dt=1i h_H+H

(t), (t)_(1.15)=1i h_H, (t)_+1i h_H

(t), 0_(1.16)where[A, B] =AB BAandonlylineartermshavebeenkept. Thedeviationof theCHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 9densitymatrixattimetcanbecalculatedbysolvingthepreviousexpression:(t) = 1i h_tei(tt

)H h[A, 0] ei(tt

)H hF(t

)dt

(1.17)TheresponseB(t)of thequantityBisstatisticallycalculatedasatraceoverBandthechangeinthedensitymatrix:B(t) = Tr ((t)B) (1.18)= 1i hTr_tei(tt

)H h[A, 0] ei(tt

)H hBF(t

)dt

(1.19)= 1i hTr_t[A, 0] B(t t

)F(t

)dt

(1.20)whereB(t)isthesolutiontoB(t) = 1ih_B(t),H_, B(0) = B.The expression above for B(t) can be considered in terms of an integral over a responsefunctionBA(t),theresponseofthesystemoveraninnitesimaltimeperiod:B(t) =_tBA(t t

)F(t

)dt

(1.21)BA(t) = 1i hTr [A, 0] B(t) (1.22)Theresponseofcurrentinthe-thdirectionwhenapulseofelectriceldisappliedinthe-thdirectionatt = 0is(t) =1i hTr ([qx, 0] q x(t)) (1.23)=1i hTr_0_q x, q x(t)__(1.24)whereqisthechargeofaparticle,and xisthepositionoperatorinthedirection.Thelastexpressionwasobtainedbycyclicinterchangeinsidethetrace:Tr ([A, 0] B) =

n,m,ln|A|m m|0|l l|B|n n|0|m m|A|l l|B|n (1.25)=

n,m,lm|0|l l|B|n n|A|m

n,m,ln|0|m m|A|l l|B|n (1.26)=

n,m,ln|0|m m|B|l l|A|n n|0|m m|A|l l|B|n (1.27)= Tr (0 [A, B]) (1.28)CHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 10Iftheappliedelectriceldisoscillatory,withanangularfrequency,thentheelectricalconductivity,(),isgivenby() =1V_0(t)eitdt (1.29)=1i hV_0Tr_0_q x, q x(t)__eitdt (1.30)where Vis the volume of the system. The quantum mechanical current operator, = q x,is related to x such that q xeitcan be replaced withi eit. The previous expressioncan thenbe rewritten interms of the current operator rather thanthe positionoperator:() =1 hV_0Tr (0 [ , (t)]) eitdt (1.31)AssumingBoltzmannstatistics, theequilibriumdensitymatrix, 0=eH/Tr_eH_where= 1/kBT.Thestaticeigenvectors of thetime-independent Hamiltonian,H, aregivenbysolvingthetime-independent Schrodinger equationH|n =En|n. Workinginthisbasistheexpressionfortheelectricalconductivitybecomes() =1 hV Q_0

n_n|eH [ , (t)] |n_eitdt (1.32)whereQisthequantummechanical partitionfunctiongivenby

n_n|eH|n_. Intro-ducingthefactthat (t) = eiHt h eiHt hthisbecomes() =1 hV Q_0

n,meEneit_ei h(EnEm)tn| |m m| |n ei h(EnEm)tn| |m m| |n_dt (1.33)Termswherem = nevidentlydonotcontributetothesumandcanberemoved.Greenwood[18]obtainedasimilarexpressionfortheconductivityinmetallicsystemsbysolvingtheBoltzmannequationforasystemofrandomlyplacedscatteringcentresinacrystal.CHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 11Thetimeintegralcanbecarriedoutanalyticallysince_0eitdt=i+i0=(). Thediagonalelementsoftheconductivityarethen() = hV Q

n,m=neEn| n| |m |2__En hEm h+ _+ _En hEm h__(1.34)The-functionsmeanthatcontributionscanonlycomewhen h= En Emwhichmeansthatwecanrewritetheexpression() = hV Q

n,m=ne2(En+Em)| n| |m |2_e2h_En hEm h+ _+ e2h_En hEm h__(1.35)Thestaticelectrical conductivityisfoundinthezero-frequencylimit, 0. Thedif-fusioncoecient is relatedtothestaticelectrical conductivitybytheNernst-Einsteinrelation,D =kBTVNq2

lim0() (1.36)whereNisthenumberofconductingparticlesinvolumeV . Thisexpressionisvalidforconductingparticlesinwhichthemotionof dierenttypesof particlesisuncorrelated.UsingtheNernst-Einsteinrelationthepreviouslycalculatedexpressionfortheelectricalconductivitycanbechangedtoanexpressionforthediusioncoecient,D:D =kBTNQhq2lim01_e2he2h_ n,m=neEn| n| |m |2_EnhEmh_(1.37)Takingthelimitthat 0thisgives:D =NQq2

n,m=neEn| n| |m |2_En hEm h_(1.38)Thecurrentoperatorissimplyrelatedtothequantummechanicalmomentumoperator, p, by =q x=qm pwheremisthemassofaparticle. Thisgivesanal expressionforthediusioncoecientofD =NQm2

n,m=neEn| n| p|m |2_En hEm h_(1.39)CHAPTER1. CALCULATINGDIFFUSIONCOEFFICIENTS 12Equation1.39 gives anexpressionthat allows the calculationof diusioncoecientsoncetheeigenstatesof asystemareknown. Ratherthanworkinginthespaceof thefull Hamiltonianof the system, it will be necessarytomake approximations byonlyconsidering certain degrees of freedom with quantum mechanics and to assume that goodresultscanstillbeobtainedusingsingle-particlewavefunctionsandenergies.Althoughthetimeintegral wasperformedtoinnity, itshouldonlybeperformedtoalargeenoughupperlimittoaverageoutmicroscopicuctuationsbutbeforemacroscopicequilibriumisreached[13][15][16]. Introducinganupperlimittothetimeintegralwouldproducesinc-likefunctionswithanitewidth. The-functioninequation1.39will bereplacedwithaGaussianfunctioninthecalculationswhichfollow.Chapter2TheoreticalMethods2.1 DensityFunctionalTheoryTheelectronicstructurecalculationsdescribedinthefollowingsectionswerecarriedoutusingthe Car-Parrinellomolecular dynamics (CPMD)[19] andCASTEP[20] packages.Thepackagesusedensityfunctionaltheory(DFT)inaplane-wavebasistocalculatetheelectronicwavefunctionsandenergiesofasystem,giventhenuclearcoordinates.InquantummechanicsthepropertiesofasystemaredescribedbyaHamiltonianforthesystem,H,andthecorrespondingsetofmany-bodywavefunctions, {N},whichsatisfythetime-independentSchrodingerequation,HN=ENN. IntheBorn-Oppenheimerapproximation[21]themany-bodywavefunctionisassumedtotaketheformN ({RI} , {ri}) = N ({RI}) n ({ri} ; {RI}) (2.1)where N ({RI}) depends onlyonthe nuclear coordinates, {RI}andn ({ri} ; {RI})depends onthe electronic coordinates,{ri}, andonlyparametricallyon {RI}. Thistendstobeagoodapproximationduetothelargedierenceinmassbetweennucleiandelectrons.The electronic wave functions, {n} are found by solving the time-independent SchrodingerequationwithaneectiveHamiltonian,Helec,whichdependsparametricallyon {RI}toobtainanelectronicenergy, En, whichalsodependsparametricallyon {RI};Helecn=En ({RI}) n.13CHAPTER2. THEORETICALMETHODS 14Theelectroniccoordinatesarethenintegratedoutandtheenergyof theentiresystemcanbecalculatedusingaHamiltonianwhichonlydependsonthenuclearcoordinates,{RI}.Within DFT the electronic energy of a system is expressed in terms of the electron density,n(r), rather than the electronic wave function. Following Kohn and Sham[22], the electrondensitycanbeexpandedinasetofone-electronorbitals,theso-calledKohn-Sham(KS)orbitals: n(r)=

i|i (r))|2. Afunctional of thedensityisthenusedtocalculatetheenergy. TheKSorbitalscanbeexpandedinanysuitablesetofbasisfunctions. Abasisofplane-wavesisusuallychosenifthecalculationsareperformedforaperiodicsystem.Ingeneral thefunctional will includeatermforthekineticenergyof anon-interactingsystemofelectrons,T,asimpleCoulombicterm,EH,givenbytheHartreeenergyfunc-tional,atermduetoanyexternalpotential,Eext,andatermcontainingthecorrectionsnecessaryduetoapproximationsinthekineticenergyandCoulombfunctionals, knownastheexchange-correlationfunctional,Exc:E [n] = T [n] + EH [n] + Eext [n] + Exc [n] (2.2)Theexchange-correlationenergyfunctional isingeneral notknownexactly, andanap-proximationtoitmustbemade. Itisoftenapproximatedwithafunctional basedonthe exchange-correlation functional for an homogeneous electron gas, with extra terms in-volvingthelocaldensity(localdensityapproximation(LDA)functionals)andoptionallythe local gradient of the density (generalised gradient approximation (GGA) functionals).TheapproximationusedinthisstudyisaGGAfunctional proposedbyPerdew, Burkeand Ernzerhof (PBE)[23] which has been shown to give accurate energies and geometriesofsystemsinthecondensedphase.Oncetheenergyfunctional isknown, theelectronicenergyisthenfoundbyminimisingthefunctionalwithrespecttotheKSorbitals. TheproblemcanbeformulatedasasetofKSequationswhichcanbesolvedfortheKSorbitals:_T+ VH+ Vext + Vxc_i(r) = ii(r) (2.3)where VH, Vext, and Vxcare the functional derivatives of EH, Eext, and Excrespectively. Ifthe potential is periodic then the equations can be solved using Bloch theory as describedinsection2.3. Alternatively, conventional CPMDperformsminimisationof theenergyfunctional byapplyingdynamical simulatedannealingtoctional dynamicsof theKSCHAPTER2. THEORETICALMETHODS 15orbitals. ThismethodtreatstheelectrondensityadiabaticallyonthezerotemperatureBorn-Oppenheimersurface.2.1.1 FreeEnergyMolecularDynamicsTheconventionalCPMDmethodisunabletocorrectlymodelmetallicsystemsatnitetemperatures. Fractionallyoccupiedstatescannotbetreatedandthetimestepsneededto accurately integrate the equations of motion are prohibitively small in metallic systems,duetothefastdynamicsoftheelectronicdegreesoffreedom.Theintroductionof afreeenergyfunctional [24] allowsforthefractional occupationofstates, andthefunctionalcanbeecientlycalculatedusinganiterativediagonalisationtechnique. Thismethodhasbeenincludedforall theCPMDcalculationsdescribedinthisstudy.Freeenergymoleculardynamics(FEMD)isbasedontheworkof Mermin[25], whoex-tendedHohenbergandKohns[26]theoryofavariationalfunctionalforthegroundstateof asystemtoanite temperature electrongas. Merminshowedthe existence of agrandpotentialfunctionalwhichisminimalforthedensityofanelectrongasinthermalequilibrium,andthatforthisdensity,thefunctionalgivesthefreeenergyofthesystem.InFEMDafreeenergyfunctional, F, is introduced, whichis stationaryat thesamedensityastheMerminfunctional,andallowsthisdensitytobecomputed,F= + N+ EII(2.4)whereisthechemicalpotential,EIIistheion-ionCoulombenergy,andisthegrandpotentialforaninteractingelectrongaswithinDFT:[n(r), RI] = 2 ln det_1 +e( H)_ 12_drdr

n(r)n(r

)|r r

|_drn(r) xcn(r) + xc(2.5)xcistheexchangecorrelationgrandpotential functional, =1/kBTewhereTeisanelectronictemperatureparameter,andHistheone-electronHamiltonianfromtheusualKohn-Shamtheory,H = 122+ Veff(r) (2.6)CHAPTER2. THEORETICALMETHODS 16whereatomicunitshavebeenusedandwhereVeff(r) =

IVeI (r RI) +_dr

n(r

)|r r

|+xcn(r)(2.7)Foragiveneectivepotential,Veff,theresultingdensity,nout(r),isgivenby:[n(r)]Veff(r)= nout(r) (2.8)andcanbedenedintermsoftheone-electronstateswhichareeigenstatesofH,i:nout(r) =

ifii(r)i(r) (2.9)wherefiarethethermalFermi-Diracoccupationnumbersgivenbyfi= 1/(1 +e(i)).Using2.8andthethermodynamicrelation(/)n(r)= Nthefunctional derivativeof Fcanbecalculatedas:Fn(r)=_dr [nout(r) n(r)]_1|r r

|+2xcn(r)n(r

)_(2.10)Thestationarypointin FandtheMerminfunctional isfoundwhenself-consistencyisreachedinthedensity(i.e. nout(r) =n(r)). Tosolvethis problem Fmust beeval-uated. This evaluationis carriedout bydiagonalisationof thedensitymatrix, ij=_riepH rj_, using Trotter factorisation to simplify the density matrix and the Krylov-space(Lanczos)iterativediagonalisationmethod, allowingmixingof inputandoutputdensitiesbetweeniterationstopreventunstablechargeoscillationsarising.CHAPTER2. THEORETICALMETHODS 172.2 PseudopotentialsDue to the rapid oscillations of the electronic wave function in the the region of a nuclearcore,andthehighkineticenergyofthetightlyboundcoreelectrons,aplane-wavebasisset ispoorlysuitedtodescribingtheelectronicwavefunctionintheregionof nuclearcores. Performinganall-electroncalculationwouldrequireaprohibitivelyhighenergycutofortheplane-wavebasisset.Since the physical properties of solids are mainly dependent on the valence electrons,thecoreelectronsandionicpotentialcanbereplacedwithonealternativeweakerpotential.Thispotentialisknownasapseudopotential. Itisalsohelpfultoremovetheoscillatorynatureof thewavefunctions inthecoreregion, andapseudopotential is constructedsothatthisisthecase. Thenewsmoothedwavefunctionsareknownaspseudo-wavefunctions. The pseudo-wave functions and pseuodopotentials are constructed so that theymatchthereal wavefunctionsandpotentialsexactlybeyondacertaincutoradius, rc(seegure2.1).The pseudopotentials usedwithCPMDwere norm-conserving potentials, that is thepseudo-wave functions andreal-wave functions have identical charge densities inthecoreregion. TheyweregeneratedaccordingtothemethodsuggestedbyTroullierandMartins[27]. ThecalculationsusingCASTEPusedtheultrasoftpseudopotentialspro-posedbyVanderbilt[28].rZ/rpseudoVpseudorcalleFigure 2.1: A schematic diagram of all-electron (solid) and pseudo (dashed) wave functionsandpotentialsCHAPTER2. THEORETICALMETHODS 182.3 BlochTheoryInaperiodicpotentialwhereaistheperiodicityinthexdirection,thesingleparticledensity, |(x)|2mustalsohavethesameperiodicity.V(x +a) = V (x) | (x +a) |2= |(x)|2(2.11)Thus a relationship between (x +a) and (x) can be obtained by splitting (x) into apartu(x)whichhasthesameperiodicityasthelatticeandaremainingphasefactor.(x) = eik.xu(x) (2.12)| (x +a) |2= (x +a) (x +a) (2.13)= eik.x+ik.aeik.xik.au (x +a) u(x +a) (2.14)= eik.xeik.xu(x)u(x) (2.15)= (x)(x) = |(x)|2(2.16)This is a statement of Blochs theorem [29]. The part of the wave function with periodicityaisknownastheBlochfunction. Thevectorkmustbeavectorinthereciprocalspaceofthesystemandcanbeusedtolabeleachwavefunctionwithoutlossofgenerality.k(x) = eik.xuk(x) (2.17)Itisoftenusefultofurtherdividethevectorkintothesumofthevectorpointingtotheclosestpointonthereciprocallattice, g, (denedbyg=2GawhereGisaninteger)andthenaremainderwhichlieswithintherstBrillouinzone, k

. Thewavefunctionisthenlabelledbygandk

. Aseig.xisperiodicinathispartcanbeabsorbedintotheBlochfunction. AdeeperdiscussionofvectorsinthereciprocallatticeofasystemisgiveninBrillouinsbookonWavePropagationinPeriodicStructures[30].k,g(x) = eik

.xeig.xuk

,g(x) = eik

.xu

k

,g(x) (2.18)Afterthispointthevectork

shall bere-labelledk, andu

re-labelledasuinordertokeepthenotationasunclutteredaspossible.The Bloch function and the corresponding single particle energy level can be found usingCHAPTER2. THEORETICALMETHODS 19thesingleparticletime-independentSchrodingerequation.12m2eik.xuk,g(x) + (V (x) k,g) eik.xuk,g(x) = 0 (2.19)12meik.x(i+k)2uk,g(x) + (V (x) k,g) eik.xuk,g(x) = 0 (2.20)12m (ik)2uk,g(x) + (V (x) k,g) uk,g(x) = 0 (2.21)AteachpointintherstBrillouinzone(i.e. eachk-point)asetofBlochequationscanbesolvedtoobtainthewavefunctionsandenergiesofthesingleparticlestateslabelledbythereciprocallatticevectorgorequivalentlythequantumnumbern.12m (ik)2uk,n(x) + (V (x) k,n) uk,n(x) = 0 (2.22)The potential and the Bloch functions have periodicity a and can be expressed in a basisof plane-waves with the same periodicity by a Fourier transform. These plane-waves havewave-vectorswhicharethereciprocallatticevectors,g.uk,n(x) =

g uk,n(g)eig.x(2.23)V (x) =

gV (g)eig.x(2.24)EnteringtheseexpressionsintotheBlochequationforuk,n(x),equation2.22,givesasetofnewequationstosolvefortheFouriercoecientsoftheBlochfunctions:12m (ik)2

g uk,n(g)eig.x+__

g

V_g

_eig

.xk,n__

g uk,n(g)eig.x= 0 (2.25)

g_12m|g +k|2k,n_ uk,n(g)eig.x+

g

g

V_g

_eig

.x uk,n(g)eig.x= 0 (2.26)These equations can be simplied by multiplying by a plane-wave with arbitrary momen-CHAPTER2. THEORETICALMETHODS 20tum,eig

.xandintegratingoverx. Takingthersttermintheequations:_a0dxeig

.x

g_12m|g +k|2k,n_ uk,n(g)eig.x=

g_12m|g +k|2k,n_ uk,n(g) (g g

) (2.27)=_12m|g

+k|2k,n_ uk,n (g

) (2.28)Thesecondtermcanbetreatedinthesameway:_a0dxeig

.x

g

g

V(g

) uk,n(g)ei(g+g

).x(2.29)=_a0dx

g

g

V(g

) uk,n(g)ei(g+g

g

).x(2.30)=

gV(g

g) uk,n(g) (2.31)InrealitytheFouriertransformsarecarriedoutasnitesumsoverthereciprocallatticevectors, g. Usuallyanupperlimittothekineticenergyischosensuchthatthesumisperformedovernmaxg-vectors. Acutoenergyischosenandaspherical cutoappliedsuchthat:12m|gmax +k|2< Ecutoff(2.32)Ingeneral, amuchhigher energycutois requiredfor convergenceof electronicwavefunctions than hydrogen wave functions, due to the mass dependence of the kinetic energy.Combiningthetwotermsbacktogethergivesthenmaxequations:_12m|g

+k|2k,n_ uk,n (g

) +

gV(g

g) uk,n(g) = 0 (2.33)andcanbesolvedbydiagonalisingthenmaxnmaxmatrixwhoseelementsaregivenby:Ug,g =12m|g +k|2g,g+V(g g

) (2.34)This gives theFourier coecients for thenmaxBlochfunctions, uk,n, andtheir corre-spondingenergies,k,n.CHAPTER2. THEORETICALMETHODS 21OncetheBlochfunctionsandenergieshavebeencalculated,themomentummatrixele-mentnecessaryforthecalculationofthediusioncoecientusingequation1.39canbeobtainedbyintegrationinFourierspace:k

,n | px|k,n = i_ g uk

,n (g) ei(g

+k

).xx

g

uk,n(g)ei(g+k).xdx (2.35)= _ g,g

uk

,n (g

) uk,n(g) (g +k) ei(g

g+k

k).xdx (2.36)=

g uk

,n (g) (g +k

) uk,n(g)k

,k(2.37)Chapter3PalladiumHydridePalladium hydride has been the subject of many experimental and theoretical studies overthe past 100 years. Extensive academic interest in the subject has arisen from its unusualpropertiesandconnectedtechnologicalapplications.Inmanyothermetals, additionofhydrogencausessevereembrittlementandthelossofelectrical conductivity. Thisisnotthecasewithpalladium. Thehighdiusionratesofhydrogenthroughpalladium, andthelargevolumesof hydrogenthatpalladiumisabletoabsorb, allowpalladiumhydridetobeusedasahydrogendiusionmembraneandapotential hydrogen storage medium. Palladium is also used extensively as a hydrogenationcatalyst.Anextensive reviewof earlyexperimental workonpalladiumhydride is presentedinLewis ThePalladiumHydrogenSystem[31]. MoreinformationcanbefoundinVolklandAlefeldsbookHydrogeninMetals[32], andamorerecentreviewwaspublishedbyFlanaganandOatesin1991[33].Thepalladiumatomsinbothbulkpalladiumandpalladiumhydride havebeenshowntohaveafacecentredcubic(FCC)structure. TwomaininterstitialsitesexistintheFCCstructure,onewithtetrahedralsymmetryandonewithoctahedralsymmetry. TheFCCunitcellwiththeinterstitialsitesmarkedcanbeseeningure3.1.Palladium hydride has been shown to exist in two concentration dependent phases. The phase occurs at low hydrogen concentrations (PdHx x < 0.1) in which the hydrogen atomsarerandomlydistributedintheoctahedral interstitial siteswithinthepalladiumlatticeandarewellseparatedfromeachother. Athigherconcentrations(x>0.6)thephase22CHAPTER3. PALLADIUMHYDRIDE 23Figure 3.1: The unit cell of a face centred cubic lattice with the tetrahedral and octahedralinterstitialsitesmarkedbyredspheres.isformedinwhichnearstoichiometric(x = 1)regionsareformedinwhichthehydrogenatomsoccupytheoctahedralinterstitialsitesformingarocksalt(NaCl)structure. Thetwophasescoexistatintermediateconcentrations.Bulkpalladiumhas alatticeconstant, a, of 3.891Aat roomtemperature[34] whichexpandswiththeadditionofhydrogentoroughly3.9Ainthephaseandbetween4.0Aand4.1Ainthephase[35]. Rosset al measuredthelatticeconstantofPdH0.99at100Ktobe4.095A[36].3.1 PreliminaryStudiesBulk palladium and stoichiometric palladium hydride (PdH) were studied using the CPMDpackage[19] toperformelectronicstructurecalculationsusingdensityfunctional theory(DFT)inaplane-wavebasisset. Calculationswereperformedusinganorm-conservingTroullier-Martins[27] pseudopotential forthepalladiumatoms. ThedensityfunctionalproposedbyPerdew, Burke, andErnzerhof (PBE)[23] withinthegeneralisedgradientapproximation(GGA)wasused. Inadditionafreeenergyfunctional [24] wasincludedtoallowtheaccuratecalculationofmetallicstates.CHAPTER3. PALLADIUMHYDRIDE 243.1.1 BulkPalladiumThe lattice constant and bulk modulus of bulk palladium in an FCC lattice were calculatedbyperformingsinglepointcalculationsofthesystemwithdierentvaluesofthelatticeconstant, a, and tting the results to a polynomial curve. The calculations were found tobeconvergedwithaplane-wavecutoof50Ryd(680eV)andwereperformedona163k-pointgrid.The lattice constant was calculatedtobe 3.97Aandthe bulkmodulus 171GPainreasonable agreement with experimental values of 3.89 A[34] and 180195 GPa[37][38][39].Density functionals using generalised gradient approximations are known to overestimatethelatticeconstantsoftransitionmetalsandunderestimatetheirbulkmoduli.3.1.2 PalladiumHydrideSimilar calculations were carriedout usingstoichiometric palladiumhydride withthehydrogenatomintheoctahedralandtetrahedralinterstitialsites. Inbothcasesaplane-wave cuto of 60 Ryd (820 eV) and a k-point mesh of 123points was found to be sucienttoobtainconvergedresults.Withthehydrogenatomintheoctahedralsitethelatticeconstant,a,wascalculatedtobe4.15Aandwiththehydrogeninthealternativetetrahedral siteitwasfoundtobe4.26 A. Experimentally the lattice constant of almost stoichiometric PdH was found to be4.095Aat100K[36]andpredictedtobe4.09AforPdH1bySchirberandMorosin[40].The DFT calculations show that the most stable conguration is found with the hydrogeninthetetrahedral siteata=4.26A. Thiscongurationwasfoundtobe70meVlowerinenergythanthatwiththehydrogenintheoctahedralsiteata=4.15A.Evenata=4.15Athetetrahedralsiteisfoundtobemorestableby11meV.The relative stabilities of the octahedral andtetrahedral sites are reversedwhenthezeropointenergy(ZPE)ofthehydrogenatomisincluded. TheZPEwasestimatedbyshiftingthehydrogenatomawayfromasiteandrecalculatingtheenergiesusingDFT.Thecurvatureof theresultingpotential wasestimatedusingnitedierencesandtheZPEofthehydrogenatomcalculatedusingtheharmonicapproximation.TheZPEof thehydrogenat theoctahedral site, witha=4.15A, was calculatedtobe68meV, theZPEof hydrogeninthetetrahedral siteat thesamelatticeconstantCHAPTER3. PALLADIUMHYDRIDE 25wascalculatedtobe194meV.IncludingtheZPEofthehydrogenatom,theoctahedralsitebecomesthemoststable,assuggestedbyexperiment[41]. Attheequilibriumlatticeconstantof 4.26AtheZPEof thehydrogenatominthetetrahedral sitedropsto169meV.ThebulkmodulusofPdHwiththehydrogenintheoctahedralsitewascalculatedtobe179 GPa, close to the experimental value of 183 GPa obtained for PdH0.76by Nygren andLeisure[38].3.2 HydrogenEnergyLevelsSeveral investigations have been made into the excitation energies between hydrogen anddeuteriumvibrationalstatesusinginelasticneutronscattering(INS)[42][36][43]. Thevi-brationspectraweremeasuredforboththeandphasesof PdHxandPdDx. Thespectrawereinterpretedbyttingtoresultscalculatedusingathree-dimensional oscil-lator. Thepotential inthedirectionof thenearest palladiumatoms was foundtobestronglyanharmonic.Alongside the experimental studies, Elsasser et al performed calculations of the vibrationalenergylevelsofhydrogenanddeuteriuminpalladiumusingabinitiotechniques[44][45].Amixedbasis of plane-waves andlocalisedfunctions was usedtocalculate the totalenergyofthesystemwiththehydrogenatomatdierentpositionswithinthesuper-cell.The calculations used DFT within the local-density-functional approximation (LDA). Thecore electrons of the palladium atom were replaced by a norm-conserving pseudopotentialandcalculationsweremadeoverasuitablylargek-pointgrid. Theequilibriumlatticeconstantwascalculatedtobe4.07A.The Fourier components of a three-dimensional adiabatic potential for the hydrogen atomwerettedtothepointscalculatedusingDFT,andthehydrogenstateswerecalculatedbysolvingBlochequationsasdescribedinsection2.3.The resulting energy levels were compared with those calculated using a harmonic approx-imationtothepotential,andapotentialinwhichanharmonicitywasincludedperturba-tively. The Fouriertechnique was foundto betterreproduce experimentalvalues andthecalculatedenergylevelswereusedtointerprettheavailabledatefromINSstudies.CHAPTER3. PALLADIUMHYDRIDE 26UsingasimilarproceduretothatemployedbyElsasseret al, apotential wasbuiltbyrepeatedlycarryingoutsinglepointelectronicstructurecalculations,withthehydrogenatomatdierentlocationswithintheFCCunitcell withalatticeconstantof 4.16A.Thehydrogenatomwas placedat thepoints ona323cubicgridwhichareunrelatedbysymmetry, andthetotal energyofthesystemcalculatedinaplane-wavebasisusingDFTwiththePBEGGAfunctional, afreeenergyfunctional, andaTroullier-Martinspseudopotential forthePdatoms. Thecalculationswerecarriedoutovera123k-pointmeshwithaplane-wavecutoof60Ryd(820meV). Acontourplotofthepotential inthe110planeofthelatticecanbeseeningure3.2. 21/2 a 0a0OTTFigure3.2: Acontour plot of thepotential inthe110planeof thePdHlattice. Thepositions of some tetrahedral (T) and octahedral (O) sites are labelled, the others can beinferredbysymmetry. Palladiumatomsarelocatedatthecornersoftheplotandatthecentreoftheupperandlowerborders. Contoursareplacedatintervalsof140meV.Oncethepotentialforthehydrogenatomwasobtainedontherealspacegrid,aFouriertransformwastakenandthehydrogenwavefunctionsandenergylevelscalculatedusingBlochtheory. Whereitwasnecessarytorepresentthewavefunctiononamoredensegridthanthepotential,theextrapointswereinterpolatedfromtheoriginalpotentialonthe323grid. Thewavefunctionsandenergieswerecalculatedatthe-point(k=0).CHAPTER3. PALLADIUMHYDRIDE 27It is assumedthat interactions betweendierent hydrogenatoms are includedinthepotential calculatedusingDFT. Therearenoother hydrogeninteractiontermsintheBlochequationssolvedforthewavefunctionsandenergies.The ground state energies (E0,|000) for1H and2D in the octahedral and tetrahedral sites,and the excitation energies in each well (eM,|) are shown in tables 3.1 and 3.2. The statesarelabelledaccordingtotheschemeusedbyElsasseret al, inwhichMisthenumberofenergyquantainvolvedintheexcitationand |showstheeigenfunctionsintowhichthequantaareplaced. ThetablesshowresultsfromINSstudies, theworkof Elsasseretal, andtheenergiescalculatedusingthemethoddescribedabove. Theenergieswerecalculated both at the experimentally determined lattice constant of 4.095 A and 4.16 A,closetothecalculatedvalue.Thestates |A, |B, |CaredenedbyElsasseretalasfollows:|A =_1/2_(|200 |020) (3.1)|B =_1/6_(|200 +|020 2 |002) (3.2)|C =_1/3_(|200 +|020 +|002) (3.3)Thegroundstateenergiesandexcitationenergiescalculatedinthisstudyarelowerthanthose calculatedbyElsasser et al whichis consistent withthe use of alarger latticeconstant, andhenceashallowerpotential. Asinthestudyof Elsasseret al excitationenergies involving double excitation in any one direction (for example e2,|200) are found tobe higher than predicted using a harmonic potential (i.e. greater than twice e1,|100). Thisis due to the anharmonicity of the potential in the direction of the surrounding palladiumatomswhichhasashallowwellwhichbecomesverysteep.The dierent states can be identied by their degeneracies and by viewing real space plotsoftherelatedwavefunctions. Plotsofrelevantwavefunctionscanbeseeningure3.3.ThestateslabelledE0,|000, e1,|100, e2,|110ande3,|111haves, p, dandf-likesymmetriesrespectively. Aspredicted, thestatelabellede2,|Cisthein-phasecombinationof threedierent2p-likestates.CHAPTER3. PALLADIUMHYDRIDE 28Table 3.1: Zero-point energies and excitation energies for1H,2D and3T in the octahedralsiteinmeV1H2D3TE0,|000a=4.16A 61 39 28a=4.095A 79Calc.[45] 78 51 40e1,|100a=4.16A 54 34 25a=4.095A 64Calc.[45] 62 40 32Expt. (phase) 60[42],56[36],57[43] 40[42]Expt. (phase) 69[42] 46.5[42]e2,|110a=4.16A 104 65 50a=4.095A 123Calc.[45] 117 78 61Expt. (phase) 110[36],115.5[43]Expt. (phase) 115[42]e2,|Ca=4.16A 123 78 60a=4.095A 142Calc.[45] 132 88 69Expt. (phase) 135[36],139[43]Expt. (phase) 137[42]e2,|A= e2,|Ba=4.16A 133 82 62a=4.095A 151Calc.[45] 147 94 73Expt. (phase) 148[36],155[43]Expt. (phase) 156[42]e3,|111a=4.16A 148 95 73a=4.095A 178Expt. (phase) 170[43]CHAPTER3. PALLADIUMHYDRIDE 29Table 3.2: Zero-point energies and excitation energies for1H,2D and3T in the tetrahedralsiteinmeV1H2D3TE0,|000a=4.16A 171 114 109a=4.095A 207Calc.[45] 213 150 123e1,|100a=4.16A 121 86 71a=4.095A 128Calc.[45] 92 77Eoct0,|000Etet0,|000e3,|111e1,|100e2,|Cetet1,|100e2,|110e2,|A/BFigure 3.3: The real part of some1H wave functions labelled in a similar way to table 3.1CHAPTER3. PALLADIUMHYDRIDE 303.3 HydrogenDiusion3.3.1 ExperimentalStudiesThetemperaturedependenceof thediusivityof hydrogenanditsheavier isotopesinpalladiumhas beenstudiedextensivelyusingmanydierent techniques. AreviewofmanyoftheearlierstudiesisprovidedbyVolklandAlefeld[46].Hydrogenandits isotopes are foundtodiuse withArrhenius behaviour inthe tem-peratureregions investigatedandthediusioncoecients canthenbettedwithanexpressionoftheform:D(T) = D0eEakBT(3.4)AcollectionofvaluesofD0andEafortsofexperimentallydetermineddiusioncoe-cientsattemperaturesabove200Kisgivenintable3.3.Inallcasesinwhichthediusionofthetwoisotopeshavebeencomparedtheactivationenergyofdeuteriumislessthanthatofhydrogen. Thisleadstoaninverseisotopeeectin which the heavier isotope, deuterium, diuses faster than the lighter, hydrogen. Thereis currentlyinsucient evidencetocomment ontherelative diusionrateof tritium,although Volkl and Alefeld[46] suggest that tritium has a lower rate of diusion than theothertwoisotopes, andtheactivationenergygivenbySickingandBuchold[47] ishighcomparedtothatobtainedfortheotherisotopesinotherstudies.There is some evidence from NMR studies that at low temperatures (< 200 K), hydrogendiuses with a much reduced activation energy. Cornell and Seymour [48] nd an activa-tionenergyof10020meVbetween100Kand195KinPdH0.7,althoughitisstressedthatthisdataisnotasreliableasthatrecordedathighertemperatures.Therehasbeenmuchdiscussionoverthemechanismofhydrogendiusioninpalladium.Bohmholdt and Wickes work[49] suggests two possible mechanisms. The rst is a simpletransitionfromone octahedral sitetothe neighbouringsite. The secondis asimilartransitioninvolvingthehydrogenatommovingthroughthetetrahedral sitein-betweenthetwooctahedralsites,butnotgettingtrappedthere.Relaxationtimes calculatedinNMRstudies suggest that onlyone process is impor-tant. A model considering only jumps between neighbouring octahedral sites ts the datawell[48][50]. StudiesusingothermethodshavesuggestedthatthediusionmechanismCHAPTER3. PALLADIUMHYDRIDE 31maybemorecomplicated. BegandRoss[51] suggest that inthe-phasediusionismodelledmoreaccuratelybyconsideringotherprocessesalongsidejumpsbetweennear-est neighbour octahedral sites, such as non-nearest neighbour jumps and jumps involvingthetetrahedralsites. SimilarcommentshavebeenmadebyRoweetal[41],KuballaandBaranowksi[52]andMajorowskiandBaranowski[53].Two paths for motion in-between two octahedral sites are shown in gure 3.4. One involvesdirect motion from one site to the next, the other involves passage through an interveningtetrahedralsite. Thehydrogenatommayormaynotbetrappedinthetetrahedralsiteasitpassesthrough.Figure3.4: Twopathsbetweenneighbouringoctahedralsites(red)Thetetrahedralsiteismarkedingreen.CHAPTER3. PALLADIUMHYDRIDE 32Table3.3:D0in103cm2s1andEainmeVfortsofthediusioncoecientofhydrogenisotopesinpalladiumtoArrheniusbehaviourfromexperimentalstudies.xisthemolarfractionofthehydrogenisotopeandpalladium.Reference1H2D3TxD0EaxD0EaxD0EaBohmholdtandWicke,1967[49]0.7435.71.42496.50.7083.50.92306.5HolleckandWicke,1967[54]0.695113.9265110.66582.824711BegandRoss,1970[51]-phase1.1147Volkletal,1971[55]0.20.62.50.52260.60.20.61.70.62060.7SickingandBuchold,1971[47]V.Low10.53.027513Seymouretal,1975[50]0.70.9(3)228(6)Katsutaetal,1979[56]0.012.830.1522513Majorowskiand0.80.9811.30.228740.80.9810.50.22685Baranowski,1982[53]CHAPTER3. PALLADIUMHYDRIDE 333.3.2 SimpleEstimationofActivationEnergiesAsimpleestimationoftheactivationenergywasmadebydraggingthehydrogenatomfromtheoctahedral sitetoaneighbouringtetrahedral siteoroctahedral siteandper-formingDFTcalculationsalongthepath. Thepalladiumatomswereheldrigidduringtheprocedure,withalatticeconstantof4.15A.Thetransitionstateforthepathfromtheoctahedral totetrahedral sitewasfoundat(0.35,0.35,0.35). Thebarrierfromtheoctahedral sitetothispoint, ignoringZPE, wascalculatedas198meV,andfromthetetrahedralsitewas209meV.TheZPEof thehydrogenatomatthetransitionstatewasestimatedbyperformingasimilar calculationtothat describedinsection3.1.2andincludingtheZPEfromthetwomodes perpendicular tothediusionpath. TheZPEat thetransitionstatewascalculatedas182meV. TheactivationenergiesincludingZPEsarethen312meVfromtheoctahedralsiteand197meVfromthetetrahedralsite.AssumingthattheZPEforadeuteriumatomis12timesthatofthehydrogenatom,asimilarcalculationgivesanactivationenergyof 279meVfromtheoctahedral siteand201meVfromthetetrahedralsite. Schematicdiagramsofthesecalculationscanbeseeningure3.5.Although the estimated activation barriers are considerably larger than the experimentalvaluesdiscussedintheprevioussection,theinverseisotopedependenceoftheactivationenergieshasbeenreproduced. TheZPEof thediusingatomatthetransitionstateishigh since the atom must pass between three close palladium atoms. This ZPE is reducedbyamuchlargeramountthantheZPEintherelativelyshallowoctahedral well whenmovingtoaheavierisotope.Asimilarprocedurewascarriedoutforthepathdirectlylinkingtwoneighbouringocta-hedralsites. Theactivationenergyforthisprocess,ignoringZPEs,wascalculatedtobe1160meV. Thisismuchhigherthanthebarriertothetetrahedral site. Thehydrogenatomisnotatatruetransitionstateatthetopofthebarriersincemotiontowardsboththe octahedral and tetrahedral sites is downhill. Only the mode involving motion towardstheneighbouringpalladiumatomshaspositivecurvature.ThecalculatedbarriersareingeneralagreementwiththebarrierscalculatedbyElsasseretalusingasimilarmethodwithaLDAfunctional[45].CHAPTER3. PALLADIUMHYDRIDE 34Thepathbetweentwooctahedral sitesviathetetrahedral siteisamuchlowerenergyprocess. Diusionismuchmorelikelytooccurasaresultofamechanisminvolvingthispaththanbydirectmotionbetweentwooctahedralsites.CHAPTER3. PALLADIUMHYDRIDE 35(a)Tet TS Oct68194182198312209197no ZPE1H, ZPE2D, ZPE(b)Tet TS Oct48137129198279209201no ZPE1H, ZPE2D, ZPEFigure3.5: Aschematicdiagramof thediusionpathfor(a)1Hand(b)2Dfromtheoctahedral sitetothetetrahedral site. CalculatedZPEsandactivationenergiesareinmeV.Thecurvefortheotherisotopehasbeenincludedforcomparison.CHAPTER3. PALLADIUMHYDRIDE 363.3.3 KuboFormulaTheenergiesandwavefunctionsforhydrogenanditsisotopescalculatedinsection3.2wereusedinequation1.39tocalculatediusioncoecientsatdierenttemperatures.ComputationalDetailsThe -function in equation 1.39 was replaced with a Gaussian function with nite width. (EN EM) 12e(ENEM)222(3.5)Thewidth,,waschosenbyperformingcalculationswithvaryingwidthsandchoosingawidth in the region where the diusion coecients varied slowly with respect to changingit. Asetof diusioncurveswithvaryingcanbeseeningure3.6. Inthefollowingcalculationsthewidthwaschosentobe1meV.The diusion coecients calculations were converged with respect to the number of plane-waves used to calculate the energies and wave functions, and the number of states includedinthesumovermomentummatrixelementsinequation1.39. Asummaryoftheparam-etersrequiredforconvergenceisgivenintable3.4.Table3.4: Convergedparametersfordiusioncoecientcalculations.1H2D3TSizeofgridusedforwavefunction 483643803Correspondingplane-wavecuto[eV] 0.68 0.61 0.63Numberofstatesincludedinsum 800 1200 3000Allcalculationswerecarriedoutatthe-point(k=0). Thestatesstudiedinsection3.2were well bound and a test calculation on a 63k-point grid showed little dierence in thediusioncoecientswhenotherk-pointswereincluded.CHAPTER3. PALLADIUMHYDRIDE 37-16-14-12-10-8-6-4 12345678910100 200 500Log10(D) [cm2s-1]1000/T [K-1]20 meV10 meV5 meV2 meV1 meV0.5 meV0.1 meVFigure 3.6: The diusion coecient of1H in PdH calculated with varying Gaussian widths,.CHAPTER3. PALLADIUMHYDRIDE 38ResultsandDiscussionThediusioncoecientsof1H,2Dand3Twerecalculatedinstoichiometricpalladiumhydride over a range of temperatures. The calculations were performed at lattice constantof4.16Aclosetothetheoreticallypredictedvalue. Theresultingdiusioncurvescanbeseeningure3.7.-16-14-12-10-8-6-4 12345678910100 200 500Log10(D) [cm2s-1]1000/T [K-1]T [K]1H2D3TFigure3.7: Thediusioncoecientsof1H,2Dand3Toveratemperaturerangeof100K1000KThediusioncurves between250Kand500Kingure3.7werettedtoArrheniusexpressionsforcomparisonwithexperiment. ThettedvaluesofD0andEacanbeseenintable3.5.CHAPTER3. PALLADIUMHYDRIDE 39Table 3.5: Parameters of an Arrhenius t of diusion coecients. between 250 K and 500K1H2D3TD0[103cm2s1] 2.7 0.35 0.23Ea[meV] 274 263 222The activation energies for1H and2D obtained from the Arrhenius ts fall within the rangeof valuesobtainedbyexperimental methods(seetable3.4). TheinverseisotopeeectintheactivationenergiesisreproducedwitharatioEa(1H)/Ea(2D)of1.04comparedto1.071.10 from the data in table 3.4. The values are closest to those given by MajorowskiandBaranowski[53]whousedsamplesneartostoichiometricpalladiumhydride.Althoughthevalueof D0fallsroughlywithintherangeof experimental datafor1HitislowerthanthatobtainedbyMajorowski andBaranowski andHolleckandWicke[54],whoobtainedsimilaractivationenergies. ThevalueofD0for2Dislowerthanwouldbeexpected. Asaresult, therateof diusionof both1Hand2Disunderestimatedwithrespect to experiment for temperatures above 250 K and the low value of D0 for2D meansthatthereisnoinverseisotopeeectinthediusioncoecientsthemselves, inspiteoftheonepresentintheactivationenergies. Aninverseisotopeisonlyseeningure3.7atlowtemperatures(3T) over most temperature regions, although3T is predicted to diuse at a fasterratethan2Dbelowroughly350K.There has beenlittle experimental researchinto the diusionof dierent isotopes ofhydrogen in the stoichiometric -phase niobium hydride. The results of this study suggeststhatthediusioncoecientsmeasuredinthe-phasewill beconsiderablylowerthanthose measured in the -phase and show large deviations away from Arrhenius behaviourbelowroomtemperature.CHAPTER4. OTHERMATERIALS 634.2 LithiumImideIt has been suggested that lithium imide (Li2NH) is a potential hydrogen storage material[68].Safe storage materials for hydrogen are necessary for the manufacture of practical hydrogen-based energy sources. Li2NH was shown to be able to absorb 6.5 wt% of hydrogen by thereactionLi2NH+H2 LiNH2+LiH.4.2.1 TheLocationoftheHydrogenAtomsInorder to control andimprove the performance of hydrogenstorage materials it isimportant tounderstandthe fundamental properties of the storage material, suchasitscrystalstructureandthelocationofthehydrogenatomswithinit.Earlyexperimental evidencesuggestedacrystal structureinwhichthenitrogenatomsformedaface centredcubic (FCC) lattice withthe lithiumatoms at the tetrahedralinterstitial sites and the hydrogen atoms at the octahedral interstitial sites[69]. A diagramofthisstructurecanbeseeningure4.7(a).Althoughthepositionofthenitrogenatomsandlithiumionshasnotbeendisputed,thepositionof thehydrogenatoms has beenquestioned. Ohoyamaet al arguedthat thenitrogen-hydrogen distance in the structure proposed by Juza and Opp was too large[70].Followinganinvestigationusingneutronpowderdiraction, twoalternativestructureswereproposed. Inbothstructuresthehydrogenatomscouldoccupyoneofseveralsitesaroundthenitrogenatoms. Thepossiblesitesofthehydrogenatomscouldbearrangedinatetrahedron,asingure4.7(b),oratruncatedoctahedron,asingure4.7(c). Thestructureinwhichoneof thetwelvesites per nitrogenatomshowningure4.7(c) isoccupied,wasalsofoundtobethemostlikelybyNoritakeetal,whousedpreciseX-raydiractiontondthestructure[71].Thewavefunctionsandenergiesof hydrogeninLi2NHwerecalculatedusingthesamemethod as that used to nd the hydrogen wave functions in palladium hydride and niobiumhydride(sections3.2and4.1.1).Thepotential wascalculatedona323real spacegridusingCASTEPtoperformDFTcalculations with the PBE functional. Ultrasoft pseudopotentials were used to model theionic cores. The plane-wave cuto was set to 330 eV and calculations performed on a 123k-pointgrid.CHAPTER4. OTHERMATERIALS 64(a) (b) (c)Figure4.7: Crystal structuresofLi2NHproposedby(a)JuzaandOpp[69] and(b),(c)Ohoyamaetal[70]Nitrogenatomsaregreen,lithiumblueandhydrogenwhite.Once the potential was calculated it was interpolated onto an 883grid and used to calculatethehydrogenwavefunctionsandenergiesbysolvingasetofBlochequationsinaplane-wavebasis. Theplane-wavecutousedforcalculatingthehydrogenwavefunctionswas1.1eV.Byperformingthiscalculationoverarangeof latticeparametersthelatticeparameterwith the lowest energy ground state was calculated to be 5.007 A, in good agreement withthevaluesof5.047A[69],5.0769A[70],and5.0742A[71]foundbyexperiment.ThehydrogenoccupationinthegroundstatewavefunctiondidnotmatcheitherofthepossiblestructuresdescribedbyOhoyamaet al andshowningure4.7. Nolowlyingstateswerefoundwithstructuressimilartothoseingure4.7.Instead, thehydrogenatomswerefoundtohaveanequal probabilityof occupyingsixsites at the vertices of an octahedron centred at each nitrogen atom. The centres of thesesiteswereseparatedfromthenitrogenatomsby 1A. Aschematicdiagramandrealspaceplotofthegroundstatewavefunctioncanbeseeningure4.8. Thegroundstatewavefunctionswerefoundat 275meVabovethepotentialminimum.In support of these ndings, DFT calculations were performed in which the position of thehydrogenatomsintheunitcell wasoptimisedwiththehydrogenatomsinitiallyplacedinthesitessuggestedbyOhoyamaetal. Inbothcasesthehydrogenwasfoundtohavemovedtotheoctahedral-typesitesattheendoftheoptimisation.Interestingly, the structure with hydrogen atoms at vertices of an octahedron was consid-ered by Noritake et al and dismissed due to a slightly worse t to the diraction data thanCHAPTER4. OTHERMATERIALS 65the model shown in gure 4.7(c). Further experimental evidence is required to determinethecorrectlocationofthehydrogenatoms.Rather thantheclassical pictureof thesites beingrandomlylledwithaprobabilityof1/6, inthequantummechanical pictureall sixsitessurroundinganitrogenatomarepartiallyoccupied.Since there are six sites around each of four distinct nitrogen atoms in the unit cell, thereare 24 dierent linear combinations of the localised vibrational states in the ground statemanifold. Thebandwidth, J, ofthismanifoldissignicant, 4meV,whichsuggestsatunnellingrateof J/ h 6ps1. Localisedorbitalscannotbeformedbycombiningdierentstatesinthemanifoldduetothesignicantgapinenergybetweenthestates.The quantumtunnellingrate is several orders of magnitude larger thanthe classicalkinetic rate constant predicted for transitions between the sites. The potential calculatedusingDFTgivesabarrierof477meVbetweendierentsitesaroundthesamenitrogen.Estimatingthe zeropoint energies at the stable site andthe transitionstate usingaharmonic approximation, the expression for the rate constant from transition state theory(equation1.12)givesanestimatedclassicalrateconstantof 6.0 108ps1at300K.This is much slower than the predicted quantum tunnelling rate, suggesting that quantumeectsmustbetakenintoaccountwhenconsideringthelocationofthehydrogenatomsintheLi2NHlattice.(a) (b)Figure4.8: (a)Thegroundstatewavefunction. Thenitrogenatompositionsareshowningreen, thelithiumionsinblue. (b)Possiblehydrogensitespredictedbycalculatingthegroundstatehydrogenwavefunctionofthesystem,colouredasingure4.7.CHAPTER4. OTHERMATERIALS 664.2.2 DiusionForLi2NHtobeapracticalhydrogenstoragematerialitmustbepossibletoabsorbanddesorbhydrogenfromitatreasonablerates.If thediusionof hydrogenatomsinLi2NHisconsideredclassically, thebarriertohy-drogentransportbetweendierentnitrogenatomsisverylarge. Thepotential usedtocalculate the wave functions previous section gives a barrier of 477 meV between dierentsites on the same nitrogen atom, and a barrier of 2.9 eV between sites on dierent nitrogenatoms.The wave functions and energies calculated in the previous section can be used in equation1.39tocalculate diusioncoecients inthe same wayas for palladiumhydride andniobiumhydride. Afterperformingthiscalculation, thetemperaturedependenceofthediusioncoecient of hydrogenfrom300Kto800KcouldbettedtoanArrheniusexpression with an activation energy of just 260 meV. This is much lower than the classicalbarriertodiusion,suggestingthatquantumeectsareverylargeinhydrogendiusioninLi2NH.4.2.3 ConclusionsThelocationof thehydrogenatomsinLi2NHisnotwell describedbytheclassical de-scriptionofrandomlyoccupiedsitesaroundthenitrogenatoms. Whatismore,thesitespredictedbydiractionexperimentsdonotmatchthelocationof hydrogencalculatedusingquantummechanics.QuantumcalculationsndthatthegroundstatehydrogenwavefunctioninLi2NHhaspartially occupied sites centred in an octahedral arrangement around each nitrogen atom.Thequantumtunnellingratebetweenthesitesaroundagivennitrogenatomismuchfaster thanthat predictedclassically. The classical barrier todiusionbetweensitesarounddierentnitrogensitesisveryhigh(2.9eV)suggestingthatquantumeectsareverylargeinhydrogendiusioninLi2NH.Chapter5TransportinOneDimension5.1 AmodelpotentialAperiodicpotential inonedimensionwas chosenfor preliminarystudies of hydrogendiusion.V (x) = 12 (EtetEoct) cos_2xL_+12 (Etet + Eoct) cos_4xL_(5.1)Thepotential isperiodicinL, thelengthof thebox, andhastwowellswhichcanbechosentohavedierentdepths,EoctandEtet,roughlymodellingthepotentialalongthepathwaybetweenoctahedralandtetrahedralsitesinafacecentredcubic(FCC)lattice.EoctisingeneraltakentobelowerinenergythanEtet. Thetwowellscanbechosentohavethesamedepth,modellingthepotentialbetweentwoequivalentsitesinalattice.Thesingle-particlewavefunctionsandenergiescanbefoundbysolvingtheusual time-independentsingleparticleSchrodingerequation.Hn= nn(5.2)Asthepotential isperiodic, thisismostsensiblydonebysolvingtheBlochequations(2.33)inaplane-wavebasis.67CHAPTER5. TRANSPORTINONEDIMENSION 68TheFouriercomponentsofV (x),V (g)aregivenbythefollowingexpression.V (g) =___14(EtetEoct) g= 114(Etet + Eoct) g= 20 otherwise___(5.3)The Fourier components of the Bloch functions at a given point in k-space, uk,n(g) are theeigenfunctions of a Hermitian matrix with elements given by equation 2.34, the eigenvaluesgivingtheenergiesofthestates.Once the wave functions and energies have been obtained, the integral over the momentumoperatorcanbecalculated.km| p|kn_=

g uk,n(g) (g + k) uk,m(g) (5.4)Foreaseofnotationtheintegralkm| p|kn_willberepresentedbyPknm.The momentum integrals and the single-particle energies can then be used in equation 1.39(reproducedhereforconvenience)tocalculatethediusioncoecientoftheparticleinthis potential at a given temperature. It is necessary to perform an integral over sucientk-pointstogiveconvergenceinthediusioncoecient.D =_dkNQm2H

n,m=nen|Pknm|2_n h m h_(5.5)5.2 ComputationalDetailsTherequiredtheorywasprogrammedinFORTRAN, thediagonalisationof theBlochmatrixcarriedoutusingaQRalgorithmprovidedbytheLAPACKlibraries.Thediusioncoecientatdierenttemperatureswascalculatedforthethreehydrogenisotopes1H,2D, and3T in dierent potentials. For each calculation the measured diusioncoecients were converged with respect to the number of plane-waves usedas a basis set(limitedbyakineticenergycuto)andthenumberofk-pointsoverwhichthediusioncoecientwasintegrated. Thewidthof theGaussianwhichreplacedthe-functioninequation5.5waschosensuchthatthediusioncoecientonlyvariedslowlywiththewidth.CHAPTER5. TRANSPORTINONEDIMENSION 695.3 ResultsandDiscussion5.3.1 AnAsymmetricDoubleWellPotentialTheresultsinthissectionwereobtainedusingaplane-wavecutoof 0.1EH, (2.7eV),213k-points,andaGaussianofwidth0.1meV.Calculationswerecarriedoutusingapotentialwhichrepeatedevery6.8a0(3.6A)Theshallowwellminimum(Etet)waskeptat-0.002EHandthedeeperwellminimum(Eoct)variedbetween-0.0065EHand-0.01EH(gure5.1). Thedependenceof thediusioncoecientsat100K,400Kand1000Kwiththedepthofthewellcanbeseeningure5.2.0.010.0080.0060.0040.002 0 0.002 0.004 0.006 0.008 024681012V(x) [EH]x [a0]EtetEoctFigure5.1: Thepotentialcurveusedatthetwoextremesofthecalculation.Thediusionofhydrogenisotopesinthispotentialshowspeaksastheminimumofthedeeperwell isvaried(seegure5.2). Thesepeaksoccurat-0.0081EHfor1H, -0.0087EHfor2D,and-0.007EHand-0.0096EHfor3T.Theisotopeeectisnotconstantoverthe region. Each of the hydrogen isotopes diuses more quickly in dierent regions of thecurve.CHAPTER5. TRANSPORTINONEDIMENSION 70-32-30-28-26-24-22-20-18-16-0.01 -0.0095 -0.009 -0.0085 -0.008 -0.0075 -0.007 -0.0065log10(D) [cm2s-1]Eoct [EH]100KHDT-10-9.5-9-8.5-8-7.5-7-6.5-6-0.01 -0.0095 -0.009 -0.0085 -0.008 -0.0075 -0.007 -0.0065log10(D) [cm2s-1]Eoct [EH]400KHDT-5.3-5.2-5.1-5-4.9-4.8-4.7-4.6-4.5-4.4-4.3-0.01 -0.0095 -0.009 -0.0085 -0.008 -0.0075 -0.007 -0.0065log10(D) [cm2s-1]Eoct [EH]1000KHDTFigure5.2: Thediusioncoecientsof1H,2Dand3Tatdierenttemperatures.CHAPTER5. TRANSPORTINONEDIMENSION 71Iftheeigenstatesatthe1Hat-0.0081EH(gure5.3(a))andthoseat-0.0075EH(gure5.3(b))arecompareditcanbeseenthattheboundstatesinthetwowells(thosebelowthe classical inter-well barrier with little dispersion) become near degenerate when Eoctis-0.0081EH,whereasat-0.0075EHthestatesindierentwellsarewellseparated.TheneardegenerateboundstateswithEoct=-0.0081EHallowcontributionstothedif-fusion coecient at much lower energies than with Eoct= -0.0075 EHwhere contributionsonlyoccuratthebandedgesof thecontinuumstatesabovetheclassical barrier. Asaresult, diusioninthepotential withEoct=-0.0081EHisfasterthanthatwithEoct=-0.0075EHatalltemperatures(gure5.4). Theclassicalbarriersareshownintable5.1.Thepeaksindiusioningure5.2arecausedwhentheeigenstatesinthedeepwellandtheeigenstatesintheshallowwellbecomecloseinenergy.The -functioninequation5.5 ensures thatcontributionstothe diusioncoecientonlyarisefrommatrixelements betweendegenerateor near degeneratestates. Suchpairsof statesthencontributetothediusioncoecientaccordingtothemagnitudeof themomentummatrixelementbetweenthem,Pnm,andaBoltzmannfactor,en.The contributions to the diusion coecient arising from dierent regions of the eigenspec-trumareplottedfordierenttemperaturesingure5.5forthetwopotentialsdiscussedabove.Therelativeratesofdiusionofthedierentisotopesofhydrogeninthesamepotentialvaryforsimilarreasons. Thetemperaturedependenceofthediusioncoecentsfor2Dand3Tinapotential withEoct=-0.0081EHareshowningure5.6, andtheseparatecontributionstodiusionfrompartsoftheeigenspectrumareshowningure5.7.The contribution from states beneath the classical barrier with1H leads to much faster dif-fusion, especially at lower temperatures than with the heavier isotopes. The contributionstothediusionof2Dand3Tonlycomesfromstatesabovetheclassicalbarrier.Althoughtunnellingbetweenstatesbeneaththeclassical barrierhasbeenshowntobeimportant, the diusion of hydrogen isotopes in this potential remains an activated processsinceinthecasesinwhichboundstatesbecomecloseenoughinenergytocontribute,thestatesinvolvedareexcitedstates. Asaresult thetemperaturedependenceof thediusion coecient remains largely Arrhenius, although at low temperatures the eectiveactivation energy is seen to decrease as higher energy excited states are no longer thermallyaccessible.CHAPTER5. TRANSPORTINONEDIMENSION 72(a)-250-200-150-100-50 0 50 100 150 200 0123456V / 10*2+ [meV]x [a0] 00.5k [/L](b)-250-200-150-100-50 0 50 100 150 200 0123456V / 10*2+ [meV]x [a0] 00.5k [/L]Figure5.3: Theprotondensitiesandbandstructureof1Hoftherst11stateswith(a)Eoct=-0.0081EH,(b)Eoct=-0.0075EH. Thepotentialenergycurveisshowninblue.CHAPTER5. TRANSPORTINONEDIMENSION 73-25-20-15-10-5 0 123456789101002005001000log10(D) [cm2s-1]1000/T [K-1]-0.0081EH-0.0075EH-0.0081EH, TST-0.0075EH, TSTFigure5.4: Thediusioncoecientsof1HplottedagainstinversetemperaturewithEoct=-0.0081EHandEoct=-0.0075EH.CHAPTER5. TRANSPORTINONEDIMENSION 74(a) 0 1e-12 2e-12 3e-12 4e-12 5e-12 6e-12 7e-12 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]100 K 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]300 K 0 100 200 300 400 500 600 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]500 K 0 1000 2000 3000 4000 5000 6000 7000 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]700 K(b) 0 5e-17 1e-16 1.5e-16 2e-16 2.5e-16 3e-16 3.5e-16 4e-16 4.5e-16 050 100150200250300350400450500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]100 K 0 0.05 0.1 0.15 0.2 0.25 0.3 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]300 K 0 50 100 150 200 250 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]500 K 0 500 1000 1500 2000 2500 3000 3500 4000 4500 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]700 KFigure5.5: Thecontributions tothediusioncoecient fromdierent regions of theeigenspectrumbelow500meVabovethegroundstatefor1H(a)Eoct=-0.0081EH,(b)Eoct=-0.0075EH.CHAPTER5. TRANSPORTINONEDIMENSION 75-25-20-15-10-5 0 123456789101002005001000log10(D) [cm2s-1] 1000/T [K-1]1H2D3T1H, TST2D, TST3T, TSTFigure 5.6: The diusion coecients of dierent hydrogen isotopes plotted against inversetemperaturewithEoct=-0.0081EH.CHAPTER5. TRANSPORTINONEDIMENSION 76(a) 0 5e-17 1e-16 1.5e-16 2e-16 2.5e-16 3e-16 3.5e-16 050 100150200250300350400450500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]100 K 0 0.02 0.04 0.06 0.08 0.1 0.12 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]300 K 0 20 40 60 80 100 120 140 160 180 200 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]500 K 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]700 K(b) 0 2e-16 4e-16 6e-16 8e-16 1e-15 1.2e-15 050 100150200250300350400450500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]100 K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]300 K 0 100 200 300 400 500 600 700 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]500 K 0 2000 4000 6000 8000 10000 12000 050100 150 200 250 300 350 400 450 500PNM2x e-(EN - EGS/kT) x (EN-EM)EN - EGS [meV]700 KFigure5.7: Thecontributions tothediusioncoecient fromdierent regions of theeigenspectrumbelow500meVabovethegroundstatefor(a)2Dand(b)3TwithEoct=-0.0081EH.CHAPTER5. TRANSPORTINONEDIMENSION 775.3.2 TransitionStateTheoryandArrheniusExpressionsA comparison can be made between the diusion coecients calculated using equation 5.5andthosecalculatedusingthefollowingsimpletransitionstatetheory(TST)expressiondiscussedinsection1.1.2.DTST=l2kBTheETSEGSkBT(5.6)whereETSistheenergyatthetopofthebarrierbetweenthetwowellsandEGSistheenergyof thelowest eigenstate, andl is thedistancebetweentwoneighbouringwells.Table5.1andgures5.4and5.6showtheresultsof thesecalculations. Thevaluesintable5.1aretheresultsofattoanArrheniusexpressionoftheformDArr= D0eEakBT(5.7)Where D0is the diusion coecient in the nite temperature limit and Eais an eectiveactivationenergy. Thediusioncurves showningures 5.4and5.6followArrheniusbehaviour except at low temperatures (