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Theoretical investigation on the point defect formation
energies in beryllium and comparison with experiments
L. Ferry, F. Virot, M. Barrachin, Y. Ferro, C. Pardanaud, D. Matveev, M.
Wensing, T. Dittmar, M. Koppen, C. Linsmeier
To cite this version:
L. Ferry, F. Virot, M. Barrachin, Y. Ferro, C. Pardanaud, et al.. Theoretical investigation on the
point defect formation energies in beryllium and comparison with experiments. Nuclear Materials and
Energy, Elsevier, 2017, 12, pp.453 - 457. �10.1016/j.nme.2017.05.012�. �hal-01793176�
ContentslistsavailableatScienceDirect
Nuclear
Materials
and
Energy
journalhomepage:www.elsevier.com/locate/nme
Theoretical
investigation
on
the
point
defect
formation
energies
in
beryllium
and
comparison
with
experiments
L.
Ferry
a,
F.
Virot
a ,∗,
M.
Barrachin
a,
Y.
Ferro
b,
C.
Pardanaud
b,
D.
Matveev
c,
M.
Wensing
c,
T.
Dittmar
c,
M.
Koppen
c,
C.
Linsmeier
ca Institut de Radioprotection et de Sûreté Nucléaire PSN-RES, SAG, LETR, Saint Paul les Durance cedex 13115, France b Aix-Marseille Université, CNRS, PIIM UMR 7345, 13397 Marseille, France
c Forschungszentrum Jülich GmbH, Institut für Energie- und Klimaforschung - Plasmaphysik, 52425 Jülich, Germany
a
r
t
i
c
l
e
i
n
f
o
Article history: Received 14 July 2016 Revised 3 May 2017 Accepted 30 May 2017 Available online 31 May 2017
a
b
s
t
r
a
c
t
Berylliumwillbeusedasaplasma-facingmaterialforITERandwillretainradioactivetritiumfuelunder normaloperatingconditions;thisposesasafetyissue.Vacanciesplayonethekeyrolesinthetrapping oftritium.Thispaperpresentsafirst-principlesinvestigationdedicatedtopointdefectinhcpberyllium. Aftershowingthe bulkproperties calculatedherein agreewellwith experimentaldata, wecalculated theformationenergyofasingle-vacancyandhenceforthproposeanestimateof0.72eV.Thisvalueis discussedwithregardtoprevioustheoreticalandexperimentalstudies.
© 2017TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
1. Introduction
ITER[1] aimstodemonstratethefeasibilityofobtainingenergy from nuclear fusion reactions in a magneticallyconfined plasma of deuterium (D) and tritium (T). The low burning efficiency of the D-Treaction impliesthatlarge amountsof hydrogenisotopes will have to be present in the plasma and consequently can be retained inthefirst-wall plasma-facingcomponents(PFCs) ofthe tokamak.Tominimizetheradiological consequencesofapossible confinementloss,theconcentrationofTinthevacuumvessel(VV) mustbelimitedduringnormaloperations.Theadministrativelimit on T-inventory inthe VV iscurrently fixed at≈700 g. Assuming that about 1g of Twill be retained aftereach plasma shot, this limitwillbe reachedafter≈700shots,correspondingtoonlytwo months of normaloperation [2] .As a result, an active control of the T in-vessel inventory is crucial for limiting the amount of T trapped in the tungsten divertor and in the beryllium tiles that cover the largestpart of the wall (about 700 m2). According to previous laboratory experiments [3,4] , baking the beryllium tiles could be an efficient waytopartly removethe implanted hydro-genisotopes.Asaconsequence,understandingthethermally acti-vated processesthat leadto tritium desorptionis ofprior impor-tance.Inthisway,Piechoczeketal.[5] proposedamodelbasedon
∗ Corresponding author.
E-mail address: francois.virot@irsn.fr (F. Virot).
reaction-diffusionequationstosimulateThermalDesorption Spec-troscopy(TDS)forlaboratoryexperimentsperformedonBe(0001) andBe(11¯20)singlecrystalspre-implantedwithdeuteriumatlow fluences of ≈3 × 1019 m−2. In their model, deuterium diffusion usingthesecondFick’slaw,formationofmono-vacanciesand self-interstitial Beatoms areintroduced assourcetermsbased on bi-narycollisionssimulationswiththeSDTrimSPcode[6] and forma-tionanddecompositionofhydrogen-vacancycomplexes(trapping and de-trapping of deuterium) are assumed to be thermally ac-tivated processes according to Arrhenius-type expressions. While mostlybasedonDFTcalculationsreportedin[7] ,Piechoczeketal. hadto correctsome parameters (self-interstitial diffusion activa-tionenergy)andintroduceanewreaction,socalledself-trapping, to properlyfit the TDSdata. This indicates that the atomic-scale mechanismsgoverning the hydrogenbehaviour in berylliummay notbeunderstoodtoday,resultinginanincompleterate-equation model.
Consequently, we have decided to focus our work on an in-depthinvestigationofhydrogenandpointdefectbehaviour.Inthis paper,we address the propertiesof empty vacancy, whereas the second topicwill be investigated inthe very next future. Vacan-cies have been shown to be the dominant defects versus inter-stitials [8] . The equilibriumconcentration of vacanciesV, cV(T) at temperature T, is a key property to simulate the hydrogen be-haviourinBeandisrelatedtothevacancyformationenergy,
fEV, through cV
(
T)
∝e−fEV/kBT, which requires an accurate value ofhttp://dx.doi.org/10.1016/j.nme.2017.05.012
454 L. Ferry et al. / Nuclear Materials and Energy 12 (2017) 453–457
fEV. However, available experimental and calculated values for
fEV arewithinarelativelybroadrange0.81-1.13eVofenergy[7– 11] .Theaimofthispaperistoclarifytheseapparentdiscrepancies andtoreducetheuncertaintyontheDFTdata.Section 2 isdevoted tothedescriptionofthecomputationaldetailsofoursimulations. InSection 3 ,acomparisonofthecalculatedpropertiesofberyllium withtheavailableexperimentaldataisprovided.Thecalculationof theformationenergyofavacancyispresentedinSection3.4and discussedin Section 4 in regard tothe previous DFT and experi-mentalstudies.
2. Computationaldetails 2.1.Nondefectiveberylliumbulk
The calculations in this work are based on DFT as imple-mentedintheQuantumEspressoPackage[12] .ThePerdew–Burke– Ernzerhoffunctional[13] builtinthegeneralizedgradient approx-imation (GGA) is used to compute the exchange and correlation energies.The same ultra-softpseudopotentials [14] (USPPs) as in [15] were usedto modeltheioniccores.Onlythe2selectrons of berylliumare explicitly considered, while the 1s2 electrons were included in the pseudo-potentials. The ionization energy of the firstcoreelectrons (≈112 eV)ismuchhigherthantheone ofthe two2s(9.3eV forthefirstionization and18.2eVforthesecond one[16] ),whichshouldinsureahightransferabilityofthe pseudo-potential.Resultsrelatedthegroundstatepropertiesofbulk beryl-liumpresentedinSection 3 validatetheusedUSPPs.
Alltheatomsareincludedintheoptimizationprocedure;they areallowedtorelaxuntiltheresidualforcefellbelow0.003 eV/˚A andthetotalenergybelow0.001eV.
The sampling of the Brillouin–Zone (BZ) was done using the
-centeredMonkhorst–Pack k-point grid[17] .[11] showedthata highnumber ofk-points is needed in orderthat the bulk prop-erties converge well. Consequently, we used the same parame-ter asGanchenkovaetal.[11] fora unit cell containing2atoms, withEcut=816 eV andk-point numberequal to 303. It was nev-erthelesspossibletoreduce the samplingto243 k-points forthe unit cellwithno loss ofaccuracy, theground state energybeing the same within 1 meV/atom. The Marzari–Vanderbilt smearing scheme[18] wasused with a broadeningof
σ
= 0.001 Ry .The calculatedlatticeparametersaswellastheequilibriumvolume0 werekeptfrozeninthenextstepsofouroptimizationprocess.
Keeping thepreviousk-meshsampling,we testedtheeffectof differentEcutvaluesontheconvergenceofthegroundstateenergy. Valuesranging from10to 60Rywere tested, leadingtoan opti-mumvalue of30Ry(408eV), forwhichthe groundstate energy isclosetothevalueobtainedforEcut=60Ry(816eV), within0.15 meV/atom.
2.2.Defectiveberylliumbulk
VacancypropertiesareknowntobehighlysensitivetotheDFT parameterused.Consequently,theconsistencyoftheseparameters was checked with regard to the formation energy of a vacancy,
fE1V,definedas:
fE1V=E
(
N− 1,1V,N0
)
−N− 1
N E
(
N,0V,N0
)
(1)whereE
(
N− 1,1V,N0
)
isthetotalenergyofthedefective super-cellwithN− 1atoms andone vacancyatvolume N0,and E(N, 0V,
0) isthe totalenergy ofthe nondefective supercellwithN atomsand
0.
Forthe calculationof
fE1V,itisrequiredthat thesize ofthe supercell is large enough to minimize the interaction of the va-cancy withits own image inthe neighbouringcells. Keeping the
Table 1
Calculated and experimental beryllium properties. Lenghts in ˚A, cohesive energy in eV and bulk modulus in GPa.
Source a c c/a E coh B
Present work 2.258 3.549 1.572 3.70 121 Experiment 2.286 a 3.584 a 1.568 3.32 b 121 c
Wachowicz d 2.230 3.510 1.573 4.20 128
Wachowicz e 2.260 3.550 1.570 3.74 115 a Experimental values from: [19] .
b[20–22] .
c extrapolated at 0 K [23] . d Calculated ones from: LDA [24] . e GGA [24] .
samegrid ofk-points andasmearing of0.001 Ryasforthe non defectivecell,itwasnotpossibletoreachconvergences on
fE1V considering supercells size up to 5 × 5 × 5. The reason is a tooloosesamplingofk-points.Wefoundawell-balanced compro-misebetweensamplingandsmearingusinga203k-pointsin con-junctionwithasmearingof0.05Rytoreachtheconvergence.This pointisdiscussedSection 3.2 .
3. Calculatedberylliumproperties
3.1. Structuralproperties,bulkmodulus,andcohesiveenergy From ambient temperature up to 1530 K at 0.1 MPa, beryl-lium has a hexagonal close-packed (hcp) structure that belongs to the P63/mmc space group. From Tα−β=1530 K to the melt-ing point, Tm=1560 K, it has a body-centred cubic (bcc) struc-ture that belongs to the Im3m space group [25] . Only the low-temperaturephase willbe considered inthe presentwork. Amo-nenko[26] performedacriticalassessmentoftheavailable lattice parametermeasurementsforthehcpphase,showingtheinfluence of impurities and uncontrolled atmosphere. He finally measured a=2.2804 ˚A andc=3.5775 ˚A under vacuum on a distilled beryl-liumsample.Lateron,thelatticeparametersofhighpurity beryl-lium were obtainedat 298.15K by means of X-raydiffraction, a = 2.2858 ˚A and c= 3.5843 ˚A [19] , in agreement with the previ-ousdetermination.Thelinearcoefficientsofthermalexpansionof berylliumpublishedbyGordon’s[27] overalargetemperature in-tervalallows extrapolationfromthepreviousdatatoestimatethe latticeparameterat0K,a≈2.281 ˚Aandc≈3.579 ˚A.Thecrystal pa-rameterwe calculated are a =2.258 ˚A andc = 3.549 ˚A ingood agreement withtheextrapolatedvalues at0K. Thecalculated c/a ratioof1.57isclosetotheexperimentaldeterminationandlower than the ideal value of 1.633 for hcp-type structure. As reported in Table 1 ,our values practically coincide withthe previous cal-culationsperformedwithintheGGAapproximation[24,28,29] .The LDAapproximation [24,28] provides rathersatisfactory lattice pa-rametersforberyllium,however,inaccordancewithaknown gen-eraltendency,theyaresystematicallylowerthantheexperimental values.
The bulkmodulus, B,has beencomputedby applyingan uni-formpressure,p,ontheunitcell.Theevolutionofthetotalenergy vs.thevolumeleadstothebulkmodulusaccordingto:
B=−
0
dp d
=
0
d2E
d
2, (2)
where
0isthevolumeattheequilibriumgeometry.
Ourvalue(121 GPa)agrees withtheextrapolateddataat0K from recent measurements carried out by resonance ultrasound spectroscopy [23] . Our results are also in agreement with other GGA-basedcalculations([30] andseethereinreferences).
Thecalculatedcohesive energyperatom isdefinedasthe dif-ference in energy between an isolated beryllium atom, Eat, and
0 25 50 75 100 125 150 175 200 225 250 number of atoms 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Δf E 1V (eV) 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 distance between two vacancies (Å)
0.8 0.9 1 1.1 1.2 1.3 Δf E 1V (eV) α1+β1r-3
Fig. 1. Monovancy formation energy as a function of the supercell size for smearing value of σ= 0.05 Ry.
thatofthecrystalperunitatom: Ecoh
N =
NEat− E
(
N,0V,N0
)
N (3)
Theobtainedvalue,3.70eV,issignificantlyhigherthanthe exper-imentalone,3.32± 0.07eVrecommendedin[20–22] .The correc-tionduetothezero-pointvibrationalenergyhastobe takeninto account consideringtherelativelyhighDebye’stemperatureofBe (
D=1453K [23] ). Thiscorrectionis equalto0.092 eV/atomin agreement withprevious estimates(0.093 eV/atomin [31] ). Ac-cordingto theDebye’smodel [32] ,it canbe estimatedas 9
8kB
D withacorrectionof0.14 eV/atom,consistentwiththecalculations. Consequently,thecorrectedEcohisequalto3.61 eV/atom. 3.2. Formationenergyofamonovacancy
Fig. 1 providestheformationenergyofamonovacancyversus thesupercellsizegiveninthenumberofatomscontainedwithin thecell.Convergenceisreachedfor128atoms(4 × 4× 4)with
fE1V =0.866eV.Theformationenergyofavacancyisalso plot-tedversusthedistancebetweentwovacanciesdinFig. 1 .A∝1/d3 behaviour seems to be identified. This is the resultof geometric and/or electronic effects, meaning the consequence of an elastic deformation ofthe unit-cellorthegeneration ofelectronic oscil-lationsaroundthevacancy.
Thepreviousvalueisthevacancyformationenergyatconstant volume. The effectof the volumerelaxation can be evaluated by minimizing theenergyofthesupercellwitha vacancy,leadingto avolumecontraction,
andadecreaseinenergy,
Erelax [33] :
Erelax≈ p
+
0 2 B
0 2
(
N− 1)
+O(3)
(4)Forp=0,thevolumecontractionisequalto
=0.54 ˚A3.It al-lowsthecalculationofthevolumeofthevacancyformation,equal to
f
1V=
0−
=7.29 ˚A3.Forp=0, thedecrease inenergy,
Erelax=4meV,isverylow.Itcanbeexplainedbythelowvalues of Band
0.The formationenthalpy ofthe vacancy(at constant pressure p=0) is then given by
fH1V =
fE1V−
Erelax. Since
fH1V doesnot reallydifferfrom
fE1V andthevariation of vol-ume,
,isverylow,enthalpyvaluesexperimentallyobtainedat atmosphericpressurearecomparabletoourcalculatedenergiesat constantsupercellvolume.
It must be underlinedthat the smearing value of
σ
=0.05Ry (0.68eV)mayappearrelativelyhighinourcalculations,eventhis value is usually not given in previous papers. When utilizing a0 0.01 0.02 0.03 0.04 0.05
σ (Ry)
0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86Δ
fE
1V(eV)
Fig. 2. Monovacancy formation energy trend in function of the smearing value σ.
Fig. 3. Configurations of considered divacancy, V aa , V ac , and V cc .
non-zerosmearingvalue,thecalculationminimizesafunctionalof thefree energyFwithrespectto thetotalelectronic density,not thetotalenergyE.Weareinfactinterestedin
fE1V
(
σ
=0)
.Additional calculations with a reduced smearing parameter werecarriedout until
σ
=0.01Ry(0.14eV);theywereperformed onthe (4 × 4× 4) supercellwithappropriate meshingin the BZto ensure convergence1.The convergenceis nearly reachedat0.01eV (Fig. 2 ) at
fE1V=0.72 eV whichcan be assumedasan upperlimit.
3.3.Divacancyformationenergy
Tocomparewithexperimentaldata,we willneedthe calcula-tionoftheformationenergyofdivacanydefinedasfollows:
fE2V =E
(
N− 2,2V,N0
)
−N− 2
N E
(
N,0V,N0
)
(5)Wehavecomputed
fE2V forthreedifferentconfigurations(aa in-plane 2V, acandcc out-of-plane 2V, Fig. 3 ) ina 4 × 4× 4 su-percellanda(73)gridofk-pointswitha0.01Rysmearing broad-ening.We found rathersimilar values, i.e.1.83, 1.94and1.75 eV respectively.
If
disH2V stands for the dissociative energyof the divacancy definedas
disH2V=2
fH1V−
fH2V correspondingtothe reac-tion2V↔1V+1V,
disH2V canbededucedfromtheprevious val-ues;itis−0.39,−0.50and−0.31eV,foraa,acandcc,inroughly agreementwithprevious calculations[11] .Thisindicates that the formationof2Vislikelynot afavourableprocessatlow tempera-ture. This seems qualitatively in agreement with “out-of equilib-rium” observations of samples irradiated by neutrons (>1 MeV) at77K [34] , witha limitedhelium concentration (<1 appm),in which vacancy clusters are not observed. A similar behaviour is describedforsamples irradiated by electrons (2–3 MeV) at20 K [34] .
1 In these calculations, as for the calculation with σ= 0.05 Ry, we used the same
456 L. Ferry et al. / Nuclear Materials and Energy 12 (2017) 453–457
Table 2
Vacancy formation energy in different studies.
Source XC-functional a Energy cutoff (eV) k-point mesh Atom number b
f E 1V (eV)
Present work PBE 408 7 × 7 × 7 128 0.72
Experiments c — — — — 0.75 ± 0.20
Krimmel d LDA — — 36 1.13
Allouche et al. e PBE 435 6 × 6 × 6 64 0.96
Zhang et al. f PW91 500 5 × 5 × 5 96 0.95
Ganchenkova et al. g GGA 300 14 × 14 × 14 200 0.81
Middleburgh et al. h PBE 450 6 × 6 × 6 200 1.09
a Exchange-correlation energy functional.
b atom number is related to the non-defective supercell. c see discussion. d[9] . e[7] . f[10] . g[11] . h[8] . 4. Discussion
4.1.Comparisonwithothercalculations
Table 2 displaystheavailabledataintheliteratureforthe for-mation energy of a vacancy. Unfortunately, neither the smearing valuesnorthebroadeningschemeisgivenintheseworks. Krim-mel[9] etal.obtainedthelargestvalue ascanbe expectedfrom LDA. The Allouche’s value [7] is significantly higher than our; it wascalculatedwitha supercell of64 atoms, which isbelow the convergencecriterionof128atomsfortheformationenergyofthe vacancy.Thedifferencewiththevalueabove1eVofMiddleburgh etal.[8] usingsimilarDFTparameters thanGanchenkovaremains unexplainedatthisstage.TheZhang’svalue[10] islargerthanour estimatetoo; the exchangeandcorrelation functional PW91they usedis very similar to PBEandshould not give markedly differ-ent results [35] ; using the same parameters as they did with a 0.05Rysmearing andthePBEfunctional,we obtained
fE1V be-tween0.87forasupercellof54atomsand0.94eVfor96atoms (Fig. 1 ),against0.95eVforZhangwiththePW91functional. 4.2.Comparisonwithexperimentaldata
The comparison between the formation energy of a vacancy
fE1V= 0.72 eV we calculatedand theexperimental value isnot straightforward.Itcanbeindirectlydeterminedfromtheactivation energyofself-diffusion(HBe
sel f),assuming thisprocesshappens ac-cordingtoamechanismdominatedbymonovacancies.Inthatcase, the following relation can be established: HBe
sel f =
fH1V+H1dV, whereH1V
d istheactivationenergyforthediffusionofthevacancy. Consequently,
fE1V canbe knownafter HBe
sel f andH1dV are deter-mined,whichis avery difficulttasksince themonovacancy con-tributionhastobeseparatedfromcontributionsofotherdefects.
Theveryscarcedata[34,36] onvacancydiffusioninhcp beryl-liumhavetobe cautiouslyconsidered inordertodetermineH1V
d . Inbothstudies,berylliumgrade(SRPechiney,≈0.1%),suppliedby the same manufacturer, was relatively low and can significantly impactH1V
d .Thevalue, usuallycited,around 0.8eV,wasdeduced fromtheactivation energy ofresistivity recovery (in the temper-ature range 220–300 K) in samples irradiated at low tempera-turesbyneutrons(>1MeV)[34,37] .Thisactivationenergyis clas-sically linked with the activation energy of the disappearance of thesursaturationofdefects,whichcanbeofseveraltypes.Nicoud [34,38] assumed an activated-monovacancy mechanism since its valuewasabouthalfofHBe
sel f determinedotherwise[39] (see there-after).Accordingtotheauthorhimself,thereis noclearevidence ofaonlyonetype-defectmechanism.Alowervalue,about0.65eV,
wasproposed byChabre[36] ,basedontheanalysisofspin relax-ationbyNMR(between300and1200K)andattributedtovacancy migrationattheLarmorfrequency.Thisvalueispreferentially re-tained,H1V
d =0.65± 0.2eVtobeconsistentwithourassessment ofself-diffusiondatathereafter.Tobeinspiteofthese uncertain-ties, the experimental values matchrelatively well with the DFT values(H1V
d,=0.72eVandH1d,V⊥=0.89eV)[8] .
Theavailableexperimentalmeasurementsofself-diffusionHBe sel f in beryllium are more numerous. Some of them, obtained by means of the 7Be radiotracer technique, [39] , show that this process has a slighly anisotropic character originating from the hcpstructure. The Arrhenius activation energiesare derived from [40] on a monocrystal over a significant interval of temperature forabetterprecision(between840and1320K).Theyare1.63and 1.71eVforperpendicularandparalleldiffusionstothebasalplane, respectively.Neverthelesslateron,lower valueswereobtainedby Chabreetal.onthebasisofNMRexperimentsperformedatlower temperatures(300–1200K):1.4± 0.1eV[41] and1.35± 0.07eV [36] .Anexplanationforthediscrepancybetweenboth experimen-taldatacouldbelinkedtothehigherpurityofberylliumsamples in[36] in comparisonwith[40] (resp.0.01%andfew 0.01%).The assessedvalueforHBe
sel f =1.4± 0.1eV.
An experimental value for
fH1V can be finally proposed on thebasis ofthisassessment,0.75± 0.2 eV.Ourcalculated value, 0.72eV,isinquantitativeagreementwiththisexperimentaldata. 5. Conclusion
Thispaperpresents aDFT ab-initio investigationofhcp beryl-lium,mainlyfocussedontheaccuratedeterminationofthe forma-tionenergyofmonovacancy.Ourbest-estimatedvalueis0.72 eV. This value well fits the assessed experimental data and is in quantitative agreement with the previous theoretical determina-tion from Ganchenkova et al. [11] . Furthermore, we have shown that theformation ofdivacanciesisnot energeticallyfavorablein respect totwo separatedmonovacancies, consistently with previ-ousstudies[11] .
Inaverynearfuture,stablepositionsofhydrogeninnon defec-tiveanddefectiveberylliumwillbere-investigatedaswellasthe diffusionpathsbetweenthesedifferentpositions.Fromthesenew data,theinterpretationoftheTDSlaboratoryexperiments[5] will beconsidered.
Acknowledgements
ThisworkissupportedbyRegionPACA.Ithasalsobeencarried out withintheframework ofthe EUROfusionConsortiumandhas
receivedfundingfromtheEuropean Union Horizon 2020 research andinnovationprogram.Theviewsandopinionsexpressedherein donot necessarilyreflectthoseoftheEuropean Commission.The authors acknowledge thecomputer time grant provided by IFER-CSC,Rokkasho,Japan,intheframeoftheBroaderApproachinthe FieldofFusionEnergyResearch.
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