Research Article First-Principles Study of the …downloads.hindawi.com/journals/acmp/2014/929750.pdfResearch Article First-Principles Study of the Structural Stability and Electronic

Research ArticleFirst-Principles Study of the Structural Stability and Electronicand Elastic Properties of Helium in 120572-Zirconium

Jian Zheng12 Huijun Zhang23 Xiaosong Zhou2 Jianhua Liang2

Liusi Sheng1 and Shuming Peng2

1 School of Nuclear Science and Technology University of Science and Technology of China Hefei 230026 China2 Institute of Nuclear Physics and Chemistry China Academy of Engineering Physics Mianyang 621900 China3 School of Mathematics and Physics University of Science and Technology Beijing Beijing 100083 China

Correspondence should be addressed to Shuming Peng pengshuminghotmailcom

Received 22 January 2014 Revised 11 April 2014 Accepted 20 April 2014 Published 5 May 2014

Academic Editor Haiyan Xiao

Copyright copy 2014 Jian Zheng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

First-principles calculations within density functional theory have been performed to investigate the behaviors of helium in 120572-zirconiumThemost favorable interstitial site for He in 120572-Zr is not an ordinary tetrahedral or octahedral site but a basal octahedralsite with a formation energy as low as 240 eV The formation energy reduces to 125 eV in the presence of preexisting vacanciesThe analysis on the density of states and the charge density has been carried out In addition the influences of He and small He-Vcomplexes on the elastic properties have been studied The He-V complexes have been found to greatly affect the elastic propertiescompared with He alone

1 Introduction

During service lifetime of materials used in nuclear energysystems the impact of helium has been considered as acritical issue Helium is almost insoluble in metals due toits inert reactivity with other elements Once introduced itcan hardly be outgassed and will have profound deleteriouseffects on the microstructures and mechanical propertiesHelium has been a topic of sustained interest during the pastseveral decades for both nuclear material engineering andfundamental research In particular numerous experimentaland theoretical studies have been dedicated to addressing theproperties of He in metals such as Ti [1 2] Fe [3 4] andW [5 6] It is known to affect the nucleation and growth ofvoids in metals causing noticeable dimensional changes andsignificant hardening [7ndash10]

Zirconiumand its alloys arewidely used in tritium storage[11ndash13] and nuclear fission energy systems [14ndash18] owingto their high tritium storage density low thermal neutronabsorption cross-section good aqueous corrosion resistanceand favorable mechanical properties And they are believedto have some potential uses in future fusion applications

[19] Under these application conditions helium can beintroduced by three principal ways tritium 120573 decay (n 120572)reactions and direct He-ion implantation [20] The tritiumstored in zirconium can decay to helium by 120573 decay in thefollowing nuclear equation 3

1T rarr 3

2He+

+ eminus + V119890 In fission

reactors because of theNi doped in zirconium alloy and the Bdoped in the primary-loop water the neutron flux may causegeneration of helium either by (a) the two-step reaction with58Ni(n 120574) and 59Ni(n 120574) or by (b) the 10B + n rarr 7Li + 120572reaction [21] In fusion reactors He++ andHe+ ions producedin the plasma can directly implant into materials

However as far as we know only a few studies havebeen centered on the influences of He on Zr mainly someexperimental works on the surface phenomenology [22 23]and He bubble formation [12 24] It is found that heliumcan easily form bubbles and these bubbles preferentiallydistribute in grain boundaries and dislocations For directHe-ion implantation blisters even exfoliations are evidentat a critical dose [22 23]

The pioneering theoretical study by Koroteev et al [25]gives us a better understanding on the structure stabilityand electronic properties of the Zr-He system However

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2014 Article ID 929750 8 pageshttpdxdoiorg1011552014929750

2 Advances in Condensed Matter Physics

Table 1 Experimental and calculated lattice constants (A) for 120572-Zr

119886 119888119886

This work 3231 1600PW91PAW [30] 3213 1605PW91UPP [31] 323 1604PW91UPP [32] 3223 1606Exp [33] 323 1593

their calculations were performed in a rather small systemcontaining only two Zr atoms and one He atom whichcorresponds to an extremely high helium concentration Aswe know the stability of self-interstitial atom in 120572-Zr isstrongly affected by the system size [26ndash29] So it is necessaryto further their studies concerning a larger system size

In the present paper through first-principles calculationswe reported heliumrsquos behavior in 120572-Zr by focusing on therelative stabilities of single He defects the electronic proper-ties of He atom and its nearest-neighbor Zr (NN-Zr) atomWe also studied the changes of elastic properties due to theintroduction of He atoms and vacancies

2 Computational Method

Thepresent calculationswere carried out usingCASTEP [39]Ultrasoft pseudo potential (USPP) [40] with a consistentcutoff energy of 450 eV based on density functional theory(DFT) was used The Perdew Burke and Ernzerhof (PBE)form [41] of generalized gradient approximation (GGA) wasapplied as the exchange-correlation potential since it has beensuggested by Domain et al [31] that GGA performs betterthan the local density approximation (LDA) to describe theexchange-correlation functional of Zr

During the relaxations the Brillouin zone (BZ) integra-tion was achieved using a cold smearing scheme (Methfessel-Paxton) with a smearing of sigma = 01 eV BZ sampling wasperformed using theMonkhorst-Pack scheme with a k-pointspacing as close as possible to 0030 Aminus1 (ie 4 times 4 times 3 k-pointmeshes for a 36-atom supercell and 4 times 4 times 2 k-point meshesfor a 54-atom supercell) The supercells were optimizedusing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) [42]algorithm allowing both atomic positions and the latticeparameters to relax The energy convergence criterion forself-consistent calculations was set as 1 times 10minus6 eV The criteriafor ionic energy minimization were set as energy derivativelt1 times 10minus5 eVatom forces on individual atoms lt001 eV Aminus1displacement of atoms derivative lt0001 A and stress com-ponent on cell lt005GPa The criteria for the calculationof elastic constants were set as energy derivative lt2 times10minus6 eVatom forces on individual atoms lt0006 eV Aminus1 anddisplacement of atoms derivativelt2times 10minus4 AThe parametersselected above ensure the convergence of formation energieswithin 0001 eVatom Spin polarization is not considered inall the calculations for the sake of time our future work willinvolve it

BT

OT

S

BO

Figure 1 Five crystallographically unique interstitial sites in 120572-zirconium (hcp structure) where the pink spheres are the hcp latticesites and the red ones are labeled as octahedral (O) basal octahedral(BO) tetrahedral (T) basal tetrahedral (BT) and substitutional (S)sites respectively

The defect formation energy is defined as

119864119891

defect = 119864119898ZrHe minus 119898119864Zr minus 119864He (1)

where 119864119898ZrHe is the total energy of an optimized supercell

containing 119898 Zr atoms and one He atom 119864Zr is the energyper Zr atom in optimized crystal and 119864He is the energy of anisolated He atom

3 Computational Results

31 Structural Stability We first investigate the lattice con-stants of an 120572-Zr crystal and the results are summarized inTable 1 compared with experimental and other calculatedresults It can be seen clearly that our calculated results arein good agreement with the corresponding experimental andother calculated values

When He atoms are introduced into a metal they mayoccupy either substitutional or interstitial lattice sites It isnecessary to figure out the locations of He atoms sincethey will influence the solubility and migration It is alsoimportant for the understanding of such effects as damagetrapping bubble nucleation embrittlement and blistering[6] It has been found that in different metals He atomspreferentially occupy different sites substitutional sites forFe Cr Mo and W [6 43] tetrahedral sites for Er [34]and FC (the center of equilateral trigonal face shared bytwo adjacent octahedrons) sites for Ti [2] To identify thelowest energy configurations of the He interstitial defects inZr and the possible influences of He concentration supercellswith 35 to 54 Zr atoms and one He atom were studied Fivecrystallographically unique interstitial sites for He atom weretaken into consideration (see Figure 1) They were denotedconventionally as octahedral (O) basal octahedral (BO)tetrahedral (T) basal tetrahedral (BT) and substitutional (S)sites

The calculated formation energies of a single vacancyand He atom in above sites are listed in Table 2 together

Advances in Condensed Matter Physics 3

Table 2 Vacancy formation energies (in eV) for a single He atompositioned in different sites All the configurations are found tobe stable except O which is unstable and transfers to BO 119864119891vrepresents the formation energy of a Zr vacancy and 1198641015840119891s representsthe formation energy of a substitutional He atom in a preexisting Zrvacancy

119864119891

v 119864119891

s 1198641015840119891

s 119864119891

T 119864119891

O 119864119891

BT 119864119891

BO

36-atom cell 197 330 133 267 301 24254-atom cell 195 320 125 264 299 240120572-Zr [25] mdash mdash mdash 308 319 mdash mdashEr [34] mdash 259 mdash 173 198 mdash mdashSc [35] mdash 298 mdash 174 206 176 197Ti [2] mdash 365 170 282 304 mdash 269

with the results reported in the literature for Zr and someother hcp-structure metals It is noticeable that in bothsupercells the configuration O is not stable and decays toconfiguration BO during relaxation Different from thoseof Fe Cr Mo and W the favorable sites in energy for Heatom in 120572-Zr are interstitial sites Among the three stableinterstitial sites the most favorable one is the BO site in bothsupercells with formation energy near 240 eV However itshould be mentioned that 119864119891s here actually represents theenergy required to form a He-V complex The formationenergy of 133 eV for a He atom in a preexisting vacancy(denoted as1198641015840119891s ) can be obtained by subtracting the formationenergy of the vacancy (denoted as 119864119891v ) It is much smallerthan 119864119891BO which means that He atoms can be trapped easilyby preexisting vacancies This is consistent with the case ofTi [2] For materials used in nuclear systems energetic par-ticles like neutrons ions or electrons can induce significantmicrostructure alteration especially the generation of largeconcentrations of vacancies which can strongly affect the Hedistribution In general the stabilities of these configurationsin descending order are preexisting vacancy BO T BT S andO Koroteev et al [25] also reported that the configuration Tis more stable than configuration O However they did nottake other configurations into consideration Andmaybe dueto the differences in calculatingmethods or system sizes theircalculated configuration O does not decay to BO This is leftto be clarified by further studies

32 Electronic Properties To further understand the relativestability of these structures with He at various sites we alsostudied the DOS of the He atoms and the NN-Zr atoms ofdifferent configurations The relative stability of the He atBO site to the other sites can be easily understood accordingto the position of the Fermi level relative to the peaks inthe DOS which determines the occupation of the statesand the nature of bonding [44] As shown in Figure 2(a)the interaction of basal octahedral He interstitial with itsNN-Zr atoms leads to the lowest DOS at the Fermi levelwhich indicates that there is a stronger bonding between Zratom and He atom at the BO site compared with other sitesThis is in accordance with the formation energy differencesas mentioned above Different from those for Fe [43] it is

also noticeable that the interaction also leads to differentchanges of the DOS at the Fermi level of the NN-Zr atoms asshown in Figure 2(b) Comparing with that of the T and BTinterstitial sites the BO sites occupy a lowerDOS at the Fermilevel The DOS at the Fermi level of the S site is somewhatparticular because of the lack of one Zr atom From Figure 3we observe similar DOS peaks of the He-119904 and Zr-119889 statesat around minus1 eV and Fermi level which suggests the stronghybridization between these states And this hybridizationbetween He and metals has also been confirmed by otherresearchers [34 35 43] It is believed that the DOS peaks ofthe He-119904 andM- (metals- like Zr Sc Er Fe etc) 119889 states willchange due to the strong hybridization This change will leadto an overall similarity in the shapes of those DOS curves

We also calculated the total charge density (120588(119903)) ofpure 120572-Zr crystal and one with preexisting vacancies Theisosurfaces of 120588(119903) = 140 enm3 and the contour plots ofthe charge density of the basal planes crossing the BO andS sites are illustrated in Figures 4(a) and 4(b) respectivelyThe isosurfaces are tubes similar to those of Ti [2] Since thecharge density inside the isosurfaces is lower than that of theoutside He atoms prefer to stay inside especially in the BOor S sites over which the tubes cross with the basal planes Itis obvious that the space inside the tube near the vacancy ismuch larger than that of the BO interstitial site And this mayexplain why He atoms prefer to stay in preexisting vacancies

33 Elastic Properties Then we studied the influences ofhelium on the elastic constants of Zr For hexagonal structurematerials there are six elastic stiffness constants five of themare independent (119862

11 11986212 11986213 11986233 and 119862

44) with 119862

66=

(11986211minus11986212)2The elastic constants were determined from the

second derivatives of the total energy with respect to suitabledeformations The bulk modulus 119861 and shear modulus 119866 ofthe polycrystal were evaluated from the elastic constants ofa single crystal by employing the Voigt-Reuss-Hill method[45ndash47] For hexagonal lattices the Reuss and Voigt bulkmodulus (119861R and 119861V) and the Reuss and Voigt shear (119866R and119866V) can be defined as [45ndash47]

119861R =11986233(11986211+ 11986212) minus 2119862

2

13

11986211+ 11986212+ 211986233minus 411986213

119861V =1

9

[2 (11986211+ 11986212) + 11986233+ 411986213]

119866R =5

2

1198624411986266[11986233(11986211+ 11986212) minus 2119862

2

13]

31198611198811198624411986266+ (11986244+ 11986266) [11986233(11986211+ 11986212) minus 2119862

2

13]

119866V =1

30

(11986211+ 11986212+ 211986233minus 411986213+ 12119862

44+ 12119862

66)

119866 =

1

2

(119866119877+ 119866119881)

119861 =

1

2

(119861119877+ 119861119881)

(2)


0 1 20000

0005

0010

0015

0020

0025

0030D

ensit

y of

stat

es

Energy (eV)

SBO

TBT

minus6 minus5 minus4 minus3 minus2 minus1

(a)

0 1 200

05

10

15

20

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(b)

Figure 2 Total DOS of (a) an interstitial He atom at various positions and (b) its NN-Zr atoms (vertical line is the Fermi level 119864119865)

0 1 20000

0005

0010

0015

0020

0025

0030

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(a)

0 100

02

04

06

08

10

12

14

Den

sity

of st

ates

Energy (eV)

SBO

TBT

minus5 minus4 minus3 minus2 minus1

(b)

Figure 3 Local DOS of a He interstitial and its NN-Zr (a) the s-projected DOS of He and (b) 119889-projected DOS of NN-Zr for the Heinterstitials at various positions (vertical line is the Fermi level 119864

119865)

Using these values Youngrsquosmodulus E can be obtained by[37]

119864 =

9119861119866

3119861 + 119866

(3)

The elastic anisotropy is expressed by two popularanisotropic indexes that is the percentage anisotropy inshear (119860

119866) [48] and the universal anisotropic index (AU)

[49] They are defined as

119860119866=

119866V minus 119866R119866V + 119866R

119860119880= 5

119866V119866R+

119861V119861Rminus 6 (4)

These values can range from zero (ie completely elasticisotropic) to 100 (ie the maximum anisotropic)

All of the calculated elastic constants results are listedin Table 3 For 120572-Zr our calculated compressive moduli 119862

11

and 11986233

and shear moduli 11986212

and 11986213

agree well withexperimental results (at 298K) showing an accuracy of aboutplusmn10 However our calculations underestimate the shearmodulus 119862

44by about 21 And these errors seem to also

exist in othersrsquo reports [31] There are at least two factorsleading to these errors one is the significance of temperaturedependence [38] since first-principle calculations are allconducted at 0 K the other one is the intrinsic featureof DFT calculations within the ultrasoft pseudo potentials


4200

3170

2140

1110

8000e minus 2

Z

YX

(a)

4200

3170

2140

1110

8000e minus 2

Z

YX

(b)

Figure 4 The charge density (120588(119903)) in a (a) perfect Zr crystal and (b) 54-atom Zr supercell with one Zr atom substituted by a vacancy Thegreen dotted surfaces are the isosurfaces of 120588(119903) = 140 enm3 and the slices are the contour plots of the charge density of the basal planescrossing the BO and S sites

Table 3 Elastic constants 119862119894119895(in GPa) bulk modulus 119861 (in GPa) shear modulus 119866 (in GPa) Youngrsquos modulus 119864 (in GPa) percentage

anisotropy in shear 119860119866 and universal anisotropic index 119860119880 of 120572-Zr and Zr-He systems as calculated in the present work compared with

experimental values

11986211

11986233

11986212

11986213

11986244

119861 119866 119864 119860119866

119860119880

120572-Zr (this work) 13822 14517 6562 7027 2514 9259 3119 8413 0016 016120572-Zr (PBEPAW) [36] 157 158 51 62 15 91 29 80120572-Zr (PW91UPP) [31] 142 164 64 64 29 92120572-Zr (PW91UPP) [32] 15938 18093 5796 6607 1752 9616 3899 10304120572-Zr (PBEUPP) [37] 1394 1627 713 663 255120572-Zr (Exp) [38] at 4 K 1554 1725 672 646 363120572-Zr (Exp) [38] at 298K 1434 1648 728 653 320Zr96He (this work) 13923 14514 6485 6960 2443 9235 3122 8417 0020 021Zr54He (this work) 14424 14703 6091 6918 2365 9261 3244 8714 0034 035Zr36He (this work) 14099 16141 6562 6474 2198 9246 3147 8480 0045 047

framework which is involved in many approximations andsimplifications for the sake of calculation time Despite theseerrors from experimental values first-principle methods arestill considered to be efficient in calculating elastic constants[32 37]

The differences of elastic constants between diagonal andoff-diagonal elements for both 120572-Zr and Zr-He systems inTable 3 indicate their anisotropy to uniaxial compression andshear biaxial compression and distortion [32] The additionof He atoms changes elastic constants in an anisotropic wayincreasing 119862

11and decreasing 119862

44and 119862

13 while changing

11986212

and 11986233

in an irregular way In particular 11986244

and11986213

change linearly with the He concentration This elasticanisotropy can also be easily seen from the119860

119866and119860119880 values

Both indexes indicate that with the increasing concentrationof He Zr host becomes more elastically anisotropic Unfor-tunately we failed to find any data on the effects of helium

concentration on elastic constants in zirconium Hence wecompared our results with those for hcp-Er [50] of which11986211 11986233 and 119862

44decrease and 119862

12increases linearly with

He concentration while11986213changes in an irregular wayThis

may be due to the crystal distortion caused by interstitial Heatoms [50] The changes of B G and E values in Table 3 arerelatively small compared with those for Er [50] Accordingto the mechanical stability criteria for hexagonal phase givenby [51]

11986244gt 0 119862

11gt100381610038161003816100381611986212

1003816100381610038161003816 (119862

11+ 211986212) 11986233gt 21198622

13

(5)

all of the systems we studied are still mechanically stableAs mentioned above large concentrations of vacancies

can be generated in nuclear structural materials So wealso studied how the interactions between He and vacancies


30

70

80

90

100

He6He5He4He3He2HeZr53V

Ani

sotro

pic i

ndex

Elas

tic m

odul

us (G

Pa)

B

G

E

000001002003004005

020304050607

AU

AG

120572-Zr

Figure 5 The elastic modulus (B G and E) and anisotropic index(AU and 119860

119866) of 120572-Zr and Zr-V-He systems calculated from Table 4

(He He2 represent Zr53VHe Zr53VHe2 resp)

Table 4 Elastic constants119862119894119895(inGPa) of120572-Zr and Zr-V-He systems

11986211

11986233

11986212

11986213

11986244

120572-Zr 13822 14517 6562 7027 2514Zr53V 14305 14901 5864 6604 2881Zr53VHe 14377 14508 5657 6492 2835Zr53VHe2 14019 14375 5950 6212 2575Zr53VHe3 14453 15328 5523 6113 2691Zr53VHe4 13659 14877 5981 6194 2163Zr53VHe5 13372 14420 6179 6091 2108Zr53VHe6 13054 15116 6297 5895 1863

affect the elastic properties Supercells with 53 Zr atoms onevacancy and several He atoms (0sim6) were studiedThe resultsare shown in Table 4 and Figure 5 Firstly when a Zr atomin a 54-atom supercell is replaced by a vacancy the elasticanisotropy does not change according to the 119860

119866and AU

values while the 11986211 11986233 and 119862

44values increase and the

11986212

and 11986213

values decrease The evaluated 119861 value of thepolycrystal also decreases a little in contrast to that of the119866 and 119864 When more He atoms are added to the vacancythe elastic anisotropy increases linearly and significantly Atthe same time the elastic moduli 119862

11 11986213 and 119862

44decrease

almost linearly with the increase in the He concentrationwhile 119862

33and 119862

12change irregularly The B G and 119864 values

also decrease linearly with the increase in the He concentra-tion which is consistent with the case of Er [50] Comparingwith the single effect of He the He-V complexes have affectedthe mechanical properties much more obviously It seemsthat the presence of He-V clusters can lead to a significantdeterioration of the mechanical properties of Zr which mayintensively limit their performances In our future workwe will promote our study by considering the synergisticeffects of He and H on Zr since these two elements always

cooperate with each other in nuclear environment producingstrengthened effects on nuclear materials

4 Conclusion

In the present study first-principles calculation has beenconducted to investigate the structural stability and electronicand mechanical properties of Zr-He systems For the perfect120572-Zr crystal the He BO interstitial site is most energeticallyfavorable while for a not perfect one with preexistingvacancies the S site is most stable The analysis on the DOSand charge density gives a better understanding about thestructural stability issue It is found that the introductionof He changes the elastic constants in an anisotropic wayleading to a significant increase in elastic anisotropy Andthis effect is further enlarged while He-V complexes formresulting in an intensive deterioration of the mechanicalproperties of Zr

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful for the financial support bythe Science Foundation of China Academy of EngineeringPhysics China (Grant no 2010A0301011) and by theNationalNatural Science Foundation of China (Grant no 91126001)

References

[1] T Schober andK Farrell ldquoHelium bubbles in120572-Ti and Ti tritidearising from tritium decay a tem studyrdquo Journal of NuclearMaterials vol 168 no 1-2 pp 171ndash177 1989

[2] Y-L Wang S Liu L-J Rong and Y-M Wang ldquoAtomisticproperties of helium in hcp titanium a first-principles studyrdquoJournal of Nuclear Materials 2010

[3] D Xu T Bus S C Glade and B D Wirth ldquoThermal heliumdesorption spectrometry of helium-implanted ironrdquo Journal ofNuclear Materials vol 367ndash370 pp 483ndash488 2007

[4] S J Zenobia LM Garrison andG L Kulcinski ldquoThe responseof polycrystalline tungsten to 30 keV helium ion implantationat normal incidence and high temperaturesrdquo Journal of NuclearMaterials vol 425 no 1ndash3 pp 83ndash92 2012

[5] E V Kornelsen and A A Van Gorkum ldquoStudy of bubblenucleation in tungsten using thermal desorptionspectrometryclusters of 2 to 100 helium atomsrdquo Journal of Nuclear Materialsvol 92 no 1 pp 79ndash88 1980

[6] C S Becquart and C Domain ldquoMigration energy of He in Wrevisited by Ab initio calculationsrdquo Physical Review Letters vol97 no 19 Article ID 196402 2006

[7] R L Simons H R Brager and W Y Matsumoto ldquoDesign ofa single variable helium effects experiment for irradiation inFFTF using alloys enriched in nickel-59rdquo Journal of NuclearMaterials vol 141ndash143 no 2 pp 1057ndash1060 1986

[8] T Ishizaki Q Xu T Yoshiie S Nagata and T Troev ldquoThe effectof hydrogen and helium on microvoid formation in iron and


nickelrdquo Journal of NuclearMaterials vol 307ndash311 no 2 pp 961ndash965 2002

[9] J D Hunn E H Lee T S Byun and L KMansur ldquoHelium andhydrogen induced hardening in 316LN stainless steelrdquo Journal ofNuclear Materials vol 282 no 2-3 pp 131ndash136 2000

[10] N Sekimura T Iwai Y Arai et al ldquoSynergistic effects ofhydrogen and helium on microstructural evolution in vana-dium alloys by triple ion beam irradiationrdquo Journal of NuclearMaterials vol 283ndash287 pp 224ndash228 2000

[11] T Schober H Trinkaus and R Lasser ldquoA TEM study of theaging of Zr tritidesrdquo Journal of Nuclear Materials vol 141ndash143no 1 pp 453ndash457 1986

[12] T Schober and R Lasser ldquoThe aging of zirconium tritidesa transmission electron microscopy studyrdquo Journal of NuclearMaterials vol 120 no 2-3 pp 137ndash142 1984

[13] T Hayashi T Suzuki and K Okuno ldquoLong-termmeasurementof helium-3 release behavior from zirconium-cobalt tritiderdquoJournal of Nuclear Materials vol 212ndash215 pp 1431ndash1435 1994

[14] B Cox ldquoPellet-clad interaction (PCI) failures of zirconium alloyfuel claddingmdasha reviewrdquo Journal of Nuclear Materials vol 172no 3 pp 249ndash292 1990

[15] B A Cheadle C E Ells and W Evans ldquoThe development oftexture in zirconium alloy tubesrdquo Journal of Nuclear Materialsvol 23 no 2 pp 199ndash208 1967

[16] A Yilmazbayhan A T Motta R J Comstock G P Sabol BLai and Z Cai ldquoStructure of zirconium alloy oxides formedin pure water studied with synchrotron radiation and opticalmicroscopy relation to corrosion raterdquo Journal of NuclearMaterials vol 324 no 1 pp 6ndash22 2004

[17] R N Singh A K Bind N S Srinivasan et al ldquoInfluence ofhydrogen content on fracture toughness of CWSR Zr-25 Nbpressure tube alloyrdquo Journal of Nuclear Materials vol 432 no1 pp 87ndash93 2012

[18] S C Lumley S T Murphy P A Burr et al ldquoThe stability ofalloying additions in Zirconiumrdquo Journal of Nuclear Materialsvol 437 no 1 pp 122ndash129 2013

[19] C B A Forty and P J Karditsas ldquoUses of zirconium alloys infusion applicationsrdquo Journal of Nuclear Materials vol 283ndash287pp 607ndash610 2000

[20] A A Lucas ldquoHelium inmetalsrdquo Physica B Physics of CondensedMatter vol 127 no 1ndash3 pp 225ndash239 1984

[21] Y Dai G R Odette and T Yamamoto ldquoThe effects of heliumin irradiated structural alloysrdquo in Comprehensive Nuclear Mate-rials pp 141ndash193 Elsevier 2012

[22] DK SoodM Sundararaman S KDeb RKrishnan andMKMehta ldquoBlistering of zirconium inconel-718 and stainless steel-316 by 2MeV helium ionsrdquo Journal of Nuclear Materials vol 79no 2 pp 423ndash425 1979

[23] R H Zee J FWatters andOMWestcott ldquoOn the critical dosefor blistering in helium irradiated zirconium andZrNbrdquo Journalof Nuclear Materials vol 115 no 1 pp 131ndash133 1983

[24] H Zheng S Liu H B Yu L B Wang C Z Liu and L Q ShildquoIntroduction of helium into metals by magnetron sputteringdeposition methodrdquoMaterials Letters vol 59 no 8-9 pp 1071ndash1075 2005

[25] Y M Koroteev O V Lopatina and I P Chernov ldquoStructurestability and electronic properties of the Zr-He system first-principles calculationsrdquo Physics of the Solid State vol 51 no 8pp 1600ndash1607 2009

[26] Q PengW Ji HHuang et al ldquoStability of self-interstitial atomsin hcp-Zrrdquo Journal of Nuclear Materials vol 429 no 1 pp 233ndash236 2012

[27] Q Peng W Ji H Huang et al ldquoAxial ratio dependence ofthe stability of self-interstitials in HCP structuresrdquo Journal ofNuclear Materials vol 437 no 1 pp 293ndash296 2013

[28] G D Samolyuk S I Golubov Y N Osetsky et al ldquoSelf-interstitial configurations in hcp Zr a first principles analysisrdquoPhilosophical Magazine Letters vol 93 no 2 pp 93ndash100 2013

[29] G Verite C Domain C C Fu et al ldquoSelf-interstitial defectsin hexagonal close packed metals revisited evidence for low-symmetry configurations in Ti Zr and Hfrdquo Physical Review Bvol 87 no 13 Article ID 134108 2013

[30] F Wang and H R Gong ldquoFirst principles study of various Zr-H phases with low H concentrationsrdquo International Journal ofHydrogen Energy vol 37 no 17 pp 12393ndash12401 2012

[31] C Domain R Besson and A Legris ldquoAtomic-scale Ab-initiostudy of the Zr-H system I Bulk propertiesrdquo Acta Materialiavol 50 no 13 pp 3513ndash3526 2002

[32] W Zhu R Wang G Shu P Wu and H Xiao ldquoFirst-principlesstudy of different polymorphs of crystalline zirconiumhydriderdquoJournal of Physical Chemistry C vol 114 no 50 pp 22361ndash22368 2010

[33] S K Sikka Y K Vohra and R Chidambaram ldquoOmega phasein materialsrdquo Progress in Materials Science vol 27 no 3-4 pp245ndash310 1982

[34] L Yang S M Peng X G Long et al ldquoAb initio study ofintrinsic H and He point defects in hcp-Errdquo Journal of AppliedPhysics vol 107 no 5 Article ID 054903 2010

[35] L Yang S M Peng X G Long et al ldquoAb initio study ofstability andmigration ofHandHe in hcp-Scrdquo Journal of PhysicsCondensed Matter vol 23 no 3 Article ID 035701 2011

[36] J Blomqvist J Olofsson A M Alvarez et al ldquoStructure andthermodynamical Properties of Zirconium hydrides from first-principlerdquo in Proceedings of the 15th International Conferenceon Environmental Degradation of Materials in Nuclear PowerSystems-Water Reactors 2012

[37] H Ikehata N Nagasako T Furuta A Fukumoto K Miwaand T Saito ldquoFirst-principles calculations for development oflow elastic modulus Ti alloysrdquo Physical Review BmdashCondensedMatter and Materials Physics vol 70 no 17 Article ID 1741132004

[38] E S Fisher and C J Renken ldquoSingle-crystal elastic moduliand the hcp rarr bcc transformation in Ti Zr and Hfrdquo PhysicalReview vol 135 no 2A article A482 1964

[39] S J Clark M D Segall C J Pickard et al ldquoFirst principlesmethods using CASTEPrdquo Zeitschrift fur Kristallographie vol220 no 5-6 pp 567ndash570 2005

[40] D Vanderbilt ldquoSoft self-consistent pseudopotentials in a gener-alized eigenvalue formalismrdquo Physical Review B vol 41 no 11article 7892 1990

[41] J P Perdew K Burke andM Ernzerhof ldquoGeneralized gradientapproximation made simplerdquo Physical Review Letters vol 77no 18 article 3865 1996

[42] B G Pfrommer M Cote S G Louie and M L CohenldquoRelaxation of crystals with theQuasi-Newtonmethodrdquo Journalof Computational Physics vol 131 no 1 pp 233ndash240 1997

[43] X T Zu L Yang F Gao et al ldquoProperties of helium defects inbcc and fcc metals investigated with density functional theoryrdquoPhysical Review BmdashCondensed Matter and Materials Physicsvol 80 no 5 Article ID 054104 2009

[44] R Pentcheva and M Scheffler ldquoInitial adsorption of Co onCu(001) a first-principles investigationrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 65 no 15 ArticleID 155418 2002


[45] W Voigt Lehrbuch der Kristallphysik Teubner Leipzig Ger-many 1928

[46] A Reuss and Z Angnew ldquoA calculation of the bulk modulus ofpolycrystalline materialsrdquoMathematical Methods vol 9 article55 1929

[47] R Hill ldquoThe elastic behaviour of a crystalline aggregaterdquoProceedings of the Physical Society A vol 65 no 5 article 3491952

[48] D H ChungW R Buessem FW Vahldiek et alAnisotropy inSingle Crystal Refractory Compounds Plenum Press New YorkNY USA 1968

[49] S I Ranganathan and M Ostoja-Starzewski ldquoUniversal elasticanisotropy indexrdquo Physical Review Letters vol 101 no 5 ArticleID 055504 2008

[50] F Kai-Min Y Li S Qing-Qiang et al ldquoFirst-principles study onelastic properties of hexagonal phase ErAx (A= H He)rdquo ActaPhysica Sinica vol 62 no 11 Article ID 116201 2013

[51] Z-J Wu E-J Zhao H-P Xiang X-F Hao X-J Liu and JMeng ldquoCrystal structures and elastic properties of superhardIr N2and Ir N

3from first principlesrdquo Physical Review Bmdash

Condensed Matter and Materials Physics vol 76 no 5 ArticleID 054115 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


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AstrophysicsJournal of


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Computational Methods in Physics

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Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

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Biophysics


ThermodynamicsJournal of


Table 1 Experimental and calculated lattice constants (A) for 120572-Zr

119886 119888119886

This work 3231 1600PW91PAW [30] 3213 1605PW91UPP [31] 323 1604PW91UPP [32] 3223 1606Exp [33] 323 1593

their calculations were performed in a rather small systemcontaining only two Zr atoms and one He atom whichcorresponds to an extremely high helium concentration Aswe know the stability of self-interstitial atom in 120572-Zr isstrongly affected by the system size [26ndash29] So it is necessaryto further their studies concerning a larger system size

In the present paper through first-principles calculationswe reported heliumrsquos behavior in 120572-Zr by focusing on therelative stabilities of single He defects the electronic proper-ties of He atom and its nearest-neighbor Zr (NN-Zr) atomWe also studied the changes of elastic properties due to theintroduction of He atoms and vacancies

2 Computational Method

Thepresent calculationswere carried out usingCASTEP [39]Ultrasoft pseudo potential (USPP) [40] with a consistentcutoff energy of 450 eV based on density functional theory(DFT) was used The Perdew Burke and Ernzerhof (PBE)form [41] of generalized gradient approximation (GGA) wasapplied as the exchange-correlation potential since it has beensuggested by Domain et al [31] that GGA performs betterthan the local density approximation (LDA) to describe theexchange-correlation functional of Zr

During the relaxations the Brillouin zone (BZ) integra-tion was achieved using a cold smearing scheme (Methfessel-Paxton) with a smearing of sigma = 01 eV BZ sampling wasperformed using theMonkhorst-Pack scheme with a k-pointspacing as close as possible to 0030 Aminus1 (ie 4 times 4 times 3 k-pointmeshes for a 36-atom supercell and 4 times 4 times 2 k-point meshesfor a 54-atom supercell) The supercells were optimizedusing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) [42]algorithm allowing both atomic positions and the latticeparameters to relax The energy convergence criterion forself-consistent calculations was set as 1 times 10minus6 eV The criteriafor ionic energy minimization were set as energy derivativelt1 times 10minus5 eVatom forces on individual atoms lt001 eV Aminus1displacement of atoms derivative lt0001 A and stress com-ponent on cell lt005GPa The criteria for the calculationof elastic constants were set as energy derivative lt2 times10minus6 eVatom forces on individual atoms lt0006 eV Aminus1 anddisplacement of atoms derivativelt2times 10minus4 AThe parametersselected above ensure the convergence of formation energieswithin 0001 eVatom Spin polarization is not considered inall the calculations for the sake of time our future work willinvolve it

BT

OT

S

BO

Figure 1 Five crystallographically unique interstitial sites in 120572-zirconium (hcp structure) where the pink spheres are the hcp latticesites and the red ones are labeled as octahedral (O) basal octahedral(BO) tetrahedral (T) basal tetrahedral (BT) and substitutional (S)sites respectively

The defect formation energy is defined as

119864119891

defect = 119864119898ZrHe minus 119898119864Zr minus 119864He (1)

where 119864119898ZrHe is the total energy of an optimized supercell

containing 119898 Zr atoms and one He atom 119864Zr is the energyper Zr atom in optimized crystal and 119864He is the energy of anisolated He atom

3 Computational Results

31 Structural Stability We first investigate the lattice con-stants of an 120572-Zr crystal and the results are summarized inTable 1 compared with experimental and other calculatedresults It can be seen clearly that our calculated results arein good agreement with the corresponding experimental andother calculated values

When He atoms are introduced into a metal they mayoccupy either substitutional or interstitial lattice sites It isnecessary to figure out the locations of He atoms sincethey will influence the solubility and migration It is alsoimportant for the understanding of such effects as damagetrapping bubble nucleation embrittlement and blistering[6] It has been found that in different metals He atomspreferentially occupy different sites substitutional sites forFe Cr Mo and W [6 43] tetrahedral sites for Er [34]and FC (the center of equilateral trigonal face shared bytwo adjacent octahedrons) sites for Ti [2] To identify thelowest energy configurations of the He interstitial defects inZr and the possible influences of He concentration supercellswith 35 to 54 Zr atoms and one He atom were studied Fivecrystallographically unique interstitial sites for He atom weretaken into consideration (see Figure 1) They were denotedconventionally as octahedral (O) basal octahedral (BO)tetrahedral (T) basal tetrahedral (BT) and substitutional (S)sites

The calculated formation energies of a single vacancyand He atom in above sites are listed in Table 2 together



119864119891

v 119864119891

s 1198641015840119891

s 119864119891

T 119864119891

O 119864119891

BT 119864119891

BO







11 11986212 11986213 11986233 and 119862

44) with 119862

66=



119861R =11986233(11986211+ 11986212) minus 2119862

2

13

11986211+ 11986212+ 211986233minus 411986213

119861V =1

9

[2 (11986211+ 11986212) + 11986233+ 411986213]

119866R =5

2

1198624411986266[11986233(11986211+ 11986212) minus 2119862

2

13]

31198611198811198624411986266+ (11986244+ 11986266) [11986233(11986211+ 11986212) minus 2119862

2

13]

119866V =1

30

(11986211+ 11986212+ 211986233minus 411986213+ 12119862

44+ 12119862

66)

119866 =

1

2

(119866119877+ 119866119881)

119861 =

1

2

(119861119877+ 119861119881)

(2)


0 1 20000

0005

0010

0015

0020

0025

0030D

ensit

y of

stat

es

Energy (eV)

SBO

TBT


(a)

0 1 200

05

10

15

20

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(b)


0 1 20000

0005

0010

0015

0020

0025

0030

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(a)

0 100

02

04

06

08

10

12

14

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(b)


119865)


119864 =

9119861119866

3119861 + 119866

(3)




119860119866=

119866V minus 119866R119866V + 119866R

119860119880= 5

119866V119866R+

119861V119861Rminus 6 (4)



11

and 11986233


and 11986213





4200

3170

2140

1110

8000e minus 2

Z

YX

(a)

4200

3170

2140

1110

8000e minus 2

Z

YX

(b)




experimental values

11986211

11986233

11986212

11986213

11986244

119861 119866 119864 119860119866

119860119880





44and 119862

13 while changing

11986212

and 11986233


and11986213


119866and119860119880 values







11986244gt 0 119862

11gt100381610038161003816100381611986212

1003816100381610038161003816 (119862

11+ 211986212) 11986233gt 21198622

13

(5)




30

70

80

90

100


Ani

sotro

pic i

ndex

Elas

tic m

odul

us (G

Pa)

B

G

E

000001002003004005

020304050607

AU

AG

120572-Zr





11986211

11986233

11986212

11986213

11986244



119866and AU



11986212

and 11986213


11 11986213 and 119862

44decrease


33and 119862




4 Conclusion




Acknowledgments


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119864119891

v 119864119891

s 1198641015840119891

s 119864119891

T 119864119891

O 119864119891

BT 119864119891

BO







11 11986212 11986213 11986233 and 119862

44) with 119862

66=



119861R =11986233(11986211+ 11986212) minus 2119862

2

13

11986211+ 11986212+ 211986233minus 411986213

119861V =1

9

[2 (11986211+ 11986212) + 11986233+ 411986213]

119866R =5

2

1198624411986266[11986233(11986211+ 11986212) minus 2119862

2

13]

31198611198811198624411986266+ (11986244+ 11986266) [11986233(11986211+ 11986212) minus 2119862

2

13]

119866V =1

30

(11986211+ 11986212+ 211986233minus 411986213+ 12119862

44+ 12119862

66)

119866 =

1

2

(119866119877+ 119866119881)

119861 =

1

2

(119861119877+ 119861119881)

(2)


0 1 20000

0005

0010

0015

0020

0025

0030D

ensit

y of

stat

es

Energy (eV)

SBO

TBT


(a)

0 1 200

05

10

15

20

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(b)


0 1 20000

0005

0010

0015

0020

0025

0030

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(a)

0 100

02

04

06

08

10

12

14

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(b)


119865)


119864 =

9119861119866

3119861 + 119866

(3)




119860119866=

119866V minus 119866R119866V + 119866R

119860119880= 5

119866V119866R+

119861V119861Rminus 6 (4)



11

and 11986233


and 11986213





4200

3170

2140

1110

8000e minus 2

Z

YX

(a)

4200

3170

2140

1110

8000e minus 2

Z

YX

(b)




experimental values

11986211

11986233

11986212

11986213

11986244

119861 119866 119864 119860119866

119860119880





44and 119862

13 while changing

11986212

and 11986233


and11986213


119866and119860119880 values







11986244gt 0 119862

11gt100381610038161003816100381611986212

1003816100381610038161003816 (119862

11+ 211986212) 11986233gt 21198622

13

(5)




30

70

80

90

100


Ani

sotro

pic i

ndex

Elas

tic m

odul

us (G

Pa)

B

G

E

000001002003004005

020304050607

AU

AG

120572-Zr





11986211

11986233

11986212

11986213

11986244



119866and AU



11986212

and 11986213


11 11986213 and 119862

44decrease


33and 119862




4 Conclusion




Acknowledgments


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 1 20000

0005

0010

0015

0020

0025

0030D

ensit

y of

stat

es

Energy (eV)

SBO

TBT


(a)

0 1 200

05

10

15

20

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(b)


0 1 20000

0005

0010

0015

0020

0025

0030

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(a)

0 100

02

04

06

08

10

12

14

Den

sity

of st

ates

Energy (eV)

SBO

TBT


(b)


119865)


119864 =

9119861119866

3119861 + 119866

(3)




119860119866=

119866V minus 119866R119866V + 119866R

119860119880= 5

119866V119866R+

119861V119861Rminus 6 (4)



11

and 11986233


and 11986213





4200

3170

2140

1110

8000e minus 2

Z

YX

(a)

4200

3170

2140

1110

8000e minus 2

Z

YX

(b)




experimental values

11986211

11986233

11986212

11986213

11986244

119861 119866 119864 119860119866

119860119880





44and 119862

13 while changing

11986212

and 11986233


and11986213


119866and119860119880 values







11986244gt 0 119862

11gt100381610038161003816100381611986212

1003816100381610038161003816 (119862

11+ 211986212) 11986233gt 21198622

13

(5)




30

70

80

90

100


Ani

sotro

pic i

ndex

Elas

tic m

odul

us (G

Pa)

B

G

E

000001002003004005

020304050607

AU

AG

120572-Zr





11986211

11986233

11986212

11986213

11986244



119866and AU



11986212

and 11986213


11 11986213 and 119862

44decrease


33and 119862




4 Conclusion




Acknowledgments


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




4200

3170

2140

1110

8000e minus 2

Z

YX

(a)

4200

3170

2140

1110

8000e minus 2

Z

YX

(b)




experimental values

11986211

11986233

11986212

11986213

11986244

119861 119866 119864 119860119866

119860119880





44and 119862

13 while changing

11986212

and 11986233


and11986213


119866and119860119880 values







11986244gt 0 119862

11gt100381610038161003816100381611986212

1003816100381610038161003816 (119862

11+ 211986212) 11986233gt 21198622

13

(5)




30

70

80

90

100


Ani

sotro

pic i

ndex

Elas

tic m

odul

us (G

Pa)

B

G

E

000001002003004005

020304050607

AU

AG

120572-Zr





11986211

11986233

11986212

11986213

11986244



119866and AU



11986212

and 11986213


11 11986213 and 119862

44decrease


33and 119862




4 Conclusion




Acknowledgments


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




30

70

80

90

100


Ani

sotro

pic i

ndex

Elas

tic m

odul

us (G

Pa)

B

G

E

000001002003004005

020304050607

AU

AG

120572-Zr





11986211

11986233

11986212

11986213

11986244



119866and AU



11986212

and 11986213


11 11986213 and 119862

44decrease


33and 119862




4 Conclusion




Acknowledgments


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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Research Article First-Principles Study of the …downloads.hindawi.com/journals/acmp/2014/929750.pdfResearch Article First-Principles Study of the Structural Stability and Electronic