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An Approximate Nodal Is Developed to Calculate the Change of •Laatio Constants Induced by Point Defect* in Hep Metals

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An Approximate Nodal Is Developed to Calculate the Change of •Laatio Constants Induced by Point Defect* in Hep Metals 1 - INTBOPUCTIOB the elastic conatants.aa well aa othernmechanieal pro partita of* IC/79/lW irradiated materiale (are vary sensitive to tne oonoantration of i- INTERNAL REPORT (Limited distribution) rradiation produced point defeota.One of the firat eatimatee of thla effect waa done by Dienea [l"J who aiaply averaged over the whole la- International Atomic Energy Agency ttice the locally changed interatoalo bonds due to the pxesense of" and the defect.With tola nodal ha predioted an inoreaee of the alaatle United Nations Educational Scientific and Cultural Organization constant* of about 10* par atonic f of interatitlala in Ou and a da. oreaee of l]t par at. % of vacanolea. INTERNATIONAL CENTRE FOE THEORETICAL PHYSICS Later on,experlaental etudiea by Konlg at al.[2]and wanal lilgar* very large decrease a of about 5O}( per at.)t of Vrenlcal dafaota.fha theory waa than iaproved in order to relate the change of a la* tic con a tan t a to the defect lnduoed change of force oonatanta and tno equivalent aethoda were devalopedi the energy-»athod of ludwig [4] CHANGE OF ELASTIC CONSTANTS and the t-aatrix method of Slllot et al.C ?]. INDUCED BY FOIMT DEFECTS IN hep CRYSTALS * Iheoretioal eetlnates for oublo oryetala have been oarrie* out by Ludwig[4] for the caae of vacancleafby Piatorlueld for intarati- Carlos Tome •• tiala and by Sederloaa et al.t?] for duabell interotitiala. International Centre for Theoretical Physics, Trieste, Italy. Re thaoretloal work baa been done ao far for hexagonal cryatmlaj and the experimental neaeurentanta (available only for Kg) ax* eona- ABSTRACT what crude t8,9i,ayan though in the last few. ye are ita teohnologlml An approximate nodal is developed to calculate the change of lnportanoa haa Inoreaeed»eapecially for nuclear applioationa. •laatio constants induced by point defect* in hep metals, sup - Hexagonal oryatala differ fron cubic crystals In that they haw posed the defect configuration la Known. two atone par unit oell,none of which ia an lnreraion eentar^and in General expressions relating the change of elastic moduli to the different lattice eymmetry whlohnreaulta in JUCL anlsotropio be- the final atoalc coordinates and to the defect force field ars haviour la toe a direction, Aa point defect* poasesa. aose of the aya- dariv«d uaing the epeeific symmetry o* the defect. metriea of the lattice their conflguratlana and propartisa are going 1 Explicit calculations are done ttr Wg.The predicted change of to be different froa thoae in oublo material*. elastic moduli -turns out to be negative for vacancies and trig- We preaent hare a flrat order (In the forca-ooatanta-ohan«aa> Ma- onal inter Btltials wlii.le for hexagonal interstltlala a positive trix) analytical approximation for the evaluation of the change in change ia predloted. Compatibility with experimental data would alaatio conatanta lnduoed by point defeote in hop orystale,The for- auggeat that the trigonal configuration is the stable one. mulation la baaed on the t-aatrix method and haa the advantage that, alnee the symmetry of toe defeat la retained,asking proper ua* of It MIRAMARE - TRIESTE the problem can be confined to a reduced non equivalent region of October 1979 the lattice and erplioit staple axpre alone are obtained f»r the ohaa* * Submitted for publication. ge In elaatio moduli aa a function ef the defeat eonfiguratlfnt .that ** Comisi6n de Investigaciones CLentfficas de la Provincia de Buenos Aires, avoida complicated numerical oaloulatione or lengthy computer aimuW boat atoms Interaction are previously known. c^ and c^ (the contracted VOigt notation has been employed).The Speelfio calculations are carried out for vacancies,hexagonal int- resulting eigenvalues are i erstltlals and trigonal lnterstltials In Mg 'based on a previous work of Tome" et al,[io](hereafter referred as I) where an empiri- cal third neighbour Interaction potential waa developed for an ideal hop Mg lattice and the configurationB of the defects wore calculated (3) by the statio-Green function method. c ~ In section 2 we Introduce the notation to be used and the affect C \i, — Of externally applied homogeneous attain* on internal displacement where I and the associated eigentensor* arei la disouteed for hop lattloee.Alao an original calculation of alas. tic eigenvalue* and related eigentansors for the specific case of hexagonal symmetry la presented there. Seftlon 3 i» devoted to develops our model in order to get a gene- ral expresion for the ohange of elastic oonBtante (section 3-a) which Is tben specified for the particular oase of defects in hep - o-iz. &^ metals talclmg proper aooflunt of symmetry (section 3-b).The ease of an ideal Kg lattice described by an empirical potential is solved in section 3-c while section 4 presents a discussion of the results £ - and »oae oonoluding remarks. ^, V (4) 2 - HOICQEKEOPa 3THAIS IB HEXAGONAL CHYSTAIiS ^ii The -proper and "natural" basle of symmetric strain tensors is obt- ained, by solving the eigenvalue equation for the elasticity tensor'4^ o © o since the resulting eigenvalues ofelaatio moduli" * can be easily \ ^ related to compressibility or shear moduli while the associated etreea-tensor = for a given = turns out to be proportional to \ o o lti o \ o it ^ t (2) We solve «q.(l) f«r the case of hexagonal symmetry, l,e. o^-o^g .-«,- O-i-T0^ «nd all other o -0 exoep* where p = - °* Indicating by £ the perfect lattloe position motor of the A (+) In what follows repeated Indexes imply summation. atom in the 1 o*ll,the most general fora ef the asBOdatad hoao- geneous displacement ist ^57 • tf* INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL,, BOIBNTIFIC AND OTJUTURAL. ORGANIZATION INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS 34J00 TRIESTE (ITALY) - P.O.B. SSO - MIHAMAKE - STKADA COSTIEHA 11 - TELEPHONES: 22*281/2/3/4/5/0 CABLE: CENT8ATOM - TKIJKX 1+60392 ICTP I 26 October 1979 CHANGE OF ELASTIC CONSTANTS INDUCED BY POIKT DEFECTS IN hep CRYSTALS Carlos Tome INTERNAL REPORT (Limited distribution) ERRATA Page 13 - lines 3 and k should read: - ,, case the predicted change in compressibility modulus is larger in absolute value than the changes ... Page 16 - Table III should read: c.c, vacancy — O.g-\ -O.83 —2 hexagonal Interstitial s -^ O--*>C £.1? 3.56 interstitial u.-cfe - ^.Z,e> 1-.bb Table III t Change of elastic moduli for Wg (in £ per at.J< of defects) fflciently far apart, one Oan suppose that tha strain fit Id laaide tha •A, crystal is still homo ganeous and that the affeot due to the preeen- oa of tha defects just averages ovar tha whole lattioe,The •la.etia 1 •hin X#/o *•• ^^ ** »tiv* displacement in tha unit nU and ia the potential energy U' la expreaaad. la terms of tha modified elastic same In •very otll for the BUM kind, of atom,while the semnd term constants o!„ • la the elastic strain.In Brevais lattices ana also in thoM non-Bra~ vais lattices whir* eaoh atom la an lnroraion center Z{u) turna £ t (B) out to be aaro.This is not tha oaaa with hop lattices, whose call oontalne two atoaa that are not Inversion oantera at the points whore T is tha volume of tha crystal and C tha homogeneous strain (0,0,0) and (a/2,a/2V5W2) fur A.-1 and/C-2 respeativaly (a,o T>e- tensor. ing the lattioe parasetore). On the other hand , the Internal energy of a strained lattice ax The JJIS\** * deduced "by imposing aero resultant fore* ovar each containing a defeot oaa also be evaluated using tha t-natrlx method atom aT the a trained lattice under tha hypotheala of email displa- of Elliot et al. [ 5"]. Comparing the latter result with (8) a relation cements* Imposing arbitrarily Zium-\\ 3 0 we get for V/^ 2} t between A c. and the microscopic parameters of the defaot is ob- ijmn tained.fha derivation is reviewed In the work of Dederioha et al. and will be omitted here.the resulting relationship iai K* (6) •*<r (9) where c is the concentration of defects,V. ia the volume of the Tha ooaetant K depend a on the foroe oostants of the lattice and unit oall an A P {£) ~ Y\ 'X.- (C + S£\<"\\ oaa be written In terms of Born and Huang* a notation ([lo) ,ehapter <* t*1 ' A| ji ** ' ia tha dipola A tensor of the defect modlfief by the strain field J^ .In the above „* i z. > 3 expression of TJ, we have omitted; the /C index in order to simpli- (7) fy the notation. The index 1 rune from one to H and labels the a- toms surrounding the defeot that are within the range of tha dsfeot- Our result ia identical for this particular symmetry to the general host interaction potential V, xY1 Indicates the coordinatee of one obtained by lapoelng nlnlnrum energy compatible with the external (r) ' * JX) axe hoaogeneoue deformation (see [ 103 )» atom 1 in the perfect lattice referred to the defeot and fC' the components of the foroe that the defect exerts over atom 1 at the final relaxed position,For a central pair interaction V, . they are given byt 3 - EIiASTIC OOSSTAWTS OF A Dr.Pi-.'CT CRYSTAL t THE htfp CASE -till -jm (00) 3-a General Formulation It is poo Bible to strain a perfect crystal homogeneouely by imoo- where tt is the displacement of atom 1 due to the foroe field Of eing proper deforaations or forces over its axirface 'but in presence the defect.The force °JlO in the variation of r due to the preoen- t ( ^ of a defeot the internal strain will not be homogeneous in ita vl- ce of a small strain flsld that shift the atone by^U,^ .They are clnltT.Neverttielesa,lf the defects! are randomly distributed and su- linked by a first order relation! (11) tenaor P E whtr* 1-1 * &**w and for the ohange of elastiot constants with (17) and UN has 'bean Bad* of (10).
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