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
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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) •* 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). Care must be taken in that the particular form (14) for C. does In writing equation (9) for the change of elaatio constants we not allow to derive £ with respect to the null components of the have neglected a contribution arising from the ohange in oAyetal tensor and therefore""for a given X the ohanga of only those elas- volume due to the presence of defeats and to the homogeneous ex- tlo constants with allowed mn indexes can be evaluated.for the pansion Induced by tha image forces. As pointed out by Dederlchs strain tensors given by (4) only the following elastic: constants are changedt in t7iteq* (9) containa tha main contribution and since we are go- ing to evaluate it through a firat order approximation the afore, mentioned tens are within the accuracy of the model and will not X-2 be included in our calculations. X-3 We now make tha assumption that the email superimposed displa— "•S oementa SiH6} induced by the external homogeneous field are well X-4 described by the homogeneous displacements y/ of eq. (5) in the vioinlty of tha defeoti tl) (13) 3 - b Application -to hep Lattices This is equivalent to identify Lr. th< t- aatrlx of Elliot with tha Tha point defect configurations In hop lattloes (namely the force constants - it belongs (la) and the label (11) Inside a given shell , we can (20) express those relations "by n. 32, =, PL and un ~ ^ - Zxx*" ^iz_ (18) Uaing these Bymnetry relation»hips In the evaluation af (17) one easily finds that the ohangeo in elavtio oonetants verify the fo- llowing relations! L) 1 where 1 -ltH. i 1.-1,1, - . and with l.-,-1l we label the etom In the Ac'* reduoed lattice while r V i1 la the matrix representation = O of any operation belonging to the syoaetry group of the defect.A* then are unitary transforaatlons they leave invariant the length (21) of the rector and then equivalent atone are equidistant frca the defect both In the perfect lattiae and In the relaxed configura- tion. Therefore they will hare identical values of a, , and B, „ as defined In (12). Therefore the variation of" the four elastio moduli c* ^' «an DS Writing (16) explieltly with the help of equations (5),(6) and expressed in terms of only six elastic constants, changest (11) tand using the new notation for the indexes one can easily nameljrt rt verify that the A..** and B, »•• factor out from the emulation a orer equivalent atone and one has to evaluate terms of the? formt (22) (19) £ r where l,J-L,2t3 and 1^-l.H^ .Evaluation of (J.g) is greatly sim- with X-5,6. plified when relations (18),are utied,and the following Identities Here i, the nuober of atom!* shells that ar. withia the range or 23) •JKL bjl JHTOJOA.ifi.held tor thf lowest STmetry pas* (O.,gr< Qft. (.*> qtience of the aotlel developed and Is valid in principle for any C I fclnd of defect having at least C and CT (23) oymaetry and inter- (24) acting with hoot atoao rim a central potential. • is) and H 3 - a Application of the Model to *& (25) As already mentioned, the defect configuration a wore calculated in -M °it "+ ^l-i °i\- '••V reference I for an "ideal" mg lattice with a o/a ratio correspon- ding to the piling of rigid spheres, using the static Green function method.The elastic postanta employed in I were those that result It should be noted that In thia Case c can be easily identified by applying the long-wave method of T*xn and Huang £lcQ to a third with a compressibility modulus slnoe C as given t>T (29) Is a hy- neighbour central interaction that holds the atoms together.The drootatic strain and °13" •ll+012~°33 (23) Relationship (23) introduces a simplification in the fens at the where elastic moduli (3) and elgentensore (4y.3inee the par aa t ter o( de- aeoond and third neighbour distances in the perfect lattice.Using the fined in (3) takes now the value £ we find! empirioal potential 7, . deaoribed in X we predict for I the value t*« -0.1286a which means that for homogeneous deformation* of the (•) The explicit explicit equations giving the elastic constants as lB th or 1 ot order of 1# the internal displacement I&- 21 °* * ** * Vk* function of the potential derivatives are given in tne work of of the lattice parameter.whloh represents a sMght ocrreotion to the Trott and Heall tiz\. effect aine# the thermally activated reorlentatlon of the defeoti The results obtained for the change of al&stia moduli in Mg are summariaed In Table HI for the three types of defects.In every la the strain field would give apparent changeB of the elaetia mo- OHM the predicted change In compressibility modulus is much lar- duli due to the large elastic deformations in their vicinity. ger In absolute value (fey about a factor of 10) than the changes Even when it is net possible te correlate exactly the macrosco- in shear moduli.Furthermore,while the changes associated with the pio moduli of a poly^-crystnl with the elastic constants of ths vacancy are all negative ae one would simply expect from th* absence oryetal (see for example Kusgrave £l3^t chapter 13),the former of some atomic lends, for the interstitial* the results are oppo- results oan at least be considered to give a trend in the ohangea site, tie Ing positire for tho hexagonal and negative for the trigo- Of elastic moduli. nal. The negative measured ohangea would be oonsia_Jpnt with our re- sults only if the trigonal configuration for the Interstitial is more stable than the hexagonal one*This result would give exper- imental support to the conclusion derived in Z concerning the sta- bility of interstitials in an hop Kg lattice.This effect does not 4 - DISCUSSION AWP O0KCI,U3IOH3 seem to be related to the fact that the trigonal configuration In the preceding secticne vre hare preoented a model for the has a negative volume change associated with it. Catting the inter- evaluation of the changes in elastic moduli induced by point de- action range of the potential just after third neighbours results fects 14 hexagonal oloee packed structures.The advantage of the in a positive change of volume while the ohangea in elastic modu- ocdel is that the symmetry of the defect is retained in the for- li become smaller but do not change sign. A weakening of the force mulation end eo considerable Bin nil float Ion in achieved,On the constants in the vicinity of the defect is not conceivable and a Other hand It presents the oversimplification of not taking in- symmetry lowering mechanism due to enharmonic forces - as propo- to account the second order terns in the ohanged-foroe constants sed by Amran Sussmarm L 1+J- has to be excluded on the basis ef matrix. computer simulations of the defect (see 1} that demonstrate that The predicted results for a calculation done on Mg are presen- the symmetry is preserved in the relaxed configuration. ted in Table III,Even if a quantitative comparison with experia- A possible explanation oan be found in the mechanism proposed ental values night not be completely valid the basic qualitative by Holder et al.[l5] and further developed by Dederlohe et al.[?] behaviour of elastic constants in irradiated materials can be de- to explain this behaviour in cubic crystale,l.e.the presence of duced from simple calculations. thermally motivated low frecuenoy modes associated with the de- Experimental measurements done at room temperature by tillchter fect. These modes wculd not affect the strong bond atretoning and Kikoin[8i in irradiated Mg gave no change of the eomweasi- constants but only the negative bond bending constants (which billty modulus. Later result* by Hillairet et al. [9") confirm would lower the frecuenoy) .The external strain field woulA couple that at this temperature the defects re combine and therefore with them giving a net lowering of the elastio oonstants.The obser- there ia no observable effect on the elastic constants.The in- ved dtecrease of elastic constants in Mg oould be an evidence of ternal friction measurements made by Hillairet on uolycrystaline the presence of these resonant nodes but further experimental Mg gave a lowering of Toung modulus of 20< per atoale < of def- data end theoretical calculations of defect resonant modes in hop ects and a lowering of 355* p. at.* of defects for the shear mo- structures is required to confirm this conjecture. dulus,these figures were obtained at 80»K and for an estimated saturation concentration of 2x10 defects per aton.Thsy notice however that these figures are probabily an overestimate of the -11- -ik- AflCTOWLEJOMEirTS Part of thl» work «»• eirritd witn an internal grant of Cooiai6n da Invastigadonaa Clantlflcae da la Provlnoia da Buenos Aires (Argentina). Tha author la grateful to K.Savlno for auggeating the problem 1-o at al.,AHst«rdantltorth Holland 1969 4 Iwdwlg V, |"Calculations of Propartiea of Vaounciea and Intera- AC titiale", Oonf.Proc. ,Hat.Bureau of Standards,Kleo.Pub. 287 151 (1966) 9 Blliot E.at al. | "Jjooaliasd Exaltations In 3olids" ,ed,R. Wallla vacancy -O.83 --12.22, H.T.Plenua Praea 1958 hexagonal 1.Z3 O.3O 6 Platoriua M. i Z.Ang.Pfcyaiic 2£ U5 (1970) interstitial trigonal 7 Dedarloha P.H.et al. j 3. Phynifc KO 155 (1975) lnteretitial — O. Z (a - zz.oe. 3 Ulditer A.I. and Klkoln A.I, j 3ov.Plyr«. JKTP ^ 772 (1957) 9 Hlllalret J.et al. | J.de Physlqua Oollotiuea j£ (2 (1971) table III t Chang* of elaetlo moduli for Mg (In f T»r at.)( 10 Born K.and Huang K.; • Thn Bjraaaloal Theory of Crystal Lat*;lcea" of defects) Clarendon Press,Oxford 1954 11 Warren J.L. j »«v.l!od. ?hya. 40 38 (1968) 12 Irott A.J.and rieald P, T. ; i)hya. st.soX. (\>) 4j6 361 (1971) 13 MUBgrave B.J. j "Crystal Acoustics", Holdon Oa/ , St,Feo.l970 th 14 Aoram Sueamann J. I ?roc. 5 Int.Conf. on "Into nal Friction and Ultrasonic Attcnuctlon in Crystallne Solids" ,«d.D. tions and K. lAe'te, Springer Verlag 1973 15 Holder J.o'; al; i Phya.Rsv. E10 363 (1974) -15- -AT- j ,_- ,-J !P r— fO ,J i/) IS r" s>' o b rJ u r °£ 8 •i \£ 1 i t 1 1 s ) 1 i i | V id r— r> T> 0 (0 -.0 •.-J T 0 4" /•• o" o i? 0 0 f« o d c 0 0 0 3 3 0 o 6 a * 8i 1 1 1 _ i v ? , T \S •r* tn r* 0 • -• Nil i<) a» •" T r-" '0 7>