Ghorai, A

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Indian J. Phys. 73B (2), 317-321 (1999)U P  B—  an in ternationa l jo u rn a lN o n -sp lit  in te r s t it ia l  fo r m a tio n  in  C u , A g , A u  a n d  P b  b a se d  sy s te m sA GhoraiA zad Physics Center, M aulana Azad C ollege,8, Rafi Ahm ed Kidw ai Road, C alcutta-700 013, IndiaA bstrac t : E xpressions for form ation energies for different non-split in terstitialsconsidering relaxation effect and their changes for im purity interstitial arc form ulatedhere using the fram ew ork o f  pseudopotential form alism  C alculations o f  these energy term s in four fee metals, viz copper, silver, gold and lead are discussed here using H eine-A barenkov m odel p o ten tia l w ith  T a y lo r’s e x ch an g e  and  c o r re la tio n  fo r o c ta h e d ra l, te trah ed ra l and crow dion  type of in terstitia lK e y w o rd s fee  m etals, n o n -sp lit in te rstitia ls , p seudopo ten tia l fo rm alismP A C S  N os. 71 .1 5 .H x , 6 1 .7 2 .J i, 6 6 .3 0  Jt1. IntroductionStudies on interstitials are interesting due to their complicated nature of formation and motion through different solid substances [ 1], [21. In Harrison’s approach [31 the perturbation in the conduction electron distribution caused by the creation of a defect is sufficiently weak to be treated in a simple linear response formalism. Any defect in the lattice changes the structure dependent lattice energy and so an algebraic difference between the energy after defect creation and that before will yield the defect formation energy when considered for the whole lattice. This structure dependent total energy depends on ion-ion, ion-electron and electron-electron interactions and is also dependent on the modified lattice wave numbers. The modification in the lattice wave numbers from its perfect lattice value is necessary to maintain the lattice volume and the number of lattice ions constant [4j. In this paper formulations for formation energies and their change in presence of impurities of different non-split interstitials, viz. octahedral, tetrahedral and crowdion, in very dilute binary alloys of copper, silver, gold and lead are derived and the results are discussed.2. FormulationsLet us consider a lattice consisting of I number of substitutional impurities and a total of N Ions. The impurities are distributed throughout the lattice so that the interactions among©19991ACS318 A Ghoraiimpurity ions are almost negligible (/ « N ). The total structure dependent energy of this lattice is given by [3], |5], [6]E = i^ \S (q ) \2U ^ ) + l i  = i \ s ( ‘f)eU,’ri + S*(q)e ^  \AU(q)If=i A -  U ( q ) -  lim{ N - n z \T J .-y o o  y y» + / z ; ,2 IW u ' ( 1)whereU(q) = limI n Z l e 1AU{q) = limTf—2x(Zt - Z H)ZHe £2Hq( 2 )V j - e  rM ’, +[tO//(^)]' t'H(q )X tM )'e v IAl) +\(»,(q)-(0H(q)]o)„(q)e„(q) x Hiq)- (3)? 2A2U(q)= l im ^ ^  Z h) e e ‘‘’,4t> +[a},(q)-(OIJ(q)]‘' e tJ(q )x H '^> (4)L 2 H qandS(q) = ^ ^ e x p ( - i q . r ) / N .  (5)The subscripts H and / are for host and impurity respectively, Z the valency, £2fJ the atomic volume, e the electronic charge, rj the convergence factor, q the wave number, S (q) the structure factor, r the position vector for the ith ion in the lattice, (Of/(q ) the pscudopotential for the host, Sff(q ) the dielectric function and x ^ q  ) perturbation characteristics. The expressions for eH(q ), x t^ q ) and o ^ q ) are given by 17, 8]£#(<?) = q  ^2 (6)3Z//4e FH(1/2) + - " /... ? 2 In%qkFHU F/l+q2k FH0)and0),(q)a ne n W(0°,(q). (8)Here, f H(q) is the exchange and correlation, eFH and kFH arc the Fermi energy and wavenumber respectively and « “(</) is the form factor for the impurity in its pure state. The structure factor for the perfect lattice S(q) is defined asNon-split interstitial formation etc S(q) = 1 for q = q0 = O f o r ? * f 0 ,319(9)q0 the lattice wave number is defined asq0 = (m, / Nl)q] + (m2 / N2)q2 + (m3 / W3) f3 (10)with / yV, ,/n2 / A^2 and w3 / W3 are integers including zero and q v q2 and$r3 arc the primitive wave numbers.Let us consider a single defect in an otherwise pure lattice (i>. /  = 0). If the point defect is a non-split interstitial then the Brillouin zone volume has to be scaled down by a factor of (N -  1 ) /N in  order to keep the lattice volume constant, i.e. the lattice wave numbers arc modified to q0 with n t = (1 -1  / 3A0- In presence of non-split interstitial the surrounding neighbours around it relax producing relaxation energy. Finally, one gets the expression for non-split formation energy using eq. ( 1), asE uFH 2(cosq0.rt - \ )U (q 0) -% d u (q0) 3 d %+ X " ' i 2(cos,0. r / -  cos,„.r, >[/<«„>•% rT |+X "'2i n 1 Jo Lsin<7|r/ ~ r  <l\rt " r/|sin^jr, - r \  sing r, - r } | sin^|rf - r ,i r - rU(q)q~dq, 01)where n, is the number of relaxed neighbours around the non-split interstitial, r '  the relaxed position of the ilh neighbour with original position vector r and r( the interstitial position.When the impurity is at an interstice it changes the interstitial formation energy. The change in this case of formation energy is given by4£,Jr = V  2(cosq0.rt - \ ) A U ( q a)s in g K ~ r, AU{q)q2dq (12)So we have two expressions for defect formation energy and its change in presence of impurity ion. Several other formulations can be deduced using equation (1) viz. (i) host interstitial formation energy in presence of substitutional impurity at an adjacent lattice site, (ii) split interstitial formation energy with one or two impurities, etc.320 A Ghorai3. DiscussionHere integration over quasi-continuous wave numbers is done by quadrature technique and the discrete sum is done over lattice wave numbers. Calculations are done using Heine- Abarenkov model potential and Taylor’s exchange and correlation (hence forth called HAT). For simplicity, relaxation effect is not considered in these calculations [ l ]. Out of six kinds of interstitials calculations, only for non-split type, viz. octahedral, tetrahedral and crowdion, are discussed here in cases of noble metals (copper, silver and gold) and lead as both host and impurity. From the above formulation, it seems that the values of formation energy and its change (EFH and AE'f ) will be large in cases of heterovalent systems in comparison to homovalent systems due to valency, pseudopotential and exchange and correlation. Table 1 gives the input parameters utilised in the present calculations while Table 2 gives the calculated values of E lFH and A It is found that HAT combination yields slightly lower values of EFH in comparison to the values calculated by 11 ]. They reported the values of EFH for octahedral, tetrahedral and crowdion type in cases of copper, silver and gold as respectively (0.25, 0.34,0.40), (0.22,0.31,0.36) and (0.21,0.30,0.35) Rydbergs. Other previous calculations were also shown in their work. Lower values in the present calculations arise due to the fact that relaxation effect is not considered. There is a systematic increase in the value of EFH from octahedral to crowdion for each metal which most probably indicates that the crowdion type is more unstable. It has been pointed out by [2] that the stable interstitial configuration in noble metals is [100] split. In case oflcad, EFH is negative for the octahedral configuration and has large positive values in comparison to those of noble metals for the other two configurations. This perhaps indicates the lower probability of occurrence of these defects in lead where vacancy plays the dominant role [9]. Hence the present calculations agree fairly well with the earlier calculations.Tabic 1. Param eters used for defect energy calculations and pseudopotcntial for pure metalsM etal Z aA UaA U^FHA U£fhR y d b e r g A UA *R y d b e r gCu 1.0 6 8219 7 9 .3 7 0 0 .7 1 9 9 0 .5 1 8 2 2 4 0 7 0 .8 3 0 8Ag 1.0 7 .7101 1 1 4 .5 8 2 0 .6 3 6 9 0 .4 0 5 7 2 .7 1 6 0 .7 3 6 4Au 1.0 7 6 9 1 2 1 13 ,742 0 6 3 8 5 0  4 0 7 7 2 7 15 0 .7 3 6 6Pb 4 .0 9 3542 2 0 4 .6 2 2 0 .8 3 3 4 0 .6 9 4 5 2 .0 1 0 9 3 .1 8 2 7* taken from  [6]The computed values of the change in the impurity-interstitial formation energy over host interstitial (AE'f ) are also shown in Table 2, and it is found that A E'f  gradually increases in magnitude from octahedral to crowdion and their orders of magnitude are in accordance with the earlier calculations [2]. Also every binary system of noble metals and its reverse (say Cu- Ag and Ag-Cu) shows complementary character. But it is not so in the case of lead based systems where Cu, Ag, and Au impurity crowdion formations show lower values than host interstitial formation. Thus they are more probable than host crowdion interstitial formation. At present, there are very few experimental data on point defect parameters. So it is difficult to say the last word about the calculated defect parameters.Non-split interstitial formation etc 321Table 2. Evaluated defect formation energies and its change (in Rydbergs) for Cu, Ag. Au and Pb based binary systems.Metal E Ul f hE U^ F H system < * 4Oct tetra crow oct tetra crowCu 0 1738 0.2580 0.3201 Cu-Ag 0.0591 0.0577 0.0578Cu-Au 0.0590 0.0575 0.0577Cu-Pb 0.6498 0.9025 1.0872Ag 0.1580 0.2326 0.2881 Ag-Cu -0.0440 -0.0426 -0.0424Ag-Au -0 0001 -0.0001 -0.0001Ag-Pb 0 4188 0.6489 0.8152Au 0.1592 0.2339 0.2895 Au-Cu -0.0440 -0  0425 -0  0423Au-Ag 0.0001 0.0001 0.0001Au-Pb 0.4253 0.6559 0.8225Pb -0.0444 0.8759 1.6859 Pb-Cu 0 8120 0.11930 -0.4915Pb-Ag 1.0040 0.3397 -0.2751Pb-Au 1.0040 0.3392 -0.2756It has been mentioned earlier that a proper choice of a pseudopotential is very important and careful attention must be paid in achieving accuracy in numerical computation in order to arrive at a meaningful result. Hence, Heine-Abarenkov model with approximate overall explanation of all atomic properties is perhaps a good choice.R eferencesfl) A K Bandyopadhyay and S K Sen Phys. Slat. Sol (b) 157 519 (1990)[2] N V Doan Phil. Mag. A58 179 (1988)[3] W A Harrison Pseudopotentials tn the Theory o f Metals (New York : Benjamin) (1966)[4] A Ghorai J . Technol. 34 47 (1997)[5] A R DuCharme and H T Weaver Phys. Rev. B5 330 (1972)[6] A Ghorai Phys. Rev. B46 5229 (1992)[7J A Ghorai Indian J . Pure Appl. Phys. 32 508 (1994)[8] M Cohen and V Heine Solid Slate Physics eds. H Ehrenreich, F Seitz and D Turnbull (Academic) 24 55 (1970)[9] S K Sen and A Ghorai Phil. Mag. AS9 707 (1989)

Non-Spilt Interstitial Formation in Cu. Ag, Au and Pb Based Systems

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Non-Spilt Interstitial Formation in Cu. Ag, Au and Pb Based Systems

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