Volume 109A, number 8 PHYSICS LETTERS 17 June 1985 EFFECT OF THREE-BODY INTERACTIONS ON THE FORMATION ENTROPY OF MONOVACANCIES IN COPPER, SILVER AND GOLD W. MAYSENHOLDER, R. BAUER and A. SEEGER Max - Planck- Institut fftr Metallforschung, Institut ff~r Physik, Heisenbergstrasse 1, D 7000 Stuttgart 80, Fed. Rep. Germany and Universiti~t Stuttgart, Institut fftr Theoretische und Angewandte Physik, Pfaffenwaldring 57, D 7000 Stuttgart 80, Fed. Rep. Germany Received 21 February 1985; revised manuscript received 12 April 1985; accepted for publication 16 April 1985 The vacancy formation entropy in Cu, Ag and Au is estimated from model calculations with explicit reatment of three-body interactions. The three-body interactions cause a rather strong relaxation around the vacancies and therefore lead to lower values for the formation entropy than usual pair-potential calculations• In a recent letter [1 ] we have reported calculations on point defects in Cu, Ag and Au based on model potentials [2] that allow explicitly for three-body interactions by including amodified Axilrod-Teller potential [3]. In contrast to the common pair-poten- tial approach, the elastic onstants c 11, Cl 2, C44 are reproduced exactly and need no longer be "adjusted" for Cauchy's relation c12 = c44 to be satisfied. We expect he inclusion of the three-body term to permit a meaningful comparison of Cu, Ag and Au. The properties calculated in ref. [1 ] include forma- tion energies and formation volumes of mono- and di-vacancies and of four interstitial configurations. • The three-body interactions led to noticeable lattice relaxations around the vacancies: The experimental value of the monovacancy relaxation volume T,-rel "1V in Cu, -(0.20 -+ 0.05)~2 [4] (I2 is atomic volume), is reproduced, whereas usual pair-potential calculations yield only about -0.0212 (see, e.g., ref. [5]). En- couraged by this result we proceed to calculate the monovacancy formation entropy sFy. In the first attempt to calculate S~V for Cu, Huntington, Shirn, and Wajda [6] arrived at 1.5k B assuming Born-Mayer epulsion between the atoms. In 1964 Schottky, Seeger, and Schmid [7] obtained 0.5k a using a (non-central) force constant model which gives a vacancy relaxation of -0.1312. In both calculations, performed before modern computers 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) were available, the underlying models could not be evaluated with the desirable accuracy. This may be seen from the calculations by Hatcher, Zeller and Dederichs [8], which demonstrated that considerable computational effort is needed in order to obtain numerically reliable results. Unfortunately, Hatcher et al. used a Morse potential ( eading to S~ v ~ 2.3kB) which yields almost no vacancy relaxation (-0.02~2). Since - as akeady stressed in ref. [8] - large relaxa- tions alter the result significantly, this potential can- not be considered adequate for the calculation of the formation entropy in Cu. Following the previous calculations we neglect electronic ontributions toSFv and consider the vibrational part only. In the quasi-harmonic approxi- marion for temperatures well above the Debye tem- perature ® this gives us 3N-6 SFv=kB n~__l ln(w(n0)/¢On), T~O, (1) where k B denotes Boltzmann's constant, N the number of atoms in the crystal and 6%, ~O(n 0) the eigenfrequen- ties of the crystal with or without vacancy. The eigen- frequencies are calculated for infinite crystals with only a limited number iV' of atoms around the vacancy allowed to vibrate. The presence of relaxations gives 393 Volume 109A, number 8 PHYSICS LETTERS 17 June 1985 rise to an additional term due to the volume change AV I of the f'mite crystal [6-8], originating from the free surface ("image forces"): S1Fv = lira S. . (N* ) + ASI , (2 ) NP,,..I. co with 3(N'+l) ()V~ ~) In to(°)/to~ O)S=(N') = kB 1 n=l ( ""** ) 3N' ' (3) [8]. The vast increase of computational requirements associated with the three-body interactions, which range up to third-nearest neighbours, prevented us to reach the 1/iV' behaviour. The following procedure was adopted: Expression (3) is calculated for the first five shells around the vacancy (N' = 12, 18, 42, 54, 78). Atoms with numbers greater than N' up to the 40th shell are treated in the Einstein approximation (i.e. for each atom a 3 × 3 matrix is diagonalized in order to obtain its Einstein frequency ton ,E)" The first five values of 4800 S**(N') = S.. (N') + k B ~ ln(to~0)/ton,E) (7) n=3N'+l ~SI = ~KAI7 I = arrrzrel v,- "1V (TE - ] ) /TE . (4) Here the subscript co refers to the inf'mite crystal; 6o~. 0) denotes the Einstein frequency of the perfect crystal, K the bulk modulus,/~ the coefficient of volume ther- mal expansion and "YE the Eshelby factor [9,10]. The factor N'/(A ~ + 1) takes into account that introducing a vacancy into an unbounded crystal reduces the number of atoms by one. The expression (4) for the "image term" AS I fol- lows from general thermodynamic arguments and is valid for all temperatures below the melting point. For high temperatures (T ~" O) AS I can also be ob- tained via the Griineisen parameter 7T for the perfect crystal, which describes the change in the frequency spectrum caused by a homogeneous volume change. The equivalent to eq. (4), ~S I = 3kB)' T AVI/~2, (5) with 7T = lim 1 3N' d lnto(n °) N'--,** ~ ~ (6) n=l din V ' will be used to obtain formation entropies entirely deduced from the model potentials without additional experimental information on the anharmonicity of the crystal [~ in (4)]. S**(N') is expected to be proportional to I[N' for large N'. For small N', however, the discrete character of the lattice dominates and makes an extrapolation N' ~ o. more difficult. An extrapolation using the 1 ]iV' behaviour may require iV'/> 400 (corresponding to 15 or more shells of atoms around the vacancy) 5.0 o • ~. / ' ' . ~ -I~ "'... 0 ~ \ "".. rlE a o -1% ? ....... / / ° I. 0 ~ ~4 0.5 s 3 0.0 , w ~ , I , LO0 0.05 Ag~ o --* AU ' \ o ; , , I , 1 ) o.lo N' Fig. 1. Results from eq. (7) for the first five shells of atoms around the vacancy obtained with model potentials including three-body interactions (Cu, Ag, Au). To illustrate the effect of relaxation the results for a rigid lattice (dotted lines, super- script "rig") and for a pair-potential c lculation (Cupp, with very little relaxation) are also shown. For extrapolation to the formation entropy S**(**) in an infinite crystal (dashed lines) see text. Open circles: Einstein approximation S**(N' = 0). 394 Volume 109A, number 8 PHYS1CSLETTERS 17 June 1985 are used for extrapolation to S**(¢~). Fig. 1 shows the five calculated S..(N') values (connected by straight lines) versus 1/N' for different potentials with or without relaxation. Cu, Ag, Au denote the model potentials including three-body interactions which have been used in ref. [1 ]. The superscript "rig" indicates the result for rigid lattices (i.e. without relaxation). Cupp denotes apair poten- tial for copper equivalent to the Morse potential used in ref. [8]. For this potential only the case of the relaxed vacancy (V 1 v = -0.02[2) is shown. From fig. I the significant effect of relaxation on the formation entropy becomes obvious: The results for CUpp and for Cu rig lie close together, whereas those for Cu (V[~ = -0.21~2) are substantially ower. Ag is very similar to Cu. For Au the effect of relaxa- tion is still more pronounced. Qualitatively, these features may be deduced from the Einstein approxi- mation S~(A f = O) indicated by open circles in fig. 1. As estimates for S**(~) and sng(~) we take the intersections of the dashed straight lines with the vertical axis, which are determined by the 3-shell value and the point halfway between the 4- and 5- shell values. The justification for this procedure is that it works in two extreme cases: (i) For CUpp (al- most no relaxation) we get 2.3kB, the value reported for the Morse potential in ref. [8]. (ii) For the Born- Mayer potential treated in ref. [8] (with very large relaxation: , i v =rlrel -0.47[2) we get from fig. 1 in ref. [8] (without he second term in (7)) 1.85k B instead of the more accurate result 1.75k B. The extrapolated values for s~g(~) and S**(~) (error probably less than 0.1k B for Cu and Ag, and less than 0.2k B for Au) and their difference AS** due to the relaxation in the in- finite crystal are shown in table 1 ~ The resulting vacan- cy formation entropy SFv in table 1 is obtained by adding S**(~) and AS I [calculated from eq. (5)]. The Griineisen parameters 7T derived from the model po- tentials and the vacancy relaxation volumes VFv al- ready given in ref. [ 1 ] are also shown. For comparison we calculate AS I from (4). All quantities in (4) are temperature d pendent, but be- cause of positive (for/3) and negative (for K and (TE -- 1)/TE) temperature coefficients the prefactor of ~ in (4) varies only little between 300 K and the melting point. We evaluate (4) with experimental values of 13 and cij at 800 K assuming for simplicity that V[~/I2(T) is independent of temperature. The image contributions AS~I °rr obtained this way are smaller by factors of 1.44, 1.21, and 1.38 for Cu, sA~lv, and Au, respectively, leading to higher values of (first and second column of table 2). Since (4) and (5) are equivalent we conclude that the GrOneisen parameters derived from the model potentials are too large by these factors, i.e. the anharmonicity changing the frequencies of the crystal in response to a homo- geneous volume change is too strong. The same effect on the frequencies can be expected for an inhomo- geneous distortion like the relaxation in the infinite crystal, and consequently we have to correct he con- tribution AS**, too. For simplicity we use the same factors already used for ASI. The reduced values ASC~ °rr cause a further increase in the formation en- tropy (last two columns of table 2). The results for SFv in columns two and four of table 2 can be regarded as lower and upper bounds, respectively (subject o the extrapolation errors men- tioned above). Column 2 gives lower bounds because only ASI is corrected and AS** is still overestimated by the too strong anharmonicity of the potentials. Column 4 gives upper bounds because the absolute Table 1 Individual contributions ( ee text) to the vacancy formation sFx,., in units of k B as determined entirely from the entropy model potentials. V[1 ~ is the vacancy relaxation volume al- ready calculated inref. [ 1 ] ; ~'T is the Griineisen parameter derived from the model potential. srig(**) S..(~*) AS** AS I Sfv rel VI V(I2) 3' T Cu 2.40 1.62 -0.78 -0.58 1.04 -0.21 2.80 Ag 2.68 1.66 -1.02 -0.76 0.90 -0.31 2.88 Au 3.45 1.17 -2.28 -0.86 0.31 -0.37 4.26 Table 2 Corrected values for AS I and AS** and final estimates for Sfv in units ofk B (1.b. = lower bound, u.b. = upper bound). Asfor Sf v Sly 1.b. u.b. Cu -0.40 1.22 -0.54 1.46 Ag -0.63 1.03 -0.84 1.21 Au -0.62 0.55 -1.65 1.18 395 Volume 109A, number 8 PHYSICS LETTERS 17 June 1985 values of AS c°rr, which cannot be justified rigorously, are very likely underestimated for the following reason: The image term AS I is related to a homogeneous re- laxation of the lattice and hence to the bulk modulus and its pressure derivative. Conversely, AS** is related to an inhomogeneous relaxation and therefore ssen- tially to the shear moduli and their pressure derivatives. Compared to experimental values the Griineisen param- eters and pressure derivatives of the bulk moduli deriv- ed from the potentials how very similar deviations, whereas the pressure derivatives of the shear moduli are in significantly better agreement with experimen- tal values, although too high as well. Hence, AS** should really be corrected by somewhat smaller fac- tors than ASI, leading to lower values of SFv . We arive at the conclusion that for calculations of vacancy formation entropies with an accuracy of about 0.1k B high-quality model potentials are re- quired. The model potentials hould be able to re- produce the vacancy relaxation volume as well as - especially in the case of large relaxations - the anharmonic properties of the crystal ike GrOneisen parameter and pressure derivatives of the elastic con- stants. The Morse potential used by Hatcher et al. [8] does not even meet the first requirement and has to be considered inappropriate for calculating S~v for Cu. Their result (2.3kB) comes out too large. Including three-body interactions in the model po- tential makes it possible to meet the ftrst require- ment. Nevertheless, the model potentials used in the present work are not completely satisfactory with respect o anharmonic properties. In particular, the model potential for Au needs to be improved. For Cu and Ag we have calculated lower and upper bounds for SFv, which differ only by approximately 0.2k B * 1. Taking into account an extrapolation error of -+0.1k B we expect experimental values within these limits. Additional uncertainties due to the neglected temperature dependence of the eigenfrequencies are presumably small. ,1 If we use the Cu potential with van der Waals exponent nvd w = 5 which yields a vacancy formation energy of 1.32 eV (instead of 1.05 eV with nvd w = 6) but also V[~ = -0.21s2 [1], we obtain very similar esults (e.g., S1Fv = 1.29k B and 1.49k B for lower and upper bound, respec- tively). References [ 1 ] R. Bauer, W. MaysenhiSlder and A. Seeger, Phys. Lett. 90A (1982) 55. [2] R. Bauer, unpublished. [3] B.M. Axilrod and E. Teller, J. Chem. Phys. 11 (1941) 299. [4] H.-G. Haubold and D. Martinsen, J. Nucl. Mater. 69 + 70 (1978) 644. [5] P.H. Dederichs, C. Lehmann, H.R. Schober, A. Seholz and R. Zeller, J. Nucl. Mater. 69 + 70 (1978) 176. [6] H.B. Huntington, G.A. Shirn and E.S. Wajda, Phys. Rev. 99 (1955) 1085. [7] G. Schottky, A. Seeger and G. Schmid, Phys. Star. Sol. 4 (1965) 439. [8] R.D. Hatcher, R. Zeller and P.H. Dederichs, Phys. Rev. B19 (1979) 5083. [9] J.D. Eshelby, J. Appl. Phys. 25 (1954) 255. [10] W. MaysenhiSlder, Phys. Lett. 100A (1984) 289. 396