Pandya, C V

Pandya, T C

Vyas, P R

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Indian J. Phys. 7SA (3). 271-272 (2001)U P  Aan international journalTemperature variation of Debye-Waller factor and mean square displacementfor some transition metalsP R Vyas, C V Pandya, T C Pai^ya and V B Gohel*Department of Physics, School of Sciences, Guiarat University.Ahtncdabad’38() 009, G|ijarat, India'tE-mail cvpandyat^ht^tmail com Received fi September 2000. accipted 12 Janaury 2001\bstract ' Debyc-Waller factors and mean square displacements at different temperatures for ten fee transition metals (Cu, Ni, Fc, Co, Ag, Pd. Klu Au, Pi and Ir) have been computed on the basis of a lattice mechanical model proposed recently by Antonov e t  a l  \Z  P h y s  H - C o n d e n s e d  M a t t e r  79 223. 233 (1990)]. The theoretical values are compared with those recently obtained from experimental density of stales by Peng e t  a l  [A c ta  C rv s t  a\ 52, 4.S6 (1996)]. A satisfactory agreement between theory and experiment has been found For Ir and Rh the theoretical values have predictivem e a n in g .Keywords Transition metals, Debye-Waller factor, mean square displacement PACS Nos. . 63 20 Dj. 63 70.-»-hRecently, Antonov et. aL[\y 2] have proposed a pseudopotential m o d e l approach for the lattice mechanical properties of fee transition metals. In this approach, the contribution of 5-like electrons is calculated in the second order perturbation theory for the local model pseudopotential while that of the <i-like electrons is taken into account by introduction of repulsive short range interatomic potential. The model was applied to ten transition metals (Cu, Ni, Fe, Co, Ag, Pd, Rh, Au, Pt and Ir). This model was used for the calculations of phonon dispersion curves, density of phonon states, clastic constants and their pressure derivatives. Interatomic interaction, variation of Debye temperature with temperature, mode Gruneisen parameters, temperature variation of Gruneisen parameter and cold equation of state with reasonable success. Encouraged by this fact, wc undertake the computations of temperature variation of Debye- Waller factors and mean square displacements for these metals. In this report, we present the results of our calculations and compare them with the recent data given by Peng et. al. [3].In the harmonic approximation, the Debye-Waller exponent 2W is directly obtained from the computed phonon density of states [4], If jg(v) is the frequency distribution function for the^Corresponding Author.phonons giving the number of vibrational modes in the frequency interval between v and v + dvy then2Wr%K^ fi ( sin 6J m n [~ J Tg(v) coth h vI k J dVy (1)where $ is the glancing angle and A is the wave length of incident radiation. M is mass of the atom and N denotes the total number of unit cells in the crystal. In the harmonic approximation, the mean square displacement of the atoms in the crystal is given by the relationt/^  =8;r^ I sin 9W r (2)The frequency spectra of the lattice vibrations of all the ten metals considered here were computed by the numerical sampling of frequencies according to Blackmann's technique [5-7]. In the present work, wc have considered a mesh of evenly spaced 64000 wave vectors in the first Brillouin zone. The details of the calculations may be found in ref [4]. The expressions for Dynamical malix and numerical values of model parameters were taken from Ref f4).Wc present in Figures 1 and 2, the Debye-Waller factors and mean square displacements for Cu, Ag, Co, Rh and Ir. We also©2001 lACS272 P R Vyas, C V Pandya, T  C Pandya and  V B G oheldisplay in the same figures, the same quantities as obtained recently from the experimental phonon density of states by Peng et al [3|. The results for other metals (Au, Ni, Pd, Pi and Fe) arc07 i^Temperature (Degree Kelvine)Figure 1. Temperature variation of Dcbye-Waller factor and mean square displacement <u*> for atoms ot Cu and Ag Continuous lines 2 and 4 represent theoretical results o f  for Ag and Cu respectively  Expcnmental results (+) and (A) arc from Peng e i  a l  [3]. Dotted lines I and 3 represent theoretical results o f <u*> for atoms of Ag and Cu Experimental results (•)  and (x) are from Peng e f a l  [31D<IIOof Dcbye-Waller factors and mean square displacement lot ir and Rh being reported for the first lime, may be considered as fairly good estimates.Table 1. The maximum deviation of calculated values of and - ii from experimental results for Cu. Ag, Au, Ni, Pd, Pt and FeD Metal Maximum %  deviation Maximum % de< for W. < u->§<0 Cu 30 30Ag 12 1 1>(P Au 6 60)O Ni 26 28Pd 12 14Pi 13 13Fe 16 25Figure 2. Temperature variation of Dcbyc-Wallcr factor and mean square displacement <u-> for atoms of Co. Ir and Rh. Continuous lines 4,5,6 represent theoretical results o f Wj. for Co, Ir and Rh respectively Dotted lines 1,2,3 represent theoretical results of <u’> for Co, Ir, Rh respectively.similar and hence we do not give these results in the form of graphs. The maximum deviations from experimental results for these metals are given in table [1]. The overall agreemcnl in each case is quite satisfactory. Particularly, the presently studied model due to Antonov eL al . [ \ ,  2] works quite well for Ag, Au, Pd & Pt metals. For Cu, Ni and Fe, the maximum deviation of about 25 “* 30% occurs only at a very small number of points. For most of the temperatures, the theoretical results agree with experiments to within about 12 -  15%. The anharmonic effects, in the case of Ni and Fe, are seen to be important beyond about 200 K. As Slated in the introduction, the pscudopotenlial approach as proposed by Antonov et al [\ ,  2] has been successful in reproducing reasonably well the phonon frequencies f 8,9] and cold equation of states of Ir and Rh right up to 40% compression ; we believe that our computed valuesWc have presented in this report, the results ofcalculaiioiK of Dcbye-Waller factors and mean square displacements ol icn fee transition metals. The model approach, wc used, is that due to Antonov etal \ \ ,2\ .  This model approach has been succcsslul in predicting many lattice mechanical properties of these inciaK Wc have extended the range of applicability of this approach b\ com puting the Debye-W aller factors and mean stju.iic displacement.Despite the importance of the Dcbyc-Waller facioi in dynamical electron diffraction calculations, accurate valuer ni the Dcbye-Waller factors arc not readily available loi most crystals. Experimentally, the Dcbyc-Waller faclors may Ik measured using such techniques as neutron or X-ray dilTraciion However, the experimental results are nonnally published at one temperature [10], which docs not correspond exactly to the experimental situation under which an electron microscop\ experiment is normally conducted. It is in this circumsianee, ;i reasonable model approach to calculate this quantity is higliK desirable. In the present report, we have successfully exploicd the possibility of u.sing for this purpose, the model approach due to Antonov et al I \ ,2].R cference.s [1] V N Antonov, V Yu Milnian, V V Nemoshkalenko and A V Zlialko Titarcnko Z. P h y s  C o n d  M a t t e r  R 7 9  223 (1990)[2] V N Antonov, V Yu. M il man. V V Nemoshkalenko and A V Zhalko-Tiiarcnko Z. P h y s .  C o n d  M a t t e r  B79 233 (1990)[3] L M Peng, G Ren, S L Dudarev and M J Whelfin Acta C ty s f A52 456 (1996)[4] A R Jani and V B Gohel Z. N a tu r fo r s c h . 38a 503 (1983)15] M  Blackman P m c  R o y  S o t .  ( iM n d o n )  A195 416 (1937)[6] M  Blackman E n c y c lo p e d e ia  o f  P h y s ic s  (Berlin : Springer-Verlag) 7 325 (1955)[7] E I Geshko V h r a in .  Fiz. Zk. 8 1058 (1963)[8] R Hcid, K P  Bohenen, K Felix, K M  Ho and W Reichardi J C im d . M a t t e r  10 7967 (1998)[9] A Eichler, K P Bohenen, W Reichardt and J Hafner P h y s  B57 324 (1998)110] N M Butt, B T M Willis and G Hegcr A c ta  C rysL  A44 396 (1988)

Temperature Variation of Debye-Waller Factor and Mean Square Displacement for some Transition Metals

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Temperature Variation of Debye-Waller Factor and Mean Square Displacement for some Transition Metals

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