A model in real space has been developed by extending the generalized form of the exponential potential known as extended generalized exponential potential (EGEP) to account for (a) the correct nature of repulsive and attractive components of forces for all the separations in general and that of small separations in particular, (b) the three-body forces such as volume forces in an indirect way in the framework of EGEP through the parameter n, (c) the dielectric screening functions in an alternative and simpler form through the parameter m. The model is employed to compute the cohesive energy, second-order elastic constants and phenon spectra for fcc platinum. The predictions show promising agreement with experimental findings.Author Affiliation: Divesh Verma
A. F. School of Engineering and Technology, Dhauj-121 004, Haryana, India
M L Verma* and A Verma
Department of Physics, GGDSD College, Palwal-121 102, Haryana, India
and
R P S Rathore
Department of Physics, B.M.A.S. Engineering College, Agra-282 002, Uttar Pradesh, India1.A. F. School of Engineering and Technology, Dhauj-121 004, Haryana, India                                         2.Department of Physics, GGDSD College, Palwal-121 102, Haryana, India                                         3.Department of Physics, B.M.A.S. Engineering College, Agra-282 002, Uttar Pradesh, Indi

Rathore, R P S

Verma, A

Verma, Divesh

Verma, M L

IACS Institutional Repository

Indian I  Phys. 78A (3). 337-341 (2004)< U P  ACohesion, e lastic constants and v ib ra tio na l mechanics o f fee p la tinumDivesh VermaA F School of Engineering and Technology, Dhauj-121 004. Haryana, IndiaM L Verma* and A VermaDcpartmeni of Physics, GGDSD Collcgf, Palwal-121 102, Haryana. Indiaa n d .R P S RathoreDepartment of Physics. B.M.A.S. Engineering College, Agra 282 002, Ultar Pradesh, India Received 30 December 2002,\ac( epted 19 June 2003Abstract : A model in real space has been developed by extending the gcncrali/x*d form of the exponential potential known as extended gi'iicialized exponential potential (EGEP) to account for <a) the correct nature o f lepulsive and attractive components of forces for all the separations m gcncial and that of small separations in particular, (b) the three-body forces such as volume foices in an indirect way in the framework of EGEP through the parameter /i, (c) the dielectric screening function.s in an alternative and simpler form through the parameter m The model is employed to compute the cohesive energy, second-order clastic constants and phenon spectra for fee platinum. The predictions show promising agreement with tApciimental findingsKeywords . Cohesion, elastic constants, phonon dispersion PA( S No.s. 63 20 Dj. 62 20 Dc1. IntroductionPlatinum (Z = 78) is a silvery white transition metal o f  VIII group in VI period with an outer electronic configuration o f 5d^ 6 s and shows high catalytic activity. Platinum is highly malleable and ductile element with high melting point, high boiling point and high density. Platinum being a transition element shows a variable valency o f two and four.Needless to say, the phonon spectra play a pivotal role in determining the mechanical, electrical and thermodynamical properties o f elements and their alloys. We have therefore, been motivated to study the co h es io n , c la stic  and vibrational behaviour o f fee platinum with a renewed interest because o f  its above stated attractive properties.The cohesion in metals has earlier been studied by many Workers [1] follow ing different approaches and using widely different approximations. Moriarty [2J has em ployed several approximations to compute the cohesive energy o f twenty two metals using a sim plified local-density theory. C les [3] hasCorresponding A uthoranalysed the role o f correlation effects in the cohesion o f  transition metals. Sethi e t  a l  [4] have employed their theory to compute cohesive energy o f  som e cubic metals. Cheliskowsky [5 ] has determined the cohesive energy o f twenty simple metals from atom ic kinetic en erg ies. The gen eralized  gradient approximation has been employed by Asada and Terakura [6 J to compute the cohesive and magnetic properties o f bcc, fee and hep iron. The uniform electron gas model for transition metals has recently been employed by Rose and Shore |7 | to calculate the cohesive energy o f  M ,  A d  and 5 d  series o f transition metals. In the light o f  widely different approaches adopted to explain cohesion in metals, we have employed the present model (EGEP) which involves the least approxim ations and em ploys the minimum number o f input parameters to predict the cohesion in metals with remarkable success [8 ].The importance o f  the study o f  elastic constants lies in the fact that they give information about the nautre of the binding forces in solids, account for their thermodynamic behaviour and leads to the determination o f the interatomic force constants o f the metals. An exercise to com pute the elastic constants has been undertaken from this point o f  view.©20041ACS338 Divesh Verrna, M L Vernm, A Venna and R P S RathoreEarlien the dynamical behaviour o f platinum has been studied by many workers using different approaches. These studies involve first principle |9 ,10] as well as phenomenological [ 11-13] calculations. The former calculations suffer from physical intractability, mathematical lediousness, conceptual obscurity and resort to various approxim ations to arrive at useful conclusions whereas the later suffers from various shortcomings such as lattice instability, combination o f short range ion-ion interactions with long range electron-ion interactions in an abrupt manner and use o f large number o f  force constants, which varies from metal to metal and from model to model. Fielek f 14] model has been employed by Singh c t  a l  [15] for lattice dynamical study o f  platinum. Agrawal and Rathore [ 16] have employed a non-central F ielek model to study the lattice vibrations in platinum, considering the equilibrium of the lattice under the combined effect o f  the volume- dependent energy o f ions, the f/“Shell electrons and the conduction electrons. The study f l 6 ] makes use o f elastic constants and zone boundary frequency, thereby introducing relative standard error. Kulshrestha and Upadhyaya |17] have computed phonon dispersion curves o f  platinum using transition metal model potential (TMMP) in local approximation with the model parameters o f Animalu [18] and the dielectric function o f Hubbard modified by Sham [ 19]. In the local approximation, the dispersion results obtained, differ widely from experimental data. The author [17] themselves improved the results by including non-local effects in the schem e o f  Eschrig and Wonn [20|. Singh e t  a l  [211 have em ployed their isotropic non-intcracting band model to calculate dielectric screening and phonon frequencies o f palladium, platinum and vanadium using number o f parameters. Prakash and Upadhyaya [2 2 ] have incorporated the effect o f  many body forces to study the dynamical behaviour o f platinum using their [23] three-body potential which helps in improving the transverse branch of dispersion curves.The outer electronic configuration of platinum 5cP 6s .suggests that the electrons occupying the ^-shells overlap with the immediate environment leading to 5-rf hybridization, which causes non-sphericity in charge distribution and hence calls for unpaired or three-body forces as pointed out by Bertoni e t  a l  [24J and recently by Verma e t a l  [25]. Although several theories [26] have been put forward for providing much insight into the relative role played by the s- and ^ f-like electrons in the bonding of the transition, but many of the details of the complex band structure are inessential to the understanding of the cohesive properties. Will and Harrison [27] have given a qualitative first principle analysis of the elastic and bonding properties of the transition metals, by computing their total energy as a function of volume and ionic configuration at constant volume, just by extending the nearly free electron theory of metals to include the effect of transition metal ^/-bands. A real space analysis of elastic and dynamical properties of transition metals have recently been investigated by Singh e t  a/l28] by means of their temperature-dependent, exponentially-damped two-bodyinteraction potential which com bines rational dielectric funciion (RD F) and H eine-A barenkov (H A ) m odel potential The com putations [27 ,28] being tedious and com plex, require enormous time, m oney and efforts.T he p resen t co m m u n ic a tio n  d e r iv e s  an empincai macroscopic potential in real space, which is an extension oi the generalized exponential potential [29], known as extended generalized exponential potential (EGEP) and explains almsoi ai! the characteristic features o f  the interatomic interactions as detailed recently [30]. The present paper aims to investigate cohesion, the elastic and dynamical behaviour o f  fee piahnum2. Theory2. /  E x te n d e d  g e n e r a l i z e d  e x p o n e n t ia l  p o t e n t i a l :The extended  gen era lized  exp on en tia l potential (KGLP representing the true and realistic nature o f the repulsive as \ult as the attractive com ponents o f  the interactions assumes the fewmwhere m  and n are the parameters which take care o f clcctmniL exchange and correlation effects and three-body forces such as volume forces in an alternative and simpler form respectively, /) is the dissociation energy, a  the hardness parameter and /„ is the equilibrium separation parameter and r is the di.stance ol they-lh atom from the origin given bywhere /, (either even or odd) are the integers of the p<isitioico-ordinates such thatEq. (1) can be put in the form to represent the cohcM'f energy at equilibrium semi-lattice constant (n^) as undermaa^ L^j/.Ml- m p i a a o r ' Z ^ y “ ‘^ "'W ,(3)(4)where P = exp(a Tq).The three defining parameters (a . and D) of the potential require for their evaluation, the precisely determined input data of equilibrium semilattice constant (Op) and bulk modulus (B) ntCohesion, elastic constants and vibrational mechanics o f fee platinum 339ihe metal only. For evaluating the three parameters a , and D  of the potential function, the condition [31J' 2 [ l f d 0 { r ) / d r ^ ] ^ ^(5)fur the equilibrium o f the crystal in the absence o f external forces IS employed which givesP m-1 m(Offln)" ( P - 6 )  ( a « o ) ‘ " ( f f  +  5 )  ’whereP  =QR =«  ' e x p i - a a ^ L j )(« /«o) ^  exp(-aaotL ^)'.'2^m a ^ l f  L /" ' ‘' e x p ( -m a a „ L j )e x p (-H ia a o /.j )riie bulk modulus can be expressed as B ^ [ r - j w ) ( d ‘ 0 l d r - )  _ ^ . (7)D = 1 8 V /B (m - l) / (A '-y ) , (8)whereP "\a  Oq ) ” \{ma Uq )^  LT/" exp(~ma OqLj )[ W ,+ 2n ( m a a 0) e x p (-m a  ) + n(« + 1 )li'A^  L]" t x p ( - m a  a o L j ) \W,  Jy = [mP(a oq )" I (a  flo f  S  exp(-a Oo ) I V2'2- 2 «(a flo ) L|,"*‘^ cxp(-of«„/.^ ) + / i ( / i - l )V>'.-X  /" exp (-a«„Z > j)2.2 The s e c o n d - o r d e r  e la s t ic  c o n s ta n ts  :(5 ) The following expressions for the second-order elastic constants (90B C ) with present interatomic interactions are used 131]:C„ = ^ { n ' a * / 2 V )  d ~  d > { r ) l ( d r - ) -C |i ~ [ n '  U i J l v )  ^ l { l l d ~  d > (r ) l id r~ )~  ,(9)( 10)where n' is the numbci of atoms per unit cell (4 for fee and 2 for bcc) and V represents the atomic volume.The value of SOEC for the metal under study has been computed by expanding the secular equation is the long wave limits ( r / - > 0 ) and the comparing with the usual Chnstoffel relation,2 3  iM tt ic e  dynam ica l b eh av iou r :The elements o f the dynamical matrix having explicit bearing on eq.( 1 ) in case o f fee platinum, are :=  2 (a , + / i ,) [ 2  -  C„(C,, + C\ )]4 4 « ,  ( 1  - C ),C \  )The parameter D  can be evaluated through the expression toi the bulk modulus following the condition given by eq. (5) toi stress-free lattice. The fo llow in g  expression  for D isuhiained :4P 2SI + 4 a 2 [ s j + S ^ y  Z ) 3 ( g ) - 2 ( / 3 , - a , ) i „ 5 p .( 1 1 )( 12)whereS„ -  sm (f/r/„/2) and C„ -  cos(</r/„ / 2 ). (13)t t ,  ^ [ U  r { d 0 / d r ) ] ^ ,  or, = [ l  / •  d-^)P, =[d^^0/ dr~]^.  / 3 3 = [ ,9 * < p /^ r - ]Jn/v (15)l^(x ^  “ component of phonon wave vector q ,  a is thelattice parameter and or,, )8 , are the force constants for the first neighbour (AO and a 2 » / ^ 2  these for the second nearest neighbour (N N )  respectively.The phonon frequencies ( u )  are obtained by solving the usual secular equation i.e.(16)where /  is the unit matrix o f 3 x  3 order and M  is the mass of the atom.340 Divesh Verma, M L Vernia, A Venna and R P S Rathore3 .  Computations and resultsThe input data for the fee platinum (Pt) i.e. the lattice constant and bulk modulus are given in Table 1, while the computed potential parameters are recorded in Table 2. The present study considers the 248 atoms extending to 12th neighbours in case o f  fee platinum to eompute potential parameters. The computed values o f  cohesive energy and second-order elastic constants are shown in Tables 3 and 4  respectively. Table 5 enlists the evaluated derivatives ap O ti for fee platinum. FigureTable 1. Input data for fee platinum (Ref 34]Metal Lattice10constantmBulk modulus 10“ N/m-Pi 3 92 2 783Table 2. Computed potential parameters of platinumn m a p DxIO-^'J FyX 10'®m0.5 2.0 3 .10466 1.584010 415.6055 11.160770 3 806628I.O 2.0 3 .13644 1.600225 1794 5240 1.092589 4.6821532.0 2.0 3.23935 1.652730 40802 .6000 0 009026 6 4236173 .0 2.0 3 35498 1 71 1751 1172507 0000 5 68E-05 S 1640804 .0 2 .0 3.42325 1.746556 3.65E+07 2 92E-07 9 .9765523. Computed values of cohesive energy of Pt [eq (3)| in eV/atomTablen m Cohesive energy Magnitudes of cohesive energyrepulsivepartattractivepartcomp exp.134]0 .5 2 .0 5 3708 11 2108 5 8400I.O 2.0 4 2207 10,0606 5 83992 .0 2.0 2.8345 8 6753 5 8390 5 843.0 2.0 2 0843 7.9243 5.84004 ,0 2.0 1 6645 7 5043 5.8398Tible 4. Computed second^ordcr elastic constants (in 10“ N/m^).Metal n c„ c„ C„ RefPt 0.5 2.0 3 .6403 2 3543 1.33531.0 2 .0 3.6568 2.3461 1.32762.0 2.0 3.6935 2.3278 1 30943 .0 2.0 3 .7350 2 .3069 1.28194 .0 2.0 3 .7830 2.2829 1.2391Exp. 3 .580 2.536 0 .774  35Table 5. Computed force constants (N/m).Metal n m 0^ 2 p ,  p .P i 3 .0  2 ,0 -3 .0 9 2 9 1 8  0 .5 7 2 0 6 1 2 57 .23967  -1 .3 7 8 5 4 5 01 depicts the computed phonon dispersion curves alongwuh the measured data o f  Dutton e t  a l  (32] for fee Pt.4* C onclusionsThe successful prediction o f  cohesive energy [eq. (3)] o f the Pi for any positive value o f  n effectively points to the eftieaiv o f the present potential (EGEP). Although the dissociation energy parameter (D) o f  the present potential appears as the consequence o f  the cohesive energy but the model parametas depend sparingly on the coh esive  energy because the saul parameters describe the mean average binding over a lai^e number o f neighbouring atoms.- ....... -  q/On..Figure 1. Phonon dispersion in fee platinum experimental findings of Dutton et al [32]present study . O ▲The extended form o f  the generalized exponential potential explains quantitatively the second-order elastic constants (SOEC) o f  fee platinum and their intimate relation with tin strength o f the metal further establishes the importance ol the present study. The study o f  the second-order elastic constants provides direct know ledge to the response o f  metallic ions to its environment and therefore, further reveals the nature of the resultant interactions.The computed values o f  second-order elastic constants j and C 2^ o f  fee platinum com pare reasonably well with the experimental values, but the computed value o f  override the experimental value as the contribution o f  attractive interactions in j 8 j , a j  and CZ2 predominates in their values calculated in  the long wave limit. The dominant values o f  , a ,  and based on reasonable nature o f  operative interactions which eventually enhances the value o f  determ ined by the slope of the dispersion curves occupying the proximity o f zone center. The strong attractive forces al zone center have also been exploited by the first p r in c ip le s  c a lc u la t io n s  as w e ll as by the phenomenological m odels.The computed phonon frequencies o f  fee platinum [Figuie 1 ] in the framework o f  extended generalized exponential potential agree satisfactorily with the experimental values o f  Dutton euil[32] and that too by em ploying the minimum number o f inp^ t^ data, arc encouraging. The computed values slightly exceed theCohesion, elastic constants and vibrational mechanics of fee platinum 341jjpcnmental values along (100 T) and f 110 Tj] except along the /(me boundaries, whereas the computed values fall short of the measured data along fill Tj the zone boundary. The results ,ilong the transverse branches can be further improved by explicit inclusion of appropriate three-body forces and the liable electronic contribution in a more direct manner. Anyway, ,,ur results are free from the relative standard error (16j, complex iormulation [22,23], various approximations |27,28) and these jaci enhances the reliability [33] of our model.Acknowledgmentsof us (MLV and AV) are thankful to N. K. Khurana of 1 iniversal Computer Training Point (UCTP), Palwal for providing computational facilities.References[Ij I r  Slater PIm 51 846 (1937 ; W Kuhn and N Rosolober i/ml 94 n i l  (1954)(2) J A Moriarty PHs Rev B19 609 (1979)(3) AM Cics Phys Rev. B23 271 (1981)(4) VC Sethi, S C Agrawal and A V S Wamco /  PIm Chem Solids 43 329 (1981)1^ 1 JR Chclikowsky Phys Rev. Lett 47 387 (1982)(61 T Asada and K Terakura Phys Rev. B46 13599 (1992)I7j j H Rose and H B Shore Phys. Rev. B49 11588 (1994)(8} ML Verma. R P S Rathore and A Verma Czech J. Phys 45 79 (1995), M L Verma, R P S Rathore and A Verma Indian J. Phys 70A 603 (1993)[9) J Singh, N Singh and S Piakash Phys. Rev. Bl« 2954 (1978)[!01 H n Tnpaathi and S Nand Phys U tt A20 341 (1979)'ll) VP Singh and M P Hemkar Phys. U tt A54 20 (1974)(121 L A Bertolo and M M Shiikla J Phvs Sor Jpn. 38 1438 (1975)m i ML Kharoo, 0  P Gupta and M P Hemkar Z. Naturforseh A32 1490 (1977)[14] B L Fielek I  PIm. F5 17 (1975) , ibid F8 577 (1978)(15] VP Singh, L P Patliak and M P Hemkar J Phys F« 2493 (1978)116] R M Agrawal and R P S Rathore Acta Crywt. A36 870 (1980)|17J 0  P Kulshreslha and J C Upadhyaya J PIm CItem Solids 38 213 (1977)[IS] A 0  E Animaiu PIm Rev B8 3542 (1973)[19] L J Sham Pme Roy Soc A283 33 (1965)|20] H Eschrig and M Wonn Pfm Stat Sol 40 163 (1970)[21] Jogindcr Singh, Natthi Singh and ,Satya Prakash PIm. Rev. B18 2954 (1978)122] 1) Prakash and J C Upadhyaya Pfm Stat Sol (h) 148 K109 (1988)123] D Prakash and J ( ' Upadhyaya y Pirn Chem Solids 49 91 (1988)(24) C M Bcriom, V B Bcriulam, C' ('alandia and E' Ni/./uli 7 PIm F4 19 (1974)(251 m l  Vmna and R P S Ralhoic PIm. Stat Sol(h) 185 93 (1994), A Verma, M L Verma and R P S Rathore Ada Pfm\ Polon A90 547 (1996)[26] V L Morru/i, A R Williams and J P Janak PIm Rev B15 2854 (1977) ; 0  K Anderson Phys Rev B12 3060 (1975) , I) G Pettilor J Phys F7 219 (1978)[27] J M Wills and W A Hariison Phys. Rev. B28 4303 (1983)[28] N Singh. N S Hanger and S P Singh PIm Rev B38 7415 (1988)[29] ML Verma, A Verma and R P S Rathore Indian J Phys 74A 67 ( 2000)[30] M 1. Verma, A Verma and R P S Ralhoie Indian J. Pfm 76A 165( 2002)[31] F Milsicin 7 Appl. Phy.s. 44 3825 (1973)[321 H H Dutton, B N Brockhouse and A P Miller Can J PIm 50 2915 (1972)[33] R S Leigh, U SzigcU and V K 'I’iwari Proe Roy. Sov (London) 320A 505 (1971)[34] r  Kind Introduction to Solid State Physics 5lh edn (New York • Wiley) (1976)[35] R E MacFarland, J A Rayne and C K Jones Phys Lett. 18 91 (1965)

Cohesion, Elastic Constants and Vibrational Mechanics of Fcc Platinum

Cohesion, elastic constants and vibrational mechanics of fcc platinumDivesh VermaA. F. School of Engineering and Technology, Dhauj-121 004, Haryana, IndiaM L Verma* and A VermaDepartment of Physics, GGDSD College, Palwal-121 102, Haryana, IndiaandR P S RathoreDepartment of Physics, B.M.A.S. Engineering College, Agra-282 002, Uttar Pradesh, IndiaReceived 30 December 2002, accepted 19 June 2003Abstract : A model in real space has been developed by extending the generalized form of the exponential potential known as extended generalized exponential potential (EGEP) to account for (a) the correct nature of repulsive and attractive components of forces for all the separations in general and that of small separations in particular, (b) the three-body forces such as volume forces in an indirect way in the framework of EGEP through the parameter n, (c) the dielectric screening functions in an alternative and simpler form through the parameter m. The model is employed to compute the cohesive energy, second-order elastic constants and phenon spectra for fcc platinum. The predictions show promising agreement with experimental findings.Keywords : Cohesion, elastic constants, phonon dispersionPACS Nos. : 63.20.Dj, 62.20.DcIndian J. Phys. 78A (3), 337-341  (2004)1. IntroductionPlatinum (Z = 78) is a silvery white transition metal of VIII group in VI period with an outer electronic configuration of 5d9 6s and shows high catalytic activity. Platinum is highly malleable and ductile element with high melting point, high boiling point and high density. Platinum being a transition element shows a variable valency of two and four.Needless to say, the phonon spectra play a pivotal role in determining the mechanical, electrical and thermodynamical properties of elements and their alloys. We have therefore, been motivated to study the cohesion, elastic and vibrational behaviour of fcc platinum with a renewed interest because of its above stated attractive properties.The cohesion in metals has earlier been studied by many workers [1] following different approaches and using widely different approximations. Moriarty [2] has employed several approximations to compute the cohesive energy of twenty two metals using a simplified local-density theory. Cles [3] © 2004 IACS* Corresponding Authorhas analysed the role of correlation effects in the cohesion of transition metals. Sethi et al [4] have employed their theory to compute cohesive energy of some cubic metals. Cheliskowsky [5] has determined the cohesive energy of twenty simple metals from atomic kinetic energies. The generalized gradient approximation has been employed by Asada and Terakura [6] to compute the cohesive and magnetic properties of bcc, fcc and hcp iron. The uniform electron gas model for transition metals has recently been employed by Rose and Shore [7] to calculate the cohesive energy of 3d, 4d and 5d series of transition metals. In the light of widely different approaches adopted to explain cohesion in metals, we have employed the present model (EGEP) which involves the least approximations and employs the minimum number of input parameters to predict the cohesion in metals with remarkable success [8].The importance of the study of elastic constants lies in the fact that they give information about the nautre of the binding forces in solids, account for their thermodynamic behaviour and leads to the determination of the interatomic force constants of the metals. An exercise to compute the elastic constants has been undertaken from this point of view.Divesh Verma 6.p65338 Divesh Verma, M L Verma, A Verma and R P S RathoreEarlier, the dynamical behaviour of platinum has been studied by many workers using different approaches. These studies involve first principle [9,10] as well as phenomenological [11-13] calculations. The former calculations suffer from physical intractability, mathematical tediousness, conceptual obscurity and resort to various approximations to arrive at useful conclusions whereas the later suffers from various shortcomings such as lattice instability, combination of short range ion-ion interactions with long range electron-ion interactions in an abrupt manner and use of large number of force constants, which varies from metal to metal and from model to model. Fielek [14] model has been employed by Singh et al [15] for lattice dynamical study of platinum. Agrawal and Rathore [16] have employed a non-central Fielek model to study the lattice vibrations in platinum, considering the equilibrium of the lattice under the combined effect of the volume- dependent energy of ions, the d-shell electrons and the conduction electrons. The study [16] makes use of elastic constants and zone boundary frequency, thereby introducing relative standard error. Kulshrestha and Upadhyaya [17] have computed phonon dispersion curves of platinum using transition metal model potential (TMMP) in local approximation with the model parameters of Animalu [18] and the dielectric function of Hubbard modified by Sham [19]. In the local approximation, the dispersion results obtained, differ widely from experimental data. The author [17] themselves improved the results by including non-local effects in the scheme of Eschrig and Wonn [20]. Singh et al [21] have employed their isotropic non-interacting band model to calculate dielectric screening and phonon frequencies of palladium, platinum and vanadium using number of parameters. Prakash and Upadhyaya [22] have incorporated the effect of many body forces to study the dynamical behaviour of platinum using their [23] three-body potential which helps in improving the transverse branch of dispersion curves.The outer electronic configuration of platinum 5d9 6s suggests that the electrons occupying the d-shells overlap with the immediate environment leading to s–d hybridization, which causes non-sphericity in charge distribution and hence calls for unpaired or three-body forces as pointed out by Bertoni et al [24] and recently by Verma et al [25]. Although several theories [26] have been put forward for providing much insight into the relative role played by the s- and d-like electrons in the bonding of the transition, but many of the details of the complex band structure are inessential to the understanding of the cohesive properties. Will and Harrison [27] have given a qualitative first principle analysis of the elastic and bonding properties of the transition metals, by computing their total energy as a function of volume and ionic configuration at constant volume, just by extending the nearly free electron theory of metals to include the effect of transition metal d-bands. A real space analysis of elastic and dynamical properties of transition metals have recently been investigated by Singh et al[28] by means of their temperature-dependent, exponentially-damped two-body interaction potential which combines rational dielectric function (RDF) and Heine-Abarenkov (HA) model potential. The computations [27,28] being tedious and complex, require enormous time, money and efforts.The present communication derives an empirical macroscopic potential in real space, which is an extension of the generalized exponential potential [29], known as extended generalized exponential potential (EGEP) and explains almsot all the characteristic features of the interatomic interactions as detailed recently [30]. The present paper aims to investigate cohesion, the elastic and dynamical behaviour of fcc platinum.2. Theory2.1  Extended generalized exponential potential :The extended generalized exponential potential (EGEP) representing the true and realistic nature of the repulsive as well as the attractive components of the interactions assumes the form   (1)where m and n are the parameters which take care of electronic exchange and correlation effects and three-body forces such as volume forces in an alternative and simpler form respectively, D is the dissociation energy,  the hardness parameter and r0 is the equilibrium separation parameter and rj is the distance of the j-th atom from the origin given by  (2)where l1, l2, l3 (either even or odd) are the integers of the position co-ordinates such that Eq. (1) can be put in the form to represent the cohesive energy at equilibrium semi-lattice constant (a0) as under   (3)where  (4)The three defining parameters ( , r0 and D) of the potential require for their evaluation, the precisely determined input data of equilibrium semilattice constant (a0) and bulk modulus (B) of the metal only. For evaluating the three parameters , r0 and D of the potential function, the condition [31]Divesh Verma 6.p65 Cohesion, elastic constants and vibrational mechanics of fcc platinum 339  = 0 (5)for the equilibrium of the crystal in the absence of external forces is employed which gives  (6)where    The bulk modulus can be expressed as  (7)The parameter D can be evaluated through the expression for the bulk modulus following the condition given by eq. (5) for stress-free lattice. The following expression for D is obtained :  (8)where    ,   2.2   The second-order elastic constants :The following expressions for the second-order elastic constants (SOEC) with present interatomic interactions are used [31] :  (9)  (10)where  is the number of atoms per unit cell (4 for fcc and 2 for bcc) and V represents the atomic volume.The value of SOEC C44 for the metal under study has been computed by expanding the secular equation is the long wave limits  and the comparing with the usual Christoffel relation.2.3  Lattice dynamical behaviour :The elements of the dynamical matrix having explicit bearing on eq.(1) in case of fcc platinum, are :  (11)  (12)where  (13)  (14)  (15) is the  - component of phonon wave vector q, a is the lattice parameter and  are the force constants for the first neighbour (N) and  are these for the second nearest neighbour (NN) respectively.The phonon frequencies ( ) are obtained by solving the usual secular equation i.e.  (16)where I is the unit matrix of 3 × 3 order and M is the mass of the atom.3. Computations and resultsThe input data for the fcc platinum (Pt) i.e. the lattice constant and bulk modulus are given in Table 1, while the computed potential parameters are recorded in Table 2. The present study considers the 248 atoms extending to 12th neighbours in case of fcc platinum to compute potential parameters. The computed values of cohesive energy and second-order elastic constants are shown in Tables 3 and 4 respectively. Table 5 enlists the evaluated derivatives  and  for fcc platinum. Figure Divesh Verma 6.p65340 Divesh Verma, M L Verma, A Verma and R P S Rathoreparameters describe the mean average binding over a large number of neighbouring atoms.Figure 1. Phonon dispersion in fcc platinum – present study ; , O,  experimental findings of Dutton et al [32].The extended form of the generalized exponential potential explains quantitatively the second-order elastic constants (SOEC) of fcc platinum and their intimate relation with the strength of the metal further establishes the importance of the present study. The study of the second-order elastic constants provides direct knowledge to the response of metallic ions to its environment and therefore, further reveals the nature of the resultant interactions.The computed values of second-order elastic constants C11 and C22 of fcc platinum compare reasonably well with the experimental values, but the computed value of C44 override the experimental value as the contribution of attractive interactions in   and  predominates in their values calculated in the long wave limit. The dominant values of   and  based on reasonable nature of operative interactions which eventually enhances the value of C44 determined by the slope of the dispersion curves occupying the proximity of zone center. The strong attractive forces at zone center have also been exploited by the first principles calculations as well as by the phenomenological models.The computed phonon frequencies of fcc platinum [Figure 1] in the framework of extended generalized exponential potential agree satisfactorily with the experimental values of Dutton et al [32] and that too by employing the minimum number of input data, are encouraging. The computed values slightly exceed the experimental values along [100 T] and [110 T2] except along the zone boundaries, whereas the computed values fall short of the measured data along [111 T] the zone boundary. The results along the transverse branches can be further improved by explicit inclusion of appropriate three-body forces and the suitable electronic contribution in a more direct manner. Anyway, our results are free from the relative standard error [16], complex formulation [22,23], various approximations [27,28] and these fact enhances the reliability [33] of our model.Table 1. Input data for fcc platinum [Ref. 34] Metal Lattice constant Bulk modulus   10–10 m 1011 N/m2 Pt 3.92 2.783Table 2. Computed potential parameters of platinum. n m  a0 ×1010m–1  D×10–21J r0×1010m 0.5 2.0 3.10466 1.584010 415.6055 11.160770 3.806628 1.0 2.0 3.13644 1.600225 1794.5240 1.092589 4.682153 2.0 2.0 3.23935 1.652730 40802.6000 0.009026 6.423617 3.0 2.0 3.35498 1.711751 1172507.0000 5.68E-05 8.164080 4.0 2.0 3.42325 1.746556 3.65E+07 2.92E-07 9.976552Table 3. Computed values of cohesive energy of Pt [eq (3)] in eV/atom. n m Cohesive energy Magnitudes of     cohesive energy   repulsive attractive comp. exp.    part part  [34] 0.5 2.0 5.3708 11.2108 5.8400 1.0 2.0 4.2207 10.0606 5.8399 2.0 2.0 2.8345 8.6753 5.8399 5.84 3.0 2.0 2.0843 7.9243 5.8400 4.0 2.0 1.6645 7.5043 5.8398Table 4. Computed second-order elastic constants (in 1011 N/m2). Metal n m C11 C12 C44 Ref. Pt 0.5 2.0 3.6403 2.3543 1.3353  1.0 2.0 3.6568 2.3461 1.3276  2.0 2.0 3.6935 2.3278 1.3094  3.0 2.0 3.7350 2.3069 1.2819  4.0 2.0 3.7830 2.2829 1.2391 Exp.   3.580 2.536 0.774 35Table 5. Computed force constants (N/m). Metal n m 1 2 1 2 Pt 3.0 2.0 –3.092918 0.5720612 57.23967 –1.37854501 depicts the computed phonon dispersion curves alongwith the measured data of Dutton et al [32] for fcc Pt.4. ConclusionsThe successful prediction of cohesive energy [eq. (3)] of the fcc Pt for any positive value of n effectively points to the efficacy of the present potential (EGEP). Although the dissociation energy parameter (D) of the present potential appears as the consequence of the cohesive energy but the model parameters depend sparingly on the cohesive energy because the said PHONON FREQUENCY v (THz) →654321[100] [110] [111]0  .2  . 4  . 6  . 8  1  . 8  . 6  4  . 2  0  . 2  4q/qmax q/qmaxLTLLT 1T 2TDivesh Verma 6.p65 Cohesion, elastic constants and vibrational mechanics of fcc platinum 341AcknowledgmentsTwo of us (MLV and AV) are thankful to N. K. Khurana of Universal Computer Training Point (UCTP), Palwal for providing computational facilities.References [1] J C Slater Phys. Rev. 51 846 (1937 ; W Kohn and N Rosotober ibid 94 1111 (1954) [2] J A Moriarty Phys. Rev. B19 609 (1979) [3] A M Cles Phys. Rev. B23 271 (1981) [4] V C Sethi, S C Agrawal and A V S Warrieo J. Phys. Chem. Solids 43 329 (1981) [5] J R Chelikowsky Phys. Rev. Lett. 47 387 (1982) [6] T Asada and K Terakura Phys. Rev. B46 13599 (1992) [7] J H Rose and H B Shore Phys. Rev. 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Cohesion, elastic constants and vibrational mechanics of fcc platinum

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Cohesion, Elastic Constants and Vibrational Mechanics of Fcc Platinum

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