Calculation of Morse Potential Parameters of bcc Crystals and Application to Anharmonic Interatomic Effective Potential, Local Force Constant

Transcription

1 VNU Journal of Scinc: Mathmatics Physics, Vol. 31, No. 3 (15) 3-3 Calculation of Mors Potntial Paramtrs of bcc Crystals and Application to Anharmonic Intratomic Effctiv Potntial, Local Forc Constant Nguyn Van Hung 1, *, Trinh Thi Hu 1, Nguyn Ba Duc 1 VNU Univrsity of Scinc, 334 Nguyn Trai, Thanh Xuan, Hanoi, Vitnam Tan Trao Univrsity, Km6, Trung Mon, Yn Son, Tuyn Quang, Vitnam Rcivd 4 Fbruary 15 Rvisd 8 April 15; Accptd 15 July 15 Abstract: In this work, Mors potntial paramtrs of bcc crystals hav bn calculatd basd on th calculation of volum pr atom and atomic numbr in ach lmntary cll, as wll as th nrgy of sublimation, th comprssibility and th lattic constant. Thy ar usd for studying th anharmonic intratomic ffctiv potntial, local forc constant in XAFS (X-ray Absorption Fin Structur) thory. Numrical rsults for F, W and Mo ar found to b in good agrmnt with xprimnt and with thos of othr thoris. Kywords: Mors potntial paramtr, ffctiv potntial, local forc constant, bcc crystals. 1. Introduction Anharmonic intratomic potntials including Mors potntial paramtrs [1,], hav bn intnsivly studid [1-17]. Thy ar usd for th calculation and analysis of th thrmodynamic paramtrs, spcially, th anharmonic ffcts containd in XAFS (X-ray Absorption Fin Structur) [1-15] which influnc on th physical information takn from ths spctra. Mors potntial is an mpirical potntial [1,] and thir paramtrs ar oftn xtractd from xprimnt [16,17]. Thrfor, calculation and analysis of Mors potntial paramtrs ar of grat intrst, spcially in XAFS thory. This work is a nxt stp of our prvious work [18] for th calculation and analysis of Mors potntial paramtrs of bcc crystals basd on th calculation of volum pr atom and atomic numbr in ach lmntary cll. This calculation of atomic numbr is our furthr dvlopmnt compard to th prvious thory [18], and du to that th prsnt mthod can b gnralizd to th calculation for th othr crystal structurs. Th nrgy of sublimation, th comprssibility and th lattic constant usd in th prsnt considrations ar availabl [19-1]. Th obtaind Mors potntial paramtrs ar applid Corrsponding author. Tl.: hungnv@vnu.du.vn 3

2 4 N.V. Hung t al. / VNU Journal of Scinc: Mathmatics Physics, Vol. 31, No. 3 (15) 3-3 to th calculation and analysis of th anharmonic intratomic ffctiv potntial, local forc constants and anharmonic ffcts in XAFS thory. Numrical rsults for F, W and Mo ar compard to xprimnt [17] and to thos of othr thory [] which show good agrmnt.. Formalism.1. Calculation of Mors potntial paramtrs Following [18] th potntial nrgy ϕ ( r i ) of two atoms i and sparatd by a distanc r i is givn in trms of th Mors function by ( ) α ( r r ) ( r o ) = i o α D i r ϕ r i, (1) whr α, D ar constants with dimnsions of rciprocal distanc and nrgy, rspctivly; r o is th quilibrium distanc of th two atoms. Sinc ϕ ( r o ) = D, D is th dissociation nrgy. In ordr to obtain th potntial nrgy of th whol crystal whos atoms ar at rst, it is ncssary to sum Eq. (1) ovr th ntir crystal. This is most asily don by choosing on atom in th lattic as an origin, calculating its intraction with all th othrs in th crystal, and thn multiplying by N /, whr N is th total atomic numbr of th crystal. Thus, th total nrgy Φ is givn by Hr quantitis whr Φ = ND 1 α r ro α r ro. () r is th distanc from th origin to th th atom. It is convnint to dfin th following 1 α [ m + n + l ] 1/ a = M a ro L = ND; β = ; r =, (3) m, n, l ar position coordinats of any atom in th lattic. Applying Eq. (3) to Eq. (), th nrgy can b rwrittn as Φ α ( a) = Lβ Lβ am αam. (4) Th first and scond drivativs of th nrgy of Eq. (4) with rspct to a ar givn by dφ αam = αlβ M + Lβα da d Φ = 4α Lβ da M αam α Lβ M αam M αam, (5). (6)

3 N.V. Hung t al. / VNU Journal of Scinc: Mathmatics Physics, Vol. 31, No. 3 (15) At absolut zro T =, th nrgy of cohsion, [ Φ / ] = whr ( ) a a is valu of a for which th lattic is in quilibrium, thn ( ) o d da a, and [ d Φ / da ] a ( a ) = U ( ) a Φ givs a o is rlatd to th comprssibility. That is, Φ, (7) U is th nrgy of sublimation at zro prssur and tmpratur, i.,., and th comprssibility is givn by 1 K dφ da a = a, (8) d U d Φ V V dv dv = =, (9) whr V is th volum at T =, and K is comprssibility at zro tmpratur and prssur. Our furthr dvlopmnt compard to th prvious calculation [18] is proposing a mthod for dtrmining th volum pr atom V a for bcc crystal V a V VEC =, (1) N n a = whr V EC = a 3 is th volum of an lmntary cll of a cubic crystal including bcc, n is th atomic numbr in this lmntary cll and a is th lattic constant. Substituting Eq. (1) in Eq. (9), th comprssibility is xprssd by Using Eq. (5) to solv Eq. (8), w obtain 1 n d Φ K 9Na da a= a =. (11) αam Consquntly, from Eqs. (4,6,7,11) w obtain th rlation 4α β αam β = M / M. (1) αam nu K Na = αam αam β M M 9 α αam, (13) which is diffrnt from that in [18] by containing th atomic numbr n in an lmntary cll. Solving th systm of Eqs. (1,13) w obtain α, β. Substituting th obtaind rsults into th scond quation of Eqs. (3), w dtrmin r. Using th obtaind α, β and Eq. (4) to solv Eq. (7), w obtain L. From this L and th first quation of Eqs. (3) w obtain D. Th obtaind Mors potntial paramtrs D and α dpnd on th comprssibility K, th nrgy of sublimation U and th lattic constant a which ar known alrady for about all crystals [18-]. Hnc, all Mors paramtrs

4 6 N.V. Hung t al. / VNU Journal of Scinc: Mathmatics Physics, Vol. 31, No. 3 (15) 3-3 dpnd on th valu n sparatd for diffrnt crystal structurs, and w will calculat it in th nxt subsction for bcc crystals... Application to calculation of anharmonic intratomic ffctiv potntial and local forc constant in XAFS thory Fig. 1 shows Fourir transform magnituds of XAFS at 93 K and 393 K, as wll as XAFS of F, masurd at Novo-Simbirk (Rissia) [17]. Thy ar diffrnt at ths tmpraturs illustratd by thir shifts which show th vidnt anharmonic ffcts in XAFS. For dscribing ths ffcts an anharmonic XAFS thory is ncssary [7-15]. Fig. 1. Fourir transform magnituds of xprimntal XAFS of F at 93 K and 393K and XAFS spctrum at 393 K [17] masurd at Novo-Simbirk (Russia). Th xprssion for th K-dg anharmonic XAFS function [11] is dscribd by R / λ( k ) (ik) χ( k) = F kr n n! iϕ( k) ( n) ( k) Im xp ikr + σ, R = r n, (14) whr F (k) is th ral atomic backscattring amplitud, φ is nt phas shift, k and λ ar th wav numbr and th man fr path of th photolctron, rspctivly, r is instantanous bond lngth btwn two immdiat nighboring atoms and σ (n) (n = 1,,3, ) ar th cumulants. For dscribing this anharmonic XAFS, an anharmonic intratomic ffctiv potntial [1,1] of th systm is drivd which in th prsnt thory is xpandd up to th 4 th ordr and givn by V ff Hr ( x) kff x + k3x + k4 x + = V ( x) + V xrˆ R ˆ M 1M R 1. i, µ =, R ˆ =. i M i M 1 + M R µ kff is ffctiv local forc constant, and k 3 is th cubic paramtr giving an asymmtry in th pair distribution function, x is dviation of instantanous bond lngth btwn th two atoms from quilibrium. Th corrlatd modl may b dfind as th oscillation of a pair of atoms with masss (15)

5 N.V. Hung t al. / VNU Journal of Scinc: Mathmatics Physics, Vol. 31, No. 3 (15) M 1 and M (.g., absorbr and backscattrr) in a givn systm. Thir oscillation is influncd by thir nighbors givn by th last trm in th lft-hand sid of Eq. (15), whr th sum i is ovr absorbr ( i = 1) and backscattrr ( i = ), and th sum is ovr all thir nar nighbors, xcluding th absorbr and backscttrr thmslvs. Th lattr contributions ar dscribd by th trm V ( x). Th advantag of this modl is that th thr-dimnsional intractions can b takn into account in th prsnt on-dimnsional modl by a simpl masurs basd on including th contributions of narst nighbors of absorbr and backscattrr in XAFS procss. For bcc crystals th anharmonic intratomic ffctiv potntial Eq. (15) has th form x x x V = ff ( x) V ( x) + V + 6V + 6V 6 6. (16) Applying Mors potntial givn by Eq. (1) xpandd up to th 4 th ordr around its minimum V ( x) = D αx αx ( ) D 1+ α x α x + α x + 1, (17) containing our calculatd Mors potntial paramtrs (MPP) to Eq. (16) and comparing that to th first quation of Eqs. (15), w obtain th anharmonic ffctiv potntial V, ffctiv local forc constant k ff, anharmonic paramtrs, k 3 4 k for bcc crystals prsntd in trms of our calculatd MPP D and α. ff 3. Numrical rsults and discussion For calculating th abov quations to obtain Mors potntial paramtrs (MPP) of bcc crystals, w calculat th atomic numbr n in ach lmntary cll of bcc crystals. Fig.. Atomic distribution in an lmntary cll of bcc crystal. From Fig. it is vidnt that 1/8 atom in ach of 8 vrtxs and on atom in th cntr ar localizd in an lmntary cll of bcc crystal. Thrfor, w obtain th valu n =. Using th drivd xprssions in th prvious sction and this calculatd paramtr n, as wll as th nrgy of

6 8 N.V. Hung t al. / VNU Journal of Scinc: Mathmatics Physics, Vol. 31, No. 3 (15) 3-3 sublimation, th comprssibility and th lattic constant from [18-], w calculatd Mors potntial paramtr D, α, r o using our cratd computing programs. Thy hav bn usd for calculating th anharmonic ffctiv local forc constants of bcc crystals. Tabl 1 show good agrmnt of th rsults calculatd using th prsnt thory with thos of L. A. Girifalco t al [] and with xprimnt of I. V. Pirog at al [17]. Tabl 1. Mors potntial paramtrs (MPP) D, α, r o calculatd using th prsnt thory and ffctiv local forc constants k ff calculatd using ths MPP for F, W, Mo compard to thos of L. A. Girifalco t al [] and to th xprimntal valus of I. V. Pirog t al [17]. Crystal D(V) α (Å -1 ) r o (Å) k ff (N/m) F, Prsnt F, Girifalco t al [] F, Expt., Pirog t al [17].4±.1 1.4± W, Prsnt W, Girifalco t al [] W, Expt., Pirog t al [17].89± ± Mo, Prsnt Mo, Girifalco t al [] Mo, Expt., Pirog t al [17].75± ± Th Mors potntials calculatd using th prsnt thory prsntd in Fig. 3 for a) F and b) W ar found to b in good agrmnt with xprimnt of I. V. Pirog t al [17] and with thos calculatd by L. A. Girifalco t al []. Thy satisfy all thir fundamntal proprtis, i.., thy dscrib th rpulsiv forc in short distanc whn atoms approach ach othr obying Pauli xclusion principl, and dscrib th attractiv forc in long distanc whn atoms go far from ach othr. Th rason of this attraction is that th atoms hav diffusion momnts which attract ach othr in long distanc. a) b) Fig. 3. Mors potntials of a) F and b) W calculatd using th prsnt thory compard to thos calculatd by L. A. Girifalco t al [] and to th xprimntal valus masurd by I. V. Pirog t al [17].

7 N.V. Hung t al. / VNU Journal of Scinc: Mathmatics Physics, Vol. 31, No. 3 (15) a) b) Fig. 4. Anharmonic intratomic ffctiv potntials for a) F and b) W calculatd using th prsnt thory and MPP prsntd in Tabl 1 and in Fig. 3, rspctivly, compard to xprimnt obtaind from MMP of I. V. Pirog t al [17] and to thos calculatd from MPP of L. A. Girifalco t al []. Figur 4 illustrats good agrmnt of th anharmonic intratomic ffctiv potntials for a) F and b) W calculatd using th prsnt thory and th MPP prsntd in Tabl 1 and in Fig. 3 with xprimnt obtaind from th masurd Mors paramtrs (MPP) of I. V. Pirog t al [17] and with thos calculatd from MPP of L. A. Girifalco t al []. Thy show strong asymmtry of ths potntials du to including th anharmonic contributions in atomic vibrations of ths bcc crystals illustratd by thir anharmonic shifting from th harmonic trms. Such anharmonic ffcts of th anharmonic intratomic ffctiv potntials lad to th shifts of th paks of Fourir transform magnituds of th xprimntal XAFS spctra of F at diffrnt tmpraturs [17] prsntd in Fig Conclusions In this work, a mthod for th calculation and analysis of Mors potntial paramtrs for bcc crystals has bn dvlopd basd on th calculation of volum pr atom and atomic numbr in ach lmntary cll, as wll as th nrgy of sublimation, th comprssibility and th lattic constant. This mthod can b gnralizd to th othr crystal structurs basd on th calculation of thir volum pr atom and atomic numbr in ach lmntary cll. Th obtaind Mors potntials satisfy all thir fundamntal proprtis and ar suitabl for th calculation and analysis of th anharmonic intratomic ffctiv potntials dscribing anharmonic ffcts in tmpratur-dpndnt XAFS thory. Th good agrmnt of th calculatd Mors potntials of F, Mo, W and th anharmonic intratomic ffctiv potntials of ths lmnts calculatd using thir obtaind Mors potntial paramtrs with xprimnt illustrat th fficincy and rliability of th prsnt thory in computing th intratomic intraction potntials, as wll as th Mors potntial paramtrs which ar important for th calculation and analysis of physical ffcts in XAFS tchniqu.

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