Thermodynamic Properties and XAFS Debye Waller Factors of Metallic Nickel

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1 Itrtiol Jourl of Modr Physics d Applictios Vol. No. 5 pp Thrmodymic Proprtis d XAFS Dby Wllr Fctors of Mtllic Nickl Nguy V Hug * Dih Quoc Vuog Dprtmt of Physics Collg of Scic VNU-Hoi Nguy Tri Thh Xu Hoi Vitm Qug Nih Eductio & Triig Dprtmt Nguy V Cu H Log Qug Nih Vitm Abstrct Thrmodymic proprtis d Dby-Wllr fctors of mtllic Nickl dscribd i trms of cumult xpsio up to th fourth ordr i X-ry bsorptio fi structur (XAFS of mtllic Nickl hv b studid bsd o hrmoic corrltd Dby modl. Alyticl xprssios for disprsio rltio corrltd Dby frucy d tmprtur d four first XAFS cumults hv b drivd which ivolv mor iformtio of phoos tk from itgrtio ovr th phoo wv umbrs vrid i th first Brilloui zo. Drivd hrmoic ffctiv pottil icluds cotributios of ll rst ighbors of bsorbr d bckscttrig toms to tk ito ccout thr-dimsiol itrctio d Mors pottil prmtrs for dscribig sigl pir tomic itrctio. Numricl rsults r foud to b i good d rsobl grmt with xprimt d with thos of th othr thoris. Kywords Thrmodymic Proprtis XAFS Dby-Wllr Fctor Cumult Expsio Rcivd: April 9 5 Accptd: April 5 5 Publishd oli: My 5 5 Th Authors. Publishd by Amric Istitut of Scic. This Op Accss rticl is udr th CC BY-NC lics. Itroductio XAFS (X-ry bsorptio fi structur hs dvlopd ito powrful prob of tomic structur d thrml ffcts of substcs [-]. Accordig to cumult xpsio [] th hrmoic XAFS fuctio icludig hrmoic cotributios of tomic vibrtio is giv by [] SN R λ ( k Im ik k F k i k χ Φ xp ikr ( T kr (! whr k d λ r th wv umbr d m fr pth of mittd photolctro rspctivly F(k is th rl tomic bckscttrig mplitud Φ ( k is t phs shift N is tomic umbr of shll S is th itrisic loss fctor du to mylctro ffcts R r with r s th isttous bod lgth btw bsorbr d bckscttrr toms r is its uilibrium vlu d ( ( r th cumults. Hc XAFS Dby-Wllr fctor is dscribd i trms of cumults whr th first cumult is th t-thrml xpsio th scod o dscribs th m sur rltiv displcmt (MSRD th third cumult chrctrizs th phs shift d th fourth o cotributs to XAFS mplitud. Th th ccurt cumults r crucil to utittiv trtmt of hrmoic XAFS spctr so tht th lck of th prcis cumults hs b o of th biggst limittios to ccurt structurl dtrmitios (.g. th coorditio umbrs d th tomic distcs from XAFS xprimt. Thrfor ivstigtio of XAFS cumults is of grt itrst. Our dvlopmt i this work is to driv mthod for th clcultio d lysis of th thrmodymic utitis d XAFS Dby-Wllr fctors dscribd i trms of cumult xpsio up to th fourth ordr of mtllic Nickl (Ni hvig fcc structur bsd o th hrmoic corrltd * Corrspodig uthor E-mil ddrss: hugv@vu.du.v (N. V. Hug

2 Itrtiol Jourl of Modr Physics d Applictios Vol. No. 5 pp. -8 Dby modl (ACDM. I Sct. th lyticl xprssios for disprsio rltio corrltd Dby frucy d tmprtur first scod third d fourth cumults of mtllic Ni hv b drivd. Th obtid cumults coti mor iformtio tk from itgrtio ovr th phoo wv umbrs vrid i th first Brilloui zo (B. Bsd o th succss of th hrmoic ffctiv pottil usd i th hrmoic corrltd Eisti modl (ACEM [-6] i XAFS thory du to tkig ito ccout thrdimsiol itrctio i simpl msurs by icludig cotributios of ll rst ighbors of bsorbr d bckscttrig toms i this work w dvlop furthr such hrmoic ffctiv pottil for highr xpsio up to th fourth ordr. Numricl rsults for Ni (Sct. r comprd to xprimt [7] d to thos clcultd usig th othr mthods [8] which show good d rsobl grmt.. Formlism.. High-Ordr Ahrmoic Effctiv Pottil To dtrmi Dby-Wllr fctors dscribd i trms of cumults it is cssry to spcify th itrtomic pottil d forc costt [-]. Lt us cosidr high-ordr xpdd hrmoic itrtomic ffctiv pottil xpdd up to th fourth ordr Vff x kff x kff x kff x x r r ( whr kff is ffctiv locl forc costt k ff d k ff r ffctiv hrmoic prmtrs givig symmtry of th hrmoic ffctiv pottil x is dvitio of th isttous bod lgth btw two immdit ighborig toms r from its uilibrium vlu r. Th ffctiv pottil E. ( is dfid bsd o ssumptio i th ctr-of-mss frm of sigl bod pir of bsorbr d bckscttrr []. For motomic crystls th msss of bsorbr d bckscttrr r th sm so tht it is giv by R R ( ˆ ˆ Vff x V x V x ij i b j b x x x V ( x V 8V 8V whr th first trm o th right cocrs oly bsorbr d bckscttrr toms th rmiig sums xtd ovr th rmiig ighbors d th scod ulity is for fcc structur of Ni. Hc this ffctiv pir pottil dscribs ot oly pir itrctio of bsorbr d bckscttrr toms thmslvs but lso ffct of thir r ighbors o such itrctio. It is th diffrc of our ffctiv pottil from th sigl-pir (SP [9] d sigl-bod (SB [] pottils which cocr oly ch pir of immdit ighborig toms.g. oly V(x without th rmiig trms o th right of Es. (. A Mors pottil is ssumd to dscrib sigl-pir tomic itrctio cotid i th ffctiv pottil E. ( d xpdd up to th fourth ordr roud its miimum 7 V x D D α x α x α x αx αx ( whr α dscribs th width of th pottil d D is dissocitio rgy. Applyig this Mors pottil E. ( to E. ( d comprig th rsults to E. ( w dtrmi th cofficits kff kff k ff of th hrmoic ffctiv pottil i trms of Mors pottil prmtrs... Disprsio Rltio Corrltd Dby Frucy d Tmprtur I this work w grliz th pir modl of E. ( to tht of lir chi with th sm ffctiv pottil i ordr to ccout for th ffcts of disprsio. Th bsd o [] for th cs of vibrtio btw bsorbr d bckscttrig toms d usig itrtomic ffctiv pottil for fcc structur of Ni th disprsio rltio is xprssd s 5D α si (5 M whr is phoo wv umbr M is mss of composit toms d is lttic costt. At th bouds of th first B of th lir chi ± th frucy hs mximum so tht w gt th corrltd Dby frucy D d tmprtur θ D i th followig form whr k B is Boltzm costt. 5D D D α θd ħ (6 M kb.. Four First XAFS Cumults W dscrib th prmtr x i trms of th displcmt of th tom u of th o dimsiol chi s x u u (7 whr th displcmt u' s r rltd to phoo displcmt oprtors A [] by

3 Nguy V Hug d Dih Quoc Vuog: Thrmodymic Proprtis d XAFS Dby Wllr Fctors of Mtllic Nickl i u ħ A NM A A A A'. (8 Applyig u from E. (8 to E. (7 th prmtr x is giv by ( i x f A f NM i ħ. (9 I ordr to iclud hrmoic ffcts Hmiltoi of th systm is writt i th summtio of hrmoic d hrmoic compots H d H rspctivly c ff ( H k x V A A A ( or i th followig form usig E. ( for th displcmt of th tom Hc kff ( u u i( k ff f ( f ( f ( A A A Comprig E. ( to E. ( d idictig. ( H H H H Hc H ( whr H cosists of cubic H c d urtic H trm. If hrmoic cotributio to th hrmoic itrtomic ffctiv pottil cosists of th cubic trm th it c b xprssd s V i N i N with N s th tomic umbr w obti ( (. ( V kff f f f Usig E. (9 d E. ( this E. ( is chgd ito i i i ħ i( ( ( ( k ff NM ( ( (. (5 Th first cumult or t thrml xpsio hs b clcultd with th id of th my-body prturbtio pproch [] usig th xprssio ( with tkig bckscttrig oly from th first shll. Substitutig ito E. (6 th rltios [] x f AS( β S( β β β S( β dτ dτt [ H( τ H( τ ]! th th H ( t H (6 (7 w obti AS ( β dτ T A ˆ H τ A (8 ( τ ˆ ˆ ˆ ˆ τ. τ τ (9 x f V d T A A A A β Usig Wick thorm for T-product i th itgrl th hrmoic phoo Gr fuctio [] G ' ( τ T A ˆ ( τ A ˆ ' ( ħ τ ħ τ { } G ' τ δ ' ( th symmtric proprtis of V( [] proprtis of fuctio δ ' th phoo dsity ( βħ β B ( xp k T

4 Itrtiol Jourl of Modr Physics d Applictios Vol. No. 5 pp s wll s from E. (5 f( from E. (9 ( from E. ( d phoo momtum cosrvtio i th first B w chg furthr E. (9 to th o i trms of Mors prmtrs by ħ z x d Dα z. (6 x ħα α si N 5 MD. ( Th third cumult hs b clcultd usig th followig xprssio For lrg N th summtio ovr c b rplcd by th corrspodig itgrl so tht th first cumult hs rsultd s ħ x d. ( Dα Now w clcult th scod cumult or MSRD form i th x x x. ( Usig E. (5 for E. (9 for x d f( E. ( for d ( E. ( for ( t d E. (7 for w clcult trms of Mors prmtrs ħ x ( x x Nα 5MD G ' x to obti th scod cumult i si. (5 ( x x x x x x x. (7 Th clcultio of bov i.. Usig S β x x is logous to th o of x ( β f f f A A A S S ( β o. (8 from E. (7with limitig oly cubic hrmoic trmth Wick thorm for T- product d th symmtric proprtis of V( [8] w clcultd of E. (8. Th product usig x x x hs b clcultd x from E. (5 d x from E. (. Substitutig th obtid x d x x ito E. (8 with pplictio of th rltio for phoo momtum cosrvtio i th first B w obti th third cumult i trms of Mors prmtrs s For lrg N th summtio ovr c b rplcd by th corrspodig itgrl so tht th scod cumult is giv ħ ( ( ( N D α ( ( ( ( ( ( ( ( βħ[ ( ( ] βħ ( 6 βħ βħ ( βħ (. (9 For lrg N th summtio ovr c b rplcd by th corrspodig itgrl so tht th third cumult is giv by ( 5ħ d df( N D α d ( F ( ( ( ( ( ( ( ( ( ( ( ( βħ[ ( ( ] βħ( 6 βħ( βħ( βħ (. ( For clcultio of th fourth cumult th Hmiltoi of th systm icluds hrmoic cotributios up to th fourth ordr so tht similr to E. ( w obti ( '( ( H kff x kff x V A A A V A A A A

5 6 Nguy V Hug d Dih Quoc Vuog: Thrmodymic Proprtis d XAFS Dby Wllr Fctors of Mtllic Nickl ' NM k V i i i i i ff ħ. ( Th fourth cumult hs b clcultd bsd o th followig xprssio ( 6 x x x x x x x x x. ( Th clcultio of x is logous to th o of x bov d usig x from E. (5 w obti x. Substitutig ths vlus ito E. ( d usig th phoo momtum cosrvtio i th first B th fourth cumult E. ( i trms of Mors prmtrs hs rsultd s D N α ħ (5 For lrg N th summtio ovr c b rplcd by th corrspodig itgrl so tht th fourth cumult E. (5 is giv by. ( d d d D α ħ (6. Numricl Rsults d Discussios Now w pply th xprssios drivd i th prvious sctios to umricl clcultios for mtllic Ni hvig fcc structur. W us th Mors prmtrs D. V α.99 Å - [5] which wr obtid usig xprimtl vlus for th rgy of sublimtio th comprssibility d th lttic costt s wll s its msurd Mors prmtrs (MMP D. V α.9 Å - [7]. Thy hv b usd for clcultig th ffctiv locl forc costt k ff corrltd Dby frucy D d tmprtur D θ of Ni. Th rsults r writt i Tbl comprd to th xprimtl vlus [7] which show rsobl grmt. Fig. illustrts good grmt of th hrmoic ffctiv pottil (Fig. d disprsio rltio (Fig. b of Ni clcultd usig th prst thory comprd to thos obtid from its MMP [7]. Th hrmoic ffctiv pottil (Fig. is shiftd from th hrmoic trm to b symmtric du to icludig th hrmoic ffcts d th mximl frucis ( (Fig. b t ± r ul to th Dby frucis of Ni writt i Tb..

6 Itrtiol Jourl of Modr Physics d Applictios Vol. No. 5 pp Fig.. Ahrmoic ffctiv itrtomic pottil d b disprsio rltio of Ni clcultd usig th prst thory comprd to xprimt (Expt. obtid from its MMP [7]. Fig.. Tmprtur dpdc of third cumult ( (T d b fourth cumult ( (T of Ni clcultd usig th prst thory comprd to th xprimtl vlus [7]. Fig.. Tmprtur dpdc of itrtomic distc R(T d b scod cumult (T of Ni clcultd usig th prst thory comprd to th xprimtl vlus [7] d to thos clcultd usig ACEM [] d PIEP [8] for (T. Tbl. Th vlus of kff D θ D for Ni clcultd usig th prst thory comprd to th xprimtl vlus [7]. Qutitis ( kff N m D ( Hz θ Ni Prst Ni Expt. [7] D K Fig. shows good grmt of th itrtomic distc R(T R( ( (T (Fig. obtid from th first cumult ( (T or t thrml xpsio d of th scod cumult (T or MSRD (Fig. b of Ni clcultd usig th prst thory with th xprimtl vlus [7] d with thos clcultd usig th ACEM [] d th pth-itgrl ffctiv pottil (PIEP mthod [8]. Furthr rsobl grmt with th xprimtl vlus [7] t diffrt tmprturs hs b prstd for th third cumult ( (T (Fig. d for th fourth cumult ( (T (Fig. b of Ni clcultd usig th prst thory. Morovr ll th umricl rsults show tht t high-tmprtur th first d scod cumults r lir to th tmprtur T th third d fourth cumults vry s T d T rspctivly s it ws obtid i th othr

7 8 Nguy V Hug d Dih Quoc Vuog: Thrmodymic Proprtis d XAFS Dby Wllr Fctors of Mtllic Nickl thoris [] d t low-tmprtur thy coti zro-poit rgy cotributios utum ffct.. Coclusios I this work mthod hs b drivd for th clcultio d lysis of th thrmodymic prmtrs d highordr xpdd XAFS Dby-Wllr fctors dscribd i trms of cumults of mtllic Ni bsd o th hrmoic corrltd Dby modl with cosidrig disprsio rltio to tk mor iformtio of phoos tk from itgrtio ovr th phoo wv umbrs vrid i th first B d th hrmoic ffctiv pottil ivolvig th cotributios of ll rst ighbors of bsorbr d bckscttrig toms to tk ito ccout thr-dimsiol itrctio. Mors pottil is usd for dscribig th sigl-pir itrtomic itrctio. Morovr sic Ni hs fcc structur th prst drivd thory c lso b grlizd to study othr mtls hvig th sm this crystl structur. Drivd lyticl xprssios for th hrmoic itrtomic ffctiv pottil disprsio rltio corrltd Dby frucy d tmprtur s wll s four first XAFS cumults stisfy ll thir fudmtl proprtis showig tht t high-tmprtur th first d scod cumults r lir to th tmprtur T th third d fourth cumults vry s T d T rspctivly d t low-tmprtur thy coti zro-poit rgy cotributios utum ffct. Th good d rsobl grmt of umricl rsults for mtllic Ni with xprimt d with thos clcultd usig th othr thoris illustrts th dvtg d fficicy of th prst thory d of usig th hrmoic ffctiv pottil i XAFS dt lysis. Ackowldgmts Th uthors thk Prof. J. J. Rhr d Prof. P. Forsii for usful commts. Rfrcs [] Crozir E. D. Rhr J. J d Iglls R.X-ry Absorptio. ditd by D. C. Koigsbrgr dr. Pris (Wily Nw York 988. [] Hug N. V. Duc N. B. Frhm R. R. A Nw Ahrmoic Fctor d EXAFS icludig Ahrmoic Cotributios. J. Phys. Soc. Jp. ; 7: 5. [] Hug N. V. d Rhr J. J. Ahrmoic corrltd Eisti modl Dby-Wllr fctors. Phys. Rv. B 99756:. [] Dil M. Ps D. M. Hug N. V. Budick J. D. Locl forc costts of trsitio mtl dopts i ickl host: Compriso to Mossbur studis. Phys. Rv. B ; 68:. [5] Hug N. V. L. H. Hug L. H. Ti T. S. Frhm R. R. Ahrmoic Effctiv Pottil Locl Forc Costt d EXAFS of HCP Crystls: Thory d Compriso to Exprimt. It. J. Mod. Phys. B 8; : 555. [6] Hug N. V. Prssur-Dpdt Ahrmoic Corrltd XAFS Dby-Wllr Fctors. J. Phys. Soc. Jp. ; 8: 8. [7] Pirog I. V. Ndosiki T. I. rubi A. I. Shuvv A.T. Ahrmoic pir pottil study i fc-ctrd cubic crystls. J. Phys.: Cods. Mttr ; : 85. [8] Yokoym T. Pth-itgrl ffctiv-pottil mthod pplid to xtdd x-ry-bsorptio fi-structur cumults. Phys. Rv. B 998; 57:. [9] Miyg T. Fujikw T. Qutum Sttisticl Approch to Dby-Wllr Fctor i EXAFS EELS d ARXPS. III. Applictio of Dby d Eisti Approximtio J. Phys. Soc. Jp. 99; 6: 68. [] Frkl A. I. d Rhr J. J. Thrml xpsio d x-rybsorptio fi-structur cumults. Phys. Rv. B 99; 8: 585. [] Str A. P. Livis d h hg. Thrml vibrtio d mltig from locl prspctiv Phys. Rv. B 99; : 885. [] Kittl C. Itroductio to Solid Stt Physics. (Joh- Wily&Sos Ic. Nw York th d. [] Mrdudi A. A. Dymicl Proprtis of Solids. d. by Horto G. K. d Mrdudi A. A. (North Holld Amstrdm 97 Vol. p.. [] G. D. Mh G. D. My-Prticl Physics (Plum Nw York 99 d d. [5] Giriflco L. A. d Wizr W. G. Applictio of th Mors Pottil Fuctio to Cubic Mtls. Phys. Rv. 959; : 687.