VNU JOURNAL OF SCIENCE, Mathematics - Physics, T.XXI, N04, 2005 THERMODYNAMIC AND CORRELATION EFFECTS IN ATOMIC VIBRATION OF BCC CRYSTALS CONTAINING DOPANT ATOM N g u y e n V a n H u n g , Ho K h a c H ie u a n d N g u y e n C o n g T o a n D epartment o f Physics, College o f Science, V NƯ Hanoi A b s tra c t: A new procedure for calculation and analysis of therm odynam ic and co rrela tio n e ffects of bcc c r y sta ls un d er in flu e n c e of d op an t atom in th e X-ray Absorption Fine S tructure (XAFS) has been developed Analytical expression for the Displacem ent-displacem ent Correlation Function (DCF) CR has been derived based on the derived Mean Square Relative Displacem ent (MSRD) Ơ2 and Mean Square Displacem ent (MSD) u of bcc crystals containing dopant atom Numerical calculations have been carried out for Fe doped by w and by Cr atom They are found to be in good agreem ent with experiment I n t r o d u c t i o n T herm odynam ic effects of atomic vibration have been oft studied by th e XAFS procedure because the em itted photoelectron is tra n s fe rre d and s c a tte re d in the vibrating atomic en v iro n m en t before in terferin g with the out going photoelectron Therefore, it is n ecessary to ta k e a th erm al av eraging ị e ^ ki an d c , / a c* - 5& - Ỉ Ế z i-'Ỉ Ế £ ệ - = Ị + u ° ! ỉ i | L i z.v X1- ^ ) °~0 «3 u ( l - z)(i + z J ’ ■ «5 (l-z.sX l + r J 0+jo (26) ) - z ,4 (27) I N g u y e n Van Hung, Ho K h a c H ie u , N g u ye n Cong T o a n 30 o ) I CT2 < J q - z A l + Z l (28) - Z.vX1 + ) If th e d o p a n t atom is tak e n from the host crystal, i., e., Dh d = d h = d > a HD=a H = a > (29) th e above o b tain ed resu lts will change into those for the pure bcc crystals [ 12 ] Dữ~\ kĨA =k 3S = &3 - -2 Da , ^ A - k s ~ kejj - 2 ho) 1+ z 3hco 1+ z u ế =u%- u = — = ; A s " 1'keff\ - z 32D a l - z (30) /Q1\ \Oi) N u m e r ic a l r e s u lts and co m p a r iso n to e x p e r im e n t Now we applv th e expressions derived in the previous section to num erical calculations for Fe doped by w and by Cr The Morse po ten tials for Fe doped by w an d by C r h a v e b e e n c a lc u la te d u s in g th e M orse p o te n tia l p a r a m e te rs D a n d a of these c ry sta ls com puted by using our procedure p resen ted in [8 ] They are shown in Figure in a good a g re e m e n t with experim ent [7] The te m p e tu re dependent values of u] , uị , Ơ2 , CR h ave been calculated a n d the re su lts a re p r e s e n te d in Table F igu re illu s tra te s the te m p e tu re dependence of our u] , uị , ơ)ak , c ỵ ’c for Fe doped by w compared to T a b l e 1: C a lc u la ted v alues of e x p e rim e n ta l values of cexp (7 exp, R [7] a t different tem p era tu re s T(K) ^ ( 2Ẳ2) uị ( 10 -2Ẳ2) °ỉ„,c (lO^Â2) < , ( ZẲ2) CcRak (10 2Ẳ2) r " p( i o 2Ẳ2) 30 0.09 0.19 0.26 0.25 0.02 0.02 50 0.10 0.19 0.26 0.26 0.02 0.02 70 0.11 0.20 0.28 0.28 0.03 100 0.14 0.22 0.32 0.32 0.03 0.04 150 0.19 0.28 0.40 0.06 0.06 200 0.24 0.50 250 0.30 0.35 0.42 0.41 0.51 0.62 0.61 0.08 0.10 0.08 0.10 300 350 0.36 0.41 0.50 0.57 0.73 0.84 0.72 0.83 0.12 0.12 0.14 0.14 400 4-50 0.47 0.65 0.96 0.94 0.17 0.17 53 1.07 1.05 0.19 0.19 500 0.59 0.64 0.73 0.81 1.12 1.08 0.21 0.21 0.88 1.30 1.27 0.23 0.96 1.41 1.38 0.23 0.25 550 600 0.70 0.04 0.25 T h e r m o d y n a m i c a n d C o r r e l a t io n Effects in A t o m i c V ib tio n o f B C C calculated M S R D ^ r ) , MSD u ị(r) and „ị(r) showing o > > u ị > u \ especially a t high tem p eratu re s The t e m p e r a t u r e dependence of our calculated D C F c, (t ) for Fe doped by w a nd by Cr is presented in Figure The functions MSRD, MSD a nd U C i contain zero-point contribution at low t e m p e r a t u r e a n d a re linearly proportional to the t e m p e r a t u r e a t high tem peratu res The t e m p e r a t u r e dependence of the ratios c„ / u] a nd CH/ for Fe doped by w are shown in Fig ure They have f a T u f0rT !° r t h e p u r e bcc c r y sta ls t 12J s a ti s f y i n g t h e s a m e p r o p e r t i e s obtained by the Debye model [2 ], They increase fastly a t low t e m p e r a t u r e s a nd approach a constan t values a t high te m p e r a tu re s (about 36% for c,f I u\ a n d 18% for c / Ơ ) tak in g from our calculated correlated E instein t e m p e r a t u r e 9; =2\1 K If These results denote th e significant rate of th e correlation effect in atomic vibration Our calculated qu an titie s shown in Table a nd in Figures - are found to be in good a gre e m e n t with the experiment [7] Figure 1: Calculated Morse potentials for doped by w a nd Cr compared to Figure 2: Calculated MSRD a nd MSD for Fe doped by w compared to experiment [7], Ĩ nalCULa te d DCF f0r Fe d0ped by Figure 4: Calculated CR l u \ a n d c „ / a w and by Cr compared to experiment [71 F T ? J , , , lor Pe doped by w co mpared to experiment [7], N g u y e n Van Hung, Ho K h a c Hieu, N g u y e n C o n g Toan 32 C o n c lu s io n s In t h is work a new procedure for study of the th erm o dy n am ic a n d correlation effects in th e atomic vibration of bcc crystals u n d e r influence of a d o p a n t a to m in XAFS h a s been developed Analytical expressions for the effective local force co n stan ts, t h e D CF ( C R), th e MSD for th e d op an t as absorber ( u\ ) a n d for the sc a t t e r e r as ho st atom ( uị ) containing a d o pant atom as n e a r e s t neig hbor a n d the r a t io s Cr ỉ u a , CR /cT2 h a v e b een derived b a s e d on t h e a n h a r m o n i c c o r r e l a t e d E in s te in model O b ta in e d expressions of the mentioned th erm od y nam ic functions show their f u n d a m e n t a l p rop erties in t e m p e r a t u r e dependence The functions CR , U2A, u 2Sì Ơ2 a re lin e arly pro portional to t e m p e r a t u r e a t h i g h - te m p e r a t u r e s a n d c o nta in zeropoint c o n tribu tio ns a t low t em p era tu re s The ratio CR / u 2A accounts for ab ou t 36% a n d th e tio CR l ab ou t 18% a t h ig h -te m p eratures Ou r developed theory for the doping bcc crystals contain the one for th e pure m a t e r i a l s a s a sp ecial case w h e n t h e d o p a n t is t a k e n aw ay T h e r e s u l t Ơ >U2S >U2A shows th e role of location of the pant (as th e central or as the neighboring atom) a n d th e correlatio n effect in stu dying atomic vibration The a g r e e m e n t of our calculated values with exp erim ent shows the efficiency of th e p r e s e n t procedure in studying therm odynam ic p a r a m e t e r s a n d correlation effects of bcc c rystals u n d e r influence of pant atom A c k n o w l e d g e m e n t s This work is suppor ted in p a r t by th e basic science res ea rch project provided by Ministry of Science and Technology a n d by th e special r e s e a r c h project of VNU Hanoi No.QG.05.04 R e fe r e n c e s c A Ashley a n d s Doniach, Phys Rev B 10(1975) 3027 G Beni a n d p M P la tz m a n , Phys Rev B 14(1976) 1514 T Yokoyama, T S a tsu k a w a , a nd T Ohta, Jpn J Appl Phys 28(1989) 1905 N V H u n g a n d J J Rehr, Phys Rev B 56(1997) 43 N V Hun g, N B Due, and R R F h m , J Phys Soc Jpn 72(2003) 1254 M Daniel, D M Pease, N Van Hung, J I Budnick, Phys Rev B 69(2004) 134414 V Pirog, T I Nedoseikina, Physica B 334(2003) 123 N V Hun g, C om m un in Phys 14(2004) N V Hung, N T T Hoai, L H Hung, V N U J o u r Science, Vol 20(2004) 8-15 10 N V Hun g, N T Van, L H Hung, V N U Jour Science, Vol 20(2005) 69 11 A A M a r a d u d i n , E w Montroll, G H Weiss, a nd I p Ipatova, Theory o f Lattice D y n a m ics in the Q uadratic A pproxim ation (Academic, New York, 1971) 12 N V Hun g, N B Trung, su b m itted to J A dvances o f N a tu l Sciences ... io n s In t h is work a new procedure for study of the th erm o dy n am ic a n d correlation effects in th e atomic vibration of bcc crystals u n d e r influence of a d o p a n t a to m in XAFS... ) tak in g from our calculated correlated E instein t e m p e r a t u r e 9; =21 K If These results denote th e significant rate of th e correlation effect in atomic vibration Our calculated... shows th e role of location of the pant (as th e central or as the neighboring atom) a n d th e correlatio n effect in stu dying atomic vibration The a g r e e m e n t of our calculated values with