DSpace at VNU: Study of interaction potential and force constants of FCC crystals containing N impurity atoms tài liệu,...
V N U JO U R N A L OF S C IE N C E , M a th e m a tics - P hysics, T.xx, N 02, 2004 STU D Y O F IN T E R A C T IO N P O T E N T IA L AN D F O R C E C O N S T A N T S O F F C C C R Y S T A L S C O N T A IN IN G N IM P U R IT Y A T O M S N guyen V an H ung, T ran T rư ng D ung, N guyen C ong T oan Department o f Physics, College o f Science - V N U Abstract A new procedure for description and calculation of the interaction potential and force constants for fee crystals containing an arbitrary number n of impurity atoms have been developed Analytical expressions for the effective atomic interaction potential, the single-bond and effective spring constants have been derived They depend on the number of the impurity atoms and approach those derived by using anharmonic correlated Einstein model, if all the impurity atoms are taken out or they replace all the host atoms Numerical results for Ni doped by Cu atoms show significant changes of the interact on potential and spring constants of the substance if the number of impurity atoms is changed I n t r o d u c t i o n Interaction potential and force constants are very im p o rta n t for studying a lot of physical properties such as therm odynam ic p a m e te rs of th e crystals r'hey art contained in the first c u m m u la n t or net th erm al expansion, th e second c u rru la n to i Debye-Waller factor, the third cum ulant, and the th erm al expansion e:pansior coefficient, which are investigated intensively in the X-ray absorption fine structure (XAFS) experim ent and theory [1-11] It is also very im p o rtan t to SU'1) therm odynam ic properties of m aterials containing im p urity atom s and of all)} system s [12 17-19] Some investigations for crystals c o n tain ing one impurity atom have been done 117-19] But more th a n one im purity atom can be dope:! into a crystal This case can lead to the development of procedures for studying therm odynam ic properties of alloys with nano stru c tu re which aie often semiconductors containing some components with different atom ic sortes The purpose of this work is to develop a new procedure for descrip ion and calculation of the interaction potential and force c o n stan ts of fee crystals containing some impurity atom s, where one im purity atom [1.7-19] is only an speciá case OÍ this theory Our developm ent is derivation of the analytical expression; for tieeffective atomic interaction potential, the single-bond and the effective spring constants for the case when the cluster involves one or more im pu rity atons Using the atomic distribution of the host (H) atoms and the d opan t (D) atom s in I cluster one can deduce the percentage of these c o nstituent ele m e n ts in the substaice or in an alloy All these expressions are different if the n u m b er of impurity atons changes so th a t one can deduce the results for the case w ith different perceitagesoi , 10 N g u y e n Van Hung T ran T run g Dung, Nguyen Cong Toa n component elem ents of which an alloy consists The re su lts in the case if all impurity atom s are ta k e n out or in the case if all host atom s are replaced by the im purity atom s are reduced to those derived by using the an harm onic correlated Einstein model [7] for th e pure m aterials Num erical calculations have been carried out for Ni crystal doped by one or more A1 im purity atom s, and the re s u lts are compared to those of the pure m aterials F o r m a l i s m We consider a fee crystal doped by some im purity atoms replace the host (H) atom s located in the centre th a t the XAFS process is taken place in the surface (indicated by D(]) in the centre and the H atom located by Hfị) as described in Figure la H atom s or dopants (D), th e D of crystal planes Supposed (001) between the D atom at the position B (indicated o QH o X ýT x , y /60" I\ r f H0 o o XhA o o H (001) HỒ a) H H DoO*X.) o / -o H ■o b) Figure Distribution and vibration of H and D atoms in fee volume (a) and in its (001) surface (b) Now move the D () atom by an am ount X D along the line A B , then the H B atom moves backw ard by an am o u n t X H so th a t the m ass centre rem a in s unchanged, the other atom s are fixed We have relations X dM d = X h M h => X H = - y —X D = e.XD , e = m h , (1) IV1H where M H, M are the m ass of H and D atom, respectively This motion leads to increasing the potential energy The contributions of the springs in the surface (001) are caused mainly by the atom s in the bond A B , and those of the springs pen p erd icu lar to AB are negligible (see Figure lb) Therefore, 11 Study of interaction p o ten tia l and force constants of t h e y consist of contributions of the spring D-HA by the value K HDX D~ / 2; OÍ the spring D-Hb by K HD( X H + X D) / 2; and of the spring betw een H h and H on the extended A B by K HHX H2 / Hence, the contribution of the atom s on the plane is given by Besides bonding w ith H atom s at verteses of the plane (001), the D {) atom is bonded with other neighboring atom s (see Figure 2) located in the centre of neighboring planes counting from to They are num bered by 1, 2, 3, and are the neighbors of the Hịị atom The rem aining atom s are the neighbors of D0, but they are also the neighbors of th e H/i atom Supposed th a t n is the total n u m b er of d opant atom s in the two neighboring lattices of D atom and among the atom s a t positions 1, 2, 3, there are n ] dopant (D) atom s, then among the positions 5, 6, 7, th e re are n> - n - - n ] dopant (D ) atom s (0 = 4: v„ = =2 - —K hdX dz + — K dd X d + —K hdX d ~ + K ddX + K hd = - A D + —K ddX d + —K hdX d (9) K DD + K hd r/ ^ d A tiff - : For n = 11, 12, 13: Vh)l - ]^Kl)nX n~ + - K 1)D4 X ị / + —K HI)X d~ + K ddX d + K DnX D + K HDX HD^D ( 10 ) = ( 4K ) 1) +K H I ) )XD~ => K v 1C - From (7, 8, 10) we obtain V 4-X V K D D + K HD "■ y {)nK HH +0U)n + vồll/, , g , s ) K D D + K HŨ ọ 12/1 +Ò13/|/ ( - n })e2K HH + ( n - l ) K DD + ^11/t^12/1 (S)13/1 + [4(c +1)^ +/Ij82 +13 — ^10n 4(e +1 )2 (11) Using this expression we can calculate the effective force co n stan t K rf{- with different num ber of im purity atom s replacing the neighboring H atom s of the D () atom located at the centre of the fee lattice Applying the Morse potential in the approxim ation for weak anharm onicity by the expansion V ( x ) = ơ(e"2tuc - 2e~iư)= d ( - + a V - a 3* +•••) for e a c h atomic p a i r from or its the equilibrium form value X by using at the (12) definition [7] y = x - a as the deviation tem p eratu re T, w here a = ( x ) , X - r - r0 , r IS in stan tan io u s bond length, and r0 is its equilibrium value V(y) = D a 2(I - Saa)y2 - D a Ầy :i + Daz(2 - 3aa)ay + Z)a2a 2(l - a.a) - D (13) 14 N g u y e n Van H un g, T ran T ru n g Dung, N g u y e n Cong Toan we obtain the following single-bond spring constants Kh h “ 2DHa 2H (l - a H( X H}) = 2DHa (14) K dd - 2DD(Xfl(l-3aD(X D)) = 2DDa D - —(&]()„ + ỗn/ỉ + ơ12/, + SỊ3/Í )aDa - (l - ỏr]n - Ô10/ỉ - ỏUn - Ỉ>ỉ2n )aD - — l +c K hd - 2DHDa H ~ D( \ 3aHD( X HD^) = 2D hdu ~ hd a HD — (15) (16) and the Morse potential p a m e te rs D IW, a HD for the case with im p u rity can be obtained by averaging those of the host atom s D H , a H and of the dopant atoms D , a D , where D //D - D ^ƠI l 'rxyDuD + D ct „3 _ D u d u -f D nCL3 Du H+ r iD y D ^2 _ ^H > u f/D ^ ^ » U//D ^ ^ £>/ / +£>D Dh +Dd u II n (17) S u b stitu tin g the values of (14-16) into (11) we obtain the effective spring constant Ì 10/1 ID Da~D 1, — a «a + ^>DHDaịi ị) * e ff - 560n D" a " /; +(Ồ]|/Ỉ + Ổ12/Í + i:j„) 4DDa'ò + òíì 0/1 ~~ u / ? w 1/1 u 12// + DHDàịfD U 13/J (18) 2fc + u ( - /il )£2ũ //a w2 3ea/ya^ , 3ttna^ + (/1 - ì)DDa n - 'D' e+ e+1 + [4(e + l) +7ZjC2 + 13-7?]DW Da 2D HirxH Now we te s t the case when there is not any im purity atom, i.e., tt = 0, we obtain: / V \ J ^ _pun*)EFF - 5DHaH Ị - —a Ha (19) This resu lt coinsides totally with the one derived by using the anharm onic correlated E instein model [7] From Eq (13) we obtain the harm onic term 15 Study of inter action p o te n tia l and force constants of or for different cases v HiAy)= \ K HDy2 ’ v HHÌy)= ^ K HHy2’ v D ũ ( y ) - K DDyz (21) and the a n h arm o n ic term of the interaction potential is given by VaJ y ) = K , y \ (2 ) K ,= -D u\ Since th is te rm is cubic power of the p a m e te r y we can use an expression sim ilar to Eq (5), for the cases n = r V'Mut = K , HDX , f + K mD( X H + X Df + K w h X h XDV n xK iDD + n ỵK ÌHD V , D + n zK;WD + (4 - n }1) K D 1)K + (4 V , - V ^ a h / XH + 4-K3HH + (A-n A' ,, = — {(8 —/2, )e: 8(c + 1) ỉ +( + (n, n, n.,)K-WD n 2)K WD +[8(e + l):i l)3 ++ n ,e 3:i + ++ «2 n2- H rt., J ]X3WƠ|;j; K , ||T = -1—- Í/1(8 - n ])c:iD Ha jj + ( n l - n2)Dd< q + [8(e + 1): + n,e + + n2 - n, ]DhdaHD S( E+ 1) 17 Dp ujj + 3D/iVa HD 16 For the case 11 - 10 we obtain K :](,n K 3eỊỊ = - and for rc = 11, 12, 13 it is given l y D p ụ + P l l D a HD 16 At the end we obtain ( - 0n " 510n ~ Sl l n ~ S1 n ~ S13n^ [ ( - ^ )e3P fíq'fí + {nx - n 2)DD0?D [8(8 + 1)3 + n}£3 + + n2 - n} ] D HDa HD 8(e + 1): \ l P Dà)j +3DHDaA HD _ 10/ 16 +8 ' 1 „ llr t +s 10*1 12n ) 19Ppttp + Q n 13/1 16 /1 (23) which for n = (w ith o u t im purity) is reduced to the resu lt derived by using the anharm onic c o rrelated E instein model [7] 16 , , N g u y e n Van H u n g T r a n Trurtg D u n g N g u y e n C on g Toan ^ 3e ff - H - ~ ~ D Ha H (24) The re m a in in g a n h a rm o n ic c o n trib u tio n ta k e n from Eq (15) is given by ^25 ) Da ( - Saa)ay ^ D a zay => K 2a = D a z which c o n ta in s 275a2 B ased on th e sim ila rity betw een K 2a of Eq (25) and K h of Eq (20) we can use Eq (18) to deduce K 2vịĩ = 5Ồ0nD Ha ị + “ 7DDazD + 3DHDaịm +(ổ]ỉn +ồl2n + Ổ13/?) 4DDa 2D +DHDa 2HD - ^ n ~ à\0n ~ ^1 ìn ~ s 12/1 ~ ^13/J 2(8 + l) ~~ n \ + ( 11 ~ U D Da D + I 4- [4(c + l)2 + riịÈ2 + 13 - n]DHDa 2HD Hence, th e to tal a n h a rm o n ic trib u tio n to th e atom ic in te rac tio n potential m u st be given by v ’, n h C y )= K -Mi-ay + K-.toỉiỳ' (2 ? ) ■ For the case n - , i., e., th e re is not any d opan t atom , from obtain Eqs (26, 28, 30) we V , M = ^ K h:FFy i + Vn, M , K eff = D a 2Ị \ - | c t a , Vnnh(y) = 5Da2ay - l ^ y (28) • (29) T hese r e s u lts coinside w ith those derived by u sin g a n h a rm o n ic correlated E instein model [7] which is considered and used widly in XAFS theory for the pure m a te ria ls [8-18] providing good a g re e m e n t w ith e x p e rim e n t even for Cu with strong a n h a rm o n ic trib u tio n s N u m e r i c a l r e s u l t s a n d d i s c u s s i o n s Now we apply th e above derived expression s to n u m e ric a l c alculatio ns for fee crystal Ni doped by several Cu atom s We calculated th e M orse potential of‘ Ni and Cu by using the pro ced ure p re se n te d in [19, 21) The r e s u lts a re illu stra te d in Figure show ing very good a g re e m e n t w ith e x p e rim e n t [15] for th e case of Ni U sing th e s e c alcu late d M orse p o ten tials we c alc u la te d single-bond and effective sp rin g c o n s ta n ts for p u re Ni a n d for Ni doped by sev eral im purity atom s Cu The re s u lts a re w ritte n in Table I The effective s p rin g c o n s ta n ts are different when Ni is doped by 11=1, 3, 5, Cu atom s Study o f inter action p o t e n t i a l a n d force co nsta n ts of 17 r (A 0) Figure Calculated Morse potential for Ni (solid), Cu (dash), and an comparison to experim ent [15] (dot) for the case of Ni T a b le I Effective spring stants of Ni doped by n - 0, 1, 3, 5, Cu atom s and of pure Cu N Cu-pure A'.rr (e V / A2) 4.1757 3.8072 3.7544 3.7016 3.6668 3.1204 K 2rtỊ( e V / Ẵ 2) 4.2389 3.8803 3.8266 3.7728 3.7358 3.1655 K:lt,n( e V / Ẵ :>) -1.5047 -1.3155 -1.3047 -1.2939 -1.3010 -1.0753 Although the v a lues of K 2efl- a re sign ifican t b u t th e te rm K 2eĩĩay contain s a very sm all factor a (ab out 0.007 Ả a t 300 K), t h a t is why th is te rm c o n trib u te s not so m uch to the effective p oten tial The effective p o te n tia ls of the system of Ni illu stra te d in figure c alcu late d by u sin g th e effective sping c o n s ta n ts of Table I are quite different from th e p a ir p o ten tial of Ni show n in figure den oting the im portance of the c o n stru c te d effective p o ten tial of th e sy stem F ig u re also shows significant ch an ges of th e effective poten tial of Ni w hen it is doped by the im p u rity Cu atom s The g r e a te r th e n u m b e r of d o p a n t atom Cu is, th e bigger th e change of the effective potential The above p ro p erties considered for one c lu s te r can be deduced for the whole crystal T hese c hang es will influence on th e th erm o d y n am ic 18 Ngu yen Van H un g, T ran T ru n g D u n g , Ngu yen Cong Toan pa m eters of the crystals like on the cu m ulants studied in the XAFS spectroscopy [7, 8, 11, 13, 19] Figure Effective potential of pure Ni and of Ni doped by ,1 , 3, 5, Cu atoms and of pure Cu C o n c l u s i o n s This work has developed a new procedure for description and calculation of the effective potential, single-bond and effective spring c o n stan ts including anharm onic contributions of a crystal doped by an a rb itra ry n u m b er n of im purity atoms Derived expressions of the considered q u a n titie s approach those derived by using the anharm onic correlated Einstein model for the pure m aterials which provides very good a g re em e n t with the experim ent and is used widly [7-18] This work also denotes the im portance of th e effective potential of a system and its relation with the p air potential, which is especially im p o rta n t for the XAFS theory [7, 8, 11, 13, 19] The above properties considered for one cluster can be deduced for the whole crystal so th a t from this procedure one can deduce a m ethod for description and calculation of the atomic interaction effective potential and force c o n stan ts of an alloy consisting of different percentage of stitu en t elem ents A c k n o w l e d g e m e n t s The a u th o rs th an k Prof D M Pease (University of Connecticut, USA) for useful discussions and comments This work is supported inpart by the basic science research program No 41.10.04 19 Study of interaction p o te n tia l and force constants of R eferen ces R B Greegor, F w Lytle, Phys Rev 1314(1979) 4902 J M T ran q u a d a, R Ingalls, Phys Rev B28(1983) 3520 E A Stern, p Livins, Zhe Zhang, Phys Rev B43(1991) 8850 N V Hung, R N V Hung, R F rah m , H Kamitsubo, J Phys Soc Jpn N V Hung, J de Physique IV(1997) C2 : 279 N V Hung, J N V Hung, N B Due, R R Frahm , J Phys Soc Jpn 72(2003) 1254 T Yokoyama, Phys R e v B57(1998) 3423 10 A.v Poiarkova, J J Rehr, Phys Rev B59(1999) 948 Frahm , Physica B 208 & 209(1995) 91 65(1996) 3571 J Rehr, Phys Rev B56(1997) 43 11 J J Rehr, R c Albers, Rewiews o f Modern Physics, Vol 72(2000) 621 12 T I Nedoseikina, A T Shuvaev, V G Vlasenko, J Phys.: Condens Matter 12(2000) 2877 13 P Fornasini, F Monti, A Sanson, J Synchrotron R adiation 8(2001) 14 Y Okamoto, M Akabori, H Motohashi, Synchrotron R a d ia t i o n , Vol 8(2001) 1191 15 I V Pirog, T I Nedoseikina, I A Zarubin, A T Shuvaev, J P h y s C o n d e n s Matter 14(2002) 1825 H Shiw aku, T 1214 Ogawa, J 16 I V Pirog, T I Nedoseikina, Physica B334(2003) 123 17 N V Hung, VNU-Jojxr Science, Vol 18, No 3(2002) 17 18 N V Hung, D X Viet, VNU-Jour Science Vol 19, No 2(2003) 19 19 M Daniel, D M Pease, N V Hung, J I Budnick, in press in Phys Rev B (2003) N V Hung, VNU-Jour Science, Vol 19, No (2003) 19 20 ... procedure one can deduce a m ethod for description and calculation of the atomic interaction effective potential and force c o n stan ts of an alloy consisting of different percentage of stitu en t... 2; and of the spring betw een H h and H on the extended A B by K HHX H2 / Hence, the contribution of the atom s on the plane is given by Besides bonding w ith H atom s at verteses of the plane... o n c l u s i o n s This work has developed a new procedure for description and calculation of the effective potential, single-bond and effective spring c o n stan ts including anharm onic contributions