DSpace at VNU: Calculation of XAFS Cumulants for FCC Crystals Containing Impurity Atom

\N1U JOURNAL OF SCIENCE, Mathematics - Physics, T.xx N03, 2004 CA LC U LA TIO N O F X A F S CƯMULANTS F O R F C C CRYSTALS C O N T A IN IN G IM P U R IT Y ATOM N g u y e n V a n H u n g , N g u y e n T h i T h u H o a i, L e H a i H u n g D e p a rtm e n t o f P h ysics, College o f Science, V N U A b s tra c t: A new procedure for calculation and evaluation of XAFS cumulants of fee crystals containing impurity atom has been developed based on the quantum statistical theory with correlated Einstein model Analytical expressions for the effective local force constants, correlated Einstein frequency and temperature, first cumulant or net thermal expansion, second cumulant or Debye Waller factor and third cumulant of fee crystals containing impurity atom have been derived Morse potential param eters of pure crystals and those with impurity included in the derived expressions have been calculated Numerical results for Cu Ni Ni-Cu are found to be in good agreement with experiment In tro d u c tio n T he c r y s ta ls w ith fee s t r u c t u r e occupies a b o u t 25 % of e le m e n ts in the )eriodical M en d eleev sy stem T h a t is why th e c a lc u la tio n of phy sical p a r a m e te r s of h e s e c ry s ta ls is very im p o rta n t To s tu d y th e r m o d y n a m ic p r o p e rtie s of a su b sta n c e t is n e c e s sa ry to in v e s tig a te its effective local force c o n s ta n ts , c o rre la te d E in s te in re q u e n c y an d t e m p e r a t u r e , net th e r m a l e x p a n sio n , m e a n s q u a re relativ e lisp la c e m e n t (MSRD) or Debye W aller factor an d th ird c u m u l a n t [1-13] which arc* o n ta in e d in th e X -ray a b s o rp tio n fine s tr u c t u r e (XAFS) [9J M oreover, th e im p u rity >r d o p a n t ato m can in fu e n ce on th e p h y sica l p a r a m e t e r s t a k e n from th e XAFS p e c tr a 110] a n d on th e efficiency of u s in g th e s e s u b s ta n c e s T h e rm o d y n a m ic >roperties of alk ali m e ta ls u n d e r in flu en c e of im p u rity h a s b e e n s tu d ie d [ 1 ] T he p u rp o se of th is w ork is to develop a m eth o d for c a lc u la tio n an d e v a lu a tio n >f th e effective local force c o n s ta n ts , c o rre la te d E in s te in frequency an d e m p e r a tu r e , firs t c u m u l a n t or n e t th e r m a l ex p an sio n , second c u m u la n t which IS •qual to MSRD or Debve W a lle r facto r an d th ird c u m u la n t of fee c r y s ta ls c o n ta in in g I d o p a n t or i m p u rity (I) ato m as a b s o rb e r in th e XAFS process Its n e a r e s t leig hb ors a re the h o st (H) a to m s T he d e riv a tio n is b a s e d on th e q u a n tu m statistical th eo ry w ith th e c o rre la te d E in s te in model [7] w hich is considered at jre s e n t as “th e b e s t th e o r e tic a l fra m e w o rk w ith which th e e x p e r im e n ta lis t can e la te force c o n s ta n ts to t e m p e r a t u r e d e p e n d e n t XAFS" [ ] Kor co m p leting th e ab nitio ca lc u la tio n p ro c e d u re th e p a r a m e t e r s of M orse p o te n tia l of p u re c ry sta ls and hose w ith im p u rity h a v e b een also c a lc u la te d N u m e ric a l c a lc u la tio n s for Cu Ni md Cu doped by Ni ato m h ave b een c a rrie d o u t in c o m p a riso n to th o se of Lh(' p u re n a t e r i a l s to show th e r m o d y n a m ic a l effects of fee c r y s ta l u n d e r influence of th e m p u rity ato m T he c a lc u la te d r e s u l ts a re found to be in good a g r e e m e n t w ith 'X perim ent for M orse p o te n tia l a n d for th e o th e r th e r m o d y n a m ic p a r a m e t e r s [13] 9 C alcu la tion of XAFS c u m u l a n t s for fee F o r m a l i s m T he ex p re ssio n for th e MSRD in XAFS th e o ry is d e riv e d b ased on th e a n h a r m o n ic c o rre la te d E in s te in model [7] acco rding to w h ich th e effective in te c tio n E in s te in p o te n tia l of the s y ste m c o n s is tin g of a n i m p u r ity (I) ato m as a b s o rb e r an d th e o th e r h o st (H) ato m s is given by ~aT x R ' 'R, i ' i'c/f(x ) - - l /■*/ M ị +M„ ( 1) H ere X is d e v ia tio n b e tw e e n th e i n s t a n t a n e o u s bond le n g th r a n d its eq u ilib riu m value r „ , kcfJ is effective local force c o n s ta n t, a n d ky th e cubic p a r a m e t e r giving an a s y m m e tr y in th e p a ir d i s tr ib u tio n function, R is bond u n i t vector T h e c o rrelated E in s te in model m ay be d efined as a oscillation of a p a i r of a to m s w ith m a ss e s M ị an d MH (e.g., of i m p u rity ato m a s a b s o rb e r a n d of h o s t a to m as b a c k s c a tte r e r ) in a given system T h e ir o scillatio n is in flu enced by th e ir n e ig h b o rs given by th e last te rm in the l e f t- h a n d side of Eq (1), w h ere th e s u m / is over a b s o rb e r (/ = 1) and b a r k s c a t t e r e r = ), a n d th e su m j is over all th e ir n e a r e s t n eig h b o rs, excluding th e a b s o rb e r a n d b a c k s c tte r e r th e m se lv e s T h e l a t t e r c o n tr ib u tio n s a re d escrib ed Ly th e te rm vm (v) For w eak a n h a r m o n ic ity in th e XAFS process th e M orse p o t e n ti a l is given by I ho e x p an sio n V(x) = d {Lr2 u ' - c ) = d ( - + a 2.x2 - e r V + ■•■) (2) for th e p u re m a te r ia l an d V/If ( v) “ D jii (-1 + Jịị -V —aj Hx H ) 0) for the case w ith im p u rity , w h e re M orse p o te n tia l p a r a m e t e r s h a v e b een obtained by a v e g in g th o se of th e p u r e m a te r ia ls a n d are given by r, D l + D ll „2 D'" ’ D l a i + D Ha H = D , * 0„ _ D l a ) + D H a hl ■ D, + D„ (1\ ■ U sin g th e d e fin itio n [2, 7] V= A' - ứ as th e d e v ia tio n from th e equilibrium v alu e of X th e Eq (1) is r e w r it t e n in th e su m of th e h a rm o n ic c o n tr ib u tio n an d tie (inh arm on ic c o n tr ib u tio n ỔV a s a p u r tu r b a t i o n Ctrl'2 +5V ■ (5) T a k in g in to ac c o u n t th e atom ic d is tr ib u tio n of fee c r y s ta l a n d u s in g th e abo/e e q u a tio n s we o b ta in th e effective local force c o n s ta n t Kir = Dm a ///[1 + 3(//,2 +/jị)] + ị o Ha ị = /.ưoị, th e cubic a n h a r m o n ic p a r a m e t e r 6, 10 N g u y e n Van H u n g , N g u y e n Th i T h u H o a i , Le H a i H u n g ky —Dm ct'm (1 + //|' + // ), ơ) the a n h a r m o n ic c o n tr ib u tio n to th e effective p o te n tia l of th e s y ste m SV(y) = DIHaj H(\ + 3( rì + v l ) ) + ^ D flaj f ay - DIHcc]H{\ + r ì + f i ị ) Ý (8) the c o rre la te d E in s te in freq u en cy I2 (9) Dm a m [1 + 1(MỈ + MÌ)] + ị DHa l [f* and th e c o rre la te d E in s te in t e m p e r a t u r e Dịh&Ĩh D + 3(//f + M i)]+ k B [// ^ H a ~H ( 10 ) M M , +M /7 (11) w here M M , +M // , //2 M, Mj + M h //2 = T h e c u m u l a n t s h av e b een d erived by a v e r a g in g p ro ced u re, u sin g th e s ta tis tic a l d e n s ity m a tr ix /9 an d th e cano nical p a r titio n fu n c tio n z in th e form < y m > = - T r ( p y m), m = ,2 ,3 ,'Zd , ( 12 ) Z = Trp, p = p +ổp, Z * Z =Trpa , (13) p =e - fiHo, H = ậ - + ị - k eíry 2, fi = \ / k BT // (14) where k B is Boltzmann constant and ỏp is neglected due to small anharmonicity in XAFS [2] U sing theÍ above re su lts we calculate the second c u m u la n t or DebyeDebye-W aller factor (15) Ơ w h ere we e x p re ss y in te r m s of a n ih ila tio n an d c re a tio n o p e r a to rs , an d (1 +, i e V = K{ « + « •); K~’ = * (16) \JLIO)e an d use h a rm o n ic o sc illa to r s ta te In) w ith e ig e n v a lu e En = nh( E (ignoring th e zero point en erg y for convenience) T herefo re, th e e x p re s s io n for second c u m u la n t (MSRD) or D eb y e-W aller factor is re s u lte d as ~= Ơ (1 + z) ( -z) Ơ PiCOr (17) 11 C alcu la tion of XAFS c u m u l a n t s for fee Now we c a lc u la te th e odd c u m u la n ts ! ^ e-0 E _ e-pE„ (18) Ơ U sin g th e c a lc u la te d m a trix e le m e n ts a n d m a th e m a tic a l fo rm u la s for differen t tr a n s f o rm a tio n s we o b ta in th e e x p re ssio n s for th e first c u m u l a n t (m —1 ) (I) i + r) J (l)uyi ) —u () _ x , V () (l-z ) 3Dw g ///(l + //|3 +/e (x 12 N g u y e n V a n H u n g , N g u y e n T h i T h u H o a i , Le H a i H u n g F ig u r e i l l u s t r a t e s o u r c a lc u la te d M o rse p o t e n t i a l s of Cu, Ni a n d of Cu d o p p ed by Ni a to m w h ic h a g r e e s w ell w ith th e e x p e r i m e n t a l r e s u l t [13] r(A°) Figure 1: Morse potentials of Cu, Ni and Cu doped by Ni atom compared to experiment [13] Figure 2: T e m p e tu r e depend ence of our calcu lated first c u m u la n t c r ^ ( r ) of Cu (dashdot), Ni (dash) and of Cu doped by Ni atom (solid) co m pared to e x p e rim e n t (dot) [13] 13 Calcu la ti on o f XA FS c u m u l a n t s for fee F igure 3: Tem perature dependence of our calculated second cu m ulan t {t ) of Cu (dash- dot), Ni (dash) and of Cu doped by Ni atom (solid) compared to experim ent (dot) [13] X1Q'4 T(K) Figure 4: T em perature dependence of our calculated third cum u lant cr '^(r) of Cu (dash” dot), Ni (dash) and of Cu doped by Ni atom (solid) compared to experim ent (dot) [13] 14 N g u y e n Van H u n g , N g u y e n T h i T h u H o a i , Le H a i H u n g T he t e m p e r a t u r e d e p e n d e n c e of o u r c a lc u la te d f ir s t c u m u l a n t or n e t th e r m a l e x p a n s io n ơ-(l)( ĩ) (F ig u re 2), second c u m u l a n t or D e b y e - W a lle r fa c to r 3) an d th ird c u m u l a n t c r ^ ( r ) 2(t ) (Figure (F ig u re 4) sh o w s s ig n if ic a n t c h a n g e s of th e s e values w h en Cu is dopped by Ni ato m a n d a r e a s o n a b le a g r e e m e n t b e tw e e n th e calcu lated by th e p r e s e n t th e o ry a n d e x p e r im e n ta l v a lu e s F i g u r e s 2, 3, show t h a t the c u m u la n ts of Cu becom e w e a k e r due to th e d o p a n t by Ni a to m T h e s e i m p u 'r ity effects are very i m p o r t a n t a n d th e y h av e to be t a k e n in to a c c o u n t in the e v a lu a tio n of th e rm o d y n a m ic p r o p e rtie s of the s u b s ta n c e s T h e c a lc u l a t e d first, second an d th ird c u m u la n t c o n ta in in g i m p u rity a to m also s a tis f y a ll i m p o r t a n t p ro p e rtie s discovered in th e o ry [7, 17] an d e x p e r im e n t [ ] T h e y c o n ta in zero -p o in t co n trib u tio n a t low t e m p e r a t u r e an d a p p r o a c h th e c la s s ic a l th e o r y r e s u l ts a t high t e m p e r a tu r e , i., e., th e p r o p o rtio n a lity to th e h ig h t e m p e r a t u r e is l i n e a r for th e first a n d second c u m u l a n t a n d q u a d r a tic a l for th e t h i r d c u m u l a n t C o n c lu sio n s A new a n a ly tic a l m eth o d for c a lc u la tio n a n d e v a l u a t i o n of th e rm o d y n a m ic p ro p e rtie s of th e fee c r y s ta ls c o n ta in in g i m p u r i ty a to m h a s b e e n d ev elop ed b ased on th e q u a n t u m s ta ti s t i c a l th e o ry w ith c o r re la te d E i n s t e i n m odel O u r d e v e lo p m e n t is th e d e riv a tio n of th e a n a l y t i c a l e x p r e s s io n s for th e local effective force c o n s ta n ts , c o rre la te d E i n s t e i n fre q u e n c y a n d t e m p e r a t u r e , th e first, second an d th ird XAFS c u m u l a n t of fee c r y s ta ls c o n t a i n i n g i m p u r i t y atom T hey are sig n ifican tly d iffe re n t from th ose of t h e p u r e m a t e r i a l s , b u t s a tis f y all s ta n d a r d p ro p e rtie s of th e se q u a n titie s T h e se d iffe re n c e s d e n o te t h e i m p u r i t y effects which discovered in e x p e r im e n t a n d th ey h a v e to be t a k e n in to a c c o u n t in th e e v a lu a tio n of th e rm o d y n a m ic p r o p e rtie s of th e s u b s ta n c e s M orse p o te n tia l p a r a m e t e r s h a v e b e e n a lso a n a l y t i c a l l y c a lc u la te d th u s co m p letin g th e ab in itio c a lc u la tio n p ro c e d u re of t h e c o n s id e re d v alu es The good a g r e e m e n t b e tw e e n th e c a lc u la te d a n d t h e e x p e r im e n t a l re s u lts d e m o n s tr a te s th e efficiency a n d p o ssib ility of u s in g th e p r e s e n t d ev elo p ed pro ced ure in XAFS d a t a a n a ly s is A c k n o w l e d g m e n t s One of th e a u t h o r s (N V H u n g ) t h a n k s Dr I V Pirog for s e n d in g Ref 13 a n d p ro v id in g th e e x p e r i m e n t a l d a t a T h i s w ork is s u p p o rte d in p a r t by th e basic science r e s e a r c h n a tio n a l p r o g r a m p r o v id e d by th e M in istry of Science a n d Technology No 41.10.04 R eferences A S te rn , p L ivins, z Z hang, P hys Rev B, (1 9 ) E A.I F r a n k e l, J J R ehr, Phys R e v B, (1 9 ) 585 T N 8550 M iy an a g a, T F u jik a w a , J Phys Soc J p n , (1 9 ) 1036 a n d 3683 V H ung, R F r a h m , P h y sica B, -2 (1 9 ) 91 15 Ca lculation o f XAFS c u rnu la nts for f e e N V H u n g , R F r a h m , H K a m its u b o , J Phys Soc J p n , 65(1996) 3571 N V H u n g , J de P h y siq u e , IV (1997) C2 : 279 N V H u n g , J J R e h r, P hys Rev B, 56(1997) 43 N V H u n g , C o m m u n P h y s., , No 1(1998) 46-54 N V H u n g , N B Due, R F r a h m , J P hys Soc J p n , 72(2003) 1254 10 M D a n ie l, D M P e a s e , e t al, a c c e p te d for p u b lic a tio n in P hys Rev B 11 N V H u n g , V N U - J o u r o f S cience, Vol 19, No 4(2003) 12 See X - r a y a b s o r p t i o n , e d ite d by D c K o n in g s b e rg e r a n d R P r in s (Wiley, N ew Y ork,1988) 13 I V P irog , T I N e d o s e ik in a , I A Z a ru b in a n d A T S h u v aev , J Phys.: C ondens M a t t e r , 14 (2002) 1825 14 L A G irifalco a n d V G W eizer, P hys Rev 114(1959) 687 15 N V H u n g , V N U - J o u r S c ie n c e , Vol 18, No 3, (2002), 17-23 16 N V H u n g , D X V iet, V N U -J o u r S cience 19, N o.2(2003) 19 17 J.M Z im a n , P r in c ip le s o f the T h eo ry of S o lid s, 2nd ed by C am b rid g e U n iv e rs ity P r e s s , 1972 18 R F F e y n m a n , S t a t i s t i c a l M e c h a n ic s (b en jam in , R e a d in g , 1972)  ... o f XA FS c u m u l a n t s for fee F igure 3: Tem perature dependence of our calculated second cu m ulan t {t ) of Cu (dash- dot), Ni (dash) and of Cu doped by Ni atom (solid) compared to experim... (dot) [13] X1Q'4 T(K) Figure 4: T em perature dependence of our calculated third cum u lant cr '^(r) of Cu (dash” dot), Ni (dash) and of Cu doped by Ni atom (solid) compared to experim ent (dot)... te in model m ay be d efined as a oscillation of a p a i r of a to m s w ith m a ss e s M ị an d MH (e.g., of i m p u rity ato m a s a b s o rb e r a n d of h o s t a to m as b a c k s c a tte