C A L C U L A T IO N O F X A F S CƯ M ULANTS F O R F C C CRYSTALS C O N T A IN IN G I M P U R I T Y A TOM
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 r tm e n t o f P h y sic s, College o f Science, V N U
A b s tr a 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 cum ulant 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.
. I n t r o d u c t i o n
T h e c r y s t a l s 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 )erio d ical M e n d e le e v s y s te m . T h a t is w hy th e c a lc u la tio n of p h y s ic a l p a r a m e t e r s of h e s e c r y s ta ls is v ery i m p o r t a n t . To s tu d y t h e r m o d y n a m i c p r o p e r t i e s of a s u b s ta n c e t is n e c e s s a r y to i n v e s t i g a t e its effective local force c o n s t a n t s , c o r r e la te d E in s te in r e q u e n c y a n d t e m p e r a t u r e , n e t t h e r m a l e x p a n s io n , m e a n s q u a r e re la tiv e lis p la c e m e n t (MSRD) or Debye W a lle r fac to r a n d t h i r d c u m u l a n t [1-13] w hich arc*
o n t a i n e d in th e X -ra y a b s o r p tio n fine s t r u c t u r e (XAFS) [9J. M o reo ver, th e im p u rity
>r d o p a n t a to m c a n in fu e n c e on t h e p h y s ic a l p a r a m e t e r s t a k e n from th e XAFS p e c t r a 110] a n d on t h e efficiency of u s in g t h e s e s u b s t a n c e s . T h e rm o d y n a m ic
>roperties of a lk a li m e t a l s u n d e r in flu e n c e of i m p u r i t y h a s b e e n s tu d i e d [1 1].
T he p u r p o s e of t h i s w ork is to develop a m e th o d for c a lc u la tio n a n d e v a lu a tio n
>f t h e effective local force c o n s t a n t s , c o r r e l a t e d E i n s t e i n freq u e n cy an d e m p e r a t u r e , f ir s t c u m u l a n t or n e t t h e r m a l e x p a n s io n , seco n d c u m u l a n t w hich IS
•qual to M SRD or Debve W a lle r fa c to r a n d t h i r d c u m u l a n t of fee c r y s t a l s c o n ta in in g I d o p a n t or i m p u r i t y (I) a to m as a b s o rb e r in th e X A FS p ro cess. Its n e a r e s t le ig h b o rs a r e th e h o s t (H) a to m s . T h e d e r iv a tio n is b a s e d on th e q u a n t u m statistic a l th e o ry w ith t h e c o r r e l a t e d E i n s t e i n m odel [7] w h ic h is c o n sid e re d at j r e s e n t as “th e b e s t t h e o r e t ic a l f ra m e w o r k w ith w hich th e e x p e r i m e n t a l i s t can e la te force c o n s t a n t s to t e m p e r a t u r e d e p e n d e n t XAFS" [1 0]. Kor c o m p le tin g th e ab nitio c a lc u la tio n p r o c e d u re th e p a r a m e t e r s of M orse p o t e n t i a l of p u r e c r y s ta ls a n d hose w ith i m p u r i t y h a v e b ee n also c a lc u la te d . N u m e r ic a l c a l c u l a t i o n s for Cu. Ni.
m d Cu doped by Ni a to m h av e b e e n c a r r ie d o u t in c o m p a r is o n to th o se of Lh(' p u re n a t e r i a l s to show t h e r m o d y n a m i c a l effects of fee c r y s t a l u n d e r in flu en c e of th e m p u r ity ato m . T h e c a l c u la te d r e s u l t s a r e found to be in good a g r e e m e n t w ith 'X perim ent for M orse p o t e n t i a l a n d for th e o t h e r t h e r m o d y n a m i c p a r a m e t e r s [13].
C al cu la ti on o f XAFS c u m u l a n t s for fee. 9
2. F o r m a l i s m
T h e e x p r e s s io n for th e M SRD in XAFS th e o r y is d e r iv e d b a s e d on th e a n h a r m o n i c c o r r e la te d E i n s t e i n model [7] acco rd in g to w h ic h th e effective i n te r a c tio n E i n s t e i n p o t e n t i a l of th e s y s te m c o n s is tin g of a n i m p u r i t y (I) a to m as a b s o rb e r a n d th e o t h e r h o s t (H) a to m s is given by
i' c/f(x ) - - l<cirx 2 + /vvY? +■■■ - ~aT xR ' 2 'R,i
1 /■*/ >
' (1)
Mị +M„
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 t h r a n d its e q u ilib riu m v alu e r „ , kcfJ is effective local force c o n s ta n t, a n d ky t h e cubic p a r a m e t e r giving an a s y m m e t r y in th e p a i r d i s t r i b u t i o n fun ctio n , R is bo nd u n i t vector. T h e c o rre la te d E i n s t e i n m odel m a y be d efin ed as a o s c illatio n of a p a i r of a t o m s w ith m a s s e s Mị
a n d MH (e.g., of i m p u r i t y a to m a s a b s o r b e r a n d of h o s t a to m as b a c k s c a t t e r e r ) in a given sy stem . T h e ir o s c illa tio n is in flu e n c e d by t h e i r n e ig h b o r s given by th e last te r m in th e l e f t - h a n d side of Eq. (1), w h e re t h e s u m / is o v er a b s o r b e r ( / = 1) and b a r k s c a t t e r e r 0 = 2), a n d t h e s u m j is over all t h e i r n e a r e s t n e ig h b o rs , excluding th e a b s o rb e r a n d b a c k s c t t e r e r th e m s e lv e s . T h e l a t t e r c o n t r i b u t i o n s a r e d e s c rib e d Ly th e te r m vm (v).
For w eak a n h a r m o n i c i t y in t h e XAFS p ro cess t h e M o rse p o t e n t i a l is given by I ho e x p a n s io n
V(x) = d{Lr2 u ' - 2 c ) = d ( - 1 + a 2.x2 - e r V + ■ • ■) (2) for th e p u r e m a t e r i a l a n d
V/If ( v) “ Djii (-1 + ơJịị -V —aj Hx H ) 0 ) for th e case w ith i m p u r ity , w h e r e M o rse p o t e n t i a l p a r a m e t e r s h a v e b e e n obtained by a v e r a g in g th o s e of t h e p u r e m a t e r i a l s a n d a r e giv en by
r, D l + D ll „ 2 D l a i + D H a H 3 _ D l a ) + D H a hl (1\
D'" ’ = D , * 0„ ■ D, + D„ ■
U s in g th e d e f in itio n [2, 7] V = A' - ứ as th e d e v ia tio n from t h e e q uilib rium v a lu e of X th e Eq. (1) is r e w r i t t e n in th e su m of th e h a r m o n ic c o n t r ib u t i o n a n d t i e (in h a rm o n ic c o n t r ib u t i o n ỔV a s a p u r t u r b a t i o n
Ctrl'2 +5V ■ (5)
T a k in g in to a c c o u n t th e a to m ic d i s t r i b u t i o n of fee c r y s t a l a n d u s in g th e abo/e e q u a t i o n s we o b ta in th e effective local force c o n s ta n t
Kir =
2 D m a ///[1 + 3(//,2+/jị)] + ị o Ha ị = /.ưoị, 6 , th e cubic a n h a r m o n i c p a r a m e t e rky — Dm ct'm(1 + //|' + //2),
th e a n h a r m o n i c c o n t r ib u t i o n to th e effective p o t e n t i a l of t h e s y s te m SV(y) = 2 D IHaj H(\ + 3( rì + v l ) ) + ^ D flaj f ay - DIHcc]H{\ + r ì + f . i ị ) Ý
ơ )
(8) th e c o r r e la te d E i n s t e i n fre q u e n c y
[f*
a n d th e c o r r e la te d E i n s t e i n t e m p e r a t u r e
Dm a m [1 + 1(MỈ + MÌ)] + ị D Ha l I 2
(9)
k B [// Dịh&Ĩh D + 3(//f + M i)]+ 2 ^ H a ~H (10) w here
M. M, M
M , +M// , / / 2 M j + Mh / / 2 =
M , +M (11)
/7
T h e c u m u l a n t s h a v e b een d e riv e d by a v e r a g i n g p ro c e d u re , u s in g th e s ta t i s t i c a l d e n s ity m a t r i x /9 a n d th e c a n o n ic a l p a r t i t i o n fu n c tio n
z
in th e form< y m > = - T r ( p y m), m = 1 ,2 ,3 ,- , 'Zd
Z = Trp, p = p 0 +ổp, Z * Z 0 =Trpa , p 0 =e - fiHo, H 0 = ậ - + ị - k eíry 2, fi = \ / k BT
(12) (13) 2// 2 (14)
where k B is Boltzmann constant and ỏp is neglected due to small anharmonicity in XAFS [2].
U sing th e above r e s u lts we calcu late the second c u m u l a n t or D ebye-W aller factor (15)
Ơ
Í above r e s u lts we calcu late the second c u m u l a n t or Debye-
w h e re we e x p r e s s y in t e r m s of a n i h i l a t i o n a n d c r e a tio n o p e r a t o r s , à a n d (1+, i. e.
{ « + « •); ’ *
V = K K~ = (16)
\JLIO)e
an d use h a r m o n ic o s c illa to r s t a t e I n) w ith e ig e n v a lu e En = nh(0E (ig n o rin g th e zero p o in t e n e rg y for c o n v e n ie n ce ).
T h e refo re , th e e x p r e s s io n for seco n d c u m u l a n t (M SRD) or D e b y e -W a lle r fac to r is r e s u lte d as
ơ ~ = Ơ (1 + z) (1-z)
PiCOr
Ơ. (17)
C al cu la ti on of XAFS c u m u l a n t s for fee. 11
Now we c a lc u la te th e odd c u m u l a n t s
! ^ e- 0 E ._ e-pE„.
Ơ
0
(18)
U s in g th e c a lc u la te d m a tr ix e le m e n ts a n d m a t h e m a t i c a l f o r m u la s for d iffe re n t t r a n s f o r m a t i o n s we o b ta in t h e e x p r e s s io n s for th e firs t c u m u l a n t (m —1)
i1 + r) J (l)- 3Dw g // / (l + //|3+/<23) ^ --- (19)
uyi ) — u () _ x , V () Ị- !
( I )
( l - z )
DiHa ĨH (1 + 3/'|2 + Ml) + 2
a n d for th e t h i r d c u m u l a n t (m=3)
n O) = (
3) (l + ioz + z2) . (
3) =
DihGIH M\ /^2 ) (20)(1-2)
DiHa ìn (1 + 3/'|2 + ) + 9D Ha~H
In th e above e x p r e s s io n s ơ i '1, <702, ơ i’ 1 a r e z e r o -p o in t c o n t r i b u t i o n s to t h e first, second a n d t h i r d c u m u l a n t s , re sp ectiv e ly . T h e y c h a r a c t e r i z e q u a n t u m effects o ccu rred by u s in g q u a n t u m t h e o r y in o u r c a lc u la tio n a n d in flu e n c e on t h e o b ta in e d r e s u lts . T h e above d e r iv e d c u m u l a n t s a r e c o n ta in e d in t h e XAFS in c lu d in g a n h a r m o n i c c o n t r ib u t i o n s [9].
3. N u m e r i c a l r e s u l t s a n d c o m p a r i s o n t o e x p e r i m e n t
Now we a p p ly th e d e riv e d e x p r e s s io n s to n u m e r ic a l c a l c u l a t i o n s for Cu, Ni a n d Cu doped by Ni a to m a s a b s o r b e r in th e XAFS p rocess. T h e i r M o rse p o t e n t i a l p a r a m e t e r s h a v e b e e n c a l c u l a t e d u s in g t h e p r o c e d u re p r e s e n t e d in [16]. T h e c a lc u la te d v a l u e s of M o rse p o t e n t i a l p a r a m e t e r s ; c o r r e 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 ; effective local force c o n s ta n t for t h e p u r e Cu, Ni a n d th o s e doped by N i a to m a r e w r i t t e n in T a b le I. T h e y a r e found to be in good a g r e e m e n t w ith e x p e r i m e n t [13].
T a b l e I C a lc u la te d v a l u e s of M o rse p o t e n t i a l p a r a m e t e r s D, a ; c o r r e la te d E i n s t e i n fre q u en cy z , a n d t e m p e r a t u r e
eE\
effective local force c o n s t a n tkeff
for N i-N i, C u-Cu, N i-C u in c o m p a r is o n to e x p e r i m e n t [13]. _____
Bond D(eV) a ( A ’) r„ ( Ả ) k ef f { N / m ) o>e (x 1013Hz )
Ni-Ni, present 0.4263 1.3819 2.8033 65.2158 3'. 64 73 278.6038
Ni-Ni, exp.[ 13] 0.4100 1.3900 2.9035 63.4596 3.5979 274.8271
Cu-Cu, present 0.3367 1.3549 2.8701 49.5156 3.0544 233.3151
Cu-Cu, exp.[13] 0.3300 1.3800 2.9802 50.3450 3.0799 235.2661
N i - C u , p r e s e n t 0.3817 1.3928 2.9537 60.1340 3.4348 262.3749
Ni-Cu, exp.[ 13] 0.3700 1.3855 2.9337 57.8621 3.3693 257.3708
F i g u r e 1 i l l u s t r a t e s o u r c a l c u l a t e d M o rse p o t e n t i a l s of C u, Ni a n d of Cu d o p p e d by Ni a t o m w h ic h a g r e e s w ell w ith t h 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 com pared to experim ent [13].
Figure 2: T e m p e r a t u r e d ep e n d e n c e of o u r ca lc u la te d first c u m u l a n t c r ^ ( r ) of Cu (dash- dot), Ni (dash) a n d of Cu doped by Ni ato m (solid) co m p a re d to e x p e r im e n t (dot) [13].
Ca lc u la ti o n o f XA FS c u m u l a n t s f o r fee. 13
F ig u re 3: T e m p e ratu re dependence of our calculated second c u m u la n t ơ {t) of Cu (dash- dot), Ni (dash) and of Cu doped by Ni atom (solid) compared to ex perim ent (dot) [13].
X 1Q'4
T(K)
Figure 4: T e m p e ra tu re dependence of our calculated th ird c u m u la n t cr '^ (r) of Cu (dash”
dot), Ni (dash) and of Cu doped by Ni atom (solid) compared to ex perim ent (dot) [13].
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 l c u l a t e d f i r s t c u m u l a n t or n e t t h 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 l l e r fa c to r ơ2(t) (F igure 3) a n d t h i r d c u m u l a n t c r ^ ( r ) (F ig u re 4) s h o w s s i g n i f i c a n t c h a n g e s of t h e s e v a lu es w h en Cu is d o p p ed by Ni a to m a n d a r e a s o n a b l e a g r e e m e n t b e t w e e n t h e c a lc u la te d by th e p r e s e n t th e o r y a n d e x p e r i m e n t a l v a lu e s . F i g u r e s 2, 3, 4 show t h a t th e c u m u l a n t s of Cu becom e w e a k e r due to t h e d o p a n t by Ni a to m . T h e s e i m p u 'r i t y effects a r e very i m p o r t a n t a n d th e y 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 t h e r m o d y n a m ic p r o p e r t i e s of th e s u b s t a n c e s . T h e c a l c u l a t e d f ir s t, seco n d a n d t h i r d c u m u l a n t c o n t a i n i n g i m p u r i t y a to m a lso s a t i s f y a l l i m p o r t a n t p r o p e rtie s disco v ered in t h e o r y [7, 17] a n d e x p e r i m e n t [1]. T h e y c o n t a i n z e ro -p o in t c o n tr ib u tio n a t low t e m p e r a t u r e a n d a p p r o a c h t h e c l a s s i c a l t h e o r y r e s u l t s a t h ig h t e m p e r a t u r e , i., e., t h e p r o p o r t i o n a l i t y 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 firs t a n d second c u m u l a n t a n d q u a d r a t i c a l for th e t h i r d c u m u l a n t .
4. C o n c l u s i o n s
A new a n a l y t i c a l m e th o d for c a lc u l a t i o n a n d e v a l u a t i o n of th e r m o d y n a m ic p r o p e r tie s of th e fee c r y s t a l s c o n t a i n i n g i m p u r i t y a t o m h a s b e e n d e v e lo p e d b ased on th e q u a n t u m s t a t i s t i c a l th e o r y w ith c o r r e l a t e 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 r iv a tio n of t h e a n a l y t i c a l e x p r e s s i o n s for t h e local effective force c o n s ta n ts , c o r r e la te d E i n s t e i n f r e q u e n c y a n d t e m p e r a t u r e , t h e first, second a n d t h i r d X AFS c u m u l a n t of fee c r y s t a l s c o n t a i n i n g i m p u r i t y ato m . T h ey a re sig n ific a n tly d iff e r e n t from th o se of t h e p u r e m a t e r i a l s , b u t s a t i s f y all s t a n d a r d p r o p e r tie s of t h e s e q u a n t i t i e s . T h e s e d iff e r e n c e s d e n o t e t h e i m p u r i t y effects w hich discov ered in e x p e r i m e n t a n d th e y 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 r m o d y n a m ic p r o p e r t i e s of th e s u b s t a n c e s .
M orse p o t e n t i a l p a r a m e t e r s h a v e b e e n a ls o a n a l y t i c a l l y c a l c u la te d th u s c o m p le tin g th e ab in itio c a l c u la tio n p r o c e d u r e of t h e c o n s id e r e d v a lu e s .
T h e good a g r e e m e n t b e tw e e n t h e c a l c u l a t e d a n d t h e e x p e r i m e n t a l r e s u lts d e m o n s t r a t e s th e efficiency a n d p o s s ib ility of u s i n g t h e p r e s e n t d e v e lo p e d p ro c e d u re 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 o rk is s u p p o rte d in p a r t by th e b a sic science r e s e a r c h n a t i o n a l p r o g r a m p r o v id e d by t h e M in is try of S cience a n d T ech n o lo g y No. 41.10.04.
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