VNU. JOURNAL O F S C IE N C E . Mathematics - Physics, T.X X i, N02, 2006
ANH ARM ONIC CO R R E LA TE D DEBYE MODEL DEBYE-W ALLER F A C T O R CO M PA R ED TO EINSTEIN MODEL RESULT
N g u y en Van H ung, N gu yen Bao T rung D ep a rtm en t o f Physics, College o f Science, VNU.
Abstract. An anharmonic correlated Debye model has been derived for vibrational amplitudes in the X-ray Absorption Fine Structure (XAFS). The model includes anharmonic effects based on Morse potential parameters. Analytical expression for Debye-Waller factor or second cumulant for different atomic shells has been derived. Numerical results for Cu approach those obtained by the anharmonic Einstein model. Both they contain zero-point contribution at low temperatures and linearly depend on the temperature at high temperatures as the classical limit values. They reflect the experimental results.
1. I n tr o d u c tio n
T h e D ebye-W aller (DW) facto r e~W(<p) accounts for th e effects of th e th e rm a l v ib ratio n of ato m in th e th e o ry of XAFS. T he d o m in a n t te rm W(p) = l p 2ơ 2 d ep en d s on th e m ean s q u a re re la tiv d e d isp lac em e n t (MSRD) [1-4], w here p is p h o to electro n wave n u m b er. A n h arm o n ic c o n trib u tio n included in th e p o ten tial yields a d d itio n a l term s in th e DW facto rs w hich if ignored can lead to non-negligible e rro rs in s tr u c tu r a l p a ra m e te r s [4-11] e x tra c te d from XAFS sp e ctra . The form alism for in clu d in g a n h a rm o n ic effects in XAFS is often b ased on c u m u la n t ex p an sio n ap p ro ach [4, 5]. M an y effo rts h av e been m ade [4-17] to include th ese an h arm o n ic c o n trib u tio n s, am o n g th em th e an h arm o n ic co rre lated E in ste in model [16] avoids com plicated c a lc u la tio n s y e t provides rea so n a b le a g re em e n t w ith experim ent.
T h is w ork is th e n e x t ste p of [15] to d erive an a n h arm o n ic c o rrelated Debye model for v ib ra tio n a l a m p litu d e s in XAFS u sin g q u a n tu m s ta tis tic a l th eo ry . The model in clu d es a n h a rm o n ic effects b ased on th e M orse p o ten tial p a ra m e te rs . A nalytical ex p ressio n s for DW factor or second c u m u lan t for different atom ic shells have been derived. N u m erical re s u lts for Cu a re com pared to those obtained by the an h arm o n ic E in ste in m odel. B oth they co n tain zero-point energy contribution a t low te m p e ra tu re s an d lin e arly depend on th e te m p e ra tu re a t high te m p e ra tu re s as the classical lim it v alu es, th u s reflectin g th e ir ex p erim en tal re s u lt behaviors [3, 6, 10].
2. Formalism
2.1. A n h a r m o n i c E in s te i n m o d e l fo r m o n a to m ic c h a in :
In th is case we u se th e M orse p o te n tia l expanded to th e 3rd order ab o u t its eq u ilib riu m
V (x ) = D ( e - 2ax - 2 e -ax) « Z) ( - 1 + a V - a3* 3 + . . . ) , (1) 40
A n h a r m o n ic C o r r e la te d D e b ye M o d e l D e b y e -W a ller F a c t o r... 41 where X is scaled relative deviation defined by X = (I - a ) I a , I is in stan tan eou s bond length between /1th and th atoms and a is the equilibrium bond len gth between these two atom s. The M orse p o ten tial p a ra m e te r a d escrib es th e w id th of the p o ten tial a n d D m ea su re s th e dissociation energy.
T ak in g in to account up to th e 3rd o rd er we o b ta in th e p o te n tia l
V(x) = - D + D a 2x 2 - D a 3Xs. (2)
U sing th e d efin itio n y = x - a or x = y + a w ith a = (X) Eq. (2) is re s u lte d as V (x) = - D + D a 2 ( y 2 +2ay + a 2)j - D a 3ị y 3 + 3a y 2 + 3 a 2y + a3 j
= ỊD a2 - 3 a D a 3 j y 2 - D a 3y3 + 2aD a2ị l - —a a j y + Ị -D + D a 2a 2 - D a 3a3 j. (3) In th e an h arm o n ic E in ste in m odel th e in te ra c tio n p o te n tia l is g iven by
v (y) = ^A s y2- *3y3 + v ,(a ). (4)
Com paring Eq. (3) to Eq. (4) we obtain the spring c o n sta n t k s, cubic p a ra m e te r k 3 ks = 2( D a 2 - 3 a D a 3 j » 2D a 2 = M o )\\ = Z)a3, (5) E in ste in frequency a n d te m p e ra tu re
(6) w here M is m ass of v ib ra tin g atom .
Atom ic v ib ratio n is q u a n tize d as phonon a n d a n h a rm o n ic ity is th e re s u lt of phonon- phonon in te ra c tio n th a t is w hy we d escrib e y by a n ih ila tio n à an d creatio n â + o p e ra to rs
(7) U sing th e c alcu latio n procedure as in [16] we o b ta in DW fa c to r or second c u m u la n t
2 2 1 + 2 2
ơ ° * = a (8)
w here 2 = e~°E' T is te m p e ra tu re v ariab le.
2.2. A n h a rm o n ic c o r r e la te d Debye m odel fo r m o n a to m ic chain:
In th is case th e M orse p o te n tia l Eq. (2) h a s th e form
V (x) = Dcc2x2 - D a 3x 3 =V0 +VC, (9) w h ere the h arm o n ic co n trib u tio n
42 Nguyen Van H u n g y Nguyen B ao T r u n g
V0 = D a * x > = D a > ỵ (u n+l- u ny , (10)
an d th e a n h a rm o n ic one
( 11) a re d escrib ed in te rm s of th e d isp lac em e n t un of n th atom . T hese d isp lac em e n ts a re re la te d to ph o n o n d isp la c e m e n t o p e ra to rs Ak as show n by [18, 19]
ikd n
A k , (12)
U n = * I 6'
2-NM k Jco{k) w h e re Ak sa tis fie s th e follow ing re la tio n
A k = A*k; [Aa ,A4. ] = 0 , (13)
a n d M is m ass of com posite ato m s, ũ)(k) is th e phonon sp ectru m of cry stal m o m en ta k, a n d N is th e n u m b e r of u n it cells in th e chain.
B ased on th e c alcu latio n pro ced u re p resen te d in [20, 21] we o b tain ed th e d isp e rsio n re la tio n co(k) as
c o (k ) = 2, 2D à
M sin \k d \ < n (14)
T he s u b s titu tio n of Eq. (12) in to Eq. (10) yields V0 = D a 2Y - ? — Y
„ 2 N M Ỷ
f ikd (n +1) p ik d n N A
f ikd(n+1) p ik d n N
e A
+
v \ l o ( k ) * y[ VSw
D a 2h Y 2JV M à D a 2h
— (el
( b \ \
ikd (n +1) _ 0 ikd n j Ịg-iA < i(7 i+ l) _ 0 ~ ikdn c o (k )
- e )A A * (15)
2 N M
N ^ 2- -. e-.. - e~ikd A kA k A la
c o (k ) M co (k) k^-k
Now we calculate DW factor for the nearest neighbor correlation denoted by index 1
f f i = ( ^ ) o =((“■•' - “- O o A pplying th e above re s u lts to Eq. (16) we obtain
w h ere [19]
(16)
(17)
A n h a r m o n ic C o r r e la te d D ebye M odel D e b y e -W a lle r F a c to r .. 43 (AkAk.)o = ( A k(0)Ak.(0)) = -G°k r (0), Ah(t) = etH°Ake-tH\
G°*.(r) =
(Ak(r)Ak,(0))0= - < w { K + + (nt )e ^ < * )Ị,
(n*) = eh<o(k)lkBT _ J ’ and is B o ltzm an n c o n stan t.
S u b s itu tin g Eqs. (19, 20) in to Eq. (18) we o b ta in (AkAk')o = {(nA + 1) + ("*)}
(18) (19) (
20
)+ 1 +
húỉ(k) e - 1>
= -<?
1 + e
hgj(k)
kaT (21)
SO th a t th e DW facto r is given by
/ ihrt w \
(e
- 1 )(e ~ 1) 1 + e
k“T _ hv 1 - cos
kd1 +
e*®T
hũỉ(k) knT h
— ỵ
2 N M Ỵ J a t k ) M - k )
2 sin2
h y
N M Ỷ
k d { 2
l - e B hứỉ(k)
N M Ỷ co(k)
knT 1 - e
„ Ỉ2D a 2 ( k d \ ho){k)
H u Sin
to1 l - e k*T
l + e
*®r ft v .
( k d ) - ^ = N ^ Ĩ 5 ^ Ỷ S m { 2 Ì1 -
e Bho)(k) l + e
*ar
(22)
hp)(k) 1 - e *B
w hich is d escribed in te rm s of U)(k) as
« * - ị i
hta(k) 2
_ J_ y
hco(k)1 + e *®r
~
N k 4 D a 2-ísíiì
1- e *-r
(23)
For th e lim it of larg e N th e su m m atio n o v er k can be rep la ce d by th e co rresp o n d in g in te rg ra l
_ ho>(k) n hcu(k)
hd ... ..
a2 = (24)
£ haỉ(k) X
dỊ . ( A c O l + e *flT JL d df
s in r ^ r — ; 7ĨT d k = — . \ _ ,
0
I
2J
1 _ “B1*•
0J
4 0«
X r~hú)(k)1 - e "fli 1 - e
T his expression for DW factor approaches Eq. (8) obtained by th e anharm onic E instein model if all atom s v ib rate by th e sam e frequency , i., e., Ct)(k) = 0)E = co n st.
I t is u su a lly to co n sid er th e high te m p e ra tu re (HT) a n d low te m p e ra tu re (LT) lim it u sin g ou r d eriv ed Debye te m p e ra tu re
44 Nguyen Van H ung, Nguyen B ao T ru n g
h(ú{k)
In th e HT lim it ( T » 0 D) sin ce e we o b tain from
kBT .. hco(k)
kBT (26)
Ơ
, hcojk) 1
2 - Ểl k g T ^ _ dr d k BT
~ 1Ĩ 54Z)a2 fico(k) ~ J ị n D a 2 k BT
kBT
2D a 2 . Í k d --- sin
M
dife, (27)
th e HT lim it e q u atio n , d k BT
Ơ H T -
2 n D a 2 k
7 d + to
0 7 ĨC C \] 2 M D
COS k d
T Vd= kB T _____2h —
0 2 D a 2 7rayj2MD
(28) 'B
2Z)a
'B 2 D a
w hich is lin e a rly p ro p o rtio n a l to th e te m p e ra tu re T.
In th e LT lim it ( r « 0 ) i t is ap p ro x im ate d e~hứ)(k)ỉkBT« 0 so t h a t DW factor is given by
/T *
_ 2 d 5f /j<y(/e) _ dr
°7T * — _ „ d k =
* 4 D a 0J
dh 2D a
4 x D a 2 V M
sin M
V 2 , d k
2 n D a 2
*0. (29) w hich c o n ta in s zero -p o in t e n e ry c o n trib u tio n .
T he d eriv ed m odel c an be g en eralized to c a lc u la te DW facto rs including co rrelatio n w ith o th e r a to m ic sh e lls p = 2, 3, 4, 5,...
(âỉ)o
(30)A pplying ex p ressio n for unto Eq. (17) we o b tain Mk+k^dn
\ 2N M f t . Jco(k)M k')
i(k+k' )an
2 N M f t , Ja>(k)Mk') h
= [ e ipkd - l ) ( e ipk'd - 1
)A*AẲ
l ) { A k A k ' ) o i p k ' d
(31)
y N M Ỷ
ho)(k)
1 - COS p k d 1 + e *flT
C ứ ( k )
1 - e
hũ)(k)
koT
w hich is rep laced by th e co rre sp o n d in g in te rg ra l for th e lim it of la rg e N
n hoj(k)
2 / 2 \ _ hd dr L - C O S p k d 1 + e ksT ơp \ p / o ~ n M J co(k) hcủ(k)
d k . (32)
1- e
A n h a r m o n ic C o r r e l a t e d D e b y e M o d e l D e b ye -W a ller F a c t o r.. 45 3. N u m e r ic a l r e s u lt s a n d d is c u s s io n
Now we a p p ly th e e x p re ssio n s derived in p rev io u s sectio n to n u m erical calcu latio n for Cu. M orse p o te n tia l p a ra m e te rs h a v e b e e n ta k e n from [22, 23].
F igure 1 show s t h a t th e second c u m u la n t or DW facto r of Cu c a lc u la te d by p re se n t an h arm o n ic c o rre la te d D ebye m odel an d by a n h a rm o n ic E in s te in m odel contain zero-point e n e rg y c o n trib u tio n a t low te m p e ra tu re s a n d lin e a rly d e p e n d s on th e te m p e ra tu re a t h ig h te m p e r a tu re as th e cla asic al lim it v a lu e s. T hey reflect fu n d a m e n ta l th e o re tic a l [4, 24] a n d e x p erim e n tal [3, 6, 10] re s u lt.
0.026
002
«*-t—
M
Ĩ 0015 4
n = 1 n « 2 n = 3 n = 4
/
F i g u r e 1: C o m p ariso n of DW factors for Cu c a lc u la te d by an h arm o n ic c o rre la te d D ebye, E in ste in m odels a n d by c la ssic a l m ethod.
F ig u re 2 illu s tr a te s th e te m p e ra tu re d e p en d e n ce of DW facto r of Cu c a lc u la te d by p re s e n t an h arm o n ic c o rre la te d D ebye m odel for different atom ic sh ells. I t is show n th a t th e DW facto r in clu d in g co rrelatio n w ith th e ato m s located fa r from th e absorbing atom in c re a se s as th e shell n u m b er n = p in c re a se s. B u t in th is case the term containing COS( p k d )
becomes a fa s tly o sc illatin g fu n ctio n of k so th a t its c o n trib u tio n to c o rre latio n effect d iscreases. I t is also c le ar th a t
ơ ị will be divergent w hen p -> 00 so that it is red u ced to th e one for vib ratio n w ith o u t c o rre latio n , th a t is why we often ta k e th e c o rre latio n only w ith n eig h b o rin g a to m s in a ‘sm all c lu ste r as in th e a n h a rm o n ic co rre lated E in stein m odel [16].
001
0 006
/ :
/ : ys
/ y
y •
0 100 203 300 400 500 600 700
T(K)
F ig u re 2: T e m p e r a tu r e dep en d en ce of DW factors ơ 2p (T) of C u c alcu late d by p re s e n t a n h a rm o n ic c o rre la te d Debye model for d iffe re n t ato m ic sh e lls
rt = p = l,2,3,4.
46 N guyen Van Hungy Nguyen B ao T ru n g 4. C o n c lu sio n s
In th is w ork a n a n h a rm o n ic c o rre la te d Debye m odel h a s b een developed in clu d in g d isp e rsio n c o n sid eratio n . A n aly tical ex p ressio n for th e DW facto r h as been derived w hich reflects b e h av io rs of fu n d a m e n ta l th e o re tic a l a n d e x p e rim e n ta l re s u lts of th is q u a n tity .
N um erical resu lts for Cu approach those obtained by th e anharm onic E instein model containing zero-point energy contribution a t low tem p eratu re, a n d a t high tem p eratu res both they linearly depend on th e tem p eratu re as th e classical lim it results.
A n h arm o n ic c o rre la te d D ebye m odel is m ore co m p licated th a n th e E in ste in one due to d isp e rsio n c o n sid era tio n , b u t i t h a s th e a d v a n ta g e for re s e a rc h w hen ato m s in th e su b s ta n c e v ib ra te by d ifferen t frequencies.
A c k n o w le d g e m e n t: T h is w ork is su p p o rte d in p a r t by th e basic science re se a rc h project No. 4 058 06.
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