They depend on the number of dopant atoms and approach those derived by anharmonic correlated Einstein model, if all dopant atoms are taken out or replacing all the host atoms.. Numerica
Trang 1VNU JOURNAL OF SCIENCE, Mathematics - Physics, T.XXII, N03, 2006
S T U D Y O F E X A FS C U M U LA N TS O F FCC CRYSTALS
C O N T A IN IN G N D O P A N T ATOM S
N g u y e n V a n H ư n g , T r a n T ru n g D ung, N g u y en C ong T o an
Department o f Physics, College o f Science, VNU
A B ST R A C T A n e w p r o c e d u r e f o r d e s c r i p t i o n a n d c a l c u l a t i o n o f t h e EXAFS (Extended X-ray Absorption Fine Structure) cumulants for fee crystals containing an abitrary number n of dopant atoms have been developed Analytical expressions for the l 8t, 2nd and 3rd cumulants have been derived They depend on the number of dopant atoms and approach those derived by anharmonic correlated Einstein model, if all dopant atoms are taken out or replacing all the host atoms Numerical results for Cu doped by Ni atoms based
on the Morse potential show significant dependence of thermodynamic parameters of the substance on the number of dopant atoms and a reasonable agreement with experiment
1 I n t r o d u c t i o n
C um ulant expansion approach has been developed [1, 2] to include anharm onic effects in th e EXAFS procedure These anharm onic effects are contained in the first cu m u lan t or net therm al expansion, the second cum ulant or Debye-W aller factors, th e th ird cum ulant, and the therm al expansion expansion coefficient, which are investigated intensively in the EXAFS experim ent and theory [1-14] It is also very im p o rtan t to study therm odynam ic properties of m aterials containing dopant atom s [10, 14, 16] Some investigations for crystals containing one dopant atom have been perform ed [10, 14, 16] But norm ally more than one atom can be doped into a crystal This case can lead to developing procedures for studying therm odynam ic properties of alloys with nano stru ctu re which are often sem iconductors containing some components with different atomic sourses The
effective interatom ic potential and local force constant for fee crystals containing n
dopant atom ts have been studied [16]
The purpose of th is work is following our previous one [16] to develop a new procedure for description and calculation of the EXAFS cum ulants and other
therm odynam ic p a ra m eters of fee crystals containing some (n) dopant atoms, where
one dopant atom [10, 14, 16] is only a special case of this theory Our development
IS derivation of the analytical expressions for the correlated E instein frequencies and tem p eratu re, for the 1st, 2nd and 3rd cum ulants, where the host atom is denoted
by the letter H an d th e dopant atom by the le tte r D All these expressions are
different if the num ber of dopant atom s changes The resu lts in the case if all the dopant atom s are tak e n out or if all the host atom s are replaced by the dopant atom s are reduced to those derived by using the anharm onic correlated Einstein model [8] for th e pure m aterials N um erical calculations have been carried out for
31
Trang 232 Nguyen Van Hung, T ran Trung D ung, Nguyen Cong Toan
Cu doped by one or m ore Ni d o p an t atom s, an d th e re s u lts a re com pared to th o se of
th e p u re m a te ria ls a n d som e to ex p erim e n t d ed u cted from th e m easu red M orse
p o ten tial p a ra m e te rs [15]
2 F o rm a lism
2.1 A n h a r m o n i c effe ctive p o t e n t i a l , l o c a l f o r c e c o n s t a n t a n d d e r i v a t i o n
o f c o r r e l a t e 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
Follow ing [16] th e an h arm o n ic c o rrelated in te ra to m ic effective p o ten tial of a
fee c ry stal doped by n ato m s from a n o th e r source is given by
as a function of th e d isp lac em e n t x = r - r 0 for r an d r0 bein g th e in sta n ta n e o u s and
eq u ilib riu m d ista n c e s b etw een ab so rb er an d b a c k s c a tte re r atom s
U sing th e d efin itio n s [8] y = X — a , a = (x) Eq (1) is changed into
co n tain in g a n effective local force c o n sta n t k ejỵ
and anharm onic effective factor k
A4HMp
M ff + M p
3 eff
(y + m ỹ
(3)
(4)
involving co n trib u tio n s of im m e d ia te atom ic neig h b o rs, w here
A = m 2(l + 4<50B- r n + ^ ) ) C = 3 ^ + <513n - ^ + i [ ( l - 4 , nX n - l ) + r n( 9 - /t)
B = ị p - 5S0n + SÍOn - ỏnn + (1 + m f (1 - y n)] + ^ [(9 - n ) ( \ - ỏ 0n- ỵ n) + m 2( 4 - 4Slĩn - Pn - 4<50n)]
(5)
A ĩ = m ĩ ạ + AS0n- y n + ị - ) , c 3 = c,
o
5 3 = 2 D - ỏ ũn + S \0n - ỏ Un + 0 + m f 0 - 7n - Ỗ0n ) ] + ị [ ( 9 - n ) ( ] - <*0» - ĩ n )
+ m \ A - A 8 nn- P n - A 5 ữn)}
(6)
t £ ' 2 ( £ n A £ , I f , r , f _ ^ D ^ O n H ^ D ) / * 7 \
ĩ n - 0 \ 0 n + ^ 1 1 n + ơ 12n + <?I3n > P n - 4 ^1 ft + 3 ^ 2 n + 2<^3n + <5An m - 777 -7 7 - ■ ( ' )
M H +Yn{MD~ M H)
Trang 3S tu d y o f EXAFS C u m u la n ts of 33
U sing th e a n h arm o n ic effective local force c o n s ta n t Eq (3) th e co rrelated
E in ste in frequency C 0 %D an d te m p e ra tu re 0j!D a re given by
w here k B is th e B o ltzm an n co n stan t.
M orse p o te n tia l p a ra m e te rs for th e case w ith doping a re o b tain ed by an
a v erag in g calcu latio n from th o se for th e h o st (H) a n d th e doping (D) cry stals expanded to th e th ird o rd er a ro u n d it equilibrium
D enoting th e eq u ilib riu m bond len g th of th e h o st atom by r0H , of th e doping atom by r0D a n d b etw een th e ho st an d th e doping atom by r0HD a n d solving th e
eq u atio n sy stem of av erag in g procedure, th e M orse p o ten tial
p a ra m e te r D hd c h a ra c te riz in g dissociation energy, th e M orse p o te n tia l p a ra m e te r
a HD c h a ra c te riz in g th e w idth of th e p o ten tial, a n d th e eq u ilib riu m bondlength
betw een H a n d D ato m s r0HD a re re su lte d as
They w ill be u se d for calcu latio n of th erm o d y n am ic p a ra m e te rs of c ry sta ls in
th e doping case
2.2 D e r i v a t i o n o f a n a l y t i c a l e x p re s si o n s f o r E XA FS c u m u l a n t s
M aking u se of q u a n tu m s ta tis tic a l m ethods [19] th e p h y sical q u a n tity is
d eterm in ed by a n a v erag in g procedure u sin g can o n ical p a rtitio n fu n ctio n z and
sta tistic a l d e n sity m a trix p , e g.,
Atomic v ib ra tio n s a re q u a n tize d in term s of 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 e x p r e s s ^ i n te rm s of phonon
a n n ih ila tio n a n d c re a tio n o p e ra to rs, à a n d a +, resp ectiv ely
(8)
V(x) = DH m ( ẽ laH^ - 2 /" » ™ * ) Dm m ( - 1 + a ị m x ‘ - a ị m x 3 + ) (9)
3
D}ja H2 + DDa D2 +
D rr 2 I ^ r°D r°H)a H2aD2DHDD(aD - a H)
u D a D +
-(1 2)
(1 3 )
Trang 434 Nguyen Van Hungy T ran T ru n g Dung, Nguyen Cong Toan
k ,ầMD
sa tisfy in g re la tio n s
[ a ,a +] = l , a + \ n > = J n + l \ n + l > , a \ n > = yfn \ n - l > , (15)
an d u se th e h a rm o n ic o scillato r s ta te I nj as th e e ig e n s ta te w ith th e eig en v alu e
E n = nhcủ£ D , ig n o rin g th e zero-point energy for convenience.
U sin g th e above re s u lts for c o rre lated ato m ic v ib ra tio n a n d th e first-o rd e r
th e rm o d y n a m ic p e rtu rb a tio n th eo ry [19] co n sid erin g th e a n h a rm o n ic term in th e
p o te n tia l Eq (2) as a p e rtu rb a tio n s v due to th e w eak a n h a rm o n ic ity in EXAFS
B ased on th e p ro ce d u re d escribed by Eqs (13-15) we d e riv e d th e c u m u la n ts
0 =<y>= ^ T r ( p y ) » -J-Tr ( ỏpy) ,
ơ (2) =< y 2 >= Ị - T r ( p y 2) « - ^ - T r ( p 0y 2) ,
ơ (3) = < y 3 > = ị T r ( p y 3) * \ - T r i ô p y3 ) ,
(18)
w h ere p0 , Z Q a re u n p e rtu rb e d q u a n titie s a n d <5p, s z th e p e rtu rb a tio n s of th e
s ta tis tic a l d e n s ity m a trix a n d th e canonical p a rtitio n fu n ctio n , resp ectiv ely
T he second c u m u lan d or M ean S q u a re R e la tiv e D isp la ce m e n t (MSRD)
d e scrib in g D ebye-W aller facto r (DWF) h a s been d eriv e d
Ơ2 = < y 2 > w Ả - T r { p ữy 2) = J ^ z n < n \ y 2 \ n > , z = e e'E IT (20)
S u b s titu tin g th e c a lcu late d m a trix e le m e n t, i e.,
< n I y 2 I n>= (2n + 1)(702 (21)
in to Eq (20) we o b tain ed
z
, 1 \ n 1 _ 2 Í 1 , 2 z \ h l + z 2 l + z - 2 _ ^ l 0 ) E D /Q Q \
„ z 0 I 1" 2 (1-z) J 2yJ{jkeJĨ 1 -2 1 -z 2kcff
T he odd m o m en ts < y > a n d < _y3 > h av e b een c a lc u la te d u sin g th e g en eral
ex p ressio n
1 1 v-n e- P E" - e - P E*
< y m > = — Tr{8pym) = — Y - - - < n \ s u E \ n'X n' \ ym \ n >, /3 = ỉ / k BT (23)
Zo z o t ? £" ~ en'
Trang 5S tu d y of EXAFS C u m u la n ts of 35
Since < y > c o n ta in s < n ' \ y \ n > w hich is d ifferen t from zero only for n' = n + 1
th a t is why from Eq (22) we o b ta in
_ 1 V
< y > = y L
^ 0 n
-hũ) e-pEn
+
hco
< n I s v 171 + 1 > < 1 Ĩ + \\ y \ n >
< n I s v \ n - l x n - l \ y \ n >
(24)
C a lc u la tin g th e m a trix e le m e n ts
< n \ y \ n + l>= ơ 0(n + 1)1/2, < n I y 3 I n +1 >= 3<703(rt + 1)3/2
an d satisfy in g th e c o n d itio n Eq (16) we o b tain ed th e 1st c u m u la n t
<7(1) = a 3 ^ 3 eff 1 + 2 _ - = CĨA -= -—— ơ , ƠQ = -(1) ^ + <2:_ 2 (1) _ 3 k 3eff
1 - 2
(25)
(26)
S in c e < y 3 > c o n ta in s < n.1 I Ai' >< n ' I ;y3 I n > w hich is d iffe re n t from zero only
for n' = n ± 1, ft’ = ft ± 3 so t h a t from Eq (22) we o b tain ed
„ e - P £ n _
^ -< n I 171 +1 >< ft + 1 1 y 3 I n >
cr(3) =< y 3 >= —
z n
-hco e-P£n _ e'^n-1 hco < n I <5V I ft - 1 >< n - 1 1 y 3 I n >
„ * - / f e » _ p - / t e „ +3
+ ^ r - z - < n I § y I ft + 3 >< ft + 3 I ;y3 I n >
-3hco
e - P £ n _ g - / t e « - 3
3hco < n I J V 171 - 3 >< 71 - 3 I y 3 I rc >
(27)
We c a lc u la te d th e m a trix e le m e n t
< n \ y 3 I n + 3 >= (cr0)3[(/i + l)(n + 2)(rc + 3)]1/2
S u b s titu tin g E qs (25, 28) in to Eq (27) we o b tain ed th e 3rd c u m u la n t as
(28)
h*h
V 1 - Z j
.(3) 1 + I 0z + z 2 (3)
( 1 - * )
2
k 3 e f f [ hcúE D ) _ k :ieffh ứ )
HD
CTq •
(29)
Note t h a t in th e above ex p ressio n s , (To , ctq3^ a re zero -p o in t c o n trib u tio n s
to th e 1st, 2nd, a n d 3rd c u m u la n ts , respectively, a n d w hen th e d o p in g a to m s a re tak e n from th e h o st m a te ria l a ll th e above ex p ressio n s will be red u c ed to th o se of
th e p u re m a te ria l [8, 20]
Trang 63 N u m e r ic a l r e s u lts a n d c o m p a r iso n to e x p e r im e n t
T a b le 1: C alcu lated v a lu e s of keff a n d k 3eff of Cu doped by n = 0, 1, 4, 8, 10, 13 atom s of Ni com pared to e x p erim e n t [15]
Now we apply ex p ressio n s derived in th e previous section to n u m erica l
calculation for Cu doped by n atom s of Ni M orse potential p aram eters for Cu and Ni
have been calculated by procedured p resented in [17, 18] They are used for calculation
of Morse p a ra m eters for Cu doped by Ni The resu lts are presented in Table 1 com pared to experim ent ex tracted from th e m easured M orse potential param eters The case n = 0 corresponds to th e pure Cu and the case n = 13 to the pure Ni because all Cu atom s are replaced by th e Ni atom s All they are found to be in good agreem ent w ith experim ent extracted from m easured M orse p a ra m e te rs [15]
Figure 1 shows te m p e ra tu re dependence of the 1st cum ulant or net th erm al expansion c r^ (r) of Cu doped by one Ni atom com pared to experim ent extracted from the m easured M orse p o ten tial p a ra m eters [15] (a) and by n =0, 1, 4, 13 Ni atom s (6)
Figure 2 illu stra te s th e tem perature dependence o f the calculated 2nd cumulant ơ 2(t) or
DW F o f Cu doped by n = 1 Ni atom com pared to experim ent extracted from the
measured M orse potential param eters [15] (a) and by n = 0, 1, 4, 13 atom s o f Ni (b)
Figure 3 shows the tem perature dependence o f the calculated 3rd cum ulant o f Cu doped
by one Ni atom com pared to experim ent extracted from m easured M orse potential
param eters [15] (a ) and by n = 0, 1 , 4 , 13 Ni atom s (6) All the above Figures contain
zero-point contributions and satisfy all their fundam ental properties, e g., a t high- tem p eratu res th e 1st an d 2nd cu m u lan ts are linearly proportional to the tem p era tu re and the 3rd cu m u lan t to th e square of tem p eratu re They provide a reasonable agreem ent w ith experim ent for th e case n = 1 doping atom
Figure 1: C alc u lated 1st c u m u la n t <x^(r) of Cu doped by n = l Ni atom com pared to
e x p e rim e n t [15] (a) a n d by n =0, 1, 4, 13 Ni atom s (b)
Trang 7S t u d y o f EXAFS Cu m u la n ts of 37
Figure 2: calcu late d 2nd c u m u la n t ơ 2( t ) or DW F of Cu doped by n = 1 Ni atom
co m p ared to e x p erim en t [15] (a) an d by n =0, 1, 4, 13 Ni ato m s (b)
Figure 3: C a lc u la ted 3rd c u m u la n t of Cu doped by n = l Ni ato m com pared to
e x p e rim e n t [15] (a) an d by n =0, 1, 4, 13 Ni ato m s (b)
4 C o n c lu ss io n s
T his w ork h a s developed a new procedure for d escrip tio n a n d calcu latio n of
th e c o rre lated E in ste in frequency and te m p e ra tu re , th e 1st, 2nd, a n d 3rd cu m u lan ts
for a fee c ry sta l doped by an a rb itr a r y n u m b er n of ato m s from a n o th e r m ateria l.
D erived ex p ressio n s of th e considered q u a n titie s a p p ro ach th o se derived by
u sin g th e a n h a rm o n ic co rre lated E in ste in m odel for th e p u re m a te ria ls w hich can
be considered as a special case of p re s e n t p rocedure T hey sa tisfy all th e ir
fu n d a m e n ta l p ro p e rtie s a n d provide a reaso n ab le a g re e m e n t w ith ex p erim en t
e x tra cted from m ea su re d M orse p o te n tia l p a ra m e te rs
T his m ethod considered for sm all clu ste r can be g e n eralize d for th e whole
c ry stal so t h a t from th e p re s e n t procedure one can develop a m ethod for description
Trang 838 Nguyen Van H ung, Tran T ru n g D ung, Nguyen Cong Toan
an d c a lc u la tio n of th e th erm o d y n am ic p a ra m e te rs of a n alloy c o n sistin g of d ifferen t
p e rc e n ta g e of c o n s titu e n t elem en ts
A c k n o w l e d g e m e n t s T he a u th o rs th a n k Prof D M P e a se (U n iv ersity of
C o n n ecticu t) for u se fu l d iscu ssio n s an d com m ents T h is w ork is su p p o rte d in p a r t by
th e Bisic S cience R esearch P roject No 4 058 06 a n d th e sp ecial re se a rc h project of VNU H anoi No Q G 05.04
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