DSpace at VNU: Thermodynamic and correlation effects in atomic vibration of bcc crystals containing dopant atom

Analytical expression for the D isplacem ent-displacem ent Correlation Function DCF CR has been derived based on the derived Mean Square Relative D isplacem ent MSRD 2 Ơ2 and Mean Square

VNU JOURNAL OF SCIENCE, Mathematics - Physics, T.XXI, N04, 2005

THERMODYNAMIC AND CORRELATION EFFECTS IN ATOMIC VIBRATION OF BCC CRYSTALS CONTAINING DOPANT ATOM

N g u y e n V a n H u n g , Ho K h a c H i e u a n d N g u y e n C o n g T o a n

D epartm ent o f Physics, College o f Science, V N Ư Hanoi

A b s tr a c t: A new procedure for calculation and analysis of therm odynam ic and

c o r re la tio n e ffe c ts o f bcc c r y s ta ls u n d er in flu e n c e of d o p a n t atom in th e X-ray

Absorption Fine S tru ctu re (XAFS) has been developed Analytical expression

for the D isplacem ent-displacem ent Correlation Function (DCF) CR has been

derived based on the derived Mean Square Relative D isplacem ent (MSRD)

2

Ơ2 and Mean Square D isplacem ent (MSD) u of bcc crystals containing dopant

atom Num erical calculations have been carried out for Fe doped by w and by

Cr atom They are found to be in good agreem ent w ith experim ent

1 I n t r o d u c t i o n

T h erm odynam ic effects of atom ic v ib ratio n have been oft stu d ied by th e XAFS procedure because th e e m itte d photoelectron is tr a n s fe r r e d an d s c a tte r e d in the

v ib ratin g atom ic e n v iro n m e n t before in te rfe rin g w ith the out going photoelectron

Therefore, it is n ecessary to ta k e a th e rm a l a v erag in g ị e ^ ki<J ^ of th e photoelectron function leadin g to th e Debye-W aller factor DWF = e ' w h e r e k is th e wave

num ber Since th is factor is m e a n t to account for th e th e r m a l v ib ra tio n s of the

atom s about th e ir e q u ilib riu m sites R °ị, someone a ssu m e t h a t th e q u a n tity Ơj is

identical w ith the MSD [1] B u t th e oscillatory motion of n e arb y a to m s is relativ e so

t h a t including c o rrelatio n effect is n ecessary [2-6] In th is case ơ 2 ị is th e MSRD

containing MSD an d DCF The doping effects have been in v estig ate d to com pare the XAFS re s u lts to th e M o ssb a u er stu d ies [6] a n d to consider th e ir influence on th e XAFS c u m u la n ts of fee [9] a n d bcc [10] crystals

The purpose of th is work is to develop a new procedure for calcu latio n and

analysis of th e DCF (Cfl), th e MSD (u2) for atom ic v ib ratio n in bcc c ry s ta ls u n d e r

influence of d o p an t atom E xpressions for th ese q u a n titie s hav e b een derived The effective in te rac tio n p o te n tia l of th e system h as been considered by ta k in g into account th e influences of n e a r e s t atomic neighbors based on th e an h arm o n ic correlated E in stein model [4] This p o ten tial c o n tain s th e M orse p o ten tial

ch arac terizin g th e in te ra c tio n of each p a ir of atom s N u m erical calcu latio n s have

been carried ou t for Fe doped by w an d by Cr The c alcu lated It2f ơ 2 y CR and ratios CR l u 2 , CK/ ơ 2 , which a re oft stu d ied in XAFS tech n iq u e [2], of th ese crystals

have been analyzed They a re found to be in good a g re e m e n t w ith th e ir

ex p erim en tal valu es d ed u cted from th e m e a su re d Morse p o ten tial p a r a m e t e r s [7]

26

2 F o r m a lis m

In th is in v es tig a tio n for XAFS process we consider th e d o p a n t (D) atom as

ab so rb er a n d th e host (H) atom as s c a tte re r so t h a t we w rite th e XAFS function in

th e form

X = Xo(e2ikA) ■ , A = R ( u 5 - U/4) , R = R / |R |, (1)

w here uv a n d a re th e s c a tte r e r or host atom a n d c e n tra l-a to m disp lacem en t respectively

To v a lu a te Eq (1) we m ake use of th e well-known re la tio n [11]

(2)

T h e r m o d y n a m i c a n d C o r r e l a t i o n Effects in A t o m i c V i b r a t i o n o f B C C 27

to o b tain Eq (1) in th e form

so t h a t th e t h e r m a l v ib ra tio n effect in XAFS is d e fin e d by Ơ 2 .

For c ry s ta ls co n ta in in g doping atom by u sin g Eq (1) th e MSRD is given by

H e re we d e fin e d th e M SD fu n c tio n for d o p a n t (D) a s a b s o rb e r (A) u] a n d for th e

h o s t (H) a to m a s s c a t te r e r U2 S h a v in g a d o p a n t a s n e a r e s t n e ig h b o r a s

SO t h a t th e D C F is g iv en by

^ c le a r t h a t a11 at0 m s v ib rate u n d e r influence of th e neighboring

e n v iro n m e n t T a k in g into account th e influences of th e n e a r e s t atom ic neighbors

th e a n h a r m o n ic effective in te rac tio n p otential for singly v ib ra tin g atom is given by (ignoring t h e o v erall constant):

- for a b s o rb e r (A):

u eff M = s V H1) (-*R o i • R ( ) ; ) = ị keffỵ2 + k *X* + • ■ • , X = r - r0 , (7 )

• for s c a t t e r e r (S):

m th r a n d r„ a s t h e in s ta n ta n e o u s a n d e q u ilib riu m bond le n g th s b e tw e e n a b s o rb e r

ind b a c k s c a tte r e r , respectively

2 8 N g u y e n Van H u n g, Ho K h a c Hieu, N g u y e n Con g T o a n

By u sin g the definitions V = X - a , a = (r - r()), we o b tain Eqs (7, 8) in th e form

Applying th e M orse p o te n tia ls e x p an d ed to th e t h ir d o rd er a b o u t its m in im u m

(10)

^ H , M ) = DHD[e~2a'">x - 2 e ~ a"»x]j = D ( - l + a 2 HI)x 2 - a ị n X * +•■•),

UH(X) = Dh [e~2a"x - 2e~aHx Ị s d (-1 + a ị x 2 - a ị x ' +•••),

Dn + D ft

f) — _— _' — — (V —

D na n + D u Ơ Du D 3 H_ H 1/2

\ D d + d h

(1 1)

(12)

for Eqs (7, 8) we o b tain th e effective local force c o n s ta n ts kejj and th e cubic

an h arm o n ic p a r a m e te r s ky for th e a b so rb er (A) an d for th e b a c k s c a tte re r (S)

keff - — DHDa~HD - M dù ) 2 d , - 2DHDa HDi

^HDa i w + T ^ H a H - M s ^ ị , ky = - ( d HDơ]fD + D //Ơ/3/ )

(13)

(14)

U sing Eqs (13, 14) we c alcu late the E in ste in frequencies an d te m p e r a tu re s for ab so rb er a n d b a c k s c a tte re r

16

DHDa HD > ~ h(0A

* S = J 2 DHDa HD + T DHa H / M s , ớ £ — hco£ / k ỊỊ J

(15)

(16)

where k B is B o ltzm an n c o n stan t, M A an d M s a re th e m asses of ab so rb er and

back sc attere r

The atom ic v ib ratio n is q u a n tiz e d as phonon, t h a t is why we express in

term s of a n n ih ila tio n a n d creatio n o p e ra to rs, a a n d a +, i e.,

hco

(» ~ + \ 2 _ ỉ ư o A,s

ma0[a + a j, aị = — T s

ZKcff

(17)

and use th e h arm o n ic oscilator s ta te I A?) as th e e ig e n sta te w ith the eigenvalue

£„ = nhcoA s , ignoring th e zero:point energy for convenience.

U sing th e q u a n tu m s ta tis tic a l method, w h ere we used th e statistica l density

m atrix z an d th e u n p e rtu r b e d canonical p a rtitio n function p {)

z = Trpo = ^ Qxp{~nf i hú)AtS)= ^ Jz A,s = -’ /3 = \ I k BT , Z A S = e ° ' s 1 , (18)

to d eterm in e th e MSD function

“i s = ( y 2) * J Tr(p0y 2 )= I £ exp(- nphcoA Sx » |^ 2|«) =

n

= 1 a 2 ữ { \ - z A S ) ỵ ^ ( \ + n) z n AS =

n

T h e r m o d y n a m i c a n d C o r r e l a t i o n Effects in A t o m i c V i b r a t i o n o f B C C , 29

hcửA,S \ + ZA,S

K 5 l ~ z ™

(19)

From Eqs (13, 14, 19) we obtain th e MSD for bcc c ry sta ls co n ta in in g dopant atom tor ab so rb er an d b a c k sc a tte re r

u 2 - ,ắ0 Ỉ + ZA 0

A ~ A 71 ’ 1 - z A A =

'bhcủ

7— > us

l - Z c

us =us

H D a HD

IhcOc

DfiDa HD + ~Df ỉ a H

Z o = e ~ ớx / T

(2 0)

(21)

J v he l rysta i each at0m vi b ra te s in th e re la tio n to th e o th e rs so t h a t the

correlation m u s t be included B ased on q u a n tu m s ta tis tic a l th eo ry w ith the

correlated E inste n model [4] the MSRD function for bcc crystals including d o p an t

a t m h as been calcu lated u sin g the procedure p re s e n te d in [10] a n d th ey a re given

2 2(1 + z) 2

~ ơ - — —(1-z) o 0 = hú)

H D U HD 1 +5 K H

(22)

2 >

H

+ — ĐHa ị

12 w H

1/2

(23)

1/2

+ — Duccjj

12

, , z = e-d* IT

(24)

(25)

M d + M h " M D + M H

v h e r e » £ , #£ a re th e c o rrelated E in ste in frequency a n d te m p e r a tu re , respectively From th e above re s u lts we o b tain ed th e DCF c „ , th e ra tio s c , /„> a n d c , / a1

c * - 5 & - Ỉ Ế z i - ' Ỉ Ế 3

«3 ■ «5 (l-z s X l + r J u 0 ( l - z)(i + z J ’

(26)

(27)

CT2 < J q 0 - z A l + Z 1 ơ l 0 - Z.vX1 + 2 )

If th e d o p a n t atom is ta k e n from th e h o st cry stal, i., e.,

th e above o b ta in e d re s u lts will change in to those for th e p u re bcc crystals [12]

^ A - k s ~ kejj - Dữ~\ kĨA =k 3S = &3 - - 2 Da , (30)

u ế =u %- u = — -= - ; - \Oi)

1'

eff

3 0 N g u y e n Van Hung, Ho K h a c H i e u, N g u y e n Cong T o a n

A s " 2 k \ - z 32D a 2 l - z

3 N u m e r ic a l r e s u l t s a n d c o m p a r is o n to e x p e r im e n t

Now we ap p lv th e expressions derived in th e previous section to n u m erical

c alcu latio n s for Fe doped by w an d by Cr The Morse p o te n tia ls for Fe doped by w

a n d by C r h a v e b e e n c a lc u la te d u s in g th e M o rse p o te n tia l p a r a m e te r s D a n d a of

th ese c ry s ta ls c o m p u ted by u sin g our procedure p re s e n te d in [8] They a re shown in Fig u re 1 in a good a g re e m e n t with e x p erim e n t [7] The t e m p e r a tu r e d e p en d e n t

valu es of u] , uị , Ơ2 , CR h ave been calcu lated a n d th e re s u lts a re p r e s e n t e d in

T able 1 F ig u re 2 i llu s tr a te s th e te m p e r a tu re d ependence of our

T a b l e 1: C a lc u la te d v alu es of u] , uị , ơ)ak , c ỵ ’c for Fe doped by w com pared to

(7 2 cex p

e x p e r im e n ta l valu es of exp, R [7] a t different te m p e ra tu re s

T(K) ^ ( 1 0 2Ẳ2) uị (10-2Ẳ2) °ỉ„,c (lO^Â2) < , ( 1 0ZẲ2) CcRak (10 2Ẳ2) r " p( i o2Ẳ2)

T h e r m o d y n a m i c a n d C o r r e l a t i o n Effects in A t o m i c V i b r a t i o n o f B C C

calculated M S R D ^ r ) , MSD u ị( r ) a nd „ ị(r) showing o > > u ị > u \ especially a t

high t e m p e r a t u re s The t e m p e r a t u r e dependence of our calculated D C F c, (t)for

Fe doped by w a n d by Cr is p re s en te d in Figure 3 The functions MSRD, MSD a n d

U C i contain zero-point contribution a t low t e m p e r a t u r e a n d a r e lin ear ly proportional to the t e m p e r a t u r e a t high t e m p e r a t u r e s The t e m p e r a t u r e de pe n d e nc e

of the ratios c„ / u] a n d CH/ ơ 2 for Fe doped by w a re shown in F i g u r e 4 They h a v e

f a T u f0rT !° r t h e p u r e bcc c r y s t a l s t 12J s a ti s f y i n g t h e s a m e p r o p e r t i e s

obtained by the Debye model [2], They increase fastly a t low t e m p e r a t u r e s a n d

approach a c o n st an t va lue s a t high t e m p e r a t u r e s (about 36% for c,f I u \ a n d 18% for

c I f / Ơ ) t a k i n g from our calculated correlated E in s te in t e m p e r a t u r e 9; =2\1 K

These res ul ts denote th e significant r a t e of t h e correlation effect in atomic vibration O ur calculated q u a n ti t ie s shown in Table 1 a n d in Fi gur es 2 - 4 a re found

to be in good a g r e e m e n t with the e x p er im e nt [7]

Figure 1: Calculated Morse potent ial s for Figure 2: Calculated MSRD a n d MSD for

doped by w a n d Cr compared to Fe doped by w compared to e x pe r im e nt [7],

Ĩ nalCULa t e d DCF f0r Fe d0ped by Figure 4: C alc u lat ed CR l u \ a n d c „ / a 2

w an d by Cr compa red to e x pe r im e nt [71 F T ? J 1 , , ,

lor Pe doped by w co m p a r e d to

e x p eri me nt [7],

32 N g u y e n Van Hun g, Ho K h a c Hieu, N g u y e n C o n g T o a n

4 C o n c l u s i o n s

In t h i s w o r k a new procedure for stu d y of the t h e r m o d y n a m i c a n d co rre la tio n effects in t h e atomic v i br atio n of bcc crystals u n d e r influence of a d o p a n t a t o m in XAFS h a s be en developed Analytical expressions for t h e effective local force

c o n s t a n t s , t h e D C F ( C R), th e MSD for th e d o p a n t as ab so r be r ( u \ ) a n d for the

s c a t t e r e r a s ho st ato m ( u ị ) co nta ini ng a d o p a n t atom as n e a r e s t n e ig h b o r a n d the

r a t i o s C r ỉ u2a , C R /cT2 h a v e b e e n d er iv ed b a s e d on t h e a n h a r m o n i c c o r r e l a t e d

E i n s t e i n model

O b t a i n e d e xp res sio ns of the menti oned th e r m o d y n a m i c functions show t h e i r

f u n d a m e n t a l p r o p e r t i e s in t e m p e r a t u r e dependence The functions CR , U2 A, u 2 Sì Ơ2

a r e l in e a r l y p ro po r t io n a l to t e m p e r a t u r e a t h i g h - t e m p e r a t u r e s a n d c o n ta in zero-

point c o n t r i b u t i o n s a t low t e m p e r a t u r e s The ratio CR / u 2 A accounts for a b o u t 36%

a n d th e ra t i o CR l ơ 2 a b o u t 18% a t h i g h - te m p e r a t u r e s

O u r developed th eor y for t h e doping bcc crystals conta in the one for t h e pu re

m a t e r i a l s a s a sp e c i a l cas e w h e n t h e d o p a n t is t a k e n aw ay T h e r e s u l t Ơ 2 >U2 S >U2 A

shows t h e role of location of the do p a n t (as th e c ent ral or as the n e ig hb o r in g atom)

a n d th e c or r e l a t io n effect in stu d y i n g atomic vibration

T he a g r e e m e n t of our calculated values wi th e x p e r i m e n t shows t h e efficiency

of t h e p r e s e n t p r oce du re in st u dy in g t h e r m o d y n a m i c p a r a m e t e r s a n d correlation effects of bcc c r y s t a l s u n d e r influence of do p a n t atom

A c k n o w l e d g e m e n t s This work is s up p or te d in p a r t by t h e basic science

r e s e a r c h project provided by M i n i st r y of Science an d Technology a n d by t h e special

r e s e a r c h project of VNU Hanoi No.QG.05.04

R e f e r e n c e s

1 c A Ashley a n d s Doniach, Phys Rev B 10(1975) 3027.

2 G Beni a n d p M P l a t z m a n , Phys Rev B 14(1976) 1514.

3 T Yok oya ma, T S a t s u k a w a , a n d T Ohta, Jp n J A ppl Phys 28(1989) 1905.

4 N V H u n g a n d J J Rehr, Phys Rev B 56(1997) 43.

5 N V H un g , N B Due, a nd R R F r a h m , J Phys Soc Jp n 72(2003) 1254.

6 M Daniel, D M Pease, N Van Hung, J I Budnick, Phys Rev B 69(2004)

134414

7 V Pirog, T I Nedoseikina, Physica B 334(2003) 123.

8 N V H un g , C o m m u n in Phys 14(2004) 7.

9 N V H u n g , N T T Hoai, L H Hung, V N U J o u r Science, Vol 20(2004) 8-15.

10 N V H un g , N T Van, L H Hung, V N U Jour Science, Vol 20(2005) 69.

11 A A M a r a d u d i n , E w Montroll, G H Weiss, a n d I p Ipatova, Theory o f

L a ttice D y n a m ic s in the Q uadratic A p p ro x im a tio n (Academic, New York, 1971).

12 N V H un g , N B Tr un g, su b m itte d to J A d va n ces o f N a tu r a l Sciences.