At low tem peratu re the harm onic theory works well [1].. But as the tem perature' increases due to anharm onic effects th e XAFS sp e ctra provide different stru ctu ral inform ation
Trang 1V N U JO U R N AL OF SCIENCE M athem atics - Physics T X V III N()3 - 2002
X A F S C U M U L A N T S A N D T H E R M A L
E X P A N S I O N O F B C C A B B I N A R Y A L L O Y S
N g u y e n V an H u n g
D e p a r tm e n t o f P h y s ic s , C ollege o f Science, V N U
A b s tr a c t : A n e w q u a n tu m s ta tis tic a l (in h a rm o n ic th eo ry has been d erived f o r cal
c u la tio n and a n a ly sis o f X A F S c u m u la n ts a nd th e rm a l earpansion o f bcc A B binary
a lloys sy ste m s T h is m o d e l is developed based on the a n h a r m o n ic vib ra tio n o f ab
so rber fr o m a to m s o rte A a n d backs c a tte re r fr o m a n o th e r a to m so rte 11, in c lu d in g
c o n trib u tio n s o f th e ir n e a re st neighbours A to m ic p a ir p o te n tia l h a s been ta ken by plu s-a vera gin g o f M o rse p o te n tia l T h e exp ressio n s have been derived f o r effective
sp rin g c o n sta n t, co rrela ted E in s te in frcq iicn cy, correlated E in s te in tem perature, fir s t
c u m u la n t or n e t th e r m a l e xp a n sio n , sec o n d c u m u la n t o r D ebye- W a ller fa c to r , third cum ulant, causing p h a se c h a n q e o f X A F S spcct.ro, a n d th e rm a l e xp a n sio n coefficient
N u m u ric a l e v a lu a tio n s h a ve been ca rried o u t fo r F eI _ s Wx T h e resu lts are te m p e r
ature dependent a n d Ttfleet tJic exp erim en t and fu n d a m e n ta l theoretical bchaviuors
o f th ese q u a n titie s.
1 I n t r o d u c t i o n
T he X-ray A bsorption F ine S tru c tu re (XAFS) spectra and their Fourier trarisfom
m agnitude provide stru c tu ra l inform ation of substances including alloys At low tem peratu re the harm onic theory works well [1] But as the tem perature' increases due to anharm onic effects th e XAFS sp e ctra provide different stru ctu ral inform ations a t different tem perares [2-7] To correct these uncertainties the cum ulant expansion approach [4] has been developed According to this theory th e XAFS function contains the factor
e ^ k \ w ( k ) = 2i<7<1> - 2 k 2a 7 - 1 » * V 3> + ,
• J
where (jt1) is the first cmnulant, or net th erm al expansion, Ơ^ is the second cum ulant or Debye-Waller factor, and is th e th ird cumulant, providing th e phase change of XAFS spectra [3] M ost of th e efforts is focused t o in terp ret the m easured anharm onic or liigh- tem perature XAFS sp ectra Some progresses have been m ade to calculate the cum ulants
of the crystals [5,6,10,11], an d recently our evaluations for fee alloys system s have been discussed [9]
T his work is our next step o f [9] deriving a quantum statistical anharrnonic proce dure to calculate the cu m u lan ts and th erm al expansion of bcc binary A n alloys system s
in XAFS theory O ur m odel is based on th e local atom ic vibration including anharm onic effects in a sm all cluster of th e ab so rb er from an atom sorte (A) an d the backscatterer from another one (B) w ith tak in g in to account of their nearest neighbors’ contributions The
Typeset by *4/VfS-ThX 17
Trang 218 N g u y e n Van H ung
creation and annihilation o p erato rs are used to describe phonon interact ion, and physical param eters have been derived by averaging calculation using the statistical density m a trix Numerical calculations have been carried o u t for bcc alloys F e \ - x w x T he results are tem perature dependent an d reflect the experim ental ones and fundam ental theoretical behaviours of the above derived quantities
2 T h e o r y
We consider allharm onic vibration between absorber as atom A and backscatterer
as atom B with taking into account the contributions of their im m ediate neighbors so th a t their interaction is characterized by an anharrnonic effective potential
^ e / / ( z ) = 2 k * f J x + k *x + • • • > X = r - r 0, (1)
where r is spontantaneous bond length between absorbing and backscattering atom s, To
is its equilibrium value, k ef f is effective spring constant., and ^3 is cubic anharm oiiicity param eter which gives an asym m etry in the pair d istrib u tio n function
It is usually sufficient to consider weak ankarm onicity, th en our derivation of the expressions of cum ulants and therm al expasion of bcc alloys system s in XAFS theory
is based on quantum statistical theory w ith quasi-harm onic approxim ation, according
to which the H am iltonian of the system is w ritten as a harm onic term with respect to the equilibrium at a given tem perature, plus an anharm onic p e rtu rb a tio n [5] Using the definition [5,12] y = X — a as the deviation from th e equilibrium value of X a t tem perature
T and a ( T ) = < r — r0 > as the net therm al expansion we express Eq ( 1) in the form
The single bond interaction potential betw een th e atom s A and B contained in the effective potential (2) of the system is o b tain ed bv an plus-averaging of Morse pair potential and is given by
V A D — 7Ị\L)A + V b ),O í AB - * -7Z -,<*AB - 2 - n — - 5 V4 )
w h e r e D a tì a n d atA.tí a r e t h e M o r s e p o t e n t i a l p a r a m e t e r s
U a , b ( x ) = D A iB( e ' 2aA BX - 2 e ~ a AB X). (5) Considering the contributions of im m ediate neighborvS of absorber and backscatterer,
as well as, the atomic distribution in bcc stru ctu re, we derive th e effective spring constant
the p ertu rb atio n potential due to anharm onicity
Trang 3X A F S c u m u la n ts and th e rm a l ex p a n sio n o f 19
O ur approach is based on a local vibration picture and a Einstein model is ap propriate FYom the above relations th e correlated Einstein frequency UJE and correlated Einstein tem p eratu re 6 e have been derived, and they are given by
1/2
(8)
<+>E — f — — \ C \ o i a b - Z C 2 a a 'A B Ỷ j
Oic — -— —\C \otA B — o C 2 a a A B \
In the Eq.(6-9) we used k[} as B oltzm ann’ s constant and the following symbols
C l = 1 + - [fify + (J?B ) , Ơ2 = 1 + n \ + /ig ,
fl ~ M a + M b ’ flA ~ M a + M b ’ ~ M A + M b '
where M a and M b are the m ass of the absorbing and backscattering atom s, respectively
T he cum ulants are derived by averaging th e value of y [5,12] Atomic vibration is quantized as phonon, and anharm onicity is the result, of phonon interaction Therefore, to calculate the m atrix elements for these interactions we express y in t erm of annihilation and creation o p erato rs ả and a + , i e ,
and use the harm onic oscillator sta te s In ) as eigenstates and E n = n h u ias eigenvalue
T he cum ulants have been derived by averaging procedure, using the statistical density
m atrix p and the canonical p a rtitio n function z in the form
where
P 0 = e - pẵio, H 0 = Ç - + ị k c f f y 2, í 3 = l / k B T , (13b)
S p = - f 0 P 0 = e ~ ^ 'H°S Ữ (P ) = e0Ho0 U e ~ 0Ho. (13c)
Jo
Using Eqs (12,13) we derived th e expressions for the averaging value of y for the even m om ents m e
n and for th e odd m om ents mo
(ymo) = ề E e~ ^ " — < n \ỏ u ( y)\n > ™0 = 1 ,3 ,5 - (15)
&0 n , n ', ton ~ tin '
Trang 4In the calculation of tran sitio n m atrix elements the selection rule has been obeyed Prom Eq (14) we derived the second cum ulant or Debye-Waller factor
Using Eq (15) and th e condition < y > = 0 we derived the first cum ulant or net therm al expansion
and the third cum ulant
(3)m C 2{ t u o E ) a ' A B 1 + 10« + 2
T he param eter a ( T ) describes an asynunetry of the pair p o ten tial or the therm al ex pansion of the bond length T a b between th e two atom s A and B due to the anharm onicity,
th a t is why from Eq.(17) we derived the therm al expansion coefficient
“ T ( r ) i ( % D A B ° \ o A B \ ( 1 - Z ) T ) 1 - (19)
To get th e above simplified form ulas several m athem atical expressions have been used
FYoin the above expressions it is easy to receive the following relation
a r r T ơ 2 3z(l + z)ln(l/z)
In order to define the behavioiưs of the above obtained therm odynam ic quantities
in tem perature dependence we derived them in the low tem p eratu re (T —> 0) and high tem perature (T —y oo) limits T he results are presented in Table I
T able I The values of cr(U ,CT2 , a T a n d ctr r T ơ 2 / ơ (3) for an alloys AB at low
tem perature ( 7’ —» 0 ) and high tem perature (7’ —>00 ) limits
a " '
*>
Ơ
a<5>
a T
a T r T ơ 2 Ơ [V)
+ 2 z ) i C ^ D ^ a ^ f t hú)E ( \ + 2 z ) 4C \D ABa AB
C i «' a b (*°> b ) 2( 1 + U z j / i e c ^ a ^
3C 2k - ịá AỊịZ ( ỉn z ) 1' ( \ + 2 z / & A B a ABrAB
3 z l n ( \ / z )
^ 2 a A B ^B ^ ■* 4 t,\ ^ A B a AB
k B I / 2Cj2
ỵ 4 C Ị
3C 2 k B a '/W 4C|2 D ju ìữ2ABr AB
1/2
Trang 5X A F S c u m u l a n t s a n d t h e r m a l e x p a n s i o n o f 21
3 N u m e r ic a l r e s u l ts
Now we apply the above derived expressions to num erical calculations for the bcc binary alloys F e \ ~ x w x T he Morse potential param eters I ) A [Ị and O la /y were taken from Ref 8 T he calculated values of D a B ì ^ a B ì a Atìi O e for the alloys F e W are given in Table II
T abic II Ihe calculated values of D AB M 4B M'AB ,k tìff (Oỵ ,&E for the alloys 1 ‘eVV.
S a m p le B o n d DM ( e V ) <*AB ( Á 2) CA3 ) k ' f f f N m) ( O g f x W* Hz ) e E( K )
T he vibration characterizing quantities for the alloys bond F e — w , calculated by present procedure are different from those for t he single crystal bond F e — F e and w — W ,
calculated by the procedure presented in Ref 14 T his can be seen in the Table III
Table III: Com parison of the values k ejg (oE.anciOE of the single crystal bond with those of
the alloys bond of FeW
Fig 1 shows th e tem perature dependence of our calculated net therm al expasion ơ-(l ' of F e W com pared w ith those of its com ponents F e and w Fig 2 shows the tem perature dependence of our calculated Debye-Waller factor Ơ2 of F e W in comparison
w ith those of its components F t and w
Fig. 7: Net therm al expansion <7(i) (Ẵ)
of FeW com pared with those
of its com ponents Fe and w
Fig 2: Debye-Waller factor Ơ 2(Ằ2)
of FeW com pared with those
of its com ponents Fe and w
Trang 622 N g u y e n Van H ung
Fig 3 inllustrates the tem p eratu re dependence of OUI* calculated third cum ulant
of F e W in comparison w ith those of its com ponents F e and w Fig 4 shows the tem perature dependence of therm al expansion coefficient Q r of F eo.88 Wo.12 compared w ith those of its com ponents F e and w Fig 5 dem onstrates the tem p era tu re dependence
of our calculated cum ulants relation o t t T ơ 2/ ơ ^ of F t W com pared w ith those of its components F e and w
Fig. 3:Third cum ulant c r5l(À3) of FeW
in comparison with those of its
com ponents Fe and w
Fig. 4 :Therm al expansion coefficient
a T ( k 1 ) of FeW in comparison with those of its com ponents Fe and w
2 0 0 3 0 0
T(K)
4 0 0 5 0 0
Fig 5: Cum ulants relation a r r T ơ 2 <7 '31 of FeW in com parison w ith those of its
com ponents Fe and w
4 D isc u ss io n s a n d c o n c lu s io n s
- In this work the expressions of XAFS cum ulants and therm al expansions of the two com ponents bcc alloys system s in tem p era tu re dependence have been derived based
on quantum statistical theory
- T he net therm al expansion Debye-Waller factor Ơ2 and th ird cum ulant Ơ
contain z cro p o in t contributions as q u an tu m effects a t low tem p eratu re, and contain the
Trang 7X A F S c u m u la n ts and th erm a l exp a n sio n o f 23 classical lim its at high tem perature, where ờ 2, ~ T , and ~ 'I'2. These behaviours are sem inlar to those of the results of theories [5,6,10] and by experim ents [2,13] for the single crystals
- T herm al expansion coefficient has the form of specific lieat which approaches
to a constant, value at high tem p eratu re as the Dulong-Petit rule and vanishes at low tem p eratu re obeying the cubic tem p eratu re rule
- T he cum ulants relation o t'yrT ơ 2Ị ơ !y?,) approaches the classical and experim ental value [2] of 1/2 at 0 g (Fig 5) T his denotes the Einstein tem p eratu re as the lim it above which th e classical approach can be applicable and below which the quantum theory m ust
be used as it was defined for the single crystals [5]
- T his approach can also be applied to the research of therm odynam ic properties of nano systems
A c k n o w le d g m e n ts This work is su pported in part, by th e basic science research pro grams No 410 801
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