VN U JO URNAL OF SCIENCE M athem atics - Physics T X V III N()3 - 2002 XA FS CUM ULANTS E X P A N S IO N OF BCC AND AB THERM AL B IN 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 sics, C ollege o f Science, V N U A b s tr a c t: A n ew q u a n tu m sta tis tic a l (in h a rm o n ic theory has been derived fo r cal cu la tio n and a n a lysis o f X A F S c u m u la n ts and th erm a l earpansion o f bcc A B binary alloys syste m s T h is m odel is developed based on the a n h a rm o n ic vib tio n o f ab sorber fr o m a to m so 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 clu d in g c o n trib u tio n s o f th e ir n ea rest neighbours A to m ic p a ir p o te n tia l h a s been taken by plus-averaging o f M o rse poten tia l T h e expressions have been derived fo r effective sp rin g c o n sta n t, correlated E in s te in frcqiicncy, correlated E in s te in tem perature, fir s t c u m u la n t or n e t th e rm a l exp a n sio n , seco n d cu m u la n t o r D ebye - W aller fa c to r, third cum ulant, causing phase chanqe o f X A F S spcct.ro, a n d th e rm a l expansion coefficient N u m u ric a l e va lu a tio n s have been carried o u t fo r F eI _ s W x T h e results are tem p er ature dependent an d Ttfleet tJic experim ent and fu n d a m en ta l theoretical bchaviuors o f these q u a ntities I n tr o d u c tio n T he X-ray Absorption Fine S tru ctu re (XAFS) spectra and their Fourier trarisfom m agnitude provide stru c tu l inform ation of substances including alloys At low tem perature the harm onic theory works well [1] But as the tem perature' increases due to anharm onic effects the XAFS sp ectra provide different structural 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 the XAFS function contains the factor e ^ k \ w ( k ) = i + , •J where (jt1) is the first cmnulant, or net therm al expansion, Ơ^ is the second cum ulant or Debye-Waller factor, and is th e th ird cumulant, providing the phase change of XAFS spectra [3] Most of th e efforts is focused t o interpret the m easured anharm onic or liightem perature XAFS spectra Some progresses have been m ade to calculate the cum ulants of the crystals [5,6,10,11], and recently our evaluations for fee alloys system s have been discussed [9] This work is our next step of [9] deriving a quantum statistical anharrnonic proce dure to calculate the cum ulants and therm al expansion of bcc binary A n alloys systems in XAFS theory O ur m odel is based on the local atom ic vibration including anharm onic effects in a small cluster of th e absorber from an atom sorte (A) and the backscatterer from another one (B) with taking into account of their nearest neighbors’ contributions The Typeset by *4/VfS-ThX 17 N g u yen Van H ung 18 creation and annihilation operators 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 out for bcc alloys F e \ - x w x T he results are tem perature dependent and reflect the experim ental ones and fundam ental theoretical behaviours of the above derived quantities T h e o ry We consider allharmonic 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 ) = 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 anharmoiiicity param eter which gives an asym m etry in the pair distribution function It is usually sufficient to consider weak ankarm onicity, then 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 erturbation [5] Using the definition [5,12] y = X —a as the deviation from the equilibrium value of X at tem perature T and a ( T ) = < r —r > as the net therm al expansion we express Eq ( ) in the form (2 ) Ueff(y) = ị k e f / y + ôư( y) The single bond interaction potential between the atom s A and B contained in the effective potential (2) of the system is obtained bv an plus-averaging of Morse pair potential and is given by \ + OiABX2 - a'A B x* - ), ƯAỉ ì ( x ) — n _ l / n VA D — , n N, 7Ị\L)A * + n D Aa 2A + D Ba ị V b ),Oí AB - (3) D Aa \ + DBa ị , * - 7Z , = we derived the first cumulant or net therm al expansion a ự ) ^ < -> (r) = g W p »i i f (,T) and the third cumulant (3)m C 2{ t u o E (r>- u c & 1+ 10« + ) a 'A B (1 - > r jfc • (18) The param eter a ( T ) describes an asynunetry of the pair potential or the therm al ex pansion of the bond length T a b between the two atom s A and B due to the anharmonicity, th at is why from Eq.(17) we derived the therm al expansion coefficient “ T (r) i ( % D AB ° \ o (19) AB\(1-Z)T)1- To get the above simplified formulas several m athem atical expressions have been used FYoin the above expressions it is easy to receive the following relation arrTơ2 3z(l + z)ln(l/z) + z ) iC ^ D ^ a ^ f t hú)E ( \ + z ) 4C \D ABa AB Ơ a k B I / 2Cj2 ỵ4CỊ C i «'a b (*°> b ) 2( + U z j / i e c ^ a ^ 3C 2k - ịá AỊịZ ( ỉn z ) 1' ( \ + z / aT a Tr T ^ a A B ^B ^ ■*4 t, \ ^ A B a AB Ơ [V) 3z ln (\ / z ) & A B a ABrAB 3C 2k B a '/W 4C|2D ju ìữ2ABr AB 1/2 X A F S c u m u la n ts and th erm a l e x p a n sio n o f 21 N u m e ric a l r e s u lts Now we apply the above derived expressions to numerical calculations for the bcc binary alloys F e \ ~ x w x The Morse potential param eters I ) A [Ị and Ola /y were taken from Ref The calculated values of D a B ì ^ a B ì a Atìi Oe for the alloys F e W are given in Table II T abic II S a m p le Ihe calculated values of D AB M 4B M'AB ,k tìff (Oỵ ,&E for the alloys 1‘eVV Bond Fe-W FeW DM ( e V ) 0.708 CA3 )