VNU JOURNAL OF SCIENCE, Mathematics - Physics, T.XXII, N03, 2006 ST U D Y O F E X A FS CUM ULANTS OF FCC CRYSTALS C O N T A IN IN G N D O PA N T ATOM S N g u y e n V an H ng, T n T ru n g D ung, N guyen C ong T oan Department o f Physics, College of Science, VNU 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 A B ST R A C T I n tr o d u c tio n C um ulant expansion approach has been developed [1, 2] to include anharm onic effects in the EXAFS procedure These anharm onic effects are contained in the first cum ulant or net therm al expansion, the second cum ulant or Debye-Waller factors, the third 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 portant to study thermodynamic 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 structure which are often semiconductors 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 this 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 aram 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 perature, for the 1st, 2nd and 3rd cum ulants, where the host atom is denoted by the letter H and the dopant atom by the letter D All these expressions are different if the num ber of dopant atom s changes The results in the case if all the dopant atom s are tak en out or if all the host atom s are replaced by the dopant atoms are reduced to those derived by using the anharm onic correlated Einstein model [8] for th e pure m aterials Num erical calculations have been carried out for 31 Nguyen Van Hung, Tran Trung Dung , Nguyen Cong Toan 32 Cu doped by one or m ore Ni dopant atom s, and th e re s u lts a re com pared to th o se of th e pure m a te ria ls an d some to exp erim en t deducted from the m easured M orse po ten tial p a m e te rs [15] F orm alism 2.1 A n h a r m o n i c effective p o t e n t i a l , l o c a l fo rce 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 in s te 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 Following [16] th e anh arm o n ic co rrelated in te rato m ic effective p o ten tial of a fee cry stal doped by n atom s from a n o th e r source is given by (1) Vx) ~ ị kejrx2 +kìeffxĩ + as a function of th e disp lacem en t x = r - r for r and r0 being th e in sta n ta n e o u s and equilibrium d istan ces betw een absorber and b a c k sc a tte re r atom s U sing th e definitions [8] y = X —a , a = (x) Eq (1) is changed into (2) Kf f GO = ~ k ejr y + Vanh ( y) Vanh i y ) = w { y ) = k efftay + K y , containing an effective local force co n stan t k ejỵ ( A k H + B k HD + C k D ) — k” ^ — ( =" K A4HMp M ff + M p h d \i ) ’ (3) and anharmonic effective factor k 3eff C-ị.kĨD) 2(^3 (4) (y + m ỹ involving co n trib u tio n s of im m ed iate atom ic neighbors, w here A = m 2(l + 4