DSpace at VNU: Calculation of Morse Potential for Diamond Crystals. Applification to AnhaiTnonic Effective Potential

VNL' Journal o f Scicncc, M athem atics - Physics 24 (2008) 125-131 Calculation o f Morse Potential for Diamond Crystals Applification to AnhaiTnonic Effective Potentia Nguyen Van Hung*, Nguyen Cong Toan, Nguyen Bao Trung, Ngo Moang Giang D e p a rtm e n t o f P iiysics C o lle g e o f S cien ce VNU, Ĩ N íỊuycìĩ Trai, Tlianh Xiiaiì H anoi Victudfii Rcccived 26 June 2008; received m revised form 15 Aimust 2008 Abstract Analytical expressions for the M orse potential param eters o f diam ond crystals ha\'c been derived T h ey contain the energy o f sublimation, the com pressibility and the lattice constant Numerical calculations have been carried out for Si and Sn, and the results reflect fundamenlul properties o f tliis potential Tlic obtained Morse potential param eters have been used for calculation o f the anharm onic correlated cffcctivc potentials o f these cp'sials in X A I'S theory showing clearly anharm onic efiects Introduction Interatomic anharmonic potential, especially Morse potential, has been studied widely [1-12 The parameters o f this potential can be extracted from the XAFS {X-ray Ahsorpdon Fine Structure) Ị 11,12J They arc also used to calculate thcrmodynamic parameters included in these spectra f4-10| This potential is successfully applied to calculating the quantities involving atomic interaction, especially, the anharmonic effects contained in XAFS [5-lOJ which influence on the physical information taken from these specfra They are also contained in the expressions o f equation o f state [IJ 'I’hcrefore, calculation o f the Morse potential is very actually desired, especially in XAFS theory The calculation o f Morse potential has been carried out for fee, bcc [1,22 J and hep [17,22] crystals The purpose o f this work is to develop a method for calculation o f Morse potential parameters Ĩ Diamond, an interesting crystal structure Analytical expressions for the parameters of this poleniial have been derived They contain the energy o f sublimation, the compressibility and the lattice constant which are known already, for example see [13,18,21], The obtained results are applied to calculating the anharmonic correlated effective potential, contained in the XAFS spectra [4-10,14], Numerical calculations have been caưied out for volume per atom, tructural parameters and Morse potential parameters o f Diamond crystals Si and Sn The obtained Morse potential parameters satisfy all fundamental properties o f this potential [19] and have been applied to calculating anharmonic correlated effective potenlials o f these crystals in XAFS theory Corresponding author E-mail: hununv@ vnu.edii.vn 125 N V H u n g et al / V N V J o u rn a l o f Science, M aihem atics - Phvsics 24 (2008) Ỉ - Ị 3Ỉ 126 Procedure for calculation of Morse potential of Diamond crystals The potential energy o f two atoms i and j separated by a distance ;-y is given in terms Í' ẹ{i-ịj) the Morse function by ( 1)

[a) = L p Ỵ ^ e - ữ(iM j j - I L P Y jC j , ^ (4) j The first and second derivatives o f the energy o f Eq (4) with respect to a are given by ^ (Ị ) IT—Ị da J jC ^ +2L/ỉaỴ^M je y —ư(ỉA'ỉ j ■) 1Y— I ' -2a-LpỴ^M ]e J —.ơ g M ịt = , and r ">i ( ) J is value o f a for which the lattice is in equilibrium, then energy o f cohesion, [ d ^ l d a \ (5) j ^ = Aa^L(3^YM ]e daAt absolute zero T = 0, a —2,ccoM :r'1 ) gives the is related to the compressibility [1J That IS, (7) where U q{cIq) is the energy o f sublimation at zero pressure and temperature, i,, c., d