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

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

VNL' Journal o f Scicncc, M ath em a tics - Physics 24 (2008) 125-131

Calculation o f M orse Potential for D iam ond Crystals

Applification to AnhaiTnonic Effective Potentia

Nguyen Van Hung*, Nguyen Cong Toan, Nguyen Bao Trung, Ngo Moang Giang

D e p a r tm e n t o f P iiy sic s C o lle g e o f S c ie n c e VNU, Ĩ 3 4 N íỊuycìĩ Trai, Tlianh Xiiaiì H a n o i Victudfii

R cccived 26 June 2008; received m revised form 15 Aimust 2008

Abstract A nalytical expressions for the M orse potential param eters o f diam o n d crystals ha\'c been derived T h e y contain the energy o f sublimation, the com pressibility and the lattice constant

N um erical calculations have been carried out for Si and Sn, and the results reflect fundamenlul properties o f tliis potential Tlic obtained M orse potential param eters have been used for calculation o f the anh a rm o n ic correlated cffcctivc potentials o f these c p 'sia ls in X A I'S theory showing clearly an harm onic efiects

1 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

0 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 X AFS spectra [4-10,14], Numerical calculations have been caư ied 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

C orresponding author E-mail: hunun v @ v n u e d ii.v n

125

126 N V H u n g et al / V N V J o u r n a l o f Science, M aihem atics - Phvsics 2 4 (2008) Ỉ 2 5 - Ị 3Ỉ

2 Procedure for calculation o f M orse potential o f Diamond crystals

The potential energy ẹ { i - ị j ) o f two atoms i and j separated by a distance ;-y is given in terms 0Í' the M orse function by

where a , D are constants with dimensions o f reciprocal distance and energy, respectively; is the

equilibrium distance o f the two atoms Since (p{r^) = - D , D is the disociation energy.

In order to obtain the potential energy o f the whole crystal whose atoms are at rest, it IS

necessary to sum Eq (1) over the entire crystal This is most easily done by choosing one a t o m

in the lattice as an orgin, calculating its interaction with all the others in the crystal, and then multiplying by N / 2 , where N IS the total atomic number m the crystal Thus the total energy o IS

given by

^2a[rj-ro] -a[rj-ro]

(2

Here r is the distance from the origin to the yth atom It is convenient to define the f o l l o w i n g quantities

where m ■ ,nJ , l j are position coordinates o f any atom in the lattice Applying Eq, (3) to Eq (2) the

energy can be rewritten as

/ \ /-«•> \I—' —2ữciAi: - ữ(iM j ,

(ĩ>[a) = L p Ỵ ^ e j - I L P Y j C ^ (4)

The first and second derivatives o f the energy o f Eq (4) with respect to a are given by

jC ^ + 2 L / ỉ a Ỵ ^ M j e

^ = A a ^ L ( 3 ^ Y M ] e ' - 2 a - L p Ỵ ^ M ] e

At absolute zero T = 0, a is value o f a for which the lattice is in equilibrium, then ) gives the

energy o f cohesion, [ d ^ l d a \ = 0 , and is related to the compressibility [1J That IS,

(5)

(6)

where U q { c I q ) is the energy o f sublimation at zero pressure and temperature, i,, c.,

d<ĩ)

and the compressibility is given by [1]

1

= v„

dV-(7)

(8)

(9)

w ị\ ỉ lu n g cl a i / I'NU J o u r n a l o f Science M athem atics - Physics 24 (2008) 125-13 Ì 11

where v\ IS the \'olunie at T = 0, and is compressibility at zero temperature and pressure 'Fhe

v o l u m e per a t o m V = V Ị N is r e l a t e d t o t h e l at ti c c c o n s t a n t a b y t h e r e l a t i o n [ 1J

V / N = ca^

Subslituting Eq (10) m Eq (9) the compressibility IS expressed by

I k r

Usintỉ Eq (5) to solve Eq (8) we obtain

[■'rom Eqs (4, 6, 7, 11) w e d e r i v e t h e r e l a t i o n

( 10 )

( 1 1 )

(12)

( 1 3 )

Solving the system o f Eqs (12, 13) we obtain a , p Substituting the obtained results into the sccond o f Hq (3) wc derive Tq Using the obtained a , ( 5 and Eq, (4) to solve Eq (7) we obtain L From this L and the first o f Eq (3) we obtain D The obtained Morse potential parameters D , a

depend on the compressibility the energy o f sublimation ƠQ and the lattice constant a These

values o f about all crystals are known already [13,16,18,21],

3 Application to calculation o f anharinonic correlated effective potential in X A FS

Figure 1 shows Fourier transform magnitudes of XAFS o f Sn (Diamond), measured at 77K and 300K at IIA SY LA B (DESY, G ennany) [20] They arc different at these temperatures by the shift o f the curvcs showing anharmonic effects in XAFS For describing these effects an anharmonic XAFS theory is necessary

R(A)

Fig 1 F ourier tran sfo rm m agnitudes o f X A F S o f S n at 7 7 K and 300K.

128 N V H u n g et a l / V N U J o u r n a l o f Science, M ath em a tic s - P hysics 24 (2008) Ỉ 2 5 - Ỉ 3 1

The expression for the K-edge anharmonic XAFS function [10] is described by

X { k ) = F { k )

where F { k ) is the real atomic backscattermg amplitude, o is net phase shift, k and /Ì arc the wave

number and the mean free path o f the p h o t o e l e c t r o n , respectively, and (n = 1,2.3 ) arc th e cumulants The expression for the cumulants in XAFS theory is derived based on the anharmonic correlated Einstein model [9] which is considered, at present, as “the best theoretical framework with which the experimentalist can relate force constants to temperature dependent XAFS"' [15J Accordinii

to this theory the effective interaction Einstein potential o f the system is given by

M, + M , , R =

R

Here is effective force constant, and the cubic parameter giving an assymmctry in the

pair distribution function, r is bondlength and Tq is its equilibrium value The con*elated Einstein

model may be defined as a oscillation o f a pair o f atoms with masses Ẳ /ị and M , (e.iz., absorber and

backscatterer) in a given system Their oscillation is influenced by their neighbors mven by the last

( / - 2 ), and the sum j is over all their near neighbors, excluding the absorber and backsctiercr

themselves The latter contributions are described by the term v { x )

Applying the Morse potential o f Eq (1) in the approximation for weak anharmonicity to the

X A FS theory by the expansion

V( x) = D e - 2 a x - 2e - a x ~ D ị - \ + a ^ x ^ - a ^ x ^ X = r

so that the anhamionic effective potential Eq (16) is fransformed as

Kff ( y) = D{c,a~ + 3c^a^a)y- + c ^ D a " y \ y ^ x - a, a = (.v) ,

with the effective local force constant and cubic parameter

= 2D{c^a~ + 'ic^a^aỴ k ^=c^Da^,

and structural parameters

c

N

J-2 COS^ a

(17)

18)

(19)

(2 0)

4 Num erical results and discussion

To calculate the above equations to obtain the Morse potential parameters, we have to calculate

the parameter c in Eq (10) The space lattice o f diamond is fee The primitive basis has tw o identical

atoms at 0 0 0 , -associated with each point o f the fee lattice Thus the conventional unit cube

4 4 4 contains eight atoms, so that we obtain the value c = 1/8 for this tructure

N v ỉ ỉ ii ỉ ỉ í ịe ta L / VNU J o u r n a l o f Science, M athem aiics - Physics 24 (2008) Ỉ 2 5 - Ỉ 3 Ỉ 129

Using the theory derived in the previous section and the calculated parameter c, as well as the energy o f sublimation, the compressibility and the lattice constant from [13,16,18,21] \vc calcLiIatcd

M orse potential parameter D, a , ỉ\ịb y our established computing programs and the results are included

in Table 1

Fig 2 N e arest neighbors o f absorber (A) and backscatterer (S) in X A F S o f d ia m o n d crystal

T abỉe 1 C alculated M orse potential param eters for D iam o n d crystals Si an d Sn

Based on the coordinates o f neighbors and central atom (Fig 2) presented in Table 2 we

T able 2 C o ordinates o f neighbors o f a central atom in d ia m o n d structure

( 21 )

calculated the structural paremeters

c, = 1 + cos^ / 2 + c o s" ỚÍ3 / 2 + cos^ a J 2 ^ 7 / 6.

= -( l-í-c o s ^ ỡ r2 /4 + c o s ^ a 3 /4 + cos^ớr_ị/4) = - 3 5 / 3 6 Hence, for diamond the anharmonic coưelated effective potential from Eq (18) is resulted as

(2 2)

Kf f { x ) = ^ D a - x - - ^ D a ^ x ^ ,

K A y ) = D 1

- a - - — a a

Figure 3 shows the calculated Morse potentials o f Si and Sn They satisfy all properties o f the Morse potential [19], 1 e., it describes repulsive force in short distance when atoms approach each other obeying Pauli exclusion principle, and describes attractive force in long distance when atoms go far

130 N V H u n ^ et aỉ / V N U Journcd o f Science M athem aỉics - P hysics 24 (200S) Ỉ 2 5 - Ỉ 3 Ỉ

from each other The reason o f this attraction is that ihe atoms have diffusion moments which attract each other in lonu distance Figure 4 illustrates the calculated anharmonic and harmonic effective potentials for Si and Sn They

potentials due to includinc anharmonic contributions in atomic vibrations o f these diamond crystals

Fig 3 C alculated M orse potentials for Si and Sn.

Fig 4 C alcu lated anharm onic and harmonic effective potentials for D ia m o n d

crystals Si (a) and Sn (b) show ing strong anha rm onic shifting.

5 Conclusions

A new procedure for calculation o f Morse potential parameters for Diamond crystals has been

developed and the obtained results are applied to calculation o f anhannonic and harmonic potential in the XA FS theory The derived expressions have been programed for the computation o f the considered physical quantities

The derived expressions for Morse potential parameters contain the energy o f sublimation, the compressibility and the lattice constant o f Diamond crystals which are available in litcralurcs

The good satisfying o f the calculated Morse potential with its fundamental properties, as well

as, the good description o f the effective potentials and the asymmetrv o f this potential due to anharmonicity show the efficiency and reliability o f the present procedure in computation of the

N V- ĩ ỉ ư n g et aỉ- / VNU J o u r n a l o f Science, M athem atics - Physics 24 (2008) Ỉ 2 5 -Ì 3 Ị

atomic interaction potential parameters as the Morse potential which arc important for calculation and analysis o f physical effccts in XAFS technique and in solving the problems involviim any type o f

atomic interaction in the Diamond crystals.

Ackno^vlcd^ments; The authors thank Dao Xuan Viet and Le Day Manh for useful conlributions to numerical calculation o f Morse potential parameters This work IS supported in part

by the basic sciencc research program No 4 058 06

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