Study of morse potential, binding energy, thermal expansion and their relation

A fter show ing the dependence of th e to tal potential on the various distance a betw een the atom s, we ca rry out the relation form ula o f lin ear therm al expansion with th[r]

S T U D Y O F M O R S E P O T E N T I A L , B I N D I N G E N E R G Y ,

T H E R M A L E X P A N S I O N A N D T H E I R R E L A T I O N

N g u y e n C o n g T o a n a n d N g u y e n V a n H u n g

D epartm ent o f Physics, College o f Science, VN U

A b s tr a c t: M orse p o te n tia l, b in d in g en erg y, th e rm a l e x p a n sio n an d th e ir rela tio n lin e a r th erm a l e x p a n s io n c o e fficie n t w ith th e M o rse p o te n tia l p a ra m e te rs h a v e

a v a ila b le e x p e rim e n ta l d a ta N um erical c a lcu la tio n s h a v e b e e n ca rrie d o u t fo r seve ra l crysta ls, a n d th e resu lts a re fo u n d to be in g o o d a g re e m e n t w ith exp erim en t

1 I n t r o d u c t io n

The pairw ise potential functions have been used widely in th e description of th e solid states One of th e m ost successful function is the M orse potential function T h e purpose of

th is work is to stu d y th e relation betw een the binding energy, th e th erm a l expansion and the p aram eters of th e M orse function Using th ese re lations we d ete rm in e th e Morse potential param eters

The p ap e r will be sta rte d w ith determ ining the average potential p e r each atom which depends on th e M orse p aram eters and th e lattice constant A fter show ing the

dependence of th e to tal potential on the various distance a betw een the atom s, we ca rry out

the relation form ula o f lin ear therm al expansion with th e crystal properties T h e calculated results are com pared to experim ent [3-7]

2 F o rm a lis m

We consider a ce rtain atom inside th e lattice The total in teraction p o ten tial of this atom w ith all th e others in th e crystal is given by th is sum:

w here <p(ri) is th e single-bond potential betw een th a t atom an d the i"' atom of crystal, i runs over all th e num bers of atom s of th e lattice (except th e original atom ) U se th e Morse function we have:

(p(ri )= D e '2a(r>'ro)- 2 D e " 2a(r'- >;>} = De2“r«e"2ai'' -2 D e ar“e _ar' = Dp v 2ar> -2 D p e ~ ar' , (2)

(1)

where, D, a , r0 are th e M orse p aram eters and

(3) Thus, we can calculate th e average potential p er each atom by:

O - ị Ẹ M V * " ' ' -2 D p e -“ > ) = Ì D p 2 j y 2“ r' - D u Ẹ e - » (4)

For th e cubic, bcc, fee, dia or hep crystals w ith th e lattice co n stan t is o, th e distance r,

can be taken by the position coordinates nij, n,, 1; (see fig 1)

j = ^ l j 2 + n ij2 + n j a a = M ; a

in which M , = J l , 2 + m ,2 + n ,2

T herefore, ẹ = 1d|52] [ V 2" m‘* - D p £ V “Mi*

We can see th a t, th e p a rtic u la r average potential (0 00) (IOC

depends on th e lattice co n stan t a: (p = <p(a) an d its form is

W hen th e la ttic e vibrates, th e re is sm all change in the bond distance a betw een the

atom s, th is lea ds to sm all change in th e potential of lattice We can expand th e potential function in term of T aylor form around th e equilibrium distance value a^:

= <p(ao ) + 0 ( a - a o)+ K 2( a - a o)z + K3( a - a 0)3 +

is the p a rtic u la r cohesion energy of th e atom s are a t rest, it is th e energy of sublim ation of one atom.,

= - D a p 2 ^T M je -2aM'a° + D a p £ M 1e - aM'a° = 0 ;

K 2 = = 2 D a 2p2^ M f e " 2aMia° - D a2p £ M ? e " “Mia°

K 3 = = -4 D a 3p2 Y M f e '2aMi"0 + D a3p y M fe"aMia°

(10)

(11)

(12)

a re th e expansion coefficients

The expression (9) shows th a t, the general potential function is anharm onic T h at is the reason w hy th e crystal h as th erm a l expansion And we see: (a ) = a 0 + Aa * a 0 , where,

Aa is th e average expansion w hich we will determ ine Define th at: x=a-a0 (x) = Aa

dtp _

dx ~

We suppose th a t, vibration is free so th a t th e average force equals zero (the atom oscillate only a ro u n d its position)

The atom vibration energy includes potential and kinetic energy By approxim ation, the vibration potential an d th e k inetic energy have th e sam e value an d they equal to one half of th e total energy

2(U) = 2(K) = (E) => (E) = 2(U> = 2{<p-ï>(a0)> = 2{K2x 2 * K :1x :3* ) « 2 K 2( x 2) (15)

-3 Kj(x2 \ ,3 K rl(E)

The th erm a l expansion coefficient is given by th e formula:

1 cl(Aa) - 3 K 3 gjE) -3 K „C

fio ^ 4 a 0K§ <5T 4 a 0K |

w here, Cr is the average p artic u la r th erm a l capacity p er each atom It can be d eterm ined by the experim ent

Now, let’s recall th e equations (9-12) and (17) They set up a com plete system of relations betw een ihe therm a l expansion coefficient, the binding energy an d th e Morse potential p aram eter

The eq (10) leads to:

From (9) we obtain:

p = 2U 0 p2] T V 2“ M'an - 2 p ^ V u,MiB|

4 a 0K | These re lations helps us determ ine the unknow n quan tities w hen we have some inform ation about th e crystal an d th e atom s T his work can be executed by a PC (Personal Computer)

3 N u m e r ic a l c a lc u l a t i o n a n d c o n c lu s io n

T he first and th e second tables give us th e re su lts calculated for som e kin d s of atom and some types of crystal likes exam ples In th ese tables we also give som e com parison between the pre sen t theory re su lts and th e experim ental values

T able 1 The thermal expansion coefficient:

T a b l e 2 T h e M orse potential paramet ers of s o m e materials.

a: reference [4]; b: reference [7].

T he ca lculated re su lts p re sen ted in th ese tables a re found to be in good agreem ent

w ith experim ent T h e M orse potentials are shown in Fig 2 an d Fig 3 com pared to experim ent [4, 7],

expr I I theory 11

Fig.2 The Morse potential function of Ni Fig.3 The Morse potential function of Cu

Acknow legm ent: T his work is suported in p a r t by th e basic science research project No

41.10.04

R e fe re n c e s

1 N V Hung, S o lid state Theory, VNU Publishing House-1999.

2 N, T Bao Ngoe, N V Nha, Solid states textbook, VNU Publishing Mouse-1998.

3 David Halliday, al.: F undam ental o f Physics, VN Educational Publisher.

4 N.I Kôskin, M.N Sừkêvich: Fundamental Physical Handbook , Mir Moscow Publishing

House-1980

5 J c Slater, Int to Chemical Physics (Me Graw-Hill Book Company, Inc New York, 1939).

6 Charles Kittel, Int to Solid State-Phys, John Wiley & Son ed., Inc N Y Chichester,

Brisbane, Toronto, Singapore 1986

7 L.A Girifalco and V.G Weizer: Phys Rev £114(1959) 687.

8 N V Hùng and J J Rehr: Phys Rev B 56(1997) 43.

9 I V Pirog, T ] Nedseikina, I A Zarubin and A T Shuvaev, Jour, o f Phys Condensed Matter

14(2002)

10 V Pirog, T I Ntídoseikina, I A Zarubin, A T Shuvaev, J Phys.: Condens Mat 14(2002)

1825

11 V Pirog, T I NedoseiMna, Phyaica B 334(2003) 123.