DSpace at VNU: Study of interaction potential and force constants of FCC crystals containing N impurity atoms

V N U . JO U R N A L OF S C IE N C E , M a th e m a tics - P hysics, T.xx, N 02, 2004

STU D Y O F IN T E R A C T IO N P O T E N T IA L AN D F O R C E C O N S T A N T S
O F F C C C R Y S T A L S C O N T A IN IN G N IM P U R IT Y A T O M S
N guyen V an H ung, T ran T rư ng D ung, N guyen C ong T oan
Department o f Physics, College o f Science - V N U
Abstract. A new procedure for description and calculation of the interaction
potential and force constants for fee crystals containing an arbitrary number n
of impurity atoms have been developed. Analytical expressions for the effective
atomic interaction potential, the single-bond and effective spring constants have
been derived. They depend on the number of the impurity atoms and approach
those derived by using anharmonic correlated Einstein model, if all the
impurity atoms are taken out or they replace all the host atoms. Numerical
results for Ni doped by Cu atoms show significant changes of the interact on
potential and spring constants of the substance if the number of impurity atoms
is changed.
1. I n t r o d u c t i o n
Interaction potential and force constants are very im p o rta n t for studying a lot
of physical properties such as therm odynam ic p a ra m e te rs of th e crystals. r'hey art
contained in the first c u m m u la n t or net th erm al expansion, th e second c u rru la n to i
Debye-Waller factor, the third cum ulant, and the th erm al expansion e:pansior
coefficient, which are investigated intensively in the X-ray absorption fine structure
(XAFS) experim ent and theory [1-11]. It is also very im p o rtan t to SU'1)
therm odynam ic properties of m aterials containing im p urity atom s and of all)}
system s [12. 17-19]. Some investigations for crystals c o n tain ing one impurity atom
have been done 117-19]. But more th a n one im purity atom can be dope:! into a
crystal. This case can lead to the development of procedures for studying
therm odynam ic properties of alloys with nano stru c tu re which aie often
semiconductors containing some components with different atom ic sortes.
The purpose of this work is to develop a new procedure for descrip ion and
calculation of the interaction potential and force c o n stan ts of fee crystals containing

some impurity atom s, where one im purity atom [1.7-19] is only an speciá case OÍ
this theory. Our developm ent is derivation of the analytical expression; for tieeffective atomic interaction potential, the single-bond and the effective spring
constants for the case when the cluster involves one or more im pu rity atons. Using
the atomic distribution of the host (H) atoms and the d opan t (D) atom s in I cluster
one can deduce the percentage of these c o nstituent ele m e n ts in the substaice or in
an alloy. All these expressions are different if the n u m b er of impurity atons
changes so th a t one can deduce the results for the case w ith different perceitagesoi
9


,

10

N g u y e n Van Hung T ran T run g Dung, Nguyen Cong Toa n

component elem ents of which an alloy consists. The re su lts in the case if all
impurity atom s are ta k e n out or in the case if all host atom s are replaced by the
im purity atom s are reduced to those derived by using the an harm onic correlated
Einstein model [7] for th e pure m aterials. Num erical calculations have been carried
out for Ni crystal doped by one or more A1 im purity atom s, and the re s u lts are
compared to those of the pure m aterials.
2. F o r m a l i s m
We consider a fee crystal doped by some im purity
atoms replace the host (H) atom s located in the centre
th a t the XAFS process is taken place in the surface
(indicated by D(]) in the centre and the H atom located
by Hfị) as described in Figure la.
H

atom s or dopants (D), th e D
of crystal planes. Supposed
(001) between the D atom
at the position B (indicated

o

QH
o

X ýT
x , y /60"
I\ r f

H0

0

o
o
XhA

o
o

0

H

(001)

HỒ

a)

H

H

DoO*X.)

o

/

-o

<_>
H

■o

b)

Figure 1. Distribution and vibration of H and D atoms in fee volume (a)
and in its (001) surface (b).
Now move the D () atom by an am ount X D along the line A B , then the H B atom
moves backw ard by an am o u n t X H so th a t the m ass centre rem a in s unchanged, the
other atom s are fixed. We have relations
X dM d = X h M h

=> X H = - y —X D = e.XD , e =
m h

,

(1)

IV1H

where M H, M 0 are the m ass of H and D atom, respectively.
This motion leads to increasing the potential energy. The contributions of the
springs in the surface (001) are caused mainly by the atom s in the bond A B , and
those of the springs pen p erd icu lar to AB are negligible (see Figure lb). Therefore,


11

Study of interaction p o ten tia l and force constants of..

t h e y consist of contributions of the spring D-HA by the value K HDX D~ / 2; OÍ the

spring D-Hb by K HD( X H + X D) 2 / 2; and of the spring betw een H h and H on the
extended A B by K HHX H2 / 2.
Hence, the contribution of the atom s on the plane is given by

Besides bonding w ith 4 H atom s at 4
verteses of the plane (001), the D {) atom is
bonded with 8 other neighboring atom s (see
Figure 2) located in the centre of 8

neighboring planes counting from 1 to 8.
They are num bered by 1, 2, 3, 4 and are the
neighbors of the Hịị atom. The rem aining
atom s are the neighbors of D0, but they are
also the neighbors of th e H/i atom. Supposed
th a t n is the total n u m b er of d opant atom s in
the two neighboring lattices of D 0 atom and
among the atom s a t positions 1, 2, 3, 4 there
are n ] dopant (D) atom s, then among the
positions 5, 6, 7, 8 th e re are n> - n - 1 - n ]
dopant (D ) atom s (0
Figure 2. The neighbors of D 0 atom.

The bond linking the D() atom with each of the atom s fom 1 to 4 builds with X D
an angle of 60°. th a t is why the effective displacem ent of th e bond from D„ to these
atom s is given by X Dcos(60°) = X D / 2 . The bond from D {) to the atom s num bered by
5, 6, 7, 8 builds with X D an angle of 120°, th a t is why the effective displacem ent of
the bond from D {) to these atom s is given by X /;c o s( l2 0 ° ) = - X D / 2 . Hence, the
potential contributed by these 8 atom s is equal to
y DO -

+ n 2 )-~^KDD(—X Dy + ( 8 - n x - n2)- —K HD(—X Dy
2

^•1 +

= ------

^2

17

2

y r

K D D A I)

2

2

~

+ -----------g

~

^

2)

TT

Y

2

(3)

2

HD^ D

The line bonding H a with 4 atom s num bered by 1, 2, 3, 4 also builds with X H
an angle of 60° and the line bonding H B with 4 other out of the surface (001)
neighbors of H h builds with X H an angle of 120°, th en the potential contributed by
these interactions is given by


12

N g u ye n Van Hung, Trail T ru n g Dung, Ngu yen Cong Toan
VH0 - ~ ^ K hdX h 2 + -—- ~ - K h„ X h ~ +4

= j K hoX „ ‘ *
Therefore,
contributions

the

total

^

i1 k

potential

„„Xh ‘ .

increase

is

(4)
the

sum

of all

the

above

V/ot =Vp + Vdo + VH0
2. 1
2 K h d ^ d + ~ k hd(x h + -^ o )2 + —K h h X h 2
n\+ n-iV
+

g

I

n \

+

g

V 2 , ( 8 - n , - n 2) ^

A D/ 5A D
Y

TS

'2.~

k hua

+

ll\ )

Y

j r

+

h

g -----------k

g

v2
hda

(5)

d

2

k h h a

h

S u b stitu tin g (1) into (5) we obtain
Vlnl = ~ K III)X I)2 + ị K HD(c + \ f x n2 + ị c 2K HHX D2
.

+

'h +>h
n,e2

+

Q

o

V

v

2 , ( 8 -/Ĩ, - n , ) „
v ,
I) +
t,
K HDA D

2

A ™ A /J

(8-n,)e2 ^
+

“

o

v

,

A ////A D

or
V
íoí

+(,"l +'h>K Dl) +\ y ,

= Ỉ j (' 12_nl
8|

+[4fc + l)- + /!,£- + 1 2 - / 1 , - n . , ] K HD\

Using X = X H + X D = (e+l).XD and com paring (6) to Vtot = —ifgffX2 the effective
force co nstant is resu lted as
K ỊỊ = ------— v ^ l 2 - n l )z1K H!i +(«1 + n , ) K nn +[4(e + l ) :i + « le2 + 1 2 -ÍỈ, - n . , \ K m I
4(v
- 77

1 ; ,...............................

,

<’ >

7 7 7 P ^ _ / l i ) c 2 ^///y + ( r c - 1 ) ^ D D +[ 4 ( c + l ) 2 +/21e 2 + 1 3 - n ] i í f / ơ Ị

4(8 + 1)“

For the case w ithout im purity, 1 . e., 6 = 1, K ol) = ÍL/yy; = AT////, we obtain
^(•IT-pure =

.

(8)

This result coinsides with the one derived by using the correlated Einstein

model [7] which is considered and used widly [8-18].


13

Study of interaction p o t e n ti a l and force constants of..

The above r e s u l t s are the harm onic potential increases due to replacing the H
atom s from num ber 1 to 9 by the D atoms (n <9). In the case of more increase of D
atom s in the surface (001) we suppose th a t the 10"' D atom is located at the place B ,
the 11"' D atom at the place A. the 12th and 13th D atom s at the rem aining places.
As it has been noted at the beginning, the contributions to th e potential increase of
the last two are (.insignificant th a t is why it is not im p o rtan t th a t they are the D or
the H atoms. Now we consider some concret cases:
For n = 10, th en n , = n-> = 4:
v„ =

=2

- —K hdX

dz

+ — K dd 4 X d 2 + —K hdX d ~ + K ddX

+ 3 K hd

2

= ------------- -------------- A D

+ —K

ddX d

+ —K hdX

d

(9)

7 K DD + 3 K hd

r/
^

d

A tiff --- ----------------:

For n = 11, 12, 13:
Vh)l - ]^Kl)nX n~ + - K 1)D4 X ị / + —K HI)X d~ + K ddX d + 9 K DnX D + 2 K HDX
HD^D

2

( 10 )

= ( 4K

1 ) 1)

+K

H I )

)XD~ => K v

1C -

From (7, 8, 10) we obtain
V

4-X

V

1 K D D + 3 K HD

"■ y à {)nK HH +0U)n

+ vồll/,

, g

, s

) 4 K D D + K HŨ
ọ

12/1 +Ò13/|/

( 1 2 - n })e2K HH + ( n - l ) K DD +
^11/t^12/1 (S)13/1
+ [4(c +1)^ +/Ij82 +13 —

^10n
4(e +1 )2

(11)

Using this expression we can calculate the effective force co n stan t K rf{- with
different num ber of im purity atom s replacing the neighboring H atom s of the D ()
atom located at the centre of the fee lattice.
Applying the Morse potential in the approxim ation for weak anharm onicity by
the expansion
V ( x ) = ơ(e"2tuc - 2e~iư)= d ( - 1 + a V - a 3* 3 +•••)
for

e a c h atomic p a i r

from

or its

the equilibrium

form

value

X

by using
at

the

(12)

definition [7] y = x - a as the deviation

tem p eratu re

T,

w here

a = ( x ) , X - r - r0 ,

r

IS

in stan tan io u s bond length, and r0 is its equilibrium value
V(y) = D a 2(I - Saa)y2 - D a Ầy :i + Daz(2 - 3aa)ay + Z)a2a 2(l - a.a) - D

(13)

14

N g u y e n Van H un g, T ran T ru n g Dung, N g u y e n Cong Toan

we obtain the following single-bond spring constants
Kh h “ 2DHa 2H (l - 3 a H( X H}) = 2DHa

(14)

K dd - 2DD(Xfl(l-3aD(X D))
= 2DDa D 1 - —(&]()„ + ỗn/ỉ + ô12/, + SỊ3/Í )aDa - (l - ỏr]n - Ô10/ỉ - ỏUn - Ỉ>ỉ2n )aD - —
l
1 +c
K hd - 2DHDa H
~ D( \

3aHD( X HD^) = 2D hdu ~
hd 1 3 a HD. —

(15)

(16)

and the Morse potential p a ra m e te rs D IW, a HD for the case with im p u rity can be
obtained by averaging

those of the host atom s D H , a H and of the dopant atoms

D 0 , a D , where
D //D -

D ^ƠI l 'rxyDuD
+ D ct . „3 _ D u d u -f D nCL3
Du
H+
r iD
y D ^2 _ ^H
0
> u f/D ^
^
» U//D ^
^
2
£>/ / +£>D
Dh +Dd
u

II

n

(17)

S u b stitu tin g the values of (14-16) into (11) we obtain the effective spring
constant
3
Ì
10/1 ID Da~D 1, —
a «a + ^>DHDaịi ị)

* e ff - 560n D" a "

2 /;

+(Ồ]|/Ỉ + Ổ12/Í + 8 i:j„) 4DDa'ò
+

1 òíì..
0/1

ồ

~~ u 1 0 / ?

w

1 1/1

u 12//

+ DHDàịfD
U 13/J

(18)

2fc + u 2
( 1 2 - /il )£2ũ //a w2

3ea/ya^ ,
9

3ttna^
+ (/1 - ì)DDa n 1 - 'D'
e+ 1
e+1

+ [4(e + l) 2 +7ZjC2 + 13-7?]DW
Da 2D
HirxH
Now we te s t the case when there is not any im purity atom, i.e., tt = 0, we
obtain:
/
V

\
J
3
^
_pun*)EFF - 5DHaH Ị 1 - —a Ha

(19)

This resu lt coinsides totally with the one derived by using the anharm onic
correlated E instein model [7]. From Eq. (13) we obtain the harm onic term


15

Study of inter action p o te n tia l and force constants of..
or for different cases
v HiAy)= \ K HDy2 ’ v HHÌy)= ^ K HHy2’ v D ũ ( y ) - 2 K DDyz

(21)

and the a n h arm o n ic term of the interaction potential is given by
VaJ y ) = K , y \

(2 2 )

K ,= -D u\

Since th is te rm is cubic power of the p a ra m e te r y we can use an expression
sim ilar to Eq. (5), for the cases n = 1 r 9
V'Mut = K , HDX , f + K mD( X H + X Df + K w h X h
XDV

n xK. iDD
+ n ỵK

ÌHD

V

2 ,

D

+ n zK;WD
+ (4 - n }1) K

D

1)K

+ (4

V

2 ,

-

V
^
a h

/

XH

+ 4-K3HH

+ (A-n<))K_

( - XA P \
V 2

y

\ * /

and from th is we deduce

g ~(Kahd - K . ahh)

K wỉỉ(c + ÌÝ = K ]HD + K :]Hn(8 + l):i +£AK :wn + - g.—{K.WI) - K :wn)+
zr> A' ,, = ------— {(8 —/2, )e:
8(c + 1)

ỉ

+(
+ (n,
n, --n.,)K-WD
n 2)K WD +[8(e + l):i
l)3 ++ n ,e 3:i + 8 ++ «2
n2- H
rt.,
.J ]X3WƠ|;j;

3
K , ||T = -------1—- Í/1(8 - n ])c:iD Ha jj + ( n l - n2)Dd<
1 q + [8(e + 1): + n,e + 8 + n2 - n, ]DhdaHD
S( E+ 1)

17 Dp ujj + 3D/iVa HD
16

For the case 11 - 10 we obtain K :](,n

K 3eỊỊ = -

and for rc = 11, 12, 13 it is given

l y D p á ụ + P l l D a HD

16

At the end we obtain
( 1 - 5 0n " 510n ~ Sl l n ~ S1 2 n ~ S13n^ [ ( 8 - ^ )e3P fíq'fí + {nx - n 2)DD0?D
[8(8 + 1)3 + n}£3 + 8 + n2 - n} ] D HDa HD
8(e + 1):

\ l P Dà)j +3DHDaA
HD
_

10/ 2

16

+8
'

1 1 „

llr t

+s
10*1

12n

) 19Ppttp +
1 Q n

13/1

16

1

/1

(23)
which for n = 0 (w ith o u t im purity) is reduced to the resu lt derived by using the
anharm onic c o rrelated E instein model [7]


16

,

,

N g u y e n Van H u n g T r a n Trurtg D u n g N g u y e n C on g Toan

^ 3e ff - H

- ~ ~ D Ha H .
4

(24)

The re m a in in g a n h a rm o n ic c o n trib u tio n ta k e n from Eq. (15) is given by
^25 )

Da ( 2 - Saa)ay ^ 2 D a zay => K 2a = 2 D a z

which c o n ta in s 275a2 . B ased on th e sim ila rity betw een K 2a of Eq. (25) and K h of
Eq. (20) we can use Eq. (18) to deduce
K 2vịĩ = 5Ồ0nD Ha ị + “

7DDazD + 3DHDaịm +(ổ]ỉn +ồl2n + Ổ13/?) 4DDa 2D +DHDa 2HD

1 - ^ 0 n ~ à\0n ~ ^1 ìn ~ s 12/1 ~ ^13/J

2(8 + l) 2

~~ n \

+ ( 11 ~ U D Da D +

I 4- [4(c + l)2 + riịÈ2 + 13 - n]DHDa 2HD

Hence, th e to tal a n h a rm o n ic con trib u tio n to th e atom ic in te rac tio n potential
m u st be given by
v ’, .n h C y )=

K -Mi-ay + K-.toỉiỳ'

(2 ? )

■

For the case n - 0 , i., e., th e re is not any d opan t atom , from
obtain

Eqs. (26, 28, 30) we

V , M = ^ K h:FFy i + Vn, M ,

K eff = 5 D a 2Ị \ - | c t a

, Vnnh(y) = 5Da2ay - l ^ y

(28)

•

(29)

T hese r e s u lts coinside w ith those derived by u sin g a n h a rm o n ic correlated
E instein model [7] which is considered and used widly in XAFS theory for the pure
m a te ria ls [8-18] providing good a g re e m e n t w ith e x p e rim e n t even for Cu with strong
a n h a rm o n ic con trib u tio n s.
3. N u m e r i c a l r e s u l t s a n d d i s c u s s i o n s
Now we apply th e above derived expression s to n u m e ric a l c alculatio ns for fee
crystal Ni doped by several Cu atom s. We calculated th e M orse potential of‘ Ni and
Cu by using the pro ced ure p re se n te d in [19, 21). The r e s u lts a re illu stra te d in
Figure 3 show ing very good a g re e m e n t w ith e x p e rim e n t [15] for th e case of Ni.

U sing th e s e c alcu late d M orse p o ten tials we c alc u la te d single-bond and
effective sp rin g c o n s ta n ts for p u re Ni a n d for Ni doped by sev eral im purity atom s
Cu. The re s u lts a re w ritte n in Table I. The effective s p rin g c o n s ta n ts are different
when Ni is doped by 11=1, 3, 5, 8 Cu atom s.


Study o f inter action p o t e n t i a l a n d force co nsta n ts of..

17

r (A 0)

Figure 3. Calculated Morse potential for Ni (solid), Cu (dash), and an comparison to
experim ent [15] (dot) for the case of Ni.
T a b le I. Effective spring con stants of Ni doped by n - 0, 1, 3, 5, 8 Cu atom s and of
pure Cu.
N

0

1

3

5

8

Cu-pure

A'.rr (e V / A2)

4.1757

3.8072

3.7544

3.7016

3.6668

3.1204

K 2rtỊ( e V / Ẵ 2)

4.2389

3.8803

3.8266

3.7728

3.7358

3.1655

K:lt,n( e V / Ẵ :>)

-1.5047

-1.3155

-1.3047

-1.2939

-1.3010

-1.0753

Although the v a lues of K 2efl- a re sign ifican t b u t th e te rm K 2eĩĩay contain s a
very sm all factor a (ab out 0.007 Ả a t 300 K), t h a t is why th is te rm c o n trib u te s not
so m uch to the effective p oten tial. The effective p o te n tia ls of the system of Ni
illu stra te d in figure 4 c alcu late d by u sin g th e effective sping c o n s ta n ts of Table I
are quite different from th e p a ir p o ten tial of Ni show n in figure 3 den oting the
im portance of the c o n stru c te d effective p o ten tial of th e sy stem . F ig u re 4 also shows
significant ch an ges of th e effective poten tial of Ni w hen it is doped by the im p u rity
Cu atom s. The g r e a te r th e n u m b e r of d o p a n t atom Cu is, th e bigger th e change of
the effective potential. The above p ro p erties considered for one c lu s te r can be
deduced for the whole crystal. T hese c hang es will influence on th e th erm o d y n am ic


18

Ngu yen Van H un g, T ran T ru n g D u n g , Ngu yen Cong Toan

pa ra m eters of the crystals like on the cu m ulants studied in the XAFS spectroscopy
[7, 8, 11, 13, 19].

Figure 4. Effective potential of pure Ni and of Ni doped by 0 ,1 , 3, 5, 8 Cu atoms and
of pure Cu.
4. C o n c l u s i o n s
This work has developed a new procedure for description and calculation of
the effective potential, single-bond and effective spring c o n stan ts including
anharm onic contributions of a crystal doped by an a rb itra ry n u m b er n of im purity
atoms.
Derived expressions of the considered q u a n titie s approach those derived by
using the anharm onic correlated Einstein model for the pure m aterials which
provides very good a g re em e n t with the experim ent and is used widly [7-18].
This work also denotes the im portance of th e effective potential of a system
and its relation with the p air potential, which is especially im p o rta n t for the XAFS
theory [7, 8, 11, 13, 19].
The above properties considered for one cluster can be deduced for the whole
crystal so th a t from this procedure one can deduce a m ethod for description and
calculation of the atomic interaction effective potential and force c o n stan ts of an
alloy consisting of different percentage of con stitu en t elem ents.
A c k n o w l e d g e m e n t s . The a u th o rs th an k Prof. D. M. Pease (University of
Connecticut, USA) for useful discussions and comments. This work is supported
inpart by the basic science research program No. 41.10.04.


19

Study of interaction p o te n tia l and force constants of..
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