VNU Joum al o f Science, M athem atics - Physics 23 (2007) 28-34

Anharmonic effective potential and XAFS cumulants for hcp
crystals containing dopant atom
Nguyen Van Hung*, Le Thi Thuy Hau, Tong Sy Tien
D epartm ent o f Physics, College o f Science, VNU
334 Nguyen Trai, Hanoi, Vietnam

Received 17 June 2007
Abstract. A nevv procedure for calculation and analysis of XAFS (X-ray Absorption Fine
Structure) cumulants of hcp crystals containing dopant atom has been derived based on quantum
statistical theory with generalized anharmonic corTelated Einstein model. Analytical expressions
for eíĩective local force constants, correlated Einstein frequency and temperature, first cumulant or
net thermal expansion, second cumulant or Debye Waller factor and third cumulant of hcp crystals
containing dopant atom have been derived. Morse potential parameters of pure crystals and those
with dopant included in the derived expressions have been calculated. Numerical results for Zn
doped by Cd are found to be in good agreement with experiment.

1

. Introduction

To study thermodynamic properties of a substance it is necessary to investigate its effective local
force constants, correlated Einstein írequency and temperature, net thermal expansion, mcan square
relative displacement (MSRD) or Debye Waller factor and third cumulant [1-14] which are contained
in thc XAFS [12]. Local force constants of transition metal dopants in a nickel host in XAFS has been
investigated but only for comparision to Mossbauer studies [10].
The purpose o f this work is to develop a method for calculation and evaluation of thc cffective
local force constants, correlated Einstein írequency and temperature, first cumulant or nct thermal
expansion, second cumulant or MSRD characterizing Debye Waller íactor and third cumulant of hcp
crystals containing a đopant (D) atom as absorber in the XAFS process. Its nearest neighbors are the

x R 12.R y
X

= VHD (X) + vun
, D K—
HD ( - K I ) + 4 VH
2J
+ 41/,HD —K

2)

A1d M„
Md +Mh

+ 4 ViHH
K

=

'1

X

2 2)

+ 4 V.HH

( 1)
1


(3)

for the pure material and
^//Z>C*) = ^M D (“ l + a Ẫ D *2 ~ a HDx * + •••)

(4 )

for the doping case, where Morse potential parameters have been obtained by averaging those of the
pure materials and they are given by
D hd -

2

„2

.

Ưsing the definition [2, 7] y =

a HD X

D d + D ịị

.

...

a ìỉD -

Dd + Dh


(8 )

g

K ) ^ H D a HD

the anharmonic contribution to the effective potential of the system
SV{y) = ^2 ( \ +3K2)DHDa HD
2 +~ỊDHa
3 "
-H
2^

( i - v--3'"
3 )dwữơ ^ 0

(9)

the correlated Einstein frequency and temperature
°>E = yjkcff 1n .

( 10)

ớE =hù)E / k

The cumulants have been derived by averaging procedure in quantum statistics, using the
statistical density matrix p and the canonical partition íunction z in the form
< y m > = j T r ( p y m), m m 1 , 2 , 3 , - ,



(15)

h

lụa,

and use harmonic oscillator State 1«) with eigenvalue En = nh(ù£ (ignoring the zero point energy for
convenience).
Thereíore, the expression for second cumulant (MSRD) or Debye-Waller factor is rcsulted as
hú)
(l + z)
ơ= ơ
(16)
(l-z)
(1 + 3K1)DHDa 1Hl) + - D Ha ị
4
Now we calculate the odd cumulants
-C-. e~PE" - e~^E"'

1

(17)

n.n'
En ~ E n'
'
1
Using the calculated matrix elements and mathematical formulas for different transíbrmations we
obtain the expressions for the íìrst cumulant (m =l)

n(1 _- 2_\2
’
)

~

(l-K-3 )DWữữtftí - ~ D Ha ị

(3 ) {hmE)
" ■ 16

(19)
( l + 3/cĩ )Df,Da 1 / D + - D n a H

4

l;rom Eq. (18) we obtain the thermal expansion coeíĩĩcient
da _ 0 z\ln(z)\'
aT —— — = a
r
r dT
0 - *)’
1

3k,

(l

K ) D hdclhd


Bond
Zn-Zn, present
Zn-Zn, exp.[13]
Cd-Cd, present
Cd-Cd, exp.[13]
Zn-Cd, present
Zn-Cd, exp.[13]

D(eV)

CC(Â' )

r0 (Ằ)

keJỵ { N / m )

0.1698
0.1685
0.1675
0.1653
0.1687
0.1669

1.7054
1.7000
1.9069
1.9053
1.8084
1.8046


174.8673

pigure 1 illustrates our calculated Morse potential (a) and anharmonic effective potential ( 6 ) for
Zn doped by Cd atom. The Morse potential for this case has the same form as for the pure material, the
anharmonic effective potential becomes asymmetric due to the third order of the potential. All they
agree well with experiment [13].


32

Nguyen Van Hung et aỉ. / VNU Journaỉ o f Science, Mathematìcs - Physics 23 (2007) 28-34

a)
b)
Figure 1. Calculated Morse potential (a) and anharmonic eíĩective potential (b) for Zn doped by Cd atom
compared to experiment [13].
Figure 2 demonstrates the calculated first cumulant or net thermal expansion ơ 0 1 (T) (a), and
second cumulant or Debye-Wal!er factor ơ 2 (T ) (b). The third cumulant ơ(3) (T) and cumulant relation
cr(l*
provided by the Ministry of Science and Technology No. 40.58.06.

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34

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