VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
MINISTRY OF EDUCATION AND TRAINING
GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY -----------------------------
CU SY THANG
STUDY OF THE THERMODYNAMIC PARAMETERS AND
CUMULANTS OF SOME MATERIALS BY ANHARMONIC
XAFS METHOD
SUMMARY OF MATERIAL SCIENCE DOCTORAL THESIS
Ha Noi - 2020
VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
MINISTRY OF EDUCATION AND TRAINING
GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY -----------------------------
CU SY THANG
STUDY OF THE THERMODYNAMIC PARAMETERS
AND CUMULANTS OF SOME MATERIALS BY
ANHARMONIC XAFS METHOD
SUMMARY OF MATERIAL SCIENCE DOCTORAL THESIS
Major: Electronic material
Code: 9 44 01 23 Supervisor: Prof. Dr. Sc. Nguyen Van Hung Dr. Le Quang Huy
Ha Noi - 2020
INTRODUCTION
The X-ray Absorption Fine Structure (XAFS) spectroscopy
technique is a modern and high precision method to be used in the
structural determination of materials. In general, this method is used
to fit the theoretical and experimental spectra to extract data or
parameters from the XAFS spectra.
Anharmonic Correlated Einstein Model (ACEM) [9] is one of the
efficiency theoretical methods [7] used to study thermodynamic
parameters of XAFS spectra. Anharmonic effective interaction
potential in ACEM has been built. In which Morse potential is
assumed to describe the single-pair atomic interaction. By using this
effective potential, the ACEM has not only overcome the limitations
of using single bond potential [8] but also simplified the many-body
system problem back to the simple one-dimensional system problem
with the contribution of many-body effects through consideration of
the interaction of neighboring atoms.
Many previous studies [10-25] showed that the numerical results
of the ACEM were good agreement with the experimental data as
well as those obtained values by other methods for several different
structural materials,... However, most of the studies focused on the
thermodynamic parameters, especially the cumulants of the XAFS
spectra without concerning the anharmonic contribution of the
second cumulant as well as the anharmonic contribution to XAFS
phase and amplitude components. Debye-Waller factor or the
second-order cumulant is an important thermodynamic parameter
that characterizes the decrease XAFS amplitude. The relationship
between the second-order cumulant and other thermodynamic
parameters and XAFS phase and amplitude needs to be continuously
studied, considered in more detail and comprehension. Therefore, I
1
choose the research topic: "Study of thermodynamic parameters
and cumulants of some materials by anharmonic XAFS method".
1. Target of thesis
To develop a method that can simplify the determination of the
thermodynamic parameters and XAFS spectra which base on the
second-order cumulant only. In particular, this method can be
applied for both theoretical calculations and experimental
measurements in the XAFS method.
2. Subject and Scope of the thesis: Subject: - Thermodynamic parameters, XAFS cumulants, XAFS spectra, and
their Fourier transform magnitudes.
- Materials: Diamond crystals (Si, Ge), fcc crystals (Cu), hcp
crystals (Zn).
Scope:
- Models and theoretical methods of XAFS: focus on anharmonic
correlated Einstein model using anharmonic effective potential in
which Morse potential is assumed to describe the singe-pair atomic
interaction.
3. Study methods
Theoretical method:
- Using the anharmonic correlated Einstein model.
- Using the anharmonic effective interaction potential method.
Experimental method: Study on documents, the configuration of
XAFS spectroscopy experimental station, the scanning process of
XAFS spectra as well as data processing methods at Synchrotron
Light Research Institute - Thailand.
Numerical calculation Program and XAFS spectra data processing
and analysis software
- Using Matlab 2014 software, Demeter 9.0.25 software.
2
4. Main contents of the thesis
- Study and establish the expressions of thermodynamic
parameters and the Debye-Waller factor of the XAFS spectra of
materials depending on the temperature.
- Study and establish a relationship between the anharmonic
contribution of the second-order cumulant and the Debye-Waller
factor of XAFS spectra of materials depending on temperature.
- Study and establish a relationship between the Debye-Waller
factor and anharmonic contribution of the XAFS phase and
amplitude of materials depending on temperature.
- Analyze and evaluate constructed theoretical calculations as
well as evaluating the agreement between theoretical results and
obtained experimental values from Synchrotron Light Research
Institute - Thailand and other theoretical methods or measurements
for fcc (Cu) and hcp (Zn) structural materials.
CHAPTER 1. OVERVIEW OF XAFS DEBYE-WALLER
FACTOR
1.1. Fundamentals of XAFS
1.1.1. Simple physical description of XAFS
X-ray absorption fine structure is the final state of interference
between photoelectric waves emitted from absorption atoms and
scattering waves by neighboring atoms.
2
2
k
2
R
/
(
k
)
2 j
j
e
2 S N e 0
j
f k ( ) j
1.1.2. XAFS equation
(k)
kR
(k)
j
j
2
sin 2
kR
j
j
(1.14)
2
w+i
2
2 j
1.1.3. XAFS Debye-Waller factor
e
ke
in (1.14) have formed as is called XAFS - The factor Debye-Waller factor [29].
3
1.1.4. XAFS cumulants
- J.J. Rehr [28,29,31] showed that XAFS Debye-Waller factor is
2 (
)
n
)
ik r R 0 j
generally complex and has a natural cumulant expansion approach in
e
n (
]
exp[
(2 ) ik n !
n
1
term of Taylor series from the generalized cumulant expression [33]: (1.22)
T a( )
(
)
(1) T ( )
jr R
r R 0 j
- With x= and lattice expansion
0 y ; cumulants are written:
0
(1)
(1)
T ( )
(
)
R
y
r
r R 0 j
j
j
R 0
(2)
2
2
Set y =x - a and
T ( )
2
(T)
(
)
y
r R 0 j
(3)
3
3
T ( )
(
)
y
r R 0 j
(1.23)
1.2. XAFS Debye-Waller factor studies methods 1.2.1. Correlated Einstein Model [1]
(
)
j
2
ii
ii
R
R
T ( )
coth
2 j
. ( ) i
2
2
j M
i
1 2
j
i
. states center at )j E The correlated Einstein model is one of the ways of calculating or fitting XAFS thermodynamic parameter values. In this model, the vibrational density of single vibrational frequency: ( 1.2.2. Equation of motion method [3,38] (1.37)
1.2.3. Statistical moment method [39-46] (1) a
(T)
x
r
(T)
a
(0)
r 0
y (T) 0
2
2
u 0
2 u i
2 u 0
u u 0 i
. R u i
2
(1.58) (1.59)
CHAPTER 2. ANHARMONIC CORRELATED EINSTEIN MODEL IN STUDY OF XAFS THERMODYNAMIC PARAMETERS 2.1. Effective potential in anharmonic correlated Einstein model The generalized expression of anharmonic effective interaction potential using in ACEM:
(x)
(x)
E
x R R 0 ij i
M
i a b j a b
,
,
i
(2.3)
4
Basing on the quantum statistical perturbation theory, we can
determine Hamiltonian of the system and can extract anharmonic
2
3
effective interaction potential expressions:
E
k a eff
k a 3
3
(2.6) a ( )
1 2 (k
y
(y) E
eff a
2 k 3 a ) y 3
k 3
2
(2.7)
(x)
a ( )
y ( )
E
E
k y eff
E
1 2
(2.9)
)
)
2 (r ij
r 0
(r ij
r 0
)
e 2
r ij(
2.2. Morse potential [53]
D e
2
2 x
x D ( )
3 3 x
)
Taylor series expansion in approximation up to the third order:
(2.13) ( 1 Table 2.2. Morse parameters of copper (Cu) and zinc (Zn) from theoretical calculation.
Materials
D (eV)
c
r0(Å)
(Å-1)
2.868 2.8669 2.793
0.3429 0.3364 0.1700
1.3588 1.5550 1.7054
Cu [20,60,61] 2 Cu [62] 2 Zn [20,15,17,22,23,59,63] 1/ 2 2.2.1. Applying Morse potential to calculate the thermodynamic parameters and effective interaction potential in anharmonic correlated Einstein model for fcc and hcp structural materials
Figure 2.3. Face centered cubic Crystal [47] Figure 2.4. Hexagonal close- packed Crystal [47]
Derive the expression of effective interaction potential which is
used in the anharmonic correlated Einstein model, we can get:
(x)
(x) 2 (
)
E
x 2
x ) 8 ( ) 8 ( 4
x 4
(2.28)
5
k
5
a
2
2
D
D
5
1
5
k
k
5
1
5
D
D
2
2
a
eff
eff
9 10
eff
2 D
5
3
D
k
k
D
3
3
3
3 4
5 4
2 D
E k
k
B
B
E E
3 y ).
2
(ay
y ( )
D
5
( ) 5 y
3 y )
(ay
2
D
3 20
1 4
E
Derive the expressions of effective local force constant, cubic anharmonic parameter as well as Einstein frequency and temperature for fcc and hcp crystals: 3 2 E (2.31); (2.32,2.34) (2.33) 2.2.2. Applying Morse potential to calculate thermodynamic parameters and interatomic effective potential in anharmonic correlated Einstein model for diamond structural materials
Figure 2.5. Diamond structural crystal [47]
(x) 3
(x) 3
(x)
x
x
x
x
E
M
M
1 3
1 3
1 6
1 6
3
3
Derive the expression of interaction effective potential which is
used in the anharmonic correlated Einstein model, we can get: (2.36) Derive the expressions of effective local force constant, cubic
anharmonic parameter as well as Einstein frequency and temperature
2
D 2
2 D
2 D
k eff
7 3
2 D
7 35 3 a 12 6
7 3
5 a 2
7 3
1
D
3
E k
k
7 3
2 D
B
B
35 36
E E
effk k 3
for fcc and hcp crystals:
(2.39); (2.40)
Morse parameters for Si[25,64]:
D=1.83 (eV); =1.56 (Å-1) và r0=2.34 (Å)
Morse parameters for Ge[25,64]:
D=1.63 (eV); =1.50 (Å-1) và r0=2.44 (Å)
6
x
( )
ijk
1
p
q
exp
a
,
a
r ij
r ij
r ij
r ij
ij
2.3. Stillinger-Weber potential [52,65] Wij (2.41)
0,
a
r ij
where the single-pair interaction potential component: A B
2
1
1
( exp
a
)
(
a
)
cos
W ijk
r ij
r ik
ij
k
1 3
(2.42) The three-body interaction potential component:
Parameters for Si[52,65]: A=7.049556277; B=0.6022245584; p=4;
q=0; a=1.80; =21.0; =1.20; =2.0951Å; =50 kcal/mol.
Parameters for Ge[52]: A=7.049556277; B=0.6022245584; p=4;
q=0; a=1.80; =31.0; =1.20; =2.181Å; =1.93 kcal/mol. 2.4. Calculating thermodynamic parameters by anharmonic correlated Einstein model 2.4.1. Calculating cumulants by anharmonic correlated Einstein
ˆ a a
0 ˆ (
)
0
2 E and ˆ ˆa a
n
with model Atomic vibration is quantized in terms of phonons, anharmonicity is the result of phonon-phonon interaction. So we can express y in term of annihilation and creation operators [68]: y
ˆ ˆ a a ,
1,
ˆ a n
n
1
n
1 ,
ˆ a n
n
1
n
1 ,
ˆ ˆ a a n
n n
The above operators have the following properties: (2.54)
m
m
y
Tr
( y ), m 1, 2,3,...
Then the averaging procedure can be calculated by statistical physics as [69]:
(2.55)
1 Z Calculating (2.55) in cases of: + m is even value:
m
n E
y
Tr
m ( y )
Tr
m y )
e
m n y n
( 0
1 Z
1 Z
1 Z
n
0
0
(2.59)
7
2
(2)
n E
We can be received the second-order cumulant:
y
e
2 n y n
1 Z
n
0
(2.60)
E n
'
E n
e
m
y
n
n
'
m n y n
'
E
+ m is odd value:
1 Z
n n ,
'
E n
e E n
'
0 We can receive the first and third-order cumulants. Finally, we can receive expressions of cumulants for fcc (Cu) and
(2.64)
(1)
2
(1)
2
hcp (Zn) structural materials:
a
)
a
)
( 0
( 0
3 4
1 1
z z
1 1
z z
(2)
2
2
(2)
)
)
( 0
( 0
z ( (1
1) z )
1) z )
2
2
z
)
3(
z
)
(3)
(3)
4 ) ( 0 2
(1 10 z 2 ) z (1
9 20 z ( (1 4 ) 0 10
(1 10 z 2 ) z (1
fcc: hcp:
2
z
2
2
( (
) )
0 0
(2.63, 2.73, 2.80) 2.4.2. Derive expressions of cumulants based on the second cumulant only in anharmonic correlated Einstein model. From the expression about the relationship between temperature variable z and mean square relative displacement given by Rabus [8,9]: 2 , replace into (2.63,2.73,2.83) we can receive
(2)
(1)
(1)
2
(2)
a
)
expressions of cumulants based on the Debye-Waller factor or the XAFS second cumulant only for fcc and hcp structural materials:
( 0
3 4
1 1
z z
3 4
(2)
(2)
2
)
2
( 0
(3)
z ( (1
1) z )
[3(
2 2 )
2((
2 2 ) ) ]
0
2
3 10
z
)
9 20 2
(3)
[3(
2 2 )
2((
2 2 ) ) ]
0
4 ) ( 0 2
(1 10 z 2 ) z (1
2
2
(2.82)
)
( 0
E 2 10 D
1
where
2
2
2
) 0 2
4 ( 3
(2.83) The relationship among cumulants is determined according to: (1) 2 (3)
8
2
) ( 0 2
Formula (2.83) showed that the ratio among cumulants related only to the second-order cumulant. This ratio is considered as the standard for the XAFS theoretical studies method regarding physics [9]. We can see that approaches this ratio will approach the classical value of ½ when
to 0 so that the classical limit is applicable. 2.4.3. Calculating thermal expansion coefficient in anharmonic
correlated Einstein model
For fcc (Cu) and hcp (Zn) structural material: The expression of
4
thermal expansion coefficient is derived base on Debye-Waller factor
)
(
2 2 )
( 0
0 T
0 T
T
0 T
3 D 9 rk 4
D rk
2 T
B
B
2
15 4 The relationship of cumulants and thermal expansion coefficient is when TE means that determined by (2.88). We see that
T r .T (3)
1 2
or the XAFS second cumulant such as: 3 with (2.87) and
2
2
1
from temperature TE, anharmonic effects are significant, we can apply the classical approximation, and when T<E, anharmonic effects are insignificant, we must use quantum theory. In particular, at temperature T=E/2, the ratio from (2.88) approaches the classical value of 1/2, so when T<E/2 we must consider anharmonic effects.
) ( 0 2
.
2
2
2 r .T T (3)
1 5 . 2
2 2 D k T B
1
) 0 2
2 ( 3
(2.88)
2.4.4. Evaluation of the calculated XAFS second cumulant results using Morse and Stillinger-Weber potential in the anharmonic correlated Einstein model for diamond structure semiconductor materials Applying the anharmonic correlated Einstein model using Morse
D
2
(ay
3 y )
E y ( )
5 12
7 3
potential: From (2.39,2.40), replace into (2.7) we can receive:
9
(1)
2
(2)
a
)
( 0
5 4
1 1
z z
5 4
(2)
2
2
)
( 0
1) z )
2
5(
z
)
(3)
[3(
2 2 )
2((
2 2 ) ) ]
0
z ( (1 4 ) 0 6
(1 10 z 2 ) z (1
5 6
Replace above expression into (2.59) and (2.64) we can determine expressions of cumulants for diamond structural materials:
(2.89)
2
)
( 0
3 E 2 D 14
where:
4
(
2 2 )
)
( 0
0 T
T
0 T
2 T
35 12
3 D rk
B
với
+ The expression of thermal expansion coefficient:
(2.90)
25
+ The expression of anharmonic factor:
T ( )
2
(T)[3
2
(T)(3
2
(T)]
5 R 4
5 R 4
2 24
(2.91)
+ The expression of anharmonic contribution:
(T)
(T)
2
(T )]= (T)[
(T)
]
2 A
2 (T)[ H
2 H
0
2 0
(2.92)
+ The expression of anharmonic contributions to XAFS spectra
2
k
(
T
)
2 A
)
2 e
AF k T ( ,
phase shift and amplitude:
3
(3)
(2.93)
)
2 [
k R
2
(T)(
)]
k
(T)
A k T ( ,
1 R
1
4 3
(2.94)
Applying the anharmonic correlated Einstein model and the
statistical moment method using Stillinger-Weber potential:
10
2.6.
Temperature
2.7.
Temperature
Figure dependence of
the second-order
the second-order
cumulant using Stillinger-Weber
cumulant using Stillinger-Weber
potential in the statistical moment
potential in the statistical moment
method for Si.
method for Ge.
Figure dependence of
Figures 2.6 and 2.7 show a good agreement of the statistical moment method using in the calculation the XAFS second cumulant values for Si and Ge diamond semiconductors, respectively. For Si, the results were compared with the obtained values given by M. Benfatto in the article [70] at 80 K, 300 K, and 500 K. For Ge, the results have an agreement with experimental values given by A.E. Stern in [71] at 300 K, G. Dalba in [72] at some temperatures and with theoretical calculation results given by J.J. Rehr in the article [4] when using the LDA method at 300 K. Moreover, the obtained results are consistent with experimental results of A.Yoshiasa in [73] in some specific temperatures, even the results are calculated from the GGA and hGGA methods given by J.J.Rehr at 300 K [4]. These results published in the article [19].
Numerical results of the second-order cumulant using Morse and Stillinger-Weber potential for Si and Ge crystals by the anharmonic correlated Einstein model were evaluated and compared in the article [18,24,25]. The anharmonic correlated Einstein model using two
potentials is consistent with experimental values as well as those obtained from other methods. Therefore, the anharmonic correlated
11
Einstein model can be applied to diamond semiconductors using Morse and Stillinger-Weber potential.
2.5. Quantum effects in low temperature limit and classical
approximation in high temperature
The obtained thermodynamic parameter formulas from quantum
theory can be applied at all temperature values. At the high-
temperatures, the formulas include the results of classical
approximation theory. At the low-temperature limit, quantum effects
express through contributions of zero-point energy.
(1)
/ k
a
(1)
(1 2 )z
k k T 33 B
eff
0
/ k
2
k T B
eff
0 (1 2 )z
(2)
2 ) / k
k k T 36 (
B
3 eff
(3)
(1 12 )z
(3)
0
r
k 33 / k B
0
2 z (ln ) (1 2 )
z
T
T z
2
2
2
(1) 2 (3)
3(1 2 ) z z 2(1 12 )
3 2
(1) (1 2 ) z 0 0 (3) z (1 12 ) 0
1 2
2
z 3 ln
0
(3)
1 2
.TT r
2 .T T r (3)
1 z
T0 T Thermodynamic Quantity
ANHARMONIC STUDY ON IN
CHAPTER 3. EXPERIMENTAL MEASUREMENT AND CORRELATED APPLICATION OF EINSTEIN MODEL XAFS THERMODYNAMIC PARAMETERS FOR HCP AND FCC STRUCTURE MATERIALS. 3.1. Synchrotron facility and XAFS spectra experimental station
The preparation for experimental samples depending on
temperature:
12
Figure 3.5. Experimental station Beamline 8. SLRI Figure 3.7. Experimental XAFS measurement depending on temperature
3.2. Experimental measurements results of the Debye-Waller factors for hcp structure material.
Experimental values are shown in figure 3.12 and table 3.1.
Figure 3.12. XAFS spectrum and Fourier transform magnitudes of
Zn at 300 K, 400 K, 500 K, and 600 K
(1)(Å)
(1)(Å)
2(Å)
2(Å)
2(Å)
(3)(Å)
(3)(Å)
T (10-5/K)
T (10-5/K)
T(K)
LT
TN
LT
MHĐH
TN
LT
TN
LT
TN
0.0139
0.0143
0.0110
0.0109
0.0113
0.0003
0.0003
1.555
1.582
300
0.0182
0.0189
0.0146
0.0143
0.0149
0.0005
0.0006
1.582
.618
400
0.022
0.0232
0.0182
0.0177
0.0185
0.0008
0.0009
1.595
1.599
500
0.0270
0.0279
0.0219
0.0211
0.0223
0.0011
0.0012
1.602
1.630
600
Table 3.1. The value of cumulants and thermal expansion coefficients of Zn: Theoreratical calculation (LT) and experimental value (TN) at temperatures. Symbol: MHĐH – Harmonic model
13
3.3. Determining thermodynamic parameters of XAFS from the
experimental values of the Debye-Waller factor or the second-
order cumulant by anharmonic correlated Einstein model for
hcp structure materials.
Figure 3.14. Temperature dependence of the first cumulant, total and
the harmonic second cumulant and experimental values
From the illustration in figure 3.14b, anharmonic correlated
Einstein model, and the harmonic correlated model [82] have certain
deviations for the second-order cumulant or Debye-Waller factor in
the high-temperature range. ACEM is more suitable for experimental
values than the harmonic correlated model. Note that the data of the
first-order cumulant is derived from the experimental value of the
second-order cumulant.
Figure 3.15. Temperature dependence of the third cumulant and
thermal expansion coefficient of Zn calculated from cumulant
experimental values.
14
Figure 3.16. Temperature dependence of cumulants ratio, the ratio
between thermal expansion coefficient and cumulants of Zn.
Similar to the first -order cumulant, we are also able to determine
the third-order cumulant and thermal expansion coefficient of zinc
(Zn) at 300 K, 400 K, 500 K, and 600 K. Figure 3.15a and 3.15b
showed the results derived from experimental measurements are very
agreement with the calculations from the theoretical model. To
assess the validity of the theoretical model, we can also check by
establishing the ratio among cumulants according to the expression
(2.83) and ratio among thermal expansion coefficient and cumulants
according to the expression (2.88). Figure 3.16 showed the above
relationships. From figure 3.16, the values are derived from
experimental values that make these ratios reach the value of ½.
These ratios are used as the standard method for assessing cumulant
studies [9,56,81,83], as well as for determining temperature when the
classical limit can be applied [9]. The theoretical results and the
results of these ratios showed that hcp structure materials,
specifically Zn, we can use classical correlation Einstein model when
the temperature is higher than Einstein temperature (E = 206 K). 3.4. Experimental results of XAFS Debye-Waller factors for fcc structure material
15
Figure 3.17. XAFS spectrum and Fourier transform magnitudes of Cu at
300 K, 400 K, 500 K
temperatures
Figure 3.18. The process of fitting the XAFS spectrum of Cu at
XAFS spectrum at temperature values after merging are fitted to the theoretical spectra by using Artemis software. The R, k variables are in R space [1-3 Å] or k space [3.00-14.023 Å-1] run to the optimal value between theoretical spectra and experimental spectra. 3.5. Determining thermodynamic parameters of XAFS from experimental values of the Debye-Waller factor or the second- order cumulant by anharmonic correlated Einstein model for fcc (Cu) structure material
Figure 3.19. Temperature dependence of the first cumulant, total and harmonic second cumulant and the experimental values.
Anharmonic correlated Einstein model and harmonic correlated
Einstein model [81] have certain deviations for the second-order
16
cumulant or Debye-Waller factor in high-temperature range (Figure
3.19). The results showed that anharmonic correlated Einstein model
is well suited to experimental values as well as obtained results of S.
a Beccara, et al. [82] for the first-order cumulant and V. Pirog, et al.
[58] for the second-order cumulant. Note that, the data of the first-
order cumulants are derived from the experimental second-order
cumulants values.
Figure 3.20. Temperature dependence of the third-cumulant and thermal expansion coefficient of Cu calculated from experimental cumulant values Similar to the first-order cumulant, we can also identify the third- order cumulant and thermal expansion coefficient of copper (Cu) at 300 K, 400 K, 500 K. Figure 3.20 showed the results derived from experiment values were very consistent with the obtained data of V. Pirog, et al [58] and T. Yokoyama, et al [88] for the third-order cumulant. Figure 3.20b indicated agreement among calculated results from the present method and the experimental results and obtained results from other documents [89] for the thermal expansion coefficient. To evaluate the validity of the theoretical model, we verified by establishing the ratio among cumulants according to expression (2.83) and the ratio among the thermal expansion coefficient and cumulants according to the expression (2.88). Figure 3.21 showed these relationships.
17
Figure 3.21. Temperature dependence of cumulants ratio, ratio between thermal expansion coefficient and cumulants of Cu
The values extracted from experiments make these ratios
approach the value of ½ (Figure 3.21). These ratios are used as the
standard method for assessing cumulant studies [9, 81, 90], as well as
for determining temperatures at which classical limits can be applied
[9]. The theoretical results and the results of these ratios showed that
hcp structure materials, specifically Cu, we can use classical
correlation Einstein model [9,81] when the temperature is higher
than Einstein temperature (E = 218 K). CHAPTER 4. ANHARMONIC CORRELATED EINSTEIN MODEL IN STUDY OF XAFS PHASE AND AMPLITUDE CONTRIBUTION OF HCP AND FCC STRUCTURE MATERIALS
4.1. Overview of anharmonic XAFS spectra
The anharmonic XAFS function is represented by cumulant
n
2 R (k)
(k)
i
n ( )
expansion approach [21,60,90,91]:
k ( )
F k ( )
Im
exp 2
ikR
2
(4.1)
(2 ) ik n !
e kR
n
e
2 i 4
(1)
3
4
W( ,
k T
)
2
ki
(T) 2
k
2 2
T ( )
ik
3
(T)
k
4
(T)
...
1
(T) k R
R (k)
4 3
2 3
3
(3)
XAFS amplitude expression [9,90-92]: (4.2)
)
k T ( ,
)
)
2 [
k R
2
(T)(
)]
k
(T)
A k T ( ,
k T ( , 0
1 R
1
4 3
(4.3)
18
2
(T)
2
(T)
2
(T ) 0
Với (4.4)
4.2. XAFS Debye-Waller factor with contribution anharmonic.
In the high-temperature range, the Debye-Waller factor includes 2
(T)
(T)
(T)
2
2 H
2 A
(4.5) components: a harmonic and anharmonic contribution component.
(T)
2
(T )]= (T)[
(T)
]
2 (T)[ H
2 H
0
2 0
(T)
(T)
2
(T)
(1
(T)
]
2 (T) A Replace (4.5) into (4.4) we receive: 2 ] 0
2 (T)[ H
2 H
2 0
2 (T)[ H
2 0
where (4.6)
With (T) is called the anharmonic factor of the XAFS second
(T)
2 G
G
V V
ln ln
E V
cumulant which depending on temperature and Grüneisen parameter.
với
4.2.1. Determination of Grüneisen parameter G
From (2.32, 2.34) we can determine lnE/T
(
R
2
)
(4.9) and lnV/T (4.10). Therefore, we can determine:
G
ln ln
E V
4(1
2 2
)
3 4 9 8
(4.11)
4.2.2. Determination of anharmonic factor (T) Determine the change in volume due to thermal expansion V/V and from (4.12) we can determine:
T ( )
2
(T)[1
2
(T)(1
2
(T)]
3 R 4
4 R
2 9 8
(4.14)
4.5. XAFS spectra with contribution anharmonic components
The Debye-Waller factor includes two components as expression
(4.5). To accurately describe the actual spectra so that the XAFS
phase and amplitude in (1.14) need to be added to the anharmonic
factors. In detail, the phase component is added to the anharmonic
k
(
T
)
2 2 A
factor:
2 e
AF k T ( ,
)
19
3
(3)
The amplitude component is added to the anharmonic factor:
)
2 [
k R
2
(T)(
)]
k
(T)
A k T ( ,
1 R
1
4 3
(4.16)
2
2
k
2
R
/
(
k
)
j
2 j
j
(k)
( )
( ,
e
kR
(k)
(k, T)
f k F k T e ) j A
j
A
j
sin 2
2 S N 0 2 kR
j
j
Then, the generalized expression of XAFS become to: (1.17)
4.6. XAFS anharmonic phase and amplitude components for HCP (Zn) structure material The anharmonic XAFS components increased with increasing temperature and k-wavenumber values (Figure 4.1).
Figure 4.1. Temperature dependence of anharmonic amplitude and phase components with the k-wave number of XAFS spectra for hcp (Zn) structural material. These components (phase and amplitude) contribute
to anharmonic XAFS spectra show in figure 4.2 in both of the theoretical calculations by the anharmonic correlated Einstein model and experimental values.
Figure 4.2. Theoretical and experimental XAFS spectra with hcp structure material (Zn) at temperatures.
20
(Zn) hcp
Figure 4.3. Comparison of Fourier transform magnitudes theoretical spectra with of results experimental XAFS for structural material at temperatures.
(T)
Figure 4.3 showed an agreement between the theoretical results of the model with obtained Fourier transform magnitudes from experimental measurements. In addition, we can see that the magnitude of spectra decreases with the increasing temperature gradually. Note that the anharmonic contribution components to the XAFS phase and amplitude are calculated base on second-order cumulant only. Moreover, by using anharmonic correlated Einstein model, we can reconstruct XAFS spectra and the Fourier transform magnitudes from the obtained experimental second-order cumulant values. This study has shown that the obtained experimental results are consistent with theoretical calculations at a temperature of 300 K, 400 K, 500 K, and 600 K for Zn. 4.7. Anharmonic contribution of XAFS phase and amplitude for FCC (Cu) structural material 4.7.1. Anharmonic contribution to the second-order cumulant and anharmonic factor
Figure 4.4. Temperature dependence of anharmonic contribution to the second-order cumulant and anharmonic factor (T) of fcc (Cu) structural material.
21
Figure 4.4a and 4.4b showed an agreement between the calculated
results from the present method and experimental values for
anharmonic contribution to the second-order cumulant and
anharmonic factor (T), respectively. These experimental values are
extracted from the experimental second-order cumulant results.
Anharmonic factor (T) is a new factor given by Nguyen Van Hung,
et al. in the article [21]. In addition, the above anharmonic
contribution is difficult to measure directly. So when using
anharmonic correlated Einstein model, we can calculate and
represent these anharmonic components that depending temperature
based on theoretical calculations or experimental measurement
values of the second-order cumulant. 4.7.2. Anharmonic contribution to XAFS phase and amplitude
For anharmonic contribution to XAFS spectra, indicated in the
figure (4.3). We can describe two components including anharmonic
)
(k, T)
A k T and phase shift
A
of XAFS contribution to amplitude F ( ,
spectra in expressions (4.16) and (4.17).
amplitude and phase with k - wave number for fcc (Cu) structure material. Anharmonic contribution to the XAFS spectra increases with increasing temperature and value of k-wavenumber (Figure 4.5). These components contribute to anharmonic of XAFS spectra presented in figure 4.6 regarding on theory of anharmonic correlated
Figure 4.5. Temperature dependence of anharmonic contribution to
22
Einstein model as well as experimental results. Figure 4.7 illustrates a good agreement of the theoretically calculated results and obtained spectra from measurements through Fourier transform magnitudes. Besides, figure 4.7 also shows that peak heights decreased and their shifts moved to the left when the temperature increased.
Figure 4.6. Theoretical and experimental XAFS spectra of fcc (Cu) structure material at temperatures.
(Zn) hcp
Figure 4.7. Comparison of Fourier transform magnitudes of the theoretical spectra with results experimental XAFS for structural material at temperatures.
Here, we note that the anharmonic contribution of XAFS phases and amplitudes are calculated base on the second-order cumulant only. Furthermore, by using anharmonic correlated Einstein model, we can reproduce XAFS spectra and transform their Fourier the obtained experimental second-order magnitudes base on cumulant values. This study showed that the obtained experimental results are consistent with calculations from the theoretical model at temperatures of 300 K, 400 K, and 500 K for Cu.
23
CONCLUSIONS AND RECOMMENDATIONS
This thesis contributed an advanced to
theory and experiment in both the progress of complement and upgrading of anharmonic correlated Einstein model a well-applied for XAFS spectroscopy. This thesis consisted of the following key findings:
1. Using calculated Morse potential parameters from theory to determine effective interacted potential in anharmonic correlated Einstein model. In this study, Stillinger-Weber potential was used additionally for diamond structure materials (Si, Ge).
2. This thesis described a procedure known as an advanced method that can simplify the calculations for thermodynamic parameters, XAFS spectra, and their Fourier transform magnitudes only through the second-order cumulant. In particular, this advanced anharmonic correlated Einstein model can apply in both theory and experiment in the XAFS field.
3. The advanced anharmonic correlated Einstein model was applied to derive, calculate and evaluate parameters of XAFS such as cumulants, thermal expansion coefficients T, XAFS spectrum, and transform magnitudes, anharmonic factor (T), their Fourier anharmonic contribution to the second-order cumulant, Grüneisen parameter, the ratio among cumulants as well as ratio among coefficient of thermal expansion and cumulants.
4. Measurements of second-order cumulant for fcc (Cu) and hcp (Zn) structure material were conducted. The obtained experimental results were evaluated and compared with theoretical values as well as results from other methods.
5. The calculated theoretical results obtained by using the advanced method were appropriate to experimental values and those obtained from other measurements. The contents of this study published through five scientific articles in which three ones belong to SCI journals.
24
LIST OF WORKS PUBLISHED
1. Nguyen Van Hung, Cu Sy Thang, Nguyen Cong Toan, Ho Khac Hieu (2014), Temperature dependence of Debye-Waller factors of semiconductors, J. Vacuum, (101), pp 63-66.
2. Nguyen Van Hung, Cu Sy Thang, Nguyen Ba Duc, Dinh Quoc Vuong (2017), Advances in theoretical and experimental XAFS studies of thermodynamic properties, anharmonic effects and structural determination of fcc crystals, The European physical Journal B, 90:256.
3. Nguyen Van Hung, Cu Sy Thang, Nguyen Ba Duc, Dinh Quoc Vuong, Tong Sy Tien (2017), Temperature dependence of theoretical and experimental Debye-Waller factor, thermal expansion and XAFS of metallic Zinc, Physica B, 521, pp 198- 203.
4. Cu Sy Thang, Nguyen Van Hung, Nguyen Bao Trung, Nguyen Cong Toan (2018). A Method for theoretical and experimental studies of thermodynamic parameters and XAFS of HCP crystals, application to metallic Zinc. Proceeding of The 5th Academic conference on natural science for young scientists, master and Ph.D. students from Asian Countries(4-7 October 2017, Da lat, Viet Nam). ISBN: 978-604-913-714-3, pp 58-65. 5. Nguyen Van Hung, Cu Sy Thang, Nguyen Bao Trung, Nguyen Cong Toan (2018). Theoretical and Experimental studies of Debye-Waller factors and XAFS of FCC crystals. Proceeding of Advances in Applied and Engineering Physics-CAEP V. ISBN: 978-604-913-232-2, pp 47-55.
