In this work, X-ray absorption fine structure (XAFS) of bcc crystals and it Fourier transformmagnitude have been studied based on the anharmonic correlated Debye model high-order expandedDebye-Waller factors. The many-body effects are taken into account in the present one-dimensionalmodel based on the anharmonic effective potential that includes interactions of absorber andbackscatterer atoms with their first shell near neighbors, where Morse potential is assumed to describethe single-pair atomic interaction. Analytical expressions of four first temperature-dependent cumulantsof bcc crystals have been derived using the many-body perturbation approach. The obtained cumulantsare applied to calculating XAFS spectra and their Fourier transform magnitudes. Numerical results forFe are found to be in good agreement with experiment

Hue, Trinh Thi

Hung, Nguyen Van

Khoa, Ha Dang

Tien, Tong Sy

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 27, No. 1 (2017), pp. 55-64
DOI:10.15625/0868-3166/27/1/8912
X-RAY ABSORPTION FINE STRUCTURE OF BCC CRYSTALS STUDIED
BASED ON HIGH-ORDER EXPANDED DEBYE-WALLER FACTORS
NGUYEN VAN HUNG1,†, TRINH THI HUE1, HA DANG KHOA2 AND TONG SY TIEN3
1Department of Physics, Hanoi University of Science. 334 Nguyen Trai, Thanh Xuan, Hanoi,
Vietnam
2School of Engineering Physics, Hanoi University of Science and Technology,
1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam
3Department of Basic Sciences, University of Fire Fighting & Prevention,
243 Khuat Duy Tien, Thanh Xuan, Hanoi, Vietnam
†E-mail: hungnv@vnu.edu.vn
Received 24 November 2016
Accepted for publication 30 March 2017
Abstract. In this work, the X-ray absorption fine structure (XAFS) spectra of bcc crystals and
their Fourier transform magnitudes have been studied based on the anharmonic correlated Debye
model high-order expanded Debye-Waller factors. The many-body effects are taken into account
in present one-dimensional model based on the anharmonic effective potential that includes in-
teractions of absorber and backscatterer atoms with their first shell near neighbors, where the
Morse potential is assumed to describe the single-pair atomic interaction. Analytical expressions
of four first temperature-dependent cumulants of bcc crystals have been derived using the many-
body perturbation approach. The obtained cumulants are applied to calculating XAFS spectra
and their Fourier transform magnitudes. Numerical results for bcc phase of Fe are found to be in
good agreement with the experimental data.
Keywords: Debye-Waller factor, effective potential, correlated Debye model, XAFS, bcc crystals.
Classification numbers: 61.05.C, 61.66.Bi, 78.70.Dm.
I. INTRODUCTION
The X-ray Absorption Fine Structure (XAFS) spectra have developed into a powerful tech-
nique for providing information on the local atomic structure and thermal effects of substances [1–
20]. The formalism for including anharmonic effects in XAFS is often based on the cumulant ex-
pansion [1] where the even cumulants contribute to the amplitude, the odd ones contribute to the
c©2017 Vietnam Academy of Science and Technology
56 X-RAY ABSORPTION FINE STRUCTURE OF BCC CRYSTALS STUDIED . . .
phase of XAFS spectra, and for small anharmonicities, it is sufficient to keep the third and fourth
cumulants [2]. The accurate Debye-Waller factors (DWFs) varying as e−W (T ) with temperature T
and wave number k (or energy) due to their exponential damping are crucial to quantitative treat-
ment of XAFS spectra. Consequently, the lack of the precise DWFs has been one of the biggest
limitations to accurate structural determinations (e.g., the coordination numbers and the atomic
distances) from XAFS experiments.
Many efforts have been made to develop procedures for the calculation and analysis of
XAFS cumulants using the classical theories [7–9] and quantum theories [1–6, 10–20]. Unfortu-
nately, there is still no theoretical result on XAFS of bcc crystals while their experimental results
are available [21].
This work is a theoretical study of XAFS spectra and their Fourier transform magnitudes of
bcc crystals using the DWFs presented in terms of cumulant expansion up to the fourth order based
on the anharmonic correlated Debye model (ACDM) [20]. In Sec. II, the analytical expressions
have been presented for four first temperature-dependent cumulants of bcc crystals derived using
the many-body perturbation approach (MBPA) [22], the dispersion relation and the anharmonic
effective potential parameters obtained based on the first shell near neighbor contribution approach
(FSNNCA), where the Morse potential is assumed to describe the single-pair atomic interaction.
The obtained cumulants are applied to calculating XAFS spectra and their Fourier transform mag-
nitudes. Numerical results for bcc phase of Fe are compared with the experimental data in [21]
which show a good agreement.
II. FORMALISM
The formalism for including anharmonic effects in XAFS is often based on the cumulant
expansion approach [1] from which the expression for anharmonic XAFS spectra has the form
χ(k) = F (k)
e−2R/λ (k)
kR2
Im
{
eiΦ(k) exp
[
2ikR+∑
n
(2ik)n
n!
σ (n)
]}
, (1)
where F(k) is the real atomic backscattering amplitude, k and λ are the wave number and the
mean free path of photoelectron, respectively, Φ is the net phase shift, R = 〈r〉 with r being the
instantaneous bond length between absorber and backscatterer atoms, and σ (n) (n = 1,2,3,4, . . .)
are the cumulants describing the high-order expanded DWFs.
If limiting to the fourth cumulant, the temperature T - and wave number k-dependent an-
harmonic contributions to the amplitude FA(k,T ) and to the phase ΦA(k,T ) of XAFS spectra are
given by
FA (k,T ) =exp
(
−2
3
k4σ (4) (T )
)
,
ΦA (k,T ) =− 4kσ
2 (T )
R
(
1+
R
λ
)
− 4
3
k3σ (3) (T ) ,
(2)
Hence, for calculating XAFS spectra and their Fourier transform magnitudes of bcc crystals
using Eqs. (1) and (2), it is necessary to calculate their DWFs expanded up to the fourth order.
In order to include the anharmonic effects, the Hamiltonian of system is written in the
summation of the harmonic and anharmonic components, H0 and Ha, respectively
NGUYEN VAN HUNG et al. 57
H = H0+Ha , Ha = Hc+Hq, (3)
where the anharmonic component Ha contains the cubic Hc and the fourth order Hq terms including
the anharmonic interatomic effective potential parameters of bcc crystals.
The anharmonic interatomic effective potential expanded up to the fourth order in the
present theory for bcc crystals is expressed as a function of the displacement x = r− r0 along the
bond direction with r and r0 being the instantaneous and equilibrium distances between absorber
and atoms, respectively
Ve f f (x)≈ 12ke f f x
2+ k3e f f x3+ k4e f f x4, (4)
where ke f f is the effective local force constant, k3e f f and k4e f f are the anharmonic parameters
giving an asymmetry of the anharmonic effective potential.
For bcc crystals each atom is bonded to 8 near neighbors. Then, the effective potential
given by Eq. (4) defined based on the FSNNCA has the form
Ve f f (x) = V (x)+2V
(
− x
2
)
+6V
( x
6
)
+6V
(
− x
6
)
, (5)
which is the sum over not only the term V(x) describing the pair-interaction between absorber and
backscatterer atoms but also the other terms describing the projections of their pair-interactions
with 14 first shell near neighbors of bcc crystals along the bond direction excluding the absorber
and backscatterer themselves whose contributions are already described by V(x).
Applying the Morse potential expanded up to the fourth order as
V (x) =D
(
e−2αx−2e−αx)
≈D
(
−1+α2x2−α3x3+ 7
12
α4x4
) (6)
to Eq. (5) and comparing the result to Eq. (4), we obtain the values of ke f f , k3e f f , k4e f f for bcc
crystals in terms of Morse potential parameters
ke f f =
11
3
Dα2 , k3e f f =−34Dα
3 , k4e f f =
1715
2592
Dα4, (7)
where α describes the width of the potential and D is dissociation energy.
Hence, the anharmonic effective potential for bcc crystals in Eq. (5) has the form
Ve f f (x)≈ 116 Dα
2x2− 3
4
Dα3x3+
1715
2592
Dα4x4, (8)
Note that the above mentioned lattice contributions based on the FSNNCA, to oscillation
between absorber and backscatterer atoms described by the projections of their pair-interactions
with 14 first shell near neighbors along the bond direction, make it possible to take into account
the many-body effects in the present one-dimensional model for bcc crystals.
The derivation of present ACDM for bcc crystals using the MBPA [22] is based on the
dualism of an elementary particle in quantum theory, its corpuscular and wave properties. Then,
we can describe the system in Debye model involving all different frequencies up to the Debye
frequency as a system consisting of many bodies or many phonons each of which corresponds to
a wave having frequency ω(q) and wave number q varied in the first Brillouin zone (BZ).
58 X-RAY ABSORPTION FINE STRUCTURE OF BCC CRYSTALS STUDIED . . .
For this purpose, the displacement u′ns in the parameter x in terms of the displacement of
nth atom un of the one dimensional chain is described by
xn = un+1−un, (9)
where the displacement un is related to the phonon displacement operators Aq [23] in the form
un =
√
h¯
2NM∑q
eiqan√
ω(q)
Aq,
Aq =A+−q,
[
Aq,Aq′
]
= 0.
(10)
Therefore, the value of xn in Eq. (9) is given by
xn =∑
q
eiqan f (q)Aq,
f (q) =
√
h¯
2NMω (q)
(
eiqa−1) , (11)
where N is the atomic number, M is the mass of composite atoms and a is the lattice constant.
The frequency ω(q) contained in Eq. (11) and then in all cumulant expressions derived for
the vibration between absorber and backscatterer atoms in XAFS process describes the dispersion
relation. Using the local force constant of the first equation of Eqs. (7), it has the form
ω (q) = 2α
√
11D
3M
∣∣∣sin(qa
2
)∣∣∣ , |q| ≤ pi
a
. (12)
Using the above results in the MBPA [22], we have derived the analytical expressions for
XAFS DWFs presented in terms of cumulant expansion up to the fourth order for bcc crystals.
The first cumulant describing the net thermal expansion or the lattice disorder in the XAFS
theory has resulted as
σ (1) (T ) = 〈x〉= σ (1)0
pi/a∫
0
ω (q)1+Z(q)1−Z(q)dq =
σ (1)0
σ20
σ2 ,
σ (1)0 =
81ah¯
484piDα , Z (q) = exp(β h¯ω (q)) , β =
1
kBT
.
(13)
Here, σ2 is the second cumulant describing the mean square relative displacement (MSRD) and
has the following form
σ2 (T ) =
〈
x2
〉
= σ20
pi/a∫
0
ω(q)
1+ z(q)
1− z(q)dq , σ
2
0 =
3h¯a
22piDα2
. (14)
NGUYEN VAN HUNG et al. 59
The third cumulant is the mean cubic relative displacement (MCRD) describing the asym-
metry of the pair distribution function in the XAFS theory and is expressed as
σ (3) (T )∼= 〈x3〉−3〈x2〉〈x〉
= σ (3)0
pi/a∫
0
dq1
pi/a−q1∫
−pi/d
dq2
ω (q1)ω (q2)ω (q1+q2)
ω (q1)+ω (q2)+ω (q1+q2)
×
{
1+6
ω (q1)+ω (q2)
ω (q1)+ω (q2)−ω (q1+q2)
eβ h¯[ω(q1)+ω(q2)]− eβ h¯ω(q1+q2)(
eβ h¯ω(q1)−1)(eβ h¯ω(q2)−1)(eβ h¯ω(q1+q2)−1)
}
,
σ (3)0 =
114×10−4h¯2a2
pi2D2α3
. (15)
The fourth cumulant describes the anharmonic contribution to the XAFS amplitude and has
the form
σ (4) (T )∼= 〈x4〉−3〈x2〉2
= σ (4)0
pi/a∫
0
dq1
pi/a−q1∫
0
dq2
pi/a−(q1+q2)∫
−pi/a
dq3
ω (q1)ω (q2)ω (q3)ω (q4)
ω (q1)+ω (q2)+ω (q3)+ω (q4)
×
{
1+8
Z (q1)Z (q2)Z (q3)−Z (q4)
(Z (q1)−1)(Z (q2)−1)(Z (q3)−1)(Z (q4)−1)
ω (q1)+ω (q2)+ω (q3)
ω (q1)+ω (q2)+ω (q3)−ω (q4)
+6
Z (q1)Z (q2)−Z (q3)Z (q4)
(Z (q1)−1)(Z (q2)−1)(Z (q3)−1)(Z (q4)−1)
ω (q3)+ω (q4)
ω (q1)+ω (q2)−ω (q3)−ω (q4)
}
,
σ (4)0 =
165×10−4h¯3a3
2pi3D3α4
, q4 =−(q1+q2+q3) . (16)
Note that in the above expressions for the cumulants of bcc crystals, σ (1)0 , σ
2
0 , σ
(3)
0 , σ
(4)
0 are
zero-point energy contributions to the first, second, third and fourth cumulant, respectively, and
these cumulant expressions have been obtained for the case of large atomic number N, when the
summation over q is replaced by the corresponding integral in the first BZ. Hence, the obtained
cumulants described in Eqs. (13) – (16) contribute to the XAFS spectra of bcc crystals given by
Eq. (1) containing the anharmonic contributions described by Eqs. (2).
III. NUMERICAL RESULTS AND DISCUSSIONS
Now the expressions derived in the previous section are applied to numerical calcula-
tions for bcc phase of Fe using its Morse potential parameters as follows: D = 0.418 eV, α =
1.397 A˚−1 [24].
Figures 1.a,b and 2.a,b show the dependences of the cumulants σ (1),σ2,σ (3) and σ (4) on
temperature for bcc phase of Fe calculated by the present theory. Our calculated results are in very
good agreement with the experimental data in [21].
60 X-RAY ABSORPTION FINE STRUCTURE OF BCC CRYSTALS STUDIED . . .
Fig. 1. The dependences of a) the first cumulant σ (1)(T) and b) the second cumulant
σ2(T) on temperature for bcc phase of Fe calculated by the present theory in the compar-
ison with the experimental data in [21].
NGUYEN VAN HUNG et al. 61
Fig. 2. The dependences of a) the third cumulant σ (3)(T ) and b) the fourth cumulant
σ (4)(T ) on temperature for bcc phase of Fe calculated by the present theory in the com-
parison with the experimental data in [21].
62 X-RAY ABSORPTION FINE STRUCTURE OF BCC CRYSTALS STUDIED . . .
Figures 3.a,b describe the dependence of anharmonic contributions FA (T,k) and ΦA (T,k)
to the XAFS spectra amplitude and phase on temperature and wave number for bcc phase of Fe
calculated by the present theory. The obtained results show that when the temperature and the
wave number increase, these anharmonic contributions also increase.
Fig. 3. The dependence of anharmonic contributions a) FA (T,k) and b)ΦA (T,k) to the
XAFS spectra amplitude and phase on temperature and wave number for bcc phase of Fe
calculated by the present theory.
NGUYEN VAN HUNG et al. 63
Fig. 4. a) XAFS spectra at 393 K and b) Fourier transform magnitudes of XAFS spectra
at 293 K and 393 K for bcc phase of Fe calculated by the present theory including the
above obtained cumulants in the comparison with the experimental data in [21].
Figures 4.a,b illustrate good agreement of XAFS spectra at 393 K and XAFS spectra Fourier
transform magnitudes at 293 K and 393 K for bcc phase of Fe calculated using Eqs. (1) and (2)
including the anharmonic contributions. The Fourier transform magnitudes are shifted when the
temperature changes from 293 K to 393 K due to the anharmonic effects described by the cumu-
lants obtained by the present theory.
64 X-RAY ABSORPTION FINE STRUCTURE OF BCC CRYSTALS STUDIED . . .
IV. CONCLUSIONS
The anharmonic XAFS spectra and their Fourier transform magnitudes of bcc crystals have
been studied using the ACDM high-order expanded Debye-Waller factors where the ACDM has
been derived based on the MBPA and the FSNNCA. The obtained results contribute to valuation
of the thermodynamic properties and anharmonic effects in XAFS that lead to getting the accurate
information on structural and other parameters of bcc crystals taken from XAFS experiments.
The advantage and efficiency of the present theory in XAFS data analysis are illustrated
by the good agreement of the numerical results for cumulants, XAFS spectra and their Fourier
transform magnitudes for bcc phase of Fe with experiments. This makes it possible to reproduce
the experimental XAFS data of bcc crystals using the present theory.
ACKNOWLEDGEMENTS
This research is funded by the Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 103.01-2015.10.
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X-ray Absorption Fine Structure of bcc Crystals Studied Based on High-order Expanded Debye-Waller Factors

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X-ray Absorption Fine Structure of bcc Crystals Studied Based on High-order Expanded Debye-Waller Factors

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