Significant pressure effects are shown by the decrease in the pressure-induced changes in the interatomic distance, EXAFS cumulants and thermal expansion coefficient, as well as by the inc
Trang 1Pressure-Dependent Anharmonic Correlated Einstein Model Extended
X-ray Absorption Fine Structure Debye –Waller Factors
Nguyen Van Hung+ Department of Physics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam (Received May 17, 2013; accepted November 13, 2013; published online January 8, 2014)
A pressure-dependent anharmonic correlated Einstein model is derived for extended X-ray absorptionfine structure
(EXAFS) Debye–Waller factors (DWFs), which are presented in terms of cumulant expansion up to the third order The
model is based on quantum thermodynamic perturbation theory and includes anharmonic effects based on empirical
potentials Explicit analytical expressions of the pressure-dependent changes in the interatomic distance, anharmonic
effective potential, thermodynamic parameters, first, second, and third EXAFS cumulants, and thermal expansion
coefficient have been derived This model avoids the use of extensive full lattice dynamical calculations, yet provides
good and reasonable agreement of numerical results for Cu with experimental results of X-ray diffraction (XRD)
analysis and pressure-dependent EXAFS Significant pressure effects are shown by the decrease in the pressure-induced
changes in the interatomic distance, EXAFS cumulants and thermal expansion coefficient, as well as by the increase in
the pressure-induced changes in the interatomic effective potential, effective spring constant, correlated Einstein
frequency, and temperature
1 Introduction
Thermal vibrations and disorder in extended X-ray
absorption fine structure (EXAFS) give rise to Debye–
Waller factors (DWFs), which are presented here in terms of
cumulant expansion1 )to include anharmonic effects EXAFS
is sensitive to pressure,2–4) which can cause changes of
information on cumulants and in the structural and
thermodynamic parameters of substances taken from
EXAFS and makes it possible for some physical effects to
occur Such effects can be seen in experimental results of
EXAFS for the Br K edge in NaBr,4) where EXAFS is
shifted and a Bragg peak appears under pressure EXAFS
has been proven to be a valuable tool in various branches of
solid state physics, but few attempts have so far been made
to use it in high-pressure physics.5 )Some efforts have been
exerted to study pressure effects in EXAFS and its
cumulants.2 – 10 ) A pressure-dependent EXAFS experiment
on Cu has been performed5 )to examine Cu as a calibrant for
EXAFS Here, its parameters such as the second cumulant
2ðPÞ or mean-square-relative displacement (MSRD) and
the third cumulantð3ÞðPÞ describing the EXAFS phase shift
due to anharmonicity, etc., as functions of pressure have
been extracted using the expressions of cumulant expansion
and Rehr’s correlated Debye model, where the ratio method
and parameterfitting are used for data analysis.5 )Moreover,
MSRD has also been measured,6,7) as well as calculated by
Monte-Carlo simulation,6)and by using Loubeyre’s model8 )
for Kr Interatomic distance as a function of pressure for
simple metals has been theoretically studied.9 )The
pressure-induced interatomic distance change of Cu was measured by
XRD analysis and the results are presented in the American
Institute of Physics Handbook.10 ) For further development
toward the use of EXAFS as a valuable tool in
high-pressure physics it is necessary to develop procedures for
the description and analysis of the pressure dependences of
physical quantities obtained from EXAFS Several methods
including anharmonic correlated Einstein model (ACEM)11)
have been successfully applied to temperature-dependent
EXAFS.11–18) Hence, further developing a model for
study-ing high-pressure EXAFS and its parameters may be useful
This work is a next step in the approach of HR (Ref 11)
to derive a pressure-dependent ACEM for approximating the pressure-dependent interatomic distance, thermodynamic parameters, and cumulants in high-pressure EXAFS In Sect 2, the pressure-dependent anharmonic effective poten-tial is developed and used for deriving an equation whose solution provides the pressure-induced interatomic distance change rðPÞ by a simple means On the basis of quantum thermodynamic perturbation theory, the explicit analytical expressions for pressure-dependent first, second, and third EXAFS cumulants, and the thermal expansion coefficient at a given temperature have been derived The Morse potential
is assumed to describe the interaction between each pair
of atoms In high-pressure physics, the pressure-induced changes in physical quantities, i.e., the difference of a quantity at pressureP 6¼ 0 from its value at P ¼ 0 are usually measured,5 , 10 ) so that in this work we also calculate and discuss such pressure-induced changes in the considered quantities, as well as in the ratios of pressure-induced changes of EXAFS cumulants Numerical results (Sect 3) for Cu, being one of the intensively studied materials,5) are compared to the results of the XRD experiment analysis,10) and pressure-dependent EXAFS,5) which show good and reasonable agreement The physical properties except for
rðPÞ depend on rðPÞ Since rðPÞ can be obtained by the XRD method, two calculations are compared in this paper, i.e., usingrðPÞ calculated on the basis of the present theory, and using rðPÞ from the experiment.10 ) Moreover, the pressure effects that appeared in all the considered quantities have been discussed in detail to show more information shown in this high-pressure EXAFS theory
2 Pressure-Dependent Anharmonic Correlated Einstein Model
We consider a pressure-dependent anharmonic correlated Einstein model characterized by an anharmonic effective potential corresponding to the case, in which under pressure
P the interatomic distance is changed byrðPÞ:
024802-1 ©2014 The Physical Society of Japan
Trang 2VeffðPÞ ¼ 12keffðrðPÞÞ2þ k3ðrðPÞÞ3þ k4ðrðPÞÞ4;
where rðPÞ is the pressure-induced change in interatomic
distance, keff is the effective spring constant, k3 and k4 are
cubic and quartic anharmonic parameters, respectively,
giving an asymmetry of the potential Equation (1) is our
definition of anharmonic effective potential for the pressure
dependence used in this work, which is different from that for
the temperature dependence11 ) by using rðPÞ instead of x
being the deviation in the instantaneous bond length between
the two atoms from equilibrium, as well as by including the
pressure dependence of the effective spring constant shown
below
To take into account all nearest neighbor interactions
for including three-dimensional interactions in the present
derived pressure-dependent ACEM, the pressure-dependent
anharmonic interatomic effective potential Eq (1) is defined
on the basis of an assumption in the center-of-mass frame of
a single-bond pair of an absorber with a mass M1 and a
backscatterer with a mass M2 as
VeffðPÞ ¼ VðPÞ þX
i¼1;2
X j6¼i
MirðPÞ ^R12 ^Rij
;
¼MM1M2
where the sum i is over absorbing and backscattering atoms,
the sum j is over their nearest neighbors, ^Rijis the unit vector
along the bond between the ith and jth atoms, and ® is the
reduced mass of M1 and M2 The first term on the right
concerns only absorbing and backscattering atoms, and the
second one describes the lattice contributions to pair
interactions and depends on the crystal structure type
observed in the potential parameterskeff,k3, andk4 obtained
by comparing Eqs (2) to (1) for each considered crystal
structure
To derive the pressure-induced interatomic distance change
rðPÞ, we consider the anharmonic interatomic effective
potential given by Eqs (1) and (2) under pressure P, which
has the definition P ¼ ð@F=@VÞT, where F is the free energy
At finite temperatures and under pressure, the effective
potential will be defined to correspond to the appropriate free
energy Hence, from the above definition of P, the change of
F under pressure P of the system corresponding to the
volume changeV is given by
If the system has N elementary cells, each of which has
volume vc, then the free-energy change under pressure P
corresponding to one elementary cell is equal to
fðrðPÞÞ ¼FN ¼ PvcVðPÞ
When a material under pressure P is compressed in one
direction, it usually tends to expand in the other two
directions perpendicular to the compression direction This
phenomenon is called the Poisson effect described by the
Poisson ratio £.19 , 20 ) The Poisson ratio is the ratio or the
fraction (or percent) of expansion to the fraction (or percent)
of compression and it is given by
¼x=xz=z ¼y=yz=z : ð5Þ From this ratio, we obtain
x
y
z
Suppose that x, y, and z are three coordinates of a sample at
P ¼ 0, then under pressure P > 0 these coordinates change intox x, y x, and z z, respectively Hence, using Eqs (5) and (6) (z=z 1), the following ratio is obtained:
VðPÞ Vð0Þ ¼
ðx xÞðy yÞðz þ zÞ xyz
xyz
¼
Atomic displacement is shown by the change of an atomic bond, which can be along the axis, along which the effective potential is directed This treats the axial compression of the atomic bond caused by atomic displacement This axial compression can also be applied to the other atomic bonds along the other axes leading to the uniform compression of the system
Suppose that z is the compression direction along the interatomic distance r so that Eq (7) is generalized into
VðPÞ Vð0Þ ¼ ð1 2Þ rðPÞrðPÞ : ð8Þ Substituting VðPÞ=Vð0Þ from Eq (8) into Eq (4), we obtain
fðrðPÞÞ ¼ ð1 2ÞPvc
rðPÞ
Since the anharmonic effective potential given in Eq (1) corresponds to the free-energy change under pressure P, as described by Eq (4) or (9), from Eqs (1) and (9), we obtain the following equation:
k4ðrðPÞÞ3þ k3ðrðPÞÞ2þ12keffðrðPÞÞ
þð1 2ÞPvc
whose solution provides the pressure-induced interatomic distance change rðPÞ and the pressure-dependent inter-atomic distancerðPÞ ¼ rð0Þ þ rðPÞ
Furthermore, on the basis of the definition of the Grüneisen parameterG¼ @ðln !EÞ=@ðln VÞ and the relation5 )
GðPÞ VðPÞ ¼
Gð0Þ
the pressure-dependent correlated Einstein frequency !EðPÞ and temperature EðPÞ, as well as the effective spring constant keffðPÞ, have been derived and are given by
!EðPÞ ¼ !Eð0Þ exp Gð0Þ 1 VðPÞ
Vð0Þ
EðPÞ ¼ h!EðPÞ=kB; keffðPÞ ¼ !2
EðPÞ: ð13Þ Making use of quantum thermodynamic perturbation theory,21) as well as of Eqs (1), (2), (12), and (13), the explicit analytical expressions for the three first EXAFS cumulants as functions of the pressure P at a given 024802-2 ©2014 The Physical Society of Japan
Trang 3temperature T have been derived They are given by, for the
first cumulant,
ð1ÞðP; TÞ ¼ ð1Þ0 ðPÞ1 þ zðP; TÞ1 zðP; TÞ¼k3k3
effðPÞ2ðP; TÞ;
ð1Þ
0 ðPÞ ¼ k3k3
for the second cumulant or MSRD,
2ðP; TÞ ¼ 2
0ðPÞ1 þ zðP; TÞ1 zðP; TÞ;
2
0ðPÞ ¼ h!EðPÞ 2keffðPÞ; zðP; TÞ ¼ exp EðPÞ
T
and for the third cumulant,
ð3ÞðP; TÞ ¼ 0ð3ÞðPÞ½3ð2ðP; TÞ=2
0ðPÞÞ2 2;
ð3Þ
0 ðPÞ ¼ 2k3
keffðPÞð02ðPÞÞ2: ð16Þ Using the obtainedfirst cumulant or net thermal expansion
Eq (14), the expression of the pressure-dependent thermal
expansion coefficient has been derived and is given by
PðP; TÞ ¼ 0
PðPÞð2ðP; TÞÞ2T ð2 2ðPÞÞ2;
0
PðPÞ ¼ 3k3
rðPÞkB
The descriptions of Eqs (14)–(17) in terms of the second
cumulant are useful because all the cumulants and thermal
expansion coefficients can be deduced when the second
cumulant or MSRD is measured or calculated This can lead
to significant reductions of the numerical calculations or
measurements of the pressure-dependent EXAFS cumulants
and thermal expansion coefficient by focusing only on the
calculations or measurements of the second cumulant in
high-pressure EXAFS
It is useful to consider the behaviors of the above derived
pressure-dependent cumulants and thermal expansion
coef-ficients in the low-pressure (LP) and high-pressure (HP)
limits In the LP (P ! 0) limit, they approach those of
P ¼ 0.11 )In the HP (P ! 1) limit, z ! 0, so that the term
z2 and higher powers can be neglected This leads to the
approximation of Eqs (14)–(17) The obtained results in the
HP limit are presented in Table I
Hence, in the HP limit the pressure dependence is defined
mainly on the basis of the pressure dependences ofð1Þ
0 ðPÞ,
2
0ðPÞ, and ð3Þ
0 ðPÞ for the EXAFS cumulants and of zðPÞ,
2
0ðPÞ, and 0
PðPÞ for the thermal expansion coefficient
Note that, in the LP limit (P ! 0) the expressions of the
EXAFS cumulants and thermal expansion coefficient, as
well as of the effective spring constant, correlated Einstein
frequency, and temperature of the present pressure-dependent
ACEM, approach those of the temperature-dependent
ACEM.11)
Moreover, the above considered physical properties except forrðPÞ depend on rðPÞ Since rðPÞ can be obtained by the XRD method, two calculations are compared in this paper (Sect 3), i.e., using rðPÞ calculated by Eq (10) and using
rðPÞ from the experiment.10 )
3 Numerical Results and Discussion Now, we apply expressions derived in the previous section
to numerical calculations for Cu, where Morse potential22)is used to describe the interaction between each pair of atoms Its parameters22 ) D ¼ 0:3429 eV, and ¼ 1:3588 Ź1 were obtained using the experimental values of the energy of sublimation, compressibility, and lattice constant They have been used for the calculation of the pressure-induced changes
in the interatomic distance, anharmonic interatomic effective potential, thermodynamic parameters, EXAFS cumulants, and thermal expansion coefficient of Cu at the given temperatureT ¼ 300 K, a temperature at which the measure-ments are often performed,3,10) as well as using Poisson’s ratio ¼ 0:355,19 ) zero-pressure Grüneisen parameter22)
Gð0Þ ¼ 2:108, and rð0Þ ¼ 2:554 Å.5 ) The pressure-induced interatomic distance change rðPÞ
of Cu calculated by the present theory using Eq (10) (Fig 1) shows good agreement with those of the XRD (AIP, DEW) experiments.10) The obtained rðPÞ values are used for calculating the pressure dependence of the pressure-induced changes in all the quantities considered in this work using the present theory The results are illustrated firstly for the effective spring constant keffðPÞ (Fig 2) calculated using the second equation of Eq (13), for the anharmonic interatomic effective potential VeffðPÞ (Fig 3) calculated using Eq (1) including keffðPÞ, for the correlated Einstein temperature EðPÞ (Fig 4) calculated using the first equation of Eq (13), and for the first cumulant ð1ÞðPÞ (Fig 5) calculated using Eq (14)
Figure 6 shows the pressure-induced change of the second cumulant or MSRD 2ðPÞ calculated using Eq (15) and
Table I Formulas of ð1Þ, 2, ð3Þ, and P in the HP ( P ! 1) limit.
P ðPÞð 2 ðPÞÞ 2 =T 2
Fig 1 Pressure dependence of pressure-induced interatomic distance change rðPÞ of Cu calculated using the present theory compared with that obtained in the experiments (XRD-AIP, XRD-DEW) 10 )
024802-3 ©2014 The Physical Society of Japan
Trang 4experimental data taken from the results of the method (A1)
and Rehr’s correlated Debye model in Ref 5 Similarly,
Fig 7 shows the pressure-induced change of the third
cumulantð3ÞðPÞ calculated using Eq (16) and
experimen-tal data taken from the results of the methods (P3) and (P4) in
Ref.5 The method (A1) is one of the four methods used in
Ref.5to obtain2ðPÞ from the amplitude ratio of the
high-pressure EXAFS datasets The methods (P3) and (P4) are two methods of fitting the phase difference of the high-pressure EXAFS datasets The parameter sets are (R, ð3Þ) for (P3) and (E, R, ð3Þ) for (P4), whereR ¼ ð1Þ, andE is
Fig 2 Pressure dependence of pressure-induced effective spring constant
change k eff ðPÞ of Cu calculated using the present theory compared with
that calculated using rðPÞ obtained from the experiments (AIP,
XRD-DEW) 10 ) at different pressures.
Fig 3 Pressure dependence of pressure-induced anharmonic interatomic
effective potential change V eff ðPÞ of Cu calculated using the present theory
compared with that calculated using rðPÞ obtained from the experiments
(XRD-AIP, XRD-DEW) 10 ) at different pressures.
Fig 4 Pressure dependence of pressure-induced correlated Einstein
temperature change E ðPÞ of Cu calculated using the present theory
compared with that calculated using rðPÞ obtained from the experiments
(XRD-AIP, XRD-DEW) 10 ) at different pressures.
Fig 5 Pressure dependence of pressure-induced first cumulant change
ð1Þ ðPÞ of Cu calculated using the present theory compared with that calculated using rðPÞ obtained from the experiments (AIP, XRD-DEW) 10 ) at different pressures.
Fig 6 Pressure dependence of pressure-induced second cumulant change
2 ðPÞ of Cu calculated using the present theory compared with the experimental results extracted from EXAFS using Rehr ’s correlated Debye model, 5 ) and the method (A1), 5 ) as well as with that calculated using rðPÞ obtained from the experiments (XRD-AIP, XRD-DEW) 10 ) at different pressures.
Fig 7 Pressure dependence of pressure-induced third cumulant change
ð3Þ ðPÞ of Cu calculated using the present theory compared with the experimental results extracted from EXAFS using the methods (P3) 5 ) and (P4), 5 ) as well as with that calculated using rðPÞ obtained from the experiments (XRD-AIP, XRD-DEW) 10 ) at different pressures.
024802-4 ©2014 The Physical Society of Japan
Trang 5the photoelectron zero-energy difference between different
datasets Both figures show reasonable agreement between
the results of the present theory and the experimental data,
where the lines of the calculated results are compared with
the average curve of the fluctuating experimental values
On the basis of Eq (14), the pressure-dependentfirst and
second cumulants (Figs 5 and 6) are proportional to each
other by the factor3k3=keffðPÞ However, the effective spring
constantkeffðPÞ is pressure-dependent (Fig 2) thus,
illustrat-ing both these cumulants can be useful for consideration of
the effect of the pressure on this proportionality Furthermore,
Fig 8 shows the pressure dependence of the
pressure-induced change in the thermal expansion coefficient PðPÞ
of Cu calculated using the present theory by Eq (17) It
approaches a constant value at very high pressures
The increases inVeffðPÞ, keffðPÞ, !EðPÞ, and EðPÞ,
as well as the decreases inð1ÞðPÞ, 2ðPÞ, ð3ÞðPÞ, and
PðPÞ under pressure denote significant pressure effects of
the anharmonic effective potential, thermodynamic
parame-ters, and EXAFS cumulants The decrease in the
pressure-induced change of the second cumulant under pressure
calculated using the present theory has also been reflected
in the experimental results for Kr.6 , 7 ) The decrease in the
pressure-induced interatomic distance change (Fig 1) under
pressure denotes that the material becomes denser as the
atoms get closer together under pressure This leads to the
enhancement in the degree of interatomic interaction shown
by the increase in the pressure-induced change of effective
spring constant keffðPÞ (Fig 2) and anharmonic effective
potential VeffðPÞ (Fig 3) under pressure From the first
equation in Eq (13), the pressure-dependent correlated
Einstein frequency is proportional to the correlated Einstein
temperature by the factor kB=h, i.e., !E ¼ kBE=h, so that
they can be determined from each other On the basis of this
proportionality one can also get from the increase in the
correlated Einstein temperature (Fig 4) making the material
harder, the increase in the correlated Einstein frequency
making the atomic vibrations faster under pressure
Con-sequently, the pressure dependence in the pressure-induced
changes in the calculated quantities shown in the above
figures affect EXAFS, causing it to shift, as observed in the
experimental results for the Br K edge in NaBr.4)Hence, the
present theory can provide more information on the pressure effects, that occur in high-pressure EXAFS
It is seen from Figs 2–8 that the results of the pressure-induced changes in the considered EXAFS parameters calculated using rðPÞ from the experiments (XRD-AIP, XRD-DEW)10 )at different pressures are found to be in good agreement with those calculated using rðPÞ obtained by
Eq (10) for all the considered quantities, as well as with the experimental second cumulant obtained using Rehr’s corre-lated Debye model; they are also in reasonable agreement with the high-pressure EXAFS experimental data5) for the second cumulant 2ðPÞ obtained by (A1) and for the third cumulantð3ÞðPÞ obtained by (P3) and (P4)
In the temperature-dependent EXAFS (P ¼ 0) the cumu-lant ratiosð1Þ2=ð3ÞandTrT2=ð3Þare often considered as standards for cumulant calculations,11,15,23) because ð1ÞðTÞ,
ð3ÞðTÞ, and TðTÞ contain the anharmonic potential parameter
k3, where these ratios approach a constant value, i.e., 1/2, at high temperatures for both classical23 ) and quantum11 , 15 ) theories In high-pressure EXAFS, the important quantities are the pressure-induced changes of pressure-dependent rðPÞ, ð1ÞðPÞ, 2ðPÞ, ð3ÞðPÞ, and PðPÞ at a given temperature Since, in the present theory, they also contain the anharmonic potential parameterk3, it is crucial to consider the ratios of the pressure-induced changes in these quantities to look for some characteristics indicating the relationship between anharmo-nicity and pressure dependence in high-pressure EXAFS Figure 9 shows the pressure dependences of the cumulant ratios ð1Þ2=ð3Þ and PrT2=ð3Þ calculated using the present theory The ratios increase markedly at low pressures and then slowly as pressure increases; they approach constant values at very high pressures We are hoping that these properties of the ratios of the pressure-induced changes
in the cumulants, thermal expansion coefficient, and inter-atomic distance will also be considered as standards for pressure-dependent cumulant calculations similarly to those
of the cumulant ratios in temperature-dependent EXAFS
4 Conclusions The present derived pressure-dependent ACEM has led to
a method of calculating and analyzing the pressure-induced changes in the interatomic distance, anharmonic effective potential, thermodynamic parameters, and three first cumu-lants in high-pressure EXAFS
Fig 8 Pressure dependence of pressure-induced thermal expansion
coef ficient change P ðPÞ of Cu calculated using the present theory
compared with that calculated using rðPÞ from the experiments (XRD-AIP,
XRD-DEW) 10 ) at different pressures.
Fig 9 Pressure dependences of cumulant ratios ð1Þ 2 = ð3Þ and
P rT 2 = ð3Þof Cu calculated using the present theory.
024802-5 ©2014 The Physical Society of Japan
Trang 6The behaviors of the EXAFS cumulants and thermal
expansion coefficient in the LP and HP limits have been
considered It is shown that the expressions of these
quantities, as well as of the effective spring constant,
correlated Einstein frequency, and temperature approach
those of the temperature-dependent ACEM, when the
pressure approaches zero (P ! 0)
Significant pressure effects have been shown by this
pressure-dependent ACEM, namely, under pressure the
material becomes denser as the interatomic distance gets
shorter, as the atoms get closer together and vibrate faster,
and as the material becomes harder This leads to the
enhancement in the degree of interatomic interaction shown
by the increases in anharmonic effective potential, and
effective spring constant, as well as to the changes in the
thermodynamic properties of the material characterized by
the thermal expansion coefficient and Debye–Waller factors
presented in terms of cumulants All these pressure effects
affect high-pressure EXAFS profiles
This model avoids the use of extensive full lattice
dynamical calculations, yet provides good and reasonable
agreement of the calculated results for Cu with the
experimental results of XRD analysis and high-pressure
EXAFS All the above results illustrate the advantage,
simplicity, and efficiency of the present theory in
high-pressure EXAFS data analysis
Acknowledgements
The author thanks J J Rehr and P Fornasini for helpful
comments This research is funded by the Vietnam National
Foundation for Science and Technology Development
(NAFOSTED) under grant number 103.01-2012.03 The
author is grateful to the Physical Society of Japan for
financial support in publication
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