Springer-Verlag 2002
Phys Chem Minerals (2002) 29: 633 – 641
DOI 10.1007/s00269-002-0257-3
ORIGINAL PAPER
T. Yamanaka Æ T. Fukuda Æ J. Tsuchiya
Bonding character of SiO2 stishovite under high pressures
up to 30 Gpa
Received: 7 January 2002 / Accepted: 6 May 2002
Abstract The charge density and bond character of the
rutile-type structure of SiO2 (stishovite) under compression to 30 GPa were investigated by X-ray diffraction study using synchrotron radiation and AgKa
rotating anode X-ray generator through a newly devised
diamond-anvil cell. The valence electron density was
determined by least-squares refinement including the j
parameter and the electron population in the X-ray
atomic scattering parameters. The oxygen j-parameter
of SiO2 is 0.94 under ambient conditions and 1.11 at
29.1 GPa and the silicon valence changes from +2.12(8)
at ambient pressure to +2.26(15) at 29.1 GPa. These
values indicate that the electron distributions are more
localized with increasing pressure. The difference Fourier map shows the deformation of the valence electron
distribution and the bonding electron population in residual electron densities. The bonding electron observed
from the X-ray diffraction study is interpreted by
molecular orbital calculations. The deformation of SiO6
octahedra and the bonding electron density of stishovite
structures are elucidated from the overlapping electron
orbits. The O–O distances of shared and unshared edge
of SiO6 octahedra change with the cation ionicity. The
repulsive force between the two cations in the adjacent
octahedron makes its shared edge shorter. The pressure
changes of the apical and equatorial Si–O interatomic
distances are explained by the electron density of state
(DOS) of Si and electron configuration.
Keywords SiO2 stishovite structure Æ Synchrotron
radiation Æ High pressure up to 30 GPa Æ
Valence electron population Æ Si–O bond character
under compression
T. Yamanaka (&) Æ T. Fukuda Æ J. Mimaki
Department of Earth and Space Science,
Graduate School of Science,
Osaka University 1–1 Machikaneyama Toyonaka,
Osaka, 560-0043 Japan
e-mail: b61400@center.osaka-u.ac.jp
Tel.:/Fax: +81-6-6850-5793
Introduction
A rutile-type SiO2 polymorph (P42/mnm Z ¼ 2) was first
synthesized by high-pressure apparatus (Stishov and
Popova 1961) and stishovite was discovered in the Meteor Crater, Arizona (Chao et al. 1962). The phase diagram of SiO2 high-pressure polymorphs has aroused
great geophysical interest, since it was confirmed to be
one of the major substances in lower mantle. Singlecrystal structure analysis was first executed under ambient conditions by Sinclair and Ringwood (1978). The
crystal-structure analyses under pressure were carried
out up to 6 GPa by Sugiyama et al. (1987) and 16 GPa
by Ross et al. (1990). Systematic single-crystal structure
analyses of IVb-group cation dioxide and metal dioxides
MO2 with rutile-type structure have been made by Baur
and Khan (1971), Bolzan et al. (1997) and Yamanaka
et al. (2000).
The electron-density distribution in stishovite
has been investigated from X-ray diffraction study
(Spackman et al. 1987). We previously investigated the
electron-density distribution by monopole refinement
(j refinement), which was originally derived by Coppens
et al. (1979), based on the variable atomic scattering
factor, and which discussed the bonding nature by the
complementary study of molecular orbital calculation
(Yamanaka et al. 2000). Molecular orbital calculation
gives a precise definition of electron density of state.
Electron orbital overlapping and bonding energy cause a
deformation of MO6 octahedra of rutile-type structures
from the bond character (Camargo 1996; Gibbs et al.
1997, 1998; Mimaki et al. 2000). Band structure and
charge distribution of rutile-type members have been
often discussed (Arlinghaus 1974; Jacquemin and
Bordure 1975; Simunek et al. 1993). Recently, Kirfel
et al. (2001) reinvestigated the electron-density distribution by extinction-free structure refinement using
high-energy synchrotron radiation, and applied multipole expansion up to 7th order. To this they compared
the result obtained from band-structure calculation.
634
The concept of electron negativity has been applied
for the approximation of the covalency/ionicity scale
(Chelikowsky and Burdett 1986). The present investigation elucidates the change in Si–O bonding character
and the valence electron distribution under high pressure
by j refinement. We devised a new type of diamondanvil cell (DAC) for single-crystal X-ray diffraction
study under compression (Yamanaka et al. 2001). The
valence electron distribution under compression was
discussed on the basis of the diffraction intensity measurement using synchrotron radiation. The pressure
change of the bonding electron observed from X-ray
diffraction study is interpreted by the optimized pair
potential and molecular orbital calculation. We aimed to
evaluate the effective charge and valence electron population of SiO2 under high pressure.
Experimental
arranged with topotactic orientation. Thus, the shear stress in both
diamond crystals is equivalent under compression. The other side
of the diamond plate is placed on the base plane of the angleadjusting steel hemisphere. The present DAC can be easily installed
on a four-circle diffractometer. Detailed specification of the new
DAC was described in Yamanaka et al. (2001).
For diffraction studies at 5.23, 9.26 and 12.3 Gpa, SiO2 a single
crystal of 40 · 60 lm wide and 40 lm thick was placed in the Re
gasket hole of 200 lm in diameter. The preindented gasket keeps a
thickness of about 100 lm. Pressure markers of ruby chips and
pressure-transmitting media were also kept in the hole. The media
at 5.23, 9.26 and 12.3 GPa was an alcohol mixture with methanol:
ethanol:H2O ¼ 16:3:1. The hydrostatic condition could be guaranteed within this pressure range. Argon gas was also used as the
transmitting medium in the case of experiments over 15 GPa in
order to preserve the hydrostatic condition. The single crystal of
40 · 20 · 20 lm (thickness) was placed in the gasket hole at 29.1
GPa with argon-transmitting media.
The new assembly greatly improves the accuracy of the structure analysis. Diamond-plate windows have the following advantages for single-crystal diffractometry: lower X-ray mass
absorption, much higher-pressure generation over 50 GPa, no
powder rings from the window and a wider observable 2h-angle
and transparent window.
Sample preparation
Single crystals of the rutile-type SiO2 were prepared under high
pressure of 12 GPa at 1300 C by a 6–8 type multianvil highpressure apparatus. Anhydrous amorphous silica was placed together with 10 mol% of flux material Li2WO4 in the anvil. The
starting sample was kept at 1300 C for 3 h and then the temperature was gradually cooled to 900 C. The grown crystals were
transparent and elongated and had a maximum size of
100 · 80 · 500 lm.
Homogeneities and chemical impurities in the synthesized
crystals were examined by EPMA and no trace element was found.
Their crystallinity was tested by optical microscope and X-ray
precession camera. Almost cube-shaped crystals with about 60 lm
in edge length were selected for the X-ray diffraction experiment.
High-pressure diffraction study using
the new diamond-anvil cell (DAC)
Single-crystal structure refinements under high pressure encountered many difficulties, such as non-hydrostaticity, a large blind
region due to the limited aperture angle of the pressure cell, large
X-ray absorption from the window and limitation of sample size.
So far, single-crystal X-ray diffraction studies of ruby at 31 GPa by
Kim-Zajonz et al. (1998) and pyrope garnet at 33 GPa by Zhang
et al. (1998) have been reported. Our new system solved these
difficulties and made diffraction study possible, enabling the discussion of electron density distribution under pressures up to 50
GPa.
Generally, beryllium hemispheres or plate windows were used
as backing plate of DAC for the diffraction intensity measurements
of single crystal, due to their very high X-ray transmittance.
However, many broad and strongly spotted powder rings from
beryllium windows often overlap the diffraction peaks of the
sample and deform the peak profiles. These obstacle rings interrupt
the intensity measurement. Further, beryllium windows cause a
limitation of pressurization because of its softness.
We tried to find windows more efficient than beryllium
(Yamanaka et al. 2001). Our new cell consists of large single-crystal
diamond plates supporting the diamond anvils and the angleadjusting steel discs. Large single crystals of (100) platy diamonds
of about 2 carats were prepared by cutting and polishing the grown
crystals. The diamond crystals had a size of 6 · 6 mm wide and
2 mm thick.
Plate windows of 2 mm thick are directly fixed on the (100)
table plane of a brilliant-cut diamond anvil with culet sizes of 400
and 600 lm, as shown in Fig. 1. Both plate windows and anvils are
Diffraction intensity measurement using
synchrotron radiation and AgKa radiation
In the present diffraction study, as shown in Table 1, intensity
measurements were conducted using an AgKa rotating anode
X-ray generator with radiation of k ¼ 0.5608 Å at 5.23, 9.26 and
12.3 GPa in our laboratory and using SR with k ¼ 0.40799 Å
(E ¼ 30.388 eV) at a pressure of 29.1 GPa at BL02B1 in SPring-8
at Nishiharima; the synchrotron radiation (SR) source with 8 GeV
and 100 mA provides a critical wavelength of 30 keV.
SR has excellent characteristics for single-crystal diffraction
study under high pressure using DAC. SR emitted from the
bending magnet provides a source intensity about 104 times greater
than that from the conventional rotating anode X-ray generator. It
has the following great advantages for high-pressure diffraction
study: high signal/noise ratio of the diffraction intensity, detection
of the weak diffraction peaks, precise determination of diffraction
intensity, very small extinction effect and short measuring period.
The incident beam was monochromated by Si (111) double crystals.
The beam is focused on the DAC by a Pt-coated mirror and a
collimator guide pipe leads to DAC to reduce the background intensity. A divergent slit of 100 lm in diameter was adopted, because the gasket hole was 200 lm in diameter and the sample size
several 10 l cross.
Diffraction intensity measurement was carried out using a fourcircle diffractometer with a scintillation counter. The new DAC together with the short wavelength allowed reflections with
d > 0.44150 Å (2h < 55) to be observed. The number of observed
reflections is four times more than those obtained from the laboratory source. A large number of reflections with precisely observed
intensities enable a more precise observation of electron density
distribution at high pressures. Intensity measurement was carried out
by the /-fixed and x-scan mode, scan speed 1 min)1, scan width 1
in x-rotation, step interval 0.01 in consideration of peak broadening
at high pressure. Reflections up to hkl ¼ 661 with dmin ¼ 0.4679 Å at
2h ¼ 51.657 were observed. A total of 147 reflections was observed
and 57 crystallographically independent reflections with Fo>3r(Fo)
were used for the least-squares refinement.
X-ray absorption correction was made for SiO2 stishovite
samples. A linear absorption coefficient of each sample is listed
in Table 2. Since all of the coefficients were negligibly small (for
example l ¼ 0.334 cm)1 at 29.1 GPa) and the sample size was
extremely small, the absorption correction of the sample was not
considered for diffraction intensity corrections. Because the total
thickness of the diamond was larger than 2 mm, the absorptions of
the diamond-anvil and diamond-plate window were taken into
account in spite of the very low linear absorption coefficient. The
635
Fig. 1 Diamond-anvil cell.
Large single-crystal diamond
plates and 1/8-carat brilliant cut
diamonds were used as the
windows and anvils, respectively. Both (100) plates of diamond single crystals were
directly fixed on the anvil table
plane in topotactic direction
absorption coefficient was l ¼ 0.1772 cm)1 in the case of the diffraction study at 29.1 GPa using SR with k ¼ 0.40799 Å.
Electron density analysis
In the first stage, conventional least-squares refinements of the five
data sets at 0.0001, 5.21, 9.20, 12.20 and 29.1 GPa were carried out
with the following variable parameters: scale factor, positional
parameter of oxygen atom, anisotropic thermal parameters and
isotropic extinction parameter Gex based on a crystalline mosaicity
(type-I model) (Becker and Coppens 1974). Higher-rank rather
than second-rank thermal parameters derived from the anharmonic
oscillator model are not considered, because SiO2 has a quite high
Debye temperature and intensity was measured at 300 K. The full
matrix least-squares refinement was carried out using the program
RADY (Sasaki and Tsukimura 1987). After the conventional
structure refinement, electron density analysis was executed by
monopole refinement introducing the j parameter in the atomic
scattering factors.
According to the pseudopotential model, inner-core electrons
are frozen with bonding effects; but valence electron clouds are
deformed due to coordination and thermal atomic vibration, because their interactions with the core electrons are relatively weak.
Accordingly, they are more sensitive to the interatomic potential
being affected by the coordination of the adjacent atoms. Since the
deformation electron densities of Si are supposed to be naturally
very small because of spherical electron orbits, except for only a
slight excitation of the d electron, a monopole refinement was applied instead of the multipole deformation density. The j parameter (Coppens et al. 1979; van der Wal and Stewart 1984) was
applied in the atomic scattering factor of Si, which is an indicator
of the radial distributions of the electron. The j parameter used in
this analysis is expressed by:
qvalence ðrÞ ¼ Pvalence j3 qvalence ðj rÞ ;
ð1Þ
where qvalence (r) is the ground-state density of the free atom.
Pvalence indicates the valence-shell population, which is the occupancy parameter of core and valence electrons. j3 is a normalization factor.
The atomic scattering factor f(s) used in the structure refinement is modified from a Hartree–Fock approximation based on the
isolated atom model. Perturbed valence electron density was
f ðs=2Þ ¼
X
Pj;core fj;core ðs=2Þ
þPj;valence fj;valence ðjj ; s=2Þ þ fj0 þ ifj00 :
ð2Þ
636
Table 1 Diffraction intensity measurement conditions
Pressure
1 atm
Diffractometer
Wave length
Energy
Monochrometer
Gasket
Pressure media
Scan mode
Crystal size (lm)
RIGAKU AFC5
Moa (0.7107 Å)
150 kV 50 mA
Graphite (002)
–
–
x)2h
50 · 60 · 80
2h angle
sinh/k
2h angle/MoKa
Ref. (observed)
Ref. (independent)
120
1.219
120
210
126
5.23 GPa
9.26 GPa
12.3 GPa
RIGAKU AFC6R
AgKa (0.5608 Å)
150 kV 50 mA
Graphite (002)
Spring steel
M:E:W=16:3:1
/-fix x-scan
60 · 40 · 40
53
0.796
69
81
25
47
0.711
61
79
25
The valence scattering of the perturbed atom at s/2 ( ¼ sinh/2k) is
given by:
fM -core ðjj ; s=2Þ ¼ fj ; M -coreðfree atomÞ ðsin h=k 1=jj Þ :
ð3Þ
A localized electron distribution as indicated by j ¼ 1.0 implies a
more ionic character in the bonding nature. The detailed formalization is discussed in Yamanaka et al. (2000).
The valence charge of the cation was introduced by the population parameter. The effective charge was determined from the j
parameter of the oxygen atomic scattering factor. The parameters
of P and j could be refined simultaneously as variable parameters.
The j parameter and population parameter were obtained by
minimizing of the reliable factor R. In this refinement, the factors
of R and wR are defined by: R ¼ S(||Fobs|)|Fcal||)/Sw|Fobs| and
Rw = [Sw(||Fobs|)|Fcal||)2/Sw|Fobs|2]1/2, where w ¼ 1/r2(|Fobs|).
The valence electrons around the atomic position and the
bonding electron distribution cannot be separately evaluated by
structure refinement. The covalency of the bond character is estimated from the effective charge. The difference Fourier synthesis
and the population parameter show an electron deformation density. The effective charge q is obtained by the spatial integration of
the difference electron density by:
Z
q ¼ 4p r2 DqðrÞdr :
ð4Þ
Then q has a correlation with the j parameter.
Results
29.1 GPa
HUBER (512.1)
SR (0.40799 Å)
8 GeV 100 mA
Si(111) double crystal
Spring steel
Ar
/-fix x-scan
40 · 60 · 20
54
0.809
70
82
26
52
1.074
99
147
57
c=a ¼ 0:6378 þ 0:7986 103 P 9:2359 106 P 2 :
ð7Þ
The lattice constant of c decreases almost straight with
pressure and indicates an opposite curvatures to the a
edge. Therefore, the increment of c/a becomes small with
increasing pressure, as shown in Fig. 2.
The change of cell volume is expressed by the following equation
VP ¼ 46:594ð1 3:0162 103 P þ 8:2944 106 P 2 Þ :
ð8Þ
The unit-cell volume at 29.1 GPa was reduced by as
much as 8.6%. Isothermal bulk modulus KT (GPa) was
calculated from the volume change using the Birch–
Murghanan equation of state, which is shown in
Table 3. The large KT value of 295(5) GPa indicates that
stishovite is noticeably hard crystal and high-pressure
polymorph of SiO2. The present value is a little smaller
than those reported by Ross et al. (1990) and Sugiyama
et al. (1987) but very similar to Andrault et al. (1998).
This may be because we observed the volume change up
to much higher pressure than the former two experiments.
Lattice constant change
The lattice constants were determined by least squares
based on the d values of 15 25 reflections in the range
20 < 2h < 30. The lattice constants, axial ratios of c/a
and unit-cell volumes are presented in Table 2. The lattice
constant ratios of a/ao and c/co are plotted in Fig. 2,
together with c/a/co/ao.
The lattice constants as a function of pressure are
represented by the following equation:
a ¼ 4:1805ð1 1:2820 103 P þ 5:5046 106 P 2 Þ
ð5Þ
c ¼ 2:6674ð1 0:4878 103 P 4:1811 106 P 2 Þ ;
ð6Þ
and the ratio of c/a changed with pressure is
Interatomic distance as a function of pressure
The converged structure parameters of SiO2 stishovite
are shown in Table 2. The data at ambient pressure are
in good agreement with the previous experiment data
(Sinclair and Ringwood 1978; Sugiyama et al. 1987;
Ross et al. 1990). Interatomic distances of stishovite
structure to 29.1 GPa are presented in Table 4. The SiO6
octahedron has a site symmetry of mmm, with four
equatorial bonds and two apical bonds, as shown in
Fig. 3. The apical Si–O bond of 1.8111(9) Å is much
longer than the equatorial bond of 1.7559(9) Å under
ambient conditions but the former is more compressive
and becomes closer to the latter with increasing pressure,
as seen in Fig. 4. The volume compression of the SiO6
octahedron is expressed by the following equation:
637
Table 2 Structure parameter.
R = S(||Fobs|)|Fcal||)/Sw|Fobs|
and Rw = [Sw(||Fobs|)|Fcal||)2/
Sw | F o b s | 2 ] 1 / 2 , where w=1/
r2(|Fobs|)
Pressure
1 atm
5.23 GPa
9.26 GPa
12.3 GPa
29.1 GPa
Radiation
a(Å)
c(Å)
c/a
V(Å3)
abs. coeff (cm)1)
No. ref.
R(F )
wR(F )
Si (000)
b11
b33
b12
O (xx0)
b11
b33
b12
MoKa
4.1812(1)
2.6662(3)
0.6377
46.61
1.463
126
0.0253
0.0243
–
0.0045(1)
0.0037(5)
0.0002(2)
0.3063(1)
0.0051(2)
0.0036(3)
)0.0009(3)
AgKa
4.152(1)
2.6590(8)
0.6404
45.84
0.759
25
0.0440
0.0234
–
0.0126(27)
0.0261(18)
0.0004(21)
0.3063(20)
0.0075(37)
0.0031(17)
0.0005(35)
AgKa
4.134(1)
2.6540(7)
0.6420
45.36
0.767
25
0.0312
0.0104
–
0.0055(21)
0.0142(12)
0.0021(14)
0.3056( 9)
0.0051(29)
0.0090(28)
0.0009(29)
AgKa
4.118(2)
2.649(1)
0.6433
44.92
0.774
26
0.0345
0.0227
–
0.0088(20)
0.0103(13)
0.0009 (12)
0.3058(19)
0.0104(21)
0.0054(18)
0.0007(35)
SR(30KeV)
4.044(6)
2.619(2)
0.6476
42.83
0.334
57
0.0330
0.0282
–
0.0035(11)
0.0131(13)
0.0019(15)
0.3039( 7)
0.0095(13)
0.0074(17)
0.0004(16)
the octahedron found in this experiment indicates a
precursor phenomenon for the post-stishovite transition.
The SiO6 octahedra are linked along the c axis with the
shared edge of O1–O2(sh). The edge is much shorter than
the unshared edge of O1–O1(unsh), which is equivalent to
the cell edge of the c axis. In spite of the shorter interatomic distance, the shared edge is more compressive than
the longer unshared edge, as seen from Table 4. Hence,
the ratio of O1–O2(sh)/O1–O1(unsh) decreases with increasing pressure, which is a peculiar phenomenon contrary to the Pauling sense. The greater shortening of the
shared edge than the unshared edge with pressure is explained by the shielding effect due to Si–Si repulsion along
the c axis (Sugiyama et al. 1987; Ross et al. 1990) and
probably by tightening the Si–O bond with pressure due to
the change in the electron density of state.
Bonding character change with pressure
Fig. 2 Axial compression. Axial ratios of a/ao and c/co are plotted as
a function of pressure. ao and co are the lattice constants at ambient
pressure. Deformation of the unit cell under compression is expressed
by c/a/co/ao
V ðSiO6 ÞP ¼ 3:6858ð1 2:7619 103 P þ 4:487 106 P 2 Þ :
ð9Þ
The volume decreased almost linearly with increasing
pressure. The octahedron is less compressible than the
unit cell in the comparison shown by Eq. (9).
SiO6 has a tendency towards the structure in which
both apical and equatorial bonds are at the same distance under compression. Stishovite transforms to a
CaCl2 structure (Pnnm, Z ¼ 2) at about 53.2 GPa by
second-order transition with oxygen displacement (Andrault et al. 1998). The CaCl2 structure is suggested as a
post-stishovite phase in several experiments (Tsuchida
and Yagi 1989; Kingma et al. 1995). The deformation of
Structure refinements from two sets of diffraction intensities obtained at ambient pressure and 29.1 GPa
provide valence electron densities in the unit cell. The
great improvement of high-pressure single-crystal
structure analysis using the SR source and newly devised
DAC enables clarification of an electron distribution of
SiO2 stishovite at 29.1 GPa. In order to estimate the
valence electron distributions from the j parameter,
the reliable factor R was minimized with optimization of
the variable of the j parameter and population parameter. The value of the j parameter for the oxygen atom
was 0.94 at ambient pressure and 1.11 at 29.1 GPa. Intensity data sets at 5.21, 9.20 and 12.20 GPa taken from
AgKa radiation cannot provide a meaningful j parameter, because the observed reciprocal space is not large
enough to determine the precise value. The j parameters
of Si and oxygen are presented in Table 5. Fourier
transform of f(s) including the population parameter (P)
of the valence electrons in Eq. (2) defines electron
density q(r) and the spatial integration of q(r) brings the
effective charge of oxygen atoms from Eq. (4).
638
Table 3 Isothermal bulk modulus
Present data
Andrault et al. (1998)
Ross et al. (1990)b
Ross et al. (1990)
Sugiyama et al. (1987)
Weidner et al. (1982)
KT (GPa)
dKT/dP
Pmax (GPa)
Data
Sample and remark
292(13)
291
302(5)
287(2)
313(4)
306(4)
6(fixed)
4.29
2.60(0.8)
6(fixed)
6(fixed)
29.1
53.2
16
5
17
6
Single crystal
Combination dataa
Single crystal
6
9
Single crystal
Brillouin scattering
a
The cell volume data are at pressures of 0.0001 15 GPa from Ross et al (1990), at 24.6 49.4 GPa
from Hemley et al. (1994) and at 48.1 53.2 GPa from Andrault et al. (1998)
when dKT/dP is fixed to 6
b
Table 4 Interatomic distances.
Abbreviations of equatorial and
apical bond are indicated by eq
and ap and those of shared and
unshared edge are by sh and
unsh, respectively
Pressure
1 atm
5.23 GPa
9.26 GPa
12.3 GPa
29.1 GPa
Si–O(eq)(Å)
Si–O(ap)(Å)
ap/eq
Vol(SiO6)(Å3)
O1–O2(sh)(Å)
O1–O1(unsh)(Å)
O1–O3(Å)
sh/unsh
1.7559(9)
1.8111(9)
1.0314
7.374
2.2906(10)
2.6662(3)
2.5226(4)
0.8591
1.750(11)
1.798(4)
1.0274
7.266
2.277(5)
2.6590(8)
2.509(10)
0.8563
1.748(8)
1.784(2)
1.0205
7.178
2.275(3)
2.6540(7)
2.498(10)
0.8572
1.742(13)
1.781(4)
1.0223
7.134
2.262(5)
2.649(1)
2.482(17)
0.8539
1.724(3)
1.738(2)
1.0081
6.806
2.242(3)
2.619(2)
2.448(2)
0.8567
Fig. 3 Stishovite structure. Although all three oxygen atoms are
crystallographically equivalent, they are distinguished for the SiO6
octahedron. Hatched section, including Si, O1 and O2 atoms, indicates
the same plane as the Fourier map in Fig. 6 and the deformation
electron density map in Fig. 8
Fig. 4 Pressure change of apical and equatorial Si–O bonds. Open
circles are from Ross et al. (1990)
Since, as expressed by Eq. (1), the electron distributions are more localized with increasing pressure, a
smaller j parameter implies more bonding electrons and
intensifies the covalent-bond nature. Our previous study
of rutile-type MO2 (M ¼ Si, Ge and Sn) indicates that
the j parameter of SiO2 stishovite has a relatively strong
covalent bond in comparison with the two other compounds (Yamanaka et al. 2000).
After the refinement using atomic scattering factors
based on the spherical electron distribution model, the
deformations of electron distributions of SiO2 at ambient pressure and 29.1 GPa are disclosed by difference
Fourier maps, as shown in Fig. 6a and b, respectively.
The maps are the section of the coplanar atoms of Si, O1
and O2 parallel to the plane (110) (Fig. 3). The map at
ambient pressure shown in Fig. 6a is very similar to that
of Hill et al. (1983) and Spackman et al. (1987). A
positive peak with height of 0.7 e Å)3 is found at almost
mid-position on the Si–O bond. Four positive residual
densities around the cation are also recognized at 0.4 Å
from the metal position. A residual electron peak position found under ambient conditions is located at 0.86 Å
from the Si atomic position and that at 29.1 GPa is
0.77 Å. The valence electron tends to be more localized
at higher pressure. The localized density implies a more
ionic character under higher compression.
The non-spherical residual electron density around
the Si site is probably induced by the overlapping orbital
of the d electron state of Si and the p electron of oxygen,
resulting in a d–p–p bond. Hence, the noticeable residual
639
Table 5 j parameter and effective charge
j parameter of oxygen
Residual electron peak
position from Si
Effective charge
Ambient
conditions
29.1 GPa
0.94
1.11
0.86 Å
+2.12(8)
0.77 Å
+2.26(15)
2.08
2.21
0.43
0.45
0.44
0.45
0.52
0.87
0.53
0.45
0.80
0.58
Mulliken population analysis
Si net charge
Overlap population
Si–O (apical)
Si–O (equatorial)
3s
3p
3d
electron density on the Si–O bond indicates a bonding
electron. A large negative density in the O1–O2 shared
edge plays a role in preventing cation repulsion. The
bonding electron distribution at 29.1 GPa is less remarkable compared with that at ambient pressure. This
feature is caused by the observed j parameter and
effective charge. By using Eq. (4), the effective charge of
Si at ambient pressure and 29.1 GPa is +2.12(8) and
+2.26(15), respectively. The dipole moment is calculated
by Eq. (5) from the effective charge and interatomic
distance. All data obtained from the charge density
analysis indicate that SiO2 stishovite becomes more ionic
with increasing pressure. The result of the apparent
relative ionicity is presented in Table 5.
Discussion
Pressure effects on the structure can be explained by the
virial theorem in statistical physics:
*
+
N X
N owij
NKB T
1 X
Pext ¼
rij
V
3V i¼1 j>i
orij
*
+
N X
N
NKB T
1 X
ð10Þ
¼
Fij rij
V
3V i¼1 j>i
where V ¼ volume, N the number of particles, wij the
interatomic potential between the i and j particles, rij the
distance between the i and j particles, Fij the force which
comes up from the j to the i particle, T the absolute
temperature and kB the Boltzman constant. Pext indicates an external pressure generated by DAC. The
compression implies the equilibrium state between external pressure and internal pressure induced from the
crystalline bonding force. The energy of PextV is the
summation of the product of interatomic forces and
interatomic distances, where V is the compressed unitcell volume. Interatomic distances can be determined
directly by diffraction study as a function of pressure.
The compression of these distances induces the compression of the lattice constants of bulk crystal. The term
Fig. 5 Shared edge/unshared edge. O1–O2(sh) and O1–O1(unsh) edge
distances of SiO6 are shown as a function of pressure. Open circles are
from Ross et al. (1990)
of interatomic force (F) can be obtained by lattice dynamic experiments. However, the X-ray diffraction
study gives the electron-density distributions, including
valence electrons and bonding electrons. The charge
density analysis based on the diffraction intensities
provides a view of the effective charge of ions.
The apparent electric dipole moment (l) of the Si–O
bond can be defined by the product of charge (q) and
mean interatomic distance (r):
l ¼ rðqSi qO Þ :
ð11Þ
The dipole moment (lobs) can be experimentally determined from the observed effective charge of Si and O. qSi
and qo are the observed value obtained from the present
j refinement in Eq. (2). The moment is lobs ¼
4.51(1) · 10)29Cm ¼ 13.54D at 1 atm and lobs ¼ 4.69
(3) · 10)29Cm ¼ 14.064D at 29.1 GPa. These results
indicate that the Si–O bond becomes more ionic with
increasing pressure.
The charge distribution reveals a significant admixture of covalency in the chemical bonds of SiO2 and the
effective charge of Si is far from a formal charge. Our
results are consistent with energy band calculations
(Svane and Antoncik 1987). The significant d-electron
population indicates some degree of non-sphericity of
valence electron distribution around the cation. The
difference Fourier map of SiO2 (Fig 6a, b) reveals apparently non-spherical electron distribution, which is
represented by the residual electron density around the
Si atom.
In order to investigate the bond character of rutiletype structures of SiO2, we carried out the cluster discrete-variational Xa (DV-Xa) method molecular orbital
calculation (Averill and Ellis 1973; Rosen et al. 1976).
The DV-Xa method is based on a self-consistent field
Hartree–Fock–Slater approximation. The cluster in the
640
Fig. 6a, b Difference Fourier
map projected on (110) plane.
Contours are at intervals of 0.2
e A3 and positive and negative
contours are expressed by solid
and broken line, respectively.
Residual valence-electron density is revealed around cation
and bonding electron distribution on the Si–O bond. The
difference Fourier map at
ambient pressure is shown in
6a and the map at 29.1 GPa
is presented in b
Fig. 8 Deformation electron density map at 29.1 GPa obtained from
molecular orbital calculation. The section is the same projection as
Fig. 6. The increment of the contours is 0.005 e in the range between
)0.04 e and +0.04 e. Solid lines and broken lines are positive and
negative electron distribution contours
Fig. 7 Electron density of state of 3s, 3p and 3d electron of Si in SiO2
stishovite at 29.1 GPa
present calculation is {SiO6Si10O38}44. Mulliken population analysis was used to analyze the local electronic
properties. It is noted that absolute values from the
analysis are dependent on the atomic basis set, but they
are meaningful when they are calculated in the same set
under different pressure conditions. The partial density
of state (PDOP) of the SiO2 valence electron under
ambient conditions and at 29.1 GPa are given in Fig. 7,
which shows a significant decrease in Si 3s and 3p PDOS
at 29.1 GPa. This indicates the accumulation of positive
charge of Si under high pressure. In comparison with Si
3s and 3p electrons, little change in the PDOS of the 3d
electron was found. This indicates that the 3d electrons
have the role of bonding electron between the Si and O
atoms. The detailed procedure of the calculation was
described in our previous paper (Mimaki et al. 2000).
Figure 8 shows the deformation density map of SiO2
stishovite at 29.1 GPa, which shows features very similar
to the difference Fourier map shown in Fig. 6b. The
bonding electrons in the map are located a little closer to
oxygen compared with the residual electron observed
from the difference Fourier.
The overlap of the electronic orbitals causes the
deformation of octahedral coordination SiO6 of the
stishovite structure and the bond character of the covalency. The d electron of cations increases the degree of
the d–p–p bond in Si–O. The ratio between the shared
and unshared edge distance has a strong relation with
641
the interatomic repulsive force between two cations Si–
Si and the degree of p bonding of Si–O. The ratio of O1–
O2(sh)/O1–O1(unsh) in Table 4 decreases with increase in
ionicity. Hence, the more negative electron density between two Si atoms revealed by the difference Fourier
map (Figs 6a, 7b) indicates the more shielding effect with
increasing pressure. The peculiar pressure changes of
bonding characters found in O–O(sh)/O–O(unsh) and Si–
O(ap)/Si–O(eq), as mentioned in the earlier section, can be
explained by the electron density of state (DOS) of Si
and the electron configuration. The electron densities of
state obtained from molecular orbital calculation are in
good agreement with the results from X-ray photoelectron spectroscopy (XPS) (Barr et al. 1991).
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