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value that is 0.2 greater than the other tetrahedral units
see
discussion below. This indicates that these two O3-fold atoms
have particularly strong bonds within the specific AlO4 tet-
rahedral units. These are the same O3-fold ions that give the
strong vibrational peak at 750 cm−1
discussed in Sec II.
The Mulliken effective charges Q

on each atom and the
bond order
BO
or the overlap population between pairs of
ions in a crystal are very useful quantities to describe inter-
atomic bonding and charge transfer. They are calculated
separately using a minimal basis set. In the OLCAO method
where the wave functions are expanded in terms of localized
atomic orbitals, the application of the Mulliken population
analysis77 to obtain Q

and BO is both convenient and natu-
ral since it involves no artificial choice of atomic radii for
different types of atoms. In some other electronic structure
methods where plane-wave basis sets are used for expansion,
the Bloch functions need to be projected onto atomic spheres
of a given radius. Such procedures introduce some uncer-
tainty since there is no specific criterion as to what radius
should be used for Altet
and Aloct
, or for O3-fold and O4-fold.
Figure 10 displays the calculated Q

of the 40 atoms
Al and
O in
-Al2O3. Figure 11 shows the distribution of the cal-
culated BO values and the bond lengths between different
Al-O pairs. These results can be succinctly summarized as

follows:
1 Q

is lower in Altet
compared to Aloct
with av-
erage values of 1.46 and 1.58 electrons, respectively;
2 The
Q

for Aloct
is fairly constant whereas Altet
has larger varia-
tions;
3 the Q

for O3-fold are slightly larger than for O4-fold
with average values of 6.99 and 6.96, respectively. There are
variations within each group.
4 The Al–O bond lengths
show two distinct values for Altet
, while for Aloct
they have
either two or three distinct values. The corresponding BO
values scale approximately inversely with the bond length.

5 The BO values for Altet
-O are higher than those of
Aloct
-O, indicating a stronger individual Al–O bond in the
tetrahedral unit than in the Al octahedral unit. These shorter
and stronger bonds, when stretched, give rise to the charac-
teristic peak at 750 cm−1
described in Sec II. However, by
adding the 4 or 6 bonds in each unit, the total BO for the
octahedral unit has an average value of 1.08 compared to the
total BO for the tetrahedral unit of 1.01. This indicates that
the octahedron is still a stronger polyhedral unit in
-Al2O3.
V. SPECTROSCOPIC PROPERTIES
The spectroscopic properties of
-Al2O3 consists of two
parts:
1 The interband optical transitions from valence band
to conduction band,
2 The core-level transitions from Al 1s,
O1s
K edges and Al 2p
L3 edge to the empty CB states.
The VB optical properties of
-Al2O3 were calculated in the
form of the frequency-dependent complex dielectric function

=
1
+i
2
 and the energy-loss function
ELF
=−Im1/

. The interband optical calcula-
tions were performed in the standard one-electron approxi-
mation within the random-phase approximation using the
electronic structures from the ground-state LDA
calculation.
47
No special adjustment for the band gap was
attempted and the calculation includes the full momentum
matrix elements between VB and CB states using ab initio
wave functions at 864 k points in the irreducible portion of
the BZ. The calculated results are shown in Fig. 12. Thecalculated
2
 has a peak at 11 eV and a broad shoulder
at about 16.7 eV. These features are comparable to the ones
obtained by Ahujal et al.
29
using a different method and a
slightly different structural model. The real part
1
h was
obtained from Kronig-Kramers conversion of the
2
.
The optical dielectric constant
1
0 has a value of 3.15. This
value compares well with
=3.11 using an entirely different
method.
33
The square root of
1
0 is frequently used to es-
timate the optical refractive index n. We obtained a value of
n=1.78 for
-Al2O3. Unfortunately, we cannot locate any
experimentally measured refractive index data to compare
with. The values of
1
0 and n for -Al2O3 in similar cal-
culations are 3.14 and 1.77, respectively.
78,79
The plasma fre-
quency p corresponds to the frequency of collective elec-
tron excitation in the bulk crystal. In the present calculation,
p was identified to be 20.5 eV from the peak position in the
energy-loss spectra. It is obvious that this peak is not as
prominent as one would like to see because of the limited
accuracy of the optical-absorption spectrum at higher transi-
tion energies. The numerical inaccuracy in the high-
frequency region is greatly amplified in the ELF
 which
is inversely proportional to

.
The XANES absorption edges
O-K, Al-K, Al-L3 of

-Al2O3 are calculated using the supercell OLCAO method
which takes the core-hole interaction into account.
80
This
method has emerged as one of the most accurate methods for
XANES/ELNES spectral calculation for inorganic crystals in
recent years. It has been successfully applied to a large num-
ber of complex crystals and their interfaces, surfaces, grain
boundaries, and other microstructures.
45,73,81–95
Since the
method of such calculations has been described in many re-
cent papers, we only outline the calculations briefly here. A
2    2    1 supercell
160 atoms for
-Al O was used in the
FIG. 12. The calculated valence-band optical spectra of

-Al2O3:
a
1
;
b
2
; and
c ELF
. The Plasmon
frequency p is indicated by the arrow.

calculation. Because all atoms in the present model of

-Al2O3 are nonequivalent, the spectra were calculated for
each of the 16 Al ions
K and L3 edges and 24 O ions
O-K
edge in the unit cell. For each target atom, the calculation
entails separate evaluations for the initial state
the core state
in the ground-state calculation and the final states
the
conduction-band states with one of the core electron pro-
moted to the bottom of the conduction band.
80 The final
spectrum is obtained as the transition probability from the
initial to the final state in accordance with the Fermi Golden
rule.
96
The dipole transition matrix elements between the ini-
tial and the final states are explicitly included using 8 k
points in the reduced BZ of the supercell to ensure accuracy.
The transition energy is obtained as the difference in the total
energies between the initial and the final-state calculations of
the supercell for each spectrum.
Figure 13 shows the calculated Al-K, Al-L3, and O-K
edges for
-Al2O3. For Al, the total edge is the weighted sum
of the spectra of Altet
and Aloct
which are also presented.
Likewise, the total spectrum for O-K edge is the weighted
sum of those of O3-fold and O4-fold. As has been pointed out in
numerous cases,
45,81,86,90,94,95 the spectral features including
the absorption edge on set, depend sensitively on the local
bonding environment of each cation or anion. It is impos-
sible to experimentally distinguish the spectra from the two
types of Al ions and two types of O ions in the actual mea-
surement. This has contributed to difficulties in the interpre-
tation of the measured XANES data. Even for ions with the
same local bonding environment say, Altet
, their spectra can
be somewhat different from each other because these atoms
are not equivalent. So, the spectra shown in Fig. 13 are the
averaged spectra of the 6
10 for Altet

Aloct
 and 12
12 for
O3-fold
O4-fold calculations. It is clear that the combined total
spectra are fundamentally different from their constitutive
groups. A more detailed inspection reveals that even within
each group of ions with similar local bonding, their edge
spectra can vary depending on the structural details such as
bond lengths and bond angles. This fact has been amply
demonstrated in a recent comprehensive study of ten inor-
ganic crystals within the Y-Si-O-N series.
36
The calculated
Al-K, Al-L3, and O-K edges in
-Al2O3 are noticeably dif-
ferent from those in -Al2O3 which has unique octahedrally
bonded Al and fourfold bonded O sites.
80
There are only a few published papers with experimen-
tally measured ELNES/XANES spectra for
-Al2O3.
15–17
These are shown in Fig. 14 for Al-L and in Fig. 15 for O-K
together with the calculated spectra. The overall agreement
between the calculated and the measured spectra is very sat-
isfactory. The main discrepancy in the Al-L edge appears to
be in the relative intensities of the double peak above the
edge on set. Kimoto et al.
17 attributed these two peaks,
which are separated by about 1.6 eV, to Altet
and Aloct
on the
basis of a similar calculation using the structure of the 
phase. The present results that are obtained by using the
correct structural model for
-Al2O3 show that these two
peaks mainly come from the Altet
site. The mixing with Aloct
only modifies the peak intensity and makes them less sharp.
It is also noted that the peak positions and the edge on-sets in
the Al-L edge from the three different experimental groups
are slightly different. This could be attributed to variations in