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Svib

V,T =  q

BZ
 i
3N
kB ln
1− e−i

V,q
/kBT

+
i

V,q

T
e−i

V,q
/kBT
1− e−i

V,q
/kBT .
6
From the Helmholtz free energy F
V,T, we can now evalu-
ate the Gibbs free energy G
P,T and enthalpy H
P,T,
G
P,T = F
V,T + PV, H
P,T = G
P,T + TS,
7
where
P
V,T =−

F/
VT
8
and S is the vibrational part of the entropy. The specific
procedures used in the GPT package to calculate the Gibbs
free energy G
P,T are outlined as follows:
1 A series of
phonon calculations are performed at different volumes;
typically, deviating from the ground-state volume by −3% to
3% at 1% increments. For each volume, the crystal structure
is fully optimized allowing both crystal shape and internal
coordinates to be adjusted. F
V,T is then calculated for that
volume.
2 The pressure P
V,T is calculated based on Eq.

8. At a given temperature T, the Helmholtz free energy
F
V,T is fitted into a fourth order polynomial function of
the volume V. The fitting process is carefully monitored as it
may indicate that either higher accuracy in the phonon cal-
culation is desirable or more phonon calculations at different
volumes are needed. The obtained polynomial function is
then used to calculate Holmholtz free energy F
V,T at a
given P and V.
3 The Gibbs free energy G
P,T and en-
thalpy H are calculated using the obtained P
V,T in the
previous step. We first obtain volume V for given pressure P
and temperature T and then calculate Gibbs free energy and
enthalpy using Eq.
7.
From the above obtained Helmholtz free energy F
V,T,
entropy S
V,T, pressure P
V,T, and Gibbs free energy
G
P,T, it is straightforward to calculate other important
thermodynamic properties. These include the constant vol-
ume specific heat Cv =T

S/
TV, volume thermal-expansion
coefficient V
T=dln V
T     /dT, isothermal bulk modulus
B
T=−1/ dln V
T /dP    , thermal Grüneisen parameter

th=VV
TB
T /CV, and constant pressure specific heat
CP=1+
thV
TCV. Figures 5 and 6 summarize some of
the calculated temperature-dependent thermodynamic func-
tions for
-Al2O3 discussed above. At ambient conditions of
P and T=295 K, the volume thermal-expansion coefficient,
volume specific heat, isothermal bulk modulus, thermal Grü-
neisen parameter, and constant pressure specific heat are
found to be 13.62    10−6
K−1
, 794 J /K kg, 195.66 GPa,
0.910, and 794 J /K kg, respectively. Compared to the cor-
responding values for corundum
16    10−6
K−1
,
775 J /K kg, 255 GPa, 1.30, and 775 J /Kkg,
55

-Al2O3
has a slightly larger heat capacity and smaller bulk modulus,
thermal-expansion coefficient and thermal Grüneisen param-
eter.III. STRUCTURAL PROPERTIES
Most of the structure properties of a crystal can be easily
derived from its elastic constants. The elastic constants of

-Al2O3 are calculated using the VASP optimized structure
and the elastic tensor module included in the GPT package
that was designed for efficient evaluation of the stress-strain
response. Details about the method have been described in a
recent paper.
56
In the actual calculation for
-Al2O3, seven
strain levels
j
ranging from −1% to 1% were applied to
each independent deformation and the stress data i
were
collected from the fully ab initio calculations. The elastic
constants Cij
were then extracted by solving the system of
linear equations,i
=j=1
6
Cij

j
. The resultant elastic constants
Cij
for
-Al2O3 are listed in Table II. Note that for the up-
right off-diagonal 3    3 matrix in the elastic tensor, only C14,
C24, and C34 are nonzero. Apparently, the elastic tensor of

-Al2O3 in the present structure shows the symmetry of a
monoclinic crystal lattice with a twofold-symmetry axis in
the x direction or a mirror plane perpendicular to the x axis.
57
From the calculated elastic constants of a single crystal, it
is possible to extract the bulk structure parameters K
bulk
modulus, G
shear modulus, E
Young’s modulus and 

Poisson’s ratio of polycrystalline
-Al2O3. Although there
are different ways of getting the bulk structural parameters,
the scheme we used is the Voigt-Reuss-Hill
VRH approxi-
mation which averages anisotropic elastic properties of the
single crystal to obtain isotropic properties of the corre-
sponding polycrystals.
58–60
The calculated elastic constants
and bulk parameters using the VRH scheme are listed in
Table III together with that of -Al2O3 for comparison. To