<s>
The	O
quasi-harmonic	B-Algorithm
approximation	I-Algorithm
is	O
a	O
phonon-based	O
model	O
of	O
solid-state	O
physics	O
used	O
to	O
describe	O
volume-dependent	O
thermal	O
effects	O
,	O
such	O
as	O
the	O
thermal	O
expansion	O
.	O
</s>
<s>
It	O
is	O
based	O
on	O
the	O
assumption	O
that	O
the	O
harmonic	B-Language
approximation	I-Language
holds	O
for	O
every	O
value	O
of	O
the	O
lattice	O
constant	O
,	O
which	O
is	O
to	O
be	O
viewed	O
as	O
an	O
adjustable	O
parameter	O
.	O
</s>
<s>
The	O
quasi-harmonic	B-Algorithm
approximation	I-Algorithm
expands	O
upon	O
the	O
harmonic	O
phonon	B-Language
model	O
of	O
lattice	O
dynamics	O
.	O
</s>
<s>
The	O
harmonic	O
phonon	B-Language
model	O
states	O
that	O
all	O
interatomic	O
forces	O
are	O
purely	O
harmonic	O
,	O
but	O
such	O
a	O
model	O
is	O
inadequate	O
to	O
explain	O
thermal	O
expansion	O
,	O
as	O
the	O
equilibrium	O
distance	O
between	O
atoms	O
in	O
such	O
a	O
model	O
is	O
independent	O
of	O
temperature	O
.	O
</s>
<s>
Thus	O
in	O
the	O
quasi-harmonic	O
model	O
,	O
from	O
a	O
phonon	B-Language
point	O
of	O
view	O
,	O
phonon	B-Language
frequencies	O
become	O
volume-dependent	O
in	O
the	O
quasi-harmonic	B-Algorithm
approximation	I-Algorithm
,	O
such	O
that	O
for	O
each	O
volume	O
,	O
the	O
harmonic	B-Language
approximation	I-Language
holds	O
.	O
</s>
<s>
where	O
Elat	O
is	O
the	O
static	O
internal	O
lattice	O
energy	O
,	O
Uvib	O
is	O
the	O
internal	O
vibrational	O
energy	O
of	O
the	O
lattice	O
,	O
or	O
the	O
energy	O
of	O
the	O
phonon	B-Language
system	O
,	O
T	O
is	O
the	O
absolute	O
temperature	O
,	O
V	O
is	O
the	O
volume	O
and	O
S	O
is	O
the	O
entropy	B-Algorithm
due	O
to	O
the	O
vibrational	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
where	O
N	O
is	O
the	O
number	O
of	O
terms	O
in	O
the	O
sum	O
,	O
is	O
introduced	O
as	O
the	O
characteristic	O
temperature	O
for	O
a	O
phonon	B-Language
with	O
wave	O
vector	O
k	O
in	O
the	O
i-th	O
band	O
at	O
volume	O
V	O
and	O
is	O
shorthand	O
for	O
the	O
number	O
of	O
(	O
k	O
,	O
i	O
)	O
-phonons	O
at	O
temperature	O
T	O
and	O
volume	O
V	O
.	O
As	O
is	O
conventional	O
,	O
is	O
the	O
reduced	O
Planck	O
constant	O
and	O
kB	O
is	O
the	O
Boltzmann	O
constant	O
.	O
</s>
<s>
The	O
first	O
term	O
in	O
Uvib	O
is	O
the	O
zero-point	O
energy	O
of	O
the	O
phonon	B-Language
system	O
and	O
contributes	O
to	O
the	O
thermal	O
expansion	O
as	O
a	O
zero-point	O
thermal	O
pressure	O
.	O
</s>
<s>
Note	O
that	O
for	O
a	O
constant	O
value	O
of	O
V	O
,	O
these	O
equations	O
corresponds	O
to	O
that	O
of	O
the	O
harmonic	B-Language
approximation	I-Language
.	O
</s>
<s>
Below	O
are	O
a	O
few	O
which	O
cannot	O
be	O
determined	O
through	O
the	O
harmonic	B-Language
approximation	I-Language
alone	O
.	O
</s>
<s>
where	O
i	O
indicates	O
a	O
phonon	B-Language
mode	O
.	O
</s>
