Geoscience Reference
In-Depth Information
crystal, which may differ from the crystal-liquid
transition.
The
Stacey Irvine melting rela-
tion
resembles Lindemann's equation, and was
derived from a simple adaptation of the Mie--
Gruneisen equation without involving the vibra-
tion amplitude assumption,
only occur in thermal boundary layers. In ther-
mal boundary layers the thermal gradient is
controlled by the conduction gradient, which is
typically 10--20
◦
C/km compared to the adiabatic
gradient of 0.3
◦
C/km. Both the melting gradient
and the adiabatic gradient decrease with depth
in the mantle.
The Lindemann law was motivated by the
observation that the product of the coefficient
of thermal expansion and the melting tempera-
ture
T
m
was very nearly a constant for a variety
of materials. This implies that
2
d
T
m
/
T
m
d
P
=
2(
γ
−
2
γ
α
T
m
)
/
K
T
The
Lindemann law
itself can be written
d
T
m
/
T
m
d
P
=
2(
γ
−
1
/
3)
/
K
T
which gives almost identical numerical values.
These can be compared with the expression for
the adiabatic gradient
d
T
m
/
T
m
/
d
P
=
2(
γ
−
1
/
3)
/
K
T
which can be written
dln
T
/
d
P
=
γ/
K
T
(d
T
m
/
T
m
)
/
d
P
=−
(
δ
ln
K
T
/δ
ln
V
)
p
/
K
T
γ
Since
is generally about 1, the melting point
gradient is steeper than the adiabatic gradient.
For
Thus, the increase of melting temperature
with pressure can be estimated from the ther-
mal and elastic properties of the solid. Typical
values for silicates give
3thereverseistrue.
If the above relations apply to the mantle,
the adiabat and the melting curve diverge with
depth. This means that melting is a shallow-
mantle phenomenon and that deep melting will
γ<
2
/
∼
6
◦
C
/
kbar
d
T
m
/
d
P
◦
C
Typical observed values are 5 to 13
/
kbar.
