Geoscience Reference
In-Depth Information
of the deep Earth through direct comparisons
between the calculated elastic properties and the
actual Earth's values (e.g., Wentzcovitch
et al
.,
2004; 2006; Kawai
et al
., 2009).
The technique is now one of the most promis-
ing in order to tightly constrain the thermochemi-
cal conditions of the deep Earth. The lower mantle
(from 660 to 2890 km depth), the largest region
in the Earth, is thought to have an ultramafic
peridotitic-like composition. But this leads to
a substantial deficiency of Si of
can be derived from the all electron Schr odinger
equation, where the exchange-correlation poten-
tial
V
XC
[
n
(
r
)] contains all the quantum many-
body effects. The local density approximation
(LDA) replaces the exchange-correlation potential
at each point by that of a homogeneous electron
gas with a density equal to the local density at the
point. The LDA works remarkably well for a wide
variety of materials, especially in the calculations
of EoS's, elastic constants, and thermodynam-
ics of silicates. Cell parameters and bulk moduli
obtained from well-converged calculations often
agree with the experimental data within a few
percent.
Attempts to improve LDA via introducing
nonlocal corrections have yielded some success.
The generalized gradient approximation (GGA;
Perdew
et al
., 1992, 1996) is a significantly
improved method over LDA for certain transition
metals (Bagno
et al
., 1989) and hydrogen-bonded
systems (Hamann, 1997; Tsuchiya
et al
., 2002,
2005a). There is some evidence, however, that
GGA improves the energetics of silicates and
oxides but the chemical bonding can be under-
bound, leading to overestimating the volumes.
The volume and bulk modulus calculated with
GGA tend to be larger and smaller, respectively,
than those measured experimentally (Hamann,
1996; Demuth
et al
., 1999). If one includes the
thermal effect with zero-point motion, LDA
provides the structural and elastic quantities
much closer (typically within a few percent) to
experimental values than those obtained with
GGA. In addition, a discrepancy of about 10
11% in the
bulk silicate Earth compared to the Mg/Si ratio
of primitive chondrite (Wade & Wood, 2005). It
is therefore still under debate whether the lower
mantle composition is pyrolitic or more silicic. In
this chapter, we review high-
P
,
T
phase relations
and elasticity of the major lower mantle phases
and some crustal compounds, and then discuss
the bulk lower mantle chemistry and cause of
heterogeneity observed in this region.
∼
7.2
Ab Initio
Methods for Mineral Physics
Ab initio
approaches are those that solve the fun-
damental equations of quantum mechanics with
a bare minimum of approximations. DFT is, in
principle, an exact theory for the ground state and
allows us to reduce the interacting many-electron
problem to a single-electron problem (the nuclei
being treated as an adiabatic background). A key
to the application of DFT in handling the inter-
acting electron gas was given by Kohn and Sham
(1965) by splitting the kinetic energy of a system
of interacting electrons into the kinetic energy
of noninteracting electrons plus some remain-
der, which can be conveniently incorporated into
the exchange-correlation energy. Using the vari-
ational principle, the one electron Schr odinger
equation, known as the Kohn-Sham equation,
−
15
GPa is usually seen in transition pressures
calculated with LDA and GGA (e.g., Hamann,
1996; Tsuchiya
et al
., 2004a; Dekura
et al
.,
2011), which provide lower and upper bounds,
respectively. Experimental transition pressures
are usually found between the values obtained
within LDA and GGA, although GGA tends
to provide the pressure with better fit to the
experimental value than LDA (e.g., Hamann,
1996; Tsuchiya
et al
., 2004a,c). The main source
of computational error can be attributed to
how to treat the exchange-correlation potential
(Yu
et al
., 2007).
∼
dr
+
V
XC
[
n
(
r
)]
ψ
i
(
r
)
h
2
2
m
+
V
ion
[
n (r)
]
+
n
(
r
)
r
|
|
r
−
=
ε
i
ψ
i
(
r
)
(7.1)
and
N
ψ
i
(
r
)
ψ
i
(
r
),
n
(
r
)
=
(7.2)
i
