Theory of the Earth
Don L. Anderson
Chapter 16. Phase Changes and Mantle Mineralogy
Boston: Blackwell Scientific Publications, c1989
Copyright transferred to the author September 2, 1998.
You are granted permission for indi
vidual, educational, research and
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Recommended citation:
Anderson, Don L. Theory of the Earth.
Boston: Blackwell Scientific Publications,
1989.
http://resolver.calt
ech.edu/CaltechBOOK:1989.001
A scanned image of the entire book may
be found at the following persistent
URL:
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Abstract:
The densities and seismic velocities of
rocks are relatively weak functions of
temperature, pressure and compositio
n unless these are accompanied by a
drastic change in mineralogy. The physical
properties of a rock depend on the
proportions and compositions of the vari
ous phases or minerals-the mineralogy.
These, in turn, depend on temperature,
pressure and composition. In general,
one cannot assume that the mineralogy is
constant as one varies temperature and
pressure. Lateral and radial variations of
physical properties in the Earth are
primarily due to changes in mineralogy. To interpret seismic velocities and
density variations requires information
about both the stable phase assemblages
and the physical properties of minerals.
se Changes
It
is my opinion
that
the Earth
is very
noble and
admirable
.
. .
and
if
it had
contained an immense globe
of
crystal,
wherein
nothing
had
ever
changed,
I
should have esteemed it
a wretched
lump
of
no
benejt
to
the
Universe.
T
he
densities
and
seismic velocities
of
rocks are
rella-
tively
weak
functions
of
temperature, pressure
and
composition unless these are accompanied
by
a drastic
change
in
mineralogy.
The physical properties
of
a rock
de
-
pend on the proportions
and
compositions
of
the various
phases or
minerals
-
the
mineralogy.
These,
in
turn,
de
-
pend on temperature, pressure
and
composition. In general,
one cannot assume that
the mineralogy
is constant
as
one
varies
temperature
and
pressure. Lateral
and
radial varia
-
,
tions
of
physical properties
in
the Earth are primarily
due
to changes
in
mineralogy.
To
interpret seismic velocities
and
density variations requires information
about both the
stable
phase
assemblages
and the
physical properties
of
minerals.
SPHERICAL
IONS
AND
CRUSTAL
S
It
is often useful
to think
of
a crystal as a
packing
of
differ
-
ent size spheres, the
small
spheres occupying interstices
in
a framework
of
larger ones. In ionic crystals each ion can
be
treated as a
ball
with
certain radius
and
charge
(Table
16
-
1). The arrangement
of
these balls, the crystal structure,
follows certain simple rules. The crystal
must
contain ions
in
such
a ratio
so
that the crystal
is
electrically neutral.
Maximum
stability is associated
with
regular arrangements
that place as
many
cations around anions
as
possible,
and
vice versa, without putting ions
with
similar charge closer
together
than
their radii
allow
while
bringing
cations
and
anions as close together as possible.
In
other words,
we
pack
the balls together
as
closely
as
possible considering
their size
and
charge.
Many
crystals are
based
on cubic
close
packing or
hexagonal close
packing
of
the
larger ions.
It is a simple matter
of
geometry
to
calculate the ratio
of
the radii
of
two
types
of
spheres,
A
and
B,
that
permits
a certain number
of
B to
fit
around
A,
and vice
versa.
If A
is very
small
compared
to
B,
taken
as
the anion,
only
two
B
ions
can
be
arranged to touch
A,
and
the coordination
number
is 2.
When
A reaches
a critical size,
three touching
B ions can surround
it in
a trigonal
-
planar group, the
only
regular threefold coordinated structure. The limiting
value
of
the radius ratio for
this packing
is
RAIRB
=
0.155
As
A grows
still further we
reach
the point
where
A can
be
surrounded
by
four
B
as
in
the
SiOi-
tetrahedron, the
basis
for
many silicates. The
R,/R,
range
for
this
arrangement is
0.225
-
0.414.
For
RAIRB
between
0.414
and
0.732,
A can
be
surrounded
by
four B
in
a square planar arrangement
or
six
B
in
an
octahedron arrangement.
Rocksalt
structures
such
as
NaCl
and
MgO
exhibit octahedral coordination, a
common
substructural element
in
silicates. Low
-
pressure
minerals commonly
have
silicon
in
tetrahedral coordination
and
the
metals (such
as
Mg,
Fe,
Ca,
Al)
in
octahedral co
-
ordination. High
-
pressure phases,
such
as
Si0,-stishovite,
MgSi0,-ilmenite
and
perovskite,
have
silicon
in
octahedral
coordination. The mineral
majorite,
v1Mg3
vl[MgSil
IVSi3Ol2
has
the silicons split
between
octahedral
and
tetrahedral
sites.
When
R,
is
almost
the size
of
R,
(RAIR,
=
0.732-
1.0),
the cation
(A)
can be
surrounded
by
eight
anions (B)
as in
a square bipyramid. The anions are at
the
corners
of
a
TABLE
16
-
1
Ionic Radii
for
Major Mineral
-
Forming Elements
(
A
)
Coordination
Ionic
Coordination
Ionic
Ion
Number
Radius
Ion
Number
Radius
~1
3+
IV
0.39
Fe
3+
IV
0.49(HS)
*
V
0.48
VI
0.55(LS)
VI
0.53
VI
0.65(HS)
Ca2+
VI
1
.OO
Mg
2+
IV
0.49
VII
1 .07
VI
0.72
VIII
1.12
VIII
0.89
IX
1.18
Fe
2+
IV
0.63(HS)
X
1.28
VI
0.61
(LS)
XI1
1.35
VI
0.77(HS)
Si
4+
IV
0.26
Ti
4+
V
0.53
VI
0.40
VI
0.61
Na
+
VI
1.02
K
+
VI
1.38
VIII
1.16
VIII
1.51
0
2-
I1
1.35
F-
I1
1 .29
I11
1.36
111
1.30
IV
1.38
IV
1.31
VI
1.40
VI
1.33
VIII
1.42
C1-
VI
1.81
*HS,
high
spin;
LS,
low
spin.
cube,
and the cation is
in
the center
of
the
cube.
CsCl
struc
-
tures
fall
in this category.
The
garnet
structure has
eight
M
2
+
ions about
each oxygen.
If
R,
=
R,,
twelve
B ions
can
surround
each A
ion in
close
packing.
The
ions can
be
arranged in
two ways,
hex
-
agonal close packing or
cubic close packing, but the coor
-
dination number
is
12
in either
case. The
ideal
perovskite
structure exhibits twelvefold coordination
of
the
M
2
+
ions
around
the
oxygen
ions.
When
A
and
B
have
the same charge,
the
anion and
cation
coordinations are
the
same because
of
the
require
-
ment
of
charge
neutrality. For AB,
compounds
neutrality
necessitates
that the coordination
numbers not
be
the
same.
Higher coordination makes
for
denser packing,
and
when ionic
crystals are compressed, structures with greater
coordination numbers tend to
form.
Common crustal
and
upper
-
mantle
minerals,
however, have such an
open
pack
-
ing
structure that
rearrangements
(phase changes)
not in
-
volving coordination changes usually
occur, leading
to
a
more
efficient packing
of
ions,
before
the
coordination-
changing
transformations can
take
place.
High
temperature
tends to
decrease
the
coordination.
In
the context
of
rigid spherical
ions, the
increase
of
coordination with pressure means that
RJR,
must
increase,
necessitating
a decrease
of
R,,
say the oxygen
ion,
or
an
increase in
RA.
The
A
-
B
distance increases as
the
coordi
-
nation
increases, since
more
A
ions must
fit
around the
B
ion,
and this is usually assigned to an increase in the
cat-
ionic
radius.
The
increase in
density is
due to
the decrease
in
B
-
B
distances from closer
packing
of
the
oxygen ions.
The
increase
of
A
-
B
and
the decrease in
B
-
B
means that
the
attractive
potential
is
decreased and
the repulsive,
or
overlap,
potential
is increased. This leads
to
an
increase
in
the
bulk modulus or incompressibility.
To
a first
approximation, then,
ionic
crystal structures,
such as
oxides
and
silicates, consist
of
relatively
large ions,
usually
the oxygens,
in
a closest
-
pack
arrangement
with
the
smaller
ions filling
some of the interstices.
The
large
ions
arrange
themselves
so
that
the
cations
do
not
"
rattle
"
in
the
interstices.
The
"
nonrattle
"
requirement
of
tangency be
-
tween ions is
another
way of
saying that ions pack
so
as
to
minimize
the
potential
energy
of
the crystal.
In
addition
to
geometric
rules
of
sphere
packing and
overall
charge
neutrality, there
are additional
rules
gov
-
erning
ionic
crystals
that have been codified by Linus
Pauling. The
considerations
discussed
so far are equivalent
to
Pauling's
first
rule. The
second
rule states that
an
ionic
structure
will be
stable
to the extent
that the sum
of
the
strengths
of
the
electrostatic bonds that reach
an
anion from
adjacent
cations equals the charge
on
the
anion.
This is the
electrostatic
valence principle
or
condition
of
local
charge
neutrality.
In general,
in
a stable ionic
crystal the charge
on
any
cation is
neutralized by adjacent
anions. Cations
with
large charges
must therefore have high
coordination num
-
bers and tend
to
occur in the large
interstices
or
holes in
the
structure.
On
the other
hand highly
charged ions are
usual'
SPHERICAL
IONS
AND
CRYSTAL
STRUCTURE
339
small
in
radius
and
thus,
on
the
basis
of
radius
ratio,
seek
to
occupy the small holes.
Which
tendency
wins
depends
on other rules.
Pauling's
third
rule states that
the
sharing of edges
and
particularly
of
faces
by two
anion polyhedra decreases
the stability
of
the crystal structure.
By
this rule, highly
charged cations
prefer to maintain
as large
a separation as
possible
and to have
anions intervening
between them
so as
to
screen them from each other. This deceases
a crystal's
potential energy
by minimizing
the replusive forces existing
between
nearby cations. Multivalent cations
tend
to
avoid
the face
-
sharing anion cubes
of
the CsCl structure
and pre
-
fer the edge
-
sharing
NaCl
structure.
The fourth rule,
an
extension
of
the third, states that
in
crystal structures containing
different
cations,
those
of
high
valency and small
coordination number
tend not to
share
polyedron elements
with
each other.
The
fifth
rule
states that
the number
of
essentially dif
-
ferent kinds
of
constituents in
a crystal tends to
be
small;
that
is,
the number
of
types
of
interstitial sites in
a periodi
-
cally regular packing
of
anions tends
to be
small.
These considerations can
be
used
to
understand
the
stability
of
crystal lattices.
For
example, magnesiowiistite,
(Mg,Fe)O,
is
a 6
-
coordinated phase,
making
this
a
low-
pressure structure.
Packing
is relatively inefficient,
having
a very large
volume per
oxygen
ion
relative to other mantle
minerals. However
(Mg,Fe)O
is stable
to
extremely
high
pressure, probably
through most
of
the
lower
mantle. The
radius ratio
of
MgO
is 0.51,
putting it
well within
the range
(0.41
-
0.73)
of
expected octahedral coordination. The
CsCl
structure,
with
8
-
coordination, is displayed
by
many
alkali
halides
and
is the high
-
pressure form
of
others that
nor
-
mally
display the
rocksalt
structure. The
packing
of
eight
cations around
an anion
occurs for
RJR,
greater than
0.732.
CsCl
itself, for example,
has
a ratio
of
0.93,
al
-
though at
high
temperature it adopts the
NaCl
structure.
Likewise
RbCl,
radius
ratio
of
0.81,
crystallizes
in
the
NaCl
structure
at
low
pressure
and
CsCl at
high
pressure,
being
close to the boundary
of
the radius
ratio
for these
structures.
If
MgO
were to
adopt the CsCl structure, the
O2-
ions
would be
in contact at the cube faces
and
the
Mg
2
+
ions
would be unshielded
across cube faces.
Each
cube
would
share
a face
with
six others. This
makes
the
CsCl
structure unattractive to multicharged ions,
such
as
Mg
2
+,
and
pressure apparently
is unable to force
MgO
to
bring its
ions into closer proximity
to
achieve
a closer packing. The
radius
ratio
V1llMg/O
is
0.63,
still outside the range
for a
CsCl structure.
In
the
NiAs
structure, another alternative
AB
structure, the octahedra share faces, whereas
in the
NaCl
structure
they
share only edges. Consequently,
the
NiAs
structure is
not favored by
ionic crystals. Thus,
MgO
has
little option
but
to
remain
in the
rocksalt
structure, in spite
of
the relatively open structure.
Garnet is another mantle mineral that is stable
over
a
large pressure range. The garnet structure consists
of
inde
-
pendent
SiO,
and
A10,
polyhedra,
which
share corners to
form
a framework
within which
each
M
2
+ ion
is surrounded
by an
irregular polyhedron (a distorted cube)
of
eight
oxy
-
gen
atoms.
In
Mg3A12Si,012
(pyrope)
two
edges
of the
sili
-
con
tetrahedron
and
six
edges
of
the
aluminum octahedron
are
shared with
the
magnesium
cube,
leaving
four
unshared
edges
in
the tetrahedron,
six
in
the
octahedron
and
six
in
the
cube.
The
high
percentage
of
shared
edges leads
to
a
tightly
packed
arrangement,
a high
density
and
an
appar
-
ently
stable
lattice.
In
deference to Pauling,
most
edges are
unshared. The
packing
of
oxygen atoms is so
efficient that
the
volume per
oxygen atom
(15.7
A3)
is less
than
in
most
other high
-
pressure silicates except ilmenite
(14.6),
perov-
skite (13.5)
and
stishovite
(1
1.6).
The
M
2
+
-
coordination
in
garnet is
8,
so
garnet can
be considered
a high
-
pressure
phase.
Because
of
its
relatively
low
density,
it probably
re
-
mains
in the upper mantle. This
is an
argument against
eclo-
gite subduction
into
the
lower
mantle.
The
polyhedra
in
garnet are considerably distorted,
giving
a wide range of
Mg-0
(in pyrope)
and
0
-
0
dis
-
tances
and
requiring
a
large
unit
cell (eight
units
of
Mf
+Mj+
(Si04)3
in
a cubic
unit
cell). This distortion
re
-
flects Pauling's
admonishment against edge sharing involv
-
ing
highly charged ions.
Local
charge balance is
a factor
in
the structure
of
pyrope,
Mg3Al2Si3OI2,
and
other garnets.
Each
O2-
ion bonds
to one
Si
4
+,
an
A1
3
+ and
two
Mg
2
+
ions. The
total of the
electrostatic
bonds
leading to
an
02-
equals
+
2.
If
Mg
2
+ were
to
occupy the
octahedral sites,
such local charge
balance would
be
impossible. The elastic
properties
of
silicate garnets are relatively insensitive to
the
nature
of
the
Vn1M2+
ion.
The garnet structure is particularly important
since
[MgSi]
and
[FeSi]
can
substitute for
[All,
at
high
pressure,
giving
a majorite
-
garnet solution
that
may be
a dominant
phase in the transition region. The large, high
-
coordination
site in garnet allows the garnet structure to accommodate
a
wide
variety
of
cations
including minor
elements
and
ele
-
ments
that are
usually termed
incompatible, particularly the
heavy
rare
-
earth elements (HREE). The crystallization
of
garnet from
a magma
can
remove
these elements, giving
a
diagnostic
HREE
depletion signature to such
magmas.
Atoms
are particularly close
packed
in
body
-
centered
(bcc)
and face
-
centered cubic
(fcc)
structures.
In
a
body-
centered cubic crystal each atom
has
eight neighbors,
and
in
a face
-
centered cubic crystal each
has twelve
neighbors.
The atoms
can
be
more
closely
packed
in
the face
-
centered
structure. The distance
between neighboring
atoms
in
the
body
-
centered
case
is
afi/2,
and in
the face
-
centered case
it is
ala
where
the
volume
of
the
unit
cell is
a
3
/2
and
a
3
/4,
respectively. Assuming spherical ions
with
radii equal
to
half
these interatomic distances, the
volumes
of
the
spheres are
fi7ra3/16
and
7ra3/(122/Z)
for
bcc
and
fcc.
The fraction
of
the unit cell occupied
by
spherical ions
is
0.68
(bcc)
and
0.742
(fcc).
The
fcc
structure has the spheres
packed
as
closely
as
possible.
A
rigid sphere
can
be
sur-
rounded
by
twelve equally spaced
neighbors since in this
case each
sphere
touches all
of
its
neighbors.
There are
two
ways
in which
one
plane
of
close
-
packed
spheres
can
fit
snugly
on
top
of
a similar plane. One
gives
the
fcc
structure,
and
the other
gives
hexagonal close
pack
(hcp) structure.
If a
is the distance
between
atoms
arranged
in
a hexagon
on
a
plane
and
c is the distance to the
next
plane above
or
below,
the
distance to the nearest neighbor
out
of
plane
is
(a
2
/3
+
~~14)~~~.
For
closest packing
this
equals
a,
giving
cla
=
1.633.
The
volume
of
the
unit
cell
is
a2cd/2
so that
the
volume per
atom is
a2cfi/4.
De
-
partures
from the
ratio
cla
=
1.633
represent departures
from
the
closest
pack.
In
simple cubic
packing
(scp)
an ion sits at
each corner
of
a cube. This
is an
open structure,
and
unusual properties
might
be
anticipated
compared
to close
-
packed structures.
The
fluorite
structures
(CaF,)
and
CsC1-structures
(CsCl,
CsI,
TlC1)
are
based on scp
of
the anions. Cations are at the
center
of
every cube for
CsCl
and
every other cube for
CaF,,
the
cations
being
surrounded in each case
by
eight
anions.
When
the cations
approach
the size
of
the anions,
the structure resembles
bcc
in
its
overall
packing
for the
CsCl
structures.
In the
rocksalt
or
NaCl
structure the cations
by
them
-
selves,
or
the anions, lie
on
a face
-
centered cubic lattice.
In
NaCl, for example,
the sodium
ions lie
in
an
fcc
lattice
and
chlorine ions
are half
-
way between
the
Na
ions
at the cen
-
ters
of
the
cube edges
and
at the center
of
the cube. The
alkali
halides and
oxides
of
magnesium, calcium, strontium
and
barium have the
NaCl
structure. The
CsCl
structure is
not
particularly important
in
mantle mineralogy.
In the
rutile
(TiO,)
structure each cation
has
six anion
neighbors,
two
in
the
plane
above,
two
in
the same plane,
and two
in
the plane
below.
Each anion
has three cation
neighbors. Stishovite is
a high
-
pressure form
of
SiO,
having
the
rutile structure. Stishovite also forms
by
the
dispropor-
tionation
of
2MgSi03
to
Mg2Si04(P
or
y)
plus
Si02(st).
As
we
go to
more
complex compounds,
we
have
a large
variety
of
possible crystal structures, but
many
of
the
more
important ones are
based
on
relatively simple packing
of
the
oxygen ions
with
the generally smaller cations fitting into
the
interstices. Some cubic minerals mimic the structures
of
perovskites
(CaTiO,),
spinels
(A12Mg04)
or
garnets
(such
as
Mg3AI2Si3Ol,-pyrope).
The
perovskite
structure,
M2+N3+o3,
has
all the at
-
oms arranged
in a cubic lattice
with
the
M
2
+ at the corners,
the
N
3
+ at the center
and
the
0's
at the
face centers. Gen
-
erally, the
M
2
+ ions (say
Mg
2
+)
and
the oxygen ions to
-
gether constitute
a cubic close
pack
structure.
All
the inter
-
atomic
distances are determined
in
terms
of
one parameter,
the side
a
of
the
unit
cell. The
M
2
+
-
0, N
3
+
-
0 and
M
2
+
-
M
3
+ distances are approximately
a/*,
a12
and
afi12,
respectively.
These are also approximately the sum
of
the
appropriate
ionic
radii.
In
MgSi0,-perovskite
there is
a considerable
range
in
the
individual distances. Although
each
magnesium
is surrounded
by
twelve
oxygens,
the
IMg-0
distances are
not
all
the
same. There
is
therefore
a
tendency
of
the structure to distort
and not
be
exactly cubic.
Some perovskites are ferroelectric: The displacements
of
the
ions from the positions that
they would have
in the cubic
structure results
in
a permanent electric dipole for the crys
-
tal.
If
the
M
2
+ ion were the
same size
as
the oxygen ions
and
precisely
fit
its twelvefold
site,
then
the line joining
the
centers
of
the oxygens
would
equal
twice
the sum
of
the
ionic radii or
1.414
times
the cube's
edge.
The
cube's
edge
in
turn equals twice the
sum
of
the oxygen
and
M
4
+ radii.
The ideal relationship
between
radii for ions
in
the
perov-
skite structure
is
R(0)
+
(M
2
+)
=
1.414
(R(0)
+
R(M
4
+)t)
with
t
=
1
(the tolerance factor).
In
perovskites
t
generally lies
between
0.8
and
1
.O.
"
High
-
temperature
"
superconductors
have
the perovskite structure,
with
con
-
ducting
layers
alternating
with
resistive
layers
of
atoms.
Spinel,
A12Mg04,
is
an
example
of
a large class
of
important
compounds, including ferrites, that
have
impor
-
tant
magnetic properties. There are eight magnesium ions
per
cube
of
side
a.
They occupy
the centers
of
four
out
of
the eight small cubes
of
side
a12
into
which
the larger
cube can
be
divided.
Each
of
the other four small cubes
contains four
aluminum
ions. There are
then
16
aluminum
ions in the cube
of
side
a.
Each aluminum
is surrounded
by
six
oxygens,
and
each
magnesium
is attached to only four
oxygens,
an
unusual coordination for magnesium.
Not
all
spinels
have
these site assignments for atoms
of
different
valencies.
In
some,
half
of
the trivalent atoms are located
in
tetrahedral sites,
and
the other
half
of
the trivalent atoms
and
the
divalent
atoms are distributed in the octahedral
sites. These are called inverse spinels. Examples
of
inverse
spinels include
the
ferrites
Fe2Mg04
and
Fe304.
The
spinel
structure
is
essentially
a cubic close
pack
(fcc)
of
oxygen
ions
with metal
cations occupying one
-
eighth
of
the tetra
-
hedral sites
and one
-
half
of
the octahedral sites.
y-Mg2Si04
can
be
viewed
approximately as the substitution
of
Mg
2
+
and
Si
4
+
for
A1
3
+
and
Mg
2
+
.
Garnets are
also
cubic minerals,
and
some
have
impor
-
tant magnetic
and
optical properties.
For
silicate garnets
(M
2
+
=
Mg, Ca,
Fe
.
.
.)
the
unit
cube contains
24
M
2
+
ions,
16
aluminums,
24
silicons and 96 oxygens. The co
-
ordinations are (for
M
2
+
=
Mg)
Many
substitutions are possible for
all
the cations,
and
gar
-
nets
in
the crust
and
mantle
are important repositories
for
trace elements, particularly
those having
ionic radii similar
to magnesium, calcium
and
aluminum. The tetrahedral sili
-
cate groups, four oxygens tetrahedrally arranged around
a
central silicon atom, are independent
of
each other. Garnet
is therefore called
an
island silicate. The elastic properties
of
garnets are
almost
independent
of
the nature
of
the
M
2
+
ion,
in
contrast
to
other silicates.
Hexagonal
and
trigonal crystals are closely related.
Calcite
(CaCO,)
and
corundum
(A1203)
are trigonal crys
-
tals,
and one
high
-
pressure form
of
MgSiO,
is similar to
ilmenite
(FeTiO,),
another trigonal crystal.
In
the calcite
structure each
M
2
+ atom is
bonded
to six
oxygens,
and
each
oxygen
is bonded
to one
M
4
+ and
two
M
2
+ atoms.
We
may
think
of
the
calcite structure
as
a distorted
NaCl
structure.
The oblate
C03
group replaces the spherical chloride ion.
Calcite
is
the
main
constituent
of
the metamorphic
rock
marble, a rock
with
strongly anisotropic properties because
of
the alignment
of
the
individual calcite crystals.
The
strong alignment
of
calcite
and
of
ice,
a hexagonal crystal,
in
natural masses suggests that
MgSi0,-ilmenite
will also
be strongly aligned in the mantle.
In
corundum
(A120,)
the oxygens occur on equilateral
triangles, similar to calcite,
but
there is
no
atom at the
cen-.
ter
of
the triangle, in the same plane. The aluminum atoms
are
not
all
in
a plane. Each aluminum
is surrounded
by
six
oxygens.
Cr203
mixes
in
all proportions
with
A1203,
Cr
3
+
and
A1
3
+ having similar radii,
and
the mixture
yields
the
gem
ruby with its
characteristic
red
color. It
has
important
optical properties.
A1203
may
be
important
in
the
lower
mantle. It is found
in
some
kimberlites.
In
ilmenite
(FeTiO,)
half
the aluminums are replaced
by
iron,
half
by
titanium. This breaks the symmetry,
and
ilmenites are expected to
be more
anisotropic
than
comn-
dums.
The
oxygens in corundum
and
ilmenite structures are
in
approximate hcp.
Some
of
the
more common
low
-
pressure silicates are
also
based
on
simple packing
of
the
oxygen
ions.
In
olivine,
for example, the oxygens are in approximate hcp.
The
transformation to the spinel form results in a slight decrease
in the
Mg-0
distance, a slight increase
in the
Si-0
distance
and
a decrease
in
the larger
of
the
0
-
0
near
-
neighbor dis
-
tances, resulting in
an
8
percent decrease in the
volume
per
oxygen
and
a change in oxygen
packing
from approxi
-
mately hcp to approximately
bcc.
The coordinations
of
the
ions remain the same.
Note that
is not
analogous
to
true
spinel
either
in
the coordination
of
magnesium or the valency
of
the ions in the tetrahedral
and
octahedral sites. Aluminate
spinels
have some
anomalous elastic properties,
presum
-
ably
related to the
IV
coordination
of
the
M
2
+ ions,
which
cannot
be assumed to
carry
over
to the silicate spinels.
In
fact,
because
of
the very small size
of
the
IVMg
ion
it must
be treated as a
different
element
than
VIMg.
In
this chapter
we
use
mineral
names such
as
spinel,
ilmenite
and
perovskite
to
refer
to
structural analogs
in
sil
-
icates rather
than to
the minerals themselves. This
has
be
-
come conventional
in
high
-
pressure petrology
and
mineral
physics,
but
it can
be
confusing
to those
trained
in
conven
-
tional mineralogy
with
no exposure
to
the high
-
pressure
world.
Interatomic Distances
in
Dense
Silicates
The elastic properties
of
minerals depend on interatomic
forces
and hence
on
bond
type,
bond length and
packing.
As
minerals undergo phase changes, the ions are rearranged,
increasing the length
of
some bonds and
decreasing others.
The interatomic distances
and
the average
volume
per
oxy
-
gen
atom are
given
in Table
16
-
2
for
many
of
the
crystal
structures that occur
in
the mantle.
For
a given coordination
the cation
-
anion distances are relatively constant. This,
in
fact, is
the basis for ionic radius estimates. Cation
-
anion
distances increase
with
coordination, as required
by
pack
-
ing
considerations.
It is clear that
the
increases
of
density
and
bulk
modulus,
K,,
are controlled
by
the increase in
packing efficiency
of
the oxygen ions.
TABLE
16
-
2
Average
Interatomic Distances
(Angstroms),
Vollume
Per
Oxygen Ion
(A31
and
Bulk
Modulus
(GPa)
in Mantle
Minerals
-
-
Mineral
or
Structure
Mg-0
Si
-
0
0
-
0
V/02-
K,
MgO
Olivine
Pyroxene
p-spinel
y
-
spinel
Ilmenite
Perovski
te
Stishovite
Garnet
*A]-0
distance
in
garnet
is
1.89.
*
*
4
shortest distances.
+Al-0
distance
is
1.91.
Crystals and Magmas
or,
for
the
very incompatible elements, remain in the
melt
to the
end,
coating the major crystals
with
exotic phases.
Up
to
this point
I
have
treated crystals as isolated entities.
Most
mantle crystals
are
formed from a melt,
and
it is in
-
structive
to
consider
them
from
this
point
of
view.
The dis
-
tribution
of
ions
and
the nature
of
the crystals formed de
-
pend
on properties
of
both
the melt
and
the solids. The
first
silicate crystals
to form from a cooling
magma usually have
the
Si
4
+ ions
as
widely
separated as possible,
in
accordance
with Pauling's
third
rule. Oxygen ions touch at
most one
Si
4
+ ion.
No
two
tetrahedra drawn about each
Si
4
+ share a
corner,
that
is,
an
oxygen. Crystal structures in
which
each
[SiO4I4
tetrahedron
is
isolated from all others are called
island
silicates
or
nesosilicates
(from
the
Greek
word
for
"
island
"
) or
orthosilicates.
Olivines
and
garnets are
such
structures.
Bloss
(1971)
gives
a good summary.
As
island
silicates crystallize from the melt, the re
-
maining liquid
is enriched in
Si
4
+,
permitting silicates
with
higher
Si
4
+ to
02-
ratios
to
form. The
Si
4
+ ions cannot
be
so
widely
spaced,
and
some
of
the oxygens touch
two
Si
4
+
ions.
P-spinel
has linked
SiO,
tetrahedra
and
is therefore a
double
-
island or
sorosilicate.
With
progressive crystalliza
-
tion
the
Sit0
ratio increases further,
and
Si
4
+ occurs
in
so
many
interstices
in
the crystals that
do
form that each tet
-
rahedron
shares a corner
with
two
others
to
form
[Si0312-
chains
as
in
the
pyroxenes,
which
are single
-
chain or
meta-
silicates.
With
increased cooling,
double
-
chain silicates
form
with
[Si,0,,I6-
units
and
a large number
of
shared
corners.
Amphiboles
are double
-
chain silicates.
At
even
lower
temperatures
the
Si
4
+ to
02-
ratio
is higher still,
and
the
(OH)-
to
02-
ratio is also high.
Layer silicates
such as
micas and
talc form under these conditions.
In some
structures, the
framework
silicates,
the pro
-
portion
of
Si
4
+ to
02-
is
S
O
high
that
all
tetrahedra share
all their corners. The various crystalline forms
of
SO2-
quartz, tridymite,
cristobalite
-
are
framework structures
that
crystallize at
low
temperature. The feldspars are also
framework
silicates,
but
A1
3
+ tetrahedra as
well
as
Si
4
+ tet
-
rahedra are involved, so
they
can also crystallize early.
Most of
the structures discussed
above have
relatively
open structures
and
are unstable
at
moderate pressures.
They also
tend
to
have low
seismic velocities and to
be
an
-
isotropic. The
increased packing efficiency
of
high
-
pressure
mineral
phases makes
their description
in
terms
of
arrange
-
ments
of
tetrahedra
and
octahedra less
useful than
a descrip
-
tion
in
terms
of
dominant sublattices
of
close
-
packed ions
with
the
remaining
ions occupying available interstices.
Nevertheless,
it is likely that
most
minerals
in
the mantle
have
crystallized from
magmas
at
low
pressure
and have
subsequently
converted
to high
-
pressure phases. Therefore,
it is of
interest
to
know
the conditions
that
initially deter
-
mine
the relative proportions
of
the various constituents
of
minerals at
low
pressure. The trace
and minor
elements fol
-
low
similar rules.
They tend to
replace major ions
of
similar
size
and
valency,
or occupy interstices
of
appropriate size
MINERALS AND PHASES
OF THE
MANTLE
As
far as physical properties
and
major elements are
concerned, the
most
important upper mantle minerals are
olivine, orthopyroxene, clinopyroxene
and
aluminum
-
rich
phases such
as
plagioclase, spinel
and
garnet.
Olivine
and
orthopyroxene are the
most
refractory phases
and
tend to
occur together,
with
only minor amounts
of
other phases,
in
peridotites. Clinopyroxene
and
garnet are
the most
fus
-
ible components
and
also
tend to
occur together as major
phases
in
rocks such
as
eclogites.
All
of
the
above
minerals are unstable
at
high pressure
and
therefore only occur in the upper part
of
the mantle.
Clinopyroxene, diopside
plus
jadeite,
may be
stable to
depths as great as 500
km.
Olivine transforms successively
to
P-spinel,
a distorted spinel
-
like structure, near 400
km
and
to y
-
spinel, a true cubic spinel, near 500
km.
At
high
pressure
it disproportionates to
(Mg,Fe)SiO,
in
the
perov-
skite structure
plus
(Mg,Fe)O,
magnesiowiistite,
which has
the
rocksalt
structure. The
FeO
component is strongly par
-
titioned into the
(Mg,Fe)O
phase.
Orthopyroxene,
(Mg,Fe)SiO,,
transforms
to
a dis
-
torted garnet
-
like phase, majorite,
with
an
increase
in
co
-
ordination
of
some
of
the
magnesium and
silicon:
vlll
(Mg
,Fe)
3V1MgV1SiV1Si30,2
where
the
Roman
numerals signify the coordination. This
can
be viewed
as
a garnet with
MgSi
replacing the
Al,.
This
is
a high
-
temperature transformation.
At
low
temperature
the following transformations occur
with
pressure:
-+
Mg2Si0,(y-sp)
+
Si02(st)
-+
2
MgSiO,
(ilmenite)
-+
2
MgSiO,
(perovskite)
A different sequence occurs for
CaMgSi,O,
(diopside
cli-
nopyroxene). The ionic radius
of
calcium is
much
greater
than
aluminum,
and this
is expected
to
make the transi
-
tion pressure
to
the garnet structure
much
higher
than
for
orthopyroxene.
In
the presence
of
A1203,
or garnet, the pyroxene gar
-
nets form solid solutions
with
ordinary aluminous garnets,
the transition pressure decreasing
with
A1303
content.
The mineralogy
in
the transition region, at normal
mantle temperatures, is expected
to
be
P-
or y
-
spinel
plus
garnet solid solutions.
At
colder temperatures, as
in
sub
-
duction zones, the mineralogy at the base
of
the transition
region is probably y
-
spinel plus ilmenite solid solution.
The garnet component of
the mantle is stable to very
PHASE
EQUILIBRIA
IN
MANTLE
SYSTEMS
343
high
pressure, becoming,
however,
less aluminous
and
more siliceous as it dissolves the pyroxenes.
At
low
tem
-
perature
and
at pressures equivalent
to
those
in
the
lower
part
of
the transition region, the garnet
as well as
the
pyrox
-
enes are probably
in
ilmenite solid solutions.
The ilmenite structure
of
orthopyroxene can be re
-
garded as a substitution
of
VIMg
VISi
for
V1A12
in
the corun
-
dum
structure. The transformation
of
CaMgSi206
clino-
pyroxene to ilmenite,
if
it occurs, is probably a higher
pressure transition.
CaMgSizO,
may
transform to the
per-
ovskite structure
without an
intervening
field
of
ilmenite:
The
ionic radii
(in
angstroms)
of
some of
the
ions
in
-
volved
in the
above
reactions are
The ionic radius
of
V1[MgSi]
is similar to
VIA1,
and
therefore
a solid
-
solution series
between
garnet
and
majorite
at rela
-
tively
low
pressure is expected. The ionic radius
of
V1[CaSi]
is
much
greater,
and
therefore diopside
is expected to
re
-
quire higher pressures for its garnet transformation
unless
only the
Mg
2
+ enters the octahedral site. The replacement
of
Al,
for
Ca,,,
Mg,,Si
in
the perovskite structure is also
expected to
be
difficult, so
it is possible that clinopyroxene
-
garnet disproportionates to calcium
-
rich perovskite
plus
A120,,
the excess
Alz03
probably combining
with
MgSiO,
x
A1203
in the garnet, ilmenite or perovskite structure,
depending on pressure. The disparity
in
ionic radii
between
vnl-xlCa,
vlll-xnMg,
IVSi
and
VIA1
probably
means
that there
will be
three separate perovskite phases in the lower mantle:
Ca
-
rich,
(Mg,
Fe)-rich
and
Al
-
rich. The presence
of
A1,0,
decreases the density
of
perovskite
but
increases the density
of
ilmenite
and
garnet.
Ferrous
iron
(Fez+)
is expected to readily substitute for
Mg
2
+ in
all the
phases
discussed so far. The relative parti
-
tioning
of
Fe
z
+ amongst
phases,
however,
is expected to
vary
with
pressure. Garnet
is
by
the far
the most
Fe
2
+
-
rich
phase in the upper mantle,
followed by
olivine,
clinopyrox-
ene
and
orthopyroxene.
As
the pyroxenes dissolve
in
the
garnet
they
dilute the
FeIMg
ratio,
and
y
-
spinel
may
be
the
most
iron
-
rich phase
in
the lower part
of
the transition re
-
gion.
In
the lower
mantle
Fez+
favors
(Mg,Fe)O
over
per-
ovskite.
When
the
Fez+
high spin
-
low spin
transition oc
-
curs,
somewhere deep in the lower mantle, solid solution
between
Fez+
and
Mg
2
+ is probably
no
longer possible
be
-
cause
of
the disparity
in
ionic radii,
and
a separate
FeO-
bearing phase,
such
as
Fe(L.S.)O,
is likely.
At
high
pres
-
sure this is expected
to
dissolve extensively
in
any molten
iron that traverses
this region
on the
way
to the
core, or
to
be
stripped out
of
any
mantle that comes into contact
with
the core in the course
of
mantle convection.
An
Fe
-
O
-
poor
lower mantle is
therefore
a distinct
possibility. The
corol
-
lary
is
an
iron-FeO
core (see Chapter
4).
As
far
as
we
know
perovskite
and
MgO
are stable
throughout the
lower
mantle,
which
is consistent
with
seismic radial
homogeneity
of
most
of
the
lower
mantle.
MgSi0,-perovskite
is
the
most
abundant mineral
in
the
mantle. The seismic properties
of
the
lower
mantle
are
broadly
consistent
with
(Mg,Fe)
Si0,-perovskite,
although
other
phases
may be
present,
such
as
(Mg,Fe)
0.
In
general
FeO
decreases
the
pressure
of
phase
transi
-
tions
in
the mantle,
including
olivine-@-spinel,
@-y-spinel
and
pyroxene
-
garnet.
A1203
widens the
stability
fields
of
garnet
and
ilmenite.
Ca
2
+ is a large
ion
and
in
simple
com
-
pounds is expected
to cause
phase
transitions at
lower
pres
-
sures
than
the equivalent
Mg
2
+ compound.
However,
the
large size
of
Ca
2
+ makes
it difficult
to substitute for
A1
3
+
(in the coupled
CaSi
substitution for
Al,).
At
high
tempera
-
ture
clinopyroxene contains excess
Mg
2
+ compared
to pure
diopside,
Therefore, orthopyroxene
probably
reacts out
of
the mantle
at shallower depths
than
clinopyroxene.
The
difficulty of
substituting
V1[CaSi]
for
V1[Al,]
sug
-
gests the following structural formula for diopside
-
garnet:
This
has
the virtue
of
requiring no coordination change for
Ca
2
+ in
going
from the diopside
to garnet structure.
PHASE EQUILIBRIA
IN
MANTLE SYSTEMS
The lateral
and
radial variations
of
seismic
velocity
and
density depend, to
first
order, on
the
stable mineral
assem
-
blages
and,
to second
order, on the variation
of
the veloci
-
ties with temperature, pressure
and
composition.
Tempera
-
ture, pressure
and
composition dictate
the
compositions
and
proportions
of
the various phases. In order to interpret ob
-
served seismic
velocity
profiles, or to
predict
the velocities
for starting composition, one
must
know
both
the expected
equilibrium assemblage
and
the properties
of
the phases.
To
a first
approximation, olivine, orthopyroxene,
cli-
nopyroxene
and
an aluminous phase
(feldspar, spinel,
gar
-
net) are stable
in the
shallow mantle.
P-spinel,
majorite,
garnet
and
clinopyroxene are stable
in
the
vicinity
of
400
km,
near the top
of
the transition region. y
-
spinel,
majorite
or y
-
spinel
plus
stishovite,
Ca-perovskite,
garnet
and
il-
menite are stable
between about
500
and
650
km.
Garnet,
ilmenite,
Mg-perovskite,
Ca-perovskite
and
magnesiowiis-