Lecture 4 (Chapter 13 in Perkins) Crystal Chemistry Part 3: Coordination of Ions Pauling

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Lecture 4 (Chapter 13 in Perkins)Crystal
ChemistryPart 3 Coordination of
IonsPaulings RulesCrystal Structures
The packing animations below are due to John
Winter, click here
Get polyhedral models from cabinet
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Coordination of Ions

• For ionic bonding, ion geometry spherical
• Spherical ions will geometrically pack
  (coordinate) oppositely charged ions around them
  as tightly as possible while maintaining charge
  neutrality
• For a particular ion, the surrounding
  coordination ions define the apices (corners) of
  a polyhedron
• The number of surrounding ions is the
  Coordination Number

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Ionic Compound Formation

• Anions negatively charged
• Larger than the un-ionized atom
• Cations positively charged
• Smaller than the un-ionized atom
• Attraction
• Anion Cation
• Repulsion
• Anion Anion
• Cation Cation

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Coordination Number and Radius Ratio
Radius Ratio is Rc (cation) / Ra (anion)
See Figure 13.3 of Perkins
See also the Ionic Radii table of Perkins,
following the inside front cover
from KD
Modified from KD
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Atomic and Ionic Radii
Can't absolutely determine e- cloud is nebulous
based on probability of encountering an e- .
In crystalline solids the center-to-center
distance bond length is accepted to sum of
ionic radii How get ionic radius of X Y in XY
compound??
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Atomic and Ionic Radii
Pure element first Native Cu. Atomic radius
1/2 bond length
X-ray d100 ? a Ionic radius
a
2
2
a
a
4
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Atomic Radii

• Absolute radius of an atom based on location of
  the maximum density of outermost electron shell
• Effective radius dependent on the charge, type,
  size, and number of neighboring atoms/ions
• - in bonds between identical atoms, this is half
  the interatomic distance
• - in bonds between different ions, the distance
  between the ions is controlled by the attractive
  and repulsive force between the two ions and
  their charges

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Charge and Attractive Force Control on Effective
Ionic Radii
Approach until Repulsive and Attractive Forces
the same
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Effect of Coordination Number and Valence on
Effective Ionic Radius
Higher coordination numbers have larger effective
ionic radius Extreme valence shells (1,6,7) have
larger effective ionic radius
Increasing Ionic radii
Decreasing Ionic radii
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Coordination Number (CN) ( of nearest
neighbors) vs. ionic radius. For cations of one
element, higher coordination numbers have larger
effective ionic radius
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Coordination with O-2 Anions
Note Sulfur can have CN 6 at great depths For
example, in the inner core
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When Rc / Ra approaches 1 a close packed
arrayforms
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Coordination Polyhedra

• We always consider coordination of anions about a
  central cation

Halite
Na
Cl
Cl
Cl
Cl
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Coordination Polyhedra

• Can predict the coordination
• by considering the radius ratio
• RC/RA
• Cations are generally smaller than anions so
  begin with maximum ratio 1.0

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Coordination Polyhedra

• Radius Ratio RC/RA 1.0 (commonly native
  elements)

Equal sized spheres Closest Packed Notice6
nearest neighbors in the plane arranged in a
hexagon Note dimples in which next layer atoms
will settle Two dimple types Type 1 upper
point NE Type 2 upper point NW They are
equivalent since you could rotate the whole
structure 60o and exchange them
2
1
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Closest Packing
Add next layer (red) Once first red atom settles
in, can only fill other dimples of that type In
this case covered all type 2 dimples, only 1s
are left
1
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Closest Packing
Third layer ? Third layer dimples again 2
types Call layer 1 A sites Layer 2 B sites (no
matter which choice of dimples is occupied) Layer
3 can now occupy A-type site (directly above
yellow atoms) or C-type site (above voids in both
A and B layers)
A
C
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Closest Packing
Third layer If occupy A-type site the layer
ordering becomes A-B-A-B and creates a hexagonal
closest packed structure (HCP) Coordination
number (nearest or touching neighbors) 12 6
coplanar 3 above the plane 3 below the plane
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Closest Packing
Third layer If occupy A-type site the layer
ordering becomes A-B-A-B and creates a hexagonal
closest packed structure (HCP)
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Closest Packing
Third layer If occupy A-type site the layer
ordering becomes A-B-A-B and creates a hexagonal
closest packed structure (HCP)
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Closest Packing
Third layer If occupy A-type site the layer
ordering becomes A-B-A-B and creates a hexagonal
closest packed structure (HCP)
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Closest Packing
Third layer If occupy A-type site the layer
ordering becomes A-B-A-B and creates a hexagonal
closest packed structure (HCP) Note top layer
atoms are directly above bottom layer atoms
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Closest Packing
Third layer Unit cell
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Closest Packing
Third layer Unit cell
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Closest Packing
Third layer Unit cell
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Closest Packing
Third layer View from top shows hexagonal unit
cell (HCP)
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Closest Packing
Third layer View from top shows hexagonal unit
cell (HCP)
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Closest Packing
Alternatively we could place the third layer in
the C-type site (above voids in both A and B
layers)
C
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Closest Packing
Third layer If occupy C-type site the layer
ordering is A-B-C-A-B-C and creates a cubic
closest packed structure (CCP) Blue layer atoms
are now in a unique position above voids between
atoms in layers A and B
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Closest Packing
Third layer If occupy C-type site the layer
ordering is A-B-C-A-B-C and creates a cubic
closest packed structure (CCP) Blue layer atoms
are now in a unique position above voids between
atoms in layers A and B
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Closest Packing
Third layer If occupy C-type site the layer
ordering is A-B-C-A-B-C and creates a cubic
closest packed structure (CCP) Blue layer atoms
are now in a unique position above voids between
atoms in layers A and B
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Closest Packing
Third layer If occupy C-type site the layer
ordering is A-B-C-A-B-C and creates a cubic
closest packed structure (CCP) Blue layer atoms
are now in a unique position above voids between
atoms in layers A and B
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Closest Packing
Third layer If occupy C-type site the layer
ordering is A-B-C-A-B-C and creates a cubic
closest packed structure (CCP) Blue layer atoms
are now in a unique position above voids between
atoms in layers A and B
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Cubic Closest Packing
View from the same side shows the cubic close
packing (CCP), also called face-centered
cubic (FCC) because of the unit cell that
results. Notice that every face of the cube has
an atom at every face center. The atoms are
slightly shrunken to aid in visualizing the
structure
A-layer
C-layer
B-layer
A-layer
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Closest Packing
Rotating toward a top view
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Closest Packing
Rotating toward a top view
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Closest Packing
You are looking at a top yellow layer A with a
blue layer C below, then a red layer B and a
yellow layer A again at the bottom
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What happens when RC/RA decreases? The center
cation becomes too small for the C.N.12 site (as
if a hard-sphere atom model began to rattle in
the 12 site) and it drops to the next lower
coordination number (next smaller site). It
will do this even if it is slightly too large for
the next lower site. It is as though it is
better to fit a slightly large cation into a
smaller site than to have one rattle about in a
site that is too large.
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The next smaller crystal site is the CUBE
Body-Centered Cubic (BCC) with cation (red) in
the center of a cube Coordination number is now
8 (corners of cube)
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A central cation will remain in 8 coordination
with decreasing RC/RA until it again reaches the
limiting situation in which all atoms mutually
touch.
Then a hard-sphere cation would rattle in the
position, and it would shift to the next lower
coordination (next smaller site). What is the
RC/RA of that limiting condition??
Set 1
arbitrary since will deal with ratios
Diagonal length then 2
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A central cation will remain in 8 coordination
with decreasing RC/RA until it again reaches the
limiting situation in which all atoms mutually
touch.
Then a hard-sphere cation would rattle in the
position, and it would shift to the next lower
coordination (next smaller site). What is the
RC/RA of that limiting condition??
Rotate
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A central cation will remain in 8 coordination
with decreasing RC/RA until it again reaches the
limiting situation in which all atoms mutually
touch.
Then a hard-sphere cation would rattle in the
position, and it would shift to the next lower
coordination (next smaller site). What is the
RC/RA of that limiting condition??
Rotate
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A central cation will remain in 8 coordination
with decreasing RC/RA until it again reaches the
limiting situation in which all atoms mutually
touch.
Then a hard-sphere cation would rattle in the
position, and it would shift to the next lower
coordination (next smaller site). What is the
RC/RA of that limiting condition??
Rotate
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A central cation will remain in 8 coordination
with decreasing RC/RA until it again reaches the
limiting situation in which all atoms mutually
touch.
Then a hard-sphere cation would rattle in the
position, and it would shift to the next lower
coordination (next smaller site). What is the
RC/RA of that limiting condition??
Rotate
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A central cation will remain in 8 coordination
with decreasing RC/RA until it again reaches the
limiting situation in which all atoms mutually
touch.
Then a hard-sphere cation would rattle in the
position, and it would shift to the next lower
coordination (next smaller site). What is the
RC/RA of that limiting condition??
Rotate
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A central cation will remain in 8 coordination
with decreasing RC/RA until it again reaches the
limiting situation in which all atoms mutually
touch.
Then a hard-sphere cation would rattle in the
position, and it would shift to the next lower
coordination (next smaller site). What is the
RC/RA of that limiting condition??
Rotate
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A central cation will remain in 8 coordination
with decreasing RC/RA until it again reaches the
limiting situation in which all atoms mutually
touch.
Then a hard-sphere cation would rattle in the
position, and it would shift to the next lower
coordination (next smaller site). What is the
RC/RA of that limiting condition??
Rotate
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A central cation will remain in 8 coordination
with decreasing RC/RA until it again reaches the
limiting situation in which all atoms mutually
touch.
Central Plane
What is the RC/RA of that limiting
condition?? 1.732 dC dA If dA 1 then
dC 0.732 dC/dA RC/RA 0.732/1 0.732
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The limits for 8 coordination are thus between
1.0 (when it would be CCP or HCP) and 0.732
Note Body Centered Cubic is not a
closest-packed oxygen arrangement.
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As RC/RA continues to decrease below the 0.732
the cation will move to the next lower
coordination 6, VI, or octahedral. The cation is
in the center of an octahedron of closest-packed
oxygen atoms
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As RC/RA continues to decrease below the 0.732
the cation will move to the next lower
coordination 6, VI, or octahedral. The cation is
in the center of an octahedron of closest-packed
oxygen atoms
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As RC/RA continues to decrease below the 0.732
the cation will move to the next lower
coordination 6, VI, or octahedral. The cation is
in the center of an octahedron of closest-packed
oxygen atoms
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As RC/RA continues to decrease below the 0.732
the cation will move to the next lower
coordination 6, VI, or octahedral. The cation is
in the center of an octahedron of closest-packed
oxygen atoms
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As RC/RA continues to decrease below the 0.732
the cation will move to the next lower
coordination 6, VI, or octahedral. The cation is
in the center of an octahedron of closest-packed
oxygen atoms
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As RC/RA continues to decrease below the 0.732
the cation will move to the next lower
coordination VI, or octahedral. The cation is in
the center of an octahedron of closest-packed
oxygen atoms
What is the RC/RA of that limiting
condition?? 1.414 dC dA If dA 1 then
dC 0.414 dC/dA RC/RA 0.414/1 0.414
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As RC/RA continues to decrease below the 0.414
the cation will move to the next lower
coordination 4, IV, or tetrahedral. The cation
is in the center of an tetrahedron of
closest-packed oxygen atoms
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As RC/RA continues to decrease below the 0.414
the cation will move to the next lower
coordination 4, IV, or tetrahedral. The cation
is in the center of an tetrahedron of
closest-packed oxygen atoms
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As RC/RA continues to decrease below the 0.414
the cation will move to the next lower
coordination 4, IV, or tetrahedral. The cation
is in the center of an tetrahedron of
closest-packed oxygen atoms
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As RC/RA continues to decrease below the 0.414
the cation will move to the next lower
coordination 4, IV, or tetrahedral. The cation
is in the center of an tetrahedron of
closest-packed oxygen atoms
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As RC/RA continues to decrease below the 0.414
the cation will move to the next lower
coordination 4, IV, or tetrahedral. The cation
is in the center of an tetrahedron of
closest-packed oxygen atoms
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As RC/RA continues to decrease below the 0.414
the cation will move to the next lower
coordination 4, IV, or tetrahedral. The cation
is in the center of an tetrahedron of
closest-packed oxygen atoms
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As RC/RA continues to decrease below the 0.414
the cation will move to the next lower
coordination 4, IV, or tetrahedral. The cation
is in the center of an tetrahedron of
closest-packed oxygen atoms
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As RC/RA continues to decrease below the 0.414
the cation will move to the next lower
coordination IV, or tetrahedral. The cation is
in the center of an tetrahedron of closest-packed
oxygen atoms
What is the RC/RA of the limiting
condition?? Center-to-corner distance of a
tetrahedron with edges of 1.0 0.6124 RC
0.6124 - 0.5 0.1124 RC/RA 0.1124/0.5
0.225
See derivation fig 4.3 c page 70
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As RC/RA continues to decrease below the 0.22 the
cation will move to the next lower coordination
III. The cation moves from the center of the
tetrahedron to the center of an coplanar
tetrahedral face of 3 oxygen atoms
What is the RC/RA of the limiting
condition?? cos 60 0.5/y y 0.5774 RC
0.5774 - 0.5 0.0774 RC/RA 0.0774/0.5
0.155
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If RC/RA decreases below 0.15 the cation will
move to the next lower coordination 2 or II. The
cation moves directly between 2 neighboring
oxygen atoms
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Paulings Rules

• Rule 1 A coordination polyhedron of anions is
  formed around each cation, where
• - the cation-anion distance is determined by the
  sum of the ionic radii, and
• - the coordination number of the polyhedron is
  determined by the cation/anion radius ratio
  (RcRa)

Linus Pauling
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Paulings Rules

• Rule 2 The electrostatic valency principle
• The strength of an ionic (electrostatic) bond
  (electrostatic valency e.v.) between a cation and
  an anion is equal to the charge of the ion (z)
  divided by its coordination number (n)
• e.v. z/n
• In a stable (neutral) structure, a charge
  balance results between the cation and its
  polyhedral anions with which it is bonded.

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Charge Balance in Halite
In Halite, Na has CN 6 and valence
1 Interpretation Each Na has 6 Cl- neighbors,
so each Cl- contributes a charge of -1/6 to the
Na 6 x -1/6 -1, so a charge balance results
between the Na cation and the six polyhedral Cl-
anions with which it bonded. NEUTRALITY IS
ACHIEVED
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Charge Balance In Fluorite
In Fluorite, Ca has CN 8 and valence 2, so the
electrostatic valency is ¼ e.v. Interpretation
Each Ca has 8 F- neighbors, so each F-
contributes a charge of -1/4 to the Ca 8 x -1/4
-2, so a charge balance results between the
Ca cation and the eight polyhedral F- anions
with which it bonded. NEUTRALITY IS ACHIEVED
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Formation of Anionic Groups

• If electronegativity of anion and cation differs
  by 2.0 or more will be ionic

C has valence 4 C.N 3 e.v. 4/3 1 1/3
S has valence 6 CN 4 electrostatic valency
6/4 1 1/2

e- for Carbon 2.5, for O 3.5 covalent e- S 2.4
so also covalent
Carbonate
Sulfate
Remaining charge on Oxygens available for bonding
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Paulings Rules

• Rule 3 Sharing of faces or edges is unstable.
• Rule 4 In structures with different types of
  cations, those cations with high valency and
  small CN tend not to share polyhedra with each
  other when they do, polyhedra are deformed to
  accommodate cation repulsion

C.N. coordination number
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Paulings Rules - principle of parsimony

• The number and types of different structural
  sites tends to be limited, even in complex
  minerals.
• Comment Different ionic elements are forced to
  occupy the same structural positions. This leads
  to solid solution.

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Ionic Compound Formation

• Stable ionic crystals
• maximize cation-anion contact
• minimize anion-anion cation-cation contact

2-dimensional illustration of the concept of
stability
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Visualizing Crystal Structure
Beryl - Be3Al2(Si6O18)
Gold colored spheres cations
Polyhedra Model
Ball and Stick Model
4-O Tetrahedral (T) and 6-O Octahedral (O)
Show polyhedral models
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Isostructural Types

• AX Compounds Halite (NaCl) structure
• Anions in Cubic Close Packing
• Cations in octahedral sites
• Rc/Ra .73-.41 so CN 6
• Examples
• Halides 1 cations (Li, Na, K, Rb) w/ anion
  charge -1 anions (F, Cl, Br, I)
• Oxides 2 cations (Mg, Ca, Sr, Ba, Ni) w/ O-2
• Sulfides 2 cations (Zn, Pb) w/ S-2

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Isostructural Types
CCP FCC close packing of the anions, small
cations in octohedral holes
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Isostructural Types

• AX Compounds Sphalerite (ZnS) structure
• RZn/RS0.60/1.840.32 (tetrahedral)

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Isostructural Types

• AX2 Compounds Fluorite (CaF2) structure
• Example CaF2 RCa / RF 1.12 / 1.31 0.75
  (cubic CN 8)
• Examples some Halides (CaF2, BaCl2...) Oxides
  (ZrO2...)

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Isostructural Types O and T sites

• ABO4 Compounds Spinel (MgAl2O4)structure
• - Oxygen anions in CCP array
• Two different cations (may be same element w two
  different valences) in tetrahedral (T) sites
  (e.g. Mg2, Fe2, Mn2, Zn2)
• or octahedral (O) sites (e.g. Al3, Cr3,
  Fe3)

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Nesosilicates
Inosilicates (double chain)
Olivine, Zircon Staurolite
Sorosilicates
Amphiboles
Epidote
Cyclosilicates
Phyllosilicates
Beryl Tourmaline
Micas, clays Serpentine Chlorite
Tectosilicates Quartz group, Feldspars Feldspathoi
ds Zeolites
Inosilicates (single chain)
Pyroxenes
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Next time

• Crystal Chemistry IV
• Compositional Variation of Minerals
• Solid Solution
• Mineral Formula Calculations
• Graphical Representation of Mineral
  Compositions