### Lecture 4

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Lecture 4
(Chapter 13 in Perkins)
Crystal Chemistry
Get polyhedral models from cabinet
Part 3:
Coordination of Ions
Pauling’s Rules
Crystal Structures
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
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
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
fromModified
K&D from K&D
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??
Atomic and Ionic Radii
Pure element first
Native Cu. Atomic radius = 1/2 bond length
X-ray d100  a
a
Ionic radius =
a
2a
4
2
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
Charge and Attractive Force Control
on Effective Ionic Radii
Approach until Repulsive and Attractive
Forces the same
Effect of Coordination Number and Valence on
Effective Ionic Radius
Increasing Ionic radii
Higher coordination numbers have
larger effective ionic radius
Extreme valence shells (1,6,7) have
larger effective ionic radius
Decreasing
Ionic radii
Coordination
Number (CN)
(# of nearest neighbors)
vs. ionic radius.
For cations of one
element,
higher coordination
numbers have
larger effective ionic
radius
Coordination
with O-2 Anions
Note: Sulfur can
have CN 6 at
great depths
For example, in
the inner core
When Rc / Ra
approaches 1
a “close
packed”
array
forms
Coordination Polyhedra

We always consider coordination of
anions about a central cation
Halite
Na
Cl
Cl
Cl
Cl
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

Coordination Polyhedra
Radius Ratio: RC/RA = 1.0
(commonly native elements)
Equal sized spheres
“Closest Packed”
Notice:6 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
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 1’s are left
1
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
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
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)
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)
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)
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
Closest Packing
Third layer:
Unit cell
Closest Packing
Third layer:
Unit cell
Closest Packing
Third layer:
Unit cell
Closest Packing
Third layer:
View from top shows
hexagonal unit cell
(HCP)
Closest Packing
Third layer:
View from top shows
hexagonal unit cell
(HCP)
Closest Packing
Alternatively we could
place the third layer in
the C-type site (above
voids in both A and B
layers)
C
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
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
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
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
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
Cubic Closest Packing
View from the same
side shows the
cubic close packing
(CCP), also called
face-centered cubic
A-layer
(FCC) because of
the unit cell that
results. Notice that
C-layer
every face of the
cube has an atom at
every face center.
B-layer
The atoms are slightly
shrunken to aid in
visualizing the
structure
A-layer
Closest Packing
Rotating toward a top
view
Closest Packing
Rotating toward a top
view
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
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.
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)
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).
Set = 1
arbitrary
since will
deal with
ratios
What is the RC/RA of
that limiting
condition??
Diagonal length then =
2
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
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
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
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
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
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
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
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.
What is the RC/RA of
that limiting
condition??
Central Plane
= 1 + 2 = 1.732
1.732 = dC + dA
=1
(arbitrary)
If dA = 1
then dC = 0.732
dC/dA = RC/RA
= 0.732/1 = 0.732
= 2
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.
= 1 + 2 = 1.732
=1
(arbitrary)
= 2
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
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
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
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
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
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??
= 2
=1
1.414 = dC + dA
If dA = 1
then dC = 0.414
dC/dA = RC/RA
= 0.414/1 = 0.414
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
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
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
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
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
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
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
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
See derivation fig 4.3 c page 70
RC = 0.6124 - 0.5 = 0.1124
RC/RA
= 0.1124/0.5 = 0.225
0.5
1
0.61
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
0.5
1
y
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
Pauling’s 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 (Rc:Ra)
Linus Pauling
Pauling’s 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.
Charge Balance
in Halite
In Halite, Na+ has CN 6 and valence +1
Interpretation: Each Na+ has 6 Cl- neighbors, so each Clcontributes 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
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 Fcontributes 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
If electronegativity of anion and cation differs by 2.0 or more will be ionic
Formation of Anionic Groups
C has valence +4
S has valence +6
C.N = 3
CN = 4
e.v. = 4/3 = 1 1/3
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
Remaining charge on Oxygens available for bonding
Sulfate
Pauling’s 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”
Pauling’s 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.
Ionic Compound Formation

Stable ionic crystals:
maximize cation-anion contact
 minimize anion-anion & cation-cation
contact

2-dimensional illustration of the concept of stability:
Visualizing Crystal Structure
Beryl - Be3Al2(Si6O18)
Ball and Stick Model
Show polyhedral models
Gold colored spheres cations
Polyhedra Model
4-O Tetrahedral (T) and 6-O Octahedral (O)
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
Isostructural Types
CCP= FCC close packing
of the anions, small cations in
octohedral “holes”
Isostructural Types

AX Compounds – Sphalerite (ZnS) structure
RZn/RS=0.60/1.84=0.32 (tetrahedral)
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...)
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+)
Nesosilicates
Olivine, Zircon
Staurolite
Sorosilicates
Epidote
Cyclosilicates
Beryl
Tourmaline
Inosilicates
(single chain)
Pyroxenes
Inosilicates
(double chain)
Amphiboles
Phyllosilicates
Micas, clays
Serpentine
Chlorite
Tectosilicates
Quartz group,
Feldspars
Feldspathoids
Zeolites
Next time

Crystal Chemistry IV
Compositional Variation of Minerals
Solid Solution
Mineral Formula Calculations
Graphical Representation of Mineral
Compositions
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