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Crystal Morphology
Remember:
Space groups for atom symmetry
Point groups for crystal face symmetry

Crystal Faces = limiting surfaces of growth
Depends in part on shape of building units & physical cond.
(T, P, matrix, nature & flow direction of solutions, etc.)
Crystal
Morphology
Observation:
The frequency with
which a given face in a
crystal is observed is
proportional to the
density of lattice nodes
along that plane
Crystal Morphology
Observation:
The frequency with
which a given face in
a crystal is observed
is proportional to the
density of lattice
nodes along that
plane
Crystal Morphology
Because faces have direct relationship to the
internal structure, they must have a direct and
consistent angular relationship to each other
Crystal Morphology
Nicholas Steno (1669): Law of Constancy of
Interfacial Angles
120o
120o
120o
Quartz
120o
120o
120o
120o
Crystal Morphology
Diff planes have diff atomic environments
Crystal Morphology
Crystal symmetry conforms to 32 point groups  32 crystal classes
in 6 crystal systems
Crystal faces act just as our homework: symmetry about the center
of the crystal so the point groups and the crystal classes are the same
Crystal System
No Center
Center
1
1
Monoclinic
2, 2 (= m)
2/m
Orthorhombic
222, 2mm
2/m 2/m 2/m
Tetragonal
4, 4, 422, 4mm, 42m
4/m, 4/m 2/m 2/m
Hexagonal
3, 32, 3m
3, 3 2/m
6, 6, 622, 6mm, 62m
6/m, 6/m 2/m 2/m
23, 432, 43m
2/m 3, 4/m 3 2/m
Triclinic
Isometric
Crystal Morphology
Crystal Axes: generally taken as parallel to the edges
(intersections) of prominent crystal faces
b
a
c
Crystal Morphology
Crystal Axes: generally taken as parallel to the edges (intersections)
of prominent crystal faces
The more faces the better  prism faces & quartz c-axis, halite
cube, etc.
We must also keep symmetry in mind: c = 6-fold in hexagonal
With x-ray crystallography we can determine the internal structure
and the unit cell directly and accurately
The crystallographic axes determined by XRD and by the face
method nearly always coincide
This is not coincidence!!
Crystal Morphology
How do we keep track of the faces of a crystal?
Crystal Morphology
How do we keep track of the faces of a crystal?
Remember, face sizes may vary, but angles can't
Note: “interfacial
angle” = the angle
between the faces
measured like this
120o
120o
120o
120o
120o
120o
120o
Crystal Morphology
How do we keep track of the faces of a crystal?
Remember, face sizes may vary, but angles can't
Thus it's the orientation & angles that are the best source
of our indexing
Miller Index is the accepted indexing method
It uses the relative intercepts of the face in question with
the crystal axes
Crystal Morphology
Given the following crystal:
2-D view
looking down c
b
a
b
c
a
Crystal Morphology
Given the following crystal:
b
a
How reference faces?
a face?
b face?
-a and -b faces?
Crystal Morphology
Suppose we get another crystal of the same mineral with
2 other sets of faces:
How do we reference them?
b
w
x
y
b
a
a
z
Miller Index uses the relative intercepts of the faces with
the axes
Pick a reference face that intersects both axes
Which one?
b
b
w
x
x
y
y
a
z
a
Which one?
Either x or y. The choice is arbitrary. Just pick one.
Suppose we pick x
b
b
w
x
x
y
y
a
z
a
MI process is very structured (“cook book”)
a
b
c
unknown face (y)
reference face (x)
1
2
1
1

1
invert
2
1
1
1
1

clear of fractions
2
1
0
b
x
Miller index of
face y using x as
the a-b reference face
y
(2 1 0)
a
What is the Miller Index of the reference face?
a
b
c
unknown face (x)
reference face (x)
1
1
1
1

1
invert
1
1
1
1
1

clear of fractions
1
1
0
b
x
(1 1 0)
Miller index of
the reference face
is always 1 - 1
y
(2 1 0)
a
What if we pick y as the reference. What is the MI of x?
a
b
c
unknown face (x)
reference face (y)
2
1
1
1

1
invert
1
2
1
1
1

clear of fractions
1
2
0
b
x
(1 2 0)
Miller index of
the reference face
is always 1 - 1
y
(1 1 0)
a
Which choice is correct?
1)
b
x = (1 1 0)
y = (2 1 0)
2)
x = (1 2 0)
y = (1 1 0)
x
The choice is arbitrary
y
What is the difference?
a
What is the difference?
b
b
unit cell
shape if
y = (1 1 0)
b
unit cell
shape if
x = (1 1 0)
a
a
x
x
b
y
y
a
a
axial ratio = a/b = 0.80
axial ratio = a/b = 1.60
The technique above requires that we graph each face
A simpler (?) way is to use trigonometry
Measure the
interfacial angles
b
b
w
x
148o
?
y
x
?
interfacial angles
141o
y
a
z
a
The technique above requires that we graph each face
A simpler (?) way is to use trigonometry
b
b
w
x
58o
148o
tan 39 = a/b = 0.801
tan 58 = a/b = 1.600
x
y
39o
141o
y
a
z
a
What are the Miller Indices of all the faces if we choose x
as the reference?
Face Z?
b
w
(1 1 0)
(2 1 0)
a
z
The Miller Indices of face z using x as the reference
b
w
a
b
c
unknown face (z)
reference face (x)
1
1

1

1
invert
1
1
1

1

clear of fractions
1
0
0
(1 1 0)
(2 1 0)
(1 0 0)
a
z
Miller index of
face z using x (or
any face) as the
reference face
b
Can you index the rest?
(1 1 0)
(2 1 0)
(1 0 0)
a
b
(0 1 0)
(1 1 0)
(1 1 0)
(2 1 0)
(2 1 0)
(1 0 0)
a
(1 0 0)
(2 1 0)
(2 1 0)
(1 1 0)
(0 1 0)
(1 1 0)
3-D Miller Indices (an unusually complex example)
a
b
c
unknown face (XYZ)
reference face (ABC)
2
1
2
4
2
3
invert
1
2
4
2
3
2
(1
4
3)
c
C
Z
clear of fractions
A
X
O
Miller index of
face XYZ using
ABC as the
reference face
Y
B
a
b
Demonstrate MI on cardboard cube model
We can get the a:b:c axial ratios from the chosen (111) face
We can also determine the true unit cell by XRD and of course
determine the a:b:c axial ratios from it
If the unit face is correctly selected, the ratios should be the
same
If not, will be off by some multiple - i.e. picked (211) and
called it (111)
Best to change it
Mineralogy texts listed axial ratios long before XRD
We had to change some after XRD developed
Form = a set of symmetrically equivalent faces
braces indicate a form {210}
b
(1 1)
(0 1)
(1 1)
(2 1)
(2 1)
(1 0)
a
(1 0)
(2 1)
(2 1)
(1 1)
(0 1)
(1 1)
Form = a set of symmetrically equivalent faces
braces indicate a form {210}
Multiplicity of a form depends on symmetry
{100} in monoclinic, orthorhombic,
tetragonal, isometric
Form = a set of symmetrically equivalent faces
braces indicate a form {210}
F. 2.36 in your text (p. 49-52)
pinacoid
related by a mirror
or a 2-fold axis
prism
related by n-fold
axis or mirrors
pyramid
dipryamid
Form = a set of symmetrically equivalent faces
braces indicate a form {210}
Quartz = 2 forms:
Hexagonal prism (m = 6)
Hexagonal dipyramid (m = 12)
Isometric forms include
Cube
Octahedron
Dodecahedron
101
011
_
110
110
_
101
_
011
_
111
111
__
111
_
111
Octahedron to Cube to
Dodecahedron
Click on image to run animation
All three combined:
001
011
_
111
101
111
_
110
010
100
__
111
_
101
110
_
111
_
011
Zone
Any group of faces || a common axis
Use of h k l as variables for a, b, c
intercepts
(h k 0) = [001]
If the MI’s of 2 non-parallel faces are
added, the result = MI of a face between
them & in the same zone
BUT doesn't say which face
(010)
(110)?
Which??
(110)?
(100)
BUT doesn't say which face
(010)
(110)
(210)
(010)
(110)?
Which??
(100)
(110)?
(010)
(100)
(120)
(110)
Either is OK
(100)