07 - Crystal Geometry and Crystallography
07 - Crystal Geometry and Crystallography
07 - Crystal Geometry and Crystallography
Lesson 7
Crystal Geometry and Crystallography, Part 1
Suggested Reading
• Chapters 2 and 6 in Waseda et al.
169
Salt crystals
Na Cl
http://healthfreedoms.org/2009/05/24/table-salt-vs-unrefined-sea-salt-a-primer/
170
What is a crystal?
What is symmetry?
171
Crystal Structure = Lattice + Motif
[Basis]
a a
x
b T = 1a+2b b
a a
• There are two non-collinear basis vectors (a and b).
T = na + pb
174
Three-Dimensional Lattice
• There are 3 non-collinear
basis vectors and 3
interaxial angles.
– T = na + pb + qc
origin
a b
175
Three-Dimensional Latticecont’d
• The Basis vectors define
the ‘shape’ of the crystal.
b
a
Axis a b c a
Inter-axial angle
177
Unit Cell Shapes
• Unit cells in crystals have specific shapes.
[*] A set of reference axes used to define the geometry of crystal and crystal structures
178
Crystal Systems
• In 2D there are only four (4). The crystal systems
1. Oblique are the only possible
2. Rectangular shapes for unit cells
3. Hexagonal
4. Square
With these shapes,
• In 3D there are only seven (7). you can fill all
1. Triclinic (anorthic) available space and
2. Monoclinic leave no voids!
3. Hexagonal
4. Rhombohedral (trigonal)
5. Orthorhombic
6. Tetragonal
7. Cubic
179
2D Crystal Systems
a≠b OBLIQUE b
γ ≠ 90° PARALLELOGRAM
γ
a≠b RECTANGLE
b
γ = 90° γ
a=b
HEXAGONAL
a γ
γ = 120°
a=b a
SQUARE
γ = 90° γ
a
180
3D Crystal Systems
Axial Interaxial
Crystal System Relationships Angles
Rhombohedral
a=b=c = = ≠ 90°
(Trigonal)
181
The mineral images are from various sites on the internet
7 crystal systems
Cubic Tetragonal Orthorhombic
a=b=c a=b≠c a≠b≠c
= β = = 90° = β = = 90° = β = = 90°
120° 120°
a
120°
183
What defines lattices and unit cells?
• Symmetry limits the number of possibilities.
184
Symmetry Operators
• Motions that allow a pattern to be transformed from an
initial position to a final position such that the initial and
final patterns are indistinguishable.
1. Translation
2. Reflection
3. Rotation
4. Inversion (center of symmetry)
5. Roto-inversion (inversion axis)
6. Roto-reflection
7. Glide (translation + reflection)
8. Screw (rotation + translation)
185
Symmetry of Crystal Systems
Crystal Axial Interaxial Minimum # of
System Relationships Angles Symmetry Elements
Four 3-fold rotation or roto-
Cubic a=b=c = = = 90° inversion axes parallel to body
diagonals
One 6-fold rotation or
Hexagonal a=b≠c = = 90°; = 120° rotoinversion axis parallel to z-
axis
Increasing symmetry
One 4-fold rotation or roto-
Tetragonal a=b≠c = = = 90° inversion axis parallel to z-axis
Symmetric Array
of Lattice Points
=
We can classify
Bravais lattices in
Bravais Lattice terms of the number
of lattice points in
the unit cell
187
2D/3D
Types of Lattices
• Primitive (P)
• Non-primitive (multiple)
– Termed “XXX-centered”
corner
interior
189
Lattice Points Per Cell in 3D
Nface Ncorner
N3D Ninterior
2 8
interior
face
corner
190
Primitive vs. Non-primitive lattices
• There are 4 crystal systems in 2D. Thus we can define 4
primitive lattices in 2D.
– 4 primitive Bravais ‘nets’ (aka. “lattices”)
– Are there more?
OF COURSE!
• There are 7 crystal systems in 3D. Thus we can define 7
primitive lattices in 3D.
– 7 primitive Bravais lattices
– Are there more? OF COURSE!
191
Primitive vs. Non-primitive lattices
• Answer: YES, if we maintain symmetry.
(“All lattice points must be equivalent.)
b
90° a
13 12
12 2D 12
rectangular
lattices
IMPOSSIBLE POSSIBLE
90° a
12
12
Mirror images of
primitive lattice
• The primitive cell is less symmetric than the centered rectangle. For
example, a mirror image of the primitive unit cell is not identical to the
original.
A mirror image of the rectangular cell with a lattice point in the center IS
identical to the original. “It has higher symmetry!”
193
Five 2D Bravais Lattices
b b a
γ 90° 120°
a a a
Primitive Centered Primitive
Oblique Rectangle Hexagonal
b a
90° 90°
a a
Primitive Primitive
Rectangle Square
194
14 Bravais Lattices (three dimensional)
primitive
– Miller indices
– Stereographic projections
– Reciprocal space
197