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17
Plane
symmetry operations. The bottom row are examples of plane Groups that correspond
to each lattice type.
5 unique 2D plane lattices + 10 2D point groups 17 plane groups Repeating
Repeating atoms or groups of atoms can be represented by a simple
array of points (lattice points). 3D Translations and Lattices
z Different ways to combine 3 nonparallel, noncoplanar axes
nonnon z Really deals with translations compatible with 32 point groups (or
crystal classes) z 32 point groups fall into 6 categories, and so do the 3D crystal
+c
lattices
lattices.
3D Lattice Types Name axes angles +a azbzc Monoclinic azbzc Orthorhombic azbzc D E J = 90 o a 1 = a2 z c D E J = 90 o Tetragonal
Hexagonal
Hexagonal (4 axes)
Rhombohedral Tridymite: Orthorhombic C cell E o Triclinic Isometric a 1 = a2 = a3 z c D z E z J z 90
o J = 90 E z 90 D o E = 90 J o o D E J z 90 a1 = a 2 = a3 D E J = 90 J D +b 120 a1 = a 2 = a3 o o Axial convention:
“right“righthand rule” c c c b
a b b P a Triclinic
Dz Ez J
azbzc Primitive P c c I=C a
Monoclinic
BodyBodycentered
D J R z E
azbzc a2
a2 a1 P
D 14 Bravais lattices P or C R Hexagonal Tetragonal
E J R a1 = a2 z c c French
mineralogist
b
Auguste Bravais a
P
(1850)
C facecentered
face a1 I D Rhombohedral
E R J R D E J z R
a1 = a2 = a3
a1 a2 z c a3 C
D F I Orthorhombic Facecentered
FaceE J R a z b z c French
mineralogist
a
Auguste Bravais 1
(1850) a2 P 14 Bravais lattices F D I Isometric
E J R a1 = a2 = a3 Monoclinic
Monoclinic Tetragonal Lattices
Tetra
Monoclinic B = P
Monoclinic I = C
Monoclinic
Monoclinic F = C
simply by a selection of
appropriate crystallographic
axes.
Monoclinic
D=JzE
azbzc
Monoclinic C P
No way to find any new
options of crystallographic
of crystallographic
axes to make C = P while
retaining a monoclinic lattice.
Not a monoclinic cell. = 0, 1, 2...
= 0.5, 1.5, 2.5 ... What’s this?
= I cell in isometric or tetragona...
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 Spring '14
 Crystallography, Crystal system, symmetry operations, BLOSS

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