Lecture 6: 3D Lattices, 3D Symmetry, Space Groups

Lecture 6:
3D Lattices, 3D Symmetry, Space Groups
Last lecture.. we discussed 2D symmetry with translation, now we
will briefly look at symmetry in 3D
3D LATTICES:
a “space” filled with points
THREE translation vectors
Unit cell vectors are commonly referred to as a, b and c axes;
lengths are given in units of Å or nm; angles between the
vectors are referred to as !, ", #
Note: Does this sound familiar? It should… these are the same as
crystallographic axes; unit cells are generally chosen to coincide with
these

3D unit cells come in the same types as 2D unit cells:
PRIMITIVE UNIT CELL (P): contains one lattice point
And also other types:
BODY-CENTERED UNIT CELL (I): doubly primitive (2 lattice
points, one extra in center of cell)
END-CENTERED UNIT CELL (A; B; C): doubly primitive (2
lattice points, extra 1/2 in each of two opposite faces)
FACE-CENTERED UNIT CELL (F): 4 lattice points (at corners +
1/2 in middle of each face)

This leads to 14 unique 3D lattices (BRAVAIS LATTICES): the only possible ways in
which points can be arranged periodically in three dimensions -- 6 of these are
PRIMITIVE - one corresponds to each of the 6 CRYSTAL SYSTEMS

Remember, the addition of translation led to the addition of a
new symmetry element, the GLIDE PLANE (glide line in 2D),
in 3D there is another new symmetry element created by
translations, the SCREW AXIS
SCREW AXIS: combination of a rotation and a translation
Screw axes are always right-handed
or left-handed (come in
enantiomorhpous pairs)

Screw Axis Notation: rotation number with subscript to indicate fraction
of a translation with each part of rotation
Ex. 2 1 = 2-fold rotation, with 1/2 t in each 180° twist
3 1 = 2-fold rotation, with 1/3 t in each 120° twist
Convention: : fractions less than or equal to 1/2 = right-handed screw;
greater than 1/2 = left-handed screw
GLIDE PLANES are notated as “a”, “b” or “c” (rather than “m” for mirror)
if they are parallel to the a, b, or c axes; they are notated as “n” if they are
1/2 a unit cell in the ab, bc, , ac or abc direction; they are notates as “d” if
they are 1/4 unit cell in the ab, bc, , ac, or abc direction

SPACE GROUPS: : symmetry in space (including translation); the various
ways in which motifs can be arranged in space in a homogeneous array
The space groups are combinations of the 14 possible BRAVAIS
LATTICES with the 32 POINT GROUPS (crystal classes), including
glide planes and screw axe = 230 SPACE GROUPS
It’s s easy to find the point group from the space group designation:
a, b, c, n, d all reduce to “m”
screw axes, like 2 1 , reduce to simple rotation axes, i.e., “2”
Ex.
Space group I2/b2/a2/m = Point group 2/m2/m2/m
Space group P2 1 /b2 1 /c2 1 /a = Point group 2/m2/m2/m