1. Lecture 2: Crystal Symmetry

2. Crystals are made of infinite number of unit cells
Unit cell is the smallest unit of a crystal, which, if
repeated, could generate the whole crystal.
A crystal’s unit cell dimensions are defined by six numbers,
the lengths of the 3 axes, a, b, and c, and the three interaxial
angles, α, β and γ.

3. A crystal lattice is a 3-D stack of unit cells
Crystal lattice is an imaginative grid system in three dimensions in
which every point (or node) has an environment that is identical to that
of any other point or node.

4. Symmetry
A state in which parts on opposite sides of a plane,
line, or point display arrangements that are related to
one another via a symmetry operation such as
translation, rotation, reflection or inversion.
Application of the symmetry operators leaves the
entire crystal unchanged.

5. Symmetry Elements
Rotation
turns all the points in the asymmetric
unit around one axis, the center of
rotation. A rotation does not change
the handedness of figures. The center
of rotation is the only invariant point
(point that maps onto itself).

6. Symmetry elements: rotation

7. Symmetry elements: rotation

8. Symmetry Elements
Translation moves all the points in the
asymmetric unit the same
distance in the same direction.
This has no effect on the
handedness of figures in the
plane. There are no invariant
points (points that map onto
themselves) under a translation.

9. Symmetry Elements
Screw axes (rotation + translation)
rotation about the axis of
symmetry by 360°/n, followed
by a translation parallel to the
axis by r/n of the unit cell length
in that direction. (r < n)

10. 120° rotation
1/3 unit cell translation

12. Symmetry Elements
Inversion, or center of symmetry
every point on one side of
a center of symmetry has a
similar point at an equal
distance on the opposite
side of the center of
symmetry.

13. Symmetry Elements
Mirror plane or Reflection
flips all points in the asymmetric unit
over a line, which is called the mirror,
and thereby changes the handedness of
any figures in the asymmetric unit.
The points along the mirror line
are all invariant points (points that map
onto themselves) under a reflection.

14. Symmetry elements:
mirror plane and inversion center
The handedness is changed.

15. Symmetry Elements
Glide reflection (mirror plane + translation)
reflects the asymmetric unit
across a mirror and then
translates parallel to the mirror.
A glide plane changes the
handedness of figures in the
asymmetric unit. There are no
invariant points (points that map
onto themselves) under a glide
reflection.

18. Symmetries in crystallography
• Crystal systems
• Lattice systems
• Space group symmetry
• Point group symmetry
• Laue symmetry, Patterson symmetry

19. Crystal system
• Crystals are grouped into seven crystal
systems, according to characteristic
symmetry of their unit cell.
• The characteristic symmetry of a crystal is a
combination of one or more rotations and
inversions.

20. 7 Crystal Systems
orthorhombic hexagonal
monoclinic trigonal
cubic tetragonal triclinic
Crystal System External Minimum Symmetry Unit Cell Properties
Triclinic None a, b, c, al, be, ga,
Monoclinic One 2-fold axis, || to b (b unique) a, b, c, 90, be, 90
Orthorhombic Three perpendicular 2-folds a, b, c, 90, 90, 90
Tetragonal One 4-fold axis, parallel c a, a, c, 90, 90, 90
Trigonal One 3-fold axis a, a, c, 90, 90, 120
Hexagonal One 6-fold axis a, a, c, 90, 90, 120
Cubic Four 3-folds along space diagonal a, a, ,a, 90, 90, 90

21. Auguste Bravais
Lattices
(1811-1863)
• In 1848, Auguste Bravais demonstrated that
in a 3-dimensional system there are fourteen
possible lattices
• A Bravais lattice is an infinite array of
discrete points with identical environment
• seven crystal systems + four lattice centering
types = 14 Bravais lattices
• Lattices are characterized by translation
symmetry

22. Four lattice centering types
No. Type Description
1 Primitive Lattice points on corners
only. Symbol: P.
2 Face Centered Lattice points on corners as
well as centered on
faces. Symbols: A (bc
faces); B (ac faces); C
(ab faces).
3 All-Face Centered Lattice points on corners as
well as in the centers of
all faces. Symbol: F.
4 Body-Centered Lattice points on corners as
well as in the center of
the unit cell body.
Symbol: I.

24. Tetragonal lattices are either primitive (P) or
body-centered (I)
C centered lattice
=
Primitive lattice

25. Monoclinic lattices are either primitive
or C centered

26. Point group symmetry
• Inorganic crystals usually have perfect shape
which reflects their internal symmetry
• Point groups are originally used to describe the
symmetry of crystal.
• Point group symmetry does not consider
translation.
• Included symmetry elements are rotation, mirror
plane, center of symmetry, rotary inversion.

27. Point group symmetry diagrams

28. There are a total
of 32 point groups

29. N-fold axes with n=5 or n>6 does
not occur in crystals
Adjacent spaces must be completely filled (no gaps, no
overlaps).

30. Laue class, Patterson symmetry
• Laue class corresponds to symmetry of
reciprocal space (diffraction pattern)
• Patterson symmetry is Laue class plus
allowed Bravais centering (Patterson map)

31. Space groups
The combination of all available symmetry operations (32
point groups), together with translation symmetry,
within the all available lattices (14 Bravais lattices) lead
to 230 Space Groups that describe the only ways in which
identical objects can be arranged in an infinite lattice.
The International Tables list those by symbol and
number, together with symmetry operators, origins,
reflection conditions, and space group projection
diagrams.

32. A diagram from International Table of Crystallography

33. Identification of the Space Group is called indexing the crystal.
The International Tables for X-ray Crystallography tell us a huge
amount of information about any given space group. For instance,
If we look up space group P2, we find it has a 2-fold rotation axis
and the following symmetry equivalent positions:
X , Y , Z
-X , Y , -Z
and an asymmetric unit defined by:
0≤x≤ 1
0≤y≤ 1
0 ≤ z ≤ 1/2
An interactive tutorial on Space Groups can be found on-line in Bernhard Rupp’s
Crystallography 101 Course: http://www-structure.llnl.gov/Xray/tutorial/spcgrps.htm

34. Space group P1
Point group 1 + Bravais lattice P1

35. Space group P1bar
Point group 1bar + Bravais lattice P1

36. Space group P2
Point group 2 + Bravais lattice “primitive monoclinic”

37. Space group P21
Point group 2 + Bravais lattice “primitive monoclinic”,
but consider screw axis

38. Coordinate triplets, equivalent positions
r = ax + by + cz,
Therefore, each point can be described by its fractional
coordinates, that is, by its coordinate triplet (x, y, z)

39. Space group determination
• Symmetry in diffraction pattern
• Systematic absences
• Space groups with mirror planes and
inversion centers do not apply to protein
crystals, leaving only 65 possible space
groups.

40. A lesson in symmetry from M. C. Escher

41. Another one:

42. Asymmetric unit
Recall that the unit cell of a crystal is the smallest 3-D geometric
figure that can be stacked without rotation to form the lattice. The
asymmetric unit is the smallest part of a crystal structure from
which the complete structure can be built using space group
symmetry. The asymmetric unit may consist of only a part of a
molecule, or it can contain more than one molecule, if the molecules
not related by symmetry.

43. Matthew Coefficient
• Matthews found that for many protein crystals the
ratio of the unit cell volume and the molecular
weight is between 1.7 and 3.5Å3/Da with most
values around 2.15Å3/Da
• Vm is often used to determine the number of
molecules in each asymmetric unit.
• Non-crystallographic symmetry related molecules
within the asymmetric unit