1. Key things to know to describe a crystal
Also---initial Steps in the Crystal Structural Determination
(1) Unit Cell Parameters (a, b, c, a, b, g)
(2) Lattice Types (P, I, F, C,…)
(3) Space Groups
(4) Positions of atoms not related by symmetry in
unit cell ---asymmetric unit
A space group consists of a set of symmetry elements
that completely describes the symmetry of a crystal.

2. • 14 Bravais lattices + 32 point groups +screw axes +glide planes
230 Space Groups
• 14 Bravais lattices + 32 point groups +screw axes +glide planes
• The symmetry of a crystal is completed described by a space group
• International Tables for Crystallography, Vol. A has complete tabulations of all 230 space groups.

3. – – – A Review of Point Group Symmetry Elements
Crystallography Spectroscopy
• Rotation around an axis n = (2, 3, 4, 6) Cn
• Improper Rotation (rotation-reflection) Sn
• Improper Rotation (rotation-inversion)
n = (2, 3, 4, 6) –
S1=m = S2= 2 1 6 4 3
S6=
S3=
S4=
• Reflection by a mirror plan s –
m = 2
• Inversion through a point 1 – i
Hermann Maugin
Schoenflies

4. Symmetry in Crystals
• A crystal lattice is infinite and has translational symmetry
that a finite-sized molecule doesn’t have.
• In terms of the number of symmetry elements, the translational
symmetry has two opposite effects.
(1) It generates some new symmetry elements such as screw axes and glide planes
(2) It reduces the number of point symmetry elements.

5. Rotation + Lattice Translation ( to rotation axis)
1) In terms of symmetry operations:
rotation • lattice translation = lattice point
R•L = L
2) Choose unit cell so that all vectors describing the lattice
translations are integral repeats of cell edges, (distance along a = one integral translation along a, etc.)then:
R must be an integer matrix and its trace must = integer

6. Rotation + Lattice Translation ( to rotation axis)
cos  sin 
- sin  cos 
Rotation operation
through angle 
Trace = ± ( cos ) = integer = n
n-1 2
cos  =
and  = 0, 60, 90, 120, or 180˚

7. Constraint of Translational Symmetry on
Point Group Symmetry in Crystals
• Only 2, 3, 4, 6-fold rotation axes possible in crystals.
4-fold
3-fold
2-fold
6-fold
8-fold also impossible
5-fold impossible to fill space
• Note that a molecule such as ferrocene can have 5-fold rotation

8. 32 Point Groups
• A point group is a set of symmetry elements that describe the
symmetry of a crystal.
• For crystals, there are only 32 unique ways to combine point
group symmetry elements.

9. Relationship between point groups and physical properties
SHG: the doubling of light frequency
Piezoelectricity: generation of dipole from mechanical stress
or conversely change of shape in an electric field.
Ferroelectricity in which there are oriented electric dipoles.can only be
observed in polar classes.

10. 2-D Plane Groups --- Escher
Homework Exercise
3.40 Huheey
2-D Plane Groups --- Escher
What symmetry elements can you find?
Answer
4 different mirror planes
Mirror plane + periodic translation, t, generates mirror plane at t/2
All four mirrors are generated by two perpendicular mirrors (mm) + two periodic translations, a and b
Plane group: pmm

11. 2-fold rotation with  translation
4 non-equivalent 2-fold axes
2b•[100]
2b•[000]
2b•[101]
2-fold + periodic translation, , generates 2-fold at
2b•[001]
All four 2-fold’s are generated by one 2-fold + two periodic translations, and
Space group: P2

12. A • t  B  A • t causes P1  P3
Rotation + Perpendicular Translation
Combination of a rotation, A with a translation t
A • t  B
In general, a rotation about an axis A through
an angle,  , followed by a translation
perpendicular to the axis, is equivalent to a
rotation through the same angle,  , in the
same sense, but about an axis B situated on
the perpendicular bisector of AA' and at a
distance (AA'/2)cot /2 from AA'.
A causes P1  P2
then, t causes P2  P3
 A • t causes P1  P3
which is equivalent to a rotation
operation B about the axis B

13. P2 What if there is only one molecule/unit cell???
2b • [001] = 2b at (0 y 1/2)
P 1 2 1
No. 3
Monoclinic c b a a
2b • [101] = 2b at (1/2 y 1/2)
Origin on 2; unique axis b
2nd Setting
Number of positions,
Wyckoff notation
and site symmetry
Co-ordinates of equivalent positions
e x,y,z; x,y,z
This is the # equivalent
positions/unit cell
What if there is only one
molecule/unit cell???
2b at (1/2 y 1/2)
d 2
, y,
2b at (1/2 y 0) c
, y, 0
b 2
0, y,
2b at (0 y 0)
a 2
0, y, 0
Molecule must have 2-fold symmetry
With it’s 2-fold coincident with one of the crystal 2-fold axes!!

14. Molecular or Formula Weight, Number of formula units, Density
Z = number of formula units (molecules) per unit cell
Generally, Z is the number of equivalent positions for the space group, but may be some simple fraction (special positions) or simple multiple of that number
Z can be derived if the density of crystal is determined:
M = atomic mass of molecule in amu
V = volume of unit cell in Å3
where M is the formula or molecular weight

15. “Depth” by Escher 3-D arrayof fish
Each fish is found at the intersection of three lines of fish, all of which cross each other at right angles.
This gives unit cells, each of which contains one “molecule” (fish). If the eyes of the fish are ignored, each fish, and  each unit cell, has C2v (mm2) symmetry. The space group would then be Pmm2

16. some matrix notation

17. Space Group P2/m four equivalent positions:
asymmetric unit = 1/4th of the unit cell

18. Symmetry in Crystals
• A crystal lattice is infinite and has translational symmetry
that a finite-sized molecule doesn’t have.
• In terms of the number of symmetry elements, the translational
symmetry has two opposite effects.
(1) It generates some new symmetry elements such as screw axes and glide planes
(2) It reduces the number of point symmetry elements.

19. Additional Symmetry Elements for Crystals
• The point group does not completely describe the
symmetry of a crystal because it does not take into
consideration translational symmetry
• In crystals, additional symmetry elements are
generated by combining point group symmetry
elements with translational symmetry
Screw axes
Glide planes

20. 21 axis, passing through origin, parallel to b axis
Screw Axes: example
21 axis, passing through origin, parallel to b axis
2b(0 1/2 0) • (x, y, z) = (x, 1/2 + y, z)
x, 1/2 + y, z
= 1/2 t
(x, y, z)
2b(0 1/2 0) operation on a cat

21. Screw Axes: example
21 axis, passing through origin, parallel to b axis
Can abbreviate this:
2b(0 1/2 0) • (x, y, z) = (x, 1/2 + y, z)

22. allowed values of the pitch interval:
COMBINATIONS OF ROTATIONS WITH PARALLEL TRANSLATIONS
allowed values of the pitch interval:
where i = integer
for 2-fold screw axis: t = 0, t/2, t
for 6-fold screw axis: t = 0, t/6, t/3, t/2, 2t/3, 5t/6, t

23. Screw axes
• Types of Screw Axes
2-fold screw axis 21
3-fold screw axis 31, 32
4-fold screw axis 41, 42, 43
6-fold screw axis 61, 62, 63, 64, 65
• Definition of Screw Axes
A screw axis, nm, is defined as a rotation around the
n-fold axis, followed by a translation of m/n along the
direction of the axis.

24. Examples of Screw Axes 32 31

25. the repetition of a point by the possible screw axes

26. Combinations of mirror planes with parallel translations
consider:
glide plane m t 
generally = 1/2 t
= 1/4 t for some d glides

28. Glide Planes
A glide plane is a reflection, followed by a translation in a direction parallel to the plane

29. Space Group P21/c

30. Space Group No. 14 b a c c b a
screw
axis

31. Homework Exercise Huheey 3.41
Among the thirteen possible monoclinic space groups are P21, P21/m, and P21/c. Compare these space groups by listing the symmetry elements for each.
These are all primitive space groups with a two fold screw axis (21). In P21/m there is a mirror plane perpendicular to 21. In P21/c there is a glide plane perpendicular to that axis with the glide translation parallel to the crystal c axis. As for the point group twofold axis, a 21 with a mirror plane or a glide plane perpendicular to it creates a center of inversion.

32. Space Group No. 14, continued

33. Space Group No. 15

34. Notes on Space Group No. 15, C2/c
2b(– 0 –) · mb(0 – 1/2) = 1(000) 1 2 3
2b(0 0 1/2) · mb(0 0 1/2) = 1(000) 4 5
2b(0 0 1/2) · 1(1/2 1/2 0) = 2b(1/2 1/2 1/2)
21 at x = z = 1/4 6
mb(0 0 1/2) · 1(1/2 1/2 0) = mb(1/2 1/2 1/2)
n-glide at y = 1/4 7
2b(0 0 1/2) · mb(1/2 1/2 1/2) = 1(1/2 1/2 0)
1 at x = y = 1/4

35. 230 Standard Space Group Symbols

36. Matrix Representation
• In actual computation, the symmetry operation is done through matrix
multiplication.
• Each symmetry element is represented as a 3 x 3 matrix, plus a
translational vector.

37. Symmetry and Asymmetric Unit
• A knowledge about the space group of a crystal greatly simplifies the
structural analysis of a crystal.
• Because of the symmetry elements, only a fraction of a unit cell
(can be as small as 1/192) is unique. Other parts of the unit cell
can be generated through symmetry.
This unique part of a unit cell is called the asymmetric unit.
• In a crystal structure determination, it is only necessary to find atoms
in the asymmetric unit.
• For example, for space group P-1, the asymmetric unit is:
0 ≤ x ≤ 1/2, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1
Only 1/2 of unit cell is unique. The other half can be generated.

38. Even though we see objects with 8-fold rotation symmetry
almost daily, such a symmetry element
is prohibited in a crystal.