The document discusses crystal systems and symmetry in crystallography. It begins by defining an asymmetric unit and how symmetry operations are used to reconstruct the full unit cell from the asymmetric unit. It then discusses the seven crystal systems, 14 Bravais lattices, 32 point groups, and 230 space groups that describe all possible symmetries of crystal structures. It also notes that the chirality of amino acids limits protein crystals to one of 65 chiral space groups. In addition, it provides an overview of X-ray crystallography instrumentation, including X-ray sources, optics, detectors, and how a rotation instrument is used to collect diffraction data.

1. Crystal Systems

2. Asymmetric unit
Which part of the unit cell can be taken?

3. As we can probably guess, an asymmetric unit contains the unit cell origin and the primary
generating symmetry element(s).
In order to reconstruct the complete unit cell from the AU, we need the unit cell dimensions
and angles to assemble the whole unit cell. The assembly information encoded in the so-called
plane group (in 2-D space) or space group symbols (in 3-D space).

4. Crystal or no crystal?

5. Symmetry in three dimensions
Any lattice point r may be described in terms of a, b
and c as
r = pa + qb + rc (where p, q are r are integers only if
a, b and c define a primitive cell).

6. Primitive and Non-primitive (centered) lattices
There are seven crystal systems. Only 14 lattices are allowed which are called the Bravais
lattices. These fall into several crystal systems depending on the overall symmetry of the unit
cell.

7. Symmetry operations and symmetry elements in three dimensions
When an operation is performed on a body with the result that the body assumes a new
disposition in space which is totally indistinguishable from the original disposition, then the
body is said to be symmetrical. Specifically, the body is symmetrical with respect to the
operation which gives rise that particular change in disposition. The operation in question is
known as a symmetry operation.
All symmetry operations involved a dispositional change in space such as rotation, reflection
in a plane. The geometrical locus about which the symmetry operation acts comprises a
symmetry element. Thus, for every symmetry operation, there is a corresponding symmetry
element.
If an axis exists within a body such that a rotation of α gives rise to a disposition
indistinguishable from the original, the body is said to contain an n-fold proper rotation axis
where n is an integer given by n=360/α.

8. The angle α must be such that its cosine is half-integral or integral i.e. cos α = 0, 1/2, -1
Thus, α = 0 , 60 , 90 , 120 , 180 or 360
For an n-fold axis, n = 360/α, Therefore n = 1, 2, 3, 4 or 6
60
six-fold rotation axis
Proper rotations:
Symmetry operation: proper rotation
Symmetry element: proper rotation axis
Symbols: n

9. Reflections:
Symmetry operation: reflection
Symmetry element: mirror plane
Symbol: m
Any plane which is such that the disposition of an object to one side of the plane is the mirror
image of that on the other is known as mirror plane.
Objects which are mirror images of each other are known as enantiomorphs if macroscopic;
microscopic mirror images are known as enantiomers.

10. Inversions:
Symmetry operation: inversion
Symmetry element: inversion centre,
center of symmetry
Symbol: i
The operation of inversion, is a rotation through 180 followed by a reflection in a plane
perpendicular to the rotation axis.

11. Screw rotations:
Symmetry operation: screw rotation
Symmetry element: screw axis
Symbol: nt
The screw rotation is a combination of a rotation and a translation parallel to the axis of
rotation. The rotation is through an angle α = 360/n. The translation is parallel to the axis of
rotation and corresponds to a fraction t/n of the lattice spacing along the direction of the
rotation axis.

12. Screw axes
Screw axes nt= n-fold rotation + t/n translation along the rotation axis
For example: 21, 31, 41, 61 screw axes.

13. Glides:
Symmetry operation: glide
Symmetry element: glide plane
Symbol: g
A glide is a combination of a reflection and a translation parallel to the plane of the mirror
plane.
Improper rotations:
Symmetry operation: improper rotation
Symmetry element: improper rotation axis
Symbol: rotoinversion and rotoreflection
An improper rotation is the combination of a rotation with either an inversion, or a reflection
in the plane perpendicular to the rotation axis. The former is know as rotoinversion and latter
is called rotoreflection.

14. Point groups
Another way of classifying symmetry groups is according to whether or not the operation
involves a translation.
Those which do not involve translation are proper rotation, reflection, inversion and improper
rotation.
Those which do include a translation are glide and screw rotation.
The significance of this categorization concerns the fact that any set of symmetry operations
which does not invoke translation may be thought of as acting at a point in space and that
point in space is unchanged by the operation.
It has been proved that in three dimensions, there are 32 different operation corresponding to
combination of proper rotations, reflections, inversions and improper rotations. These 32
arrangements are called the 32 point groups.

15. For example, a twofold axis in a monoclinic crystal is indicated by the symbol 2. A mirror
plane orthogonal to the twofold axis is indicated by the symbol 2/m.
In orthorhombic crystals, the presence of twofold axes parallel to the three unit cell axes
would be indicated by the symbol 222 and likewise three orthogonal mirror planes by the
symbol mmm.
In tetragonal crystals, the presence of fourfold axis parallel to z is indicated by the symbol 4
and a mirror plane perpendicular to this axis would be indicated by the symbol 4/m. A
tetragonal crystal with two fold axes perpendicular to the tetrad axis would be indicated by the
symbol 422 and the presence of mirror planes perpendicular these axes would generate the
point group 4/mmm.
A hexagonal crystal with only a sixfold axis would belong to the point group 6 and one with a
mirror plane perpendicular to the hexad axis would belong to the point group 6/m. A
hexagonal crystal with twofold axes perpendicular to the hexad would belong to the point
group 622 and addition of a mirror plane perpendicular to the hexad would give the point
group 6/mmm.
Laue groups

16. Space groups
When we consider the effect of the translational symmetry of the lattice and the two
operations which include translation (glides and screw rotations), how these operations
combine with each of the 32 point groups.
The effect of the translations is to generate motifs in all space, so we may refer to the types of
pattern built up as space groups.
It has been shown that there are 17 plane space groups and 230 three-dimensional space
groups.
Any pattern whatsoever in two dimensions must correspond to one these 17 plane groups and
likewise, any regular pattern in three dimensions must fall into one of the 230 space groups.

17. There is a nomenclature and symbolism for each of the space groups.
A very common space group for the protein is P21, which signifies a 21 screw axis along they
axis.
Another common crystal system for proteins is the orthorhombic system and the associated
primitive space groups are symbolized as P222, P21212, P2221 and P212121. The numbers refer
to the symmetry elements parallel to the crystallography x, y and z axes, respectively.
So, for example, P21212 has 21 screw axes parallel to x and y and a proper two fold axis
parallel to z.
Tetragonal space groups are defined by a representation such as P43212 in which the first
symbol refers to the tetrad, which is always parallel to z, the second symbol refers to the x or
y axis and the third symbol refers to the diagonal direction at 45 to x and y.
Space groups nomenclature

18. Possible Symmetry for Protein Crystals
The fact that the amino acids are chiral has an important consequence. It means that proteins
can crystallize only in one of the enantiomorphic space groups – namely those space groups
which do not involve mirror or inversion symmetry elements. This reduces the number of
space groups that proteins can crystallize in down to 65.
Why can not we build the remaining 12 plane structures?
As any symmetry operation must lead to an identical copy of the original object, the absence
of symmetry in the motif places limits the possibilities of their arrangement in two ways:
1. A symmetry operation must generate an identical copy of the motif
2. An asymmetric motif can not be located on a non-translational symmetry element.

19. Limitations resulting from asymmetry of the motif

20. The 65 chiral space groups
Lattice
properties
Minimum
internal symmetry
Crystal system Point
group
M Bravais
type
B Lattice
type
Chiral space groups z, M
a≠b≠c
α≠β≠γ≠90°
None Triclinic 1 1 P 1 aP P1 1
a≠b≠c
α=γ=90°
β≠90°
2-fold parallel to b Monoclinic 2 2 P 1 mP P2, P21 2
C 2 mC C2 4
a≠b≠c
α=β=γ=90°
3 perpendicular non-intersecting
2-fold
Orthorhombic 222 4 P 1 oP P222, P2221, P21212, P212121 4
I 2 oI I222, I212121 8
C 2 oC C2221, C222 8
F 4 oF F222 16
a=b≠c
α=β=γ=90°
4-fold parallel to c Tetragonal 4 4 P 1 tP P4, P41, P42, P43 4
I 2 tI I4, I41 8
422 8 P
I
1 tP P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212 8
2 tI I422, I4122 16
a=b≠c
α=β=90°
γ=120°
3-fold parallel to c Trigonal 3 3 P 1 hP P3, P31, P32 3
R 3 hR R3 6
32 6 P 1 hP P312, P321, P3112, P3121, P3212, P3221 9
R 3 hR R32 18
6-fold parallel to c Hexagonal 6 6 P 1 hP P6, P61, P65, P62, P64, P63 6
622 12 P 1 hP P622, P6122, P6522, P6222, P6422, P6322 12
a=b=c
α=β=γ=90°
Four 3-fold along diagonal Cubic 23 12 P 1 cP P23, P213 12
I 2 cI I23, I213 24
F 4 cF F23 48
432 12 P 1 cP P432, P4232, P4332, P4132 24
I 2 cI I432, I4132 48
F 4 cF F432, F4132 96

21. Space group preferences Plain rotation axes Vs screw axes

23. The instrument used for X-ray data collection is a diffractometer, measuring the position
(angular distribution) and intensity of the diffracted X-rays.
X-ray instrumentation
The basic components of a diffractometer
are:
1. An intense source of hard X-rays in the 5-
25 keV range.
2. Suitable optics to select monochromatic
X-rays and to focus or collimate them into
a brilliant beam of X-rays.
3. A mechanical device, the goniostate, to
orient the crystal in the primary X-ray
beam.
4. A detector for the diffracted X-rays,
generally a 2-dimensional area detector.

24. X-ray Sources
Sealed tubes
Rotating anodes

25. Energy and Wavelength
The electron energy is e (accelerating voltage V), where e is the electron charge. The photon
energy is hν = h(c/λ), where h is Planck’s constant, ν is the frequency of the radiation, c is the
speed of light, and λ is the wavelength. Therefore,
where V is in kilovolts. At V = 40 kV the cutoff edge
is at 0.31 Å.
For the emission of the CuKα line, V should be at
least 8 kV. If a higher voltage is applied, the intensity
of the line is stronger with respect to the continuous
radiation. The intensity of the line is also
proportional to the tube current.
A normal setting is V = 40 kV with a tube current of
37 mA for a 1.5-kW tube.

26. X-ray optics

27. Selected characteristic X-ray wavelengths
Anode element z Emission line Wavelength (Å) Energy (eV) Line width (eV)
Cr 24 Kα2 2.293652 5405.20 2.4
Kα1 2.289755 5414.42 2.0
Kα (avg) 2.291048 5411.34
Kβ 2.084912 5946.36
Cu 29 Kα2 1.544414 8027.40 3.0
Kα1 1.540593 8047.32 2.4
Kα (avg) 1.541867 8040.67
Kβ 1.392246 8904.78
Mo 42 Kα2 0.713607 17373.2 6.7
Kα1 0.709317 17478.3 6.4
Kα (avg) 0.710747 17443.1
Kβ1 0.632303 19607.1

28. 3
59.5
E
r
c
r: diameter in meters
E: energy in GeV
Synchrotron Radiation
Synchrotrons are devices for circulating electrically charged particles (negatively charged
electrons or positively charged positrons) at nearly the speed of light.
The European Synchrotron Radiation Facility (ESRF) in Grenoble has a circumference of
844.39 m, is operated with an energy of 6 GeV.

29. Properties of Synchrotron Radiation
Intensity
The main advantage of synchrotron radiation for X-ray diffraction is its high intensity, which
is two orders of magnitude stronger than for a conventional X-ray tube. Another advantage is
the low divergence of the beam resulting in sharper diffraction spots.
Tunability
Any suitable wavelength in the spectral range can be selected with a monochromator.
Time Structure
Synchrotron radiation, in contrast to X-ray tube radiation, is produced in flashes by the
circulating bunches of charged particles. The ESRF operates in a single-bunch or multibunch
mode with a bunch length in the picosecond range. This allows structural changes in the
nanosecond timescale to be observed.

30. Detectors
Single-Photon Counters Photographic Film Image Plates
Area Detectors (CCD Cameras)

31. The Rotation (Oscillation) Instrument