Crystal Systems

Asymmetric unit

Which part of the unit cell can be taken?

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).

Crystal or no crystal?

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).

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.

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.

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 axisProper rotations:

Symmetry operation: proper rotation

Symmetry element: proper rotation axis

Symbols: n

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.

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.

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.

Screw axes

Screw axes nt= n-fold rotation + t/n translation along the rotation axis

For example: 21, 31, 41, 61 screw axes.

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.

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.

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

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.

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

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.

Limitations resulting from asymmetry of the motif

The 65 chiral space groups

Lattice properties

Minimum internal symmetry

Crystal system Point group

M Bravais type

B Latticetype

Chiral space groups z, M

a≠b≠cα≠β≠γ≠90°

None Triclinic 1 1 P 1 aP P1 1

a≠b≠cα=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≠cα=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≠cα=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 PI

1 tP P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212 8

2 tI I422, I4122 16

a=b≠cα=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=cα=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

Space group preferences Plain rotation axes Vs screw axes

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.

X-ray Sources

Sealed tubes

Rotating anodes

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.

X-ray optics

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

359.5

E

rc

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.

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.

Detectors

Single-Photon Counters Photographic Film Image Plates

Area Detectors (CCD Cameras)

The Rotation (Oscillation) Instrument