Crystal Systems

What is a crystal

How is the structure of crystal related to the fundamental repeating unit?

The crystalline state of matter is characterized by a long-range, three-dimensional order .

We call the structure that is regularly repeated in space the motif and the conceptual array of

points which defines the geometrical relation between the motifs is called the lattice.

The motif is the structural unit which is repeated regularly in space. The lattice is a conceptual

array of points in space which serves to define the geometrical relationship between the motifs

in a structure.

For a two-dimensional system, we speak of a plane lattice. There are two particular

features of the motif and lattice which are important:

(a) The position of the lattice points with respect to the motif is arbitrary, provided that we are

consistent.

(b) The motif may be quite complex structurally.

In three dimensions, any crystal has a well-defined repeat unit, the motif and each motif is

associated with a lattice point so that the overall lattice is a three-dimensional array, usually

called a space lattice. Thus, a crystal is of the form

Crystal structure = lattice x motif (where x means associated with).

Characterization of the Crystals

If crystals of sufficient size (≥ 0.1 mm) have been grown, they must first be characterized:

The quality of the crystals depends on the ordering of the molecules in the unit cell. Because

of thermal vibrations and static disorder, the positions of the atoms are not strictly fixed. As a

consequence, the intensities of the X-ray reflections drop at higher diffraction angles. The

diffracted X-ray beams can be considered as being reflected against lattice planes and that the

relation between the lattice plane distance d and the diffraction angle is given by 2dsinθ = nλ.

Diffraction patterns with maximal observed resolution corresponding to a lattice spacing of 5

Å can be regarded as poor, of 2.0–2.5 Å as normal, and of 1.0–1.5 Å as high quality.

What is their quality?

Most crystals cannot be considered as ideal single crystals because the regular repetition of

the unit cells is interrupted by lattice defects. The diffraction pattern of such crystals can be

regarded as the sum of the diffraction patterns originating from mosaic blocks with slightly

different orientations. The mosaicity in good quality protein crystals is moderate, between

0.2 and 0.5 . The cell dimensions can easily be derived from the diffraction pattern collected

with an image plate or area detector.

An estimation of the number of molecules per unit cell (Z) can be made by a method proposed

by Matthews. He found that for many protein crystals the ratio of the unit cell volume and the

molecular weight is between 1.7 and 3.5 Å3/Dalton, with most values around 2.15 Å3/Dalton.

This number is called Matthews Coefficient (VM).

VM (specific volume) is calculated using the formula: VM = V/(n*M)

where, V = volume of the unit cell, n = the number of asymmetric units, M = molecular

weight of contents of the asymmetric unit

Example:

Suppose a crystal belongs to space group C2 and has a unit cell volume of 3,19,000 Å3. The

molecular weight Mr of the protein is known to be 32,100 Da. Then, for Z = 2, 4, or 8, VM is

respectively 5, 2.5, or 1.25 Å3/Dalton. This crystal most likely has four molecules in the unit

cell. Space group C2 has four asymmetric units and, therefore, there is one protein molecule

per asymmetric unit.

How many protein molecules are in the unit cell and in one asymmetric unit?

Characterization of the Crystals

What are the unit cell dimensions?

To what space group do they belong?

Characterization of the Crystals

It is the lattice that determines the overall structure of the crystal so it is the geometry of the

lattice that is used to categorize crystal structures.

What is the structure of the molecules within a crystal? and

What is the nature of the geometrical array which defines the way in which the

molecules are arranged in space?

Basic concepts of crystal assembly

Lattice and Symmetry

One of the most important properties of well-ordered, regular geometrical objects is that they

are symmetrical.

Periodic lattices

Let us see how to arrange tiles into periodic, 2-D patterns in order to fit a new floor in our X-

ray laboratory. We are provided with a collection of arbitrary white tiles with no discernible

pattern painted on them (i.e. they harbor no motif) and examine if we can cover the floor with

a periodically repeating pattern of these tiles.

The description of the plane lattice

We may define these directions by two vectors a and b, called fundamental lattice vectors.

They will define a square, which we may call the fundamental lattice cell. Thus, we may

express any lattice point r in terms of a and b as

r = pa + qb (where p and q are integers corresponding to the number of steps required to

reach r from the origin.

One immediately recognizes that only certain types of tiles can fill the plane. The property

common to the oblique, rectangular and square tiles is that their shape is defined by two sets

of intersecting, equidistant parallel edges or lines. These lines form the lattice of each tile

pattern.

Another important property of the fundamental lattice cell is that the cell chosen must have

the property that a close-packed assembly of cells fills all space with no gaps between cells.

An array of pentagons just cannot fit together to fill all space and so no two-dimensional

lattice based on a pentagonal primitive cell may exist.

Oblique P (a≠b, γ≠90 )

γba

90

a

b

Square P (a=b, γ=90 )

120°ba

Hexagonal P (a=b, γ=120 )

90

a

b

Rectangular P (a≠b, γ=90 )

90

a

b

Rectangular I (a≠b, γ=90 )

The five plane lattices

The five plane lattices may be divided into four systems, depending on the shape of the unit

cell. Once lattice is based on the square, one on the hexagon (or equilateral triangle), one on a

general oblique parallelogram and two on the rectangle.

Lattice

system

Lattice type Conventional

representation

Representative

points

Oblique P a ≠ b, γ ≠ 90 (0,0)

Square P a = b, γ = 90 (0,0)

Hexagonal P a = b, γ = 120 (0,0)

Rectangular P a ≠ b, γ = 90 (0,0)

I a ≠ b, γ = 90 (0,0), (1/2, 1/2)

Primitive cells: A means of categorizing fundamental lattice cells is to determine the number

of lattice points associated with a given fundamental lattice cell. All fundamental lattice cells

associated with a single lattice point are called Primitive cells (1/4 x 4 = 1 for 2D cell).

Non-primitive cells: In general, fundamental lattice cells associated with more than once

lattice pint are called multiple cells or Non-primitive cells (1/4 x 4 + 1 = 2).

Periodic patterns, motifs and unit cells

Crystals are translationally periodic arrangement of unit cells. Once we have created the unit

cells, we can stack them into a repeating arrangement that forms the actual 2-D crystal

corresponding to the tiles floor of the imaginary laboratory.

The crystal is derived from an oblique lattice from 2-D crystal system. The unit cell does not

have any additional internal symmetry and contains only one molecule, hence it is thus called

Primitive (P). Thus, we can classify out unit cell as belonging to plane group p1.

Alternate unit cell origins

The structure itself remains unaffected by a different choice of the p1 lattice origin in the

crystal. Only primitive p1 structures and their 3D relative p1 allow an entirely arbitrary choice

of origin, but many others allow multiple specific choices of origin.

Symmetry within the unit cell

One of the most basic ways to create a second copy of the motif on one tile would be to rotate

the molecule by 180 .

This manipulation is called a symmetry operation and known as a 2-fold rotation (or in short a

2-fold rotation with the axis being perpendicular to the tile or plane).

The multiplicity n of a plain rotation axis and its rotation angle φ are related by φ = (360/n) .

A 3-fold symmetry operation around a 3-fold rotation axis or in short a 3-fold rotation thus

involves three consecutive rotations by 120 , a 4-fold rotation axis four consecutive rotations

by 90 , and a 6-fold rotation six consecutive rotations by 60 , with each rotation successively

creating additional copies of the original molecule.

In periodically repeating systems such as crystals, specific limitations exist as to what

operations can be used to created additional copies of the motif in unit cell.

1. The application of a symmetry operation to a motif can not generate any changes within

the motif.

2. As a consequence of the closure requirement and the reasonable limitation that only a

finite number of symmetry operations are allowed, a plain rotation can only involve

operations by (360/n) .

3. Such cyclic operation generates, after a certain number of repeated applications, again the

original motif in its original position (i.e. at the same point, therefore such rotations are

also called point group operations).

Are all kind of internal symmetry operations allowed?

4. As an example in a 4-fold rotation we reach the starting position after four consecutive

rotations of 90 , in the process creating three more symmetry related copies of the original

motif (called symmetry equivalents of the motif).

5. The allowable symmetry operations must be compatible with the translational

requirements of the specific lattice. As a consequence, only certain symmetry operations

can occur in crystals.

6. Most fundamentally, only 2-, 3-, 4- and 6- fold rotations are allowed.

Two-fold rotations

Multiple unit-cell choices

Four-fold rotations

Note that this structure has a large solvent channel in the center of the chosen cell, and a

somewhat smaller channel at the corners. This situation is in fact quite typical for crystals

with higher symmetry axes and for protein oligomers of high rotational symmetry.

Three-fold and six-fold rotations

The hexagon-shaped tile does not fit into a basic grid system spanned by two basis vectors,

although it can fill a plane without gaps. The this still works is that-given proper 3- or 6-fold

internal symmetry-the hexagonal tile can be represented by three equivalent tiles one third the

size of the hexagon. These unit cells now satisfy the translational requirements along a plane

lattice.

A two-dimensional lattice with 3-fold symmetry axes perpendicular to the plane of the figure.

A closer look reveals that in order to be able to assemble the hexagon from the new unit cell

tiles we need at leas a 3-fold axis in the center of the hexagon, which automatically generates

the triads in the unit cell corners and two additional ones on the unit cell diagonal.

The plane groups

Lattice

properties

Minimum

internal

symmetry

2-D crystal

system

Cell

centering

Plane groups

a≠b, γ≠90 None Oblique p p1, p2

a≠b, γ=90 2-fold Rectangular p pm, pg, p2mm, p2mg, p2gg

a=b, y=90 4-fold Square p p4, p4mm, p4gm

a=b, y=120 3-fold Trigonal p p3, p3m1, p31m

6-fold Hexagonal p p6, p6mm

The combination of symmetry elements with lattice translation in two dimensions is called

plane group.

Why can not we build the remaining 12 plane structures?