Page 80 - Macromolecular Crystallography

FIRST ANALYSIS OF MACROMOLECULAR CRYS TALS 69

system is chosen to contain at least one lattice point. (I; German for Innenzentrierte), where a unit cell
A unit cell that has only one lattice point at each of contains a second lattice point at the centre of the cell,
its corners is called primitive (P). Note that each cor- face-centred (F) having an additional lattice point
ner of a unit cell in a crystal is shared by eight other at the centre of each of its six faces and C-centred
unit cells; therefore, a primitive cell contains only (C) having an additional lattice point at only one
one lattice point. However, at times it is advanta- of the six faces (by definition the ab face). Centring
geous to select a unit cell that contains more than expands the volume of the unit cell. Whereas C-
one lattice point. These cells are called centred cells; and I-centring double the volume relative to that
there are three such cells. They are body-centred the P cell, F-centring quadruples the volume. When
centring is included, the number of unique lattices
expands to 14 (Table 4.3 and Fig. 4.8). These 14 lat-
tices are known as the Bravais lattices; they were
first described by Frankheimer and Bravais in the
mid-nineteenth century.
c A unique combination of crystal symmetry
α β elements and centring is called a space group.
γ There are only 230 possible space groups. However,
b
this number is reduced to 65 for biological macro-
a
molecules because the chirality of their biological
Figure 4.7 A unit cell, showing the six parameters known as lattice building blocks. The 65 biologically relevant space
constants. groups are listed in Table 4.4.

Table 4.2 Elements of crystal symmetry

Symmetry element Description

◦
Rotation axes Counterclockwise rotation of 360 /n about an axis, where n is 1, 2, 3, 4 or 6
2-fold axis is rotation by 180 ◦
3-fold axis is rotation by 120 ◦
4-fold axis is rotation by 90 ◦
6-fold axis is rotation by 60 ◦
Screw axes Same as rotation axis, but followed by a translation of p/n along the rotation axis,
where p is an integer <n
◦
2 1 screw axis is rotation by 180 followed by translation of 1/2 of a unit cell
◦
3 1 screw axis is rotation by 120 followed by translation of 1/3 of a unit cell
3 2 screw axis is rotation by 120 followed by translation of 2/3 of a unit cell
◦
◦
4 1 screw axis is rotation by 90 followed by translation of 1/4 of a unit cell
◦
4 2 screw axis is rotation by 90 followed by translation of 1/2 of a unit cell
◦
4 3 screw axis is rotation by 90 followed by translation of 3/4 of a unit cell
6 1 screw axis is rotation by 60 followed by translation of 1/6 of a unit cell
◦
6 2 screw axis is rotation by 60 followed by translation of 1/3 of a unit cell
◦
6 3 screw axis is rotation by 60 followed by translation of 1/2 of a unit cell
◦
◦
6 4 screw axis is rotation by 60 followed by translation of 2/3 of a unit cell
◦
6 5 screw axis is rotation by 60 followed by translation of 5/6 of a unit cell
Inversion centre All points inverted through a centre of symmetry
Mirror plane Reflection through a plane
Glide plane Same as mirror plan, but followed by a translation of half the unit cell parallel to the plane; glide planes
are not relevant in macromolecular crystallography due to the chirality of the biological building blocks