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Properties of Crystals and Crystal Symmetry:

Good Crystals

Diffract X-rays to reasonably high resolution (at least 3 ) Diffracts in 3 dimensions Isotropic diffraction Single (not twinned, split or attached to another crystal) Reproducible Stable Appearance is unimportant

A crystal is an ordered array of molecules

Solvent content: 25-90%

Molecules are linked by

Ionic interactions H-bonds Hydrophobic interactions

To study positions of all atoms we have to have a set of reference aces and an origin

This creates a box around the molecule known as the unit cell

The unit cell is defined as containing at least one repeating unit of the crystal

within it

The crystal can be reconstructed by translating the unit cell by multiples of the

axes lengths a, b, c.

a, b, c = lengths of the unit cell edges []

E (b & c), F (a & c), K (a & b) = angles between the edges

These parameters are measured from the data7 Crystal systems all are mathematically defined

Triclinic Monoclinic Orthorhombic Tetragonal Trigonal Hexagonal Cubic

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There can often be more than one molecule / unit cell

Asymmetric unit is the smallest unit from which the crystal can be created by applying crystal symmetry operations

(rotations & translations) to create the unit cell.

A two-fold rotation of the asymmetric unit will create

the unit cell shown on the right.

An object has symmetry if symmetry operations can be applied that leave the object

indistinguishable from the object in its original orientation.

Rotations or translations do not change the hand of an object, inversions or reflections will.

A pair of molecules with different hands are known as enantiomorphs

Proteins are composed only of L-amino acids; only rotations and translations are allowed in protein

crystals mirror plane symmetry is not involved (which is good because its very complicated)

Space Group of a crystal is a description of crystal symmetry relating to the

molecules within the unit cell. Each asymmetric unit within the unit cell is related to

others in the unit cell by symmetry operations.

Pure rotations rotations only, e.g. 2-fold 180r rotation Screw axes rotations combined with translation

o 21 screw axis = 180r, then translate unit cell dimension Centring a 2ndrepeating unit on 1 or more faces of the crystal

Several of these symmetry elements exist in the unit cell, giving rise to complex space

group definitions

e.g. P212121 Primitive lattice, a 21screw axis in each dimension

CentringPrimitive (P) lattice: one lattice point at each corner

I: extra lattice point at the centre of the unit cell

Face centred (F): an extra lattice point at the centre of the unit cell

A, B, C centred: an extra copy on one face

Pure rotational symmetry

No symmetry: P1

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2-fold rotation: P2 4-fold rotation: P4

There are a total of 230 space groups, however only 65 of these occur in proteins and large biomolecules because

these molecules are chiral mirror planes are not possible.

Symmetry is important, as the symmetry of a crystal affects the diffraction pattern

Symmetry axes within a crystal will lead to symmetry in the diffraction pattern

Some diffraction spots will be the same, as they are related by symmetry, and can be averaged

If symmetry can be described of all the molecules in a unit cell, the atom positions do not all need to be described.

The number of parameters (X,Y,Z coordinates) required to describe structure is 3 the number of atoms. If unit cells

contains 2 asymmetric units, then you only need half of this.