Mark Farnworth - Pilkington European Technical Centre

1. State, with a very brief explanation, which of the following items are (i) due to the packing of atoms/molecules in a crystal, (ii) chosen by the crystallographer, and (iii) neither (i) or (ii):

   (a) The Unit Cell,
   (b) The Crystal Lattice,
   (c) The Origin in Reciprocal Space,
   (d) The Origin in the Crystal,
   (e) The Space-Group Symmetry of the Crystal.

   [5 Marks]

(a) The Unit Cell

Chosen by the crystallographer. The 'Unit cell' in 3D space is a parallelepiped. It is a repeat unit which when attached to itself many times over, forms the 3D structure of the material. Cubic, hexagonal and trigonal structures have very different unit cells.

(b) The Crystal Lattice

Due to the packing of atoms/molecules. The crystal lattice is the consequence of how atoms/molecules pack together throughout the crystal structure. The crystal lattice can be drawn and it shows the positions of the atoms/molecules in the crystal. They are represented as points. The lattice can be thought of as an infinite set of equally spaced points.

(c) The Origin in Reciprocal Space

Not chosen by the crystallographer, nor due to the packing of atoms/molecules. It corresponds to the natural point that the 'straight-through' X-ray beam contacts the Ewald sphere. The origin is assigned the hkl values (0,0,0).

(d) The Origin in the Crystal

The origin of the crystal lattice and the origin of the unit cell are chosen by the crystallographer. The origin of the unit cell is chosen to conveniently describe the contents of the unit cell e.g. points of high symmetry such as on an axis or where two or more axes intersect. Unit cells are also chosen to contain the smallest number of lattice points.

(e) The Space-Group Symmetry of the Crystal

Due to the packing of atoms/molecules. The 230 Space Groups describe all the combinations of rotation and screw axes, mirror and glide planes, plus points of inversion that can be applied to the different crystal types. Therefore, the Space-Group symmetry is a consequence of the the crystal structure and hence the packing of its atoms/molecules.

2. Briefly explain, with two examples, what is meant by the term Laue Class.

   [5 Marks]

The Laue Class is the name used to describe a class of symmetry point groups that are associated with a crystal system.

A Point Group may be defined as any group of symmetry operators that leave a point in space unmoved. The simplest point groups are due to rotation axis (point groups 1, 2, 3, 4 and 6) and rotary-inversion axes (point groups -1, m(-2), -3, -4 and -6). All of the other 32 crystallographic point group are combinations of these 10 point groups. For each Laue Class there can be two or more point groups (each representing a specific symmetry element.

Of the 32 crystallographic point groups, 11 possess a centre of inversion and are called centrosymmetric. These are known as the Laue classes and describe 3-dimensional diffraction symmetry (the symmetry applied to diffraction data).

Crystal System	Laue Classes
Triclinic	-1
Monoclinic	2/m
Orthorhombic	mmm
Tetragonal	4/m	4/mmm
Trigonal	-3	-3m
Hexagonal	6/m	6/mmm
Cubic	m-3	m-3m

Example 1:

Monoclinic Crystal System - Laue class 2/m

This has one mirror plane perpendicular to a two-fold rotation axis.The trans isomer of E-1,2-dibromoethylene has a Laue class of 2/m.

CHBr=CHBr

Top view down 2 axis	Side view

Example 2:

Orthorhombic Crystal System - Laue class mmm

This has 3 mirror planes, one parallel to each of the x,y,z axes, and thus perpendicular to each other.The oxalate anion has 3 mutually perpendicular mirror planes, 3 twofold rotation axes, and a point of inversion and has a Laue class of mmm.

C₂O₄⁼

Top view	Side view