4.2: Unit Cells

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Introduction

The construction of a simple powder diffractometer was first described by Hull in 1917¹ which was shortly after the discovery of X-rays by Wilhelm Conrad Röntgen in 1895². Diffractometer measures the angles at which X-rays get reflected and thus get the structural information they contains. Nowadays resolution of this technique get significant improvement and it is widely used as a tool to analyze the phase information and solve crystal structures of solid-state materials.

Since the wavelength of X-rays is similar to the distance between crystal layers, incident X-rays will be diffracted, interacting with certain crystalline layers and diffraction patterns containing important structural information about the crystal can be obtained. The diffraction pattern is considered the fingerprint of the crystal because each crystal structures produce unique diffraction patterns and every phase in a mixture produces its diffraction pattern independently. We can grind bulk samples into fine powders, which are typical under 10 µm,²as samples in powder X-ray Diffraction (XRD). Unlike single crystal X-ray diffraction (X-ray Crystallography) technique, the sample will distribute evenly at every possible orientation and powder XRD collects one-dimensional information, which is a diagram of diffracted beam intensity vs. Bragg angle θ, rather than three-dimensional information.

Unit cells

“Crystals are built up of regular arrangements of atoms in three dimensions; these arrangements can be represented by a repeat unit or motif called the unit cell.”² In crystallography, all the crystal unit cells can be classified into 230 space groups. Some basic knowledge about crystallography is necessary for a well understanding of powder XRD technique. In crystallography, the basic possible classifications are: 6 crystal families, 7 crystal systems, 5 centering position, 14 Bravais lattices and 32 crystal classes.

Based on the angles and the length of the axes sides, unit cell can be divided into 6 crystal families, which are cubic, tetragonal, hexagonal, orthorhombic, monoclinic and triclinic. As the hexagonal family can have two different appearances, we can divide it into two systems which are trigonal lattice and hexagonal lattice. That is how the 7 crystal systems generate. If forget the shape of lattice and just consider the atoms' positions, we can divide the lattices into primitive lattices and non-primitive ones. A primitive lattice (also defined as simple) is the lattice with the smallest possible atomic coordination number,² e.g. when eight atoms lie in the eight corners. And all the other lattices are called non-primitive lattice. Based on the three-dimension position of the atoms in the unit cell, we can divided the non-primitive lattice into three types: face centered (F), side centered (C), body centered (I) and based centered(R).

Crystal family	Lattice system	Point group	14 Bravais lattices
Crystal family	Lattice system	Point group	Primitive (P)	Side-centered (C)	Body-centered (I)	Face-centered (F)
Triclinic (a)		C_i	aP
Monoclinic (m)		C_2h	mP	mS
Orthorhombic (o)		D_2h	oP	oS	oI	oF
Tetragonal (t)		D_4h	tP		tI
Hexagonal (h)	Rhombohedral	D_3d	hR
Hexagonal (h)	Hexagonal	D_6h	hP
Cubic (c)		O_h	cP		cI	cF

The 32 crystal classes refer to 32 crystallographic point group classified by the possible symmetric operations, which are rotation, reflection and inversion. You may wonder why only 32 possible point groups. The answer is crystallographic restriction, which means crystal systems can only have 5 kinds of rotation axes: 1-fold, 2-fold, 3-fold, 4-fold and 6-fold. Only the permissible rotation axes allow unit cells to grow uniformly without any openings among them.

The 230 space groups are combinations of 14 Bravais lattice and 32 crystal classes. Those space groups are generated from translations of related Bravais lattice and glide planes and/or screw axes of relative crystal classes. They are represented by Hermann-Mauguin notation. For example, space group 62, P_nma is derived from D_2h crystal class. P indicated it is primitive structure and n, m, a stand for a diagonal glide plane, a mirror plane and a axial glide planes. The space group belongs to orthorhombic crystal family.