Crystallography

Before the 20th century, the study of crystals was based on physical measurements of their geometry using a goniometer.[4] This involved measuring the angles of crystal faces relative to each other and to theoretical reference axes (crystallographic axes), and establishing the symmetry of the crystal in question. The position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to each face is plotted on the net. Each point is labelled with its Miller index. The final plot allows the symmetry of the crystal to be established.[5][6]

The discovery of X-rays and electrons in the last decade of the 19th century enabled the determination of crystal structures on the atomic scale, which brought about the modern era of crystallography. The first X-ray diffraction experiment was conducted in 1912 by Max von Laue, while electron diffraction was first realized in 1927 in the Davisson–Germer experiment. They later developed into the two main branches of crystallography, X-ray crystallography and electron crystallography. The quality and throughput of solving crystal structures were greatly increased in the second half of the 20th century, with the developments of customized instruments and phasing algorithms. Nowadays, crystallography is an interdisciplinary field, supporting theoretical and experimental discoveries in various domains.[7] The modern-day scientific instruments for crystallography vary from laboratory-sized equipments, such as diffractometers and electron microscopes, to dedicated large facilities, such as photoinjectors, synchrotron light sources and free-electron lasers.

Crystallographic methods depend mainly on analysis of the diffraction patterns of a sample targeted by a beam of some type. X-rays are most commonly used; other beams used include electrons or neutrons. Crystallographers often explicitly state the type of beam used, as in the terms X-ray diffraction, neutron diffraction and electron diffraction. These three types of radiation interact with the specimen in different ways.

• X-rays interact with the spatial distribution of electrons in the sample.
• Neutrons are scattered by the atomic nuclei through the strong nuclear forces, but in addition, the magnetic moment of neutrons is non-zero. They are therefore also scattered by magnetic fields. When neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels. However, the material can sometimes be treated to substitute deuterium for hydrogen. Because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies.
• Electrons are charged particles and therefore interact with the total charge distribution of both the atomic nuclei and the electrons of the sample.

It is hard to focus x-rays or neutrons, but since electrons are charged they can be focused and are used in electron microscope to produce magnified images. There are many ways that transmission electron microscopy and related techniques such as scanning transmission electron microscopy, high-resolution electron microscopy can be used to obtain images with in many cases atomic resolution from which crystallographic information can be obtained. There are also other methods such as low-energy electron diffraction, low-energy electron microscopy and reflection high-energy electron diffraction which can be used to obtain crystallographic information about surfaces.

• Coordinates in square brackets such as [100] denote a direction vector (in real space).
• Coordinates in angle brackets or chevrons such as <100> denote a family of directions which are related by symmetry operations. In the cubic crystal system for example, <100> would mean [100], [010], [001] or the negative of any of those directions.
• Miller indices in parentheses such as (100) denote a plane of the crystal structure, and regular repetitions of that plane with a particular spacing. In the cubic system, the normal to the (hkl) plane is the direction [hkl], but in lower-symmetry cases, the normal to (hkl) is not parallel to [hkl].
• Indices in curly brackets or braces such as {100} denote a family of planes and their normals. In cubic materials the symmetry makes them equivalent, just as the way angle brackets denote a family of directions. In non-cubic materials, <hkl> is not necessarily perpendicular to {hkl}.

The International Tables for Crystallography[10] is an eight-book series that outlines the standard notations for formatting, describing and testing crystals. The series contains books that covers analysis methods and the mathematical procedures for determining organic structure through x-ray crystallography, electron diffraction, and neutron diffraction. The International tables are focused on procedures, techniques and descriptions and do not list the physical properties of individual crystals themselves. Each book is about 1000 pages and the titles of the books are:

Vol A - Space Group Symmetry,

Vol A1 - Symmetry Relations Between Space Groups,

Vol B - Reciprocal Space,

Vol C - Mathematical, Physical, and Chemical Tables,

Vol D - Physical Properties of Crystals,

Vol E - Subperiodic Groups,

Vol F - Crystallography of Biological Macromolecules, and

Vol G - Definition and Exchange of Crystallographic Data.

• American Crystallographic Association
• Learning Crystallography
• Web Course on Crystallography
• Crystallographic Space Groups