1.1 Historical remarks
Crystals can easily be distinguished from all other types of matter by optical inspection. They are generally transparent with sharp edges and plain faces, properties which make them unique in nature. For these samples the Greeks introduced the notation κρνσταλλοζ (kristallos = ice) because they were comparable in shape with ice crystals. Over the centuries crystals were the subject of detailed examination in mineralogy or admired as precious stones. The microscopic nature of the regularities which could be observed macroscopically was originally not known.
It was a milestone in crystallography when in 1912 Max von Laue together with Friedrich and Knipping performed the first X-ray diffraction experiment on a crystal [1]. This experiment did not only confirm the character of X-rays as electromagnetic waves, but also proved the periodic arrangement of matter in a crystal. This opened up a rapidly developing field of research in crystallography, the X-ray structure analysis.
In the first months this field saw an exciting development. Still in 1912 X-ray diffraction was described theoretically by von Laue himself, Ewald [2] and their English colleagues W.H. Bragg and W.L. Bragg (father and son). The famous “Bragg equation” was reported on a conference in November 1912 and already in 1913 the first X-ray crystal structure, namely of sodium chloride, was published by the Braggs [3].
Then, however, for about 40–50 years X-ray structure analysis was confronted with severe problems. There was on one hand the so-called “phase problem” (see Chapters 2 and 7), which could be solved only in exceptional cases at that time, and there was on the other hand the need to carry out huge numerical calculations especially for larger structures, which was impossible in a time where computers were practically not available.
The decisive breakthrough came in the 1960s based on two developments which happened almost at the same time. The so-called “Direct Methods” for the solution of the crystallographic phase problem became available through sophisticated and easy-to-use computer programs, and the introduction of always faster and more and more powerful computers allowed almost unlimited numerical calculations.
Figure 1.1 shows the rapid development of crystal structure research in the last 50 years, illustrated by the exponentially increasing number of entries into the two most important databases of worldwide published crystal structures. While it took 50 years for the first 1000 structures, we see now more than 40 000 structures per year. In 2009 the International Union of Crystallography (IUCr) announced the 500 000th entry into the Cambridge Structural Data Base (CSD) [7].
It follows that nowadays crystal structure analysis is a reliable, accurate and one of the fastest methods to determine three-dimensional atomic structures in the solid state. Applications take place in various disciplines like chemistry/biochemistry, biology, pharmaceutical research, solid state physics, material sciences etc.
Figure 1.1: Entries of published crystal structures of organic and organometallic compounds into the Cambridge Structural Data Base (CSD) [4] in five year intervals. Insert: Corresponding entries of macromolecular structures into the Protein Data Bank (PDB) [5]. For inorganic structures, a third international database, the Inorganic Crystal Structure Data Base exists [6].
1.2 The crystal lattice: Basic definitions
1.2.1 Periodicity, lattice constants
A single crystal is a sample of material where the distribution of matter q(r) (the nature of q(r) will be specified later in detail, see Section 2.1) is periodic in three dimensions; that is (Figure 1.2), there are three non-coplanar vectors a, b, c with
The volume element defined by a, b, c is called the unit cell. It represents the nonperiodic unit. The whole crystal lattice is obtained by periodic sequences of unit cells in all three dimensions. Generally the unit...
