The description of crystal structures

The faces of a crystal, irrespective of the overall shape of the crystal, could always be labelled with respect to the crystal axes. Each face was given a set of three integers, (h k l), called Miller indices. These are such that the crystal face in question made intercepts on the three axes of a/h, b/k and c/l. A crystal face that intersected the axes in exactly the axial ratios was given importance as the parametral plane, with indices (111). [Miller indices are now used to label any plane, internal or external, in a crystal, as described in Chapter 2, and the nomenclature is not just confined to the external faces of a crystal.] 

The application of Miller indices allowed crystal faces to be labelled in a consistent fashion. This, together with accurate measurements of the angles between crystal faces, allowed the morphology of crystals to be described in a reproducible way, which, in itself, lead to an appreciation of the symmetry of crystals. Symmetry was broken down into a combination of symmetry elements. These were described as mirror planes, axes of rotation, and so on, that, when taken in combination, accounted for the external shape of the crystal. The crystals of a particular mineral, regardless of its precise morphology, were always found to possess the same symmetry elements. 

Symmetry elements are operators. That is, each one describes an operation, such as reflection. When these operations are applied to the crystal, the external form is reproduced. It was found that all crystals fell into one or another of 32 different groups of symmetry operations. These were called crystal classes. Each crystal class could be allocated to one of the six crystal families. These symmetry elements and the resulting crystal classes are described in detail in Chapters 3 and 4. 

The descriptions above were made using optical techniques, especially optical microscopy. However, the absolute arrangement of the atoms in a crystal cannot be determined in this way. This limitation was overcome in the early years of the 20th century, when it was discovered that X-rays were scattered, or diffracted, by crystals in a way that could be interpreted to yield the absolute arrangement of the atoms in a crystal, the crystal structure. X-ray diffraction remains the most widespread technique used for structure determination, but diffraction of electrons and neutrons is also of great importance, as these reveal features that are complementary to those observed with X-rays. 

The physics of diffraction by crystals has been worked out in detail. It is found that the incident radiation is scattered in a characteristic way, called a diffraction pattern. The positions and intensities of the diffracted beams are a function of the arrangements of the atoms in space and some other atomic properties, such as the atomic number of the atoms. Thus, if the positions and the intensities of the diffracted beams are recorded, it is possible to deduce the arrangement of the atoms in the crystal and their chemical nature. The determination of crystal structures by use of the diffraction of radiation is outlined in Chapter 6. 

---

For the description of crystal structures one adds to the preceding molecular parameters an appropriate combination of rotation angles and translations 6 parameters or less according the space group symmetry. 

Ternary two anions compounds - References of the second column are concerned with the description of compounds. References of the third column are concerned with the description of crystal structures. 

Aside from the conventions mentioned for the cell choice, further rules have been developed to achieve standardized descriptions of crystal structures [36], They should be followed to assure a systematic and comparable documentation of the data and to facilitate the inclusion in databases. However, contraventions of the standards are rather frequent, not only from negligence or ignorance of the rules, but often for compelling reasons, for example when the relationships between different structures are to be pointed out. 

Pauling, L. (1960). The Nature of the Chemical Bond, 3rd ed. Cornell University Press, Ithaca, NY. A classic book that presents a good description of crystal structures and bonding in solids. 

Although it has often proved useful as a mnemonic, this approach has led to a number of misconceptions about relative atomic sizes and the origin of close-packing geometry, to some of which we allude below. More relevant in the present context is the observation that one natural and simple description of crystal structures has been. overlooked, and an unnecessarily complicated and opaque one used instead. We will provide many examples throughout this article. 

Table 2 records some examples of this phenomenon in which oxygen arrays in oxides are the same as metal atom arrays in alloys. Recognition of this fact has been exploited to simplify the description of complex alloys (see especially Andersson ), which is essentially the reverse of what we propose to do here, namely to simplify the description of oxide structures by giving them in terms of known alloy structures. Nevertheless Tables 1 and 2 provide striking evidence of Nature s parsimony in the use of patterns in crystal structures. 

Figure 8.50 A comparison of the performance of atom-atom potentials using the UNI method80 and PIXEL potentials in the description of the energy landscape for 133 naphthalene crystal structures. The experimental crystal structure is represented by a cluster of 5 points representing very similar structures with different unit cell settings. Energies are given on the abscissa in kj mol 1. The plot shows the usual way of representing the results of crystal structure calculations with the expectation that the most stable structure should be at the lowest energy and exhibit the highest density. (Reproduced with permission from The Royal Society of Chemistry).

In the description of crystals and crystal structures the two terms/om and habit have very specific and very different meanings. Form refers to the internal crystal structure and etymologically is the descendant of the Greek morph. Hence, polymorph refers to a number of different crystal modifications or different crystal structures, and the naming of different structures as Form F or a Form follows directly from this definition and usage. As we have seen above, the difference in crystal structure is very much, although not exclusively, a function of thermodynamics. Certainly, only the structures which are thermodynamically accessible can ever exist, but there often is a question of thermodynamic vs kinetic control over which particular structure may be obtained under any particular set of crystal growth conditions. 

The perturbation (modulation) function used in the description of aperiodic structures is obtained by associating interatomic distances (or larger fragments in the crystal structure) with length ratio x to 1 to letters L... 

The structural stability and the biological activity of protein molecules are dependent upon the interactions of the protein with solvent. Protein crystals typically contain 50% solvent and this component needs to be accounted for in the calculation of structure factors. In turn, the refinement of crystal structures and their solvent content leads to a description of the ordered water molecules and the bulk solvent continuum. 

A structure model must be based on a noncontradictory, closed and complete definition. A definition is closed if it does not contain indefinite elements and notions, and it is complete if it includes the description of all structure elements. Thus, for instance, the model in which the amorphous structure is considered as a dislocationally disordered crystal [6.21, 22] becomes not closed if the dislocation structure (in particular, the one of dislocation core) is not defined. At high density of dislocations when their cores may overlap and their structure becomes very indefinite, the model is not closed. The free-volume model [6.23 25], in which the question about geometry and topology of atomic configurations is put aside, is not complete. 

Transition metals are important materials with intriguing properties and they have been studied with ever improved methods. A major difficulty is posed by the standard one-electron models where the tight-binding model seems appropriate for the narrow, so-called d-bands while near-plane-wave crystal orbitals are adequate for the conduction bands. Canonical Hartree-Fock solutions are awkward starting points for the description of magnetic structures and the use of spin-polarized versions destroys basic symmetry properties. 

Present knowledge of the details of the conformation of proteins is based almost exclusively on results of studies of protein crystals by x-ray diffraction. Protein crystals contain anywhere from 20 to 80% solvent (1 ) (dilute buffer, often containing a high molarity of salt or organic precipitant). While some solvent molecules can be discerned as discrete maxima of the electron density distribution calculated from the x-ray results, the majority of the solvent molecules cannot be located in this manner most of the solvent appears to be very mobile and to have a fluctuating structure perhaps similar to that of liquid water. Many additional distinct locations near which a solvent molecule is present during much of the time have been identified in the course of crystallographic refinement of several small proteins (2,3,4,5, 6), but in all cases the description of solvent structure in the crystal is incomplete probably because only a statistical description is inherently appropriate. 

Several modular structures and related series are described in the section devoted to the modelling of crystal structures. To underline the utility of a modular description of the crystal structures (when practicable, of course), we first provide examples of compounds based on perovskite modules and important for materials science. 

The atomic surface order is described in terms of a simple unit cell and techniques for the preparation of surfaces with defined atomic order are well established. The description of mesoscopic structures is not as straightforward for a single-crystal surface mesoscopic properties can be, e. g., terrace widths and step densities, for dispersed electrodes the size and distribution of particles. Real-space information under in-situ electrochemical conditions is required for the characterization of such mesoscopic properties. This information can only be derived from the application of scanning probe techniques, which were introduced to electrochemistry in the mid-1980s and give high-resolution real-space images of electrode siufaces under in-situ electrochemical conditions. 

The systematic description of crystal structures is presented primarily in the well-known Structurbericht. The classification of crystals by the Structurbericht does not reflect their crystal class, the Bravais lattice, but is based on the crystaUochemical type. This makes it inconvenient to use the Structurbericht categories for comparison of some individual crystals. Thus, there have been several attempts to provide a more convenient classification of crystals. Table 5 presents a compilation of different classifications which allows the reader to correlate the Structurbericht type with the international and Schoenflies point and space groups and with Pearsons symbols, based on the Bravais lattice and chemical composition of the class prototype. The information included in Table 5 has been chosen as an introduction to a more detailed crystal-lophysical and crystaUochemical description of solids. 

---