Atomic positions in crystals

Electron crystallography provides two major advantages over X-ray crystallography for determination of atomic positions in crystal structures extremely small samples can be analysed and the crystallographic structure factor phases can be determined from images. The crystallographic structure factor phases must be known in order to arrive at a structure model,... 

Homometric structure A structure with a uniquely different arrangement of atoms from another, but having the same sets of interatomic vectors, and hence the same Patterson map. Examples to date consist of homometric heavy-atom positions in crystal structures that also contain lighter atoms. The total crystal structure is not, however, homometric. 

X-ray diffraction studies of the molecular structure of solid proteins may be divided conveniently into two categories (1) investigations made directly on protein material, both fibrous and crystalline, and (2) determinations of atomic positions in crystals of amino acids and other compounds related to proteins. The former have been reported and discussed in some detail in a recent volume of this series (12), and will be but briefly mentioned here the latter constitute the subject matter of the present paper. The attack on the constitution and configuration of protein molecules and on the forces which hold them together in natural proteins is thus being carried out both from the top and from the bottom. Eecent advances in experimental techniques and in theoretical interpretations encourage the hope that the time is not far distant when these two complementary programs will meet and the detailed structure of many protein molecules will be known and understood. 

Table 2. Values of the Electrostatic potentials (V) at the nuclear positions in crystals and free atoms and corresponding mean inner potentials ((po) ...

In a second step d5mamical effects must be taken into account. For dynamical refinement we used a multi-slice least squares (MSLS) procedure [21], where the centre of laue circle, crystal thickness and scaling factor were refined for each zone separately with fixed atom positions. In the case of non-centrosymmetric space groups the enantiomorph used for refinement can be chosen for each zone, additionally. After the basic parameters have been derived satisfactorily, the atom positions can be refined. The R-values (see Eq. 2) refined by MSLS usually dropped down to 6-13%. 

From a single crystal, it is possible to measure the position and intensity of the hkl reflections accurately and from this data determine not only the unit cell dimensions and space group, but also the precise atomic positions. In most cases, this can be done with speed and accuracy, and it is one of the most powerful structural techniques available to a chemist. 

Another important contribution by Landau is related to symmetry changes accompanying phase transitions. In second-order or structural transitions, the symmetry of the crystal changes discontinuously, causing the appearance (or disappearance) of certain symmetry elements, unlike first-order transitions, where there is no relation between the symmetries of the high- and low-temperature phases. If p(x, y, z) describes the probability distribution of atom positions in a crystal, then p would reflect the symmetry group of the crystal. This means that for T> T p must be consistent with... 

One consequence of the insensitivity of the topology to the details of the electron density is that one does not need to know the exact electron density in order to determine its topology. It is sufficient, for example, to use the proatom density, namely the electron density obtained by placing free atoms or free ions at the observed atomic positions in the crystal. The topology is therefore not sensitive to the relaxation of the electrons that occurs on compound formation. 

The exact disposition of the side chains in a globular protein is difficult to define in solution. Although it is likely that the peptide main chain (backbone) of the protein is relatively rigid, the side chains have been shown to be undergoing motion of several different types (see lysozyme, peroxidase, and carboxypeptidase). This means that the full definition of atomic positions in the structure requires a knowledge of the time dependence of their coordinates. The motion of side chains is likely to be different in the crystal and solution states, but this difference may well... 

At present the X-ray diffraction method is used mostly to determine atomic positions in a crystal. However, X-ray scattering amplitudes depend directly on the electron-density distribution in a crystal, from which atomic positions can be derived on the assumption of coincidence of the nuclear positions and the center of gravities of total electron densities around atomic nuclei. 

To describe the contents of a unit cell, it is sufficient to specify the coordinates of only one atom in each equivalent set of atoms, since the other atomic positions in the set are readily deduced from space group symmetry. The collection of symmetry-independent atoms in the unit cell is called the asymmetric unit of the crystal structure. In the International Tables, a portion of the unit cell (and hence its contents) is designated as the asymmetric unit. For instance, in space group P2 /c, a quarter of the unit cell within the boundaries 0asymmetric unit. Note that the asymmetric unit may be chosen in different ways in practice, it is preferable to choose independent atoms that are connected to form a complete molecule or a molecular fragment. It is also advisable, whenever possible, to take atoms whose fractional coordinates are positive and lie within or close to the octant 0 < x < 1/2,0 < y < 1/2, and 0 < z < 1 /2. Note also that if a molecule constitutes the asymmetric unit, its component atoms may be related by non-crystallographic symmetry. In other words, the symmetry of the site at which the molecule is located may be a subgroup of the idealized molecular point group. 

Solid-state reaction in all cases implies the introduction of disorder into a crystal. Even for a perfectly homogeneous reaction the initial product sites are randomly distributed. Thus it is frequently not possible to refine atomic positions in the structures of crystalline reaction intermediates. In rare cases where atomic refinement of intermediates can be done it gives extremely precise information about how a solid-state reaction occurs. Examples of such cases will be discussed below [70,71]. 

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