Crystal structure statistics

For hydrogen bonds in crystal structures, statistical analyses are possible only if neutron and X-ray data are combined. Tb obtain the metrical information described in this chapter, it is necessary to combine the limited amount of accurate neutron diffraction data with the much more abundant, but less accurate or reliable, data from the X-ray crystal structure analyses. This combination is possible if the X-H covalent-bond lengths, which appear notoriously shortened in the X-ray analyses, are normalized to standard neutron diffraction values. 

At the beginning of the century, nobody knew that a small proportion of atoms in a crystal are routinely missing, even less that this was not a mailer of accident but of thermodynamic equilibrium. The recognition in the 1920s that such vacancies had to exist in equilibrium was due to a school of statistical thermodynamicians such as the Russian Frenkel and the Germans Jost, Wagner and Schollky. That, moreover, as we know now, is only one kind of point defect an atom removed for whatever reason from its lattice site can be inserted into a small gap in the crystal structure, and then it becomes an interstitial . Moreover, in insulating crystals a point defect is apt to be associated with a local excess or deficiency of electrons. 

Another recent database, still in evolution, is the Linus Pauling File (covering both metals and other inorganics) and, like the Cambridge Crystallographic Database, it has a "smart software part which allows derivative information, such as the statistical distribution of structures between symmetry types, to be obtained. Such uses are described in an article about the file (Villars et al. 1998). The Linus Pauling File incorporates other data besides crystal structures, such as melting temperature, and this feature allows numerous correlations to be displayed. 

Figure 6.4 Crystal structure of ar-tetragonal boron. This was originally thought to be B50 (4Bi2 + 2B) but is now known to be either B50C2 or B50N2 in which the 2C (or 2N) occupy the 2(b) positions the remaining 2B are distributed statistically at other vacant sites in the lattice. Note that this reformulation solves three problems which attended the description of the or-tetragonal phase as a crystalline modification of pure B ...

The crystal structure of 2-bromo-l,4-phenylenediyl bis(tran5-4-n-prop-ylcyclohexanoate) was determined by Hartung and Winter [114]. The molecules exhibit pseudo-centrosymmetry in consequence of a special kind of disorder within the crystal lattice. The peculiarity of the crystal structure is the disorder of the molecules with respect to the position of their bromine atoms which occupy the 2- or 5-position of the phenyl ring in a statistical manner. 

Statistical, Pharmacophore, and Homology Models and Crystal Structures of Drug-Metabolizing Enzymes... 

STATISTICAL, PHARMACOPHORE, AND HOMOLOGY MODELS AND CRYSTAL STRUCTURES OF DRUG-METABOLIZING ENZYMES... 

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

The scheme in Figure 2 illustrates a possible alternative explanation for the observation that bond lengths in m-enol systems are intermediate between single and double bonds. If the molecules have statistically disordered enol systems, the hydrogen atoms of the hydrogen bond will be distributed over two positions in the crystal structure. Indeed this was the case for the C polymorph of naphtazarin above 110 K at this temperature there is a second-order transition to a state with an ordered enol hydrogen [2],... 

Brunger AT (1992) Free R-value - a novel statistical quantity for assessing the accuracy of crystal-structures. Nature 355(6359) 472 t75... 

In the case of cisPI (A = A = H and B = CH3 and B = H), the two conformations corresponding to the two nonequivalent energy minima in the map of Figure 2.16c,68 (A+dsA T) [s(2/l) symmetry] and (A d.vA TA cis A T) (tc symmetry), correspond to the conformations of the chains in the crystal structure of cisPI proposed by Nyburg76 and Bunn,77 respectively. The X-ray diffraction data are, however, better accounted for by a disordered conformation, proposed by Corradini and co-workers,72,78 characterized by a statistical succession of monomeric units which assume these two energy minimum conformations. 

Note A typical example is provided by the up-down statistical coexistence of anticlined chains in the same crystal structure. 

It is impossible to directly measure phases of diffracted X-rays. Since phases determine how the measured diffraction intensities are to be recombined into a three-dimensional electron density, phase information is required to calculate an electron density map of a crystal structure. In this chapter we discuss how prior knowledge of the statistical distribution of the electron density within a crystal can be used to extract phase information. The information can take various forms, for example ... 

---