Density, from crystal data

It is interesting to note that there are, in the body-centred cubic lattice unit, nine atoms, of which only one, the centre one, belongs entirely to the lattice unit itself. The eight corner atoms are each in turn shared by eight adjacent unit cubes, so that only one-eighth of each of them really belongs to the original unit. The unit therefore contains 1 + 8 x atoms, i.e. 2 atoms. It is thus possible to calculate density from crystal structure data. In the case of a body-centred cubic metal it can be seen that... 

Spectral data from a (111) face of a crystal of magnesium stannide are given in Table I. Using the value of 3.591 for the density,2 these data place nz/m = 0.248 for the first reflection. No reflections were found on the Laue photographs with values of n less than 0.26 A. U., calculated for the unit containing four Mg2Sn, with n — 1, and di0o = 6.78 0.02 A. U. 

The shortest cation-anion distance in an ionic compound corresponds to the sum of the ionic radii. This distance can be determined experimentally. However, there is no straightforward way to obtain values for the radii themselves. Data taken from carefully performed X-ray diffraction experiments allow the calculation of the electron density in the crystal the point having the minimum electron density along the connection line between a cation and an adjacent anion can be taken as the contact point of the ions. As shown in the example of sodium fluoride in Fig. 6.1, the ions in the crystal show certain deviations from spherical shape, i.e. the electron shell is polarized. This indicates the presence of some degree of covalent bonding, which can be interpreted as a partial backflow of electron density from the anion to the cation. The electron density minimum therefore does not necessarily represent the ideal place for the limit between cation and anion. 

Once a suitable crystal is obtained and the X-ray diffraction data are collected, the calculation of the electron density map from the data has to overcome a hurdle inherent to X-ray analysis. The X-rays scattered by the electrons in the protein crystal are defined by their amplitudes and phases, but only the amplitude can be calculated from the intensity of the diffraction spot. Different methods have been developed in order to obtain the phase information. Two approaches, commonly applied in protein crystallography, should be mentioned here. In case the structure of a homologous protein or of a major component in a protein complex is already known, the phases can be obtained by molecular replacement. The other possibility requires further experimentation, since crystals and diffraction data of heavy atom derivatives of the native crystals are also needed. Heavy atoms may be introduced by covalent attachment to cystein residues of the protein prior to crystallization, by soaking of heavy metal salts into the crystal, or by incorporation of heavy atoms in amino acids (e.g., Se-methionine) prior to bacterial synthesis of the recombinant protein. Determination of the phases corresponding to the strongly scattering heavy atoms allows successive determination of all phases. This method is called isomorphous replacement. 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

The overall density of surface functional groups can be determined from crystallographic data provided that the specific surface area of the sample and the crystal morphology can be determined accurately. The extent of development of the different crystal faces can be found by means of electron microscopy, but only if the crystal morphology is sufficiently well expressed. 

Figure 5.16 Resolved shear stress as a function of dislocation density for copper. Data are for polycrystalline copper O single-crystal copper with one slip system operative 0 single-crystal copper with two slip systems operative and A single-crystal copper with six slip systems operative. From K. M. Rails, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission John Wiley Sons, Inc.

OH- (g). The heat of formation of OH (g) can be estimated by the method of lattice energies. Using the approximate formula for the lattice energy, with rough estimates of the crystal constants of NaOH, KOH, and RbOH from their densities and molecular weights, we have calculated from the data on these substances, for OH- (g), Qf= — 64, —58, and —56, respectively. Lederle1 calculated a value of —81. 

In the following sections we shall focus on the structure and properties of the two-dimensional phases formed by the bent-core liquid crystals. In Sect. 2 we describe the structure studies by the X-ray diffraction (XRD) method, optical studies, and the response of different structures to the external electric field. In Sect. 3 we give theoretical models of the director and layer structure in 2D modulated phases and discuss how to reconstruct electron density maps from XRD data. 

Lithium iodide crystallizes in the NaCl lattice in spite of the fact that r+/r is less than 0.414. Its density is 3.49 g/cm3. Calculate from these data the ionic radius of the iodide ion. 

Thallium(I) bromide crystallizes in the CsCl lattice. Its density is 7557 kg/m3 and its unit cell edge-length, a, is 397 pm. From these data, estimate Avogadro s number. 

As shown by Eq. (54), growth rate G can be obtained from the slope of a plot of the log of population density against crystal size nucleation rate B° can be obtained from the same data by using the relationship given by Eq. (57), with n° being the intercept of the population density plot. Nucleation rates obtained by these procedures should be checked by comparison with values obtained from a mass balance (see the later discussion of Eq. (66)). 

---