Unit cell: content, density

Two helices are packed antiparallel in the orthorhombic unit cell. Association of the helices occurs through a series of periodic carboxylate potassium water - carboxylate interactions. An axial projection of the unit-cell contents (Fig. 23b) shows that the helices and guest molecules are closely packed. This is the first crystal structure of a polysaccharide in which all the guest molecules in the unit cell, consistent with the measured fiber density, have been experimentally located from difference electron-density maps. The final / -value is 0.26 for 54 reflections, of which 43 are observed, and it is based on normal scattering factors.15... 

The theoretical density of a crystal can be obtained from the volume of the unit cell and the mass of the unit cell contents. The results of an X-ray diffraction structure determination gives both of these data, as the unit cell dimensions are accurately measured and the type and number of formula units in the unit cell are also determined. An example of this type of calculation for FeO follows ... 

If the mass of the unit cell contents is Mand the unit cell volume is 7then the density, p is given by... 

The crude molecular image seen in the F0 map, which is obtained from the original indexed intensity data (IFobsI) and the first phase estimates (a calc), serves now as a model of the desired structure. A crude electron density function is devised to describe the unit-cell contents as well as they can be observed in the first map. Then the function is modified to make it more realistic in the light of known properties of proteins and water in crystals. This process is called, depending on the exact details of procedure, density modification, solvent leveling, or solvent flattening. 

The weight of the unit cell contents can be calculated if the unit cell dimensions and the density of the crystal are known. The results ma indicate the presence of solvent of crystallization in the crystal it may also indicate that the crystal cannot possibly contain what it was expected to. Therefore, crystal density measurements are essential in the early stages of a crystal structure analysis. 

Measure the density of flotation in a mixture of liquids. Check the weight of the unit-cell contents. 

When the angles are all 90° (so that their cosines are 0), this formula reduces to the simple result V = abc for the volume of a rectangular box. If the mass of the unit cell contents is known, the theoretical cell density can be computed. This density must come close to the measured density of the crystal, a quantity that can be... 

The content of the unit cell is defined by specifying the positions of all the atoms it contains. It is more convenient, however, to specify the unit cell content by the distribution pu(r) of the appropriate scattering length density, so that the same expression can be used in discussing both x-ray and neutron diffraction. The convolution product... 

Biological apatite crystals are too small to allow direct measurement of their density. However, the calculated density from estimated unit cell contents (see later section on atomic structure of biological apatites) is about 5% less than HAP (Elliott 1997). 

As mentioned at the beginning of this review, the very large surface area of the crystals in bone makes it very problematical to try to determine the unit cell contents from the chemical composition of the mineral. With the much smaller surface area of the crystals in enamel, it is much more feasible and several formulae have been published (Elliott 1997). As the density of enamel apatite is not known, the absolute contents cannot be determined, so it is usually assumed that the number of ions in the phosphate sites... 

Table 1.9. Unit cell contents, space group, unit cell parameters" and calculated density for model compounds for PANI at 23 1°C. (Reproduced from ref 340 with kind permission. Copyright (1988) Elsevier Science S.A.)...

Syntheses have been made of energetically substituted cubanes. For example, 1,3,5,7-tetranitrocubane [10] (shown in Fig. 26) has been synthesized. As noted previously, its density is 1.814 g/cc. Another is the flu-dinitroethy] ester of tetracarboxycubane [36] (Fig. 27). The density is 1.762 g/cc. An energetic ammonium perchlorate substitution of cubone has been made in the form of 1,4-bis-cubanediammonium perchlorate hydrate [40] (Fig. 28). The dashed lines in Fig. 28 represent some of the many hydrogen bonds that occur in the crystal whose density is 1.7S5 g/ce. An estimate of the dry density extrapolated from this work is 1.83 g/cc- It was obtained by subtracting the volume and mass of water from the unit cell contents. The volume of H2O was taken to be the same as that in liquid water. 

Fig. 9.23. Six different choices (I-VI) of unit cell content (motifs) for a linear chain (LiH)oo- Each cell has the same length a = 6.3676 a.u. There are two nuclei Li + and H" " and two Gaussian doubly occupied is atomic orbitals (denoted by xi and X2, with exponents 1.9815 and 0.1677, respectively) per cell. Motif I corresponds to a common sense situation both nuclei and electron distribution determined by xi and X2 are within the section (0, a). The other motifs (II-VI), all corresponding to the same unit cell (0, a) of length a are very strange. Each motif is characterized by the symbol (k,l,m,n), which means that the Li nucleus, H nucleus, xi and X2 are shifted to the right by ka, la, ma, and na, respectively. All the unit cells with their contents (motifs) are fully justed, equivalent from the mathematical point of view, and, therefore, legal from the point of view ofphysics. Note that the nuclear framework and the electronic density coiresponding to a cell are very different for all the choices.

Regarding the free parameters, for exarr tle a, the notation 8.174(1) in Table 1.3-5 means 8.174 0.001. The chemical formula and the unit cell contents can easily be calculated from the site multiplicities (given by the Wyckoff positions) and the occupancies. So can the (crystallographic) density, using the appropriate atomic masses. 

From these, the density of states (DOS) can be derived, either for the whole unit cell content or for certain atoms or atom types. This is shown in Figure 6. The DOS curves for the isostmctural Dyl2 and Prl2-IV show that the 4f and 5d states are much more separated for Dyl2 for Prl2-IV, they... 

Before inclusion, the data are carefully checked for internal consistency and plausibility. Automatic checks comprise tests of the unit cell content as derived from the chemical formula and from the site occupation, experimental and calculated density, and charge balance. The interatomic distances are checked against the sum of the ionic radii. In case of unusually short interatomic distances or other irregularities, an expert rechecks the data for misprints or other sources of errors, for example by comparing the calculated with the published distances. Some of the problems find an immediate solution in other cases an e-mail message is sent to the author if an address is available. Finally flags are set to document the test results. 

X-ray diffraction allows the dimensions of the unit cell to be accurately measured. If the structure type of the material is known, the ideal cell contents are also known. Thus, the unit cell of a crystal of composition M2O3 that adopts the corundum structure contains 12 M atoms and 18 O atoms (Supplementary Material, SI). This readily allows the theoretical density of a solid to be calculated. The weights of all of the atoms in the cell are added, and this is divided by the cell volume. 

The crystallinity of organic pigment powders makes X-ray diffraction analysis the single most important technique to determine crystal modifications. The reflexions that are recorded at various angles from the direction of the incident beam are a function of the unit cell dimensions and are expected to reflect the symmetry and the geometry of the crystal lattice. The intensity of the reflected beam, on the other hand, is largely controlled by the content of the unit cell in other words, since it is indicative of the structural amplitudes and parameters and the electron density distribution, it provides the basis for true structural determination [32],... 

From the cubic unit cell dimension a, we can calculate the volume of the unit cell, Elf the density, p, of the crystals are known, then the mass of the contents of the unit cell, M, can also be calculated... 

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