Unit cells, of crystals

T — equilibrium melting temperature A//f = heat of fusion a, b — unit cell of crystal a = surface free energy of crystal surface free energy of crystal Q = energy to form chain folding in crystal. h from Thomas-Stavely relationship a = 0AAHf(ab)l/2. 

Unit Cell of Crystals of Poly (o -f luorosty rene)... 

Figure 3.7. The volume of unit cell of crystallizing polypropylene vs. concentration of nucleating agent (sodium benzoate). [Adapted, by permission, from Xu, T Lei, H Xie, C S, Mater. Design, 24, 227-30,2003.]...

Figure 1. The unit cell of crystal structure of molecule I (monoclinic space group = P21/n a = 9.435A b = 18.105A c = 14.373A gt= 90.00° 103.62° 3 90.00°).

The rocksalt stmcture is illustrated in figure Al.3.5. This stmcture represents one of the simplest compound stmctures. Numerous ionic crystals fonn in the rocksalt stmcture, such as sodium chloride (NaCl). The conventional unit cell of the rocksalt stmcture is cubic. There are eight atoms in the conventional cell. For the primitive unit cell, the lattice vectors are the same as FCC. The basis consists of two atoms one at the origin and one displaced by one-half the body diagonal of the conventional cell. 

When atoms, molecules, or molecular fragments adsorb onto a single-crystal surface, they often arrange themselves into an ordered pattern. Generally, the size of the adsorbate-induced two-dimensional surface unit cell is larger than that of the clean surface. The same nomenclature is used to describe the surface unit cell of an adsorbate system as is used to describe a reconstructed surface, i.e. the synmietry is given with respect to the bulk tenninated (unreconstructed) two-dimensional surface unit cell. 

It is relatively straightforward to detemiine the size and shape of the three- or two-dimensional unit cell of a periodic bulk or surface structure, respectively. This infonnation follows from the exit directions of diffracted beams relative to an incident beam, for a given crystal orientation measuring those exit angles detennines the unit cell quite easily. But no relative positions of atoms within the unit cell can be obtained in this maimer. To achieve that, one must measure intensities of diffracted beams and then computationally analyse those intensities in tenns of atomic positions. 

Ewald summation was invented in 1921 [7] to permit the efl5.cient computation of lattice sums arising in solid state physics. PBCs applied to the unit cell of a crystal yield an infinite crystal of the appropriate. symmetry performing... 

Physical Properties. The absorption of x-rays by iodine has been studied and the iodine crystal stmcture deterrnined (12,13). Iodine crystallizes in the orthorhombic system and has a unit cell of eight atoms arranged as a symmetrical bipyramid. The cell constants at 18°C (14) are given in Table 1, along with other physical properties. Prom the interatomic distances of many iodine compounds, the calculated effective radius of the covalently bound iodine atom is 184 pm (15). 

As a result of having two chiral centers, four stereoisomers of ascorbic acid are possible (Table 1) (Fig. 2). Besides L-ascorbic acid (Activity = 1), only D-araboascorbic acid (erythorbic acid (9)) shows vitamin C activity (Activity = 0.025-0.05). The L-ascorbic acid stmcture (1) in solution and the soHd state are almost identical. Ascorbic acid crystallizes in the space group P2 with four molecules in the unit cell. The crystal data are summarized in Table 2. 

The a-tetragonal form of boron has a unit cell B qC2 or B qN2 it always has a carbon or nitrogen in the crystal. The cell is centered a single-boron atom is coordinated to four icosahedrons (4Bj2 + 2B). The -tetragonal form has a unit cell of 192 boron atoms but is not, as of this writing, totally defined. 

Fig. 3. A cross-section of a nearly square cellulose microfibril, with the individual molecular chains shown as rectangles. Also shown are the one- and two-chain unit cells of la and ip. This view of the microfibril is parallel to the long axis. The chains are arranged so that the edges of the crystal correspond...

The pyrimidine ring is virtually flat. Its corrected bond lengths, as determined by a least-squares analysis of the crystal structure data for a unit cell of four molecules, are shown in formula (2) (60AX80), and the bond angles derived from these data show good agreement with those (3) derived by other means (63JCS5893) for comparison, each bond... 

Figure 18.1 A crystal is built up from many billions of small identical units, or unit cells. These unit cells are packed against each other in three dimensions much as identical boxes are packed and stored in a warehouse. The unit cell may contain one or more than one molecule. Although the number of molecules per unit cell is always the same for all the unit cells of a single crystal, it may vary between different crystal forms of the same protein. The diagram shows in two dimensions several identical unit cells, each containing two objects packed against each other. The two objects within each unit cell are related by twofold symmetry to illustrate that each unit cell in a protein cr) stal can contain several molecules that are related by symmetry to each other. (The pattern is adapted from a Japanese stencil of unknown origin from the nineteenth century.)...

How is the diffraction pattern obtained in an x-ray experiment such as that shown in Figure 18.5b related to the crystal that caused the diffraction This question was addressed in the early days of x-ray crystallography by Sir Lawrence Bragg of Cambridge University, who showed that diffraction by a crystal can be regarded as the reflection of the primary beam by sets of parallel planes, rather like a set of mirrors, through the unit cells of the crystal (see Figure 18.6b and c). 

MIR), requires the introduction of new x-ray scatterers into the unit cell of the crystal. These additions should be heavy atoms (so that they make a significant contribution to the diffraction pattern) there should not be too many of them (so that their positions can be located) and they should not change the structure of the molecule or of the crystal cell—in other words, the crystals should be isomorphous. In practice, isomorphous replacement is usually done by diffusing different heavy-metal complexes into the channels of preformed protein crystals. With luck the protein molecules expose side chains in these solvent channels, such as SH groups, that are able to bind heavy metals. It is also possible to replace endogenous light metals in metal-loproteins with heavier ones, e.g., zinc by mercury or calcium by samarium. 

Amorphous stereotactic polymers can crystallise, in which condition neighbouring chains are parallel. Because of the unavoidable chain entanglement in the amorphous state, only modest alignment of amorphous polymer chains is usually feasible, and moreover complete crystallisation is impossible under most circumstances, and thus many polymers are semi-crystalline. It is this feature, semicrystallinity, which distinguished polymers most sharply from other kinds of materials. Crystallisation can be from solution or from the melt, to form spherulites, or alternatively (as in a rubber or in high-strength fibres) it can be induced by mechanical means. This last is another crucial difference between polymers and other materials. Unit cells in crystals are much smaller than polymer chain lengths, which leads to a unique structural feature which is further discussed below. 

Figure 4.17 Crystal and molecular structure of (LiMe)4 showing (a) the unit cell of lithium methyl, (b) the LijCj skeleton of the tetramer viewed approximately along one of the threefold axes, (c) the 7-coordinate environment of each C atom, and (d) the (4 -I- 3 -I- 3)-coordinate environment of each Li atom. After ref. 93, modified to include Li—H contacts.

Fig. 20.25 Unit cells of (a) the face-centred cubic (f.c.c.), (b) the close-packed hexagonal (c.p.h.) and (c) the body-centred cubic (b.c.c.) crystal structures...

Figure 16-16. Molecular packing of Oocl-OPV5 in the crystal lattice. Lett oblique view of (he unit cell of Oocl-OPV5 right projection of the unit cell on a plane perpendicular to the ci-axis.

The unit cell of the hydrates crystallizing in Structure II is rather complicated, and for a detailed description the reader is referred to the original publications.6 48 Its composition is characterized by ... 

The unit cell of MgB2 has six boron atoms in the center of a hexagonal array of magnesium atoms. The superconductivity appears to stem from the high-energy vibrational modes of the planes of boron atoms that extend throughout the crystal. 

Krypton crystallizes with a face-centered cubic unit cell of edge 559 pm. (a) What is the density of solid krypton (b) What is the atomic radius of krypton (c) What is the volume of one krypton atom (d) What percentage of the unit cell is empty space if each atom is treated as a hard sphere ... 

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