P unit cell

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes, a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry... 

HT high temperature modification P unit cell angle (in degrees)... 

Figure 1 Perspective drawings of ball models for the cluster structures. N = 7 pentagonal bipyramid (right op) N = 13 h.c.p. unit cell, unstable, (left top) N = 13 f.c.c. cubo-octahedron, unstable, (left middle)-, N = 13 icosahedron, stable, (right middle), N = 55 f.c.c. cubo-octahedron, unstable, (left bottom)-, N = 55 icosahedron, stable (right bottom)...

Take the simplest plane group, p 1, (number 1), as an example. Recall that there is only one lattice point in a primitive (p) unit cell. The motif can be placed anywhere in the unit cell. The unit cell has no symmetry elements present, (Figure 3.11a), and hence the motif is not replicated, (Figure 3.11b). Note that the cell is divided up into quarters, and motifs in the surrounding unit cells are also shown to demonstrate the pattern of repetition. Although the unit cell contains just one lattice point, the motif... 

The wavefunctions describing the chain 13.7 may be generated by seeing how the wavefunctions of the finite ring change when N becomes large. As before we define x (r — Rp) as the atomic orbital wavefunction located on the atom in the p unit cell, namely, Rp = (() l)o- From equation I2.I0, the wavefunctions of the N atom chain are given by... 

Let pj be a vector from the origin of the m,n,p unit cell to the nucleus of the/th atom in the unit cell and let p be a vector from the nucleus to a point in its electron cloud. Let (/r(p) be the radial electron density function of the atom. The vector r extends from the crystal origin to p so that r = Tm p + py + p as shown in Figure 6.8. 

Xu = degree of crystallinity (%) p = sample density p, = unit cell density Pa = amorphous density... 

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

Iron(III) fluoride ttihydrate [15469-38-2] FeF3-3H2 0, crystallizes from 40% HF solution ia two possible crystalline forms. At low temperature the a-form, which is isostmctural with a-AlF 3H2O, is favored. High temperatures favor P-FeF 3H2O, the stmcture of which consists of fluoride-bridged octahedra with one water of hydration per unit cell. 

Aminophenol. This compound forms white plates when crystallized from water. The base is difficult to maintain in the free state and deteriorates rapidly under the influence of air to pink-purple oxidation products. The crystals exist in two forms. The a-form (from alcohol, water, or ethyl acetate) is the more stable and has an orthorhombic pyramidal stmcture containing four molecules per unit cell. It has a density of 1.290 g/cm (1.305 also quoted). The less stable P-form (from acetone) exists as acicular crystals that turn into the a-form on standing they are orthorhombic bipyramidal or pyramidal and have a hexamolecular unit (15,16,24) (see Tables 3—5). 

The a-rhombohedral form of boron has the simplest crystal stmcture with slightly deformed cubic close packing. At 1200°C a-rhombohedral boron degrades, and at 1500°C converts to P-rhombohedral boron, which is the most thermodynamically stable form. The unit cell has 104 boron atoms, a central B 2 icosahedron, and 12 pentagonal pyramids of boron atom directed outward. Twenty additional boron atoms complete a complex coordination (2). 

The crystallographic requirement for tire formation of G-P zones is that the material within the zones shall have an epitaxial relationship with the maUix, and tlrus the eventual precipitate should have a similar unit cell size in one direction as tha maUix. In dre Al-Cu system, the f.c.c. structure of aluminium has a lattice parameter of 0.4014 nm, and the tetragonal CuAl2 compound has lattice parameters a — 0.4872 and b — 0.6063 nm respectively. 

Let us now look at the c.p.h. unit cell as shown in Fig. 5.4. A view looking down the vertical axis reveals the ABA stacking of close-packed planes. We build up our c.p.h. crystal by adding hexagonal building blocks to one another hexagonal blocks also stack so that they fill space. Here, again, we can use the unit cell concept to open up views of the various types of planes. 

At small strains the cell walls at first bend, like little beams of modulus E, built in at both ends. Figure 25.10 shows how a hexagonal array of cells is distorted by this bending. The deflection can be calculated from simple beam theory. From this we obtain the stiffness of a unit cell, and thus the modulus E of the foam, in terms of the length I and thickness t of the cell walls. But these are directly related to the relative density p/ps= t/lY for open-cell foams, the commonest kind. Using this gives the foam modulus as... 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

The structures of boron-rich borides (e.g. MB4, MBfi, MBio, MB12, MBe6) are even more effectively dominated by inter-B bonding, and the structures comprise three-dimensional networks of B atoms and clusters in which the metal atoms occupy specific voids or otherwise vacant sites. The structures are often exceedingly complicated (for the reasons given in Section 6.2.2) for example, the cubic unit cell of YB e has ao 2344 pm and contains 1584 B and 24 Y atoms the basic structural unit is the 13-icosahedron unit of 156 B atoms found in -rhombohedral B (p. 142) there are 8 such units (1248 B) in the unit cell and the remaining 336 B atoms are statistically distributed in channels formed by the packing of the 13-icosahedron units. 

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