Space lattice volume

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

Now that we know how to find the cell volume, we can use some previous information to calculate an important property of a material, namely, its density, which we represent with the lowercase Greek letter rho, p. For example, aluminum has an FCC space lattice. Recall that there are fonr atoms in the FCC unit cell. We know that each aluminum atom has an atomic weight of 27 g/mol. From Table 1.11, the cnbic lattice parameter for alnminnm is 4.05 A, or 0.405 nm (4.05 x 10 cm). This gives ns a volume of = 6.64 x 10 cm. You should confirm fhaf fhe fheorefical densify for aluminum is then ... 

The concept of polymer free volume is illustrated in Figure 2.22, which shows polymer specific volume (cm3/g) as a function of temperature. At high temperatures the polymer is in the rubbery state. Because the polymer chains do not pack perfectly, some unoccupied space—free volume—exists between the polymer chains. This free volume is over and above the space normally present between molecules in a crystal lattice free volume in a rubbery polymer results from its amorphous structure. Although this free volume is only a few percent of the total volume, it is sufficient to allow some rotation of segments of the polymer backbone at high temperatures. In this sense a rubbery polymer, although solid at the macroscopic level, has some of the characteristics of a liquid. As the temperature of the polymer decreases, the free volume also decreases. At the glass transition temperature, the free volume is reduced to a point at which the... 

Matter is composed of spherical-like atoms. No two atomic cores—the nuclei plus inner shell electrons—can occupy the same volume of space, and it is impossible for spheres to fill all space completely. Consequently, spherical atoms coalesce into a solid with void spaces called interstices. A mathematical construct known as a space lattice may be envisioned, which is comprised of equidistant lattice points representing the geometric centers of structural motifs. The lattice points are equidistant since a lattice possesses translational invariance. A motif may be a single atom, a collection of atoms, an entire molecule, some fraction of a molecule, or an assembly of molecules. The motif is also referred to as the basis or, sometimes, the asymmetric unit, since it has no symmetry of its own. For example, in rock salt a sodium and chloride ion pair constitutes the asymmetric unit. This ion pair is repeated systematically, using point symmetry and translational symmetry operations, to form the space lattice of the crystal. 

X-ray Data.—The structure of black phosphorus has been calculated from the X-ray reflection spectrum, using the powder method of Debye and Scherrer. It is a rhombohedral space-lattice having a characteristic angle of 60° 47, and a side of 5-96 A. The unit cell contains 8 atoms, and therefore the volume of the unit molecular aggregate is ... 

The Volume of Phosphorus in Liquid Compounds under Conditions of Maximum Contraction.—The volumes which liquid compounds would occupy if they remained in this state at temperatures not far removed from the absolute zero represent the closest packing possible at ordinary external pressure and under the influence of the internal or intrinsic pressure alone of non-oriented molecules, i.e. those which are not arranged in a space-lattice. These volumes can be obtained by shorter or longer extrapolations from the actual observed liquid volumes. 

The reciprocal lattice of a BCC real-space lattice is an FCC lattice. The Wigner-Seitz cell of the FCC lattice is the rhombic dodecahedron in Figure A. b. The volume enclosed by this polyhedron is the first BZ for the BCC real-space lattice. The high symmetry points are shown in Table 4.4. 

BCC real-space lattices are completely determined by the condition that each inner vector, k, go over into another by all the symmetry operations. This is not the case for the tmncated octahedron. The surface of the Wigner-Seitz cell is only fixed at the truncating planes, not the octahedral planes. Nonetheless, the volume enclosed by the truncated octahedron is taken to be the first BZ for the FCC real-space lattice (Bouckaert et ak, 1936). The special high-symmetry points are shown in Table 4.5. 

Figure 1.10 Illustration of the region of reciprocal space that is accessible in a powder measurement. The smaller circle represents the Ewald sphere. As shown in Figure 1.9, in a powder the reciprocal lattice is rotated to sample all orientations. An equivalent operation is to rotate the Ewald sphere in all possible orientations around the origin of reciprocal space. The volume swept out (area in the figure) is the region of reciprocal space accessible in the experiment.

A plane which passes through three lattice points (and hence through an infinite number of lattice points) is a lattice plane. Planes equivalent by translation form a family of regularly spaced lattice planes. The greater the distance between the planes, the smaller is the area of the primitive two-dimensional cell because all the primitive cells of the lattice have the same volume. Figure 1.10 shows a family of parallel planes numbered consecutively with plane 0 passing... 

The theoretical density, p, expressed in kg.m of a crystal having a number Z of entities with atomic (or molecular) molar mass M, expressed in kg.mol , placed in a space lattice structure having a unit cell of volume Y, expressed in m is given by the following equation, where AT, is Avogadro s number (i.e., 6.0221367 x 10 mol ) ... 

Where Q is the volume of the unit cell, G is the reciprocal lattice such that the plane-wave has the periodicity of the real space lattice, and c (G) are the Fourier coefficients stored on the grid of G. 

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