Body-centered lattice

Examination of the Laue data of Table II shows that with only one exception first-order reflections occurred only from planes with h -(- k -(-even. This would indicate the body-centered lattice T", except for the... 

Using data from rotation and Laue photographs, it is shown that the unit of structure of sodalite, containing < NaiAlzSiiOi2Gl, has a0 = 8.87 A. The lattice is the simple cubic one, Fc the structure closely approximates one based on a body centered lattice, however. The atomic arrangement has... 

The structure is based on a body-centered lattice. At each lattice point there is a small atom (Zn, Al). It is surrounded by an icosahedron of twelve atoms (Fig. 11-14). This group is then surrounded by 20 atoms, at the corners of a pentagonal dodecahedron, each atom lying directly out from the center of one of the 20 faces of the icosahedron. The next 12 atoms lie out from the centers of the pentagonal faces of the dodecahedron this gives a complex of 45 atoms, the outer 32 of which lie at the corners of a rhombic triacontahedron. The next shell consists of 60 atoms, each directly above the center of a... 

The body-centered lattice implies that each unit cell... 

Figure 2. Schematic of xanthan fibril. Small circle represents xanthan molecule or its assembly. The rectangle indicates the unit cell of the body-centered lattice in which molecules are packed.

The valence electron density of the tetragonal-phase polymer is shown in Fig. 10a [37]. It is evident from the figures that this tetragonal phase should have different in-plane lattice constants (a and b) if the stacking is a simple AA type with a body-centered lattice. It has been reported recently that it is actually the case in this polymer, and the material has a pseudo-tetragonal orthorhombic lattice [38]. 

First symbol refers to the Bravais lattice P = primitive lattice C = centered lattice F = face-centered lattice I = body-centered lattice... 

For a body-centered lattice I, some of the lattice points have coordinates that are expressible as integers mnp and some have coordinates that mrrst be expressed as half integers m +, n +, p +. For the latter,... 

FIGURE 21.9 Centered lattices, like all lattices, have lattice points at the eight corners of the unit cell. A body-centered lattice has an additional lattice point at the center of the cell, a face-centered lattice has additional points at the centers of the six faces, and a side-centered lattice has points at the centers of two parallel sides of the unit cell. (Note The colored dots in the lattice diagrams represent lattice points, not atoms.)... 

Further discussion of primitive, face-centered, and body-centered lattices will be found in Chapter 4. 

In the monoclinic crystal system, the body-centered lattice can be converted into a base-centered lattice (C), which is standard. The face-... 

The latter example is illustrated in Figure 1.27, where a tetragonal face-centered lattice is reduced to a tetragonal body-centered lattice, which has the same symmetry but half the volume of the unit cell. The reduction is carried out using the transformations of basis vectors as shown in Eqs. 1.6 through 1.8. 

Consider a body-centered lattice, in which every atom has a symmetrically equivalent atom shifted by (1/2, 1/2, 1/2). The two matrices A (Eq. 2.109) for every pair of the symmetrically identical atoms are ... 

This property, which is introduced by the presence of a translational symmetry, is called the systematic absence (or the systematic extinction). Therefore, in a body-centered lattice only Bragg reflections in which the sums of all Miller indices are even (i.e. h + k + l = 2n and = 1, 2, 3,. ..) may have non-zero intensity and be observed. It is worth noting that some (but not all) of the Bragg reflections with h + k + I = 2n may become extinct because their intensities are too low to be detected due to other reasons, e.g. a specific distribution of atoms in the unit cell, which is not predetermined by symmetry. 

Thus, the column labeled is obtained by rounding the values from the previous column and multiplying them by 2. The corresponding hkl triplets confirm a body-centered lattice (h + k + I = 2n), and the last column contains the values of the lattice parameter calculated from the individual Bragg peaks using Eq. 5.17. 

Figure 5.16. Alternative axes selection in the monoclinic crystal system. Open and hatched points represent lattice points. The open points are located in the plane, while the hatched points are raised by 1/2 of the full translation in the direction perpendicular to the plane of the projection. Unit cells based on the vectors a and c or a and d correspond to a base-centered (C) lattice, while the unit cell based on the vectors c and d corresponds to a body-centered lattice.

First, 1174 compounds containing Mo and O were found. Second, the list was narrowed to 151 structures that have monoclinic base-centered or body-centered lattices. Third, M07O22 text in the chemical formula was searched. Searching all ICSD records for a specific text would be very slow. 

Cll = tetragonal body-centered lattice with hexagonal (110) plane... 

In Ha-mm-urabrs Babylon iron was the next most expensive element after silver two shekels of silver cost eight of iron and 120-140 shekels of copper. Hie iron column near Delhi is more than 1500 years old, is 7.66 m in height and weighs 6 t. It consists of 99.72 % pure iron (as well as traces of C, Mn, S and P) and has retained its purity throughout the centuries. And it is symbolic that the Atomiiim built in Brussels la 1958 consists of nine iron spheres which represent the cubic body-centered lattice structure of the stable modification a-iron. 

The number of atoms per unit cell in any crystal is partially dependent on its Bravais lattice. For example, the number of atoms per unit cell in a crystal based on a body-centered lattice must be a multiple of 2, since there must be, for any atom in the cell, a corresponding atom of the same kind at a translation of from the first. The number of atoms per cell in a base-centered lattice must also be a multiple of 2, as a result of the base-centering translations. Similarly, the number of atoms per cell in a face-centered lattice must be a multiple of 4. 

Consider reflections from the (001) planes which are shown in profile in Fig. 4-2. For the base-centered lattice shown in (a), suppose that the Bragg law is satisfied for the particular values of A and 0 employed. This means that the path difference ABC between rays 1 and 2 is one wavelength, so that rays 1 and 2 are in phase and diffraction occurs in the direction shown. Similarly, in the body-centered lattice shown in (b), rays 1 and 2 are in phase, since their path difference ABC is one wavelength. However, in this case, there is another plane of atoms midway between the (001) planes, and the path difference DEF between rays 1 and 3 is exactly half of ABC, or one-half wavelength. Thus rays 1 and 3 are completely out of phase and annul each other. Similarly, ray 4 from the next plane down (not shown) annuls ray 2, and so on throughout the crystal. There is no 001 reflection from the body-centered lattice. 

Mathematics (Hassel, 1830) has shown that there are only 32 combinations of symmetry operations (rotation, inversion, and reflection) that are consistent with a three-dimensional crystal lattice. These 32 point groups, or crystal classes, can be grouped into one of the seven crystal systems given in Table 2.1. There are four types of crystal lattices primitive (P), end-centered (C, B, and A), face-centered (/O, and body-centered (/). The primitive lattice contains a lattice point at each comer of the unit cell, the end-centered lattice has an additional lattice point on one of the lattice faces, the face-centered lattice has an extra lattice on each of the lattice faces, and the body-centered lattice has an extra lattice point at the center of the crystal lattice. By combining the seven crystal systems with the four lattice types (P, C, I, F), 14 unique crystal lattices, also known as Bravais lattices (Bravais, 1849), are produced. 

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