Simple lattice arrangements

In spite of the huge number of different P compounds prepared in the twentieth century, as yet, compounds based solely on P and C have not generally been well defined. On the other hand, nitrogen carbides such as C3N4 and C4N3 are believed to exist - probably with defect structures based on simple lattice arrangements [1]. 

At temperatures only slightly below the liquefaction temperatures, the liquids freeze. The solids are all simple crystals in which the atoms are close-packed in a regular lattice arrangement. The narrow temperature range over which any one of these liquids can exist suggests that the forces holding the crystal together are very much like the forces in the liquid. 

Yet another common crystal lattice based on the simple cubic arrangement is known as the face-centered cubic structure. When four atoms form a square, there is open space at the center of the square. A fifth atom can fit into this space by moving the other four atoms away from one another. Stacking together two of these five-atom sets creates a cube. When we do this, additional atoms can be placed in the centers of the four faces along the sides of the cube, as Figure 11-28 shows. 

A crystal lattice is made up of identical repeating unit cells that give the crystal its characteristic shape. Figure 4.21 shows a three-dimensional representation of a simple cubic arrangement of unit cells. 

Fig. 6. (A) The simple lattice myosin filament arrangement that occurs in all the...

The A-band lattices in different kinds of striated muscles have distinct arrangements. As shown in Fig. 3 and reproduced in simpler form in Fig. 10A and B, vertebrate striated muscle A-bands have actin filaments at the trigonal points of the hexagonal myosin filament array. As discussed prevously, this array also occurs in two types, the simple lattice and superlattice. The ratio of actin filaments to myosin filaments in each unit cell is 2 1. In both cases the center-to-center distance between adjacent myosin filaments is 70 A, but this varies as a function of overlap, becoming smaller as the sarcomere lengthens, giving an almost constant volume to the sarcomere (April et al, 1971). 

That is, among the pure iron forms, only ferrite (a, bcc) is magnetic. This is intriguing, as the 5-Fe form also exhibits a body-centered cubic crystal structure. This must indicate that in addition to the simple 3D arrangement of lattice iron atoms, their individual magnetic dipoles must also be suitably aligned in order to yield a particular magnetic behavior. 

Lattice arrangements SC, simple cubic BCC, body-centered cubic FCC, face-centered cubic HCP, hexagonal close packing. 

The simple geometrical arrangement of the reciprocal lattice, Ewald s sphere, and three vectors (ko, ki, and d hki) in a straightforward and elegant fashion yields Braggs equation. From both Figure 2.27 and Figure 2.28, it is clear that vector ki is a sum of two vectors, ko and d hki ... 

Most solid substances are crystalline— their component atoms are arranged in a regular array which is called a lattice. A lattice is a continuous structure, capable of indefinite extension in at least two dimensions, rather like the pattern on a wallpaper that is repeated again and again. This repeat pattern is called a unit cell. Even in a simple lattice such as that of NaCl, however, there are no individual molecules of NaCl as such no single atom can be said to belong entirely to any other atom it is shared between all its neighbours. 

A structure with a coordination number of 8 is known as the CsCl structure. It can be described as two interpenetrating simple cubic lattices with the anions on the comers of one lattice and the cations located on the comers of the second interpenetrating lattice arranged such that they sit in the centers of the cubes of the anion lattice so that each ion is surroimded by 8 coimterions. The Madelung constant for this structure is equal to 1.763. 

Figure A2.5.18. Body-centred cubic arrangement of (3-brass (CiiZn) at low temperature showing two interpenetrating simple cubic superlattices, one all Cu, the other all Zn, and a single lattice of randomly distributed atoms at high temperature. Reproduced from Hildebrand J H and Scott R L 1950 The Solubility of Nonelectrolytes 3rd edn (New York Reinliold) p 342.

A similar effect occurs in highly chiral nematic Hquid crystals. In a narrow temperature range (seldom wider than 1°C) between the chiral nematic phase and the isotropic Hquid phase, up to three phases are stable in which a cubic lattice of defects (where the director is not defined) exist in a compHcated, orientationaHy ordered twisted stmcture (11). Again, the introduction of these defects allows the bulk of the Hquid crystal to adopt a chiral stmcture which is energetically more favorable than both the chiral nematic and isotropic phases. The distance between defects is hundreds of nanometers, so these phases reflect light just as crystals reflect x-rays. They are called the blue phases because the first phases of this type observed reflected light in the blue part of the spectmm. The arrangement of defects possesses body-centered cubic symmetry for one blue phase, simple cubic symmetry for another blue phase, and seems to be amorphous for a third blue phase. 

The reason for the formation of a lattice can be the isotropic repulsive force between the atoms in some simple models for the crystalhzation of metals, where the densely packed structure has the lowest free energy. Alternatively, directed bonds often arise in organic materials or semiconductors, allowing for more complicated lattice structures. Ultimately, quantum-mechanical effects are responsible for the arrangements of atoms in the regular arrays of a crystal. 

---