Lattices primitive lattice cell

Equation (8.4.2) suggests that a wavefunction uk(r) needs to be found by standard quantum-chemical means for only the atoms or molecules in the one direct-lattice primitive unit cell. For each of the Avogadro s number s worth of fermions in a solid, the factor exp(ik R) in Eq. (8.4.2) provides a new quantum "number," the wavevector k, that guarantees the fermion requirement of a unique set of quantum numbers. The Bloch waves were conceived to explain the behavior of conduction electrons in a metal. 

Bravais postulated that there were fourteen different ways of arranging the lattice points in three-dimensional space. These are consistent with seven crystal systems that are listed in Table 2.1. The primitive lattice cell (P) has a lattice point only at the corner of the cell. Face centred (F) involves a lattice point at the centre of the opposite pairs of faces, while base centred (C) has a lattice point at the centres of the basal planes of the cell. Finally, body centred (I) involves a lattice point at the centre of the cell. The idealized Bravais structures are shown in Figure 2.3. 

Fig. 1.8. Body-centred cubic lattice, primitive unit cell (thick lines) and conventional unit cell (thin lines). Both cells are centred on one atom...

The FCC structure is illustrated in figure Al.3.2. Metallic elements such as calcium, nickel, and copper fonu in the FCC structure, as well as some of the inert gases. The conventional unit cell of the FCC structure is cubic with the lengdi of the edge given by the lattice parameter, a. There are four atoms in the conventional cell. In the primitive unit cell, there is only one atom. This atom coincides with the lattice pomts. The lattice vectors for the primitive cell are given by... 

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. 

Fig. 1. Two-dimensional honeycomb lattice of graphene primitive lattice vectors R and R2 are depicted outlining primitive unit cell.

Transfer matrix calculations of the adsorbate chemical potential have been done for up to four sites (ontop, bridge, hollow, etc.) or four states per unit cell, and for 2-, 3-, and 4-body interactions up to fifth neighbor on primitive lattices. Here the various states can correspond to quite different physical systems. Thus a 3-state, 1-site system may be a two-component adsorbate, e.g., atoms and their diatomic molecules on the surface, for which the occupations on a site are no particles, an atom, or a molecule. On the other hand, the three states could correspond to a molecular species with two bond orientations, perpendicular and tilted, with respect to the surface. An -state system could also be an ( - 1) layer system with ontop stacking. The construction of the transfer matrices and associated numerical procedures are essentially the same for these systems, and such calculations are done routinely [33]. If there are two or more non-reacting (but interacting) species on the surface then the partial coverages depend on the chemical potentials specified for each species. 

The size-dependence of the intensity of single shake-up lines is dictated by the squares of the coupling amplitudes between the Ih and 2h-lp manifolds, which by definition (22) scale like bielectron integrals. Upon a development based on Bloch functions ((t>n(k)), a LCAO expansion over atomic primitives (y) and lattice summations over cell indices (p), these, in the limit of a stereoregular polymer chain consisting of a large number (Nq) of cells of length ao, take the form (31) ... 

Primitive cubic crystal lattice. One unit cell is marked... 

Two modifications are known for polonium. At room temperature a-polonium is stable it has a cubic-primitive structure, every atom having an exact octahedral coordination (Fig. 2.4, p. 7). This is a rather unusual structure, but it also occurs for phosphorus and antimony at high pressures. At 54 °C a-Po is converted to /3-Po. The phase transition involves a compression in the direction of one of the body diagonals of the cubic-primitive unit cell, and the result is a rhombohedral lattice. The bond angles are 98.2°. 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

An infinite three-dimensional crystal lattice is described by a primitive unit cell which generates the lattice by simple translations. The primitive cell can be represented by three basic lattice vectors such as and h defined above. They may or may not be mutually perpendicular, depending on the crystal... 

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 dye molecules are positioned at sites along the linear channels. The length of a site is equal to a number ns times the length of c, so that one dye molecule fits into one site. Thus ns is the number of unit cells that form a site we name the ns-site. The parameter ns depends on the size of the dye molecules and on the length of the primitive unit cell. As an example, a dye with a length of 1.5 nm in zeolite L requires two primitive unit cells, therefore ns = 2 and the sites are called 2-site. The sites form a new (pseudo) Bravais lattice with the primitive vectors a, b, and ns c in favorable cases. 

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

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... 

The extension of the quantum-mechanic interpretation of the vibrational motion of atoms to a crystal lattice is obtained by extrapolating the properties of the diatomic molecule. In this case there are 3 ( independent harmonic oscillators (9l is here the number of atoms in the primitive unit cell—e.g., fayalite has four... 

As shown in Table E.l, there is only one centered lattice, oc. It is easy to show that for monoclinic, orthorhombic, and hexagonal cases, the centered lattice reduces to primitive lattices with halved unit cells. 

Unit cells, such as (la) and (lb) in Figure 1.17 and (b) in Figure 1.18, have a lattice point at each corner. However, they each contain one lattice point because four adjacent unit cells share each lattice point. They are known as primitive unit cells and are given the symbol P. The unit cell marked (a) in Figure 1.18 contains... 

The primitive unit cell—symbol P—has a lattice point at each corner. 

Thus, the reciprocal lattice of a simple cubic lattice is also simple cubic. It is shown in Fig. 5.7 in the xy plane, where it is clear that the bisectors of the first nearest-neighbour (100) reciprocal lattice vectors from a closed volume about the origin which is not cut by the second or any further near-neighbour bisectors. Hence, the Brillouin zone is a cube of volume (2n/a)2 that from eqn (2.38) contains as many allowed points as there are primitive unit cells in the crystal. The second, third, and fourth zones can... 

---