Bravais cubic

It is perhaps also worth noting that the behavior of the relaxation spectrum H(r) of a cubic network is due to its three-dimensional connectivity character but not to the details of the particular network structure, hi the fractal framework the spectral dimension of all these networks is 3, and the relaxation behavior is universal In this regard the work of Denneman et al [64] is very instructive. The authors considered Hookean springs cross-linked into the three Bravais cubic lattices, namely simple cubic (sc) (this corresponds to the network considered above), body-centered cubic (bcc), and face-centered cubic (fee). They succeeded in finding analytical expressions for the eigenvalues of the sc lattice (they coincide with Eq. 74) but not for the bcc and fee lattices, which were treated numerically. It tiuns out (in agreement with the statement above), that the dynamic modulus for all... 

Fig. 3.8 Some basic Bravais lattices (a) simple cubic, (b) body-centred cubic, (c) face-centred cubic and (d) simple hexagonal close-packed. (Figure adapted in part from Ashcroft N V and Mermin N D 1976. Solid State Physics.

In two dimensions, five different lattices exist, see Fig. 5.6. One recognizes the hexagonal Bravais lattice as the unit cell of the cubic (111) and hep (001) surfaces, the centered rectangular cell as the unit cell of the bcc and fee (110) surfaces, and... 

With the exception of He, solid noble gases crystallize in the face-centred-cubic Bravais lattice and their cohesive energies (eV atom ) are 0.02 for Ne, 0.08 for Ar, 0.11 for Kr and 0.17 for Xe (Ashcroft Mermin, 1976 p. 401). VanderWaals-Londonforces areexpressedby theLennard-Jonespotential f/u a — CaZ> ,... 

When these four types of lattice are combined with the 7 possible unit cell shapes, 14 permissible Bravais lattices (Table 1.3) are produced. (It is not possible to combine some of the shapes and lattice types and retain the symmetry requirements listed in Table 1.2. For instance, it is not possible to have an A-centred, cubic, unit cell if only two of the six faces are centred, the unit cell necessarily loses its cubic symmetry.)... 

It is possible to characterize the type of Bravais lattice present by the pattern of systematic absences. Although our discussion has centred on cubic crystals, these absences apply to all crystal systems, not just to cubic, and are summarized in Table 2.3 at the end of the next section. The allowed values of are listed in Table 2.2 for... 

Nickel crystallizes in a cubic crystal system. The first reflection in the powder pattern of nickel is the 111. What is the Bravais lattice ... 

X-ray powder data for NaCl is listed in Table 2.5. Determine the Bravais lattice, assuming that it is cubic. 

The Bravais or space lattice does not distinguish between different types of local atomic environments. For example, neighbouring aluminium and silicon both take the same face-centred cubic Bravais lattice, designated cF, even though one is a close-packed twelve-fold coordinated metal, the other... 

Figure 6. TEM images of STAC-1 viewed down the a axis of a hexagonal unit cell (indicated by [M/]h) or the [110] direction of a cubic unit cell (indicated by [M/]c). The crystal is dominated by ABCABC close packing (indicated on (a)) with one stacking fault (marked by a horizontal line). A Fourier transform optical diffraction pattern with both Miller-Bravais indices to the hexagonal unit cell and Miller indices (in parentheses) to the cubic unit cell is inserted in (b). Simulated images based on a proposed model (right) are also inserted with specimen thickness of 30 nm, and lens focuses of—30 nm (a) and —10 nm (b).

It is not always possible to choose a unit cell which makes every pattern point translationally equivalent, that is, accessible from O by a translation a . The maximum set of translationally equivalent points constitutes the Bravais lattice of the crystal. For example, the cubic unit cells shown in Figure 16.2 are the repeat units of Bravais lattices. Because nt, n2, and w3 are integers, the inversion operator simply exchanges lattice points, and the Bravais lattice appears the same after inversion as it did before. Hence every Bravais lattice has inversion symmetry. The metric M = [a, a ] is invariant under the congruent transformation... 

Figure 16.12. Brillouin zones, with symmetry points marked, of (a) the primitive cubic Bravais lattice and (b) the cubic close-packed or fee Bravais lattice.

Figure 17.3. Energy bands for the simple cubic Bravais lattice in the free-electron approximation at A on rx. The symmetry of the eigenfunctions at T and at X given in the diagram satisfy compatibility requirements (Koster et al. (1963)). Degeneracies are not marked, but may be easily calculated from the dimensions of the representations.

The Seven Systems of Crystals are shown in Figure 2.2. The relationship between the trigonal and rhombohedral systems is shown in Figure B.la. The possibilities of body-centered and base-centered cells give the 14 Bravais Lattices, also shown in Figure 2.2. A face-centered cubic (fee) cell can be represented as a 60° rhombohedron, as shown in Figure B.lb. The fee cell is used because it shows the high symmetry of the cube. 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

The 14 possible Bravais lattices for crystals of a monoatomic molecule. The full designation shown here bears a numerical prefix—for example, 23Ffor face-centered cubic. When space groups are generated from the Bravais lattices, then this numerical prefix is dropped (e.g., the 23P cubic Bravais lattice reappears simply as P), because the other numbers or letters that follow the P will identify the space group uniquely. 

In positronium forming materials without optical phonons much longer Ps slowing-down times should be found. This may be tested by AMOC measurements in solid rare gases, crystallizing in the face centered cubic (fee) structure which, being a Bravais lattice, does not have optical phonon branches. 

The only possible cells in two dimensions are oblique (p only), rectangular (p and c) and hexagonal (p). For each of the seven three-dimensional crystal systems primitive and centred cells can be chosen, but centring is not advantageous in all cases. In the case of triclinic cells no centred cell can have higher symmetry than the primitive and is therefore avoided. In all there are 14 different lattice types, known as the Bravais lattices Triclinic (P), Monoclinic (P,C), Orthorhombic (P,C,I,F), Trigonal (R), Tetragonal (P,I), and Cubic (P,I,F). 

Within a given crystal system, there are in some cases several different types of crystal lattice, depending upon the type of minimum-size unit cell that corresponds to a choice of axes appropriate to the given crystal system. This unit cell may be primitive P or in certain cases body-centered I, face-centered F, or end-centered A, B, or Q depending on which pair of end faces of the unit cell is centered. The lattices are designated as primitive, body-centered, face-centered, or end-centered depending on whether the smallest possible unit cell that corresponds to the appropriate type of axes is primitive, body-centered, face-centered, or end-centered. There are in all 14 types of lattice, known as Bravais lattices. In the cubic system there are three primitive, body-centered, and face-centered these are shown in Fig. 2. 

The body-centered position in the cubic Bravais lattice (a = fa = c) is an inversion center and for this reason is taken as the origin in space group Im 3 m. The position possesses three-fold rotoinversion symmetry with three perpendicular mirror planes. 

Figure 3.11. The cubic cesium-chloride unit cell is not a body-centered cubic Bravais lattice since there are two nonequivalent lattice points.

It must be stressed that this result holds only for uncorrelated jumps on a cubic Bravais lattice. It is possible to include a correlation factor and the interested reader is referred to the book by Heitjans and Karger (Heitjans and Karger, 2005). 

However, this numbering sequence only holds for primitive cnbic unit cells. Certain systematic absences occur for centered cells of which we need consider only I and F in the cnbic system (the allowed Bravais Lattices). These absences arise in the following way. Let us assume that we have a cubic unit cell with a = 5.00 A. Normally we expect the 100 reflection to have zfioo = 5.00 A for which 29 = 17.74°. The angle of incidence of the X-ray beam for this plane is 0, 8.87°. However, there is an exactly similar plane of atoms 200 at (5.00 A/2) = 2.50 A. When the angle of incidence is 8.87°... 

For hexagonal crystal planes, a slightly different indexing nomenclature is used relative to cubic crystals. To index a plane in the hexagonal system, four axes are used, called Miller-Bravais indices. In addition to both a and b axes, another axis... 

---