Body-centred cubic lattice examples

The origin of the absences is due to destructive interference occurring between the diffracted waves, meaning that the intensity cancels out. For example. Figure 3.12 shows diffraction from the 100 plane of a body-centred cubic lattice. 

With increasing temperature the order of dipoles in each sublattice decreases and, at a certain temperature, a phase transition into the paraelectric phase occurs. It may be either second or first order transition. In the paraelectric phase local polarization Pq vanishes. The nature of the spontaneous polarization is similar in solid ferro- and antiferroelectrics. In both cases, the dipole-dipole interactions are dominant. For example, if dipoles are situated in the points of the body-centred cubic lattice, they preferably orient parallel to each other and such a structure is ferroelectric. However, the same dipoles placed into the points of a simple cubic... 

If we consider the Peierls force from section 6.2.9 as obstacle, it can also be overcome by thermal activation. This is especially relevant if the Peierls force is large i. e., when slip is along planes that are not close-packed, for example in body-centred cubic lattices. For this reason, the yield strength of body-centred cubic lattices is strongly dependent on the temperature, different from face-centred cubic metals (figure 6.29). The Peierls stress can reach values of up to several hundred megapascal. 

The majority of elemental substances, and a large number of compounds, have metallic properties (see Section 3.3). Metallic elemental substances are characterised by three-dimensional structures with high coordination numbers. For example, Na(s) has a body-centred cubic (bcc) structure in which each atom is surrounded by eight others at the corners of a cube, each at a distance of 371.6pm from the atom at the centre. The Na atom also has six next-nearest neighbours in the form of an octahedron, with Na-Na distances of 429.1pm. A fragment of this lattice is shown in Fig. 7.14. These distances may be compared with the Na-Na bond length of 307.6 pm in the Na2 molecule, which can be studied in the gas phase by vaporisation of sodium metal. 

At high concentrations, amphiphiles can self-assemble into lyotropic liquid crystalline phases. As discussed in Chapter 5, a liquid crystalline phase is one that lacks the full three-dimensional translational order of molecules on a crystal lattice. Lyotropic refers to the fact that such phases are formed by amphiphiles as a function of concentration (as well as temperature). Lyotropic phases with one-dimensional translational order consisting of bilayers of amphiphiles separated by solvent are called lamellar phases. A two-dimensional structure is formed by the hexagonal packing of rod-like micelles. Cubic phases are formed by packing micelles into body-centred cubic or face-centred cubic arrays, for example. The bicontinuous cubic phases are more complex structures, where space is partitioned into two continuous labyrinths (usually a surfactant bilayer separating two congruent subvolumes of water). 

From these results and Example 3.3, it becomes clear that closed-packed cubic lattice has the best space economy (best packing, least empty space), followed by the body-centred lattice, whereas the simple cubic packing has the lowest space economy with the highest fraction of unoccupied space. 

Look at the Example 3.8 and Self-Test 3.8 if you have problems determining the composition of the unit ceils. Using the same approach as outlined in these, you can find that the cubic F lattice has indeed four points, and that body-centred has two. 

Not all types of lattice are allowable within each crystal system, because the symmetrical relationships between cell parameters mean a smaller cell could be drawn in another crystal system. For example a C-centred cubic unit cell can be redrawn as a body-centred tetragonal cell. The fourteen allowable combinations for the lattices are given in Table 1.4. These lattices are called the Bravais lattices. 

The experimental techniques outlined in the previous sections allow the lattice parameters of a crystal to be determined. However, the determination of the appropriate crystal lattice, face-centred cubic as against body-centred, for example, requires information on the intensities of the diffracted beams. More importantly, in order to proceed with a determination of the complete crystal structure, it is vital to understand the relationship between the intensity of a beam diffracted from a set of (hkl) planes and the atoms that make up the planes themselves. 

Using the expressions derived above, one can calculate the complete phase diagram of a two-component alloy with an arbitrary crystal structure and with many-body atomic interactions of arbitrary orders and effective radii of action ". As an example and with the aim to study the numerical accuracy of the ring approximation, we considered the model case appropriate to the face-centred cubic (f.c.c.) crystal. lattice with... 

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