Solid State Physics. Philip Hofmann

Figure 1.4 (a) Simple cubic structure; (b) body‐centered cubic structure; and (c) face‐centered cubic structure. Note that the spheres are depicted much smaller than in the situation of most dense packing and not all of the spheres on the faces of the cube are shown in (c).

      1.2.1 Cubic Structures

We begin with one of the simplest crystal structures possible, the simple cubic structure shown in Figure 1.4a. This structure is not very common among elemental solids, but it is an important starting point for understanding many other structures. Only one chemical element (polonium) is found to crystallize in the simple cubic structure. The structure is unfavorable because of its openness – there are many voids, if we think of the atoms as solid spheres in contact with each other. In metals, which are the most common elemental solids, directional bonding is not important, and a close packing of the atoms is usually favored. We will learn more about this in the next chapter. For covalent solids, on the other hand, directional bonding is important, but six bonds extending from the same atom in an octahedral configuration is highly uncommon in elemental solids.

The packing density of the cubic structure is improved in the body‐centered cubic (bcc) and face‐centered cubic (fcc) structures that are also depicted in Figure 1.4. In fact, the fcc structure has the highest possible packing density for identical spheres, as we shall see later. These two structures are very common – 17 elements crystallize in the bcc structure and 24 elements in the fcc structure. Note that the simple cubic structure ist the only one for which the cube is identical with the Bravais lattice. While the cube is also a unit cell for the bcc and fcc lattices, ist it not the primitive unit cell in these cases. Still, both structures are Bravais lattices with a basis containing one atom, but the vectors spanning these Bravais lattices are not the edges of the cube.

Cubic structures with a more complex basis than a single atom are also important. Figure 1.5 shows the structures of the ionic crystals CsCl and NaCl, which are both cubic with a basis containing two atoms. For CsCl, the structure can be thought of as two simple cubic structures stacked into each other. For NaCl, it consists of two fcc lattices stacked into each other. Which structure is preferred for such ionic crystals depends on the relative size of the positive and negative ions.

Figure 1.5 Structures of CsCl and NaCl. The spheres are depicted much smaller than in the situation of dense packing, but the relative size of the different ions in each structure is correct.

      1.2.2 Close‐Packed Structures

Many metals prefer structural arrangements where the atoms are packed as closely as possible. In two dimensions, the closest possible packing of atoms (i.e. spheres) is the hexagonal structure shown on the left‐hand side of Figure 1.6. To build a three‐dimensional close‐packed structure, one adds a second layer as in the middle of Figure 1.6. Now there are two possibilities, however, for adding a third layer. We can either put the atoms in the “holes” just on top of the first‐layer atoms, or we can put them into the other type of “holes.” The result are two different crystal structures. The first has an ABABAB… layer stacking sequence, the second an ABCABCABC… layer stacking sequence. Both have exactly the same packing density with the spheres filling about 74% of the total volume. The former structure is called the hexagonal close‐packed structure (hcp), and the latter turns out to be the fcc structure we already know. An alternative sketch of the hcp structure is shown in Figure 1.16b. The fcc and hcp structures are very common in elemental metals, 36 chemical elements crystallizing in hcp and 24 in fcc lattices. These structures also maximize the number of nearest neighbors for a given atom, the so‐called coordination number. For both the fcc and the hcp lattices, the coordination number is 12.

Figure 1.6 Close packing of spheres leading to the hcp and fcc structures.

It is as yet an unresolved question why not all metals crystallize in the fcc or hcp structures, if coordination is indeed so important. Whereas a prediction of the actual structure for a given element is not possible on the basis of simple arguments, we can identify some factors that play a role. For example, structures that are not optimally packed, such as the bcc structure, have a lower coordination number, but they bring the second‐nearest neighbors much closer to a given atom than in the close‐packed structures. Another important consideration is that the bonding situation is often not quite so simple, particularly in transition metals. In these, bonding is not only achieved through the delocalized s and p valence electrons as in simple metals, but also by the more localized d electrons. Bonding through the latter results in a much more directional character so that not only the close packing of the atoms is important.

      The structures of many ionic solids can also be viewed as “close‐packed” in some sense. One can derive these structures by treating the ions as hard spheres that have to be packed as closely to each other as possible.

      1.2.3 Structures of Covalently Bonded Solids

In covalent structures, the valence electrons of the atoms are not completely delocalized but shared between neighboring atoms, and bond lengths and directions are far more important than the packing density. Prominent examples are graphene, graphite, and diamond as displayed in Figure 1.7. Graphene is a single sheet of carbon atoms in a honeycomb lattice structure. It is a truly two‐dimensional solid with a number of remarkable properties – so remarkable, in fact, that their discovery has lead to the 2010 Nobel prize in physics being awarded to A. Geim and K. Novoselov. The carbon atoms in graphene are connected through a network of spСкачать книгу