Packing and the PTOT System

When we begin a third layer there are two choices  

The sequence and spacings given for close packing are not artificial descriptions or approximations, as these are determined by geometry. The PTOT system is the most detailed and definitive treatment presented for close-packed structures, and many other structures can be described in this system. 

The objective is to use simple, clear, and informative notation for designating the close-packing type and layers occupied, including partial occupancy. From this information and knowledge of close-packed structures we can determine coordination numbers and local symmetries. For these purposes there are two parts of the notation the index, and the layers occupied (L)  

The index a is usually the number 2 or 3, indicating hep or ccp packing, respectively, and b is a number such that the product a-b gives the total number of close-packed layers in the unit cell. The total number of close-packed layers includes P, T, and O layers. The index b can be a fraction, but the product a-b must be a whole number. 

The index usually indicates the sequence cubic (3), hexagonal (2), or double hexagonal (4) for the ABAC sequence. The Pearson symbols (Table 2.5) can clarify cases such as hexagonal structures with an ABC sequence and a = 3 for the index. The symbols t for tetragonal, o for orthorhombic, m for monoclinic, and h for hexagonal, rhombohedral, or trigonal indicate the type of distortion of an idealized structure. Without distortion, a = 3 is for a cubic structure and a = 2 is for a hexagonal (or rhombohedral) structure. 

---

All P, O, and T layers have the same hexagonal close-packed arrangement within each layer. The two T layers are equivalent for ccp and hep, and for ccp, only P and O layers are interchangeable, and together they are equivalent to the two T layers (considered together). Because of these similarities, ccp, hep, the simple cubic structure, and even bcc structures can be handled in the PTOT system. It also applies to much more complex structures. The PTOT system provides a framework for considering the mechanism of formation and transformation of crystal structures. The transformations of structures of metals, ccp, hep, and bcc, are of particular interest. These are considered in detail in Chapter 4. 

In this book we are particularly interested in simple descriptions of structures that are easily visualized and providing information of the chemical environment of the ions and atoms involved. For metals, there is an obvious pattern of structures in the periodic table. The number of valence electrons and orbitals are important. These factors determine electron densities and compressibilities, and are essential for theoretical band calculations, etc. The first part of this book covers classical descriptions and notation for crystals, close packing, the PTOT system, and the structures of the elements. The latter and larger part of the book treats the structures of many crystals organized by the patterns of occupancies of close-packed layers in the PTOT system. 

In the introductory chapters the examples using the PTOT notation have been simple inorganic compounds. Most of these structures have a close-packed arrangement of anions with cations occupying T and / or O sites. Most of the elements are monatomic, and close packing is expected. The structures of most elements with diatomic molecules and even those with larger molecules can be described also in the PTOT system. 

The structure of p-quartz is unique and much different from those of tridymite and cristobalite. It does not follow the rules for the PTOT system, but the notation can be applied with some qualification. In general, for close-packed structures between packing layers there are two T layers, and half way between the P layers there is an O layer. The... 

---