Crystal structure  (Page 6/9)

Coordination number

The coordination number of an atom or ion within an extended structure is defined as the number of nearest neighbor atoms (ions of opposite charge) that are in contact with it. A slightly different definition is often used for atoms within individual molecules: the number of donor atoms associated with the central atom or ion. However, this distinction is rather artificial, and both can be employed.

The coordination numbers for metal atoms in a molecule or complex are commonly 4, 5, and 6, but all values from 2 to 9 are known and a few examples of higher coordination numbers have been reported. In contrast, common coordination numbers in the solid state are 3, 4, 6, 8, and 12. For example, the atom in the center of body-centered cubic lattice has a coordination number of 8, because it touches the eight atoms at the corners of the unit cell, while an atom in a simple cubic structure would have a coordination number of 6. In both fcc and hcp lattices each of the atoms have a coordination number of 12.

Octahedral and tetrahedral vacancies

As was mentioned above, the packing fraction in both fcc and hcp cells is 74.05%, leaving 25.95% of the volume unfilled. The unfilled lattice sites (interstices) between the atoms in a cell are called interstitial sites or vacancies. The shape and relative size of these sites is important in controlling the position of additional atoms. In both fcc and hcp cells most of the space within these atoms lies within two different sites known as octahedral sites and tetrahedral sites. The difference between the two lies in their “coordination number”, or the number of atoms surrounding each site. Tetrahedral sites (vacancies) are surrounded by four atoms arranged at the corners of a tetrahedron. Similarly, octahedral sites are surrounded by six atoms which make-up the apices of an octahedron. For a given close packed lattice an octahedral vacancy will be larger than a tetrahedral vacancy.

Within a face centered cubic lattice, the eight tetrahedral sites are positioned within the cell, at the general fractional coordinate of ( ⁿ / ₄ , ⁿ / ₄ , ⁿ / ₄ ) where n = 1 or 3, e.g., ( ¹ / ₄ , ¹ / ₄ , ¹ / ₄ ), ( ¹ / ₄ , ¹ / ₄ , ³ / ₄ ), etc. The octahedral sites are located at the center of the unit cell ( ¹ / ₂ , ¹ / ₂ , ¹ / ₂ ), as well as at each of the edges of the cell, e.g., ( ¹ / ₂ ,0,0). In the hexagonal close packed system, the tetrahedral sites are at (0,0, ³ / ₈ ) and ( ¹ / ₃ , ² / ₃ , ⁷ / ₈ ), and the octahedral sites are at ( ¹ / ₃ , ¹ / ₃ , ¹ / ₄ ) and all symmetry equivalent positions.

Important structure types

The majority of crystalline materials do not have a structure that fits into the one atom per site simple Bravais lattice. A number of other important crystal structures are found, however, only a few of these crystal structures are those of which occur for the elemental and compound semiconductors and the majority of these are derived from fcc or hcp lattices. Each structural type is generally defined by an archetype, a material (often a naturally occurring mineral) which has the structure in question and to which all the similar materials are related. With regard to commonly used elemental and compound semiconductors the important structures are diamond, zinc blende, Wurtzite, and to a lesser extent chalcopyrite. However, rock salt, β-tin, cinnabar and cesium chloride are observed as high pressure or high temperature phases and are therefore also discussed. The following provides a summary of these structures. Details of the full range of solid-state structures are given elsewhere.