The mineral chalcopyrite CuFeS2 is the archetype of this structure. (iii) Since, the number of octahedral holes in ccp structure is equal to the number of anions, every octahedral hole is occupied by Na+ ions. Thus, the number of NaCl units per unit cell is 4. There are two systems of coordinates commonly in use, which can cause some confusion. The converse of an interstitial impurity is when there are not enough atoms in a particular area of the lattice. 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. Antisite defects are a particular form of substitution defect, and are unique to compound semiconductors. In the triclinic lattice none of the sides of the unit cell are equal, and none of the angles within the unit cell are equal to 90°. Extended defects may be created either during crystal growth or as a consequence of stress in the crystal lattice. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Small atoms, such as carbon, will prefer to occupy these interstitial sites. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. These usually have the units of Angstroms and relate to the distance in each direction between the origin of the cell and the atom. The atom was 1.5 Å in the a direction away from the origin. Some metals with hexagonal close-packed crystal structures include cobalt, cadmium, zinc, and the α phase of titanium. As with crystal directions, Miller indices directions may be grouped in families. It is important to distinguish the characteristics of each of the individual systems. For example, a vector that intercepts the center of the top face of the unit cell has the coordinates x = 1/2, y = 1/2, z = 1. In addition, cations and anions will have a different number of electrons in their valence shells, so this substitution will alter the local electron concentration and the electronic properties of this area of the semiconductor. (ii) Only half the alternate tetrahedral voids are occupied by Zn+2 ions. The vector is indicated by the notation [hkl], where h, k, and l are reciprocals of the point at which the vector exits the unit cell. In metals, and in many other solids, the atoms are arranged in regular arrays called crystals. An example of a material that takes on each of the Bravais lattices is shown in Table $$\PageIndex{2}$$. That is each point would be surrounded by an identical set of points as any other point, so that all points would be indistinguishable from each other. If the mismatch is significant epitaxial growth is not energetically favorable, causing a textured film or polycrystalline untextured film to be grown. A number of inter-atomic distances may be calculated for any material with a zinc blende unit cell using the lattice parameter (a). Details of the full range of solid-state structures are given elsewhere. The structure is tetragonal (a = b ≠ c, α = β = γ = 90°, and is essentially a superlattice on that of zinc blende. The cesium chloride structure is found in materials with large cations and relatively small anions. If two close packed layers A and B are placed in contact with each other so as to maximize the density, then the spheres of layer B will rest in the hollow (vacancy) between three of the spheres in layer A. These are multiplied by a scalar to insure that is in the simple ratio of whole numbers. The unit cell of cubic close packed structure is actually that of a face-centered cubic (fcc) Bravais lattice. The cubic lattice is the most symmetrical of the systems. (a) If the anions (B-) constitute the crystal lattice and all octahedral voids are occupied by cations (A+), then the formula of the ionic solid is AB. Table 1: Crystal Structure for some Metals (at room temperature) Aluminum FCC Nickel FCC Cadmium HCP Niobium BCC Chromium ... Zinc HCP Lead FCC Zirconium HCP Magnesium HCP Unit cell structures determine some of the properties of metals. Mathematician Auguste Bravais discovered that there were 14 different collections of the groups of points, which are known as Bravais lattices. The rock salt unit cell is shown in Figure $$\PageIndex{11}$$. of Zn+2 ions as well as S−2 ions is 4. An illustration of this along with the (111) and (110) planes is given in Figure $$\PageIndex{3}$$. Among the two ions, constituting the binary compounds, the anions usually constitute the space lattice with hcp or ccp type of arrangements whereas the cations, occupy the interstitial voids. However, this information is sometimes insufficient to allow for an understanding of the true structure in three dimensions. A face-centered cubic crystal structure will exhibit more ductility (deform more readily under load before breaking) than a body-centered cubic structure. For example, the direction along the a-axis according to this scheme would be [100] because this has a component only in the a-direction and no component along either the b or c axial direction. Thus, there are 6 formula units per unit cell. $C-C\ =\ a \frac{\sqrt{3} }{4} \approx \ 0.422 a \label{1}$. The “packing fraction” in a hexagonal close packed cell is 74.05%; that is 74.05% of the total volume is occupied. Planes in a crystal can be specified using a notation called Miller indices.