How many anions per unit cell for

The three illustrations show a the cubic hole that is in the center of a simple cubic lattice of anions, b the locations of the octahedral holes in a face-centered cubic lattice of anions, and c the locations of the tetrahedral holes in a face-centered cubic lattice of anions. Many ionic compounds with relatively large cations and a cation:anion ratio have this structure, which is called the cesium chloride structure Figure Solid-state chemists tend to describe the structures of new compounds in terms of the structure of a well-known reference compound.

Notice in Figure The cesium chloride structure is most common for ionic substances with relatively large cations, in which the ratio of the radius of the cation to the radius of the anion is in the range shown in Table Such cross-sections often help us visualize the arrangement of atoms or ions in the unit cell more easily. Table Very large cations occupy cubic holes, cations of intermediate size occupy octahedral holes, and small cations occupy tetrahedral holes in the anion lattice.

In contrast, a face-centered cubic fcc array of atoms or anions contains two types of holes: octahedral holes, one in the center of the unit cell plus a shared one in the middle of each edge part b in Figure As shown in Table Very large cations occupy cubic holes in a cubic anion lattice, cations of intermediate size tend to occupy the octahedral holes in an fcc anion lattice, and relatively small cations tend to occupy the tetrahedral holes in an fcc anion lattice.

In general, larger cations have higher coordination numbers than small cations. The most common structure based on a fcc lattice is the sodium chloride structure Figure The result is an electrically neutral unit cell and a stoichiometry of NaCl. As shown in Figure The sodium chloride structure is favored for substances with two atoms or ions in a ratio and in which the ratio of the radius of the cation to the radius of the anion is between 0. It is observed in many compounds, including MgO and TiC.

The structure shown in Figure It results when the cation in a substance with a cation:anion ratio is much smaller than the anion if the cation:anion radius ratio is less than about 0. If all the tetrahedral holes in an fcc lattice of anions are occupied by cations, what is the stoichiometry of the resulting compound? Use the ionic radii given in Figure 7.

A Figure B Because the tetrahedral holes are located entirely within the unit cell, there are eight cations per unit cell. We calculated previously that an fcc unit cell of anions contains a total of four anions per unit cell. The stoichiometry of the compound is therefore M 8 Y 4 or, reduced to the smallest whole numbers, M 2 Y.

According to Figure 7. If only half the octahedral holes in an fcc lattice of anions are filled by cations, what is the stoichiometry of the resulting compound? Answer : MX 2 ; an example of such a compound is cadmium chloride CdCl 2 , in which the empty cation sites form planes running through the crystal. We examine only one other structure of the many that are known, the perovskite structure.

The oxides are in the centers of the square faces part a in Figure The stoichiometry predicted from the unit cell shown in part a in Figure The Ti and Ca atoms have coordination numbers of 6 and 12, respectively. We will return to the perovskite structure when we discuss high-temperature superconductors in Section Two equivalent views are shown: a a view with the Ti atom at the center and b an alternative view with the Ca atom at the center.

For MgCl 2 the lattice must be an array of chloride anions with only half that number of magnesium ion. The packing arrangement adopted by an ionic compound is determined by the comparative sizes of the ions.

Consider a lattice in which the anions assume a cubic array. The diagram below shows four spheres represent some anions of a part of a cubic layer.

The dashed circle represents the anions below and above the plane. The shaded circle shows the interstitial space available for a cation to fit between the six anions. The cation has to be the size of the shaded circle. Using geometry, we can work out the ideal radius ratio for perfect packing. Using the Pythagorean theorem, the optimum ratio of cation radius to anion radius is 0. If the cation is too large to give the optimum 0. When the radius ratio exceeds 0. When the ratio is less than 0.

Cesium chloride forms a lattice in which the chloride anions adopt a simple cubic packing arrangement, with each cesium cation occupying the center of a cube. Most metal crystals are one of the four major types of unit cells.

Note that there are actually seven different lattice systems, some of which have more than one type of lattice, for a total of 14 different types of unit cells. We leave the more complicated geometries for later in this module. This is called a body-centered cubic BCC solid. Atoms in the corners of a BCC unit cell do not contact each other but contact the atom in the center. Any atom in this structure touches four atoms in the layer above it and four atoms in the layer below it. Thus, an atom in a BCC structure has a coordination number of eight.

Each atom touches four atoms in the layer above it and four atoms in the layer below it. Elements or compounds that crystallize with the same structure are said to be isomorphous. This arrangement is called a face-centered cubic FCC solid. The atoms at the corners touch the atoms in the centers of the adjacent faces along the face diagonals of the cube.

Because the atoms are on identical lattice points, they have identical environments. This structure is also called cubic closest packing CCP. In CCP, there are three repeating layers of hexagonally arranged atoms.

Each atom contacts six atoms in its own layer, three in the layer above, and three in the layer below. In this arrangement, each atom touches 12 near neighbors, and therefore has a coordination number of Atoms in a CCP structure have a coordination number of 12 because they contact six atoms in their layer, plus three atoms in the layer above and three atoms in the layer below.

By rotating our perspective, we can see that a CCP structure has a unit cell with a face containing an atom from layer A at one corner, atoms from layer B across a diagonal at two corners and in the middle of the face , and an atom from layer C at the remaining corner. This is the same as a face-centered cubic arrangement. Because closer packing maximizes the overall attractions between atoms and minimizes the total intermolecular energy, the atoms in most metals pack in this manner.

Both consist of repeating layers of hexagonally arranged atoms. Rotate the body-centered cubic bcc unit cell. As the name suggests it contains an ion or atom in the center of the cube.

If all the spheres have the same radius, like in metals, then the spheres centered on the lattice points do not make contact with each other. Another way of looking at the layout is using 2-dimensional layer diagrams. The bcc has 3 layers along the z-axis , which would look like this:. Illustrated left is the face-centered cubic fcc unit cell.

It has a particle in the middle of each of the six faces of the cube.