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| The Simplest Repeating Unit in a Crystal | A Three-Dimensional Graph |
| NaCl and ZnS | Measuring the Distance Between Particles |
| Determining the Unit Cell of a Crystal | Calculating Metallic or Ionic Radii |
The structure of solids can be described as if they were three-dimensional analogs of a piece of wallpaper. Wallpaper has a regular repeating design that extends from one edge to the other. Crystals have a similar repeating design, but in this case the design extends in three dimensions from one edge of the solid to the other.
We can unambiguously describe a piece of wallpaper by specifying the size, shape, and contents of the simplest repeating unit in the design. We can describe a three-dimensional crystal by specifying the size, shape, and contents of the simplest repeating unit and the way these repeating units stack to form the crystal.
The simplest repeating unit in a crystal is called a unit cell. Each unit cell is defined in terms of lattice pointsthe points in space about which the particles are free to vibrate in a crystal.
The structures of the unit cell for a variety of salts are shown below.
In 1850, Auguste Bravais showed that crystals could be divided into 14 unit cells, which meet the following criteria.
- The unit cell is the simplest repeating unit in the crystal.
- Opposite faces of a unit cell are parallel.
- The edge of the unit cell connects equivalent points.
The 14 Bravais unit cells are shown in the figure below.
These unit cells fall into seven categories, which differ in the three unit-cell edge lengths (a, b, and c) and three internal angles (a, � and g), as shown in the table below.
The Seven Categories of Bravais Unit Cells
| Category | Edge Lengths | Internal Angles |
| Cubic | (a = b = c) | (a = �/i> = g = 90o) |
| Tetragonal | (a = bc) | (a = �/i> = g = 90o) |
| Monoclinic | (abc) | (a = �/i> = 90o g) |
| Orthorhombic | (abc) | (a = �/i> = g = 90o) |
| Rhombohedral | (a = b = c) | (a = �/i> = g 90o) |
| Hexagonal | (a = bc) | (a = �/i> = 90o, g = 120o) |
| Triclinic | (abc) | (a�/i> g 90o) |
We will focus on the cubic category, which includes the three types of unit cellssimple cubic, body-centered cubic, and face-centered cubicshown in the figure below.
These unit cells are important for two reasons. First, a number of metals, ionic solids, and intermetallic compounds crystallize in cubic unit cells. Second, it is relatively easy to do calculations with these unit cells because the cell-edge lengths are all the same and the cell angles are all 90.
The simple cubic unit cell is the simplest repeating unit in a simple cubic structure. Each corner of the unit cell is defined by a lattice point at which an atom, ion, or molecule can be found in the crystal. By convention, the edge of a unit cell always connects equivalent points. Each of the eight corners of the unit cell therefore must contain an identical particle. Other particles can be present on the edges or faces of the unit cell, or within the body of the unit cell. But the minimum that must be present for the unit cell to be classified as simple cubic is eight equivalent particles on the eight corners.
The body-centered cubic unit cell is the simplest repeating unit in a body-centered cubic structure. Once again, there are eight identical particles on the eight corners of the unit cell. However, this time there is a ninth identical particle in the center of the body of the unit cell.
The face-centered cubic unit cell also starts with identical particles on the eight corners of the cube. But this structure also contains the same particles in the centers of the six faces of the unit cell, for a total of 14 identical lattice points.
The face-centered cubic unit cell is the simplest repeating unit in a cubic closest-packed structure. In fact, the presence of face-centered cubic unit cells in this structure explains why the structure is known as cubic closest-packed. App uninstaller.
The lattice points in a cubic unit cell can be described in terms of a three-dimensional graph. Because all three cell-edge lengths are the same in a cubic unit cell, it doesn't matter what orientation is used for the a, b, and c axes. For the sake of argument, we'll define the a axis as the vertical axis of our coordinate system, as shown in the figure below.
The b axis will then describe movement across the front of the unit cell, and the c axis will represent movement toward the back of the unit cell. Furthermore, we'll arbitrarily define the bottom left corner of the unit cell as the origin (0,0,0). The coordinates 1,0,0 indicate a lattice point that is one cell-edge length away from the origin along the a axis. Similarly, 0,1,0 and 0,0,1 represent lattice points that are displaced by one cell-edge length from the origin along the b and c axes, respectively.
Thinking about the unit cell as a three-dimensional graph allows us to describe the structure of a crystal with a remarkably small amount of information. We can specify the structure of cesium chloride, for example, with only four pieces of information.
- CsCl crystallizes in a cubic unit cell.
- The length of the unit cell edge is 0.4123 nm.
- There is a Cl- ion at the coordinates 0,0,0.
- There is a Cs+ ion at the coordinates 1/2,1/2,1/2.
Because the cell edge must connect equivalent lattice points, the presence of a Cl- ion at one corner of the unit cell (0,0,0) implies the presence of a Cl- ion at every corner of the cell. The coordinates 1/2,1/2,1/2 describe a lattice point at the center of the cell. Because there is no other point in the unit cell that is one cell-edge length away from these coordinates, this is the only Cs+ ion in the cell. CsCl is therefore a simple cubic unit cell of Cl- ions with a Cs+ in the center of the body of the cell.
Unit Cells: NaCl and ZnS
NaCl should crystallize in a cubic closest-packed array of Cl- ions with Na+ ions in the octahedral holes between planes of Cl- ions. We can translate this information into a unit-cell model for NaCl by remembering that the face-centered cubic unit cell is the simplest repeating unit in a cubic closest-packed structure.
There are four unique positions in a face-centered cubic unit cell. These positions are defined by the coordinates: 0,0,0; 0,1/2,1/2; 1/2,0,1/2; and 1/2,1/2,0. The presence of an particle at one corner of the unit cell (0,0,0) requires the presence of an equivalent particle on each of the eight corners of the unit cell. Because the unit-cell edge connects equivalent points, the presence of a particle in the center of the bottom face (0,1/2,1/2) implies the presence of an equivalent particle in the center of the top face (1,1/2,1/2). Similarly, the presence of particles in the center of the 1/2,0,1/2 and 1/2,1/2,0 faces of the unit cell implies equivalent particles in the centers of the 1/2,1,1/2 and 1/2,1/2,1 faces.
The figure below shows that there is an octahedral hole in the center of a face-centered cubic unit cell, at the coordinates 1/2,1/2,1/2. Any particle at this point touches the particles in the centers of the six faces of the unit cell.
The other octahedral holes in a face-centered cubic unit cell are on the edges of the cell, as shown in the figure below.
If Cl- ions occupy the lattice points of a face-centered cubic unit cell and all of the octahedral holes are filled with Na+ ions, we get the unit cell shown in the figure below.
We can therefore describe the structure of NaCl in terms of the following information.
- NaCl crystallizes in a cubic unit cell.
- The cell-edge length is 0.5641 nm.
- There are Cl- ions at the positions 0,0,0; 1/2,1/2,0; 1/2,0,1/2; and 0,1/2,1/2.
- There are Na+ ions at the positions 1/2,1/2,1/2; 1/2,0,0; 0,1/2,0; and 0,0,1/2.
Placing a Cl- ion at these four positions implies the presence of a Cl- ion on each of the 14 lattice points that define a face-centered cubic unit. Placing a Na+ ion in the center of the unit cell (1/2,1/2,1/2) and on the three unique edges of the unit cell (1/2,0,0; 0,1/2,0; and 0,0,1/2) requires an equivalent Na+ ion in every octahedral hole in the unit cell.
ZnS crystallizes as cubic closest-packed array of S2- ions with Zn2+ ions in tetrahedral holes. The S2- ions in this crystal occupy the same positions as the Cl- ions in NaCl. The only difference between these crystals is the location of the positive ions. The figure below shows that the tetrahedral holes in a face-centered cubic unit cell are in the corners of the unit cell, at coordinates such as 1/4,1/4,1/4. An atom with these coordinates would touch the atom at this corner as well as the atoms in the centers of the three faces that form this corner. Although it is difficult to see without a three-dimensional model, the four atoms that surround this hole are arranged toward the corners of a tetrahedron.
Because the corners of a cubic unit cell are identical, there must be a tetrahedral hole in each of the eight corners of the face-centered cubic unit cell. If S2- ions occupy the lattice points of a face-centered cubic unit cell and Zn2+ ions are packed into every other tetrahedral hole, we get the unit cell of ZnS shown in the figure below.
The structure of ZnS can therefore be described as follows.
- ZnS crystallizes in a cubic unit cell.
- The cell-edge length is 0.5411 nm.
- There are S2- ions at the positions 0,0,0; 1/2,1/2,0; 1/2,0,1/2; and 0,1/2,1/2.
- There are Zn2+ ions at the positions 1/4,1/4,1/4; 1/4,3/4,3/4; 3/4,1/4,3/4; and 3/4,3/4,1/4.
Note that only half of the tetrahedral holes are occupied in this crystal because there are two tetrahedral holes for every S2- ion in a closest-packed array of these ions.
Nickel is one of the metals that crystallize in a cubic closest-packed structure. When you consider that a nickel atom has a mass of only 9.75 x 10-23 g and an ionic radius of only 1.24 x 10-10 m, it is a remarkable achievement to be able to describe the structure of this metal. The obvious question is: How do we know that nickel packs in a cubic closest-packed structure?
The only way to determine the structure of matter on an atomic scale is to use a probe that is even smaller. One of the most useful probes for studying matter on this scale is electromagnetic radiation.
In 1912, Max van Laue found that x-rays that struck the surface of a crystal were diffracted into patterns that resembled the patterns produced when light passes through a very narrow slit. Shortly thereafter, William Lawrence Bragg, who was just completing his undergraduate degree in physics at Cambridge, explained van Laue's resultswith an equation known as the Bragg equation, which allows us to calculate the distance between planes of atoms in a crystal from the pattern of diffraction of x-rays of known wavelength.
n = 2d sin T
The pattern by which x-rays are diffracted by nickel metal suggests that this metal packs in a cubic unit cell with a distance between planes of atoms of 0.3524 nm. Thus, the cell-edge length in this crystal must be 0.3524 nm. Knowing that nickel crystallizes in a cubic unit cell is not enough. We still have to decide whether it is a simple cubic, body-centered cubic, or face-centered cubic unit cell. This can be done by measuring the density of the metal.
Unit Cells: Determining the Unit Cell of a Crystal
Atoms on the corners, edges, and faces of a unit cell are shared by more than one unit cell, as shown in the figure below. An atom on a face is shared by two unit cells, so only half of the atom belongs to each of these cells. An atom on an edge is shared by four unit cells, and an atom on a corner is shared by eight unit cells. Thus, only one-quarter of an atom on an edge and one-eighth of an atom on a corner can be assigned to each of the unit cells that share these atoms.
If nickel crystallized in a simple cubic unit cell, there would be a nickel atom on each of the eight corners of the cell. Because only one-eighth of these atoms can be assigned to a given unit cell, each unit cell in a simple cubic structure would have one net nickel atom.
Simple cubic structure:
