Efficient Packing of Balls
Suppose you are given a large number of tennis balls and asked to pack them together in the most efficient fashion. What is the most efficient packing strategy? One could toss all the balls together in a box and shack the box to induce the balls to settle. The resulting packing of the balls is called a random closest-packed structure. Not surprisingly it is not the most efficient way to pack the tennis balls.
Although there are a variety of factors that influence how atoms pack together in crystals, atoms generally seek the most efficient packing structure in order to maximumize intermolecular attractions. Metals provide the simplest packing case, because these atoms can generally be regarded as uniform spheres.
The two most efficient packing arrangements are the hexagonal closest-packed structure (hcp) and the cubic closest-packed structure (ccp). This exercise focuses on the hexagonal closest-packed structure, and the next exercise deals with the cubic closest-packed structure.
In a crystal the atoms are arranged in a regular repeating pattern. The smallest repeating unit is called the unit cell. The entire structure can be reconstructed from knowledge of the unit cell. The unit cell is characterized by three lengths and three angles. The quantities a and b are the lengths of the sides of the base of the cell and γ is the angle between these two sides. The quantity c is the height of the unit cell. The angles α and β describe the angles between the base and the vertical sides of the unit cell.
In the hexagonal closest-packed structure, a = b = 2r and c = 4(2/3)1/2 r, where r is the atomic radius of the atom. The sides of the unit cell are perpendicular to the base, thus α = β = 90o. The base has a diamond (hexagonal) shape corresponding with γ = 120o.
How might one characterize the efficiency of the packing of atoms in a crystal?
The volume of the unit cell is readily calculated from knowledge of a, b, c, α, β, and γ. The volume of the hexagonal unit cell, which described the hexagonal closest-packed structure, is V = 8(2)1/2 r3. The volume of an individual atom is Va = 4 π r3/3 and there are two atoms in the unit cell for the hexagonal closest-packed structure, thus the volume occupied by the atoms is 2 Va = 8 π r3/3.
The packing efficiency, f, is the fraction of the volume of the unit cell actually occupied by atoms. For the hexagonal closest-packed structure f = π/(18)1/2 = 74.05%. The cubic closest-packed structure has the same packing efficiency, and this value is the highest efficiency that can be achieved.
Hexagonal Closest-Packed Structure
The virtual reality display illustrates the packing of atoms in the cubic closest-packed structure. This display requires Java3D. If the display is not visible, consult the Java3D FAQ. Drag with the left mouse button to rotate, the center button to zoom, and the right button to move the object.
Follow the suggested steps to visualized the structure, which consist of a 4x4x4 array of 64 atoms. All of the atoms are identical; however, the atoms have been colored red and green to illustrate which rows have an identical position in the xy plane. The layers of atoms in the hexagonal closest-packed structure follow an ABABAB pattern.
Use the "+" button to start adding atoms in the first layer. Before adding each atom, think about the best location for each atom to achieve the most efficient packing (there are multiple positions). Pay special attention to how adjacent rows are positioned.
After the first layer is complete, use the "+" button to add atoms to the second layer. Before adding the first atom, think about the best position in which to place the atom.
After the second layer is complete, use the "+" button to add atoms to the third layer. Before adding the first atom, carefully examine the completed layers. Notice that there are two different positions to place the first atom in the third layer. In the hexagonal closest-packed structure, the third layer lies directly above the first layer.
Remove all atoms. Display the unit cell and the lattice positions. (Each atoms is centered on a lattice position.) Add atoms to each layer and observe the arrangement of atoms in the unit cell.