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Packing Efficiency

Posted by RAHMA CAHYANINGRUM pada Maret 9, 2010

The packing efficiency is the fraction of the crystal (or unit cell) actually occupied by the atoms. It must always be less than 100% because it is impossible to pack spheres (atoms are usually spherical) without having some empty space between them.

P.E. = (volume of spheres within the unit cell) / (volume of cell)

A. Side by side structure

1. Simple cubic

Simple cubic (sc) is the simplest and often is observed for metals where too close packing encounters greater repulsion of cations. In the unit cell below, the atoms at the corners are cut because only a portion (in this case 1/8) belongs to that cell. The rest of the atom belongs to neighboring cells.

Although we have shown space between the spheres, this is only to “open up” the structure to view. In the real crystal, the spheres touch as shown in the unit cell and lattice below. Thus, the edge length of the cell is 2x the sphere radius.

Since each unit cell contains (8 corner atoms x 1/8 =) 1 atom and 1 interstitial site, the number of atoms and interstitial sites is the same.

a = 2r

a = the edge length of the cube

r = radius

Volume of atom = 1 x (4/3 πr3) = 4/3 πr3

Volume of cube = a3 = (2r)3 = 8r3

For a simple cubic lattice, this is:

P.E. = Volume of atom/volume of cube = (4/3 πr3)/8r3 = π/6 = 0.52 = 52%

The arrangement about any single atom in the simple cubic lattice is given by 6 neighboring atoms, so the atomic coordination number is 6. The coordination geometry is octahedral (an octahedron has 6 corners).

Since both the coordination number and packing efficiency are low, a simple cubic lattice uses space inefficiently. Very few examples of simple cubic lattices are known (alpha – polonium is one of the few known simple cubic lattices).

2. Body Centered Cubic (Bcc)

We can think of this unit cell as made by stuffing another atom into the center of the simple cubic lattice, slightly spreading the corners. Thus, the corner spheres no longer quite touch one another, but do touch the center. The diagonal through the body of the cube is 4x (sphere radius).

a√3 = 4r  –>  a = 4/√3 r

Each unit cell contains (1/8 x 8 corner atoms) + 1 interior atoms = 2 atoms.


Volume of the atoms = 2 x (4/3 πr3) = 8/3 πr3

Volume of cube = a3 = (4/√3 r)3 = 64/3√3 r3

P. E. = Volume of atom/volume of cube = (8/3 πr3)/(64/3√3 r3) = √3/8 π = 0.68 = 68%

The packing efficiency of a bcc lattice is considerably higher than that of a simple cubic:   68 %

The higher coordination number and packing efficiency mean that this lattice uses space more efficiently than simple cubic.

BCC lattices are very common in metals (iron, chromium, tungsten, and sodium, for example).

B. Closed Packed Stucture

There are two possibilities for closed packed structures: Cubic Close Packed (ccp) and Hexagonal Close Packed (hcp)

1. Cubic close Packed

Cubic Close Packed (ccp) also comes with a different name, Face Centered Cubic (fcc).

This cell has an additional atom in each face of the simple cubic lattice – hence the “face centered cubic” name. In the picture below various colors are used to help viewing how cells stack in the solid (the atoms are all the same).

Each unit cell contains (1/8 x 8 corner atoms) + (1/2 x 6 face atoms) = 1+3 = 4 atoms.

b = 4r   –> b = a√2

a√2 = 4r   –> a = 4/√2 r = 2√2 r

Volume of the atoms = 4 x (4/3 πr3) = 16/3 πr3

Volume of cube = a3 = (2√2 r)3 = 16√2 r3

P. E. = Volume of atom/volume of cube = (16/3 πr3)/(16√2 r3) = π/3√2 = 0.74 = 74%

Actually, the corner atoms touch the one in the center of the face. The name “close packed” refers to the packing efficiency of 74%. No other packing can exceed this efficiency (although there are others with the same packing efficiency).

Examples of fcc/ccp metals include nickel, silver, gold, copper, and aluminum.

2. Hexagonal Cubic Packed

The coordination number of hcp and fcc is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74. Examples of hcp metals include zinc, titanium, and cobalt.



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