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Chapter 12: Problem 27

Compute the atomic packing factor for the diamond cubic crystal structure (Figure 12.16). Assume that bonding atoms touch one another, that the angle between adjacent bonds is 109.5, and that each atom internal to the unit cell is positioned a/4 of the distance away from the two nearest cell faces (a is the unit cell edge length).

Short Answer

Expert verified
The atomic packing factor (APF) for the diamond cubic crystal structure is approximately 0.3401, or 34.01%.

Step by step solution

01

Determine the number of atoms in the unit cell

In the diamond cubic crystal structure, there are 8 corner atoms and 6 face atoms per FCC sublattice. Since there are two interpenetrating sublattices, the total number of atoms in the diamond unit cell is: Number of atoms = (8 corner atoms + 6 face atoms) * 2 sublattices = 28 atoms
02

Calculate the volume of the atoms in the unit cell

Assuming all atoms touch one another, the diameter of an atom can be determined by the distance between two nearest cell faces (given as a/4): Diameter = 2 * a/4 = a/2 To find the volume of an individual atom, we will use the following equation for the volume of a sphere: V_atom = (4/3) * pi * (radius)^3 = (4/3) * pi * ((a/4)^3) = (pi * a^3) / 64 As there are 28 atoms in the unit cell, the total volume occupied by the atoms is: V_atoms = 28 * V_atom = 28 * (pi * a^3) / 64
03

Calculate the volume of the unit cell

The diamond cubic crystal structure consists of a cube with an edge length of 'a'. The volume of the unit cell (V_unitcell) can be calculated as: V_unitcell = a^3
04

Calculate the atomic packing factor (APF)

The atomic packing factor (APF) is the ratio of the volume occupied by atoms in the unit cell to the total volume of the unit cell. Thus, we have: APF = V_atoms / V_unitcell = (28 * (pi * a^3) / 64) / a^3 = (28 * pi) / 64 = 0.3401 The atomic packing factor for the diamond cubic crystal structure is approximately 0.3401, or 34.01%.

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