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Chapter 3: Problem 6

Show that the atomic packing factor for HCP is 0.74.

Short Answer

Expert verified
Short Answer: The atomic packing factor (APF) for a hexagonal close-packed (HCP) structure is 0.74. This means that 74% of the total volume of the HCP unit cell is occupied by atoms.

Step by step solution

01

1. Calculate the total number of atoms in HCP structure

To calculate the total number of atoms in the HCP unit cell, we need to consider the number of atoms at different positions within the unit cell. There are 6 atoms at the corners, 12 on the edges, and 3 in the interior. Total number of atoms in a HCP unit cell = (6 x 1/6) + (12 x 1/2) + 3 = 1 + 6 + 3 = 10
02

2. Calculate the volume of atoms in a unit cell

The volume of a single atom (sphere) can be calculated using the formula for the volume of a sphere, which is V = (4/3)πr^3. Since there are 10 atoms in a HCP unit cell, the total volume occupied by atoms in a unit cell can be calculated by multiplying the volume of a single atom with the total number of atoms: V_atoms = 10 x (4/3)πr^3 = (40/3)πr^3
03

3. Calculate the volume of the HCP unit cell

To calculate the volume of the HCP unit cell, we need to first find the height (c) and the length of the side (a) of the unit cell. The side length (a) can be determined in terms of the atomic radius (r) as follows: a = 2r To determine the height (c), consider a right triangle formed by the height, side of the unit cell, and the distance between the center of an atom in the interior to the nearest base plane (which is r). By the Pythagorean theorem, the height can be expressed as: c = 2√(2)r Now, the volume of the HCP unit cell (V_HCP) can be calculated using the formula for the volume of a hexagonal prism: V = (3√3/2)a^2c V_HCP = (3√3/2)(2r)^2(2√(2)r) = (3√3/2)(4r^2)(2√(2)r) = 12√2(√3/2)r^3
04

4. Calculate the atomic packing factor (APF)

Now that we have calculated the volume occupied by atoms (V_atoms) and the total volume of the unit cell (V_HCP), we can calculate the APF for the HCP structure using the following formula: APF = (Volume occupied by atoms) / (Total volume of unit cell) = (V_atoms) / (V_HCP) APF = ((40/3)πr^3) / (12√2(√3/2)r^3) = ((40/3)π) / (12√2(√3/2)) = 0.74 Hence, the atomic packing factor for HCP is 0.74.

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