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Chapter 3: Problem 5

Show that the atomic packing factor for \(\mathrm{BCC}\). is \(0.68\).

Short Answer

Expert verified
Question: Show that the atomic packing factor (APF) for a body-centered cubic (BCC) structure is 0.68. Answer: The atomic packing factor (APF) for a body-centered cubic (BCC) structure is approximately 0.68, as calculated through a series of steps: defining APF, determining the number of atoms per unit cell in BCC, calculating the volume of atoms in a unit cell, finding the volume of the unit cell, and finally, calculating the APF.

Step by step solution

01

Define the atomic packing factor (APF)

The atomic packing factor (APF) is the ratio of the total volume of the atoms in a unit cell to the total volume of the unit cell. The APF gives us an idea of how efficiently the atoms are packed in a given structure.
02

Determine the number of atoms per unit cell for BCC

In a body-centered cubic (BCC) structure, there is one atom at each corner of the cubic unit cell and one atom in the center. Since there are 8 corners and each corner atom is shared by 8 adjacent unit cells, the total contribution from the corner atoms is \(\frac{1}{8}\times 8 = 1\) atom. Additionally, there is 1 central atom which is totally inside the unit cell. Therefore, there are a total of \((1+1)=2\) atoms in a BCC unit cell.
03

Calculate the volume of the atoms in a unit cell

To calculate the volume of the atoms present in a unit cell, we need to find the volume of a single atom and then multiply it by the number of atoms per unit cell. The volume of a single atom can be determined using the formula for the volume of a sphere:$$V_\text{atom} = \frac{4}{3}\pi r^3$$Since there are 2 atoms in a BCC unit cell, the total volume of the atoms in the unit cell is$$V_\text{atoms} = 2\left(\frac{4}{3}\pi r^3\right) = \frac{8}{3}\pi r^3$$
04

Calculate the volume of the unit cell

In a BCC structure, the side length of the unit cell (a) can be expressed in terms of the atomic radius (r) as$$a = 4r / \sqrt{3}$$The volume of the unit cell is given by$$V_\text{unit cell} = a^3 = \left( \frac{4r}{\sqrt{3}} \right)^3 = \frac{64r^3}{3\sqrt{3}}$$
05

Calculate the atomic packing factor (APF)

To calculate the atomic packing factor (APF) for BCC, we need to find the ratio of the volume of the atoms in the unit cell to the total volume of the unit cell. Using the values we have found above, the APF for BCC can be calculated as follows:$$\text{APF} = \frac{V_\text{atoms}}{V_\text{unit cell}} = \frac{\frac{8}{3}\pi r^3}{\frac{64r^3}{3\sqrt{3}}} = \frac{8\pi}{64\sqrt{3}} = \frac{\pi}{8\sqrt{3}} \approx 0.68$$Thus, the atomic packing factor for the BCC structure is approximately 0.68.

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Most popular questions from this chapter

Zirconium has an HCP crystal structure and a density of \(6.51 \mathrm{~g} / \mathrm{cm}^{3}\). (a) What is the volume of its unit cell in cubic meters? (b) If the \(c / a\) ratio is \(1.593\), compute the values of \(c\) and \(a\).

For the HCP crystal structure, show that the ideal \(c / a\) ratio is \(1.633\).

Determine the expected diffraction angle for the first-order reflection from the (113) set of planes for FCC platinum when monochromatic radiation of wavelength \(0.1542 \mathrm{~nm}\) is used.

The metal iridium has an FCC crystal structure. If the angle of diffraction for the (220) set of planes occurs at \(69.22^{\circ}\) (first-order reflection) when monochromatic \(\mathrm{x}\)-radiation. having a wavelength of \(0.1542 \mathrm{~nm}\) is used, compute (a) the interplanar spacing for this set of planes and (b) the atomic radius for an, iridium atom.

Sketch a unit cell for the body-centered orthorhombic crystal structure.
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