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Chapter 11: Problem 5

In the body centered cubic unit cell and simple cubic unit cell, the radius of atom in terms of edge length (a) of the unit cell is respectively: (a) \(\frac{a}{2}, \frac{a}{2 \sqrt{2}}\) (b) \(\frac{a}{\sqrt{2}}, \frac{a}{2}\) (c) \(\frac{a}{2 \sqrt{2}}, \frac{a}{2}\) (d) \(\frac{\sqrt{3} a}{4}, \frac{a}{2}\)

Short Answer

Expert verified
Option (d): \(\frac{\sqrt{3} a}{4}, \frac{a}{2}\) is correct.

Step by step solution

01

Understanding the Body-Centered Cubic (BCC) Unit Cell

In a body-centered cubic unit cell, the atom at the center is in contact with the eight corner atoms. This means that the diagonal of the cube's face ( oot ext{3} imes a oot) contains the diameter of two atoms. It's critical to determine the relationship between the edge length, \(a\), and the atomic radius,\(r\).
02

Deriving the Body-Centered Cubic (BCC) Radius Formula

For the BCC structure, consider the body diagonal: the body diagonal is \( oot{3}{a}\), which is equal to 4 times the radius (2 diameters). The equation is given by: \( oot{3}{a} = 4r\). Thus, solving for the radius, we get: \[r = \frac{\sqrt{3}a}{4}\].
03

Understanding the Simple Cubic (SC) Unit Cell

In a simple cubic unit cell, the atoms are located only at the corners of the cube. Atoms will touch each other along the edge of the cube. This establishes a direct relationship between the edge length, \(a\), and the atomic radius, \(r\).
04

Deriving the Simple Cubic (SC) Radius Formula

In a simple cubic structure, the edge length \(a\) is twice the atomic radius, because the atom at each corner touches the next one. Therefore, the relationship is: \( a = 2r \), so the radius \(r\) can be given by: \[ r = \frac{a}{2} \].
05

Matching the Options to the Calculated Results

The BCC radius \( r = \frac{\sqrt{3}a}{4} \) and SC radius \( r = \frac{a}{2} \) match the options provided in choice (d). Therefore, the correct answer is the option (d): \(\frac{\sqrt{3} a}{4}, \frac{a}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Body-Centered Cubic
The body-centered cubic (BCC) unit cell is a structure where one atom is positioned at the center of the cube, apart from the corner atoms. This means a full set of nine atoms: one in the center while each corner contributes a fraction. Connections between the center atom and each of the corner atoms form a diagonal line across the cube. In crystallography, understanding this is valuable as the properties of metals like iron, chromium, and tungsten are often influenced by their BCC structure.
  • The body diagonal contains the diameter of two atoms.
  • Relationship: The body diagonal \[ \sqrt{3}a = 2 imes (2r) \] leads to the formula for atomic radius, \[ r = \frac{\sqrt{3}a}{4} \].
  • Useful in determining packing efficiency and density of materials.
Simple Cubic Unit Cell
The simple cubic (SC) unit cell is the most basic type of crystal structure. Each atom is situated only at the cube's corners, touching its neighboring atoms along the edge. This simplicity makes it a useful starting point for understanding more complex structures but is relatively rare due to its inefficient packing.
  • Atoms touch along the cube's edge.
  • Relationship: The edge length is equal to twice the atomic radius, \[ a = 2r \] resulting in the atomic radius, \[ r = \frac{a}{2} \].
  • Examples include polonium which crystallizes in this structure.
Atomic Radius
The atomic radius is a crucial measurement in crystallography, giving insight into how atoms pack together in a crystal lattice. It is half the distance between the nuclei of two adjacent atoms, and it is instrumental in predicting the properties of elements and compounds. Understanding atomic radius is key in comparing different unit cells:
  • For BCC structures: \[ r = \frac{\sqrt{3}a}{4} \].
  • For SC structures: \[ r = \frac{a}{2} \].
  • Atomic radius affects material properties such as strength and ductility.
Edge Length
Edge length in crystal structures refers to the length of each side of the unit cell cube. It's one of the primary metrics defining how atoms are arranged within a crystal. The edge length can vary based on the type of unit cell and affects the material's properties, like density and stability.
  • For BCC: Relates to the atomic radius as \[ \sqrt{3}a = 4r \].
  • For SC: Simply twice the atomic radius, or \[ a = 2r \].
  • Critical for calculating the weight and volume of crystalline materials.
Crystallography
Crystallography is the study of how atoms are arranged in solid matter. It combines principles of chemistry, physics, and mathematics to explain and predict the most effective and stable atomic arrangements. Crystallographers utilize unit cell models like BCC and SC to describe and analyze different materials.
  • Essential for material science, helping in the development of new alloys and compounds.
  • Informs on the physical and chemical behavior of crystalline substances.
  • Leads to the discovery of materials with targeted properties for specific applications.

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Most popular questions from this chapter

Which of the following statements for crystals having Schottky defect is not correct? (a) Schottky defect arises due to the absence of a cation or anion from the position which it is expected to occupy. (b) Schottky defect are more common in ionic compounds with high co-ordination numbers. (c) The density of the crystals having Schottky defect is larger than that of the perfect crystal. (d) The crystal having Schottky defect is electrical neutral as a whole.

Ferrous oxide has a cubic structure and edge length of the unit cell is \(5.0 \mathrm{~A}\). Assuming the density of ferrous oxide to be \(3.84 \mathrm{~g} / \mathrm{cm}^{3}\), the number of \(\mathrm{Fe}^{2+}\) andO \(^{2 \prime}\) ions present in each unit cell be: (use \(\mathrm{N}_{\mathrm{A}}=6\) \(\left.\times 10^{23}\right)\) (a) \(4 \mathrm{Fe}^{2+}\) and \(4 \mathrm{O}^{2-}\) (b) \(2 \mathrm{Fe}^{2+}\) and \(2 \mathrm{O}^{2-}\) (c) \(1 \mathrm{Fe}^{2+}\) and \(1 \mathrm{O}^{2-}\) (d) \(3 \mathrm{Fe}^{2+}\) and \(4 \mathrm{O}^{2-}\)

First three nearest neighbour distances for body centered cubic lattice are respectively: (a) \(\sqrt{2} a, a, \sqrt{3} a\) (b) \(\frac{a}{\sqrt{2}}, a, \sqrt{3} a\) (c) \(\frac{\sqrt{3} a}{2}, a, \sqrt{2} a\) (d) \(\frac{\sqrt{3} a}{2}, a, \sqrt{3} a\)

When heated above \(916^{\circ} \mathrm{C}\), iron changes its bec crystalline form to foc without the change in the radius of atom. The ratio of density of the crystal before heating and after heating is: (a) \(1.069\) (b) \(0.918\) (c) \(0.725\) (d) \(1.231\)

The limiting radius ratio of the complex \(\left[\mathrm{Ni}(\mathrm{CN})_{4}\right]^{2-}\) is (a) \(0.225-0.414\) (b) \(0.414-0.732\) (c) \(0.155-0.225\) (d) None
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