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Chapter 12: Problem 27

What is the minimum number of atoms that could be contained in the unit cell of an element with a body-centered cubic lattice? $(\mathbf{a}) 1,(\mathbf{b}) 2,(\mathbf{c}) 3,(\mathbf{d}) 4,(\mathbf{e}) 5 .$

Short Answer

Expert verified
The minimum number of atoms that could be contained in the unit cell of an element with a body-centered cubic lattice is \(2\). This includes the contributions of \(1\) atom from the 8 corner atoms and \(1\) atom from the center. Therefore, the correct answer is \((\mathbf{b}) 2\).

Step by step solution

01

Understand the body-centered cubic lattice structure

A body-centered cubic lattice has atoms at the eight corners of the unit cell and one atom at the center. It's important to note that each of the corner atoms is shared by 8 adjacent unit cells. Therefore, only one-eighth of each corner atom belongs to the unit cell.
02

Calculate the total number of atoms in the unit cell

To find the total number of atoms in a unit cell of a BCC lattice, first, we need to find the contribution of atoms from the corners and the center: 1. Atoms at the corners: There are 8 corners, and each corner atom is shared by 8 unit cells. So, the contribution of corner atoms to the unit cell is \(8 \times \frac{1}{8} = 1\). 2. Atom at the center: The center atom is entirely inside the unit cell. Therefore, its contribution to the unit cell is equal to 1.
03

Add the contributions of corner and center atoms

The total number of atoms in the unit cell of a body-centered cubic lattice can be found by adding the contributions of corner atoms and the center atom: Total atoms in unit cell = corner atoms contribution + center atom contribution = 1 + 1 = 2 The minimum number of atoms that could be contained in the unit cell of an element with a body-centered cubic lattice is 2. So, the correct answer is: \((\mathbf{b}) 2\).

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Most popular questions from this chapter

Indicate whether each statement is true or false: (a) Substitutional alloys tend to be more ductile than interstitial alloys. (b) Interstitial alloys tend to form between elements with similar ionic radii. (c) Nonmetallic elements are never found in alloys.

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Besides the cubic unit cell, which other unit cell(s) has edge lengths that are all equal to each other? (a) Orthorhombic, \((\mathbf{b})\) hexagonal, \((\mathbf{c})\) rhombohedral, (d) triclinic, (e) both rhombohedral and triclinic.

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