Materials Science and Engineering: An Introduction

Published by Wiley
ISBN 10: 1118324579
ISBN 13: 978-1-11832-457-8

Chapter 3 - The Structure of Crystalline Solids - Questions and Problems - Page 97: 3.3


$a = \frac{4R}{\sqrt{3}}$

Work Step by Step

For a BCC structure, the corner atoms do not touch one another, but all touch the center BCC atom. Therefore, we can draw a line from one corner of the unit cell to the other, as this is where the atoms are touching. First, we will find the distance from one corner of the unit cell, to the other, in terms of the unit cell edge length, $a$. We begin by finding the corner to corner distance on one face, $b$ of the unit cell, using Pythagorean's theorem. $a^2 + a^2 = b^2$ $b = \sqrt{2}a$ Next we can use Pythagorean's theorem again to now find the distance from one corner to the other of the entire unit cell, denoted as $c$ since we know the distance of one face in terms of $a$. $(\sqrt{2}a)^2 + a^2 = c^2$ $c = \sqrt{3}a$ Since the corner atoms overlap with adjacent unit cells, if we draw a line from one corner of the unit cell to the other, this runs through 1 corner atom, the entire center atom, and then 1 corner atom on the opposite side. This is a total of 4 radii. $4R = c$ $4R = \sqrt{3}a$ $$a = \frac{4R}{\sqrt{3}}$$
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