Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 3

Show for the body-centered cubic crystal structure that the unit cell edge length \(a\) and the atomic radius \(R\) are related through \(a=4 R / \sqrt{3}\)

Short Answer

Expert verified
Answer: In a BCC crystal structure, the unit cell edge length, \(a\), is related to the atomic radius, \(R\), through the equation \(a= \frac{4R}{\sqrt{3}}\).

Step by step solution

01

Understanding the BCC structure

In a body-centered cubic structure, there are atoms at each corner of the cubic unit cell and one atom at the center. To show the relationship between \(a\) and \(R\), we can consider a body diagonal that connects two opposite corners of the unit cell. Along this diagonal, there are 2 corner atoms and the central atom. The total length of the body diagonal will be the sum of 4 atomic radii, as the central atom is in contact with both corner atoms.
02

Finding the body diagonal length

To find the length of the body diagonal, we can use the Pythagorean theorem in three dimensions. This theorem tells us that for a right-angled parallelepiped (which the BCC unit cell is), the square of the body diagonal length is equal to the sum of the squares of the edge lengths: \(D^2 = a^2 + a^2 + a^2\) Since the body diagonal length, \(D\), can be written as \(4R\), we get: \((4R)^2 = a^2 + a^2 + a^2\)
03

Solve for \(a\) in terms of \(R\)

Now, we can solve the equation to find the relationship between \(a\) and \(R\). \((4R)^2 = 3a^2\) Take the square root of both sides: \(4R = a \sqrt{3}\) Finally, divide by \(\sqrt{3}\): \(a = \frac{4R}{\sqrt{3}}\) This shows the relationship between the unit cell edge length, \(a\), and the atomic radius, \(R\), as required.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Within a cubic unit cell, sketch the following directions: (a) \([110]\) (e) \(\left[\begin{array}{ll}1 & 1\end{array}\right]\) (b) \([\overline{1} \overline{2} 1]\) (f) \([\overline{1} 22]\) (c) \([0 \overline{1} 2]\) (g) \([123]\) (d) \([133]\) (h) \([103]\)

Sketch the (1\overline{1101) and (1120) planes in a hexag- } onal unit cell.

Cite the indices of the direction that results from the intersection of each of the following pairs of planes within a cubic crystal: (a) the (100) and (010) planes, (b) the (111) and \((11 \overline{1})\) planes, and \((\mathbf{c})\) the \((10 \overline{1})\) and \((001)\) planes.

The unit cell for tin has tetragonal symmetry, with \(a\) and \(b\) lattice parameters of \(0.583\) and \(0.318 \mathrm{~nm}\), respectively. If its density, atomic weight, and atomic radius are \(7.27 \mathrm{~g} / \mathrm{cm}^{3}\), \(118.71 \mathrm{~g} / \mathrm{mol}\), and \(0.151 \mathrm{~nm}\), respectively, compute the atomic packing factor.

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

Recommended explanations on Physics Textbooks

Work Energy and Power

Read Explanation

Electricity and Magnetism

Read Explanation

Physical Quantities And Units

Read Explanation

Medical Physics

Read Explanation

Nuclear Physics

Read Explanation

Fluids

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free