91Ó°ÊÓ

Using the Molecule Definition Utility found in both "Metallic Crystal Structures and Crystallography" and "Ceramic Crystal Structures" modules of \(V M S E\), located on the book's web site [www.wiley.com/ college/callister (Student Companion Site)], generate (and print out) a three-dimensional unit cell for titanium dioxide, \(\mathrm{TiO}_{2}\), given the following: (1) The unit cell is tetragonal with \(a=0.459 \mathrm{~nm}\) and \(c=0.296 \mathrm{~nm},(2)\) oxygen atoms are located at the following point coordinates: \(\begin{array}{llllll}0.356 & 0.356 & 0 & 0.856 & 0.144 & \frac{1}{2} \\\ 0.664 & 0.664 & 0 & 0.144 & 0.856 & \frac{1}{2}\end{array}\) and (3) Ti atoms are located at the following point coordinates: \(\begin{array}{llllll}0 & 0 & 0 & & 1 & 0 & 1 \\ 1 & 0 & 0 & & 0 & 1 & 1 \\\ 0 & 1 & 0 & & 1 & 1 & 1 \\ 0 & 0 & 1 & & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ 1 & 1 & 0 & & & & \end{array}\)

Step by step solution

01

Access Molecule Definition Utility

Go to the website mentioned in the exercise, www.wiley.com/college/callister, and find the Molecule Definition Utility located in either the "Metallic Crystal Structures and Crystallography" or "Ceramic Crystal Structures" modules.

02

Set up the unit cell parameters

Input the unit cell parameters for titanium dioxide. Enter the following values into the appropriate fields: System: Tetragonal, a-axis: 0.459 nm, c-axis: 0.296 nm.

03

Input the oxygen atom positions

Add oxygen atoms to the crystal by entering the given point coordinates for each oxygen atom: \(0.356,~0.356,~0, ~0.856,~0.144,~\dfrac{1}{2}\) \\ \(0.664,~0.664,~0, ~0.144,~0.856,~\dfrac{1}{2}\)

04

Input the Ti atom positions

Add titanium atoms to the crystal by entering the given point coordinates for each titanium atom: \(0,~0,~0,~~ 1,~0,~1\) \\ \(1,~0,~0,~~ 0,~1,~1\) \\ \(0,~1,~0, ~~1,~1,~1\) \\ \(0,~0,~1, ~~\dfrac{1}{2},~\dfrac{1}{2},~\dfrac{1}{2}\) \\ \(1,~1,~0\)

05

Generate the unit cell and visualize

After entering all the necessary inputs, click "Generate" in the Molecule Definition Utility. The three-dimensional unit cell for titanium dioxide should appear on the screen. You can manipulate and rotate the cell to visualize its structure in different orientations.

06

Print out the unit cell

To create a printout of the unit cell, click on "Print" in the Molecule Definition Utility. This will open up a preview of the unit cell, allowing you to print it out or save it as an image.

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Key Concepts

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

Crystallography

Crystallography is the scientific study of crystals and their structure. It involves analyzing the arrangement of atoms within a crystalline solid. One way to understand a crystal's structure is by examining its unit cell, the smallest repetitive pattern that shows the entire structure of the crystal. For example, the exercise deals with the tetragonal crystal system, a type of unit cell characterized by two equal lateral dimensions ()

Ceramic Crystal Structures

Ceramic crystal structures are composed of both metallic and non-metallic elements, bonded together through ionic and/or covalent bonds. The arrangement of atoms in ceramics influences properties like brittleness, hardness, and electrical insulation. The exercise focuses on the crystal structure of titanium dioxide ()

Metallic Crystal Structures

Metallic crystal structures refer to the arrangement of atoms in metals. These atoms are arranged in a highly ordered pattern, forming structures such as body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed (HCP). While titanium dioxide isn't a metal, the exercise involves understanding metallic crystal structures through the Molecule Definition Utility, which can model both ceramic and metallic structures.