Chapter 10, Problem 10.90SP

Interpretation Introduction

Interpretation:

The closest cubic cell where crystallization of Titanium takes places has to be identified.

Concept Introduction:

Body-centered unit cell:

A body – centered cubic cell is a type of unit cell with atoms arranged in all the vertices of the unit cell and one atom at the center of the cube.

Figure 1

A body – centered cubic unit cell has its all the eight vertices arranged with atoms in it.  Eight atoms are arranged in a simple cubic unit cell in eight vertices.  Each atom in a vertex is shared by eight neighboring unit cells such that each cubic unit cell has $\frac{\text{1}}{\text{8}}\text{th}$ of an atom at each vertex.  Further, each body – centered cubic unit cell has one atom at the center.  So 8 vertices have $\text{8}\text{×}\frac{\text{1}}{\text{8}}=1\text{atom}\text{+}\text{1}$ atom at the body of the cube gives 2 atoms per body – centered cubic unit cell.

Face-centered cubic:

In face-centered cubic unit cell, each of the six corners is occupied by every single atom.  Each face of the cube is occupied by one atom.

Each atom in the corner is shared by eight unit cells and each atom in the face is shared by two unit cells.  Thus the number of atoms per unit cell in FCC unit cell is,

$\text{8}\text{×}\frac{\text{1}}{\text{8}}\text{atoms}\text{in}\text{corners}\text{+}\text{6}\text{×}\frac{\text{1}}{\text{2}}\text{atoms}\text{in}\text{faces}\text{=}\text{1}\text{+}\text{3}\text{=}\text{4}\text{atoms}$

Simple cubic unit:

A simple cubic unit cell is the simplest form of a cubic unit cell.  A cube has eight vertices, twelve edges and six faces.  Similarly a cubic unit cell has eight vertices, twelve edges and six faces.  If in a cubic unit cell, the components occupy only the eight vertices, then the unit cell is known as simple cubic unit cell.

Figure 2

A simple cubic unit cell has its all the eight vertices arranged with atoms in it.  Thus eight atoms are arranged in a simple cubic unit cell in eight vertices.  Each atom in a vertex is shared by eight neighboring unit cells.  Each layer of a lattice has four unit cells and the upper or lower to that layer has four unit cells.  So, each simple cubic unit cell has $\frac{\text{1}}{\text{8}}\text{th}$ of an atom at each vertex.  So 8 vertices have $\text{8}\text{×}\frac{\text{1}}{\text{8}}=1\text{atom}$ per unit cell.