Chapter 15, Problem 15A.2ST

Interpretation Introduction

Interpretation:

The separation of the $\left\{133\right\}$ and $\left\{399\right\}$ planes in the given lattice has to be calculated.

Concept Introduction:

The separation of planes, d, of the $\left\{hkl\right\}$ planes in an orthogonal lattice is given by,

  $\frac{1}{d^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}+\frac{l^2}{c^2}$

Where,

  a, b and c are dimensions of the unit cell.

Explanation of Solution

Given data:

The dimensions of the unit cell are $a=0.82\text{ nm, }b\text{ = 0}\text{.94 nm and }c\text{ = 0}\text{.75 nm}$

The planes of orthorhombic lattice is $\left\{133\right\}$ and $\left\{399\right\}$

Separation of planes:

The separation of planes is determined using the below formula.

  $\frac{1}{d^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}+\frac{l^2}{c^2}$

Consider $\left\{hkl\right\}$ indices as $\left\{133\right\}$

Substitute the given data in the below equation.

  $\begin{array}{l}\frac{1}{d^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}+\frac{l^2}{c^2}\\ \frac{1}{d^2}=\frac{1^2}{{(0.82\text{ nm)}}^2}+\frac{3^2}{{(0.94\text{ nm)}}^2}+\frac{3^2}{{(0.75\text{ nm)}}^2}\\ \frac{1}{d^2}=27.6728{\text{ nm}}^{\text{2}}\\ d^2=\frac{1}{27.6728}{\text{ nm}}^{\text{2}}\\ =0.0361{\text{ nm}}^{\text{2}}\\ d=0.19\text{ nm}\end{array}$

Now, consider $\left\{hkl\right\}$ indices as $\left\{399\right\}$

Substitute the given data in the below equation.

  $\begin{array}{l}\frac{1}{d^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}+\frac{l^2}{c^2}\\ \frac{1}{d^2}=\frac{{(3\times 1)}^2}{{(0.82\text{ nm)}}^2}+\frac{{(3\times 3)}^2}{{(0.94\text{ nm)}}^2}+\frac{{(3\times 3)}^2}{{(0.75\text{ nm)}}^2}\\ \frac{1}{d^2}=\frac{{(3)}^2}{{(0.82\text{ nm)}}^2}+\frac{{(9)}^2}{{(0.94\text{ nm)}}^2}+\frac{{(9)}^2}{{(0.75\text{ nm)}}^2}\\ \frac{1}{d^2}=249.0553{\text{ nm}}^{\text{2}}\\ d^2=\frac{1}{249.0553}{\text{ nm}}^{\text{2}}\\ =0.004015{\text{ nm}}^{\text{2}}\\ d=0.63\text{ nm}\end{array}$

Therefore,

The separation of the $\left\{133\right\}$ planes in the given lattice is $0.19\text{ nm}$

The separation of the $\left\{399\right\}$ planes in the given lattice is $0.63\text{ nm}$