Chapter 3, Problem 3.63P

Interpretation Introduction

(a)

Interpretation:

The directions for the hexagonal unit cell for $\left[\overline{2}110\right]$ needs to be sketched.

Concept Introduction:

Miller indices for plane pass through the three points at the edge of the hexagonal lattice. It uses four-digit system [h, k, i, l]. The additional indices are used to show the direction in three-dimensional vectors.

Answer to Problem 3.63P

Directions within the hexagonal unit cell for $\left[\overline{2}110\right]$ is shown below

  

Explanation of Solution

Procedure for drawing direction for $\left[\overline{2}110\right]$ :

1. First draw empty hexagonal unit cell.
2. Draw the line in opposite direction of a₁ by 2 units which indicates $\overline{2}$ .
3. Draw the line in a₂ direction by 1 unit.
4. Draw the line in a₃ direction by 1 units.
5. There is no displacement along z-axis as coordinate is zero.
6. Then join the origin to the point on a₁ to obtain desired direction_.

  

Interpretation Introduction

(b)

Interpretation:

The directions for the hexagonal unit cell for $\left[11\overline{2}1\right]$ needs to be sketched.

Concept Introduction:

Miller indices for plane pass through the three points at the edge of the hexagonal lattice. It uses four-digit system [h, k, i, l]. The additional indices are used to show the direction in three-dimensional vectors.

Answer to Problem 3.63P

Directions within the hexagonal unit cell for $\left[11\overline{2}1\right]$ is shown below

  

Explanation of Solution

Procedure for drawing direction for $\left[11\overline{2}1\right]$ :

1. First draw empty hexagonal unit cell.
2. Draw the line in direction of a₁ by 1 units which indicates 1.
3. Draw the line in a₂ direction by 1 unit.
4. Draw the line in opposite direction of a₃ by $\overline{2}$ units.
5. Draw the line along z-axis by 1 unit..
6. Then join the origin to the point along z-axis to obtain desired direction_.

  

Interpretation Introduction

(c)

Interpretation:

The directions for the hexagonal unit cell for $\left[10\overline{1}0\right]$ needs to be sketched.

Concept Introduction:

Miller indices for plane pass through the three points at the edge of the hexagonal lattice. It uses four-digit system [h, k, i, l]. The additional indices are used to show the direction in three-dimensional vectors.

Answer to Problem 3.63P

Directions within the hexagonal unit cell for $\left[10\overline{1}0\right]$ is shown below

  

Explanation of Solution

Procedure for drawing direction for $\left[10\overline{1}0\right]$ :

1. First draw empty hexagonal unit cell.
2. Draw the line in direction of a₁ by 1 units which indicates 1.
3. There is no displacement along a₂ direction as coordinate is zero.
4. Draw the line in opposite direction of a₃ by $\overline{1}$ units.
5. There is no displacement along z-axis as coordinate is zero.
6. Then join the origin to the point on basal plane to obtain desired direction_.

  

Interpretation Introduction

(d)

Interpretation:

The planes for the hexagonal unit cell for $\left[1\overline{2}10\right]$ needs to be sketched.

Concept Introduction:

Miller indices for plane pass through the three points at the edge of the hexagonal lattice. It uses four-digit system [h, k, i, l]. The additional indices are used to show the direction in three-dimensional vectors.

Answer to Problem 3.63P

Plane within the hexagonal unit cell for $\left[1\overline{2}10\right]$ is shown below

  

Explanation of Solution

Procedure for drawing planes for $\left[1\overline{2}10\right]$ :

1. Converting four index miller indices of $\left[1\overline{2}10\right]$_. into three indices.
$\begin{array}{l}\left(hkil\right)= \left(1\overline{2}10\right)\\ h=1\\ k=\overline{2}\\ i=1\\ l=0\end{array}$
2. Now taking reciprocals of plane and putting lattice parameters
$\begin{array}{l}{a}_1=a\\ a_2=\frac{-a}{2}\\ a_3=a\\ z=\infty \end{array}$

  

Interpretation Introduction

(e)

Interpretation:

The planes for the hexagonal unit cell for $\left[\overline{1}\overline{1}22\right]$ needs to be sketched.

Concept Introduction:

Miller indices for plane pass through the three points at the edge of the hexagonal lattice. It uses four-digit system [h, k, i, l]. The additional indices are used to show the direction in three-dimensional vectors.

Answer to Problem 3.63P

Plane within the hexagonal unit cell for $\left[\overline{1}\overline{1}22\right]$ is shown below

  

Explanation of Solution

Procedure for drawing planes for $\left[\overline{1}\overline{1}22\right]$ :

1. Converting four index miller indices of $\left[\overline{1}\overline{1}22\right]$_. into three indices.
$\begin{array}{l}\left(hkil\right)= \left( \overline{1}\overline{1}22\right)\\ h=\overline{1}\\ k=\overline{1}\\ i=2\\ l=2\end{array}$
2. Now taking reciprocals of plane and putting lattice parameters
$\begin{array}{l}{a}_1=-a\\ a_2=-a\\ a_3=\frac{a}{2}\\ z=\frac{a}{2}\end{array}$

  

Interpretation Introduction

(f)

Interpretation:

The planes for the hexagonal unit cell for $\left[12\overline{3}0\right]$ needs to be sketched.

Concept Introduction:

Miller indices for plane pass through the three points at the edge of the hexagonal lattice. It uses four-digit system [h, k, i, l]. The additional indices are used to show the direction in three-dimensional vectors.

Answer to Problem 3.63P

Plane within the hexagonal unit cell for $\left[12\overline{3}0\right]$ is shown below

  

Explanation of Solution

Procedure for drawing planes for $\left[12\overline{3}0\right]$ :

1. Converting four index miller indices of $\left[12\overline{3}0\right]$_. into three indices.
$\begin{array}{l}\left(hkil\right)= \left(12\overline{3}0\right)\\ h=1\\ k=2\\ i=\overline{3}\\ l=0\end{array}$
2. Now taking reciprocals of plane and putting lattice parameters
$\begin{array}{l}{a}_1=a\\ a_2=\frac{a}{2}\\ a_3=-\frac{a}{3}\\ z=\infty \end{array}$