Chapter 3, Problem 3.61P

Interpretation Introduction

(a)

Interpretation:

The plane within the cubic cell for $\left[1\overline{1}0\right]$ needs to be sketched.

Concept Introduction:

Miller indices are a group of numbers which shows the orientation of the plane in given directions within a crystal structure.

Answer to Problem 3.61P

The figure shows direction within a cubic cell unit for $\left[1\overline{1}0\right]$

  

Explanation of Solution

Procedure to draw $\left[1\overline{1}0\right]$ direction within cubic unit cell:

	x	y	z
Reduction of Integer	1	$\overline{1}$	0
Reduction of scale	-	-	-
Projection	a	-b	0c

Coordinates of points are $\left(100\right),\left(0-10\right),\left(000\right)$

  

Interpretation Introduction

(b)

Interpretation:

The plane within the cubic cell for $\left[\overline{2}\overline{2}1\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure. We have to show the directions and planes in the cubic cell as per the Miller index for each example.

Answer to Problem 3.61P

The figure show direction within a cubic cell unit for $\left[\overline{2}\overline{2}1\right]$

  

Explanation of Solution

Procedure to draw $\left[\overline{2}\overline{2}1\right]$ direction within cubic unit cell:

	x	y	z
Reduction of Integer	-2	-2	1
Reduction of scale	-1	-1	$\frac{1}{2}$
Projection	-a	-b	$\frac{1}{2}$ c

Coordinates of points are $\left(-200\right),\left(0-20\right),\left(001\right)$

  

Interpretation Introduction

(c)

Interpretation:

The plane within the cubic cell for $\left[410\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure show direction within a cubic cell unit for $\left[410\right]$

  

Explanation of Solution

Procedure to draw $\left[410\right]$ direction within cubic unit cell:

	x	y	z
Reduction of Integer	4	1	0
Reduction of scale	1	$\frac{1}{4}$	0
Projection	a	$\frac{1}{4}$ b	0c

Coordinates of points are $\left(400\right),\left(010\right),\left(000\right)$

  

Interpretation Introduction

(d)

Interpretation:

The plane within the cubic cell for $\left[0\overline{1}2\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure show direction within a cubic cell unit for $\left[0\overline{1}2\right]$

  

Explanation of Solution

Procedure to draw $\left[0\overline{1}2\right]$ direction within cubic unit cell:

	x	y	z
Reduction of Integer	0	-1	2
Reduction of scale	0	- $\frac{1}{2}$	1
Projection	0a	- $\frac{1}{2}$b	c

Coordinates of points are $\left(000\right),\left(0-10\right),\left(002\right)$

  

Interpretation Introduction

(e)

Interpretation:

The plane within the cubic cell for $\left[\overline{3}\overline{2}1\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure show direction within a cubic cell unit for $\left[\overline{3}\overline{2}1\right]$

  

Explanation of Solution

Procedure to draw $\left[\overline{3}\overline{2}1\right]$ direction within cubic unit cell:

	x	y	z
Reduction of Integer	-3	-2	1
Reduction of scale	- $\frac{3}{2}$	$-1$	$\frac{1}{2}$
Projection	- $\frac{3}{2}$ a	-b	$\frac{1}{2}$ c

Coordinates of points are $\left(-300\right),\left(0-20\right),\left(001\right)$

  

Interpretation Introduction

(f)

Interpretation:

The plane within the cubic cell for $\left[11\overline{1}\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure shows the plane within a cubic cell unit for $\left[11\overline{1}\right]$

  

Explanation of Solution

Procedure to draw $\left[11\overline{1}\right]$ plane within cubic unit cell:

1. Remove indices from parenthesis and taking reciprocals of the indices
Here, 1 indicates that plane intersects at 1 on x-axis, 1 indicates plane intersects at 1on y-axis and -1 indicates that plane intersects at 1 on negative direction on z-axis.
2. Coordinates of points are $\left(100\right),\left(010\right),\left(001\right)$

  

Interpretation Introduction

(g)

Interpretation:

The plane within the cubic cell for $\left[1\overline{1}1\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure show direction within a cubic cell unit for $\left[1\overline{1}1\right]$

  

Explanation of Solution

Procedure to draw $\left[1\overline{1}1\right]$ direction within cubic unit cell:

	x	y	z
Reduction of Integer	1	-1	1
Reduction of scale	1	-1	1
Projection	A	-b	c

Coordinates of points are $\left(100\right),\left(0-10\right),\left(001\right)$

  

Interpretation Introduction

(h)

Interpretation:

The plane within the cubic cell for $\left[01\overline{1}\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure show direction within a cubic cell unit for $\left[01\overline{1}\right]$

  

Explanation of Solution

Procedure to draw $\left[01\overline{1}\right]$ direction within cubic unit cell:

	x	y	z
Reduction of Integer	0	1	-1
Reduction of scale	-	-	-
Projection	0a	B	-c

Coordinates of points are $\left(000\right),\left(010\right),\left(00\overline{1}\right)$

  

Interpretation Introduction

(i)

Interpretation:

The plane within the cubic cell for $\left[030\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure shows the plane within a cubic cell unit for $\left[030\right]$

  

Explanation of Solution

Procedure to draw $\left[030\right]$ plane within cubic unit cell:

1. Remove indices from parenthesis and taking reciprocals of the indices
Here, $\infty$ indicates that plane is parallel to x-axis, 3 indicates plane intersects at 3 on y-axis and $\infty$ indicates that plane is parallel to z-axis.
2. Coordinates of points are $\left(000\right),\left(030\right),\left(000\right)$

  

Interpretation Introduction

(j)

Interpretation:

The plane within the cubic cell for $\left[\overline{1}21\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure shows the plane within a cubic cell unit for $\left[\overline{1}21\right]$

  

Explanation of Solution

Procedure to draw $\left[\overline{1}21\right]$ plane within cubic unit cell:

1. Remove indices from parenthesis and taking reciprocals of the indices
Here, -1 indicates that plane intersects at 1 on negative direction of x-axis, 2 indicates plane intersects at 2 on y-axis and 1 indicates that plane intersects at 1 on z-axis.
2. Coordinates of points are $\left(-100\right),\left(020\right),\left(001\right)$

  

Interpretation Introduction

(k)

Interpretation:

The plane within the cubic cell for $\left[11\overline{3}\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure shows the plane within a cubic cell unit for $\left[11\overline{3}\right]$

  

Explanation of Solution

Procedure to draw $\left[11\overline{3}\right]$ plane within cubic unit cell:

1. Remove indices from parenthesis and taking reciprocals of the indices
Here, 1 indicates that plane intersects at 1 on x-axis, 1 indicates plane intersects at 1on y-axis and -3 indicates that plane intersects at 3 on negative direction on z-axis.
2. Coordinates of points are $\left(100\right),\left(010\right),\left(00-3\right)$

  

Interpretation Introduction

(l)

Interpretation:

The plane within the cubic cell for $\left[0\overline{4}1\right]$ needs to be sketched.

Concept Introduction:

Miller indices are the groups of numbers which indicate the orientation of a group of numbers given directions within a crystal structure.

Answer to Problem 3.61P

The figure shows the plane within a cubic cell unit for $\left[0\overline{4}1\right]$

  

Explanation of Solution

Procedure to draw $\left[0\overline{4}1\right]$ plane within cubic unit cell:

1. Remove indices from parenthesis and taking reciprocals of the indices
Here, $\infty$ indicates that plane is parallel to x-axis, -4 indicates plane intersects at 4 on negative direction of y-axis and 1 indicates that plane intersects at 1 on z-axis.
2. Coordinates of points are $\left(00\right),\left(0-40\right),\left(001\right)$