Chapter 21, Problem 21.64E

Interpretation Introduction

(a)

Interpretation:

The angles of diffraction for a cubic crystal for the given incoming X radiation are to be calculated.

Concept introduction:

A unit cell of the crystal is the three-dimensional arrangement of the atoms present in the crystal. The unit cell is the smallest and simplest unit of the crystal which on repetition forms an entire crystal. The parameters of a crystal can be obtained experimentally by X-ray diffraction technique.

Answer to Problem 21.64E

The table that represents the miller indices and corresponding value of diffraction angle is shown below as,

Miller indices $\left(hkl\right)$	Diffraction angle $\left(\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)\right)$
$\left(100\right)$	$6.853688$
$\left(110\right)$	$9.715979$
$\left(111\right)$	$11.92863$
$\left(200\right)$	$13.80798$
$\left(210\right)$	$15.4763$
$\left(211\right)$	$16.9962$
$\left(220\right)$	$19.72637$
$\left(300\right),\left(221\right)$	$20.97761$
$\left(310\right)$	$22.17077$
$\left(311\right)$	$23.31509$
$\left(222\right)$	$24.41774$
$\left(320\right)$	$25.48445$
$\left(321\right)$	$26.51988$
$\left(400\right)$	$28.51165$

Explanation of Solution

The wavelength of the given X-ray is $1.5418\stackrel{\circ }{\text{A}}$.

The given lattice parameter is $6.46\stackrel{\circ }{\text{A}}$.

The value of $n$ for first-order diffractions is $1$.

The Bragg equation for diffraction of X rays is given by an expression as shown below.

$n\lambda =2\left(\frac{a}{\sqrt{h^2+k^2+l^2}}\right)\sin \theta$ …(1)

Where,

• $h,k,l$are the Miller indices.

• $a$ is the side of the unit cell.

• $\lambda$ is the wavelength of the incident ray.

• $\theta$ is the Bragg’s angle.

• $n$ is the order of diffraction.

Rearrange the equation (1) for the value of $\theta$.

$\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)$ …(2)

The value of $\lambda$, $a$, $n$ and different Miller indices are substituted in the equation (2). The table that represents the miller indices and corresponding value of diffraction angle is shown below as,

Miller indices $\left(hkl\right)$	$\left(h^2+k^2+l^2\right)$	Diffraction angle $\left(\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)\right)$
$\left(100\right)$	$1$	$6.853688$
$\left(110\right)$	$2$	$9.715979$
$\left(111\right)$	$3$	$11.92863$
$\left(200\right)$	$4$	$13.80798$
$\left(210\right)$	$5$	$15.4763$
$\left(211\right)$	$6$	$16.9962$
$\left(220\right)$	$8$	$19.72637$
$\left(300\right),\left(221\right)$	$9$	$20.97761$
$\left(310\right)$	$10$	$22.17077$
$\left(311\right)$	$11$	$23.31509$
$\left(222\right)$	$12$	$24.41774$
$\left(320\right)$	$13$	$25.48445$
$\left(321\right)$	$14$	$26.51988$
$\left(400\right)$	$16$	$28.51165$

Conclusion

The table that represents the miller indices and corresponding value of diffraction angle is shown below as,

Miller indices $\left(hkl\right)$	Diffraction angle $\left(\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)\right)$
$\left(100\right)$	$6.853688$
$\left(110\right)$	$9.715979$
$\left(111\right)$	$11.92863$
$\left(200\right)$	$13.80798$
$\left(210\right)$	$15.4763$
$\left(211\right)$	$16.9962$
$\left(220\right)$	$19.72637$
$\left(300\right),\left(221\right)$	$20.97761$
$\left(310\right)$	$22.17077$
$\left(311\right)$	$23.31509$
$\left(222\right)$	$24.41774$
$\left(320\right)$	$25.48445$
$\left(321\right)$	$26.51988$
$\left(400\right)$	$28.51165$

Interpretation Introduction

(b)

Interpretation:

The diffractions that would be absent if the given crystal were body-centered cubic or face centered cubic are to be determined.

Concept introduction:

A unit cell of the crystal is the three-dimensional arrangement of the atoms present in the crystal. The unit cell is the smallest and simplest unit of the crystal which on repetition forms an entire crystal. The parameters of a crystal can be obtained experimentally by X-ray diffraction technique.

Answer to Problem 21.64E

The diffractions that would be absent if the given crystal was body-centered cubic are represented in the table shown below.

Miller indices $\left(hkl\right)$	Diffraction angle $\left(\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)\right)$
$\left(100\right)$	$6.853688$
$\left(111\right)$	$11.92863$
$\left(210\right)$	$15.4763$
$\left(300\right),\left(221\right)$	$20.97761$
$\left(311\right)$	$23.31509$
$\left(320\right)$	$25.48445$

The diffractions that would be absent if the given crystal was face-centered cubic are represented in the table shown below.

Miller indices $\left(hkl\right)$	Diffraction angle $\left(\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)\right)$
$\left(100\right)$	$6.853688$
$\left(110\right)$	$9.715979$
$\left(210\right)$	$15.4763$
$\left(211\right)$	$16.9962$
$\left(300\right),\left(221\right)$	$20.97761$
$\left(310\right)$	$22.17077$
$\left(320\right)$	$25.48445$
$\left(321\right)$	$26.51988$

Explanation of Solution

From Table $21.3$,

The Miller indices of diffractions those are present in body centered cubic crystal are $\left(110\right)$, $\left(200\right)$, $\left(211\right)$, $\left(220\right)$, $\left(310\right)$, $\left(222\right)$, $\left(321\right)$ and $\left(400\right)$.

Therefore, the diffractions that would be absent if the given crystal was body-centered cubic are represented in the table shown below as,

Miller indices $\left(hkl\right)$	$\left(h^2+k^2+l^2\right)$	Diffraction angle $\left(\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)\right)$
$\left(100\right)$	$1$	$6.853688$
$\left(111\right)$	$3$	$11.92863$
$\left(210\right)$	$5$	$15.4763$
$\left(300\right),\left(221\right)$	$9$	$20.97761$
$\left(311\right)$	$11$	$23.31509$
$\left(320\right)$	$13$	$25.48445$

From Table $21.3$,

The Miller indices of diffractions those are present in face centered cubic crystal are $\left(111\right)$, $\left(200\right)$, $\left(220\right)$, $\left(311\right)$, $\left(222\right)$, and $\left(400\right)$.

Therefore, the diffractions that would be absent if the given crystal was face-centered cubic are represented in the table shown below as,

Miller indices $\left(hkl\right)$	$\left(h^2+k^2+l^2\right)$	Diffraction angle $\left(\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)\right)$
$\left(100\right)$	$1$	$6.853688$
$\left(110\right)$	$2$	$9.715979$
$\left(210\right)$	$5$	$15.4763$
$\left(211\right)$	$6$	$16.9962$
$\left(300\right),\left(221\right)$	$9$	$20.97761$
$\left(310\right)$	$10$	$22.17077$
$\left(320\right)$	$13$	$25.48445$
$\left(321\right)$	$14$	$26.51988$

Conclusion

The diffractions that would be absent if the given crystal was body-centered cubic are represented in the table shown below.

Miller indices $\left(hkl\right)$	Diffraction angle $\left(\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)\right)$
$\left(100\right)$	$6.853688$
$\left(111\right)$	$11.92863$
$\left(210\right)$	$15.4763$
$\left(300\right),\left(221\right)$	$20.97761$
$\left(311\right)$	$23.31509$
$\left(320\right)$	$25.48445$

The diffractions that would be absent if the given crystal was face-centered cubic are represented in the table shown below.

Miller indices $\left(hkl\right)$	Diffraction angle $\left(\theta ={\sin }^{-1}\left(\frac{n\lambda \sqrt{h^2+k^2+l^2}}{2a}\right)\right)$
$\left(100\right)$	$6.853688$
$\left(110\right)$	$9.715979$
$\left(210\right)$	$15.4763$
$\left(211\right)$	$16.9962$
$\left(300\right),\left(221\right)$	$20.97761$
$\left(310\right)$	$22.17077$
$\left(320\right)$	$25.48445$
$\left(321\right)$	$26.51988$