Chapter 3, Problem 3.98P

Interpretation Introduction

(a)

Interpretation:

The crystal structure of the metal is to be determined from the given x-ray diffraction results.

Concept Introduction:

X-ray diffraction (XRD) is a rapid analytical technique which is primarily used to identify phase of a crystalline material and it can provide information about the dimensions of the unit cell. The material to be analyzed is finely ground, homogenized, and its average bulk composition is determined.

Answer to Problem 3.98P

The crystal structure of the metal is FCC.

Explanation of Solution

Given information:

The figure which shows the results of the x-ray diffraction experiment is in the form of the intensity of the diffracted peak versus the $2\theta$ diffraction angle. The wavelength of the x-ray used is of $0.15418\text{ nm}$.

  

The graph between intensity of the diffracted peak versus the $\text{2θ}$ diffracting angle is as follows:

  

It is required to tabulate the different values of lattice parameter $\left({\text{a}}_{\text{o}}\right)$, inter planar space $\left(\text{d}\right)$, peak for the given XRD pattern. Determine the $\text{2θ}$ values from the given XRD pattern.

The table with different values is as follows:

Peak	$\text{2θ}(°)$	${\text{sin}}^2\text{θ}$	$\frac{{\text{sin}}^2\text{θ}}{0.0077}$	${\text{h}}^{\text{2}}{\text{+k}}^{\text{2}}{\text{+l}}^{\text{2}}$	Planar indices $\left(\text{hkl}\right)$
1.	17.5	0.023	3	$\sqrt{3}$	111
2.	20.5	0.032	4	4	200
3.	28.5	0.061	8	8	220
4.	33.5	0.083	11	$\sqrt{11}$	311
5.	35.5	0.093	12	$\sqrt{12}$	222
6.	41	0.123	16	16	400
7.	45	0.146	19	$\sqrt{19}$	331
8.	46.5	0.156	20	$\sqrt{20}$	420

The values of $\frac{{\text{sin}}^2\text{θ}}{0.0077}$ is calculated to find the appropriate pattern or sequence.

Observe the sequence of $\frac{{\text{sin}}^2\text{θ}}{0.0077}$, the sequence is $\text{3, 4, 8, 11, 12, 16, 19 and 20}$. So, the crystal structure of the metal is FCC.

Interpretation Introduction

(b)

Interpretation:

The indices of the planes which produce each of the given peaks in the figure are to be determined.

Concept Introduction:

X-ray diffraction (XRD) is a rapid analytical technique which is primarily used to identify phase of a crystalline material and it can provide information about the dimensions of the unit cell. The material to be analyzed is finely ground, homogenized, and its average bulk composition is determined.

Answer to Problem 3.98P

The planar indices are $\text{111, 200, 220, 311, 222, 400, 331 and 420}\text{.}$

Explanation of Solution

Given information:

The figure which shows the results of the x-ray diffraction experiment is in the form of the intensity of the diffracted peak versus the $2\theta$ diffraction angle. The wavelength of the x-ray used is of $0.15418\text{ nm}$.

  

From part (a), the tabulated data for the lattice parameter $\left({\text{a}}_{\text{o}}\right)$, inter planar space $\left(\text{d}\right)$, peak for the given XRD pattern is shown below as:

Peak	$\text{2θ}(°)$	${\text{sin}}^2\text{θ}$	$\frac{{\text{sin}}^2\text{θ}}{0.0077}$	${\text{h}}^{\text{2}}{\text{+k}}^{\text{2}}{\text{+l}}^{\text{2}}$	Planar indices $\left(\text{hkl}\right)$
1.	17.5	0.023	3	$\sqrt{3}$	111
2.	20.5	0.032	4	4	200
3.	28.5	0.061	8	8	220
4.	33.5	0.083	11	$\sqrt{11}$	311
5.	35.5	0.093	12	$\sqrt{12}$	222
6.	41	0.123	16	16	400
7.	45	0.146	19	$\sqrt{19}$	331
8.	46.5	0.156	20	$\sqrt{20}$	420

The planar indices are indicated on the $8^{\text{th}}$ column of the table. The planar indices are $\text{111, 200, 220, 311, 222, 400, 331 and 420}\text{.}$

Interpretation Introduction

(c)

Interpretation:

The lattice parameter of the metal is to be determined.

Concept Introduction:

In a crystal lattice, the physical dimension of unit cells is indicated by Lattice Parameter. It is also called Lattice Constant. Lattices in three dimensions generally have three lattice parameters, they are h, k, and l.

Bragg's Law equation is as follows:

  $\sin \text{θ}=\frac{\lambda }{2{\text{d}}_{\text{hkl}}}$   ......(1)

Here,

  $\text{θ}$ is the half angle between the diffracted bean and the original beam direction.

  $\lambda$ is the wavelength of x-rays.

  ${\text{d}}_{\text{hkl}}$ is the interplanar spacing between the planes.

The formula to calculate the lattice parameter $\left({\text{a}}_{\text{o}}\right)$ is:

  ${\text{a}}_{\text{o}}=d_{\text{hkl}}\times \sqrt{h^2+k^2+l^2}$   ......(2)

Answer to Problem 3.98P

The lattice parameter of the metal is $0.8804\text{ nm}$.

Explanation of Solution

Given information:

The figure which shows the results of the x-ray diffraction experiment is in the form of the intensity of the diffracted peak versus the $2\theta$ diffraction angle. The wavelength of the x-ray used is of $0.15418\text{ nm}$.

  

From part (a), the tabulated data for the lattice parameter $\left({\text{a}}_{\text{o}}\right)$, inter planar space $\left(\text{d}\right)$, peak for the given XRD pattern is shown below as:

Peak	$\text{2θ}(°)$	${\text{sin}}^2\text{θ}$	$\frac{{\text{sin}}^2\text{θ}}{0.0077}$	${\text{h}}^{\text{2}}{\text{+k}}^{\text{2}}{\text{+l}}^{\text{2}}$	Planar indices $\left(\text{hkl}\right)$
1.	17.5	0.023	3	$\sqrt{3}$	111
2.	20.5	0.032	4	4	200
3.	28.5	0.061	8	8	220
4.	33.5	0.083	11	$\sqrt{11}$	311
5.	35.5	0.093	12	$\sqrt{12}$	222
6.	41	0.123	16	16	400
7.	45	0.146	19	$\sqrt{19}$	331
8.	46.5	0.156	20	$\sqrt{20}$	420

To calculate the lattice parameter, it is required to choose any peak from the table. Choose peak 6 which has 400 planar indices.

The lattice parameter, ${\text{a}}_{\text{o}}$ of the metal can be calculated using Bragg's Law equation as:

Here,

  $\begin{array}{c}h=4\\ k=0\\ l=0\end{array}$

Substitute the above values in the equation 1,

  $\begin{array}{c}\sin \text{θ}=\frac{\lambda }{2{\text{d}}_{\text{hkl}}}\\ \sin \left(\frac{41°}{2}\right)=\frac{0.15418\text{ nm}}{2\times {\text{d}}_{400}}\\ {\text{d}}_{400}=\frac{0.15418\text{ nm}}{2\times \sin \left(\frac{41°}{2}\right)}\end{array}$

Using this value of $d_{400}$, calculate the lattice parameter ${\text{a}}_{\text{o}}$ as:

  $\begin{array}{c}{\text{a}}_{\text{o}}=d_{400}\times \sqrt{h^2+k^2+l^2}\\ =\frac{0.15418\text{ nm}}{2\times \sin \left(\frac{41°}{2}\right)}\times \sqrt{4^2+0^2+0^2}\\ =\frac{0.15418\text{ nm}}{2\times \sin \left(\frac{41°}{2}\right)}\times 4\\ {\text{a}}_{\text{o}}=0.8804\text{ nm}\end{array}$

Hence, the Lattice Parameter is $0.8804\text{ nm}$.