Question

Question: Polonium is the only element known to crystallize in the simple cubic lattice.

(a) What is the distance between nearest neighbor polonium atoms if the first-order diffraction of X-rays with $\mathbf{\lambda }\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{785}\mathbf{A}\mathbf{°}$ from the parallel faces of its unit cells appears at an angle of $\mathbf{2}\mathbf{\theta }\mathbf{=}\mathbf{30}\mathbf{.}\mathbf{96}\mathbf{°}$ from these planes?

(b) What is the density of polonium in this crystal (in gcm^-3 )?

Short answer

Answer

1. The distance between the nearest neighbour is 3.34 A⁰
2. The density of polonium is $9.30 g.c{m}^{-3}$ .

Step-by-step solution

Bragg’s equation

For X-rays of a particular wavelength $\lambda$and for a given value of d, maximum reinforcement can be noticed at a certain value of $\theta$. By knowing the values of $\lambda and\theta$ , the distance “d” between planes in crystal can be calculated.

is Bragg’s equation.

Subpart (a)

Use Bragg’s equation and calculate the “d” value:

Given, $\lambda =1.785 A°$

$$\begin{gathered} 2\theta =30.96° \\ \theta =15.48° \end{gathered}$$

Substitute these values in Bragg’s equation:

$$\begin{gathered} n\lambda =2d \sin \theta \\ 1\times 1.785A°=2\times d \times \sin 15.48° \\ d=3.34 A° \end{gathered}$$

Subpart (b)

For a simple cubic, the number of atoms(n) per unit cell is one.

The density of Polonium:

$$\begin{gathered} density=\frac{n\times Molar mass}{a^3\times N} \\ =\frac{1 atom\times 208.9 g/mol}{{\left(3.34\times {10}^{-8} cm\right)}^3\times \left(6.023\times {10}^{23}atoms/mol\right)} \\ =9.30 g.c{m}^{-3} \end{gathered}$$