A diffraction experiment is performed on a polonium crystal with a primitive cubic lattice with a lattice constant of $a = 3.359$ Å.

a) Calculate the primitive lattice vectors of the reciprocal space lattice.

b) Calculate the length of the reciprocal lattice vector 123 $(h =1,\, k = 2,\, l =4)$. Give the units in your answer.

c) At which scattering angle 2θ is the diffraction of the 123 reflection is observed, if the wavelength of the X-rays is 1.1 Å?

d) Determine the square of the structure factor of the 123 reflection using the structure factor app.

Solution

<hr /> <p>(a) The reciprocal lattice vectors of a simple cubic lattice with lattice constant $a=3.359$ Å are,</p> $$\vec{b}_1 = 2\pi \frac{(\vec{a}_2\times \vec{a}_3)}{\vec{a}_1(\vec{a}_2\times \vec{a}_3)} = \frac{2\pi}{a} \begin{bmatrix} 1\\ 0 \\ 0 \end{bmatrix}= 1.870\begin{bmatrix} 1\\ 0 \\ 0 \end{bmatrix} \text{A}^{-1}, \\ \vec{b}_2 = 2\pi \frac{(\vec{a}_3\times \vec{a}_1)}{\vec{a}_1(\vec{a}_2\times \vec{a}_3)} = \frac{2\pi}{a} \begin{bmatrix} 0\\ 1 \\ 0 \end{bmatrix}= 1.870\begin{bmatrix} 1\\ 0 \\ 0 \end{bmatrix} \text{A}^{-1}, \\ \vec{b}_3 = 2\pi \frac{(\vec{a}_1\times \vec{a}_2)}{\vec{a}_1(\vec{a}_2\times \vec{a}_3)} = \frac{2\pi}{a} \begin{bmatrix} 0\\ 0 \\ 1 \end{bmatrix}= 1.870\begin{bmatrix} 1\\ 0 \\ 0 \end{bmatrix} \text{A}^{-1}.$$ <p>(b) The reciprocal lattice vector $\vec{G}_{124} = \vec{b}_1 + 2\vec{b}_2 + 4\vec{b}_3$. The length of this vector is $|\vec{G}_{124}| = \frac{2\pi}{3.359}\sqrt{1^2+2^2+3^2}$ Å$^{-1}$ = 6.999 Å$^{-1}$.</p> <p>(c) The diffraction condition is $\Delta\vec{k}=\vec{G}$ and $2\theta$ is the angle between $\vec{k}$ and $\vec{k}'$. Since the condition for elastic scattering $|\vec{k}|=|\vec{k}'|$ is satisfied, we can construct a right triangle where,</p> $$\sin(\theta) = \frac{\frac{|G_{124}|}{2}}{|\vec{k}|}$$ <p>Using the definition of wavenumber, $|\vec{k}| = \frac{2\pi}{\lambda}$, $\theta$ can then be computed to be,</p> <p style="text-align:center">$\theta = 37.78$°,   $2\theta = 75.56$°.</p> <p>(d) The structure factor is,</p> \begin{equation} |S_{123}|^2=2246 \end{equation}