Question

Find the lattice parameter \(a\) for a simple cubic crystal refracting an X-Ray \((\lambda = 1.54\ \text{\AA})\) at an angle \(45^\circ\) from a plane having Miller indices \((1,1,0)\) at order \((n=1)\).

• A. \(1.54\ \text{\AA}\)
• B. \(3.08\ \text{\AA}\)
• C. \(0.77\ \text{\AA}\)
• D. \(2.12\ \text{\AA}\)

Hint:

For cubic crystals, first find \(d\) using Bragg's law, then use \(d=\frac{a}{\sqrt{h^2+k^2+l^2}}\).

Answer 1

The Correct Option is A

Concept:
For X-ray diffraction, Bragg's law is used: \[ n\lambda = 2d\sin\theta \] For a cubic crystal, interplanar spacing is: \[ d=\frac{a}{\sqrt{h^2+k^2+l^2}} \]

Step 1: Use Bragg's law. \[ n=1,\quad \lambda=1.54\ \text{\AA},\quad \theta=45^\circ \] \[ 1\times 1.54 = 2d\sin45^\circ \] \[ 1.54 = 2d\left(\frac{1}{\sqrt{2}}\right) \] \[ 1.54 = \sqrt{2}d \] \[ d=\frac{1.54}{\sqrt{2}} \]

Step 2: Use cubic crystal spacing formula.
For plane \((1,1,0)\), \[ h=1,\quad k=1,\quad l=0 \] \[ d=\frac{a}{\sqrt{1^2+1^2+0^2}} \] \[ d=\frac{a}{\sqrt{2}} \]

Step 3: Compare both values of \(d\). \[ \frac{a}{\sqrt{2}}=\frac{1.54}{\sqrt{2}} \] \[ a=1.54\ \text{\AA} \] \[ \therefore \text{Correct Answer is (A)} \]