Question

A first order reflection of X-ray from the 220 plane of copper crystal is observed at a glancing angle of 22°. The wavelength of the X-ray used is ................ pm. (Round off to one decimal place) 
[Given: Copper forms an fcc crystal with a unit cell edge length of 361 pm.]
 

Hint:

Use Bragg's law to calculate the wavelength of X-rays based on the diffraction angle and the spacing between planes in the crystal.

Answer 1

Correct Answer: 95

Step 1: Use Bragg's Law. 
Bragg's law for diffraction is given by: \[ n\lambda = 2d\sin\theta \] where \(n\) is the order of reflection, \(\lambda\) is the wavelength, \(d\) is the distance between planes in the crystal, and \(\theta\) is the glancing angle. 
Step 2: Calculate \(d\) for the 220 planes. 
For an fcc crystal, the interplanar spacing \(d\) for hkl planes is given by: \[ d = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \] where \(a\) is the unit cell edge length, and \(h, k, l\) are the Miller indices of the plane. For 220 planes: \[ d = \frac{361}{\sqrt{2^2 + 2^2}} = \frac{361}{\sqrt{8}} = 128.4 \, \text{pm} \] Step 3: Apply Bragg's Law. 
For first order reflection (\(n = 1\)) and \(\theta = 22^\circ\): \[ \lambda = \frac{2 \times 128.4 \times \sin(22^\circ)}{1} \] \[ \lambda = \frac{2 \times 128.4 \times 0.3746}{1} = 1.2 \, \text{pm} \] Step 4: Conclusion. 
Thus, the wavelength of the X-ray used is 1.2 pm.