Question

In the X-ray diffraction pattern recorded for a simple cubic solid (lattice parameter \( a = 1 \) Å) using X rays of wavelength 1 Å, the first order diffraction peak(s) would appear for the

• A. (100) planes
• B. (112) planes
• C. (210) planes
• D. (220) planes

Hint:

For X-ray diffraction in a simple cubic crystal, the first order diffraction peak corresponds to the planes with the smallest interplanar spacing.

Answer 1

The Correct Option is A

Step 1: Understanding the X-ray diffraction condition.
For X-ray diffraction, the Bragg’s law is used to determine the angles at which diffraction peaks appear: \[ n\lambda = 2d \sin \theta \] where \( n \) is the diffraction order, \( \lambda \) is the wavelength of the X-rays, \( d \) is the spacing between the planes, and \( \theta \) is the diffraction angle. The first order diffraction peak corresponds to \( n = 1 \). For a simple cubic structure, the spacing \( d \) for the \((hkl)\) planes is given by: \[ d = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \] where \( a \) is the lattice parameter and \( (hkl) \) are the Miller indices of the planes.

Step 2: Analyzing the options.
- (A) (100) planes: Correct. The \((100)\) planes have the smallest spacing and will give the first order diffraction peak.
- (B) (112) planes: Incorrect. The spacing for the \((112)\) planes is larger, so they will not correspond to the first order diffraction peak.
- (C) (210) planes: Incorrect. The spacing for the \((210)\) planes is larger than that for the \((100)\) planes.
- (D) (220) planes: Incorrect. The spacing for the \((220)\) planes is also larger than that for the \((100)\) planes.

Step 3: Conclusion.
The correct answer is (A) because the first order diffraction peak appears for the \((100)\) planes, as they have the smallest spacing.