FCC Interplanar Spacing Calculation
For cubic crystal systems, the interplanar spacing ($d_{hkl}$) between parallel planes with Miller indices (hkl) is related to the lattice parameter ($a$) by the formula:
$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$
In X-ray diffraction, the peaks correspond to allowed reflections. For an FCC (Face-Centered Cubic) structure, the allowed reflections occur when h, k, and l are all odd or all even. The sequence of reflections based on increasing $(\sqrt{h^2 + k^2 + l^2})$ value determines the order of diffraction peaks.
Identifying First Two FCC Peaks
- The first allowed reflection in FCC corresponds to the Miller indices (111). The value of $\sqrt{h^2 + k^2 + l^2}$ is $\sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$.
- The second allowed reflection corresponds to the Miller indices (200). The value of $\sqrt{h^2 + k^2 + l^2}$ is $\sqrt{2^2 + 0^2 + 0^2} = \sqrt{4} = 2$.
Calculating Interplanar Spacing Ratio
Let $d_1$ be the interplanar spacing for the first peak (111) and $d_2$ be the spacing for the second peak (200).
- $d_1 = d_{111} = \frac{a}{\sqrt{3}}$
- $d_2 = d_{200} = \frac{a}{\sqrt{4}} = \frac{a}{2}$
The question asks for the ratio of the interplanar spacing from the first two peaks. Based on the options, this refers to the ratio $\frac{d_1}{d_2}$.
Ratio = $\frac{d_1}{d_2} = \frac{a/\sqrt{3}}{a/2}$
Ratio = $\frac{2}{\sqrt{3}}$
Calculating the value:
Ratio $\approx \frac{2}{1.732} \approx 1.1547$
This value is approximately 1.15.
Important Questions from Crystallography Stereographic Projection
- Residual stress present in a material can be determined by which one of the following techniques:
A schematic of X-ray diffraction pattern of a single phase cubic polycrystal is given below. The miller indices of peak A is
In a powder diffraction experiment on BCC iron, the first peak occurs at $2\theta = 68.7^\circ$. The wavelength of X-rays is ________ (in nm to three decimal places).
Given: The lattice parameter of iron = $0.287 \text{ nm}$
X-ray diffraction pattern from an elemental metal with a FCC crystal structure shows the first peak at a Bragg angle $\theta = 24.65^\circ$. The lattice parameter of this metal is ____________ nm.
Given, wavelength of the X-ray used is $0.1543$ nm.- If d is the inter-planar spacing of the planes {h k l}, the inter-planar spacing of the planes {nh nk nl}, n being an integer, is
