The second peak in the powder X-ray diffraction pattern of a FCC metal occurs at a Bragg angle $\theta$ (in degrees) = ____________ (answer up to two decimal places) (Given: $\lambda_{CuK\alpha} = 0.154$ nm; lattice parameter of the metal = $0.36$ nm)
Understanding FCC Diffraction Peaks
For a Face-Centered Cubic (FCC) crystal structure, allowed diffraction peaks correspond to Miller indices (hkl) where h, k, and l are either all even or all odd. The sequence of diffraction peaks is determined by the sum $S = h^2 + k^2 + l^2$.
- 1st peak: (111), $S = 1^2 + 1^2 + 1^2 = 3$
- 2nd peak: (200), $S = 2^2 + 0^2 + 0^2 = 4$
- 3rd peak: (220), $S = 2^2 + 2^2 + 0^2 = 8$
The question asks for the second peak in the powder X-ray diffraction pattern, which corresponds to the (200) reflection.
Calculating Interplanar Spacing (d)
The interplanar spacing $d_{hkl}$ for a cubic crystal system is calculated using the formula: $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$ Here, $a$ is the lattice parameter. For the second peak (200), the value of $h^2 + k^2 + l^2$ is 4.
Given the lattice parameter $a = 0.36$ nm, the interplanar spacing $d_{200}$ is: $d_{200} = \frac{0.36 \text{ nm}}{\sqrt{4}} = \frac{0.36 \text{ nm}}{2} = 0.18 \text{ nm}$
Applying Bragg's Law
Bragg's Law relates the wavelength of the X-rays ($\lambda$), the interplanar spacing ($d$), the order of diffraction ($n$), and the Bragg angle ($\theta$): $n\lambda = 2d \sin(\theta)$ For typical powder diffraction analysis, we consider the first-order diffraction ($n=1$): $\lambda = 2d \sin(\theta)$
To find the Bragg angle $\theta$ for the second peak, we rearrange Bragg's Law: $\sin(\theta) = \frac{\lambda}{2d}$
Substitute the given wavelength $\lambda = 0.154$ nm and the calculated spacing $d_{200} = 0.18$ nm: $\sin(\theta) = \frac{0.154 \text{ nm}}{2 \times 0.18 \text{ nm}}$ $\sin(\theta) = \frac{0.154}{0.36} \approx 0.4278$
Calculate the Bragg angle $\theta$: $\theta = \arcsin(0.4278)$ $\theta \approx 25.37^\circ$
The calculated Bragg angle of approximately $25.37^\circ$ confirms the provided answer range (between 24 and 26 degrees).
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 = $0.287 \text{ nm}$
- For an FCC metal, the ratio of interplanar spacing obtained from the first two peaks of the X-ray diffraction pattern is
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.
