The crystalline structure of CuSO4( s) is orthorhombic with unit cell dimensions of a=488.2 pm,

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The crystalline structure of CuSO4( s) is orthorhombic with...

The crystalline structure of $\mathrm{CuSO}_{4}(\mathrm{~s})$ is orthorhombic with unit cell dimensions of $a=488.2 \mathrm{pm}, b=665.7 \mathrm{pm}$, and $c=831.6 \mathrm{pm}$. Calculate the value of $\theta$, the first-order Bragg diffraction angle, from the 100 planes, the 110 planes, and the 111 planes if $\mathrm{CuSO}_{4}(\mathrm{~s})$ is irradiated with $\mathrm{X}$-rays with $\lambda=154.433 \mathrm{pm}$.

Key Concepts

Bragg's Law

Bragg's Law is a fundamental principle in X?ray diffraction which relates the wavelength of radiation, the interplanar spacing in a crystal, and the angle of incidence at which constructive interference occurs. The law is expressed as n? = 2d sin ?, where n is the diffraction order, ? is the X-ray wavelength, d is the interplanar spacing, and ? is the Bragg angle.

Miller Indices

Miller indices (h, k, l) are a notation system in crystallography for designating the orientation of crystal planes. They are derived from the intercepts of the plane with the crystal axes and are essential for calculating interplanar spacings and understanding diffraction patterns in various crystal systems.

Interplanar Spacing in an Orthorhombic Lattice

In an orthorhombic crystal, the interplanar spacing d for a plane with Miller indices (h, k, l) is calculated using the relation 1/d² = (h²/a²) + (k²/b²) + (l²/c²), where a, b, and c are the unit cell dimensions. This formula is crucial for determining the distance between specific sets of lattice planes which in turn affects the diffraction angle according to Bragg's Law.

Crystalline Structure and Unit Cell Parameters

Understanding the crystalline structure and unit cell parameters is fundamental in crystallography. The unit cell dimensions (a, b, and c) define the geometry of the lattice in orthorhombic crystals, influencing properties such as the interplanar spacing and the resulting diffraction patterns when the crystal is irradiated with X-rays.

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