The unit cell of topaz is orthorhombic with a=839 pm, b=879 pm, and c=465 pm. Calculate the valu

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The unit cell of topaz is orthorhombic with a=839 pm, b=879...

The unit cell of topaz is orthorhombic with $a=839 \mathrm{pm}, b=879 \mathrm{pm}$, and $c=465 \mathrm{pm}$. Calculate the values of the Bragg $\mathrm{X}$-ray diffraction angles from the $110,101,111$, and 222 planes. Take the wavelength of the $\mathrm{X}$-radiation to be $\lambda=154.433 \mathrm{pm}$.

Key Concepts

Bragg's Law

Bragg's Law is a fundamental principle in X?ray diffraction that relates the angles at which X-rays are diffracted by the crystal planes to the wavelength of the X-rays and the spacing between the planes. According to the law, constructive interference occurs when 2d sin ? equals an integer multiple of the wavelength, making it a crucial tool for determining the crystal structure through the analysis of diffraction patterns.

Miller Indices

Miller indices are a set of three numbers that denote the orientation of crystal planes in a lattice. They are defined as the reciprocals of the fractional intercepts that the plane makes with the crystallographic axes and are essential for identifying specific sets of parallel planes in a crystal, which directly affect the diffraction pattern.

Interplanar Spacing in Orthorhombic Crystals

For an orthorhombic crystal system, the interplanar spacing for a set of planes indexed by (hkl) is calculated using the formula 1/d² = (h²/a²) + (k²/b²) + (l²/c²), where a, b, and c are the unit cell parameters. This relationship is vital when applying Bragg's Law to determine the diffraction angles corresponding to specific crystallographic planes.

X-ray Diffraction

X-ray diffraction is an experimental technique used to study the arrangement of atoms within a crystal. By measuring the angles and intensities of diffracted X-rays, researchers can infer the distances between planes of atoms and, ultimately, the overall structure of the crystal. This method relies heavily on Bragg's Law and the geometric relationships defined by the crystal's unit cell parameters and Miller indices.

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