A single crystal of NaCl is oriented such that the incident...

A single crystal of $\mathrm{NaCl}$ is oriented such that the incident X-rays are perpendicular to the a axis of the crystal. The distance between the spots corresponding to diffraction from the 000 and 100 planes is $14.8 \mathrm{~mm}$, and the detector is located $52.0 \mathrm{~mm}$ from the crystal. Calculate the value of $a$, the length of the unit cell along the a axis. Take the wavelength of the X-radiation to be $\lambda=154.433 \mathrm{pm}$.

Key Concepts

Interplanar Spacing

Interplanar spacing refers to the distance between parallel planes of atoms in a crystal. For cubic crystals, the spacing d for a set of crystallographic planes can be easily related to the unit cell parameter a. For example, the spacing for the (100) planes in a simple cubic lattice is equivalent to a. Knowledge of interplanar spacing is crucial when applying Bragg's Law to infer lattice parameters from diffraction data.

Diffraction Geometry

The geometry of X-ray diffraction involves relating the positions of diffraction spots on a detector to the angles at which X-rays are scattered. By measuring the displacement of diffraction spots and knowing the distance from the crystal to the detector, one can calculate the diffraction angle. This geometric relationship is key to translating experimental measurements into values for the interplanar spacing using Bragg's Law, and ultimately into lattice parameters.

Bragg's Law

Bragg's Law provides the condition for constructive interference of X-rays scattered by a crystalline lattice. It is expressed as n? = 2d sin?, where n is an integer, ? is the wavelength of the incident X-ray, d is the spacing between the crystal planes, and ? is the angle of incidence. This law is fundamental for determining the arrangement of atoms in a crystal by relating X-ray diffraction patterns to the geometric structure of the lattice.

Crystal Lattice and Unit Cell

A crystal lattice is a periodic arrangement of atoms in space, and the unit cell is the smallest repeating unit that contains the complete set of structural information for the crystal. In cubic crystals like NaCl, the unit cell is defined by its edge length a. Understanding the unit cell is essential because the lattice parameters determine the spacing between atomic planes, which directly affects the diffraction pattern observed in experiments.

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