Silver crystallizes as a face-centered cubic structure with...

Silver crystallizes as a face-centered cubic structure with a unit cell length of $408.6 \mathrm{pm}$. The single crystal of silver is oriented such that the incident X-rays are perpendicular to the c axis of the crystal. The detector is located $29.5 \mathrm{~mm}$ from the crystal. What is the distance between the diffraction spots from the 001 and 002 planes on the face of the detector for (a) the $\lambda=154.433$-pm line of copper, and (b) the $\lambda=70.926$-pm line of a molybdenum X-ray source? Which X-ray source gives you the best spatial resolution between the diffraction spots?

Key Concepts

Bragg's Law

Bragg's Law is fundamental to understanding X-ray diffraction in crystalline materials. It relates the X-ray wavelength, the interplanar spacing in the crystal, and the incident angle at which constructive interference occurs. This law is used to calculate diffraction angles for various sets of crystal planes, which then determine where diffraction spots will appear on a detector.

Crystal Structure and Lattice Parameters

The crystal structure, such as the face-centered cubic (FCC) arrangement, and its unit cell dimensions are central to determining the interplanar spacings. These spacings, which depend on the Miller indices of the planes, play a critical role in interpreting diffraction data and calculating the angular distribution of the diffracted beams.

Interplanar Spacing

Interplanar spacing is the distance between parallel crystal planes and is determined by the crystal structure and the Miller indices of the planes. It is a key parameter in applying Bragg's Law, as small differences in spacing (such as between 001 and 002 planes) result in slight variations in diffraction angles that are essential for precise diffraction spot measurements.

Diffraction Geometry

The geometry of the diffraction experiment, including the positioning of the crystal, the incident X-ray beam, and the detector, influences how the diffraction angles map onto positions on the detector. Understanding the spatial relation between the diffracted rays and the detector distance is crucial for converting angular differences into measurable distances between diffraction spots.

X-ray Wavelength and Spatial Resolution

The wavelength of the incident X-rays directly affects the diffraction angle according to Bragg's Law. Shorter wavelengths tend to produce smaller diffraction angles, potentially increasing the spatial separation between diffracted spots on the detector. This concept is important when considering the resolution of distinct diffraction features, as a better separation (higher spatial resolution) can be achieved with an optimal choice of wavelength.

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