Gold crystallizes in a fcc structure. When X-radiation of...

Gold crystallizes in a fcc structure. When X-radiation of 0.70926 Å wavelength from molybdenum is used to determine the structure of metallic gold, the minimum diffraction angle of $X$ -rays by the gold is 8.6838 . Calculate the spacing between parallel layers of gold atoms.

Key Concepts

Bragg's Law

Bragg's Law is a fundamental principle in crystallography that relates the wavelength of incident X-rays to the diffraction angle and the distance between parallel planes in a crystal. It is mathematically expressed as n? = 2d sin?, where n is an integer denoting the order of diffraction, ? is the wavelength, d is the interplanar spacing, and ? is the angle of incidence. This law underpins the analysis of diffraction patterns to determine crystal structures.

X-ray Diffraction

X-ray diffraction is an analytical technique used to study the structural arrangement of atoms within a crystal. By measuring the angles and intensities of X-rays diffracted by the crystal lattice, scientists can infer the spacing between lattice planes and deduce the overall three-dimensional arrangement of atoms. It is a critical method in materials science for characterizing crystal structures.

Interplanar Spacing

Interplanar spacing refers to the distance between two parallel crystallographic planes within a crystal. This spacing is a crucial parameter in diffraction studies, as it directly influences the angles at which X-rays are diffracted according to Bragg's Law. Accurate determination of the interplanar spacing helps in understanding the atomic structure and symmetry of a crystalline material.

Face-centered Cubic (fcc) Structure

The face-centered cubic (fcc) structure is a common crystal lattice arrangement in which atoms are positioned at each of the cube's corners and at the center of each face of the cube. This highly symmetric structure affects the pattern of diffraction peaks observed in X-ray diffraction and is essential for interpreting the relationships between lattice parameters and interplanar spacing.

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