The density of tantallum at 20^∘ C is 16.69 g ·cm^-3, and its unit cell is cubic. Given that th

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The density of tantallum at 20^∘ C is 16.69 g ·cm^-3, and...

The density of tantallum at $20^{\circ} \mathrm{C}$ is $16.69 \mathrm{~g} \cdot \mathrm{cm}^{-3}$, and its unit cell is cubic. Given that the first five observed Bragg diffraction angles are $\theta=19.31^{\circ}, 27.88^{\circ}, 34.95^{\circ}, 41.41^{\circ}$, and $47.69^{\circ}$, find the type of unit cell and its length. Take the wavelength of the X-radiation to be $\lambda=154.433 \mathrm{pm}$.

Key Concepts

Density in Crystallography

The density of a crystalline material is related to the mass of atoms in the unit cell, the number of atoms per unit cell, and the volume of the cell. In crystallography, the density formula, ? = (n·m) / (a³·N_A), where n is the number of atoms per cell, m is the atomic mass, a is the lattice parameter, and N_A is Avogadro's number, can be combined with diffraction data to confirm or determine the structure type and dimensions of the unit cell.

Bragg's Law

Bragg's Law relates the angle at which X-rays are diffracted by the lattice planes of a crystal to the distance between those planes and the wavelength of the incident radiation. It is expressed as n? = 2d sin?, where n is the order of reflection, ? is the wavelength, d is the spacing between the crystallographic planes, and ? is the angle of incidence. This law provides the fundamental principle for interpreting X-ray diffraction data in crystallography.

Cubic Crystal Lattice

A cubic crystal lattice is one wherein the unit cell is in the shape of a cube, and its properties are determined by a single lattice parameter (the edge length). In cubic systems, the interplanar spacing for a set of planes defined by Miller indices (hkl) can be calculated using d = a/?(h² + k² + l²), linking the observed diffraction peaks directly to the lattice constant.

Miller Indices

Miller indices are a notation system in crystallography for designating the orientation of lattice planes. They are essential for identifying which planes cause diffraction in X-ray experiments. The values of h, k, and l affect the interplanar spacing and, in a cubic system, lead to specific allowed reflections that can be matched with observed diffraction angles to determine the type of cubic structure, such as primitive, body-centered, or face-centered.

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