The density of silver at 20^∘ C is 10.50 g ·cm^-3, and its unit cell is cubic. Given that the f

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The density of silver at 20^∘ C is 10.50 g ·cm^-3, and its...

The density of silver at $20^{\circ} \mathrm{C}$ is $10.50 \mathrm{~g} \cdot \mathrm{cm}^{-3}$, and its unit cell is cubic. Given that the first five observed Bragg diffraction angles are $\theta=19.10^{\circ}, 22.17^{\circ}, 32.33^{\circ}, 38.82^{\circ}$, and $40.88^{\circ}$, 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 and Atomic Packing in Crystals

Density in crystalline materials is determined by the mass of the atoms contained in the unit cell divided by the volume of the cell. By knowing the density, the atomic mass, and Avogadro's number, one can calculate the number of atoms per unit cell. This information, in turn, helps to identify the type of unit cell (e.g., FCC has 4 atoms per cell, BCC has 2, and SC has 1) and to determine the cell dimensions, offering insights into the material’s atomic packing and structure.

Bragg's Law

Bragg's Law is the fundamental principle used to analyze X-ray diffraction by crystals. It relates the wavelength of the incident X-rays, the angle of diffraction, and the distance between crystal planes (interplanar spacing) through the equation n? = 2d sin ?. This law allows one to determine the spacing between planes in the crystal lattice by measuring the angles at which constructive interference occurs.

Unit Cell Geometry in Crystallography

The concept of the unit cell in crystallography refers to the smallest repeating unit in a crystal lattice that, when repeated in all directions, creates the entire crystal structure. Cubic unit cells, as one important class, include different types such as simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC). The geometry and symmetry of the unit cell play a critical role in determining the physical properties of the material and can be inferred from diffraction data and density calculations.

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