A metal crystallises into two cubic phases, face centered...

A metal crystallises into two cubic phases, face centered cubic (FCC) and body centred cubic (BCC), whose unit cell lengths are $3.5$ and $3.0$ $\AA$, respectively. Calculate the ratio of densities of $\mathrm{FCC}$ and $\mathrm{BCC}$.

Key Concepts

Crystallographic Density Computation

The density of a crystal can be determined by dividing the total mass of atoms within a unit cell by the volume of the unit cell. In comparing different crystal structures, the ratio of their densities can be analyzed by considering both the number of atoms per unit cell and the differences in the cell volumes due to varying lattice parameters.

Unit Cell Volume Calculation

In crystallography, the unit cell is the smallest repeating unit that describes the entire crystal lattice. For cubic systems, this volume is computed as the cube of the unit cell side length (a^3). This calculation is key to determining properties like density when combined with the number of atoms per cell.

Body Centered Cubic (BCC) Structure

The BCC structure is characterized by atoms positioned at each corner of a cube with one atom at the very center. This configuration results in two atoms per unit cell, and although it is less densely packed compared to FCC, it is common in metals and affects the material's density and other physical characteristics.

Face Centered Cubic (FCC) Structure

The FCC structure is a type of crystal lattice in which atoms are located at each of the corners and the centers of all the faces of the cube. This results in a high packing efficiency and four atoms per unit cell, making it distinct in terms of its geometric and physical properties.

Crystal Structures

Crystal structures refer to the ordered arrangement of atoms in a crystalline solid. They determine many physical properties of the material, such as density and packing efficiency. Understanding the types of crystal lattices is fundamental to solid state physics and materials science.

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