Chromium is either a face-centered cubic or a body-centered...

Chromium is either a face-centered cubic or a body-centered cubic crystalline solid. Given that it has the following observed successive values of $d: 203.8 \mathrm{pm}, 144.2 \mathrm{pm}$ $117.7 \mathrm{pm}, 102.0 \mathrm{pm}, 91.20 \mathrm{pm},$ and $83.25 \mathrm{pm},$ determine the type of cubic unit cell, the length of the unit cell, and the density. (Hint: See the table in Problem $29-38$.)

Key Concepts

Density Calculation in Crystals

Density calculation in crystalline materials involves relating the mass of atoms within a unit cell to its volume. This is typically done using the number of atoms per cell, the atomic or molecular weight, and Avogadro’s number. This calculation is essential for verifying structural models and understanding material properties.

Crystalline Structure

Crystalline structure refers to the ordered arrangement of atoms in a solid where a repeated pattern forms a lattice. This concept is key to understanding how materials exhibit various physical properties, including mechanical strength and density, and is the basis for techniques such as X-ray diffraction that probe the arrangement of atoms within a crystal.

Cubic Unit Cells

Cubic unit cells are the simplest three-dimensional lattices with equal axes and 90° angles between them. They include different types, notably face?centered cubic (FCC) and body?centered cubic (BCC), which differ in the arrangement and number of atoms per cell; this difference significantly influences the material’s density and its diffraction pattern.

Interplanar Spacing and Miller Indices

Interplanar spacing is the distance between parallel planes of atoms within a crystal, and Miller indices (h, k, l) are used to denote these planes. In cubic crystals, the spacing is related to the lattice parameter and the Miller indices by geometric relations, making it a critical tool for identifying the type of crystal structure from diffraction data.

Unit Cell Parameters

The unit cell parameter, often symbolized as 'a', is the edge length of the basic repeating unit in a crystal lattice. Determining this parameter from experimental data, such as interplanar spacings, helps in fully characterizing the crystalline structure and calculating other properties like density.

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