Titanium forms hexagonal close-packed crystals. Given the atomic radius of 146 pm, what are the

step 1 For a hexagonal structure, we can denote the relationship between A and B by this equation. A is

Titanium forms hexagonal close-packed crystals. Given the...

Key Concepts

Hexagonal Close?Packed (HCP) Crystal Structure

The hexagonal close?packed structure is one of the most efficient arrangements of atoms in solid state materials. It is characterized by a hexagonal unit cell with two lattice parameters: the edge length of the base (a) and the height (c). The atoms are arranged in layers such that each atom in the middle layer touches three atoms in the layer below and three in the layer above, leading to a high coordination number and a typical c/a ratio of about 1.633 in an ideal structure.

Unit Cell Dimensions and Lattice Parameters

The unit cell for an HCP crystal is defined by two parameters: the basal plane edge length (a) and the vertical height (c) of the cell. These dimensions are directly related to the atomic radius; for instance, in ideal conditions, the edge length a is approximately twice the atomic radius. The height c is then determined from the ideal c/a ratio, which arises from the close packing geometry. Calculating the unit cell volume also involves the geometry of a hexagon-based prism.

Atomic Radius and Its Role in Crystal Geometry

The atomic radius is the measure of the size of an atom, which directly influences the lattice parameters of a crystal. In close-packed structures like HCP, the distance between the centers of adjacent atoms is approximately equal to the sum of their radii, allowing one to derive the lattice constants (a and c) from the known atomic radius. This relationship is fundamental in transitioning from atomic-scale measurements to the macroscopic dimensions of the unit cell.

Density Calculation of Crystals

The density of a crystalline material is determined by the mass of the atoms within a unit cell divided by the volume of that unit cell. This calculation requires knowledge of the number of atoms per unit cell (which in the HCP structure is typically 6 atoms per unit cell when properly accounted for), the atomic mass (converted from atomic weight using Avogadro’s number), and the unit cell volume computed from the lattice parameters. This concept connects microscopic structural parameters to macroscopic physical properties like density.

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