For the HCP crystal structure, show that the ideal c / a ratio is 1.633.

Name: Problem 4, Chapter 3: Materials Science and Engineering: An Introduction
Uploaded: 2021-11-30T21:58:07-08:00
Duration: 4 min 41 s
Channel: Abhinav Roy
Description: For the HCP crystal structure, show that the ideal c / a ratio is 1.633.

step 1 In this problem we are given an HCP unit cell. So this is the HCP unit cell and It consists of s

For the HCP crystal structure, show that the ideal c / a...

Key Concepts

Ideal c/a Ratio

The ideal c/a ratio in the HCP structure is the optimal ratio of the height (c) to the basal plane edge (a) that allows atoms to be packed most efficiently. Geometric considerations—taking into account the contacts between atoms in adjacent layers—lead to this ratio, which, when derived, is found to be approximately 1.633. This value ensures that atoms in different layers are in contact, maximizing the packing density.

Hexagonal Close-Packed (HCP) Structure

The HCP structure is a type of crystal structure in which atoms are arranged in densely packed layers that follow an ABAB... stacking sequence. This structure is one of the most efficient ways to pack equal spheres in three dimensions and is characterized by a hexagonal unit cell defined by two lattice parameters: a, the in-plane distance between atoms, and c, the vertical distance between equivalent layers.

Unit Cell Geometry

Unit cell geometry refers to the spatial arrangement and dimensions of the repeating unit in a crystal. In the HCP structure, the unit cell is defined by the lattice parameters a and c, where a represents the distance between nearest-neighbor atoms in the basal plane, and c is the height of the cell. The ratio between these parameters is critical for maintaining ideal atomic packing and symmetry.

Geometric Derivation

The derivation of the ideal c/a ratio involves analyzing the geometry of atom arrangements within the unit cell. By considering the tetrahedral configuration formed by three atoms in one layer and one atom in the layer above, one can apply the principles of Euclidean geometry to relate the interatomic distances. This approach leads to the mathematical relationship c/a = ?(8/3), which numerically approximates to 1.633, indicating the optimized separation in a perfectly close-packed HCP structure.

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