## Crystal Structure: Packing Spheres

#### Learning Objective

• Illustrate how atoms or molecules pack in crystalline materials.

#### Key Points

• The coordination number of a given atom in a crystalline substance identifies the number of atoms that directly neighbor the given atom.
• The atomic spheres of crystalline substances pack into unit cells, which are the fundamental building blocks of crystal lattices.
• An understanding of how spheres are packed within a given crystalline material contributes to researchers’ understanding of the properties of the given material.

#### Terms

• crystal structureThe unique three-dimensional arrangement of atoms or molecules in a crystalline solid.
• coordination numberThe total number of atoms that directly neighbor a central atom in a molecule or ion.
• unit cellIn a crystal, the smallest repeating structure (parallelepiped) of atoms from which the structure of the complete crystal can be inferred.

## Three-Dimensional Lattice

The unique arrangement of atoms or molecules within a crystalline solid is referred to as the crystal structure of that material. A crystal structure reflects the periodic pattern of the atoms which compose a crystalline substance. Crystalline materials are so highly ordered that a crystal lattice arises from repetitions along all three spatial dimensions of the same pattern. The crystal lattice represents the three-dimensional structure of the crystal’s atomic/molecular components.

## The Unit Cell

The structure seen within the crystalline lattice of a material can be described in a number of ways. The most common way to describe a crystal structure is to refer to the size and shape of the material’s characteristic unit cell, which is the simplest repeating unit within the crystal. In principle, one can reconstruct the structure of an entire crystal by repeating the unit cell so as to create a three-dimensional lattice.

## Packing of Atoms Within a Unit Cell

Within a crystalline material, each atom can be thought of as a sphere. These spheres are packed into unit cells. Each sphere that participates in a crystal structure has a coordination number, which corresponds to the number of spheres within the crystalline structure that touch the sphere that is being evaluated. For a sphere in the interior of a crystal lattice, the number of spheres contacting the sphere that is being evaluated is known as the bulk coordination number. For a sphere at the surface of a crystal, the number of spheres contacting the sphere being evaluated is known as the surface coordination number.

By considering how atomic spheres are arranged relative to one another, their coordination numbers, and the dimensions of the unit cell, it is possible to form a general view of the structure and complexity of particular crystal structures.