## Hexagonal Unit Cell

### Hexagonal Closest-Packed Structure

The unit cell for the hexagonal closest-packed structure has a diamond-shaped or hexagonal base with sides of equal length. The base is perpendicular to the longest side of the unit cell. An atom is centerd on each corner of the unit cell. In addition, an atom is centered inside the unit cell, and there are two other atoms whose centers lie outside the unit cell but whose bodied penetrate the unit cell.

The unit cell completely describes the structure of the solid, which can be regarded as an almost endless repetition of the unit cell.

The volume of the unit cell is readily calculated from its shape and dimensions. The volume is the product of the area of the base and the height of the cell. The lengths of the sides of the base are the same: a = b. The height of the unit cell is c. Because the atoms are in "physical contact", these lengths are directly related to the atomic radius, r. Atoms on adjacent corners of the base are in contact, thus a = b = 2r. The height of the unit cell, which more challenging to calculate, is c = 4r(2/3)1/2. See if you can derive this value!

Atoms, of course, do not have well-defined bounds, thus the radius of an atom is somewhat ambiguous. In the context of crystal structures, the diameter (2r) of an atom can be defined as the center-to-center distance between two atoms packed as tightly together as possible. This provides a type of effective radius for the atom and is sometime called the atomic radius.

A more challenging task is to determine the number of atoms that lie in the unit cell. As described above, an atom is centered on each corner, there is an atom centered inside the unit cell (but this atom extends outside the unit cell), and there are two atoms centered outside the unit cell that extend into the unit cell. Part of each atom lies within the unit cell and the remainder lies outside the unit cell. In determining the number of atoms inside the unit cell, one must count only that portion of an atom that actually lies within the unit cell.

The density of a solid is the mass of all the atoms in the unit cell divided by the volume of the unit cell.

### Properties of Magnesium

Magnesium crystallizes in a hexagonal closest-packed structure. The unit cell for this structure is shown in the virtual reality image below . The positions of the individual magnesium nuclei are shown by small dots. The magnesium atoms or sections of magnesium atoms are shown by the spheres or sphere sections.

The atomic mass of magnesium is 24.305 and the atomic radius of magnesium is 1.60 angstroms.

Use this structure to answer the following questions.

1. What are the lengths a, b, and c for the unit cell for magnesium?

2. What is the volume of the unit cell?

3. What is the volume of a magnesium atom (based upon the atomic radius)?

4. How many magnesium atoms are contained in the unit cell?

5. What fraction of the volume of the unit cell is "occupied" by magnesium atoms?

6. What is the density (g/cm3) of magnesium metal?

The animation controls at the left of the display perform the following operations.

 Reset Resets the animation to the unit cell containing the various sections of silver atoms. Show/Hide Sides Shows or hides the solid sides of the unit cell. Clear Unit Cell Shows only the outline of the unit cell. Show Sections Shows the sections of the silver atoms that lie within the unit cell. Show Atoms Shows the silver atoms that lie in the unit cell (no atom lies completely within the unit cell in this case). View Sections The sections of silver atoms that lie within the unit cell are spread out for display and examination. The user may step through this process. Form Atoms The sections of silver atoms that lie within the unit cell are pieced together to form whole atoms.

To run through the animation in sequence,

1. Select Show Sections to display the portions of the atoms in the unit cell.
2. Select View Sections or step through this sequence to spread the atom sections across the display for viewing.
3. Select Form Atoms or step through this sequence to assemble the atom sections into full atoms.

 HexagonalUnit Cell

Unit Cells:     Simple Cubic     Body-Centered Cubic     Face-Centered Cubic     Hexagonal Closest-Packed