The hexagonal closest-packed structure is described by a hexagonal unit cell, which has a diamond shaped or hexagonal base with sides of equal length (a = b). The base is perpendicular to the longest side (length c)) of the unit cell. An atom is centered on each corner of the unit cell. An atom is also centered inside the unit cell, and two atoms whose center lie outside the unit cell extended into 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 of the hexagonal unit cell is the product of the area of the base and the height of the cell. For a closest-packed structure, the atoms at the corners of base of the unit cell are in contact, thus a = b = 2 r. The height (c) of the unit cell, which is more challenging to calculate, is c = 2 a (2/3)1/2 = 4 r (2/3)1/2. See if you can derive this value!
Atoms, of course, do not have well-defined bounds, and the radius of an atom is somewhat ambiguous. In the context of crystal structures, the diameter (2 r) of an atom can be defined as the center-to-center distance between two atoms packed as tightly together as possible. This provides an 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 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.
Magnesium crystallizes in a hexagonal closest-packed structure. The unit cell for the hcp structure is the hexagonal cell, which is illustrated in the virtual reality display shown 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. Consult the Description of Controls or simply experiment with the features of the display.
The atomic mass of magnesium is 24.305 and the atomic radius of magnesium is 1.60 Å.
Use the hexagonal unit cell to answer the following questions.
- What are the lengths a, b, and c for the unit cell for magnesium?
- What is the volume of the unit cell?
- What is the volume of a magnesium atom (based upon the atomic radius)?
- How many magnesium atoms are contained in the unit cell?
- What fraction of the volume of the unit cell is "occupied" by magnesium atoms? (This fraction is the packing efficiency.)
- What is the density (g/cm3) of magnesium metal?
This virtual reality display requires Java3D. If the display is not visible, consult the Java3D FAQ. Dragging an object with the left mouse button rotates the object. Dragging with the center mouse buttons expands the display, and dragging with the right mouse button moves the display.
Unit Cells: Simple Cubic Body-Centered Cubic Face-Centered Cubic Hexagonal Closest-Packed