The simple cubic unit cell is a cube (all sides of the same length and all face perpendicular to each other) with an atom at each corner of 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. This calculation is particularly easy for a unit cell that is cubic. In this example, the atoms are in contact with each other along the edges of the unit cell. Thus the side of the unit cell has a length of 2 r, where r is the radius of an atom.
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. In this case, however, none of these atoms lies completely within the 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.
The virtual reality image below illustrates the simple cubic unit cell, which is the unit cell that describes the structure of polonium metal. The positions of the individual polonium nuclei are shown by small dots. The polonium atoms or sections of polonium atoms are shown by the spheres or sphere sections. Consult the Description of Controls or simply experiment with the features of the display.
The most stable isotope of polonium is Po-209, which has an atomic mass of 208.982.
Crystallographic data shows the length of a side of the unit cell to be 3.34 Å.
Use the simple cubic unit cell to answer the following questions.
- What is the atomic radius of polonium?
- What is the volume of the unit cell?
- How many polonium atoms are contained in the unit cell?
- What is the volume of a polonium atom (based upon the atomic radius)?
- What fraction of the volume of the unit cell is "occupied" by polonium atoms? (This fraction is the packing efficiency.)
- What is the density (g/cm3) of polonium?
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