Simple Cubic Unit Cell

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. The dimensions of the unit cell are dictated by the size of the atoms in the cell. In the case of the simple cubic unit cell the atom lying on a corner is in contact with each atom on a neighboring corner. Thus the side of the unit cell has a length of 2r, where r is the radius of an atom.

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. 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.

Properties of Polonium

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.

The atomic mass of polonium is 209 and the crystallographic data shows the length of a side of the unit cell to be 3.34 angstroms.

Use the simple cubic unit cell to answer the following questions.

1. What is the atomic radius of polonium?

2. What is the volume of the unit cell?

1. How many polonium atoms are contained in the unit cell?

4. What is the volume of a polonium atom (based upon the atomic radius)?

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

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

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,

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

Simple Cubic
Unit Cell

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

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