The body-centered cubic unit cell is cubic (all sides of the same length and all face perpendicular to each other) with an atom at each corner of the unit cell and an atom in the center 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 body-centered cubic unit cell the atoms lying the diagonal of the cube are in contact with each other. Thus the diagonal of the unit cell has a length of 4r, 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 and in the middle of each face of the body-centered cubic unit cell. The atom at the center of the unit cell lies completely within the unit cell. The atoms located on the corners, however, exist partially inside the unit cell and partially 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 body-centered cubic unit cell, which is the unit cell that describes the structure of sodium metal. The positions of the individual sodium nuclei are shown by small dots. The sodium atoms or sections of sodium atoms are shown by the spheres or sphere sections.
The atomic mass of sodium is 22.9898 and the density of sodium metal is 0.971 g/cm3.
Use the body-centered cubic unit cell to answer the following questions.
1. How many sodium atoms are contained in the unit cell?
2. What is the volume of the unit cell?
3. What is the atomic radius of a silver atom?
4. What is the volume of a sodium atom (based upon the atomic radius)?
5. What fraction of the volume of the unit cell is "occupied" by silver atoms?
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,
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Body-Centered Cubic Unit Cell |
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