In an end-centered unit cell, 2 atoms are present.
Understanding the End-Centered Unit Cell
An end-centered unit cell, often referred to as an end-centered cubic (ECC) unit cell in the context of common crystal systems, is a type of unit cell in crystallography. It is characterized by atoms located at all eight corners of the cube and at the center of two opposite faces.
Atomic Contribution Calculation
To determine the total number of atoms effectively belonging to a single end-centered unit cell, we must consider the fractional contribution of each atom based on its position:
- Corner Atoms: Each corner atom is shared by eight adjacent unit cells. Therefore, only one-eighth (1/8) of an atom at a corner belongs to the specific unit cell. Since there are eight corners in a cubic unit cell, their total contribution is $8 \times (1/8)$.
- Face-Centered Atoms (End Faces): In an end-centered unit cell, atoms are present at the centers of two opposite faces. Each atom located at the center of a face is shared by two adjacent unit cells. Thus, one-half (1/2) of a face-centered atom belongs to the specific unit cell. With two such faces, their total contribution is $2 \times (1/2)$.
Let's summarize these contributions:
| Atom Position | Number of Atoms | Contribution per Atom | Total Contribution |
|---|---|---|---|
| Corners | 8 | 1/8 | $8 \times (1/8) = 1$ |
| Centers of End Faces | 2 | 1/2 | $2 \times (1/2) = 1$ |
Total Number of Atoms
Adding the contributions from the corners and the end faces gives the total number of atoms in an end-centered unit cell:
Total Atoms = (Contribution from corners) + (Contribution from end faces)
Total Atoms = 1 + 1
Total Atoms = 2
This calculation shows that despite appearing to have many atoms distributed throughout its structure, an end-centered unit cell effectively contains two atoms. This configuration is less common than simple cubic, body-centered cubic, or face-centered cubic arrangements but is crucial for certain crystal structures.
