What is the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice? (a) 1, (b) 2, (c) 3, (d) 4, (e) 5.

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Understand that a face-centered cubic (FCC) lattice is a type of crystal structure where atoms are located at each of the corners and the centers of all the faces of the cube.

In a face-centered cubic unit cell, there are 8 corner atoms and 6 face-centered atoms.

Each corner atom is shared by 8 adjacent unit cells, so each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.

Each face-centered atom is shared by 2 adjacent unit cells, so each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell.

Calculate the total number of atoms in the unit cell by adding the contributions from the corner atoms and the face-centered atoms: \( 8 \times \frac{1}{8} + 6 \times \frac{1}{2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. It defines the arrangement of atoms in the crystal and can vary in shape and size. Understanding the unit cell is crucial for determining the properties of the material, including its density and packing efficiency.

Face-Centered Cubic (FCC) Lattice

A face-centered cubic lattice is a type of crystal structure where atoms are located at each corner of the cube and at the center of each face. This arrangement leads to a high packing efficiency, with each unit cell containing four atoms. The FCC structure is common in metals like copper and aluminum, influencing their physical properties.

Atomic Coordination Number

The coordination number refers to the number of nearest neighbor atoms surrounding a central atom in a crystal structure. In a face-centered cubic lattice, the coordination number is 12, indicating that each atom is in contact with 12 others. This concept is important for understanding the stability and reactivity of the material.