Determining the Plane with the Highest Density of Atoms in FCC Structure:

In an FCC (Face-Centered Cubic) crystal structure, there are several planes with different densities of atoms. To determine which plane has the highest density of atoms, we need to analyze the arrangement of atoms in the FCC structure.

1. FCC Crystal Structure:
The FCC crystal structure consists of a cube with atoms positioned at each corner and in the center of each face. This arrangement creates a closely-packed structure with a high packing efficiency.

2. Miller Indices:
Miller indices are used to identify crystallographic planes in a crystalline material. These indices are denoted by three integers (hkl), where 'h', 'k', and 'l' represent the intercepts of the plane with the crystallographic axes.

3. Calculating Density of Atoms:
To determine the density of atoms in a specific plane, we can use the formula:
Density of atoms = (Number of atoms in the unit cell) / (Volume of the unit cell)

Explanation for Copper (Cu) as an Example:

Copper is a commonly used material with an FCC crystal structure. Let's evaluate the density of atoms for copper as an example.

1. FCC Structure of Copper:
Copper has an FCC crystal structure, where each corner of the cube contains one copper atom, and each face-centered position also has one copper atom.

2. Plane with the Highest Density:
To determine the plane with the highest density of atoms, we need to consider the Miller indices (hkl) of the planes. The (111) plane has the highest density in an FCC structure because it intersects the most number of atoms.

3. Calculating the Density:
The number of atoms in the FCC unit cell of copper is 4. The volume of the FCC unit cell can be calculated using the formula:
Volume of the unit cell = (4 * radius^3) / 3

4. Copper Density Calculation:
To calculate the density of copper, we need to know the atomic radius of copper (which is approximately 0.128 nm). Using this information, we can calculate the volume of the unit cell and the density of copper.

Density of copper = (Number of atoms in the unit cell) / (Volume of the unit cell)
= 4 / [(4 * (0.128 nm)^3) / 3]

By substituting the values into the formula, we can calculate the density of copper.

In conclusion, by analyzing the FCC crystal structure and considering the Miller indices, we can determine that the (111) plane has the highest density of atoms in the FCC structure. For copper, the density of atoms can be calculated using the number of atoms in the unit cell and the volume of the unit cell.