Assume that the aluminum atoms can be represented as spheres, as shown in the drawing (Part C figure) . If each Al atom has a radius of 1.43 angstroms, what is the length of a side of the unit cell?
To calculate the length of a side of the unit cell, we need to understand the concept of a unit cell in crystal structures. A unit cell is the smallest repeating unit of a crystal lattice that demonstrates the overall structure of the crystal.
In this case, we are given the radius of an aluminum (Al) atom, which is 1.43 angstroms. It is important to note that the atomic radius represents the distance from the nucleus of the atom to the outermost electron.
The unit cell for aluminum (Al) atoms is face-centered cubic (FCC). In an FCC structure, each corner of the cube has an atom, and there is an additional atom at the center of each face. This arrangement provides a total of 4 atoms per unit cell.
To find the length of a side of the unit cell, we can use the concept of the body diagonal, which passes through the center of the unit cell and connects opposite corners. By calculating the body diagonal, we can determine the length of the side.
To calculate the body diagonal, we can use the Pythagorean theorem. In an FCC structure, the body diagonal passes through the face diagonal of the cube (the distance between two opposite corners of a face).
The length of a face diagonal can be found by calculating the distance between two opposite corners of a face. Since there are four atoms in the FCC unit cell, we can consider the face diagonal as the distance between one corner atom to the center atom of the same face, forming a right-angled triangle.
Using the Pythagorean theorem, the length of the face diagonal (d) can be calculated as follows:
d^2 = (2 * radius)^2 + (2 * radius)^2
Substituting the given radius of 1.43 angstroms:
d^2 = (2 * 1.43)^2 + (2 * 1.43)^2
Simplifying the equation:
d^2 = 4 * 1.43^2 + 4 * 1.43^2
d^2 = 16 * (1.43^2)
Taking the square root of both sides to find the length of the face diagonal:
d = √[16 * (1.43^2)]
Calculating the value:
d ≈ 4.04 angstroms
Since the face diagonal connects two opposite corners of the cube, we can consider it as the side length of the unit cell.
Therefore, the length of a side of the unit cell is approximately 4.04 angstroms.