Determine for solid manganese (Mn) the following:

What is the atomic density (atoms/m) along the [110] direction of the Mn crystal?

What are the number of nearest neighbors around an atom in the Mn crystal?

Calculate the interplanar spacing (m) between {220} planes in the Mn crystal.

I'm obtaining a lattice constant of 6.251×10^-9 centimeters which is too small for Mn

To determine the atomic density along the [110] direction of solid manganese (Mn), we need to consider the crystal structure of Mn and the arrangement of atoms within it.

Manganese crystallizes in a face-centered cubic (FCC) structure, meaning each lattice point is surrounded by 12 equidistant nearest neighboring atoms.

To find the atomic density along the [110] direction, we first need to understand the Miller indices notation. In this notation, the [110] direction of a crystal represents a direction that runs parallel to the [1, 1, 0] vector in the crystal lattice.

In an FCC crystal, the atomic density along a specific direction can be calculated using the formula:

Atomic density = (Number of atoms in one unit cell) / (Length of the unit cell along the given direction)

For the [110] direction, we need to determine the length of the unit cell along this direction. In an FCC crystal, the length along the [110] direction is equal to the edge length of the unit cell, which can be calculated using the formula:

Edge length = (4 * radius) / sqrt(2)

Now, let's calculate the atomic density along the [110] direction:

1. Find the radius of the Mn atom. The atomic radius of Mn is approximately 0.139 nm.

2. Calculate the edge length of the FCC unit cell along the [110] direction using the formula:
Edge length = (4 * 0.139 nm) / sqrt(2)

3. Calculate the atomic density using the formula:
Atomic density = (Number of atoms in one unit cell) / (Length of the unit cell along the [110] direction)

In an FCC unit cell, there are 4 atoms. Therefore, the atomic density along the [110] direction of the Mn crystal is:
Atomic density = 4 / (Edge length)

Next, let's determine the number of nearest neighbors around an atom in the Mn crystal.

In an FCC crystal structure, each atom has 12 equidistant nearest neighbors. Therefore, the number of nearest neighbors in the Mn crystal is 12.

Finally, let's calculate the interplanar spacing between the {220} planes in the Mn crystal.

For an FCC crystal, the interplanar spacing between {hkl} planes is given by the formula:

Interplanar spacing = (Edge length) / sqrt(h^2 + k^2 + l^2)

In this case, we need to calculate the interplanar spacing between the {220} planes in the Mn crystal.

Using the edge length we obtained earlier and substituting h = 2, k = 2, and l = 0 into the formula, we can calculate the interplanar spacing.