Please help!

Answer the following for a body centered unit cell.

Crystalline Solids Program
Answers should be numerical,
set r = 4.0.

edge in terms of r, the lattice pt radius:
face diagonal in terms of r, the lattice pt radius:
body diagonal in terms of r, the lattice pt radius:

To answer these questions, we first need to understand the body-centered unit cell. In a body-centered unit cell, there is an additional lattice point in the center of the unit cell in addition to the lattice points at the corners.

Now, let's consider the different lengths in terms of the radius of the lattice points (r).

1. Edge in terms of r:
For a body-centered unit cell, the length of the edge (a) can be calculated by considering the distance between two adjacent lattice points along an edge of the unit cell.
In this case, the distance between two adjacent lattice points along an edge can be broken down into two parts: the radius of one lattice point (r) and the radius of the adjacent lattice point (r).

So, the total length (a) of the edge will be:
a = r + r = 2r

Therefore, the edge length in terms of the lattice point radius (r) is 2r.

2. Face diagonal in terms of r:
The face diagonal is the length between two opposite corners of a face of the unit cell. To calculate this length (d), we can use the Pythagorean theorem in three dimensions.

In a body-centered unit cell, the face diagonal can be broken down into three parts: the length of the edge (a), the length from one corner to the center of the unit cell (r), and the length from the center of the unit cell to the opposite corner (r).

Using the Pythagorean theorem, we can calculate the face diagonal (d) as follows:
d^2 = a^2 + r^2 + r^2
d^2 = 2r^2 + 4r^2
d^2 = 6r^2

Taking the square root of both sides, we get:
d = √(6r^2)
d = √(6) * r

Therefore, the face diagonal in terms of the lattice point radius (r) is √(6) * r.

3. Body diagonal in terms of r:
The body diagonal is the length between two opposite corners of the unit cell. To calculate this length (D), we can use the Pythagorean theorem in three dimensions.

In a body-centered unit cell, the body diagonal can be broken down into three parts: the length of the edge (a), the length from one corner to the center of the unit cell (r), and the length from the center of the unit cell to the opposite corner (2r).

Using the Pythagorean theorem, we can calculate the body diagonal (D) as follows:
D^2 = a^2 + r^2 + (2r)^2
D^2 = 4r^2 + 5r^2
D^2 = 9r^2

Taking the square root of both sides, we get:
D = √(9r^2)
D = 3r

Therefore, the body diagonal in terms of the lattice point radius (r) is 3r.

To summarize:
- Edge length in terms of r: 2r
- Face diagonal in terms of r: √(6) * r
- Body diagonal in terms of r: 3r