Question about unit cell and edge length?

Question

A hypothetical metal has a body-centered cubic unit cell and its atomic radius is 1.172 Å. Calculate the edge length of the unit cell.

Answer 1

Before you begin reading, look at the following picture.
http://chemistry.umeche.maine.edu/CHY132/Cubic.jpg

A couple of important points:
1. Look for a way to go from one corner to another going through only atoms.
2. If an atom is sitting on a corner, the distance from the corner to the edge of the atomic sphere is the atomic radius.

With that in mind,

Draw a line from the top right corner to the bottom left corner, so that it passes through the middle sphere. The picture does not show it well (I'll put a better one in the source if I can find one), but this line has a length of 4r (1r for one corner, 2r for the center sphere, and another 1r for the second corner).

Let's make this line the hypotenuse of a right triangle. Use an edge (say the right edge) as one side, which means the other side is a diagonal through the bottom face.

(4r)^2 = edge^2 + face^2 = e^2 + f^2.
The face is the hypotenuse of a triangle in which both sides are edges, so
f^2 = e^2 + e^2 = 2e^2. Substitute this into the first equation:
(4r)^2 = e^2 + 2e^2
(4r)^2 = 3e^2. Take the square root
e = 4r/sqrt(3).
The edge for your hypothetical metal is 4 * 1.172 / sqrt(3) = 2.71 angstroms.

PS - How did you get the Angstrom symbol in the textbox?

Answer 2

The diagonal of the unit cube would be x√3 where x is the unit length. The diagonal is equal to 4 times the radius.

x√3 = 4 * 1.172

x = 4*1.172/(x√3) = 2.707 angstroms