A tetrahedral site in a close-packed lattice is formed by four spheres at the corners of a regular tetrahedron. This is equivalent to placing the spheres at alternate corners of a cube. In such a close-packed arrangement the spheres are in contact and if the spheres have a radius r, the diagonal of the face of the cube is 2r. The tetrahedral hole is inside the middle of the cube. Find the body diagonal of this cube.

Question

A tetrahedral site in a close-packed lattice is formed by four spheres at the corners of a regular tetrahedron. This is equivalent to placing the spheres at alternate corners of a cube. In such a close-packed arrangement the spheres are in contact and if the spheres have a radius r, the diagonal of the face of the cube is 2r. The tetrahedral hole is inside the middle of the cube. Find the body diagonal of this cube.

Honestly just a formula or procedure on how to solve this will be helpful. My professor does not explain anything to us. I have looked in the book and still have no idea.

Answer 1

I made a typo so I erased the page and reposted this.

I can't visualize it and my attempted drawings doin't help. I found this site on the web and appears to be what you are describing if you look at the pink one (I think three from the left) and the"clear" on the far right.
If done this right, and I don't make any promises because my 3D is not all that good. If the face diagonal is 2r then the face diagonal and the bottom side and right side form a triangle and the py theorem gives a2 + b2 = c2
a2 + a2 = 4r^2 and
the sides then are a and I think that is 2a2 = 4r^2 and a = r(2)^1/2

Now if my 3D is working, the body diagonal appears to consist of a triangle consisting of 1 side of the cube, the face diagonal of the cube and the hypotenuse (which is the body diagonal).
The side is r(2)^1/2. The face diagonal is 2r. Solve for the hypotenuse. If you have an answer you will know if this is right or not.
https://www.google.com/search?q=detrahedron+inside+cube&ie=utf-8&oe=utf-8

Answer 2

To find the body diagonal of the cube formed by placing spheres at the alternate corners of a regular tetrahedron, we can use the Pythagorean theorem.

Step 1: Determine the length of the edge (a) of the cube
Since the spheres are in contact, the diagonal of the face of the cube is 2r. Therefore, the edge of the cube (a) can be expressed as a = 2r.

Step 2: Calculate the length of the body diagonal (d) of the cube
The body diagonal (d) of a cube can be found using the formula:
d = √(a^2 + a^2 + a^2)

Substituting the value of a from Step 1 into the formula:
d = √(2r)^2 + (2r)^2 + (2r)^2)
= √(4r^2 + 4r^2 + 4r^2)
= √(12r^2)
= √12(r^2)
= 2√3r

Therefore, the body diagonal of the cube is 2√3r.