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.
No because i have no idea where your "answer below" is
To find the body diagonal of the cube, we need to understand the relationship between the radius of the spheres and the diagonal of the face of the cube.
Let's consider a close-packed arrangement of spheres in a cube. Each sphere has a radius r, and the spheres are in contact with each other. The diagonal of the face of the cube is given as 2r.
Now, let's focus on a tetrahedral hole in the middle of the cube. This hole is formed by placing four spheres at the alternate corners of the cube, forming a regular tetrahedron. To find the body diagonal of this cube, we need to find the length of the diagonal inside the tetrahedral hole.
Here's how you can solve for the body diagonal of the cube:
Step 1: Determine the length of the diagonal of the tetrahedron:
The diagonal of a regular tetrahedron with edges of length a can be found using the formula: diagonal = a√2.
Since the edge length of the tetrahedron is equal to the diameter of the spheres, which is 2r, the diagonal of the tetrahedron can be written as: diagonal = (2r)√2.
Step 2: Find the length of the body diagonal of the cube:
The body diagonal of the cube is the length of the line passing through the center of the cube from one corner to the opposite corner.
In the given close-packed arrangement, the tetrahedral hole is in the middle of the cube. Hence, the body diagonal of the cube is equal to the diagonal of the tetrahedron plus twice the radius of the spheres.
Therefore, the body diagonal of the cube = (2r)√2 + 2r.
Simplifying the expression, we get:
Body diagonal of the cube = 2r(√2 + 1).
So, the formula to find the body diagonal of the cube, given the radius of the spheres, is:
Body diagonal = 2r(√2 + 1).
Please note that this formula is specific to a close-packed arrangement of spheres with a tetrahedral hole in the middle of the cube.