Answer the following for a body centered unit cell. Answers should be numerical, set r = 7.0.
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 find the answers for a body-centered unit cell, we need to understand the relationship between the edge length and the radii of the lattice points. In a body-centered unit cell, there is an additional lattice point located at the center of the cell.
1. Edge in terms of r, the lattice point radius:
In a body-centered unit cell, there are two lattice points at opposite corners of the cell connected by an edge. If we denote the edge length as "a," we can determine its length based on the radii of the lattice points.
The length of the edge can be calculated as follows:
a = 2r + 2r (taking into account the diameters of the two lattice points)
= 2r + 2r
= 4r
Therefore, the edge length in terms of the lattice point radius (r) is 4r.
2. Face diagonal in terms of r, the lattice point radius:
The face diagonal connects two opposite corners of a face of the unit cell. To find its length, we can use the Pythagorean theorem by considering the right triangle formed by the face diagonal, the edge, and the body diagonal. The edge length is 4r (as calculated in the previous question), and the body diagonal can be found in the next question.
Using the Pythagorean theorem:
Face Diagonal^2 = Edge^2 + Edge^2
Face Diagonal^2 = (4r)^2 + (4r)^2
Face Diagonal^2 = 16r^2 + 16r^2
Face Diagonal^2 = 32r^2
Taking the square root of both sides, we get:
Face Diagonal = √(32r^2)
Face Diagonal ≈ 5.6569r
Therefore, the face diagonal in terms of the lattice point radius (r) is approximately 5.6569r.
3. Body diagonal in terms of r, the lattice point radius:
The body diagonal connects two lattice points located at opposite corners of the unit cell. Again, we can use the Pythagorean theorem to find its length.
Using the Pythagorean theorem:
Body Diagonal^2 = Edge^2 + Edge^2 + Edge^2
Body Diagonal^2 = (4r)^2 + (4r)^2 + (4r)^2
Body Diagonal^2 = 16r^2 + 16r^2 + 16r^2
Body Diagonal^2 = 48r^2
Taking the square root of both sides, we get:
Body Diagonal = √(48r^2)
Body Diagonal ≈ 6.9282r
Therefore, the body diagonal in terms of the lattice point radius (r) is approximately 6.9282r.