Answer the following for a primitive cubic 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=?
For r=7.0
7(2)= 14
14(squ root 2) = 19.7989887
14(squ root 3) = 24.24871131
To answer these questions, we need to understand the geometry of a primitive cubic unit cell and its relationship to the lattice point radius.
In a primitive cubic unit cell, the lattice points are arranged at the corners of a cube. The edge length of the cube (a) is related to the lattice point radius (r) as follows:
a = 2r
Therefore, in terms of r, the lattice point radius, the edge length of the primitive cubic unit cell is 2r.
Now, let's move on to the second question: the face diagonal length in terms of r, the lattice point radius.
In a cubic unit cell, the face diagonal (d) connects two opposite corners of a face of the cube. To find the length of the face diagonal, we can use the Pythagorean theorem.
The length of the face diagonal of a cube with edge length a is given by:
d = sqrt(2) * a
Substituting the value of a = 2r (from the previous question), we get:
d = sqrt(2) * (2r) = 2 * sqrt(2) * r
Therefore, in terms of r, the lattice point radius, the face diagonal length of the primitive cubic unit cell is 2 * sqrt(2) * r.
Finally, let's move on to the third question: the body diagonal length in terms of r, the lattice point radius.
The body diagonal (D) connects two opposite corners of the cube, passing through its center. To find the length of the body diagonal, we can again use the Pythagorean theorem.
The length of the body diagonal of a cube with edge length a is given by:
D = sqrt(3) * a
Substituting the value of a = 2r (from the first question), we get:
D = sqrt(3) * (2r) = 2 * sqrt(3) * r
Therefore, in terms of r, the lattice point radius, the body diagonal length of the primitive cubic unit cell is 2 * sqrt(3) * r.
So, to summarize the answers:
- Edge length in terms of r, the lattice point radius: 2r
- Face diagonal length in terms of r, the lattice point radius: 2 * sqrt(2) * r
- Body diagonal length in terms of r, the lattice point radius: 2 * sqrt(3) * r
For this specific scenario with r = 7.0, you can substitute r = 7.0 into the equations to get the numerical values of the edge length, face diagonal length, and body diagonal length.