Answer: (a) The distance between nearest neighbors in silicon can be calculated using the formula:
d = a/√2
where d is the distance between nearest neighbors, and a is the lattice constant, which is the distance between adjacent lattice points in a crystal lattice. For silicon, the lattice constant is 5.43 Å (angstroms).
Substituting the values, we get:
d = 5.43 Å/√2 ≈ 3.82 Å
Therefore, the distance between nearest neighbors in silicon is approximately 3.82 angstroms.
(b) The number of atoms per square centimeter in a crystal lattice can be calculated using the formula:
N = (1/d^2) x Z x A
where N is the number of atoms per square centimeter, d is the distance between nearest neighbors, Z is the number of atoms in the unit cell, and A is the area of the unit cell.
For silicon, the crystal structure is face-centered cubic (FCC), and the number of atoms in the unit cell is 4. The area of the unit cell in each plane can be calculated based on the Miller indices of the plane.
For the (100) plane, the Miller indices are [100]. The area of the unit cell in the (100) plane can be calculated using the formula:
A = a^2
where a is the lattice constant. Substituting the values, we get:
A = (5.43 Å)^2 ≈ 29.53 Å^2
Substituting the values in the formula for N, we get:
N = (1/(3.82 Å)^2) x 4 x 29.53 Å^2
N ≈ 5.00 x 10^14 atoms/cm^2
For the (110) plane, the Miller indices are [110]. The area of the unit cell in the (110) plane can be calculated using the formula:
A = a^2/2
Substituting the values, we get:
A = (5.43 Å)^2/2 ≈ 14.76 Å^2
Substituting the values in the formula for N, we get:
N = (1/(3.82 Å)^2) x 4 x 14.76 Å^2
N ≈ 1.25 x 10^15 atoms/cm^2
For the (111) plane, the Miller indices are [111]. The area of the unit cell in the (111) plane can be calculated using the formula:
A = (3^(1/2)/2) x a^2
Substituting the values, we get:
A = (3^(1/2)/2) x (5.43 Å)^2 ≈ 25.08 Å^2
Substituting the values in the formula for N, we get:
N = (1/(3.82 Å)^2) x 4 x 25.08 Å^2
N ≈ 6.14 x 10^14 atoms/cm^2
Therefore, the number of atoms per square centimeter in silicon in the (100), (110), and (111) planes are approximately 5.00 x 10^14 atoms/cm^2, 1.25 x 10^15 atoms/cm^2, and 6.14 x 10^14 atoms/cm^2, respectively.