Final answer:
To derive the planar density expressions for bcc (100) and (110) planes in terms of the atomic radius r, you can use the number of atoms on the plane, atomic radius, and the area of the plane. Comparing the planar density values will help understand the difference between the two planes.
Step-by-step explanation:
For a body-centered cubic (bcc) crystal lattice, the (100) plane is represented by a square with atoms located at the corners and one atom at the center of the square. The planar density can be calculated by dividing the area of the atoms in the plane by the area of the square. The atoms in the corners of the square contribute to the planar density because they are shared with adjacent squares, while the atom at the center of the square only contributes to one plane. The planar density is given by:
Planar density of (100) plane = (number of atoms on the plane * atomic radius) / (area of the plane)
Similarly, for the (110) plane in a bcc crystal lattice, the plane is represented by a rhombus shape with the atoms located at the corners. The planar density can be calculated in the same way:
Planar density of (110) plane = (number of atoms on the plane * atomic radius) / (area of the plane)