Final answer:
Miller indices are used to describe the orientation and spacing of crystal planes within a unit cell. In a simple cubic unit cell, the Miller indices for any plane that intersects the axes at 1 unit would be (111). In a body-centered cubic unit cell, the Miller indices for a plane passing through the center of the unit cell would be (2, -1, 1). In a face-centered cubic unit cell, the Miller indices for a plane passing through the face centers would be (001).
Step-by-step explanation:
Miller indices are used to describe the orientation and spacing of crystal planes within a unit cell. They are denoted by three integers (hkl), known as Miller indices, which indicate the intercepts of the plane with the three axes of the unit cell. To determine the Miller indices for the planes shown in a unit cell, you need to identify the intercepts of the plane with the axes and take the reciprocals of these intercepts.
For example, in a simple cubic unit cell, each face intersects with the x, y, and z axes at a distance of 1 unit. Therefore, the Miller indices for any plane that intersects the axes at 1 unit would be (111), since the reciprocals of 1 are also 1.
In a body-centered cubic unit cell, each face still intersects with the x, y, and z axes at a distance of 1 unit. However, there is also an additional plane passing through the center of the unit cell. This plane intersects the x, y, and z axes at -1, 1, and 1 units, respectively. Therefore, the Miller indices for this plane would be (2, -1, 1).
In a face-centered cubic unit cell, each face still intersects with the x, y, and z axes at a distance of 1 unit. Additionally, there are planes passing through the face centers. These planes intersect the axes at 0, 1, and 1 units, respectively. Therefore, the Miller indices for this plane would be (001).