Final answer:
The question asks to sketch directions such as [101], [211], and [301] within a cubic unit cell, which represent movements along the axes of the cube from an origin point. The indices indicate the number of units to move in each of the x, y, and z directions to determine the end point of the direction in the 3D lattice structure.
Step-by-step explanation:
The question pertains to the direction indices within a cubic unit cell. Direction indices are a set of three numbers denoted in square brackets, such as [101]. These indices describe directions in the crystal lattice and are vital for understanding the crystallography of materials.
To explain the concept using a simple cubic lattice as an example, let's consider a cube with edges of equal length. The direction index [101] means moving one unit in the x-direction, zero units in the y-direction, and one unit in the z-direction from the origin of the cube. Similarly, for [211], it means moving two units in the x-direction, one unit in the y-direction, and one unit in the z-direction. The [301] direction would involve moving three units in the x-direction, zero units in the y-direction, and one unit in the z-direction.
To sketch these directions, you would start at one corner of the cubic cell (the origin) and draw a line to the respective ending point. For example, for [101], you would draw a diagonal line from one corner of the cube to the opposite edge on the same face. For [211], the line would end on the opposing edge but not on the same face, and for [301], the diagonal would extend to an edge that is three units over and one unit up.