(a) Draw an orthorhombic unit cell, and within that cell a \((02 \overline{1})\) plane. (b) Draw a monoclinic unit cell, and within that cell a (200) plane.

The orthorhombic unit cell is fundamental in the study of crystallography. Imagine this cell as a three-dimensional box where each of the edges meets at right angles, akin to the room you might be sitting in. However, unlike the regular volume of your room, the orthorhombic unit cell's sides, labeled as a, b, and c, differ in length. These varying sides represent the axes of the crystal lattice and form the basis for understanding the cell's spatial dimensions.

When you're drawing this cell, envision it as a 3D rectangle where each corner has a connection to the others through the edges, and each face represents a plane in the crystal. This mental model is essential for later visualizing planes that cut through the cell, which are described using a unique notation system known as the Miller indices.

For those who find visualizing this challenging, a helpful exercise is to build a physical model using something as simple as straws or sticks cut to different lengths. This hands-on approach can solidify understanding of the orthorhombic unit cell's structure.

Miller indices serve as a sort of 'code' that uniquely defines a plane within a crystal structure. These indices, made up of three integers, represent the orientation of crystal planes. Understanding how to interpret and apply these indices can be tricky, but it's much like reading a map's coordinates. The three numbers \(hkl\) correspond to the intercepts of the plane along the crystal axes.

The indices articulate a plane's orientation by denoting the reciprocal of its intersection points with the axes. If a plane does not intersect an axis, like the '0' in our \(02\overline{1}\), the reciprocal is infinite, and thus the index is zero. This explains why the \(02\overline{1}\) plane never intersects the x-axis.

Why Negative Indices?

A bar over a number represents a negative index, indicating the plane's intersection behind the origin point on an axis. It's crucial to grasp this concept as it significantly expands the types of planes that can be described within a cell.

Imagine drawing straight lines using the indices as a guide, cutting the cell at precise points and angles. Each line intersects axes based on these codes, with negative values indicating directions. This visualization helps greatly in understanding how Miller indices map the inner geometry of a crystal.

The monoclinic unit cell might remind you of the orthorhombic cell at first glance, yet it holds a twist - literally. With edges a, b, and c, each a different length like its orthorhombic counterpart, the monoclinic cell's uniqueness lies in its angles. Two of its angles are 90 degrees, but the third – between the a and c axes, known as angle β – deviates from a right angle.

This leaning structure defines the monoclinic cell's skewed look, presenting new angles and faces for planes to intersect. Drawing it requires imagining a 3D object that seems to have been pushed over slightly, causing one of its angles to become obtuse (or acute).

A Closer Look at the \(200\) Plane

When applying our \(200\) Miller indices to the monoclinic cell, you'll find that the plane slices the crystal cleanly halfway along the x-axis but doesn't touch the y or z axes at all—hinted at by the zeroes in the indices. Remember, the indices dictate where a plane will hit or miss the axes, providing a blueprint for sketching it within the cell.

Visual aids can again be beneficial here; using a model or even software to draw the cells and planes can help students overcome the abstract nature of these concepts, transforming them into tangible examples.