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Chapter 3: Problem 39

Sketch within a cubic unit cell the following planes: (a) \((10 \overline{1})\) (b) \((2 \overline{1} 1)\) (c) (012) (d) \((3 \overline{1} 3)\) (e) \((\overline{1} 1 \overline{1})\) (f) \((\overline{2} 12)\) (g) \((3 \overline{1} 2)\) (h) (301)

Short Answer

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Question: Sketch the following planes within a cubic unit cell using their Miller indices: (a) (10\(\overline{1}\)), (b) (2\(\overline{1}\)1), (c) (012), (d) (3\(\overline{1}\)3), (e) (\(\overline{1}\)1\(\overline{1}\)), (f) (\(\overline{2}\)12), (g) (3\(\overline{1}\)2), and (h) (301). Answer: To sketch each plane, first determine the intercepts from the Miller indices, then connect the intercepts on the corresponding axes within the cubic unit cell. For example, for plane (a) with Miller indices (10\(\overline{1}\)), the intercepts are 1, 0, and -1 on the x, y, and z-axes respectively. Sketch the plane diagonally across the unit cell connecting these intercepts. Repeat this process for the remaining planes to visualize them within the cubic unit cell.

Step by step solution

01

Understand Miller indices

Miller indices (hkl) represent the reciprocal of the intercepts along the x, y, and z axes, respectively. If hkl is negative, we show it using (h,k,l) notation with a negative sign, such as \(\overline{1}\). If any of the indices is infinity, we represent it as 0.
02

Determine intercepts from Miller indices

To determine the intercepts on the three axes, we need to find the reciprocal of each index and simplify it into a fraction. For example, if the plane is \((2 \overline{1} 1)\), the intercepts along the x, y, and z axes are \(\frac{1}{2}, -1, \text{ and } 1\), respectively.
03

Sketch the planes within the cubic unit cell

Once we have determined the intercepts, we can sketch each plane within the unit cell. Connect the intercepts on the corresponding axes to draw the plane. Sketching the given planes: (a) \((10 \overline{1})\) Intercepts: \(1, 0, -1\) The plane intersects the x-axis at 1 and z-axis at -1. It does not intersect the y-axis. Sketch the plane diagonally across the unit cell. (b) \((2 \overline{1} 1)\) Intercepts: \(\frac{1}{2}, -1, 1\) The plane intersects the x-axis at 0.5, y-axis at -1, and z-axis at 1. Sketch the plane accordingly within the unit cell. (c) (012) Intercepts: \(\infty, \frac{1}{2}, \frac{1}{2}\) The plane does not intersect the x-axis, but it intersects the y and z-axes at 0.5. Sketch the plane parallel to the x-axis inside the unit cell, halfway along the y and z directions. (d) \((3 \overline{1} 3)\) Intercepts: \(\frac{1}{3}, -1, \frac{1}{3}\) The plane intersects the x and z-axes at 1/3, and the y-axis at -1. Sketch the plane within the unit cell with this intercepts. (e) \((\overline{1} 1 \overline{1})\) Intercepts: \(-1, 1, -1\) Each intercept is at -1 or 1 on each axis. Sketch the plane diagonally across the unit cell. (f) \((\overline{2} 1 2)\) Intercepts: \(-\frac{1}{2}, 1, \frac{1}{2}\) The plane intersects the x-axis at -0.5, the y-axis at 1, and the z-axis at 0.5. Sketch the plane accordingly within the unit cell. (g) \((3 \overline{1} 2)\) Intercepts: \(\frac{1}{3}, -1, \frac{1}{2}\) The plane intersects the x-axis at 1/3, y-axis at -1, and z-axis at 1/2. Sketch the plane within the unit cell with these intercepts. (h) (301) Intercepts: \(\infty, -1, \frac{1}{3}\) The plane does not intersect the x-axis but intersects the y-axis at -1, and z-axis at 1/3. Sketch the plane parallel to the x-axis inside the unit cell.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Crystallographic Planes
Crystallographic planes are fundamental concepts in the study of crystal structures. They are flat surfaces that pass through the crystal lattice, a three-dimensional array of points depicting the positions of atoms, ions, or molecules. In a crystal, many such planes can be identified, each distinguished by its orientation within the crystal structure. To describe these orientations unambiguously, we use a notation system called Miller indices. These indices are determined by the intercepts the plane makes with the crystal axes and are crucial for understanding facets of a crystal, as well as for predicting and interpreting many of its physical properties, including cleavage patterns, diffraction patterns, and growth forms. The ability to visualize and sketch these planes helps students gain a deeper insight into crystal geometry, aiding them in fields like material science, mineralogy, and solid-state physics.

When we talk about crystallographic planes, it's important to remember they are an idealized concept that helps in the visualization and mathematical representation of the solid matter in a crystalline state. While they may not always represent physical surfaces in a crystal, they provide a basis for understanding and characterizing the directional properties of crystals.
Cubic Unit Cell
The cubic unit cell is a type of unit cell, which is the smallest repeating unit of a crystal lattice that still maintains the overall symmetry of the entire crystal. Among the seven crystal systems in three-dimensional space, the cubic system is one of the simplest and most symmetric. It is characterized by three axes of equal lengths that are perpendicular to each other. In the context of crystallography, the cubic unit cell can be classified into three distinct lattice types: simple cubic, body-centered cubic, and face-centered cubic. Each type has different arrangements of atoms at its corners and, in the case of body-centered and face-centered variants, within the cell itself.

The cubic unit cell is fundamental for teaching students about crystalline structures as it represents an ideal model enabling them to grasp more complex structures. An understanding of cubic unit cells is essential for interpreting X-ray diffraction patterns and for the calculation of material properties such as density and packing efficiency.
Plane Intercepts
Plane intercepts refer to the points at which a crystallographic plane intersects the axes of the crystallographic coordinate system in a unit cell. In the Miller indices notation, these intercepts are not written directly as distances; instead, their reciprocals are taken (with the exception of zero, which denotes an intercept at infinity), and these values are then used to calculate the Miller indices. To understand planes better, one must be able to translate the Miller indices back into axis intercepts, which is where students often require additional assistance.

To better support learner development in this area, it's helpful to emphasize that if an intercept is a fraction, it indicates that the plane cuts between the origin and the next lattice point along that axis. Conversely, a negative intercept signifies that the plane cuts in the negative direction from the origin. Giving examples with both positive and negative intercepts, as well as using intercepts of infinity, offers students a more robust understanding of the relationship between Miller indices and plane intercepts.
Crystal Lattice Sketches
Crystal lattice sketches are visual representations of the periodic structure of a crystal's unit cells in two or three dimensions. Sketching these lattices helps with visualizing how planes intersect with unit cells and provides concrete examples for a better understanding of the abstract numerical Miller indices. Steps to create such sketches generally start with drawing the unit cell, then marking the points of intersection according to the intercepts, and finally drawing the plane itself.

In an educational setting, it is important to guide students through the process of sketching crystal lattices by providing clear instructions and examples. One should encourage students to start with simple cubic lattices before advancing to more complicated structures. Using graph paper or isometric dot paper can greatly facilitate the accuracy of these sketches. Also, the usage of 3D models or computer simulations can tremendously enhance spatial understanding, making the abstract concepts of crystal lattices more tangible.

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Most popular questions from this chapter

Below are listed the atomic weight, density, and atomic radius for three hypothetical alloys. For each determine whether its crystal structure is FCC, BCC, or simple cubic and then justify your determination. A simple cubic unit cell is shown in Figure 3.23 $$\begin{array}{lccc} \hline & \text {Atomic} & & \text {Atomic} \\ & \text {Weight} & \text {Density} & \text {Radius} \\ \text {Alloy} & \text { (g/mol) } & \left(\mathrm{g} / \mathrm{cm}^{3}\right) & (\boldsymbol{n m}) \\ \hline \mathrm{A} & 43.1 & 6.40 & 0.122 \\ \mathrm{B} & 184.4 & 12.30 & 0.146 \\ \mathrm{C} & 91.6 & 9.60 & 0.137 \\ \hline \end{array}$$

Magnesium has an HCP crystal structure, a \(c / a\) ratio of \(1.624,\) and a density of \(1.74 \mathrm{g} / \mathrm{cm}^{3}\) Compute the atomic radius for \(\mathrm{Mg}.\)

Molybdenum has a BCC crystal structure, an atomic radius of \(0.1363 \mathrm{nm},\) and an atomic weight of \(95.94 \mathrm{g} / \mathrm{mol}\). Compute and compare its theoretical density with the experimental value found inside the front cover.

Within a cubic unit cell, sketch the following directions: (a) [101] (b) [211] (c) \([10 \overline{2}]\) (d) \([3 \overline{1} 3]\) (e) \([\overline{1} 1 \overline{1}]\) (f) \([\overline{2} 12]\) (g) \([3 \overline{1} 2]\) (h) [301]

(a) Derive planar density expressions for FCC (100) and (111) planes in terms of the atomic radius \(R.\) (b) Compute and compare planar density values for these same two planes for aluminum.
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