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lab report with calculations Crystal Structure Activity Learning Objectives After this activity, students should be able to 1. Identify different layering patterns that lead to the cubic unit cells, determine coordination numbers, and compute packing efficiencies for atomic solids. 2. Determine the empirical formula of an ionic compound from its crystal structure. Overview Use the visualization tool found at https://atom.calpoly.edu/crystal/ and answer the following questions. Many of the functions in the simulation are bound to keys; look at Key Controls for the list. The simulation starts by default with the Simple cubic lattice screen. The drop-down menu allows you to view other lattice structures. You can rotate the structure and view it from different sides by holding the mouse and dragging the structure. You can also zoom in and out with the mouse wheel. There are two important modes that are controlled with the Expansion slider at the bottom of the screen. In Layering mode, you can see how the 3D crystal lattice can be made by stacking layers of atoms. In Unit Cell mode, you can see how the 3D lattice is composed of repeating unit cells with fractional atoms. Lattice Structures of Atomic Solids Layering We will begin this activity by looking at the layering pattern of particles that gives rise to each of the cubic unit cells. A unit cell is the smallest unit in a repetitive pattern that makes the 3-dimensional lattice structure. As shown in Figure 1, there are two basic 2D patterns for layers of atoms. The atoms in each layer can be packed in a square array, or “close-packed” with a rhombus representing the simplest repeating pattern. When multiple layers of a particular 2D pattern are stacked together, they can generate a variety of 3D patterns, depending on how the layers are shifted relative to each other. If the layers repeat identically as they stack, this can be described as “aa” stacking. If the second layer is staggered relative to the first layer, but the third layer is stacked directly above the first layer, this stacking pattern is described as “aba.” You can explore this layering effect by selecting Layering on the left of the visualization tool and using the Expansion slider. Figure 1. Square and rhombic unit cells in 2D layers. For each of the cubic lattices (simple cubic, body-centered cubic, and face-centered cubic), answer the following questions. Use the visualization tool to help. https://atom.calpoly.edu/crystal/ 1. What type of 2D unit cell exists in each layer, square or rhombic? (See Figure 1). 2. What is the stacking pattern in the corresponding lattice structure? (use letters a, b, c, etc. to label different layers). Unit Cells Once atoms are stacked into a 3D crystal lattice, the simplest repeating geometric pattern—the unit cell—will usually contain fractions of atoms. While only whole atoms exist in the crystal, the geometric representation of the unit cell will have atoms split between multiple neighboring unit cells. To find a unit cell, we take the smallest repeating pattern and “slice” the shared parts off, to make it look like a cube (here we are exploring cubic unit cells, but there are shapes for unit cells as well). With Unit Cell selected on the left, use the Expansion slider to see how multiple unit cells together makes up an entire lattice. To highlight a single unit cell within the crystal lattice, press “t” on the keyboard to toggle the translucency. For each of the cubic lattices, answer the following questions. 1. Which part(s) of a 3D unit cell do the atoms occupy (corner, edge, center, face)? 2. What fraction of an atom does each contribute to the unit cell? 3. What is the total number of atoms per unit cell? Coordination Number The coordination number is the number of closest neighbors an atom has in the lattice, including atoms in the adjacent unit cells. For the following questions, you can use the Coordination mode in the visualization tool to verify your answer. 1. Determine the coordination number for the simple cubic lattice. 2. Determine the coordination number for the grey atoms in a body-centered cubic (bcc) lattice. 3. Determine the coordination number for the red atoms in a bcc lattice. 4. Explain why the coordination number for all the atoms in the bcc lattice is the same. 5. Determine the coordination number for the face-centered cubic lattice. Packing Efficiency Since the layering pattern in all of the lattices leaves empty space between the particles, the unit cell is not completely occupied by atoms (here we are treating atoms like hard spheres). The packing efficiency, which is the percentage of occupied space in the cube, is not 100%. The packing efficiency is not the same for all 3 cubic lattices. A more densely packed unit cell will have a higher packing efficiency than a less densely packed one. The packing efficiency of a lattice structure measures how well the space inside of a unit cell is utilized. It is the percent ratio of volume occupied by the particles in a unit cell to its total volume. 𝑃𝑎𝑐𝑘𝑖𝑛𝑔 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = 𝑉𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑 𝑉𝑡𝑜𝑡𝑎𝑙 × 100 The occupied volume is related to the number of particles occupying the cell and their location within the cell. The edge length of each unit cell is derived using the trigonometric relationships shown in Figure 2. Figure 2. Geometric relationships showing how the edge length is related to the atomic radius for simple cubic, body-centered cubic, and face-centered cubic unit cells. Unit Cell Edge length in terms of radius Simple cubic 𝑙 = 2𝑟 Body-centered cubic 𝑙 = 4𝑟 √3 Face-centered cubic 𝑙 = 2√2𝑟 𝑉𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑 = (# 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠) × 4 3 𝜋𝑟3 𝑉𝑡𝑜𝑡𝑎𝑙 = 𝑙 3 Answer the following questions. Assume that the lattice consists of only one type of atom, and the radius of this atom is r. 1. Assume an atom is a perfect sphere. In terms of r, what volume of the simple cubic unit cell is occupied by atoms? 2. What is the total volume of the simple cubic unit cell? 3. Determine the packing efficiency of a simple cubic unit cell. Use your answers from the previous two questions. 4. Determine the packing efficiency for a body-centered cubic unit cell. 5. Determine the packing efficiency for a face-centered cubic unit cell. 6. Observe the difference in stacking patterns of the unit cells and note how they are related to the packing efficiency. Summary 2D layer pattern (square vs rhombic) Stacking Pattern (e.g. aba) Number of Atoms per Unit Cell Coordination Number Packing Efficiency Simple Cubic Body-Centered Cubic Face-Centered Cubic Lattice Structures of Ionic Compounds Now we will look at a few examples of ionic solids. The Legend button will show the ion coloring scheme. The ions are roughly scaled to their relative ionic radii within each of the lattices. Sodium Chloride 1. Determine the number of sodium ions per unit cell. 2. Determine the number of chloride ions per unit cell. 3. What is the empirical formula of sodium chloride based on the relative number of each ion in the unit cells? 4. Is the empirical formula determined from the lattice structure in agreement with the one predicted by the typical ion charges? 5. Are either of the ions arranged in one of the basic cubic unit cells (simple, body-centered, face- centered)? Calcium Fluoride 1. Determine the number of calcium ions per unit cell. 2. Determine the number of fluoride ions per unit cell. 3. What is the empirical formula of calcium fluoride based on the relative number of each ion in the unit cells? 4. Is the empirical formula determined from the lattice structure in agreement with the one predicted by the typical ion charges? 5. Are either of the ions arranged in one of the basic cubic unit cells (simple, body-centered, face- centered)?