Earth Materials Lab 2 - Lattices and the Unit Cell Unit Cell Minerals are crystallographic solids and therefore are made of atoms arranged into lattices. The average size hand specimen is made of more than10 23 atoms, so the lattice of a single crystal is spatially extensive. To simplify, we need to only show enough of the lattice to understand the pattern. Thus most minerals are defined structurally by a simple subset of the lattice known as the unit cell. 1. The unit cell is a subset of the entire lattice that shows all of the features of the structure. Below are several examples of 2D lattices. Outline three possible unit cells for each except the one on the right it only has one. Note that although there are many possibilities for subdividing most of these diagrams into simple subsets that define the lattice. However, if you want to show elements of symmetry, your choices are limited. For crystals, it will be important to choose unit cells that reflect the highest order of symmetry possible. For example, the rightmost diagram has the small atoms that make it different from the leftmost. For both, you can choose unit cells that can be rotated 90º and look the same. This is known as four-fold rotational symmetry. Of the two, only the leftmost can be reflected in half, vertically, horizontally, and along both diagonals, to produce the same image. This is known as mirror symmetry. Much more on symmetry next week. 2. Minerals have atoms in three-dimensional lattices, and that provides additional complications. We have examined one 3D model in class the halite structure. Please examine the model and draw a perspective representation of a unit cell for this structure. As it turns out, there aren t that many ways to bring atoms together. There are three basic structures and a series of derivative structures associated with them. These can be easily modeled as rigid spheres. 3. You need to build three models with the model spheres: Simple cubic packing (SCP), closest cubic packing (CCP), and hexagonal closest packing (HCP). After building each one, have your instructor sign off below. SCP CCP (Demonstrate the orientation with hexagonal symmetry) HCP By these packing arrangements, we are modeling atomic arrangements as hard spheres. For some crystals, the atoms involved are largely spherical in the space they occupy. Others have electronic interactions that are lengthened. So theses packing models are the closest possible spacing with some lattices we might observe elongation in one or two directions, as well as offsets from right or 120º angles seen in our models.
Earth Materials 2 Lab 2 The 14 arrangements of atoms known as the Bravais Lattices. P is the primitive form (atoms at corners only). I is internal, also known as body-centered. C is center of a face, or face-centered. R is rhombohedral.
Earth Materials 3 Lab 2 Note too that we can also think of spaces within these simple and closest packing models. Atoms are not all the same size, and in some instances the packing model best describes one of the elements of a polyatomic compound. We may consider configurations where the cations are modeled by imaginary spheres within some of the interstices between the spheres. By omitting certain atoms and/or elongating the structure, you can generate the 14 lattice configurations known as the Bravais lattices. As we will examine next week, these also define all possible symmetries for the arrangement of atoms in nature. Orientation Look carefully at the Bravais lattice diagrams on the previous page. Note that the spatial axes a, b, and c are conveniently altered to fit the space needed to define the lattice. For example, the triclinic axes are defined to be non-perpendicular; the value of each angle depends on the lattice arrangement. In the hexagonal example, we may even invoke three axes normal to the up-down c axis to more easily define the relationships between the atoms. In fact, once we recognize the differences in distances between the atoms, we may scale each axis to conveniently fit the atomic spacing, so that each atom is one unit apart on its respective axis. Additionally, because in the systems that have the same spacing in two or more directions (tetragonal, hexagonal, rhombohedral, and cubic), we may refer to the equivalent axes all as a axes, because you can t tell them apart. All of this means we can oriented ourselves quickly regardless of which lattice we are discussing. At left is an example of labeling atoms for the Cubic I (bodycentered cubic) lattice. All of the labels follow an (a,b,c) or (a 1, a 2, a 3 ) format. 4. Label all of the atoms in the Bravais lattice diagram (previous page). Start with the origin atom (0,0,0) and label others in (a,b,c) format. For the hexagonal and rhombohedral systems, use a (a 1, a 2, a 3, c) format. The Planes more orientation The sides of the lattices form planes, and there are planes to be found by connecting the dots in a number of directions. The planes turn out be incredibly useful features for two reasons: 1). Lattice planes may manifest themselves at the macroscopic scale as facets and cleavage planes (ooh! You saw this last week!), and 2). X-rays will diffract at specific angles as controlled by the internal planar structure of the mineral. We ll do more with both attributes later. For now, we need to learn to reference these planes.
Earth Materials 4 Lab 2 The Miller index system uses the Bravais lattice orientation to define the planar features of a crystal. Let s start with the 2D plot of a P Cube on the a 3 =0 plane. For Miller indices, we need to record the intercepts of the plane with the three axes. These are known as the Weiss indices. The plane intercepts a 1 at 1 unit, but doesn t intercept the a 2 or a 3 (it is parallel to both). Mathematically, the intercept is infinity. To determine the Miller index, we take the reciprocal value of the intercept, for a 1 it is 1 and for a 2 and a 3 it is 0 each (the reciprocal of a 1, a 2, and a 3 is known as h, k and l, respectively). We write the index as (hkl), and for this example we write the index as (100). Above are two other examples using the same lattice. The left shows intercepts at a 1 = infinity, a 2 = 1, and a 3 = infinity; the right diagram shows intercepts at a 1 = 1, a 2 = 1, and a 3 = infinity. Planes may intercept negative space as well. In the example at left, the a 1 intercept is -1, so the reciprocal is also -1. However, in writing it as a Miller index, we denote its negative value by putting a bar over the top. Verbally, we convey its index in (hkl) as bar-one, one, zero. Because the selection of an origin is arbitrary and could be placed on any corner of any unit cell within a mineral lattice, all parallel planes are indexed identically (thankfully eliminating the need for fractional (hkl)). You can determine Weiss indices that are greater than 1, but in converting to Miller indices you should always multiply the reciprocal through by the common demeanor. Example, if we had intercepts of 2, 1, and 1, and then the reciprocals would be ½, 1, and 1. The common denominator would be 2, multiplying we get (122) for the plane. Note this would be the same as a parallel plane with a Weiss index of (1 ½ ½).
Earth Materials 5 Lab 2 5. Write the Miller indices for the following planes projected as lines onto a 3 = 0 (assume all are parallel to the a 3 or c). 6. In similar fashion to the examples above, draw the following lattices and their planes. This time plot and project on the a or a 1 =0 plane. a. Cubic C (011) b. Cubic I (00 1) c. Hexagonal (0101) d. Monoclinic (011) e. Orthorhombic (031) 7. Index the planes in the figure at the beginning of the orientation section. Three dimensions your personal computing experience We ve largely stuck with working in two of the three dimensions thus far. Some information is typically lost projecting three-dimensional objects on to two-dimensional surfaces. However, there are several computing programs that permit you to explore x, y, and z space to a fuller extent. Crystalmaker is one such program (and is the industry standard for crystal structure rendering, both organic and inorganic). Even with the academic discount, the program is still pricey. However, the demo version is powerful even with some of the features removed, and the price is right. Download and install the demo program for your OS from http://www.crystalmaker.com/crystalmaker/index.html. Next go to the class website s tools page, download the two structure models, halite.cif and rutile.cif. These are mineral structures for NaCl and TiO 2 as presented in Wyckoff s 1963 treatise on mineral structures. Go to file on the menu bar and select import to bring in the cif files.
Earth Materials 6 Lab 2 8. What is the Bravais lattice for each structure (Hint, in halite, unselect Vis for Cl; in rutile unselect Vis for O)? 9. On the menu bar, find model and select model options. Select the unit cell button and change the axial vectors to abc (to be consistent with the descriptions above). Select the Labels button and change the Type to fractional coordinates. 10. On the atom info menu, select Lbl for both elements on each model. 11. On the menu bar, find transform, select Lattice Plane and edit. For each structure, show the following planes (111), (210), (010), and (141). Print each and turn them in attached to your lab report (note, to save ink/toner, consider changing background to white also under model options). 12. Pick your favorite view of each crystal lattice. Select rendering from the menu bar and select stereo pair. Print out each model in this view and bring to the next lab session. 13. Spend at least half an hour getting comfortable with the program. We will use it again (and again).