Chrystal Structures Lab Experiment 1 Professor Greene Mech 496-02 Submitted: 4 February, 2009 Max Nielsen Trevor Nickerson Ben Allen Kushal Sherpa
Abstract: The study of materials science requires an understanding of the structural makeup of a solid on an atomic level. This laboratory experiment introduces the unit cell and several relevant properties. To develop a spatial understanding, models of unit cells shall be constructed and compared against ideal, calculated models. The results yield circa 15% difference in APF, with the HCP only at 0.22%. Objectives: Build three unit cells of the most common crystalline structures: BCC, FCC, HCP Calculate the atomic packing factor for each model and contrast against scientifically accepted values Introduction: A unit cell, also known as the hard sphere model, utilizes the fewest number of atoms while maintaining the structural makeup of a material. It is effectively a representation of the physical arrangement of atoms in a solid. The unit cell is also defined by its ability to be stacked to form larger and larger blocks of material. This model assumes that the atom is akin to a sphere, and therefore the unit cell is often tightly packed. Several properties have been developed to describe the unit cell and consequently the structural makeup of the material they represent. The atomic packing constant, for example, is the ratio of atom over void by volume, as described by equation 1. Theoretical density is a figure which defines the density of a pure, perfectly packed chunk of material. Finally, the lattice 2
constant (a 0 ) describes the length of the side of a unit cell, usually defined as a multiple of the radius of an atom in the unit cell. APC = volume of atom/volume of unit cell (1) Where Avogadro s # = 6.022 x 10 2 atoms/mole Three common arrangements can describe most materials: The Body Centric Cubic (BCC), Face Centered Cubic (FCC), and the Hexagonal Close Pack (HCP). The Body Centric Cubic unit cell is the smallest and simplest of the common configurations. The center of a sphere resides at each of the corners of a cube and a ninth sphere of equal size fills the gap between them. Figure 1 demonstrates this configuration. Figure 1 Body Centric Cubic Only two atoms make up this unit cell, making it the smallest of the unit cells in this lab. Atomic contact, where atoms make contact with one another, occurs diagonally. Only through the center of each unit cell do the atoms anchor together. Despite its seemingly compact dimensions, this unit cell is only 68% atom by volume. Thus its atomic packing factor is 0.68. The lattice constant of a body centric cubic is described by equation 2.
4r a 0 = (2) The Face Centered Cubic is roughly similar to the body centered cubic, in which six spheres are centered at each corner of a cube. That is where the similarities end however, as a sphere resides on each face between its four adjacent corner-centered spheres. This is an extremely efficient packing structure, as the APF is 0.74: the highest ratio possible. Four atoms make up one unit. The lattice constant is described by equation. Figure 2 is a representation of the face centered cubic. Figure 2 Face Centered Cubic a 0 = 2r 2 () Tied for the highest APF at 0.74 is the hexagonal close pack. This arrangement is unlike the two former because it is, as the name implies, hexagonal in shape. Six spheres surround a central sphere in the first and third of three layers. Three spheres triangulate in the second layer between the first and third. Figure represents this complex structure. 4
Figure Hexagonal Close Pack The hexagonal close pack unit cell demonstrates its complexity with 6 atoms per unit cell. Atomic contact is made across all adjacent face atoms. Because of this, the lattice constant is twice the radius, demonstrated by equation 4. Ideally, the hexagonal close pack cell height is 1.6 times its width. a 2r 0 = (4) Procedure: 40 marbles were gathered to create all three modules. Using the lab handout as a guide and the angled jigs, all three of the models where created. Using a caliper, the sides of the structures where measured, from the top of one marble to the bottom on the corner of the model, and the atomic packing factor was calculated using the equation above. 5
Experimental Data: After the three different structures were made, measurements were taken of the size of the models. The measurements were recorded and are shown in Table 1 below. BBC FCC HCP Height 0.761cm 0.929cm A C 0.621cm 1.017cm Atoms/unit cell 2.000 4.000 6.000 Table 1 Experimental Data Sample Calculations: Volume of atoms / unit cell of BCC using the volume of a sphere equation: Volume of BCC unit cell: 4 π *0.761 Atomic Packing Factor using Equation 1: *2 cm =. 761 =.44cm.25cm.25cm.44cm =.567 Lattice Constant using Equation 4: 4(.105) =.7170 6
Percent Difference:.567.680 *100 = 16.2%.680 Experimental Results: One of the main points of this laboratory experiment is to calculate the atomic packing factor, and compare that of the models that were made to the published values for the molecule. In order to do that, the volume of the atoms and the volume of the unit cells are calculated. Table 2 below shows the calculations that were made using the equations and the data previously shown. BBC FCC HCP Volume of atoms in unit cell 0.251cm 0.502cm 0.752cm Volume of unit cell 0.441cm 0.802cm 1.019cm Calculated APF 0.569 0.626 0.78 Known APF 0.680 0.740 0.740 Percent Difference of APF 16.2% 15.46% 0.22% Lattice Constant 0.717 0.878 0.621 Percent Difference of Lattice Constant 24.2% Table 2 Calculated Results 7
Diagram: Figure 4 below shows the models that were created and used to make measurements for this laboratory experiment. Figure 4 Marble Structure Models Discussion of Results: The calculated atomic packing factor yielded reasonable results even though the percent errors where above 10%. The ideal atomic packing factor was not achieved because of the crude material used to create the structures resulting in the inability to create the angles needed. There is a great chance of inaccuracy of making the models due to inconsistencies in the size of the marbles used (they all varied in size), and the angles they are placed at. If all the side where measured and an average of measurements was used for the calculations, the percent error may have been reduced. There also could have been simple measurement errors that could have increased the percent error. If the unit cube of the BBC model where to be expanded to fit all of the atoms, the cube would need to be 1.82 cm. This was found simply by adding the radius of the marble to each side of the cube. 8
Conclusion: Making sample models of common metallic crystal structures such as BCC, FCC and the HCP crystal structure gave a great visual representation. Along with experimental examples of the similarities and differences among these three structures. From the sample models, the APF can also be calculated relatively accurately. The HCP model was very close to the theoretical value and FCC and BCC was little over 10%. Physical structures of FCC and BCC crystal structures are very similar mainly because of their shape. It can be seen from the data that the atomic radius of the HCP crystal structure would have the largest volume because the unit cell contains more atoms. Some of the common metals with FCC crystal structure would be Gold, Silver and Copper. Some common metals with the HCP structure would be Titanium and Cobalt and lastly Iron and Chromium for the BCC crystal structure. From the periodic table, metals with like properties are set together. The crystal structures of metals, the can have a correlation with the mechanical, electrical and thermal properties. For example, copper gold and silver are good electric conductors, and all have an FCC crystal structure. 9