Download Chrystal Structures Lab Experiment 1 Professor

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 (a0) 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 1023 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.
3 a0 =
4r
3
(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 3. Figure 2 is a representation of
the face centered cubic.
Figure 2 – Face Centered Cubic
a 0 = 2r 2
(3)
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 3 represents this complex structure.
4 Figure 3 – 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
1.633 times its width.
a 0 = 2r
(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
Height
FCC
HCP
0.761cm 0.929cm
A
0.621cm
C
1.017cm
Atoms/unit cell
2.000
4.000
Table 1 – Experimental Data
6.000
Sample Calculations:
Volume of atoms / unit cell of BCC using the volume of a sphere equation:
⎛4
3
3⎞
3
⎜ π * 0.761 cm ⎟ * 2 = .25cm
⎝3
⎠
Volume of BCC unit cell:
.7613 = .44cm 3
Atomic Packing Factor using Equation 1:
.25cm 3
= .567
.44cm 3
Lattice Constant using Equation 4:
4(.3105)
3
= .7170
6 Percent Difference:
⎛ .567 − .680 ⎞
⎜
⎟ * 100 = 16.23%
⎝ .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
Volume of
atoms in unit
cell
Volume of unit
cell
Calculated
APF
FCC
0.251cm3 0.502cm3 0.752cm3
0.441cm3 0.802cm3 1.019cm3
0.569
0.626
0.738
Known APF
0.680
0.740
Percent
Difference of
APF
16.32%
15.46%
Lattice
Constant
0.717
0.878
Percent
Difference of
Lattice
Constant
Table 2 – Calculated Results
0.740
7 HCP
0.22%
0.621
24.32%
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 1.382 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.
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