Section: __________________________________________
Solid State Structure, Week 1
Lab Report
1. A cube (see below) has _______ corners, _______ edges & _______ faces.
2. Structure A below shows how a unit cell may be drawn where one choice of unit cell is shown in bold lines. In Structures B, C and D below, draw the outline(s) of their 2-D unit cells (two-dimensional repeating patterns). A 2-D unit cell is a parallelogram that encloses a portion of the structure.
If the unit cell is moved in the X,Y-plane in directions parallel to its sides and in distance increments equal to the length of its sides, it has the property of duplicating the original structural pattern of circles as well as spaces between circles. Can a structure have more than one type of unit cell? ________
3. If the circle segments enclosed inside each of the bold-faced parallelograms shown below were cut out and taped together, how many whole circles could be constructed for each one of the patterns:
_____________________ _____________________ _____________________
4. Shown below is a 3-D unit cell for a structure of packed spheres. The center of each of 8 spheres is at a corner of the cube, and the part of each that lies in the interior of the cube is shown. If all of the sphere segments enclosed inside the unit cell could be glued together, how many whole spheres could be constructed?
number of whole spheres: ________
5. For each of the figures shown at the right, determine the number of corners and faces. Identify and name each as one of the regular geometric solids.
Number of corners: Number of faces: Name of the shape of this object:
a. How would you designate the sc stacking - aa, ab, abc, or some other?
b. If the radius of each atom in this cell is r, what is the equation that describes the volume of the cube generated in terms of r? (Note that the face of the cube is defined by the position of the rods and does not include the whole sphere.)
c. Draw the face of the extended structure and outline each unit cell.
d. To how many different cells does a corner atom which is exactly in the center of the extended structure belong?
e. How many corner spheres does a single unit cell possess?
f. If you consider the sharing of atoms by adjacent unit cells, what is the fractional contribution of a single corner atom to a particular unit cell?
g. What is the net number of atoms in a unit cell [e x f]?
h. Pick an interior sphere in the extended array. How many spheres are touching it? This is called the coordination number (CN) of the atoms in the sc structure.
i.What is the formula for the volume of a sphere with radius r?
j. Calculate the packing efficiency of a simple cubic unit cell (the % volume or space occupied by atomic material in the unit cell). Hint: Consider the net number of atoms per simple cubic unit cell (question h) the volume of one sphere (question j), and the volume of the cube (question b).
a. Draw the z diagrams for the layers.
b. What is the number of corner atoms in one unit cell?
c. What is the fractional contribution of each corner atom to the unit cell ?
d. What is the total number of atoms in the unit cell due to presence of atoms at the corners of the unit cell?
e. What is the number of body-centered atoms in one unit cell?
f. What is the fractional contribution of each body-centered atom to the unit cell?
g. What is the total number of atoms in the unit cell due to the presence of atoms at the body-center of the unit cell?
h. What is the total number of atoms in the unit cell?
i. Look at the stacking of the layers. How are they arranged if we call the layers a, b, c, etc.?
j.What is the coordination number of an atom at the corner of the unit cell?
k.What is the coordination number of the atom at the center of the unit cell?
l. If the radius of each atom in this cell is r, what is the formula for the volume of the cube generated in terms of the radius of the atom? (See diagrams below.)
m. Calculate the packing efficiency of the bcc cell, that is: find the volume occupied by the net number of spheres per unit cell if the radius of each sphere is r; then calculate the volume of the cube using r of the sphere to find the diagonal of the cube and the Pythagorean theorem.
a. What is the number of corner atoms in one unit cell ?
b. What is the fractional contribution of each corner atom to the unit cell?
c. What is the total number of atoms in the unit cell due to presence of atoms at the corners of the unit cell?
d. What is the number of face-centered atoms in one unit cell?
e. What is the fractional contribution of each face-centered atom to the unit cell?
f. What is the total number of atoms in the unit cell due to presence of atoms at the face-centers of the unit cell?
g. What is the total number of atoms in the unit cell?
h. Using a similar procedure to that applied in Part B above, calculate the packing efficiency of the face-centered cubic unit cell.
Parts A and B.
a. Compare the hexagonal and cubic close-packed structures.
b. Identify the layers which consist of close-packed spheres.
c. How do the arrangements of these close-packed layers differ in the hexagonal and cubic
close-packed structures?
d. Locate the interior sphere in the layer of seven next to the new top layer. For this interior sphere, determine the following:
No. of touching spheres in layer
below No. of touching spheres in same layer
No. of touching spheres in layer
above TOTAL Coordination number of the
interior sphere
Sphere packing that has this number of adjacent and touching nearest neighbors is referred to as close-packed. Non-close-packed structures will have lower coordination numbers.
C.
a. Are the two unit cells the identical?
b. If they are the same, how are they related? If they are different, what makes them different?
c. Is the face-centered cubic unit cell aba or abc layering? Explain.