Calculate the maximum packing fraction of the unit cell volume that can be filled by hard spheres in the SC, BCC, FCC, Diamond and Hexagonal structures. Which structure most efficiently fills space?
- Calculate the maximum packing fraction of the unit cell volume that can be filled by hard spheres in the SC, BCC, FCC, Diamond and Hexagonal structures. Which structure most efficiently fills space?
SC: It is easy to determine that there is 1 lattice point per unit cell. The maximum radius that a hard sphere can have is 
, where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch each other. Hence the packing fraction is:
, where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch each other. Hence the packing fraction is:
B.C.C: It is easy to determine that there is 2 lattice points per unit cell. The maximum radius that a hard sphere can have is 
, where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch the body-centered sphere. Hence the packing fraction is:
, where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch the body-centered sphere. Hence the packing fraction is:
F.C.C: It is easy to determine that there is 4 lattice points per unit cell. The maximum radius that a hard sphere can have is 
, where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch the face-centered sphere. Hence the packing fraction is:
, where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch the face-centered sphere. Hence the packing fraction is:
Diamond: It is easy to determine that there is 8 lattice points per unit cell. The maximum radius that a hard sphere can have is 
, where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch the nearest neighbor sphere. Hence the packing fraction is:
, where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch the nearest neighbor sphere. Hence the packing fraction is:
Hexagonal: It is easy to determine that there is 3 lattice points per unit cell. If 
, then the maximum radius that a hard sphere can have is 
, where a is the lattice constant. The volume of the hexagonal unit cell if 
is 
. Hence the packing fraction is:
, then the maximum radius that a hard sphere can have is , where a is the lattice constant. The volume of the hexagonal unit cell if is . Hence the packing fraction is:
It is seen that the FCC structure most efficiently fills space.
- Calculate the volume density of Si atoms (number of atoms/cm3) given that the lattice constant of Si is 5.43 angstroms. Calculate the areal density of atoms (number/cm2) on the (100) plane and (111) plane.
Solution: Because Si has a Diamond structure, there are 8 Si atoms per conventional unit cell with a volume of 
. Hence the volume density is:
. Hence the volume density is:
For the areal density of atoms on the (100) plane, there are 2 Si atoms per 
area. Hence the areal density is:
area. Hence the areal density is:
For the areal density of atoms on the (111) plane, there are 2 Si atoms per 
area. Hence the areal density is:
area. Hence the areal density is:
- An engineer wants to grow GaxIn1-xAs on an InP substrate. What value of x (the fraction of Ga and the fraction of In) is needed so that he can grow a relatively thick film without defects occurring?
where the terms aGaAs = 5.6533 angstroms and aInAs = 6.0584 angstroms are the lattice constants of the two semiconductors. Also, aInP = 5.8686 angstroms)
Solution: The problem itself is fairly simple. All you have to do is choose the correct composition of GaxIn1-xAs that is lattice matched to the InP substrate. Hence we have the following condition:
This gives usand hence we need to grow
.
- The ratio of the lengths of the reciprocal lattice vectors for the S.C., B.C.C., F.C.C. structures are given below:
Ratio Table
|
S.C. Reciprocal Lattice
|
B.C.C. Reciprocal Lattice
|
F.C.C. Reciprocal Lattice
|
 |
1
|
1
|
1
|
 |  |  |  |
 |  |  |  |
 |
2
|  |
2
|
 |  |
2
|  |
 |  |  |  |
 |  |  |  |
 |
3
|  |  |
a. Now calculate the set of ratios for the Diamond structure. You will have to review the concept of the structure factor in my notes online in order to do this problem
Note: There are several ways to do this problem. One way that may be easier than other ways is to consider the diamond structure as a simple cubic crystal structure with a basis of eight atoms
Solution: You need to calculate the structure factor for this case. I will use the method that considers the Diamond structure a SC lattice with a basis of eight atoms. The eight atoms have the following displacement vectors
:
Note: There are several ways to do this problem. One way that may be easier than other ways is to consider the diamond structure as a simple cubic crystal structure with a basis of eight atoms
Solution: You need to calculate the structure factor for this case. I will use the method that considers the Diamond structure a SC lattice with a basis of eight atoms. The eight atoms have the following displacement vectors
:
Atom
|
FCC 1
|
FCC 2
|
1 and 5
|  |  |
2 and 6
|  |  |
3 and 7
|  |  |
4 and 8
|  |  |
The structure factor for 
:
:
The summation over 4 atoms is the structure factor for the FCC and the summation over 2 atoms is an additional structure factor caused by the fact that there for the diamond structure there are two atoms at every FCC lattice point.
For the FCC structure, the structure factor is:
If all the integers n1, n2, n3 are either all even or all odd then the structure factor is 
. If any integer is odd while the other two are even or if one is even while the other two are odd then 
.
. If any integer is odd while the other two are even or if one is even while the other two are odd then .
For the summation over two atoms, the structure factor is:
If 
, then 
. If 
, then 
. If 
, then 
. Let us compare the FCC and Diamond structure factors as a function of 
.
, then . If , then . If , then . Let us compare the FCC and Diamond structure factors as a function of . |
FCC
|
Diamond
|
0,0,0
|
4
|
4
|
1,0,0
|
0
|
0
|
1,1,0
|
0
|
0
|
1,1,1
|
4
|  |
2,0,0
|
4
|
0
|
2,1,0
|
0
|
0
|
2,1,1
|
0
|
0
|
2,2,0
|
4
|
8
|
2,2,1
|
0
|
0
|
3,0,0
|
0
|
0
|
3,1,0
|
0
|
0
|
3,1,1
|
4
|  |
2,2,2
|
4
|
0
|
We see that the reciprocal lattice of the FCC is a BCC lattice with a lattice constant of
. The reciprocal lattice of the diamond structure is similar the BCC structure but with several of the reciprocal lattice points vanishing including the 
and the 
reciprocal lattice point which are the BCC reciprocal lattice translation vectors that have the second and fifth shortest translation vectors (excluding 
). Hence our new ratio table with the Diamond structure data is:
Ratio Table
|
F.C.C. Lattice
(i.e. Reciprocal lattice of the BCC)
|
B.C.C. Lattice
(i.e. Reciprocal lattice of the FCC)
|
Diamond Reciprocal Lattice
|
 |
1
|
1
|
1
|
 |  =1.41 |  =1.15 |
Missing
|
 |  =1.73 |  =1.63 |  =1.63 |
 |
2
|  =1.91 |  =1.91 |
 |  =2.24 |
2
|
Missing
|
 |  =2.45 |  =2.30 |
---
|
 |  =2.65 |  =2.52 |
---
|
 |  =2.82 |  =2.58 |
---
|
b. Now armed with these ratios let us move on to the problem in Ashcroft and Mermin. Identify the F.C.C., B.C.C. and Diamond structure and calculate the lattice constant. 
Using the equation for the magnitude of the reciprocal lattice vector 
, we have:
, we have:
Sample A
 |  |
Sample B
 |  |
Sample C
 |  |
42.2
|
1
|
28.8
|
1
|
42.8
|
1
|
49.2
|
1.156
|
41.0
|
1.408
|
73.2
|
1.634
|
72.0
|
1.633
|
50.8
|
1.725
|
89.0
|
1.921
|
87.3
|
1.917
|
59.6
|
2
|
115.00
|
2.311
|
We compare the experimental ratios and the theoretical ratios and we see that Sample A has a FCC real space lattice, Sample B has a BCC real space lattice and Sample C has a Diamond real space lattice.
5. The Kronig-Penny model for a periodic potential leads to the concept of energy bandgaps. Let us approach the situation in a slightly different way by looking at a particle in a 1D infinite box of length d that has a small periodic perturbation in the bottom of the well (see figure below).
I want you to investigate a couple of results of this perturbation on the energy and wavefunctions of the electron states in this potential well.
- First solve for the normalized wavefunctions and their energies for the unperturbed infinite well of width d. Graph the energies of the states as a function of the wavevector
.
Solution: Using the Schrödinger equation, the boundary conditions and the normalization condition it is easy to solve for the electron wavefunctions and associated quantized energy levels:
n = integer
- Now introduce a small periodic perturbation at the bottom of the potential well such that the shape of the bottom of the well is now given by the equation:
whereand
is the strength of the periodic perturbation. We are now going to use time independent perturbation (TIP) theory to evaluate the effects on the electron states. To do this, you need to note that every state has a two-fold energy degeneracy because
, hence you have to use the form of TIP that can handle this degeneracy. I want you to do the following:
1. Calculate the first order energy corrections to the electron states.
Solution: You have to use the form of TIP that can handle degeneracy because. To do this, you have to calculate the following:
In the case of two-fold degeneracy that we have in this problem, we have to solve the following eigenvalue/eigenvector equation for each:
The matix elementsare easily calculated to be:
Hence the eigenvalue/eigenvector problem becomes:
The solutions to this is:
Thus, unless, the perturbation does not cause any effect. But at
, there is a first-order energy correction of
.
2. How are the wavefunctions changed? Are any of the wavefunctions hybridized?
Solution: When, the perturbation does not cause any first-order change in the wavefunctions. But at
, the wavefunctions are hybridized resulting in the new wavefunctions:
3. At the Brillouin zone boundaries of, what is the value of the energy gap that is produced from the two initially degenerate energy levels? That is, the two states with
initially had the same energy but now the states are changed slightly resulting in two new states with a slightly increased energy for one and a slightly decreased energy for the other. The difference between these two new states will be the energy bandgap.
Solution: The bandgap that is opened up is
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