Problem 1
(a) How would you calculate the rotational energy levels of CH4? How could you measure these energy levels in an experiment?
(b) How many molecular orbitals are filled in the ground state of CH4?
(c) How would you calculate the energy that is released when CH4 burns in O2 to form CO2 and H2O?
(a) How would you calculate the rotational energy levels of NH3? How could you measure these energy levels in an experiment?
(b) How many molecular orbitals are filled in the ground state of NH3?
(c) How would you calculate the energy that is released when NH3 burns in O2 to form NO2 and H2O?
Problem 2
A diffraction experiment is performed on a single crystal of Ni using electrons with an energy of 100 eV. The netplanes are indexed using the unit vectors of the conventional unit cell. This means that even though Ni is fcc, it is considered to be simple cubic with four atoms in the basis. The lattice constant is 3.52 Å.
(a) What is the length of the $\vec{G}_{120}$ reciprocal lattice vector?
(b) At what angle $2\theta$ would a diffraction peak appear for the 120 reflection?
(c) The atomic form factor is of the 120 reflection is $f_{120}$. What is the structure factor?
A diffraction experiment is performed on a single crystal of Na using electrons with an energy of 100 eV. The netplanes are indexed using the unit vectors of the conventional unit cell. This means that even though Na is bcc, it is considered to be simple cubic with two atoms in the basis. The lattice constant is 4.29 Å.
(a) What is the length of the $\vec{G}_{110}$ reciprocal lattice vector?
(b) At what angle $2\theta$ would a diffraction peak appear for the 110 reflection?
(c) The atomic form factor is of the 110 reflection is $f_{110}$. What is the structure factor?
Problem 3
An indirect bandgap semiconductor has an zincblend crystal structure and a band gap of 1.4 eV. The effective mass is $m_e$ for the electrons, $2m_e$ for the light holes, and $4m_e$ for the heavy holes. Here $m_e$ is the mass of a free electron. The maximum of the valence band is at $\Gamma$. The minimum of the conduction band is along $\Gamma - L$.
(a) Plot the band structure ($E$ vs. $k$) along $L-\Gamma - X$ indicating the position of the chemical potential for an intrinsic semiconductor.
(b) Plot the electronic density of states and the phonon density of states.
(c) Plot the resistivity of this semiconductor as a function of temperature.
A direct bandgap semiconductor has an wurzite crystal structure and a band gap of 2.3 eV. The effective mass is $m_e$ for the electrons, $m_e$ for the light holes, and $3m_e$ for the heavy holes. Here $m_e$ is the mass of a free electron. The maximum of the valence band is at $\Gamma$. The minimum of the conduction band is along $\Gamma - L$.
(a) Plot the band structure ($E$ vs. $k$) along $K - \Gamma -M$ indicating the position of the chemical potential for an intrinsic semiconductor.
(b) Plot the electronic density of states and the phonon density of states.
(c) Plot the resistivity of this semiconductor as a function of temperature.
Quantity | Symbol | Value | Units | |
| electron charge | e | 1.60217733 × 10-19 | C | |
| speed of light | c | 2.99792458 × 108 | m/s | |
| Planck's constant | h | 6.6260755 × 10-34 | J s | |
| reduced Planck's constant | $\hbar$ | 1.05457266 × 10-34 | J s | |
| Boltzmann's constant | kB | 1.380658 × 10-23 | J/K | |
| electron mass | me | 9.1093897 × 10-31 | kg | |
| Stefan-Boltzmann constant | σ | 5.67051 × 10-8 | W m-2 K-4 | |
| Bohr radius | a0 | 0.529177249 × 10-10 | m | |
| atomic mass constant | mu | 1.6605402 × 10-27 | kg | |
| permeability of vacuum | μ0 | 4π × 10-7 | N A-2 | |
| permittivity of vacuum | ε0 | 8.854187817 × 10-12 | F m-1 | |
| Avogado's constant | NA | 6.0221367 × 1023 | mol-1 |