PHY.K03 Solid State Physics Exercise Exam
29.04.2025

Useful constants and conversion factors
1 a.m.u. (atomic mass unit) = $1.66 \cdot 10^{-24}$ kg

(a) Write down the many-electron Hamiltonian (i.e. the total Hamiltonian) for carbonmonoxide ($Z_C = 6$, $M_C$ = 12 a.m.u., $Z_O = 8$,$M_O$ = 16 a.m.u. ) neglecting Slater's rules (i.e., without accouting for screening or an effective charge $Z_{eff}$. (20% of the points)

(b) How many molecular orbitals are filled in the ground state of negatively charged CO (oxygen has 8 protons and 8 neutrons, carbon 6 protons and 6 neutrons) (20% of the points)?

(c) The many electron Hamiltonian is difficult to solve. Write down an approximate many electron wavefunction for the ground state of neutral carbonmonoxide. This wavefunction should be properly antisymmetrized. (20% of the points)

(d) How could you determine the bond length of CO from the rotational spectrum? (20% of the points)

(e) How could you determine the effective spring constant of an CO₂ from the vibrational spectrum? (20% of the points)

Solution

(a)

$H_{\text{total}}= -\frac{\hbar}{2 M_C} \nabla_C^2 - \frac{\hbar}{2 M_O} \nabla_O^2 - \sum \limits_i^{14} \frac{\hbar^2}{2m}\nabla_i^2  - \sum \limits_i^{14} \frac{6e^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{R}_C|} - \sum \limits_i^{14} \frac{8e^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{R}_O|} +\sum \limits_{i \lt j} \frac{e^2}{4\pi \epsilon_0 |\vec{r}_i-\vec{r}_j|} + \frac{e^2}{4 \pi \epsilon_0} \frac{6 \cdot 8}{|\vec{R}_C - \vec{R}_O| } $.

(b) 15 spin-orbitals are filled. Alternatively: 7 orbitals are doubly filled, and one is filled with a single electron

(c) A many-electron wavefunction can be written as a Slater determinant.

Note: For the sake of space, the first two orbitals / 4 electrons were omitted. If you include those, make sure to also adjust the normalization constant accordingly.

$\psi\left(r_5,r_5, .., r_{14} \right) = \frac{1}{\sqrt{9!}}\begin{vmatrix} 
 3 \sigma(\vec{r}_5)\uparrow  & 3 \sigma(\vec{r}_5)\downarrow & 4 \sigma(\vec{r}_5)\uparrow  & 4 \sigma(\vec{r}_5)\downarrow & 1 \pi (\vec{r}_5)\uparrow  & 1 \pi (\vec{r}_5)\downarrow & 2 \pi (\vec{r}_5)\uparrow  & 2 \pi (\vec{r}_5)\downarrow  & 5 \sigma(\vec{r}_5)\uparrow  & 5 \sigma(\vec{r}_5)\downarrow \\
 3 \sigma(\vec{r}_6)\uparrow  & 3 \sigma(\vec{r}_6)\downarrow & 4 \sigma(\vec{r}_6)\uparrow  & 4 \sigma(\vec{r}_6)\downarrow & 1 \pi (\vec{r}_6)\uparrow  & 1 \pi (\vec{r}_6)\downarrow & 2 \pi (\vec{r}_6)\uparrow  & 2 \pi (\vec{r}_6)\downarrow  & 5 \sigma(\vec{r}_6)\uparrow  & 5 \sigma(\vec{r}_6)\downarrow  \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots

\end{vmatrix}
$

(d) The rotational energy levels are,

$E_J= \frac{\hbar^2}{2I}J(J+1)= BJ(J+1)$ .

The lines in the rotational spectrum have a spacing of $2B$. From $B$ we could calculate the moment of inertia $I=\frac{m_am_b}{m_a+m_b}r_0^2$. Since the mass of the oxygen and the carbon atom are known, $m_a = M_C$ andi $m_b = m_O$, we can calculate the bond length.

(e) Photons emitted due to a transition between vibrational levels have an energy $\hbar\omega = \hbar\sqrt{\frac{k}{m_r}}$ where $k$ is the effective spring constant and $m_r = \frac{m_O \cdot m_C}{M_O + M_C}$ is the reduced mass of the molecule. Since the mass of the atoms are known, the effective spring constant can be calculated.

Problem 2

The potential energy of a diatomic molecule as a function of the interatomic distance $r$ is,

$$U(r)=\frac{-A}{r^6} + \frac{B}{r^{12}}$$

The equilibrium interatomic distance was experimentally determined to be 2.5 Å and the binding energy was experimentally determined to be 55 meV.

(a) Sketch the bond potential and indicate the two experimentally determined quantities in the drawing. (50% of the points)

(b) Show that the equilibrium interatomic distance is $r_e=\left(\frac{2B}{A}\right)^{1/6}$. (20% of the points)

(c) Given the model potential above, what is likely to be the dominating bond archetype in this diatomic molecule? (30% of the points)

Solution

Problem 3

Magnesium Oxide (MgO) is an insulator that crystallize in a rock-salt form (a=4.121 Å).

(a) What is the Bravais Lattice? (30% of the total points)

(b) How many atoms there are in the basis? What are the positions of the atoms of the basis given in fractional coordinates of the conventional (cubic) unit cell? Please use the reference system indicated in the picture. (30% of the total points)

(c) Indicate the (100) plane in the sketch above (please use the reference system indicated in the picture) and draw a top-view of it. (20% of the total points)

(d) In the sketch above, mark [001] and [010] directions. (please use the reference system indicated in the picture) (20% of the total points)

Solution

PHY.K03 Solid State Physics Exercise Exam29.04.2025

PHY.K03 Solid State Physics Exercise Exam
29.04.2025