Title: ETH_Sem5 / Solid_State_Intro / Woche_03

Q: Typical lattice constant?
A: 1 nm

Q: What are the bounds for a wavevector to perform scattering?
A: at least \(\pi/a\)

Q: Wave vector transfer in scattering
A: \(|q| = 4\pi \sin \theta/\lambda\)

Q: Advantages/Disadvantage of Powder XRD
A: Advantage:  all Bragg peaks are measured in a single intensity-vs.-2θ scan .
Disadvantage: different Bragg peaks with similar wave numbers will overlap

Q: What is Laue XRD?
and what are the advantages/disadvantages?
A: Scattering with a white X-Ray beam and omidirectional detection of scattering.
Advantage: Sample does not need to be rotated and all peaks are measured at the same time
Disadvantage: Low power per peak

Q: Derivation VdW potential
A: - Approximate dipole fluctuation as \(\langle p^2 \rangle \approx e^2r_0^2\)
- \(E \approx k_e p/R^3\)
- \(U \approx -p_{ind}\cdot E = -k_e \frac{p p_{ind}}{R^3} = -k_e e^2 r_0^5 \frac{1}{R^6}\)

Q: Motivation for Lennard-Jones potential
A: Attractive \(R^6 \) VdW term and repulsive \(R^{12}\) electron confinement term

Q: Formula Lennard-Jones potential
A: \(U(R) = 4 U_0 \left[ (\frac{\sigma}{R})^{12} - (\frac{\sigma}{R})^6 \right]\)

Q: Cohesion energy of ionic crystal using Madelung's constant
A: \(U_M = - \frac{e^2 \alpha}{4\pi\varepsilon_0 R_{opt}}\)

Q: Typical Madelung's constants
A: - NaCl: \(\alpha = 1.747...\)
- CsCl: \(\alpha = 1.76...\)

Q: How to fix boiling point of Ionic Crystals?
A: Realize that cohesion energy is used for atomisation and we can get lower boiling point by subtracting energy due to ionisation

Q: Ground state of hydrogen atom
A: \(\psi_0 = 2 e^{-r/r_B}\)
with
\(r_B = \frac{4\pi \varepsilon \hbar^2}{m_ee^2} = 5.29\cdot10^{-11} m\)
the Boltzman radius

Q: Energy eigenstates of \(H_2^+ \) molecule
A: \(\psi_+ = (2 + 2S)^{-1/2} (\psi_1 + \psi_2)\)
\(\psi_- = (2 - 2S)^{-1/2} (\psi_1 - \psi_2)\)
with the overlap integral
\(S = \braket{\psi_1|\psi_2}\)

Q: Hamiltonian of \(H_2^+\) molecule in eigenbasis
A: \(diag(\frac{1}{1+S} [E_0(1+S) - \delta - \Delta], \frac{1}{1-S} [E_0(1-S) - \delta + \Delta])\)
With \(\delta = e^2 /4\pi|R_1-R_2|\)

Orbital picture for \(H_2^+\) molecule