Title: ETH_Sem5 / Solid_State_Intro / Woche_02

Q: Definition of the primitive translation vectors in the reciprocal lattice :
A: \[\vec{a^*} = \frac{2 \pi}{V} \vec{b} \times \vec{c} \]
\[\vec{b^*} = \frac{2 \pi}{V} \vec{c} \times \vec{a}\]
\[\vec{c^*} = \frac{2 \pi}{V} \vec{a} \times \vec{b}\]
\[V = \vec{a} \cdot (\vec{b}\times \vec{c})\]

Q: Describe the construction of a Wigner-Seitz  primitive unit cell.
A: - Pick starting node and connect it to neighbouring nodes.
- For each segment draw a perpendicular plane through its middle.
- The shape enclosed is the Wigner-Seitz unit cell.

Q: Dot product of real-space vector \(\vec{r}\) and reciprocal-space vector \(\vec{q}\)
A: \[\vec{r} \cdot \vec{q} = 2 \pi (hx + ky + lz)\]

Q: What are the Miller indexes for a plane that runs through points \(3 \vec{a}, \; -2\vec{b}, \;\; 2\vec{c}\) ?
A: \[h = \frac{\text{lcm}(n_x, n_y,n_z)}{n_x}, \;\; k = \frac{\text{lcm}(n_x, n_y,n_z)}{n_y}, \;\; l = \frac{\text{lcm}(n_x, n_y,n_z)}{n_y}\]\[h = \frac{6}{3}, \;\;k = \frac{6}{-2}, \;\;l = \frac{6}{2}\]
\[(2, \overline{3}, 3)\]

Q: Formula for Miller indexes
A: lcm = least common multiple
\[h = \frac{\text{lcm}(n_x, n_y,n_z)}{n_x}, \;\; k = \frac{\text{lcm}(n_x, n_y,n_z)}{n_y}, \;\; l = \frac{\text{lcm}(n_x, n_y,n_z)}{n_z}\]

Q: Definition of the Brillouin zone:
A: The Wigner-Seitz primitive unit cell transformed in the reciprocal lattice

Q: Definition of the scattering vector:  [...]
How does the intensity change with the scattering vector?
A: \(\vec{q} = \vec{k_i} - \vec{k_f}\)
Back Extra: \(I(\vec{q}) \propto \bigl| \int d\vec{r} \; \exp(i\vec{q}\vec{r}) \; \rho(\vec{r}) \;\bigl|^2 \)

Q: What are the solutions of \(I(\vec{q}) \propto \bigl| \int d\vec{r} \; \exp(i\vec{q}\vec{r}) \; \rho(\vec{r}) \;\bigl|^2 \) in the case of a periodic scattering density \(\rho(\vec{r}) = \rho(\vec{r}+\vec{T})\)?
A: \(I(\vec{q})= \begin{cases} \displaystyle \mid F(\vec{q}) \mid^2, \; \text{if} \; \vec{q} \; \text{is a reciprocal lattice vector }\\ 0, \; \text{otherwise} \end{cases} \)
where \(F(\vec{q})\) is the structure factor
\(F(\vec{q}) = \int_{\text{unitcell}}^{} d\vec{R} \; \exp(i\vec{q}\vec{R}) \; \rho(\vec{R}) \)

Q: State the phase problem:
A: The measured intensity contains the absolute value of the Fourier components, while the phase remains undetermined. This makes a simple inverse-Fourier transform from the reciprocal to the real lattice impossible.