Title: ETH_Sem5 / Solid_State_Intro / Woche_05

Q: [...] is characterized through:
- [...]
- [...]
A: Elastic scattering; \(q = k_i - k_f\) is a reciprocal lattice vector; \(\omega_{in} = \omega_{out}\)

Q: How to measure inelastic scattering?
A: For fixed \(q\) we plot intensity vs \(\omega = \omega_i - \omega_f\) (frequency/energy transfer)
This plot will have some distinct peaks.
We plot the peak positions vs varying \(q\) to obtain the dispersion curves

Q: Where are possible \(q\) at RT if we allow inelastic scattering?
A: Everywhere!

Q: How do we model lattice vibrations in a solid using Quantum Mechanics?
A: Every normal mode is considered as a harmonic oscilator with energies \(E_n = \hbar \omega(n+1/2)\)

Q: What is the idea behind phonons?
A: We take an ideal gas of quasiparticles, where each normal mode is a type of particle. And the occupation of the normal mode is the number of those particles.
We need to note that the particle number is not conserved

Q: Energy in most general phonon model
A: \(E=\sum_j\sum_{\vec{k}}\hbar\omega_j(\vec{k})\left(n_j(\vec{k}) + 1/2\right)\)

Q: What is conserved for phonons in a crystal?
A: Quasi-momentum (not real momentum!)
-> Momentum modulo \(\hbar\vec\tau\) with \(\vec\tau\) a lattice vector

Q: Definition: Specific Heat
A: \(C_V = (\dede{E}{T})_V\)

Q: Bose-Einstein Distribution:
A: \((\exp(\hbar\omega\beta)-1)^{-1}\)
Describes the distribution of bosons

Q: What are the assumptions for the einstein solid?
A: There is only one specific vibrational energy \(\varepsilon_E\)

Q: Heat capacity and expected energy for einstein solid:
\(C_V = \)[...]
\(\langle E\rangle = \)[...]
A: \(3Nk_B(\varepsilon\beta)^2 \frac{e^{\varepsilon\beta} }{(e^{\varepsilon\beta}-1)^2} \); \(\varepsilon_E (e^{-\varepsilon \beta}-1)^{-1} \cdot 3mN\)

Q: What is an Umklapp process?
A: A scattering process wherein a particle is scattered out of the first brioulin zone.
This causes a seeming violation of conservation of momentum (as we get projected back to the first brillouin zone)
It's the motivation for the introduction of quasimomentum.

Q: What is a monochromator, how does it work?
A: A single crystal with a strong Bragg peak. It can be irradiated with white xrays to yield a well defined wave vector \(k_i\) according to bragg scattering.

Q: Explain the 3-axis spectrometer
A: Using a monochromator the Xrays/Neutrons are selected for correct \(k_i\). The sample then scatters in all directions. We pick one direction where we filter again for correct \(k_f\) using an additional monochromator.

Q: Define DOS, how would you calculate it?
A: Density Of States: \(\nu(\omega)\)
Has the property that the number of independent normal modes of any type per energy/frequency interval \(d\omega\) is given by \(\nu(\omega)d\omega\)
It can be calculated by integrating a thin shell of a reciprocal space sphere of radius \(\omega\).
The trick is to realize the thickness of the layer is given as \(\frac{1}{|\nabla \omega(k)|}\)