Title: ETH_Sem5 / Solid_State_Intro / Woche_04

Q: Definition of harmonic approximation
Why do 1st order terms vanish?
A: \(\Phi \approx \Phi_0 +\frac{1}{2} \sum_{\vec{i},j,\alpha} \sum_{\vec{i'},j',\beta} \frac{\partial^2\Phi}{\partial\rho_{\vec{i},j,\alpha} \;\partial\rho_{\vec{i'},j',\beta}}\)
Is a 2nd order taylor expansion  of the potential energy around the equilibrium position.
Because the equilibrium state must be a minimum of the energy functional, the 1st order terms are all zero.
\(\frac{\partial\Phi}{\partial\rho_{\vec{i},j,\alpha}} = 0 \)

Q: General number of distinct normal modes for any given wave vector.
A: At most 3*m distinct normal modes. But due to symmetry some may have equal frequencies and are thus degenerate.

Q: There are precisely three normal modes whose frequency goes to [...] as \(\vec{q}\) goes to zero.
These modes are called [...].
A: zero; acoustic modes

Q: There are \(3*m -3\) normal modes which have a [...] frequency as \(\vec{q}\) goes to 0.
These modes are called [...].
A: non-zero; optical modes

Q: What is described by the dispersion relation
A: The wave vector dependence of the frequency for a given normal mode.
\(\omega(\vec{q}) \equiv \omega_{\vec{q}}\)

Q: Due to [...] , only wave vectors within [...] define physically distinct normal modes.
A: the periodicity of the crystal; the first Brillouin zone

Q: In a sample of N unit cells there are precisely [...] wave vectors allowed in the first Brillouin zone.
Each such wave vector occupies a reciprocal space volume of [...]
A: N equally spaced; \(\frac{(2\pi)^3}{V}\)

Q: How is the phonon dispersion of measured? Explain using graphs:
A: - Monochromatic incident beam with \(\vec{k_i}\) aimed at sample.
- Measure intensity vs. scattering angle ( as for diffraction )
- Additionaly measure frequency of scattered wave \(\vec{k_f}\)
- At a fixed momentum transfer \(\vec{q} = \vec{k_i} - \vec{k_f}\) plot intensity vs. frequency transfer \(\omega := \omega_i - \omega_f\)
- Position of intensity maxima gives frequency of corresponding mode \(\omega_\vec{q}\)
- Repeat for all \(\vec{q}\) to map out dispersion curves \(\omega(\vec{q})\)