61 lines
3.7 KiB
TeX
61 lines
3.7 KiB
TeX
\Part[
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\eng{Condensed matter physics}
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\ger{Festkörperphysik}
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]{cm}
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\TODO{Bonds, hybridized orbitals}
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\TODO{Lattice vibrations, van hove singularities, debye frequency}
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\begin{formula}{dos}
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\desc{Density of states (DOS)}{}{\QtyRef{volume}, $N$ number of energy levels, \QtyRef{energy}}
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\desc[german]{Zustandsdichte (DOS)}{}{\QtyRef{volume}, $N$ Anzahl der Energieniveaus, \QtyRef{energy}}
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\eq{D(E) = \frac{1}{V}\sum_{i=1}^{N} \delta(E-E(\vec{k_i}))}
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\end{formula}
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\begin{formula}{dos_parabolic}
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\desc{Density of states for parabolic dispersion}{Applies to \fqSecRef{cm:egas}}{}
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\desc[german]{Zustandsdichte für parabolische Dispersion}{Bei \fqSecRef{cm:egas}}{}
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\eq{
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D_1(E) &= \frac{1}{2\sqrt{c_k(E-E_0)}} && (\text{1D}) \\
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D_2(E) &= \frac{\pi}{2c_k} && (\text{2D}) \\
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D_3(E) &= \pi \sqrt{\frac{E-E_0}{c_k^3}}&& (\text{3D})
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}
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\end{formula}
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\Section[
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\eng{Lattice vibrations}
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\ger{Gitterschwingungen}
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]{vib}
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\begin{formula}{dispersion_1atom_basis}
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\desc{Phonon dispersion of a lattice with a one-atom basis}{same as the dispersion of a linear chain}{$C_n$ force constants between layer $s$ and $s+n$, $M$ \qtyRef{mass} of the reference atom, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u$ Ansatz for the atom displacement}
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\desc[german]{Phonondispersion eines Gitters mit zweiatomiger Basis}{gleich der Dispersion einer linearen Kette}{$C_n$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M$ \qtyRef{mass} des Referenzatoms, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u$ Ansatz für die Atomauslenkung}
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\eq{
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\omega^2 = \frac{4C_1}{M}\left[\sin^2 \left(\frac{qa}{2}\right) + \frac{C2}{C1} \sin^2(qa)\right] \\
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\intertext{\GT{with}}
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u_{s+n} = U\e^{-i \left[\omega t - q(s+n)a \right]}
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}
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\fig{img/cm_phonon_dispersion_one_atom_basis.pdf}
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\end{formula}
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\TODO{Plots}
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\begin{formula}{dispersion_2atom_basis}
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\desc{Phonon dispersion of a lattice with a two-atom basis}{}{$C$ force constant between layers, $M_i$ \qtyRef{mass} of the basis atoms, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u, v$ Ansatz for the displacement of basis atom 1 and 2, respectively}
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\desc[german]{Phonondispersion eines Gitters mit einatomiger Basis}{}{$C$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M_i$ \qtyRef{mass} der Basisatome, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u, v$ jeweils Ansatz für die Atomauslenkung des Basisatoms 1 und 2}
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\eq{
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\omega^2_{\txa,\txo} = C \left(\frac{1}{M_1}+\frac{1}{M_2}\right) \mp C \sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2 - \frac{4}{M_1M_2} \sin^2 \left(\frac{qa}{2}\right)}
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\intertext{\GT{with}}
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u_{s} = U\e^{-i \left(\omega t - qsa \right)}, \quad
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v_{s} = V\e^{-i \left(\omega t - qsa \right)}
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}
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\fig{img/cm_phonon_dispersion_two_atom_basis.pdf}
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\end{formula}
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\Subsection[
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\eng{Debye model}
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\ger{Debye-Modell}
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]{debye}
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\begin{ttext}
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\eng{Atoms behave like coupled \hyperref[sec:qm:hosc]{quantum harmonic oscillators}. The finite sample size leads to periodic boundary conditio. The finite sample size leads to periodic boundary conditions for the vibrations.}
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\ger{Atome verhalten sich wie gekoppelte \hyperref[sec:qm:hosc]{quantenmechanische harmonische Oszillatoren}. Die endliche Ausdehnung des Körpers führt zu periodischen Randbedingungen. }
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\end{ttext}
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