diff --git a/src/cm/cm.tex b/src/cm/cm.tex index 6f5f2fb..83a2315 100644 --- a/src/cm/cm.tex +++ b/src/cm/cm.tex @@ -20,54 +20,3 @@ } \end{formula} - \Section[ - \eng{Lattice vibrations} - \ger{Gitterschwingungen} - ]{vib} - - \begin{formula}{dispersion_1atom_basis} - \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} - \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} - \begin{gather} - \omega^2 = \frac{4C_1}{M}\left[\sin^2 \left(\frac{qa}{2}\right) + \frac{C2}{C1} \sin^2(qa)\right] \\ - \intertext{\GT{with}} - u_{s+n} = U\e^{-i \left[\omega t - q(s+n)a \right]} - \end{gather} - \newFormulaEntry - \fig{img/cm_phonon_dispersion_one_atom_basis.pdf} - \end{formula} - \begin{formula}{dispersion_2atom_basis} - \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} - \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} - \begin{gather} - \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)} - \intertext{\GT{with}} - u_{s} = U\e^{-i \left(\omega t - qsa \right)}, \quad - v_{s} = V\e^{-i \left(\omega t - qsa \right)} - \end{gather} - \newFormulaEntry - \fig{img/cm_phonon_dispersion_two_atom_basis.pdf} - \end{formula} - - \begin{formula}{branches} - \desc{Vibration branches}{}{} - \desc[german]{Vibrationsmoden}{}{} - \ttxt{\eng{ - \textbf{Acoustic}: 3 modes (1 longitudinal, 2 transversal), the two basis atoms oscillate in phase. - \\\textbf{Optical}: 3 modes, the two basis atoms oscillate in opposition. A dipole moment is created that can couple to photons. - }\ger{ - \textbf{Akustisch}: 3 Moden (1 longitudinal, 2 transversal), die zwei Basisatome schwingen in Phase. - \\ \textbf{Optisch}: 3 Moden, die zwei Basisatome schwingen gegenphasig. Das dadurch entstehende Dipolmoment erlaubt die Wechselwirkung mit Photonen. - }} - \end{formula} - - \Subsection[ - \eng{Debye model} - \ger{Debye-Modell} - ]{debye} - \begin{ttext} - \eng{Atoms behave like coupled \fRef[quantum harmonic oscillators]{sec:qm:hosc}. The finite sample size leads to periodic boundary conditio. The finite sample size leads to periodic boundary conditions for the vibrations.} - \ger{Atome verhalten sich wie gekoppelte \fRef[quantenmechanische harmonische Oszillatoren]{sec:qm:hosc}. Die endliche Ausdehnung des Körpers führt zu periodischen Randbedingungen. } - \end{ttext} - - diff --git a/src/cm/crystal.tex b/src/cm/crystal.tex index 3b8c018..60cf5a0 100644 --- a/src/cm/crystal.tex +++ b/src/cm/crystal.tex @@ -17,7 +17,7 @@ \eng[bravais_lattices]{Bravais lattices} \ger[bravais_lattices]{Bravais Gitter} - \newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img/bravais/#1.pdf}\end{center}} + \newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img_static/bravais/#1.pdf}\end{center}} \renewcommand\tabularxcolumn[1]{m{#1}} \newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X} @@ -170,18 +170,34 @@ \begin{formula}{zincblende} \desc{Zincblende lattice}{}{} \desc[german]{Zinkblende-Struktur}{}{} - \ttxt{ - \includegraphics[width=0.5\textwidth]{img/cm_zincblende.png} - \eng{Like \fRef{:::diamond} but with different species on each basis} - \ger{Wie \fRef{:::diamond} aber mit unterschiedlichen Spezies auf den Basen} + \fsplit{ + \centering + \includegraphics[width=0.9\textwidth]{img/cm_crystal_zincblende.png} + }{ + \ttxt{ + \eng{Like \fRef{:::diamond} but with different species on each basis} + \ger{Wie \fRef{:::diamond} aber mit unterschiedlichen Spezies auf den Basen} + } + } + \end{formula} + \begin{formula}{rocksalt} + \desc{Rocksalt structure}{\elRef{Na}\elRef{Cl}}{} + \desc[german]{Kochsalz-Struktur}{}{} + \fsplit{ + \centering + \includegraphics[width=0.9\textwidth]{img/cm_crystal_NaCl.png} + }{ + } \end{formula} \begin{formula}{wurtzite} \desc{Wurtzite structure}{hP4}{} \desc[german]{Wurtzite-Struktur}{hP4}{} - \ttxt{ - \includegraphics[width=0.5\textwidth]{img/cm_wurtzite.png} - \TODO{Placeholder} + \fsplit{ + \centering + \includegraphics[width=0.9\textwidth]{img/cm_crystal_wurtzite.png} + }{ + } \end{formula} diff --git a/src/cm/superconductivity.tex b/src/cm/superconductivity.tex index a432a74..4f13dbd 100644 --- a/src/cm/superconductivity.tex +++ b/src/cm/superconductivity.tex @@ -27,6 +27,7 @@ Has a single critical magnetic field, $\Bcth$. \\$B < \Bcth$: \fRef{:::meissner_effect} \\$B > \Bcth$: Normal conductor + \\ Very small usable current density because current only flows within the \fRef{cm:super:london:penetration_depth} of the surface. }} \end{formula} @@ -38,6 +39,7 @@ \\$B < B_\text{c1}$: \fRef{:::meissner_effect} \\$B_\text{c1} < B < B_\text{c2}$: \fRef{:::shubnikov_phase} \\$B > B_\text{c2}$: Normal conductor + \\ In \fRef{:::shubnikov_phase} larger usable current density because current flows within the \fRef{cm:super:london:penetration_depth} of the surface and the penetrating flux lines. }} \end{formula} @@ -75,7 +77,9 @@ \desc{Shubnikov phase}{in \fRef{::type2}}{} \desc[german]{Shubnikov-Phase}{}{} \ttxt{\eng{ - + Mixed phase in which some magnetic flux penetrates the superconductor. + }\ger{ + Gemischte Phase in der der Supraleiter teilweise von magnetischem Fluss durchdrungen werden kann. }} \end{formula} @@ -92,9 +96,6 @@ \eng{London Theory} \ger{London-Theorie} ]{london} - \begin{ttext} - \end{ttext} - \begin{formula}{description} \desc{Description}{}{} \desc[german]{Beschreibung}{}{} @@ -148,7 +149,27 @@ \eq{\lambda_\txL(T) = \lambda_\txL(0) \frac{1}{\sqrt{1- \left(\frac{T}{T_\txc}\right)^4}}} \end{formula} - \TODO{macroscopic wavefunction approach, energy-phase relation, current-phase relation} + \Subsubsection[ + \eng{Macroscopic wavefunction} + \ger{Makroskopische Wellenfunktion} + ]{macro} + \begin{formula}{ansatz} + \desc{Ansatz}{}{} + \desc[german]{Ansatz}{}{} + \ttxt{\eng{Alternative derivation of London equations by assuming a macroscopic wavefunction which is uniform in space}\ger{Alternative Herleitung der London-Gleichungen durch Annahme einer makroskopischen Wellenfunktion, welche nicht Ortsabhängig ist}} + \eq{\Psi(\vecr,t) = \Psi_0(\vecr,t) \e^{\theta(\vecr,t)}} + \end{formula} + \begin{formula}{energy-phase_relation} + \desc{Energy-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, $\phi_\text{el}$ \qtyRef{electric_scalar_potential}, \QtyRef{chemical_potential}} + \desc[german]{Energie-Phase Beziehung}{}{} + \eq{\hbar \pdv{\theta(\vecr,t)}{t} = - \left(\frac{m_\txs}{n_\txs^2 q_\txs^2} \vecj_\txs^2(\vecr,t) + q_\txs\phi_\text{el}(\vecr,t) + \mu(\vecr,t)\right)} + \end{formula} + \begin{formula}{current-phase_relation} + \desc{Current-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, \QtyRef{magnetic_vector_potential}} + \desc[german]{Strom-Phase Beziehung}{}{} + \eq{\vecj_\txs(\vecr,t) = \frac{q_\txs^2 n_\txs(\vecr,t)}{m_\txs} \left(\frac{\hbar}{q_\txs} \Grad\theta(\vecr,t) - \vecA(\vecr,t)\right) } + \end{formula} + \Subsubsection[ \eng{Josephson Effect} @@ -167,8 +188,8 @@ \end{formula} \begin{formula}{coupling_energy} - \desc{Josephson coupling energy}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, \fRef[critical current density]{::1st_relation}, $\phi$ phase differnce accross junction} - \desc[german]{Josephson}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, \fRef[kritische Stromdichte]{::1st_relation}, $\phi$ Phasendifferenz zwischen den Supraleitern} + \desc{Josephson coupling energy}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[critical current density]{::1st_relation}, $\phi$ phase differnce accross junction} + \desc[german]{Josephson}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[kritische Stromdichte]{::1st_relation}, $\phi$ Phasendifferenz zwischen den Supraleitern} \eq{\frac{E_\txJ}{A} = \frac{\Phi_0 \vecj_\txc}{2\pi}(1-\cos\phi)} \end{formula} @@ -217,7 +238,7 @@ % \desc[german]{}{}{} \begin{multline} g_\txs = g_\txn + \alpha \abs{\Psi}^2 + \frac{1}{2}\beta \abs{\Psi}^4 + - \\ \frac{1}{2\mu_0}(\vecB_\text{ext} -\vecB_\text{inside})^2 + \frac{1}{2m_\txs} \abs{ \left(\frac{\hbar}{t}\Grad - q_\txs \vecA\right)\Psi}^2 + \dots + \\ \frac{1}{2\mu_0}(\vecB_\text{ext} -\vecB_\text{inside})^2 + \frac{1}{2m_\txs} \abs{ \left(-\I\hbar\Grad - q_\txs \vecA\right)\Psi}^2 + \dots \end{multline} \end{formula} @@ -254,9 +275,9 @@ \end{formula} \begin{formula}{boundary_energy} - \desc{Boundary energy}{}{$\Delta E_\text{boundary}$ \TODO{TODO}, $\Delta E_\text{cond}$ \fRef{:::condensation_energy}} - \desc[german]{Grenzflächenenergie}{}{} - \eq{\Delta E_\text{boundary} = \Delta E_\text{con} - \Delta E_\txB = (\xi_\gl - \lambda) \frac{B_\text{c,th}^2}{2\mu_0}} + \desc{Boundary energy}{Negative for \fRef{:::type2}, positive for \fRef{:::type1}}{$\Delta E_\text{B}$ energy gained by expelling the external magnetic field, $\Delta E_\text{cond}$ \fRef{:::condensation_energy}} + \desc[german]{Grenzflächenenergie}{Negativ für \fRef{:::type2}, positiv für \fRef{:::type1}}{} + \eq{\Delta E_\text{boundary} = \Delta E_\text{con} - \Delta E_\txB = (\xi_\gl - \lambda_\gl) \frac{B_\text{c,th}^2}{2\mu_0}} \end{formula} \begin{formula}{parameter} @@ -276,14 +297,20 @@ \abs{\Psi(x)}^2 &= \frac{n_\txs(x)}{n_\txs(\infty)} = \tanh^2 \left(\frac{x}{\sqrt{2}\xi_\gl}\right) \\ B_z(x) &= B_z(0) \Exp{-\frac{x}{\lambda_\gl}} } - \fig{img/cm_sc_n_s_boundary.pdf} - \TODO{plot, slide 106} + \fig{img/cm_super_n_s_boundary.pdf} + % \TODO{plot, slide 106} + \end{formula} + + \begin{formula}{bcth} + \desc{Thermodynamic critical field}{}{} + \desc[german]{Thermodynamisches kritisches Feld}{}{} + \eq{\Bcth = \frac{\Phi_0}{2\pi \sqrt{2} \xi_\gl \lambda_\gl}} \end{formula} \begin{formula}{bc1} - \desc{Lower critical magnetic field}{Above $B_\text{c1}$, flux starts to penetrate the superconducting phase}{\ConstRef{flux_quantum}, $\lambda\gl$ \fRef{::penetration_depth} $\kappa$ \fRef{::parameter}} + \desc{Lower critical magnetic field}{Above $B_\text{c1}$, flux starts to penetrate the superconducting phase}{\ConstRef{flux_quantum}, $\lambda_\gl$ \fRef{::penetration_depth} $\kappa$ \fRef{::parameter}} \desc[german]{Unteres kritisches Magnetfeld}{Über $B_\text{c1}$ dringt erstmals Fluss in die supraleitende Phase ein}{} - \eq{B_\text{c1} = \frac{\Phi_0}{4\pi\lambda\gl^2}(\ln\kappa+0.08) = \frac{1}{\sqrt{2}\kappa}(\ln\kappa + 0.08) \Bcth} + \eq{B_\text{c1} = \frac{\Phi_0}{4\pi\lambda_\gl^2}(\ln\kappa+0.08) = \frac{1}{\sqrt{2}\kappa}(\ln\kappa + 0.08) \Bcth} \end{formula} \begin{formula}{bc2} @@ -297,7 +324,6 @@ % \desc[german]{}{}{} \ttxt{\eng{ Superconductor wavefunction extends into the normal conductor or isolator - \TODO{clarify} }} \end{formula} @@ -343,6 +369,7 @@ \end{itemize} }} \end{formula} + \def\BCS{{\text{BCS}}} \begin{formula}{hamiltonian} \desc{BCS Hamiltonian}{for $N$ interacting electrons}{ @@ -388,11 +415,11 @@ \end{formula} \begin{formula}{gap_at_t0} - \desc{BCS Gap at $T=0$}{}{\QtyRef{debye_frequency}, $V_0$ \fRef{::potential}, $D$ \qtyRef{dos}} + \desc{BCS Gap at $T=0$}{}{\QtyRef{debye_frequency}, $V_0$ \fRef{::potential}, $D$ \qtyRef{dos}, \TODO{gamma}} \desc[german]{BCS Lücke bei $T=0$}{}{} \eq{ \Delta(T=0) &= \frac{\hbar\omega_\txD}{\Sinh{\frac{2}{V_0\.D(E_\txF)}}} \approx 2\hbar \omega_\txD\\ - \frac{\Delta(T=0)}{\kB T_\txc} &= = \frac{\pi}{\e^\gamma} = 1.764 + \frac{\Delta(T=0)}{\kB T_\txc} &= \frac{\pi}{\e^\gamma} = 1.764 } \end{formula} @@ -400,30 +427,53 @@ \begin{formula}{cooper_pair_binding_energy} \desc{Binding energy of Cooper pairs}{}{$E_\txF$ \absRef{fermi_energy}, \QtyRef{debye_frequency}, $V_0$ retarded potential, $D$ \qtyRef{dos}} \desc[german]{Bindungsenergie von Cooper-Paaren}{}{} - \eq{E \approx 2E_\txF - 2\hbar\omega_\txD \Exp{-\frac{4}{V_0D(E_t\txF)}}} + \eq{E \approx 2E_\txF - 2\hbar\omega_\txD \Exp{-\frac{4}{V_0 D(E_\txF)}}} \end{formula} \Subsubsection[ - \eng{Excitations in BCS} - % \ger{} + \eng{Excitations and finite temperatures} + \ger{Anregungen und endliche Temperatur} ]{excite} + \begin{formula}{description} + \desc{Description}{}{} + \desc[german]{Beschreibung}{}{} + \ttxt{\eng{ + The ground state consists of \fRef{cm:super:micro:cooper_pairs} and the excited state of Bogoliubov quasi-particles (electron-hole pairs). + The states are separated by an energy gap $\Delta$. + }\ger{ + Den Grundzustand bilden \fRef{cm:super:micro:cooper_pairs} und den angeregten Zustands Bogoloiubons (Elektron-Loch Quasipartikel). + Die Zustände sind durch eine Energielücke $\Delta$ getrennt. + }} + \end{formula} \begin{formula}{bogoliubov-valatin} - \desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:super:micro:bcs:hamiltonian} to derive excitation energies}{} + \desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:super:micro:bcs:hamiltonian} to derive excitation energies}{ + $\xi_\veck = \epsilon_\veck-\mu$ Energy relative to the \qtyRef{chemical_potential}, + \\ $E_\veck$ \fRef{::excitation_energy}, + \\ $\Delta$ Gap + \\ $g_\veck$ \fRef{::pairing_amplitude}, + \\ $\alpha / \beta$ create and destroy symmetric/antisymmetric Bogoliubov quasiparticles + } \desc[german]{Bogoliubov-Valatin transformation}{}{} \eq{ - \hat{H}_\BCS - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck a_\veck^\dagger a_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big] + \hat{H}_\BCS - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck \alpha_\veck^\dagger \alpha_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big] } \end{formula} \begin{formula}{pairing_amplitude} - \desc{Pairing amplitude}{Excitation energy}{\TODO{gamma}} - \desc[german]{Paarungsamplitude}{Anregungsenergie}{} + \desc{Pairing amplitude}{}{} + \desc[german]{Paarungsamplitude}{}{} + \eq{g_\veck \equiv \Braket{\hat{c}_{-\veck\downarrow} \hat{c}_{\veck\uparrow}}} + \end{formula} + + \begin{formula}{excitation_energy} + \desc{Excitation energy}{}{} + \desc[german]{Anregungsenergie}{}{} \eq{E_\veck = \pm \sqrt{\xi^2_\veck + \abs{\Delta_\veck}^2}} \end{formula} \begin{formula}{coherence_factors_energy} - \desc{Energy dependance of the \fRef{:::coherence_factors}}{}{$E_\veck$ \fRef{::pairing_amplitude}, \GT{see} \fRef{:::coherence_facotrs}} - \desc[german]{Energieabhängigkeit der \fRef{:::coherence_factors}}{}{} + \desc{Energy dependance of the \fRef{:::bcs:coherence_factors}}{}{$E_\veck$ \fRef{::pairing_amplitude}, \GT{see} \fRef{:::bcs:coherence_factors}} + \desc[german]{Energieabhängigkeit der \fRef{:::bcs:coherence_factors}}{}{} \eq{ \abs{u_\veck}^2 &= \frac{1}{2} \left(1+\frac{\xi_\veck}{E_\veck}\right) \\ \abs{v_\veck}^2 &= \frac{1}{2} \left(1-\frac{\xi_\veck}{E_\veck}\right) \\ @@ -431,17 +481,18 @@ } \end{formula} + \begin{formula}{gap_equation} - \desc{BCS-gap equation}{}{} - \desc[german]{BCS Energielückengleichung}{}{} - \eq{\Delta_\veck^* = -\sum_\veck^\prime V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)} + \desc{Self-consistend gap equation}{}{} + \desc[german]{Selbstkonsitente Energielückengleichung}{}{} + \eq{\Delta_\veck^* = -\sum_{\veck^\prime} V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)} \end{formula} \begin{formula}{gap_t} \desc{Temperature dependence of the BCS gap}{}{} \desc[german]{Temperaturabhängigkeit der BCS-Lücke}{}{} - \eq{\frac{\Delta(T)}{\Delta(T=0)} \approx 1.74 \frac{1}{\sqrt{1-\frac{T}{T_\txC}}}} + \eq{\frac{\Delta(T)}{\Delta(T=0)} \approx 1.74 \sqrt{1-\frac{T}{T_\txC}}} \end{formula} \begin{formula}{dos} @@ -456,8 +507,40 @@ \end{formula} \begin{formula}{Bcth_temp} - \desc{Temperature dependance of the crictial magnetic field}{}{} - \desc[german]{Temperaturabhängigkeit des kritischen Magnetfelds}{}{} + \desc{Temperature dependance of the crictial magnetic field}{Jump at $T_\txc$, then exponential decay}{} + \desc[german]{Temperaturabhängigkeit des kritischen Magnetfelds}{Sprung bei $T_\txc$, denn exponentieller Abfall}{} \eq{ \Bcth(T) = \Bcth(0) \left[1- \left(\frac{T}{T_\txc}\right)^2 \right] } - \TODO{empirical relation, relate to BCS} + % \TODO{empirical relation, relate to BCS} + \end{formula} + + \begin{formula}{heat_capacity} + \desc{Heat capacity in superconductors}{}{} + \desc[german]{Wärmekapazität in Supraleitern}{}{} + \fsplit{ + \fig{img/cm_super_heat_capacity.pdf} + }{ + \eq{c_\txs \propto T^{-\frac{3}{2}} \e^{\frac{\Delta(0)}{\kB T}}} + } + \end{formula} + + \Subsubsection[ + \eng{Flux pinning} + \ger{Haftung von Flusslinien} + ]{pinning} + \begin{formula}{description} + \desc{Description}{}{} + \desc[german]{Beschreibung}{}{} + \ttxt{\eng{ + If a current flows in a \fRef{cm:super:type2}s in the \fRef{cm:super:shubnikov_phase} perpendicular to the penetrating flux lines, + the lines experience a Lorentz force. This leads to ohmic behaviour of the superconductor. + The flux lines can be pinned to defects, in which the superconducting order parameter is reduced. + To move the flux line out of the defect, work would have to be spent overcoming the \fRef{cm:super:micro:pinning:potential}. + This restores the superconductivity. + }\ger{ + Wenn ein Strom in einem \fRef{cm:super:type2}s in der \fRef{cm:super:shubnikov_phase} senkrecht zu den eindringenden Flusslinien fließt, erfahren die Linien eine Lorentzkraft. + Dies führt zu einem ohmschen Verhalten des Supraleiters. + Die Flusslinien können an Defekten festgehalten werden, in denen der supraleitende Ordnungsparameter reduziert ist. + Um die Flusslinie aus dem Defekt zu bewegen, müsste Arbeit aufgewendet werden, um das \fRef{cm:super:micro:pinning:potential} zu überwinden. + Dies stellt die Supraleitfähigkeit wieder her. + }} \end{formula} diff --git a/src/cm/techniques.tex b/src/cm/techniques.tex index 20a07e6..801705a 100644 --- a/src/cm/techniques.tex +++ b/src/cm/techniques.tex @@ -55,7 +55,7 @@ \begin{minipage}{0.45\textwidth} \begin{figure}[H] \centering - % \includegraphics[width=0.8\textwidth]{img/cm_amf.pdf} + % \includegraphics[width=0.8\textwidth]{img_static/cm_amf.pdf} % \caption{\cite{Bian2021}} \end{figure} \end{minipage} @@ -97,7 +97,7 @@ \begin{minipage}{0.45\textwidth} \begin{figure}[H] \centering - \includegraphics[width=0.8\textwidth]{img/cm_amf.pdf} + \includegraphics[width=0.8\textwidth]{img_static/cm_amf.pdf} \caption{\cite{Bian2021}} \end{figure} \end{minipage} @@ -122,7 +122,7 @@ \begin{minipage}{0.45\textwidth} \begin{figure}[H] \centering - \includegraphics[width=0.8\textwidth]{img/cm_stm.pdf} + \includegraphics[width=0.8\textwidth]{img_static/cm_stm.pdf} \caption{\cite{Bian2021}} \end{figure} \end{minipage} @@ -168,7 +168,7 @@ \end{minipagetable} \begin{minipage}{0.45\textwidth} \centering - \includegraphics[width=\textwidth]{img/cm_cvd_english.pdf} + \includegraphics[width=\textwidth]{img_static/cm_cvd_english.pdf} \end{minipage} \end{bigformula} diff --git a/src/cm/vib.tex b/src/cm/vib.tex new file mode 100644 index 0000000..5d0a0b7 --- /dev/null +++ b/src/cm/vib.tex @@ -0,0 +1,102 @@ +\Section[ + \eng{Lattice vibrations} + \ger{Gitterschwingungen} +]{vib} + + \begin{formula}{dispersion_1atom_basis} + \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} + \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} + \begin{gather} + \omega^2 = \frac{4C_1}{M}\left[\sin^2 \left(\frac{qa}{2}\right) + \frac{C2}{C1} \sin^2(qa)\right] \\ + \intertext{\GT{with}} + u_{s+n} = U\e^{-i \left[\omega t - q(s+n)a \right]} + \end{gather} + \newFormulaEntry + \fig{img/cm_vib_dispersion_one_atom_basis.pdf} + \end{formula} + \begin{formula}{dispersion_2atom_basis} + \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} + \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} + \begin{gather} + \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)} + \intertext{\GT{with}} + u_{s} = U\e^{-i \left(\omega t - qsa \right)}, \quad + v_{s} = V\e^{-i \left(\omega t - qsa \right)} + \end{gather} + \newFormulaEntry + \fig{img/cm_vib_dispersion_two_atom_basis.pdf} + \end{formula} + + \begin{formula}{branches} + \desc{Vibration branches}{}{} + \desc[german]{Vibrationsmoden}{}{} + \ttxt{\eng{ + \textbf{Acoustic}: 3 modes (1 longitudinal, 2 transversal), the two basis atoms oscillate in phase. + \\\textbf{Optical}: 3 modes, the two basis atoms oscillate in opposition. A dipole moment is created that can couple to photons. + }\ger{ + \textbf{Akustisch}: 3 Moden (1 longitudinal, 2 transversal), die zwei Basisatome schwingen in Phase. + \\ \textbf{Optisch}: 3 Moden, die zwei Basisatome schwingen gegenphasig. Das dadurch entstehende Dipolmoment erlaubt die Wechselwirkung mit Photonen. + }} + \end{formula} + + \Subsection[ + \eng{Einstein model} + \ger{Einstein-Modell} + ]{einstein} + \begin{formula}{description} + \desc{Description}{}{} + \desc[german]{Beschreibung}{}{} + \ttxt{\eng{ + All lattice vibrations have the \fRef[same frequency]{:::frequency}. + Underestimates the \fRef{:::heat_capacity} for low temperatures. + }\ger{ + Alle Gittereigenschwingungen haben die \fRef[selbe Frequenz]{:::frequency} + Sagt zu kleine \fRef[Wärmekapazitäten]{:::heat_capacity} für tiefe Temperaturen voraus. + }} + \end{formula} + \begin{formula}{frequency} + \desc{Einstein frequency}{}{} + \desc[german]{Einstein-Frequenz}{}{} + \eq{\omega_\txE} + \end{formula} + \begin{formula}{heat_capacity} + \desc{\qtyRef{heat_capacity}}{according to the Einstein model}{} + \desc[german]{}{nach dem Einstein-Modell}{} + \eq{C_V^\txE = 3N\kB \left( \frac{\hbar\omega_\txE}{\kB T}\right)^2 \frac{\e^{\frac{\hbar\omega_\txE}{\kB T}}}{ \left(\e^{\frac{\hbar\omega_\txE}{\kB T}} - 1\right)^2}} + \end{formula} + + \Subsection[ + \eng{Debye model} + \ger{Debye-Modell} + ]{debye} + \begin{formula}{description} + \desc{Description}{}{} + \desc[german]{Beschreibung}{}{} + \ttxt{\eng{ + Atoms behave like coupled \fRef[quantum harmonic oscillators]{sec:qm:hosc}. The finite sample size leads to periodic boundary conditio. The finite sample size leads to periodic boundary conditions for the vibrations. + }\ger{ + Atome verhalten sich wie gekoppelte \fRef[quantenmechanische harmonische Oszillatoren]{sec:qm:hosc}. Die endliche Ausdehnung des Körpers führt zu periodischen Randbedingungen. + }} + \end{formula} + + \begin{formula}{phonon_dos} + \desc{Phonon density of states}{}{\QtyRef{volume}, $v$ \qtyRef{speed_of_sound} of the phonon mode, $\omega$ phonon frequency} + \desc[german]{Phononenzustandsdichte}{}{\QtyRef{volume}, $v$ \qtyRef{speed_of_sound} des Dispersionszweigs, $\omega$ Phononfrequenz} + \eq{D(\omega) \d \omega = \frac{V}{2\pi^2} \frac{\omega^2}{v^3} \d\omega} + \end{formula} + + \begin{formula}{debye_frequency} + \desc{Debye frequency}{Maximum phonon frequency}{$v$ \qtyRef{speed_of_sound}, $N/V$ atom density} + \desc[german]{Debye-Frequenz}{Maximale Phononenfrequenz}{$v$ \qtyRef{speed_of_sound}, $N/V$ Atomdichte} + \eq{\omega_\txD = v \left(6\pi^2 \frac{N}{V}\right)^{1/3}} + \hiddenQuantity{\omega_\txD}{\per\s}{s} + \end{formula} + + \begin{formula}{heat_capacity} + \desc{\qtyRef{heat_capacity}}{according to the Debye model}{$N$ number of atoms, \ConstRef{boltzmann}, \QtyRef{debye_frequency}} + \desc[german]{}{nach dem Debye-Modell}{$N$ Anzahl der Atome, \ConstRef{boltzmann}, \QtyRef{debye_frequency}} + \eq{C_V^\txD = 9N\kB \left(\frac{\kB T}{\hbar \omega_\txD}\right)^3 \int_0^{\frac{\hbar\omega_\txD}{\kB T}} \d x \frac{x^4 \e^x}{(\e^x-1)^2} } + \end{formula} + + + diff --git a/src/img/cm_crystal_NaCl.png b/src/img/cm_crystal_NaCl.png index 737268f..76eb20b 100644 Binary files a/src/img/cm_crystal_NaCl.png and b/src/img/cm_crystal_NaCl.png differ diff --git a/src/img/cm_crystal_wurtzite.png b/src/img/cm_crystal_wurtzite.png index 7e276b6..402ee89 100644 Binary files a/src/img/cm_crystal_wurtzite.png and b/src/img/cm_crystal_wurtzite.png differ diff --git a/src/img/cm_crystal_zincblende.png b/src/img/cm_crystal_zincblende.png new file mode 100644 index 0000000..11a56d7 Binary files /dev/null and b/src/img/cm_crystal_zincblende.png differ diff --git a/src/main.tex b/src/main.tex index a089903..06fb97d 100644 --- a/src/main.tex +++ b/src/main.tex @@ -3,7 +3,7 @@ \documentclass[11pt, a4paper]{article} % SET LANGUAGE HERE \usepackage[english]{babel} -\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} +\usepackage[left=1.6cm,right=1.6cm,top=2cm,bottom=2cm]{geometry} % ENVIRONMENTS etc \usepackage{adjustbox} \usepackage{colortbl} % color table @@ -145,6 +145,7 @@ \Input{cm/crystal} \Input{cm/egas} \Input{cm/charge_transport} +\Input{cm/vib} \Input{cm/superconductivity} \Input{cm/semiconductors} \Input{cm/misc} diff --git a/src/pkg/mqformula.sty b/src/pkg/mqformula.sty index a654a33..229b30f 100644 --- a/src/pkg/mqformula.sty +++ b/src/pkg/mqformula.sty @@ -1,7 +1,7 @@ \ProvidesPackage{mqformula} \def\descwidth{0.3\textwidth} -\def\eqwidth{0.69\textwidth} +\def\eqwidth{0.65\textwidth} \RequirePackage{mqfqname} \RequirePackage{mqconstant} @@ -16,14 +16,11 @@ % [1]: minipage width % 2: fqname of name % 3: fqname of a translation that holds the explanation -\newcommand{\NameWithDescription}[3][\descwidth]{ +\newcommand{\NameWithDescription}[3][\descwidth]{% \begin{minipage}{#1} - \IfTranslationExists{#2}{ - \raggedright - \GT{#2} - }{\detokenize{#2}} - \IfTranslationExists{#3}{ - \\ {\color{fg1} \GT{#3}} + \raggedright\GT{#2}% + \IfTranslationExists{#3}{% + \\ {\color{fg1} \GT{#3}}% }{} \end{minipage} } @@ -152,12 +149,16 @@ } \newcommand{\fsplit}[3][0.5]{ - \begin{minipage}{##1\linewidth} - ##2 - \end{minipage} - \begin{minipage}{\luavar{0.99-##1}\linewidth} - ##3 - \end{minipage} + \begingroup + \renewcommand{\newFormulaEntry}{} + \begin{minipage}{##1\linewidth} + ##2 + \end{minipage} + \begin{minipage}{\luavar{0.99-##1}\linewidth} + ##3 + \end{minipage} + \endgroup + \newFormulaEntry } }{ \mqfqname@leave diff --git a/src/pkg/mqperiodictable.sty b/src/pkg/mqperiodictable.sty index 3281d5a..c7d949b 100644 --- a/src/pkg/mqperiodictable.sty +++ b/src/pkg/mqperiodictable.sty @@ -10,7 +10,8 @@ elements = {} elementsOrder = {} function elementAdd(symbol, nr, period, column) - elementsOrder[nr] = symbol + --elementsOrder[nr] = symbol + table.insert(elementsOrder, symbol) elements[symbol] = { symbol = symbol, atomic_number = nr, diff --git a/src/pkg/mqref.sty b/src/pkg/mqref.sty index 4015061..0a8901f 100644 --- a/src/pkg/mqref.sty +++ b/src/pkg/mqref.sty @@ -119,7 +119,7 @@ } \newcommand{\fRef}[2][]{ - \directlua{hyperref(translateRelativeFqname(\luastring{#2}), \luastring{#1})} + \directlua{hyperref(translateRelativeFqname(\luastring{#2}), \luastring{#1})}% } % [1]: link text % 2: number of steps to take up @@ -145,7 +145,7 @@ \newrobustcmd{\qtyRef}[2][]{% % \edef\tempname{\luaDoubleFieldValue{quantities}{"#1"}{"fqname"}}% % \hyperref[qty:#1]{\GT{\tempname}}% - \directlua{hyperref(quantityGetFqname(\luastring{#2}), \luastring{#1})} + \directlua{hyperref(quantityGetFqname(\luastring{#2}), \luastring{#1})}% } % \newrobustcmd{\QtyRef}[2][]{% @@ -156,7 +156,7 @@ \newrobustcmd{\constRef}[2][]{% % \edef\tempname{\luaDoubleFieldValue{constants}{"#1"}{"linkto"}}% % \hyperref[const:#1]{\GT{\tempname}}% - \directlua{hyperref(constantGetFqname(\luastring{#2}), \luastring{#1})} + \directlua{hyperref(constantGetFqname(\luastring{#2}), \luastring{#1})}% } % \newrobustcmd{\ConstRef}[2][]{% diff --git a/src/qm/atom.tex b/src/qm/atom.tex index 2ee369c..09643b7 100644 --- a/src/qm/atom.tex +++ b/src/qm/atom.tex @@ -4,7 +4,7 @@ \Section[ \eng{Hydrogen Atom} \ger{Wasserstoffatom} - ]{h} +]{h} \begin{formula}{reduced_mass} \desc{Reduced mass}{}{} @@ -188,8 +188,8 @@ \eq{f &= j \pm i \\ m_f &= -f,-f+1,\dots,f-1,f} \end{formula} \begin{formula}{constant} - \desc{Hyperfine structure constant}{}{$B_\textrm{HFS}$ hyperfine field, $\mu_\textrm{K}$ nuclear magneton, $g_i$ nuclear g-factor \ref{qm:h:lande}} - \desc[german]{Hyperfeinstrukturkonstante}{}{$B_\textrm{HFS}$ Hyperfeinfeld, $\mu_\textrm{K}$ Kernmagneton, $g_i$ Kern-g-Faktor \ref{qm:h:lande}} + \desc{Hyperfine structure constant}{}{$B_\textrm{HFS}$ hyperfine field, $\mu_\textrm{K}$ nuclear magneton, $g_i$ nuclear g-factor \fRef{qm:h:lande}} + \desc[german]{Hyperfeinstrukturkonstante}{}{$B_\textrm{HFS}$ Hyperfeinfeld, $\mu_\textrm{K}$ Kernmagneton, $g_i$ Kern-g-Faktor \fRef{qm:h:lande}} \eq{A = \frac{g_i \mu_\textrm{K} B_\textrm{HFS}}{\sqrt{j(j+1)}}} \end{formula} \begin{formula}{energy_shift} diff --git a/src/qm/qm.tex b/src/qm/qm.tex index 2fce0a5..d792a9f 100644 --- a/src/qm/qm.tex +++ b/src/qm/qm.tex @@ -289,7 +289,7 @@ \ger{Ehrenfest-Theorem} ]{ehrenfest_theorem} \absLink{}{ehrenfest_theorem} - \GT{see_also} \ref{sec:qm:basics:schroedinger_equation:correspondence_principle} + \GT{see_also} \fRef{qm:se:time:ehrenfest_theorem:correspondence_principle} \begin{formula}{ehrenfest_theorem} \desc{Ehrenfest theorem}{applies to both pictures}{} \desc[german]{Ehrenfest-Theorem}{gilt für beide Bilder}{} @@ -386,8 +386,6 @@ \eq{E_n = \hbar\omega \Big(\frac{1}{2} + n\Big)} \end{formula} - \GT{see_also} \ref{sec:qm:hosc:c_a_ops} - \Subsection[ \ger{Erzeugungs und Vernichtungsoperatoren / Leiteroperatoren} \eng{Creation and Annihilation operators / Ladder operators} @@ -525,8 +523,8 @@ \ger{Symmetrien} ]{symmetry} \begin{ttext}[desc] - \eng{Most symmetry operators are unitary \ref{sec:linalg:unitary} because the norm of a state must be invariant under transformations of space, time and spin.} - \ger{Die meisten Symmetrieoperatoren sind unitär \ref{sec:linalg:unitary}, da die Norm eines Zustands invariant unter Raum-, Zeit- und Spin-Transformationen sein muss.} + \eng{Most symmetry operators are \fRef[unitary]{math:linalg:matrix:unitary} because the norm of a state must be invariant under transformations of space, time and spin.} + \ger{Die meisten Symmetrieoperatoren sind \fRef[unitär]{math:linalg:matrix:unitary}, da die Norm eines Zustands invariant unter Raum-, Zeit- und Spin-Transformationen sein muss.} \end{ttext} \begin{formula}{invariance} \desc{Invariance}{$\hat{H}$ is invariant under a symmetrie described by $\hat{U}$ if this holds}{} @@ -587,6 +585,7 @@ } \end{formula} + \begin{formula}{slater_det} \desc{Slater determinant}{Construction of a fermionic (antisymmetric) many-particle wave function from single-particle wave functions}{} \desc[german]{Slater Determinante}{Konstruktion einer fermionischen (antisymmetrischen) Vielteilchen Wellenfunktion aus ein-Teilchen Wellenfunktionen}{} diff --git a/src/readme.md b/src/readme.md new file mode 100644 index 0000000..c4de8e7 --- /dev/null +++ b/src/readme.md @@ -0,0 +1,18 @@ +# Formulary tex source + +## Special directories +- `pkg`: My custom Latex packages +- `img`: Images generated by `../scripts/` +- `img_static`: Downloaded or other images not generated by me +- `img_static_svgs`: Downloaded or other images not generated by me that need to be converted to pdf + + +## Subject directories + +- `bib`: bibliography files +- `ch`: chemistry +- `cm`: condensed matter +- `comp`: computational +- `ed`: electrodynamics +- `math`: mathematics +- `qm`: quantum mechanics diff --git a/src/statistical_mechanics.tex b/src/statistical_mechanics.tex index 0f207ac..2ee2faf 100644 --- a/src/statistical_mechanics.tex +++ b/src/statistical_mechanics.tex @@ -219,19 +219,19 @@ \eng{Material properties} \ger{Materialeigenschaften} ]{props} - \begin{formula}{heat_cap} + \begin{formula}{heat_capacity} \desc{Heat capacity}{}{\QtyRef{heat}} \desc[german]{Wärmekapazität}{}{} \quantity{c}{\joule\per\kelvin}{} \eq{c = \frac{Q}{\Delta T}} \end{formula} - \begin{formula}{heat_cap_V} + \begin{formula}{heat_capacity_V} \desc{Isochoric heat capacity}{}{\QtyRef{heat}, \QtyRef{internal_energy} \QtyRef{temperature}, \QtyRef{volume}} \desc[german]{Isochore Wärmekapazität}{}{} \eq{c_v = \pdv{Q}{T}_V = \pdv{U}{T}_V} \end{formula} - \begin{formula}{heat_cap_p} + \begin{formula}{heat_capacity_p} \desc{Isobaric heat capacity}{}{\QtyRef{heat}, \QtyRef{enthalpy} \QtyRef{temperature}, \QtyRef{pressure}} \desc[german]{Isobare Wärmekapazität}{}{} \eq{c_p = \pdv{Q}{T}_p = \pdv{H}{T}_p} @@ -832,7 +832,7 @@ } \end{formula} - \begin{formula}{heat_cap} + \begin{formula}{heat_capacity} \desc{Heat capacity}{\gt{low_temps}}{differs from \fRef{td:TODO:petit_dulong}} \desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \fRef{td:TODO:petit_dulong}} \fig{img/td_fermi_heat_capacity.pdf} diff --git a/src/util/colors.tex b/src/util/colors.tex index 3021e39..e144079 100644 --- a/src/util/colors.tex +++ b/src/util/colors.tex @@ -30,8 +30,8 @@ \hypersetup{ colorlinks=true, - linkcolor=fg-purple, + linkcolor=fg-blue, citecolor=fg-green, - filecolor=fg-blue, + filecolor=fg-purple, urlcolor=fg-orange }