find . -name \*.txt -type f -exec /Users/ruess/Downloads/pandoc-2.7.2/bin/pandoc -f mediawiki -t markdown -o {}.md {} \;
then replace all .txt.md files by .txt
and finally run git mv on all .txt files to convert to .md files

scattering/theory.md

+###### Perform Impurity calculations using kkrflex {#perform_impurity_calculations_using_kkrflex}
+
+##### Theory
+
+#### Motivation
+
+STS (Scanning tunnelling spectroscopy) is an experimental technique
+which provides information about the electrons charge density resolved
+on energy and position. This allows to extract a compound surface
+properties such its surface states and densities. One main features STS
+allows to access are the standing waves patterns which correspond to
+charge oscillations generated by an impurity and which decays with a
+power law from it. Due to the presence of many impurities at different
+positions and the superposition of different standing waves the
+topographic data is very difficult to interpret. For this reason a
+Fourier transform (FT) is performed in order to access main scattering
+processes.
+
+#### Stationary phase approximation (SPA) {#stationary_phase_approximation_spa}
+
+The spin, energy and position resolved density in presence of an
+impurity is: \\\\ \$\\Delta n (\\vec{r}
+\\sigma;E)=-\\frac{1}{\\pi}\\mathrm{Im} \\left\\langle \\vec{r}\\sigma
+\\left\| G\_{imp}(E)-G\_{host}(E) \\right\| \\vec{r} \\sigma
+\\right\\rangle\$. \\\\ In a non-degenerate case this Green\'s function
+difference can be written under the form: \\\\ \$\\Delta G(\\vec{r} +
+\\vec{R}\_n,\\vec{r} + \\vec{R}\_n;E) = - \\frac{1}{\\Omega\_{rec}\^2}
+\\int\\limits\_{0}\^{+\\infty} dt \\ dt\' \\int d\^2\\vec{k} \\
+d\^2\\vec{k}\' \\Psi\_{\\vec{k}}(\\vec{r})
+\\mathcal{T}\_{\\vec{k}\\vec{k}\'} \\Psi\^{\*}\_{\\vec{k}\'}(\\vec{r})
+e\^{i\\Phi}\$ \\\\ where \$\\Phi=(E-\\epsilon\_{\\vec{k}}+\\eta)t +
+(E-\\epsilon\_{\\vec{k}\'}+\\eta\')t\' +
+(\\vec{k}-\\vec{k}\')\\cdot\\vec{R}\_n\$.
+
+The main idea behind the SPA is to suppress the quick \$\\vec{k}\$ and
+\$t\$ oscillations in the above integral. The waves having the same
+phase will sum constructively while the ones with different phases will
+sum incoherently cancelling their contribution.This can be done looking
+for points \$\\{ \\vec{k}\_0,\\vec{k}\_0\',t\_0,t\_0\' \\}\$ which makes
+the phase \$\\Phi\$ stationary. This gives the following set of
+equations: \\\\ \$\\left\\{\\begin{array}{ll}\\vec{R}\_n &= -
+v\_{\\vec{k}\_0\'}t\_0\'\\\\\\vec{R}\_n &= v\_{\\vec{k}\_0}t\_0 \\\\E &=
+\\epsilon\_{\\vec{k}\_0} \\\\E &=
+\\epsilon\_{\\vec{k}\_0\'}\\end{array}\\right.\$ \\\\
+
+This means that the velocities at \$\\vec{k}\_0\$ and \$\\vec{k}\_0\'\$
+must be parallel but with an opposite direction. Furthermore the states
+defined by the previous wave vectors must lie in the same energy shell.
+Within these conditions one can obtain the difference in the density
+performing the integrals and trace calculation on spins, which results
+in : \\\\
+
+\$\\Delta n
+(\\vec{R}\_n+\\vec{r};E)=\|A\_{\\vec{k}\_0,\\vec{k}\_0\'}(\\vec{r})\|
+\\frac{\\sin((\\vec{k}\_0-\\vec{k}\_0\')\\cdot
+\\vec{R}\_n+\\phi\_{\\vec{k}\_0,\\vec{k}\_0\'})}{\|\\vec{R}\_n\|}\$
+
+where \$\|A\_{\\vec{k}\_0,\\vec{k}\_0\'}(\\vec{r})\| \\propto
+\|\\mathcal{T}\_{\\vec{k}\_0,\\vec{k}\_0\'}\|\$. The above relation
+shows that the density oscillates with respect to the position of the
+site n. It is also observed that this oscillation on the density has an
+\$1/\|\\vec{R}\_n\|\$ envelope. More generally the envelope has a power
+law shape which depends on the type of impurity (point defect, edge
+defect, magnetic defect \... ) and of the dimensionality of the crystal:
+\$\\frac{1}{R}\$ for a surface and \$\\frac{1}{R\^2}\$ for the bulk.
+This envelope also rely on the symmetries connecting the scattered
+states, in particular TRS. On Q. Liu et al.((Liu et al. Phys. Rev. B 85
+2012)) established a chart depicting the shape of the envelop with
+respect to these last considerations.
+
+#### Accessing scattering processes beyond JDOS : the extended joint density of states (exJDOS) {#accessing_scattering_processes_beyond_jdos_the_extended_joint_density_of_states_exjdos}
+
+STS experiments allow to observe standing waves in the density however
+mainly because of the presence of multiple impurities and because of the
+superposition of distinctive standing waves with different scattering
+vectors it\'s difficult to extract information from the raw density map.
+For this reason a FT is performed on the density map highlighting the
+main scattering processes. \\\\
+
+There is a general method exhibiting directly the FT depicted above and
+is commonly named joint density of states (JDOS). The formula describing
+this method gives the contribution of all scattering processes, having a
+scattering vector \$\\vec{q}=\\vec{k}\_f-\\vec{k}\_i\$, lying in the
+same energy shell : \\begin{equation}\\label{eq:JDOS formula}
+\\mbox{JDOS}(\\vec{q})=\\int d\^2 \\vec{k} \\ A\_{\\vec{k}}(E)
+A\_{\\vec{k}+\\vec{q}}(E) % \\quad \\mbox{with} \\quad %
+A\_{\\vec{k}}(E)=\\int\\limits\_{surface} d\^2 \\vec{r} \\
+\|\\Psi\_{\\vec{k}}(\\vec{r})\|\^2 \\delta(E-\\epsilon\_{\\vec{k}})
+\\end{equation} However according to this formula all scattering
+processes have the same weight which is in general not the case.
+Furthermore this formula leads to homogeneous quasiparticle
+interferences which is erroneous. In order to highlight the real
+scattered processes the extended joint density of states (exJDOS) method
+is used. Within this method the scattering processes are weighted as
+following:
+
+` * by the transition rate between states thanks to "Fermi's golden rule" and the $\mathcal{T}_{\vec{k},\vec{k}'}$ components: $\mbox{P}_{\vec{k},\vec{k}'}=2\pi |\mathcal{T}_{\vec{k},\vec{k}'}|^2\delta(\epsilon_{\vec{k}}-\epsilon_{\vec{k}'})$`\
+` * by the velocity-dependent scattering probability.States having velocities with opposite directions are most likely to scatter: $1-\cos(\vec{v}_{\vec{k}},\vec{v}_{\vec{k}'})$`
+
+The first point gives the probability of transition from one state to
+another in presence of a perturbation (here an impurity in the host
+system). The second point stems directly from the SPA. With the above
+considerations the convenient quantity exhibiting the FT of the
+quasiparticle interference map is : \\begin{multline}\\label{eq:exJDOS
+formula} \\mbox{exJDOS}(\\vec{q})=\\int d\^2 \\vec{k} \\
+A\_{\\vec{k}}(E) M\_{\\vec{k},\\vec{k}+\\vec{q}}
+A\_{\\vec{k}+\\vec{q}}(E) \\\\ \\quad \\mbox{with} \\quad %
+M\_{\\vec{k},\\vec{k}+\\vec{q}}=\\mbox{P}\_{\\vec{k},\\vec{k}+\\vec{q}}
+% \\left( 1-\\cos(\\vec{v}\_{\\vec{k}},\\vec{v}\_{+\\vec{q}}) \\right)
+\\end{multline}
+
+The method called JDOS-\$\\gamma\$STM only takes into account the group
+velocity contribution of exJDOS. In the case where the transition rate
+is not known there exists a phenomenological result, obtained by Rushan,
+and valid in topological insulators surfaces which asserts that states
+having spins with same directions are most likely to scatter :
+\$\\mathrm{P}\_{\\vec{k},\\vec{k}\'} \\approx \\left(
+1-\\cos(\\vec{s}\_{\\vec{k}},\\vec{s}\_{\\vec{k}+\\vec{q}}) \\right)\$.
+This approach is called spin-dependent scattering probability (SSP).
+
+##### A key object : the \$\\mathcal{T}\$-matrix {#a_key_object_the_mathcalt_matrix}
+
+In the calculation of electron scattering properties induced by defects
+within the KKR method, the transition matrix \$T\_{kk\'}\$ (where
+\$k=(\\mathbf{k},\\nu)\$ comprises the crystal momentum \$\\mathbf{k}\$
+and another index \$\\nu\$ for additional degeneracies) takes the
+central role. It is defined as \\begin{equation}
+
+` T_{k'k} =\int{ \mathrm{d} \mathbf{x} \, \mathrm{d} \mathbf{x}\,' ~ \boldsymbol{\psi}^{\dagger}_{k'}(\mathbf{x}) \, \Delta \mathbf{V}(\mathbf{x},\mathbf{x}\,') \, \boldsymbol{\psi}^{\mathrm{imp}}_{k}(\mathbf{x}\,') }~,`
+
+\\end{equation} and, using the expansions that are made in the KKR
+formalism, takes the form \\begin{equation}
+
+` T_{k'k} = \sum_{i,i'} \sum_{\Lambda,\Lambda'}  {c^i}^*_{k',\Lambda} T^{ii'}_{\Lambda \Lambda'} ~ c^{i'}_{k,\Lambda'}~.`
+
+\\end{equation} with \\begin{equation}
+
+` T^{ii'}_{\Lambda \Lambda'} = \sum_{\Lambda`*`}`` ``\Delta^{i}_{\Lambda`` ``\Lambda`*`} \, \left(  \delta_{ii'} \, \delta_{\Lambda'' \Lambda'} + \sum_{\Lambda`**`}`` ``G^{\mathrm{imp},ii'}_{\Lambda''`` ``\Lambda`**`} ~ \Delta t^{\mathrm{imp},i'}_{\Lambda''' \Lambda'} \right)~.`
+
+\\end{equation} The matrix \$T\^{ii\'}\_{\\Lambda \\Lambda\'}\$ is
+independent on the state vector \$k\$, but it depends only on the energy
+(often equal to the Fermi energy). All \$k\$-dependencies enter via the
+.
+
+In order to calculate scattering properties off defects, several steps
+have to be performed.
+
+` - converge a potential of an ideal host system with the Jülich-München code`\
+` - calculate the Fermi surface and wave-functions on the Fermi surface `\
+` - converge a potential of the defect (=impurity cluster) with the KKRimp code`