scattering/theory.md

-###### Perform Impurity calculations using kkrflex {#perform_impurity_calculations_using_kkrflex}
+# Perform Impurity calculations using kkrflex
 
-##### Theory
+## Theory
 
-#### Motivation
+### Motivation
 
 STS (Scanning tunnelling spectroscopy) is an experimental technique
 which provides information about the electrons charge density resolved
@@ -16,77 +16,83 @@ 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}
+### 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
+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\'\$
+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 : \\\\
+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
+$`\Delta n
+(\vec{R}_n+\vec{r};E)=|A_{\vec{k}_0,\vec{k}_0\'}(\vec{r})|
+\frac{\sin((\vec{k}_0-\vec{k}_0\')\cdot
+\vec{R}_n+\phi_{\vec{k}_0,\vec{k}_0\'})}{|\vec{R}_n|}`$
+
+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
+$`1/|\vec{R}_n|`$ 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.
+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}
+### 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. \\\\
+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
+scattering vector $`\vec{q}=\vec{k}_f-\vec{k}_i`$, lying in the
+same energy shell :
+```math
+\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}})
+```
+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
@@ -94,57 +100,58 @@ 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: $1-\cos(\vec{v}_{\vec{k}},\vec{v}_{\vec{k}'})$`
+ * 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
+quasiparticle interference map is :
+```math
+\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)
+```
+with
+```
+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)
+```
+
+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)\$.
+$`\mathrm{P}_{\vec{k},\vec{k}\'} \approx \left(
+1-\cos(\vec{s}_{\vec{k}},\vec{s}_{\vec{k}+\vec{q}}) \right)`$.
 This approach is called spin-dependent scattering probability (SSP).
 
-##### A key object : the \$\\mathcal{T}\$-matrix {#a_key_object_the_mathcalt_matrix}
+## A key object : the $`\mathcal{T}`$-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
-.
+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
+```math
+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}\,') }~,`
+```
+and, using the expansions that are made in the KKR
+formalism, takes the form
+```math
+T_{k'k} = \sum_{i,i'} \sum_{\Lambda,\Lambda'}  {c^i}^*_{k',\Lambda} T^{ii'}_{\Lambda \Lambda'} ~ c^{i'}_{k,\Lambda'}~.`
+```
+with
+```math
+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)~.`
+```
+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`
+ - 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