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###### KKRimp program: Theory {#kkrimp_program_theory}
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# KKRimp program: Theory
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This page should give you a brief introduction to the theoretical
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background of the impurity calculations. To find more details on the
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theory and a detailed description on the method follow the following
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links:
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` * PhD thesis of D. Bauer (https://publications.rwth-aachen.de/record/229375)`\
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` * Dederichs et al., MRS Proceedings 253, 185 (1991)`
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##### Green function based DFT calculations {#green_function_based_dft_calculations}
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Starting point: Schrödinger equation \$\$ \\mathcal{H}\|\\psi\\rangle =
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E\|\\psi\\rangle \$\$ This can formally be written with the Green
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function \$\\mathcal{G}\$ as \$\$ (\\epsilon
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-\\mathcal{H})\\mathcal{G}(\\epsilon)=1, \\quad \\epsilon=E+i\\delta,
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\\quad \\delta\>0 \$\$ Thus by knowing the GF one can extract the
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systems properties, i.e. the expectation value \$\\langle
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\\mathcal{A}\\rangle\$ of an operator \$\\hat{\\mathcal{A}}\$: \$\$
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\\langle\\mathcal{A}\\rangle=-\\frac{1}{\\pi}\\mathrm{Im}\\int\_{-\\infty}\^{E\_F}\\mathrm{d}E\\mathrm{Tr}\\left\[
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\\hat{\\mathcal{A}}\\mathcal{G}(E) \\right\]\$\$ For example the density
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\$\\rho\$ can be calculated via
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\$\$\\rho(\\vec{x})=-\\frac{1}{\\pi}\\mathrm{Im}\\int\_{-\\infty}\^{E\_F}\\mathrm{d}E\\mathcal{G}(\\vec{x},\\vec{x};E)\$\$
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* PhD thesis of D. Bauer (https://publications.rwth-aachen.de/record/229375)
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* Dederichs et al., MRS Proceedings 253, 185 (1991)
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## Green function based DFT calculations
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Starting point: Schrödinger equation
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```math
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\mathcal{H}\|\psi\rangle = E\|\psi\rangle
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```
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This can formally be written with the Green
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function $`\mathcal{G}`$ as
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```math
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(\epsilon -\mathcal{H})\mathcal{G}(\epsilon)=1, \quad \epsilon=E+i\delta,
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\quad \delta\>0
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```
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Thus by knowing the GF one can extract the
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systems properties, i.e. the expectation value $`\langle
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\mathcal{A}\rangle`$ of an operator $`\hat{\mathcal{A}}`$:
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```math
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\langle\mathcal{A}\rangle=-\frac{1}{\pi}\mathrm{Im}\int_{-\infty}^{E_F}\mathrm{d}E\mathrm{Tr}\left\[
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\hat{\mathcal{A}}\mathcal{G}(E) \right\]
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```
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For example the density
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$`\rho`$ can be calculated via
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```math
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\rho(\vec{x})=-\frac{1}{\pi}\mathrm{Im}\int_{-\infty}^{E_F}\mathrm{d}E\mathcal{G}(\vec{x},\vec{x};E)
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```
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which gives the exression
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\$\$\\rho=-\\frac{1}{\\pi}\\int\\mathrm{d}\\vec{x}\\rho(\\vec{x})=-\\frac{1}{\\pi}\\sum\_{n}\\int\\mathrm{d}\\vec{r}
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\\sum\_{L=(l,m)}\\rho\^n\_L(r) Y\_L(\\hat{r})\$\$ where the second
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```math
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\rho=-\frac{1}{\pi}\int\mathrm{d}\vec{x}\rho(\vec{x})=-\frac{1}{\pi}\sum_{n}\int\mathrm{d}\vec{r}
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\sum_{L=(l,m)}\rho^n_L(r) Y_L(\hat{r})
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```
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where the second
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expression is the formulation in the KKR formalism where all quantities
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are expressed in atom centered (at \$\\vec{R}\^n\$) voronoi cells and
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therein expanded in real spherical harmonics \$Y\_L(\\hat{r})\$. {{
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:kkrimp/voronoi\_cells.png?600 \|Division of space in KKR formalism}}
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are expressed in atom centered (at $`\vec{R}^n`$) voronoi cells and
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therein expanded in real spherical harmonics $`Y_L(\hat{r})`$. {{
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:kkrimp/voronoi_cells.png?600 \|Division of space in KKR formalism}}
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##### Impurity embedding {#impurity_embedding}
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## Impurity embedding
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The KKR formalism relies on the multiple scattering principle. The basic
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idea is to divide the problem of calculating the Green function of a
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| ... | ... | @@ -38,57 +53,70 @@ crystal into three steps: (i) compute the scattering properties of free |
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Lippmann-Schwinger equation, (ii) construct the multiple scattering via
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the structural Green function from systems geometry and (iii) construct
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the Green from the in (i) obtained wavefunctions and single-site
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\$t\$-matrix and the in thep (ii) computed structural GF.
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$`t$`-matrix and the in thep (ii) computed structural GF.
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#### (i) Single-site problem {#i_single_site_problem}
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### (i) Single-site problem
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Solving the single site problem is done by computing the solution of the
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Lippmann-Schwinger equation
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\$\$R\^n\_L(\\vec{r})=J\_L(\\vec{r})+\\sqrt{E}\\sum\_{L''}H\_{L''}(\\vec{r})t\_{L'',L}(E)\$\$
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With the singel-site \$t\$-matrix that is defined as
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\$\$\\underline{\\underline{t}}=V+V\\underline{\\underline{g}}\_{ref}\\underline{\\underline{t}}\$\$
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This contains the atomic potential \$V\$ and the analytically known
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reference GF \$g\_{ref}\$ which traditionally used to be the free space
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```math
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R^n_L(\vec{r})=J_L(\vec{r})+\sqrt{E}\sum_{L''}H_{L''}(\vec{r})t_{L'',L}(E)
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```
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With the singel-site $`t`$-matrix that is defined as
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```math
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\underline{\underline{t}}=V+V\underline{\underline{g}}_{ref}\underline{\underline{t}}
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```
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This contains the atomic potential $`V`$ and the analytically known
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reference GF $`g_{ref}`$ which traditionally used to be the free space
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green function by nowadays for numerical reasons is the Green function
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of repulsive muffin-tin potentials.
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#### (ii) Structural Green function {#ii_structural_green_function}
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### (ii) Structural Green function
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The structural Green function takes the system's geometry via the
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analytically known reference green function
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\$\\underline{\\underline{g}}\_{ref}\^{nn'}\$ and includes the multiple
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$`\underline{\underline{g}}_{ref}^{nn'}`$ and includes the multiple
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scattering properties in theis lattice by solving the Dyson equation
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\$\$\\underline{\\underline{G}}\^{nn'}=\\underline{\\underline{g}}\_{ref}\^{nn'}+\\sum\_{n''}
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\\underline{\\underline{g}}\_{ref}\^{nn''}\\underline{\\underline{t}}\^{n''}\\underline{\\underline{G}}\^{n''n'}\$\$
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Where the atomic scattering properties at site \$n''\$ enter via the
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single-site \$t\$ matrix \$\\underline{\\underline{t}}\^{n''}\$.
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#### (iii) Calculation of the full Green function {#iii_calculation_of_the_full_green_function}
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```math
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\underline{\underline{G}}^{nn'}=\underline{\underline{g}}_{ref}^{nn'}+\sum_{n''}
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\underline{\underline{g}}_{ref}^{nn''}\underline{\underline{t}}^{n''}\underline{\underline{G}}^{n''n'}
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```
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Where the atomic scattering properties at site $`n''`$ enter via the
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single-site $`t`$ matrix $`\underline{\underline{t}}^{n''}`$.
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### (iii) Calculation of the full Green function
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The last step is to combine the two steps that were explained above and
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compute the Green function of the system that contains a single-site
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contribution and a multiple scattering or backscattering contribution.
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Then the charge density at the atomic site \$n\$ is given by
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\$\$\\rho\_L\^{n}(r)=-\\frac{1}{\\pi}\\int\_{-\\infty}\^{E\_F}\\mathrm{d}E\\left\[
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\\underline{R}\_L\^n(r;E)\\underline{\\overline{S}}\_L\^n(r;E)+\\underline{R}\_L\^n(r;E)\\underline{\\underline{G}}\^{nn}\\underline{\\overline{G}}\_L\^n(r;E)
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\\right\]\$\$ Where the expression in square brackets is the Trace over
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Then the charge density at the atomic site $`n`$ is given by
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```math
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\rho_L^{n}(r)=-\frac{1}{\pi}\int_{-\infty}^{E_F}\mathrm{d}E\left\[
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\underline{R}_L^n(r;E)\underline{\overline{S}}_L^n(r;E)+\underline{R}_L^n(r;E)\underline{\underline{G}}^{nn}\underline{\overline{G}}_L^n(r;E)
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\right\]
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```
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Where the expression in square brackets is the Trace over
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the Green function and the first part is called single-site contribution
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and the latter multiple-scattering or backscattering contribution.
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##### Self-consistent algorithm {#self_consistent_algorithm}
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## Self-consistent algorithm
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To calculate the impurity Green function we make use of the Dyson
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equation. The idea is that an impurity locally changes the potential.
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This perturbation extends only over a small area that typically
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containes the impurity and the first few shells of neighbors. Then
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outside this impurity region the potentials do not change any more and
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consequently the difference in the single-site \$t\$-matrices of these
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positions vanish, i.e. \$\\Delta t=0\$. This allows us to embed the
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consequently the difference in the single-site $t$-matrices of these
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positions vanish, i.e. $`\Delta t=0`$. This allows us to embed the
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impurity in a finite real space cluster into the host system. The
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calcualtion is then done as follows.
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` - Compute the host systems Green function and write out the structural Green function as well as the single-site $t$ matrix of the host $t_{\mathrm{host}}$.`\
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` - From the impurity potential compute the $t$-matrix of the impurity and construct $\Delta t=t_{\mathrm{host}}-t_{\mathrm{imp}}$`\
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` - Solve the impurity Dyson equation: $\underline{\underline{G}}=\underline{\underline{G}}_{\mathrm{host}}+\underline{\underline{G}}_{\mathrm{host}}\Delta t\underline{\underline{G}}$`\
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` - Compute the new impurity potential from the Green function and update the input potential`\
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` - Repeat steps 2.-4. until the impurity potential converges` |
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- Compute the host systems Green function and write out the structural Green function as well as the single-site $`t`$ matrix of the host $`t_{\mathrm{host}}`$.
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- From the impurity potential compute the $t$-matrix of the impurity and construct $`\Delta t=t_{\mathrm{host}}-t_{\mathrm{imp}}`$
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- Solve the impurity Dyson equation: $`\underline{\underline{G}}=\underline{\underline{G}}_{\mathrm{host}}+\underline{\underline{G}}_{\mathrm{host}}\Delta t\underline{\underline{G}}`$
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- Compute the new impurity potential from the Green function and update the input potential
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- Repeat steps 2.-4. until the impurity potential converges |
