diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index fe907b6..6d6e187 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -481,7 +481,11 @@ The single-particle scattering problem at frequency \end_inset . - Inside this volume, the electric field can be expanded as + Inside +\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$ +\end_inset + +, the electric field can be expanded as \begin_inset Note Note status open @@ -770,7 +774,7 @@ literal "false" its (maximum) refractive index. \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout \begin_inset Formula @@ -1281,7 +1285,7 @@ In practice, the multiple-scattering problem is solved in its truncated \begin_inset Formula $l\le L_{p}$ \end_inset -, laeving only +, leaving only \begin_inset Formula $N_{p}=2L_{p}\left(L_{p}+2\right)$ \end_inset @@ -1428,11 +1432,7 @@ Let \end_inset -where an explicit formula for the (regular) -\emph on -translation operator -\emph default - +where an explicit formula for the regular translation operator \begin_inset Formula $\tropr$ \end_inset @@ -1547,7 +1547,7 @@ reference "eq:regular vswf translation" , \begin_inset Formula \[ -\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{q}\right) +\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(k\left(\vect r-\vect r_{q}\right)\right) \] \end_inset @@ -1579,7 +1579,12 @@ reference "eq:regular vswf coefficient translation" \end_inset -(note the reversed indices; TODO redefine them in +(note the reversed indices +\begin_inset Note Note +status open + +\begin_layout Plain Layout +; TODO redefine them in \begin_inset CommandInset ref LatexCommand eqref reference "eq:regular vswf translation" @@ -1593,7 +1598,12 @@ reference "eq:singular vswf translation" \end_inset -? Similarly, if we had only outgoing waves in the original expansion around +? +\end_layout + +\end_inset + +) Similarly, if we had only outgoing waves in the original expansion around \begin_inset Formula $\vect r_{p}$ \end_inset diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx index 181d8fa..5c93352 100644 --- a/lepaper/infinite.lyx +++ b/lepaper/infinite.lyx @@ -329,6 +329,16 @@ noprefix "false" \begin_layout Standard As in the case of a finite system, eq. + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Multiple-scattering problem unit cell" +plural "false" +caps "false" +noprefix "false" + +\end_inset + can be written in a shorter block-matrix form, \begin_inset Formula \begin{equation} @@ -526,7 +536,17 @@ noprefix "false" \begin_inset Formula $\left|\vect k\right|=\sqrt{\epsilon\mu}\omega/c_{0}$ \end_inset - (modulo lattice points; TODO write this a clean way). + (modulo reciprocal lattice points +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO write this in a clean way +\end_layout + +\end_inset + +). A somehow challenging step is to distinguish the different bands that can all be very close to the empty lattice approximation, especially if the particles in the systems are small. @@ -687,7 +707,7 @@ translation operator for spherical waves originating in \end_inset is in fact a function of a single 3d argument, -\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$ +\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$ \end_inset . @@ -701,7 +721,7 @@ reference "eq:W integral" can be rewritten as \begin_inset Formula \[ -W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)} +W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0))\left(\vect k\right)} \] \end_inset @@ -735,10 +755,10 @@ reference "eq:Dirac comb uaFt" (REF?) for the Fourier transform of Dirac comb) \begin_inset Formula \begin{eqnarray} -W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\ - & = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\ - & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\ - & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\nonumber +W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)(\vect k)\nonumber \\ + & = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)\left(\vect k\right)\nonumber \\ + & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\ + & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\nonumber \end{eqnarray} \end_inset @@ -840,8 +860,8 @@ reference "eq:W sum in reciprocal space" \begin_inset Formula \begin{eqnarray} W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\ -W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\ -W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition} +W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\ +W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition} \end{eqnarray} \end_inset @@ -879,7 +899,7 @@ CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS \begin_inset Formula \begin{equation} -\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect 0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums} +\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums} \end{equation} \end_inset diff --git a/lepaper/symmetries.lyx b/lepaper/symmetries.lyx index c7f11f2..de25f9c 100644 --- a/lepaper/symmetries.lyx +++ b/lepaper/symmetries.lyx @@ -98,11 +98,11 @@ If the system has nontrivial point group symmetries, group theory gives \end_layout \begin_layout Standard -As an example, if our system has a +As an example, if the system has a \begin_inset Formula $D_{2h}$ \end_inset - symmetry and our truncated + symmetry and the corresponding truncated \begin_inset Formula $\left(I-T\trops\right)$ \end_inset @@ -961,7 +961,7 @@ where \begin_inset Formula $1$ \end_inset - through + to \begin_inset Formula $d_{n}$ \end_inset @@ -969,7 +969,7 @@ where \begin_inset Formula $i$ \end_inset - goes from 1 through the multiplicity of irreducible representation + goes from 1 to the multiplicity of irreducible representation \begin_inset Formula $\Gamma_{n}$ \end_inset @@ -1328,8 +1328,8 @@ horizontal the same unit cell, e.g. \begin_inset Formula \begin{align*} -\outcoeffp{\vect 0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0E},\\ -\outcoeff_{\vect 0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0C}, +\outcoeffp{\vect0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0E},\\ +\outcoeff_{\vect0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0C}, \end{align*} \end_inset @@ -1374,8 +1374,8 @@ vertical , \begin_inset Formula \begin{align*} -\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\ -\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C}, +\outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\ +\outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C}, \end{align*} \end_inset @@ -1385,22 +1385,22 @@ but we want \end_inset to operate only inside one unit cell, so we use the Bloch condition -\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$ +\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$ \end_inset : in this case, we have -\begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect 0\alpha}e^{i0}=\outcoeffp{\vect 0\alpha}$ +\begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect0\alpha}e^{i0}=\outcoeffp{\vect0\alpha}$ \end_inset , -\begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect 0\alpha}=e^{i\pi}\outcoeffp{\vect 0\alpha}=-\outcoeffp{\vect 0\alpha},$ +\begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect0\alpha}=e^{i\pi}\outcoeffp{\vect0\alpha}=-\outcoeffp{\vect0\alpha},$ \end_inset so \begin_inset Formula \begin{align*} -\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0E},\\ -\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0C}. +\outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0E},\\ +\outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0C}. \end{align*} \end_inset @@ -1439,19 +1439,19 @@ the original rotation, as an example we have \begin_inset Formula \begin{align*} -\outcoeffp{\vect 0A} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(0,-1\right)E}=e^{2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0E},\\ -\outcoeff_{\vect 0C} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)A}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0A},\\ -\outcoeff_{\vect 0B} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)B}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0B}, +\outcoeffp{\vect0A} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(0,-1\right)E}=e^{2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect0E},\\ +\outcoeff_{\vect0C} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)A}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect0A},\\ +\outcoeff_{\vect0B} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)B}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect0B}, \end{align*} \end_inset because in this case, the Bloch condition gives -\begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect 0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect 0\alpha}e^{2\pi i/3}=\outcoeffp{\vect 0\alpha}$ +\begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect0\alpha}e^{2\pi i/3}=\outcoeffp{\vect0\alpha}$ \end_inset , -\begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect 0\alpha}.$ +\begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect0\alpha}.$ \end_inset