diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 53ae262..14efb8c 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -1548,56 +1548,270 @@ outside. \end_layout \begin_layout Standard -In our convention, the regular translation operator can be expressed explicitly - as (TODO CHECK CAREFULLY FOR POSSIBLE -\begin_inset Formula $(-1)^{m'}$ +In our convention, the regular translation operator elements can be expressed + explicitly as +\begin_inset Formula +\begin{align} +\tropr_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\nonumber \\ +\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator} +\end{align} + \end_inset - AND SIMILAR FACTORS AND REWRITE IN TERMS OF SPHERICAL HARMONICS) +and analogously the elements of the singular operator +\begin_inset Formula $\trops$ +\end_inset + +, having spherical Hankel functions ( +\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$ +\end_inset + +) in the radial part instead of the regular bessel functions, +\begin_inset Formula +\begin{align} +\trops_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\nonumber \\ +\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator singular} +\end{align} + +\end_inset + +where the constant factors in our convention read (TODO CHECK ONCE AGAIN + CAREFULLY FOR POSSIBLE PHASE FACTORS FACTORS) \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout -Teďka jsou tam zkopírovány výrazy pro C a D z Kristenssona, chybějí fase -\end_layout - +Original Kristensson's +\begin_inset Formula $C,D's$ \end_inset - +from F.7: \begin_inset Formula -\begin{multline} -\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sqrt{\frac{\varepsilon_{m}\varepsilon_{m'}}{4}}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ \times\begin{pmatrix}l & l' & \lambda\\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ m & -m' & m'-m \end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\ \times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\ -\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=-i\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +D_{ml,m'l'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sqrt{\frac{\varepsilon_{m}\varepsilon_{m'}}{4}}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ \times\begin{pmatrix}l & l' & \lambda-1\\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ m & -m' & m'-m \end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\ -\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'.\label{eq:translation operator} -\end{multline} +\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'. +\end{multline*} \end_inset -The singular operator -\begin_inset Formula $\trops$ +where I have found a +\begin_inset Formula $-i$ \end_inset - for re-expanding outgoing waves into regular ones has the same form except - the regular spherical Bessel functions -\begin_inset Formula $j_{l}$ + factor in the +\begin_inset Formula $\tau\ne\tau'$ \end_inset - in are replaced with spherical Hankel functions -\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$ + coefficients, so I force it here: +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +\times\begin{pmatrix}l & l' & \lambda\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\ +\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\ +D_{ml,m'l'}\left(\vect d\right)=-i\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +\times\begin{pmatrix}l & l' & \lambda-1\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\ +\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'. +\end{multline*} + +\end_inset + + TODO check influence of the +\begin_inset Formula $\varepsilon_{m}$ +\end_inset + +s, whether they can be just removed as above. + If we take our definition of spherical harmonics, +\begin_inset Formula +\[ +\ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right) +\] + +\end_inset + +so +\begin_inset Formula +\[ +\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}=\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right) +\] + +\end_inset + +and taking into account that we use the CS phase +\begin_inset Formula $\dlmfFer lm\left(\cos\theta\right)=\left(-1\right)^{m}P_{l}^{m}$ +\end_inset + +, and that +\begin_inset Formula $\left(-1\right)^{m+m'}=\left(-1\right)^{m-m'}$ +\end_inset + + we have +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\frac{1}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +\times\begin{pmatrix}l & l' & \lambda\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\ +\times j_{\lambda}\left(d\right)\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\ +D_{ml,m'l'}\left(\vect d\right)=-i\frac{1}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +\times\begin{pmatrix}l & l' & \lambda-1\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\ +\times j_{\lambda}\left(d\right)\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'. +\end{multline*} + +\end_inset + + +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\frac{1}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +\times\begin{pmatrix}l & l' & \lambda\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\ +\times j_{\lambda}\left(d\right)\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right),\\ +D_{ml,m'l'}\left(\vect d\right)=-i\frac{1}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +\times\begin{pmatrix}l & l' & \lambda-1\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\ +\times j_{\lambda}\left(d\right)\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'. +\end{multline*} + +\end_inset + + +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\frac{1}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ +\times\begin{pmatrix}l & l' & \lambda\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\ +\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\\ +D_{ml,m'l'}\left(\vect d\right)=-i\frac{1}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ +\times\begin{pmatrix}l & l' & \lambda-1\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\ +\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'. +\end{multline*} + +\end_inset + +and finally +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ +\times\begin{pmatrix}l & l' & \lambda\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\ +\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\\ +D_{ml,m'l'}\left(\vect d\right)=-i\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ +\times\begin{pmatrix}l & l' & \lambda-1\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\ +\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'. +\end{multline*} + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{multline*} +C_{lm;l'm'}^{\lambda}=\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ +\times\begin{pmatrix}l & l' & \lambda\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right),\\ +D_{lm;l'm'}^{\lambda}=-i\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ +\times\begin{pmatrix}l & l' & \lambda-1\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}. +\end{multline*} + +\end_inset + +Here +\begin_inset Formula $\begin{pmatrix}l_{1} & l_{2} & l_{3}\\ +m_{1} & m_{2} & m_{3} +\end{pmatrix}$ +\end_inset + + is the +\begin_inset Formula $3j$ +\end_inset + + symbol defined as in +\begin_inset CommandInset citation +LatexCommand cite +after "§34.2" +key "NIST:DLMF" +literal "false" + \end_inset . + Importantly for practical calculations, these rather complicated coefficients + need to be evaluated only once up to the highest truncation order, +\begin_inset Formula $l,l'\le L$ +\end_inset + +. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO write more here. +\end_layout + +\end_inset + + \end_layout \begin_layout Standard @@ -1618,12 +1832,11 @@ todo different notation for the complex conjugation without transposition??? or in the per-particle matrix notation, \begin_inset Formula \begin{equation} -\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}\label{eq:regular translation unitarity} +\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}.\label{eq:regular translation unitarity} \end{equation} \end_inset -. Note that truncation at finite multipole degree breaks the unitarity, \begin_inset Formula $\truncated{\troprp qp}L^{-1}\ne\truncated{\troprp pq}L=\truncated{\troprp qp^{\dagger}}L$ \end_inset diff --git a/lepaper/p6m_kpoint.png b/lepaper/p6m_kpoint.png new file mode 100644 index 0000000..125f46a Binary files /dev/null and b/lepaper/p6m_kpoint.png differ diff --git a/lepaper/p6m_mpoint.png b/lepaper/p6m_mpoint.png new file mode 100644 index 0000000..eb4ead0 Binary files /dev/null and b/lepaper/p6m_mpoint.png differ diff --git a/lepaper/symmetries.lyx b/lepaper/symmetries.lyx index aaed97f..01b306a 100644 --- a/lepaper/symmetries.lyx +++ b/lepaper/symmetries.lyx @@ -712,15 +712,15 @@ This means that the field expansion coefficients transform as \begin_inset Formula \begin{align} -\rcoeffptlm p{\tau}lm & \mapsto\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\ -\outcoeffptlm p{\tau}lm & \mapsto\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation} +\rcoeffptlm p{\tau}lm & \overset{g}{\longmapsto}\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\ +\outcoeffptlm p{\tau}lm & \overset{g}{\longmapsto}\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation} \end{align} \end_inset Obviously, the expansion coefficients belonging to particles in different orbits do not mix together. - As before, we introduce a short-hand block-matrix notation for + As before, we introduce a short-hand pairwise matrix notation for \begin_inset CommandInset ref LatexCommand eqref reference "eq:excitation coefficient under symmetry operation" @@ -730,14 +730,23 @@ noprefix "false" \end_inset - (TODO avoid notation clash here in a more consistent and readable way!) + (TODO avoid notation clash here in a more consistent and readable way! +\begin_inset Formula +\begin{align} +\rcoeffp p & \overset{g}{\longmapsto}\tilde{J}\left(g\right)\rcoeffp{\pi_{g}^{-1}(p)},\nonumber \\ +\outcoeffp p & \overset{g}{\longmapsto}\tilde{J}\left(g\right)\outcoeffp{\pi_{g}^{-1}(p)},\label{eq:excitation coefficient under symmetry operation matrix form} +\end{align} + +\end_inset + +and also a global block-matrix form \end_layout \begin_layout Standard \begin_inset Formula \begin{align} -\rcoeff & \mapsto J\left(g\right)a,\nonumber \\ -\outcoeff & \mapsto J\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation block form} +\rcoeff & \overset{g}{\longmapsto}J\left(g\right)a,\nonumber \\ +\outcoeff & \overset{g}{\longmapsto}J\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation global block form} \end{align} \end_inset @@ -1134,7 +1143,8 @@ The transformation to the symmetry adapted basis \begin_inset Formula $J(g)$ \end_inset -. +: this can happen if the point group symmetry maps some of the scatterers + from the reference unit cell to scatterers belonging to other unit cells. This is illustrated in Fig. \begin_inset CommandInset ref @@ -1147,14 +1157,232 @@ noprefix "false" \end_inset . + Fig. + +\begin_inset CommandInset ref +LatexCommand ref +reference "Phase factor illustration" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +a shows a hexagonal periodic array with +\begin_inset Formula $p6m$ +\end_inset + + wallpaper group symmetry, with lattice vectors +\begin_inset Formula $\vect a_{1}=\left(a,0\right)$ +\end_inset + + and +\begin_inset Formula $\vect a_{2}=\left(a/2,\sqrt{3}a/2\right)$ +\end_inset + +. + If we delimit our representative unit cell as the Wigner-Seitz cell with + origin in a +\begin_inset Formula $D_{6}$ +\end_inset + + point group symmetry center (there is one per each unit cell). + Per unit cell, there are five different particles placed on the unit cell + boundary, and we need to make a choice to which unit cell the particles + on the boundary belong; in our case, we choose that a unit cell includes + the particles on the left as denoted by different colors. + If the Bloch vector is at the upper +\begin_inset Formula $M$ +\end_inset + +point, +\begin_inset Formula $\vect k=\vect M_{1}=\left(0,2\pi/\sqrt{3}a\right)$ +\end_inset + +, it creates a relative phase of +\begin_inset Formula $\pi$ +\end_inset + + between the unit cell rows, and the original +\begin_inset Formula $D_{6}$ +\end_inset + + symmetry is reduced to +\begin_inset Formula $D_{2}$ +\end_inset + +. + The +\begin_inset Quotes eld +\end_inset + +horizontal +\begin_inset Quotes erd +\end_inset + + mirror operation +\begin_inset Formula $\sigma_{xz}$ +\end_inset + + maps, acording to our boundary division, all the particles only inside + 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}, +\end{align*} + +\end_inset + +as in eq. + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:excitation coefficient under symmetry operation" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + However, both the +\begin_inset Quotes eld +\end_inset + +vertical +\begin_inset Quotes erd +\end_inset + + mirroring +\begin_inset Formula $\sigma_{yz}$ +\end_inset + + and the +\begin_inset Formula $C_{2}$ +\end_inset + + rotation map the boundary particles onto the boundaries that do not belong + to the reference unit cell with +\begin_inset Formula $\vect n=\left(0,0\right)$ +\end_inset + +, so we have, explicitly writing down also the lattice point indices +\begin_inset Formula $\vect n$ +\end_inset + +, +\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}, +\end{align*} + +\end_inset + +but we want +\begin_inset Formula $J(g)$ +\end_inset + + to operate only inside one unit cell, so we use the Bloch condition +\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$ +\end_inset + +: in this case, we have +\begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect 0\alpha}e^{i0}=\outcoeffp{\vect 0\alpha}$ +\end_inset + +, +\begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect 0\alpha}=e^{i\pi}\outcoeffp{\vect 0\alpha}=-\outcoeffp{\vect 0\alpha},$ +\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}. +\end{align*} + +\end_inset + +If we set instead +\begin_inset Formula $\vect k=\vect K=\left(4\pi/3a,0\right),$ +\end_inset + +the original +\begin_inset Formula $D_{6}$ +\end_inset + + point group symmetry reduces to +\begin_inset Formula $D_{3}$ +\end_inset + + and the unit cells can obtain a relative phase factor of +\begin_inset Formula $e^{-2\pi i/3}$ +\end_inset + + (blue) or +\begin_inset Formula $e^{2\pi i/3}$ +\end_inset + + (red). + The +\begin_inset Formula $\sigma_{xz}$ +\end_inset + + mirror symmetry, as in the previous case, acts purely inside the reference + unit cell with our boundary division. + However, for a counterclockwise +\begin_inset Formula $C_{3}$ +\end_inset + + 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}, +\end{align*} + +\end_inset + +because in this case, the Bloch condition gives +\begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect 0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect 0\alpha}e^{2\pi i/3}=\outcoeffp{\vect 0\alpha}$ +\end_inset + +, +\begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect 0\alpha}.$ +\end_inset + + +\end_layout + +\begin_layout Standard \begin_inset Float figure placement document alignment document wide false sideways false -status open +status collapsed \begin_layout Plain Layout +\align center +\begin_inset Graphics + filename p6m_mpoint.png + width 100col% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Graphics + filename p6m_kpoint.png + width 100col% + +\end_inset + \end_layout @@ -1174,10 +1402,6 @@ name "Phase factor illustration" \end_inset -\end_layout - -\begin_layout Plain Layout - \end_layout \end_inset