Merge branch 'article' of necada:~/repo/qpms into article
Former-commit-id: 82c5c429c2b2925d57979ced34dbab264b007917
This commit is contained in:
Marek Nečada
commit
44b8146fe2
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@ -1548,56 +1548,270 @@ outside.
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\end_layout
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\begin_layout Standard
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In our convention, the regular translation operator can be expressed explicitly
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as (TODO CHECK CAREFULLY FOR POSSIBLE
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\begin_inset Formula $(-1)^{m'}$
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In our convention, the regular translation operator elements can be expressed
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explicitly as
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\begin_inset Formula
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\begin{align}
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\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 \\
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\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}
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\end{align}
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\end_inset
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AND SIMILAR FACTORS AND REWRITE IN TERMS OF SPHERICAL HARMONICS)
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and analogously the elements of the singular operator
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\begin_inset Formula $\trops$
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\end_inset
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, having spherical Hankel functions (
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\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$
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\end_inset
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) in the radial part instead of the regular bessel functions,
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\begin_inset Formula
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\begin{align}
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\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 \\
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\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}
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\end{align}
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\end_inset
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where the constant factors in our convention read (TODO CHECK ONCE AGAIN
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CAREFULLY FOR POSSIBLE PHASE FACTORS FACTORS)
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\begin_inset Note Note
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status open
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status collapsed
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\begin_layout Plain Layout
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Teďka jsou tam zkopírovány výrazy pro C a D z Kristenssona, chybějí fase
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\end_layout
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Original Kristensson's
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\begin_inset Formula $C,D's$
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\end_inset
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from F.7:
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\begin_inset Formula
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\begin{multline}
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\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\\
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\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}},\\
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\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\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\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}
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\end{multline}
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\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'.
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\end{multline*}
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\end_inset
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The singular operator
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\begin_inset Formula $\trops$
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where I have found a
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\begin_inset Formula $-i$
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\end_inset
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for re-expanding outgoing waves into regular ones has the same form except
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the regular spherical Bessel functions
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\begin_inset Formula $j_{l}$
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factor in the
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\begin_inset Formula $\tau\ne\tau'$
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\end_inset
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in are replaced with spherical Hankel functions
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\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$
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coefficients, so I force it here:
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\begin_inset Formula
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\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}},\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\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'.
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\end{multline*}
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\end_inset
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TODO check influence of the
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\begin_inset Formula $\varepsilon_{m}$
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\end_inset
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s, whether they can be just removed as above.
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If we take our definition of spherical harmonics,
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\begin_inset Formula
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\[
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\ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right)
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\]
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\end_inset
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so
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\begin_inset Formula
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\[
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\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}=\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right)
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\]
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\end_inset
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and taking into account that we use the CS phase
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\begin_inset Formula $\dlmfFer lm\left(\cos\theta\right)=\left(-1\right)^{m}P_{l}^{m}$
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\end_inset
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, and that
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\begin_inset Formula $\left(-1\right)^{m+m'}=\left(-1\right)^{m-m'}$
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\end_inset
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we have
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\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}},\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\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'.
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\end{multline*}
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\end_inset
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\begin_inset Formula
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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||||
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\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),\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\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'.
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\end{multline*}
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\end_inset
|
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|
||||
|
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\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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
|
||||
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
|
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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),\\
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||||
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\\
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||||
\times\begin{pmatrix}l & l' & \lambda-1\\
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||||
0 & 0 & 0
|
||||
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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||||
m & -m' & m'-m
|
||||
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
|
||||
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
|
||||
m & -m' & m'-m
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||||
\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
|
||||
|
|
|
|||
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|
After Width: | Height: | Size: 136 KiB |
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|
After Width: | Height: | Size: 136 KiB |
|
|
@ -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
|
||||
|
|
|
|||
Loading…
Reference in New Issue