540 lines
18 KiB
Plaintext
540 lines
18 KiB
Plaintext
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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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1D in 3D Ewald sum
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\end_layout
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\ud}{\mathrm{d}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\mathbf{#1}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\uvec}[1]{\hat{\mathbf{#1}}}
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\end_inset
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\lang english
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\begin_inset FormulaMacro
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\newcommand{\ush}[2]{Y_{#1}^{#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ushD}[2]{Y'_{#1}^{#2}}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\vsh}{\vect A}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\vshD}{\vect{A'}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wfkc}{\vect y}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wfkcout}{\vect u}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wfkcreg}{\vect v}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wckcreg}{a}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wckcout}{f}
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\end_inset
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\end_layout
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\begin_layout Section
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General formula
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\end_layout
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\begin_layout Standard
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We need to find the expansion coefficient
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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\tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{i}{\kappa j_{l'}\left(\kappa\left|\vect r\right|\right)}\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)\ushD{l'}{m'}\left(\uvec r\right).\label{eq:tau extraction formula}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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[Linton, (2.24)] with slightly modified notation and setting
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\begin_inset Formula $d_{c}=2$
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\end_inset
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:
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\begin_inset Formula
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\[
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G_{\Lambda}^{(1;\kappa)}\left(\vect r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect r}\int_{1/\eta}^{\infty e^{i\pi/4}}e^{-\kappa^{2}\gamma^{2}t^{2}/4}e^{-\left|\vect r^{\bot}\right|^{2}/t^{2}}t^{1-d_{c}}\ud t
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\]
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\end_inset
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or, evaluated at point
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\begin_inset Formula $\vect s+\vect r$
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\end_inset
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instead
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\begin_inset Formula
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\[
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G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\left(\vect s+\vect r\right)}\int_{1/\eta}^{\infty e^{i\pi/4}}e^{-\kappa^{2}\gamma^{2}t^{2}/4}e^{-\left|\vect s^{\bot}+\vect r^{\bot}\right|^{2}/t^{2}}t^{1-d_{c}}\ud t
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\]
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\end_inset
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The integral can be by substitutions taken into the form
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\lang english
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\begin_inset Formula
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\[
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G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{2\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta}^{\infty\exp\left(i\pi/4\right)}e^{-\kappa^{2}\gamma_{m}^{2}\zeta^{2}/4}e^{-\left|\vect r_{\bot}\right|^{2}/\zeta^{2}}\zeta^{1-d_{c}}\ud\zeta
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\]
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\end_inset
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Try substitution
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\begin_inset Formula $t=\zeta^{2}$
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\end_inset
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: then
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\begin_inset Formula $\ud t=2\zeta\,\ud\zeta$
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\end_inset
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(
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\begin_inset Formula $\ud\zeta=\ud t/2t^{1/2}$
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\end_inset
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) and
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\begin_inset Formula
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\[
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G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\kappa^{2}\gamma_{m}^{2}t/4}e^{-\left|\vect r_{\bot}\right|^{2}/t}t^{\frac{-d_{c}}{2}}\ud t
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\]
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\end_inset
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Try subst.
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\begin_inset Formula $\tau=k^{2}\gamma_{m}^{2}/4$
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\lang english
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\begin_inset Formula
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\[
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G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\left(\frac{\kappa\gamma_{m}}{2}\right)^{d_{c}}\int_{\kappa^{2}\gamma_{m}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{m}^{2}/4\tau}\tau^{\frac{-d_{c}}{2}}\ud\tau
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\]
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\end_inset
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\end_layout
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\end_inset
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\begin_inset Formula
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\[
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G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\left(\vect s+\vect r\right)}\int_{\kappa^{2}\gamma_{m}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{m}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}}\ud\tau
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Foot
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status open
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\begin_layout Plain Layout
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[Linton, (2.25)] with slightly modified notation:
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\begin_inset Formula
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\[
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G_{\Lambda}^{(1;\kappa)}\left(\vect r\right)=-\frac{1}{\sqrt{4\pi}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect r}\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\left|\vect r^{\bot}\right|^{2j}}{j!}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2j-1}\Gamma_{j\vect K}
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\]
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\end_inset
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We want to express an expansion in a shifted point, so let's substitute
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\begin_inset Formula $\vect r\to\vect s+\vect r$
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\end_inset
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\begin_inset Formula
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\[
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G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)=-\frac{1}{\sqrt{4\pi}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\left(\vect s+\vect r\right)}\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\left|\vect s^{\bot}+\vect r^{\bot}\right|^{2j}}{j!}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2j-1}\Gamma_{j\vect K}
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\]
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\end_inset
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\end_layout
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\end_inset
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Let's do the integration to get
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\begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$
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\end_inset
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\begin_inset Formula
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\[
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\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\left(\vect s+\vect r\right)}\int_{\kappa^{2}\gamma_{\vect K}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}}\ud\tau
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\]
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\end_inset
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The
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\begin_inset Formula $\vect r$
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\end_inset
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-dependent plane wave factor can be also written as
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\begin_inset Formula
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\begin{align*}
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e^{i\vect K\cdot\vect r} & =e^{i\left|\vect K\right|\vect r\cdot\uvec K}=4\pi\sum_{lm}i^{l}\mathcal{J}'_{l}^{m}\left(\left|\vect K\right|\vect r\right)\ush lm\left(\uvec K\right)\\
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& =4\pi\sum_{lm}i^{l}j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\ushD lm\left(\uvec{\vect r}\right)\ush lm\left(\uvec K\right)
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\end{align*}
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\end_inset
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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or the other way around
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\begin_inset Formula
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\[
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e^{i\vect K\cdot\vect r}=4\pi\sum_{lm}i^{l}j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect r}\right)\ushD lm\left(\uvec K\right)
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\]
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\end_inset
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\end_layout
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\end_inset
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so
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\begin_inset Formula
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\[
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\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\frac{1}{2\pi\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\ushD lm\left(\uvec r\right)\ush lm\left(\uvec K\right)\int_{\kappa^{2}\gamma_{\vect K}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}}\ud\tau
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\]
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||
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|
|
||
|
|
\end_inset
|
||
|
|
|
||
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|
|
||
|
|
\end_layout
|
||
|
|
|
||
|
|
\begin_layout Standard
|
||
|
|
We also have
|
||
|
|
\begin_inset Formula
|
||
|
|
\begin{align*}
|
||
|
|
e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau} & =e^{-\left(\left|\vect s_{\bot}\right|^{2}+\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\\
|
||
|
|
& =e^{-\left|\vect s_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\sum_{n=0}^{\infty}\frac{1}{n!}\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}}{4\tau}\right)^{n},
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
hence
|
||
|
|
\begin_inset Formula
|
||
|
|
\begin{align*}
|
||
|
|
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right) & =-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\ushD lm\left(\uvec r\right)\ush lm\left(\uvec K\right)\sum_{n=0}^{\infty}\frac{1}{n!}\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}}{4}\right)^{n}\underbrace{\int_{\kappa^{2}\gamma_{\vect K}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}-n}\ud\tau}_{\Delta_{n}^{\left(d\right)}}\\
|
||
|
|
& =-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\ush lm\left(\uvec K\right)\sum_{n=0}^{\infty}\frac{\Delta_{n}^{\left(d\right)}}{n!}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}}{4}\right)^{n}\\
|
||
|
|
& =-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\ush lm\left(\uvec K\right)\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{n!}\Delta_{n}^{\left(d\right)}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2n}\sum_{k=0}^{n}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left|\vect r_{\bot}\right|^{2(n-k)}\left(2\vect r_{\bot}\cdot\vect s_{\bot}\right)^{k}
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
If we label
|
||
|
|
\begin_inset Formula $\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|\cos\varphi\equiv\vect r_{\bot}\cdot\vect s_{\bot}$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
, we have
|
||
|
|
\begin_inset Formula
|
||
|
|
\[
|
||
|
|
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\ush lm\left(\uvec K\right)\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{n!}\Delta_{n}^{\left(d\right)}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2n}\sum_{k=0}^{n}\left(2\left|\vect s_{\bot}\right|\right)^{k}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left|\vect r_{\bot}\right|^{2n-k}\left(\cos\varphi\right)^{k}
|
||
|
|
\]
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
and if we label
|
||
|
|
\begin_inset Formula $\left|\vect r\right|\sin\vartheta\equiv\left|\vect r_{\bot}\right|$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
|
||
|
|
\begin_inset Formula
|
||
|
|
\[
|
||
|
|
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\ush lm\left(\uvec K\right)\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{n!}\Delta_{n}^{\left(d\right)}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2n}\sum_{k=0}^{n}\left|\vect r\right|^{2n-k}\left(2\left|\vect s_{\bot}\right|\right)^{k}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\sin\vartheta\right)^{2n-k}\left(\cos\varphi\right)^{k}
|
||
|
|
\]
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
Now let's put the RHS into
|
||
|
|
\begin_inset CommandInset ref
|
||
|
|
LatexCommand eqref
|
||
|
|
reference "eq:tau extraction formula"
|
||
|
|
plural "false"
|
||
|
|
caps "false"
|
||
|
|
noprefix "false"
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
and try eliminating some sum by taking the limit
|
||
|
|
\begin_inset Formula $\left|\vect r\right|\to0$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
.
|
||
|
|
We have
|
||
|
|
\begin_inset Formula $j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\sim\left(\left|\vect K\right|\left|\vect r\right|\right)^{l}/\left(2l+1\right)!!$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
; the denominator from
|
||
|
|
\begin_inset CommandInset ref
|
||
|
|
LatexCommand eqref
|
||
|
|
reference "eq:tau extraction formula"
|
||
|
|
plural "false"
|
||
|
|
caps "false"
|
||
|
|
noprefix "false"
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
behaves like
|
||
|
|
\begin_inset Formula $j_{l'}\left(\kappa\left|\vect r\right|\right)\sim\left(\kappa\left|\vect r\right|\right)^{l'}/\left(2l'+1\right)!!.$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
The leading terms are hence those with
|
||
|
|
\begin_inset Formula $\left|\vect r\right|^{l-l'+2n-k}$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
.
|
||
|
|
So
|
||
|
|
\begin_inset Formula
|
||
|
|
\[
|
||
|
|
\tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa^{1+l'}}\left(2l'+1\right)!!\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{lm}4\pi i^{l}\frac{\left|\vect K\right|^{l}}{\left(2l+1\right)!!}\ush lm\left(\uvec K\right)\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{n!}\Delta_{n}^{\left(d\right)}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2n}\sum_{k=0}^{n}\delta_{l'-l,2n-k}\left(2\left|\vect s_{\bot}\right|\right)^{k}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\sin\vartheta\right)^{l'-l}\left(\cos\varphi\right)^{k}.
|
||
|
|
\]
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
Let's now focus on rearranging the sums; we have
|
||
|
|
\begin_inset Formula
|
||
|
|
\[
|
||
|
|
S(l')\equiv\sum_{l=0}^{\infty}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\delta_{l'-l,2n-k}f(l',l,n,k)=\sum_{l=0}^{\infty}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\delta_{l'-l,2n-k}f(l',l,n,2n-l'+l)
|
||
|
|
\]
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
We have
|
||
|
|
\begin_inset Formula $0\le k\le n$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
, hence
|
||
|
|
\begin_inset Formula $0\le2n-l'+l\le n$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
, hence
|
||
|
|
\begin_inset Formula $-2n\le-l'+l\le-n$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
, hence also
|
||
|
|
\begin_inset Formula $l'-2n\le l\le l'-n$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
, which gives the opportunity to swap the
|
||
|
|
\begin_inset Formula $l,n$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
sums and the
|
||
|
|
\begin_inset Formula $l$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
-sum becomes finite; so also consuming
|
||
|
|
\begin_inset Formula $\sum_{k=0}^{n}\delta_{l'-l,2n-k}$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
we get
|
||
|
|
\begin_inset Formula
|
||
|
|
\[
|
||
|
|
S(l')=\sum_{n=0}^{\infty}\sum_{l=\max(0,l'-2n)}^{l'-n}f(l',l,n,2n-l'+l).
|
||
|
|
\]
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
Finally, we see that the interval of valid
|
||
|
|
\begin_inset Formula $l$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
becomes empty when
|
||
|
|
\begin_inset Formula $l'-n<0$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
, i.e.
|
||
|
|
|
||
|
|
\begin_inset Formula $n>l'$
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
; so we get a finite sum
|
||
|
|
\begin_inset Formula
|
||
|
|
\[
|
||
|
|
S(l')=\sum_{n=0}^{l'}\sum_{l=\max(0,l'-2n)}^{l'-n}f(l',l,n,2n-l'+l).
|
||
|
|
\]
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
Applying rearrangement,
|
||
|
|
\begin_inset Formula
|
||
|
|
\[
|
||
|
|
\tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa^{'}}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{n=0}^{l'}\frac{\left(-1\right)^{n}}{n!}\Delta_{n}^{\left(d\right)}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2n}\sum_{l=\max\left(0,l'-2n\right)}^{l'-n}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2n-l'+l}\frac{\left|\vect K\right|^{l}}{\left(2l+1\right)!!}\sum_{m=-l}^{l}\ush lm\left(\uvec K\right)\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\sin\vartheta\right)^{l'-l}\left(\cos\varphi\right)^{2n-l'+l},
|
||
|
|
\]
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
or replacing the anles with their original definition,
|
||
|
|
\begin_inset Formula
|
||
|
|
\[
|
||
|
|
\tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa^{'}}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{n=0}^{l'}\frac{\left(-1\right)^{n}}{n!}\Delta_{n}^{\left(d\right)}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2n}\sum_{l=\max\left(0,l'-2n\right)}^{l'-n}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2n-l'+l}\frac{\left|\vect K\right|^{l}}{\left(2l+1\right)!!}\sum_{m=-l}^{l}\ush lm\left(\uvec K\right)\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\frac{\left|\vect r_{\bot}\right|}{\left|\vect r\right|}\right)^{l'-l}\left(\frac{\vect r_{\bot}\cdot\vect s_{\bot}}{\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|}\right)^{2n-l'+l},
|
||
|
|
\]
|
||
|
|
|
||
|
|
\end_inset
|
||
|
|
|
||
|
|
|
||
|
|
\end_layout
|
||
|
|
|
||
|
|
\end_body
|
||
|
|
\end_document
|