diff --git a/notes/ewald_23_z_nonzero.lyx b/notes/ewald_23_z_nonzero.lyx index cd23fc7..3e2fe3c 100644 --- a/notes/ewald_23_z_nonzero.lyx +++ b/notes/ewald_23_z_nonzero.lyx @@ -241,6 +241,64 @@ For \end_inset +\end_layout + +\begin_layout Section + +\lang english +Ewald long range integral +\end_layout + +\begin_layout Standard + +\lang english +Linton has (2.24): +\begin_inset Formula +\[ +G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{2\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}\int_{1/\eta}^{\infty\exp\left(i\pi/4\right)}e^{-\kappa^{2}\gamma_{m}^{2}\zeta^{2}/4}e^{-\left|\vect r_{\bot}\right|^{2}/\zeta^{2}}\zeta^{1-d_{c}}\ud\zeta +\] + +\end_inset + +Try substitution +\begin_inset Formula $t=\zeta^{2}$ +\end_inset + +: then +\begin_inset Formula $\ud t=2\zeta\,\ud\zeta$ +\end_inset + + ( +\begin_inset Formula $\ud\zeta=\ud t/2t^{1/2}$ +\end_inset + +) and +\begin_inset Formula +\[ +G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}\int_{1/\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\kappa^{2}\gamma_{m}^{2}t/4}e^{-\left|\vect r_{\bot}\right|^{2}/t}t^{\frac{-d_{c}}{2}}\ud t +\] + +\end_inset + +Try subst. + +\begin_inset Formula $\tau=k^{2}\gamma_{m}^{2}/4$ +\end_inset + + +\end_layout + +\begin_layout Standard + +\lang english +\begin_inset Formula +\[ +G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}\left(\frac{\kappa\gamma_{m}}{2}\right)^{d_{c}}\int_{\kappa^{2}\gamma_{m}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{m}^{2}/4\tau}\tau^{\frac{-d_{c}}{2}}\ud\tau +\] + +\end_inset + + \end_layout \end_body diff --git a/notes/kambe_linton_dict.lyx b/notes/kambe_linton_dict.lyx index f8f6996..3840643 100644 --- a/notes/kambe_linton_dict.lyx +++ b/notes/kambe_linton_dict.lyx @@ -371,5 +371,16 @@ Let's look at the Ewald parameter (integration limits) \end_layout +\begin_layout Standard +\begin_inset Formula +\[ +\Linton{\eta}=\sqrt{\frac{1}{2\Kambe{\omega}}} +\] + +\end_inset + +(Based on comparison of some function arguments, not checked.) +\end_layout + \end_body \end_document