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zccZFx|G)TXf8%@m<F<Uwdi-~P*|vFkHT8n==O0Ja*WS5i+>@U%X|3_!(z$EXrS3PC zZaeq!`u!VK(>|{K*Sxl0;{GhwZGJ}k-rT=wb^XUT)!S9uK2CkJ`KxFBU+!pi>G-EA z(bIpumfkimxn^tp?*pavd#_$A?tTAzZcgpt_<CFM&pq$oAIgb+xPJe;&oz&Kw{t~@ z|9owKvnsOj<KNF|(fdDK=iL^UQDeU|?e+D4zh0Mn8t#j)%g)}raSx~Y;(IG^^jR00 zzgqjCc13xue}dvX$Gzs&jNKoMrah3|)6c(=?W^AQf6|Y3zLfi9JN@;4#=8gn;->4V TD9LjhSX!8Hsj9mAyKw;kTOTp+ diff --git a/Doc/DevRef/Multilayers.tex b/Doc/DevRef/Multilayers.tex index d9347c67ec8..27af97d8a50 100644 --- a/Doc/DevRef/Multilayers.tex +++ b/Doc/DevRef/Multilayers.tex @@ -540,7 +540,7 @@ the continuity conditions~\cref{Econtcond} take the form +A^-_l &+A^+_l &=& +A^-_{l-1}\delta_{l-1} &+A^+_{l-1}\delta_{l-1}^{-1}, - \\ + \\[1.3ex] -A^-_l \kappa_l &+A^+_l \kappa_l &=& -A^-_{l-1}\delta_{l-1} \kappa_{l-1} &+A^+_{l-1}\delta_{l-1}^{-1} \kappa_{l-1}. @@ -561,9 +561,9 @@ Parratt \cite{Par54}, unaware of Abelès, expressed the same relation as a recur \index{Parratt recursion}% } \begin{equation}\label{EMil} - M_l \coloneqq \Delta_{l-1} S_l + M_l \coloneqq \Delta_{l-1} S_l, \end{equation} -where +which we write using the phase rotation matrix \begin{equation}\label{DmatD} \Delta_l \coloneqq @@ -572,14 +572,14 @@ where 0 & \delta_{l} \end{array}\right) \end{equation} -and +and the refraction matrix \begin{equation}\label{DmatS} S_l \coloneqq \frac{1}{2} \left(\begin{array}{cc} s^+_l&s^-_l\\ - s^-_l&s^+_l) + s^-_l&s^+_l \end{array}\right) \end{equation} with coefficients diff --git a/Doc/DevRef/Roughness.tex b/Doc/DevRef/Roughness.tex index 3e211a147f7..9190031845c 100644 --- a/Doc/DevRef/Roughness.tex +++ b/Doc/DevRef/Roughness.tex @@ -49,7 +49,7 @@ Graded interfaces have a smooth SLD profile, i.e.\ the function $\overline{v}(z)$ or $\kappa^2(z)$ evolves continuously from one bulk value to the other. Among the SLD profiles that can be solved analytically, -the tanh profile is particularly important. +the tanh (\cref{Ftanhc}a) profile is particularly important. A good summary of the solution can be found in Ch.~2.5 of Lekner \cite{Lek16}.\footnote {He credits Eckart (1930) and Epstein (1930) for the solution. For a short summary, see also \cite[\S~25, exercise~3]{LL3}.} @@ -83,10 +83,6 @@ The transmission coefficient $t_{ab}$ is communicated in \cite{AnMR88}. Using various properties of the Gamma and sinh functions, one can verify flux conservation~\cref{EConservation}. -%==================================================================================================% -\subsection{Fresnel coefficients with roughness factor} -%==================================================================================================% - In the limit $\tau\to0$, the phase factor $\varphi$ in \cref{ErTanh} goes to zero. For simplicity, we let $\varphi=0$ throughout. This approximation is equivalent to an adjustment of the interface position~$z_{ab}$ @@ -97,7 +93,7 @@ by an amount that can be expected to be small compared to the interface thicknes \includegraphics[width=0.41\textwidth]{fig/funcplot/tanhc.ps} \hfill \includegraphics[width=0.41\textwidth]{fig/funcplot/FresnelReductionTanh.ps} -\caption{(a) Function tanhc. (b) Reflectivity reduction factor, +\caption{(a) Functions tanh and tanhc. (b) Reflectivity reduction factor, obtained by dividing \cref{ErTanh2} through the Fresnel reflectivity~\cref{EtFresnel}, as function of $\kappa_a \tau$ for ratios $\kappa_b / \kappa_a$ of 0.1, 0.2, 0.4, 0.9, 1.1, 2, and~5.} @@ -140,15 +136,9 @@ Tests confirm that flux conservation is fulfilled at machine precision.\footnote Eq.~\cref{ESab2}, which relates the refraction matrix to transmission and reflection coefficients, remains valid for our modified coefficients \cref{ErTanh2} and~\cref{EtTanh2}. -Accordingly, \cref{ESab} must be replaced by -\begin{equation}\label{ESabR} - S_{ab} - \coloneqq - \frac{1}{2\kappa_a} - \left(\begin{array}{ll} - (R_{ab}^{-1}\kappa_a+R_{ab}\kappa_b)&(R_{ab}^{-1}\kappa_a-R_{ab}\kappa_b)\\ - (R_{ab}^{-1}\kappa_a-R_{ab}\kappa_b)&(R_{ab}^{-1}\kappa_a+R_{ab}\kappa_b) - \end{array}\right). +Accordingly, the coefficients~\cref{Dslpm} must be replaced by +\begin{equation}\label{EslpmTanh} + s^\pm_l = R_{l,l-1}^{-1} \pm R_{l,l-1}\kappa_{l-1}/\kappa_l. \end{equation} %==================================================================================================% @@ -175,28 +165,23 @@ This interpretation is mentioned by N\'evot et al. \cite{NePC88}. More questionable is the simultaneous modification of the transmission coefficient. Currently BornAgain uses \begin{align}\label{EtNC} -\tilde t_{ab} &= t_{ab}\, \e^{+\left( k_a - k_b \right)^2 \sigma_{ab}^2/2}, +\tilde t_{ab} &= t_{ab}\, \e^{+\left( k_a - k_b \right)^2 \sigma^2/2}, \end{align} where $t_{ab}$ is the Fresnel coefficient \cref{EtFresnel}. This is the result obtained by Tolan \cite[Eq.~2.35]{Tol99}, and is also given by de Boer \cite{BoLe96} as a result from formal perturbation theory in the limit of very small lateral correlation length. -With \cref{ESab2}, we obtain the refraction matrix -\begin{align}\label{ESabGibaud} -\tilde S_{ab} &= \frac{1}{2\kappa_a} \begin{pmatrix} - \left( \kappa_a + \kappa_b \right) \NCm & - \left( \kappa_a - \kappa_b \right) \NCp\\[.2cm] - \left( \kappa_a - \kappa_b \right) \NCp & - \left( \kappa_a + \kappa_b \right) \NCm -\end{pmatrix}, -\end{align} +With \cref{ESab2}, we obtain the coefficients to replace~\cref{Dslpm}, +\begin{equation}\label{EslpmNC} + s^\pm_l = (1 \pm \kappa_{l-1}/\kappa_l) \exp(-(\kappa_{l-1}\mp\kappa_l)^2\sigma^2/2), +\end{equation} which shows that the above is also consistent with \cite[Eq.~3.114]{GiVi09}. However, the total reflected and transmitted flux $\kappa_a|\tilde r_{ab}|^2+\kappa_b|\tilde t_{ab}|^2$, computed as in \cref{EConservation}, is \emph{greater} than the incoming flux~$\kappa_a$. -This takes all credibility from \cref{EtNC} and \cref{ESabGibaud}. +This takes all credibility from \cref{EtNC} and \cref{EslpmNC}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Scattering by a rough interface}\label{Sroughscatter} diff --git a/Doc/DevRef/fig/funcplot/tanhc.ps b/Doc/DevRef/fig/funcplot/tanhc.ps index 70c14e68ffc..8effb78624d 100644 --- a/Doc/DevRef/fig/funcplot/tanhc.ps +++ b/Doc/DevRef/fig/funcplot/tanhc.ps @@ -1,5 +1,5 @@ %!PS-Adobe-1.0 EPSF-1.0 -%%BoundingBox: 14 604 258 788 +%%BoundingBox: 7 604 259 788 %%Comment: Bounding box extracted by bboxx %%+: A program by Dov Grobgeld 2003 %%Comment: Bounding box extracted by bboxx @@ -1630,11 +1630,10 @@ WuGdict18a begin { 13 1 0 1. 1. pset 6 ipCol } ] def -{ 3 aCol1 iColA } /icCol x bind def % number of colours and colour style +{ 7 aCol3 iColA } /icCol x bind def % number of colours and colour style /cStyles [ - { 1. [] lset 0 icCol } - { 1. [] lset 1 icCol } - { 1. [] lset 2 icCol } + { 2. [] lset 0 icCol } + { 2. 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2.8112 wx 0.3531567 wy + 6.35452 6.81397 0.00000 t % 2.885733 wx 0.3443801 wy + 6.38796 6.76817 0.00000 t % 2.962241 wx 0.3357822 wy + 6.42140 6.72219 0.00000 t % 3.040778 wx 0.3273639 wy + 6.45485 6.67603 0.00000 t % 3.121398 wx 0.3191258 wy + 6.48829 6.62971 0.00000 t % 3.204155 wx 0.3110679 wy + 6.52174 6.58325 0.00000 t % 3.289106 wx 0.3031898 wy + 6.55518 6.53667 0.00000 t % 3.376309 wx 0.2954905 wy + 6.62207 6.44317 0.00000 t % 3.557713 wx 0.2806231 wy + 6.68896 6.34932 0.00000 t % 3.748863 wx 0.266452 wy + 6.75585 6.25520 0.00000 t % 3.950283 wx 0.2529589 wy + 6.82274 6.16088 0.00000 t % 4.162526 wx 0.2401223 wy + 6.88963 6.06641 0.00000 t % 4.386172 wx 0.2279186 wy + 7.02341 5.87719 0.00000 t % 4.870158 wx 0.205308 wy + 7.29097 5.49825 0.00000 t % 6.004237 wx 0.166547 wy +10.00000 1.65951 0.00000 tf % 50 wx 0.02 wy { black 0 -3 13 1.65 NewList - {(fit_grid1x7)} TxLine - {( cc tanh(t)/t # z from grid1x7)} TxLine + {(curve0)} TxLine + {( cca tanh(t))} TxLine + {( tanh(t))} TxLine + {( weighing: with reciprocal variance (data and curve))} TxLine + {( j oc chi^2 1-R^2)} TxLine + 1 {( 0 0 1 )} CvTxLine + {(curve1)} TxLine + {( cca tanh(t)/t)} TxLine {( tanh(t)/t)} TxLine - {( data file: 3, weighing: constant)} TxLine - {( j z0 oc chi^2 1-R^2)} TxLine - 1 {( 0.1 0 0 1 )} CvTxLine + {( weighing: with reciprocal variance (data and curve))} TxLine + {( j oc chi^2 1-R^2)} TxLine + 2 {( 0 0 1 )} CvTxLine {(plot -> /home/jwu/gnew/L1.ps)} TxLine } oooinfo 1 eq { exec } { pop } ifelse -- GitLab