diff --git a/Doc/DevRef/DevRef.pdf b/Doc/DevRef/DevRef.pdf
index 9e9d911b180c3db1f4eb78357d25ae402b6ce15b..fc66d330f16feb85225709aef1277c0aa005350e 100644
Binary files a/Doc/DevRef/DevRef.pdf and b/Doc/DevRef/DevRef.pdf differ
diff --git a/Doc/DevRef/Multilayers.tex b/Doc/DevRef/Multilayers.tex
index d9347c67ec813d69f0ce85399725b3881c9fa8c4..27af97d8a50b81c5774b3db1f95df05dc9009eba 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 3e211a147f77bb922f54913dc61044e9efbb69a7..9190031845c691990f763d9ea1179b026219c55c 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 70c14e68ffc08d95183feaa1137f0bb7e854e5d8..8effb78624df0a9bc9c013317cac8509fca9dd98 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. [] lset 1 icCol }
] def
Resets
@@ -1646,8 +1645,8 @@ BoxBackground
%% output created by Frida version post-2.4.4a
-1 0.03 30 xSetCoord
-1 0.02 2 ySetCoord
+1 0.02 50 xSetCoord
+1 0.008 2 ySetCoord
% x axis:
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@@ -1655,258 +1654,449 @@ BoxBackground
1 wx {(1)}
10 wx {(10)}
] SetTacVec
-0.01 wx 100 wx 5 9 SetTicVecLog
+0.01 wx 100 wx 5 3 SetTicVecLog
0 10 0 0 0 90 OneAxx Axx Tic Tac xNumL %% low x axis
0 10 0 10 0 270 OneAxx Axx Tic Tac %% top x axis
{(x)} xCL
% y axis:
[
+ 0.01 wy {(0.01)}
0.1 wy {(0.1)}
1 wy {(1)}
] SetTacVec
-0.01 wy 10 wy 4 9 SetTicVecLog
+0.001 wy 10 wy 5 9 SetTicVecLog
0 10 0 0 90 0 OneAxx Axx Tic Tac yNumL %% left y axis
0 10 10 0 90 180 OneAxx Axx Tic Tac %% right y axis
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+{(f(x))} yCL
+24 setown
+3 5 50 {(tanh x)} rtextCM
+7 5 -50 {(tanhc x)} rtextCM
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{ 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