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Doc/DevRef/Roughness.tex

@@ -754,24 +754,23 @@ thoroughly analysed and criticised by Pynn \cite{Pyn92}:
 neglect of phase factor makes approximation irrelevant for reflectometry.
 
 \B{N\'evot \& Croce 1980 \cite{NeCr80}:} Experimental X-ray study.
-Often cited (Google Scholar March 2020: $>1500$ citations)
-for the attenuation of the reflected beam,
-described by the \emph{Névot-Croce factor} [Eqn.~3].
+Highly cited.
+Attenuation of the reflected beam described by the \emph{Névot-Croce factor} [Eqn.~3].
 Theoretical section is hard to read;
 starts from previous results of Croce et al;
 claims to be self-consistent (auto-cohérente, p.~764).
 The key results of this work are rederived in much shorter, clearer, and more standard ways
-by Pynn \cite{Pyn92} who also explicates the approximations made by Névot \& Croce.
+by Pynn \cite{Pyn92} who also explicates which approximations were made.
 
 \B{Beckmann and Spizzichino 1987 \cite{BeSp87}:}
 Book about radar reflections; almost entirely concerned with wavelengths
 shorter than local radius of curvature, irrelevant for reflectometry \cite{Pyn92}.
 
 \B{Sinha et al 1988 \cite{SiSG88}:}
-Top-cited paper (Google Scholar March 2020: almost 2800 citations).
+Top-cited paper.
 Sects.\ II and~III are in Born Approximation,
 with application e.~g.\ to powders.
-Application to liquid interface (possibly interesting to our reactor collegues).
+Application to liquid interface.
 They consider only single interfaces.
 Grazing incidence and DWBA come in Sect.~IV.
 For $q_z\ge q_c$, a small-$q$ expansion reproduces the Névot-Croce factor.
@@ -790,50 +789,45 @@ Also discusses correlated interfaces.
 \B{Hol\'y \& al 1993 \cite{HoKO93}:}
 Concerned with multilayer reflectivity and diffuse scattering.
 Very readable summary and extension of Sinha theory.
-They write the perturbation Hamiltonian of a multilayer system as a sum of single-layer contributions.
+They write the perturbation Hamiltonian of a multilayer system
+as a sum of single-layer contributions.
 This splits up into a sum of four terms, similar to the expression Walter uses.
-They have only four terms as they assume that the fields are identical directly below and above the interface.
-Walter drops this condition and hence gets twice the terms with different averaging below and above the interface.
+They have only four terms
+as they assume that the fields are identical directly below and above the interface.
+Walter drops this condition and hence gets twice the terms
+ with different averaging below and above the interface.
 As a consequence of writing the Hamiltonian as a sum of single interface contributions,
 diffuse scattering leads to a double sum with the covariances appearing.
-Here the correlation model then comes in, they use two different models,
-one without and the second with vertical correlation.
+Here correlation models come in.
+They introduce two of them: one without and the second with vertical correlation.
 
 Summarized and extended to periodic multilayers by Hol\'y \& Baumbach \cite{HoBa94}.
 Hol\'y \& al also contributed to the book \cite{PiHB04};
 in particular, chapter 11 could be interesting.
 
-\B{Caticha 1995 \cite{Cat95}:}
-Studies graded interface with roughness.
-
-\B{Rauscher et al 1995 \cite{RaSS95}:}
-Combine the roughness theory of Sinha et al \cite{SiSG88}
-with bulk density fluctuations for different geometries,
-thereby specializing the generic formalism of Dietrich and Haase \cite{DiHa95}.
-
-
 \B{de Boer 1994 \cite{Boe94}:}
 Purely theoretical description of specular reflectivity on single rough interfaces.
 Second order DWBA calculations of the reflection and transmission coefficients
 in the $T$-matrix formalism.
 The resulting expressions include the lateral correlation and are shown to
-have the N\'evot-Croce factor as a limiting value for small correlation length (i.e.
-negligible diffuse scattering).
-For large correlation length, the well-known Debye-Waller factor is recovered,
+have the N\'evot-Croce factor as a limiting value for small correlation length
+(i.e.\ negligible diffuse scattering).
+For large correlation length, the Debye-Waller like factor is recovered,
 while for intermediate correlation length an interpolation factor needs to be
 evaluated.
 This factor includes a two-dimensional surface integral.
-For a suitably chosen correlation function, this be reduced to a
-one-dimensional integral enabling it's numerical evaluation.
-
+For a suitably chosen correlation function, it can be reduced to a one-dimensional integral
+which facilitates numeric evaluation.
 
 \B{de Boer \& Leenaers 1996 \cite{BoLe96}:}
-Essentially a summary article that briefly gives results from several other articles.
-Explains limiting conditions when results are applicable and they give formulas of the
+Survey article that briefly summarizes results from several other articles.
+Explains under which limiting conditions results are applicable.
+Gives formulae of the
 Fresnel coefficients for both reflection and transmission on a single interface.
-N\'evot-Croce recovered as limit of small correlation lengths, can also serve for multilayer calculations.
+N\'evot-Croce recovered as limit of small correlation lengths,
+can also serve for multilayer calculations.
 This limit corresponds to weak scattering and can be compared to graded interfaces,
-i.e. the numerical approximation via Slicing that completely neglects diffuse scattering.
+i.e.\ the numerical approximation via Slicing that completely neglects diffuse scattering.
 Mentions DWBA leading to intensities greater than unity below the critical angle,
 introduce Rayleigh method to circumvent this.
 This leads to another expression for the Fresnel coefficients for large correlation lengths,
@@ -854,31 +848,39 @@ Other potentially interesting papers from the same author: \cite{Boe91} \cite{Bo
 The lateral correlation function implemented in BornAgain is taken from the paper \cite{Boe95}.
 
 \B{de Boer 1996 \cite{Boe96}:}
-In this paper, the author deals with multilayers and considers the effects of roughness
+Deals with multilayers and considers the effects of roughness
 in both specular reflectivity and diffuse scattering.
 Employs the $T$-matrix formalism to compute corrections in the DWBA up to second order,
 rather hard to understand and result not easily usable (for me, rb).
-The author uses flat interfaces as the starting point for perturbation theory in Section~II and
+Uses flat interfaces as the starting point for perturbation theory in Section~II and
 graded interfaces in Section~III.
 The latter is rather vague and hard to grasp.
 Results are presented, dominantly for x-ray fluorescence.
 
-The author concludes that as a starting point for the DWBA graded interfaces should be used,
+Concludes that as a starting point for the DWBA graded interfaces should be used,
 if both the reflectivity as well as the roughness are reasonably large.
-Furthermore, they suggest an interpolation method for the fields as a starting point for the DWBA,
+Suggests an interpolation method for the fields as a starting point for the DWBA,
 as the field obtained from N\'evot-Croce factors are wrong in the vicinity of interfaces.
 
-The author claims that the second-order term for diffuse scattering is generally negligible,
+Claims that the second-order term for diffuse scattering is generally negligible,
 except when the reflectivity is large as well as for large lateral correlation length and roughness.
-For specular reflection, he mentions the first and second order contribution to be of the same order
+For specular reflection, mentions the first and second order contribution to be of the same order
 in the roughness and is hence only negligible for small roughnesses.
-However, he stresses that the DWBA is only valid for small roughness values or far above the critical wave vector.
+However, stresses that the DWBA is only valid for small roughness values or far above the critical wave vector.
 Their way of extrapolating \cite{Boe94} the results is only valid for single interfaces or
 very large perpendicular correlation.
 
-At the same time, he also mentions, that he is not aware of any samples,
+Also mentions that he is not aware of any samples,
 where the second order contribution has to be considered and that the theory is completely untested.
 
+\B{Caticha 1995 \cite{Cat95}:}
+Studies graded interface with roughness.
+
+\B{Rauscher et al 1995 \cite{RaSS95}:}
+Combine the roughness theory of Sinha et al \cite{SiSG88}
+with bulk density fluctuations for different geometries,
+thereby specializing the generic formalism of Dietrich and Haase \cite{DiHa95}.
+
 \B{Ogura \& Takahashi 1996 \cite{OgTa96}:}
 Scattering and reflection from a random surface in the
 language of mathematical physics, using It\^o\ stochastic functionals.
@@ -897,18 +899,6 @@ This paper drew our attention to the optical theorem,
 but is most probably made obsolete by other publications
 that work out more details.
 
-\B{Chukhovskii 2011 \& 2012 \cite{Chu11,Chu12}:}
-Claims that DWBA is inapplicable for large roughness rms~$\sigma$.
-As an alternative, develops \E{self-consistent wave approximation} (SCWA).
-Starting from a Green function \cite[Eqn.~4]{Chu11},
-the derivation of the scattering cross section \cite[Eqn.~20]{Chu11}
-and of the reflected intensity \cite[Eqn.~19]{Chu11}
-looks relatively straightforward.
-Subsequent averages of random functions for the standard Gaussian surface model,
-however, lead to very long expressions \cite[Eqns.~22,23]{Chu11}.
-The optical theorem is only satisfied in the limit of large
-surface correlation lengths ($k\xi\vartheta^2\gg1$) \cite{Chu12}.
-
 \B{Fuji 2010: \cite{Fuj11}:}
 Seems to be the initial claim, that the Parrat formalism as it is currently used
 with roughness included in the Fresnel coefficients assumes flux conservation
@@ -947,6 +937,22 @@ Obtains an expression that again resembles equations (2.40) and (2.41)
 in the book by Tolan\cite{Tol99}, however, with an effective roughness.
 Claims good agreement with AFM measurements.
 
+\B{Chukhovskii 2011 \& 2012 \cite{Chu11,Chu12}:}
+Claims that DWBA is inapplicable for large roughness rms~$\sigma$.
+As an alternative, develops \E{self-consistent wave approximation} (SCWA).
+Starting from a Green function \cite[Eqn.~4]{Chu11},
+the derivation of the scattering cross section \cite[Eqn.~20]{Chu11}
+and of the reflected intensity \cite[Eqn.~19]{Chu11}
+looks relatively straightforward.
+Subsequent averages of random functions for the standard Gaussian surface model,
+however, lead to very long expressions \cite[Eqns.~22,23]{Chu11}.
+The optical theorem is only satisfied in the limit of large
+surface correlation lengths ($k\xi\vartheta^2\gg1$) \cite{Chu12}.
+
+TODO https://doi.org/10.1107/S2053273315016666 (2015)
+
+TODO https://www.nature.com/articles/s41598-020-68326-2 (2020)
+
 \B{Chukhovskii \& Roshchin 2015 \cite{ChRo15}:}
 Yet another alternative to DWBA: expansion in q-eigenfunctions of the plane-surface problem.