text:
\section{Scattering}
\label{sec:scattering}
The use of scattering techniques to probe soft condensed matter systems is commonplace.
This work will focus on the use of small angle scattering\footnote{SAS.} and reflectometry techniques.
These are particularly appropriate for application to soft condensed matter systems due to the length scales capable of being probed being similar to the persistence length of the soft condensed matter systems.
The length scale covered for such techniques is from around \SIrange{1}{300}{\nano\metre}, as is shown in Figure~\ref{fig:lengths}.
The focus is on the equilibrium structure(s) of a material, and therefore there is no interest in the system dynamics, meaning that exclusively elastic scattering techniques may be used, where there is no energy transfer between the probing radiation and the material.
This is in contrast to inelastic scattering where energy transfer occurs; facilitating the measurement of system dynamics, such as the dynamical modes of polymers and phospholipid bilayers\autocite{garcia_sakai_quasielastic_2009, farago_recent_2009}
%
\begin{figure}[t]
    \forceversofloat
    \centering
    \includegraphics[width=\textwidth]{theory/length}
    \caption{A representation of how different techniques can be used to probe various length scales. Reprinted, with permission of Oxford University Press, from \cite{sivia_elementary_2011}.}
    \label{fig:lengths}
\end{figure}
%

Both X-ray and neutron scattering techniques are discussed and used in this work.
From an experimental viewpoint, there are significant differences between an X-ray scattering and a neutron scattering experiment.
However, there is little variation in terms of the data analysis, where the differences are limited to; the nature of the scattering lengths\footnote{See Section~\ref{convar}.} and the higher background that is present in the neutron scattering experiments.

\subsection{The scattering vector}
The scattering of some probing radiation,\footnote{Not just X-rays and neutrons, but any wave.} by some sample, can be represented as shown in Figure~\ref{fig:scat}.
Since only elastic scattering is being considered, there will be no change in the frequency of the radiation, $\omega_i = \omega_f$.
This means that only the wavevector, $\mathbf{k}$, can change, $\mathbf{k}_i\neq \mathbf{k}_f$.
The difference between the incident and final wavevectors is the scattering vector, $\mathbf{q}$, where,
%
\begin{equation}
    \mathbf{q} = \mathbf{k}_i - \mathbf{k}_f.
\end{equation}
%
%
\begin{marginfigure}
    \centering
    \includegraphics[width=\linewidth]{theory/scat}
    \caption{A schematic of the scattering of some probing radiation by a sample (blue circle). Adapted, with permission of Oxford University Press, from \cite{sivia_elementary_2011}.}
    \label{fig:scat}
\end{marginfigure}
%
The scattering vector strictly has units of \si{\per\meter}, however it is often more practical to use \si{\per\nano\meter} or \si{\per\angstrom}.\footnote{Throughout this work, units of \si{\per\angstrom} will be wherever possible.}
Since the frequency of the probing radiation does not change during an elastic scattering event, the wavelength, $\lambda$, will also not change, meaning that the moduli of the incident and final wavevectors are,
%
\begin{equation}
    |\mathbf{k}_i| = |\mathbf{k}_f| = \frac{2\pi}{\lambda}.
    \label{equ:wavevec}
\end{equation}
%
This also means that only the angle will change during the elastic scattering event.
The vector diagram in Figure~\ref{fig:scatvec} can be used to describe the geometry of an elastic scattering event.
From this, and Equation~\ref{equ:wavevec}, the value of $q$, where $q = |\mathbf{q}|$ can be shown as,
%
\begin{equation}
    q = \frac{4\pi\sin{\theta}}{\lambda}.
    \label{equ:theq}
\end{equation}
%
%
\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth]{theory/scatvec}
    \caption{A vector diagram describing an elastic scattering event, where $\mathbf{k}_i$ is the incident wavevector, $\mathbf{k}_f$ is the final wavevector, $2\theta$ is the scattering angle, and $\mathbf{q}$ is the scattering vector. Adapted, with permission of Oxford University Press, from \cite{sivia_elementary_2011}.}
    \label{fig:scatvec}
\end{figure}
%
However, this fails to fully capture the three dimensional nature of the scattering event.
Hence, it is necessary to describe the scattering with spherical coordinates, $2\theta$, and $\phi$, such that the incoming and outgoing radiation can be described as,
%
\begin{equation}
    \begin{aligned}
        \mathbf{k}_i & = \bigg(0, 0, \frac{2\pi}{\lambda}\bigg), \\
        \mathbf{k}_f & = \frac{2\pi}{\lambda}(\sin{2\theta}\cos{\phi}, \sin{2\theta}\sin{\phi}, \cos{2\theta}),
    \end{aligned}
\end{equation}
%
where, $|\mathbf{k}_f| = \sfrac{2\pi}{\lambda}$. This allows the scattering vector to be written,
%
\begin{equation}
    \mathbf{q} = \frac{4\pi\sin{\theta}}{\lambda}(-\cos{\theta}\cos{\phi}, -\cos{\theta}\sin{\phi},\sin{\theta}).
\end{equation}
%
For an isotropic scattering pattern, it is the magnitude of the scattering vector, $q$, that is measured.
In practical terms, the scattering vector allows for easy comparison of measurements made at different radiation wavelengths.

The basic quantity measured in a scattering experiment is the differential cross section, $\sfrac{\text{d}\sigma(q)}{\text{d}\Omega}$.
This is the fraction of particles of probing radiation that is scattered with a particular set of polar coordinates, $2\theta$ and $\phi$,
%
\begin{equation}
    \frac{\text{d}\sigma(q)}{\text{d}\Omega} = \frac{R_i(2\theta,\phi)}{NV\Phi\Delta \Omega},
    \label{equ:dsc}
\end{equation}
%
where, $R_i(2\theta,\phi)$ is the rate of arrival of the scattered particles at the position $2\theta$, $\phi$, $V$ is the illuminated volume of the sample, $\Phi$ is incident flux, $\Delta \Omega$ is some small solid angle, and $N$ is the number of scattering particles of interest, in the case of elastically scattered radiation, $N = N\%_{\text{el}}$, where $\%_{\text{el}}$ is the fraction of elastically scattered radiation.

\subsection{Scattering from a single fixed particle}
It is possible to describe a steady stream X-ray photons or neutrons of wavelength, $\lambda$, travelling through space as follows,
%
\begin{equation}
    \psi_i = \psi_o \exp{(\mathbf{i} kz)},
    \label{equ:wave}
\end{equation}
%
where, $z$ is the direction of travel, $k=\sfrac{2\pi}{\lambda}$, and the incident flux is the magnitude of the wave squared, $\Phi = |\psi_o|^2$.
This wave then interacts with a single fixed particle elastically, propagating the wave radially outwards, as shown in Figure~\ref{fig:singleatom}.
This propagation is centred on the atom, therefore the wavevector, $\mathbf{k_f}$ is parallel to the displacement vector, $\mathbf{r}$, and the following holds,
%
\begin{equation}
    \exp{(\mathbf{i}\mathbf{k}_f\cdot \mathbf{r})} = \exp{(\mathbf{i}kr)}.
\end{equation}
%
This final wave is no longer collimated and therefore diminishes with distance, $r$.
Hence the final scattered wave has the form,
%
\begin{equation}
    \psi_f = \psi_o b\frac{\exp{(\mathbf{i}kr)}}{r},
\end{equation}
%
where, $b$ is the scattering length discussed in Section~\ref{convar}.
%
\begin{marginfigure}
    \centering
    \includegraphics[width=\linewidth]{theory/singleatom}
    \caption{A schematic showing the propagation of the wave of probing radiation (green lines) radially outwards following the scattering event, where $r$ is the magnitude of the displacement vector. Adapted, with permission of Oxford University Press, from \cite{sivia_elementary_2011}.}
    \label{fig:singleatom}
\end{marginfigure}
%

\subsection{Scattering from multiple particles}
\label{sec:multiscat}

It is important to consider how the probing radiation would interact with a real system, consisting of many particles.
If the incident beam has the form of Equation~\ref{equ:wave}, with the wavevector $\mathbf{k}_i = (0, 0, k)$, each particle, $j$, will contribute the following to the total scattered wave, $\psi_f$, made up of the scattering from all, $N$, atoms,
%
\begin{equation}
    [\delta\psi_f]_j = \psi_o\exp{(\mathbf{ik}_i\cdot \mathbf{R}_j)}b_j\frac{\exp{\big\{\mathbf{ik}_f\cdot (\mathbf{r}-\mathbf{R}_j)\big\}}}{|\mathbf{r}-\mathbf{R}_j|},
\end{equation}
%
where, $\mathbf{R}_j$ is the position of particle $j$, $\mathbf{r}$ is some arbitrary position, and $\mathbf{k}_f$ is the wavevector of the scattered wave, described graphicall in Figure~\ref{fig:multiatom}.
This allows the total scattered wave to be defined as a summation of the contributions from the individual waves,
%
\begin{equation}
    \psi_f = \psi_o \exp{(\mathbf{ik}_f\cdot\mathbf{r})}\sum_{j=1}^{N}\bigg\{b_j \frac{\exp{(\mathbf{iq}\cdot \mathbf{R}_j)}}{|\mathbf{r}-\mathbf{R}_j|}\bigg\}.
    \label{equ:scatter}
\end{equation}
%
Equation~\ref{equ:scatter} holds true, within the Born approximation, where the scattered wave has no impact on the incident wave and each wave is scattered only once.
%
\begin{figure}[t]
    \forcerectofloat
    \centering
    \includegraphics[width=\textwidth]{theory/multiatom}
    \caption{A schematic showing the interaction of radition scattered by two particles that are separated by the vector $\mathbf{R}_j$. Adapted, with permission of Oxford University Press, from \cite{sivia_elementary_2011}.}
    \label{fig:multiatom}
\end{figure}
%

The sample-detector distance is usually much larger than the typical particle size, allowing for the following approximation,
%
\begin{equation}
    |\mathbf{r} - \mathbf{R}_j| = |\mathbf{r}| = r.
\end{equation}
%
This is termed the Fraunhofer, or far-field limit, and allows Equation~\ref{equ:scatter} to be simplified,
%
\begin{equation}
    |\psi_f|^2 = \frac{\Phi}{r^2}\Bigg|\sum_{j=1}^{N}b_j\exp{(\mathbf{iq}\cdot\mathbf{R}_j)}\Bigg|^2.
\end{equation}
%
In the scattering experiment, radiation is deflected elastically into a detector with a small area, $\delta A$, at the polar coordinates, $2\theta$ and $\phi$, at a rate of $R_{\text{el}}$,
%
\begin{equation}
    R_{\text{el}}(2\theta,\phi) = |\psi_f|^2\delta A = \Phi\delta\Omega\Bigg|\sum_{j=1}^{N}b_j\exp{(\mathbf{iq}\cdot\mathbf{R}_j)}\Bigg|^2,
\end{equation}
%
where, $\delta\Omega = \sfrac{\delta A}{r^2}$.
Therefore, the differential cross section, defined in Equation~\ref{equ:dsc} can be related to the scattering from the sample as,
%
\begin{equation}
    \bigg(\frac{\text{d}\sigma(q)}{\text{d}\Omega}\bigg)_{\text{el}} = \frac{1}{V} \Bigg|\sum_{j=1}^{N}b_j\exp{(\mathbf{iq}\cdot\mathbf{R}_j)}\Bigg|^2.
    \label{equ:sca}
\end{equation}
%

\subsection{Scattering length density}
\label{sec:sld}
While it may be helpful to consider the scattering from multiple particles individually, where each particle has a scattering length, $b$.
In practice, due to low experimental resolution at small angles, it is more common to consider the SLD of the system,
%
\begin{equation}
    \text{SLD} = \frac{1}{V}\sum_{i=0}^{N} b_i,
\end{equation}
%
where $N$ is the total number of particles in the volume $V$.
A result of this equation is the ability to rewrite Equation~\ref{equ:sca} as,
%
\begin{equation}
    \bigg(\frac{\text{d}\sigma(q)}{\text{d}\Omega}\bigg)_{\text{el}} = \frac{1}{V} \Bigg|\iiint \limits_V \text{SLD}\exp{(\mathbf{iq}\cdot\mathbf{R})}\text{d}^3\mathbf{R}\Bigg|^2.
    \label{equ:sldsca}
\end{equation}
%
This equation shows that the scattering differential cross-section from some object is related to the SLD profile of that object by a Fourier transform.

\subsection{Model-dependent analysis}
\label{sec:moddep}
All types of scattering patterns can be analysed by two analysis methods; model independent and model-dependent.
Model-independent analysis is where there is no \emph{a priori} information used in the analysis, when there are no assumptions made about the underlaying structure of the sample.
However, model-dependent analysis is when reasonable assumptions are made about the structure before the analysis is considered.
The nature of this work means that it will focus on model-dependent analysis methods.\footnote{With the model usually being derived from some atomistic, or coarse-grained simulation.}
Model-dependent analysis has significant benefits over model-independent methods, such as improved resolution and more detailed information about the structure.
However, the necessity of the inclusion of \emph{a priori} information within model-dependent analysis may act to bias the result.
While this is undesirable, these assumptions can, and should, be educated based on the chemical information present.\sidecite[such as the propensity for twin-tailed phospholipid molecules to form monolayers at an air-water interface or small surfactants to form micelles in solution.]{mccluskey_model-dependent_2018}

The scattering from the model system is determined, using technique specific methods that are discussed in detail in later sections.
This is then compared with the experimental data using some figure of merit, the model is then varied to find the best possible model for the data provided.\footnote{This typically uses some optimisation algorithm to determine the best solution, the particular algorithms used in this work are discussed in Section~\ref{sec:optimisation}.}
In order to accurately reproduce the experimental measurement, it is necessary to include some instrumental resolution function, $res(q)$, in the modelling procedure.
This is instrument-specific, although it may be approximated by convolving the experimental dataset with some Gaussian smearing function, the modelled intensity can then be determined from,\autocite{nelson_towards_2013-1,nelson_towards_2014}
%
\begin{equation}
    I(q) = res(q) * \frac{\text{d}\sigma(q)}{\text{d}\Omega},
\end{equation}
%
where, $\sfrac{\text{d}\sigma(q)}{\text{d}\Omega}$ is the differential cross-section, a measure of the number of scattering particles hitting a given solid angle of the detector.

The aim of model-dependent analysis is to obtain a model for the system which agrees well with the experimentally measured scattering data while producing something that is chemically and physically relevant.

\subsection{Reflectometry}
\label{sec:refltheory}

Reflectometry involves the interaction of the probing radiation with some interface, from which the radiation is reflected.
The geometry of a reflectometry experiment is shown in Figure~\ref{fig:refgeo}, where the reflectometry instrument is in the horizontal configuration, ideal for the study of liquid interfaces.\footnote{Such at those investigated in Chapters~\ref{reflectometry1} and \ref{reflectometry2}.}
Reflectometry measurements give information about the structure perpendicular to the interface, the \emph{z}-dimension in Figure~\ref{fig:refgeo}, and therefore the analysis of reflectometry data is founded on the assumption that the layers will be completely homogenous in the plane of the interface, the \emph{xy}-plane in Figure~\ref{fig:refgeo}.
In reality, since the layers are usually not completely homogeneous, an average is obtained for the area in the radiation beam.
%
\begin{figure}[t]
    \forcerectofloat
    \centering
    \includegraphics[width=0.8\textwidth]{theory/reflectgeo}
    \caption{A schematic showing the geometry of a typical specular reflectometry experiment from a layered sample. Reprinted, with permission of Oxford University Press, from \cite{sivia_elementary_2011}.}
    \label{fig:refgeo}
\end{figure}
%

A reflectometry instrument operates by measuring the intensity of specular reflected radiation at a series of different angles, $\theta$, or wavelengths, $\lambda$.
The reflected intensity is described in terms of the $q$-vector\footnote{Using Equation~\ref{equ:theq}}, and is defined as follows,
%
\begin{equation}
    R(q) = \frac{\text{specular reflected radiation intensity at }q}{\text{incident radiation intensity}}.
    \label{equ:refl}
\end{equation}
%
It is clear from Equation~\ref{equ:refl} that the value of the measured reflectometry cannot be greater than one, as this would mean that more particles of probing radiation were being reflected than were incident.

There are two model-dependent analysis techniques that can be applied to the understanding of a reflectometry dataset.
The first is the kinematic approach, which can be described with Equation~\ref{equ:sca}, from the assumption that $q_x = 0$ and $q_y = 0$, as only the specular scattering is being measured.
This approach models the reflectometry as a function of the SLD profile in the \emph{z}-dimension, $\text{SLD}(z)$,
%
\begin{equation}
    R(q) \approx \frac{16\pi^2}{q^4}\bigg|\int_{-\infty}^{+\infty}\frac{\text{d}\text{SLD}(z)}{\text{d}z}\exp{(-\mathbf{i}zq_z)}\text{d}z\bigg|^2,
    \label{kine}
\end{equation}
%
where, $\sfrac{\text{d}\text{SLD}(z)}{\text{d}z}$ is the first derivative of the SLD profile.
However, this method has a significant problem, which can be demonstrated by applying Equation~\ref{kine} to the SLD profile of a bare silicon substrate, which can be modelled as a Heaviside function, as shown in Figure~\ref{fig:kine}(a),
%
\begin{equation}
    \text{SLD}(z) =
  \begin{cases}
    0, & \text{where}\ z < 0 \\
    \text{SLD}_{\text{Si}}, & \text{otherwise}
  \end{cases}
\end{equation}
%
where, $\text{SLD}_{\text{Si}}$ is the SLD of pure silicon.\footnote{This is \SI{2.1e-6}{\angstrom^{-2}} for neutrons.}
The derivative of a stepwise Heaviside function is a scaled $\delta$-function, as shown in Figure~\ref{fig:kine}(b),
%
\begin{equation}
    \text{SLD}'(z) = \text{SLD}_{\text{Si}}\delta(z).
\end{equation}
%
Then, as in Equation~\ref{kine}, the Fourier transform of this $\delta$-function is taken,
%
\begin{equation}
    \text{SLD}_{\text{Si}}\int_{-\infty}^{+\infty}\delta(z)\exp{(-\mathbf{i}zq_z)}\text{d}z = \text{SLD}_{\text{Si}}\exp(0) = \text{SLD}_{\text{Si}}.
    \label{equ:rhosi}
\end{equation}
%
This means that, using Equations~\ref{equ:rhosi} and \ref{kine}, the reflectometry profile could be calculated from the following relationship,
%
\begin{equation}
    R(q)\approx \frac{16\pi^2\text{SLD}_{\text{Si}}^2}{q_z^4}.
\end{equation}
%
The curve from this relationship is shown in Figure~\ref{fig:kine}, where it is clear that the agreement with an experimental profile would be poor as $q \rightarrow 0$.
It can be seen that for low values of $q$ the calculated reflectometry is greater than 1, which violates the physical constraint imposed with Equation~\ref{equ:refl}.
This break down of the kinematic approach is due to the assumption present in this approach that the Born approximation\footnote{Mentioned previously in Section~\ref{sec:multiscat}.} will hold.
However, in the reflectometry scattering geometry, this is no longer true rendering the kinematic approach invalid.
%
\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth]{theory/kine}
    \caption{A graphical representation of the kinematic approach; (a) the Heaviside function describing the SLD profile of a bare silicon substrate, (b) the $\delta$-function arising from the first derivative of the function in (a), and (c) the reflectometry profile resulting from the kinemtic approach, where the orange line at $R=1$ identifies the break down between experimental and theory in the kinematic approach. Adapted, with permission of Oxford University Press, from \cite{sivia_elementary_2011}.}
    \label{fig:kine}
\end{figure}
%

This breakdown of the kinematic approach has led to the application of the Abel\`{e}s, or Parratt, model for the reflection of light at a given number of stratified interfaces.\sidecite[also known as dynamical theory]{abeles_sur_1948,parratt_surface_1954}
This method involves considering the system as a layered structure at the interfaces of which, the probing radiation can either be reflected or refracted, by some refractive index, $n_i$.
Figure~\ref{fig:reflrefr} shows this process for a system of two layers, where the layer \num{0} is the air or vacuum above the sample, it is clear to see how the two waves labelled $r$ could interfere constructively or destructively depending on the thickness of layer \num{1}, $d$.
This means that for a single interface,\footnote{Such as that between layers \num{0} and \num{1} in Figure~\ref{fig:reflrefr}.} the reflectometry can be described by the Fresnel equation,
%
\begin{equation}
    R(q) = \bigg| \frac{n_0\sin{\theta_0} - n_1\sin{\theta_1}}{n_0\sin{\theta_0} + n_1\sin{\theta_1}} \bigg|^2.
\end{equation}
%
Additionally at the point of total reflection, where $\theta_0 = \theta_{\text{c}}$, the critical angle, there will be no transmitted wave so,
%
\begin{equation}
    n_1\sin{\theta_1} = 0,
\end{equation}
%
and therefore the reflected radiation will never be greater than 1, while the critical angle can be defined as,
%
\begin{equation}
    \cos^2{\theta_{\text{c}}} = \frac{n_1^2}{n_0^2}.
\end{equation}
%
This is the angle below which a reflectometry profile will be measured.
%
\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth]{theory/reflrefr}
    \caption{A schematic diagram showing the reflected ($r$) and transmitted ($t$) waves when an incident ($i$) wave enters an interface of thickness $d$, where the refractive indices of each layer are $n_0$, $n_1$, and $n_2$. Adapted, with permission of Elsevier, from \cite{foglia_studies_2015}.}
    \label{fig:reflrefr}
\end{figure}
%

%
\begin{figure}[t]
    \forceversofloat
    \centering
    \includegraphics[width=\textwidth]{theory/dyna}
    \caption{A comparison of the kinematic approach (blue solid line), and the dynamical approach (green dashed line), to determine the reflected intensity from the material with the SLD profile given in Figure~\protect\ref{fig:kine}(a). It is clear that at low $q$, there is a noticable deviation between the two.}
    \label{fig:dyna}
\end{figure}
%
The above method can then be generalised to a structure of an arbitrary number of layers, as shown in Code Block~\ref{cb:refl}.\footnote{The purpose of the Code Blocks throughout this work is to ensure transparency and reproducibility, these are written as Python functions.}
For each value of $q$ for which the reflectometry is to be calculated, the system is considered in terms of $n_{\text{max}}$ layers.
The incident radiation beam will be refracted by each of the layers, giving wavevectors values for each layer, $k_n$,
%
\begin{equation}
    \label{equ:knsld}
    k_n = \sqrt{k_0^2 + 4\pi (\text{SLD}_n - \text{SLD}_0)},
\end{equation}
%
where, $k_0=\sfrac{q}{2}$.
The Fresnel equation coefficient between layers $n$ and $n+1$, $r_{n,n+1}$ can then be found along with the phase factor, $\beta_n$, as the Born approximation holds for SAS, which is dependent on the thickness of the layer, $d_n$,
%
\begin{equation}
    r_{n,n+1} = \frac{k_n - k_{n+1}}{k_n + k_{n+1}},
    \label{equ:fres}
\end{equation}
%
%
\begin{equation}
    \label{equ:nowthick}
    \beta_n = k_n d_n.
\end{equation}
%
The means that a matrix can be evaluated for each layer, $M_n$,
%
\begin{equation}
    M_n =
    \begin{bmatrix}
        \exp{\beta_n} & r_{n,n+1}\exp{-\beta_n} \\ r_{n,n+1}\exp{\beta_n} & \exp{-\beta_n}
    \end{bmatrix}
\end{equation}
%
The resultant matrix, $B$, is then found as a product of the matrix from each layer,
%
\begin{equation}
    B = \prod_{n=0}^{n_{\text{max}}} M_n,
\end{equation}
%
and from this the reflected intensity at the given value of $q$ can be found,
%
\begin{equation}
    R(q_z) = \frac{B_{1,2}}{B_{1,1}}.
\end{equation}
%
This algorithm models the layers as perfectly flat layers, which will not be strictly true.\footnote{Particularly for soft matter systems.}
This resulted in the correction term being added to Equation~\ref{equ:fres} to account for the roughness of the layers.
This adapts Equation~\ref{equ:fres} to the form,
%
\begin{equation}
    r_{n,n+1} = \frac{k_n - k_{n+1}}{k_n + k_{n+1}}\exp{(-2k_nk_{n+1}\sigma^2_{n,n+1})},
\end{equation}
%
where, $\sigma_{n,n+1}$ is the interfacial roughness between layers $n$ and $n+1$.\sidecite<-2\baselineskip>{nevot_caracterisation_1980}
This has the effect of Gaussian broadening the layers into each other, as a result.
This method\sidenote[][-1.5\baselineskip]{That is given programmatically in Code Block~\protect\ref{cb:refl}.} is currently implemented in a variety of reflectometry modelling software packages, such as \texttt{refnx}, MOTOFIT, RasCAL, and Aurore \sidecite<-2\baselineskip>{nelson_refnx_2019,nelson_co-refinement_2006,hughes_rascal_nodate,gerelli_aurore_2016-1,gerelli_aurore_20162}.
Applying this method to the SLD profile shown in Figure~\ref{fig:kine} gives the reflectometry profile shown with the dashed green line in Figure~\ref{fig:dyna}.
%
\begin{listing}[t]
    \forcerectofloat
    \centering
    \caption{An example Python code block for the Abel\`{e}s method for the calculation of reflectometry, adapted from \cite{nelson_refnx_2019}. The input variables are \texttt{q\_values} which are the $q$-vectors at which the reflected intensity should be calculated, \texttt{sld} which is the array of scattering length densities for the layers, and \texttt{d} which is the array of thicknesses for the layers. This will return an array of floats that is the same size as the \texttt{q\_values} and contains the reflected intensities.}
    \lstinputlisting[nolol]{reports/code_blocks/reflectometry.py}
    \label{cb:refl}
\end{listing}
%

\subsection{Small angle scattering}
\label{sec:sasanal}

Equation~\ref{equ:sldsca} identified that the scattering differential cross-section for some object was related to the SLD by a Fourier transform, which is shown graphically in Figure~\ref{fig:scales}.
This figure shows that there is a reciprocal relationship between the size of the object and the scattered intensity, decaying significantly up to values of $2\pi/d_x$, where $d_x$ is the size of the object.
This means that in order to probe large-scale structural features that are of interest in the study of soft materials, it is necessary to consider small values of $q$.
When considering the nature of $q$ in Equation~\ref{equ:theq}, it is clear that such experiments would benefit from small values of $\theta$ and large values of $\lambda$.
Hence, the use of scattering at small angles.
%
\begin{marginfigure}
    \centering
    \includegraphics[width=\linewidth]{theory/scales}
    \caption{The effect of a Fourier transform (a) the SLD profile for some object with a width of \SI{10}{\angstrom}, (b) the Fourier transform of this object showing the minima in the differential cross section at values of $\sfrac{2n\pi}{10}$, where $n$ is some integer.}
    \label{fig:scales}
\end{marginfigure}
%

A SAS experiment generally involves some sample being placed in the path of the probing radiation; the scattering pattern that results from this transmission is measured at some distance, as is shown in Figure~\ref{fig:sasgeo} for the D22 SANS instrument of the ILL.
SAS instruments are usually very large, due to the large post sample flight path that is necessary to reach the small angles being measured.\footnote{A longer flight path allows more space for angular divergence.}
Transmission SAS can provide information about the size, shape and orientation of the sample's components.\autocite{willis_experimental_2009}
The range of $q$ that is typically covered by a SAS instrument is usually around \SIrange{2e-3}{0.5}{\per\angstrom}, which corresponds to \SIrange{10}{3000}{\angstrom} in real-space.
The neutron or X-ray detector of a SAS instrument is often two-dimensional, meaning that for an isotropic scattering profile, the detector image is radially averaged to give an $I(q)$ scattering profile.
It is possible to increase the $q$-range of a SAS instrument through the introduction of wide-$q$ detector banks close to the sample or small-$q$ detector banks further away.
This allows the SANS2D instrument, at the ISIS Neutron and Muon Source, to have a total range from \SIrange{2e-3}{2}{\per\angstrom}.\autocite{noauthor_isis_nodate}
Furthermore, the SANS2D instrument may leverage the ToF method discussed in Section \ref{sec:neutrons} to allow for a much shorter post sample flight path than is present at D22.
%
\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth]{theory/d22}
    \caption{A schematic of the D22 instrument of the ILL. Reprinted with permission of Springer Nature Customer Service Centre GmbH: Springer Nature from \cite{grillo_small-angle_2008}.}
    \label{fig:sasgeo}
\end{figure}
%

A radially averaged SAS pattern can be considered as consisting of two sections that arise from the form and structure factors for the scattering species.
The form factor gives information about the average shape of the scattering particle, while the structure factor is a measure of the interaction present between the objects.
It is often possible to control the presence of the structure factor by changing the concentration of the sample, eventually, the concentration will be so low that all interparticle interaction is screened by the solvent.\autocite{edler_combining_2015}
This method is frequently applied in biological SAXS applications, where the interactions between the protein molecules are of less interest than the overall structure of the complex.
However, with micelles, it is not always possible to remove the structure factor, as the critical micelle concentration may be higher than the minimum concentration at which the structure factor is present.
It is possible to deconvolute the structure and form factors for a micellar solution by studying different concentrations, assuming that the form of the micelle is concentration independent, over the measured concentration range.

The rigorous, model-independent method for the analysis of SAS involves taking the inverse Fourier transform of the scattering profile, to give an auto-correlation function of the average particle in the system, which following a deconvolution procedure will resolve the radially averaged SLD profile.
However, this is often cumbersome and has a low information density, when compared to model-dependent techniques.
Additionally, if the experimental data lacks information at wide enough $q$ to cover all features of the sample, artefacts may be present in the inverse Fourier transform of the scattering.

There are two common and straight-forward analysis procedures that can be used to give an understanding of the scattering species structure.
The first is the Guinier approximation, which is used in the determination of the radius of gyration, $R_g$, of the scattering species at ``infinite dilution''.
This scattering law is only valid at very small values of $q$, where $q < R_g^{-1}$,\autocite{sivia_elementary_2011}
%
\begin{equation}
    \ln[I(q)] = \ln[I(0)] - \Bigg(\frac{R_g^2}{3}\Bigg)q^2.
\end{equation}
%
This relationship allows the radius of gyration to be found by plotting the scattering profile transformed into $\ln[I(q)]$ vs. $q^2$, and evaluating the gradient at low $q$.
The Guinier plot for the scattering from a sphere with a radius of \SI{20}{\angstrom} is shown in Figure~\ref{fig:rg}, where the radius of gyration correlates with the radius of the sphere, $R_s$, as follows,
%
\begin{equation}
    R_g = \sqrt{\frac{3}{5}}R_s.
\end{equation}
%
The Guinier analysis is very common in the study of proteins by SAS, as it allows for the determination of the protein size in the native, solution phase.\autocite{skou_synchrotron-based_2014}
Another common analysis of SAS data comes in the form of Porod's law, which states that for large values of $q$, the scattering intensity becomes proportional to $Sq^{-4}$, where $S$ is the surface area of the sample.
This means that by plotting $I(q)q^4$ vs. $q$ and extrapolating to $q \rightarrow \infty$, it is possible to determine the external surface area of the system.\autocite{willis_experimental_2009}
Using the surface area, it is then possible to qualitatively determine the ``roughness'' of the system based on the relation of the surface area to the particle size.
%
\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth]{theory/rg}
    \caption{The Guinier plot, (a) the ideal scattering profile from a sphere of radius \SI{20}{\angstrom}, (b) the associated Guinier plot, with a straight line (orange) at low-$q$ showing the radius of gyration to be \SI{\sim15.5}{\angstrom}.}
    \label{fig:rg}
\end{figure}
%

In the calculation of a SAS pattern, both the structure and form factors will contribute.
Therefore, when the pattern is modelled, the differential cross-section, which is measured experimentally, has the following form.\footnote{When the system is centrosymmetric.}
%
\begin{equation}
    \frac{\text{d}\sigma(q)}{\text{d}\Omega} = N_\rho\Delta\text{SLD}^2V_p^2 P(q)S(q),
\end{equation}
%
where $N_\rho$ is the number density of the particles, $\Delta\text{SLD}$ is the difference in scattering length density between the particles and the solvent, $V_p$ is the particle volume, $P(q)$ is the particle form factor, and $S(q)$ is the structure factor.\autocite{pedersen_monte_2002}
Therefore, it is necessary to understand the form and structure factors individually.

The most common method for the modelling of the form factor is by using very coarse shapes; such as spheres, cylinders, or ellipses.
This involves the evaluation of analytical or quasi-analytical solutions for the scattering, which have been derived for many common shapes.
The solution for a sphere was solved in the early 19th century by Lord Rayleigh.\autocite{pedersen_monte_2002}
%
\begin{equation}
    P(q) = \Bigg\{\frac{3\big\{\sin(qR_s) - qR\cos(qR_s)\big\}}{(qR_s)^3}\Bigg\}^2,
    \label{equ:sphere}
\end{equation}
%
where $R_s$ is the radius of the sphere.
A comparison between a possible experimental scattering pattern and the scattering generated from Equation~\ref{equ:sphere} is shown in Figure~\ref{fig:sphere}.
Analytical form factors exist for a wide variety of shapes; these can be found in software such as SASView and SASFit.\autocite{noauthor_sasview_nodate, noauthor_sasfit_nodate}
%
\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth]{theory/sphere}
    \caption{The SANS profile of a micelle of \ce{C_{16}TAB} with radius $\SI{50(3)}{\angstrom}$ (circles, generated using SASView \cite{noauthor_sasview_nodate}, with instrumental smearing) compared with a curve of Equation~\protect\ref{equ:sphere}, where $R = \SI{50}{\angstrom}$ (solid line).}
    \label{fig:sphere}
\end{figure}
%

The structure factor accounts for the scattering interference that arises from the interaction of different particles.
This is modelled using expressions which depend on the nature of the scattering particles; hard-sphere, sticky hard-spheres, screened Coulomb, etc.
Structure factor expressions are generated as solutions to the Ornstein-Zernike Equation.\autocite{klein_interacting_2002}
The most relevant structure factor in terms of micelle modelling is probably the Hayter-Penfold Mean Spherical Approximation, \autocite{hayter_analytic_1981} this is where the micelles are modelled as like-charged, soft spheres and is valid for dilute solutions.
Again, a whole range of these structure factor functions are built into the SASView package.\autocite{noauthor_sasview_nodate}

In order to evaluate the scattering profile from an atomic structure,\footnote{Or any coordinate system, such as a coarse-grained simulation.} it is possible to use the Debye equation.\autocite{debye_zerstreuung_1915}
This is an analytical relationship for the determination of the scattering profile based on the atomic positions, $\mathbf{r}_i$,
%
\begin{equation}
    I(q) = \sum_{i}^{N}\sum_{j}^{N} b_ib_j\frac{\sin{(q|\mathbf{r}_i-\mathbf{r}_j|)}}{q|\mathbf{r}_i-\mathbf{r}_j|},
\end{equation}
%
where, $N$ is the number of particles, $q$ is the scattering vector, and $b_i$ and $b_j$ are the scattering lengths of particles $i$ and $j$ respectively.\footnote{In the case of a coarse-grained particle, this can be obtained from summing the scattering lengths of the constituent atoms.}
The Debye equation is powerful, however, it is not intrinsically parallelisable and scales as $\mathcal{O}(N^2)$.
In order to improve the efficiency of the calculation of the scattering profile, a variety of methods have been developed that offer a sufficiently accurate approximation.\autocite{svergun_solution_1994,watson_rapid_2013}
The Golden Vector method, developed by Watson and Curtis,\autocite{watson_rapid_2013} scales as $\mathcal{O}(Nn)$, where $n$ is the number of scattering vectors used in the calculation.
In this method, the scattering amplitude is calculated numerically for a single $\mathbf{q}$-vector,
%
\begin{equation}
    I(\mathbf{q}) = \Bigg[\sum_{i}^{N}b_i\cos{(\mathbf{q}\cdot\mathbf{r}_i)}\Bigg]^2 + \Bigg[\sum_{i}^{N}b_i\sin{(\mathbf{q}\cdot\mathbf{r}_i)}\Bigg]^2.
\end{equation}
%
This is carried out for $n$ scattering vectors that are selected in an orientationally averaged fashion from a quasi-uniform lattice on a sphere.
This was first developed such that $n$ is a number from the Fibonacci sequence,\autocite{svergun_solution_1994} however for the Golden Vector method, $n$ may be any positive integer.
This leads to the scattering vectors being calculated as,
%
\begin{equation}
    \begin{aligned}
        q_x^{(k)} & = q\cos\Bigg[\sin^{-1}\bigg(\frac{2k}{n}\bigg)\Bigg]\cos\bigg(\frac{2\pi k}{\Phi}\bigg), \\
        q_y^{(k)} & = q\cos\Bigg[\sin^{-1}\bigg(\frac{2k}{n}\bigg)\Bigg]\sin\bigg(\frac{2\pi k}{\Phi}\bigg), \\
        q_z^{(k)} & = \frac{2 k q}{n},
    \end{aligned}
\end{equation}
%
where, $k$ runs from $-(n-1)/2,\ldots,0,\ldots,(n-1)/2$ and $\Phi$ is the golden ratio,
%
\begin{equation}
    \Phi = \frac{1+\sqrt{5}}{2}.
\end{equation}
%
The approximate orientationally averaged scattering is then found as an average of the scattering from each of ther individual $q$-vectors,
%
\begin{equation}
    I(q) = \frac{1}{n}\Bigg\{\sum_{k=(1-n)/2}^{(n-1)/2} I[\mathbf{q}^{(k)}]\Bigg\}.
\end{equation}
%
The accuracy of this calculation increases with $n$, however agreement, comparible to the Debye equation, between experiment and simulation has been shown for $n < 100$ even for highly anisotropic systems.\autocite{watson_rapid_2013}


text:
\documentclass[a4paper, 11pt]{article}

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\title{Gender Gap on GitHub}

\begin{document}

\date{\today}
\author{E. Jobst}
\maketitle

\begin{abstract}
Female participation, which is already bad in the computer science field, is even worse regarding contributions to FLOSS projects. In this work the current state-of-the-art on the gender gap on the example of GitHub is reviewed. The barriers women face on contributing, range from the consequences of descriptive and prescriptive stereotypes, to women acting differently on GitHub in order to avoid backlash. One major issue is how competent women often lack the self-esteem to contribute. Finally, possible solutions to these problems are presented, with a considerable focus on supporting newcomer.
\end{abstract}

\section{Introduction} \label{sec:intro}
Female participation in the software world is already sparse. Using Austria as an example only 26,7\% of the female university graduates in the year 2016/17 have chosen a technical study. This is already a slight improvement to the statistics that were conducted 10 years prior, where 20,5\% graduated in a technical field \cite{education-austria}. Nevertheless, these numbers decline at a rapid rate regarding contributors of FLOSS projects \cite{competence-confidence-gap}. \newline

In the version-control platform GitHub only 5\% of the projects are owned by women. Of the 5000 top-most contributor only around 3\% are female \cite{competence-confidence-gap}. This may seem frustrating as diverse team-structures are very beneficial for a project. Ortu et al. proved that with a gender-diverse team the amount of time spent on solving an issue on GitHub drops significantly \cite{how-diverse-team}\cite{gender-tenure-diversity}. \newline

With these numbers present it makes someone wonder on the reasons why such a small number of women participate in the FLOSS community with such an, assumingly, small amount of time. The purpose of this paper is collecting issues and barriers that women face on the example of GitHub. The found issues will be contrasted with possible solutions to them.

\section{Gender Gap on GitHub} \label{sec:gender-gap-github}

\subsection{Contribution Differences of Women and Men}
Robles et al. highlight the differences in contribution type between women and men. In their study they analysed it with splitting the types into three categories called „code, programming“, „other“ and „both“. The category „other“ is defined with tasks, that are necessary, however, outside the realm of adding new features or fixing bugs, like documentation, testing or translating. It’s noteworthy that men mostly contributed via source-code contributions, however, women performed the other tasks more frequently \cite{women-floss}.

\subsection{Pull-Request Acceptance Rate}
Terrell et al. made an in-depth investigation on the pull-request acceptance rate of women. In general, women have a higher pull-request acceptance rate than men. However, they found possible bias in certain circumstances: When the gender is unidentifiable their acceptance rate is higher by 12\%, than in comparison to when their gender is known.The rate drops for men as well, however, only by 3,8\%. This was compared by splitting the data up in “Insider”, which are the owner or collaborator of the project and “Outsider”, which are regular contributor. This decline in the pull-request acceptance rate has only been observed with contributor, meaning for owner or collaborator such a decline wasn’t perceived. This suggests a possible bias against newcomers and unregular contributor, with an emphasis on women \cite{pull-acceptance-women}.

\subsection{Competence-Confidence Gap}
The competence-confidence gap describes the phenomenon, where womens competence doesn’t directly translate to confidence. As an additional variable women require the support of the community - this is dubbed as “social attraction”. Even very accomplished women might suffer from it, Wang et al. name the Facebook COO Sheryl Sandberg as one example. Through this lack of confidence they lose many opportunities, as confident people are more likely to be hired or might be more reluctant in taking chances \cite{competence-confidence-gap}. \newline

Wang et al. hypothesized women will increase in contributions once their follower count increases, insinuating that event will give them the necessary push in confidence. For that they sampled the top female contributor on GitHub and the same amount of randomly selected male contributor. They were able to prove that a correlation between surge in follower and opening pull-requests exist for women \cite{competence-confidence-gap}.

\subsection{Descriptive and Prescriptive Stereotypes as Barriers}
Descriptive stereotypes define how women and men are perceived to be, while prescriptive stereotypes describe how they are expected to be. Such descriptive stereotypes associate communal domains with women and agentic domains with men. Prescriptive stereotypes act in a similar way as women are expected to be caring and men are expected to be assertive. What’s even more troubling is that the prescriptive stereotypes also define how women and men are not expected to be \cite{gender-stereotypes}. \newline

According to an experiment conducted by Rudman et al. showing assertiveness during the hiring process is disadvantageous for women. Furthermore in their study, the only applicant who didn’t experience bias was one exhibiting both traits: kindness and expertise. Rudman et al. have used the analogy of this being like walking on a tightrope, as women have to compensate their agency with communal traits \cite{prescriptive-gender-stereotypes}. \newline

Imtiaz et al. go a step further and hypothesize that this tightrope of socially-accepted ways of conducting oneself influence how women interact on GitHub. They were able to prove their hypothesis where women, amongst other things, avoided overly emotional reactions and swear words \cite{gender-bias-github}.

\subsection{Social and Cultural Barriers for Newcomers}
Steinmacher et al. defined six categories of barriers newcomers face when they desire to contribute to a OSS project: they deal with the responses of other contributors, newcomers‘ characteristics, onboarding newcomers, issues with the documentation, cultural differences and technical problems. Four categories out of them are social/cultural issues and thus shall be explained further \cite{social-barriers-newcomer}:

\begin{itemize}
	\item The issues found with the responses of the other contributors dealt with the way collaborator replied: the messages were either lacking, delayed or rude. However, some of them were also too complex and could only be understood by veterans of the field.
	\item The newcomers themselves can be a barrier, as they often start contributing before completely knowing the process or having enough knowledge about the technology. However, they can also behave impatient or lack engagement.
	\item For newcomers it might at first be overwhelming. Due to missing documentation or lack of a mentoring system they often don‘t know where or how to start.
	\item With the collaboration on OSS projects newcomers can meet people from all over the world. With this, however, different cultures can clash. Messages, that are meant to be honest and direct, can in a different culture be considered rude \cite{social-barriers-newcomer}.
\end{itemize}

\subsection{Tools and Infrastructure Barriers for Female Newcomers}
As tools and infrastructure barriers Mendez et al. called technological barriers newcomers face. Inadequate documentation and technical problems, as observed in the study conducted by Steinmacher et al., are part of tools and infrastructure barriers \cite{open-source-barriers}\cite{social-barriers-newcomer}. Mendez et al. hypothesize these issues to be a reason why newcomers are quick to drop-out of OSS projects \cite{open-source-barriers}. \newline

The process they have used in analysing their collected data is called GenderMap - the Gender Inclusiveness Magnifier. The theory behind this is that women and men have a tendency on what kind of approach they use to solve a problem. Consequently, when a tool isn‘t supporting approaches associated with both genders it can be a disadvantage for either women or men. The process builds on four key-differences \cite{open-source-barriers}:

\begin{itemize}
	\item Women are more likely to use technology in achieving a goal, while men simply use it because they like it
	\item Women tend to have lower self-esteem in their computer-skills
	\item Women are more likely to avoid risks
	\item Women and men process information differently: women primarily gather all available information, while men take the first possible path. Furthermore, women are less likely to learn new features through trying them out; they learn these features through a process \cite{open-source-barriers}
\end{itemize}

In their study they use a diverse setup of team consisting of mixed teams and teams consisting only of one gender. These teams had to work through certain use-cases, some examples being „Use GitHub issue tracker to find an issue“ or „Find help with pull requests on GitHub“. The problems the different teams ran into varied from unclear terminology in the documentation to missing information or lacking guides on how to contribute. In total, 73\% of the found barrier-types don‘t support methods women are likely to use and thus are biased against them \cite{open-source-barriers}.

\subsection{Conclusion}
The previous section focused on issues and barriers women commonly face on GitHub. It‘s notable that women are more likely to work on the documentation of a project, than implementing new features. In addition to that, women have little confidence in their skills. These are some of the issues found, however, a common factor is how newcomers face many struggles in their first experiences with GitHub and women could be easier discouraged by them.

\section{Possible Solutions for GitHub}
\subsection{Supporting Female Newcomer through an Onboarding-Process}

One crucial issue mentioned in the previous section is how newcomers feel overwhelmed with, amongst other things, the processes, technology and terminology. These are likely reasons why they stop contributing at a very early stage. Most of these barriers can be overcome with a proper onboarding process. One approach could be assigning mentors to them. These mentors should be veterans of the field or at least regular contributor, who are familiar enough with the workflow. Assigning mentors should be the project owners responsibility. They might have a good grip on possible candidates. It is crucial that this onboarding should tackle the social and technical barriers. Making a guide, which should be short and precise, is one way of solving some technical barriers. It would at least make them familiar with the terminology. However, there‘s only so much that could be explained and some issues just come up on first-hand experience. That‘s why the newcomer should work together on their first task with the mentor. Through this the mentor could support them in crucial, however necessary, issues, like setup of the local environment, or where guides and guidelines can be found. \newline

The previously mentioned method could be used for women and men. However, the low participation of women need to be considered separately as well. The owner of the repository need to be aware of this problem and recruit able women as mentors. This woman could be regarded as a role model by the newcomer. The lack of female role models is hypothesized to be a reason why women avoid the computer science field \cite{computer-science-women}. Working on an issue collaboratively has been shown to be beneficial for women \cite{pair-programming}. While in that study the focus was on pair-programming, what the proposed onboarding-strategy and pair-programming have in common is the collaborative work on an issue. However, that problem could be resolved with the introduction of videoconferences.

\subsection{Documentation and Guidelines}
As inadequate documentation is a reason why newcomers feel overwhelmed, once they join a new project, there should be more effort put into keeping it up-to-date. This part, as well, could be part of the onboarding. In addition, to the mentor and newcomer working on their first programming task together, they could also focus on providing a proper documentation for it. This documentation can be in the form of well-formulated pull-requests and inline comments in the code, where necessary. Furthermore, a well-structured code with well-defined names for the files, classes, methods and fields could also work as documentation. \newline

On another note, the mentor and newcomer should also work on a documentation-task together. This method would have similar benefits to the collaborative work on a coding-task, where the newcomer gets familiar with the structure for the documentation. Furthermore, through the questions and input of the newcomer the mentor quickly becomes aware of the places where the documentation is lacking. \newline

When resources for possible mentors are scarce, the owner and collaborator could write guidelines for newcomers. In these guidelines they should write the most important things that should be considered when one wants to contribute to their project. This set of guidelines should appear to the newcomer the moment they first want to open a pull-request in a project. Furthermore, this message should contain some kind of contact person, that the newcomer could write in case of open questions. An additional bonus would be if the contact person would additionally message the newcomer, as shy people would be more likely to talk about their issues once asked.

\section{Conclusion}
In this paper, the current state-of-the-art regarding gender gap on GitHub was reviewed. On that platform women are sparsely represented. The amount of women owning repositories or being very active members is low. One major problem is the confidence-competence gap of women. \newline

It is suspected the reason for the low participation is how women still suffer under prescriptive and descriptive stereotypes. In one study it was found women avoid overreaction or swear words in their comments. Furthermore, the conversation culture on GitHub leaves room for improvement. Newcomers are faced with impolite, delayed or no responses to their queries. Aside from that, newcomers themselves often start collaborating before getting familiar with the procedures or technology. Using GenderMap, Mendez et al. have concluded that GitHub lacks support in the preferred problem-solving approach of women. \newline

Finally, as a possible solution an onboarding-process, which would ideally assign mentors to newcomers is proposed. For female newcomers using women as mentors might lower the dropout rate. Furthermore, the same could be achieved through keeping the documentation up-to-date and clean and provide guides for newcomers with a contact person.

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\subsection{Specific function results}

\subsubsection{Examples of utility functions}

\subsubsection{Cobb Douglas}

\(U=f(x,y)\)

\(U=x^\alpha y^\beta \)

\(MU_x=\alpha x^{\alpha -1}y^\beta \)

\(MU_x=\alpha \dfrac{y^\beta }{x^{1- \alpha }}\)

\(MU_y=\beta \dfrac{x^\alpha}{y^{1- \beta }}\)

So:

\(\dfrac{MU_x}{MU_y}=\dfrac{\alpha \dfrac{y^\beta }{x^{1- \alpha }} }{\beta \dfrac{x^\alpha}{y^{1- \beta }}}\)

\(\dfrac{MU_x}{MU_y}=\dfrac{\alpha }{\beta }\dfrac{y}{x}\)

\subsubsection{Constant elasticity of substitution}

\(U=f(x,y)\)

\(U=(ax^b+(1-a)y^b)^{\dfrac{1}{b}}\)

\(MU_x=\dfrac{1}{b}abx^{b-1}(ax^b+(1-a)y^b)^{\dfrac{1}{b}-1}\)

\(MU_y=\dfrac{1}{b}(1-a)by^{b-1}(ax^b+(1-a)y^b)^{\dfrac{1}{b}-1}\)

So:

\(\dfrac{MU_x}{MU_y}=\dfrac{\dfrac{1}{b}abx^{b-1}(ax^b+(1-a)y^b)^{\dfrac{1}{b}-1}}{\dfrac{1}{b}(1-a)by^{b-1}(ax^b+(1-a)y^b)^{\dfrac{1}{b}-1}}\)

\(\dfrac{MU_x}{MU_y}=\dfrac{ax^{b-1}}{ (1-a)y^{b-1}}\)

\subsubsection{Constant elasticity of substitution}

\(U=f(x,y)\)

\(U=ax+by\)

\(MU_x=a\)

\(MU_y=b\)

So

\(\dfrac{MU_x}{MU_y}=\dfrac{a}{b}\)

\subsubsection{Constant elasticity of substitution}

\(U=f(x,y)\)

\(U=\alpha .ln(x)+\beta .ln(y)\)

\(MU_x=\dfrac{\alpha }{x}\)

\(MU_y=\dfrac{\beta }{y}\)

So:

\(\dfrac{MU_x}{MU_y}=\dfrac{\alpha }{\beta }\dfrac{y}{x}\)

This is the same as Cobb Douglas. This reflects that the logarithmic function preseres positive inflections.

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across both endpoints.\vspace{1em}
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Qt{} implementation.}
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\end{minipage}}
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\begin{lightquadblock}{How Cloud Back-Ends Enhance Native Front Ends}
{\Large\begin{itemize}
\sqitem {\lsep}  Cloud Backup {\MyOct} Share Data between Users {\MyOct} Collaborative Editing \vspace{1em}
\sqitem {\lsep}  \parbox[t]{14cm}{Maintain users' application state across different
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text:
\section{The Usual Counterterm}\label{sec:counterterm/spherical}

The spherical integrals $I_D$ are cut off by a radius of $N/2$.
In \eqref{spherical quantization} a variety of dimensionful parameters are separated from the integral itself.
For convenience, the integrals that appear in $S_D^\spherical$ for $x>0$ itself are given by
\begin{equation}
    \label{eq:spherical S improved regulator}
     \Omega_D \int_0^{N/2}  \mathcal P \frac{ n^{D-1} \mathrm{d}n}{n^2-x} = \begin{cases}
        4\pi\left( \frac{N}{2} - \sqrt{x} \tanh\inverse \frac{\sqrt{x}}{N/2}\right)
            &   (D=3)\\[0.25cm]
        \pi \log\left( \frac{(N/2)^2}{x} - 1\right)
            &   (D=2)\\[0.25cm]
        -\frac{2}{\sqrt{x}} \tanh\inverse \frac{\sqrt{x}}{N/2}                     &   (D=1)
    \end{cases}.
\end{equation}

The regulating behavior needed to cancel the divergence in the sum in $S_D^\spherical$ can be found by expanding these integrals around the large-$N$ behavior,
\begin{equation}
     \Omega_D\int_0^{N/2} \mathcal P \frac{ n^{D-1} \mathrm{d}n}{n^2-x} \goesto \begin{cases}
        4\pi \left(\frac{N}{2}\right) - \frac{4 \pi x}{N/2} + \order{\left(\frac{N}{2}\right)^{-2}}
            &   (D=3)\\[0.25cm]
        \pi \log \left(\frac{(N/2)^2}{x}\right) - \frac{\pi x}{(N/2)^2} + \order{\left(\frac{N}{2}\right)^{-4}}
            &   (D=2)\\[0.25cm]
        -\frac{2}{(N/2)} - \frac{2 x}{3 (N/2)^3} + \order{\left(\frac{N}{2}\right)^{-4}}
            &   (D=1)
    \end{cases}
\end{equation}
However, rather than including only the leading divergent behavior, one can use the exact integral values in \eqref{spherical S improved regulator}, accelerating the convergence to the large-$N$ limit.

In two dimensions, we can rewrite separate the large-$N$ behavior of the infinite-volume integral
\begin{align}
    \Omega_D\int_0^{N/2} \mathcal P \frac{ n^{D-1} \mathrm{d}n}{n^2-x}
    &=
    \pi \log \left(\frac{(N/2)^2}{x}\right) + \pi \log\left(1-\frac{x}{(N/2)^2}\right)
    &
    (D&=2)
\end{align}
but cannot separate out an $x$-independent counterterm.

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text:
\section{FFYAFF}
\label{FFYAFF}
\begin{ipifield}{}%
{Uses a Yaff force field to compute the forces.}%
{}%
{\ipiitem{name}%
{Mandatory. The name by which the forcefield will be identified in the System forces section.}%
{data type: string; }%
\ipiitem{pbc}%
{Applies periodic boundary conditions to the atoms coordinates before passing them on to the driver code.}%
{default:  True ; data type: boolean; }%
}
\begin{ipifield}{latency}%
{The number of seconds the polling thread will wait between exhamining the list of requests.}%
{default:  0.01 ; data type: float; }%
{}
\end{ipifield}
\begin{ipifield}{yaffsys}%
{This gives the file name of the Yaff input system file.}%
{default: `system.chk'; data type: string; }%
{}
\end{ipifield}
\begin{ipifield}{parameters}%
{The parameters of the force field}%
{default:  \{ \} ; data type: dictionary; }%
{}
\end{ipifield}
\begin{ipifield}{activelist}%
{List with indexes of the atoms that this socket is taking care of.    Default: all (corresponding to -1)}%
{default:
      [-1]; data type: integer; }%
{\ipiitem{shape}%
{The shape of the array.}%
{default:  (1,) ; data type: tuple; }%
\ipiitem{mode}%
{If 'mode' is 'manual', then the array is read from the content of 'cell' takes a 9-elements vector containing the cell matrix (row-major). If 'mode' is 'abcABC', then 'cell' takes an array of 6 floats, the first three being the length of the sides of the system parallelopiped, and the last three being the angles (in degrees) between those sides. Angle A corresponds to the angle between sides b and c, and so on for B and C. If mode is 'abc', then this is the same as for 'abcABC', but the cell is assumed to be orthorhombic. 'pdb' and 'chk' read the cell from a PDB or a checkpoint file, respectively.}%
{default: `manual'; data type: string; options: `manual', `file'; }%
}
\end{ipifield}
\begin{ipifield}{yafflog}%
{This gives the file name of the Yaff output log file.}%
{default: `yaff.log'; data type: string; }%
{}
\end{ipifield}
\begin{ipifield}{alpha\_scale}%
{This gives the alpha parameter in the Ewald summation based on the real-space cutoff: alpha = alpha\_scale / rcut. Higher values for this parameter imply a faster convergence of the reciprocal terms, but a slower convergence in real-space.}%
{default:  3.5 ; data type: float; }%
{}
\end{ipifield}
\begin{ipifield}{rcut}%
{This gives the real space cutoff used by all pair potentials in atomic units.}%
{default:  18.8972613392 ; data type: float; }%
{}
\end{ipifield}
\begin{ipifield}{gcut\_scale}%
{This gives the reciprocale space cutoff based on the alpha parameter: gcut = gcut\_scale * alpha. Higher values for this parameter imply a better convergence in the reciprocal space.}%
{default:  1.1 ; data type: float; }%
{}
\end{ipifield}
\begin{ipifield}{smooth\_ei}%
{This gives the flag for smooth truncations for the electrostatic interactions.}%
{default:  False ; data type: boolean; }%
{}
\end{ipifield}
\begin{ipifield}{yaffpara}%
{This gives the file name of the Yaff input parameter file.}%
{default: `parameters.txt'; data type: string; }%
{}
\end{ipifield}
\begin{ipifield}{skin}%
{This gives the skin parameter for the neighborlist.}%
{default:  0 ; data type: integer; }%
{}
\end{ipifield}
\begin{ipifield}{reci\_ei}%
{This gives the method to be used for the reciprocal contribution to the electrostatic interactions in the case of periodic systems. This must be one of 'ignore' or 'ewald'. The 'ewald' option is only supported for 3D periodic systems.}%
{default: `ewald'; data type: string; }%
{}
\end{ipifield}
\end{ipifield}

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