text:
\section{Introduction}
The studies of real materials through diffraction are of undeniably great value and allow obtaining structural information on the atomic distribution as well as on the distribution, size and shape of inhomogeneities present within materials. Substantial efforts have been made in the treatment of the problems relative to diffraction from scattering systems in order to determine the materials structural parameters such as the atomic distribution, the sizes and shapes of small particles etc. resulting in a very rich
diffraction-related literature, for instance naming just a few \cite{Friedrich Knipping and Laue 1913}-\cite{Grawert and Svergun 2020}. Here, the total scattered intensity distribution for a material system whose homogeneity has been disturbed through substitution of groups of atoms by other groups of atoms, hereafter termed, respectively, deficient particles and substitute particles, of different chemical nature and different atomic distribution is obtained and partially studied. Effectively, the expression of the total elastic SID, also referred to as total SID, is obtained as an explicit function of the scattering components originating (i) from the different structural sub-systems, namely the atomic, deficient-particles and substitute-particles networks, that make up the material system and (ii) from the different couplings between these sub-systems. The calculations were carried out for real atomic scattering factors and within the frame of the kinematic approach to diffraction making use of no special kind-of-material-related structural assumption. Therefore, the obtained results should hold for the wide range of material systems for which the atomic scattering factors can be assumed to be real. The derived expression for the total SID conveniently gives the modifications that are introduced on the scattering distribution from a material system in its homogeneous state by the presence of structural disturbances, or, which is equivalent, of inhomogeneities. Therefore this expression may prove to be quite useful for obtaining the effects on the diffracted intensity due to particular kinds of disturbances or even for obtaining the diffracted intensity relative to situations such as those in which the material systems consist only of groups of atoms of different sorts, that is of groups of atoms which are not embedded in an otherwise homogeneous material system. An example of these can be the particular case of particles possessing a center of symmetry, with the note that the idea of `a center of symmetry' was first used by Guinier, \cite{Guinier (1939)}, in scattering studies connected with ensembles of particles (groups of atoms), see also \cite{Guinier and Fournet (1955)}. As a contribution to this case a full section, the fifth, is devoted to the expression for the SID from inhomogeneities that possess a center of symmetry. The implications for the small angle scattering (SAS) of possible structural differences in chemical nature and atomic distribution between the deficient and substitute particles have been given a particular attention here. This has led to a shape of the small angle SID that, in some cases of structural disturbances, does not obey the very attractive ``Gaussian'' Guinier law, see \cite{Guinier (1939), Guinier and Fournet (1955), Guinier (1963)} for the detailed description of this law. Consequently, the techniques of analysis which can be used to extract the so-called radius of gyration, e.g. see \cite{Guinier (1939), Guinier and Fournet (1955), Guinier (1963), Hadji and Craven (1988)} for these techniques, from real substances are not valid in some cases and, thus, new techniques will be welcome for these cases.\\
The main expression for the diffracted intensity obtained here applies to a wide range of material systems, which makes it eligible for use to immediately infer diffraction intensity expressions relevant to these systems. It is thus that, as examples, (a) a number of, already existing and non existing, SAS-related results for the diffracted intensity are deduced from the main expression, given below by (\ref{4}) and therefore by (\ref{5}), and presented under a table sort of scheme and (b) a full section is dedicated to the case of identical interacting inhomogeneities.\\
No particular study of the modifications that are introduced on the medium and large angle scattering is considered here.\\
Only time-independent, static, diffraction is considered here.
\section{General situation: the expression of the total static elastic SID}
In this first section, the broad situation generated by inhomogeneities present within real materials is described and the different resulting components of the expression of the total elastic scattering intensity separated.\\
The usual expression for the scattered intensity distribution from a material system is (e.g. \cite{Guinier and Fournet (1955)}, \cite{Guinier (1963)}, \cite{Warren (1990)}, \cite{Cowley (1992-1993)})
\begin{equation}
I(\vec{S})=\sum_{P=1}^{Nr}\sum_{Q=1}^{Nr}f_{P}(\vec{S})f_{Q}(\vec{S})\cos (%
\vec{S}.\vec{r}_{PQ})=\sum_{P=1}^{Nr}\sum_{Q=1}^{Nr}J_{PQ}\text{,}  \label{1}
\end{equation}
where the summations are over all atoms of the material ($Nr$ in total). $f_{P(Q)}$ is the scattering factor of atom $P$($Q$); $\vec{r}_{PQ}$ is the vector that gives the relative positions of atoms $P$ and $Q$; $\vec{S} =(4\pi /\lambda $)$\sin \theta $.$\vec{u}$ ($\lambda =$ wavelength, $2\theta $ is the scattering angle and $\vec{u}$ the unit vector that defines the direction of the scattering vector $\vec{S}$). Defining, on the one hand, that: 1) $N$ is the total number of atoms in the homogeneous material before the substitutions take place; 2) $n$ is the total number of the deficient particles. And, assuming, on the other hand, (a) that the $i^{th}$ deficient and the $i^{th}$ substitute particles are made up of, respectively, $M_{2i}$ and
$M_{1i}$ atoms, and (b) for simplicity reasons, that the total number of the deficient particles is the same as that of the substitute particles allows writing the total number, $Nr$, of atoms in the inhomogeneous material as
\begin{equation}
Nr=N+\sum_{i=1}^{n}M_{1i}-\sum_{i=1}^{n}M_{2i} \text{.}  \label{2}
\end{equation}
And noticing that
\[
\cos (\vec{S}.\vec{r}_{PQ})=\cos (\vec{S}.(-\vec{r}_{QP}))=\cos (\vec{S}.%
\vec{r}_{QP})
\]
allows writing
\begin{equation}
J_{PQ}=J_{QP}.  \label{3}
\end{equation}
This means that $J_{PQ}$ is an invariant with respect to the permutation of the labels $P$ and $Q$ and that we can write (\ref{1}) as (M.13), see proof at the end of the present section, or as
\begin{eqnarray}
I(\vec{S})
&=&\sum_{P=1}^{N}\sum_{Q=1}^{N}J_{PQ}+2\sum_{P=1}^{N}\left\lbrace \sum_{i=1}^{n}\left[ %
\sum_{q_{i}=1}^{M_{1i}}J_{Pq_{i}}-\sum_{q_{i}^{\prime
}=1}^{M_{2i}}J_{Pq_{i}^{\prime }}\right] \right\rbrace   \nonumber \\
&&+\sum_{i=1}^{n}\sum_{j=1}^{n}\left\lbrace \sum_{p_{i}^{\prime
}=1}^{M_{2i}}\sum_{q_{i}^{\prime }=1}^{M_{2i}}J_{p_{i}^{\prime
}q_{j}^{\prime }}-2\sum_{p_{i}^{\prime
}=1}^{M_{2i}}\sum_{q_{j}=1}^{M_{1j}}J_{p_{i}^{\prime }q_{j}}
+\sum_{p_{i}=1}^{M_{1i}}\sum_{q_{j}=1}^{M_{1j}}J_{p_{i}q_{j}}\right\rbrace \text{,}
\label{4}
\end{eqnarray}
where, now, with the new notations, $P$ and $Q$ are the labels of the atoms of the material in its homogeneous state, if no inhomogeneity were present, $p_{i(j)}$ and $q_{i(j)}$ are the labels of the atoms of the $i(j)^{th}$ substitute particle, $p_{i(j)}^{\prime }$ and $q_{i(j)}^{\prime }$ are the labels of the atoms of the $i(j)^{th}$ deficient particle, $i$ and $j$ being the labels of both types of particles.\\
Also, noticing that
\begin{eqnarray*}
\sum_{k_{u}=1}^{N_{u}}\sum_{m_{v}=1}^{N_{v}}J_{k_{u}m_{v}}
&=&\sum_{k_{u}=1}^{N_{u}}\sum_{m_{v}=1}^{N_{v}}f_{k_{u}}f_{m_{v}}\cos \vec{S}%
.\vec{r}_{k_{u}m_{v}} \\
&=&\left[ \sum_{m_{v}=1}^{N_{v}}f_{m_{v}}\exp (-i\vec{S}.\vec{r}%
_{m_{v}})\right] \left[ \sum_{k_{u}=1}^{N_{u}}f_{k_{u}}\exp (i\vec{S}.\vec{r}%
_{k_{u}})\right] \\
&=&\digamma _{v}\digamma _{u}^{*}
\end{eqnarray*}
is the product of the structure factor $\digamma _{v}$ {\large (}e.g. of the $v^{th}$ deficient (or substitute) particle (with $N_{v}=M_{1(2)}$) or of the system in its homogeneous state (with $N_{v}=N$){\large )} and its complex conjugate $\digamma _{u}^{*}$ {\large [}e.g. of the $u^{th}$ deficient (or substitute) particle (with $N_{u}=M_{1(2)}$) or of the system in its homogeneous state (with $N_{u}=N$){\large ]} allows rewriting (\ref{4}) as
\begin{equation}
I(\vec{S}) =\digamma _{H}\digamma _{H}^{*}+2\sum_{i=1}^{n}\left[ \digamma
_{i}-\digamma _{i}^{\prime }\right] \digamma
_{H}^{*}+\sum_{i=1}^{n}\sum_{j=1}^{n}\left\lbrace \left[ \digamma _{j}^{\prime }-\digamma
_{j}]\digamma _{i}^{\prime *}
+\digamma _{j}[\digamma _{i}^{*}-\digamma _{i}^{\prime *}\right] \right\rbrace \text{,}
\label{5}
\end{equation}
where $i$ and $j$ are relative to the different substitute and deficient particles.\\
Moreover, we may write (\ref{4}), and therefore (\ref{5}), as:
\begin{equation}
I(\vec{S})=I_{HH}(\vec{S})+I_{ad}(\vec{S})=I_{HH}(\vec{S})+I_{HD}(\vec{S}%
)+I_{DD}(\vec{S})\text{,}  \label{6}
\end{equation}
where $I_{HH}(\vec{S})$, corresponds to the first terms of equations (\ref{4}) (and (\ref{5})) and represents the scattered intensity in absence of inhomogeneities; $I_{ad}(\vec{S})$ corresponds to the remaining terms of (\ref{4}), and therefore of (\ref{5}), and represents the resulting effect of the generation of inhomogeneities in the material system. The total scattered intensity, as expressed by equations (\ref{4}) and (\ref{5}), may, therefore, be thought of as the scattering, $I_{HH}(\vec{S})$, from the material in its homogeneous state taken as reference modified by an additional term, $I_{ad}(\vec{S})$. This term represents the deviation from homogeneity of the real material. $I_{ad}(\vec{S})$, made up of two terms $%
I_{HD}(\vec{S})$ and $I_{DD}(\vec{S})$, see (\ref{6}), may be expressed as:
\[
I_{ad}(\vec{S})=I_{HD}(\vec{S})+I_{DD}(\vec{S})\text{,}
\]
where
\begin{eqnarray}
I_{HD}(\vec{S}) &=&I_{HD1}(\vec{S})+I_{HD2}(\vec{S})  \nonumber \\
&=&2\sum_{P=1}^{N}\left\lbrace \sum_{i=1}^{n}\left[ \sum_{q_{i}=1}^{M_{1i}}f_{P}f_{q_{i}}\cos
(\vec{S}.\vec{r}_{Pq_{i}})-\sum_{q_{i}^{\prime
}=1}^{M_{2i}}f_{P}f_{q_{i}^{\prime }}\cos (\vec{S}.\vec{r}_{Pq_{i}^{\prime
}})\right] \right\rbrace   \label{7}
\end{eqnarray}
is the contribution from the pairing of atoms ($P$) of the system in its homogeneous state with atoms $q_{i}$ and $q_{i}^{\prime }$ of the ensemble of the local deviations ($i$). Each of these deviations is represented by one deficient and one substitute particles. It may also be thought of as representing the coupling between the material in its homogeneous state and the ensemble of the local deviations present in the real material.\\ $I_{DD}(\vec{S}$), given by:
\begin{eqnarray}
I_{DD}(\vec{S}) &=&\sum_{i=1}^{n}\sum_{j=1}^{n}\left\lbrace \sum_{p_{i}^{\prime
}=1}^{M_{2i}}\sum_{q_{j}^{\prime }=1}^{M_{2j}}f_{p_{i}^{\prime
}}f_{q_{j}^{\prime }}\cos (\vec{S}.\vec{r}_{p_{i}^{\prime }q_{j}^{\prime }})\right.
\nonumber \\
&& \left.-2\sum_{p_{i}^{\prime }=1}^{M_{2i}}\sum_{q_{j}=1}^{M_{1j}}f_{p_{i}^{\prime
}}f_{q_{j}}\cos (\vec{S}.\vec{r}_{p_{i}^{\prime
}q_{j}})+\sum_{p_{i}=1}^{M_{1i}}\sum_{q_{j}=1}^{M_{1j}}f_{p_{i}}f_{q_{j}}%
\cos (\vec{S}.\vec{r}_{p_{i}q_{j}})\right\rbrace \text{,}  \label{8}
\end{eqnarray}
does not involve any atom of the ``homogeneous system'' and is associated with the local deviations acting as if they were totally isolated from the host material. These considerations allow saying that the total scattering from inhomogeneous materials may not be considered as a simple sum of the scattering from the ideal network [that which is associated with the material in its homogeneous state ($I_{HH}(\vec{S})$)] and the scattering from the inhomogeneities in absence of the host material: the insertion of perturbations results in the appearance of extra terms that are inherent in the total scattered intensity. These extra terms are:
\begin{itemize}
\item $I_{HD}(\vec{S})$, the term that represents the coupling between the homogeneous system and the
deviations,
\item $I_{DD1}(\vec{S})$, this is due to the deficient particles acting as is they were isolated from the rest of the material and \item $I_{DD2}(\vec{S})$ which represents the coupling between the deficient and the substitute particles.
\end{itemize}
The formulation of the SID from a real material system as given by (\ref{6}) may prove to be particularly useful in comparative studies of amorphous materials that have identical atomic networks. The subtraction, see \cite{Hadji et al (1987)}, of $I_{1}(\vec{S})$, the scattering profile obtained from an amorphous specimen 1, from $I_{2}(\vec{S})$, obtained from a different area within specimen 1 or from a specimen 2 that has the same network and the same thickness as specimen 1, will reduce the contributions that are equal, e.g. those from the homogeneous networks, to nothing leaving a difference that is due to the existence of dissimilarities in the distributions of inhomogeneities present within the specimens. \\
(\ref{6}) is the general expression of the static elastic scattering intensity for an \emph{arbitrary} inhomogeneous material system; and we believe that it could be helpful to keep it within arm's reach as it can be quite useful when the interest is in rapidly getting the particular intensity expression corresponding to the particular inhomogeneous material system in hand. This is for the reason that this (\ref{6}), constituting the central result of the whole work presented here, can be seen to represent at least a large part of the 'kinematic approach to diffraction'-related results already existing in the literature, for instance see \cite{Guinier (1939)}, \cite{Guinier and Fournet (1955)}), \cite{Guinier (1963)}, \cite{Warren (1990)}-\cite{Grawert and Svergun 2020}, as well as, possibly, results for quite a number of not yet studied heterogeneous material systems. It is thus that the remaining part of this work is dedicated to applying the third component, given by (\ref{8}), of this (\ref{6}) to several not complicated particulate systems. In the immediately following section, examples of diffraction intensities for particular such systems cases are inferred using (\ref{8}). The section just after the next one, Sec. \ref{Examples2}, deals with the so called small angle scattering and the one after, Sec. \ref{Examples3}, deals with the case of the diffraction from identical centrosymmetric interacting local deviations.\\
\rule[0mm]{\linewidth}{3.pt}
\paragraph{{\large A useful intensity-related result:}}
Here we are concerned with the splitting of a rather complex double sum of the type of
\begin{equation}
A=\sum_{P=1}^{N_{r}}\sum_{Q=1}^{N_{r}}J_{PQ}  \tag{M.1}
\end{equation}
into its different components to have a much easier to deal with form, the form given by (M.13) below. In (M.1) the labels $P$ and $Q$ take the same values and refer to the atoms within the \emph{inhomogeneous} specimen, the total number of these being $N_{r}$ and given by:
\begin{equation}
N_{r}=N-\sum_{i=1}^{n}M_{2i}+\sum_{i=1}^{n}M_{1i}\text{,}  \tag{M.2}
\end{equation}
where $ N $ is the total number of atoms within the specimen in its \emph{homogeneous} state, $ n $ is the total number of inhomogeneities within the specimen and the $i^{th}$ inhomogeneity consists of $M_{2i}$ deficient atoms and $M_{1i}$ replacement atoms. But in facts, the specimen consists of a matrix and inclusion groups of atoms; the specimen matrix is made up of $N-\sum_{i=1}^{n}M_{2i}$ atoms and the $i^{th}$ inclusion group of atoms is made up of $M_{1i}$ atoms.
\\
\textbf{{\large Proof:}} Writing symbolically $Nr=N-M$, where the ``symbol'' $M$ is given by:
\begin{equation}
M=\sum_{i=1}^{n}M_{2i}-\sum_{i=1}^{n}M_{1i}\text{,}  \tag{M.3}
\end{equation}
allows rewriting (M.1) concisely as:
\begin{equation}
A=\sum_{P=1}^{N-M}\sum_{Q=1}^{N-M}J_{PQ}\text{,}  \tag{M.4}
\end{equation}
and then as
\begin{equation}
A=\sum_{P=1}^{N-M}\left( \sum_{Q=1}^{N}J_{PQ}-\sum_{Q=(N-(M-1))}^{N}J_{PQ}\right) \text{,}
\tag{M.6}
\end{equation}
where the sum $\sum_{Q=(N-(M-1))}^{N}$ is over the atoms whose number is represented symbolically by $M$, see (M.3), and therefore can be rewritten to reflect this fact, for instance making use of the the following substitutions $Q\longrightarrow Q_{1}=Q-(N-M)=1$ and $N\longrightarrow M=N-(N-M)$, as $\sum_{Q_{1}=1}^{M}$. Thus (M.6) can be written as:
\begin{equation}
A=\sum_{P=1}^{N-M}\sum_{Q=1}^{N}J_{PQ}-\sum_{P=1}^{N-M}\sum_{Q_{1}=1}^{M}J_{PQ_{1}}\text{,}  \tag{M.7}
\end{equation}
and can be split further to be:
\begin{equation*}
A=\left\lbrace \sum_{P=1}^{N}\sum_{Q=1}^{N}J_{PQ} - \sum_{P=N-(M-1)}^{N}\sum_{Q=1}^{N} J_{PQ} \right\rbrace \
-\left\lbrace \sum_{P=1}^{N}\sum_{Q_{1}=1}^{M}J_{PQ_{1}}-\sum_{P=N-(M-1)}^{N}\sum_{Q_{1}=1}^{M}J_{PQ_{1}} \right\rbrace \tag{M.8}
\end{equation*}
where the sum $\sum_{P=(N-(M-1))}^{N}$ is also over the atoms whose number is represented symbolically by $M$ and can, also, be rewritten to reflect this fact, e.g. as $\sum_{P_{1}=1}^{M}$. This leads to:
\begin{equation*}
A=\left\lbrace \sum_{P=1}^{N}\sum_{Q=1}^{N}J_{PQ} - \sum_{P_{1}=1}^{M}\sum_{Q=1}^{N} J_{P_{1}Q} \right\rbrace
-\left\lbrace \sum_{P=1}^{N}\sum_{Q_{1}=1}^{M}J_{PQ_{1}}-\sum_{P_{1}=1}^{M}\sum_{Q_{1}=1}^{M}J_{P_{1}Q_{1}} \right\rbrace \tag{M.9}
\end{equation*}
where therefore $Q_{1}$ and $P_{1}$ take the same values and are both labels that refer to the same atoms, those of the inhomogeneities of the specimen.\\
On the other hand, one can write the second term of the second member of this (M.9), and therefore the second term of the second member of (M.8), as, since $P$ and $Q$ take the same values and $Q_{1}$ and $P_{1}$ take the same values:
\[
\sum_{P_{1}=1}^{M}\sum_{Q=1}^{N} J_{P_{1}Q}=\sum_{P=N-(M-1)}^{N}\sum_{Q=1}^{N} J_{PQ}=\sum_{Q_{1}=1}^{M}\sum_{P=1}^{N} J_{Q_{1}P}=\sum_{P=1}^{N} \sum_{Q_{1}=1}^{M}J_{Q_{1}P}
\]
and compare it with the third term of the second member of, also, (M.9) - and therefore with the third term of the second member of (M.8) - to observe that, provided $J_{Q_{1}P}=J_{PQ_{1}}$ or equivalently $J_{QP}=J_{PQ}$, i.e. the permutation of $P$ and $Q_{1}$ leaves $J_{PQ_{1}}$ unchanged or equivalently the permutation of $P$ and $Q$ leaves $J_{PQ}$ unchanged, the second and the third terms of the second member of (M.9) are equal to one another, i.e.
\begin{equation*}
\sum_{P_{1}=1}^{M}\sum_{Q=1}^{N} J_{P_{1}Q}= \sum_{P=1}^{N}\sum_{Q_{1}=1}^{M}J_{PQ_{1}}
\end{equation*}
which, in turn, allows writing (M.9) as:
\begin{equation*}
A= \sum_{P=1}^{N}\sum_{Q=1}^{N}J_{PQ} - 2 \sum_{P=1}^{N}\sum_{Q_{1}=1}^{M}J_{PQ_{1}}+\sum_{P_{1}=1}^{M}\sum_{Q_{1}=1}^{M}J_{P_{1}Q_{1}}  \tag{M.10}
\end{equation*}
where the first term of the second member represents the homogeneous material, i.e. the material in absence of inhomogeneities, the second term represents the coupling between the homogeneous material and the inhomogeneities and the third term represents the coupling between the inhomogeneities, the coupling between the local deviations from homogeneity. \\
We concludes from this that, provided the permutation of the labels in $J_{PQ}$ leaves $J_{PQ}$ unchanged, any equation of the type of (M.4) may be expressed as a sum of three components and be given by a relation of the sort of (M.10).\\
Thereafter, replacing $M$ by its expression (M.2) in the second term of (M.10) and successively applying the previous result, allows us to express this term as a sum of the type:
\begin{equation}
A_{2} =-2\sum_{P=1}^{N}\sum_{Q_{1}=1}^{M}J_{PQ_{1}}=-2\left[ \sum_{P=1}^{N}\sum_{i=1}^{n}\sum_{q_{1}^{\prime
}=1}^{M_{2i}}J_{Pq_{1}^{\prime }}-\sum_{P=1}^{N}\sum_{i=1}^{n}\sum_{q_{i}=1}^{M_{1i}}J_{Pq_{1}}\right] \text{.} \tag{M.11}
\end{equation}
A similar procedure leads to expressing the third term of (M.10) as:
\begin{equation}
A_{3} = \sum_{P_{1}=1}^{M}\sum_{Q_{1}=1}^{M}J_{P_{1}Q_{1}}=\sum_{i=1}^{n}%
\sum_{j=1}^{n}\sum_{p_{1}^{\prime }=1}^{M_{2i}}\sum_{q_{j}^{\prime
}=1}^{M_{2j}}J_{p_{i}^{\prime }q_{j}^{\prime }}
-2\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{p_{1}^{\prime
}=1}^{M_{2i}}\sum_{q_{j}=1}^{M_{1j}}J_{p_{i}^{\prime
}q_{j}}+\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{p_{1}=1}^{M_{1i}}%
\sum_{q_{j}=1}^{M_{1j}}J_{p_{i}q_{j}}\text{.}  \tag{M.12}
\end{equation}
Consequently and finally, provided the permutation of the labels in $J_{PQ}$ leaves $J_{PQ}$ unchanged, we may express (M.1) as follows:
\begin{equation}
A=\sum_{P=1}^{N}\sum_{Q=1}^{N}J_{PQ}+A_{2}\text{[equation(M.11)]}+A_{3}\text{%
[equation(M.12)].}  \tag{M.13}
\end{equation}
\rule[0mm]{\linewidth}{3.pt}
\section{Examples 1: diffraction intensities inferred}
\label{Examples1}
\subsection{Diffraction intensity from local deviations each of which possesses a center of symmetry}
\label{subDifLocDev}
At this level, the scattering from local deviations acting as if they were totally isolated from the host matrix, given by (\ref{8}), is expressed in terms of factors that are associated with the different, deficient and substitute, particles. The only assumption made here is that which was first made and used by Guinier, see \cite{Guinier (1939)}, in connection with centrosymmetrical particles. This assumption considers that each particle, here of either deficient or substitute type, possesses a center of symmetry. This is not a complex case.
\begin{figure}
\begin{center}
\includegraphics[scale=0.66]{Figure_1_inhomogeneous.png}
\textbf{\caption{Two, $i$ and $j$, local deviations from homogeneity.}}
\begin{minipage}[t]{14cm}
Two local deviations, $i$ and $j$, are shown. O$_{i}$ and O$_{i}^{\prime }$ are the centers of symmetry of, respectively, the substitute group $i$ and the deficient group $i$. O$_{j}$ and O$_{j}^{\prime }$ are similar to O$_{i}$ and O$_{i}^{\prime }$ but are relative to the local deviation $j$. $\vec{R}_{i}^{O} $ and $\vec{R} _{j}^{O}$ are the vectors giving the positions of the points O$_{i}$ and O$_{j}$ with respect to the arbitrary origin O; $\vec{R}_{i}^{\prime O}$ and $\vec{R}_{j}^{\prime O}$ locate the points O$_{i}^{\prime }$ and O$_{j}^{\prime }$, these are not represented in the figure.
\end{minipage}
\label{FigureInhom01}
\end{center}
\end{figure}
\noindent \underline{Substitute particle:} The center of symmetry for the $i^{th}$ substitute particle, the point $O_{i}$ in figure \ref{FigureInhom01}, may be defined as, since in a centrosymmetrical group of atoms an atom at a position given by $\vec{r}$ relative to the center of symmetry is coupled to another, identical, atom located at $-\vec{r}$ by the interatomic distance $\vert \vec{r} -(-\vec{r})\vert$ the middle point of which is the center of symmetry:
\[
\sum f_{p_{i}}\overrightarrow{O_{i}M_{p_{i}}}=\sum f_{p_{i}}\vec{r}_{p_{i}}=0%
\text{,}
\]
where $\vec{r}_{p_{i}}=\overrightarrow{O_{i}M_{p_{i}}}$ is the vector which defines the position of the atom $p_{i}$ of the $i^{th}$ substitute particle with respect to $O_{i}$, its center of symmetry;\\
\underline{Deficient particle:} For the $i^{th}$ deficient particle the center of symmetry is defined by:
\[
\sum f_{p_{i}^{\prime }}\overrightarrow{O_{i}^{\prime }M_{p_{i}^{\prime }}}%
=\sum f_{p_{i}^{\prime }}\vec{r}_{p_{i}^{\prime }}=0\text{,}
\]
where $\vec{r}_{p_{i}^{\prime }}=\overrightarrow{O_{i}^{\prime }M_{p_{i}^{\prime }}}$ gives the position of atom $p_{i}^{\prime }$ of the $ i^{th}$ deficient particle with respect to the center of symmetry $ O_{i}^{\prime }$. \\
We note that, according to figure 1, $O_{i}^{\prime }$ and $O_{i}$ do not necessarily coincide and $\vec{r}_{p_{i}q_{j}}$ may be written as
\begin{equation}
\vec{r}_{p_{i}q_{j}}=\vec{r}_{q_{j}}^{O_{j}}-\vec{r}_{p_{i}}^{O_{i}}+%
\vec{R}_{j}^{O}-\vec{R}_{i}^{O}=\vec{r}_{q_{j}}-\vec{r}_{p_{i}}+\vec{R}_{ij}%
\text{,}  \label{9}
\end{equation}
where
\[
\vec{R}_{ij}=\vec{R}_{j}^{O}-\vec{R}_{i}^{O}
\]
is the vector which gives the relative positions of the substitute particles $j$ and $i$, $O$ is the arbitrary origin. Similarly, one may write for the deficient particles:
\begin{equation}
\vec{r}_{p_{i}^{\prime }q_{j}^{\prime }}=\vec{r}_{q_{j}^{\prime }}-\vec{r}%
_{p_{i}^{\prime }}+\vec{R}_{ij}^{\prime }  \label{10}
\end{equation}
and for the deficient-substitute particle coupling:
\begin{equation}
\vec{r}_{p_{i}^{\prime }q_{j}}=\vec{r}_{q_{j}}-\vec{r}_{p_{i}^{\prime }}+(%
\vec{R}_{j}-\vec{R}_{i}^{\prime })  \label{11}
\end{equation}
where $\vec{R}_{j}-\vec{R}_{i}^{\prime }=\overrightarrow{R_{j}^{O}}-\overrightarrow{R_{i}^{\prime O}}$ gives the relative positions of the substitute ($j$) and the deficient ($i$) particles.\\
\textit{\textbf{Calculation of the terms of $I_{DD}(\vec{S})$ using }}(\ref{8}).
Inserting $\vec{r}_{p_{i}q_{j}}$, as given by (\ref{9}), into the cosine factor of the third term of (\ref{8}) gives:
\[
\cos (\vec{S}.\vec{r}_{p_{i}q_{j}})=\cos\vec{S}.(\vec{r}_{q_{j}}-\vec{r}_{p_{i}}+\vec{R}_{ij})
\]
and expanding gives:
\[
\cos (\vec{S}.\vec{r}_{p_{i}q_{j}})=\cos \vec{S}.(\vec{r}_{q_{j}}-\vec{r}%
_{p_{i}})\cos (\vec{S}.\vec{R}_{ij})-\sin \vec{S}.(\vec{r}_{q_{j}}-\vec{r}%
_{p_{i}})\sin (\vec{S}.\vec{R}_{ij})
\]
and, therefore:
\begin{eqnarray}
\cos (\vec{S}.\vec{r}_{p_{i}q_{j}}) &=&[\cos \vec{S}.\vec{r}_{q_{j}}\cos
\vec{S}.\vec{r}_{p_{i}}+\sin \vec{S}.\vec{r}_{q_{j}}\sin \vec{S}.\vec{r}%
_{p_{i}}]\cos \vec{S}.\vec{R}_{ij}  \nonumber \\
&&-[\sin \vec{S}.\vec{r}_{q_{j}}\cos \vec{S}.\vec{r}_{p_{i}}-\cos \vec{S}.%
\vec{r}_{q_{j}}\sin \vec{S}.\vec{r}_{p_{i}}]\sin \vec{S}.\vec{R}_{ij}\text{.}
\label{12}
\end{eqnarray}
So the third term of (\ref{8}) is obtained by inserting $\cos (\vec{S}.\vec{r}_{p_{i}q_{j}})$, as given by (\ref{12}), into this term and summing over all atoms $p_{i(j)}$ of all substitute particles $i(j)$. This contribution represents the scattering from the substitute particles only, i.e. the scattering in the ``absence of the host material''; writing it as $I_{DD3}(\vec{S})$ gives:
\begin{eqnarray}
I_{DD3}(\vec{S})
&=&\sum_{i=1}^{n}\sum_{j=1}^{n}\{(F_{1j}F_{1i}+F_{1j}^{\prime
}F_{1i}^{\prime })\cos \vec{S}.\vec{R}_{ij}  \nonumber \\
&&-(F_{1j}^{\prime }F_{1i}-F_{1j}F_{1i}^{\prime })\sin \vec{S}.\vec{R}%
_{ij}\},  \label{13}
\end{eqnarray}
where:
\[
F_{1j}=\sum_{q_{j}=1}^{M_{1j}}f_{q_{j}}\cos \vec{S}.\vec{r}_{q_{j}}\text{;
and }F_{1i}=\sum_{p_{i}=1}^{M_{1i}}f_{p_{i}}\cos \vec{S}.\vec{r}_{p_{i}}
\]
are, as defined by Guinier, see \cite{Guinier (1939)} and \cite{Guinier and Fournet (1955)}, for the centrosymmetric particles, the structure factors of, respectively, the $j^{th}$ and the $i^{th}$ substitute particles. On the other hand, the factors $F_{1j}^{\prime }$ and $F_{1i}^{\prime }$ are, respectively, given by
\[
F_{1j}^{\prime }=\sum_{q_{j}=1}^{M_{1j}}f_{q_{j}}\sin \vec{S}.\vec{r}_{q_{j}}%
\text{, and }F_{1i}^{\prime }=\sum_{p_{i}=1}^{M_{1i}}f_{p_{i}}\sin \vec{S}.%
\vec{r}_{p_{i}}\text{ .}
\]
Proceeding in the same way we find the first term, $I_{DD1}(\vec{S})$, of (\ref{8}) to be:
\begin{eqnarray}
I_{DD1}(\vec{S})
&=&\sum_{i=1}^{n}\sum_{j=1}^{n}\{(F_{2j}F_{2i}+F_{2j}^{\prime
}F_{2i}^{\prime })\cos \vec{S}.\vec{R}_{ij}  \nonumber \\
&&-(F_{2j}^{\prime }F_{2i}-F_{2j}F_{2i}^{\prime })\sin \vec{S}.\vec{R}_{ij}\}
\label{14}
\end{eqnarray}
where $F_{2j}$ and $F_{2i}$ are, respectively, the structure factors of the $j^{th}$ and the $i^{th}$ deficient centrosymmetric particles and are given by:
\[
F_{2j}=\sum_{q^{\prime}_{j}=1}^{M_{2j}}f_{q^{\prime}_{j}}\cos \vec{S}.\vec{r}_{q^{\prime}_{j}}\text{
and }F_{2i}=\sum_{p^{\prime}_{i}=1}^{M_{2i}}f_{p^{\prime}_{i}}\cos \vec{S}.\vec{r}_{p^{\prime}_{i}}%
\text{.}
\]
The $F_{2j}^{\prime }$ and $F_{2i}^{\prime }$ factors are as follows:
\[
F_{2j}^{\prime }=\sum_{q^{\prime}_{j}=1}^{M_{2j}}f_{q^{\prime}_{j}}\sin \vec{S}.\vec{r}_{q^{\prime}_{j}}%
\text{ and }F_{2i}^{\prime }=\sum_{p^{\prime}_{i}=1}^{M_{2i}}f_{p^{\prime}_{i}}\sin \vec{S}.%
\vec{r}_{p^{\prime}_{i}}\text{.}
\]
Equation (\ref{14}) represents the scattering from the deficient particles alone.\\
Finally, we find the second term of (\ref{8}) is given by:
\begin{eqnarray}
I_{DD2}(\vec{S})
&=&-2n\sum_{i=1}^{n}\sum_{j=1}^{n}\{(F_{1j}F_{2i}+F_{1j}^{\prime
}F_{2i}^{\prime })\cos \vec{S}.(\vec{R}_{j}-\vec{R}_{i}^{\prime })  \nonumber
\\
&&-(F_{1j}^{\prime }F_{2i}-F_{1j}F_{2i}^{\prime })\sin \vec{S}.(\vec{R}_{j}-%
\vec{R}_{i}^{\prime })\}  \label{15}
\end{eqnarray}
This term, $I_{DD2}(\vec{S})$, represents the coupling of the deficient and the substitute particles. $F_{1j}$, $F_{2i}$, $F_{1j}^{\prime }$ and $F_{2i}^{\prime }$ are as defined above and together with the other $F_{2j}$, $F_{2j}^{\prime }$ etc. will be referred to, in the remaining part of this text, as the ``$F$'' factors.\\
The scattered contribution $I_{DD}(\vec{S})$, see (\ref{8}), to the total scattered intensity from the ensemble of the local deviations ``behaving as if they were totally isolated from the host matrix'' is obtained by combining equations (\ref{13}), (\ref{14}) and (\ref{15}).\\
No structure-related assumption has been made so far apart from allowing ($\alpha $) the different atomic structure factors to be real and ($\beta $) all the particles to have a center of symmetry. Therefore, equations (\ref{13}), (\ref{14}) and (\ref{15}) are expected to hold in any structural situation, that satisfies ($\alpha $) and ($\beta $), for which the relaxation effects that may result from the particle substitutions are negligible. In particular, they should hold in the case of non-interacting as well as in the case of interacting local deviations. A limited treatment of these two cases is given in the next subsections.
\subsection{Scattering from \textit{n} non-interacting local deviations}
\subsubsection{The not necessarily identical deviations}
Widely separated deviations could still have some interacting effects. However, these are commonly assumed to be negligible and ignored. The main consequence of this approximation is to ignore, in the expression of the
scattered intensity, all terms that are not due to the self-pairing of the different local deviations. This is achieved by setting $i=j$ in all equations that describe the scattered intensity. Consequently:
\[
\vec{R}_{ij}=\vec{R}_{ij}^{\prime }=\vec{R}_{ii}=\vec{R}_{ii}^{\prime }=\vec{%
0}\text{ and }\vec{R}_{j}-\vec{R}_{i}^{\prime }=\vec{R}_{i}-\vec{R}_{i}=%
\overrightarrow{O^{\prime }O_{i}}\text{,}
\]
where $\overrightarrow{O^{\prime }O_{i}}$ is the vector which gives the relative positions of the centers of symmetry of the substitute and the deficient particles of the $i^{th}$ local deviation, therefore:
\[
F_{1j}=F_{1i}\text{; }F_{1j}^{\prime }=F_{1i}^{\prime }\text{; }F_{2j}=F_{2i}%
\text{; }F_{2j}^{\prime }=F_{2i}^{\prime }\text{; }M_{1j}=M_{2i}\text{ and }%
M_{2j}=M_{2i}\text{.}
\]
Using these results together with equations (\ref{13}), (\ref{14}) and (\ref{15}) leads to the following expression for the contribution from non-interacting local deviations, $I_{DD}(\vec{S})$:
\begin{eqnarray}
I_{DD}(S) &=&\sum_{i=1}^{n}\left\lbrace F_{1i}^{2}+F_{1i}^{\prime
2}+F_{2i}^{2}+F_{2i}^{\prime 2}+2[F_{1i}F_{2i}+F_{1i}^{\prime
}F_{2i}^{\prime }]\cos \vec{S}.\overrightarrow{O^{\prime }O_{i}}\right.  \nonumber
\\
&&\left.-2[F_{1i}^{\prime }F_{2i}-F_{1i}F_{2i}^{\prime }]\sin \vec{S}.%
\overrightarrow{O^{\prime }O_{i}}\right\rbrace \text{.}  \label{16}
\end{eqnarray}
The form of this relation suggests that it may be written as a sum of $n$ terms, $I_{iDD}(\vec{S})$, $i = 1, 2, \cdots, n$, each, $i$, of which represents the contribution from a single local deviation:
\[
I_{DD}(\vec{S})=\sum_{i=1}^{n}I_{iDD}(\vec{S})\text{.}
\]
\subsubsection{The non-interacting and identical local deviations}
In the case where all local deviations are identical, or assumed to be, on average, identical, one may expect, as a consequence, that all deficient particles are similar and that all substitute particles are similar. This would imply that particles of the same type consist of atoms of the same chemical natures which are distributed according to a same atomic distribution function, therefore the number of atoms $M_{2i}$ is expected to be the same for all deficient particles ($M_{2i}=M_{2}$ whatever $i$) and, similarly, $M_{2i}=M_{2}$ whatever $i$ for the substitute particles. Although these considerations leave the expression for the scattered component $I_{DD}(\vec{S})$, see (\ref{16}), formally unchanged, only $M_{1i}$ and $M_{2i}$ are set equal to, respectively, $M_{1}$ and $M_{2}$, they greatly ease the calculations for two limiting cases: the case of identical and randomly oriented local deviations and the case of identical local deviations with a preferred orientation. These two particular cases are considered next.
\paragraph{The case of identical and randomly oriented local deviations}
The first term of $I_{DD}(\vec{S})$, (\ref{8}), corresponds to the contribution from the substitute particles ``behaving as if they were isolated from the host matrix''. The third term, on the other hand, corresponds to the contribution from the deficient particles acting as if they were alone. These two terms are similar to one another and to the expression for the relevant corresponding system, or ensemble, of groups of atoms, or particles, considered and dealt with elsewhere by Guinier, see \cite{Guinier (1939)} and \cite{Guinier and Fournet (1955)}. But the second term of this equation is the result of the coupling between the deficient particles and the substitute particles. This is similar in form to the scattering from a set of simple particles, the difference is that $\vec{r}_{p^{\prime }q}$ does not necessarily take zero values.\footnote{The exception is for the case where the atomic distribution functions for both deficient and substitute particles are the same.} For randomly oriented identical deviations, we find, through angularly averaging this second term over the $4\pi$ solid angle as done by Debye to get the equation bearing his name, i.e. the Debye equation of scattering also known as the Debye equation, see \cite{Debye 1915}, \cite{Guinier (1939)}, \cite{Guinier and Fournet (1955)}, that $I_{DD}(\vec{S})$ is given by:
\begin{eqnarray}
I_{DD}(S) &=&n\left\lbrace \sum_{p=1}^{M_{1}}\sum_{q=1}^{M_{1}}f_{p}f_{q}\frac{%
\sin Sr_{pq}}{Sr_{pq}}-2\sum_{p^{\prime
}=1}^{M_{2}}\sum_{q=1}^{M_{1}}f_{p^{\prime }}f_{q}\frac{\sin Sr_{p^{\prime
}q}}{Sr_{p^{\prime }q}} \right.  \nonumber \\
&&\left.+\sum_{p^{\prime }=1}^{M_{2}}\sum_{q^{\prime }=1}^{M_{2}}f_{p^{\prime
}}f_{q^{\prime }}\frac{\sin Sr_{p^{\prime }q^{\prime }}}{Sr_{p^{\prime
}q^{\prime }}}\right\rbrace   \label{17}
\end{eqnarray}
where $p^{\prime }$ and $q^{\prime }$ are the labels of the atoms of one of the deficient particles, and $p$ and $q$ are the labels of the atoms of one of the substitute particles. For monatomic systems, $I_{DD}(\vec{S})$ is obtained by setting $f_{p^{\prime }}=f_{q^{\prime }}$, whatever $p^{\prime }$ and $q^{\prime }$, and $f_{p}=f_{q}$, whatever $p$ and $q$.
\paragraph{Case of identical local deviations that are oriented in a same direction}
In this case the scattered component $I_{DD}(\vec{S})$ is found to be given by:
\begin{eqnarray}
I_{DD}(\vec{S}) &=&n\left\lbrace  \sum_{p=1}^{M_{1}}\sum_{q=1}^{M_{1}}f_{p}f_{q}\cos
\vec{S}.\vec{r}_{pq}-2\sum_{p^{\prime
}=1}^{M_{2}}\sum_{q=1}^{M_{1}}f_{p^{\prime }}f_{q}\cos \vec{S}.\vec{r}%
_{p^{\prime }q}\right.  \nonumber \\
&&\left. +\sum_{p^{\prime }=1}^{M_{2}}\sum_{q^{\prime }=1}^{M_{2}}f_{p^{\prime
}}f_{q^{\prime }}\cos \vec{S}.\vec{r}_{p^{\prime }q^{\prime }}\right\rbrace   \label{18}
\end{eqnarray}
\section{Examples 2: the simplest ``Small Angle Scattering'' cases}
\label{Examples2}
SAS has found application in both material science and biology. Effectively, small-angle X-ray scattering (SAXS), sensitive to inhomogeneities in electron density in the 10-250-nm range, is very useful for the study of materials such as the organic-templated mesostructured chalcogenides, \cite{Brant et al (2013)}, and, together with small-angle neutron scattering (SANS), is useful for investigating the nanoscale structure and interactions in materials with disorder at the atomic scale level, see \cite{Eds Duncan et al 2014} plus references therein. Also, SAXS is, at present and for various reasons, the standard tool for biologists, see \cite{Grawert and Svergun 2020}.
In this section we consider two SAS cases. One is relative to identical non-interacting and randomly oriented local deviations, see  \ref{ROIandN-ILD}, and the other one is relative to identical non-interacting with ``a preferred orientation'' local deviations, see \ref{IN-IwPOLD}.
The scattering which appears at small angles, around the direction of the unscattered beam, may be dominated by the scattering from finite sized particles embedded in an otherwise homogeneous material, see \cite{Guinier (1939)}, \cite{Guinier and Fournet (1955)}. We, therefore, may expect the small angle scattering from a real specimen to be essentially coming from the ensemble of its local deviations and given by the scattering component which is represented by one of the two (\ref{17}) and (\ref{18}) according to whether the deviations are randomly oriented or oriented in a well defined direction. The two terms corresponding, one, to the deficient and, the other one, to the substitute particles, i.e. the first and the third terms of either of (\ref{17}) and (\ref{18}), acting as if they were alone are similar to the original cases considered by Guinier, \cite{Guinier (1939)}, and therefore have been dealt with elsewhere, -see also \cite{Guinier and Fournet (1955)}-, and were found to approximate a Gaussian form at small angles in the case of randomly oriented non-interacting particles. The approximate expression for the second term of (\ref{17}) is obtained here by applying the technique by Guinier, see \cite{Guinier (1939)}, and found to be, see next subsection, a bit different from the Guinier law: it contains in it an extra local deviation-related parameter.
\subsection{Randomly oriented identical and non-interacting local deviations}
\label{ROIandN-ILD}
Applying the method by Guinier for randomly oriented identical non-interacting local deviations, the deficient-substitute coupling term, $I_{DD2}(S)$ as given by the second term of (\ref{17}), is expanded in a power series of $Sr_{p^{\prime }q}$ to give, when $Sr_{p^{\prime }q}$ $<<1$:
\begin{eqnarray}
I_{DD2}(S) &\thickapprox &-2n\left[ \sum_{p^{\prime
}=1}^{M_{2}}\sum_{q=1}^{M_{1}}f_{p^{\prime }}f_{q}-\sum_{p^{\prime
}=1}^{M_{2}}\sum_{q=1}^{M_{1}}f_{p^{\prime }}f_{q}\frac{S^{2}}{6}%
r_{p^{\prime }q}^{2}\right.  \nonumber \\
&&\left.+\sum_{p^{\prime }=1}^{M_{2}}\sum_{q=1}^{M_{1}}f_{p^{\prime }}f_{q}\frac{%
S^{4}}{120}r_{p^{\prime }q}^{4}- \cdots \right]   \label{19}
\end{eqnarray}
where $r_{p^{\prime }q}=\mid \vec{r}_{q}-\vec{r}_{p}^{\prime }+\overrightarrow{O^{\prime }O\mid }$, which inserted into this limited expansion of $I_{DD2}(S)$, leads to a sum of a quite large number of terms. However, consequent to the allowance for a center of symmetry to exist for either of the deficient and substitute particles, this number reduces to ten, since each vector $\vec{r}$, e.g. $\vec{r}_{p}$, is dual to another vector $-\vec{r}$, e.g. $-r_{p}$, thus causing all terms involving odd powers of $\vec{r}$ to vanish. For instance the term which is related to $ \vec{r}_{q}.(\vec{r}_{p^{\prime }})^{3}$ is given by:
\begin{equation}
8n\frac{S^{4}}{120}\sum_{q=1}^{M_{1}}\sum_{p^{\prime
}=1}^{M_{2}}f_{q}f_{p^{\prime }}\vec{r}_{q}(\vec{r}_{p^{\prime }})^{3}=8n\frac{%
S^{4}}{120}\left( \sum_{q=1}^{M_{1}}f_{q}\vec{r}_{q}\right) \left(
\sum_{p^{\prime }=1}^{M_{2}}f_{p^{\prime }}(\vec{r}_{p^{\prime }})^{3}\right)
=0\text{,}  \label{20}
\end{equation}
where each of the sums in brackets is zero. And thus $I_{DD2}(S)$ reduces to:
\begin{eqnarray}
I_{DD2}(S) &\thickapprox &-2n\left( \sum_{p^{\prime }=1}^{M_{2}}f_{p^{\prime
}}\right) \left( \sum_{q=1}^{M_{1}}f_{q}\right) \{1-\frac{S^{2}}{6}%
(R^{2}+R^{\prime 2}+O^{\prime }O^{2})  \nonumber \\
&&+\frac{S^{4}}{120}[R^{4}+R^{\prime 4}+O^{\prime }O^{4}+6(R^{2}R^{\prime
2}+R^{2}O^{\prime }O^{2}  \nonumber \\
&&\qquad \qquad \quad +R^{\prime 2}O^{\prime }O^{2})]-...\}\text{,}
\label{21}
\end{eqnarray}
[i.e. after dividing and multiplying by $(\sum_{p^{\prime }=1}^{M_{2}}f_{p^{\prime }})(\sum_{q=1}^{M_{1}}f_{q})$] where:
\begin{equation}
R^{\prime 2}=\frac{\sum_{p^{\prime }=1}^{M_{2}}f_{p^{\prime }}\vec{r}%
_{p^{\prime }}^{2}}{\sum_{p^{\prime }=1}^{M_{2}}f_{p^{\prime }}}\text{ \quad
and\quad }R^{2}=\frac{\sum_{q=1}^{M_{1}}f_{q}\vec{r}_{q}^{2}}{%
\sum_{q=1}^{M_{1}}f_{q}}\text{,}  \label{22}
\end{equation}
$R$ and $R^{\prime }$ are the radii of gyration, as defined by Guinier, see \cite{Guinier (1939)} and \cite{Guinier and Fournet (1955)}. These are associated with, respectively, the substitute and the deficient particles and are dimensions dependent structural parameters. Thus, doing as Guinier did, \cite {Guinier (1939)}, \cite{Guinier and Fournet (1955)}, and provided the contribution from the terms in $S^{4}$ and higher are not significant we may write (\ref{19}), at small angles, as;
\begin{equation}
I_{DD2}(S)\thickapprox -2n\left( \sum_{p^{\prime }=1}^{M_{2}}f_{p^{\prime
}}\right) \left( \sum_{q=1}^{M_{1}}f_{q}\right) \exp \left( -\frac{%
R^{2}+R^{\prime 2}+O^{\prime }O^{2}}{6}S^{2}\right)  \label{23}
\end{equation}
where $O^{\prime }O$ is the distance between the centers of symmetry of the deficient and the substitute particles that pertain to the same local deviation. It is to be stressed, at this point, that this expression is only valid over the angular range where:
\[
(R^{2}+R^{\prime 2}+O^{\prime }O^{2})S^{2}/6<<1\text{,}
\]
for then it makes a good approximation to the exact expression, i.e. the second term of (\ref{17}). Thus, the total small angle scattering from $n$ identical local deviations that are randomly oriented is approximately given by:
\begin{eqnarray}
I_{SAS}(S) &\thickapprox &n\left( \sum_{p=1}^{M_{1}}f_{p}\right) ^{2}\exp
\left( -\frac{S^{2}R^{2}}{3}\right) +n\left( \sum_{p^{\prime
}=1}^{M_{2}}f_{p^{\prime }}\right) ^{2}\exp \left( -\frac{S^{2}R^{\prime 2}}{%
3}\right)  \nonumber \\
&&-2n\left( \sum_{p=1}^{M_{1}}f_{p}\right) \left( \sum_{p^{\prime
}=1}^{M_{2}}f_{p^{\prime }}\right) \exp \left( -\frac{S^{2}(R^{2}+R^{\prime
2}+O^{\prime }O^{2}}{6}\right) \text{,}  \label{24}
\end{eqnarray}
where:
\[
\frac{S^{2}R^{\prime 2}}{3}<<1\text{; }\frac{S^{2}R^{2}}{3}<<1\text{ and }%
\frac{S^{2}(R^{2}+R^{\prime 2}+O^{\prime }O^{2})}{6}<<1\text{.}
\]
The shape of (\ref{24}) is different from the Gaussian shape. Therefore the application of the Guinier law is not possible in this case.\\
Also, it is important to notice that (\ref{24}) is an \emph{approximate} expression for the corresponding exact expression, given by (\ref{17}), which is suitable for use when and only when the conditions imposed by $S^{2}R^{\prime 2}/3<<1\text{; }S^{2}R^{2}/3<<1\text{ and }S^{2}(R^{2}+R^{\prime 2}+O^{\prime }O^{2})/6<<1$ are fully satisfied. These conditions totally specify the angular range of validity of (\ref{24}); and within this range (\ref{24}) varies rather like $y=a-bS^{2}+\cdots$. Consequently, if (\ref{24}) is applied properly, i.e. really within the angular range over which it is effectively valid, then the exponential decays it is made up of will introduce no apparent correlation peak within this range.\\
If furthermore the host matrix and the substitute particles are monatomic then, in this equation, $(\sum_{p=1}^{M_{1}}f_{p})^{2}$, $(\sum_{p^{\prime}=1}^{M_{2}}f_{p^{\prime }})^{2}$ and $(\sum_{p=1}^{M_{1}}f_{p})(\sum_{p^{\prime }=1}^{M_{2}}f_{p^{\prime }})$ become, respectively, $(M_{1}f_{B})^{2}$, $(M_{2}f_{M})^{2}$ and $M_{1}M_{2}f_{M}f_{B}$, where $f_{B}$ and $f_{M}$ are the atomic scattering factors for, respectively, the substitute and the host matrix atoms.\\
We note that $OO^{\prime }=O^{\prime }O$ is expected to be different from zero in situations where the structural differences, in size and in shape, and therefore in dimensions, between the deficient and the substitute particles are such that their respective centers of symmetry do not coincide. For instance, this can be so when the size of the substitute atoms is comparatively large or comparatively small with respect to the size of the deficient atoms.\\
This subsection on small angle scattering is closed with a table of some particular expression cases deduced from (\ref{24}). Therefore, all these expression cases deduce from the mother expression for the partial intensity, $I_{DD}(\vec{S})$, given by (\ref{8}) and all of them rely on the approximations used by Guinier to get the law bearing his name, the Ginier law. These approximations, in number of two, are: that assuming the randomness considered by Debye, \cite{Debye 1915}, to get the formula bearing his name, i.e. the Debye formula of scattering, and that assuming the product of $S$ and $r$ is very small compared with unity considered by Guinier, \cite{Guinier (1939)}, e.g. $Sr_{p^{\prime }q}$ $<<1$. Therefore the application of all of these expressions are subject to the same sort of restrictions as the Guinier law. The approximation considered by Guinier is particularly restricting as regards to the angular range over which his law applies.
\subsubsection{A table of SAS intensity expressions for particular inhomogeneity cases obtained from (\ref{24})}
\paragraph{Case 1}
Case where the centers of symmetry of the deficient and the substitute particles coincide, i.e. $OO^{\prime }=0$;
\begin{equation}
I_{SAS}(S)\thickapprox n\left\{ \left[ \sum_{p=1}^{M_{1}}f_{p}\right] \exp
\left( -\frac{S^{2}R^{2}}{6}\right) -\left[ \sum_{p^{\prime
}=1}^{M_{2}}f_{p^{\prime }}\right] \exp \left( -\frac{S^{2}R^{\prime 2}}{6}%
\right) \right\} ^{2}\text{.}  \label{25}
\end{equation}
This (\ref{25}) resembles, but is fundamentally different from, the equation given at the bottom of page 342 in the Guinier's 1963 book, \cite{Guinier (1963)}, which is quite normal, since the two equations correspond to two different material systems.
\paragraph{Case 2}
Case where the atomic distributions of the deficient and the substitute particles are identical, but with different atomic scattering factors. In this situation we may expect the number of atoms to be the same ($M$) for both types of particles. Therefore, we may expect the radii of gyration to be the same ($R=R^{\prime }$) for both types of particles and write
\begin{equation}
I_{SAS}(S)\thickapprox n\left\{ \left[ \sum_{p=1}^{M}f_{p}\right] -\left[
\sum_{p^{\prime }=1}^{M}f_{p^{\prime }}\right] \right\} ^{2}\exp \left( -%
\frac{S^{2}R^{2}}{3}\right) \text{.}  \label{26}
\end{equation}
If, furthermore, in this particular case, we assume the scattering is associated with some average scattering power densities
\[
\rho _{s}=\frac{\sum_{p=1}^{M}f_{p}}{V}
\]
for the substitute particles and
\[
\rho _{d}=\frac{\sum_{p^{\prime }=1}^{M}f_{p^{\prime }}}{V}
\]
for the deficient particles; where $V$ is the volume that is occupied by one local deviation, then (\ref{26}) becomes
\begin{equation}
I_{SAS}(S)\thickapprox nV^{2}[\rho _{s}-\rho _{d}]^{2}\exp \left( -\frac{%
S^{2}R^{2}}{3}\right) \text{.}  \label{27}
\end{equation}
\paragraph{Case 3}
In situations where the local deviations are associated with pure deficiencies, e.g. when dealing with pure materials, the small angle scattering approximate distribution is obtained by setting equal to zero the
scattering factors of the substitute atoms in (\ref{24}) to get to the form expressing the very well known Guinier law, see \cite {Guinier (1939)}, \cite{Guinier and Fournet (1955)}, \cite {Guinier (1963)}:
\begin{equation}
I_{SAS}(S)\thickapprox n\left[ \sum_{p^{\prime }=1}^{M_{2}}f_{p^{\prime
}}\right] ^{2}\exp \left( -\frac{S^{2}R^{\prime 2}}{3}\right) \text{,}
\label{28}
\end{equation}
$R^{\prime }$ is the radius of gyration for the deficient particles.
\subsection{Identical non-interacting with ``a preferred orientation'' local deviations}
\label{IN-IwPOLD}
For identical non-interacting and ``all oriented in the same direction'' local deviations the scattered intensity component $I_{DD}(S)$, as given by (\ref{18}), may be approximated using the method given in the book by Guinier and Fournet, see \cite{Guinier and Fournet (1955)}, to give:
\begin{eqnarray}
I_{SAS}(S) &\thickapprox
&n\sum_{p=1}^{M_{1}}\sum_{q=1}^{M_{1}}f_{p}f_{q}\exp [-S^{2}D^{2}(\vec{u} )]+n\sum_{p^{\prime }=1}^{M_{2}}\sum_{q^{\prime }=1}^{M_{2}}f_{p^{\prime }}f_{q^{\prime }}\exp [-S^{2}D^{\prime 2}(\vec{u})]  \nonumber \\
&&-2n\sum_{p^{\prime }=1}^{M_{2}}\sum_{q=1}^{M_{1}}f_{p}f_{q}\exp \left[ -\frac{S^{2}(D^{2}(\vec{u})+D^{\prime 2}(\vec{u})+(\vec{u}.OO^{\prime })^{2})}{2}\right] \text{.}  \label{29}
\end{eqnarray}
Where:
\begin{equation}
D^{2}(\vec{u})=\frac{\sum_{q=1}^{M_{1}}f_{q}(\vec{u}.\vec{r}_{q})^{2}}{%
\sum\nolimits_{q=1}^{M_{1}}f_{q}}\text{\quad and\quad }D^{\prime 2}(\vec{u})=%
\frac{\sum\nolimits_{p^{\prime }=1}^{M_{2}}f_{p^{\prime }}(\vec{u}.\vec{r}%
_{p^{\prime }q})^{2}}{\sum\nolimits_{p^{\prime }=1}^{M2}f_{p^{\prime }}}%
\text{.}  \label{30}
\end{equation}
$D^{2}(\vec{u})$ is the square of the average inertial distance $D(\vec{u})$, see, for instance, the book by Guinier and Fournet \cite{Guinier and Fournet (1955)}, of the substitute particle with respect to the plane perpendicular to $\vec{u}$, the unit vector which defines the direction of the scattering vector $\vec{S}$, that passes through the center of symmetry $O$ of the substitute particle. $D^{\prime }(\vec{u})$ is the square of the average inertial distance $D^{\prime 2}(\vec{u})$ associated with the deficient particle with respect to the plane perpendicular to $\vec{u}$ and which passes through the center of symmetry $O^{\prime }$.
\section{Example 3: identical centrosymmetric interacting local deviations}
\label{Examples3}
In this section we deduce from (\ref{8}), as another example of application, the partial diffraction intensity expression, given by (\ref{34}), relative to identical centrosymmetric interacting local deviations which was originally obtained by Guinier and Fournet, see \cite{Guinier and Fournet (1955)}. The deduction requires passing through different successive intensity expressions each of which is, therefore, relative to a different stage corresponding to a different material case, the complexity of the material decreasing progressively.

The total scattered intensity from an inhomogeneous system that may be described as being made up of $n$ identical interacting local deviations may be obtained by firstly adding together ($\delta $) the intensity contributions originating from the coupling of the local deviations with the ``homogeneous system'', ($\epsilon $) the intensity contributions that are due to the local deviations in the absence of the host matrix and ($\varepsilon $) the scattered intensity from the homogeneous system, i.e. in absence of the local disturbances, and secondly setting (i) $M_{1i(j)}$, the number of atoms which make up the $i(j)^{th}$ substitute particle, equal to the same number $M_{1}$ for all substitute particles and (ii) $M_{2i(j)}$ equal to the same number $M_{2}$ for all deficient particles. This leaves the total scattered intensity expression formally unchanged. However, it is considerably simplified in the particular case of monatomic systems, i.e. when the inclusion particles as well as the host matrix are monatomic.

The assumption, see \cite{Guinier (1939)}, \cite{Guinier and Fournet (1955)} that the particles possess a center of symmetry allows writing: $F_{2i}^{\prime }=F_{2j}^{\prime }=F_{1j}^{\prime }=F_{1i}^{\prime }=0$, since, in this situation, each vector $\vec{r}$ of any particle possesses a dual vector $-\vec{r}$ with respect to the center of symmetry resulting in the zeroing of the primed $F$ factors. This leads, in a first stage, to writing the contribution to the total scattered intensity that is due to $n$ local interacting deviations, this is given by the sum of the three (\ref{13}), (\ref{14}) and (\ref{15}), as:
\begin{eqnarray}
I_{DD}(\vec{S}) &=&\sum_{i=1}^{n}\sum_{j=1}^{n}\left\lbrace [F_{1j}F_{1i}]\cos \vec{S}.\vec{R}_{ij}+[F_{2j}F_{2i}]\cos \vec{S}.\vec{R}_{ij}^{\prime }\right. \nonumber \\
&&\left.-2[F_{1j}F_{2i}]\cos \vec{S}.(R_{j}-\vec{R}_{i}^{\prime })\right\rbrace \text{,}
\label{31}
\end{eqnarray}
where the different $F_{1i(j)}$ and $F_{2i(j)}$ are structure factors. Further simplifications may be obtained for the scattering from identical local deviations if the centers of symmetry of the substitute and the deficient particles coincide. In this case, and a second stage, $\vec{R}_{j}=\vec{R}_{j}^{\prime }$ whatever $j$, and (\ref{31}) simplifies to lead to an $I_{DD}(\vec{S})$ given by:
\begin{equation}
I_{DD}(\vec{S})=\sum_{i=1}^{n}%
\sum_{j=1}^{n}[F_{1j}F_{1i}-2F_{1j}F_{2i}+F_{2j}F_{2i}]\cos \vec{S}.\vec{R}%
_{ij}\text{.}  \label{32}
\end{equation}
For pure materials, i.e. when no particle is included, therefore when the local deviations are of pure deficiency type, the scattering component $I_{DD}(\vec{S})$ is obtained by setting to zero the scattering factors of the substitute particles, $F_{1i(j)}$, in (\ref{32}) which, then and in a third and last stage, leads to an $I_{DD}(\vec{S})$ given by:
\begin{equation}
I_{DD}(\vec{S})=\sum_{i=1}^{n}\sum_{j=1}^{n}[F_{2j}F_{2i}]\cos \vec{S}.\vec{R%
}_{ij}  \label{33}
\end{equation}
and has the same form as (\ref{1}). This similarity suggests that, as for atoms, diffraction peaks may be generated by particles present in the material system, i.e. in favorable cases. The $F$ factors play here a role that is very much similar to that of the atomic form factors $f$ in (\ref{1}) and are particle shape and size, and therefore dimensions, dependent.
Finally, using the expressions for $F_{2i(j)}$, see \ref{subDifLocDev}, allows getting:
\begin{equation}
I_{DD}(\vec{S})=\sum_{i=1}^{n}\sum_{j=1}^{n}\left[ \sum_{p_{j}^{\prime }=1}^{M_{2}}f_{p_{j}^{\prime }}\cos \vec{S}.\vec{r}_{p_{j}^{\prime }}\right] \left[ \sum_{q_{j}^{\prime }=1}^{M_{2}}f_{q_{j}^{\prime }}\cos \vec{S}.\vec{r}_{q_{j}^{\prime }}\right] \cos \vec{S}.\vec{R}_{ij}\text{,}  \label{34}
\end{equation}
which is, as it should be, identical to the corresponding expression that is given in the book by Guinier and Fournet, \cite{Guinier and Fournet (1955)}, and that is relative to ``a group of identical interacting particles''.
\section{ Conclusion}
It has been found that the introduction of structural disturbances on an otherwise homogeneous material system yields a total scattered intensity distribution that is the sum of two main components. The first of these is that which one would expect if the material were in its homogeneous state, and the other, an additional contribution due to the ``particle substitution operation'', represents the deviation from homogeneity of the material system. The introduction of the structural disturbances, through substitutions of groups of atoms (small particles) by other groups of atoms of a different kind, allows varying at will the two important degrees of freedom that are the chemical nature and the atomic distribution. The additional scattering is found to consist of two separate groups of components. One of these represents the scattering from the local deviations acting as if they were totally isolated from the host matrix and the other represents the coupling between the local deviations and the ``host homogeneous state''. The presence of this last component indicates that the scattering from inhomogeneous systems may not be assumed to consist of the simple sum of the intensity from the ideal network, on the one hand, and of that from the inhomogeneities supposed totally isolated from the host material, on the other hand. Thus, this conceptional representation of a real material has resulted in a formulation of the scattered intensity that may prove to be useful for comparative studies of material systems with the same, or, at least, on average the same, sort of network. This is for the reason that the parts of the scattered intensity distribution which are due to the particles present within the material system are very much dependent on their dimensions.
It has also resulted in shapes of the scattered intensity distribution at small angles, i.e. from inhomogeneous material systems, that do not obey the very attractive Guinier law in some cases. As a consequence of this, the techniques of analysis [such as the Guinier plot \cite{Guinier (1939)},
\cite{Guinier and Fournet (1955)}, \cite {Guinier (1963)}, the tangent technique, \cite{Guinier and Fournet (1955)}, \cite{Guinier (1963)}, and the derivative of the logarithm of the small angle scattering intensity distribution technique, \cite{Hadji and Craven (1988)}], which can be used to extract the radius of gyration, first defined in \cite{Guinier (1939)}, are not applicable in cases where the chemical nature and the atom distribution of the inclusion particles do not allow representing the intensity at small angles of scattering using a Gaussian form. New techniques will, therefore, be most welcome.
\addcontentsline{toc}{section}{References}

redpajama_set_name: RedPajamaArXiv


Playstation 1 Ultimate Emulator Guide - Max Settings!
Launchbox - Review And Setup Guide!
Pubg Mobile (english Version) - On Pc Plus Mouse And Keyboard!
3ds Emulator Full Guide - Controllers, Game Fixes And Pokemon Hacks!


It's past midday in the dead of summer here in the nation's capital and with the NASDAQ hovering right around 1900 and interest rates falling on 10 YR TRES, we can't help extending our mind half way around the globe. After all, way back in the year 1900, there must have been contrarian thinkers out there ready to predict the meteoric rise of the US economy to a position of world dominance. Nobody (but their relatives) had to tell the streams of immigrants coming from Poland, Russia, Italy, Ireland, the Ukraine --you name it-- that factory jobs were opening up in hard to pronounce places called Waterbury, Bridgeport, the Bowery, Trenton, Buffalo, Toledo, Chicago, Seattle etc. Nobody had to tell Tesla or Marconi where to bring their ideas.
In a world dominated by the global colonial powers the equation was quite simple: the poor countries provided foodstuffs and raw materials to the rich countries which turned them into usable items that were then exported at a premium back to the same suppliers. Of course, like everything else in life, it wasn't quite that simple but in taking a look at a fading photo of a meeting of the world's leading physicists, Conseil de Physique Solvay, held in Brussels (see brownish pic) at the Hotel Metropole in 1911 not only is there no attendee from Asia, there isn't one American although names like Plank, Curie, Einstein, Rutherford would later take on international significance.
Most Americans, we'd wager, couldn't give a damn about any of this in July or any other month but even among those who do, it's unlikely many have grasped just how much of the US trade picture is third worldly. To a large degree we export agricultural and capital goods and import manufactured goods. Last year, for instance, the only sector in which the US had a trade surplus, was agricultural products, according to the US Census Bureau. In the single month of May of 2004, in a recent but wholly typical example we exported, to China alone, a little less than $3 billion and imported a little more than $15 billion for a gap of more than $12 billion. The capital goods segment is important because it actually underlines the trend. Capital goods, is the term economist's use for machinery that is used to make manufactured goods. In simple terms, capital goods means machines that American factories might be using to provide manufacturing jobs that are instead being sent to Asia, Mexico and other cheap labor cost areas.
China, like Japan and the US and Europe before it, has become a massive siphon of raw materials like iron ore, copper, petroleum and other agricultural goods and manufacturing know-how that has primarily benefited neighboring countries like Australia, New Zealand and Japan.
Lately, the Chinese government, sensing an impending bubble, has started to cause a slowing mainly by restricting lending activity.
On October 30, 2003 Greg Mankiw, the Bush Administration's Chairman of the Council of Economic Advisers testified to Congress on the subject "China's Trade and US Manufacturing Jobs" The speech is famous because in it Mankiw argues that despite the enormous imbalance, trade with China is a positive with little impact on US manufacturing jobs.
"Imports from China are one of many factors that influence manufacturing employment. The five industries that have contributed most significantly to manufacturing job losses since July 2000 are: computer and electronic equipment (16% of all manufacturing job losses), machinery (10.8%) transportation equipment (10.7%), fabricated metal products (10.7%), and semiconductor and electronic components (7.5%). These are export-intensive industries for the United States where imports from China are small. This suggests that US job losses are more closely related to declines in domestic investment and weak exports than to import competition."
What are we supposed to make of these comments?..... Perhaps, that the real losses in jobs to China are okay because we are also losing jobs to other parts of the world, (Europe, airplanes, Singapore, chips) in more vital areas?
So what's going to happen now that it's clear to anybody who knows how to read (and isn't a member of the band) that the much anticipated present US recovery is nearly stillborn despite --or perhaps, because of-- the massive dose of steroids it's been given by the Fed for the last three years? One thing is for sure --and the bond market knows it-- it's going to be very, very hard for the Fed to raise rates any more than the measly 25 basis points it got back last month. Like Japan, after its big bust, the Fed finds itself with absolutely no room down on interest rates. Given, the upcoming election sensitivities, they will probably stand pat.
But after the election, there's going to be noise all over the place no matter who wins. The dollar is already nearly back to its lows against the Euro and Pound and gold has crept back up over $400. Suddenly, we can expect to see a lot of attention given to the Chinese RMB by the Americans. A devaluation against the RMB would slow Chinese imports and raise Chinese buying power vis a vis the dollar, they will exclaim, even as the noise alone drives the dollar further down against the Euro, Pound and Yen. That will, if history is any example, rekindle interest in gold.
But even under intense pressure from its major trading partner, will the Chinese government accede to a revaluation? This is a tricky question. The Chinese do not want to give up their competitive advantage, which is working out just fine for them. They face enormous pressures from the massive populations in their own hinterlands to create their own Waterbury's and Akron's and San Jose's.
One thing we do think: Alan Greeenspan will not stay out the full six years of this present term he just got appointed to. The kitchen is going to get a lot hotter.
The US trade imbalance and what used to be called back in the quaint old days, "the dollar overhang", have been pet subjects here at DW. You might ask why we are so obsessed with a situation that seems, after all, to work pretty well. In short, the American consumer gets to borrow and do what he does best, consume; the Asians make and service, OPEC pumps oil, and the dollars get recycled back in the form of direct investment and government notes.
What's the alternative?, you might ask. All this global economic activity is pretty good, isn't it? and prices for goods and services are holding pretty steady, aren't they? You're not suggesting some sort of protectionism, are you?
The answer, of course, is that we have no belief in building artificial barriers whether it is around the trade of goods and services or the movement of bits and bytes. It's as if we had just discovered that two wrongs don't make a right or some other profound old saw. How about, "if it ain't broke why fix it?" We say, ask Ken Lay! Just because it ain't collapsed don't mean it won't collapse.
Here's the scary thought in our mind: everybody acknowledges that the present economy rests on the shoulders of the American consumer. Great news, you say, why just last month the university that measures such things noted that consumer confidence was up! Having borrowed against her house, maxed out her plastic, this gal was coming back for more. She can even get a new SUV with nothing down and 0% interest. GM is even offering $5,000 back on some of its more macho models. In other words, it's a dream come true, it's almost like, like... you get paid for buying. Who wouldn't have confidence?
And so the Fed prints more dollars even as it bumps up interest rates a quarter of a point. And the world continues to buy our notes.
Sounds good, n'est-ce-pas? So how does that reconcile with a public education system that has lost the confidence of middle class families in the cities and the suburbs? How does it reconcile with a world order in which growing masses of left-behind peoples reject edgy angst-ridden "entertainment" --a major export of the "knowledge" society-- for a return to the relative security of a pre-Galilean world view? Of course, to put it mildly, there is no shortage of contradictions.
Perhaps you can run ever greater trade deficits year after year. Perhaps the world will never tire of taking more newly minted greenbacks. Perhaps Trenton will take and take and take ad infinitum. And so with our dominance we can pretty much get what we want. If you feel that way there is little to worry about.
As for us, we're a little more old fashioned. We actually think there may come a day when investors and government bankers around the world start taking a harder look at all those dollars accumulating in their vaults. So here's a thought, instead of concentrating on imports that fill our need for tchotckes; we might note that only half the current accounts balance is based on imports, the other half registers in a countervailing manner, exports. In other words, if we can get exports up to all those newly minted Chinese and Indian shoppers not to mention all those Europeans we can offset rising imports.
Now let's look at what is certain highly valuable US intellectual property, our brand names. Here's some rhetorical questions we will pose to the current administration: does dropping out of highly charged international conventions like the Kyoto Treaty help or enhance US brands abroad, does rejecting the rule of the World Bank over US behavior, does unilateral US foreign policy, rejection of the role of the UN, insults to old allies, a cavalier attitude that looks like bullying, etc. enhance the world view of US brands? Fade left as we imagine that shining city on the hill topped by giant golden arches.
US foreign corporate earnings have been somewhat masked in the last couple of years by a falling dollar that quite automatically boosts incoming profits denominated in Yen and Euros on US corporate balance sheets. If the real cutting edge of US business is measured in the extension of US brands into the new and old countries whose goods we buy, can we count on government and corporate decision-makers in these countries purchasing proprietary mission-critical systems from companies like Oracle, Cisco and Microsoft whose intellectual property has now become, along with Hollywood's, not only key to our prosperity but also a major concern of Congress?
One of the reasons we have put a major focus on the debates raging around Open Systems and digital rights management (DRM) here at DW has a lot to do with the above. After all, as Microsoft has itself acknowledged in its Trustworthy Computing initiative, it all comes down to Trust, doesn't it? And Trust is in mighty short supply these days.


text:
Home / Sports / Football reaches out to prevent the spread of coronavirus
Football reaches out to prevent the spread of coronavirus
By Paul Seaton
Tags: Coronavirus, English Premier League, football
While football is suspended indefinitely, many fans thoughts have started out being focused on those who have contracted Coronavirus, and how quickly they have overcome the virus. From the Arsenal manager, Mikel Arteta, to Zlatan Ibrahimovich, there are dozens of players who have contracted the virus and the sport has seen every major league now suspended after the Australian leagues closed down today.
It's got so serious that the World Health Organisation (WHO) have put out a new video showing football fans – and anyone else – which five steps you can take to prevent Coronavirus.
https://www.youtube.com/watch?v=ZTl-5AjDb48
Video can't be loaded because JavaScript is disabled: Pass the message: Five steps to kicking out coronavirus (https://www.youtube.com/watch?v=ZTl-5AjDb48)
There have already been a great number of touching, humorous and helpful videos featuring footballers arriving in cyberspace during these worrying times. Everton manager Carlo Ancelotti called up reached out to a Toffees fan called Mark who suffers with Motor Neurone Disease and who had been living off Netflix for a few days in isolation until he got a surprise call:
https://www.youtube.com/watch?v=PFLPQi3PAP8
Video can't be loaded because JavaScript is disabled: CARLO ANCELOTTI CALLS ISOLATED FAN WITH MOTOR NEURONE DISEASE (https://www.youtube.com/watch?v=PFLPQi3PAP8)
There are plenty of efforts being made to help out anyone in isolation, but many of us who love football are thinking of the NHS workers who are tirelessly trying to help those with Coronavirus or suffering with any other ailment in these difficult times.
Former Manchester United players Gary Neville and Ryan Giggs, who collectively own two hotels in the Manchester area, have revealed that they are helping NHS staff in their area by putting them up in the hotel for free, with volunteer members of the two hotels Hotel Football, located around the corner from Old Trafford and The Stock Exchange in the city centre attending to help out the doctors and nurses that they can.
https://www.youtube.com/watch?v=_xNFS4l33-0
Video can't be loaded because JavaScript is disabled: Coronavirus: Gary Neville says NHS staff will stay at his hotels for free (https://www.youtube.com/watch?v=_xNFS4l33-0)
While some of the efforts have appealed to the practical and emotional needs of the great British public, there has been no shortage of humour from the footballers, with the 'Stay at Home Challenge' of juggling toilet rolls taking off, with players such as Odion Ighalo, Peter Crouch and Tammy Abraham all taking part.
https://www.youtube.com/watch?v=EKmD7Ir58uk
Video can't be loaded because JavaScript is disabled: Footballers find inventive ways to beat COVID-19 lockdown boredom (https://www.youtube.com/watch?v=EKmD7Ir58uk)
However long the Coronavirus crisis goes on, football has a key role to play in reaching fans and spreading helpful messages, from teaching the public how to avoid spreading the virus to informing us all of how we can all reach out and connect with those we love.
From managers such as Carlo Ancelotti to players like Lionel Messi, donations from Cristiano Ronaldo to advice from Gary Lineker, it turns out footballers aren't bad at educating us when it comes to saving our lives and the lives of others in dealing with threat of COVID-19 and kicking it firmly into touch as soon as we get the chance.
Premier League Review – Gameweek #25
The Secret Coach: Seven free players who could move to Premier League this summer

redpajama_set_name: RedPajamaCommonCrawl


Oh wow! All of you ladies did a fantastic job! So hard to choose but I think it's a toss up for me between Tee and LaTonya! Now I'm inspired to make one of these trench coats for myself!
Totally the best year of this challenge yet! Some serious competition among these ladies!! Everyone did an amazing job and I'm astounded by the craftsmanship in such a short period of time. Bravo!
I have just reviewed quickly all the entries and am totally blown away. This is the first time I was aware of the contest and took a look at the last challenge winners' projects and was totally wowed. This is going to be hard. I will read every word the entrants wrote about their project and deliberate and write comments. I planned my day around this so I would have time to do the job--a good one I hope. Congrats ladies on a job amazingly done. You all have your own style!! Let's get started Melissa! Julie, this is a great challenge you've created.
Wow, I am VERY impressed with what these ladies turned out in such a short time!! Honestly... every time I'd read about a contestant and see what she made, I would think "Oh, well she needs to stay in the contest!" and then I ran out of contestants... Beautiful and interesting coats.
I really like LaToyna's coat in the red, but I didn't see any pictures on the construction, just the final product - so that would make me choose someone else as the winner.
so yay! I so love it..How I wish I can do the sewing you know.
All contestants have to be commended for turning out these garments in just a week, that is no mean feat! For me, it's a toss up between the red and the burgundy.
Impressive makes for one week! Kudos to everyone.
All of the trench coats are awesome! Each participant shared a little about their personalities and their trench coats seems to match who they are. A lot of work went into the construction of each coat. It is definitely a hard decision to make.