\subsubsection{LimitSurface}
\label{LimitSurface}
The \textbf{LimitSurface} PostProcessor is aimed to identify the transition
zones that determine a change in the status of the system (Limit Surface).

\ppType{LimitSurface}{LimitSurface}

\begin{itemize}
  \item \xmlNode{parameters}, \xmlDesc{comma separated string, required field},
  lists the parameters that define the uncertain domain and from which the LS
  needs to be computed.
  \item \xmlNode{tolerance}, \xmlDesc{float, optional field}, sets the absolute
  value (in CDF) of the convergence tolerance.
 %
  This value defines the coarseness of the evaluation grid.
 %
 \default{1.0e-4}
  \item \xmlNode{side}, \xmlDesc{string, optional field}, in this node the user can specify
  which side of the limit surface needs to be computed. Three options are available:
  \\ \textit{negative},  Limit Surface corresponding to the goal function value of ``-1'';
  \\ \textit{positive}, Limit Surface corresponding to the goal function value of ``1'';
  \\ \textit{both}, either positive and negative Limit Surface is going to be computed.
  %
  %
\default{negative}
  % Assembler Objects
  \item \textbf{Assembler Objects} These objects are either required or optional
  depending on the functionality of the Adaptive Sampler.
  %
  The objects must be listed with a rigorous syntax that, except for the xml
  node tag, is common among all the objects.
  %
  Each of these nodes must contain 2 attributes that are used to map those
  within the simulation framework:
   \begin{itemize}
    \item \xmlAttr{class}, \xmlDesc{required string attribute}, is the main
    ``class'' of the listed object.
    %
    For example, it can be ``Models,'' ``Functions,'' etc.
    \item \xmlAttr{type}, \xmlDesc{required string attribute}, is the object
    identifier or sub-type.
    %
    For example, it can be ``ROM,'' ``External,'' etc.
    %
  \end{itemize}
  The \textbf{LimitSurface} post-processor requires or optionally accepts the
  following objects' types:
   \begin{itemize}
    \item \xmlNode{ROM}, \xmlDesc{string, optional field}, body of this xml
    node must contain the name of a ROM defined in the \xmlNode{Models} block
    (see section \ref{subsec:models_ROM}).
    \item \xmlNode{Function}, \xmlDesc{string, required field}, the body of
    this xml block needs to contain the name of an External Function defined
    within the \xmlNode{Functions} main block (see section \ref{sec:functions}).
    %
    This object represents the boolean function that defines the transition
    boundaries.
    %
    This function must implement a method called
    \textit{\_\_residuumSign(self)}, that returns either -1 or 1, depending on
    the system conditions (see section \ref{sec:functions}).
    %
    \end{itemize}
\end{itemize}

\textbf{Example:}
\begin{lstlisting}[style=XML,morekeywords={name,subType,debug,class,type}]
<Simulation>
 ...
 <Models>
  ...
    <PostProcessor name="computeLimitSurface" subType='LimitSurface' verbosity='debug'>
      <parameters>x0,y0</parameters>
      <ROM class='Models' type='ROM'>Acc</ROM>
      <!-- Here, you can add a ROM defined in Models block.
           If it is not Present, a nearest neighbor algorithm
           will be used.
       -->
      <Function class='Functions' type='External'>
        goalFunctionForLimitSurface
      </Function>
    </PostProcessor>
    ...
  </Models>
  ...
</Simulation>
\end{lstlisting}


\section{HashPipe Prototype in P4}
\label{sec:prototype}

%- add details of p4 prototype
%- code fragments?
%- some details of barefoot compiler; and configurations there.
%- p4 to OVS compiler?

We built a prototype of \TheSystem using P4 version 1.1~\cite{p4-v1.1-spec}.  We
verified that our prototype produced the same results as our HashPipe simulator
by running a small number of artificially generated packets on the switch
behavioral model \cite{p4-bm} as well as our simulator, and ensuring that the
hash table is identical in both cases at the end of the measurement interval.

\begin{figure}
 \begin{lstlisting}[basicstyle=\footnotesize, caption=
  HashPipe stage with insertion of new flow. Fields prefixed with m are metadata fields., label=q1:p4-code, captionpos=b, basicstyle=\footnotesize, breaklines = true,
numbers=left, xleftmargin=2em,frame=single,framexleftmargin=2.0em]
 action doStage1(){
   mKeyCarried = ipv4.srcAddr;
   mCountCarried = 0;
  modify_field_with_hash_based_offset(mIndex, 0, stage1Hash, 32);

   // read the key and value at that location
   mKeyTable = flowTracker[mIndex];
   mCountTable = packetCount[mIndex];
   mValid = validBit[mIndex];

   // check for empty location or different key
   mKeyTable = (mValid == 0) ? mKeyCarried : mKeyTable;
   mDif = (mValid == 0) ? 0 : mKeyTable -  mKeyCarried;

   // update hash table
   flowTracker[mIndex] = ipv4.srcAddr;
   packetCount[mIndex] = (mDif == 0) ? mCountTable+1: 1;
   validBit[mIndex] = 1;

   // update metadata carried to the next table stage
   mKeyCarried = (mDif == 0) ? 0 : mKeyTable;
   mCountCarried = (mDif == 0) ? 0 : mCountTable;
}
 \end{lstlisting}
 \end{figure}

At a high level, \TheSystem uses a match-action stage in the switch pipeline for
each hash table. In our algorithm, each match-action stage has a single default
action---the algorithm execution---that applies to every packet. Every stage
uses its own P4 \emph{register arrays}---stateful memory that persists across
successive packets---for the hash table. The register arrays maintain the flow
identifiers and associated counts.
%% In this context, every stage's P4
%% registers act as a hash table where the flow identifiers and the associated
%% counts are maintained.
The P4 action blocks for the first two stages are
presented in Listings \ref{q1:p4-code} and \ref{q2:p4-code}; actions for
further stages are identical to that of stage $2$. The remainder
of this
section walks through the P4 language constructs with specific references to
their usage in our \TheSystem prototype.\\

\noindent \textbf{Hashing to sample locations:} The first step of the action is
to hash on the flow identifier (source IP address in the listing) with a custom
hash function, as indicated in line $4$ of Listing \ref{q1:p4-code}. The result
is used to pick the location where we check for the key. The P4
behavioral model \cite{p4-bm} allows customized hash function definitions. We
use hash functions of the type $(a_i\cdot x + b_i)\%p$ where the chosen $a_i,
b_i$ are co-prime to ensure independence of the hash functions across
stages. Hash functions of this sort are implementable on hardware and have
been used in prior work~\cite{univmon,li2016flowradar}.
%% If the key is absent,
%% the flow is used as one of the samples for approximating the minimum.

\noindent \textbf{Registers to read and write flow statistics:} The flows are
tracked and updated using three registers: one to track the flow identifiers,
one for the packet count, and one to test validity of each table index. The
result of the hash function is used to index into the registers for reads and
writes. Register reads occur in lines $6$-$9$ and register writes occur in lines
$15$-$18$ of Listing \ref{q1:p4-code}. When a particular flow identifier is read
from the register, it is checked against the one currently being
carried. Depending on whether there is a match, either the value $1$ is written
back or the current value is incremented by $1$.

\noindent \textbf{Packet metadata for tracking current minimum:} The values read
from the registers are placed in packet metadata since we cannot test conditions
directly on the register values in P4. This enables us to compute the minimum of
the carried key and the key in the table before writing back into the register
(lines $11$-$13$ of Listing \ref{q1:p4-code} and lines $3$, $6$, and $9$ of
Listing \ref{q2:p4-code}). Packet metadata also plays a crucial role in
conveying state (the current minimum flow id and count, in this case) from one
stage to
another. The metadata is later used to compute the sample minimum. The updates
that
set these
metadata across stages are similar to lines $20$-$22$ of Listing
\ref{q1:p4-code}.

\begin{figure}
 \begin{lstlisting}[basicstyle=\footnotesize, caption=
HashPipe stage with rolling minimum. Fields prefixed with m are metadata fields., label=q2:p4-code, captionpos=b, basicstyle=\footnotesize, breaklines=true,
numbers=left, xleftmargin=2em,frame=single,framexleftmargin=2.0em,escapeinside={(*}{*)}]
action doStage2{
 (*$\cdots$*)
 mKeyToWrite =  (mCountInTable <  mCountCarried) ? mKeyCarried : mKeyTable));
 flowTracker[mIndex] = (mDif == 0) ? mKeyTable : mKeyToWrite;

 mCountToWrite =  (mCountTable < mCountCarried) ? mCountCarried : mCountTable;
 packetCount[mIndex] = (mDif == 0) ? (mCountTable + mCountCarried) :  mCountToWrite;

 mBitToWrite = (mKeyCarried == 0) ? 0 : 1);
 validBit[mIndex] = (mValid == 0) ? mBitToWrite : 1);
 (*$\cdots$*)
}
\end{lstlisting}
\end{figure}

\noindent \textbf{Conditional state updates to store local maximum:} The first
pipeline stage involves a conditional update to a register to distinguish a hit
and a miss for the incoming packet key in the table (line 17,
Listing~\ref{q1:p4-code}).
%
Subsequent stages must also write back different values to the table key and
count registers, depending on the result of the lookup (hit/miss) and a
comparison of the flow counts.
%
Accordingly, we perform a conditional write into the flow id and count registers
(lines 4 and 7, Listing~\ref{q2:p4-code}).
%
%This ensure than an update happens if and only if the flow in our metadata is
%larger than the flow currently in the table, the flow matches the entry in the
%table or if the location in the table is currently empty.
%
Such conditional state updates are feasible at line rate~\cite{domino,minions}.
%% If an update is not needed, the metadata simply continues to the next stage
%% in the pipeline. These kinds of conditional updates are possible at line
%% rate~\cite{domino,minions}.

%Our implementation uses version 1.1 of P4 since we use ternary operators for our conditional updates that aren't supported in previous versions. There is no compiler, currently, to compile down P4 version 1.1 to a hardware target. However, the only addition in version 1.1 from the supported version 1.0 other than syntactic sugar around assignment statements, is the ability to perform a read, test on a condition and a consequent update to a register location as opposed to a simple read, modify, write. However, these conditional updates are possible at line-rate (\cite{domino}, \cite{tpp}) and there will be a compiler to compile them down to hardware in the near future \vls{@Srinivas,is this needed?}.

\iffalse
\begin{figure}[h]
\lstinputlisting[language=c]{stage2.c}
\caption{Second and subsequent stages in our scheme where incoming flow is compared against the flow in the location that it hashes to and a conditional update is made depending on which of the two is larger in size currently; The code is identical to the action block from Stage 1 except that the update to the hash table (lines 18 - 22) is replaced with the conditional update in this figure}
\label{fig:Stage2P4}
\lstset{numbers=left, numberstyle=\tiny, stepnumber=1, numbersep=5pt, escapechar=`,linewidth=8cm}
\end{figure}
\fi

%Once again, the empty location condition is subsumed under the match condition since we merely need to increment the counter with the number of packets carried in either case. However, we do need a conditional update to determine whether we need to overwrite the key in the location or not in the event of a mismatch, which is why our algorithm cannot be implemented on P4 supported hardware at the moment, but will be implementable in the near future.

\iffalse
\begin{figure}[h]
\lstinputlisting[language=c]{stage1.c}
\caption{First stage in our scheme where incoming flow is placed in the location it hashes to with either value 1 (key mismatch) or current count + 1 (key match)}
\label{fig:Stage1P4}
\end{figure}
\fi

\iffalse
\begin{figure*}[h]
 \includegraphics[width=\textwidth,height=12cm]{figures/Code.pdf}
 \caption{First stage in our scheme where incoming flow is placed in the location it hashes to with either value 1 (key mismatch) or current count + 1 (key match)}
\label{fig:P4}
\end{figure*}
\fi


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\title{Tower of operators}
\author[1,2]{Thomas Luu}
\affil[1]{Institute for Advanced Simulation 4\\
Forschungszentrum J\"ulich, Germany}
\affil[2]{Rheinische Friedrich-Williams-Universit\"at Bonn, Germany}

%\email{t.luu@fz-juelich.de}
\date{}                                           % Activate to display a given date or no date

\begin{document}
\maketitle
\begin{center}
email: \href{mailto:t.luu@fz-juelich.de}{t.luu@fz-juelich.de}
\end{center}
\abstract{
Here I try to determine the scattering amplitude for an interaction of the type
\begin{displaymath}
V(p',p)=C_0+\frac{C_2}{2}\left(p'^2+p^2\right)
\end{displaymath}
and I assume a hard-cutoff regulator.
}

\thispagestyle{fancy}

\clearpage{}
%\tableofcontents
%\newpage

\section{Interaction}
I assume an interaction of the form
\begin{equation}\label{eqn:potential}
V(p',p)=\left(C_0+C_2\frac{\left(p'^2+p^2\right)}{2}\right)f_\Lambda(p')f_\Lambda(p)
\end{equation}
with an implicit hard-cutoff regulator at some scale $\Lambda$,
\begin{equation}
f_\Lambda(p)=\Theta\left(|p|-\Lambda\right)\ .
\end{equation}
 Note that this interaction is a sum of separable terms, and there is NO delta function (we are in momentum space).  The coefficients $C_n$ are bare parameters and must be ``renormalized"

 \section{T-matrix}
 The T-matrix is defined as
 \begin{equation}
 T(p',p;E)\equiv V(p',p)+\int \frac{d^3q}{(2\pi)^3} V(p',q) G_0(q;E) T(q,p;E)\ ,
 \end{equation}
 where
 \begin{equation}
 G_0(q;E)=\frac{1}{E-q^2/m+i\epsilon}\ .
 \end{equation}
 Another relation, which is actually more useful when deriving the T-matrix, is the expression for the fully interacting propagator, which I write in operator form
 \begin{equation}\label{eqn:full G}
 \hat G(E)\equiv \frac{1}{E-\hat H_0-\hat V+i\epsilon}=\hat G_0(E)+\hat G_0(E)\hat T(E)\hat G_0(E)\ .
 \end{equation}

 \subsection{Summing the T-matrix}
I can sum the T-matrix to all orders in $V(p',p)$ since $V(p',p)$ is a sum of separable terms.   I will present my derivations (based off eq.~\eqref{eqn:full G}) for the full T-matrix in later sections, but only express the result here
\begin{multline}\label{eqn:T-matrix 1}
T(p',p;E)=\\
\frac{\mathbb{C}_0+\mathbb{C}_2\frac{\left(p'^2+p^2\right)}{2}+\mathbb{X}_4\  p'^2p^2}
{1-C_0I_0(E)-C_2I_2(E)-\frac{1}{4}C^2_2\left[I_0(E)I_4(E)-I_2(E)^2\right]}f_\Lambda(p')f_\Lambda(p)\ ,
\end{multline}
where
\begin{align}
\mathbb{C}_0&=C_0+\frac{1}{4}C^2_2I_4(E)\\
\mathbb{C}_2&=C_2-\frac{1}{2}C^2_2I_2(E)\\
\mathbb{X}_4&=\frac{1}{4}C^2_2I_0(E)\ ,
\end{align}
and\footnote{I assume we are in the infinite-volume, continuum limit, and therefore perform the integrals in a spherical basis.}
\begin{equation}
I_n(E)\equiv \frac{1}{2\pi^2}\int_0^\Lambda dq\frac{q^{2+n}}{E-q^2/m+i\epsilon}\ .
\end{equation}
Note that the solution to the T-matrix has an \emph{induced} higher-order term $\mathbb{X}_4p'^2p^2$.

We can also relate all terms $I_{n>0}(E)$ to $I_0(E)$,
\begin{align}
I_2(E)&=-\frac{m}{6\pi^2}\Lambda^3+[mE]\  I_0(E)\\
I_4(E)&=-\frac{m}{10\pi^2}\Lambda^5-[mE]\ \frac{m}{6\pi^2}\Lambda^3+[mE]^2\ I_0(E)\ .
\end{align}
And finally, we have that
\begin{equation}
I_0(E)=-\frac{m}{2\pi^2}\Lambda -i \frac{m}{4\pi}\sqrt{mE}+\frac{m}{2\pi^2\Lambda}mE+\mathcal{O}(\Lambda^{-3})\ .
\end{equation}
With these relations, the \emph{denominator} in eq.~\eqref{eqn:T-matrix 1} becomes
\begin{multline}\label{eqn:denominator}
1-C_0 I_0(E)-C_2 \left([mE]\ I_0(E) -\frac{\Lambda ^3 m}{6 \pi ^2}\right)\\
+C_2^2 \left(I_0(E) \left(\frac{\Lambda ^5 m}{40 \pi
   ^2}-\frac{\Lambda ^3 m\ [mE]}{24 \pi ^2}\right)+\frac{\Lambda ^6 m^2}{144 \pi
   ^4}\right)
   \end{multline}

\section{Scattering amplitude near threshold}
Near threshold I set everything on-shell, $p'=p=\sqrt{mE}$.  In this limit, the \emph{numerator} of eq.~\eqref{eqn:T-matrix 1} becomes
\begin{equation}\label{eqn:numerator}
C_0+C_2 p^2+C_2^2 \left(\frac{\Lambda ^3 m p^2}{24 \pi ^2}-\frac{\Lambda ^5 m}{40 \pi
   ^2}\right)\ .
   \end{equation}
% and one has that
%\begin{equation}
%T(p'=p; E=p^2/m)\equiv T(p)=\frac{4\pi/m}{p\cot \delta -ip}
%\end{equation}
%or equivalently
%\begin{equation}
%p\cot \delta -ip=\frac{4\pi}{m\ T(p)}\ .
%\end{equation}
%Plugging everything in, I get
%\begin{multline}\label{eqn:matching 1}
%p\cot \delta -ip=\\
%\frac{4\pi}{m}\frac{1-(C_0 +C_2 p^2)I_0(E)+C_2 \frac{\Lambda ^3 m}{6 \pi ^2}
%+C_2^2 \left(I_0(E) \left(\frac{\Lambda ^5 m}{40 \pi
%   ^2}-\frac{\Lambda ^3 mp^2}{24 \pi ^2}\right)+\frac{\Lambda ^6 m^2}{144 \pi
%   ^4}\right)}{C_0+C_2 p^2+C_2^2 \left(\frac{\Lambda ^3 m p^2}{24 \pi ^2}-\frac{\Lambda ^5 m}{40 \pi
%   ^2}\right)}
%   \end{multline}
%
Now I define the renormalized coefficients,
\begin{align}
C_0(\Lambda)&=C_0-C_2^2\frac{\Lambda^5 m}{40 \pi^2}\\
C_2(\Lambda)&=C_2\left(1+C_2\frac{\Lambda^3 m}{24\pi^2}\right)\ ,
\end{align}
which means eq.~\eqref{eqn:numerator} becomes
\begin{equation}
\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}\ .
\end{equation}
The denominator, eq.~\eqref{eqn:denominator}, becomes
\begin{multline}
1-I_0(E)\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}+\frac{\Lambda ^3 m}{6 \pi ^2}C_2 \left(1+
C_2 \frac{\Lambda ^3 m}{24 \pi
   ^2}\right)\\
   =1-I_0(E)\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}+C_2(\Lambda) \frac{\Lambda ^3 m}{6 \pi ^2}\ .
\end{multline}
So the full on-shell T-matrix near threshold is
\begin{equation}
T(E=p^2/m)=-\frac{4\pi/m}{p\cot \delta -ip}=\frac{\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}}{1-I_0(E)\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}+C_2(\Lambda) \frac{\Lambda ^3 m}{6 \pi ^2}}\ .
\end{equation}
This expression is \emph{almost} like eq.(1) of Beane et al. (hep-lat0312004) ``Two-Nucleons on a Lattice",
\begin{displaymath}
\mathcal{A}=\frac{\sum C_{2 n}(\mu) p^{2 n}}{1-I_{0} \sum C_{2 n}(\mu) p^{2 n}}\ ,
\end{displaymath}
except for the extra \emph{induced} term $C_2(\Lambda) \frac{\Lambda ^3 m}{6 \pi ^2}$ in the denominator.

\section{So do we have a ``Tower of Interactions"?}
I \emph{suspect} what happens (this is by no means a proof) is that as we include more terms in our potential, eq.~\eqref{eqn:potential}, the induced term gets pushed to higher and higher order.  One must be consistent in adding more terms into the potential, however (i.e. derivative expansion w/ Galilean invariance, etc. . .--essentially pionless EFT).

For example, if we include a $C_4$ term into our potential, then I suspect we would get for the scattering amplitude
 \begin{equation}
 \frac{\sum_{n=0,1,2}C_{2n}(\Lambda)p^{2n}}{1-I_0(E)\sum_{n=0,1,2}C_{2n}(\Lambda)p^{2n}+C_4(\Lambda) \mathcal{C}\Lambda^5}\ ,
 \end{equation}
 where $\mathcal{C}$ is some coefficient.  If this is true (and I'm not going to even attempt to prove this), then only in the limit of an ``infinite tower of interactions" do we recover Beane et al.'s expression,
\begin{equation}
\frac{\sum C_{2 n}(\Lambda) p^{2 n}}{1-I_{0}(E) \sum C_{2 n}(\Lambda) p^{2 n}}\ ,
\end{equation}

\section{Summing the T-matrix}
Later. . .

%Then eq.~\eqref{eqn:matching 1} becomes
%\begin{align}\label{eqn:matching 2}
%p\cot \delta -ip&=
%\frac{4\pi}{m}\frac{1-I_0(E)\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}+C_2 \frac{\Lambda ^3 m}{6 \pi ^2}\left(1+
%C_2 \frac{\Lambda ^3 m}{24 \pi
%   ^2}\right)}{\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}}\\\
%   &=\frac{4\pi}{m}\left(\frac{1}{\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}}-I_0(E)+\frac{C_2 \frac{\Lambda ^3 m}{6 \pi ^2}\left(1+
%C_2 \frac{\Lambda ^3 m}{24 \pi
%   ^2}\right)}{\sum_{n=0,1}C_{2n}(\Lambda)p^{2n}}\right)
%\end{align}
%\subsection{}

%\newpage
%\appendix

%\clearpage
%\bibliography{references}

\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Kind Inference for Datatypes}
\label{chap:kindinference}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In recent years, languages like Haskell have seen a dramatic surge of new
features that significantly extends the expressive power of their type systems.
With these features, the challenge of \emph{kind inference} for datatype
declarations has presented itself and become a worthy research problem on its
own.

In this chapter, we apply promotion to kind inference for datatypes. Inspired by
previous research on type-inference, we offer declarative specifications for
what datatype declarations should be accepted, both for \hne and for a more
advanced system we call \tit, based on the extensions in modern Haskell,
including a limited form of dependent types. We believe these formulations to be
novel and without precedent, even for \hne. These specifications are
complemented with implementable algorithmic versions. We study \emph{soundness},
\emph{completeness} and the existence of \emph{principal kinds} in these
systems, proving the properties where they hold. This work can serve as a guide
both to language designers who wish to formalize their datatype declarations and
also to implementors keen to have principled inference of principal types.

\input{Gen/KindInference/introduction.lhstex}
\input{Gen/KindInference/haskell98.lhstex}
\input{Gen/KindInference/haskell98_algo.lhstex}
\input{Gen/KindInference/haskell98_tparm.lhstex}
\input{Gen/KindInference/typeintype.lhstex}
\input{Gen/KindInference/typeintype_algo.lhstex}
\input{Gen/KindInference/extension.lhstex}

%%% Local Variables:
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\subsubsection{Application of "while" and "if" in image processing.}
Now, we try solving a problem with image thresholding by an application of
\ilcom{while} loop in a macro. Open image \textbf{mt\_darkening.tif} in the sample image you downloaded.
This is a stack, so you could slide the bar at the bottom of the window to see what is happening:
the image gets darker and darker, as frame number increases. When you study fluorescence images,
you will find such effect very often, because fluorescence bleaches due to the irradiated excitation light
for the acquisition.
When you want to segment this structure (a microtubule), you might use image-thresholding as follows.

%figure
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.6]{fig/fig23441_mtStack.png}
\caption{A stack with darkening microtubule}
\label{fig:MTstack}
\end{center}
\end{figure}

Go back to the first frame and do \ijmenu{[Image -> Adjust -> Thresholding\ldots]}. The image is then automatically adjusted with threshold level. and it seems Ok that the structure is well segmented. But the problem appears as you slid the bar at the bottom. Since image is darkening, area where highlighted decreases.

%figure
\begin{figure}[htbp]
 \centering
 \subfloat[]{\label{fig:frame1Th}\includegraphics[height = 20mm]{fig/fig23442a_frame1threshold.png}}
 \subfloat[]{\label{fig:framelastTh}\includegraphics[height = 20mm]{fig/fig23442b_frameLastthreshold.png}}
 \caption{Adjusted with threshold level first frame (a) and the last frame (b)}
 \label{fig:degradingThreshold}
\end{figure}

This is because the threshold minimum and the maximum is kept constant while the intensity of the image is decreasing. To segment the structure while the image darkening is occurring, we must adjust the threshold intensity range as the frame progresses.

The macro below finds the minimum value for the thresholding, that the highlighted area in each frame in a stack is approximately similar to the first frame. \ilcom{while} is used to loop the adjustment until the highlighted area is constant.  Then the threshold is applied to the image to convert the stack to a binary stack.

\lstinputlisting[morekeywords={*, getThreshold, getSliceNumber, getImageID, getResult, while, setThreshold}]{code/code14.ijm}

%figure
\begin{figure}[htbp]
 \centering
 \subfloat[]{\label{fig:binMTorg}\includegraphics[height = 20mm]{fig/fig23443a_framelastbinOrg.png}}
 \subfloat[]{\label{fig:binMTproc}\includegraphics[height = 20mm]{fig/fig23443b_framelastbinProc.png}}
 \caption{Binarized last frame without threshold adjustment (a) and with adjustment using macro (b).}
 \label{fig:ThresholdAdjustResults}
\end{figure}

\begin{itemize}
\item Lines 3 to 5 Check if the active window is a stack. If \ilcom{nSlices==1} (meaning that the image is not a stack), macro is terminated.

\item Lines 6 to 9: Get the threshold parameter from image and check if the image is adjusted with threshold level. If not, both upper and lower values are -1. In this case, macro is terminated.

\item Lines 10 to 14: Get stack information. \ilcom{getSliceNumber()} returns the current frame in the stack. Adjusted with threshold level area in this frame (first frame) will be used as the reference area. \ilcom{getImageID()} returns a number that specifically identifies the active window. This ImageID will be used later,by \ilcom{selectImageID(ImageID)} to re-activate the window.
%\end{itemize}

\begin{indentCom}
\textbf{getImageID()}\\
Returns the unique ID (a negative number) of the active image. Use the selectImage(id), isOpen(id) and isActive(id) functions to activate an image or to determine if it is open or active.
\end{indentCom}
%\begin{itemize}
\item Line 16: clears the results table without saving.

\item Line 17: sets the measurement parameter Area, and limits the measurement to the adjusted with threshold level region.

\item Line 18: Do the measurements. Result is recorded in the first row of the Results table.

\item Line 19: The measured area is stored in the variable ref\_area.

\item Line 20: temp\_area will be used later in the while loop.

\item Line 21: the variable ilcom{tol} is a tolerance ratio of error against the reference area. So the adjusted with threshold level area in each frame should be between 97 and 103\% of the reference area.

\item Line 22: Create a destination stack, where adjusted with threshold level images will be pasted.

\item Line 23: get the Image ID of newly created image.

\item Line 25: Loop for the frames starts.

\item Lines 26, 7: Select the original stack and sets the frame number according to the loop number. \ilcom{selectImageID} works with \ilcom{getImageID} function in line 14.

\begin{indentCom}
\textbf{selectImage(id)}\\
Activates the image with the specified ID (a negative number). If id is greater than zero, activates the idth image listed in the Window menu. With ImageJ 1.33n and later, id can be an image title (a string).
\end{indentCom}

\item Line 28: Copy the full frame.

\item Lines 29, 30: creates a temporally single frame image and the image copied in line 28 is pasted.

\item Lines 31 to 37: While loop. temp\_area is evaluated if the area is outside 97 and 103\% of the reference area. If true, then loop continues. Initial temp\_area value is 0 so the loop is at least one time. Set Threshold with lower and upper (line 32). Measure the adjusted with threshold level area, and then lower is incremented -1. The area is evaluated, and if it does not meet the criteria set in line 31, then the loop continues with wider threshold range.

\item Lines 38 to 40: The adjusted with threshold level image will be converted to black \& white image and then copied. The single frame temporary image is closed.

\item Line 41, 42: destination stack is activated and the same frame as the source stack is set.

\item Line 43: Binarized image in the clipboard is pasted into the destination stack.

\item Line 44: returns to Line 25 until all stack frames are processed.

\item Line 45: Terminates the macro.
\end{itemize}


\section{Performance Measurement}

In this section, we will outline some basics of several of the more useful
profilers available in \thispackage.

\subsection{flips and flops}

\thispackage offers two high-level utilities for measuring floating point
performance data: \code{system.flips()} and \code{system.flops()}.
The former captures floating point instruction measurements; a \textit{flip} is
the rate of floating point instructions (flpins), or the number of flpins per
second.  Perhaps the more well-known measurement is the rate of floating point
\emph{operations}.  Like its cousin \code{system.flips()}, the \thispackage
function \code{system.flops()} will measure both the number of floating point
operations as well as their rate --- the number of floating point operations
per second, or flops.

Generally, reports of flops (or flips) are not given, but Mega-flops (Mflops);
as the name implies, 1 Mflop is 1,000,000 flops.

\paragraph{Theoretical flops}

A processor has a theoretical peak number of flops, which is typically much
higher than what is found experimentally.  Still, understanding the peak flops
of a system can be useful in understanding ``good'' flops performance of a
program (just understand that it will always be lower in practice than the
theoretical peak).

The equation below demonstrates how to compute the peak Mflops of a processor:
\begin{align*}
\text{Mflops} = \text{(\# cores)} * \text{(Speed in Mhz)} *
\text{(\# of SSE units per core)} *  \text{(\# SSE operations per cycle)}
\end{align*}

single precision Mflops (divide by 2 for double precision).  So for this Intel
Sandy Bridge Core i7 as a reference, the theoretical peak is:
\begin{align*}
\text{Mflops} &= (4) * (2800 Mhz) * 2 * 2
\end{align*}

which is roughly 45 single precision Gflops, or 22.5 double precision Gflops
--- 22,500,000,000 floating point operations per second!

See where your computer stacks up against the fastest supercomputers in the
world at \href{http://www.top500.org/}{top500.org}.  For instance, again using
this laptop as a reference, and using theoretical flops as a proxy for running
the Linpack benchmark (which is not really fair, we grant), we would crush
every supercomputer on the list from June 1997, but wouldn't make the cut for
June 1999.

\subsection{Cache Misses and Cache Hits}

\paragraph{Memory and Cache}

Computers operate at \emph{billions} of cycles per second.  Of course, those
operations occur on data.  A useful abstraction we use in thinking about
processing data is you load the stuff up into RAM and then the processor does
things to it.  This is usually fine, or at least convenient, but it's not
accurate, as you are probably aware.

\begin{figure}[ht]
  \centering
  \includegraphics[scale=.54]{./include/pics/memory}
  \caption{Computer Memory Hierarchy}
  \label{fig:mem}
\end{figure}
Another more accurate abstraction is that shown in Figure~\ref{fig:mem}.  In a
sense, the magic really happens when things get into the CPU registers.  But
something that's in RAM that you want to operate on, as it's headed to the CPU,
gets cached into various levels of (comparatively) fast access storage along
the way.  Understanding this behavior, and writing your code to take advantage
of it, can have \emph{tremendous} impacts on performance.

\begin{figure}[ht]
  \centering
  \includegraphics[scale=.54]{./include/pics/cache}
  \caption{Detailed Computer Memory Hierarchy}
  \label{fig:mem2}
\end{figure}
A more detailed presentation of Figure~\ref{fig:mem} is that found in
Figure~\ref{fig:mem2}.  Here we see a more accurate presentation; for example,
L3 cache (if it exists) is shared by all cores on a moder CPU, with L2 private,
and L1 split into data and instruction caches.  The lookup costs in terms of
processor cycles are provided for each type of memory.  These ``times'' are
presented in terms of the CPU cycle for mostly historical reasons, though there
is benefit to thinking in these terms (for example, when trying to minimize
wait cycles).  Though note that a faster processor could actually
\emph{increase} these costs, specifically RAM to L3 lookups.

If you are unfamiliar with the importance (or existence) of cache, I would
strongly encourage you to experiment with this great
\href{http://www.overbyte.com.au/misc/Lesson3/CacheFun.html}%
{interactive visualization}
showing (relative) speeds of cache misses. It too involves some simplifications
of how modern hardware actually works, so if this is at all confusing, let us
all take a moment to pity the tragic life of the computer engineer.  Another
great resource is this
\href{http://www.eecs.berkeley.edu/~rcs/research/interactive_latency.html}%
{interactive visualization} showing memory and cache latency numbers by year.

\paragraph{Cache Misses} Fundamentally, a cache miss occurs when the cache
needs some piece of data to pass along to registers, but it isn't immediately
available and the computer has to go digging through RAM (or god help you,
disk) to get it.  Cache misses are bad and reduce performance.  You can't get
rid of them completely, unless your entire problem --- copies and all ---
comfortably fits into cache, but you can eliminate \emph{unnecessary} cache
misses being aware of how your data and algorithms interact with cache.  See
the demo \code{cache_access.r} in \thispackage for an example of good versus
bad cache access.


\section{{\Large$\langle j_0\rangle$ }Form factors for 3d transition elements and their ions}
\begin{table}[H]
\caption{\noindent $\langle j_0\rangle$ Form factors for 3d transition elements and their ions\hfill}
\label{3dj0}
\vspace{2mm}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Sc0 &$0.2512$ &$90.0296$ &$0.3290$ &$39.4021$ &$0.4235$ &$14.3222$ &$-0.0043$ \\
Sc1 &$0.4889$ &$51.1603$ &$0.5203$ &$14.0764$ &$-0.0286$ &$0.1792$ &$0.0185$ \\
Sc2 &$0.5048$ &$31.4035$ &$0.5186$ &$10.9897$ &$-0.0241$ &$1.1831$ &$0.0000$ \\
Ti0 &$0.4657$ &$33.5898$ &$0.5490$ &$9.8791$ &$-0.0291$ &$0.3232$ &$0.0123$ \\
Ti1 &$0.5093$ &$36.7033$ &$0.5032$ &$10.3713$ &$-0.0263$ &$0.3106$ &$0.0116$ \\
Ti2 &$0.5091$ &$24.9763$ &$0.5162$ &$8.7569$ &$-0.0281$ &$0.9160$ &$0.0015$ \\
Ti3 &$0.3571$ &$22.8413$ &$0.6688$ &$8.9306$ &$-0.0354$ &$0.4833$ &$0.0099$ \\
V0 &$0.4086$ &$28.8109$ &$0.6077$ &$8.5437$ &$-0.0295$ &$0.2768$ &$0.0123$ \\
V1 &$0.4444$ &$32.6479$ &$0.5683$ &$9.0971$ &$-0.2285$ &$0.0218$ &$0.2150$ \\
V2 &$0.4085$ &$23.8526$ &$0.6091$ &$8.2456$ &$-0.1676$ &$0.0415$ &$0.1496$ \\
V3 &$0.3598$ &$19.3364$ &$0.6632$ &$7.6172$ &$-0.3064$ &$0.0296$ &$0.2835$ \\
V4 &$0.3106$ &$16.8160$ &$0.7198$ &$7.0487$ &$-0.0521$ &$0.3020$ &$0.0221$ \\
Cr0 &$0.1135$ &$45.1990$ &$0.3481$ &$19.4931$ &$0.5477$ &$7.3542$ &$-0.0092$ \\
Cr1 &$-0.0977$ &$0.0470$ &$0.4544$ &$26.0054$ &$0.5579$ &$7.4892$ &$0.0831$ \\
Cr2 &$1.2024$ &$-0.0055$ &$0.4158$ &$20.5475$ &$0.6032$ &$6.9560$ &$-1.2218$ \\
Cr3 &$-0.3094$ &$0.0274$ &$0.3680$ &$17.0355$ &$0.6559$ &$6.5236$ &$0.2856$ \\
Cr4 &$-0.2320$ &$0.0433$ &$0.3101$ &$14.9518$ &$0.7182$ &$6.1726$ &$0.2042$ \\
Mn0 &$0.2438$ &$24.9629$ &$0.1472$ &$15.6728$ &$0.6189$ &$6.5403$ &$-0.0105$ \\
Mn1 &$-0.0138$ &$0.4213$ &$0.4231$ &$24.6680$ &$0.5905$ &$6.6545$ &$-0.0010$ \\
Mn2 &$0.4220$ &$17.6840$ &$0.5948$ &$6.0050$ &$0.0043$ &$-0.6090$ &$-0.0219$ \\
Mn3 &$0.4198$ &$14.2829$ &$0.6054$ &$5.4689$ &$0.9241$ &$-0.0088$ &$-0.9498$ \\
Mn4 &$0.3760$ &$12.5661$ &$0.6602$ &$5.1329$ &$-0.0372$ &$0.5630$ &$0.0011$ \\
Fe0 &$0.0706$ &$35.0085$ &$0.3589$ &$15.3583$ &$0.5819$ &$5.5606$ &$-0.0114$ \\
Fe1 &$0.1251$ &$34.9633$ &$0.3629$ &$15.5144$ &$0.5223$ &$5.5914$ &$-0.0105$ \\
Fe2 &$0.0263$ &$34.9597$ &$0.3668$ &$15.9435$ &$0.6188$ &$5.5935$ &$-0.0119$ \\
Fe3 &$0.3972$ &$13.2442$ &$0.6295$ &$4.9034$ &$-0.0314$ &$0.3496$ &$0.0044$ \\
Fe4 &$0.3782$ &$11.3800$ &$0.6556$ &$4.5920$ &$-0.0346$ &$0.4833$ &$0.0005$ \\
Co0 &$0.4139$ &$16.1616$ &$0.6013$ &$4.7805$ &$-0.1518$ &$0.0210$ &$0.1345$ \\
Co1 &$0.0990$ &$33.1252$ &$0.3645$ &$15.1768$ &$0.5470$ &$5.0081$ &$-0.0109$ \\
Co2 &$0.4332$ &$14.3553$ &$0.5857$ &$4.6077$ &$-0.0382$ &$0.1338$ &$0.0179$ \\
Co3 &$0.3902$ &$12.5078$ &$0.6324$ &$4.4574$ &$-0.1500$ &$0.0343$ &$0.1272$ \\
Co4 &$0.3515$ &$10.7785$ &$0.6778$ &$4.2343$ &$-0.0389$ &$0.2409$ &$0.0098$ \\
Ni0 &$-0.0172$ &$35.7392$ &$0.3174$ &$14.2689$ &$0.7136$ &$4.5661$ &$-0.0143$ \\
Ni1 &$0.0705$ &$35.8561$ &$0.3984$ &$13.8042$ &$0.5427$ &$4.3965$ &$-0.0118$ \\
Ni2 &$0.0163$ &$35.8826$ &$0.3916$ &$13.2233$ &$0.6052$ &$4.3388$ &$-0.0133$ \\
Ni3 &$0.0012$ &$34.9998$ &$0.3468$ &$11.9874$ &$0.6667$ &$4.2518$ &$-0.0148 $ \\
Ni4 &$-0.0090$ &$35.8614$ &$0.2776$ &$11.7904$ &$0.7474$ &$4.2011$ &$-0.0163$ \\
Cu0 &$0.0909$ &$34.9838$ &$0.4088$ &$11.4432$ &$0.5128$ &$3.8248$ &$-0.0124$ \\
Cu1 &$0.0749$ &$34.9656$ &$0.4147$ &$11.7642$ &$0.5238$ &$3.8497$ &$-0.0127$ \\
Cu2 &$0.0232$ &$34.9686$ &$0.4023$ &$11.5640$ &$0.5882$ &$3.8428$ &$-0.0137$ \\
Cu3 &$0.0031$ &$34.9074$ &$0.3582$ &$10.9138$ &$0.6531$ &$3.8279$ &$-0.0147$ \\
Cu4 &$-0.0132$ &$30.6817$ &$0.2801$ &$11.1626$ &$0.7490$ &$3.8172$ &$-0.0165$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_0\rangle$ }Form factors for 4d transition elements and their ions}
\begin{table}[H]
\caption{
$\langle j_0\rangle$ form factors for 4d atoms and ions}\vspace{2mm}
\label{4dj0}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Y0 &$0.5915$ &$67.6081$ &$1.5123$ &$17.9004$ &$-1.1130$ &$14.1359$ &$0.0080$ \\
Zr0 &$0.4106$ &$59.9961$ &$1.0543$ &$18.6476$ &$-0.4751$ &$10.5400$ &$0.0106$ \\
Zr1 &$0.4532$ &$59.5948$ &$0.7834$ &$21.4357$ &$-0.2451$ &$9.0360$ &$0.0098$ \\
Nb0 &$0.3946$ &$49.2297$ &$1.3197$ &$14.8216$ &$-0.7269$ &$9.6156$ &$0.0129$ \\
Nb1 &$0.4572$ &$49.9182$ &$1.0274$ &$15.7256$ &$-0.4962$ &$9.1573$ &$0.0118$ \\
Mo0 &$0.1806$ &$49.0568$ &$1.2306$ &$14.7859$ &$-0.4268$ &$6.9866$ &$0.0171$ \\
Mo1 &$0.3500$ &$48.0354$ &$1.0305$ &$15.0604$ &$-0.3929$ &$7.4790$ &$0.0139$ \\
Tc0 &$0.1298$ &$49.6611$ &$1.1656$ &$14.1307$ &$-0.3134$ &$5.5129$ &$0.0195$ \\
Tc1 &$0.2674$ &$48.9566$ &$0.9569$ &$15.1413$ &$-0.2387$ &$5.4578$ &$0.0160$ \\
Ru0 &$0.1069$ &$49.4238$ &$1.1912$ &$12.7417$ &$-0.3176$ &$4.9125$ &$0.0213$ \\
Ru1 &$0.4410$ &$33.3086$ &$1.4775$ &$9.5531$ &$-0.9361$ &$6.7220$ &$0.0176$ \\
Rh0 &$0.0976$ &$49.8825$ &$1.1601$ &$11.8307$ &$-0.2789$ &$4.1266$ &$0.0234$ \\
Rh1 &$0.3342$ &$29.7564$ &$1.2209$ &$9.4384$ &$-0.5755$ &$5.3320$ &$0.0210$ \\
Pd0 &$0.2003$ &$29.3633$ &$1.1446$ &$9.5993$ &$-0.3689$ &$4.0423$ &$0.0251$ \\
Pd1 &$0.5033$ &$24.5037$ &$1.9982$ &$6.9082$ &$-1.5240$ &$5.5133$ &$0.0213$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_0\rangle$ }Form factors for 5d transition elements and their ions}
\begin{table}[H]
 \caption{$\langle j_0\rangle$ form factors for the 5d electrons of transition atoms and ions from Hf to Re.\cite{kob:11}}
\label{5dj0} \vspace{1ex}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E }\\
\hline
Hf2 & 6s05d2 &$0.4229$ &$50.465$ &$0.7333$ &$23.865$ &$-0.3798$ &$4.051$ &$0.2252$ &$2.497$ &$-0.0018$ \\
Hf3 & 6s05d1 &$0.3555$ &$40.954$ &$0.8483$ &$21.726$ &$-0.4116$ &$4.305$ &$0.2101$ &$2.349$ &$-0.0023$ \\
Ta2 & 6s05d3 &$0.3976$ &$45.095$ &$0.7746$ &$21.028$ &$-0.6098$ &$3.471$ &$0.4395$ &$2.570$ &$-0.0020$ \\
Ta3 & 6s05d2 &$0.3611$ &$36.921$ &$0.8579$ &$19.195$ &$-0.4945$ &$3.857$ &$0.2781$ &$2.303$ &$-0.0026$ \\
Ta4 & 6s05d1 &$0.3065$ &$31.817$ &$0.9611$ &$17.749$ &$-0.5463$ &$3.979$ &$0.2816$ &$2.232$ &$-0.0030$ \\
W0 & 6s05d6 &$0.3990$ &$73.810$ &$0.7138$ &$22.815$ &$-2.0436$ &$2.710$ &$1.9319$ &$2.559$ &$-0.0023$ \\
W0 & 6s15d5 &$0.3811$ &$62.707$ &$0.7523$ &$21.434$ &$-12.5449$ &$2.702$ &$12.4130$ &$2.674$ &$-0.0023$ \\
W0 & 6s25d4 &$0.3653$ &$53.965$ &$0.7926$ &$20.078$ &$-0.8142$ &$3.030$ &$0.6581$ &$2.476$ &$-0.0023$ \\
W0 & 6s05d5 &$0.4077$ &$51.367$ &$0.7436$ &$20.256$ &$-9.8283$ &$2.780$ &$9.6788$ &$2.740$ &$-0.0021$ \\
W0 & 6s15d4 &$0.3834$ &$46.233$ &$0.7890$ &$19.278$ &$-1.4650$ &$2.947$ &$1.2945$ &$2.628$ &$-0.0022$ \\
W2 & 6s05d4 &$0.3876$ &$40.340$ &$0.8008$ &$18.621$ &$-1.3911$ &$2.995$ &$1.2048$ &$2.627$ &$-0.0023$ \\
W3 & 6s05d3 &$0.3610$ &$33.519$ &$0.8717$ &$17.176$ &$-0.6183$ &$3.445$ &$0.3883$ &$2.276$ &$-0.0028$ \\
W4 & 6s05d2 &$0.3221$ &$29.047$ &$0.9574$ &$15.979$ &$-0.6287$ &$3.597$ &$0.3525$ &$2.174$ &$-0.0033$ \\
W5 & 6s05d1 &$0.2725$ &$25.966$ &$1.0558$ &$14.954$ &$-0.6990$ &$3.643$ &$0.3745$ &$2.145$ &$-0.0037$ \\
Re0 & 6s05d7 &$0.3807$ &$63.042$ &$0.7497$ &$19.967$ &$-6.5300$ &$2.501$ &$6.4013$ &$2.451$ &$-0.0028$ \\
Re0 & 6s15d6 &$0.3691$ &$53.934$ &$0.7837$ &$18.790$ &$-9.1491$ &$2.558$ &$8.9983$ &$2.517$ &$-0.0027$ \\
Re0 & 6s25d5 &$0.3548$ &$47.108$ &$0.8210$ &$17.769$ &$-9.8674$ &$2.599$ &$9.6938$ &$2.556$ &$-0.0027$ \\
Re0 & 6s05d6 &$0.3944$ &$45.427$ &$0.7742$ &$17.948$ &$-3.1692$ &$2.653$ &$3.0028$ &$2.521$ &$-0.0026$ \\
Re0 & 6s15d5 &$0.3736$ &$41.151$ &$0.8160$ &$17.158$ &$-7.0396$ &$2.642$ &$6.8523$ &$2.577$ &$-0.0026$ \\
Re2 & 6s05d5 &$0.3825$ &$36.336$ &$0.8218$ &$16.636$ &$-8.7220$ &$2.657$ &$8.5201$ &$2.601$ &$-0.0026$ \\
Re3 & 6s05d4 &$0.3585$ &$30.671$ &$0.8863$ &$15.527$ &$-0.8682$ &$3.047$ &$0.6263$ &$2.280$ &$-0.0030$ \\
Re4 & 6s05d3 &$0.2974$ &$27.372$ &$0.9826$ &$14.807$ &$-1.8869$ &$2.840$ &$1.6100$ &$2.476$ &$-0.0031$ \\
Re5 & 6s05d2 &$0.3143$ &$23.522$ &$1.0276$ &$13.505$ &$-0.7438$ &$3.393$ &$0.4059$ &$2.030$ &$-0.0041$ \\
Re6 & 6s05d1 &$0.2146$ &$22.496$ &$1.1616$ &$13.064$ &$-1.0455$ &$3.162$ &$0.6734$ &$2.196$ &$-0.0041$ \\
\hline
\end{tabular}
}
\end{table}
\begin{table}[H]
 \caption{$\langle j_0\rangle$ form factors for the 5d electrons of transition atoms and ions from Os to Au.\cite{kob:11}}
 \label{5dj0b}
 \vspace{1ex}
{ \tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E}\\
\hline
Os0 & 6s05d8 &$0.3676$ &$54.835$ &$0.7793$ &$17.716$ &$-2.0669$ &$2.418$ &$1.9224$ &$2.247$ &$-0.0034$ \\
Os0 & 6s15d7 &$0.3571$ &$47.458$ &$0.8123$ &$16.770$ &$-1.2072$ &$2.556$ &$1.0404$ &$2.211$ &$-0.0033$ \\
Os0 & 6s25d6 &$0.3467$ &$41.778$ &$0.8458$ &$15.918$ &$-5.6370$ &$2.459$ &$5.4472$ &$2.381$ &$-0.0032$ \\
Os0 & 6s05d7 &$0.3837$ &$40.665$ &$0.8006$ &$16.096$ &$-3.5305$ &$2.487$ &$3.3488$ &$2.366$ &$-0.0030$ \\
Os0 & 6s15d6 &$0.3666$ &$36.997$ &$0.8390$ &$15.425$ &$-2.6944$ &$2.537$ &$2.4916$ &$2.360$ &$-0.0031$ \\
Os2 & 6s05d6 &$0.3786$ &$33.005$ &$0.8412$ &$14.990$ &$-7.0632$ &$2.503$ &$6.8462$ &$2.433$ &$-0.0030$ \\
Os3 & 6s05d5 &$0.3557$ &$28.222$ &$0.9002$ &$14.140$ &$-2.5972$ &$2.601$ &$2.3444$ &$2.376$ &$-0.0032$ \\
Os4 & 6s05d4 &$0.3337$ &$24.723$ &$0.9655$ &$13.288$ &$-0.9653$ &$2.906$ &$0.6698$ &$2.117$ &$-0.0037$ \\
Os5 & 6s05d3 &$0.3055$ &$22.152$ &$1.0395$ &$12.529$ &$-0.9158$ &$3.016$ &$0.5750$ &$2.032$ &$-0.0042$ \\
Os6 & 6s05d2 &$0.2714$ &$20.218$ &$1.1211$ &$11.851$ &$-0.9773$ &$3.050$ &$0.5894$ &$2.005$ &$-0.0046$ \\
Os7 & 6s05d1 &$0.2101$ &$19.108$ &$1.2240$ &$11.347$ &$-1.2543$ &$2.933$ &$0.8250$ &$2.088$ &$-0.0048$ \\
Ir0 & 6s05d9 &$0.3564$ &$48.464$ &$0.8049$ &$15.923$ &$-2.5258$ &$2.265$ &$2.3675$ &$2.121$ &$-0.0040$ \\
Ir0 & 6s15d8 &$0.3492$ &$42.195$ &$0.8350$ &$15.113$ &$-5.1496$ &$2.279$ &$4.9686$ &$2.201$ &$-0.0038$ \\
Ir0 & 6s25d7 &$0.3400$ &$37.499$ &$0.8675$ &$14.402$ &$-2.3703$ &$2.370$ &$2.1661$ &$2.177$ &$-0.0037$ \\
Ir0 & 6s05d8 &$0.3744$ &$36.764$ &$0.8240$ &$14.576$ &$-8.8616$ &$2.303$ &$8.6664$ &$2.255$ &$-0.0035$ \\
Ir0 & 6s15d7 &$0.3604$ &$33.570$ &$0.8597$ &$13.993$ &$-2.1686$ &$2.412$ &$1.9518$ &$2.188$ &$-0.0036$ \\
Ir2 & 6s05d7 &$0.3802$ &$30.032$ &$0.8550$ &$13.567$ &$-1.6185$ &$2.488$ &$1.3866$ &$2.162$ &$-0.0035$ \\
Ir3 & 6s05d6 &$0.3678$ &$25.828$ &$0.9065$ &$12.788$ &$-0.8587$ &$2.745$ &$0.5883$ &$1.960$ &$-0.0040$ \\
Ir4 & 6s05d5 &$0.3969$ &$22.050$ &$0.9310$ &$11.768$ &$-0.7090$ &$3.017$ &$0.3857$ &$1.778$ &$-0.0047$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large$\langle j_0\rangle$ }Form factors for Rare earth ions}
\begin{table}[H]
\caption{$\langle j_0\rangle$ form factors for the 4f electrons of rare earth ions}
\label{re4fj0}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Ce2 &$0.2953$ &$17.6846$ &$0.2923$ &$6.7329$ &$0.4313$ &$5.3827$ &$-0.0194$ \\
Nd2 &$0.1645$ &$25.0453$ &$0.2522$ &$11.9782$ &$0.6012$ &$4.9461$ &$-0.0180$ \\
Nd3 &$0.0540$ &$25.0293$ &$0.3101$ &$12.1020$ &$0.6575$ &$4.7223$ &$-0.0216$ \\
Sm2 &$0.0909$ &$25.2032$ &$0.3037$ &$11.8562$ &$0.6250$ &$4.2366$ &$-0.0200$ \\
Sm3 &$0.0288$ &$25.2068$ &$0.2973$ &$11.8311$ &$0.6954$ &$4.2117$ &$-0.0213$ \\
Eu2 &$0.0755$ &$25.2960$ &$0.3001$ &$11.5993$ &$0.6438$ &$4.0252$ &$-0.0196$ \\
Eu3 &$0.0204$ &$25.3078$ &$0.3010$ &$11.4744$ &$0.7005$ &$3.9420$ &$-0.0220$ \\
Gd2 &$0.0636$ &$25.3823$ &$0.3033$ &$11.2125$ &$0.6528$ &$3.7877$ &$-0.0199$ \\
Gd3 &$0.0186$ &$25.3867$ &$0.2895$ &$11.1421$ &$0.7135$ &$3.7520$ &$-0.0217$ \\
Tb2 &$0.0547$ &$25.5086$ &$0.3171$ &$10.5911$ &$0.6490$ &$3.5171$ &$-0.0212$ \\
Tb3 &$0.0177$ &$25.5095$ &$0.2921$ &$10.5769$ &$0.7133$ &$3.5122$ &$-0.0231$ \\
Dy2 &$0.1308$ &$18.3155$ &$0.3118$ &$7.6645$ &$0.5795$ &$3.1469$ &$-0.0226$ \\
Dy3 &$0.1157$ &$15.0732$ &$0.3270$ &$6.7991$ &$0.5821$ &$3.0202$ &$-0.0249$ \\
Ho2 &$0.0995$ &$18.1761$ &$0.3305$ &$7.8556$ &$0.5921$ &$2.9799$ &$-0.0230$ \\
Ho3 &$0.0566$ &$18.3176$ &$0.3365$ &$7.6880$ &$0.6317$ &$2.9427$ &$-0.0248$ \\
Er2 &$0.1122$ &$18.1223$ &$0.3462$ &$6.9106$ &$0.5649$ &$2.7614$ &$-0.0235$ \\
Er3 &$0.0586$ &$17.9802$ &$0.3540$ &$7.0964$ &$0.6126$ &$2.7482$ &$-0.0251$ \\
Tm2 &$0.0983$ &$18.3236$ &$0.3380$ &$6.9178$ &$0.5875$ &$2.6622$ &$-0.0241$ \\
Tm3 &$0.0581$ &$15.0922$ &$0.2787$ &$7.8015$ &$0.6854$ &$2.7931$ &$-0.0224$ \\
Yb2 &$0.0855$ &$18.5123$ &$0.2943$ &$7.3734$ &$0.6412$ &$2.6777$ &$-0.0213$ \\
Yb3 &$0.0416$ &$16.0949$ &$0.2849$ &$7.8341$ &$0.6961$ &$2.6725$ &$-0.0229$ \\
Pr3 &$0.0504$ &$24.9989$ &$0.2572$ &$12.0377$ &$0.7142$ &$5.0039$ &$-0.0219$ \\
\hline\\[-2ex]
\end{tabular}
}
\vspace{2mm}
\end{table}
\begin{table}[H]
 \caption{$\langle j_0\rangle$ form factors for the 5d electrons of rare earth ions.\cite{kob:12}}
 \vspace{1ex}
 \label{re5dj0}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E}\\
\hline
La2 & 4f05d1 &$0.5488$ &$63.822$ &$0.7238$ &$34.429$ &$-6.0375$ &$7.092$ &$5.7655$ &$6.839$ &$-0.0008$ \\
Ce2 & 4f15d1 &$0.4959$ &$63.797$ &$0.7571$ &$34.334$ &$-5.9903$ &$6.595$ &$5.7381$ &$6.370$ &$-0.0009$ \\
Ce3 & 4f05d1 &$1.2395$ &$35.447$ &$-1.6420$ &$4.939$ &$1.9467$ &$3.715$ &$-0.5481$ &$2.671$ &$0.0020$ \\
Pr2 & 4f25d1 &$0.4568$ &$63.765$ &$0.7795$ &$34.094$ &$-5.1096$ &$6.186$ &$4.8743$ &$5.950$ &$-0.0011$ \\
Pr3 & 4f15d1 &$0.7735$ &$40.670$ &$0.6284$ &$21.775$ &$-7.2962$ &$6.114$ &$6.8955$ &$5.866$ &$-0.0015$ \\
Pr4 & 4f05d1 &$1.3076$ &$27.762$ &$-3.2363$ &$4.278$ &$4.3424$ &$3.471$ &$-1.4168$ &$2.758$ &$0.0018$ \\
Nd2 & 4f35d1 &$1.2972$ &$27.071$ &$-2.4336$ &$4.208$ &$3.2415$ &$3.271$ &$-1.1081$ &$2.568$ &$0.0016$ \\
Nd3 & 4f25d1 &$0.6179$ &$41.924$ &$0.7344$ &$23.743$ &$-5.2140$ &$5.726$ &$4.8631$ &$5.427$ &$-0.0016$ \\
Nd4 & 4f15d1 &$1.2972$ &$27.071$ &$-2.4336$ &$4.208$ &$3.2415$ &$3.271$ &$-1.1081$ &$2.568$ &$0.0016$ \\
Pm3 & 4f35d1 &$0.5149$ &$42.825$ &$0.8077$ &$24.559$ &$-3.6851$ &$5.432$ &$3.3642$ &$5.053$ &$-0.0018$ \\
Sm2 & 4f55d1 &$0.3908$ &$63.383$ &$0.8062$ &$32.974$ &$-5.5271$ &$5.174$ &$5.3313$ &$5.015$ &$-0.0013$ \\
Sm3 & 4f45d1 &$0.4725$ &$42.826$ &$0.8301$ &$24.562$ &$-3.0608$ &$5.170$ &$2.7600$ &$4.755$ &$-0.0019$ \\
Eu2 & 4f65d1 &$0.3827$ &$63.023$ &$0.8044$ &$32.481$ &$-1.8540$ &$5.066$ &$1.6682$ &$4.618$ &$-0.0014$ \\
Eu3 & 4f55d1 &$0.3784$ &$44.430$ &$0.9025$ &$25.126$ &$-3.1521$ &$4.871$ &$2.8731$ &$4.511$ &$-0.0020$ \\
Gd0 & 4f85d1 &$0.4982$ &$104.440$ &$0.5928$ &$44.451$ &$-14.5061$ &$4.692$ &$14.4155$ &$4.664$ &$-0.0008$ \\
Gd0 & 6s14f75d1 &$0.4132$ &$75.211$ &$0.7389$ &$33.606$ &$-1.8951$ &$4.755$ &$1.7441$ &$4.411$ &$-0.0013$ \\
Gd2 & 4f75d1 &$0.3745$ &$62.755$ &$0.8026$ &$32.071$ &$-8.6327$ &$4.644$ &$8.4569$ &$4.560$ &$-0.0014$ \\
Gd3 & 4f65d1 &$0.3587$ &$44.300$ &$0.9082$ &$24.875$ &$-2.6769$ &$4.664$ &$2.4120$ &$4.274$ &$-0.0020$ \\
Gd4 & 4f55d1 &$0.4512$ &$32.712$ &$0.9183$ &$19.602$ &$-2.8665$ &$4.569$ &$2.4997$ &$4.098$ &$-0.0027$ \\
Tb0 & 4f95d1 &$0.5057$ &$104.335$ &$0.5796$ &$43.869$ &$-1.6061$ &$4.572$ &$1.5212$ &$4.339$ &$-0.0008$ \\
Tb0 & 6s14f95d1 &$0.4131$ &$74.925$ &$0.7311$ &$33.131$ &$-1.2648$ &$4.612$ &$1.1216$ &$4.133$ &$-0.0013$ \\
Tb2 & 4f85d1 &$0.3734$ &$62.276$ &$0.7955$ &$31.562$ &$-3.4780$ &$4.492$ &$3.3105$ &$4.298$ &$-0.0014$ \\
Tb3 & 4f75d1 &$0.3223$ &$44.892$ &$0.9303$ &$24.826$ &$-3.9948$ &$4.375$ &$3.7443$ &$4.140$ &$-0.0021$ \\
Tb4 & 4f65d1 &$0.3614$ &$33.730$ &$0.9844$ &$19.945$ &$-2.6536$ &$4.355$ &$2.3105$ &$3.892$ &$-0.0028$ \\
Dy2 & 4f95d1 &$0.3754$ &$61.715$ &$0.7859$ &$31.049$ &$-1.7452$ &$4.388$ &$1.5853$ &$4.024$ &$-0.0014$ \\
Dy3 & 4f85d1 &$0.3014$ &$45.110$ &$0.9392$ &$24.622$ &$-3.2549$ &$4.203$ &$3.0163$ &$3.936$ &$-0.0021$ \\
Ho3 & 4f95d1 &$0.2974$ &$44.755$ &$0.9335$ &$24.262$ &$-2.3286$ &$4.079$ &$2.0998$ &$3.728$ &$-0.0021$ \\
Er3 & 4f105d1 &$0.2916$ &$44.550$ &$0.9304$ &$23.933$ &$-1.3854$ &$4.035$ &$1.1655$ &$3.468$ &$-0.0021$ \\
Tm2 & 4f125d1 &$0.4067$ &$59.305$ &$0.7386$ &$29.168$ &$-0.3795$ &$4.593$ &$0.2356$ &$2.956$ &$-0.0015$ \\
Tm3 & 4f115d1 &$0.3302$ &$42.750$ &$0.8905$ &$23.057$ &$-0.5894$ &$4.397$ &$0.3708$ &$2.923$ &$-0.0022$ \\
Tm4 & 4f105d1 &$0.2406$ &$35.005$ &$1.0453$ &$19.560$ &$-4.5727$ &$3.572$ &$4.2895$ &$3.383$ &$-0.0027$ \\
Yb2 & 4f135d1 &$0.4170$ &$58.627$ &$0.7233$ &$28.615$ &$-0.3308$ &$4.553$ &$0.1919$ &$2.742$ &$-0.0015$ \\
Yb3 & 4f125d1 &$0.3498$ &$41.749$ &$0.8666$ &$22.455$ &$-0.4923$ &$4.434$ &$0.2780$ &$2.671$ &$-0.0022$ \\
Lu2 & 4f145d1 &$0.4216$ &$58.262$ &$0.7131$ &$28.218$ &$-0.3048$ &$4.446$ &$0.1715$ &$2.593$ &$-0.0015$ \\
Lu3 & 4f135d1 &$0.3217$ &$42.404$ &$0.8845$ &$22.448$ &$-0.4726$ &$4.258$ &$0.2686$ &$2.566$ &$-0.0022$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large$\langle j_0\rangle$ }Form factors for Actinide ions}
\begin{table}[H]
\caption{$\langle j_0\rangle$ Form factors for 5f electrons of actinide ions}\vspace{2mm}
\label{acj0}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
U3 &$0.5058$ &$23.2882$ &$1.3464$ &$7.0028$ &$-0.8724$ &$4.8683$ &$0.0192$ \\
U4 &$0.3291$ &$23.5475$ &$1.0836$ &$8.4540$ &$-0.4340$ &$4.1196$ &$0.0214$ \\
U5 &$0.3650$ &$19.8038$ &$3.2199$ &$6.2818$ &$-2.6077$ &$5.3010$ &$0.0233$ \\
Np3 &$0.5157$ &$20.8654$ &$2.2784$ &$5.8930$ &$-1.8163$ &$4.8457$ &$0.0211$ \\
Np4 &$0.4206$ &$19.8046$ &$2.8004$ &$5.9783$ &$-2.2436$ &$4.9848$ &$0.0228$ \\
Np5 &$0.3692$ &$18.1900$ &$3.1510$ &$5.8500$ &$-2.5446$ &$4.9164$ &$0.0248$ \\
Np6 &$0.2929$ &$17.5611$ &$3.4866$ &$5.7847$ &$-2.8066$ &$4.8707$ &$0.0267$ \\
Pu3 &$0.3840$ &$16.6793$ &$3.1049$ &$5.4210$ &$-2.5148$ &$4.5512$ &$0.0263$ \\
Pu4 &$0.4934$ &$16.8355$ &$1.6394$ &$5.6384$ &$-1.1581$ &$4.1399$ &$0.0248$ \\
Pu5 &$0.3888$ &$16.5592$ &$2.0362$ &$5.6567$ &$-1.4515$ &$4.2552$ &$0.0267$ \\
Pu6 &$0.3172$ &$16.0507$ &$3.4654$ &$5.3507$ &$-2.8102$ &$4.5133$ &$0.0281$ \\
Am2 &$0.4743$ &$21.7761$ &$1.5800$ &$5.6902$ &$-1.0779$ &$4.1451$ &$0.0218$ \\
Am3 &$0.4239$ &$19.5739$ &$1.4573$ &$5.8722$ &$-0.9052$ &$3.9682$ &$0.0238$ \\
Am4 &$0.3737$ &$17.8625$ &$1.3521$ &$6.0426$ &$-0.7514$ &$3.7199$ &$0.0258$ \\
Am5 &$0.2956$ &$17.3725$ &$1.4525$ &$6.0734$ &$-0.7755$ &$3.6619$ &$0.0277$ \\
Am6 &$0.2302$ &$16.9533$ &$1.4864$ &$6.1159$ &$-0.7457$ &$3.5426$ &$0.0294$ \\
Am7 &$0.3601$ &$12.7299$ &$1.9640$ &$5.1203$ &$-1.3560$ &$3.7142$ &$0.0316$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 3d transition elements and their ions}
\label{3dj2}
\begin{table}[H]
\caption{
$\langle j_2\rangle$ form factors for 3d transition elements and their ions}
 \vspace{2mm}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Sc0 &$10.8172$ &$54.3270$ &$4.7353$ &$14.8471$ &$0.6071$ &$4.2180$ &$0.0011$ \\
Sc1 &$8.5021$ &$34.2851$ &$3.2116$ &$10.9940$ &$0.4244$ &$3.6055$ &$0.0009$ \\
Sc2 &$4.3683$ &$28.6544$ &$3.7231$ &$10.8233$ &$0.6074$ &$3.6678$ &$0.0014$ \\
Ti0 &$4.3583$ &$36.0556$ &$3.8230$ &$11.1328$ &$0.6855$ &$3.4692$ &$0.0020$ \\
Ti1 &$6.1567$ &$27.2754$ &$2.6833$ &$8.9827$ &$0.4070$ &$3.0524$ &$0.0011$ \\
Ti2 &$4.3107$ &$18.3484$ &$2.0960$ &$6.7970$ &$0.2984$ &$2.5476$ &$0.0007$ \\
Ti3 &$3.3717$ &$14.4441$ &$1.8258$ &$5.7126$ &$0.2470$ &$2.2654$ &$0.0005$ \\
V0 &$3.8099$ &$21.8313$ &$2.4026$ &$7.5458$ &$0.4464$ &$2.6628$ &$0.0017$ \\
V1 &$4.7474$ &$23.3226$ &$2.3609$ &$7.8082$ &$0.4105$ &$2.7063$ &$0.0014$ \\
V2 &$3.4386$ &$16.5303$ &$1.9638$ &$6.1415$ &$0.2997$ &$2.2669$ &$0.0009$ \\
V3 &$2.3005$ &$14.6821$ &$2.0364$ &$6.1304$ &$0.4099$ &$2.3815$ &$0.0014$ \\
V4 &$1.8377$ &$12.2668$ &$1.8247$ &$5.4578$ &$0.3979$ &$2.2483$ &$0.0012$ \\
Cr0 &$3.7600$ &$20.1267$ &$2.1006$ &$6.8020$ &$0.4266$ &$2.3941$ &$0.0019$ \\
Cr1 &$3.7768$ &$20.3456$ &$2.1028$ &$6.8926$ &$0.4010$ &$2.4114$ &$0.0017$ \\
Cr2 &$2.6422$ &$16.0598$ &$1.9198$ &$6.2531$ &$0.4446$ &$2.3715$ &$0.0020$ \\
Cr3 &$1.6262$ &$15.0656$ &$2.0618$ &$6.2842$ &$0.5281$ &$2.3680$ &$0.0023$ \\
Cr4 &$1.0293$ &$13.9498$ &$1.9933$ &$6.0593$ &$0.5974$ &$2.3457$ &$0.0027$ \\
Mn0 &$2.6681$ &$16.0601$ &$1.7561$ &$5.6396$ &$0.3675$ &$2.0488$ &$0.0017$ \\
Mn1 &$3.2953$ &$18.6950$ &$1.8792$ &$6.2403$ &$0.3927$ &$2.2006$ &$0.0022$ \\
Mn2 &$2.0515$ &$15.5561$ &$1.8841$ &$6.0625$ &$0.4787$ &$2.2323$ &$0.0027$ \\
Mn3 &$1.2427$ &$14.9966$ &$1.9567$ &$6.1181$ &$0.5732$ &$2.2577$ &$0.0031$ \\
Mn4 &$0.7879$ &$13.8857$ &$1.8717$ &$5.7433$ &$0.5981$ &$2.1818$ &$0.0034$ \\
Fe0 &$1.9405$ &$18.4733$ &$1.9566$ &$6.3234$ &$0.5166$ &$2.1607$ &$0.0036$ \\
Fe1 &$2.6290$ &$18.6598$ &$1.8704$ &$6.3313$ &$0.4690$ &$2.1628$ &$0.0031$ \\
Fe2 &$1.6490$ &$16.5593$ &$1.9064$ &$6.1325$ &$0.5206$ &$2.1370$ &$0.0035$ \\
Fe3 &$1.3602$ &$11.9976$ &$1.5188$ &$5.0025$ &$0.4705$ &$1.9914$ &$0.0038$ \\
Fe4 &$1.5582$ &$8.2750$ &$1.1863$ &$3.2794$ &$0.1366$ &$1.1068$ &$-0.0022$ \\
Co0 &$1.9678$ &$14.1699$ &$1.4911$ &$4.9475$ &$0.3844$ &$1.7973$ &$0.0027$ \\
Co1 &$2.4097$ &$16.1608$ &$1.5780$ &$5.4604$ &$0.4095$ &$1.9141$ &$0.0031$ \\
Co2 &$1.9049$ &$11.6444$ &$1.3159$ &$4.3574$ &$0.3146$ &$1.6453$ &$0.0017$ \\
Co3 &$1.7058$ &$8.8595$ &$1.1409$ &$3.3086$ &$0.1474$ &$1.0899$ &$-0.0025$ \\
Co4 &$1.3110$ &$8.0252$ &$1.1551$ &$3.1792$ &$0.1608$ &$1.1301$ &$-0.0011$ \\
Ni0 &$1.0302$ &$12.2521$ &$1.4669$ &$4.7453$ &$0.4521$ &$1.7437$ &$0.0036$ \\
Ni1 &$2.1040$ &$14.8655$ &$1.4302$ &$5.0714$ &$0.4031$ &$1.7784$ &$0.0034$ \\
Ni2 &$1.7080$ &$11.0160$ &$1.2147$ &$4.1031$ &$0.3150$ &$1.5334$ &$0.0018$ \\
Ni3 &$1.4683$ &$8.6713$ &$1.1068$ &$3.2574$ &$0.1794$ &$1.1058$ &$-0.0023$ \\
Ni4 &$1.1612$ &$7.7000$ &$1.0027$ &$3.2628$ &$0.2719$ &$1.3780$ &$0.0025$ \\
Cu0 &$1.9182$ &$14.4904$ &$1.3329$ &$4.7301$ &$0.3842$ &$1.6394$ &$0.0035$ \\
Cu1 &$1.8814$ &$13.4333$ &$1.2809$ &$4.5446$ &$0.3646$ &$1.6022$ &$0.0033$ \\
Cu2 &$1.5189$ &$10.4779$ &$1.1512$ &$3.8132$ &$0.2918$ &$1.3979$ &$0.0017$ \\
Cu3 &$1.2797$ &$8.4502$ &$1.0315$ &$3.2796$ &$0.2401$ &$1.2498$ &$0.0015$ \\
Cu4 &$0.9568$ &$7.4481$ &$0.9099$ &$3.3964$ &$0.3729$ &$1.4936$ &$0.0049$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 4d transition elements}
\begin{table}[H]
\caption{$\langle j_2\rangle$ form factors for 4d atoms and ions}\vspace{2mm}
\label{4dj2}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Y0 &$14.4084$ &$44.6577$ &$5.1045$ &$14.9043$ &$-0.0535$ &$3.3189$ &$0.0028$ \\
Zr0 &$10.1378$ &$35.3372$ &$4.7734$ &$12.5453$ &$-0.0489$ &$2.6721$ &$0.0036$ \\
Zr1 &$11.8722$ &$34.9200$ &$4.0502$ &$12.1266$ &$-0.0632$ &$2.8278$ &$0.0034$ \\
Nb0 &$7.4796$ &$33.1789$ &$5.0884$ &$11.5708$ &$-0.0281$ &$1.5635$ &$0.0047$ \\
Nb1 &$8.7735$ &$33.2848$ &$4.6556$ &$11.6046$ &$-0.0268$ &$1.5389$ &$0.0044$ \\
Mo0 &$5.1180$ &$23.4217$ &$4.1809$ &$9.2080$ &$-0.0505$ &$1.7434$ &$0.0053$ \\
Mo1 &$7.2367$ &$28.1282$ &$4.0705$ &$9.9228$ &$-0.0317$ &$1.4552$ &$0.0049$ \\
Tc0 &$4.2441$ &$21.3974$ &$3.9439$ &$8.3753$ &$-0.0371$ &$1.1870$ &$0.0066$ \\
Tc1 &$6.4056$ &$24.8243$ &$3.5400$ &$8.6112$ &$-0.0366$ &$1.4846$ &$0.0044$ \\
Ru0 &$3.7445$ &$18.6128$ &$3.4749$ &$7.4201$ &$-0.0363$ &$1.0068$ &$0.0073$ \\
Ru1 &$5.2826$ &$23.6832$ &$3.5813$ &$8.1521$ &$-0.0257$ &$0.4255$ &$0.0131$ \\
Rh0 &$3.3651$ &$17.3444$ &$3.2121$ &$6.8041$ &$-0.0350$ &$0.5031$ &$0.0146$ \\
Rh1 &$4.0260$ &$18.9497$ &$3.1663$ &$6.9998$ &$-0.0296$ &$0.4862$ &$0.0127$ \\
Pd0 &$3.3105$ &$14.7265$ &$2.6332$ &$5.8618$ &$-0.0437$ &$1.1303$ &$0.0053$ \\
Pd1 &$4.2749$ &$17.9002$ &$2.7021$ &$6.3541$ &$-0.0258$ &$0.6999$ &$0.0071$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 5d Transition elements}
\begin{table}[H]
 \caption{$\langle j_2\rangle$ form factors for the 5d electrons of transition atoms and ions from Hf to Re.\cite{kob:11}}
 \label{5dj2}
 \vspace{1ex}
 {\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E}\\
\hline
Hf2 & 6s05d2 &$9.6670$ &$33.435$ &$5.2429$ &$13.529$ &$-0.5533$ &$1.402$ &$0.4934$ &$1.254$ &$-0.0033$ \\
Hf3 & 6s05d1 &$7.5646$ &$27.367$ &$5.0743$ &$12.402$ &$-0.4133$ &$1.742$ &$0.3163$ &$1.437$ &$-0.0012$ \\
Ta2 & 6s05d3 &$8.1746$ &$29.871$ &$4.9405$ &$12.188$ &$-1.1294$ &$1.254$ &$1.0658$ &$1.181$ &$-0.0046$ \\
Ta3 & 6s05d2 &$6.5998$ &$25.026$ &$4.7913$ &$11.303$ &$-0.7801$ &$1.529$ &$0.6820$ &$1.382$ &$-0.0019$ \\
Ta4 & 6s05d1 &$5.5048$ &$21.620$ &$4.6734$ &$10.556$ &$-0.2181$ &$2.100$ &$0.0810$ &$1.049$ &$-0.0016$ \\
W0 & 6s05d6 &$8.4216$ &$64.652$ &$7.6983$ &$23.124$ &$2.3605$ &$8.583$ &$-0.3876$ &$4.620$ &$0.0033$ \\
W0 & 6s15d5 &$6.9289$ &$53.425$ &$7.2157$ &$20.016$ &$5.7754$ &$6.543$ &$-4.2114$ &$5.854$ &$0.0036$ \\
W0 & 6s25d4 &$5.4017$ &$47.110$ &$6.9113$ &$18.760$ &$7.0586$ &$6.363$ &$-5.4525$ &$5.802$ &$0.0040$ \\
W0 & 6s05d5 &$5.6854$ &$44.743$ &$6.8903$ &$18.753$ &$7.8523$ &$6.308$ &$-6.3447$ &$5.842$ &$0.0038$ \\
W0 & 6s15d4 &$4.5118$ &$41.281$ &$6.7136$ &$17.964$ &$7.6331$ &$6.269$ &$-6.0365$ &$5.758$ &$0.0042$ \\
W2 & 6s05d4 &$7.0301$ &$26.990$ &$4.6604$ &$11.083$ &$-0.8220$ &$1.145$ &$0.7581$ &$1.042$ &$-0.0068$ \\
W3 & 6s05d3 &$5.8036$ &$22.969$ &$4.5243$ &$10.361$ &$-0.7897$ &$1.388$ &$0.6927$ &$1.248$ &$-0.0031$ \\
W4 & 6s05d2 &$4.9006$ &$20.117$ &$4.4360$ &$9.765$ &$-2.0009$ &$1.497$ &$1.8714$ &$1.431$ &$-0.0014$ \\
W5 & 6s05d1 &$4.1973$ &$17.967$ &$4.3791$ &$9.255$ &$-1.8830$ &$1.603$ &$1.7205$ &$1.518$ &$-0.0004$ \\
Re0 & 6s05d7 &$6.7574$ &$55.529$ &$6.7931$ &$20.125$ &$2.3113$ &$7.529$ &$-0.5004$ &$4.412$ &$0.0037$ \\
Re0 & 6s15d6 &$5.5830$ &$46.852$ &$6.4516$ &$17.855$ &$5.0609$ &$6.037$ &$-3.5427$ &$5.312$ &$0.0040$ \\
Re0 & 6s25d5 &$4.4322$ &$41.798$ &$6.2063$ &$16.844$ &$9.8763$ &$5.751$ &$-8.3294$ &$5.406$ &$0.0043$ \\
Re0 & 6s05d6 &$4.7231$ &$40.130$ &$6.2139$ &$16.902$ &$8.8240$ &$5.766$ &$-7.3608$ &$5.396$ &$0.0042$ \\
Re0 & 6s15d5 &$3.7875$ &$37.254$ &$6.0587$ &$16.264$ &$6.9896$ &$5.786$ &$-5.4457$ &$5.279$ &$0.0045$ \\
Re2 & 6s05d5 &$3.0708$ &$33.896$ &$5.9941$ &$15.817$ &$6.7816$ &$5.768$ &$-5.2056$ &$5.231$ &$0.0047$ \\
Re3 & 6s05d4 &$1.7870$ &$31.248$ &$5.9068$ &$15.160$ &$17.4262$ &$5.551$ &$-15.6856$ &$5.339$ &$0.0053$ \\
Re4 & 6s05d3 &$4.4033$ &$18.733$ &$4.1971$ &$9.047$ &$-2.3106$ &$1.352$ &$2.1866$ &$1.299$ &$-0.0025$ \\
Re5 & 6s05d2 &$3.8162$ &$16.843$ &$4.1400$ &$8.605$ &$-4.0882$ &$1.443$ &$3.9333$ &$1.408$ &$-0.0012$ \\
Re6 & 6s05d1 &$3.3349$ &$15.341$ &$4.1015$ &$8.213$ &$-2.3511$ &$1.531$ &$2.1651$ &$1.459$ &$-0.0003$ \\
\hline
\end{tabular}
}
\end{table}
\begin{table}[H]
 \caption{$\langle j_\rangle$ form factors for the 5d electrons of transition atoms and ions from Os to Au.\cite{kob:11}}
\label{5dj2b} \vspace{1ex}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E }\\
\hline
Os0 & 6s05d8 &$5.5418$ &$48.893$ &$6.0803$ &$17.984$ &$2.2542$ &$6.853$ &$-0.5285$ &$4.095$ &$0.0040$ \\
Os0 & 6s15d7 &$4.6511$ &$41.610$ &$5.8194$ &$16.062$ &$6.7172$ &$5.447$ &$-5.2689$ &$4.986$ &$0.0043$ \\
Os0 & 6s25d6 &$3.7421$ &$37.491$ &$5.6137$ &$15.256$ &$10.0488$ &$5.293$ &$-8.5683$ &$4.991$ &$0.0046$ \\
Os0 & 6s05d7 &$4.0236$ &$36.272$ &$5.6349$ &$15.338$ &$8.6568$ &$5.318$ &$-7.2505$ &$4.980$ &$0.0044$ \\
Os0 & 6s15d6 &$3.2611$ &$33.818$ &$5.4945$ &$14.808$ &$5.7610$ &$5.391$ &$-4.2811$ &$4.827$ &$0.0048$ \\
Os2 & 6s05d6 &$2.7289$ &$30.900$ &$5.4357$ &$14.396$ &$18.6546$ &$5.151$ &$-17.1708$ &$4.993$ &$0.0050$ \\
Os3 & 6s05d5 &$4.5913$ &$19.692$ &$4.0615$ &$8.862$ &$-0.9950$ &$1.086$ &$0.9081$ &$0.986$ &$-0.0077$ \\
Os4 & 6s05d4 &$3.9724$ &$17.514$ &$3.9817$ &$8.421$ &$-0.7719$ &$1.264$ &$0.6552$ &$1.107$ &$-0.0042$ \\
Os5 & 6s05d3 &$3.4764$ &$15.826$ &$3.9241$ &$8.032$ &$-0.6854$ &$1.397$ &$0.5394$ &$1.184$ &$-0.0024$ \\
Os6 & 6s05d2 &$3.0642$ &$14.479$ &$3.8847$ &$7.686$ &$-4.7235$ &$1.384$ &$4.5485$ &$1.352$ &$-0.0012$ \\
Os7 & 6s05d1 &$2.7164$ &$13.366$ &$3.8554$ &$7.370$ &$-5.0211$ &$1.439$ &$4.8166$ &$1.405$ &$-0.0004$ \\
Ir0 & 6s05d9 &$4.6102$ &$43.878$ &$5.4892$ &$16.394$ &$2.1707$ &$6.423$ &$-0.4761$ &$3.722$ &$0.0043$ \\
Ir0 & 6s15d8 &$3.9372$ &$37.508$ &$5.2846$ &$14.661$ &$3.7267$ &$5.263$ &$-2.3158$ &$4.416$ &$0.0045$ \\
Ir0 & 6s25d7 &$3.2263$ &$33.922$ &$5.1086$ &$13.921$ &$6.5993$ &$4.978$ &$-5.1841$ &$4.549$ &$0.0048$ \\
Ir0 & 6s05d8 &$3.4956$ &$32.991$ &$5.1369$ &$13.998$ &$8.3991$ &$4.924$ &$-7.0561$ &$4.612$ &$0.0047$ \\
Ir0 & 6s15d7 &$2.8732$ &$30.809$ &$5.0094$ &$13.522$ &$6.8656$ &$4.933$ &$-5.4669$ &$4.526$ &$0.0050$ \\
Ir2 & 6s05d7 &$2.4419$ &$28.356$ &$4.9470$ &$13.222$ &$4.7478$ &$5.029$ &$-3.3259$ &$4.393$ &$0.0052$ \\
Ir3 & 6s05d6 &$1.5883$ &$25.969$ &$4.8472$ &$12.711$ &$5.6507$ &$4.949$ &$-4.1190$ &$4.388$ &$0.0056$ \\
Ir4 & 6s05d5 &$3.5964$ &$16.439$ &$3.7872$ &$7.873$ &$-1.2303$ &$1.092$ &$1.1232$ &$1.003$ &$-0.0068$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 4f electrons of rare earth ions}
\begin{table}[H]
\caption{$\langle j_2\rangle$ form factors for 4f electrons of rare-earth ions}
\vspace{2mm}
\label{re4fj2}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Ce2 &$0.9809$ &$18.0630$ &$1.8413$ &$7.7688$ &$0.9905$ &$2.8452$ &$0.0120$ \\
Nd2 &$1.4530$ &$18.3398$ &$1.6196$ &$7.2854$ &$0.8752$ &$2.6224$ &$0.0126$ \\
Nd3 &$0.6751$ &$18.3421$ &$1.6272$ &$7.2600$ &$0.9644$ &$2.6016$ &$0.0150$ \\
Sm2 &$1.0360$ &$18.4249$ &$1.4769$ &$7.0321$ &$0.8810$ &$2.4367$ &$0.0152$ \\
Sm3 &$0.4707$ &$18.4301$ &$1.4261$ &$7.0336$ &$0.9574$ &$2.4387$ &$0.0182$ \\
Eu2 &$0.8970$ &$18.4429$ &$1.3769$ &$7.0054$ &$0.9060$ &$2.4213$ &$0.0190$ \\
Eu3 &$0.3985$ &$18.4514$ &$1.3307$ &$6.9556$ &$0.9603$ &$2.3780$ &$0.0197$ \\
Gd2 &$0.7756$ &$18.4695$ &$1.3124$ &$6.8990$ &$0.8956$ &$2.3383$ &$0.0199$ \\
Gd3 &$0.3347$ &$18.4758$ &$1.2465$ &$6.8767$ &$0.9537$ &$2.3184$ &$0.0217$ \\
Tb2 &$0.6688$ &$18.4909$ &$1.2487$ &$6.8219$ &$0.8888$ &$2.2751$ &$0.0215$ \\
Tb3 &$0.2892$ &$18.4973$ &$1.1678$ &$6.7972$ &$0.9437$ &$2.2573$ &$0.0232$ \\
Dy2 &$0.5917$ &$18.5114$ &$1.1828$ &$6.7465$ &$0.8801$ &$2.2141$ &$0.0229$ \\
Dy3 &$0.2523$ &$18.5172$ &$1.0914$ &$6.7362$ &$0.9345$ &$2.2082$ &$0.0250$ \\
Ho2 &$0.5094$ &$18.5155$ &$1.1234$ &$6.7060$ &$0.8727$ &$2.1589$ &$0.0242$ \\
Ho3 &$0.2188$ &$18.5157$ &$1.0240$ &$6.7070$ &$0.9251$ &$2.1614$ &$0.0268$ \\
Er2 &$0.4693$ &$18.5278$ &$1.0545$ &$6.6493$ &$0.8679$ &$2.1201$ &$0.0261$ \\
Er3 &$0.1710$ &$18.5337$ &$0.9879$ &$6.6246$ &$0.9044$ &$2.1004$ &$0.0278$ \\
Tm2 &$0.4198$ &$18.5417$ &$0.9959$ &$6.6002$ &$0.8593$ &$2.0818$ &$0.0284$ \\
Tm3 &$0.1760$ &$18.5417$ &$0.9105$ &$6.5787$ &$0.8970$ &$2.0622$ &$0.0294$ \\
Yb2 &$0.3852$ &$18.5497$ &$0.9415$ &$6.5507$ &$0.8492$ &$2.0425$ &$0.0301$ \\
Yb3 &$0.1570$ &$18.5553$ &$0.8484$ &$6.5403$ &$0.8880$ &$2.0367$ &$0.0318$ \\
Pr3 &$0.8734$ &$18.9876$ &$1.5594$ &$6.0872$ &$0.8142$ &$2.4150$ &$0.0111$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
%
\section{{\large $\langle j_2\rangle$} Form factors for 5d electrons of rare earth elements}
\begin{table}[H]
 \caption{$\langle j_2\rangle$ form factors of the 5d electrons of rare earth ions.\cite{kob:12}}
 \vspace{1ex}
 \label{re5dj2}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E}\\
\hline
La2 & 4f05d1 &$12.9681$ &$48.694$ &$9.8018$ &$22.831$ &$-0.5443$ &$4.354$ &$0.2584$ &$2.743$ &$-0.0001$ \\
Ce2 & 4f15d1 &$12.6550$ &$47.222$ &$9.3025$ &$21.878$ &$-0.6896$ &$3.762$ &$0.4405$ &$2.849$ &$0.0000$ \\
Ce3 & 4f05d1 &$8.7013$ &$35.824$ &$8.5894$ &$18.242$ &$-0.6110$ &$4.306$ &$0.2181$ &$2.336$ &$-0.0003$ \\
Pr2 & 4f25d1 &$12.4487$ &$45.891$ &$8.8239$ &$21.010$ &$-17.0630$ &$3.120$ &$16.8417$ &$3.095$ &$0.0001$ \\
Pr3 & 4f15d1 &$8.5203$ &$34.877$ &$8.2069$ &$17.602$ &$-0.5841$ &$3.958$ &$0.2347$ &$2.302$ &$-0.0002$ \\
Pr4 & 4f05d1 &$6.3910$ &$28.309$ &$7.6746$ &$15.204$ &$-0.6973$ &$4.010$ &$0.2304$ &$2.119$ &$-0.0004$ \\
Nd2 & 4f35d1 &$6.2640$ &$27.628$ &$7.3661$ &$14.731$ &$-0.6560$ &$3.743$ &$0.2354$ &$2.073$ &$-0.0003$ \\
Nd3 & 4f25d1 &$8.3541$ &$34.036$ &$7.8675$ &$17.031$ &$-0.5998$ &$3.584$ &$0.2896$ &$2.326$ &$0.0000$ \\
Nd4 & 4f15d1 &$6.2640$ &$27.628$ &$7.3661$ &$14.731$ &$-0.6560$ &$3.743$ &$0.2354$ &$2.073$ &$-0.0003$ \\
Pm3 & 4f35d1 &$8.2028$ &$33.286$ &$7.5637$ &$16.518$ &$-0.8756$ &$3.105$ &$0.6007$ &$2.486$ &$0.0002$ \\
Sm2 & 4f55d1 &$12.0956$ &$42.678$ &$7.6076$ &$18.857$ &$-1.4355$ &$2.749$ &$1.2720$ &$2.547$ &$0.0001$ \\
Sm3 & 4f45d1 &$8.1212$ &$32.502$ &$7.2340$ &$15.990$ &$-5.8639$ &$2.698$ &$5.6142$ &$2.630$ &$0.0003$ \\
Eu2 & 4f65d1 &$12.0033$ &$41.860$ &$7.2896$ &$18.289$ &$-1.4882$ &$2.592$ &$1.3411$ &$2.425$ &$0.0000$ \\
Eu3 & 4f55d1 &$8.0653$ &$31.753$ &$6.9151$ &$15.484$ &$-5.5701$ &$2.578$ &$5.3406$ &$2.515$ &$0.0003$ \\
Gd0 & 4f85d1 &$22.2510$ &$65.913$ &$7.5925$ &$23.045$ &$-3.5803$ &$1.952$ &$3.5355$ &$1.930$ &$-0.0010$ \\
Gd0 & 6s14f75d1 &$14.0295$ &$48.586$ &$7.3596$ &$18.927$ &$-5.1053$ &$2.242$ &$5.0055$ &$2.211$ &$-0.0004$ \\
Gd2 & 4f75d1 &$11.9202$ &$41.129$ &$7.0021$ &$17.772$ &$-5.3982$ &$2.392$ &$5.2663$ &$2.354$ &$-0.0001$ \\
Gd3 & 4f65d1 &$8.0082$ &$31.074$ &$6.6281$ &$15.022$ &$-1.9932$ &$2.515$ &$1.7825$ &$2.354$ &$0.0003$ \\
Gd4 & 4f55d1 &$5.8839$ &$25.328$ &$6.3297$ &$13.143$ &$-1.1650$ &$2.597$ &$0.8829$ &$2.219$ &$0.0004$ \\
Tb0 & 4f95d1 &$22.4050$ &$65.581$ &$7.3158$ &$22.565$ &$-4.9513$ &$1.745$ &$4.9149$ &$1.732$ &$-0.0014$ \\
Tb0 & 6s14f85d1 &$14.0011$ &$48.000$ &$7.0994$ &$18.468$ &$-0.6304$ &$2.188$ &$0.5431$ &$1.960$ &$-0.0007$ \\
Tb2 & 4f85d1 &$11.8504$ &$40.480$ &$6.7412$ &$17.303$ &$-11.6794$ &$2.248$ &$11.5615$ &$2.233$ &$-0.0002$ \\
Tb3 & 4f75d1 &$7.9534$ &$30.456$ &$6.3665$ &$14.596$ &$-0.8418$ &$2.508$ &$0.6482$ &$2.145$ &$0.0002$ \\
Tb4 & 4f65d1 &$5.8406$ &$24.788$ &$6.0831$ &$12.775$ &$-2.0431$ &$2.400$ &$1.7828$ &$2.221$ &$0.0004$ \\
Dy2 & 4f95d1 &$11.7905$ &$39.898$ &$6.5024$ &$16.874$ &$-11.5993$ &$2.115$ &$11.4943$ &$2.102$ &$-0.0003$ \\
Dy3 & 4f85d1 &$7.8866$ &$29.902$ &$6.1361$ &$14.212$ &$-12.5093$ &$2.229$ &$12.3329$ &$2.210$ &$0.0002$ \\
Ho3 & 4f95d1 &$7.8279$ &$29.392$ &$5.9224$ &$13.854$ &$-3.0002$ &$2.151$ &$2.8393$ &$2.080$ &$0.0001$ \\
Er3 & 4f105d1 &$7.7700$ &$28.922$ &$5.7263$ &$13.523$ &$-4.1104$ &$2.035$ &$3.9640$ &$1.990$ &$0.0000$ \\
Tm2 & 4f125d1 &$11.6672$ &$38.468$ &$5.8888$ &$15.774$ &$-4.8013$ &$1.717$ &$4.7296$ &$1.697$ &$-0.0012$ \\
Tm3 & 4f115d1 &$7.7142$ &$28.488$ &$5.5450$ &$13.214$ &$-5.7987$ &$1.923$ &$5.6660$ &$1.895$ &$-0.0002$ \\
Tm4 & 4f105d1 &$5.6664$ &$23.000$ &$5.2790$ &$11.534$ &$-3.2389$ &$1.988$ &$3.0496$ &$1.919$ &$0.0002$ \\
Yb2 & 4f135d1 &$11.6437$ &$38.074$ &$5.7103$ &$15.455$ &$-2.9992$ &$1.582$ &$2.9369$ &$1.554$ &$-0.0017$ \\
Yb3 & 4f125d1 &$7.6609$ &$28.088$ &$5.3775$ &$12.926$ &$-5.9455$ &$1.817$ &$5.8255$ &$1.793$ &$-0.0004$ \\
Lu2 & 4f145d1 &$11.6308$ &$37.717$ &$5.5427$ &$15.158$ &$-6.0682$ &$1.432$ &$6.0147$ &$1.420$ &$-0.0024$ \\
Lu3 & 4f135d1 &$7.6106$ &$27.715$ &$5.2212$ &$12.656$ &$-2.1969$ &$1.729$ &$2.0887$ &$1.671$ &$-0.0007$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large $\langle j_2\rangle$} Form factors for 5f electrons  of actinide ions}
\begin{table}[H]
\caption{$\langle j_2\rangle$ Form factors for actinide ions}\vspace{2mm}
\label{acj2}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
U3 &$4.1582$ &$16.5336$ &$2.4675$ &$5.9516$ &$-0.0252$ &$0.7646$ &$0.0057$ \\
U4 &$3.7449$ &$13.8944$ &$2.6453$ &$4.8634$ &$-0.5218$ &$3.1919$ &$0.0009$ \\
U5 &$3.0724$ &$12.5460$ &$2.3076$ &$5.2314$ &$-0.0644$ &$1.4738$ &$0.0035$ \\
Np3 &$3.7170$ &$15.1333$ &$2.3216$ &$5.5025$ &$-0.0275$ &$0.7996$ &$0.0052$ \\
Np4 &$2.9203$ &$14.6463$ &$2.5979$ &$5.5592$ &$-0.0301$ &$0.3669$ &$0.0141$ \\
Np5 &$2.3308$ &$13.6540$ &$2.7219$ &$5.4935$ &$-0.1357$ &$0.0493$ &$0.1224$ \\
Np6 &$1.8245$ &$13.1803$ &$2.8508$ &$5.4068$ &$-0.1579$ &$0.0444$ &$0.1438$ \\
Pu3 &$2.0885$ &$12.8712$ &$2.5961$ &$5.1896$ &$-0.1465$ &$0.0393$ &$0.1343$ \\
Pu4 &$2.7244$ &$12.9262$ &$2.3387$ &$5.1633$ &$-0.1300$ &$0.0457$ &$0.1177$ \\
Pu5 &$2.1409$ &$12.8319$ &$2.5664$ &$5.1522$ &$-0.1338$ &$0.0457$ &$0.1210$ \\
Pu6 &$1.7262$ &$12.3240$ &$2.6652$ &$5.0662$ &$-0.1695$ &$0.0406$ &$0.1550$ \\
Am2 &$3.5237$ &$15.9545$ &$2.2855$ &$5.1946$ &$-0.0142$ &$0.5853$ &$0.0033$ \\
Am3 &$2.8622$ &$14.7328$ &$2.4099$ &$5.1439$ &$-0.1326$ &$0.0309$ &$0.1233$ \\
Am4 &$2.4141$ &$12.9478$ &$2.3687$ &$4.9447$ &$-0.2490$ &$0.0215$ &$0.2371$ \\
Am5 &$2.0109$ &$12.0534$ &$2.4155$ &$4.8358$ &$-0.2264$ &$0.0275$ &$0.2128$ \\
Am6 &$1.6778$ &$11.3372$ &$2.4531$ &$4.7247$ &$-0.2043$ &$0.0337$ &$0.1892$ \\
Am7 &$1.8845$ &$9.1606$ &$2.0746$ &$4.0422$ &$-0.1318$ &$1.7227$ &$0.0020$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors for 3d atoms and ions}
\begin{table}[H]
\caption{$\langle j_4\rangle$ form factors for 3d atoms and ions}\vspace{2mm}
\label{3dj4}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Sc0 &$1.3420$ &$10.2000$ &$0.3837$ &$3.0786$ &$0.0468$ &$0.1178$ &$-0.0328$ \\
Sc1 &$7.1167$ &$15.4872$ &$-6.6671$ &$18.2692$ &$0.4900$ &$2.9917$ &$0.0047$ \\
Sc2 &$-1.6684$ &$15.6475$ &$1.7742$ &$9.0624$ &$0.4075$ &$2.4116$ &$0.0042$ \\
Ti0 &$-2.1515$ &$11.2705$ &$2.5149$ &$8.8590$ &$0.3555$ &$2.1491$ &$0.0045$ \\
Ti1 &$-1.0383$ &$16.1899$ &$1.4699$ &$8.9239$ &$0.3631$ &$2.2834$ &$0.0044$ \\
Ti2 &$-1.3242$ &$15.3096$ &$1.2042$ &$7.8994$ &$0.3976$ &$2.1562$ &$0.0051$ \\
Ti3 &$-1.1117$ &$14.6349$ &$0.7689$ &$6.9267$ &$0.4385$ &$2.0886$ &$0.0060$ \\
V0 &$-0.9633$ &$15.2729$ &$0.9274$ &$7.7315$ &$0.3891$ &$2.0530$ &$0.0063$ \\
V1 &$-0.9606$ &$15.5451$ &$1.1278$ &$8.1182$ &$0.3653$ &$2.0973$ &$0.0056$ \\
V2 &$-1.1729$ &$14.9732$ &$0.9092$ &$7.6131$ &$0.4105$ &$2.0391$ &$0.0067$ \\
V3 &$-0.9417$ &$14.2045$ &$0.5284$ &$6.6071$ &$0.4411$ &$1.9672$ &$0.0076$ \\
V4 &$-0.7654$ &$13.0970$ &$0.3071$ &$5.6739$ &$0.4476$ &$1.8707$ &$0.0081$ \\
Cr0 &$-0.6670$ &$19.6128$ &$0.5342$ &$6.4779$ &$0.3641$ &$1.9045$ &$0.0073$ \\
Cr1 &$-0.8309$ &$18.0428$ &$0.7252$ &$7.5313$ &$0.3828$ &$2.0032$ &$0.0073$ \\
Cr2 &$-0.8930$ &$15.6641$ &$0.5590$ &$7.0333$ &$0.4093$ &$1.9237$ &$0.0081$ \\
Cr3 &$-0.7327$ &$14.0727$ &$0.3268$ &$5.6741$ &$0.4114$ &$1.8101$ &$0.0085$ \\
Cr4 &$-0.6748$ &$12.9462$ &$0.1805$ &$6.7527$ &$0.4526$ &$1.7999$ &$0.0098$ \\
Mn0 &$-0.5452$ &$15.4713$ &$0.4406$ &$4.9024$ &$0.2884$ &$1.5430$ &$0.0059$ \\
Mn1 &$-0.7947$ &$17.8673$ &$0.6078$ &$7.7044$ &$0.3798$ &$1.9045$ &$0.0087$ \\
Mn2 &$-0.7416$ &$15.2555$ &$0.3831$ &$6.4693$ &$0.3935$ &$1.7997$ &$0.0093$ \\
Mn3 &$-0.6603$ &$13.6066$ &$0.2322$ &$6.2175$ &$0.4104$ &$1.7404$ &$0.0101$ \\
Mn4 &$-0.5127$ &$13.4613$ &$0.0313$ &$7.7631$ &$0.4282$ &$1.7006$ &$0.0113$ \\
Fe0 &$-0.5029$ &$19.6768$ &$0.2999$ &$3.7762$ &$0.2576$ &$1.4241$ &$0.0071$ \\
Fe1 &$-0.5109$ &$19.2501$ &$0.3896$ &$4.8913$ &$0.2810$ &$1.5265$ &$0.0069$ \\
Fe2 &$-0.5401$ &$17.2268$ &$0.2865$ &$3.7422$ &$0.2658$ &$1.4238$ &$0.0076$ \\
Fe3 &$-0.5507$ &$11.4929$ &$0.2153$ &$4.9063$ &$0.3468$ &$1.5230$ &$0.0095$ \\
Fe4 &$-0.5352$ &$9.5068$ &$0.1783$ &$5.1750$ &$0.3584$ &$1.4689$ &$0.0097$ \\
Co0 &$-0.4221$ &$14.1952$ &$0.2900$ &$3.9786$ &$0.2469$ &$1.2859$ &$0.0063$ \\
Co1 &$-0.4115$ &$14.5615$ &$0.3580$ &$4.7170$ &$0.2644$ &$1.4183$ &$0.0074$ \\
Co2 &$-0.4759$ &$14.0462$ &$0.2747$ &$3.7306$ &$0.2458$ &$1.2504$ &$0.0057$ \\
Co3 &$-0.4466$ &$13.3912$ &$0.1419$ &$3.0110$ &$0.2773$ &$1.3351$ &$0.0093$ \\
Co4 &$-0.4091$ &$13.1937$ &$-0.0194$ &$3.4169$ &$0.3534$ &$1.4214$ &$0.0112$ \\
Ni0 &$-0.4428$ &$14.4850$ &$0.0870$ &$3.2345$ &$0.2932$ &$1.3305$ &$0.0096$ \\
Ni1 &$-0.3836$ &$13.4246$ &$0.3116$ &$4.4619$ &$0.2471$ &$1.3088$ &$0.0079$ \\
Ni2 &$-0.3803$ &$10.4033$ &$0.2838$ &$3.3780$ &$0.2108$ &$1.1036$ &$0.0050$ \\
Ni3 &$-0.4014$ &$ 9.0462$ &$0.2314$ &$3.0753$ &$0.2192$ &$1.0838$ &$0.0060$ \\
Ni4 &$-0.3509$ &$8.1572$ &$0.2220$ &$2.1063$ &$0.1567$ &$0.9253$ &$0.0065$ \\
Cu0 &$-0.3204$ &$15.1324$ &$0.2335$ &$4.0205$ &$0.2312$ &$1.1957$ &$0.0068$ \\
Cu1 &$-0.3572$ &$15.1251$ &$0.2336$ &$3.9662$ &$0.2315$ &$1.1967$ &$0.0070$ \\
Cu2 &$-0.3914$ &$14.7400$ &$0.1275$ &$3.3840$ &$0.2548$ &$1.2552$ &$0.0103$ \\
Cu3 &$-0.3671$ &$14.0816$ &$-0.0078$ &$3.3149$ &$0.3154$ &$1.3767$ &$0.0132$ \\
Cu4 &$-0.2915$ &$14.1243$ &$-0.1065$ &$4.2008$ &$0.3247$ &$1.3516$ &$0.0148$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors 4d atoms and ions}
\begin{table}[H]
\noindent\caption{$\langle j_4\rangle$ form factors for 4d atoms and ions}\
\label{4dj4}
 \vspace{1ex}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Y0 &$-8.0767$ &$32.2014$ &$7.9197$ &$25.1563$ &$1.4067$ &$6.8268$ &$-0.0001$ \\
Zr0 &$-5.2697$ &$32.8680$ &$4.1930$ &$24.1833$ &$1.5202$ &$6.0481$ &$-0.0002$ \\
Zr1 &$-5.6384$ &$33.6071$ &$4.6729$ &$22.3383$ &$1.3258$ &$5.9245$ &$-0.0003$ \\
Nb0 &$-3.1377$ &$25.5948$ &$2.3411$ &$16.5686$ &$1.2304$ &$4.9903$ &$-0.0005$ \\
Nb1 &$-3.3598$ &$25.8202$ &$2.8297$ &$16.4273$ &$1.1203$ &$4.9824$ &$-0.0005$ \\
Mo0 &$-2.8860$ &$20.5717$ &$1.8130$ &$14.6281$ &$1.1899$ &$4.2638$ &$-0.0008$ \\
Mo1 &$-3.2618$ &$25.4862$ &$2.3596$ &$16.4622$ &$1.1164$ &$4.4913$ &$-0.0007$ \\
Tc0 &$-2.7975$ &$20.1589$ &$1.6520$ &$16.2609$ &$1.1726$ &$3.9427$ &$-0.0008$ \\
Tc1 &$-2.0470$ &$19.6830$ &$1.6306$ &$11.5925$ &$0.8698$ &$3.7689$ &$-0.0010$ \\
Ru0 &$-1.5042$ &$17.9489$ &$0.6027$ &$9.9608$ &$0.9700$ &$3.3927$ &$-0.0010$ \\
Ru1 &$-1.6278$ &$18.5063$ &$1.1828$ &$10.1886$ &$0.8138$ &$3.4180$ &$-0.0009$ \\
Rh0 &$-1.3492$ &$17.5766$ &$0.4527$ &$10.5066$ &$0.9285$ &$3.1555$ &$-0.0009$ \\
Rh1 &$-1.4673$ &$17.9572$ &$0.7381$ &$9.9444$ &$0.8485$ &$3.1263$ &$-0.0012$ \\
Pd0 &$-1.1955$ &$17.6282$ &$0.3183$ &$11.3094$ &$0.8696$ &$2.9089$ &$-0.0006$ \\
Pd1 &$-1.4098$ &$17.7650$ &$0.7927$ &$9.9991$ &$0.7710$ &$2.9297$ &$-0.0006$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors for 5d atoms and ions}
\begin{table}[H]
 \caption{$\langle j_4\rangle$ form factors for the 5d electrons of transition atoms and ions from Hf to Re.\cite{kob:11}}
\label{5dj4}
 \vspace{1ex}
 {\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E }\\
\hline
Hf2 & 6s05d2 &$-2.5342$ &$43.826$ &$1.8466$ &$10.393$ &$0.7761$ &$4.888$ &$-0.0327$ &$1.589$ &$0.0017$ \\
Hf3 & 6s05d1 &$-2.3574$ &$32.651$ &$1.8717$ &$8.476$ &$0.6367$ &$3.953$ &$-0.1133$ &$2.169$ &$0.0017$ \\
Ta2 & 6s05d3 &$-2.1974$ &$38.294$ &$1.6220$ &$8.838$ &$0.6836$ &$4.212$ &$-0.0539$ &$1.746$ &$0.0016$ \\
Ta3 & 6s05d2 &$-2.0884$ &$29.531$ &$1.7145$ &$7.385$ &$1.1809$ &$2.994$ &$-0.7705$ &$2.577$ &$0.0016$ \\
Ta4 & 6s05d1 &$-2.0226$ &$24.035$ &$1.7084$ &$6.594$ &$1.1799$ &$2.775$ &$-0.8384$ &$2.430$ &$0.0020$ \\
W0 & 6s05d6 &$-2.1307$ &$79.955$ &$1.4055$ &$11.876$ &$0.8808$ &$4.969$ &$-0.0120$ &$0.419$ &$0.0051$ \\
W0 & 6s15d5 &$-1.9667$ &$60.069$ &$1.3446$ &$9.992$ &$0.7861$ &$4.554$ &$-0.0197$ &$1.087$ &$0.0021$ \\
W0 & 6s25d4 &$-1.8575$ &$47.554$ &$1.3868$ &$8.437$ &$0.6537$ &$3.939$ &$-0.0570$ &$1.722$ &$0.0015$ \\
W0 & 6s05d5 &$-2.0231$ &$46.962$ &$1.3937$ &$9.263$ &$0.7420$ &$4.332$ &$-0.0288$ &$1.327$ &$0.0018$ \\
W0 & 6s15d4 &$-1.9122$ &$39.952$ &$1.4385$ &$8.019$ &$0.6336$ &$3.725$ &$-0.0819$ &$1.868$ &$0.0015$ \\
W2 & 6s05d4 &$-1.9355$ &$33.935$ &$1.5020$ &$7.541$ &$0.6453$ &$3.375$ &$-0.1601$ &$2.110$ &$0.0015$ \\
W3 & 6s05d3 &$-1.8752$ &$26.706$ &$1.5440$ &$6.585$ &$5.9785$ &$2.575$ &$-5.6111$ &$2.516$ &$0.0018$ \\
W4 & 6s05d2 &$-1.8309$ &$22.142$ &$1.5605$ &$5.935$ &$1.4898$ &$2.482$ &$-1.1930$ &$2.266$ &$0.0022$ \\
W5 & 6s05d1 &$-1.7958$ &$18.987$ &$1.5913$ &$5.419$ &$2.0498$ &$2.278$ &$-1.8262$ &$2.156$ &$0.0026$ \\
Re0 & 6s05d7 &$-1.8013$ &$63.944$ &$1.1773$ &$9.808$ &$0.7912$ &$4.380$ &$-0.0155$ &$0.795$ &$0.0027$ \\
Re0 & 6s15d6 &$-1.7056$ &$49.628$ &$1.2209$ &$8.231$ &$0.6637$ &$3.836$ &$-0.0443$ &$1.514$ &$0.0015$ \\
Re0 & 6s25d5 &$-1.6402$ &$40.319$ &$1.3192$ &$7.064$ &$0.8659$ &$2.901$ &$-0.4299$ &$2.306$ &$0.0013$ \\
Re0 & 6s05d6 &$-1.7723$ &$40.683$ &$1.2795$ &$7.798$ &$0.6385$ &$3.630$ &$-0.0656$ &$1.686$ &$0.0015$ \\
Re0 & 6s15d5 &$-1.6968$ &$34.939$ &$1.3535$ &$6.851$ &$1.7292$ &$2.681$ &$-1.3118$ &$2.445$ &$0.0014$ \\
Re2 & 6s05d5 &$-1.7305$ &$30.305$ &$1.3808$ &$6.606$ &$1.5787$ &$2.630$ &$-1.1785$ &$2.378$ &$0.0015$ \\
Re3 & 6s05d4 &$-1.6969$ &$24.325$ &$1.4088$ &$5.901$ &$1.5067$ &$2.424$ &$-1.1834$ &$2.209$ &$0.0019$ \\
Re4 & 6s05d3 &$-1.6679$ &$20.454$ &$1.4439$ &$5.357$ &$0.8175$ &$2.321$ &$-0.5685$ &$1.976$ &$0.0024$ \\
Re5 & 6s05d2 &$-1.6427$ &$17.722$ &$1.4880$ &$4.916$ &$0.5419$ &$2.189$ &$-0.3694$ &$1.781$ &$0.0029$ \\
Re6 & 6s05d1 &$-1.6211$ &$15.673$ &$1.5419$ &$4.548$ &$1.9205$ &$1.833$ &$-1.8287$ &$1.766$ &$0.0034$ \\
\hline
\end{tabular}
}
\end{table}
\begin{table}[H]
 \caption{$\langle j_4\rangle$ form factors for the 5d electrons of transition atoms and ions from Os to Au.\cite{kob:11}}
\label{5dj4b} \vspace{1ex}
{\tablesize
\begin{tabular}{llrrrrrrrrr}
\hline
\multicolumn{1}{c}{ Ion}&\multicolumn{1}{c}{ Config}&\multicolumn{1}{c}{ A }&\multicolumn{1}{c}{  a }&\multicolumn{1}{c}{B }&\multicolumn{1}{c}{ b }&\multicolumn{1}{c}{ C }&\multicolumn{1}{c}{ c }&\multicolumn{1}{c}{ D }&\multicolumn{1}{c}{ d }&\multicolumn{1}{c}{E }\\
\hline
Os0 & 6s05d8 &$-1.5677$ &$53.075$ &$1.0631$ &$8.143$ &$0.6808$ &$3.771$ &$-0.0308$ &$1.243$ &$0.0017$ \\
Os0 & 6s15d7 &$-1.5109$ &$42.193$ &$1.1910$ &$6.850$ &$2.4597$ &$2.602$ &$-2.0163$ &$2.444$ &$0.0011$ \\
Os0 & 6s25d6 &$-1.4734$ &$34.814$ &$1.2105$ &$6.165$ &$0.9468$ &$2.534$ &$-0.5773$ &$2.131$ &$0.0015$ \\
Os0 & 6s05d7 &$-1.5777$ &$35.746$ &$1.2222$ &$6.650$ &$1.7260$ &$2.590$ &$-1.2942$ &$2.360$ &$0.0012$ \\
Os0 & 6s15d6 &$-1.5274$ &$30.891$ &$1.2364$ &$6.044$ &$1.4100$ &$2.431$ &$-1.0480$ &$2.188$ &$0.0016$ \\
Os2 & 6s05d6 &$-1.5637$ &$27.292$ &$1.2627$ &$5.880$ &$1.3526$ &$2.393$ &$-1.0026$ &$2.146$ &$0.0017$ \\
Os3 & 6s05d5 &$-1.5453$ &$22.300$ &$1.3058$ &$5.301$ &$0.7208$ &$2.294$ &$-0.4480$ &$1.881$ &$0.0022$ \\
Os4 & 6s05d4 &$-1.5267$ &$18.972$ &$1.3619$ &$4.834$ &$3.0121$ &$1.938$ &$-2.8240$ &$1.877$ &$0.0027$ \\
Os5 & 6s05d3 &$-1.5094$ &$16.573$ &$1.4158$ &$4.458$ &$0.8448$ &$1.797$ &$-0.7349$ &$1.638$ &$0.0033$ \\
Os6 & 6s05d2 &$-1.4938$ &$14.751$ &$1.4678$ &$4.149$ &$0.8091$ &$1.535$ &$-0.7720$ &$1.442$ &$0.0041$ \\
Os7 & 6s05d1 &$-0.0341$ &$37.994$ &$-1.4680$ &$13.159$ &$1.5216$ &$3.898$ &$-0.0308$ &$0.550$ &$0.0083$ \\
Ir0 & 6s05d9 &$-1.3913$ &$45.243$ &$1.0627$ &$6.722$ &$2.5141$ &$2.534$ &$-2.0510$ &$2.383$ &$0.0009$ \\
Ir0 & 6s15d8 &$-1.3605$ &$36.399$ &$1.0953$ &$5.990$ &$1.7223$ &$2.353$ &$-1.3416$ &$2.162$ &$0.0014$ \\
Ir0 & 6s25d7 &$-1.3382$ &$30.628$ &$1.1376$ &$5.420$ &$1.4261$ &$2.160$ &$-1.1282$ &$1.974$ &$0.0019$ \\
Ir0 & 6s05d8 &$-1.4233$ &$31.680$ &$1.1221$ &$5.872$ &$2.2721$ &$2.301$ &$-1.8973$ &$2.163$ &$0.0015$ \\
Ir0 & 6s15d7 &$-1.3875$ &$27.660$ &$1.1508$ &$5.362$ &$0.6586$ &$2.302$ &$-0.3554$ &$1.810$ &$0.0020$ \\
Ir2 & 6s05d7 &$-1.4233$ &$24.796$ &$1.1799$ &$5.246$ &$3.2548$ &$2.072$ &$-2.9649$ &$1.997$ &$0.0020$ \\
Ir3 & 6s05d6 &$-1.4149$ &$20.563$ &$1.2388$ &$4.761$ &$1.1780$ &$1.928$ &$-0.9708$ &$1.761$ &$0.0026$ \\
Ir4 & 6s05d5 &$-1.4039$ &$17.664$ &$1.2993$ &$4.371$ &$3.5599$ &$1.661$ &$-3.4340$ &$1.625$ &$0.0033$ \\
\hline
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors for 4f electrons of rare earth ions}
\begin{table}[H]
\caption{$\langle j_4\rangle$ form factors for rare earth ions}\vspace{2mm}
\label{rej4}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Ce2 &$-0.6468$ &$10.5331$ &$0.4052$ &$5.6243$ &$0.3412$ &$1.5346$ &$0.0080$ \\
Nd2 &$-0.5744$ &$10.9304$ &$0.4210$ &$6.1052$ &$0.3124$ &$1.4654$ &$0.0081$ \\
Nd2 &$-0.5416$ &$12.2043$ &$0.3571$ &$6.1695$ &$0.3154$ &$1.4847$ &$0.0098$ \\
Nd3 &$-0.4053$ &$14.0141$ &$0.0329$ &$7.0046$ &$0.3759$ &$1.7074$ &$0.0209$ \\
Sm2 &$-0.4150$ &$14.0570$ &$0.1368$ &$7.0317$ &$0.3272$ &$1.5825$ &$0.0192$ \\
Sm3 &$-0.4288$ &$10.0525$ &$0.1782$ &$5.0191$ &$0.2833$ &$1.2364$ &$0.0088$ \\
Eu2 &$-0.4145$ &$10.1930$ &$0.2447$ &$5.1644$ &$0.2661$ &$1.2054$ &$0.0065$ \\
Eu3 &$-0.4095$ &$10.2113$ &$0.1485$ &$5.1755$ &$0.2720$ &$1.2374$ &$0.0131$ \\
Gd2 &$-0.3824$ &$10.3436$ &$0.1955$ &$5.3057$ &$0.2622$ &$1.2032$ &$0.0097$ \\
Gd3 &$-0.3621$ &$10.3531$ &$0.1016$ &$5.3104$ &$0.2649$ &$1.2185$ &$0.0147$ \\
Tb2 &$-0.3443$ &$10.4686$ &$0.1481$ &$5.4156$ &$0.2575$ &$1.1824$ &$0.0104$ \\
Tb3 &$-0.3228$ &$10.4763$ &$0.0638$ &$5.4189$ &$0.2566$ &$1.1962$ &$0.0159$ \\
Dy2 &$-0.3206$ &$12.0714$ &$0.0904$ &$8.0264$ &$0.2616$ &$1.2296$ &$0.0143$ \\
Dy3 &$-0.2829$ &$9.5247$ &$0.0565$ &$4.4292$ &$0.2437$ &$1.0665$ &$0.0092$ \\
Ho2 &$-0.2976$ &$9.7190$ &$0.1224$ &$4.6345$ &$0.2279$ &$1.0052$ &$0.0063$ \\
Ho3 &$-0.2717$ &$9.7313$ &$0.0474$ &$4.6378$ &$0.2292$ &$1.0473$ &$0.0124$ \\
Er2 &$-0.2975$ &$9.8294$ &$0.1189$ &$4.7406$ &$0.2116$ &$1.0039$ &$0.0117$ \\
Er3 &$-0.2568$ &$9.8339$ &$0.0356$ &$4.7415$ &$0.2172$ &$1.0281$ &$0.0148$ \\
Tm2 &$-0.2677$ &$9.8883$ &$0.0925$ &$4.7838$ &$0.2056$ &$0.9896$ &$0.0124$ \\
Tm3 &$-0.2292$ &$9.8948$ &$0.0124$ &$4.7850$ &$0.2108$ &$1.0071$ &$0.0151$ \\
Yb2 &$-0.2393$ &$9.9469$ &$0.0663$ &$4.8231$ &$0.2009$ &$0.9651$ &$0.0122$ \\
Yb3 &$-0.2121$ &$8.1967$ &$0.0325$ &$3.1533$ &$0.1975$ &$0.8842$ &$0.0093$ \\
Pr3 &$-0.3970$ &$10.9919$ &$0.0818$ &$5.9897$ &$0.3656$ &$1.5021$ &$0.0110$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_4\rangle$} Form factors for 4f electrons of actinide ions}
\begin{table}[H]
\caption{$\langle j_4\rangle$ Form factors for actinide ions}\vspace{2mm}
\label{acj4}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
U3 &$-0.9859$ &$16.6010$ &$0.6116$ &$6.5147$ &$0.6020$ &$2.5970$ &$-0.0010$ \\
U4 &$-1.0540$ &$16.6055$ &$0.4339$ &$6.5119$ &$0.6746$ &$2.5993$ &$-0.0011$ \\
U5 &$-0.9588$ &$16.4851$ &$0.1576$ &$6.4397$ &$0.7785$ &$2.6402$ &$-0.0010$ \\
Np3 &$-0.9029$ &$16.5858$ &$0.4006$ &$6.4699$ &$0.6545$ &$2.5631$ &$-0.0004$ \\
Np4 &$-0.9887$ &$12.4415$ &$0.5918$ &$5.2941$ &$0.5306$ &$2.2625$ &$-0.0021$ \\
Np5 &$-0.8146$ &$16.5809$ &$-0.0055$ &$6.4751$ &$0.7956$ &$2.5623$ &$-0.0004$ \\
Np6 &$-0.6738$ &$16.5531$ &$-0.2297$ &$6.5055$ &$0.8513$ &$2.5528$ &$-0.0003$ \\
Pu3 &$-0.7014$ &$16.3687$ &$-0.1162$ &$6.6971$ &$0.7778$ &$2.4502$ &$0.0000$ \\
Pu4 &$-0.9160$ &$12.2027$ &$0.4891$ &$5.1274$ &$0.5290$ &$2.1487$ &$-0.0022$ \\
Pu5 &$-0.7035$ &$16.3601$ &$-0.0979$ &$6.7057$ &$0.7726$ &$2.4475$ &$0.0000$ \\
Pu6 &$-0.5560$ &$16.3215$ &$-0.3046$ &$6.7685$ &$0.8146$ &$2.4259$ &$0.0001$ \\
Am2 &$-0.7433$ &$16.4163$ &$0.3481$ &$6.7884$ &$0.6014$ &$2.3465$ &$0.0000$ \\
Am3 &$-0.8092$ &$12.8542$ &$0.4161$ &$5.4592$ &$0.5476$ &$2.1721$ &$-0.0011$ \\
Am4 &$-0.8548$ &$12.2257$ &$0.3037$ &$5.9087$ &$0.6173$ &$2.1881$ &$-0.0016$ \\
Am5 &$-0.6538$ &$15.4625$ &$-0.0948$ &$5.9971$ &$0.7295$ &$2.2968$ &$0.0000$ \\
Am6 &$-0.5390$ &$15.4491$ &$-0.2689$ &$6.0169$ &$0.7711$ &$2.2970$ &$0.0002$ \\
Am7 &$-0.4688$ &$12.0193$ &$-0.2692$ &$7.0415$ &$0.7297$ &$2.1638$ &$-0.0011$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large $\langle j_6\rangle$} Form factors for 4f electrons of rare earth ions}
\begin{table}[H]
\caption{$\langle j_6\rangle$ Form factors for rare earth ions}\vspace{2mm}
\label{rej6}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
Ce2 &$-0.1212$ &$7.9940$ &$-0.0639$ &$4.0244$ &$0.1519$ &$1.0957$ &$0.0078$ \\
Nd2 &$-0.1600$ &$8.0086$ &$0.0272$ &$4.0284$ &$0.1104$ &$1.0682$ &$0.0139$ \\
Nd3 &$-0.0416$ &$8.0136$ &$-0.1261$ &$4.0399$ &$0.1400$ &$1.0873$ &$0.0102$ \\
Sm2 &$-0.1428$ &$6.0407$ &$0.0723$ &$2.0329$ &$0.0550$ &$0.5134$ &$0.0081$ \\
Sm3 &$-0.0944$ &$6.0299$ &$-0.0498$ &$2.0743$ &$0.1372$ &$0.6451$ &$-0.0132$ \\
Eu2 &$-0.1252$ &$6.0485$ &$0.0507$ &$2.0852$ &$0.0572$ &$0.6460$ &$0.0132$ \\
Eu3 &$-0.0817$ &$6.0389$ &$-0.0596$ &$2.1198$ &$0.1243$ &$0.7639$ &$-0.0001$ \\
Gd2 &$-0.1351$ &$5.0298$ &$0.0828$ &$2.0248$ &$0.0315$ &$0.5034$ &$0.0187$ \\
Gd3 &$-0.0662$ &$6.0308$ &$-0.0850$ &$2.1542$ &$0.1323$ &$0.8910$ &$0.0048$ \\
Tb2 &$-0.0758$ &$6.0319$ &$-0.0540$ &$2.1583$ &$0.1199$ &$0.8895$ &$0.0051$ \\
Tb3 &$-0.0559$ &$6.0311$ &$-0.1020$ &$2.2365$ &$0.1264$ &$1.1066$ &$0.0167$ \\
Dy2 &$-0.0568$ &$6.0324$ &$-0.1003$ &$2.2396$ &$0.1401$ &$1.1062$ &$0.0109$ \\
Dy3 &$-0.0423$ &$6.0376$ &$-0.1248$ &$2.2437$ &$0.1359$ &$1.2002$ &$0.0188$ \\
Ho2 &$-0.0725$ &$6.0453$ &$-0.0318$ &$2.2428$ &$0.0738$ &$1.2018$ &$0.0252$ \\
Ho3 &$-0.0289$ &$6.0504$ &$-0.1545$ &$2.2305$ &$0.1550$ &$1.2605$ &$0.0177$ \\
Er2 &$-0.0648$ &$6.0559$ &$-0.0515$ &$2.2303$ &$0.0825$ &$1.2638$ &$0.0250$ \\
Er3 &$-0.0110$ &$6.0609$ &$-0.1954$ &$2.2242$ &$0.1818$ &$1.2958$ &$0.0149$ \\
Tm2 &$-0.0842$ &$4.0699$ &$0.0807$ &$0.8492$ &$-0.2087$ &$0.0386$ &$0.2095$ \\
Tm3 &$-0.0727$ &$4.0730$ &$0.0243$ &$0.6888$ &$3.9459$ &$0.0023$ &$-3.9076$ \\
Yb2 &$-0.0739$ &$5.0306$ &$0.0140$ &$2.0300$ &$0.0351$ &$0.5080$ &$0.0174$ \\
Yb3 &$-0.0345$ &$5.0073$ &$-0.0677$ &$2.0198$ &$0.0985$ &$0.5485$ &$-0.0076$ \\
Pr3 &$-0.0224$ &$7.9931$ &$-0.1202$ &$3.9406$ &$0.1299$ &$0.8938$ &$0.0051$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}
\section{{\large$\langle j_6\rangle$} Form factors for 4f electrons of actinide ions}
\begin{table}[H]
\caption{$\langle j_6\rangle$ Form factors for actinide ions}\vspace{2mm}
\label{acj6}
{\tablesize
\begin{tabular}{lrrrrrrr}
\hline
Ion&
\multicolumn{1}{c}{A}&\multicolumn{1}{c}{a}&
\multicolumn{1}{c}{B}&\multicolumn{1}{c}{b}&
\multicolumn{1}{c}{C}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{D}\\
\hline\\[-2ex]
U3 &$-0.3797$ &$9.9525$ &$0.0459$ &$5.0379$ &$0.2748$ &$1.6072$ &$0.0016$ \\
U4 &$-0.1793$ &$11.8961$ &$-0.2269$ &$5.4280$ &$0.3291$ &$1.7008$ &$0.0030$ \\
U5 &$-0.0399$ &$11.8909$ &$-0.3458$ &$5.5803$ &$0.3340$ &$1.6448$ &$0.0029$ \\
Np3 &$-0.2427$ &$11.8444$ &$-0.1129$ &$5.3774$ &$0.2848$ &$1.5676$ &$0.0022$ \\
Np4 &$-0.2436$ &$9.5988$ &$-0.1317$ &$4.1014$ &$0.3029$ &$1.5447$ &$0.0019$ \\
Np5 &$-0.1157$ &$9.5649$ &$-0.2654$ &$4.2599$ &$0.3298$ &$1.5494$ &$0.0025$ \\
Np6 &$-0.0128$ &$9.5692$ &$-0.3611$ &$4.3035$ &$0.3419$ &$1.5406$ &$0.0032$ \\
Pu3 &$-0.0364$ &$9.5721$ &$-0.3181$ &$4.3424$ &$0.3210$ &$1.5233$ &$0.0041$ \\
Pu4 &$-0.2394$ &$7.8367$ &$-0.0785$ &$4.0243$ &$0.2643$ &$1.3776$ &$0.0012$ \\
Pu5 &$-0.1090$ &$7.8188$ &$-0.2243$ &$4.1000$ &$0.2947$ &$1.4040$ &$0.0015$ \\
Pu6 &$-0.0001$ &$7.8196$ &$-0.3354$ &$4.1439$ &$0.3097$ &$1.4027$ &$0.0020$ \\
Am2 &$-0.3176$ &$7.8635$ &$0.0771$ &$4.1611$ &$0.2194$ &$1.3387$ &$0.0018$ \\
Am3 &$-0.3159$ &$6.9821$ &$0.0682$ &$3.9948$ &$0.2141$ &$1.1875$ &$-0.0015$ \\
Am4 &$-0.1787$ &$7.8805$ &$-0.1274$ &$4.0898$ &$0.2565$ &$1.3152$ &$0.0017$ \\
Am5 &$-0.0927$ &$6.0727$ &$-0.2227$ &$3.7840$ &$0.2916$ &$1.3723$ &$0.0026$ \\
Am6 &$0.0152$ &$6.0788$ &$-0.3549$ &$3.8610$ &$0.3125$ &$1.4031$ &$0.0036$ \\
Am7 &$0.1292$ &$6.0816$ &$-0.4689$ &$3.8791$ &$0.3234$ &$1.3934$ &$0.0042$ \\
\hline\\[-2ex]
\end{tabular}
}
\end{table}