appendix_fparameters.tex

\chapter{Electron Elastic Scattering Factor of Select Elements}
\label{ap: fparameters}

From Peng~et~al~\cite{Ren1996}, a convenient and robust parameterization
of the electron elastic scattering factor is given by
%
\begin{equation}
  \begin{aligned}
    f_Z(\boldsymbol{q}) & = \sum_{n = 1}^6 a_n \mathrm{e}^{- b_n s^2}
  \end{aligned}
\end{equation}
%
where $4 \unslant[-.2]\pi s = q$ and
the fitted values for the parameters are tabulated in the two tables here.
The original values upon which the fitting is performed, with the exception of those for hydrogen,
are the result of relativistic Hartree-Fock calculations.
%
\begin{table}[ht!]
  \centering
  \renewcommand*{\arraystretch}{1.5}{
    \begin{tabular}{ C{1.5cm} C{0.5cm} C{1.4cm} C{1.4cm} C{1.4cm} C{1.4cm} C{1.4cm} }
      \toprule
      Element & $Z$ & $a_1$  & $a_2$  & $a_3$  & $a_4$  & $a_5$  \\
      \midrule
      H       & 1   & 0.0349 & 0.1201 & 0.1970 & 0.0573 & 0.1195 \\
      C       & 6   & 0.0893 & 0.2563 & 0.7570 & 1.0487 & 0.3575 \\
      N       & 7   & 0.1022 & 0.3219 & 0.7982 & 0.8197 & 0.1715 \\
      O       & 8   & 0.0974 & 0.2921 & 0.6910 & 0.6990 & 0.2039 \\
      F       & 9   & 0.1083 & 0.3175 & 0.6487 & 0.5846 & 0.1421 \\
      P       & 15  & 0.2548 & 0.6106 & 1.4541 & 2.3204 & 0.8477 \\
      S       & 16  & 0.2497 & 0.5628 & 1.3899 & 2.1865 & 0.7715 \\
      Fe      & 26  & 0.3946 & 1.2725 & 1.7031 & 2.3140 & 1.4795 \\
      Sb      & 51  & 0.7106 & 1.9247 & 2.6149 & 1.8899 & 3.8322 \\
      \bottomrule
    \end{tabular}
  }
  \caption{Parameter $a_n$ for $q$ up to $12.6$~\AA$^{-1}$.}
  \label{tab: fparameters_a}
\end{table}
%
\begin{table}[ht!]
  \centering
  \renewcommand*{\arraystretch}{1.5}{
    \begin{tabular}{ C{1.5cm}  C{0.5cm} C{1.4cm} C{1.4cm} C{1.4cm} C{1.4cm} C{1.4cm} }
      \toprule
      Element & $Z$ & $b_1$  & $b_2$  & $b_3$   & $b_4$   & $b_5$   \\
      \midrule
      H       & 1   & 0.5347 & 3.5867 & 12.3471 & 18.9525 & 38.6269 \\
      C       & 6   & 0.2465 & 1.7100 & 6.4094  & 18.6113 & 50.2523 \\
      N       & 7   & 0.2451 & 1.7481 & 6.1925  & 17.3894 & 48.1431 \\
      O       & 8   & 0.2067 & 1.3815 & 4.6943  & 12.7105 & 32.4726 \\
      F       & 9   & 0.2057 & 1.3439 & 4.2788  & 11.3932 & 28.7881 \\
      P       & 15  & 0.2908 & 1.8740 & 8.5176  & 24.3434 & 63.2996 \\
      S       & 16  & 0.2681 & 1.6711 & 7.0267  & 19.5377 & 50.3888 \\
      Fe      & 26  & 0.2717 & 2.0443 & 7.6007  & 29.9714 & 86.2265 \\
      Sb      & 51  & 0.2562 & 1.9646 & 6.8852  & 24.7648 & 68.9168 \\
      \bottomrule
    \end{tabular}
  }
  \caption{Parameter $b_n$ for $q$ up to $12.6$~\AA$^{-1}$.}
  \label{tab: fparameters_b}
\end{table}