\section{Introduction}

Additional electromagnetic physics processes for photons, electrons, hadrons 
and ions have been implemented in Geant4 in order to extend the validity 
range of particle interactions to lower energies than those available in the 
standard Geant4 electromagnetic processes~\cite{ov-leg4,ov-had,ov-ion}.
Because atomic shell structure is more important in most cases at low energies
than it is at higher energies, the low energy processes make direct use of 
shell cross section data.  The standard processes, which are optimized for 
high energy physics applications, rely on parameterizations of these data.
 
The low energy processes include the photo-electric effect, Compton scattering,
Rayleigh scattering, gamma conversion, bremsstrahlung and ionization.  
Fluorescence of excited atoms is also considered.  

Some features common to all low energy processes currently implemented in
Geant4 are summarized in this section.  Subsequent sections provide more 
detailed information for each process.

\subsection{Physics}

The low energy processes of Geant4 represent electromagnetic interactions 
at lower energies than those covered by the equivalent Geant4 standard 
electromagnetic processes.

The current implementation of low energy processes is valid for energies 
down to 250 eV (and can be used up to approximately 100 GeV), unless
differently specified.  It covers elements with atomic number between 1 and 
99.

All processes involve two distinct phases: 
\begin{itemize}
\item the calculation and use of total cross sections, and
\item the generation of the final state. 
\end{itemize}
Both phases are based on the theoretical models and on
exploitation of evaluated data.

\subsection{Data Sources}

The data used for the determination of cross-sections and 
for sampling of the final state are extracted 
from a set of publicly distributed evaluated data
libraries: 
\begin{itemize}
\item
EPDL97 (Evaluated Photons Data Library)~\cite{ov-EPDL97};
\item
EEDL (Evaluated Electrons Data Library)~\cite{ov-EEDL};
\item
EADL (Evaluated Atomic Data Library)~\cite{ov-EADL};
\item
stopping power data~\cite{ov-Ziegler771,ov-Ziegler774,ov-Ziegler85,ov-ICRU49};
\item
binding energy values based on data of Scofield~\cite{ov-bindingEnergies}.
\end{itemize}

Evaluated data sets are produced through the process of critical comparison,
selection, renormalization and averaging of the available experimental data,
normally complemented by model calculations.
%Evaluated Libraries are computer files of evaluated data which, appropriately
%processed, form the input data to computations for a wide variety of nuclear
%science and technology applications. Each of these evaluated libraries
%may consist of individual evaluated data sets for several hundred isotopes
%or elements (commonly referred to as 'materials')~\cite{ov-NEA}.
These libraries provide the following data relevant for the simulation of 
Geant4 low energy processes:

\begin{itemize}

\item 
total cross-sections for photoelectric effect, Compton scattering, 
Rayleigh scattering, pair production and bremsstrahlung,
\item 
subshell integrated cross sections for photo-electric effect and ionization,
\item 
energy spectra of the secondaries for electron processes,
\item 
scattering functions for the Compton effect,
\item 
form factors for Rayleigh scattering,
\item 
binding energies for electrons for all subshells,
\item 
transition probabilities between subshells for fluorescence and the Auger 
effect, and
\item
stopping power tables.
\end{itemize}

The energy range covered by the data libraries extends from 100 GeV 
down to 1 eV for Rayleigh and Compton effects, down to the lowest binding 
energy for each element for photo-electric effect and ionization, and
down to 10 eV for bremsstrahlung.

%\subsubsection{Data validity and accuracy}
%For a rough estimate of the accuracy of EPDL97 data libraries in different 
%energy regimes, see \cite{ov-data-est}. 
%The data are only applicable to cold, neutral, 
%ground-state and isolated atoms, 
%i.e. solid-state effects, plasmas, 
%external electric or magnetic fields are not taken into
%account although they could have significant 
%effects on the cross-sections near photoeffect
%thresholds of inner-shell electrons, 
%and completely alter the cross-sections near the
%thresholds of outer-shells~\cite{ov-EPDL97}.

%The precision of the binding energies is estimated 
%to be better than 1\%~\cite{ov-EPDL97}
%which is small compared to the differences expected due 
%to elements combining into radicals or
%compounds. 
%For example, the K shell binding energy of 
%H is 13.6 eV, and that of H2 is 16.4 eV,
%about 21\% higher. This will effect accuracy 
%over narrow energy ranges near photoionisation
%edges.

%Precision estimates on the data can be 
%found in~\cite{ov-EPDL97},~\cite{ov-EEDL} and~\cite{ov-EADL}. 

\subsection{Distribution of the Data Sets}

The author of EPDL97~\cite{ov-EPDL97}, who is also responsible for the 
EEDL~\cite{ov-EEDL} and EADL~\cite{ov-EADL} data libraries, Dr. Red Cullen, 
has kindly permitted the libraries and their related documentation to be 
distributed with the Geant4 toolkit.  The data are reformatted for Geant4 
input.  They can be downloaded from the source code section of the Geant4 
page:  http://cern.ch/geant4/geant4.html. 

The EADL, EEDL and EPDL97 data-sets are also available from several 
public distribution centres in a format different 
from the one used by Geant4~\cite{ov-NEA}.

Stopping power data are taken from 
publications~\cite{ov-Ziegler771,ov-Ziegler774,ov-Ziegler85,ov-ICRU49}.

\subsection{Calculation of Total Cross Sections}\label{subsubsigmatot}

The energy dependence of the total cross section is derived for each 
process from the evaluated data libraries.  Since the libraries provide 
cross sections for a set of discrete incident energies, the total cross
section at a given energy, $E$, is obtained by interpolation according to 
the formula~\cite{ov-stepanek}: 
\begin{equation}
  \label{eqloglog}
  \log(\sigma(E)) = {log(\sigma_1)log(E_2/E) + log(\sigma_2)log(E/E_1)
                      \over log(E_2/E_1)}
\end{equation}
where $E_1$ and $E_2$ are respectively the closest lower and higher energy  
for which data ($\sigma_1$ and $\sigma_2$) are available.

For each process a production threshold energy is defined;  by default it 
is set to the low end of the energy validity range of the process (250~eV 
in the current implementation), but a higher or lower value can be set by 
the user.  

For a particle of energy $E$, the mean free path for interacting via a 
given process is calculated as:
\begin{equation}
  \lambda = {1\over{\Sigma_i\\ \sigma_{i}(E)\cdot n_i}}
\end{equation}
where $\sigma_{i}(E)$ is the microscopic 
integrated cross-section of the process considered at energy $E$, and $n_i$ 
is the atomic density of the $i-th$ element contributing to the material 
composition.  The sum runs over all the elements of which the material is 
composed.  The cross sections are determined as described in this section.  
An exception to this method is the implementation of the chemical effect on 
hadron/ion stopping powers for a set of materials.


\subsection{Sampling of Relevant Physics Quantities}

The final state products of the processes are generated by sampling
relevant physical quantities, such as energies and angular distributions 
of secondaries, from distributions derived from theoretical models and 
evaluated data.  The energy dependence of the parameters which characterize 
the distributions is taken into account either by direct interpolation of 
the data available in the libraries, or by interpolation of values obtained 
from fits to the data.

When generating the final state, an atom of the material in which the 
interaction occurs is randomly selected and atomic de-excitation is 
simulated.

Secondaries which would be produced with energies below their user defined 
production threshold are not created and their energy is deposited locally.


\subsection{Status of the document}

\noindent
30.09.1999 created by Alessandra Forti\\
07.02.2000 modified by V\'eronique Lef\'ebure\\
08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia\\
04.12.2001 reviewed by Vladimir Ivanchenko\\
26.01.2003 minor re-write by D.H. Wright .


\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{ov-leg4}
 ``Geant4 Low Energy Electromagnetic Models for Electrons and Photons",
   J.Apostolakis et al., CERN-OPEN-99-034(1999), INFN/AE-99/18(1999) 
\bibitem{ov-had}
V.N.~Ivanchenko et al., GEANT4 Simulation of 
Energy Losses of Slow Hadrons, CERN-99-121, INFN/AE-99/20, (September 1999).
\bibitem{ov-ion}
S.~Giani et al., GEANT4 Simulation of 
Energy Losses of Ions, CERN-99-300, INFN/AE-99/21, (November 1999).
\bibitem{ov-EPDL97} 
  %http://reddog1.llnl.gov/homepage.red/photon.htm
 ``EPDL97: the Evaluated Photon Data Library, '97 version",
  D.Cullen, J.H.Hubbell, L.Kissel,
  UCRL--50400, Vol.6, Rev.5
\bibitem{ov-EEDL} 
  %http://reddog1.llnl.gov/homepage.red/Electron.htm
  ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from
  the LLNL Evaluated Electron Data Library (EEDL), Z=1-100"
  S.T.Perkins, D.E.Cullen, S.M.Seltzer,
  UCRL-50400 Vol.31
\bibitem{ov-EADL} 
  %http://reddog1.llnl.gov/homepage.red/ATOMIC.htm
  ``Tables and Graphs of Atomic Subshell and Relaxation Data Derived from
  the LLNL Evaluated Atomic Data Library (EADL), Z=1-100"
  S.T.Perkins, D.E.Cullen, M.H.Chen, J.H.Hubbell, J.Rathkopf, J.Scofield,
  UCRL-50400 Vol.30
\bibitem{ov-Ziegler771}
H.H.~Andersen and J.F.~Ziegler, 
The Stopping
and Ranges of Ions in Matter. Vol.3, Pergamon Press, 1977.
\bibitem{ov-Ziegler774}
J.F.~Ziegler, The Stopping
and Ranges of Ions in Matter. Vol.4, Pergamon Press, 1977.
\bibitem{ov-Ziegler85}
J.F.~Ziegler, J.P.~Biersack, 
U.~Littmark, The Stopping
and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985.
\bibitem{ov-ICRU49}
ICRU (A.~Allisy et al),
Stopping Powers and Ranges for Protons and Alpha
Particles,
ICRU Report 49, 1993.
\bibitem{ov-bindingEnergies}
  J.H. Scofield, ``Radiative Transitions", in ``Atomic Inner-Shell Processes",
  B.Crasemann ed. (Academic Press, New York, 1975),pp.265-292.
%\bibitem{ov-data-est}
%  http://reddog1.llnl.gov/homepage.red/DOCUMENT/EPDL97/epdl97.htm
\bibitem{ov-NEA}
  http://www.nea.fr/html/dbdata/nds\_evaluated.htm
\bibitem{ov-stepanek}
   ``New Photon, Positron and Electron Interaction Data 
for Geant in Energy Range
   from 1 eV to 10 TeV",
   J. Stepanek, Draft to be submitted for publication

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item
 ``Geant4 Low Energy Electromagnetic Models for Electrons and Photons",
   J.Apostolakis et al., CERN-OPEN-99-034(1999), INFN/AE-99/18(1999) 
\item
V.N.~Ivanchenko et al., GEANT4 Simulation of 
Energy Losses of Slow Hadrons, CERN-99-121, INFN/AE-99/20, (September 1999).
\item
S.~Giani et al., GEANT4 Simulation of 
Energy Losses of Ions, CERN-99-300, INFN/AE-99/21, (November 1999).
\item
  %http://reddog1.llnl.gov/homepage.red/photon.htm
 ``EPDL97: the Evaluated Photon Data Library, '97 version",
  D.Cullen, J.H.Hubbell, L.Kissel,
  UCRL--50400, Vol.6, Rev.5
\item 
  %http://reddog1.llnl.gov/homepage.red/Electron.htm
  ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from
  the LLNL Evaluated Electron Data Library (EEDL), Z=1-100"
  S.T.Perkins, D.E.Cullen, S.M.Seltzer,
  UCRL-50400 Vol.31
\item 
  %http://reddog1.llnl.gov/homepage.red/ATOMIC.htm
  ``Tables and Graphs of Atomic Subshell and Relaxation Data Derived from
  the LLNL Evaluated Atomic Data Library (EADL), Z=1-100"
  S.T.Perkins, D.E.Cullen, M.H.Chen, J.H.Hubbell, J.Rathkopf, J.Scofield,
  UCRL-50400 Vol.30
\item
H.H.~Andersen and J.F.~Ziegler, 
The Stopping
and Ranges of Ions in Matter. Vol.3, Pergamon Press, 1977.
\item
J.F.~Ziegler, The Stopping
and Ranges of Ions in Matter. Vol.4, Pergamon Press, 1977.
\item
J.F.~Ziegler, J.P.~Biersack, 
U.~Littmark, The Stopping
and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985.
\item
ICRU (A.~Allisy et al),
Stopping Powers and Ranges for Protons and Alpha
Particles,
ICRU Report 49, 1993.
\item
  J.H. Scofield, ``Radiative Transitions", in ``Atomic Inner-Shell Processes",
  B.Crasemann ed. (Academic Press, New York, 1975),pp.265-292.
%\bibitem{ov-data-est}
%  http://reddog1.llnl.gov/homepage.red/DOCUMENT/EPDL97/epdl97.htm
\item
  http://www.nea.fr/html/dbdata/nds\_evaluated.htm
\item
   ``New Photon, Positron and Electron Interaction Data 
for Geant in Energy Range
   from 1 eV to 10 TeV",
   J. Stepanek, Draft to be submitted for publication
\end{enumerate}

\end{htmlonly}