\section{Hadron and Ion Ionisation} \label{le_had_ion}
 

 The class {\tt G4hLowEnergyIonisation} calculates the continuous energy loss
due to ionisation and simulates the $\delta$-ray production by charged hadrons
or ions.  This represents an extension of the Geant4 physics models down to 
low energy \cite{hlei.prepHadr,hlei.prepIon}.

\subsection{Delta-ray production}

In Geant4, $\delta$-rays are generated generally only above a threshold 
energy, $T_c$, the value of which depends on atomic parameters and the cut 
value, $T_{cut}$, calculated from the unique {\em cut in range} parameter
for all charged particles in all materials.  The total cross-section
for the production of a $\delta$-ray electron of kinetic energy $T > T_c$ 
by a particle of kinetic energy $E$ is:
\begin{equation}
\label{hlei.a}
\sigma (E,T_{c}) = \int_{T_{c}}^{T_{max}} \frac{d \sigma (E,T)}{dT} dT
\hspace{5mm} \mbox{with } T_c = \min(\max(I,T_{cut}),T_{max})
\end{equation}
where $I$ is the mean excitation potential of the atom (the formulae of 
this charter are precise if $T \gg I$),
$T_{max}$ is the maximum energy transferable to the free electron
\begin{equation}
\label{hlei.a1}
T_{max} =\frac{2m_e c^2 (\gamma^2 -1)} {1+2\gamma (m_e/M) + (m_e/M)^2}
\end{equation}
with $m_e$ the electron mass, $M$ the mass of the incident particle, and
$\gamma$ is the relativistic factor. 
For heavy charged particles the differential cross-section per atom
can be written as \cite{hlei.pdg,hlei.rossi52}:
\begin{eqnarray}
\label{hlei.bbb}
\mbox{for spin 0} &\frac {d\sigma }{dT} = & K Z \frac {Z^2_h}{\beta^2 T^2}
\left[ 1- \beta^2 \frac{T} { T_{max} }\right] 
\\ \nonumber 
\mbox{for spin 1/2} &\frac{d \sigma} {dT} = & K Z \frac {Z^2_h} {\beta^2 T^2}
\left[1- \beta^ 2 \frac{T}{T_{max} }+ \frac{T^2} {2E^2}
 \right] 
\\  \nonumber 
\mbox{for spin 1} &\frac{d \sigma} {dT}= & K Z \frac {Z^2_h}{\beta^2 T^2}
\left[\left(1- \beta^ 2 \frac{T}{T_{max} }\right)
\left(1 + \frac{T}{3Q_c} \right)
+ \frac{T^2} {3E^2}\left(1+\frac{T}{2Q_c}\right)
 \right] 
\end{eqnarray}
where $Z$ is the atomic number,
$Z_{h}$ is the effective charge of 
the incident particle in units
of positron charge, $\beta$ is the relativistic velocity, and
$Q_c=(M c^2)^2/m_e c^2$. The
factor $K$ is expressed as 
$K = 2\pi r^2_e m_e c^2$,
where $r_e$ is the classical electron
radius. 
The integration of
formula (\ref{hlei.a}) gives the total cross-section,
which
for particles with spin 0 and 1/2 are the following :
\begin{eqnarray}
\label{hlei.c}
\mbox{for spin 0} &\sigma (Z,E,T_{c})  = &
K Z  \frac{Z^2_h}{\beta^2} \left (
   \frac{1-\tau+\beta^2 \tau\ln \tau}{T_{c}} \right )
\\ \nonumber
\mbox{for spin 1/2} &\sigma (Z,E,T_{c})  = &
K Z  \frac{Z^2_h}{\beta^2} \left ( \frac{1-\tau+\beta^2 \tau\ln \tau}{T_{c}}
+ \frac{T_{max}-T_{c}} {2E^2} \right ) 
\end{eqnarray}
where  $\tau = T_c/T_{max}$.

\noindent 
The average energy transfer $\Delta E_{\delta}$
of a particle with spin 0
to $\delta$-electrons with $T > T_c$
can be expressed as:  
\begin{equation}
\Delta E_{\delta} = 
N_{el}\frac{Z^2_h}{\beta^2} \left (-
\ln{\tau} - 
\beta^2(1-\tau) \right )
\label{hlei.del}
\end{equation}
where  $N_{el}$ is the electron density of the medium.
Using (\ref{hlei.bbb}) one finds that
the correction to (\ref{hlei.del}) for particles with spin 1/2
is $(T^2_{max}-T^2_c)/4E^2$. 
This value is very small for low energy and can be neglected.
The same conclusion can be drawn for particles with spin 1.

\noindent
The mean free path of the particle 
is tabulated during initialisation
as a function of the material and of the energy for 
all the charged hadrons and static ions. Note, that for
low energy $T_c = T_{max}$, cross-section is zero and
the mean free path is set to infinity, compatible with the
machine precision.


\subsection{Energy Loss of Fast Hadrons}


The energy lost in soft
ionising collisions producing $\delta$-rays below ${T_c}$
are included in the continuous energy loss. 
 The mean value of the energy loss 
is given by the restricted Bethe-Bloch formula \cite{hlei.bethe,hlei.pdg} :
\begin{eqnarray}
\left.\frac{dE}{dx} \right]_{T<T_c} &=& K N_{el}\frac{Z^2_{h}}{\beta^2}L_0
\\ \nonumber 
&=& K N_{el}\frac{Z^2_{h}}{\beta^2} \left [
\ln{\frac{2m_e c^2 \beta^2\gamma^2T_{max}}{I^2}} -
\beta^2 \left ( 1 +\frac{T_c}{T_{max}} \right )
 - \delta - \frac{2C_e}{Z} \right ]
\label{hlei.d}
\end{eqnarray}
where  $N_{el}$ is the electron density of the medium, 
 $\delta$ is the density correction term, and $C_e/Z$ is the shell correction 
term.

\noindent
The density effect becomes important at high 
energies because of the long-range polarisation 
of the medium by a relativistic charged particle. The shell correction term
takes into account the fact that, at low energies for light elements,
and at all energies for heavy ones, the probability of 
hadron interaction with inner atomic shells becomes small.
The accuracy of the Bethe-Bloch formula with the correction terms mentioned
above is estimated as 1~\% for
energies between 6~MeV and 6~GeV \cite{hlei.pdg}. 
Using (\ref{hlei.bbb}) one can find out that
the correction to $L_0$ for particles with the spin 1/2
is $T^2_c/4E^2$. This value is very small and can be neglected.

\noindent
There exists a variety of phenomenological approximations for 
parameters in the Bethe-Bloch formula.
 In Geant4 the tabulation of
the ionisation potential from Ref.\cite{hlei.ICRU37} 
is implemented for all the 
elements. For the density
effect the formulation of Sternheimer \cite{hlei.sternheimer}
is used:
\input{electromagnetic/utils/densityeffect}

\noindent
The semi-empirical formula due to Barkas, which is applicable to all 
materials, is used for the shell correction term\cite{hlei.bark62}:
\begin{equation}
C_e(I, \beta\gamma) = \frac{a(I)}{(\beta\gamma)^2}
                     +\frac{b(I)}{(\beta\gamma)^4}
                     +\frac{c(I)}{(\beta\gamma)^6}
\end{equation}
The functions a(I), b(I), c(I) can be found in the source code. \\
This formula breaks down at low energies, and it only applies for $\beta 
\gamma > 0.13$ (e.g. $T > 7.9$ MeV for a proton). 
For $\beta \gamma \leq 0.13$ the shell correction term is calculated as:
$$
\left . C_{e}(I,\beta \gamma) \rule{0mm}{5mm} \right |_{\beta \gamma \leq
0.13} = C_{e}(I,\beta \gamma=0.13)\frac{\ln (T/T_{2l})}
         {\ln (7.9 \mbox{ MeV}/T_{2l})}
$$
hence the correction becomes progressively smaller from $T=7.9$
MeV to $T=T_{2l}=2 \mbox{ MeV}$.

\noindent
Since $M \gg m_e$,
the ionisation loss does not depend on the hadron
mass, but on its velocity.
Therefore the energy loss of a charged hadron 
with kinetic energy, $T$, is the same as
the energy loss of a proton with the same velocity. The corresponding
kinetic energy of the proton $T_p$ is
\begin{equation}
T_{proton} = \frac{M_{proton}}{M} \ T.
\label{hlei.e}
\end{equation}  

\noindent
At initialisation stage of Geant4 the $dE/dx$ tables 
and range tables for all materials
are calculated only for protons and antiprotons.
During run time the energy loss and the range of any hadron or ion are
recalculated using the scaling relation (\ref{hlei.e}).


\subsection{Barkas and Bloch effects}


The accuracy of 
the Bethe-Bloch stopping power formula 
(\ref{hlei.e}) can be improved
if the higher order terms are taken into account:
\begin{equation}
-\frac{dE}{dx} = K \frac{Z^2_{h}}{\beta^2}(L_0 +Z_{h}L_1+Z^2_{h}L_2), 
\label{hlei.f}
\end{equation}
where $L_1$ is the Barkas term \cite{hlei.bark56}, 
describing the difference
between ionisation of positively and negatively charged particles, and
$L_2$ is the Bloch term.

The Barkas effect for kinetic energy of 
protons or antiprotons greater than $500 keV$ can be described as 
\cite{hlei.arb72}:
\begin{equation}
L_1=\frac{F\left ( b / \sqrt{x}\right ) }{\sqrt{Z x^3}}, \,\,\,
x=\frac{\beta^2c^2}{Zv_0^2},\,\,\,
b=0.8 Z^{\frac 16}\left( 1+6.02Z^{-1.19}\right), 
\label{hlei.g}
\end{equation}
where 
$v_0$ is the Bohr velocity (corresponding to proton energy $T_p=25 keV$), and 
the function $F$ is  tabulated according to \cite{hlei.arb72}.

The Bloch term \cite{hlei.bloch}
can be expressed in the following way:
\begin{equation}
Z^2_{h}L_2 = - y^2 \sum^{\inf}_{j=1} \frac{1}{j(j^2 + y^2)},\,\,\,
y=\frac{Z_{h}}{137\beta}.
\label{hlei.h}
\end{equation}
Note, that for $y \ll 1$ the simplified expression 
$Z^2_{h}L_2=-1.202y^2$ can be used.

Both the Barkas and Bloch terms break scaling of ionisation losses
if the absolute value of particle charge is different from unity,
because the particle charge $Z_h$ is not factorised
in the formula (\ref{hlei.f}).
To take these terms into account correction is made at
each step of the simulation for the value of $dE/dx$
re-calculated from the proton or antiproton tables.
There is the possibility to switch off the calculation
of these terms.

\subsection{Energy losses of slow positive hadrons}

At low energies the total energy loss is usually described 
in terms of {\it electronic stopping power} $S_e = - dE/dx$.
For charged hadron with velocity $\beta < 0.05$ (corresponding 
to 1~MeV for protons), formula (\ref{hlei.d}) becomes inaccurate.
In this case the velocity of the incident 
hadron is comparable to the velocity
of atomic electrons. At very low energies, when 
$\beta < 0.01$, the model of a free electron gas \cite{hlei.Lindhard}
predicts the stopping power to be proportional to
the hadron velocity,   
but it is not as accurate as the Bethe-Bloch formalism.
The intermediate region $0.01 < \beta < 0.05$ is not covered 
by precise theories. In this energy 
interval the Bragg peak of ionisation loss occurs.

To simulate slow proton energy loss
the  following
parametrisation from the review  \cite{hlei.Ziegler771} was implemented:
\begin{eqnarray}
S_e & = &  A_1E^{1/2}, \; \; \; \; \; \; \; \; \hspace{46mm}
 1~keV < T_p < 10~keV, \nonumber \\
S_e & = & \frac{S_{low}S_{high}}{S_{low}+S_{high}}, \hspace{46mm}
                       10~keV < T_p < 1~MeV, \nonumber \\
S_{low}  & = & A_2E^{0.45},  \nonumber\\
S_{high} & = & \frac{A_3}{E}\ln{\left(1 + \frac{A_4}{E} + A_5E \right)},  
             \nonumber \\
S_e & =  & \frac{A_6}{\beta^2} \left [\ln{\frac{A_7\beta^2}{1-\beta^2}}
-\beta^2 - \sum^{4}_{i=0} A_{i+8}(\ln{E})^i \right ],  
                      \;   1~MeV < T_p < 100~MeV, \nonumber \\
\label{hlei.i}
\end{eqnarray}  
where $S_e$ is the stopping power
in $[eV/10^{15}atoms/cm^2]$, $E=T_p/M_p [keV/amu]$, $A_i$ are twelve 
fitting parameters found individually for each atom for
atomic numbers from 1 to 92.
This parametrisation is used
in the interval of proton kinetic energy:
\begin{equation}
T_1 < T_p < T_2,
\label{hlei.j}
\end{equation}  
where $T_1 = 1~keV$ is the minimal kinetic energy of protons 
in the tables of Ref.\cite{hlei.Ziegler771}, 
 $T_2$ is an arbitrary value 
between 2~MeV and 100~MeV, since in this range
 both the  parametrisation (\ref{hlei.i})
and the Bethe-Bloch formula (\ref{hlei.e}) 
have practically the same accuracy and
are close to each other. 
Currently the value  $T_2 = 2~MeV$ is chosen.


To avoid problems in computation and 
to provide a continuous $dE/dx$ function, the factor
\begin{equation}
F = \left (1 + B\frac{T_2}{T_p} \right )
\label{hlei.r}
\end{equation}  
is multiplied by the value of  $dE/dx$ for $T_p > T_{2}$.
The parameter $B$ is determined for each element of the material 
in order to
provide continuity at $T_p=T_2$. The value of $B$ for all
atoms is less than 0.01. For the
simulation of the stopping power of very slow protons the model of a
free electron gas \cite{hlei.Lindhard} is used:
\begin{equation}
S_e = A \sqrt{T_p}, \; \; T_p < T_{1}.
\label{hlei.k}
\end{equation}  
The parameter $A$ is defined for each atom
by requiring the stopping power to be continuous
  at $T_p=T_{1}$. Currently the value used is $T_1=1~keV$.

Note that
if the cut kinetic energy is small ($T_c < T_{max}$), then the average
energy deposit giving
rise to $\delta$-electron production  (\ref{hlei.del})
is subtracted from the
value of the stopping power $S_e$, which is calculated by formula
(\ref{hlei.i}).


Alternative parametrisations of proton energy loss
are also available within Geant4 (Table \ref{hlei.tab0}).
The parameterisation formulae
in Ref.\cite{hlei.ICRU49} are the same 
as in Ref.(\cite{hlei.Ziegler771}) 
for the kinetic
energy of protons $T_p < 1~MeV$, but 
the values of the parameters are different.
The type of parameterisation is optional and
can be chosen by the user separately for each particle
at the initialisation stage of Geant4.


\begin{table*}
\caption{The list of parameterisations available.}
%\vspace {2pt}
\label{hlei.tab0}
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Name  & Particle & Source \\ 
\hline
{\bf Ziegler1977p} & proton & J.F.~Ziegler parameterisation 
    \cite{hlei.Ziegler771} \\
{\bf Ziegler1977He} & $He^4$ & J.F.~Ziegler parameterisation 
    \cite{hlei.Ziegler774}\\
{\bf Ziegler1985p} & proton & TRIM'85 parameterisation \cite{hlei.Ziegler85} \\
{\bf ICRU\_R49p} & proton & ICRU parameterisation \cite{hlei.ICRU49} \\
{\bf ICRU\_R49He} & $He^4$ & ICRU parameterisation \cite{hlei.ICRU49} \\
\hline
\end{tabular} 
\end{center}
\end{table*}


\subsection{Energy loss of alpha particles}

The accuracy of the data for the ionisation losses of $\alpha$-particles 
in all elements \cite{hlei.ICRU49,hlei.Ziegler774} 
is comparable to the accuracy
of the data for proton energy loss \cite{hlei.Ziegler771,hlei.ICRU49}. 
In the GEANT4 energy loss model for $\alpha$-particles  
the Bethe-Bloch formula is used for kinetic energy 
$T > T_2$, where $T_2$ is the arbitrary parameter, currently set to $8~MeV$. 
For lower energies a parameterisation is performed.
In the energy range of the Bragg peak, 
$1~keV < T < 10~MeV$, the 
parameterisation is:  
\begin{eqnarray}
S_e & = & \frac{S_{low}S_{high}}{S_{low}+S_{high}},  \nonumber \\
S_{low}  & = & A_1T^{A_2},  \nonumber\\
S_{high} & = & \frac{A_3}{T}\ln{\left(1 + \frac{A_4}{T} + A_5T \right)},  
             \nonumber \\
\label{hlei.l}
\end{eqnarray}  
where $S_e$ is the electronic stopping power
in $[eV/10^{15}atoms/cm^2]$, $T$ is the kinetic energy of $\alpha$-particles in
$MeV$,
$A_i$ are the five fitting
parameters fitted individually for each atom for
atomic numbers from 1 to 92. 

For higher energies $T > 10~MeV$, another
 parametrisation \cite{hlei.Ziegler774} is applied
\begin{equation}
S_e= exp \left(A_6+A_7E+A_8E^2+A_9E^3 \right ), \; E=ln(1/T).
\label{hlei.m}
\end{equation}  
To ensure a continuous $dE/dx$ function from the energy range of the
Bethe-Bloch formula to the energy range of the parameterisation, the factor
\begin{equation}
F = \left (1 + B\frac{T_2}{T} \right )
\label{hlei.n}
\end{equation}  
is multiplied by the value of  $S_e$ as predicted by the Bethe-Bloch formula
for $T > T_{2}$.
The parameter $B$ is determined for each element of the material in order to
ensure continuity at $T_p=T_2$. The value of $B$ for different atoms is
usually less than 0.01.

For kinetic energies of $\alpha$-particles $T < 1~keV$ the model
of free electron gas  \cite{hlei.Lindhard} is used
\begin{equation}
S_e = A \sqrt{T},
\label{hlei.o}
\end{equation}  
The parameter $A$ is defined for each atom by requiring the stopping power to be
continuous at $T=1~keV$.


\subsection{Effective charge of ions}

For hadrons or ions
the scaling relation can be written as
\begin{equation}
S_{ei}(T) = Z_{eff}^2\cdot S_{ep}(T_p),
\label{hlei.sei}
\end{equation}  
where $S_{ei}$ is the ion stopping power,
 $S_{ep}$ is the proton stopping power at the energy scaled
according (\ref{hlei.e}), and  
$Z_{eff}$ is effective charge of the particle, which has to be used in 
all expressions in place of $Z_h$.
For fast particles it is equal to the particle charge $Z_h$,
but for slow ions it differs significantly because  
 a slow ion
picks up electrons from the medium.
The ion effective charge is expressed via 
the ion charge $Z_h$ and the
fractional effective charge of ion $\gamma_i$:
\begin{equation}
Z_{eff} = \gamma_i Z_h.
\label{hlei.p}
\end{equation}  

For helium ions 
fractional effective charge
is parameterised for all 
elements with good accuracy \cite{hlei.Ziegler85} according to:
\begin{eqnarray}
(\gamma_{He})^2 & = &\left (1-\exp\left [-\sum_{j=0}^5{C_jQ^j}\right ]\right)
\left ( 1  + \frac{ 7 + 0.05  Z }{1000} \exp( -(7.6-Q)^2 ) \right )^2, 
\nonumber \\
 Q & = & \max ( 0, \ln T_p) ,  
\label{hlei.q} 
\end{eqnarray}  
where the coefficients $C_j$ are the same for all elements, and the
helium ion kinetic energy is in $keV/amu$.


The following expression is used for heavy ions \cite{hlei.BK}:  
\begin{equation}
\gamma_i = \left ( q + \frac{1-q}{2} \left (\frac{v_0}{v_F} \right )^2 
\ln {\left ( 1 + \Lambda^2 \right )} \right )
\left ( 1 + \frac{(0.18+0.0015Z)\exp(-(7.6-Q)^2)}{Z_i^2} \right ),
\label{hlei.s}
\end{equation}  
where $q$ is 
the fractional average charge of the ion,
$v_0$ is the Bohr velocity,
$v_F$ is the Fermi velocity of  
the electrons in the target medium, and $\Lambda$ is 
the term taking into account the screening effect. In Ref.~\cite{hlei.BK},
$\Lambda$ is estimated to be:
\begin{equation}
\Lambda = 10 \frac{v_F}{v_0} \frac{(1-q)^{2/3}}{Z_i^{1/3}(6+q)}.
\label{hlei.t}
\end{equation}  
The Fermi velocity of the medium is of the same order as the Bohr velocity, and
its exact value depends on the detailed electronic structure of the medium.
Experimental data on the Fermi velocity are taken from
Ref.\cite{hlei.Ziegler85}.
The expression for the fractional average charge of the ion is the following:
\begin{equation}
q = [1 -\exp(0.803y^{0.3}-1.3167y^{0.6}-0.38157y-0.008983y^2)],
\label{hlei.u}
\end{equation}  
where $y$ is a parameter that depends on the ion velocity $v_i$
\begin{equation}
y = \frac{v_i}{v_0Z^{2/3}} \left ( 1 +\frac {v_F^2}{5v_i^2} \right ).
\label{hlei.v}
\end{equation}  

The parametrisation described in this chapter is only valid
if the reduced kinetic energy of the ion is higher than the lower limit
of the energy:
\begin{equation}
T_p > \max \left ( 3.25~keV, \frac{25~keV}{Z^{2/3}} \right ).
\label{hlei.x}
\end{equation}  
If the ion energy is lower, then the free electron gas model (\ref{hlei.o})
is used  to calculate the stopping power.


\subsection{Energy losses of slow negative particles}

At low energies, e.g. below a few MeV for protons/antiprotons, the
Bethe-Bloch formula is no longer accurate in describing the energy 
loss of charged hadrons and higher $Z$ terms should be taken in 
account. 
Odd terms in $Z$ lead to a significant difference between energy 
loss of positively and negatively charged particles. 
The energy loss of negative hadrons is scaled from that 
of antiprotons. 
The antiproton energy loss is calculated in the following way:
\begin{itemize}
\item 
if the material is elemental, the quantum harmonic oscillator model is used, as
described in \cite{hlei.sigmund} and references therein.
The lower limit of applicability of the model is chosen for all
materials at $50~keV$. Below this value stopping power is set to constant
equal to the $dE/dx$ at $50~keV$.
\item 
if the material is not elemental, the energy loss is calculated 
down to $500~keV$ using the Barkas correction (\ref{hlei.n})
 and at lower energies fitting the 
proton energy loss curve.
\end{itemize}



   
\subsection{Energy losses of hadrons in compounds}

To obtain energy losses in 
a mixture or compound, 
the absorber can be thought of as made up of thin 
layers of pure elements with weights proportional to the electron 
density of the element in the absorber (Bragg's rule):
\begin{equation}
\frac{dE}{dx}=\sum_i{\left (\frac{dE}{dx} \right )_i},
\label{hlei.y}
\end{equation}  
where the sum is taken over all elements of the absorber, $i$ is
the number of the element,
$(\frac{dE}{dx})_i$ is energy loss in the pure $i$-th element.

Bragg's rule is very accurate for relativistic particles
when the interaction of electrons with a nucleus is negligible.
But at low energies the accuracy of Bragg's rule is limited
because the energy loss to the electrons in any material
depends on the detailed orbital
and excitation structure of the material.
In the description of Geant4 materials there is a special
attribute: the chemical formula.
It is used in the
following way: 
\begin{itemize}
\item
if the data on the stopping power for a compound
as a function of the proton kinetic energy
  is available (Table \ref{hlei.tab1}), then the
direct parametrisation of the data for this material is performed;
\item
if the data on the stopping power for a compound
  is available for only one incident 
energy (Table \ref{hlei.tab2}), then 
the computation is
performed based on Bragg's rule and the chemical
factor for the compound is taken into account;
\item
if there are no data for the compound, the computation is
performed based on Bragg's rule.
\end{itemize}
\noindent
In the review \cite{hlei.Ziegler88} the parametrisation stopping
power data are presented as
\begin{equation}
S_e(T_p)= S_{Bragg}(T_p)\left [1 + \frac{f(T_p)}{f(125~keV)}
\left (\frac{S_{exp}(125~keV)}{S_{Bragg}(125~keV)}-1 \right ) \right ],
\label{hlei.z}
\end{equation}  
where $S_{exp}(125~keV)$ is the experimental value of the energy loss
for the compound
for $125~keV$ protons or  the
reduced experimental value for He ions, $S_{Bragg}(T_p)$ is
a value of energy loss calculated according to Bragg's
rule, and $f(T_p)$ is a universal function, which describes
the disappearance of deviations from Bragg's rule
for higher kinetic energies according to:
\begin{equation}
f(T_p)=\frac{1}{1+\exp \left [1.48(\frac{\beta(T_p)}
{\beta(25~keV)}-7.0) \right ]},
\label{hlei.fun}
\end{equation}  
where $\beta(T_p)$ is the relative velocity of the proton with
kinetic energy $T_p$.  


\begin{table*}
\caption{The list of chemical formulae of compounds for which 
 parametrisation of stopping power as a function
of kinetic energy is in Ref.\cite{hlei.ICRU49}.}
%\vspace {2pt}
\label{hlei.tab1}
\begin{center}
\begin{tabular}{|l|l|}
\hline
Number & Chemical formula \\ 
\hline
1.     &  AlO  \\
2.     &  C\_2O \\
3.     &  CH\_4  \\
4.     &  (C\_2H\_4)\_N-Polyethylene \\
5.     &  (C\_2H\_4)\_N-Polypropylene \\
6.     &  (C\_8H\_8)\_N \\  
7.     &  C\_3H\_8 \\
8.     &  SiO\_2 \\
9.     &  H\_2O \\
10.    &  H\_2O-Gas \\
11.    &  Graphite \\
\hline
\end{tabular} 
\end{center}
\end{table*}

\begin{table*}
\caption{The list of chemical formulae of compounds for which 
the {\it chemical factor} is calculated from the data 
 of Ref.\cite{hlei.Ziegler88}.}
%\vspace {2pt}
\label{hlei.tab2}
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Number & Chemical formula & Number & Chemical formula \\ 
\hline
1.   &   H\_2O    & 28. & C\_2H\_6 \\
2.   &   C\_2H\_4O    & 29. & C\_2F\_6 \\
3.   &   C\_3H\_6O    & 30.   & C\_2H\_6O \\
4.   &   C\_2H\_2    & 31.   & C\_3H\_6O \\
5.   &   C\_H\_3OH    & 32. & C\_4H\_10O \\
6.   &   C\_2H\_5OH    & 33. & C\_2H\_4 \\
7.   &   C\_3H\_7OH       & 34. & C\_2H\_4O \\
8.   &   C\_3H\_4    & 35.      & C\_2H\_4S \\
9.   &   NH\_3                 & 36.   & SH\_2 \\
10. & C\_14H\_10    & 37.   & CH\_4 \\
11.   & C\_6H\_6      & 38.    & CCLF\_3 \\
12. & C\_4H\_10     & 39. & CCl\_2F\_2 \\
13.  & C\_4H\_6    & 40.       & CHCl\_2F \\
14.  & C\_4H\_8O       & 41.   & (CH\_3)\_2S \\
15.        & CCl\_4    & 42. & N\_2O \\
16.   & CF\_4      & 43.    & C\_5H\_10O \\
17.   & C\_6H\_8      & 44. & C\_8H\_6 \\
18.   & C\_6H\_12    & 45.         & (CH\_2)\_N \\
19.   & C\_6H\_10O     & 46.   & (C\_3H\_6)\_N \\
20.    & C\_6H\_10    & 47. & (C\_8H\_8)\_N \\
21.   & C\_8H\_16     & 48. & C\_3H\_8 \\
22. & C\_5H\_10     & 49.  & C\_3H\_6-Propylene \\
23.  & C\_5H\_8    & 50. & C\_3H\_6O \\
24.  & C\_3H\_6-Cyclopropane    & 51.   & C\_3H\_6S \\
25.  & C\_2H\_4F\_2    & 52. & C\_4H\_4S \\
26.   & C\_2H\_2F\_2    & 53.  & C\_7H\_8 \\
27. & C\_4H\_8O\_2     & & \\
\hline
\end{tabular} 
\end{center}
\end{table*}


\subsection{Nuclear stopping powers}

Low energy ions transfer their energy not only to electrons of a medium 
but also to the nuclei of the medium  due to the elastic Coulomb
collisions. 
This contribution to the energy loss is called {\it
nuclear stopping power}.  
It is parametrised \cite{hlei.Ziegler774,hlei.Ziegler85,hlei.ICRU49}
 using a universal parameterisation for  reduced 
ion energy, $\epsilon$, which depends on ion parameters and on
the charge, $Z_t$, and the mass, $M_t$, of the target nucleus: 
\begin{equation}
\epsilon = \frac{32.536TM_t}{Z_{eff}Z_t(M+M_t)
\sqrt{Z_{eff}^{0.23}+Z_t^{0.23}}}.
\label{hlei.ep}
\end{equation}
The universal reduced nuclear stopping power, $s_n$, is determined
by J.~Moliere in the framework of Thomas-Fermi potential \cite{hlei.mol}.
The corresponding tabulation from Ref.\cite{hlei.ICRU49}
is implemented. 
To transform the value of
nuclear stopping power from reduced units to
 $[eV/10^{15}atoms/cm^2]$ the following formula is used:
\begin{equation}
S_n = s_n \frac{8.462Z_iZ_tM_i}{(M_i+M_t)\sqrt{Z_i^{0.23}+Z_t^{0.23}}}.
\label{hlei.re}
\end{equation}
The effect of nuclear stopping power is very small at high energies, but
it is of the same order of magnitude as electronic stopping power
for very slow ions (e.g. for protons, $T_p < 1 keV$).

\subsection{Fluctuations of energy losses of hadrons} 

The total continuous energy loss of charged particles is a stochastic 
quantity with a distribution described in terms of a straggling function.
The straggling is partially taken into account by the simulation
of energy loss by the production of $\delta$-electrons with energy
$T > T_c$. However, continuous energy loss also has fluctuations.  Hence 
in the current GEANT4 implementation two different models of fluctuations
are applied depending on the value of the parameter $\kappa$ which is the 
lower limit of the number of interactions of the particle in the step.
The default value chosen is $\kappa = 10$.  To select a model for thick 
absorbers the following boundary conditions are used:
\begin{equation}
\Delta E > T_c\kappa)\;\; or \;\; T_c < I\kappa,
\label{le_cond}
\end{equation}
where $\Delta E$ is the mean continuous energy loss in a track segment of 
length $s$, $T_c$ is the cut kinetic energy of $\delta$-electrons, and $I$ 
is the average ionisation potential of the atom.
   
For long path lengths the straggling function 
approaches the Gaussian distribution with Bohr's variance \cite{hlei.ICRU49}:
\begin{equation}
\Omega^2 = K N_{el}\frac{Z_h^2}{\beta^2} T_c s f
\left(1 - \frac{\beta^2}{2} \right),
\label{sig}
\end{equation}
where $f$ is a screening factor, which is equal to unity for fast particles,
whereas for slow positively charged 
ions with $\beta^2 < 3Z (v_0/c)^2$
$f = a + b/Z^2_{eff}$, where parameters $a$ and $b$
are parametrised for all atoms \cite{hlei.Yang,hlei.Chu}.

For short path lengths, when the condition \ref{le_cond} is not satisfied,
the model described in the charter \ref{gen_fluctuations} is applied.

\subsection{Sampling}

At each step for a charged hadron or ion in an absorber,
the step limit is calculated using range tables
for protons or antiprotons depending on the particle charge.
If the reduced particle energy $T_p < T_2$ the step limit is
forced to be not longer than $\alpha R(T_2)$, where $R(T_2)$ 
is the range of the particle with the reduced energy $T_2$,
$\alpha$ is an arbitrary coefficient, which is currently set to 0.05
in order to provide at least 20 steps for particles
in the Bragg peak energy range.
\noindent   
In each step continuous energy loss of the particle
is calculated using loss tables for protons or antiprotons
for $T_p > T_2$. For lower energies, continuous energy loss
is calculated using the effective charge approach, chemical
factors, and nuclear stopping powers.
\noindent
If the step of the particle is limited by the ionisation process
the sampling of $\delta$-electron production is performed.
(A short overview of the method is given in Chapter \ref{secmessel}.) \\
Apart from the normalisation, the cross-section 
(\ref{hlei.bbb}) can be written as :
\begin{eqnarray}
\frac{d\sigma}{dT} \sim f(T) \ g(T) &with& T \in [T_{c}, \ T_{max}]
\end{eqnarray}
with :
\begin{eqnarray*}
f(T) &=& \left(\frac{1}{T_{c}} -\frac{1}{T_{max}}\right)
\frac{1}{T^2} \\
g(T) &=& 1 - \beta^2\frac{T}{T_{max}} + S(T),
\end{eqnarray*}
where $S(T)$ is a spin dependent term (\ref{hlei.bbb}). 
For a spin-0 particle this term is zero, for 
a spin-$\frac{1}{2}$ particle $S(T)=T^2/2E^2$,
whilst for spin-1 the expression is more complicated.
\\
The energy, $T$, is sampled by :
\begin{enumerate}
\item Sample $T$ from $f(T)$.
\item Calculate the rejection function $g(T)$ and accept the
sampled $T$ with a probability of $g(T)$.
\end{enumerate}
After the successful sampling of the energy, the polar angles of the 
emitted electron are generated with respect to the direction of the
incident particle.  The azimuthal angle, $\phi$, is generated isotropically;
the polar angle $\theta$ is calculated from the energy momentum conservation.
This information is used to calculate the energy and momentum of both
particles and to transform them into the {\it global} coordinate system.

\subsection{PIXE}
PIXE is simulated by calculating cross-sections according to 
\cite{hlei.Gryzinski1} and \cite{hlei.Gryzinski2} to identify the primary 
ionised shell, and generating the subsequent atomic relaxation as described 
in \ref{relax}.  Sampling of excitations is carried out for both the 
continuous and the discrete parts of the process.


\subsection{Status of this document}

\noindent
21.11.2000 Created by V.Ivanchenko \\
30.05.2001 Modified by V.Ivanchenko \\
23.11.2001 Modified by M.G. Pia to add PIXE section. \\
19.01.2002 Minor corrections (mma) \\
13.05.2002 Minor corrections (V.Ivanchenko) \\
28.08.2002 Minor corrections (V.Ivanchenko)

\begin{latexonly}

\begin{thebibliography}{599}

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of 
Energy Losses of Slow Hadrons, CERN-99-121, INFN/AE-99/20, (September 1999).
\bibitem{hlei.prepIon}S.~Giani et al., GEANT4 Simulation
of 
Energy Losses of Ions, CERN-99-300, INFN/AE-99/21, (November 1999).
\bibitem{hlei.pdg} D.E.~Groom et al., Eur. 
Phys. Jour. C15 (2000) 1. 
\bibitem{hlei.rossi52} B.~Rossi, High Energy
Particles, Pretice-Hall, Inc., Englewood Cliffs, NJ, 1952.
\bibitem{hlei.bethe}H.~Bethe, Ann. Phys.  5 (1930) 325.
\bibitem{hlei.ICRU37} (A.~Allisy et al),
Stopping Powers for Electrons and Positrons,
ICRU Report 37, 1984.
\bibitem{hlei.sternheimer}
  R.M.~Sternheimer. Phys.Rev. B3 (1971) 3681.
\bibitem{hlei.bark62}
  W.H.~Barkas. Technical Report 10292,UCRL, August 1962.
\bibitem{hlei.bark56}
W.H.~Barkas, W.~Birnbaum, F.M.~Smith, Phys. Rev. 
101 (1956) 778.
\bibitem{hlei.arb72}  
J.C.~Ashley, R.H.~Ritchie and W.~Brandt,
Phys. Rev. B5 (1972) 1.
\bibitem{hlei.bloch}F.~Bloch, Ann. Phys. 16 (1933) 285.
\bibitem{hlei.Lindhard}
J.~Linhard and A.~Winther, Mat. Fys. Medd. Dan. Vid. Selsk. 
 34, No 10 (1963). 
\bibitem{hlei.Ziegler771}H.H.~Andersen and J.F.~Ziegler, 
The Stopping
and Ranges of Ions in Matter. Vol.3, Pergamon Press, 1977.
\bibitem{hlei.ICRU49}ICRU (A.~Allisy et al),
Stopping Powers and Ranges for Protons and Alpha
Particles,
ICRU Report 49, 1993.
\bibitem{hlei.Ziegler774}J.F.~Ziegler, The Stopping
and Ranges of Ions in Matter. Vol.4, Pergamon Press, 1977.
\bibitem{hlei.Ziegler85}J.F.~Ziegler, J.P.~Biersack, U
.~Littmark, The Stopping
and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985.
\bibitem{hlei.BK}
W.~Brandt and M.~Kitagawa, Phys. Rev. B25 (1982) 5631.
\bibitem{hlei.sigmund}
P.~Sigmund, Nucl. Instr. and Meth.
B85 (1994) 541.
\bibitem{hlei.Ziegler88} J.F.~Ziegler and 
J.M.~Manoyan, Nucl. Instr. and Meth.  
B35 (1988) 215.
\bibitem{hlei.mol}G.~Moliere, 
Theorie der Streuung schneller geladener Teilchen I;
 Einzelstreuungam abbgeschirmten Coulomb-Feld, Z. f. Naturforsch, A2
 (1947) 133.
\bibitem{hlei.GEANT3} GEANT3 manual, 
CERN Program Library Long Writeup
W5013 (October 1994).
\bibitem{hlei.Yang} Q.~Yang, 
D.J.~O'Connor, Z.~Wang, Nucl. Instr. and Meth.  
B61 (1991) 149.
\bibitem{hlei.Chu} W.K.~Chu, in: Ion Beam Handbook for
Material Analysis, edt. J.W.~Mayer and E.~Rimini,
Academic Press, NY, 1977. 
\bibitem{hlei.Gryzinski1} M. Gryzinski, Phys. Rev. A 135 (1965) 305.
\bibitem{hlei.Gryzinski2} M. Gryzinski, Phys. Rev. A 138 (1965) 322.
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\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 D.E.~Groom et al., Eur. 
Phys. Jour. C15 (2000) 1. 
\item B.~Rossi, High Energy
Particles, Pretice-Hall, Inc., Englewood Cliffs, NJ, 1952.
\item H.~Bethe, Ann. Phys.  5 (1930) 325.
\item (A.~Allisy et al),
Stopping Powers for Electrons and Positrons,
ICRU Report 37, 1984.
\item
  R.M.~Sternheimer. Phys.Rev. B3 (1971) 3681.
\item
  W.H.~Barkas. Technical Report 10292,UCRL, August 1962.
\item
W.H.~Barkas, W.~Birnbaum, F.M.~Smith, Phys. Rev. 
101 (1956) 778.
\item  
J.C.~Ashley, R.H.~Ritchie and W.~Brandt,
Phys. Rev. B5 (1972) 1.
\item F.~Bloch, Ann. Phys. 16 (1933) 285.
\item
J.~Linhard and A.~Winther, Mat. Fys. Medd. Dan. Vid. Selsk. 
 34, No 10 (1963). 
\item H.H.~Andersen and J.F.~Ziegler, 
The Stopping
and Ranges of Ions in Matter. Vol.3, Pergamon Press, 1977.
\item ICRU (A.~Allisy et al),
Stopping Powers and Ranges for Protons and Alpha
Particles,
ICRU Report 49, 1993.
\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
W.~Brandt and M.~Kitagawa, Phys. Rev. B25 (1982) 5631.
\item
P.~Sigmund, Nucl. Instr. and Meth.
B85 (1994) 541.
\item J.F.~Ziegler and 
J.M.~Manoyan, Nucl. Instr. and Meth.  
B35 (1988) 215.
\item G.~Moliere, 
Theorie der Streuung schneller geladener Teilchen I;
 Einzelstreuungam abbgeschirmten Coulomb-Feld, Z. f. Naturforsch, A2
 (1947) 133.
\item GEANT3 manual, 
CERN Program Library Long Writeup
W5013 (October 1994).
\item Q.~Yang, 
D.J.~O'Connor, Z.~Wang, Nucl. Instr. and Meth.  
B61 (1991) 149.
\item W.K.~Chu, in: Ion Beam Handbook for
Material Analysis, edt. J.W.~Mayer and E.~Rimini,
Academic Press, NY, 1977. 
\item M. Gryzinski, Phys. Rev. A 135 (1965) 305.
\item M. Gryzinski, Phys. Rev. A 138 (1965) 322.
\end{enumerate}

\end{htmlonly}


