\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 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.pp} \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} \bibitem{hlei.prepHadr}V.N.~Ivanchenko et al., GEANT4 Simulation 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}