[1208] | 1 | \section{Hadron and Ion Ionisation} \label{le_had_ion} |
---|
| 2 | |
---|
| 3 | |
---|
| 4 | The class {\tt G4hLowEnergyIonisation} calculates the continuous energy loss |
---|
| 5 | due to ionisation and simulates the $\delta$-ray production by charged hadrons |
---|
| 6 | or ions. This represents an extension of the Geant4 physics models down to |
---|
| 7 | low energy \cite{hlei.prepHadr,hlei.prepIon}. |
---|
| 8 | |
---|
| 9 | \subsection{Delta-ray production} |
---|
| 10 | |
---|
| 11 | In Geant4, $\delta$-rays are generated generally only above a threshold |
---|
| 12 | energy, $T_c$, the value of which depends on atomic parameters and the cut |
---|
| 13 | value, $T_{cut}$, calculated from the unique {\em cut in range} parameter |
---|
| 14 | for all charged particles in all materials. The total cross-section |
---|
| 15 | for the production of a $\delta$-ray electron of kinetic energy $T > T_c$ |
---|
| 16 | by a particle of kinetic energy $E$ is: |
---|
| 17 | \begin{equation} |
---|
| 18 | \label{hlei.a} |
---|
| 19 | \sigma (E,T_{c}) = \int_{T_{c}}^{T_{max}} \frac{d \sigma (E,T)}{dT} dT |
---|
| 20 | \hspace{5mm} \mbox{with } T_c = \min(\max(I,T_{cut}),T_{max}) |
---|
| 21 | \end{equation} |
---|
| 22 | where $I$ is the mean excitation potential of the atom (the formulae of |
---|
| 23 | this charter are precise if $T \gg I$), |
---|
| 24 | $T_{max}$ is the maximum energy transferable to the free electron |
---|
| 25 | \begin{equation} |
---|
| 26 | \label{hlei.a1} |
---|
| 27 | T_{max} =\frac{2m_e c^2 (\gamma^2 -1)} {1+2\gamma (m_e/M) + (m_e/M)^2} |
---|
| 28 | \end{equation} |
---|
| 29 | with $m_e$ the electron mass, $M$ the mass of the incident particle, and |
---|
| 30 | $\gamma$ is the relativistic factor. |
---|
| 31 | For heavy charged particles the differential cross-section per atom |
---|
| 32 | can be written as \cite{hlei.pdg,hlei.rossi52}: |
---|
| 33 | \begin{eqnarray} |
---|
| 34 | \label{hlei.bbb} |
---|
| 35 | \mbox{for spin 0} &\frac {d\sigma }{dT} = & K Z \frac {Z^2_h}{\beta^2 T^2} |
---|
| 36 | \left[ 1- \beta^2 \frac{T} { T_{max} }\right] |
---|
| 37 | \\ \nonumber |
---|
| 38 | \mbox{for spin 1/2} &\frac{d \sigma} {dT} = & K Z \frac {Z^2_h} {\beta^2 T^2} |
---|
| 39 | \left[1- \beta^ 2 \frac{T}{T_{max} }+ \frac{T^2} {2E^2} |
---|
| 40 | \right] |
---|
| 41 | \\ \nonumber |
---|
| 42 | \mbox{for spin 1} &\frac{d \sigma} {dT}= & K Z \frac {Z^2_h}{\beta^2 T^2} |
---|
| 43 | \left[\left(1- \beta^ 2 \frac{T}{T_{max} }\right) |
---|
| 44 | \left(1 + \frac{T}{3Q_c} \right) |
---|
| 45 | + \frac{T^2} {3E^2}\left(1+\frac{T}{2Q_c}\right) |
---|
| 46 | \right] |
---|
| 47 | \end{eqnarray} |
---|
| 48 | where $Z$ is the atomic number, |
---|
| 49 | $Z_{h}$ is the effective charge of |
---|
| 50 | the incident particle in units |
---|
| 51 | of positron charge, $\beta$ is the relativistic velocity, and |
---|
| 52 | $Q_c=(M c^2)^2/m_e c^2$. The |
---|
| 53 | factor $K$ is expressed as |
---|
| 54 | $K = 2\pi r^2_e m_e c^2$, |
---|
| 55 | where $r_e$ is the classical electron |
---|
| 56 | radius. |
---|
| 57 | The integration of |
---|
| 58 | formula (\ref{hlei.a}) gives the total cross-section, |
---|
| 59 | which |
---|
| 60 | for particles with spin 0 and 1/2 are the following : |
---|
| 61 | \begin{eqnarray} |
---|
| 62 | \label{hlei.c} |
---|
| 63 | \mbox{for spin 0} &\sigma (Z,E,T_{c}) = & |
---|
| 64 | K Z \frac{Z^2_h}{\beta^2} \left ( |
---|
| 65 | \frac{1-\tau+\beta^2 \tau\ln \tau}{T_{c}} \right ) |
---|
| 66 | \\ \nonumber |
---|
| 67 | \mbox{for spin 1/2} &\sigma (Z,E,T_{c}) = & |
---|
| 68 | K Z \frac{Z^2_h}{\beta^2} \left ( \frac{1-\tau+\beta^2 \tau\ln \tau}{T_{c}} |
---|
| 69 | + \frac{T_{max}-T_{c}} {2E^2} \right ) |
---|
| 70 | \end{eqnarray} |
---|
| 71 | where $\tau = T_c/T_{max}$. |
---|
| 72 | |
---|
| 73 | \noindent |
---|
| 74 | The average energy transfer $\Delta E_{\delta}$ |
---|
| 75 | of a particle with spin 0 |
---|
| 76 | to $\delta$-electrons with $T > T_c$ |
---|
| 77 | can be expressed as: |
---|
| 78 | \begin{equation} |
---|
| 79 | \Delta E_{\delta} = |
---|
| 80 | N_{el}\frac{Z^2_h}{\beta^2} \left (- |
---|
| 81 | \ln{\tau} - |
---|
| 82 | \beta^2(1-\tau) \right ) |
---|
| 83 | \label{hlei.del} |
---|
| 84 | \end{equation} |
---|
| 85 | where $N_{el}$ is the electron density of the medium. |
---|
| 86 | Using (\ref{hlei.bbb}) one finds that |
---|
| 87 | the correction to (\ref{hlei.del}) for particles with spin 1/2 |
---|
| 88 | is $(T^2_{max}-T^2_c)/4E^2$. |
---|
| 89 | This value is very small for low energy and can be neglected. |
---|
| 90 | The same conclusion can be drawn for particles with spin 1. |
---|
| 91 | |
---|
| 92 | \noindent |
---|
| 93 | The mean free path of the particle |
---|
| 94 | is tabulated during initialisation |
---|
| 95 | as a function of the material and of the energy for |
---|
| 96 | all the charged hadrons and static ions. Note, that for |
---|
| 97 | low energy $T_c = T_{max}$, cross-section is zero and |
---|
| 98 | the mean free path is set to infinity, compatible with the |
---|
| 99 | machine precision. |
---|
| 100 | |
---|
| 101 | |
---|
| 102 | \subsection{Energy Loss of Fast Hadrons} |
---|
| 103 | |
---|
| 104 | |
---|
| 105 | The energy lost in soft |
---|
| 106 | ionising collisions producing $\delta$-rays below ${T_c}$ |
---|
| 107 | are included in the continuous energy loss. |
---|
| 108 | The mean value of the energy loss |
---|
| 109 | is given by the restricted Bethe-Bloch formula \cite{hlei.bethe,hlei.pdg} : |
---|
| 110 | \begin{eqnarray} |
---|
| 111 | \left.\frac{dE}{dx} \right]_{T<T_c} &=& K N_{el}\frac{Z^2_{h}}{\beta^2}L_0 |
---|
| 112 | \\ \nonumber |
---|
| 113 | &=& K N_{el}\frac{Z^2_{h}}{\beta^2} \left [ |
---|
| 114 | \ln{\frac{2m_e c^2 \beta^2\gamma^2T_{max}}{I^2}} - |
---|
| 115 | \beta^2 \left ( 1 +\frac{T_c}{T_{max}} \right ) |
---|
| 116 | - \delta - \frac{2C_e}{Z} \right ] |
---|
| 117 | \label{hlei.d} |
---|
| 118 | \end{eqnarray} |
---|
| 119 | where $N_{el}$ is the electron density of the medium, |
---|
| 120 | $\delta$ is the density correction term, and $C_e/Z$ is the shell correction |
---|
| 121 | term. |
---|
| 122 | |
---|
| 123 | \noindent |
---|
| 124 | The density effect becomes important at high |
---|
| 125 | energies because of the long-range polarisation |
---|
| 126 | of the medium by a relativistic charged particle. The shell correction term |
---|
| 127 | takes into account the fact that, at low energies for light elements, |
---|
| 128 | and at all energies for heavy ones, the probability of |
---|
| 129 | hadron interaction with inner atomic shells becomes small. |
---|
| 130 | The accuracy of the Bethe-Bloch formula with the correction terms mentioned |
---|
| 131 | above is estimated as 1~\% for |
---|
| 132 | energies between 6~MeV and 6~GeV \cite{hlei.pdg}. |
---|
| 133 | Using (\ref{hlei.bbb}) one can find out that |
---|
| 134 | the correction to $L_0$ for particles with the spin 1/2 |
---|
| 135 | is $T^2_c/4E^2$. This value is very small and can be neglected. |
---|
| 136 | |
---|
| 137 | \noindent |
---|
| 138 | There exists a variety of phenomenological approximations for |
---|
| 139 | parameters in the Bethe-Bloch formula. |
---|
| 140 | In Geant4 the tabulation of |
---|
| 141 | the ionisation potential from Ref.\cite{hlei.ICRU37} |
---|
| 142 | is implemented for all the |
---|
| 143 | elements. For the density |
---|
| 144 | effect the formulation of Sternheimer \cite{hlei.sternheimer} |
---|
| 145 | is used: |
---|
| 146 | \input{electromagnetic/utils/densityeffect} |
---|
| 147 | |
---|
| 148 | \noindent |
---|
| 149 | The semi-empirical formula due to Barkas, which is applicable to all |
---|
| 150 | materials, is used for the shell correction term\cite{hlei.bark62}: |
---|
| 151 | \begin{equation} |
---|
| 152 | C_e(I, \beta\gamma) = \frac{a(I)}{(\beta\gamma)^2} |
---|
| 153 | +\frac{b(I)}{(\beta\gamma)^4} |
---|
| 154 | +\frac{c(I)}{(\beta\gamma)^6} |
---|
| 155 | \end{equation} |
---|
| 156 | The functions a(I), b(I), c(I) can be found in the source code. \\ |
---|
| 157 | This formula breaks down at low energies, and it only applies for $\beta |
---|
| 158 | \gamma > 0.13$ (e.g. $T > 7.9$ MeV for a proton). |
---|
| 159 | For $\beta \gamma \leq 0.13$ the shell correction term is calculated as: |
---|
| 160 | $$ |
---|
| 161 | \left . C_{e}(I,\beta \gamma) \rule{0mm}{5mm} \right |_{\beta \gamma \leq |
---|
| 162 | 0.13} = C_{e}(I,\beta \gamma=0.13)\frac{\ln (T/T_{2l})} |
---|
| 163 | {\ln (7.9 \mbox{ MeV}/T_{2l})} |
---|
| 164 | $$ |
---|
| 165 | hence the correction becomes progressively smaller from $T=7.9$ |
---|
| 166 | MeV to $T=T_{2l}=2 \mbox{ MeV}$. |
---|
| 167 | |
---|
| 168 | \noindent |
---|
| 169 | Since $M \gg m_e$, |
---|
| 170 | the ionisation loss does not depend on the hadron |
---|
| 171 | mass, but on its velocity. |
---|
| 172 | Therefore the energy loss of a charged hadron |
---|
| 173 | with kinetic energy, $T$, is the same as |
---|
| 174 | the energy loss of a proton with the same velocity. The corresponding |
---|
| 175 | kinetic energy of the proton $T_p$ is |
---|
| 176 | \begin{equation} |
---|
| 177 | T_{proton} = \frac{M_{proton}}{M} \ T. |
---|
| 178 | \label{hlei.e} |
---|
| 179 | \end{equation} |
---|
| 180 | |
---|
| 181 | \noindent |
---|
| 182 | At initialisation stage of Geant4 the $dE/dx$ tables |
---|
| 183 | and range tables for all materials |
---|
| 184 | are calculated only for protons and antiprotons. |
---|
| 185 | During run time the energy loss and the range of any hadron or ion are |
---|
| 186 | recalculated using the scaling relation (\ref{hlei.e}). |
---|
| 187 | |
---|
| 188 | |
---|
| 189 | \subsection{Barkas and Bloch effects} |
---|
| 190 | |
---|
| 191 | |
---|
| 192 | The accuracy of |
---|
| 193 | the Bethe-Bloch stopping power formula |
---|
| 194 | (\ref{hlei.e}) can be improved |
---|
| 195 | if the higher order terms are taken into account: |
---|
| 196 | \begin{equation} |
---|
| 197 | -\frac{dE}{dx} = K \frac{Z^2_{h}}{\beta^2}(L_0 +Z_{h}L_1+Z^2_{h}L_2), |
---|
| 198 | \label{hlei.f} |
---|
| 199 | \end{equation} |
---|
| 200 | where $L_1$ is the Barkas term \cite{hlei.bark56}, |
---|
| 201 | describing the difference |
---|
| 202 | between ionisation of positively and negatively charged particles, and |
---|
| 203 | $L_2$ is the Bloch term. |
---|
| 204 | |
---|
| 205 | The Barkas effect for kinetic energy of |
---|
| 206 | protons or antiprotons greater than $500 keV$ can be described as |
---|
| 207 | \cite{hlei.arb72}: |
---|
| 208 | \begin{equation} |
---|
| 209 | L_1=\frac{F\left ( b / \sqrt{x}\right ) }{\sqrt{Z x^3}}, \,\,\, |
---|
| 210 | x=\frac{\beta^2c^2}{Zv_0^2},\,\,\, |
---|
| 211 | b=0.8 Z^{\frac 16}\left( 1+6.02Z^{-1.19}\right), |
---|
| 212 | \label{hlei.g} |
---|
| 213 | \end{equation} |
---|
| 214 | where |
---|
| 215 | $v_0$ is the Bohr velocity (corresponding to proton energy $T_p=25 keV$), and |
---|
| 216 | the function $F$ is tabulated according to \cite{hlei.arb72}. |
---|
| 217 | |
---|
| 218 | The Bloch term \cite{hlei.bloch} |
---|
| 219 | can be expressed in the following way: |
---|
| 220 | \begin{equation} |
---|
| 221 | Z^2_{h}L_2 = - y^2 \sum^{\inf}_{j=1} \frac{1}{j(j^2 + y^2)},\,\,\, |
---|
| 222 | y=\frac{Z_{h}}{137\beta}. |
---|
| 223 | \label{hlei.h} |
---|
| 224 | \end{equation} |
---|
| 225 | Note, that for $y \ll 1$ the simplified expression |
---|
| 226 | $Z^2_{h}L_2=-1.202y^2$ can be used. |
---|
| 227 | |
---|
| 228 | Both the Barkas and Bloch terms break scaling of ionisation losses |
---|
| 229 | if the absolute value of particle charge is different from unity, |
---|
| 230 | because the particle charge $Z_h$ is not factorised |
---|
| 231 | in the formula (\ref{hlei.f}). |
---|
| 232 | To take these terms into account correction is made at |
---|
| 233 | each step of the simulation for the value of $dE/dx$ |
---|
| 234 | re-calculated from the proton or antiproton tables. |
---|
| 235 | There is the possibility to switch off the calculation |
---|
| 236 | of these terms. |
---|
| 237 | |
---|
| 238 | \subsection{Energy losses of slow positive hadrons} |
---|
| 239 | |
---|
| 240 | At low energies the total energy loss is usually described |
---|
| 241 | in terms of {\it electronic stopping power} $S_e = - dE/dx$. |
---|
| 242 | For charged hadron with velocity $\beta < 0.05$ (corresponding |
---|
| 243 | to 1~MeV for protons), formula (\ref{hlei.d}) becomes inaccurate. |
---|
| 244 | In this case the velocity of the incident |
---|
| 245 | hadron is comparable to the velocity |
---|
| 246 | of atomic electrons. At very low energies, when |
---|
| 247 | $\beta < 0.01$, the model of a free electron gas \cite{hlei.Lindhard} |
---|
| 248 | predicts the stopping power to be proportional to |
---|
| 249 | the hadron velocity, |
---|
| 250 | but it is not as accurate as the Bethe-Bloch formalism. |
---|
| 251 | The intermediate region $0.01 < \beta < 0.05$ is not covered |
---|
| 252 | by precise theories. In this energy |
---|
| 253 | interval the Bragg peak of ionisation loss occurs. |
---|
| 254 | |
---|
| 255 | To simulate slow proton energy loss |
---|
| 256 | the following |
---|
| 257 | parametrisation from the review \cite{hlei.Ziegler771} was implemented: |
---|
| 258 | \begin{eqnarray} |
---|
| 259 | S_e & = & A_1E^{1/2}, \; \; \; \; \; \; \; \; \hspace{46mm} |
---|
| 260 | 1~keV < T_p < 10~keV, \nonumber \\ |
---|
| 261 | S_e & = & \frac{S_{low}S_{high}}{S_{low}+S_{high}}, \hspace{46mm} |
---|
| 262 | 10~keV < T_p < 1~MeV, \nonumber \\ |
---|
| 263 | S_{low} & = & A_2E^{0.45}, \nonumber\\ |
---|
| 264 | S_{high} & = & \frac{A_3}{E}\ln{\left(1 + \frac{A_4}{E} + A_5E \right)}, |
---|
| 265 | \nonumber \\ |
---|
| 266 | S_e & = & \frac{A_6}{\beta^2} \left [\ln{\frac{A_7\beta^2}{1-\beta^2}} |
---|
| 267 | -\beta^2 - \sum^{4}_{i=0} A_{i+8}(\ln{E})^i \right ], |
---|
| 268 | \; 1~MeV < T_p < 100~MeV, \nonumber \\ |
---|
| 269 | \label{hlei.i} |
---|
| 270 | \end{eqnarray} |
---|
| 271 | where $S_e$ is the stopping power |
---|
| 272 | in $[eV/10^{15}atoms/cm^2]$, $E=T_p/M_p [keV/amu]$, $A_i$ are twelve |
---|
| 273 | fitting parameters found individually for each atom for |
---|
| 274 | atomic numbers from 1 to 92. |
---|
| 275 | This parametrisation is used |
---|
| 276 | in the interval of proton kinetic energy: |
---|
| 277 | \begin{equation} |
---|
| 278 | T_1 < T_p < T_2, |
---|
| 279 | \label{hlei.j} |
---|
| 280 | \end{equation} |
---|
| 281 | where $T_1 = 1~keV$ is the minimal kinetic energy of protons |
---|
| 282 | in the tables of Ref.\cite{hlei.Ziegler771}, |
---|
| 283 | $T_2$ is an arbitrary value |
---|
| 284 | between 2~MeV and 100~MeV, since in this range |
---|
| 285 | both the parametrisation (\ref{hlei.i}) |
---|
| 286 | and the Bethe-Bloch formula (\ref{hlei.e}) |
---|
| 287 | have practically the same accuracy and |
---|
| 288 | are close to each other. |
---|
| 289 | Currently the value $T_2 = 2~MeV$ is chosen. |
---|
| 290 | |
---|
| 291 | |
---|
| 292 | To avoid problems in computation and |
---|
| 293 | to provide a continuous $dE/dx$ function, the factor |
---|
| 294 | \begin{equation} |
---|
| 295 | F = \left (1 + B\frac{T_2}{T_p} \right ) |
---|
| 296 | \label{hlei.r} |
---|
| 297 | \end{equation} |
---|
| 298 | is multiplied by the value of $dE/dx$ for $T_p > T_{2}$. |
---|
| 299 | The parameter $B$ is determined for each element of the material |
---|
| 300 | in order to |
---|
| 301 | provide continuity at $T_p=T_2$. The value of $B$ for all |
---|
| 302 | atoms is less than 0.01. For the |
---|
| 303 | simulation of the stopping power of very slow protons the model of a |
---|
| 304 | free electron gas \cite{hlei.Lindhard} is used: |
---|
| 305 | \begin{equation} |
---|
| 306 | S_e = A \sqrt{T_p}, \; \; T_p < T_{1}. |
---|
| 307 | \label{hlei.k} |
---|
| 308 | \end{equation} |
---|
| 309 | The parameter $A$ is defined for each atom |
---|
| 310 | by requiring the stopping power to be continuous |
---|
| 311 | at $T_p=T_{1}$. Currently the value used is $T_1=1~keV$. |
---|
| 312 | |
---|
| 313 | Note that |
---|
| 314 | if the cut kinetic energy is small ($T_c < T_{max}$), then the average |
---|
| 315 | energy deposit giving |
---|
| 316 | rise to $\delta$-electron production (\ref{hlei.del}) |
---|
| 317 | is subtracted from the |
---|
| 318 | value of the stopping power $S_e$, which is calculated by formula |
---|
| 319 | (\ref{hlei.i}). |
---|
| 320 | |
---|
| 321 | |
---|
| 322 | Alternative parametrisations of proton energy loss |
---|
| 323 | are also available within Geant4 (Table \ref{hlei.tab0}). |
---|
| 324 | The parameterisation formulae |
---|
| 325 | in Ref.\cite{hlei.ICRU49} are the same |
---|
| 326 | as in Ref.(\cite{hlei.Ziegler771}) |
---|
| 327 | for the kinetic |
---|
| 328 | energy of protons $T_p < 1~MeV$, but |
---|
| 329 | the values of the parameters are different. |
---|
| 330 | The type of parameterisation is optional and |
---|
| 331 | can be chosen by the user separately for each particle |
---|
| 332 | at the initialisation stage of Geant4. |
---|
| 333 | |
---|
| 334 | |
---|
| 335 | \begin{table*} |
---|
| 336 | \caption{The list of parameterisations available.} |
---|
| 337 | %\vspace {2pt} |
---|
| 338 | \label{hlei.tab0} |
---|
| 339 | \begin{center} |
---|
| 340 | \begin{tabular}{|l|l|l|} |
---|
| 341 | \hline |
---|
| 342 | Name & Particle & Source \\ |
---|
| 343 | \hline |
---|
| 344 | {\bf Ziegler1977p} & proton & J.F.~Ziegler parameterisation |
---|
| 345 | \cite{hlei.Ziegler771} \\ |
---|
| 346 | {\bf Ziegler1977He} & $He^4$ & J.F.~Ziegler parameterisation |
---|
| 347 | \cite{hlei.Ziegler774}\\ |
---|
| 348 | {\bf Ziegler1985p} & proton & TRIM'85 parameterisation \cite{hlei.Ziegler85} \\ |
---|
| 349 | {\bf ICRU\_R49p} & proton & ICRU parameterisation \cite{hlei.ICRU49} \\ |
---|
| 350 | {\bf ICRU\_R49He} & $He^4$ & ICRU parameterisation \cite{hlei.ICRU49} \\ |
---|
| 351 | \hline |
---|
| 352 | \end{tabular} |
---|
| 353 | \end{center} |
---|
| 354 | \end{table*} |
---|
| 355 | |
---|
| 356 | |
---|
| 357 | \subsection{Energy loss of alpha particles} |
---|
| 358 | |
---|
| 359 | The accuracy of the data for the ionisation losses of $\alpha$-particles |
---|
| 360 | in all elements \cite{hlei.ICRU49,hlei.Ziegler774} |
---|
| 361 | is comparable to the accuracy |
---|
| 362 | of the data for proton energy loss \cite{hlei.Ziegler771,hlei.ICRU49}. |
---|
| 363 | In the GEANT4 energy loss model for $\alpha$-particles |
---|
| 364 | the Bethe-Bloch formula is used for kinetic energy |
---|
| 365 | $T > T_2$, where $T_2$ is the arbitrary parameter, currently set to $8~MeV$. |
---|
| 366 | For lower energies a parameterisation is performed. |
---|
| 367 | In the energy range of the Bragg peak, |
---|
| 368 | $1~keV < T < 10~MeV$, the |
---|
| 369 | parameterisation is: |
---|
| 370 | \begin{eqnarray} |
---|
| 371 | S_e & = & \frac{S_{low}S_{high}}{S_{low}+S_{high}}, \nonumber \\ |
---|
| 372 | S_{low} & = & A_1T^{A_2}, \nonumber\\ |
---|
| 373 | S_{high} & = & \frac{A_3}{T}\ln{\left(1 + \frac{A_4}{T} + A_5T \right)}, |
---|
| 374 | \nonumber \\ |
---|
| 375 | \label{hlei.l} |
---|
| 376 | \end{eqnarray} |
---|
| 377 | where $S_e$ is the electronic stopping power |
---|
| 378 | in $[eV/10^{15}atoms/cm^2]$, $T$ is the kinetic energy of $\alpha$-particles in |
---|
| 379 | $MeV$, |
---|
| 380 | $A_i$ are the five fitting |
---|
| 381 | parameters fitted individually for each atom for |
---|
| 382 | atomic numbers from 1 to 92. |
---|
| 383 | |
---|
| 384 | For higher energies $T > 10~MeV$, another |
---|
| 385 | parametrisation \cite{hlei.Ziegler774} is applied |
---|
| 386 | \begin{equation} |
---|
| 387 | S_e= exp \left(A_6+A_7E+A_8E^2+A_9E^3 \right ), \; E=ln(1/T). |
---|
| 388 | \label{hlei.m} |
---|
| 389 | \end{equation} |
---|
| 390 | To ensure a continuous $dE/dx$ function from the energy range of the |
---|
| 391 | Bethe-Bloch formula to the energy range of the parameterisation, the factor |
---|
| 392 | \begin{equation} |
---|
| 393 | F = \left (1 + B\frac{T_2}{T} \right ) |
---|
| 394 | \label{hlei.n} |
---|
| 395 | \end{equation} |
---|
| 396 | is multiplied by the value of $S_e$ as predicted by the Bethe-Bloch formula |
---|
| 397 | for $T > T_{2}$. |
---|
| 398 | The parameter $B$ is determined for each element of the material in order to |
---|
| 399 | ensure continuity at $T_p=T_2$. The value of $B$ for different atoms is |
---|
| 400 | usually less than 0.01. |
---|
| 401 | |
---|
| 402 | For kinetic energies of $\alpha$-particles $T < 1~keV$ the model |
---|
| 403 | of free electron gas \cite{hlei.Lindhard} is used |
---|
| 404 | \begin{equation} |
---|
| 405 | S_e = A \sqrt{T}, |
---|
| 406 | \label{hlei.o} |
---|
| 407 | \end{equation} |
---|
| 408 | The parameter $A$ is defined for each atom by requiring the stopping power to be |
---|
| 409 | continuous at $T=1~keV$. |
---|
| 410 | |
---|
| 411 | |
---|
| 412 | \subsection{Effective charge of ions} |
---|
| 413 | |
---|
| 414 | For hadrons or ions |
---|
| 415 | the scaling relation can be written as |
---|
| 416 | \begin{equation} |
---|
| 417 | S_{ei}(T) = Z_{eff}^2\cdot S_{ep}(T_p), |
---|
| 418 | \label{hlei.sei} |
---|
| 419 | \end{equation} |
---|
| 420 | where $S_{ei}$ is the ion stopping power, |
---|
| 421 | $S_{ep}$ is the proton stopping power at the energy scaled |
---|
| 422 | according (\ref{hlei.e}), and |
---|
| 423 | $Z_{eff}$ is effective charge of the particle, which has to be used in |
---|
| 424 | all expressions in place of $Z_h$. |
---|
| 425 | For fast particles it is equal to the particle charge $Z_h$, |
---|
| 426 | but for slow ions it differs significantly because |
---|
| 427 | a slow ion |
---|
| 428 | picks up electrons from the medium. |
---|
| 429 | The ion effective charge is expressed via |
---|
| 430 | the ion charge $Z_h$ and the |
---|
| 431 | fractional effective charge of ion $\gamma_i$: |
---|
| 432 | \begin{equation} |
---|
| 433 | Z_{eff} = \gamma_i Z_h. |
---|
| 434 | \label{hlei.p} |
---|
| 435 | \end{equation} |
---|
| 436 | |
---|
| 437 | For helium ions |
---|
| 438 | fractional effective charge |
---|
| 439 | is parameterised for all |
---|
| 440 | elements with good accuracy \cite{hlei.Ziegler85} according to: |
---|
| 441 | \begin{eqnarray} |
---|
| 442 | (\gamma_{He})^2 & = &\left (1-\exp\left [-\sum_{j=0}^5{C_jQ^j}\right ]\right) |
---|
| 443 | \left ( 1 + \frac{ 7 + 0.05 Z }{1000} \exp( -(7.6-Q)^2 ) \right )^2, |
---|
| 444 | \nonumber \\ |
---|
| 445 | Q & = & \max ( 0, \ln T_p) , |
---|
| 446 | \label{hlei.q} |
---|
| 447 | \end{eqnarray} |
---|
| 448 | where the coefficients $C_j$ are the same for all elements, and the |
---|
| 449 | helium ion kinetic energy is in $keV/amu$. |
---|
| 450 | |
---|
| 451 | |
---|
| 452 | The following expression is used for heavy ions \cite{hlei.BK}: |
---|
| 453 | \begin{equation} |
---|
| 454 | \gamma_i = \left ( q + \frac{1-q}{2} \left (\frac{v_0}{v_F} \right )^2 |
---|
| 455 | \ln {\left ( 1 + \Lambda^2 \right )} \right ) |
---|
| 456 | \left ( 1 + \frac{(0.18+0.0015Z)\exp(-(7.6-Q)^2)}{Z_i^2} \right ), |
---|
| 457 | \label{hlei.s} |
---|
| 458 | \end{equation} |
---|
| 459 | where $q$ is |
---|
| 460 | the fractional average charge of the ion, |
---|
| 461 | $v_0$ is the Bohr velocity, |
---|
| 462 | $v_F$ is the Fermi velocity of |
---|
| 463 | the electrons in the target medium, and $\Lambda$ is |
---|
| 464 | the term taking into account the screening effect. In Ref.~\cite{hlei.BK}, |
---|
| 465 | $\Lambda$ is estimated to be: |
---|
| 466 | \begin{equation} |
---|
| 467 | \Lambda = 10 \frac{v_F}{v_0} \frac{(1-q)^{2/3}}{Z_i^{1/3}(6+q)}. |
---|
| 468 | \label{hlei.t} |
---|
| 469 | \end{equation} |
---|
| 470 | The Fermi velocity of the medium is of the same order as the Bohr velocity, and |
---|
| 471 | its exact value depends on the detailed electronic structure of the medium. |
---|
| 472 | Experimental data on the Fermi velocity are taken from |
---|
| 473 | Ref.\cite{hlei.Ziegler85}. |
---|
| 474 | The expression for the fractional average charge of the ion is the following: |
---|
| 475 | \begin{equation} |
---|
| 476 | q = [1 -\exp(0.803y^{0.3}-1.3167y^{0.6}-0.38157y-0.008983y^2)], |
---|
| 477 | \label{hlei.u} |
---|
| 478 | \end{equation} |
---|
| 479 | where $y$ is a parameter that depends on the ion velocity $v_i$ |
---|
| 480 | \begin{equation} |
---|
| 481 | y = \frac{v_i}{v_0Z^{2/3}} \left ( 1 +\frac {v_F^2}{5v_i^2} \right ). |
---|
| 482 | \label{hlei.v} |
---|
| 483 | \end{equation} |
---|
| 484 | |
---|
| 485 | The parametrisation described in this chapter is only valid |
---|
| 486 | if the reduced kinetic energy of the ion is higher than the lower limit |
---|
| 487 | of the energy: |
---|
| 488 | \begin{equation} |
---|
| 489 | T_p > \max \left ( 3.25~keV, \frac{25~keV}{Z^{2/3}} \right ). |
---|
| 490 | \label{hlei.x} |
---|
| 491 | \end{equation} |
---|
| 492 | If the ion energy is lower, then the free electron gas model (\ref{hlei.o}) |
---|
| 493 | is used to calculate the stopping power. |
---|
| 494 | |
---|
| 495 | |
---|
| 496 | \subsection{Energy losses of slow negative particles} |
---|
| 497 | |
---|
| 498 | At low energies, e.g. below a few MeV for protons/antiprotons, the |
---|
| 499 | Bethe-Bloch formula is no longer accurate in describing the energy |
---|
| 500 | loss of charged hadrons and higher $Z$ terms should be taken in |
---|
| 501 | account. |
---|
| 502 | Odd terms in $Z$ lead to a significant difference between energy |
---|
| 503 | loss of positively and negatively charged particles. |
---|
| 504 | The energy loss of negative hadrons is scaled from that |
---|
| 505 | of antiprotons. |
---|
| 506 | The antiproton energy loss is calculated in the following way: |
---|
| 507 | \begin{itemize} |
---|
| 508 | \item |
---|
| 509 | if the material is elemental, the quantum harmonic oscillator model is used, as |
---|
| 510 | described in \cite{hlei.sigmund} and references therein. |
---|
| 511 | The lower limit of applicability of the model is chosen for all |
---|
| 512 | materials at $50~keV$. Below this value stopping power is set to constant |
---|
| 513 | equal to the $dE/dx$ at $50~keV$. |
---|
| 514 | \item |
---|
| 515 | if the material is not elemental, the energy loss is calculated |
---|
| 516 | down to $500~keV$ using the Barkas correction (\ref{hlei.n}) |
---|
| 517 | and at lower energies fitting the |
---|
| 518 | proton energy loss curve. |
---|
| 519 | \end{itemize} |
---|
| 520 | |
---|
| 521 | |
---|
| 522 | |
---|
| 523 | |
---|
| 524 | \subsection{Energy losses of hadrons in compounds} |
---|
| 525 | |
---|
| 526 | To obtain energy losses in |
---|
| 527 | a mixture or compound, |
---|
| 528 | the absorber can be thought of as made up of thin |
---|
| 529 | layers of pure elements with weights proportional to the electron |
---|
| 530 | density of the element in the absorber (Bragg's rule): |
---|
| 531 | \begin{equation} |
---|
| 532 | \frac{dE}{dx}=\sum_i{\left (\frac{dE}{dx} \right )_i}, |
---|
| 533 | \label{hlei.y} |
---|
| 534 | \end{equation} |
---|
| 535 | where the sum is taken over all elements of the absorber, $i$ is |
---|
| 536 | the number of the element, |
---|
| 537 | $(\frac{dE}{dx})_i$ is energy loss in the pure $i$-th element. |
---|
| 538 | |
---|
| 539 | Bragg's rule is very accurate for relativistic particles |
---|
| 540 | when the interaction of electrons with a nucleus is negligible. |
---|
| 541 | But at low energies the accuracy of Bragg's rule is limited |
---|
| 542 | because the energy loss to the electrons in any material |
---|
| 543 | depends on the detailed orbital |
---|
| 544 | and excitation structure of the material. |
---|
| 545 | In the description of Geant4 materials there is a special |
---|
| 546 | attribute: the chemical formula. |
---|
| 547 | It is used in the |
---|
| 548 | following way: |
---|
| 549 | \begin{itemize} |
---|
| 550 | \item |
---|
| 551 | if the data on the stopping power for a compound |
---|
| 552 | as a function of the proton kinetic energy |
---|
| 553 | is available (Table \ref{hlei.tab1}), then the |
---|
| 554 | direct parametrisation of the data for this material is performed; |
---|
| 555 | \item |
---|
| 556 | if the data on the stopping power for a compound |
---|
| 557 | is available for only one incident |
---|
| 558 | energy (Table \ref{hlei.tab2}), then |
---|
| 559 | the computation is |
---|
| 560 | performed based on Bragg's rule and the chemical |
---|
| 561 | factor for the compound is taken into account; |
---|
| 562 | \item |
---|
| 563 | if there are no data for the compound, the computation is |
---|
| 564 | performed based on Bragg's rule. |
---|
| 565 | \end{itemize} |
---|
| 566 | \noindent |
---|
| 567 | In the review \cite{hlei.Ziegler88} the parametrisation stopping |
---|
| 568 | power data are presented as |
---|
| 569 | \begin{equation} |
---|
| 570 | S_e(T_p)= S_{Bragg}(T_p)\left [1 + \frac{f(T_p)}{f(125~keV)} |
---|
| 571 | \left (\frac{S_{exp}(125~keV)}{S_{Bragg}(125~keV)}-1 \right ) \right ], |
---|
| 572 | \label{hlei.z} |
---|
| 573 | \end{equation} |
---|
| 574 | where $S_{exp}(125~keV)$ is the experimental value of the energy loss |
---|
| 575 | for the compound |
---|
| 576 | for $125~keV$ protons or the |
---|
| 577 | reduced experimental value for He ions, $S_{Bragg}(T_p)$ is |
---|
| 578 | a value of energy loss calculated according to Bragg's |
---|
| 579 | rule, and $f(T_p)$ is a universal function, which describes |
---|
| 580 | the disappearance of deviations from Bragg's rule |
---|
| 581 | for higher kinetic energies according to: |
---|
| 582 | \begin{equation} |
---|
| 583 | f(T_p)=\frac{1}{1+\exp \left [1.48(\frac{\beta(T_p)} |
---|
| 584 | {\beta(25~keV)}-7.0) \right ]}, |
---|
| 585 | \label{hlei.fun} |
---|
| 586 | \end{equation} |
---|
| 587 | where $\beta(T_p)$ is the relative velocity of the proton with |
---|
| 588 | kinetic energy $T_p$. |
---|
| 589 | |
---|
| 590 | |
---|
| 591 | \begin{table*} |
---|
| 592 | \caption{The list of chemical formulae of compounds for which |
---|
| 593 | parametrisation of stopping power as a function |
---|
| 594 | of kinetic energy is in Ref.\cite{hlei.ICRU49}.} |
---|
| 595 | %\vspace {2pt} |
---|
| 596 | \label{hlei.tab1} |
---|
| 597 | \begin{center} |
---|
| 598 | \begin{tabular}{|l|l|} |
---|
| 599 | \hline |
---|
| 600 | Number & Chemical formula \\ |
---|
| 601 | \hline |
---|
| 602 | 1. & AlO \\ |
---|
| 603 | 2. & C\_2O \\ |
---|
| 604 | 3. & CH\_4 \\ |
---|
| 605 | 4. & (C\_2H\_4)\_N-Polyethylene \\ |
---|
| 606 | 5. & (C\_2H\_4)\_N-Polypropylene \\ |
---|
| 607 | 6. & (C\_8H\_8)\_N \\ |
---|
| 608 | 7. & C\_3H\_8 \\ |
---|
| 609 | 8. & SiO\_2 \\ |
---|
| 610 | 9. & H\_2O \\ |
---|
| 611 | 10. & H\_2O-Gas \\ |
---|
| 612 | 11. & Graphite \\ |
---|
| 613 | \hline |
---|
| 614 | \end{tabular} |
---|
| 615 | \end{center} |
---|
| 616 | \end{table*} |
---|
| 617 | |
---|
| 618 | \begin{table*} |
---|
| 619 | \caption{The list of chemical formulae of compounds for which |
---|
| 620 | the {\it chemical factor} is calculated from the data |
---|
| 621 | of Ref.\cite{hlei.Ziegler88}.} |
---|
| 622 | %\vspace {2pt} |
---|
| 623 | \label{hlei.tab2} |
---|
| 624 | \begin{center} |
---|
| 625 | \begin{tabular}{|l|l|l|l|} |
---|
| 626 | \hline |
---|
| 627 | Number & Chemical formula & Number & Chemical formula \\ |
---|
| 628 | \hline |
---|
| 629 | 1. & H\_2O & 28. & C\_2H\_6 \\ |
---|
| 630 | 2. & C\_2H\_4O & 29. & C\_2F\_6 \\ |
---|
| 631 | 3. & C\_3H\_6O & 30. & C\_2H\_6O \\ |
---|
| 632 | 4. & C\_2H\_2 & 31. & C\_3H\_6O \\ |
---|
| 633 | 5. & C\_H\_3OH & 32. & C\_4H\_10O \\ |
---|
| 634 | 6. & C\_2H\_5OH & 33. & C\_2H\_4 \\ |
---|
| 635 | 7. & C\_3H\_7OH & 34. & C\_2H\_4O \\ |
---|
| 636 | 8. & C\_3H\_4 & 35. & C\_2H\_4S \\ |
---|
| 637 | 9. & NH\_3 & 36. & SH\_2 \\ |
---|
| 638 | 10. & C\_14H\_10 & 37. & CH\_4 \\ |
---|
| 639 | 11. & C\_6H\_6 & 38. & CCLF\_3 \\ |
---|
| 640 | 12. & C\_4H\_10 & 39. & CCl\_2F\_2 \\ |
---|
| 641 | 13. & C\_4H\_6 & 40. & CHCl\_2F \\ |
---|
| 642 | 14. & C\_4H\_8O & 41. & (CH\_3)\_2S \\ |
---|
| 643 | 15. & CCl\_4 & 42. & N\_2O \\ |
---|
| 644 | 16. & CF\_4 & 43. & C\_5H\_10O \\ |
---|
| 645 | 17. & C\_6H\_8 & 44. & C\_8H\_6 \\ |
---|
| 646 | 18. & C\_6H\_12 & 45. & (CH\_2)\_N \\ |
---|
| 647 | 19. & C\_6H\_10O & 46. & (C\_3H\_6)\_N \\ |
---|
| 648 | 20. & C\_6H\_10 & 47. & (C\_8H\_8)\_N \\ |
---|
| 649 | 21. & C\_8H\_16 & 48. & C\_3H\_8 \\ |
---|
| 650 | 22. & C\_5H\_10 & 49. & C\_3H\_6-Propylene \\ |
---|
| 651 | 23. & C\_5H\_8 & 50. & C\_3H\_6O \\ |
---|
| 652 | 24. & C\_3H\_6-Cyclopropane & 51. & C\_3H\_6S \\ |
---|
| 653 | 25. & C\_2H\_4F\_2 & 52. & C\_4H\_4S \\ |
---|
| 654 | 26. & C\_2H\_2F\_2 & 53. & C\_7H\_8 \\ |
---|
| 655 | 27. & C\_4H\_8O\_2 & & \\ |
---|
| 656 | \hline |
---|
| 657 | \end{tabular} |
---|
| 658 | \end{center} |
---|
| 659 | \end{table*} |
---|
| 660 | |
---|
| 661 | |
---|
| 662 | \subsection{Nuclear stopping powers} |
---|
| 663 | |
---|
| 664 | Low energy ions transfer their energy not only to electrons of a medium |
---|
| 665 | but also to the nuclei of the medium due to the elastic Coulomb |
---|
| 666 | collisions. |
---|
| 667 | This contribution to the energy loss is called {\it |
---|
| 668 | nuclear stopping power}. |
---|
| 669 | It is parametrised \cite{hlei.Ziegler774,hlei.Ziegler85,hlei.ICRU49} |
---|
| 670 | using a universal parameterisation for reduced |
---|
| 671 | ion energy, $\epsilon$, which depends on ion parameters and on |
---|
| 672 | the charge, $Z_t$, and the mass, $M_t$, of the target nucleus: |
---|
| 673 | \begin{equation} |
---|
| 674 | \epsilon = \frac{32.536TM_t}{Z_{eff}Z_t(M+M_t) |
---|
| 675 | \sqrt{Z_{eff}^{0.23}+Z_t^{0.23}}}. |
---|
| 676 | \label{hlei.ep} |
---|
| 677 | \end{equation} |
---|
| 678 | The universal reduced nuclear stopping power, $s_n$, is determined |
---|
| 679 | by J.~Moliere in the framework of Thomas-Fermi potential \cite{hlei.mol}. |
---|
| 680 | The corresponding tabulation from Ref.\cite{hlei.ICRU49} |
---|
| 681 | is implemented. |
---|
| 682 | To transform the value of |
---|
| 683 | nuclear stopping power from reduced units to |
---|
| 684 | $[eV/10^{15}atoms/cm^2]$ the following formula is used: |
---|
| 685 | \begin{equation} |
---|
| 686 | S_n = s_n \frac{8.462Z_iZ_tM_i}{(M_i+M_t)\sqrt{Z_i^{0.23}+Z_t^{0.23}}}. |
---|
| 687 | \label{hlei.re} |
---|
| 688 | \end{equation} |
---|
| 689 | The effect of nuclear stopping power is very small at high energies, but |
---|
| 690 | it is of the same order of magnitude as electronic stopping power |
---|
| 691 | for very slow ions (e.g. for protons, $T_p < 1 keV$). |
---|
| 692 | |
---|
| 693 | \subsection{Fluctuations of energy losses of hadrons} |
---|
| 694 | |
---|
| 695 | The total continuous energy loss of charged particles is a stochastic |
---|
| 696 | quantity with a distribution described in terms of a straggling function. |
---|
| 697 | The straggling is partially taken into account by the simulation |
---|
| 698 | of energy loss by the production of $\delta$-electrons with energy |
---|
| 699 | $T > T_c$. However, continuous energy loss also has fluctuations. Hence |
---|
| 700 | in the current GEANT4 implementation two different models of fluctuations |
---|
| 701 | are applied depending on the value of the parameter $\kappa$ which is the |
---|
| 702 | lower limit of the number of interactions of the particle in the step. |
---|
| 703 | The default value chosen is $\kappa = 10$. To select a model for thick |
---|
| 704 | absorbers the following boundary conditions are used: |
---|
| 705 | \begin{equation} |
---|
| 706 | \Delta E > T_c\kappa)\;\; or \;\; T_c < I\kappa, |
---|
| 707 | \label{le_cond} |
---|
| 708 | \end{equation} |
---|
| 709 | where $\Delta E$ is the mean continuous energy loss in a track segment of |
---|
| 710 | length $s$, $T_c$ is the cut kinetic energy of $\delta$-electrons, and $I$ |
---|
| 711 | is the average ionisation potential of the atom. |
---|
| 712 | |
---|
| 713 | For long path lengths the straggling function |
---|
| 714 | approaches the Gaussian distribution with Bohr's variance \cite{hlei.ICRU49}: |
---|
| 715 | \begin{equation} |
---|
| 716 | \Omega^2 = K N_{el}\frac{Z_h^2}{\beta^2} T_c s f |
---|
| 717 | \left(1 - \frac{\beta^2}{2} \right), |
---|
| 718 | \label{sig} |
---|
| 719 | \end{equation} |
---|
| 720 | where $f$ is a screening factor, which is equal to unity for fast particles, |
---|
| 721 | whereas for slow positively charged |
---|
| 722 | ions with $\beta^2 < 3Z (v_0/c)^2$ |
---|
| 723 | $f = a + b/Z^2_{eff}$, where parameters $a$ and $b$ |
---|
| 724 | are parametrised for all atoms \cite{hlei.Yang,hlei.Chu}. |
---|
| 725 | |
---|
| 726 | For short path lengths, when the condition \ref{le_cond} is not satisfied, |
---|
| 727 | the model described in the charter \ref{gen_fluctuations} is applied. |
---|
| 728 | |
---|
| 729 | \subsection{Sampling} |
---|
| 730 | |
---|
| 731 | At each step for a charged hadron or ion in an absorber, |
---|
| 732 | the step limit is calculated using range tables |
---|
| 733 | for protons or antiprotons depending on the particle charge. |
---|
| 734 | If the reduced particle energy $T_p < T_2$ the step limit is |
---|
| 735 | forced to be not longer than $\alpha R(T_2)$, where $R(T_2)$ |
---|
| 736 | is the range of the particle with the reduced energy $T_2$, |
---|
| 737 | $\alpha$ is an arbitrary coefficient, which is currently set to 0.05 |
---|
| 738 | in order to provide at least 20 steps for particles |
---|
| 739 | in the Bragg peak energy range. |
---|
| 740 | \noindent |
---|
| 741 | In each step continuous energy loss of the particle |
---|
| 742 | is calculated using loss tables for protons or antiprotons |
---|
| 743 | for $T_p > T_2$. For lower energies, continuous energy loss |
---|
| 744 | is calculated using the effective charge approach, chemical |
---|
| 745 | factors, and nuclear stopping powers. |
---|
| 746 | \noindent |
---|
| 747 | If the step of the particle is limited by the ionisation process |
---|
| 748 | the sampling of $\delta$-electron production is performed. |
---|
| 749 | (A short overview of the method is given in Chapter \ref{secmessel}.) \\ |
---|
| 750 | Apart from the normalisation, the cross-section |
---|
| 751 | (\ref{hlei.bbb}) can be written as : |
---|
| 752 | \begin{eqnarray} |
---|
| 753 | \frac{d\sigma}{dT} \sim f(T) \ g(T) &with& T \in [T_{c}, \ T_{max}] |
---|
| 754 | \end{eqnarray} |
---|
| 755 | with : |
---|
| 756 | \begin{eqnarray*} |
---|
| 757 | f(T) &=& \left(\frac{1}{T_{c}} -\frac{1}{T_{max}}\right) |
---|
| 758 | \frac{1}{T^2} \\ |
---|
| 759 | g(T) &=& 1 - \beta^2\frac{T}{T_{max}} + S(T), |
---|
| 760 | \end{eqnarray*} |
---|
| 761 | where $S(T)$ is a spin dependent term (\ref{hlei.bbb}). |
---|
| 762 | For a spin-0 particle this term is zero, for |
---|
| 763 | a spin-$\frac{1}{2}$ particle $S(T)=T^2/2E^2$, |
---|
| 764 | whilst for spin-1 the expression is more complicated. |
---|
| 765 | \\ |
---|
| 766 | The energy, $T$, is sampled by : |
---|
| 767 | \begin{enumerate} |
---|
| 768 | \item Sample $T$ from $f(T)$. |
---|
| 769 | \item Calculate the rejection function $g(T)$ and accept the |
---|
| 770 | sampled $T$ with a probability of $g(T)$. |
---|
| 771 | \end{enumerate} |
---|
| 772 | After the successful sampling of the energy, the polar angles of the |
---|
| 773 | emitted electron are generated with respect to the direction of the |
---|
| 774 | incident particle. The azimuthal angle, $\phi$, is generated isotropically; |
---|
| 775 | the polar angle $\theta$ is calculated from the energy momentum conservation. |
---|
| 776 | This information is used to calculate the energy and momentum of both |
---|
| 777 | particles and to transform them into the {\it global} coordinate system. |
---|
| 778 | |
---|
| 779 | \subsection{PIXE} |
---|
| 780 | PIXE is simulated by calculating cross-sections according to |
---|
| 781 | \cite{hlei.Gryzinski1} and \cite{hlei.Gryzinski2} to identify the primary |
---|
| 782 | ionised shell, and generating the subsequent atomic relaxation as described |
---|
| 783 | in \ref{relax}. Sampling of excitations is carried out for both the |
---|
| 784 | continuous and the discrete parts of the process. |
---|
| 785 | |
---|
[1222] | 786 | \subsection{ICRU 73-based energy loss model} |
---|
| 787 | The ICRU 73 \cite{hlei.ICRU73} report contains stopping power tables |
---|
| 788 | for ions with atomic numbers 3--18 and 26, covering a range of different |
---|
| 789 | elemental and compound target materials. The stopping powers derive from |
---|
| 790 | calculations with the PASS code \cite{hlei.sigm02}, which implements the |
---|
| 791 | binary stopping theory described in \cite{hlei.sigm02,hlei.sigm00}. Tables |
---|
| 792 | in ICRU 73 extend over an energy range up to 1 GeV/nucleon. All stopping |
---|
| 793 | powers were incorporated into Geant4 and are available through a |
---|
| 794 | parameterisation model ({\tt G4IonParametrisedLossModel}). For a few |
---|
| 795 | materials revised stopping powers were included (water, water vapor, nylon type |
---|
| 796 | 6 and 6/6 from P. Sigmund et al \cite{hlei.sigm09a} and copper from P. Sigmund |
---|
| 797 | \cite{hlei.sigm09b}), which replace the corresponding tables of the original |
---|
| 798 | ICRU 73 report. |
---|
[1208] | 799 | |
---|
[1222] | 800 | To account for secondary electron production above $T_{c}$, the continuous |
---|
| 801 | energy loss per unit path length is calculated according to |
---|
| 802 | \begin{equation} |
---|
| 803 | \label{hlei.rstp} |
---|
| 804 | \frac{dE}{dx}\bigg|_{T<T_C} = \bigg(\frac{dE}{dx}\bigg)_{ICRU73} - |
---|
| 805 | \bigg(\frac{dE}{dx}\bigg)_{\delta} |
---|
| 806 | \end{equation} |
---|
| 807 | where $(dE/dx)_{ICRU73}$ refers to stopping powers obtained by interpolating |
---|
| 808 | ICRU 73 tables and $(dE/dx)_{\delta}$ is the mean energy transferred to |
---|
| 809 | $\delta$-electrons per path length given by |
---|
| 810 | \begin{equation} |
---|
| 811 | \bigg(\frac{dE}{dx}\bigg)_{\delta} = \sum_{i} n_{at,i} \int_{T_c}^{T_{max}} |
---|
| 812 | \frac{d\sigma_i(T)}{dT} T dT |
---|
| 813 | \label{} |
---|
| 814 | \end{equation} |
---|
| 815 | where the index $i$ runs over all elements composing the material, $n_{at,i}$ |
---|
| 816 | is the number of atoms of the element $i$ per volume, $T_{max}$ is the maximum |
---|
| 817 | energy transferable to an electron according to formula (\ref{hlei.a1}) and |
---|
| 818 | $d\sigma_i/dT$ specifies the differential cross section per atom for producing |
---|
| 819 | an $\delta$-electron following equation (\ref{hlei.bbb}). |
---|
| 820 | |
---|
| 821 | For compound targets not considered in the ICRU 73 report, the first term on |
---|
| 822 | the rightern side in equation (\ref{hlei.rstp}) is computed by applying Bragg's |
---|
| 823 | additivity rule \cite{hlei.ICRU49} if tables for all elemental components are |
---|
| 824 | available in ICRU 73. |
---|
| 825 | |
---|
[1208] | 826 | \subsection{Status of this document} |
---|
| 827 | |
---|
| 828 | \noindent |
---|
| 829 | 21.11.2000 Created by V.Ivanchenko \\ |
---|
| 830 | 30.05.2001 Modified by V.Ivanchenko \\ |
---|
| 831 | 23.11.2001 Modified by M.G. Pia to add PIXE section. \\ |
---|
| 832 | 19.01.2002 Minor corrections (mma) \\ |
---|
| 833 | 13.05.2002 Minor corrections (V.Ivanchenko) \\ |
---|
[1222] | 834 | 28.08.2002 Minor corrections (V.Ivanchenko) \\ |
---|
| 835 | 11.12.2009 Modified by A. Lechner to add ICRU 73 section |
---|
[1208] | 836 | |
---|
| 837 | \begin{latexonly} |
---|
| 838 | |
---|
| 839 | \begin{thebibliography}{599} |
---|
| 840 | |
---|
| 841 | \bibitem{hlei.prepHadr}V.N.~Ivanchenko et al., GEANT4 Simulation |
---|
| 842 | of |
---|
| 843 | Energy Losses of Slow Hadrons, CERN-99-121, INFN/AE-99/20, (September 1999). |
---|
| 844 | \bibitem{hlei.prepIon}S.~Giani et al., GEANT4 Simulation |
---|
| 845 | of |
---|
| 846 | Energy Losses of Ions, CERN-99-300, INFN/AE-99/21, (November 1999). |
---|
| 847 | \bibitem{hlei.pdg} D.E.~Groom et al., Eur. |
---|
| 848 | Phys. Jour. C15 (2000) 1. |
---|
| 849 | \bibitem{hlei.rossi52} B.~Rossi, High Energy |
---|
| 850 | Particles, Pretice-Hall, Inc., Englewood Cliffs, NJ, 1952. |
---|
| 851 | \bibitem{hlei.bethe}H.~Bethe, Ann. Phys. 5 (1930) 325. |
---|
| 852 | \bibitem{hlei.ICRU37} (A.~Allisy et al), |
---|
| 853 | Stopping Powers for Electrons and Positrons, |
---|
| 854 | ICRU Report 37, 1984. |
---|
| 855 | \bibitem{hlei.sternheimer} |
---|
| 856 | R.M.~Sternheimer. Phys.Rev. B3 (1971) 3681. |
---|
| 857 | \bibitem{hlei.bark62} |
---|
| 858 | W.H.~Barkas. Technical Report 10292,UCRL, August 1962. |
---|
| 859 | \bibitem{hlei.bark56} |
---|
| 860 | W.H.~Barkas, W.~Birnbaum, F.M.~Smith, Phys. Rev. |
---|
| 861 | 101 (1956) 778. |
---|
| 862 | \bibitem{hlei.arb72} |
---|
| 863 | J.C.~Ashley, R.H.~Ritchie and W.~Brandt, |
---|
| 864 | Phys. Rev. B5 (1972) 1. |
---|
| 865 | \bibitem{hlei.bloch}F.~Bloch, Ann. Phys. 16 (1933) 285. |
---|
| 866 | \bibitem{hlei.Lindhard} |
---|
| 867 | J.~Linhard and A.~Winther, Mat. Fys. Medd. Dan. Vid. Selsk. |
---|
| 868 | 34, No 10 (1963). |
---|
| 869 | \bibitem{hlei.Ziegler771}H.H.~Andersen and J.F.~Ziegler, |
---|
| 870 | The Stopping |
---|
| 871 | and Ranges of Ions in Matter. Vol.3, Pergamon Press, 1977. |
---|
| 872 | \bibitem{hlei.ICRU49}ICRU (A.~Allisy et al), |
---|
| 873 | Stopping Powers and Ranges for Protons and Alpha |
---|
| 874 | Particles, |
---|
| 875 | ICRU Report 49, 1993. |
---|
| 876 | \bibitem{hlei.Ziegler774}J.F.~Ziegler, The Stopping |
---|
| 877 | and Ranges of Ions in Matter. Vol.4, Pergamon Press, 1977. |
---|
| 878 | \bibitem{hlei.Ziegler85}J.F.~Ziegler, J.P.~Biersack, U |
---|
| 879 | .~Littmark, The Stopping |
---|
| 880 | and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985. |
---|
| 881 | \bibitem{hlei.BK} |
---|
| 882 | W.~Brandt and M.~Kitagawa, Phys. Rev. B25 (1982) 5631. |
---|
| 883 | \bibitem{hlei.sigmund} |
---|
| 884 | P.~Sigmund, Nucl. Instr. and Meth. |
---|
| 885 | B85 (1994) 541. |
---|
| 886 | \bibitem{hlei.Ziegler88} J.F.~Ziegler and |
---|
| 887 | J.M.~Manoyan, Nucl. Instr. and Meth. |
---|
| 888 | B35 (1988) 215. |
---|
| 889 | \bibitem{hlei.mol}G.~Moliere, |
---|
| 890 | Theorie der Streuung schneller geladener Teilchen I; |
---|
| 891 | Einzelstreuungam abbgeschirmten Coulomb-Feld, Z. f. Naturforsch, A2 |
---|
| 892 | (1947) 133. |
---|
| 893 | \bibitem{hlei.GEANT3} GEANT3 manual, |
---|
| 894 | CERN Program Library Long Writeup |
---|
| 895 | W5013 (October 1994). |
---|
| 896 | \bibitem{hlei.Yang} Q.~Yang, |
---|
| 897 | D.J.~O'Connor, Z.~Wang, Nucl. Instr. and Meth. |
---|
| 898 | B61 (1991) 149. |
---|
| 899 | \bibitem{hlei.Chu} W.K.~Chu, in: Ion Beam Handbook for |
---|
| 900 | Material Analysis, edt. J.W.~Mayer and E.~Rimini, |
---|
| 901 | Academic Press, NY, 1977. |
---|
| 902 | \bibitem{hlei.Gryzinski1} M. Gryzinski, Phys. Rev. A 135 (1965) 305. |
---|
| 903 | \bibitem{hlei.Gryzinski2} M. Gryzinski, Phys. Rev. A 138 (1965) 322. |
---|
[1222] | 904 | \bibitem{hlei.ICRU73} |
---|
| 905 | Stopping of Ions Heavier Than Helium, |
---|
| 906 | ICRU Report 73, Oxford University Press (2005). |
---|
| 907 | \bibitem{hlei.sigm02} |
---|
| 908 | P.~Sigmund and A.~Schinner, |
---|
| 909 | Nucl. Instr. Meth. in Phys. Res. B 195 (2002) 64. |
---|
| 910 | \bibitem{hlei.sigm00} |
---|
| 911 | P.~Sigmund and A.~Schinner, |
---|
| 912 | Eur. Phys. J. D 12 (2000) 425. |
---|
| 913 | \bibitem{hlei.sigm09a} |
---|
| 914 | P.~Sigmund, A.~Schinner and H.~Paul, |
---|
| 915 | Errata and Addenda for ICRU Report 73, Stopping of Ions Heavier |
---|
| 916 | than Helium (2009). |
---|
| 917 | \bibitem{hlei.sigm09b} |
---|
| 918 | Personal communication with P.~Sigmund (2009). |
---|
[1208] | 919 | \end{thebibliography} |
---|
| 920 | |
---|
| 921 | \end{latexonly} |
---|
| 922 | |
---|
| 923 | \begin{htmlonly} |
---|
| 924 | |
---|
| 925 | \subsection{Bibliography} |
---|
| 926 | |
---|
| 927 | \begin{enumerate} |
---|
| 928 | \item V.N.~Ivanchenko et al., GEANT4 Simulation of |
---|
| 929 | Energy Losses of Slow Hadrons, CERN-99-121, INFN/AE-99/20, (September 1999). |
---|
| 930 | \item S.~Giani et al., GEANT4 Simulation of |
---|
| 931 | Energy Losses of Ions, CERN-99-300, INFN/AE-99/21, (November 1999). |
---|
| 932 | \item D.E.~Groom et al., Eur. |
---|
| 933 | Phys. Jour. C15 (2000) 1. |
---|
| 934 | \item B.~Rossi, High Energy |
---|
| 935 | Particles, Pretice-Hall, Inc., Englewood Cliffs, NJ, 1952. |
---|
| 936 | \item H.~Bethe, Ann. Phys. 5 (1930) 325. |
---|
| 937 | \item (A.~Allisy et al), |
---|
| 938 | Stopping Powers for Electrons and Positrons, |
---|
| 939 | ICRU Report 37, 1984. |
---|
| 940 | \item |
---|
| 941 | R.M.~Sternheimer. Phys.Rev. B3 (1971) 3681. |
---|
| 942 | \item |
---|
| 943 | W.H.~Barkas. Technical Report 10292,UCRL, August 1962. |
---|
| 944 | \item |
---|
| 945 | W.H.~Barkas, W.~Birnbaum, F.M.~Smith, Phys. Rev. |
---|
| 946 | 101 (1956) 778. |
---|
| 947 | \item |
---|
| 948 | J.C.~Ashley, R.H.~Ritchie and W.~Brandt, |
---|
| 949 | Phys. Rev. B5 (1972) 1. |
---|
| 950 | \item F.~Bloch, Ann. Phys. 16 (1933) 285. |
---|
| 951 | \item |
---|
| 952 | J.~Linhard and A.~Winther, Mat. Fys. Medd. Dan. Vid. Selsk. |
---|
| 953 | 34, No 10 (1963). |
---|
| 954 | \item H.H.~Andersen and J.F.~Ziegler, |
---|
| 955 | The Stopping |
---|
| 956 | and Ranges of Ions in Matter. Vol.3, Pergamon Press, 1977. |
---|
| 957 | \item ICRU (A.~Allisy et al), |
---|
| 958 | Stopping Powers and Ranges for Protons and Alpha |
---|
| 959 | Particles, |
---|
| 960 | ICRU Report 49, 1993. |
---|
| 961 | \item J.F.~Ziegler, The Stopping |
---|
| 962 | and Ranges of Ions in Matter. Vol.4, Pergamon Press, 1977. |
---|
| 963 | \item J.F.~Ziegler, J.P.~Biersack, U |
---|
| 964 | .~Littmark, The Stopping |
---|
| 965 | and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985. |
---|
| 966 | \item |
---|
| 967 | W.~Brandt and M.~Kitagawa, Phys. Rev. B25 (1982) 5631. |
---|
| 968 | \item |
---|
| 969 | P.~Sigmund, Nucl. Instr. and Meth. |
---|
| 970 | B85 (1994) 541. |
---|
| 971 | \item J.F.~Ziegler and |
---|
| 972 | J.M.~Manoyan, Nucl. Instr. and Meth. |
---|
| 973 | B35 (1988) 215. |
---|
| 974 | \item G.~Moliere, |
---|
| 975 | Theorie der Streuung schneller geladener Teilchen I; |
---|
| 976 | Einzelstreuungam abbgeschirmten Coulomb-Feld, Z. f. Naturforsch, A2 |
---|
| 977 | (1947) 133. |
---|
| 978 | \item GEANT3 manual, |
---|
| 979 | CERN Program Library Long Writeup |
---|
| 980 | W5013 (October 1994). |
---|
| 981 | \item Q.~Yang, |
---|
| 982 | D.J.~O'Connor, Z.~Wang, Nucl. Instr. and Meth. |
---|
| 983 | B61 (1991) 149. |
---|
| 984 | \item W.K.~Chu, in: Ion Beam Handbook for |
---|
| 985 | Material Analysis, edt. J.W.~Mayer and E.~Rimini, |
---|
| 986 | Academic Press, NY, 1977. |
---|
| 987 | \item M. Gryzinski, Phys. Rev. A 135 (1965) 305. |
---|
| 988 | \item M. Gryzinski, Phys. Rev. A 138 (1965) 322. |
---|
[1222] | 989 | \item |
---|
| 990 | Stopping of Ions Heavier Than Helium, |
---|
| 991 | ICRU Report 73, Oxford University Press (2005). |
---|
| 992 | \item |
---|
| 993 | P.~Sigmund and A.~Schinner, |
---|
| 994 | Nucl. Instr. Meth. in Phys. Res. B 195 (2002) 64. |
---|
| 995 | \item |
---|
| 996 | P.~Sigmund and A.~Schinner, |
---|
| 997 | Eur. Phys. J. D 12 (2000) 425. |
---|
| 998 | \item |
---|
| 999 | P.~Sigmund, A.~Schinner and H.~Paul, |
---|
| 1000 | Errata and Addenda for ICRU Report 73, Stopping of Ions Heavier than Helium (2009). |
---|
| 1001 | \item |
---|
| 1002 | Personal communication with P.~Sigmund (2009). |
---|
[1208] | 1003 | \end{enumerate} |
---|
| 1004 | |
---|
| 1005 | \end{htmlonly} |
---|
| 1006 | |
---|
| 1007 | |
---|