1 | |
---|
2 | \section{Energy loss fluctuations} \label{gen_fluctuations} |
---|
3 | |
---|
4 | |
---|
5 | \subsection{Fluctuations in thick absorbers} |
---|
6 | |
---|
7 | The total continuous energy loss of charged particles is a stochastic |
---|
8 | quantity with a distribution described in terms of a straggling function. |
---|
9 | The straggling is partially taken into account in the simulation |
---|
10 | of energy loss by the production of $\delta$-electrons with energy |
---|
11 | $T > T_c$. However, continuous energy loss also has fluctuations. |
---|
12 | Hence in the current GEANT4 implementation two different models of |
---|
13 | fluctuations are applied depending on the value of the parameter $\kappa$ |
---|
14 | which is the lower limit of the number of interactions of the particle in |
---|
15 | a step. The default value chosen is $\kappa = 10$. In the case of a high |
---|
16 | range cut (i.e. energy loss without delta ray production) |
---|
17 | for thick absorbers the following condition should be fulfilled: |
---|
18 | \begin{equation} |
---|
19 | \Delta E > \kappa \ T_{max} |
---|
20 | \label{cond} |
---|
21 | \end{equation} |
---|
22 | where $\Delta E$ is the mean continuous energy loss in a track segment of |
---|
23 | length $s$, and $T_{max}$ is the maximum kinetic energy that can be |
---|
24 | transferred to the atomic electron. If this condition holds the fluctuation |
---|
25 | of the total (unrestricted) energy loss follows a Gaussian distribution. It |
---|
26 | is worth noting that this condition can be true only for heavy particles, |
---|
27 | because for electrons, $T_{max}=T/2$, and for positrons, $T_{max}=T$, where |
---|
28 | $T$ is the kinetic energy of the particle. |
---|
29 | In order to |
---|
30 | simulate the fluctuation of the continuous (restricted) energy loss, the |
---|
31 | condition should be modified. After a study, the following conditions |
---|
32 | have been chosen: |
---|
33 | \begin{equation} |
---|
34 | \Delta E > \kappa \ T_c |
---|
35 | \label{cond2} |
---|
36 | \end{equation} |
---|
37 | |
---|
38 | and |
---|
39 | |
---|
40 | \begin{equation} |
---|
41 | T_{max} <= 2 \ T_c |
---|
42 | \label{cond3} |
---|
43 | \end{equation} |
---|
44 | where $T_c$ is the cut kinetic energy of $\delta$-electrons. For thick |
---|
45 | absorbers the |
---|
46 | straggling function approaches the Gaussian distribution with Bohr's |
---|
47 | variance \cite{eloss.ICRU49}: |
---|
48 | \begin{equation} |
---|
49 | \Omega^2 = 2\pi r^2_e m_e c^2 N_{el}\frac{Z_h^2}{\beta^2} T_c s |
---|
50 | \left(1 - \frac{\beta^2}{2} \right), |
---|
51 | \label{sig.fluc} |
---|
52 | \end{equation} |
---|
53 | where $r_e$ is the classical electron radius, $N_{el}$ is the electron |
---|
54 | density of the medium, $Z_{h}$ is the charge of the incident particle in |
---|
55 | units of positron charge, and $\beta$ is the relativistic velocity. |
---|
56 | |
---|
57 | |
---|
58 | \subsection{Fluctuations in thin absorbers} |
---|
59 | If the conditions \ref{cond2} and \ref{cond3} are not satisfied the model of |
---|
60 | energy |
---|
61 | fluctuations in thin absorbers is applied. The formulae used to compute |
---|
62 | the energy loss fluctuation (straggling) are based on a very simple physics |
---|
63 | model of the atom. It is assumed that the atoms have only two energy levels |
---|
64 | with binding energies $E_1$ and $E_2$. The particle-atom interaction can be |
---|
65 | an excitation with energy loss $E_1$ or $E_2$, or ionization with energy |
---|
66 | loss distributed according to a function $g(E) \sim 1/E^2$ : |
---|
67 | \begin{equation} \label{fluct.eqn0} |
---|
68 | \int_{E_0}^{T_{up}} g(E)\ dE = 1 \Longrightarrow |
---|
69 | g(E) = \frac{E_0 T_{up}}{T_{up}-E_0} \frac{1}{E^2} . |
---|
70 | \end{equation} |
---|
71 | |
---|
72 | \noindent |
---|
73 | The macroscopic cross section for excitation $(i=1,2)$ is |
---|
74 | \begin{equation} \label{fluct.eqn1} |
---|
75 | \Sigma_i = C \frac{f_i}{E_i} |
---|
76 | \frac{\ln[2mc^2 \ (\beta\gamma)^2/E_i]-\beta^2} |
---|
77 | {\ln[2mc^2 \ (\beta\gamma)^2/I]-\beta^2}\ (1-r) |
---|
78 | \end{equation} |
---|
79 | and the ionization cross section is |
---|
80 | \begin{equation} \label{fluct.eqn2} |
---|
81 | \Sigma_3 = C \frac{T_{up}-E_0} |
---|
82 | {E_0 T_{up}\ln(\frac{T_{up}}{E_0})}\ r |
---|
83 | \end{equation} |
---|
84 | where $E_0$ denotes the ionization energy of the atom, $I$ is the mean |
---|
85 | ionization energy, $T_{up}$ is the |
---|
86 | production threshold for delta ray production (or the maximum energy |
---|
87 | transfer if this value smaller than the |
---|
88 | production threshold), $E_i$ and $f_i$ are the energy levels and |
---|
89 | corresponding oscillator strengths of the atom, and $C$ and $r$ are model |
---|
90 | parameters. |
---|
91 | \noindent |
---|
92 | The oscillator strengths $f_i$ and energy levels $E_i$ should satisfy the |
---|
93 | constraints |
---|
94 | \begin{equation} \label{fluct.eqn3} |
---|
95 | f_1 + f_2 = 1 |
---|
96 | \end{equation} |
---|
97 | \begin{equation} \label{fluct.eqn4} |
---|
98 | f_1 \cdotp lnE_1 + f_2 \cdotp lnE_2 = lnI . |
---|
99 | \end{equation} |
---|
100 | The cross section formulae \ref{fluct.eqn1},\ref{fluct.eqn2} and the sum |
---|
101 | rule equations \ref{fluct.eqn3},\ref{fluct.eqn4} can be found e.g. in |
---|
102 | Ref. \cite{straggling.bichsel}. |
---|
103 | \noindent |
---|
104 | The model parameter $C$ can be defined in the following way. The numbers of |
---|
105 | collisions ($n_i$, $i=1,2$ for excitation and $3$ for ionization) |
---|
106 | follow the Poisson distribution with a mean value $\langle n_i \rangle$. |
---|
107 | In a step of length $\Delta x$ the mean number of collisions is given by |
---|
108 | \begin{equation} |
---|
109 | \langle n_i \rangle = \Delta x \ \Sigma_i |
---|
110 | \end{equation} |
---|
111 | The mean energy loss in a step is the sum of the excitation and ionization |
---|
112 | contributions and can be written as |
---|
113 | \begin{equation} |
---|
114 | \frac{dE}{dx} \cdotp \Delta x = |
---|
115 | \left \{ \Sigma_1 E_1 + \Sigma_2 E_2 + |
---|
116 | \int_{E_0}^{T_{up}} E g(E) dE \right \} \Delta x . |
---|
117 | \end{equation} |
---|
118 | From this, using eq. \ref{fluct.eqn1} - \ref{fluct.eqn4}, one can see that |
---|
119 | \begin{equation} |
---|
120 | C = dE/dx . |
---|
121 | \end{equation} |
---|
122 | |
---|
123 | \noindent |
---|
124 | The other parameters in the fluctuation model have been chosen |
---|
125 | in the following way. $Z \cdotp f_1$ and $Z \cdotp f_2$ represent in |
---|
126 | the model the number of loosely/tightly bound electrons |
---|
127 | \begin{equation} |
---|
128 | f_2 = 0 \hspace {15 pt} for \hspace {15 pt} Z \leq 2.8 |
---|
129 | \end{equation} |
---|
130 | \begin{equation} |
---|
131 | f_2 = 2.8/Z \hspace {15 pt} for \hspace {15 pt} Z > 2.8 |
---|
132 | \end{equation} |
---|
133 | \begin{equation} |
---|
134 | E_2 = 10 \mbox{ eV } Z^2 |
---|
135 | \end{equation} |
---|
136 | \begin{equation} |
---|
137 | E_0 = 10 \mbox{ eV } . |
---|
138 | \end{equation} |
---|
139 | Using these parameter values, $E_2$ corresponds approximately to the |
---|
140 | K-shell energy of the atoms. |
---|
141 | The parameters $f_1$ and $E_1$ can be obtained from Eqs.~ \ref{fluct.eqn3} |
---|
142 | and \ref{fluct.eqn4}. |
---|
143 | \noindent |
---|
144 | The parameter $r$ is the only variable in the model which can be tuned. |
---|
145 | This parameter determines the relative contribution of ionization and |
---|
146 | excitation to the energy loss. Based on comparisons of simulated energy |
---|
147 | loss distributions to experimental data, its value has been parametrized |
---|
148 | as |
---|
149 | \begin{equation} |
---|
150 | r = 0.03 + 0.23 \cdotp ln(ln(\frac{T_{up}}{I})) |
---|
151 | \end{equation} |
---|
152 | |
---|
153 | This simple parametrization in the model gives good |
---|
154 | values for the most probable energy loss. The width (FWHM) of the energy loss |
---|
155 | distribution is good too, if the absorber layer is not too thin. |
---|
156 | In order to get good FWHM in thin absorbers a width correction has been |
---|
157 | applied. The form of this correction is |
---|
158 | |
---|
159 | \begin{equation} |
---|
160 | w = \sqrt{\frac{10 E_1}{\Delta E}} \hspace{25 mm} for \Delta E > 10 E_1 |
---|
161 | \end{equation} |
---|
162 | \begin{equation} |
---|
163 | w = 1 \hspace{35 mm} for \Delta E \leq 10 E_1 |
---|
164 | \end{equation} |
---|
165 | |
---|
166 | and the transformation |
---|
167 | $\langle n_1 \rangle \longrightarrow \langle n_1 \rangle \cdot w$ , $E_1 \longrightarrow \frac{E_1}{w}$ should be applied. |
---|
168 | This correction is a kind of scaling for the energy of the lower level and for the mean |
---|
169 | number of excitations with lower energy. |
---|
170 | |
---|
171 | \paragraph{Sampling the energy loss.} |
---|
172 | The energy loss is computed in the model under the assumption that |
---|
173 | the step length (or relative energy loss) is small and, in consequence, the |
---|
174 | cross section can be considered constant along the step. The loss due to |
---|
175 | the excitation is |
---|
176 | \begin{equation} |
---|
177 | \Delta E_{exc} = n_1 E_1 + n_2 E_2 |
---|
178 | \end{equation} |
---|
179 | where $n_1$ and $n_2$ are sampled from a Poisson distribution. The energy |
---|
180 | loss due to ionization can be generated from the distribution $g(E)$ |
---|
181 | by the inverse transformation method : |
---|
182 | \begin{equation} |
---|
183 | u = F(E) = \int_{E_0}^E g(x) dx |
---|
184 | \end{equation} |
---|
185 | \begin{equation} |
---|
186 | E = F^{-1}(u) = \frac {E_0}{1-u \frac{T_{up}-E_0}{T_{up}}} |
---|
187 | \end{equation} |
---|
188 | where $u$ is a uniformly distributed random number $\in [0,\ 1]$. |
---|
189 | The contribution coming from the ionization will then be |
---|
190 | \begin{equation} |
---|
191 | \Delta E_{ion} = \sum_{j = 1}^{n3} \frac {E_0} |
---|
192 | {1-u_j \frac{T_{up}-E_0}{T_{up}}} |
---|
193 | \end{equation} |
---|
194 | where $n_3$ is the number of ionizations sampled from the Poisson |
---|
195 | distribution. The total energy loss in a step will be |
---|
196 | $ \Delta E = \Delta E_{exc} + \Delta E_{ion}$ and the energy loss |
---|
197 | fluctuation comes from fluctuations in the number of collisions $n_i$ |
---|
198 | and from the sampling of the ionization loss. |
---|
199 | |
---|
200 | \paragraph{Thick layers} |
---|
201 | If the mean energy loss and step are in the range of validity of the |
---|
202 | Gaussian approximation of the fluctuation, the much faster Gaussian |
---|
203 | sampling is used to compute the actual energy loss. |
---|
204 | |
---|
205 | \paragraph{Conclusions} |
---|
206 | This simple model of energy loss fluctuations is rather fast and can |
---|
207 | be used for any thickness of material. This has been verified by performing |
---|
208 | many simulations and comparing the results with experimental data, such |
---|
209 | as those in Ref.\cite{straggling.lassila}. \\ |
---|
210 | As the limit of validity of Landau's theory is approached, the loss |
---|
211 | distribution approaches the Landau form smoothly. |
---|
212 | |
---|
213 | \subsection{Status of this document} |
---|
214 | 30.01.02 created by L. Urb\'an.\\ |
---|
215 | 28.08.02 updated by V.Ivanchenko.\\ |
---|
216 | 17.08.04 moved to common to all charged particles (mma) \\ |
---|
217 | 04.12.04 spelling and grammar check by D.H. Wright \\ |
---|
218 | 04.05.05 updated by L. Urb\'an.\\ |
---|
219 | 09.12.05 updated by V.Ivanchenko.\\ |
---|
220 | 29.03.07 updated by L. Urb\'an.\\ |
---|
221 | |
---|
222 | \begin{latexonly} |
---|
223 | |
---|
224 | \begin{thebibliography}{99} |
---|
225 | \bibitem{straggling.bichsel} H.Bichsel |
---|
226 | {\em Rev.Mod.Phys. 60 (1988) 663} |
---|
227 | \bibitem{straggling.lassila} K.Lassila-Perini, L.Urb\'an |
---|
228 | {\em Nucl.Inst.Meth. A362(1995) 416} |
---|
229 | \bibitem{eloss.geant3} {\sc geant3} manual |
---|
230 | {\em Cern Program Library Long Writeup W5013 (1994)} |
---|
231 | \bibitem{eloss.ICRU49}ICRU (A.~Allisy et al), |
---|
232 | Stopping Powers and Ranges for Protons and Alpha |
---|
233 | Particles, |
---|
234 | ICRU Report 49, 1993. |
---|
235 | \end{thebibliography} |
---|
236 | |
---|
237 | \end{latexonly} |
---|
238 | |
---|
239 | \begin{htmlonly} |
---|
240 | \subsection{Bibliography} |
---|
241 | |
---|
242 | \begin{enumerate} |
---|
243 | \item H.Bichsel {\em Rev.Mod.Phys. 60 (1988) 663} |
---|
244 | \item K.Lassila-Perini, L.Urb\'an |
---|
245 | {\em Nucl.Inst.Meth. A362(1995) 416} |
---|
246 | \item {\sc geant3} manual |
---|
247 | {\em Cern Program Library Long Writeup W5013} (1994). |
---|
248 | \item ICRU (A.~Allisy et al), |
---|
249 | Stopping Powers and Ranges for Protons and Alpha |
---|
250 | Particles, ICRU Report 49, 1993. |
---|
251 | \end{enumerate} |
---|
252 | |
---|
253 | \end{htmlonly} |
---|
254 | |
---|