Low energy extensions

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 according to the quantum harmonic oscillator model is used, as described in [eal05] and references therein. The lower limit of applicability of the model is chosen for all materials at 10 keV. Below this value stopping power is set to constant equal to the \(dE/dx\) at 10 keV.

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):

(186)\[ \frac{dE}{dx}=\sum_i{\left (\frac{dE}{dx} \right )_i},\]

where the sum is taken over all elements of the absorber, \(i\) is the number of the element, \((dE/dx)_i\) 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:

if the data on the stopping power for a compound as a function of the proton kinetic energy is available (Table 26), then the direct parametrisation of the data for this material is performed;
if the data on the stopping power for a compound is available for only one incident energy (Table 27), then the computation is performed based on Bragg’s rule and the chemical factor for the compound is taken into account;
if there are no data for the compound, the computation is performed based on Bragg’s rule.

In the review [ZM88] the parametrisation stopping power data are presented as

(187)\[S_e(T_p)= S_{Bragg}(T_p)\left [1 + \frac{f(T_p)}{f(125~keV)} \left (\frac{S_{exp}(125~keV)}{S_{Bragg}(125~keV)}-1 \right ) \right ],\]

where \(S_{exp}(125~\mbox{keV})\) 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:

(188)\[f(T_p)=\frac{1}{1+\exp \left [1.48(\frac{\beta(T_p)} {\beta(25~keV)}-7.0) \right ]},\]

where \(\beta(T_p)\) is the relative velocity of the proton with kinetic energy \(T_p\).

Table 26 Stopping Power Compounds Paremeterized vs. Energy
Number	Chemical formula
1	AlO
2	C₂O
3	CH₄
4	(C₂H₄)_N-Polyethylene
5	(C₂H₄)_N-Polypropylene
6	C₈H₈)_N
7	C₃H₈
8	SiO₂
9	H₂O
10	H₂O-Gas
11	Graphite

Table 27 Stopping Power Compounds Data for Fixed Energy
Number	Chemical formula	Number	Chemical formula
1	H₂O	28	C₂H₆
2	C₂H₄O	29	C₂F₆
3	C₃H₆O	30	C₂H₆O
4	C₂H₂	31	C₃H₆O
5	CH₃OH	32	C₄H₁₀O
6	C₂H₅OH	33	C₂H₄
7	C₃H₇OH	34	C₂H₄O
8	C₃H₄	35	C₂H₄S
9	NH₃	36	SH₂
10	C₁₄H₁₀	37	CH₄
11	C₆H₆	38	CCLF₃
12	C₄H₁₀	39	CCl₂F₂
13	C₄H₆	40	CHCl₂F
14	C₄H₈O	41	(CH₃)₂S
15	CCl₄	42	N₂O
16	CF₄	43	C₅H₁₀O
17	C₆H₈	44	C₈H₆
18	C₆H₁₂	45	(CH₂)_N
19	C₆H₁₀O	46	(C₃H₆)_N
20	C₆H₁₀	47	(C₈H₈)_N
21	C₈H₁₆	48	C₃H₈ C_3H_8
22	C₅H₁₀	49	C₃H₆-Propylene
23	C₅H₈	50	C₃H₆O
24	C₃H₆-Cyclopropane	51	C₃H₆S
25	C₂H₄F₂	52	C₄H₄S
26	C₂H₂F₂	53	C₇H₈
27	C₄H₈O₂

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:

(189)\[\Delta E > T_c\kappa \;\; \mbox{or} \;\; T_c < I\kappa,\]

where \(\Delta E\) is the mean continuous energy loss in a track segment of length \(s\), \(T_c\) is the kinetic energy cut 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 [eal93]:

(190)\[\Omega^2 = K N_{el}\frac{Z_h^2}{\beta^2} T_c s f \left(1 - \frac{\beta^2}{2} \right),\]

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 [QY91, WKC77].

For short path lengths, when the condition (189) is not satisfied, the model described in Energy Loss Fluctuations is applied.

ICRU 73-based energy loss model

The ICRU 73 [eal05] report contains stopping power tables for ions with atomic numbers 3–18 and 26, covering a range of different elemental and compound target materials. The stopping powers derive from calculations with the PASS code [SS02], which implements the binary stopping theory described in [SS00, SS02]. Tables in ICRU 73 extend over an energy range up to 1 GeV/nucleon. All stopping powers were incorporated into Geant4 and are available through a parameterisation model (G4IonParametrisedLossModel). For a few materials revised stopping powers were included (water, water vapor, nylon type 6 and 6/6 from P. Sigmund et al. [PSP09] and copper from P. Sigmund [PSigmund09]), which replace the corresponding tables of the original ICRU 73 report.

To account for secondary electron production above \(T_{c}\), the continuous energy loss per unit path length is calculated according to

(191)\[\frac{dE}{dx}\bigg|_{T<T_C} = \bigg(\frac{dE}{dx}\bigg)_{ICRU73} - \bigg(\frac{dE}{dx}\bigg)_{\delta}\]

where \((dE/dx)_{ICRU73}\) refers to stopping powers obtained by interpolating ICRU 73 tables and \((dE/dx)_{\delta}\) is the mean energy transferred to \(\delta\)-electrons per path length given by

(192)\[\bigg(\frac{dE}{dx}\bigg)_{\delta} = \sum_{i} n_{at,i} \int_{T_c}^{T_{max}} \frac{d\sigma_i(T)}{dT} T dT\]

where the index \(i\) runs over all elements composing the material, \(n_{at,i}\) is the number of atoms of the element \(i\) per volume, \(T_{max}\) is the maximum energy transferable to an electron according to formula and \(d\sigma_i/dT\) specifies the differential cross section per atom for producing an \(\delta\)-electron.

For compound targets not considered in the ICRU 73 report, the first term on the right hand side in Eq.(191) is computed by applying Bragg’s additivity rule [eal93] if tables for all elemental components are available in ICRU 73.

Bibliography

eal93(1,2): M.J. Berger et al. Report 49. Journal of the International Commission on Radiation Units and Measurements, os25(2):NP–NP, may 1993. ICRU Report 49. URL: https://doi.org/10.1093/jicru/os25.2.Report49, doi:10.1093/jicru/os25.2.report49.
eal05(1,2): P. Sigmund et al. Stopping of ions heavier than helium. Journal of the International Commission on Radiation Units and Measurements, jun 2005. ICRU Report 73. URL: https://doi.org/10.1093/jicru/ndi001, doi:10.1093/jicru/ndi001.
PSP09: A. Schinner P. Sigmund and H. Paul. Stopping of ions heavier than helium. Technical Report Errata and Addenda for ICRU Report 73, 2009.
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WKC77: in: W.K. Chu. Ion Beam Handbook for Material Analysis, edt. J.W. Mayer and E. Rimini. Academic Press, NY, edition, 1977.
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