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The class G4hLowEnergyIonisation calculates the continuous energy loss due to ionisation and simulates the -ray production by charged hadrons or ions. This represents an extension of the Geant4 physics models down to low energy [1,2].

## Delta-ray production

In Geant4, -rays are generated generally only above a threshold energy, , the value of which depends on atomic parameters and the cut value, , calculated from the unique cut in range parameter for all charged particles in all materials. The total cross-section for the production of a -ray electron of kinetic energy by a particle of kinetic energy is:

 (11.24)

where is the mean excitation potential of the atom (the formulae of this charter are precise if ), is the maximum energy transferable to the free electron
 (11.25)

with the electron mass, the mass of the incident particle, and is the relativistic factor. For heavy charged particles the differential cross-section per atom can be written as [3,4]:
 (11.26)

where is the atomic number, is the effective charge of the incident particle in units of positron charge, is the relativistic velocity, and . The factor is expressed as , where is the classical electron radius. The integration of formula (11.24) gives the total cross-section, which for particles with spin 0 and 1/2 are the following :
 (11.27)

where .

The average energy transfer of a particle with spin 0 to -electrons with can be expressed as:

 (11.28)

where is the electron density of the medium. Using (11.26) one finds that the correction to (11.28) for particles with spin 1/2 is . This value is very small for low energy and can be neglected. The same conclusion can be drawn for particles with spin 1.

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 , cross-section is zero and the mean free path is set to infinity, compatible with the machine precision.

## Energy Loss of Fast Hadrons

The energy lost in soft ionising collisions producing -rays below are included in the continuous energy loss. The mean value of the energy loss is given by the restricted Bethe-Bloch formula [5,3] :

 (11.29)

where is the electron density of the medium, is the density correction term, and is the shell correction term.

The density effect becomes important at high energies because of the long-range polarisation of the medium by a relativistic charged particle. The shell correction term takes into account the fact that, at low energies for light elements, and at all energies for heavy ones, the probability of hadron interaction with inner atomic shells becomes small. The accuracy of the Bethe-Bloch formula with the correction terms mentioned above is estimated as 1 % for energies between 6 MeV and 6 GeV [3]. Using (11.26) one can find out that the correction to for particles with the spin 1/2 is . This value is very small and can be neglected.

There exists a variety of phenomenological approximations for parameters in the Bethe-Bloch formula. In Geant4 the tabulation of the ionisation potential from Ref.[6] is implemented for all the elements. For the density effect the formulation of Sternheimer [7] is used:
is a kinetic variable of the particle : ,
and is defined by

 (11.30)

where the matter-dependent constants are calculated as follows:
 (11.31)

For condensed media

and for gaseous media

The semi-empirical formula due to Barkas, which is applicable to all materials, is used for the shell correction term[8]:

 (11.32)

The functions a(I), b(I), c(I) can be found in the source code.
This formula breaks down at low energies, and it only applies for (e.g. MeV for a proton). For the shell correction term is calculated as:

hence the correction becomes progressively smaller from MeV to .

Since , 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, , is the same as the energy loss of a proton with the same velocity. The corresponding kinetic energy of the proton is

 (11.33)

At initialisation stage of Geant4 the 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 (11.33).

## Barkas and Bloch effects

The accuracy of the Bethe-Bloch stopping power formula (11.33) can be improved if the higher order terms are taken into account:

 (11.34)

where is the Barkas term [9], describing the difference between ionisation of positively and negatively charged particles, and is the Bloch term.

The Barkas effect for kinetic energy of protons or antiprotons greater than can be described as [10]:

 (11.35)

where is the Bohr velocity (corresponding to proton energy ), and the function is tabulated according to [10].

The Bloch term [11] can be expressed in the following way:

 (11.36)

Note, that for the simplified expression 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 is not factorised in the formula (11.34). To take these terms into account correction is made at each step of the simulation for the value of re-calculated from the proton or antiproton tables. There is the possibility to switch off the calculation of these terms.

## Energy losses of slow positive hadrons

At low energies the total energy loss is usually described in terms of electronic stopping power . For charged hadron with velocity (corresponding to 1 MeV for protons), formula (11.30) becomes inaccurate. In this case the velocity of the incident hadron is comparable to the velocity of atomic electrons. At very low energies, when , the model of a free electron gas [12] 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 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 [13] was implemented:

where is the stopping power in , , 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:
 (11.37)

where is the minimal kinetic energy of protons in the tables of Ref.[13], is an arbitrary value between 2 MeV and 100 MeV, since in this range both the parametrisation (11.37) and the Bethe-Bloch formula (11.33) have practically the same accuracy and are close to each other. Currently the value is chosen.

To avoid problems in computation and to provide a continuous function, the factor

 (11.38)

is multiplied by the value of for . The parameter is determined for each element of the material in order to provide continuity at . The value of 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 [12] is used:
 (11.39)

The parameter is defined for each atom by requiring the stopping power to be continuous at . Currently the value used is .

Note that if the cut kinetic energy is small (), then the average energy deposit giving rise to -electron production (11.28) is subtracted from the value of the stopping power , which is calculated by formula (11.37).

Alternative parametrisations of proton energy loss are also available within Geant4 (Table 11.1). The parameterisation formulae in Ref.[14] are the same as in Ref.([13]) for the kinetic energy of protons , 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.

 Name Particle Source Ziegler1977p proton J.F. Ziegler parameterisation [13] Ziegler1977He J.F. Ziegler parameterisation [15] Ziegler1985p proton TRIM'85 parameterisation [16] ICRU_R49p proton ICRU parameterisation [14] ICRU_R49He ICRU parameterisation [14]

## Energy loss of alpha particles

The accuracy of the data for the ionisation losses of -particles in all elements [14,15] is comparable to the accuracy of the data for proton energy loss [13,14]. In the GEANT4 energy loss model for -particles the Bethe-Bloch formula is used for kinetic energy , where is the arbitrary parameter, currently set to . For lower energies a parameterisation is performed. In the energy range of the Bragg peak, , the parameterisation is:

where is the electronic stopping power in , is the kinetic energy of -particles in , are the five fitting parameters fitted individually for each atom for atomic numbers from 1 to 92.

For higher energies , another parametrisation [15] is applied

 (11.40)

To ensure a continuous function from the energy range of the Bethe-Bloch formula to the energy range of the parameterisation, the factor
 (11.41)

is multiplied by the value of as predicted by the Bethe-Bloch formula for . The parameter is determined for each element of the material in order to ensure continuity at . The value of for different atoms is usually less than 0.01.

For kinetic energies of -particles the model of free electron gas [12] is used

 (11.42)

The parameter is defined for each atom by requiring the stopping power to be continuous at .

## Effective charge of ions

For hadrons or ions the scaling relation can be written as

 (11.43)

where is the ion stopping power, is the proton stopping power at the energy scaled according (11.33), and is effective charge of the particle, which has to be used in all expressions in place of . For fast particles it is equal to the particle charge , 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 and the fractional effective charge of ion :
 (11.44)

For helium ions fractional effective charge is parameterised for all elements with good accuracy [16] according to:

 (11.45)

where the coefficients are the same for all elements, and the helium ion kinetic energy is in .

The following expression is used for heavy ions [17]:

 (11.46)

where is the fractional average charge of the ion, is the Bohr velocity, is the Fermi velocity of the electrons in the target medium, and is the term taking into account the screening effect. In Ref. [17], is estimated to be:
 (11.47)

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.[16]. The expression for the fractional average charge of the ion is the following:
 (11.48)

where is a parameter that depends on the ion velocity
 (11.49)

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:

 (11.50)

If the ion energy is lower, then the free electron gas model (11.44) is used to calculate the stopping power.

## 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 terms should be taken in account. Odd terms in 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:

• if the material is elemental, the quantum harmonic oscillator model is used, as described in [18] and references therein. The lower limit of applicability of the model is chosen for all materials at . Below this value stopping power is set to constant equal to the at .
• if the material is not elemental, the energy loss is calculated down to using the Barkas correction (11.43) and at lower energies fitting the proton energy loss curve.

## 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):

 (11.51)

where the sum is taken over all elements of the absorber, is the number of the element, is energy loss in the pure -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 11.2), 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 11.3), 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 [19] the parametrisation stopping power data are presented as
 (11.52)

where is the experimental value of the energy loss for the compound for protons or the reduced experimental value for He ions, is a value of energy loss calculated according to Bragg's rule, and is a universal function, which describes the disappearance of deviations from Bragg's rule for higher kinetic energies according to:
 (11.53)

where is the relative velocity of the proton with kinetic energy .

 Number Chemical formula 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

 Number Chemical formula Number Chemical formula 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

## 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 nuclear stopping power. It is parametrised [15,16,14] using a universal parameterisation for reduced ion energy, , which depends on ion parameters and on the charge, , and the mass, , of the target nucleus:

 (11.54)

The universal reduced nuclear stopping power, , is determined by J. Moliere in the framework of Thomas-Fermi potential [20]. The corresponding tabulation from Ref.[14] is implemented. To transform the value of nuclear stopping power from reduced units to the following formula is used:
 (11.55)

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, ).

## 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 -electrons with energy . 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 which is the lower limit of the number of interactions of the particle in the step. The default value chosen is . To select a model for thick absorbers the following boundary conditions are used:

 (11.56)

where is the mean continuous energy loss in a track segment of length , is the cut kinetic energy of -electrons, and is the average ionisation potential of the atom.

For long path lengths the straggling function approaches the Gaussian distribution with Bohr's variance [14]:

 (11.57)

where is a screening factor, which is equal to unity for fast particles, whereas for slow positively charged ions with , where parameters and are parametrised for all atoms [22,23].

For short path lengths, when the condition 11.58 is not satisfied, the model described in the charter 7.2 is applied.

## 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 the step limit is forced to be not longer than , where is the range of the particle with the reduced energy , 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. In each step continuous energy loss of the particle is calculated using loss tables for protons or antiprotons for . For lower energies, continuous energy loss is calculated using the effective charge approach, chemical factors, and nuclear stopping powers. If the step of the particle is limited by the ionisation process the sampling of -electron production is performed. (A short overview of the method is given in Chapter 2.)
Apart from the normalisation, the cross-section (11.26) can be written as :

 (11.58)

with :

where is a spin dependent term (11.26). For a spin-0 particle this term is zero, for a spin- particle , whilst for spin-1 the expression is more complicated.
The energy, , is sampled by :
1. Sample from .
2. Calculate the rejection function and accept the sampled with a probability of .
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, , is generated isotropically; the polar angle 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 global coordinate system.

## PIXE

PIXE is simulated by calculating cross-sections according to [24] and [25] to identify the primary ionised shell, and generating the subsequent atomic relaxation as described in 11.9. Sampling of excitations is carried out for both the continuous and the discrete parts of the process.

## Status of this document

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)

## Bibliography

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