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].
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:
The average energy transfer
of a particle with spin 0
to -electrons with
can be expressed as:
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.
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] :
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 . 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.
is implemented for all the
elements. For the density
effect the formulation of Sternheimer 
is a kinetic variable of the particle : ,
and is defined by
The semi-empirical formula due to Barkas, which is applicable to all
materials, is used for the shell correction term:
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
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).
The accuracy of
the Bethe-Bloch stopping power formula
(11.33) can be improved
if the higher order terms are taken into account:
The Barkas effect for kinetic energy of
protons or antiprotons greater than can be described as
The Bloch term 
can be expressed in the following way:
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.
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  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
parametrisation from the review  was implemented:
To avoid problems in computation and
to provide a continuous function, the factor
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. are the same as in Ref.() 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.
|Ziegler1977p||proton||J.F. Ziegler parameterisation |
|Ziegler1977He||J.F. Ziegler parameterisation |
|Ziegler1985p||proton||TRIM'85 parameterisation |
|ICRU_R49p||proton||ICRU parameterisation |
|ICRU_R49He||ICRU parameterisation |
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,
For higher energies , another
parametrisation  is applied
For kinetic energies of -particles the model
of free electron gas  is used
For hadrons or ions
the scaling relation can be written as
For helium ions
fractional effective charge
is parameterised for all
elements with good accuracy  according to:
The following expression is used for heavy ions :
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:
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:
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):
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:
|Number||Chemical formula||Number||Chemical formula|
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
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:
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:
For long path lengths the straggling function
approaches the Gaussian distribution with Bohr's variance :
For short path lengths, when the condition 11.58 is not satisfied, the model described in the charter 7.2 is applied.
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 :
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)