1. Geant4 Low Energy Compton Profile
Gerardo Depaola*
* National University of Córdoba (Argentina)

2. Compton Profile
The Compton atomic DDCS obtained from the IA is given by:
Where:
R and R’ are function of E, E’, pz , q and 
(p) is the electron momentum distribution

3. Zi: number of e- in the ith shell
For an isotropic distribution ->
Compton excitations are allowed only if the target electron is promoted to a free state, i.e. if the energy transfer E - E’ is large than Ui.=> DDCS is:
The most probable values of |pz| is zero => pz << mc => E’ -> Ec, so that, X reduce to Klein-Nishina X factor:
where:

4. 4 approaches, from the more easy to the more complex.
Proposal:
4 approaches, from the more easy to the more complex.
“On the multiple scattering of linearly polarized gamma rays”
Fridhelm Bell NIM B86 (1994).
“Implementation of the Doppler broading of a Compton-scatterd photon into
the EGS4 code”. Y. Namito, S. Ban, H. Hirayama. NIM A349 (1994).
“Fast sampling algorithm for the simulation of photon Compton Scattering”
D. Brusa, G. Stutz, j. A. Riveros, J. M. Fernandez Vares and F. Salvat.
NIM A379 (1996).
“The EGSncr code system”. I. Kawrakow and D. W. O. Rogers.
“Función de dispersión incoherente para electrones K”
G. Stutz. SARX’98 ISSN (1998).

5. Brief Description of each method:
1) The Bell method.
Write the DCS as:
x = cos ; f(x) is the Klein Nishina unpolarized cross section; g(x) is the polarization function -> g(x) = 1 - x2, P1 and P3 are the linear and circular degree of polarization respectively.
Due to the Doppler boarding, Ec change to E’.
Adopt for simplicity a Lorentzial shaped Compton profile => sample E’ from:
where: , pz* and  are adjustable parameters
which allow a rough representation of the total Compton profile.

6. 2) Namito et al. -> EGS4 code.
Set pz= 0 in the F equation => F = E’/EC which is really only correct for E’=EC, then they integrated over E’ and obtain:
where
Incoherent scattering function of the i-th shell electron with converges to the number of electron in each sub-shell for pi,max ->
The cross section for the whole atom is obtained by summing over all sub-shells.
The pi,max is calculated by putting E’= E - Ui in the pz equation. The pz is sample in the interval [0,100] using a normalized cumulative density function of Ji(x):

7. 3) Brusa et al. -> PENELOPE Code
Approach F by the first-order term of the Taylor expansion:
where:
; naming
pz values are obtained by solving the sampling equation:
Associate with the incoherent scattering function in the IA
Use a approximate one electron profiles of the form:

8. Solving for pz
Where:
In conclusion, for sample the Compton profile we need create a data base witch contain, for each Z:
Zi : number of electron in the ith shell.
Ui : Ionization energy of the ith shell.
Ji(0) :Hartree-Fock Compton profile for pz=0 for each ith shell.
All this tables are in the literature.

9. 4) Kawrakow et al. -> EGSnrc code.
Basically is the same method of PENELOPE, the difference consist in that they used a better approximation for the F function. They checked, by numerical integration, that the incoherent scattering function calculated with the approximation agrees to better that 0.3% with the incoherent scattering function calculated using the exact expression.