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Geant4 Low Energy Compton Profile

Geant4 Low Energy Compton Profile. Gerardo Depaola *. * National University of Córdoba (Argentina). Compton Profile. The Compton atomic DDCS obtained from the IA is given by:. Where:. R and R ’ are function of E, E ’ , p z , q and . (p) is the electron momentum distribution.

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Geant4 Low Energy Compton Profile

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  1. Geant4 Low EnergyCompton 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. 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. • http://www.irs.inms.nrs.ca/inms/irs/EGSnrc/EGSnrc.html

  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, P1and 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’/ECwhich is really only correct for E’=EC, then they integrated over E’ and obtain: Incoherent scattering function of the i-th shell electron with converges to the number of electron in each sub-shell for pi,max -> where 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 pzis 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: ; naming where: 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.

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