Study Notes on A Low Energy Bound Atomic Electron Compton Scattering Model for Geant4

A Low Energy Bound Atomic Electron Compton Scattering Model for Geant4 by J.M.C. Brown et al.

Article Information

• Authors: J.M.C. Brown, M.R. Dimmock, J.E. Gillam, D.M. Paganin
• Affiliations:
  a. School of Physics, Monash University, Victoria 3800, Australia
  b. Department of Medical Imaging and Radiation Science, School of Biomedical Science, Monash University, Victoria 3800, Australia
  c. IFIC CSIC Universitat de Valencia, Valencia E-46071, Spain
  d. Health Sciences, University of Sydney, NSW 2006, Australia
• Article History:
  • Received: 2 February 2014
  • Revised: 16 June 2014
  • Accepted: 31 July 2014
  • Available online: 29 August 2014
• Keywords: Compton scattering, Geant4, Radiation transport modelling, Monte Carlo method

Abstract

• Development of an alternative Compton scattering computational model using a two-body, fully relativistic, three-dimensional scattering framework, termed the Monash University Compton scattering model (MUCSM).
• Features:
  • Utilizes conservation of energy and momentum under the relativistic impulse approximation.
  • Develops energy and directional algorithms for both scattered photon and ejected Compton electron from first principles.
  • Addressed the limitations of existing models based on Ribberfors’ Compton scattering frameworks, specifically focusing on the directionality of ejected Compton electrons.
• MUCSM implementation in Geant4 low energy electromagnetic physics class (G4LowEPComptonModel).
• Performance assessment performed via:
  1. Comparison with standard Geant4 Compton scattering classes: G4LivermoreComptonModel and G4PenelopeComptonModel.
  2. Experimental data comparison with Compton electron kinetic energy spectra of 662 keV photons scattering from gold's K-shell.
• Conclusions indicate MUCSM successfully replicates the Compton scattering differential cross-section for gold within experimental uncertainties and serves as a viable alternative to existing models.

1. Introduction

• Overview of Compton Scattering:
  • Compton scattering is relevant in various fields, including gamma-ray astronomy and medical imaging.
• Ribberfors’ Compton Scattering Model:
  • The Double Differential Compton Scattering Cross-Section (DDCS) in the Relativistic Impulse Approximation (RIA) has been a foundation for most Monte Carlo models.
  • Framework based on Quantum Electrodynamics (QED) validated through simulation and experimental comparisons.
  • Addresses complications such as electron binding energies and pre-collision momentum effects.
• Geant4 Development:
  • Geant4 has low energy Compton scattering classes released since 2008, which integrate the effects of Doppler broadening in Compton scattering, specifically through G4LivermoreComptonModel and G4PenelopeComptonModel.
• Limitations of Existing Models:
  • Existing models only incorporate components of pre-collision momentum in the photon plane, constraining ejection directions into two dimensions.
  • Different methods used by G4Livermore (free-electron assumption) and G4Penelope (photon momentum transfer vector assumption) model the Compton electron's emission direction.

2. The Monash University Compton Scattering Model

2.1. Scattering Model

• The MUCSM employs special-relativistic transforms for modeling the impact of a bound, non-stationary electron similar to unpolarised inverse Compton scattering.
• Compton scattering is represented through momentum conservation:
  • $P + Q = P<em>0 + Q</em>0$
    where:
  • $P$ = Four-momentum of the incident photon
  • $Q$ = Four-momentum of the target electron
  • $P_0$ = Four-momentum of the scattered photon
  • $Q_0$ = Four-momentum of the recoil electron
• Four-Momenta in Spherical Coordinates:
  1. Incident photon: P = ( -c rac{E}{c}, 1, 0, 0)
  2. Target electron: $Q = (mc, u, a, b)$
  3. Scattered photon: P0 = ( -c rac{E0}{c}, 1, h, 0)
  4. Recoil electron: $Q<em>0 = (c</em>0 m, c, u_0, /, w)$
• Energy Relationships:
  1. Energy conservation pre- and post-collision generated by:
     $E<em>0 = E + E</em>k - E_0$
     where
  • $E_k$ = Kinetic energy of target electron
  • $E_0$ = Scattered photon energy.

2.2. Overview of G4LowEPComptonModel

• Developed following Geant4 Low Energy Electromagnetic Physics Working Group guidelines:

1. Determine target atomic element in composite material.
2. Sample photon scattering angles based on target's Klein-Nishina cross-section.
3. Identify shell and momentum of the electron in the target.
4. Calculate scattered photon energy using derived equations.
5. Verify photon energy transfer exceeds target electron's binding energy.
6. Compute Compton electron energy and ejection direction.
7. Update photon energy, direction and track the ejected electron.

2.2.1. Scattering Model Implementation

• Analytical methodology cited from relevant literature (e.g., Lightman et al.) for energy determination and directionality in Compton scattering.
• Use of previously developed G4 algorithms facilitates parameter sampling and interaction calculation.

3. Comparison and Experimental Validation

3.1. Energy Spectra Comparison

• Experiments performed using Compton scattering of photons off Carbon (C), Copper (Cu), and Lead (Pb) at photon energies between 10 keV to 10 MeV.
• Observations highlighted:
  • High agreement in energy spectra and NCCC assessments between Monash model and standard models except when energies drop below 50 keV.
  • Differences in energy spectra identified spikes and peaks attributed to electronic binding effects in specific target materials.

3.2. Directionality Analysis

• Examines polar and azimuthal angle distributions of ejected electrons:
  • Monash model spans a wider range, compared to constraints seen in Livermore and Penelope models.
• Key insights:
  • As the atomic number increases, Monash’s distribution becomes less forward-focused.
  • With higher photon energy, the Monash directionality aligns more closely with the other models.

4. Experimental Comparison with Ljubicˇic´ et al.

• Comparison of Monash Compton scattering model against measured K-shell TDCS data.
• Significance of measurements:
  • Analysis performed for distinct angles; observed similarities and discrepancies analyzed.
  • Demonstrated consistency within experimental uncertainties for the Compton electron models.

5. Discussion

• Highlights the comparative analysis findings between Monash and established Geant4 models, emphasizing Monash's broadened ejection direction prediction.
• Improved model complexity resulting in an increase in simulation time, but justified by enhanced accuracy in low-energy photon transport simulations.

6. Conclusion

• MUCSM developments demonstrate potential applicability as an alternative to Ribberfors adaptation models.
• Conclusively validated against experimental data, maintaining a high degree of correlation over specific energy ranges.

Acknowledgements

• Thanks to contributors and supporting institutions for their assistance in model development and validation.

References

• Provide citations for previous work that informed the model development and verification processes.

Note: In-depth investigations are ongoing to improve model precision and function, including potential integration of photon polarization effects and adapting momentum density functions.