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Screened Thin-Target Bremsstrahlung with Partially-Ionized High-Z Species

Salomon Guinchard, Yves Savoye-Peysson, Joan Decker

TL;DR

This paper tackles accurate thin-target electron–ion bremsstrahlung in partially ionized high‑Z targets for electron energies up to tens of MeV. It constructs a fully analytic screening model by representing the atomic form factor as a multi‑Yukawa sum, $F_{Z_{s',s}}(\bar{q})$, and combining it with the Bethe–Heitler cross section using the Olsen–Maximon–Wergeland additivity rule to yield a compact doubly differential cross section $d^{2}\sigma/(d\bar{k}d\Omega_k)$. The main contributions are (i) an analytic expression for the screened DDCS valid for arbitrary ionization, (ii) systematic comparisons with experimental data showing good forward-angle accuracy, and (iii) a demonstration of nonmonotonic screening effects arising from ionization‑dependent AFFs. The framework supports fast integration into transport solvers for fusion, safety, and astrophysics, and the authors discuss limitations of the Furry–Sommerfeld–Maue approach at high momentum transfer and potential extensions to fully relativistic Dirac treatments.

Abstract

Bremsstrahlung emission remains a cornerstone process in the characterization of electron dynamics in diverse high-energy environments. In particular, the accurate description of thin-target electron-ion bremsstrahlung in the presence of high-$Z$ species requires careful treatment of atomic screening effects, especially when atoms are partially ionized. We present a fully analytic screening model based on a multi-Yukawa representation of the atomic potential, enabling the calculation of bremsstrahlung cross sections for arbitrary nuclear charge and ionization state, and electron energies up to a few tens of MeV. This framework extends prior treatments of neutral atoms to include partially ionized high-$Z$ elements in a fully analytic framework.

Screened Thin-Target Bremsstrahlung with Partially-Ionized High-Z Species

TL;DR

This paper tackles accurate thin-target electron–ion bremsstrahlung in partially ionized high‑Z targets for electron energies up to tens of MeV. It constructs a fully analytic screening model by representing the atomic form factor as a multi‑Yukawa sum, , and combining it with the Bethe–Heitler cross section using the Olsen–Maximon–Wergeland additivity rule to yield a compact doubly differential cross section . The main contributions are (i) an analytic expression for the screened DDCS valid for arbitrary ionization, (ii) systematic comparisons with experimental data showing good forward-angle accuracy, and (iii) a demonstration of nonmonotonic screening effects arising from ionization‑dependent AFFs. The framework supports fast integration into transport solvers for fusion, safety, and astrophysics, and the authors discuss limitations of the Furry–Sommerfeld–Maue approach at high momentum transfer and potential extensions to fully relativistic Dirac treatments.

Abstract

Bremsstrahlung emission remains a cornerstone process in the characterization of electron dynamics in diverse high-energy environments. In particular, the accurate description of thin-target electron-ion bremsstrahlung in the presence of high- species requires careful treatment of atomic screening effects, especially when atoms are partially ionized. We present a fully analytic screening model based on a multi-Yukawa representation of the atomic potential, enabling the calculation of bremsstrahlung cross sections for arbitrary nuclear charge and ionization state, and electron energies up to a few tens of MeV. This framework extends prior treatments of neutral atoms to include partially ionized high- elements in a fully analytic framework.

Paper Structure

This paper contains 14 sections, 55 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Theoretical Framework for Electron–Ion Bremsstrahlung
  3. Bethe--Heitler Cross Section
  4. Sauter--Elwert Cross Section
  5. Roche--Ducos--Proriol Cross Section
  6. Screened Electron–Ion Bremsstrahlung: The Multi-Yukawa Model
  7. Atomic Form Factor
  8. Screened Cross Section
  9. Cross Section Results
  10. Multi-Yukawa and Ionized Cross Sections
  11. Comparison with Data for the Doubly Differential Cross Section
  12. Conclusion and Outlooks
  13. Olsen--Maximon--Wergeland Additivity Rule
  14. Higher-Energy Results

Figures (10)

  • Figure 1: Comparison of the DDCS obtained from the closed form Eq. \ref{['eq:DDCS_MY']} (solid lines) and from a numerical integration of Eq. \ref{['eq:dsigmabrem3']} (markers), for several ionized states of copper, and two Yukawa exponentials.
  • Figure 2: Top: Atomic form factor for a neutral copper atom ($Z=29$) as a function of the recoil momentum, calculated with one, two, and three Yukawa exponentials, and compared with the reference form factor obtained from DFT. Bottom: Neutral doubly differential cross section (DDCS), normalized by the inverse photon energy, for one, two, and three Yukawa exponentials and for two emission angles.
  • Figure 3: Top: Ionized doubly differential cross section (DDCS) for gold. Every third ionization level is shown; the red line represents Sauter’s model. Vertical lines indicate the photon energies for which the DDCS is plotted as a function of ionization level in the bottom inset. Bottom: DDCS for gold at the four photon energies indicated above.
  • Figure 4: Atomic form factor for several ionization states of gold, and one Yukawa exponential. The form factor exhibits some crossings of the ionized curves, as emphasized by a zoom in the red rectangle.
  • Figure 5: Top: Bound-electron density profiles for all ionization states of gold, calculated using DFT with the Gaussian code GAUSSIAN. The upper blue curve corresponds to the neutral atom and the ionization degree increases from right to left. Bottom: Largest intersection radius between each reference density (highlighted in the legend) and all higher ionization states for a gold atom.
  • ...and 5 more figures