[Paper Review] Modeling heavy ion ionization loss in the MARS15 code

Igor Rakhno, N. Mokhov+1 more|ArXiv.org|May 5, 2005

Particle Detector Development and Performance14 references3 citations

TL;DR

This paper presents an advanced ionization energy loss model in the MARS15 Monte Carlo code, incorporating Lindhard-Sørensen corrections for finite nuclear size and low/high-energy effects such as Barkas and shell corrections. The model achieves excellent agreement with experimental data and NIST tabulated values, particularly within 1–2% for protons and up to 10% for super-heavy ions like lead and uranium across a broad energy range from keV/A to GeV/A.

ABSTRACT

The needs of various accelerator and space projects stimulated recent developments to the MARS Monte Carlo code. One of the essential parts of those is heavy ion ionization energy loss. This paper describes an implementation of several corrections to dE/dx in order to take into account the deviations from the Bethe theory at low and high energies as well as the effect of a finite nuclear size at ultra-relativistic energies. Special attention is paid to the transition energy region where the onset of the effect of a finite nuclear size is observed. Comparisons with experimental data and NIST data are presented.

Motivation & Objective

• To improve the accuracy of ionization energy loss calculations for heavy ions in the MARS15 Monte Carlo code.
• To address deviations from Bethe theory at low and high energies, especially for relativistic and super-heavy ions.
• To implement finite nuclear size effects in ultra-relativistic regimes, replacing older Mott and Bloch corrections.
• To validate the model against experimental data and NIST tabulated values across a wide energy range.
• To support radiation transport simulations for accelerator, space, and shielding applications involving heavy ions.

Proposed method

• The model uses a three-region approach: Bethe theory for energies >10 MeV/A, tabulated proton stopping power below 1 MeV/A, and a mix-and-match interpolation in between.
• The Lindhard-Sørensen formalism is applied for relativistic corrections, accounting for finite nuclear size via relativistic Coulomb phase shifts and partial wave summation.
• The asymptotic ultra-relativistic limit is used when γmₑcR ≫ ħ/2, yielding L_ultra = ln(2c/Rωₚ) - 0.2.
• Low-energy corrections include the Barkas effect (z³ term) via a Thomas-Fermi-based function F(V), and shell corrections (ΔL_shell = -C/Z) for slow projectiles.
• The model combines these corrections into a unified ionization logarithm L(β) = L₀(β) + δ/2 + ΣΔLᵢ, with effective charge effects adjusted for target Z.
• A mix-and-match interpolation procedure ensures smooth transition between low-energy, intermediate, and high-energy regimes.

Experimental results

Research questions

• RQ1 How can ionization energy loss for heavy ions be accurately modeled across the full energy range from keV/A to GeV/A?
• RQ2 What is the impact of finite nuclear size on ionization loss at ultra-relativistic energies?
• RQ3 How do Barkas and shell corrections improve agreement with experimental data at low energies?
• RQ4 How does the Lindhard-Sørensen formalism compare to traditional Mott and Bloch corrections in relativistic regimes?
• RQ5 To what extent does the MARS15 model reproduce NIST tabulated stopping power for protons and alpha particles?

Key findings

• The MARS15 model achieves within 1.3% agreement with NIST data for protons across the entire energy range, improving on MCNP5’s 3% discrepancy above 4 MeV.
• For alpha particles, the model shows 10–15% deviation below 400 keV/A, consistent with known theory-experiment discrepancies in that region.
• The Lindhard-Sørensen correction reduces systematic underestimation of ionization loss by 2–3% compared to the BMA (Bloch-Mott-Ahlen) approach, especially for high-Z ions like Xe.
• For lead and uranium ions, the model shows good agreement within 10% at low energies and correctly captures the saturation of ionization loss due to finite nuclear size at ultra-relativistic energies.
• At 160 GeV/u, the model matches experimental data for lead ions in argon, confirming the finite-size effect suppresses the logarithmic rise seen in point-like models.
• The model successfully reproduces the density dependence of ionization loss, with lowest-density targets showing highest energy loss for ultra-relativistic uranium ions.