1. by Santosh Bhatt & Lawrence W. Townsend The University of Tennessee, Knoxville, TN 37996-2300

2. Introduction Transport Formalism The abrasion-ablation model Abrasion cross section Ablation cross section Results Questions/ Comments

3. To establish protection from high-energy space radiation during long term space exploration, accurate and precise radiation transport models and relevant nuclear cross section databases are needed. An abrasion-ablation model has been successfully used to describe the high-energy, heavy-ion particle scattering for these applications.

4. Unlike the collision with atomic electrons, nuclear collisions result in loss of mass and charge of the ion and the struck nucleus, with many secondary particles being produced. The secondary particles (light ion fragments and nucleons) produced will have longer ranges and mean free paths, thus resulting into greater penetration depths. These secondaries will further undergo nuclear interactions, producing more secondaries, which can penetrate deeper into the material. Accurate models for transport of high energy heavy ions and their secondaries are needed to for appropriate shielding design.

5. It is a two staged process and incorporates a simple idea of interaction between two relativistic heavy particles, projectile P and target T. The first stage is abrasion process. This stage can be visualized as projectile P, moving with some initial momentum colliding with target T and thus the overlapping volumes being sheared off due to collision. The remaining piece of the projectile or target, called a prefragment or spectator, remains in its original trajectory with its pre-collision velocity. The final states are reached when the excited prefragements decay by emitting gamma rays and/or they disintegrate into fragments or nucleons.

6. Figure 1: Schematic of abrasion-ablation model. (Giacomelli et al. [4])

7. The abrasion part of the model is based on Glauber multiple scattering theory which applies the expansion of scattering terms using eikonal differential approximations. The eikonal expansions in the Glauber model are considered in terms of small angle, high energy approximations in the center of momentum system. Further they expansion also assumes that the total eikonal phase is equal to the sum of individual eikonal phases for scattering of projectile off each target constituent. The assumption above are only valid for heavy charge particles at high energies where forward scattering dominates and may not necessarily be true at lower energies and for secondaries. In this study, we have extended the current abrasion model by including the next two higher order correction terms in eikonal expansion using Wallace’s perturbation method with out making any small angle approximation.

8. The cross-section for abrading n nucleons from the projectile nucleus is given by and the total absorption cross section is Where

9. The phase factor has been approximated by Glauber as where V is the spherically symmetric potential And the scattering amplitude as the partial sum for a fixed wave number k, in impact parameter representation, is given by

10. Using Wallace’s perturbation approach to derive higher order terms for the correction of Glauber’s small angle approximation, we find Wallace’s perturbation method develops the eikonal expansions using direct conversion of the partial wave sum to Fourier-Bessel integral based on an expansion of Legendre polynomials. We can see that the first term in approximation is exactly same as Glauber’s model.

11. And the imaginary parts of the phase function after performing the integral can be given as: Recall that:

12. One of the concerns in the above given eikonal expansions is flux conservation, which requires Secondly, it is of importance to discuss the convergence of the series and also validity of results with regards to quantum mechanical results. Carstoiu et al conclude that the series shows good convergence properties and is in reasonable agreement with quantum mechanical results. Further, they also conclude that these expansions can be used for calculations of total cross section even at low energies

13. In the ablation stage, after the projectile target collision, the prefragment nuclei give up their excess energies by evaporating fragment nucleons, light ion clusters and gamma rays while decaying into their ground states. Hence the total cross-section for nucleon production from the ablation process is where P i (j, k) is the probability of a prefragment emitting a particle E j

16. Addition of two correction terms show a significant increase in cross- section, which could be pivotal in dose calculations and shielding design considerations for space missions For light ion projectiles (A<20), the cross-sections were higher until 200 MeV and seem to converge as we go higher. For heavy ions (A >20), there was significant jump at lower energies. The cross-sections converged at around 5000 MeV Future work includes comparisons of projectile-target systems for various different projectile-target combinations. Determination of number of higher order correction terms needed Implementing the model into current codes

17. Questions / Comments

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