Modeling Ultra-High-Energy Cosmic Rays propagation using the input from Configuration Interaction Shell Model

This paper employs the Configuration Interaction Shell Model to predict photon strength functions for light nuclei ($A=7$–$40$) and evaluates their impact on the propagation modeling of ultra-high-energy cosmic rays, specifically addressing the needs of the PANDORA project.

The Gist

Imagine the universe is a vast, dark ocean, and Ultra-High-Energy Cosmic Rays (UHECRs) are like giant, super-fast submarines traveling through it. These aren't normal submarines; they are atomic nuclei (like tiny solar systems made of protons and neutrons) moving at speeds so close to the speed of light that they carry more energy than a baseball thrown by a professional pitcher, but packed into something smaller than a grain of sand.

As these cosmic "submarines" travel across the galaxy, they don't just glide smoothly. They crash into the "fog" of the universe: the Cosmic Microwave Background (CMB). This fog is made of ancient, low-energy photons (light particles) left over from the Big Bang.

The Problem: The "Fog" is Dangerous

When a cosmic ray hits a photon from the fog, it's like a high-speed car hitting a pebble. The collision is so violent that it can shatter the cosmic ray, knocking pieces off (like neutrons or protons). This process is called photodisintegration.

To predict how far these cosmic rays can travel before they break apart, scientists need to know exactly how "fragile" or "strong" the atomic nuclei are when hit by light. This strength is described by something called the Photon Strength Function (PSF). Think of the PSF as the nucleus's "armor rating" against light.

The Old Way vs. The New Way

For a long time, scientists used two main ways to guess this armor rating:

1. The Smooth Curve (Phenomenological Models): Imagine drawing a smooth, gentle hill to represent the armor. It's an average guess that works okay for big, heavy nuclei but misses the tiny bumps and cracks.
2. The Linear Response (Microscopic Models): Imagine trying to predict the armor by looking at how the nucleus vibrates like a drum. It's better, but it assumes the drum vibrates in a simple, predictable way.

The Problem: For light nuclei (small atoms like Carbon, Oxygen, or Calcium), the real world isn't smooth or simple. The nucleus is more like a complex, chaotic dance floor where every particle is jostling and interacting in unique ways. The old "smooth hill" and "simple drum" models fail to capture this chaos, leading to wrong predictions about how far cosmic rays can travel.

The Solution: The "Configuration Interaction Shell Model" (CI-SM)

This paper introduces a new, super-detailed method called the Configuration Interaction Shell Model (CI-SM).

The Analogy:
Imagine you want to predict how a specific, complex Lego castle will break if you throw a ball at it.

• The Old Models would say, "Well, it's a castle, so it will break in a generic way based on its average size."
• The CI-SM is like having a computer simulation that tracks every single Lego brick and every single connection between them. It simulates exactly how the bricks wiggle, shake, and interact with each other before the ball hits.

Because the CI-SM tracks every single interaction (correlation) between the protons and neutrons, it reveals that the "armor" isn't a smooth hill. Instead, it's fragmented—it has sharp peaks and valleys, like a jagged mountain range. Some parts are very weak, others are very strong.

What Did They Find?

The researchers applied this "Lego-brick-by-brick" simulation to light nuclei (from mass 7 to 40). Here are the key takeaways:

1. It's Messier Than We Thought: The new model shows that the nuclear "armor" is much more broken up and complex than the old smooth models suggested.
2. Better Accuracy: When they compared their jagged, detailed model to real-world experiments, it matched the data better than the other theoretical models, even without needing to "tweak" the numbers to fit.
3. Impact on Cosmic Travel: When they used this new, jagged armor data to simulate the journey of a Calcium-40 cosmic ray:
   • The old "smooth" models and the new "jagged" model agreed that the ray could travel a certain distance.
   • However, one of the other popular models (D1M+QRPA) predicted the ray would break apart much sooner. This was because that model thought the nucleus was weaker at the wrong energy levels.

Why Does This Matter?

If we use the wrong "armor rating" for cosmic rays, we might think they come from closer to home than they actually do, or vice versa. By using the CI-SM method, scientists can now build a more accurate map of the universe.

The Bottom Line:
This paper is like upgrading from a blurry, low-resolution map of the universe to a high-definition, 3D model. By understanding the tiny, chaotic details of how light nuclei react to light, we can finally trace the true origins of the most energetic particles in the universe. The authors are now working to expand this "high-definition" view to even heavier nuclei, completing the map for the next generation of cosmic ray explorers.

Technical Summary

1. Problem Statement

The propagation of Ultra-High-Energy Cosmic Rays (UHECRs) through intergalactic space is governed by their interactions with the Cosmic Microwave Background (CMB). These interactions are dominated by photoabsorption, where CMB photons (boosted to the Giant Dipole Resonance, GDR, energy region in the nucleus rest frame) cause photodisintegration of the cosmic ray nuclei.

Accurate modeling of this propagation requires precise knowledge of the Photon Strength Function (PSF), specifically the electric dipole (E1) response, for light to medium-mass nuclei ($A \le 60$).

• The Gap: While phenomenological models (e.g., Modified Lorentzian) and linear-response microscopic models (e.g., QRPA) are standard for mid-to-heavy nuclei, they often fail to reproduce the complex fragmentation of the PSF in light nuclei.
• The Need: The PANDORA project aims to understand UHECR mass distribution, necessitating a systematic, microscopic evaluation of E1 strength distributions for light nuclei ($p$ and $sd$-shells) that goes beyond the harmonic approximation of standard linear-response theories.

2. Methodology

The authors employed the Configuration Interaction Shell Model (CI-SM), a large-scale shell-model approach, to calculate E1 photoabsorption strength functions.

• Theoretical Framework:
  • Hamiltonian: Diagonalized the nuclear Hamiltonian in specific valence spaces using effective interactions:
    • $p$-shell nuclei ($A=7-16$): Used the WBP effective Hamiltonian in the $s-p-sd-pf$ model space.
    • $sd$-shell nuclei ($A=17-40$): Used the PSDPF interaction.
  • Excitations: Calculated transitions from $0\hbar\omega$ ground states to $1\hbar\omega$ excited states.
  • Strength Calculation: Used the Lanczos strength-function method to determine reduced transition probabilities $B(E1)$ and the resulting strength distributions.
• Normalization and Smearing:
  • Renormalization: To account for correlations outside the model space, the E1 operator was renormalized using effective charges. A global reduction factor $Q^2$ was applied, derived from the ratio of the calculated Energy-Weighted Sum Rule (EWSR) to the Thomas-Reiche-Kuhn (TRK) sum rule.
  • Continuum Coupling: The discrete $B(E1)$ distributions were folded with a generalized Lorentzian to simulate coupling to the continuum (escape width).
    • Width $\Gamma = 1$ MeV for most $sd$-shell nuclei.
    • Width $\Gamma = 3.5$ MeV for $p$-shell nuclei and $^{40}$Ca (to account for strength bunching due to closed shells).
• Application to UHECR:
  • The calculated PSFs were input into the TALYS reaction code to generate photonuclear cross-sections ($\sigma_{\gamma,x}$).
  • These cross-sections were used in a nuclear reaction network to simulate the propagation of a $^{40}$Ca seed nucleus through the CMB, tracking the evolution of the mean mass number ($\langle A \rangle$) and mass dispersion ($\sigma_A$) over distance.

3. Key Contributions

1. Systematic CI-SM Database: The study provides the first systematic CI-SM evaluation of E1 PSFs for all long-lived nuclei in the $p$-shell and $sd$-shell ($A=7$ to $40$).
2. Benchmarking: The results were rigorously compared against:
   • Available experimental photoabsorption data.
   • Phenomenological models (SMLO).
   • Microscopic linear-response models: D1M+QRPA (Gogny force) and RQFAMz (Relativistic Quasi-particle Finite Amplitude Method).
3. UHECR Propagation Impact: The paper quantifies how the choice of nuclear model (CI-SM vs. QRPA vs. SMLO) affects the predicted survival distance and mass evolution of UHECRs.

4. Key Results

A. Nuclear Structure Properties

• Fragmentation: The CI-SM approach captures nucleonic correlations beyond the harmonic approximation, resulting in significantly more fragmented strength distributions compared to the smooth, single-peak structures predicted by D1M+QRPA and RQFAMz.
• Centroids and Widths:
  • CI-SM centroids show stronger variations and are generally closer to experimental values than D1M+QRPA (which systematically underestimates centroids for light nuclei).
  • CI-SM widths are more scattered and lack the smooth mass dependence seen in QRPA models, reflecting the specific shell structure of individual nuclei.
• Pygmy Dipole Resonance (PDR): In neutron-rich isotopes (e.g., Oxygen and Neon chains), CI-SM predicts a distinct accumulation of low-lying dipole strength (PDR) that is more pronounced and structured than in linear-response models.

B. Comparison with Other Models

• Accuracy: Without empirical adjustments to centroids or widths, CI-SM predictions for $sd$-shell nuclei showed a Root-Mean-Square (RMS) deviation of 0.85 MeV for centroids and 0.50 MeV for widths against experimental data. This is superior to D1M+QRPA (1.71 MeV centroid deviation) and comparable to or slightly better than RQFAMz (0.89 MeV centroid deviation).
• Consistency: The CI-SM results are compatible with the recently developed RQFAMz approach but show significant variance with D1M+QRPA, particularly regarding the energy position of the GDR peak.

C. Impact on UHECR Propagation

• Survival Distance: Simulations of a $^{40}$Ca source revealed that the D1M+QRPA model predicts a significantly shorter survival distance for UHECRs compared to CI-SM, SMLO, and RQFAMz.
  • Reason: D1M+QRPA predicts lower energy centroids for the E1 strength, leading to a larger overlap with the CMB photon density (Eq. 10), thereby increasing the photodisintegration rate.
• Mass Evolution: The CI-SM, SMLO, and RQFAMz models yield similar predictions for the evolution of the mean mass number $\langle A \rangle$ and its dispersion. The D1M+QRPA model deviates significantly, predicting faster mass loss.
• Conclusion on Models: The CI-SM results align closely with RQFAMz and SMLO, suggesting that the D1M+QRPA approach may overestimate photodisintegration rates for light nuclei due to its centroid shift.

5. Significance and Future Work

• Validation of CI-SM: The study demonstrates that CI-SM is a viable and valuable alternative to linear-response models for generating PSFs required in astrophysical applications, particularly for light nuclei where shell effects and fragmentation are critical.
• PANDORA Project Support: The work directly supports the PANDORA initiative by providing a robust theoretical database for light nuclei, essential for interpreting UHECR composition data.
• Limitations and Next Steps:
  • The current study covers nuclei up to $A=40$ ($sd$-shell). Since UHECR sources may contain nuclei up to $^{56}$Fe, the authors note that CI-SM predictions currently cover only ~50% of the necessary nuclei.
  • Future Work: The authors are developing approaches to extend CI-SM calculations to the $pf$-shell (up to $^{50}$Fe) to complete the systematics required for full UHECR modeling. They also plan to investigate the impact of these nuclear models in non-thermal radiation environments (e.g., Gamma-Ray Bursts).

In summary, this paper establishes the Configuration Interaction Shell Model as a high-fidelity tool for predicting photon strength functions in light nuclei, revealing that standard linear-response models may introduce systematic biases in the theoretical modeling of Ultra-High-Energy Cosmic Ray propagation.