Energy-differential measurement of the $^{\mathrm{nat}}$C(n,p) and $^{\mathrm{nat}}$C(n,d) reactions at the n_TOF facility at CERN

This paper presents energy-differential cross-section measurements for the $^{\mathrm{nat}}$C(n,p) and $^{\mathrm{nat}}$C(n,d) reactions up to 25 MeV at the n_TOF facility, revealing significant discrepancies with major evaluation libraries while showing unexpected agreement with TALYS-2.0 calculations, particularly for the (n,p) reaction.

The Gist

Imagine you are trying to understand how a specific type of "bullet" (a neutron) interacts with a very common target: a block of carbon (like the graphite in a pencil). When these bullets hit the carbon, they sometimes knock out smaller pieces, like tiny marbles (protons) or slightly heavier marbles (deuterons).

Scientists at CERN's n_TOF facility (a giant machine that shoots neutrons at targets) decided to measure exactly how often this happens and how much energy is involved. They focused on two specific reactions:

1. The (n,p) reaction: A neutron hits carbon, and a proton flies out.
2. The (n,d) reaction: A neutron hits carbon, and a deuteron (a proton and neutron stuck together) flies out.

Here is the story of what they did, how they did it, and what they found, explained simply.

The Setup: A High-Speed Camera and a Carbon Pencil

The scientists didn't just use a regular camera; they used a "time-of-flight" technique. Imagine a race track that is 182.5 meters long.

• They fired a burst of protons at a lead target, creating a spray of neutrons.
• These neutrons raced down the long track.
• Because they are fast, the time it took them to reach the end told the scientists exactly how much energy they had. Faster neutrons = more energy.

In the middle of this track, they placed a very thin slice of natural carbon (about as thick as a human hair). Surrounding this slice were two sets of silicon telescopes. Think of these telescopes as high-tech sandwich detectors.

• Layer 1 (The Thin Slice): A very thin layer of silicon that measures how much energy a particle loses just by passing through (like a speed bump).
• Layer 2 (The Thick Slice): A thicker layer that catches the particle and measures its total remaining energy.

By comparing the "speed bump" energy to the "total energy," the scientists could tell the difference between a proton and a deuteron, even though they look very similar. It's like telling the difference between a ping-pong ball and a golf ball by how they bounce off a wall.

The Challenge: Sorting the Messy Data

The data they collected was a chaotic mix. When a neutron hits carbon, it doesn't just produce one clean result. It can leave the remaining carbon nucleus in a state of "excitement" (an excited state), similar to how a bell rings with a specific tone after being struck.

• The nucleus could be in its "calm" state (ground state) or in various "excited" states.
• Each state produces particles with slightly different energies and directions.

To make sense of this, the scientists had to use a computer model (TALYS-2.0). Think of this model as a sophisticated recipe book that predicts how the carbon nucleus behaves. They didn't just use one recipe; they tried 480 different variations of the recipe to see how much the results changed. This was crucial because if the recipe was wrong, their measurements would be wrong.

They also used Artificial Intelligence (Neural Networks). Since the particles were so close together in the data, a human eye couldn't easily separate the protons from the deuterons. They trained a computer to recognize the unique "fingerprint" of each particle type, acting like a very smart bouncer at a club who knows exactly who belongs in which line.

The Big Discovery: The "Missing" Energy

When the scientists finally calculated the results, they found something surprising.

The "Library" vs. The "Real World"
Scientists usually rely on "libraries" of data (like a library of physics books) that tell them what to expect when neutrons hit carbon. These libraries are used to design nuclear reactors, medical equipment, and space shields.

• The Expectation: The libraries said the reaction should happen a certain amount of times (a specific "cross-section").
• The Reality: The n_TOF team found that the reaction happened significantly more often than the libraries predicted, especially for the proton reaction.

It's like if a weather forecast said there was a 10% chance of rain, but when you stepped outside, it was pouring. The existing "forecasts" (the data libraries) were underestimating the storm.

The Silver Lining
Interestingly, their new, more detailed measurements matched the predictions of the TALYS-2.0 computer model very well. This suggests that the computer model was actually right all along, but the "libraries" (the books scientists use) had outdated or incorrect information.

Why Does This Matter?

The paper explains that this isn't just a theoretical game. Carbon is everywhere:

• In our bodies: It's a main part of our tissues.
• In medicine: It's used in cancer treatments (hadrontherapy).
• In space: It's used in shielding for satellites.

When high-energy neutrons hit carbon in these environments, they create secondary particles. If we don't know exactly how often this happens, we can't accurately calculate the radiation dose a patient receives or how well a spaceship shield will work.

The Conclusion

The team successfully measured these reactions with high precision, from the moment the reaction starts (about 14-15 MeV) up to 25 MeV.

• They proved that the reaction happens more frequently than current standard data suggests.
• They confirmed that their results agree with a specific computer model (TALYS-2.0) but disagree with the major data libraries used by engineers and doctors today.

In short, they took a very thin slice of carbon, shot it with high-speed neutrons, used AI and super-computers to sort the debris, and discovered that the "rulebook" for how carbon reacts to neutrons needs a major update.

Technical Summary

1. Problem Statement and Motivation

Neutron-induced reactions on light nuclei, specifically carbon, are critical for fundamental nuclear physics, medical applications (hadrontherapy), radiation shielding, and the calibration of neutron-sensitive diamond detectors.

• The Gap: While the $^{nat}$C(n,p) and $^{nat}$C(n,d) reactions have been studied previously, the cross-section data in the energy range just above the reaction thresholds (10–25 MeV) remain poorly constrained. Existing experimental data show significant inconsistencies, and major nuclear data evaluation libraries (e.g., ENDF/B-VIII.1, TENDL-2023) often disagree with each other and with integral measurements.
• The Trigger: A previous integral cross-section measurement at n_TOF yielded values significantly higher than any available evaluation, suggesting a potential systematic error in existing libraries or a need for more precise differential data.
• Objective: To perform a high-precision, energy-differential measurement of the $^{nat}$C(n,p) and $^{nat}$C(n,d) reactions up to 25 MeV to resolve discrepancies and provide benchmark data.

2. Methodology

Experimental Setup

• Facility: The experiment was conducted at the n_TOF (neutron Time-Of-Flight) facility at CERN, specifically in the EAR1 (Experimental Area 1) with a flight path of 182.5 m. This long path ensures high energy resolution at high neutron energies.
• Target: A natural carbon ($^{nat}$C) graphite sample (5 cm $\times$ 5 cm, 0.25 mm thick) tilted at 45°, resulting in an effective thickness of 0.35 mm. Natural carbon consists of ~98.9% $^{12}$C and ~1.1% $^{13}$C.
• Detectors: Two position-sensitive $\Delta E$-$E$ silicon telescopes were placed outside the beam (covering angles from 20° to 140°).
  • Each telescope comprises a 20 $\mu$m thin $\Delta E$ layer and a 300 $\mu$m thick $E$ layer.
  • The detectors were segmented into 64 strips total (32 per telescope), allowing for granular detection.
• Beam: A pulsed 20 GeV proton beam from the CERN Proton Synchrotron strikes a lead target, producing a white neutron spectrum via spallation.

Data Analysis and Reconstruction

The analysis involved four main steps:

1. Neutron Energy Calibration: Neutron energy ($E_n$) was determined via time-of-flight (TOF) relative to the proton pulse (synchronized via a Wall Current Monitor). The resolution function of the beam was modeled using a Gaussian distribution for energies >4 MeV.
2. Particle Discrimination: A Machine Learning (Neural Network) approach was employed to distinguish between protons, deuterons, tritons, and background particles based on their $\Delta E$-$E$ signatures. This was necessary because the patterns for protons and deuterons are very close in parameter space.
3. Cross-Section Reconstruction:
   • The problem was formulated as an inverse problem (Fredholm integral equation of the first kind) relating observed counts to the underlying cross-sections.
   • Model Inputs: Since the angular distributions and branching ratios for excited states in the daughter nuclei ($^{11}$B, $^{12}$B, $^{13}$B) are not directly measured, the analysis relied on TALYS-2.0 nuclear reaction code calculations.
   • Uncertainty Quantification: To address model dependence, the authors performed 480 different TALYS model sets (varying parameters like level density models, optical models, and gamma strength functions). They also tested "artificial" distributions (isotropic angular distributions and simplified branching ratios) to establish conservative systematic uncertainties.
   • Isotope Separation: Due to the low abundance of $^{13}$C and the inability to distinguish reactions by isotope, the final results were reconstructed for natural carbon ($^{nat}$C), weighted by natural abundance.

3. Key Contributions

• High-Precision Differential Data: Provided the first high-resolution, energy-differential cross-section data for $^{nat}$C(n,p) and $^{nat}$C(n,d) from threshold up to 25 MeV.
• Advanced Analysis Techniques: Successfully applied neural networks for particle identification in a challenging $\Delta E$-$E$ regime and utilized a rigorous statistical framework (least-squares minimization with Lagrange multipliers) to reconstruct cross-sections while accounting for the neutron beam resolution function.
• Systematic Uncertainty Assessment: Conducted an exhaustive sensitivity analysis using 384 TALYS model sets and artificial distributions, demonstrating that model-related uncertainties are below 10% for (n,p) and below 20% for (n,d).
• Validation of TALYS: Demonstrated that while standard evaluation libraries fail, specific TALYS-2.0 configurations can accurately reproduce the experimental data, provided the correct model parameters are chosen.

4. Key Results

• Cross-Section Values:
  • $^{nat}$C(n,p): The measured cross-sections are significantly higher (by a large margin) than those predicted by major evaluation libraries (TENDL-2023, ENDF/B-VIII.1, JEFF). The values peak around 75 mb at 22 MeV.
  • $^{nat}$C(n,d): The data also show discrepancies with libraries, though the agreement is better for some datasets (e.g., TENDL-2015) compared to the (n,p) reaction.
• Comparison with Libraries:
  • The n_TOF results are largely inconsistent with TENDL-2023, ENDF/B-VIII.1, and JEFF-3.3.
  • The results show unexpected agreement with specific TALYS-2.0 calculations (specifically those using the back-shifted Fermi gas model for level density).
• Comparison with Past Data:
  • The new data aligns well with older measurements by Kreger & Kern (1959) and Bobyr et al. (1972).
  • It contradicts the basis for current libraries (e.g., Rimmer & Fisher data).
  • It confirms the earlier integral measurement from n_TOF, validating that the high cross-section is a real physical phenomenon and not an artifact of the integral method.

5. Significance and Implications

• Nuclear Data Evaluation: The findings necessitate a revision of the $^{nat}$C(n,p) and (n,d) evaluations in major nuclear data libraries. The current underestimation of these cross-sections could lead to errors in radiation transport simulations.
• Medical Applications (Hadrontherapy): Accurate cross-sections are vital for calculating secondary neutron doses in healthy tissues during proton and ion therapy. The higher measured cross-sections imply that current dose estimates for secondary neutrons may be underestimated.
• Detector Development: The results are crucial for the design and calibration of diamond neutron detectors, which rely on precise knowledge of reaction thresholds and cross-sections in the fast-neutron range.
• Astrophysics and Fusion: Improved data aids in shielding calculations for fusion facilities (like IFMIF-DONES) and space dosimetry.

In conclusion, this work provides a definitive experimental benchmark that challenges existing nuclear data libraries and highlights the importance of combining high-precision differential measurements with advanced theoretical modeling (TALYS) to resolve long-standing discrepancies in light-nucleus reaction physics.