Random close packing of binary hard spheres favors the stability of neutron-rich atomic nuclei

Abstract: In spite of the success of the Bethe-Weizs\"acker mass formula in its modern numerical and predictive implementations, the common-knowledge principle that it is electrostatics which, ultimately, favors neutron-rich nuclei still presents unclear aspects. For example, while it is true that the Coulomb interaction promotes the tendency towards neutron-rich nuclei, the opposite effects of Majorana exchange forces and Pauli exclusion are known to counteract this tendency. We show that a recent analytical progress in the mathematical description of random close packing of spheres with different sizes provides a missing contribution to the theoretical description of the $Z$ versus $N$ slope in the nuclides chart. In particular, the theory suggests, on geometric grounds and with a physically-reasoned assumption that the excluded-volume size of neutrons is 20\% larger than that of protons, that the most stable nuclei are those with ratio $Z/N\approx 0.75$. This new `geometric'' random-packing contribution to the semi-empirical mass formula may be the missing aspect of nuclear structure that tilts the balance towards neutron-rich nuclei in the Segr\e stability chart.

• The paper demonstrates that random close packing of binary hard spheres predicts an optimal proton ratio of approximately 43% (N/Z ~1.33) for neutron-rich nuclei.
• It utilizes analytical advancements to extend RCP models to binary mixtures, accounting for the size difference between protons and neutrons.
• These insights offer a geometric complement to conventional nuclear physics models and align with empirical observations of nuclear stability.

Random Close Packing of Binary Hard Spheres and Neutron-Rich Atomic Nuclei Stability

The paper, titled "Random close packing of binary hard spheres favors the stability of neutron-rich atomic nuclei," investigates a geometric approach to understanding the stability of neutron-rich atomic nuclei through the random close packing (RCP) concept. The authors propose that the stability of such nuclei can be significantly influenced by the packing geometry of protons and neutrons, extending beyond the traditional electrostatic and nuclear force explanations.

Main Concepts and Findings

1. Random Close Packing Analogy: The paper explores the analogy between nucleons in atomic nuclei and random close packing of hard spheres with two different sizes (neutrons and protons). This concept draws inspiration from the liquid drop model of nuclei, which has successfully described certain aspects of nuclear fission and structure previously.
2. RCP of Binary Spheres: By utilizing recent analytical advancements in describing RCP, the authors extend this theory to binary mixtures of spheres, representing protons and neutrons. For this, they consider protons and neutrons having distinct sizes, with neutrons having a larger effective diameter due to their charge neutral nature.
3. Analytical Prediction: The model predicts that the most stable nuclear configurations arise when the ratio of protons to the total number of nucleons (Z/A) is approximately 0.43, equating to a neutron-to-proton ratio (N/Z) of about 1.33. This prediction aligns with empirical data showing the preference for neutron-rich nuclei as observed in the Segrè chart.
4. Physical Implications: The RCP framework suggests a geometric mechanism contributing to the stability of neutron-rich nuclei. It proposes that the optimal stability is achieved when the nucleons are packed to maximize the volume fraction, reducing the space and minimizing the energy due to more efficient packing.
5. Empirical Validation: The theory accurately predicts the maximal random close packing density when taking into account size variations between neutrons and protons, providing a compelling geometrical explanation for the Z/N ratio observed in stable nuclides.

Theoretical and Practical Implications

This approach highlights a structural aspect of nucleon interactions that may complement traditional nuclear physics models. The findings suggest that by considering nucleon packing geometries, one can achieve theoretical predictions that align closely with experimental observations without solely relying on complex quantum mechanical computations.

For practical applications, the results inform nuclear physics research, particularly in modeling nuclear stability and predicting the properties of neutron-rich isotopes and superheavy elements. The geometric insights could also be beneficial in nuclear material engineering, aiding in the development of stable isotopic configurations for energy applications.

Future Directions

The research opens avenues for further exploration in nuclear structure theory, suggesting that incorporating geometric considerations can enhance the predictive capabilities of nuclear models. Future work could focus on extending this model to include more complex geometries and considering its applicability in other areas of physics, such as the structure of neutron stars where similar dense packing phenomena are encountered.

In conclusion, this paper provides a novel perspective on nucleon interactions within atomic nuclei, emphasizing the significant role of packing geometry in influencing nuclear stability, particularly for neutron-rich isotopes. The geometrical approach offers complementary insights to conventional nuclear theories, potentially informing future research and applications in nuclear physics and material science.