Delaunay-Like Compact Equilibria in the Liquid Drop Model

Area of Science:

• Nuclear physics and mathematical modeling
• Theoretical physics and computational mathematics

Background:

• The liquid drop model, developed by Gamow, Bohr, and Wheeler, explains atomic nuclei by balancing surface tension and proton repulsion (Coulomb force).
• The model's energy functional involves surface area and a Coulombic repulsion term, subject to a fixed nuclear volume constraint.

Purpose of the Study:

• To investigate the existence of non-spherical, critical surface configurations for the liquid drop model's energy functional.
• To explore solutions beyond the typical spherical nuclei, particularly for large nuclear volumes.

Main Methods:

• Formulating the problem as finding critical points of an energy functional involving surface area and electrostatic repulsion under a volume constraint.
• Analyzing the associated Euler-Lagrange equation, which relates mean curvature to the electrostatic potential and a Lagrange multiplier.
• Investigating solutions that deviate from the standard spherical equilibrium.

Main Results:

• Spherical nuclei are confirmed as solutions and are energy minimizers for small volumes.
• A novel class of compact, embedded solutions with large volumes was discovered.
• These new solutions exhibit a 'pearl necklace' geometry, approximating Delaunay's unduloid surfaces.

Conclusions:

• The existence of non-spherical, stable nuclear configurations is demonstrated for large volumes.
• These findings expand the understanding of nuclear shapes beyond simple spheres, offering new insights into nuclear structure.
• The results highlight the complex interplay between surface tension and Coulomb forces in determining nuclear equilibria.