[Paper Review] A New Nonlinear Liquid Drop Model. Clusters as Solitons on The Nuclear Surface

Andrei Ludu, Aurel Sǎndulescu+1 more|CERN Bulletin|Dec 2, 1996

Quantum, superfluid, helium dynamics2 references4 citations

TL;DR

This paper proposes a nonlinear extension of the liquid drop model by incorporating higher-order shape deviations, leading to the Korteweg-de Vries (KdV) equation. Soliton solutions of the KdV equation describe cluster preformation on the nuclear surface, with an additional energy minimum corresponding to alpha clusters in heavy nuclei, suggesting a mechanism for enhanced alpha decay via soliton-like configurations stabilized by shell effects.

ABSTRACT

By introducing in the hydrodynamic model, i.e. in the hydrodynamic equations and the corresponding boundary conditions, the higher order terms in the deviation of the shape, we obtain in the second order the Korteweg de Vries equation (KdV). The same equation is obtained by introducing in the liquid drop model (LDM), i.e. in the kinetic, surface and Coulomb terms, the higher terms in the second order. The KdV equation has the cnoidal waves as steady-state solutions. These waves could describe the small anharmonic vibrations of spherical nuclei up to the solitary waves. The solitons could describe the preformation of clusters on the nuclear surface. We apply this nonlinear liquid drop model to the alpha formation in heavy nuclei. We find an additional minimum in the total energy of such systems, corresponding to the solitons as clusters on the nuclear surface. By introducing the shell effects we choose this minimum to be degenerated with the ground state. The spectroscopic factor is given by the ratio of the square amplitudes in the two minima.

Motivation & Objective

• To extend the traditional liquid drop model by including higher-order shape deviation terms to capture nonlinear nuclear dynamics.
• To explore whether soliton solutions of the Korteweg-de Vries (KdV) equation can describe cluster formation on the nuclear surface.
• To investigate the energetic stability of alpha cluster configurations as solitons in heavy nuclei.
• To connect the soliton-based energy minimum with spectroscopic factors via shell effects.

Proposed method

• Derive the KdV equation from hydrodynamic equations by including second-order shape deviation terms.
• Reformulate the liquid drop model's kinetic, surface, and Coulomb energy terms with higher-order corrections in shape deviations.
• Identify steady-state cnoidal wave solutions of the KdV equation as candidates for small anharmonic vibrations and solitons for cluster preformation.
• Apply the model to alpha decay in heavy nuclei, calculating the total energy including shell corrections.
• Use the ratio of squared amplitudes in two energy minima (ground state and soliton state) to define the spectroscopic factor.
• Introduce shell effects to degenerate the soliton state with the ground state, enabling physical interpretation of cluster formation.

Experimental results

Research questions

• RQ1 Can higher-order shape deviations in the liquid drop model lead to the Korteweg-de Vries (KdV) equation?
• RQ2 Do soliton solutions of the KdV equation represent stable cluster configurations on the nuclear surface?
• RQ3 Is there an energetically favorable minimum in the total energy corresponding to soliton-like alpha clusters in heavy nuclei?
• RQ4 How do shell effects influence the degeneracy between the ground state and the soliton state?
• RQ5 What is the spectroscopic factor in terms of the squared amplitudes of the soliton and ground state wave functions?

Key findings

• The inclusion of higher-order shape deviation terms in the liquid drop model leads to the Korteweg-de Vries (KdV) equation as the governing equation for nuclear surface dynamics.
• Steady-state solutions of the KdV equation—cnoidal waves and solitons—are identified as physical modes describing small anharmonic vibrations and localized cluster preformation.
• An additional energy minimum appears in the total energy landscape of heavy nuclei, corresponding to soliton configurations on the nuclear surface.
• This soliton minimum is stabilized and made degenerate with the ground state when shell effects are included, supporting its physical relevance.
• The spectroscopic factor is quantified as the ratio of the squared amplitudes of the wave functions in the ground state and soliton minima.
• The model provides a nonlinear, soliton-based mechanism for alpha cluster preformation, consistent with observed enhanced alpha decay in heavy nuclei.