Low energy Lagrangian and energy levels of deformed nuclei Eduardo A. Coello Perez

Symmetry of the system  For intrinsically deformed nuclei, the symmetry of the Lagrangian is “spontaneously broken”.  The ground state of the system is invariant under axial rotations denoted by h.

Deformed nuclei

Low energy modes  Any rotation r in SO(3) can be written as the product of two rotations gh. In terms of the Euler angles  The degrees of freedom of g( α, β ) are the degrees of freedom of the low energy or Nambu-Goldstone modes

Dynamics  The dynamics of the system can be studied in terms of  Under a general rotation r

Dynamics  According to the Baker-Campbell-Hausdorff formula  These functions behave properly under rotations around the z axis. Also

Lagrangian  A Lagrangian can be constructed from the previous functions.  The energy spectra for this Lagrangian is of the form

Charge  Under a small rotation given by ω  A comparison between the expressions leads to

Charge  Since the Lagrangian is invariant under rotations  From here

Real data  As an example consider the low energy level scheme of 156 Sm.  The energy levels given by the constructed Lagrangian are

Real data  Calculated energies for 156 Gd are

Summary  The identification of the degrees of freedom of the low enery modes lead to the construction of a low energy Lagrangian for deformed nuclei.  The energy level scheme predicted by the Lagrangian fits the low energy level scheme of deformed nuclei.

References  1. Papenbrock, Thomas, Effective theory for deformed nuclei,  2. Varshalovich, D. A., Quantum theory of angular momentum, 1988.