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, 2010.

2. Varshalovich, D. A., Quantum theory of angular momentum, 1988.