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.