Impact of the Center of Mass Fluctuations on the Ground State Properties of Nuclei

Ground state properties across the entire nuclear chart are described predominantly and rather accurately within the density functional theory (DFT). DFT however breaks many symmetries, among them the most important being the translational, rotational, and gauge symmetries. The translational symmetry breaking is special, since it is broken for all nuclei, unlike the rotational and gauge symmetries. The center-of-mass (CoM) correction most commonly used in the literature [see Vautherin and Brink, Phys. Rev. C {\bf 5}, 626 (1972) and Bender {\it et al.}, Rev. Mod. Phys. {\bf 75}, 121 (2003)] leads to a gain of 15,…,19 MeV, which varies rather weakly for medium and heavy mass nuclei. A better approximation to the CoM correction was suggested by Butler {\it et al.}, Nu cl. Phys. A {\bf 422}, 157 (1984) and its magnitude varies between 10 and 5 MeV from light to heavy nuclei, a correction which is also significantly larger than the RMS energy error in the Bethe-Weizsäcker mass formula, initially proposed by Gamow, Proc. Phys. Soc. A {\bf 126}, 157 (1930), which is at most 3.5 MeV, and which for heavy nuclei corresponds to about 0.2% of their mass. ….

💡 Research Summary

This paper, “Impact of the Center of Mass Fluctuations on the Ground State Properties of Nuclei,” presents a critical analysis and an improved solution for a persistent problem in nuclear structure theory: the accurate treatment of center-of-mass (CoM) motion within the framework of Density Functional Theory (DFT).

The core issue stems from the mean-field approximation inherent in DFT, which breaks fundamental symmetries, including translational invariance. Because the many-body wave function is not an eigenstate of total momentum, the calculated ground-state energy spuriously includes the kinetic energy of the CoM motion, artificially reducing the binding energy. Correcting for this spurious CoM energy (E_CoM) is essential for precise mass predictions. The paper notes that the magnitude of E_CoM, ranging from several to tens of MeV, is significantly larger than the typical error of semi-empirical mass formulas (~3.5 MeV), making it a major source of error in ab initio calculations.

The authors first review two commonly used approximate prescriptions: the mass renormalization method by Vautherin and Brink (Eq. 2), which yields a nearly mass-independent correction of 15-19 MeV, and the method by Butler et al. (Eq. 5), based on a harmonic oscillator model, which gives a correction decreasing from ~10 MeV in light nuclei to ~5 MeV in heavy nuclei. However, the paper identifies a fundamental flaw in these approaches: they evaluate the expectation value of the CoM kinetic energy using the symmetry-broken mean-field wave function. This process can contaminate the result with contributions from excited states that have non-zero CoM momentum, meaning the correction is not purely for the ground state.

To overcome this limitation, the paper advocates for the rigorous symmetry restoration method proposed by Peierls and Yoccoz (PY). The PY method (Eq. 10) is a projection technique. It constructs a translationally invariant intrinsic wave function by superposing copies of the original mean-field wave function shifted by all possible displacement vectors. This procedure is mathematically equivalent to angular momentum projection used for restoring rotational symmetry and is the only method guaranteed to extract the genuine CoM correction for the ground state without contamination from excited states.

The main results are summarized in Figure 1. The authors calculate E_CoM using the PY projection method with the SeaLL1 Energy Density Functional (black solid line and dots). These results are compared against values from Da Costa et al. (red triangles, using Eq. 1), Dobaczewski (blue line, using a combination of PY and Lipkin methods), and the Butler et al. prescription (green squares). The PY-based E_CoM values are generally larger than those from other methods, indicating a more complete removal of CoM fluctuations. Furthermore, the PY correction shows a mass dependence of approximately E_CoM ∝ A^(-0.131), which differs from the A^(-1) dependence suggested elsewhere and the near-constant value from the Vautherin-Brink prescription.

The implications are profound. An accurate E_CoM is not merely about getting a better mass number; it directly influences the calculated nuclear surface tension. Since surface tension is a key ingredient in determining fission barrier heights, an improved CoM correction can significantly impact predictions in nuclear fission dynamics and astrophysical processes like the r-process nucleosynthesis. The paper also briefly touches on relativistic considerations, correctly noting that the physical nuclear mass M_A = A*m - BE/c^2 should be used in the CoM kinetic energy operator, a detail often overlooked in non-relativistic treatments.

In conclusion, this work rigorously demonstrates that the conventional approximate treatments of CoM corrections in nuclear DFT are inadequate because they mix in excited-state contributions. It establishes the Peierls-Yoccoz projection method as the theoretically sound approach for restoring translational invariance and provides updated, more reliable estimates for E_CoM across the nuclear chart. Implementing this correction is crucial for achieving the level of precision required for predictive studies in fundamental nuclear physics and its applications.