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# The impact of nuclear shape on the emergence of the neutron dripline

## Abstract

Atomic nuclei are composed of a certain number of protons Z and neutrons N. A natural question is how large Z and N can be. The study of superheavy elements explores the large Z limit1,2, and we are still looking for a comprehensive theoretical explanation of the largest possible N for a given Z—the existence limit for the neutron-rich isotopes of a given atomic species, known as the neutron dripline3. The neutron dripline of oxygen (Z = 8) can be understood theoretically as the result of single nucleons filling single-particle orbits confined by a mean potential, and experiments confirm this interpretation. However, recent experiments on heavier elements are at odds with this description. Here we show that the neutron dripline from fluorine (Z = 9) to magnesium (Z = 12) can be predicted using a mechanism that goes beyond the single-particle picture: as the number of neutrons increases, the nuclear shape assumes an increasingly ellipsoidal deformation, leading to a higher binding energy. The saturation of this effect (when the nucleus cannot be further deformed) yields the neutron dripline: beyond this maximum N, the isotope is unbound and further neutrons ‘drip’ out when added. Our calculations are based on a recently developed effective nucleon–nucleon interaction4, for which large-scale eigenvalue problems are solved using configuration-interaction simulations. The results obtained show good agreement with experiments, even for excitation energies of low-lying states, up to the nucleus of magnesium-40 (which has 28 neutrons). The proposed mechanism for the formation of the neutron dripline has the potential to stimulate further thinking in the field towards explaining nucleosynthesis with neutron-rich nuclei.

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## Data availability

All data relevant to this study are shown in the paper, but if more details are needed, they are available from the corresponding author upon reasonable request.

## Code availability

Several codes for the conventional shell-model (configuration-interaction) calculation are available, of which we used KSHELL33 in the present work. Reasonable inquiries about the MCSM code will be responded to by the corresponding author.

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## Acknowledgements

N.T., T.O. and N.S. acknowledge partial support from MEXT via the “Priority Issue on post-K computer” (Elucidation of the Fundamental Laws and Evolution of the Universe) grants (hp160211, hp170230, hp180179, hp190160), partial support from MEXT via the “Program for Promoting Researches on the Supercomputer Fugaku” (Simulation for basic science: from fundamental laws of particles to creation of nuclei) grant (hp200130) and from Joint Institute for Computational Fundamental Science (JICFuS). This work was supported in part by JSPS KAKENHI grant JP19H05145. N.S. and T.S. acknowledge JSPS KAKENHI grants JP17K05433 and JP19K03855, respectively. T.O. thanks P. Van Duppen for valuable comments and Y. Aritomo and H. Koura for information. We thank T. Abe and Y. Tsunoda for help. N.T., T.O. and K.T. are grateful to M. H.-Jensen for collaboration.

## Author information

Authors

### Contributions

N.T. derived the EEdf1 interaction and performed many of the configuration-interaction calculations; T.O. supervised the whole study; K.T. derived the EKK method and contributed to in-depth discussions; N.S. wrote most of the computer codes and performed some calculations; T.S. calculated the Fujita–Miyazawa three-nucleon interaction; Y.U. contributed to detailed discussions; S.Y. calculated the χEFT three-nucleon interactions, H.U. suggested this project at the initial stage; and T.O. wrote the manuscript. All authors discussed the results and commented on the manuscript.

### Corresponding author

Correspondence to Takaharu Otsuka.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Calvin Johnson and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data figures and tables

### Extended Data Fig. 1 Dripline nucleus in each chain of isotopes as a function of Δε.

The single-particle energies are shifted from their original values by the same amount, Δε, which is the vertical axis. The dark pink belt indicates the range of Δε suggested by the recent experiment39 identifying 31F, 34Ne and 39Na as dripline nuclei, while the light pink belt is obtained similarly but by assuming 37Na is the dripline nucleus of Na isotopes.

### Extended Data Fig. 2 Dependences of the ground-state energy on the cutoff parameter of the Vlowk approach.

The ground-state expectation value of the effective NN interaction originating in the χEFT NN forces by Entem and Machleidt22,53 (Methods) are shown for the Ne and Mg isotopes. a, Ne; b, Mg. For each isotope, the second (orange), third (red) and fourth (grey) columns depict, respectively, this quantity obtained with the cutoff parameter ΛVlowk = 1.8 fm−1, 2.0 fm−1 and 2.2 fm−1 by the second-order $$\hat{Q}$$-box calculation in the EKK process. For comparison, the third-order $$\hat{Q}$$-box result with ΛVlowk = 2.0 fm−1 is shown by the first (green) column. All values are shown relative to their corresponding N = 16 values.

### Extended Data Fig. 3 Dependences of the rest interaction contribution on the cutoff parameter of the Vlowk approach.

Cutoff dependences of the ground-state expectation value of the rest (such as quadrupole) term of the effective NN interaction originating in the χEFT NN forces by Entem and Machleidt22,53 (Methods) are shown for the Ne and Mg isotopes. a, Ne; b, Mg. For each isotope, the orange, red and grey lines depict, respectively, this quantity obtained with the cutoff parameter ΛVlowk = 1.8 fm−1, 2.0 fm−1 and 2.2 fm−1 by the second-order $$\hat{Q}$$-box calculation in the EKK process. For comparison, the third-order $$\hat{Q}$$-box result with ΛVlowk = 2.0 fm−1 is shown by the green line. All values are shown relative to their corresponding N = 16 values. The dripline isotopes suggested by the present work are indicated by star symbols.

### Extended Data Fig. 4 Ground-state expectation value of the 3NF for the Ne and Mg isotopes.

a, Ne; b, Mg. For each isotope, the second (orange), third (green) and fourth (red) columns depict, respectively, this quantity obtained with the 3NF of Gazit et al.80, that of Hebeler et al.77 and that of Hebeler et al. with single-particle energy shift (labelled ‘Hebeler et al.−0.5N’) (Methods). For comparison, the same quantity by the Fujita–Miyazawa 3NF is shown by the first (blue) column.

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Tsunoda, N., Otsuka, T., Takayanagi, K. et al. The impact of nuclear shape on the emergence of the neutron dripline. Nature 587, 66–71 (2020). https://doi.org/10.1038/s41586-020-2848-x

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• DOI: https://doi.org/10.1038/s41586-020-2848-x

• ### α-Clustering in atomic nuclei from first principles with statistical learning and the Hoyle state character

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