Skip to main content

Advertisement

SpringerLink
Log in

The shape of gold

  • Regular Article - Theoretical Physics
  • Published:
The European Physical Journal A Aims and scope Submit manuscript

Abstract

Having a detailed theoretical knowledge of the low-energy structure of the heavy odd-mass nucleus \(^{197}\hbox {Au}\) is of prime interest as the structure of this isotope represents an important input to theoretical simulations of collider experiments involving gold ions performed at relativistic energies. In the present article, therefore, we report on new results on the structure of \(^{197}\hbox {Au}\) obtained from state-of-the-art multi-reference energy density functional (MR-EDF) calculations. Our MR-EDF calculations were realized using the Skyrme-type pseudo-potential SLyMR1, and include beyond mean-field correlations through the mixing, in the spirit of the Generator Coordinate Method (GCM), of particle-number and angular-momentum projected triaxially deformed Bogoliubov quasi-particle states. Comparison with experimental data shows that the model gives a reasonable description of \(^{197}\hbox {Au}\) with in particular a good agreement for most of the spectroscopic properties of the \(3/2_1^+\) ground state. From the collective wave function of the correlated state, we compute an average deformation \(\bar{\beta }(3/2_1^+)=0.13\) and \(\bar{\gamma }(3/2_1^+)=40^\circ \) for the ground state. We use this result to construct an intrinsic shape of \(^{197}\hbox {Au}\) representing a microscopically-motivated input for precision simulations of the associated collider processes. We discuss, in particular, how the triaxiality of this nucleus is expected to impact \(^{197}\hbox {Au}\)+\(^{197}\hbox {Au}\) collision experiments at ultrarelativistic energy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data availability

This manuscript has no associated data or the data will not be deposited [Authors’ comment: There is no open data associated with this article. Data from the theoretical calculations is available upon request to the authors.]

Notes

  1. When considering axial deformations, this corresponds to a step of \(\varDelta \beta \approx 0.05\).

  2. Note that all the extrema discussed in this article are computed from an interpolation based on the results obtained at the points on the discretized deformation mesh.

  3. We mention in passing that some authors argue that using a \(^{198}\)Hg core provides a better global description of experimental data [6].

  4. We mention that the \(B(E2:2_1^+ \rightarrow 0_1^+)\) values are 40.6(20) and 28.8(4) W.u. for \(^{196}\)Pt and \(^{198}\)Hg, respectively [46, 61, 62].

  5. Provided that we interpret the \(3/2_2^+\) and \(3/2_3^+\) states as being inverted in our calculations compared to experimental data.

  6. More sophisticated calculations based on nuclear configurations obtained from ab initio nuclear theory have also been recently performed [65,66,67,68]. For the moment, they are limited to the description of collisions of \(^{16}\)O ions.

  7. The trial one-quasi-particle state is built by blocking a single-particle state originating from the spherical \(2\text {d}_{3/2}\) shell.

  8. All other non-vanishing multipole moments authorized by the symmetries of our calculations are let free to adopt a value that minimizes the total energy of the trial quasi-particle state.

  9. We safely neglect the effect of the very small hexadecapolarity of the nucleus, \(\beta _4^{\textrm{WS}}=-0.023\), in these simulations.

References

  1. H. Geiger, E. Marsden, On a diffuse reflection of the \(\alpha \)-particles. Proc. R. Soc. London. Series A 82(557), 495–500 (1909). https://doi.org/10.1098/rspa.1909.0054

    Article  ADS  Google Scholar 

  2. E. Rutherford, The scattering of \(\alpha \) and \(\beta \) particles by matter and the structure of the atom. London Edinburgh Dublin Philos. Magaz. J. Sci. 21(125), 669–688 (1911). https://doi.org/10.1080/14786440508637080

    Article  MATH  Google Scholar 

  3. X. Huang, C. Zhou, Nuclear data sheets for A=197. Nuclear Data Sheets 104(2), 283–426 (2005). https://doi.org/10.1016/j.nds.2005.01.001https://www.sciencedirect.com/science/article/pii/S0090375205000025

  4. R.M. Elliott, J. Wulff, Hyperfine structure of gold. Phys. Rev. 55, 170–175 (1939). https://doi.org/10.1103/PhysRev.55.170

  5. C.F. Cook, C.M. Class, J.T. Eisinger, Electric excitation of \(^{197}\)Au. Phys. Rev. 96, 658–663 (1954). https://doi.org/10.1103/PhysRev.96.658

    Article  ADS  Google Scholar 

  6. H. Bolotin, D. Kennedy, B. Linard, A. Stuchbery, S. Sie, I. Katayama, H. Sakai, Measured lifetimes of states in \(^{197}\)Au and a critical comparison with the weak-coupling core-excitation model. Nucl. Phys. A 321(1), 231–249 (1979). https://doi.org/10.1016/0375-9474(79)90696-1https://www.sciencedirect.com/science/article/pii/0375947479906961

    Article  ADS  Google Scholar 

  7. A. Stuchbery, L. Wood, H. Bolotin, C. Doran, I. Morrison, A. Byrne, G. Lampard, Measured gyromagnetic ratios and the low-excitation spectroscopy of \(^{197}\)Au. Nucl. Phys. A 486(2), 374–396 (1988). https://doi.org/10.1016/0375-9474(88)90242-4https://www.sciencedirect.com/science/article/pii/0375947488902424

    Article  ADS  Google Scholar 

  8. N. Fotiades, R.O. Nelson, M. Devlin, K. Starosta, J.A. Becker, L.A. Bernstein, P.E. Garrett, W. Younes, States in \(^{197}\rm Au \) from the (\(n,{n}^{^{\prime }}\gamma \)) reaction. Phys. Rev. C 71, 064314 (2005). https://doi.org/10.1103/PhysRevC.71.064314

  9. C. Wheldon, J.J. Valiente-Dobón, P.H. Regan, C.J. Pearson, C.Y. Wu, J.F. Smith, A.O. Macchiavelli, D. Cline, R.S. Chakrawarthy, R. Chapman, M. Cromaz, P. Fallon, S.J. Freeman, W. Gelletly, A. Görgen, A.B. Hayes, H. Hua, S.D. Langdown, I.Y. Lee, X. Liang, Z. Podolyák, G. Sletten, R. Teng, D. Ward, D.D. Warner, A.D. Yamamoto, Observation of an isomeric state in \(^{197}\rm Au \). Phys. Rev. C 74, 027303 (2006). https://doi.org/10.1103/PhysRevC.74.027303

    Article  ADS  Google Scholar 

  10. N. Fotiades, R.O. Nelson, M. Devlin, S. Holloway, T. Kawano, P. Talou, M.B. Chadwick, J.A. Becker, P.E. Garrett, Feeding of the \(11/{2}^{-}\) isomers in stable Ir and Au isotopes. Phys. Rev. C 80, 044612 (2009). https://doi.org/10.1103/PhysRevC.80.044612

  11. N.J. Stone, Table of recommended magnetic dipole moments: Part I, Long-lived states, INDC(NDS)-0794 (2019). https://www-nds.iaea.org/publications/indc/indc-nds-0794.pdf

  12. N.J. Stone, Table of recommended magnetic dipole moments: Part II, Short-lived states, INDC(NDS)-0816 (2020). https://www-nds.iaea.org/publications/indc/indc-nds-0816.pdf

  13. Å. Bohr, B. R. Mottelson, Nuclear structure, Volume I and II (World Scientific Publishing, Singapore, 1998)

  14. I. Hamamoto, B.R. Mottelson, Further examination of prolate-shape dominance in nuclear deformation. Phys. Rev. C 79, 034317 (2009). https://doi.org/10.1103/PhysRevC.79.034317

  15. J. Dechargé, D. Gogny, Hartree-Fock-Bogolyubov calculations with the \(D1\) effective interaction on spherical nuclei. Phys. Rev. C 21, 1568–1593 (1980). https://doi.org/10.1103/PhysRevC.21.1568

  16. J. Berger, M. Girod, D. Gogny, Time-dependent quantum collective dynamics applied to nuclear fission. Comput. Phys. Commun. 63(1), 365–374 (1991). https://doi.org/10.1016/0010-4655(91)90263-Khttps://www.sciencedirect.com/science/article/pii/001046559190263K

    Article  ADS  MATH  Google Scholar 

  17. S. Hilaire, M. Girod, Large-scale mean-field calculations from proton to neutron drip lines using the D1S Gogny force. Eur. Phys. J. A 33(2), 237–241 (2007). https://doi.org/10.1140/epja/i2007-10450-2

    Article  ADS  Google Scholar 

  18. S.T.A.R. Collaboration, Experimental and theoretical challenges in the search for the quark-gluon plasma: The STAR collaboration’s critical assessment of the evidence from rhic collisions. Nucl. Phys. A 757(1), 102–183 (2005). https://www.sciencedirect.com/science/article/pii/S0375947405005294

  19. PHENIX Collaboration, Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: Experimental evaluation by the PHENIX collaboration. Nucl. Phys. A 757(1), 184–283 (2005). https://doi.org/10.1016/j. nuclphysa.2005.03.086

  20. PHOBOS Collaboration, The phobos perspective on discoveries at RHIC, Nucl. Phys. A 757(1), 28–101 (2005), first Three Years of Operation of RHIC. https://www.sciencedirect.com/science/article/pii/S0375947405005282https://doi.org/10.1016/j.nuclphysa.2005.03.084

  21. BRAHMS Collaboration, Quark-gluon plasma and color glass condensate at RHIC? the perspective from the BRAHMS experiment, Nucl. Phys. A 757(1), 1–27 (2005). https://www.sciencedirect.com/science/article/pii/S0375947405002770https://doi.org/10.1016/j.nuclphysa.2005.02.130

  22. M.L. Miller, K. Reygers, S.J. Sanders, P. Steinberg, Glauber modeling in high-energy nuclear collisions. Ann. Rev. Nucl. Particle Sci. 57(1), 205–243 (2007). https://doi.org/10.1146/annurev.nucl.57.090506.123020

    Article  ADS  Google Scholar 

  23. W. Busza, K. Rajagopal, W. van der Schee, Heavy ion collisions: the big picture and the big questions. Ann. Rev. Nucl. Particle Sci. 68(1), 339–376 (2018)

    Article  ADS  Google Scholar 

  24. S.T.A.R. Collaboration, Azimuthal anisotropy in \(\rm U +\rm U \) and \(\rm Au +\rm Au \) collisions at RHIC. Phys. Rev. Lett. 115, 222301 (2015). https://doi.org/10.1103/PhysRevLett.115.222301

    Article  ADS  Google Scholar 

  25. ALICE Collaboration, Anisotropic flow in Xe-Xe collisions at \(\sqrt{s_{NN}}=\) 5.44 TeV, Phys. Lett. B 784, 82–95 (2018). https://doi.org/10.1016/j.physletb.2018.06.059

  26. CMS Collaboration, Charged-particle angular correlations in XeXe collisions at \(\sqrt{{s}_{NN}}=5.44\) TeV, Phys. Rev. C 100, 044902 (2019). https://doi.org/10.1103/PhysRevC.100.044902

  27. ATLAS Collaboration, Measurement of the azimuthal anisotropy of charged-particle production in \(\rm Xe+\rm Xe\rm \) collisions at \(\sqrt{{s}_{NN}}=5.44\) TeV with the ATLAS detector, Phys. Rev. C 101 , 024906 (2020). https://doi.org/10.1103/PhysRevC.101.024906

  28. STAR Collaboration, Search for the chiral magnetic effect with isobar collisions at \(\sqrt{{s}_{NN}}=200\) GeV by the STAR collaboration at the BNL Relativistic Heavy Ion Collider, Phys. Rev. C 105, 014901 (2022). https://doi.org/10.1103/PhysRevC.105.014901

  29. B. Bally, J. D. Brandenburg, G. Giacalone, U. Heinz, S. Huang, J. Jia, D. Lee, Y.-J. Lee, W. Li, C. Loizides, M. Luzum, G. Nijs, J. Noronha-Hostler, M. Ploskon, W. van der Schee, B. Schenke, C. Shen, V. Somá, A. Timmins, Z. Xu, Y. Zhou, Imaging the initial condition of heavy-ion collisions and nuclear structure across the nuclide chart (2022). https://arxiv.org/abs/2209.11042

  30. B. Bally, M. Bender, G. Giacalone, V. Somà, Evidence of the triaxial structure of \(^{129}\rm Xe \) at the large hadron collider. Phys. Rev. Lett. 128, 082301 (2022). https://doi.org/10.1103/PhysRevLett.128.082301

    Article  ADS  Google Scholar 

  31. ATLAS Collaboration, Correlations between flow and transverse momentum in Xe+Xe and Pb+Pb collisions at the LHC with the ATLAS detector: a probe of the heavy-ion initial state and nuclear deformation (4 2022). arXiv:2205.00039

  32. L. Morrison, K. Hadyńska-Klȩk, Z. Podolyák, D. T. Doherty, L. P. Gaffney, L. Kaya, L. Próchniak, J. Samorajczyk-Pyśk, J. Srebrny, T. Berry, A. Boukhari, M. Brunet, R. Canavan, R. Catherall, S. J. Colosimo, J. G. Cubiss, H. De Witte, C. Fransen, E. Giannopoulos, H. Hess, T. Kröll, N. Lalović, B. Marsh, Y. M. Palenzuela, P. J. Napiorkowski, G. O’Neill, J. Pakarinen, J. P. Ramos, P. Reiter, J. A. Rodriguez, D. Rosiak, S. Rothe, M. Rudigier, M. Siciliano, J. Snäll, P. Spagnoletti, S. Thiel, N. Warr, F. Wenander, R. Zidarova, M. Zielińska, Quadrupole deformation of \(^{130}\rm Xe\) measured in a coulomb-excitation experiment. Phys. Rev. C 102, 054304 (2020). https://doi.org/10.1103/PhysRevC.102.054304

  33. S. Kisyov, C.Y. Wu, J. Henderson, A. Gade, K. Kaneko, Y. Sun, N. Shimizu, T. Mizusaki, D. Rhodes, S. Biswas, A. Chester, M. Devlin, P. Farris, A.M. Hill, J. Li, E. Rubino, D. Weisshaar, Structure of \(^{126,128}\rm Xe \) studied in coulomb excitation measurements. Phys. Rev. C 106, 034311 (2022). https://doi.org/10.1103/PhysRevC.106.034311

  34. B. Bally, G. Giacalone, M. Bender, Structure of \(^{128,129,130}\)Xe through multi-reference energy density functional calculations. Eur. Phys. J. A 58(9), 187 (2022). https://doi.org/10.1140/epja/s10050-022-00833-4

  35. M. Bender, P.-H. Heenen, P.-G. Reinhard, Self-consistent mean-field models for nuclear structure. Rev. Mod. Phys. 75, 121–180 (2003). https://doi.org/10.1103/RevModPhys.75.121

    Article  ADS  Google Scholar 

  36. N. Schunck (Ed.), Energy Density Functional Methods for Atomic Nuclei, 2053-2563, IOP Publishing, (2019). https://doi.org/10.1088/2053-2563/aae0ed

  37. M. Bender, P.-H. Heenen, Configuration mixing of angular-momentum and particle-number projected triaxial Hartree-Fock-Bogoliubov states using the Skyrme energy density functional. Phys. Rev. C 78, 024309 (2008). https://doi.org/10.1103/PhysRevC.78.024309

    Article  ADS  Google Scholar 

  38. T.R. Rodríguez, J.L. Egido, Triaxial angular momentum projection and configuration mixing calculations with the Gogny force. Phys. Rev. C 81, 064323 (2010). https://doi.org/10.1103/PhysRevC.81.064323

    Article  ADS  Google Scholar 

  39. J.M. Yao, J. Meng, P. Ring, D. Vretenar, Configuration mixing of angular-momentum-projected triaxial relativistic mean-field wave functions. Phys. Rev. C 81, 044311 (2010). https://doi.org/10.1103/PhysRevC.81.044311

    Article  ADS  Google Scholar 

  40. B. Bally, B. Avez, M. Bender, P.-H. Heenen, Beyond mean-field calculations for odd-mass nuclei. Phys. Rev. Lett. 113, 162501 (2014). https://doi.org/10.1103/PhysRevLett.113.162501

    Article  ADS  Google Scholar 

  41. M. Borrajo, J.L. Egido, Ground-state properties of even and odd magnesium isotopes in a symmetry-conserving approach. Phys. Lett. B 764, 328–334 (2017). https://doi.org/10.1016/j.physletb.2016.11.037https://www.sciencedirect.com/science/article/pii/S0370269316307006

    Article  ADS  Google Scholar 

  42. B. Bally, M. Bender, Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states. Phys. Rev. C 103, 024315 (2021). https://doi.org/10.1103/PhysRevC.103.024315

    Article  ADS  Google Scholar 

  43. J. Sadoudi, T. Duguet, J. Meyer, M. Bender, Skyrme functional from a three-body pseudopotential of second order in gradients: Formalism for central terms. Phys. Rev. C 88, 064326 (2013). https://doi.org/10.1103/PhysRevC.88.064326

    Article  ADS  Google Scholar 

  44. R. Jodon, Ph.D. thesis, Université Claude Bernard - Lyon 1 (2014)

  45. D. Baye, P.-H. Heenen, Generalised meshes for quantum mechanical problems. J. Phys. A: Math. General 19(11), 2041–2059 (1986). https://doi.org/10.1088/0305-4470/19/11/013

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Data retrieved from the IAEA online nuclear data services. https://www-nds.iaea.org/

  47. A. de Shalit, Core excitations in nondeformed, odd-\(A\), nuclei. Phys. Rev. 122, 1530–1536 (1961). https://doi.org/10.1103/PhysRev.122.1530

    Article  ADS  Google Scholar 

  48. A. Braunstein, A. De-Shalit, New evidence for core excitation in Au197. Phys. Lett. 1(7), 264–266 (1962). https://doi.org/10.1016/0031-9163(62)91375-6https://www.sciencedirect.com/science/article/pii/0031916362913756

    Article  ADS  Google Scholar 

  49. A. De-Shalit, Core excitation: Au197 revisited. Phys. Lett. 15(2), 170–172 (1965). https://doi.org/10.1016/0031-9163(65)91327-2https://www.sciencedirect.com/science/article/pii/0031916365913272

    Article  ADS  Google Scholar 

  50. F. McGowan, W. Milner, R. Robinson, P. Stelson, Couloub excitation of \(^{197}\)Au and a core-excitation analysis. Ann. Phys. 63(2), 549–561 (1971). https://doi.org/10.1016/0003-4916(71)90028-5https://www.sciencedirect.com/science/article/pii/0003491671900285

    Article  ADS  Google Scholar 

  51. R. Powers, P. Martin, G. Miller, R. Welsh, D. Jenkins, Muonic \(^{197}\)Au: A test of the weak-coupling model. Nucl. Phys. A 230(3), 413–444 (1974). https://doi.org/10.1016/0375-9474(74)90147-Xhttps://www.sciencedirect.com/science/article/pii/037594747490147X

    Article  ADS  Google Scholar 

  52. R.D. Lawson, J.L. Uretsky, Center-of-gravity theorem in nuclear spectroscopy. Phys. Rev. 108, 1300–1304 (1957). https://doi.org/10.1103/PhysRev.108.1300

    Article  ADS  Google Scholar 

  53. M. Borrajo, T.R. Rodríguez, J.L. Egido, Symmetry conserving configuration mixing method with cranked states. Phys. Lett. B 746, 341–346 (2015). https://doi.org/10.1016/j.physletb.2015.05.030http://www.sciencedirect.com/science/article/pii/S0370269315003676

    Article  ADS  Google Scholar 

  54. J.L. Egido, M. Borrajo, T.R. Rodríguez, Collective and single-particle motion in beyond mean field approaches. Phys. Rev. Lett. 116, 052502 (2016). https://doi.org/10.1103/PhysRevLett.116.052502

    Article  ADS  Google Scholar 

  55. I. Angeli, K. Marinova, Table of experimental nuclear ground state charge radii: an update. Atomic Data Nucl. Data Tables 99(1), 69–95 (2013). https://doi.org/10.1016/j.adt.2011.12.006https://www.sciencedirect.com/science/article/pii/S0092640X12000265

    Article  ADS  Google Scholar 

  56. W. Huang, M. Wang, F. Kondev, G. Audi, S. Naimi, The AME 2020 atomic mass evaluation (I). Evaluation of input data, and adjustment procedures. Chin. Phys. C 45(3), 030002 (2021). https://doi.org/10.1088/1674-1137/abddb0

    Article  ADS  Google Scholar 

  57. M. Wang, W. Huang, F. Kondev, G. Audi, S. Naimi, The AME 2020 atomic mass evaluation (II). tables, graphs and references. Chin. Phys. C 45(3), 030003 (2021). https://doi.org/10.1088/1674-1137/abddaf

    Article  ADS  Google Scholar 

  58. N. Stone, Table of nuclear electric quadrupole moments. Atomic Data Nucl. Data Tables 111(112), 1–28 (2016). https://doi.org/10.1016/j.adt.2015.12.002https://www.sciencedirect.com/science/article/pii/S0092640X16000024

    Article  ADS  Google Scholar 

  59. P. Schwerdtfeger, R. Bast, M.C.L. Gerry, C.R. Jacob, M. Jansen, V. Kellö, A.V. Mudring, A.J. Sadlej, T. Saue, T. Söhnel, F.E. Wagner, The quadrupole moment of the \(3/2+\) nuclear ground state of \(^{197}\)Au from electric field gradient relativistic coupled cluster and density-functional theory of small molecules and the solid state. J. Chem. Phys. 122(12), 124317 (2005). https://doi.org/10.1063/1.1869975

    Article  ADS  Google Scholar 

  60. W.M. Itano, Quadrupole moments and hyperfine constants of metastable states of \({\rm Ca }^{+}\), \({\rm Sr }^{+}\), \({\rm Ba }^{+}\), \({\rm Yb }^{+}\), \({\rm Hg }^{+}\), and Au. Phys. Rev. A 73, 022510 (2006). https://doi.org/10.1103/PhysRevA.73.022510

    Article  ADS  Google Scholar 

  61. H. Xiaolong, Nuclear data sheets for A = 196. Nuclear Data Sheets 108(6), 1093–1286 (2007). https://doi.org/10.1016/j.nds.2007.05.001https://www.sciencedirect.com/science/article/pii/S0090375207000488

    Article  ADS  Google Scholar 

  62. X. Huang, M. Kang, Nuclear data sheets for A = 198. Nuclear Data Sheets 133, 221–416 (2016). https://doi.org/10.1016/j.nds.2016.02.002https://www.sciencedirect.com/science/article/pii/S0090375216000156

    Article  Google Scholar 

  63. P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, New York, 1980)

    Book  Google Scholar 

  64. J.P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire, S. Péru, N. Pillet, G.F. Bertsch, Structure of even-even nuclei using a mapped collective hamiltonian and the D1S Gogny interaction. Phys. Rev. C 81, 014303 (2010). https://doi.org/10.1103/PhysRevC.81.014303

    Article  ADS  Google Scholar 

  65. S. H. Lim, J. Carlson, C. Loizides, D. Lonardoni, J. E. Lynn, J. L. Nagle, J. D. Orjuela Koop, J. Ouellette, Exploring new small system geometries in heavy ion collisions. Phys. Rev. C 99, 044904 (2019). https://doi.org/10.1103/PhysRevC.99.044904

  66. M. Rybczyński, W. Broniowski, Glauber monte carlo predictions for ultrarelativistic collisions with \(^{16}\rm O \). Phys. Rev. C 100, 064912 (2019). https://doi.org/10.1103/PhysRevC.100.064912

    Article  ADS  Google Scholar 

  67. N. Summerfield, B.-N. Lu, C. Plumberg, D. Lee, J. Noronha-Hostler, A. Timmins, \(^{16}\rm O ^{16}\rm O \) collisions at energies available at the bnl relativistic heavy ion collider and at the CERN large hadron collider comparing \(\alpha \) clustering versus substructure. Phys. Rev. C 104, L041901 (2021). https://doi.org/10.1103/PhysRevC.104.L041901

    Article  ADS  Google Scholar 

  68. G. Nijs, W. van der Schee, Predictions and postdictions for relativistic lead and oxygen collisions with the computational simulation code trajectum. Phys. Rev. C 106, 044903 (2022). https://doi.org/10.1103/PhysRevC.106.044903

    Article  ADS  Google Scholar 

  69. W. Ryssens, G. Giacalone, B. Schenke, C. Shen, Evidence of the hexadecapole deformation of uranium-238 at the relativistic heavy ion collider. arXiv:2302.13617

  70. STAR Collaboration, Tomography of Ultra-relativistic Nuclei with Polarized Photon-gluon Collisions (4 2022). arXiv:2204.01625

  71. H. Mäntysaari, B. Schenke, Evidence of strong proton shape fluctuations from incoherent diffraction. Phys. Rev. Lett. 117, 052301 (2016). https://doi.org/10.1103/PhysRevLett.117.052301

    Article  ADS  Google Scholar 

  72. G. Giacalone, B. Schenke, C. Shen, Constraining the nucleon size with relativistic nuclear collisions. Phys. Rev. Lett. 128, 042301 (2022). https://doi.org/10.1103/PhysRevLett.128.042301

    Article  ADS  Google Scholar 

  73. H. Mäntysaari, B. Schenke, C. Shen, W. Zhao, Bayesian inference of the fluctuating proton shape. Phys. Lett. B 833, 137348 (2022). https://doi.org/10.1016/j.physletb.2022.137348https://www.sciencedirect.com/science/article/pii/S0370269322004828

    Article  MATH  Google Scholar 

  74. G. Nijs, W. van der Schee, The hadronic nucleus-nucleus cross section and the nucleon size (6 2022). arXiv:2206.13522

  75. G. Giacalone, There and Sharp Again: The Circle Journey of Nucleons and Energy Deposition (8 2022). arXiv:2208.06839

  76. D. Everett et al., Multisystem bayesian constraints on the transport coefficients of QCD matter. Phys. Rev. C 103, 054904 (2021). https://doi.org/10.1103/PhysRevC.103.054904

    Article  ADS  Google Scholar 

  77. H. De Vries, C. De Jager, C. De Vries, Nuclear charge-density-distribution parameters from elastic electron scattering. Atomic Data and Nuclear Data Tables 36(3), 495–536 (1987). https://doi.org/10.1016/0092-640X(87)90013-1https://www.sciencedirect.com/science/article/pii/0092640X87900131

    Article  ADS  Google Scholar 

  78. P. Möller, A. Sierk, T. Ichikawa, H. Sagawa, Nuclear ground-state masses and deformations: FRDM(2012). Atomic Data Nucl. Data Tables 109(110), 1–204 (2016). https://doi.org/10.1016/j.adt.2015.10.002https://www.sciencedirect.com/science/article/pii/S0092640X1600005X

    Article  ADS  Google Scholar 

  79. Q. Shou, Y. Ma, P. Sorensen, A. Tang, F. Videbæk, H. Wang, Parameterization of deformed nuclei for Glauber modeling in relativistic heavy ion collisions. Phys. Lett. B 749, 215–220 (2015). https://doi.org/10.1016/j.physletb.2015.07.078https://www.sciencedirect.com/science/article/pii/S0370269315005985

    Article  ADS  Google Scholar 

  80. H. jie Xu, H. Li, X. Wang, C. Shen, F. Wang, Determine the neutron skin type by relativistic isobaric collisions. Phys. Lett. B 819, 136453 (2021). https://doi.org/10.1016/j.physletb.2021.136453https://www.sciencedirect.com/science/article/pii/S0370269321003932

    Article  Google Scholar 

  81. J. Jia, C.-J. Zhang, Scaling approach to nuclear structure in high-energy heavy-ion collisions (11 2021). arXiv:2111.15559

  82. G. Giacalone, Observing the deformation of nuclei with relativistic nuclear collisions. Phys. Rev. Lett. 124, 202301 (2020). https://doi.org/10.1103/PhysRevLett.124.202301

    Article  ADS  Google Scholar 

  83. G. Giacalone, Constraining the quadrupole deformation of atomic nuclei with relativistic nuclear collisions. Phys. Rev. C 102, 024901 (2020). https://doi.org/10.1103/PhysRevC.102.024901

    Article  ADS  Google Scholar 

  84. J. Jia, S. Huang, C. Zhang, Probing nuclear quadrupole deformation from correlation of elliptic flow and transverse momentum in heavy ion collisions. Phys. Rev. C 105, 014906 (2022). https://doi.org/10.1103/PhysRevC.105.014906

    Article  ADS  Google Scholar 

  85. P. Bożek, Transverse-momentum-flow correlations in relativistic heavy-ion collisions. Phys. Rev. C 93, 044908 (2016). https://doi.org/10.1103/PhysRevC.93.044908

    Article  ADS  Google Scholar 

  86. J.S. Moreland, J.E. Bernhard, S.A. Bass, Alternative ansatz to wounded nucleon and binary collision scaling in high-energy nuclear collisions. Phys. Rev. C 92, 011901 (2015). https://doi.org/10.1103/PhysRevC.92.011901

    Article  ADS  Google Scholar 

  87. F. G. Gardim, G. Giacalone, M. Luzum, J.-Y. Ollitrault, Effects of initial state fluctuations on the mean transverse momentum, Nucl. Phys. A 1005 (2021) 121999. https://www.sciencedirect.com/science/article/pii/S0375947420303092https://doi.org/10.1016/j.nuclphysa.2020.121999

  88. G. Giacalone, F.G. Gardim, J. Noronha-Hostler, J.-Y. Ollitrault, Correlation between mean transverse momentum and anisotropic flow in heavy-ion collisions. Phys. Rev. C 103, 024909 (2021). https://doi.org/10.1103/PhysRevC.103.024909

    Article  ADS  Google Scholar 

  89. ATLAS Collaboration, Measurement of flow harmonics correlations with mean transverse momentum in lead–lead and proton–lead collisions at \(\sqrt{s_{NN} = 5.02}\) TeV with the ATLAS detector, Eur. Phys. J. C 79(12), 985 (2019). https://doi.org/10.1140/epjc/s10052-019-7489-6

  90. ALICE Collaboration, Characterizing the initial conditions of heavy-ion collisions at the LHC with mean transverse momentum and anisotropic flow correlations, Phys. Lett. B 834, 137393 (2022). https://www.sciencedirect.com/science/article/pii/S0370269322005275https://doi.org/10.1016/j.physletb.2022.137393

  91. J. Jia, Probing triaxial deformation of atomic nuclei in high-energy heavy ion collisions, Phys. Rev. C 105(4), 044905 (2022). https://doi.org/10.1103/PhysRevC.105.044905

  92. C. Zhang, An overview of new measurements of flow, chirality, and vorticity from STAR experiment, In: 10th International Conference on New Frontiers in Physics, (2022). arXiv:2203.13106

  93. G. Giacalone, J. Jia, C. Zhang, Impact of nuclear deformation on relativistic heavy-ion collisions: Assessing consistency in nuclear physics across energy scales. Phys. Rev. Lett. 127, 242301 (2021) https://doi.org/10.1103/PhysRevLett.127.242301

  94. J. Sadoudi, M. Bender, K. Bennaceur, D. Davesne, R. Jodon,T. Duguet, Pseudo-potential based EDF parametrization for spuriousity-free MR EDF calculations. Physica Scripta T 154, 014013 (2013). https://doi.org/10.1088/0031-8949/11002013/t154/014013

  95. M. Borrajo, J.L. Egido, Symmetry conserving configuration mixing description of odd mass nuclei. Phys. Rev. C 98, 044317 (2018). https://doi.org/10.1103/PhysRevC.98.044317

    Article  ADS  Google Scholar 

  96. G. Giacalone, F.G. Gardim, J. Noronha-Hostler, J.-Y. Ollitrault, Skewness of mean transverse momentum fluctuations in heavy-ion collisions. Phys. Rev. C 103, 024910 (2021). https://doi.org/10.1103/PhysRevC.103.024910

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank Chunjian Zhang for help with the entropy-to-multiplicity conversion used in Fig. 6, and Wouter Ryssens for useful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 839847. M.B. acknowledges support by the Agence Nationale de la Recherche, France, under grant No. 19-CE31-0015-01 (NEWFUN). G.G. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC2181/1-390900948 (the Heidelberg STRUCTURES Excellence Cluster), within the Collaborative Research Center SFB1225 (ISOQUANT, Project-ID 273811115). The calculations were performed by using HPC resources from CIEMAT (Turgalium), Spain (FI-2021-3-0004, FI-2022-1-0004).

Author information

Authors and Affiliations

  1. ESNT, IRFU, CEA, Université Paris-Saclay, 91191, Gif-sur-Yvette, France

    B. Bally

  2. Departamento de Física Teórica, Universidad Autónoma de Madrid, E-28049, Madrid, Spain

    B. Bally

  3. Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120, Heidelberg, Germany

    G. Giacalone

  4. IP2I Lyon, UMR 5822, 4 rue Enrico Fermi, Université de Lyon, Université Claude Bernard Lyon 1, CNRS/IN2P3, F-69622, Villeurbanne, France

    M. Bender

Authors
  1. B. Bally
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Giacalone
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Bender
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Bally.

Additional information

Communicated by Thomas DUGUET.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bally, B., Giacalone, G. & Bender, M. The shape of gold. Eur. Phys. J. A 59, 58 (2023). https://doi.org/10.1140/epja/s10050-023-00955-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epja/s10050-023-00955-3

Search

Navigation