Accessing the shape of atomic nuclei with relativistic collisions of isobars
AAccessing the shape of atomic nuclei with relativistic collisions of isobars
Giuliano Giacalone, Jiangyong Jia,
2, 3 and Vittorio Som`a Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Department of Chemistry, Stony Brook University, Stony Brook, NY 11794, USA Physics Department, Brookhaven National Laboratory, Upton, NY 11976, USA IRFU, CEA, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France
Nuclides sharing the same mass number (isobars) are observed ubiquitously along the stabilityline. While having nearly identical radii, stable isobars can differ in shape, and present in particulardifferent quadrupole deformations. We show that even small differences in these deformations canbe probed by relativistic nuclear collisions experiments, where they manifest as deviations fromunity in the ratios of elliptic flow coefficients taken between isobaric systems. Collider experimentswith isobars represent, thus, a unique means to obtain quantitative information about the geometricshape of atomic nuclei.
Introduction.
A remarkable connection betweenlow- and high-energy nuclear physics has been recentlyestablished in collider experiments conducted at the BNLRelativistic Heavy Ion Collider (RHIC) and at the CERNLarge Hadron Collider (LHC) with the realization thatthe output of relativistic nuclear collisions is strongly af-fected by the deformation of the colliding ions.The key observable driving this observation is ellipticflow, the quadrupole deformation (second Fourier har-monic) of the azimuthal distribution of hadrons detectedin the final state of relativistic nuclear collisions [1]: V ∝ (cid:90) detector f ( ϕ ) e i ϕ , (1)where f ( ϕ ) is the distribution of azimuthal angles (inmomentum space) collected in a collision event.In nucleus-nucleus collisions, elliptic flow emerges as aresponse to the quadrupole asymmetry (ellipticity) of thesystem created, right after the interaction takes place, inthe plane transverse to the beam direction [2]: E ∝ (cid:90) overlap area (cid:15) ( r, φ ) r e i φ , (2)where ( r, φ ) parametrize the transverse plane (for sim-plicity at z = 0), and (cid:15) is the density of energy depositedin the overlap. In full generality, E (cid:54) = 0 ⇒ V (cid:54) = 0. Asillustrated in the left panel of Fig. 1, any collision oc-curring at finite impact parameter presents an overlaparea which carries an elliptical deformation, i.e., E (cid:54) = 0,explaining in particular the observation that V growssteeply with the collision impact parameter.However, for the majority of isotopes, even in the limitof vanishing impact parameter, one expects E (cid:54) = 0, andthus V (cid:54) = 0 from nuclear structure arguments. Most ofnuclei are in fact deformed objects, presenting a nonva-nishing intrinsic quadrupole moment, i.e., an ellipsoidaldeformation [3]: Q ∝ (cid:90) nucleus ρ ( r, Θ , Φ) r Y (Θ , Φ) , (3) where ρ ( r, Θ , Φ) represents the nucleon density in the in-trinsic frame of the nucleus. Ultrarelativistic collisionstake snapshots of randomly-oriented configurations of nu-cleons at the time of interaction, so that, if the collidingions present Q (cid:54) = 0, regions of overlap such as that pro-posed in the right panel of Fig. 1 can be produced. Theselead to E (cid:54) = 0 at zero impact parameter.The bottom line is that in nucleus-nucleus collisions: Q (cid:54) = 0 = ⇒ E (cid:54) = 0 = ⇒ V (cid:54) = 0 . (4)The importance of this statement has been recently clari-fied in the context of U+ U collisions at RHIC, withthe realization that observables based on V (and on thehadron mean transverse momentum, (cid:104) p t (cid:105) ) are essentiallydominated by effects due to the deformed shape of ura-nium nuclei [4, 5], and at LHC with the measurement ofan abnormally large V in Xe+
Xe collisions com-pared to
Pb+
Pb collisions [6–8].These discoveries naturally trigger the question ofwhether one can use the great resolving power of high-energy colliders to infer something new about the low-energy structure of nuclei, and provide a new means totest the state-of-the-art approaches applied to the nuclear y x impact parameter
FIG. 1. Anisotropic overlap regions in nuclear collisions. Left:a collision of spherical nuclei breaks anisotropy in the trans-verse plane due to the finite impact parameter. Right: acentral collision of deformed nuclei breaks anisotropy due tothe non-spherical shape of the colliding bodies. a r X i v : . [ nu c l - t h ] F e b many-body problem. In this Letter, we show that thisis possible, and we delineate an experimental program topursue this goal. The idea.
We exploit the seemingly uninterestingfact that a large number of stable nuclides belong to pairsof isobars, i.e., that for a given nuclide X one can oftenfind a different nuclide Y that contains the same numberof nucleons. This feature has an important implicationfor high-energy collisions. If X and Y are isobars, thenX+X collisions produce a system which has the sameproperties (volume, density) as that produced in Y+Ycollisions. As a consequence, X+X and Y+Y systemspresent the same geometry, the same dynamical evolu-tion, and thus the same elliptic flow in the final state.This leads us to our main point. Given two isobars, Xand Y, we ask the following: v { } X+X v { } Y+Y ? = 1 (5)where v { } represents the usual rms measure of themagnitude of V in a given multiplicity class. As ar-gued above, the ratio should be equal to 1. Experimen-tally, once a number of minimum bias collisions of order10 is available, the ratio at small centralities can beobtained free of statistical error, while systematic uncer-tainties cancel in the ratio if the detector conditions ofthe X+X and Y+Y runs are the same [9]. Corrections tothe ratio in Eq. (5) can further appear if the system pro-duced in X+X and Y+Y collisions have different sizes,for instance, due to the fact that X and Y present differ-ent neutron numbers. However, two stable isobars candiffer in matter radius by at most 0.5% [10], leading to anegligible correction to Eq. (5), of order of few per mille.Significant deviations ( > v coefficients are instead caused by the differ-ent deformations of the chosen isobars. Deformation re-flects the collective organization of nucleons in the nu-clear ground state and can quickly vary with proton andneutron numbers. In general, then, one does not expecttwo stable isobars to present the same deformed shape.The point we want to make in this Letter is the following: Given two isobars, X and Y, if one mea-sures v { } X+X v { } Y+Y > , with a deviation fromunity of order 1% or larger, then one mustconclude that nuclide X has a larger intrinsicquadrupole deformation than nuclide Y. This statement is based on two facts: ( i ) that ellipticflow emerges from the elliptic anisotropy of the overlaparea; ( ii ) that nuclei in their ground states typically havenonvanishing intrinsic quadrupole moments. These areestablished features of nuclear physics that do not relyon any specific approximation or model.Therefore, through measurements of the ratio inEq. (5) one obtains an information about the relative deformation of the isobars. In the following, we showthat this qualitative information can in fact be turnedinto a quantitative one, as even small differences in thequadrupole deformation of two isobars give rise to unam-biguous and detectable effects in the ratio of the v coef-ficients. As mentioned above, this observable is virtuallydevoid of experimental error and systematically accessi-ble, under the same experimental conditions, throughoutthe Segr`e chart. Such features are hardly attainable inlow-energy nuclear structure experiments. These mea-surements are thus expected to challenge the predictionsof nuclear models tuned to low-energy experimental datain an unprecedented way. Application.
So far we have illustrated our ideathrough conceptual arguments. To get some intuitionabout the kind of results that will be obtained in colliderexperiments, we now perform quantitative calculationsof Eq. (5) by choosing some models, namely, a standardparametrization for the deformed nuclear matter density,and a Glauber-type model for the collision process.A common parametrization of the nucleon density isthe 2-parameter Fermi (2pF) distribution: ρ ( r, Θ , Φ) ∝
11 + exp ([ r − R (Θ , Φ)] /a ) , (6)where a denotes the surface diffuseness and the half-density radius R carries information about the deformedshape. We characterize R through a spherical harmonicexpansion: R (Θ , Φ) = R (cid:20) βY , (Θ , Φ) (cid:21) , (7)truncated, for the present application, at the axialquadrupole, Y , . We shall neglect, then, potential cor-rections due to triaxial, Y , ± , and hexadecapole, Y , ,deformations, which can be systematically added in fu-ture. We stress, however, that for well-deformed nuclei,the most important for our analysis, such corrections aresubleading, and will not alter our conclusions. The coef-ficient β quantifies the ellipsoidal shape of the nucleus: β (cid:39) π (cid:82) ρ ( r, Θ , Φ) r Y (Θ , Φ) (cid:82) ρ ( r, Θ , Φ) r . (8)Well-deformed nuclei, such as U, or stable nuclideswith 150 < A < β ≈ . We assume here that nuclei have a fixed quadrupole deforma-tion, while, in reality, shape fluctuations are normally present(to a small or large extent, depending on the “softness” of thenuclear system, see e.g. Ref. [11]). While the qualitative state-ments discussed in the previous section are independent of thisfeature, we have checked that the quantitative results presentedin this section do not change significantly if a distribution in β is considered instead of the fixed value stated below [12]. Two nuclei, described as randomly oriented, deformedbatches of nucleons sampled independently according tothe 2pF distribution given in Eq. (6), are then collided atultrarelativistic energy. On a collision-by-collision basis,the energy density deposited in the process possesses anonvanishing eccentricity, ε ≡ |E | , which triggers thedevelopment of elliptic flow during the expansion of thesystem, resulting in the observed momentum anisotropy, V . Dubbing v ≡ | V | , at a given multiplicity (central-ity) one has: v = κ ε , where κ is a real coefficient thatdepends on the properties of the system (e.g., equationof sate and viscosity in a hydrodynamic model [1]). Now,as anticipated, isobaric systems share the same phys-ical properties, so that a crucial simplification occurs: κ [X + X] = κ [Y + Y]. As soon as we take a ratio be-tween v coefficients calculated in two different isobaricsystems, then, the response factor κ drops out. This inturn implies that: v { } X+X v { } Y+Y = ε { } X+X ε { } Y+Y ? = 1 . (9)The question of whether or not the measured ratio of v coefficients is equal to unity boils down to whether thetwo isobaric system possess the same fluctuations of ε .This reformulation of our question in terms of ε fluc-tuations allows us now to employ a collision model tocalculate Eq. (9), and thus to perform a quantitativeevaluation of the ratio of the v coefficients to be mea-sured at colliders. To do so, we use the default TRENTomodel [13], which has proven able to capture with goodaccuracy the effects of the quadrupole deformation of nu-clei on elliptic flow data collected in U+ U collisions.We perform this analysis for two pairs of isobars: • A pair of well-deformed rare-earth nuclei withnearly identical deformations, namely,
Sm and
Gd. For the 2pF density profile, we assumethat the matter distribution is well described withparameters extracted from the measured chargedensity, a good approximation for stable nuclides.For both nuclei we employ R = 5 .
975 fm and a = 0 .
59 fm, motivated by the results of Ref. [14].For the deformation parameters, we adopt valuesinferred from the measured transition probabili-ties of the electric quadrupole operator from theground state to the first 2 + state, tabulated, e.g.,in Ref. [15]. One finds β = 0 .
34 for
Sm and β = 0 .
31 for
Gd. Note that the definition of these “experimental” β relies on modelapproximations (e.g., a sharp nuclear surface) that are not com-pletely consistent with the use of Eq. (6). We neglect here thispossible mismatch, which is typically of order 5-10% (see Ref. [16]for more details). • A pair of lighter nuclei, Zr and Ru. Thesespecies are of particular relevance because a run ofboth Zr+ Zr and Ru+ Ru collisions has beenperformed at RHIC in 2018 [9], and experimentalresults will be released shortly. For the 2pF of thesenuclei we set R = 5 .
06 fm for Zr and R = 5 .
03 fmfor Ru, taking into account the fact that Zr hasan excess of 4 neutrons, while a = 0 .
52 fm for both.The deformation parameters are instead β = 0 . Ru, and β = 0 .
06 for Zr [15].For each system we perform 5 million minimum biascollisions. We sort events into centrality classes accord-ing to the entropy created in the process, following thedefault TRENTo model prescription. In each centralitybin, we evaluate ε { } , and by subsequently taking ra-tios between isobaric systems, as in Eq. (9), we obtainthe results displayed in Fig. 2.The shaded band corresponds to a departure fromunity smaller than 1%. Any deviation falling outsidethe band can be interpreted as a genuine signature ofthe different quadrupole deformations carried by the twoisobars, although we caution that for precise comparisonswith future experimental data, potential corrections dueto the imperfect centrality definition of our model willhave to be carefully addressed [17, 18]. The black solidline represents our result for the systems collided in 2018at RHIC, and can be confronted with upcoming data.For our choice of the deformation parameters, we observethat the splitting between the flow coefficients in centralcollisions is well above 1%, consistent with the fact that Ru has a larger quadrupole deformation in these sim-ulations. The same behavior is observed for the pair ofheavier nuclei (red dashed line). A deviation from unityof order 5% emerges in central collisions, due to the factthat
Sm nuclei are more deformed.The fact that both pairs return a similar (small) split-ting between v coefficients can be understood as follows.At a given small centrality, one expects [19]: ε { } = a + a β . (10)The coefficient a is the eccentricity due to quantum fluc-tuations (nucleon positions), while the term proportionalto β represents the contribution from fluctuations of thegeometry of the system driven by the random orientationof the deformed ions. Systems X+X and Y+Y have thesame size and number of participant nucleons, therefore,they present the same coefficients a and a . Now, inthe TRENTo results we find that, even for A = 154 and β ≈ .
3, the contribution from a in Eq (10) is largerthan the contribution from a β . Inserting Eq. (10) inEq. (9), expanding the ratio around unity, and keepingthe leading correction, equation (9) becomes: ε { } X+X ε { } Y+Y (cid:39) c (cid:0) β − β (cid:1) , (11) centrality (%) . . . . . . . v { } X + X / v { } Y + Y X= Ru ( β = 0 . Zr ( β = 0 . Sm ( β = 0 . Gd ( β = 0 . FIG. 2. Rms elliptic flow in X+X collisions divided by therms elliptic flow in Y+Y collisions as a function of collisioncentrality. The ratio of flow coefficients is estimated followingEq. (9) and the TRENTo model. The shaded band representsa 1% deviation from unity. Any deviation from unity whichfalls outside the shaded band can be considered as a significantsignature that β X (cid:54) = β Y . where c ∼ O (1). The order of the deviation from unity isdriven by the difference β X − β Y . For both pairs of isobarsconsidered in our application one has β X − β Y ≈ . c ≈ A = 154. The contribution from the nuclear defor-mation in Eq. (10) is quadratic in β , therefore, it is muchmore important for well-deformed nuclei, β ≈ .
3. Forthis reason, and as demonstrated by the results in Fig. 2,for well-deformed nuclei even differences at the level offew percents in the values of β will leave visible signaturesin the ratio of flow coefficients. As anticipated, then, inthis scenario the qualitative statement that X is more de-formed than Y turns into a nontrivial quantitative issue,driven by small differences in the shape of the isobars. Conclusion & Outlook.
Thanks to the great imag-ing power of high-energy colliders, relativistic collisionexperiments involving stable isobars, such as Zr+ Zrand Ru+ Ru collisions recently run at RHIC, yieldratios of v coefficients between isobaric systems that arenot simply equal to one, but rather look like the curvesshown in Fig. 2. A given ratio falling outside the shadedband indicates that the geometric shapes of the collidingions are different. This information is virtually free ofexperimental error, and has to be confronted with ourknowledge of nuclear physics across energy scales.Within the present paradigm, deviations from unityin the proposed ratio inform us about the relative A isobars A isobars A isobars36 Ar, S 106 Pd, Cd 148 Nd, Sm40 Ca, Ar 108 Pd, Cd 150 Nd, Sm46 Ca, Ti 110 Pd, Cd 152 Sm, Gd48 Ca, Ti 112 Cd, Sn 154 Sm, Gd50 Ti, V, Cr 113 Cd, In 156 Gd, Dy54 Cr, Fe 114 Cd, Sn 158 Gd, Dy64 Ni, Zn 115 In, Sn 160 Gd, Dy70 Zn, Ge 116 Cd, Sn 162 Dy, Er74 Ge, Se 120 Sn, Te 164 Dy, Er76 Ge, Se 122 Sn, Te 168 Er, Yb78 Se, Kr 123 Sb, Te 170 Er, Yb80 Se, Kr 124 Sn, Te, Xe 174 Yb, Hf84 Kr, Sr, Mo 126 Te, Xe 176 Yb, Lu, Hf86 Kr, Sr 128 Te, Xe 180 Hf, W87 Rb, Sr 130 Te, Xe, Ba 184 W, Os92 Zr, Nb, Mo 132 Xe, Ba 186 W, Os94 Zr, Mo 134 Xe, Ba 187 Re, Os96 Zr, Mo, Ru 136 Xe, Ba, Ce 190 Os, Pt98 Mo, Ru 138 Ba, La, Ce 192 Os, Pt100 Mo, Ru 142 Ce, Nd 198 Pt, Hg102 Ru, Pd 144 Nd, Sm 204 Hg, Pb104 Ru, Pd 146 Nd, SmTABLE I. Pairs and triplets of stable isobars (half-life longerthan 10 y ). A total of 139 nuclides are listed. The regionhighlighted in red contains large well-deformed nuclei ( A ≥ β > . quadrupole deformation of the colliding species. In theregime of well-deformed nuclei, the mere statement thatnucleus X is more deformed than nucleus Y translatesinto precise, quantitative information, which can be ac-cessed systematically across the nuclide chart thanks tothe abundance of stable isobars found in Nature.Physicists should take advantage of this great opportu-nity. All the pairs and triplets of isobars which are stableenough to be used in potential future collider experimentsare listed in Tab. I. Well-deformed nuclei are highlightedin a red box in the table. A recent study [20] furthersuggests that , , Nd and
Sm may present an oc-tupole deformation ( Q ∝ (cid:82) r Y (Θ , Φ) ρ ( r, Θ , Φ) (cid:54) = 0)in their ground state. A small octupole deformationwould be visible in high-energy collisions as an enhance-ment of the fluctuations of triangular flow, v , as ex-plicitly shown in a recent application to Pb+
Pbcollisions [21]. Nd and Sm isotopes are therefore idealcandidates for such a study.We stress that these experiments can be repeated forseveral pairs of isotopes in identical conditions and pro-vide us with information that does not rely on specificnuclear structure details. Ultrarelativistic collisions thusrepresent an outstanding tool that is truly complemen-tary to modern low-energy experiments. They offer aunique way to test the goodness of existing nuclear mod-els for a wide range of species, and consequently pose asolid baseline for the next generation of theory-to-datacomparisons including ab-initio frameworks of nuclearstructure currently under intense development [22].
Acknowledgments.
We would like to thank theparticipants of the Initial Stages 2021 conference, in par-ticular Jaki Noronha-Hostler, S¨oren Schlichting, and Pe-ter Steinberg for their feedback and inspiring commentson the topic of this manuscript. G.G. acknowledges use-ful discussions with Benjamin Bally, Michael Bender, andMatt Luzum. The work of G.G. is supported by theDeutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) under Germany’s Excellence Strat-egy EXC 2181/1 - 390900948 (the Heidelberg STRUC-TURES Excellence Cluster), SFB 1225 (ISOQUANT)and FL 736/3-1. The work of J.J is supported by DOEDEFG0287ER40331 and NSF PHY-1913138. [1] U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. ,123-151 (2013) doi:10.1146/annurev-nucl-102212-170540[arXiv:1301.2826 [nucl-th]].[2] D. Teaney and L. Yan, Phys. Rev. C , 064904 (2011)doi:10.1103/PhysRevC.83.064904 [arXiv:1010.1876[nucl-th]].[3] A. Bohr and B. Mottelson, Nuclear Structure , Vol. I (W.A. Benjamin Inc., 1969; World Scientific, 1998).[4] L. Adamczyk et al. [STAR], Phys. Rev. Lett. , no.22,222301 (2015) doi:10.1103/PhysRevLett.115.222301[arXiv:1505.07812 [nucl-ex]].[5] J. Jia, contribution to the VI th International Conferenceon the Initial Stages of High-Energy Nuclear Collisions(IS21), Virtual Meeting, January 2021, https://indico.cern.ch/event/854124/contributions/4135480/ .[6] S. Acharya et al. [ALICE Collaboration], Phys. Lett.B , 82 (2018) doi:10.1016/j.physletb.2018.06.059[arXiv:1805.01832 [nucl-ex]].[7] A. M. Sirunyan et al. [CMS Collaboration],Phys. Rev. C , no. 4, 044902 (2019)doi:10.1103/PhysRevC.100.044902 [arXiv:1901.07997[hep-ex]]. [8] G. Aad et al. [ATLAS Collaboration], Phys. Rev. C ,no. 2, 024906 (2020) doi:10.1103/PhysRevC.101.024906[arXiv:1911.04812 [nucl-ex]].[9] J. Adam et al. [STAR], [arXiv:1911.00596 [nucl-ex]].[10] H. Li, H. j. Xu, Y. Zhou, X. Wang, J. Zhao, L. W. Chenand F. Wang, Phys. Rev. Lett. , no.22, 222301 (2020)doi:10.1103/PhysRevLett.125.222301 [arXiv:1910.06170[nucl-th]].[11] A. Poves, F. Nowacki and Y. Alhassid, Phys. Rev. C ,no. 5, 054307 (2020) doi:10.1103/PhysRevC.101.054307[arXiv:1906.07542 [nucl-th]].[12] B. Bally, M. Bender, G. Giacalone, V. Som`a, in prepara-tion (2021).[13] J. S. Moreland, J. E. Bernhard and S. A. Bass,Phys. Rev. C , no. 1, 011901 (2015)doi:10.1103/PhysRevC.92.011901 [arXiv:1412.4708[nucl-th]].[14] H. De Vries, C. W. De Jager and C. De Vries, Atom.Data Nucl. Data Tabl. , 495 (1987). doi:10.1016/0092-640X(87)90013-1[15] NuDat database, National Nuclear Data Center(NNDC), [16] Q. Y. Shou, Y. G. Ma, P. Sorensen, A. H. Tang,F. Videbæk and H. Wang, Phys. Lett. B , 215 (2015)doi:10.1016/j.physletb.2015.07.078 [arXiv:1409.8375[nucl-th]].[17] J. Jia, C. Zhang and J. Xu, Phys. Rev. Res. , no.2,023319 (2020), [arXiv:2001.08602 [nucl-th]], M. Zhouand J. Jia, Phys. Rev. C , no.4, 044903 (2018),[arXiv:1803.01812 [nucl-th]].[18] M. Aaboud et al. [ATLAS], JHEP , 051 (2020)doi:10.1007/JHEP01(2020)051 [arXiv:1904.04808 [nucl-ex]].[19] G. Giacalone, Phys. Rev. C , no. 2, 024910 (2019)doi:10.1103/PhysRevC.99.024910 [arXiv:1811.03959[nucl-th]].[20] Y. Cao, S. E. Agbemava, A. V. Afanasjev, W. Nazarewiczand E. Olsen, Phys. Rev. C , no. 2, 024311 (2020)doi:10.1103/PhysRevC.102.024311 [arXiv:2004.01319[nucl-th]].[21] P. Carzon, S. Rao, M. Luzum, M. Sievert and J. Noronha-Hostler, Phys. Rev. C , no. 5, 054905 (2020)doi:10.1103/PhysRevC.102.054905 [arXiv:2007.00780[nucl-th]].[22] H. Hergert, Front. in Phys.8