Low-Energy Coulomb Excitation for the Shell Model

Low-energy Coulomb excitation is capable of providing unique information on static electromagnetic moments of short-lived excited nuclear states, including non-yrast states. The process selectively populates low-lying collective states and is, therefore, ideally suited to study phenomena such as shape coexistence and the development of exotic deformation (triaxial or octupole shapes). Historically, these experiments were restricted to stable isotopes. However, the advent of new facilities providing intense beams of short-lived radioactive species has opened the possibility to apply this powerful technique to a much wider range of nuclei. The paper discusses the observables that can be measured in a Coulomb-excitation experiment and their relation to the nuclear structure parameters with an emphasis on the nuclear shape. Recent examples of Coulomb-excitation studies that provided outcomes relevant for the Shell Model are also presented.


Introduction
Among the multitude of experimental techniques used in nuclear-structure physics, low-energy Coulomb excitation is one of the oldest and, still to this day, one of the most widely employed. The reason for its success is twofold. On the one hand, this technique requires ion beams with relatively low energy (a few MeV/A) and the large cross sections of the Coulomb-excitation process can compensate for low beam intensity. For these reasons, it was widely used for experimental nuclear-structure studies in their early days and, at present, leads the way at new-generation ISOL radioactive ion beam (RIB) facilities. On the other hand, low-energy Coulomb excitation is particularly sensitive to nuclear collective properties, such as the shape. Specifically, this method can be used to determine reduced transition probabilities between low-lying states, and their spectroscopic quadrupole moments. As it relies on the well-known electromagnetic interaction, all these observables can be extracted in a model-independent way. Furthermore, the unique and model-independent information on relative signs of E2 matrix elements, achievable solely with this technique, makes it possible to link transitional and diagonal E2 matrix elements to Hill-Wheeler parameters (β 2 , γ) describing a quadrupole shape, via non-energy weighted quadrupole sum rules [1]. Hence, low-energy Coulomb excitation constitutes a powerful tool to study phenomena such as shape coexistence, shape transitions, superdeformation, and octupole collectivity (see Refs. [2][3][4] for recent examples).
This paper aims to outline how the results of low-energy Coulomb-excitation measurements can be used to benchmark the Shell Model and inspire further theoretical developments. In the next section, we briefly introduce the method and discuss firstand higher-order effects giving rise to sensitivity to transitional and diagonal electromagnetic matrix elements. The following section presents examples of low-energy Coulombexcitation experiments that provided outcomes particularly relevant for the Shell Model. We do not aim to provide a comprehensive review of low-energy Coulomb-excitation studies, which can be found elsewhere (see for instance Refs. [5,6]).
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0 + g.s.  importance of the triaxial degree of freedom. By measuring reduced transition probabili-297 ties and spectroscopic quadrupole moments the authors were able to assess that the two 298 nuclei are, indeed, collective, with b 2 ⇡ 0.15 and g ⇡ 30 . Several theoretical models 299 were applied to interpret the results. In both cases the nuclei show several characteristics  which became closer to the energy at the limit with the Cline's safe distance criterion, 292 the number of counts for multi-step excitations was comparable despite the facto ⇡ 10 293 of difference in the experiment duration. And this considering also that the reduced 294 transition probabilities are much larger in 140 Sm.

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Both the experiments showed similarities between the two nuclei in terms of the 296 importance of the triaxial degree of freedom. By measuring reduced transition probabili-297 ties and spectroscopic quadrupole moments the authors were able to assess that the two  h2 of difference in the experiment duration. And this considering also that the reduced 294 transition probabilities are much larger in 140 Sm.

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Both the experiments showed similarities between the two nuclei in terms of the 296 importance of the triaxial degree of freedom. By measuring reduced transition probabili-297 ties and spectroscopic quadrupole moments the authors were able to assess that the two Both the experiments showed similarities between the two nuclei in terms of the 296 importance of the triaxial degree of freedom. By measuring reduced transition probabili-297 ties and spectroscopic quadrupole moments the authors were able to assess that the two 298 nuclei are, indeed, collective, with b 2 ⇡ 0.15 and g ⇡ 30 . Several theoretical models 299 were applied to interpret the results. In both cases the nuclei show several characteristics 300 typicall of triaxial nuclei, with a certain difussness in both beta and gamma. Shell Model 301 calculations were performed in large model spaces consisting of a 100 Sn inert core and 302 all the orbitals up to N = Z = 82. The GCN50:82 interaction was used in for both nuclei, 303 based on a realistic G matrix derived from the CD-Bonn potential with two-body matrix 304 elements modified by normalizing to sets of experimental excitation energies in even-305 even and even-odd semimagic nuclei, even-odd Sb isotopes and N = 81 isotones, and 306 some known odd-odd nuclei around 132 Sn. The calculations were undertaken using the 307 KSHELL program and Nathan shell-model code. In the case of 130 Xe, also the SN100PN 308 interaction was used, based on the jj55pna Hamil-tonian. 309 ransition rates were calculated using effective charges of 0.65e and 1.65e for neutrons 310 and protons, respectively. based on a realistic G matrix derived from the CD-Bonn potential with two-body matrix 304 elements modified by normalizing to sets of experimental excitation energies in even-305 even and even-odd semimagic nuclei, even-odd Sb isotopes and N = 81 isotones, and 306 some known odd-odd nuclei around 132 Sn. The calculations were undertaken using the 307 KSHELL program and Nathan shell-model code. In the case of 130 Xe, also the SN100PN 308 interaction was used, based on the jj55pna Hamil-tonian. 309 ransition rates were calculated using effective charges of 0.65e and 1.65e for neutrons 310 and protons, respectively.
interaction was used, based on the jj55pna Hamil-tonian. 309 ransition rates were calculated using effective charges of 0.65e and 1.65e for neutrons 310 and protons, respectively. and protons, respectively. and protons, respectively. and protons, respectively.
and protons, respectively.
ransition rates were c 310 and protons, respectively.  and protons, respectively.   The following abbreviations are used in this manuscript:

Conflicts of Interest:
The authors declare no conflict of interest.

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The following abbreviations are used in this manuscript:

Conflicts of Interest:
The authors declare no conflict of interest.

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The following abbreviations are used in this manuscript:

Conflicts of Interest:
The authors declare no conflict of interest.

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The following abbreviations are used in this manuscript:

Conflicts of Interest:
The authors declare no conflict of interest.

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The following abbreviations are used in this manuscript:

Basics of Low-Energy Coulomb Excitation
Coulomb excitation is an inelastic scattering process, in which the two colliding nu-1 clei are excited via the mutually-generated, time-dependent electromagnetic field. If the 2 distance between the collision partners is sufficiently large, the short-range nuclear inter-3 action has a negligible influence on the excitation process, which is governed solely by 4 the well-known electromagnetic interaction. This condition can be quantified using the 5 Cline's safe distance criterion [7], appropriate for heavy nuclei, which states that if the 6 distance of closest approach between the surfaces of the collision partners exceeds 5 fm,  While Coulomb-excitation cross sections can be calculated using a full quantum-22 mechanical treatment, a semi-classical approach is typically employed to overcome diffi- 23 culties arising from the long-range of the Coulomb interaction and complex level schemes 24 of the colliding nuclei. In this approach, introduced by K. Alder and A. Winther [9], 25 the relative motion of collision partners is described using classical equations, and the 26 quantal treatment is limited to the excitation process. The validity of this procedure, 27 which provides a significant simplification of the calculations without a relevant loss of 28 accuracy, stems from the fact that the interaction in the Coulomb-excitation process is 29 dominated by the Rutherford term. For the semi-classical approximation to be valid, the de 30 Broglie wavelength associated with the projectile must be small compared to the distance of closest approach, and the energy transferred in the excitation process must be small 32 compared with the total kinetic energy in the centre-of-mass reference system. These two 33 conditions are well satisfied in low-energy Coulomb-excitation experiments involving 34 heavy ions, but when light nuclei are involved (i.e. protons, deuterons, α particles), a full 35 quantum-mechanical analysis is required. 36

37
If the interaction between the colliding nuclei is weak, i.e. the excitation probability is 38 1, Coulomb-excitation amplitudes can be calculated within the first-order perturbation 39 theory. In the first order, the cross section for the excitation of the state I f from the ground 40 state I g.s. is proportional to the square of the transitional matrix element I f ||EL||I g.s. .

41
Therefore, from the measured I g.s. → I f Coulomb-excitation cross section it is possible to 42 extract the reduced transition probability B(EL; I g.s. → I f ). 43 The excitation process strongly depends on the kinematics and the mass and atomic

52
If the electromagnetic field acting between the collision partners is strong enough and 53 the collision process lasts a sufficiently long time, multi-step excitation becomes a possibility 54 and higher-order contributions have to be taken into account. These contributions give 55 rise to the experimental sensitivity to relative signs of transitional matrix elements and 56 spectroscopic quadrupole moments of excited states, as described in the following. 57

58
To understand the importance of multi-step excitation it is useful to consider the 59 population of two excited states I π = 0 + 2 , 4 + 1 in an even-even nucleus (see Figure 1). 60 Because Coulomb excitation via an E0 transition is strictly forbidden, two-step excitation 61 is the only way to populate the 0 + 2 state. The 4 + 1 state can be Coulomb-excited in two 62 ways: directly from the ground state, via an E4 excitation, or with an E2 two-step excitation 63 through the first excited state. Since the probability of Coulomb-exciting a given state 64 through an E4 transition is much smaller than through the E2 excitation [8], the two- 65 step excitation is typically dominant 1 . Consequently, by measuring the intensities of the 66 4 + 1 → 2 + 1 , 0 + 2 → 2 + 1 γ-ray transitions with respect to the 2 + 1 → 0 + 1 decay, and relating them 67 to excitation cross sections, it is possible to extract the B(E2; 4 + 1 → 2 + 1 ) and B(E2; 0 values. 69 In some cases, single-and multi-step excitations are comparable in magnitude; an 70 example is the 2 + 2 state in an even-even nucleus (see Figure 1). This state can be populated 71 by a direct E2 transition from the ground state and by a two-step excitation through the 72 first excited state. The total excitation probability for the 2 + 2 state can be written as: 1 In the extreme case of 150 Nd, characterised by an enhanced E4 strength ( 4 + 1 ||E4||0 + g.s. = 0.22(12) eb 2 [10]), the contribution from E4 excitation is expected to reach ≈ 30% of the total cross section to populate the 4 + 1 state in backscattering of 150 Nd on a 208 Pb target. Typical contributions are much lower.
2 ), one related to two-step 75 excitation ( 2 + 2 ||E2||2 + 1 2 2 + 1 ||E2||0 + g.s. 2 ) and the interference term In this last term, at variance with all the others, the matrix elements are not squared. As 77 the total Coulomb-excitation cross section will be different for a negative (destructive) and 78 a positive (constructive) interference term, its sign becomes an observable. 79 More complex interference terms can influence the Coulomb-excitation cross sections 80 if states are populated through several excitation patterns involving multiple intermediate

99
The reorientation effect [11] is another second-order effect in Coulomb excitation, (see Figure 1). For a given state I π , reorientation produces a second-order correction to its 104 Coulomb-excitation cross section, which is proportional to the diagonal matrix element 105 I π ||E2||I π , i.e. to Q s (I π ). Since this matrix element, and not its square, appears in the 106 expression for cross section, also its sign is an observable. In favourable conditions, the 107 reorientation effect may have a considerable influence on the measured γ-ray intensities.

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For example, in a recent study of 74  The nuclear shape can be inferred indirectly from transition probabilities or spectro-127 scopic quadrupole moments, but this approach is not always unambiguous and generally 128 depends on comparisons with models. An alternative model-independent approach, pro-129 posed by K. Kumar and D. Cline [1,7], exploits the specific properties of the electromagnetic 130 multipole operators. As these operators are spherical tensors, their zero-coupled products 131 are rotationally invariant. The expectation values of these products are observables, and 132 they are strictly related to the parameters describing the shape of the charge distribution.

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The electric quadrupole operator in the principal axis system can be represented 134 using the variables Q and δ, whose expectation values are equivalent to the Hill-Wheeler 135 parameters (β 2 , γ) describing the quadrupole shape. The simplest invariants read: The expectation values of these invariants for a state I n can be expressed through E2 matrix 137 elements defined in the laboratory system. For instance: where M ab ≡ I a ||E2||I b and the expression in curly brackets is a 6j coefficient. Higher-139 order invariants can be defined, such as Q 4 , which can be linked to the dispersion in A similar definition applies to σ(Q 3 cos 3δ). In principle, this approach can be extended to 142 more complex, non-quadrupole shapes.

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The invariants obtained from quadrupole sum rules provide a model-independent de-144 scription of the nuclear shape in the intrinsic reference system. vergence issue [14][15][16]. The contributions of individual products of matrix elements to the 164 experimentally determined invariants have also been analysed in some cases [15,[17][18][19].  The potential of Coulomb excitation as a tool to study superdeformation has been 174 demonstrated in the very first experiment using the AGATA γ-ray tracking array [20].

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The superdeformed (SD) structure in 42 Ca was populated following Coulomb excitation 176 of a 42 Ca beam on 208 Pb and 197 Au targets [3,19]. From the measured γ-ray intensities, 177 magnitudes and relative signs of numerous E2 matrix elements coupling the low-lying 178 states in 42 Ca were determined. In particular, two key pieces of information were obtained 179 for the first time, which confirm that the band built on the 0 + 2 state in 42 Ca has a SD character 180 at low spin: the spectroscopic quadrupole moment of the 2 + 2 state, which corresponds to 181 β 2 = 0.48 (16), as well as the enhanced B(E2; 2 + 2 → 0 + 2 ) = 15 +6 −4 W.u. value. As discussed 182 in Ref. [13], even though the 2 + 2 → 0 + 2 transition is too weak to be observed and prior 183 to the study of Refs. [3,19] only an upper limit for the branching ratio was known, the state with respect to that for the ground state was attributed to the mixing of the 2 + states.

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Additionally, the triaxiality parameter cos 3δ obtained for the 0 + 2 state, corresponding  number. The 0 + 2 state in 70 Ge is more deformed than the ground state [24], in 72 Ge both 231 states seem to have comparable overall deformations and considerable triaxiality [25], 232 while those for the 0 + 2 states in 74,76 Ge point to nearly spherical shapes [26,27]. Based on 233 the similarity of the 0 + 2 energy systematics in Ge and Zn nuclei (see Fig. 2b), one could 234 speculate that shape coexistence is present also in the latter isotopic chain. First hints of 235 the intruder character of the 0 + 2 states in the Zn isotopes came from E0 measurements in 236 the stable even-even 64−68 Zn isotopes [28], a feature further supported by the results of 237 multi-step Coulomb-excitation experiments on 66,68 Zn [15,29]. However, only for 68 Zn has 238 the key 2 + 3 E2 0 + 2 matrix element been determined which, when combined with other 239 matrix elements involving the 0 + 2 state, leads to a Q 2 invariant significantly different from 240 that of the ground state [29]. On the other hand, multiple low-energy Coulomb-excitation 241 studies of stable Ge and Zn isotopes [15,26,29] pointed to the importance of the triaxial 242 degree of freedom in their structure, which was also evoked for the neighbouring 76 deformation. This is particularly important considering that 76 Ge is a candidate for searches 246 of neutrinoless double-β decay, and the nuclear shape is predicted to play a significant role 247 in this process [33,34]. suggested [41,42]  of triaxiality also in neutron-rich Zn isotopes, whose ground states were suggested to be 285 rather diffuse in the γ degree of freedom [43]. Furthermore, the energy of the first excited 286 state in 80   certainties, and the Q s (2 + 1 ) value for 104 Cd significantly changes if a previously measured 408 lifetime of the 2 + 1 state is used as an additional constraint in the Coulomb-excitation data 409 analysis.

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Quadrupole deformation of light Cd isotopes was explored in an LSSM study [16] 411 using a modified v3sb effective interaction [70] in the π(2p 1/2 , 1g 9/2 ), ν(2d 5/2 , 3s 1/2 , 2d 3/2 , 412 1g 7/2 , 1h 11/2 ) model space. The calculated E2 matrix elements provide a good reproduction 413 of the experimental B(E2; 2 + 1 → 0 + 1 ) and B(E2; 4 + 1 → 2 + 1 ) values, and were analysed in 414 terms of quadrupole invariants Q 2 and Q 3 cos 3δ pointing to a predominantly prolate 415 character of 100−108 Cd with both β and γ increasing with N. Very recently, Coulomb 416 excitation of 106 Cd was performed [71] at the NSCL ReA3 facility. Quadrupole moments of 417 the 2 + 1 , 4 + 1 , 6 + 1 and 2 + 2 states were obtained, as well as the Q 2 and Q 3 cos 3δ invariants 418 for the ground state, which suggest its considerable triaxiality. This feature does not 419 emerge from the LSSM calculations reported in Ref. [71], which also used a G-matrix-420 renormalized CD-Bonn nucleon-nucleon potential and the same model space as those 421 of Ref. [16], but allowed at most two neutrons in the 1h 11/2 orbital. While they well 422 reproduced the experimental Q 2 invariant, the shapes that they predict for light Cd 423 isotopes are decidedly prolate. The difference with respect to a more γ-soft behaviour 424 suggested by Ref. [16] was attributed to the different 1h 11/2 single-particle energies, as 425 well as the adopted truncation. However, none of these calculations are able to explain 426 the observed pattern of spectroscopic quadrupole moments in the light Cd nuclei, which 427 hopefully will trigger more experimental and theoretical work aiming at understanding 428 their quadrupole properties. is observed at ongoing experiments, such as CUORE [77] and SNO+ [78], the relevant ββ 460 nuclear matrix elements will need to be calculated in order to extract the Majorana mass.

461
Such calculations are under way, also within the Shell Model [79]. Further low-energy

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Coulomb-excitation studies should help to elucidate the nuclear structure at A ≈ 130 − 140.

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A 130 Xe beam could be delivered by a stable ion beam facility with a much higher intensity 464 than that available in Ref. [17], and the use of a heavier target (e.g. 208 Pb) would increase  theoretical developments will bring forward our understanding of nuclear structure, while 479 also being relevant for cross-disciplinary fields such as astrophysics, neutrino physics, and 480 physics of (and beyond) the Standard Model [4,34,81]. In this context, a precise under-481 standing of the nuclear shape can bring us closer to answering long-standing questions in 482 physics, such as how heavy elements originate in cataclysmic stellar events and the reason 483 for the matter-antimatter asymmetry in the Universe.

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Thanks to the constant development of powerful computational resources, and refine-485 ments of Shell-Model codes and methods, this theoretical approach can now be extended 486 to vast regions of the nuclear chart. It can be anticipated that this progress will be com-487 plemented and inspired by the availability of high-precision spectroscopic data and that 488 low-energy Coulomb excitation will continue to play an important role in future studies 489 throughout the nuclear chart. We emphasize, however, as in the case of 98,100 Mo that the 490 combination of data from a variety of techniques that probe both collective and single-491 particle degrees of freedom will provide perhaps the most demanding tests of Shell-Model 492 calculations, and studies in that direction should be pursued.