Electric dipole polarizability of low-lying excited states in atomic nuclei

Novel equations for the electric dipole polarizability $\alpha_{_{E1}}$ of low-lying excited states in atomic nuclei -- and the related $(-2)$ moment of the total photo-absorption cross section, $\sigma_{_{-2}}$ -- are inferred in terms of electric dipole and quadrupole matrix elements. These equations are valid for arbitrary angular momenta of the initial/ground and final/excited states and have been exploited in fully converged 1$\hbar\omega$ shell-model calculations of selected {\it p-} and {\it sd-}shell nuclei that consider configuration mixing; advancing previous knowledge from $^{17}$O to $^{36}$Ar, where thousands of electric dipole matrix elements are computed from isovector excitations which include the giant dipole resonance region. Our results are in reasonable agreement with previous shell-model calculations and follow -- except for $^{6,7}$Li and $^{17,18}$O -- Migdal's global trend provided by the combination of the hydrodynamic model and second-order non-degenerate perturbation theory. Discrepancies in $^{6,7}$Li and $^{17}$O arise as a result of the presence of $\alpha$-cluster configurations in odd-mass nuclei, whereas the disagreement in $^{18}$O comes from the mixing of intruder states, which is lacking in the shell-model interactions. More advanced \emph{ab initio} calculations of the dipole polarizability for low-lying excited states covering all the isovector states within the giant dipole resonance region are missing and could be very valuable to benchmark the results presented here and shed further light on how atomic nuclei polarize away from the ground state


MOTIVATION
The bulk of knowledge on the nuclear electric dipole (E1) polarizability, α E1 , concerns the ground states of nuclei and arises from photo-absorption cross-section data [1][2][3], where most of the absorption (and emission) of photons is provided by the giant dipole resonance (GDR) [4].The latter is understood macroscopically as the collective motion of inter-penetrating proton and neutron fluids out of phase [5][6][7], whereas is described microscopically by the shell-model (SM) interpretation of a system of independent nucleons or particlehole excitations plus configuration mixing [8][9][10].Data predominantly involve photo-neutron cross sections, although photo-proton contributions are relevant for light and N = Z self-conjugate nuclei [11].To a much lesser extend, α E1 has been determined from several experiments using radioactive ion beams [12], inelastic proton scattering [13][14][15][16][17] and virtual photons [18].The latter are also responsible for the polarization of atoms and molecules [19].
The understanding of how α E1 evolves as a function of excitation energy is relevant for nuclear structure physics [18] and nuclear astrophysics [20].Average properties can be extracted from GDRs built on excited states by fitting the GDR energy and width parameters to data [20], using the second-sound hydrodynamic model [21] and assuming the validity of the Brink-Axel hypothesis [22,23].The latter seems validated below critical temperatures of T T c = 0.7 + 37.5/A MeV and angular momenta J J c = 0.6A 5/6 [24], where excited GDRs present similar parameters to their groundstate counterparts [25,26].There is bountiful information for T ≫1 MeV from heavy-ion fusion-evaporation reactions [25][26][27][28], some in the range 0.7 T 1 MeV [29][30][31][32][33][34][35], and hardly anything for 0 T < 0.7 MeV.At T ≈ 0 MeV, α E1 has been determined from Coulombexcitation reactions for only a couple of favorable cases with excited states J = 1/2 -7 Li [36][37][38] and 17 O [39]  -where the spectroscopic or static quadrupole moment is zero, Q S (J = 1/2) = 01 .Inasmuch as α E1 , Q S (J) is a second-order effect in Coulomb-excitation perturbation theory [40,[43][44][45] that provides a measure of the extent to which the nuclear charge distribution in the laboratory frame acquires an ellipsoidal deformation.The empirical disentanglement of α E1 and Q S values for excited states with J = 1/2 requires increasing experimental accuracy and has never been done.
For ground states, α E1 can be deduced using nondegenerate perturbation theory by means of the energyshift of nuclear levels arising from the quadratic Stark effect [46], and has been investigated with SM calcula-tions [47][48][49][50][51][52] using, where the sum extend over |n intermediate states connecting the initial/ground state |i with isovector E1 transitions [53], 2J i + 1 is the normalization constant arising from the Wigner-Eckart theorem [54,55] -validating Eq. 1 for arbitrary J i ground states -and σ −2 the (−2) moment of the total photo-absorption cross section, σ total (E γ ), defined by [56,57], which is generally integrated between neutron threshold S n and the experimentally available upper limit for monochromatic photons, E max γ ≈ 20 − 50 MeV [1].An upper limit of E max γ ≈ 50 MeV approximates the σ −2 asymptotic value for light and medium-mass nuclei [58].For heavy nuclei with atomic mass number A = N + Z 50, σ −2 values generally follow the empirical power-law formula [59,60], in agreement with Migdal's original calculation [5], where κ is the dipole polarizability parameter that accounts for deviations (κ = 1) of the hydrodynamic model from the actual GDR effects.Calculations can be benchmarked with available photo-absorption cross-section data [61,62].
In this work, we further explore how the E1 polarizability evolves from the ground state to the first excitation of selected p-and sd-shell nuclides.Where possible, we perform 1 ω SM calculations, compare with available data and explore deviations from the hydrodynamic model [59,60,[63][64][65].Similar SM calculations of the E1 polarizability for ground states have already been published in Ref. [52].

SHELL-MODEL CALCULATIONS
Firstly, we deduce new polarization equations for excited states on the same footing as ground states.Applying second-order non-degenerate perturbation theory to the Coulomb-excitation process shown in Fig. 1, the second-order transition amplitude b (2) i→f from |i -again, the ground state -to a final excited state |f is given by [44] b where b ( states of the interference term between first-order and second-order transitions, which is proportional to where [66] -with λ in = λ nf = 1 for E1 multipolarity and λ = λ in + λ nf2 -and the sum extends over the intermediate J n states connecting both |i and |f states with isovector E1 transitions following the general isospin selection rule for electromagnetic transitions: ∆T = 0, ±1 [53].Particularly, for self-conjugate N = Z nuclei the isovector contribution arises only from ∆T = 1 transitions3 , while both ∆T = 0 and ∆T = 1 have to be considered otherwise. Furthermore, S(E1) is connected to Eq. 1 through the reference parameter η 0 [36,43,44,[67][68][69], where the relation is defined by association with the excitation amplitude n b inf [44].For simplicity, Eichler originally assumed the closure approximation [43] -i.e.η 0 = 1 for closed-shell nuclei -but smaller η 0 < 1 values are expected because of the random phase of the off-diagonal matrix elements in the numerator of Eq. 6.For instance, η 0 ≃ 1/12 is determined assuming a 2 + 1 rotational state and the characteristic energy relation of the GDR double peak for a strongly-deformed prolate nucleus [67], reaching smaller η 0 ≤ 1/12 values when considering the triaxial degree of freedom.A slightly larger value of η 0 ≈ 0.3 is determined using the dynamic collective model for a stronglydeformed nucleus [68].Generally, the dynamic collective model by Danos and Greiner [70], later extended to spherical nuclei by Weber [71,72], can be used to calculate η 0 values, which allows for rotations, surface quadrupole vibrations and higher-energy giant resonance oscillations.As stated by de Boer and Eichler, Eq. 7 may also provide a useful estimate for the more general case, since the polarization is essentially a nuclear-size effect.Moreover, the electric quadrupole i Ê2 f matrix element in Eq. 7 naturally arises from the interference between first-order (E2) and second-order (E1) transitions [44].Now using Eqs. 1, 5, 6 and 7, new relations for α E1 and σ −2 values can be deduced for excited states with arbitrary J, α E1 = 1.11 in units of fm 3 and fm 2 /MeV, respectively.For instance, assuming a two-step processes of the type 0 Further, the polarizability parameter κ can also be determined in terms of E1 and E2 matrix elements, as originally done by Häusser and collaborators [36], For the usual case of a final 2 Additional Racah coefficients of W (1 1 3/2 1/2, 2 3/2) = 15/2 1/3 1/5 1/15 1/30, ( 16) were used for 6 Li (1 . Accordingly, Eqs. 8, 9 and 12 allow the general calculation of α E1 , σ −2 and κ values for excited states using E1 and E2 matrix elements computed by various theoretical models, comparison with sum rules and Coulomb-excitation measurements.Here, accurate Coulomb-excitation measurements of second-order contributions to the inelastic cross sections could be used to benchmark the polarizability of excited states [43]. Shell-model calculations of σ −2 and κ values for first low-lying excitations were previously performed in light nuclei [47][48][49][50][51]74] -as shown in Fig. 2 -and used in the analysis of Coulomb-excitation studies in order to treat adequately the GDR effect [36-39, 51, 74, 77-80].Large κ > 1 values were predicted for the J = 1/2 1 excited state in 7 Li and 17 O [47,48,50], in agreement with Coulomb-excitation measurements [36][37][38][39]77].In more detail, the first SM calculations in 0sd-shell nuclei [47] performed for 17 O [48] and 18 O [49] assumed the closure approximation (up triangles in Fig. 2), in which all the E1 strength is concentrated at the GDR energy.A more exhaustive approach considered a realistic Hamiltonian with harmonic-oscillator and Woods-Saxon (squares in Fig. 2) single-particle wave functions restricted to the lowest configuration [50], which included configuration mixing only for the simpler case of 17 O and presented a better correlation with experimental values.
In the present work, full 1 ω SM calculations of the dipole polarizability for the first excitation in selected p− and sd-shell nuclei have been performed with the OXBASH code [92] using the WBP [93] and FSU [94][95][96] Hamiltonians and the spsdpf model space.Shell model calculations for A ≤ 12 nuclides arise from the WBP interaction whereas for A ≥ 17 we quote results using the FSU interaction.Indeed, the FSU Hamiltonian starts with the WBP Hamiltonian, fitting over 270 experimental levels from 13 C to 51 Ti, and additionally includes particle-hole states originating from cross-shell excitations that give rise to intruder states.In essence, both Hamiltonians are the same near 16 O, with the WBP interaction presenting -as for ground states [52] slightly smaller polarizability values in the middle and end of the sd shell.
The general procedure involves the calculation of all the E1 and E2 matrix elements following Eqs.9 and 12. Similar SM calculations were performed for ground states and are explained in detail in Ref. [52].For self-conjugate nuclei, we calculate E1 matrix elements connecting all the intermediate GDR states with ∆T = 1; for instance, a total of 6770 0 36 Ar.Similarly, we calculate all possible E1 matrix elements from the various GDR intermediate states for other nuclei, including ∆T = 0, 1 isovector transitions.For consistency, all i Ê2 f matrix elements required in the 0 ω calculation of σ = 0.06522 eb.This discrepancy encourages further NCSM calculations using a new generation of χEF T interactions, the Lanczos-continued-fraction algorithm and reaching higher excitation energies and larger N max basis sizes [102].As shown in Table I

DISCUSSION AND CONCLUSIONS
Overall, σ SM −2 and κ SM values calculated for the first excited states in selected light nuclei align with the smooth, global trend predicted by Eq. 3 (dotted κ = 1 lines in Fig. 2), in agreement with Migdal's original calculation arising from the combination of the hydrodynamic model and second-order non-degenerate perturbation theory [5]; hence, validating modern Coulomb-excitation codes [108] used to extract collective properties of p-and sd-shell nuclei [109].
Deviations from simple hydrodynamic-model estimates are, nonetheless, observed at the beginning of the p-and sd-shells, where anomalously large σ −2 and κ values are calculated for the excited states of 6,7 Li and 17,18 O.For the odd-mass nuclei, this may be associated with the slightly unbound particle [50,110] -whose wave function extends far apart from the α-cluster configurations, i.e. α + d, α + t, 4α + n in 6 Li, 7 Li and 17 O, respectively -as inferred from the dipole resonances observed at relatively low excitation energies [111,112].Such cluster structures were avoided in the 1 ω cross-shell fits to the WBP and FSU interactions because of potential distortion in the A = 5 − 9 region.These deviations from the GDR effect are not surprising considering the fragmentation of the GDR spectrum into different 1p-1h states [113], which include the possibility of α-cluster configurations [114,115] and the virtual breakup into the continuum [116,117]; the latter supporting the breakup into the α-t continuum as the the main contribution to the polarizability in 7 Li.
The reasons behind the large σ SM −2 and κ SM values computed in 17,18 O deserve further investigation.For 18 O, this is probably related to the aforementioned mixing of intruder states lacking in the sd-shell wave functions.However, the anomalously large κ value measured for the first excitation of 17 O [39] suggests an alternative physical origin.As suggested by Kuehner et al. [39] and Ball et al. [118], a value slightly larger than κ = 1 attributed to the first 2 + 1 excitation at 1.982 MeV in 18 O could explain the long-standing ≈ 10% discrepancy between the smaller B(E2; 0 + 1 → 2 + 1 ) = 0.00421(9) e 2 b 2 determined from seven Coulomb-excitation measurements [119] and the larger one, 0.00476(11) e 2 b 2 , extracted from a highprecision lifetime measurement [118].The overall lack of high-sensitive measurements of the nuclear polarizability of excited states prevent further conclusions.
In conclusion, we present novel equations of σ −2 and α E1 values for excited states on equal footing to ground states by calculating E1 and E2 matrix elements.We apply this framework and perform full 1 ω SM calculations of σ SM −2 and κ SM values up to 36 Ar, with reasonable agreement with previous SM calculations done up to 20 Ne.Except the anomalous cases of 17,18 O, results follow the global trend predicted by Migdal [5].Dedicated Coulomb-excitation measurements with increasing sensitivity are relevant in order to elucidate the reasons behind large dipole polarizabilities that may affect quadrupole collective properties [18,40,120].These measurements could be done with the new GAMKA array at iThemba LABS in South Africa and elsewhere.More detailed ab initio calculations with uncertainty quantification above 12 C -the heaviest nucleus where the dipole polarizability for excited states has been computed from first principles -are needed to benchmark our results.
SM −2 (column 5) and κ SM (column 10) values of first-excited states in selected p − sd shell nuclei together with their corresponding S(E1) SM and i Ê2 f SM (columns 6 and 7).Experimental i Ê2 f exp [76] and κ exp values (columns 8 an 9, respectively) as well as previous SM calculations (column 11) are listed for comparison.
SM values are also computed using OXBASH with isoscalar E2 effective charges of e p ef f + e n ef f = 1.55 and e p ef f + e n ef f = 1.81 for the p [97] and sd [98] shells, respectively.Results arising from the WBP interaction have been benchmarked for compatibility with available NCSM calculations in 12 C [51] using the N N + 3N 350 − srg2.0 [81-83, 86] and N N N 4 N LO500 − srg2.4 [84-86] χEF T interactions, yielding a difference of ≈ 27% for the product of E1 matrix elements in Eq. 5. Nevertheless, the highest 1 − state calculated with OXBASH is at ≈ 65 MeV, whereas for the NCSM it corresponds to ≈ 30 MeV.Below 30 MeV, the sum of E1 strengths reach converging values of ≈ 1.3 e 2 fm 2 and ≈ 0.16 e 2 fm 2 for 0 +