Precision measurement of the E2 transition strength to the 2 + 1 state of 12 C
A. D'Alessio, T. Mongelli, M. Arnold, S. Bassauer, J. Birkhan, M. Hilcker, T. Hüther, J. Isaak, L. Jürgensen, T. Klaus, P. von Neumann-Cosel, N. Pietralla, V. Yu. Ponomarev, P. C. Ries, R. Roth, M. Singer, G. Steinhilber, K. Vobig, V. Werner
PPrecision measurement of the E transition strength to the 2 +1 state of C A. D’Alessio, ∗ T. Mongelli, † M. Arnold, S. Bassauer, J. Birkhan, M. Hilcker,T. H¨uther, J. Isaak, L. J¨urgensen, T. Klaus, P. von Neumann-Cosel, N. Pietralla, ‡ V. Yu. Ponomarev, P. C. Ries, R. Roth, § M. Singer, G. Steinhilber, K. Vobig, and V. Werner
Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany (Dated: May 13, 2020)The form factor of the electromagnetic excitation of C to its 2 +1 state was measured at extremelylow momentum transfers in an electron scattering experiment at the S-DALINAC. A combinedanalysis with the world form factor data results in a reduced transition strength B ( E
2; 2 +1 → +1 ) = 7 . fm with an accuracy improved to 2.5%. In-Medium-No Core Shell Model resultswith interactions derived from chiral effective field theory are capable to reproduce the result. Aquadrupole moment Q (2 +1 ) = 5 . can be extracted from the strict correlation with the B (E2) strength emerging in the calculations. I. INTRODUCTION
Alpha clustering dominates the structure features ofmany light nuclei, especially of so-called α -like nucleiwith mass numbers A = 4 n , where n is an integer [1]. Thenucleus C is a prime example with the first excited 0 + state (the Hoyle state) showing pronounced cluster fea-tures [2]. Accordingly, a variety of microscopically basedcluster models have been developed (see Ref. [1] and ref-erences therein). There, the B ( E
2) transition strengthto the 2 +1 state plays a special role because it determinesthe degree of α clustering in the ground state (g.s.) wavefunction and many properties of rotational and vibra-tional states built on it. A particular example are alge-braic models exploiting geometrical symmetries [3].On the other hand, the nucleus C is a crucial testingground for ab-initio calculations in modern theoreticalnuclear physics. The No Core Shell Model (NCSM), aswell as importance truncated no-core shell model (IT-NCSM) calculations and other theoretical approacheslike coupled cluster methods [4–22] focus on describ-ing and predicting g.s. properties, excitation energiesand spectroscopic quantities in p - and sd -shell nuclei.Since the model space increases strongly with the num-ber of nucleons, the NCSM can be used for light nucleionly. To overcome this limitation, the In-Medium Sim-ilarity Renormalization Group (IM-SRG)[23] has beencombined with the NCSM forming the In-Medium NoCore Shell Model (IM-NCSM)[20], which allows to im-prove significantly the convergence behaviour. Observ-ables that react sensitively to long-range correlations ofthe wave function, such as radii, the quadrupole momentor the B ( E
2) strength, converge more slowly than, for ex-ample, excitation energies. This makes them importantfor setting boundary conditions for calculations. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] A remarkable correlation between the B ( E
2; 2 +1 → +1 )strength and the quadrupole moment Q (2 +1 ) in C wasobserved recently for a wider range of chiral effectivefield theory (EFT) interactions [24]. Experimentally, thevalue of the 2 +1 quadrupole moment of 6(3) efm [25]was poorly known only. Therefore, a Coulomb-excitationreorientation-effect measurement was recently carried out[26]. Based on the then available information for the B ( E
2) strength, the oblate g.s. deformation expectedfrom the cluster models could be confirmed but the over-all uncertainty was only slightly improved to about 35%.The reorientation of the magnetic sub-states of the 2 +1 state is a second-order process and in order to extract Q (2 +1 ) from the experimental data it is necessary to knowthe first order process (i.e. the B ( E
2) strength) as preciseas possible to further improve the uncertainty.Considering the impact on the above problem and thegeneral importance as a benchmark for the structure cal-culations, an improved value of the B(E2; 2 +1 → +1 )transition strength in C is clearly of interest and vari-ous experimental approaches are currently being pursuedincluding nuclear resonance fluorescence selfabsorptionexperiments [27] and the ( e, e (cid:48) ) experiment presented inthis paper.
II. ELECTRON SCATTERING EXPERIMENT
The form factor measurements of the transition tothe 2 +1 state of the C nucleus were performed withthe LINTOTT spectrometer [28] using an electron beamof 42.5 MeV from the S-DALINAC [29] impinging on a100 mg/cm natural carbon target (98.9% abundance of C). The LINTOTT spectrometer was placed at anglesof 69 ◦ , 81 ◦ and 93 ◦ with respect to the incoming electronbeam, allowing measurements at extremely low momen-tum transfers of q (cid:39) (0 . − .
32) fm − . The low- q datapermit an improved extrapolation of the form factor ofthe 2 +1 state to the photon point ( k = E x / ¯ hc ) as dis-cussed below.Since elastic scattering cross sections in C are knownwith high precision [30–34], the form factor of the excited a r X i v : . [ nu c l - e x ] M a y -0.5 0.0 0.5 1.0E x (MeV)10 C o un t s C (e,e')E = 42.5 MeV = 69° 0 +1 x (MeV)6008001000 C o un t s +1 FIG. 1. Elastic electron scattering spectrum taken at a beamenergy of 42.5 MeV and a scattering angle of 69 ◦ . The insetshows the excitation of the 2 +1 state. The red lines display afit using Eq. (1) and the blue lines a linear background. +1 state was determined in a relative measurement. Atthe low beam energy, the momentum acceptance of thespectrometer of 2% is not sufficient to observe the g.s.and the excited state transition with the same magneticfield settings. However, the fields can be set in such a waythat the peaks of the ground state and of the 2 +1 state ap-pear in the same channels of the silicon strip focal planedetector [28] minimizing solid angle and efficiency uncer-tainties of the detector system. An example of the elasticscattering data is shown in Fig. 1. The inset presents acorresponding measurement of the excitation of the 2 +1 state.In order to further reduce the systematic uncertainties,the data taking for the inelastic transition was stoppedin regular intervals and intermittent measurements of theelastic line were performed. Thus, variations due to pos-sible changes in beam position and/or beam energy werereduced by averaging over the ratio of the peak areasnormalized to the collected charge. The elastic scatter-ing data were sliced into spectra with 50000 counts intotal before the area-over-charge ratio was determined.Typical fluctuations (blue circles) and the uncertainty-weighted average (red bands) for the 69 ◦ data as an ex-ample are presented in Fig. 2 for elastic (main figure)and inelastic (inset) scattering. The weighted averagevalues are 4 . · − Counts/nC for the inelastic scattering data.The peak areas were determined by a fit using the phe-nomenological parameterization [35] y ( x ) = y exp (cid:2) − ln 2 · ( x − x ) / ∆ x (cid:3) x < x exp (cid:2) − ln 2 · ( x − x ) / ∆ x (cid:3) x < x ≤ x + η ∆ x A/ ( B + x − x ) γ x > x + η ∆ x (1) with x denoting the peak energy, y the count rate at x ,and ∆ x , the half widths at half maximum for E x < x and E x > x , respectively. The parameters η , A, B, and γ describe the radiative tail. A possible instrumental A r e a / C h a r g e A r e a / C h a r g e
1e 3
FIG. 2. Area/Charge ratios for slices of the elastic line mea-surements at 69 ◦ (blue circles) and uncertainty weighted av-erage (red band). The inset shows corresponding values forthe excitation of the 2 +1 state. background was allowed for, approximated by a linearfunction. The peak area was determined by integrationof the deduced line shape from x − x to x + 5∆ x .Then, the form factor of the inelastic transition to the2 +1 state can be determined from the relation | F ( q ) | +1 = | F ( q ) | A +1 A g.s. , (2)where A g.s. and A +1 denote the areas under the peaksnormalized to the collected charge of the respective mea-surement. The results are summarized in Tab. I.Extensive form factor data have been measured for thistransition over a wide range of momentum transfers, butnot below q = 0 .
405 fm − [36–39]. In Ref. [21] an ana-lytic, global, and model-independent analysis of transi-tion form factors of exited states was introduced. F ( q ) = 1 Z e − ( bq ) n max (cid:88) n =1 c n ( bq ) n , (3)with Z being the charge of the probed nucleus, q the mo-mentum transfer of the electron, and b , c n fit parameters.As illustrated in Ref. [21] for the example of the transi-tion to the 0 +2 state (the Hoyle state), inclusion of low- q data is essential for a minimization of uncertainties.Since Eq. (3) holds in plane wave Born approximationonly, the experimental data corresponding to distorted TABLE I. Experimental form factors for the transition to the2 +1 state of C from the present experiment. E Θ lab q | F ( q ) | (MeV) (deg) (fm − ) (10 − )42.5 93 ◦ ◦ ◦ q (fm )10 | F ( q ) | photon point 10 fit to dataworld dataS-DALINAC data 10 -505 residuals FIG. 3. Experimental form factor of the transition to the2 +1 state in C after the DWBA corrections described in thetext. Data from the present work are shown as blue squaresand previous measurements [36–39] as green circles. Most ofthe error bars are smaller than the displayed data points. Thered band shows a fit of Eq. (3) with a 1 σ uncertainty. Thearrow indicates the photon point. wave Born approximation (DWBA) form factors must becorrected as outlined in Ref. [21]. The theoretical tran-sition density of the 2 +1 state needed as starting point ofthe iterative procedure stems from a NCSM calculation.Figure 3 presents the corrected experimental form factordata together with a fit of Eq. (3) shown as red band. Theresults of Ref. [30] at very high q with incident energiesof 600 −
800 MeV were not taken into account as it wasnot possible to calculate a DWBA correction for thesedata and their contribution to the extrapolation of thetransition form factor to the photon point is negligible.The fit provides a value | F ( q ) | = 1 . · − atthe photon point. The impact of the current experimentcan be seen from the corresponding result obtained with-out the low- q data points | F ( q ) | = 1 . · − witha four times larger relative uncertainty. Using the rela-tion [40] B ( E
2; 2 +1 → +1 ) = 45 Z πq lim q → k | F ( q ) | (4)we derive a transition strength of 7.63(19) e fm . Thisagrees with the literature value 7.94(40) e fm [41] withinerror bars but improves the uncertainty from currently5.5% to 2.5%. III. IN-MEDIUM NCSM CALCULATIONS
For the theoretical description of the spectroscopy of C we use the IM-NCSM introduced in Ref. [20]. Thisnovel ab initio method combines NCSM [42, 43] with anIM-SRG [44–46] decoupling of the many-body Hamilto-nian, which drastically accelerates the model-space con-vergence of the NCSM. This is particularly relevant for the description of electric quadrupole observables for nu-clei in the upper p -shell and above, as these observablescannot be fully converged within the standard NCSM orthe IT-NCSM [8, 24, 47].The IM-NCSM calculation is a four-step process: Ina first step, an optimized single-particle basis is con-structed for the nucleus and interaction under considera-tion, using natural orbitals for a perturbatively improvedone-body density matrix [48]. In the second step, the ref-erence state for the IM-SRG decoupling is obtained froma NCSM calculation in a small N refmax model space. Thethird step then uses a multi-reference version of the IM-SRG using the White generator [49] to decouple the ref-erence space from all excitations. We employ the Magnusformulation of the flow equations, which enables a consis-tent and efficient transformation of the Hamiltonian andall other operators, including the electric quadrupole op-erator [50]. In the final step, the IM-SRG-transformedoperators are used in a NCSM calculation for moder-ate N max . The two model-space truncation parameters, N refmax and N max , will be used later on for the quantifica-tion of uncertainties in this many-body approach.All calculations build on a new family of chiral two-plus three-nucleon interactions presented in Ref. [51].Starting from the accurate chiral two-nucleon interac-tions by Entem, Machleidt, and Nosyk [52] with non-local regulators up to N LO for three different cutoffsΛ = 450 MeV/c, 500 MeV/c, and 550 MeV/c, we supple-ment chiral three-body forces at N LO and N LO withthe same regulators and cutoff values. The low-energyconstants in the three-nucleon sector are determined fromthe H and the O ground state energies. This leads to afamily of interactions that provides a good simultaneousdescription of ground state energies and charge radii upinto the medium-mass regime and, at the same time, agood description of excitation spectra of light nuclei [51].The Hamiltonian is evolved in a free-space SRG evolu-tion at the three-body level with a flow-parameter α =0.04 fm [53, 54]. We note that for the E B ( E
2) value, smallerthan our present theory uncertainties, but are eventuallyneeded for a fully consistent description.To illustrate the superior convergence behavior and theuncertainties of the IM-NCSM calculation, Fig. 4 depictsthe excitation spectrum, the B ( E , +1 → +1 ) strength,and the electric quadrupole moment Q (2 +1 ) as a functionof N max for different values of N refmax . Obviously, the re-sults for all observables are very stable with increasing N max , showing that the final NCSM calculation is fullyconverged even for these small model spaces. The depen-dence on the reference-space size N refmax , which indirectlyprobes the effect of omitted normal-ordered three-bodyterms in the IM-SRG, is also quite small. We estimatethe uncertainties of the many-body treatment based onthe differences of the observables for successive values of E * [ M e V ] +1 +1 +2 +1 N refmax = 0 +1 +1 +2 +1 N refmax = 2 +1 +1 +2 +1 N refmax = 4 +1 +1 +1 +1 +2 Exp B ( E ) [ e f m ] +1 +1 +1 +1 +1 +1 +1 +1 N max Q [ e f m ] +1 N max +1 N max +1 +1 FIG. 4. Excitation spectrum, B ( E
2) transition strength, andquadrupole moment for C obtained in the IM-NCSM for dif-ferent reference-space truncations N refmax (panels left to right)as function of N max . All calculations are performed with thechiral two- plus three-body interaction at N LO with cutoffΛ = 500 MeV/c. The error bars indicate the many-bodyuncertainties (see text). N max and N refmax and we also include a variation of the IM-SRG flow parameter by a factor of two. The maximum ofthese three differences gives the many-body uncertaintyinducted by the error bars in Fig. 4. Note that in allcases, the change of N refmax determines this maximum and,thus, the total many-body uncertainty. For the interac-tion employed in Fig. 4, the chiral interaction at N LOwith Λ = 500 MeV/c the agreement of the 2 +1 excitationenergies, the B ( E
2) strength, and the quadrupole mo-ment with experiment is remarkable. Moreover, the newfamily of chiral interactions gives us the opportunity tostudy the robustness of the results under variation of thechiral order. This is illustrated in Fig. 5 for the inter-actions from NLO to N LO with cutoff Λ = 500 MeV/c.Given the complete convergence with N max we only showthe results for N max = 4 with error bars indicating themany-body uncertainties as described before. From theorder-by-order behavior of the individual observables wecan extract the uncertainties caused by the truncationof the chiral expansion. We use a simple prescriptiondescribed in Ref. [51], which goes back to Refs. [55–57],using the differences of subsequent orders weighted bypowers of the expansion parameter. These interactionuncertainties at N LO and N LO are indicated by shadedbands in Fig. 5. We observe that the results for the2 +1 excitation energy and the B ( E , +1 → +1 ) strengthrobustly agree with experiment within uncertainties atN LO and N LO. Furthermore, we obtain an accurateprediction for the quadrupole moment with theory un-certainties that are almost an order of magnitude smallerthan the present experimental uncertainties [26].Finally, we combine the results for B ( E , +1 → +1 ) E * [ M e V ] +1 +1 +1 +1 N refmax = 0 +1 +1 +1 +1 N refmax = 2 +1 +1 +1 +1 N refmax = 4 +1 +1 +1 +1 Exp B ( E ) [ e f m ] +1 +1 +1 +1 +1 +1 +1 +1 NLO N LON LOChiral Order0510 Q [ e f m ] +1 NLO N LO N LOChiral Order +1 NLO N LO N LOChiral Order +1 +1 FIG. 5. Excitation spectrum, B ( E
2) transition strength, andquadrupole moment for C obtained in the IM-NCSM for dif-ferent reference-space truncations N refmax (panels left to right)with interactions from NLO to N LO with cutoff Λ = 500MeV/c. The error bars represent many-body uncertaintiesand the shaded bars indicate the interaction uncertainties (seetext). and Q (2 +1 ) in a correlation plot shown in Fig. 6. We in-clude the N LO and N LO interactions for all three val-ues of the cutoff with error bars reflecting the combinedmany-body and interaction uncertainties. Here we onlyshow the IM-NCSM calculations for the largest modelspace with N max = N refmax = 4. The results for all 6interactions fall onto a single line, as was already ob-served in Ref. [24] for various first-generation chiral in-teractions. While N LO interactions show a larger cutoffdependence, the N LO results bracket the experimen-tal B ( E
2) value and show a reduced cutoff dependence,as summarized in Tab. II. The various microscopic re-sults can be fit by a simple rotor-model correlation. Thetwo lines show the correlation predicted by a rigid rotor(dashed) and the fitted rotor model with a ratio of theintrinsic quadrupole moments Q ,t /Q ,s = 0.967 (solid).Details can be found in Ref. [24], where almost the sameratio of the transition and static intrinsic quadrupole mo-ments was found based on a completely different set ofinteractions. TABLE II. Electric quadrupole obervables obtained with theIM-NCSM for N max = N refmax = 4 using the N LO interac-tions with three different cutoffs Λ. The uncertainties includemany-body and interaction uncertainties.Λ E x (2 +1 ) B ( E , +1 → +1 ) Q (2 +1 )(MeV/c) (MeV) ( e fm ) ( e fm )450 3.96(20) 7.14(53) 5.86(15)500 4.41(30) 8.68(79) 6.28(29)550 4.45(27) 8.18(108) 6.12(41) Q ( +1 ) [e fm ]456789 B ( E , + + ) [ e f m ]
450 MeV/c500 MeV/c550 MeV/cN LON LO FIG. 6. Correlation of the quadrupole observables B ( E , +1 → +1 ) and Q (2 +1 ) for C obtained with N LO andN LO interactions for three different cutoffs. All IM-NCSMcalculations are performed with N max = N refmax = 4. The errorbars indicate the combined many-body and interaction un-certainties. The lines show the prediction of a simple rigidrotor (dashed) and a fitted (solid) rotor model, see text. Thehorizontal and vertical red shaded bands indicate the exper-imental B ( E
2) value and the Q (2 +1 ) value derived from theintersection with the model correlation. The grey and redareas indicate the experimental limits from literature values[26, 41] and from the present work, respectively. We can combine this correlation with the new exper-imental value for the B ( E , +1 → +1 ) to obtain anaccurate value for the quadrupole moment Q (2 +1 ) =5 . e fm , where the uncertainties include the av-erage many-body and interaction uncertainties of theN LO calculations for the quadrupole moment and theexperimental uncertainties of the transition strength propagated via the correlation. This value is compatiblewithin uncertainties with the Q (2 +1 ) computed directly inthe IM-NCSM with the N LO interactions for all threecutoffs, as seen in Tab. II. The red area in Fig. 6 indi-cates the new experimental value of the B ( E
2) and thequadrupole moment of the 2 +1 state in C extracted fromthe correlation analysis, both with their uncertainties, incomparison to the literature values [26, 41] (grey area).
IV. SUMMARY
The present work reports a new measurement of theelectron scattering form factor of the transition to the 2 +1 state in C at very low momentum transfers. Combinedwith the world data this permits an extraction of the B ( E
2) strength based on the model-independent analy-sis introduced in Ref. [21] with a much improved relativeuncertainty of 2.5%. This highly precise value is used tobenchmark a new family of chiral two- plus three-nucleoninteractions [51] and test the convergence properties ofcalculations with the novel ab initio
IM-NCSM method[20]. Very good agreement is obtained. The correlationbetween the B ( E
2) and Q (2 +1 ) values in the model re-sults, which can be described by a simple rotor model,permits an extraction of the hard-to-measure quadrupolemoment [26] with a precision improved by almost an or-der of magnitude. ACKNOWLEDGMENTS
This work was funded by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) un-der grant No. SFB 1245 (project ID 279384907) andGRK 2128 (project ID 264883531). The ab initio cal-culations were performed on the LICHTENBERG highperformance cluster at the computing center of the TUDarmstadt. [1] M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, andU.-G. Meißner, Rev. Mod. Phys. , 035004 (2018).[2] M. Chernykh, H. Feldmeier, T. Neff, P. von Neumann-Cosel, and A. Richter, Phys. Rev. Lett. , 032501(2007).[3] R. Bijker and F. Iachello, Prog. Part. Nucl. Phys. ,103735 (2020).[4] E. Evgeny, H. Krebs, D. Lee, and U.-G. Meißner, Phys.Rev. Lett. , 192501 (2011).[5] E. Epelbaum, H. Krebs, T. A. L¨ahde, D. Lee, and U.-G.Meißner, Phys. Rev. Lett. , 252501 (2012).[6] W. R. Zimmerman, M. W. Ahmed, B. Bromberger, S. C.Stave, A. Breskin, V. Dangendorf, T. Delbar, M. Gai,S. S. Henshaw, J. M. Mueller, C. Sun, K. Tittelmeier,H. R. Weller, and Y. K. Wu, Phys. Rev. Lett. , 152502(2013). [7] K. Kravvaris and A. Volya, Phys. Rev. Lett. , 062501(2017).[8] C. Forss´en, R. Roth, and P. Navr´atil, J. Phys. G: Nucl.Part. Phys. , 055105 (2013).[9] S. C. Pieper, Nucl. Phys. A , 516 (2005).[10] P. Maris, J. P. Vary, A. Calci, J. Langhammer, S. Binder,and R. Roth, Phys. Rev. C , 014314 (2014).[11] B. R. Barrett, P. Navrtil, and J. P. Vary, Prog. Part.Nucl. Phys. , 131 (2013).[12] P. Navr´atil, V. G. Gueorguiev, J. P. Vary, W. E. Ormand,and A. Nogga, Phys. Rev. Lett. , 042501 (2007).[13] P. Navr´atil, S. Quaglioni, I. Stetcu, and B. R. Barrett, J.Phys. G: Nucl. Part. Phys. , 083101 (2009).[14] R. Roth and P. Navrtil, Phys. Rev. Lett. (2007).[15] R. Roth, Phys. Rev. C , 064324 (2009). [16] A. Tichai, J. M¨uller, K. Vobig, and R. Roth, Phys. Rev.C , 034321 (2019).[17] E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner,Phys. Rev. Lett. , 192501 (2011).[18] T. Neff and H. Feldmeier, J. Phys.: Conf. Series ,012062 (2014).[19] Y. Yoshida and Y. Kanada-En’yo, Prog. Theor. Exp.Phys. , 123D04 (2016).[20] E. Gebrerufael, K. Vobig, H. Hergert, and R. Roth, Phys.Rev. Lett. , 152503 (2017).[21] M. Chernykh, H. Feldmeier, T. Neff, P. von Neumann-Cosel, and A. Richter, Phys. Rev. Lett. , 022501(2010).[22] M. Freer and H. Fynbo, Prog. Part. Nucl. Phys. , 1(2014).[23] H. Hergert, S. K. Bogner, S. Binder, A. Calci, J. Lang-hammer, R. Roth, and A. Schwenk, Phys. Rev. C ,034307 (2013).[24] A. Calci and R. Roth, Phys. Rev. C , 014322 (2016).[25] W. Vermeer, M. Esat, J. Kuehner, R. Spear, A. Baxter,and S. Hinds, Phys. Lett. B , 23 (1983).[26] M. Kumar Raju, J. N. Orce, P. Navrtil, G. C. Ball,T. E. Drake, S. Triambak, G. Hackman, C. J. Pearson,K. J. Abrahams, E. H. Akakpo, H. Al Falou, R. Church-man, D. S. Cross, M. K. Djongolov, N. Erasmus, P. Fin-lay, A. B. Garnsworthy, P. E. Garrett, D. G. Jenkins,R. Kshetri, K. G. Leach, S. Masango, D. L. Mavela, C. V.Mehl, M. J. Mokgolobotho, C. Ngwetsheni, G. O’Neill,E. T. Rand, S. K. L. Sjue, C. S. Sumithrarachchi, C. E.Svensson, E. R. Tardiff, S. J. Williams, and J. Wong,Phys. Lett. B , 250 (2018).[27] C. Romig, D. Savran, J. Beller, J. Birkhan, A. Endres,M. Fritzsche, J. Glorius, J. Isaak, N. Pietralla, M. Scheck,L. Schnorrenberger, K. Sonnabend, and M. Zweidinger,Phys. Lett. B , 369 (2015).[28] A. Lenhardt, U. Bonnes, O. Burda, P. von Neumann-Cosel, M. Platz, A. Richter, and S. Watzlawik, Nucl. In-strum. Methods Phys. Res. A , 320 (2006).[29] N. Pietralla, Nucl. Phys. News , 4 (2018).[30] H. Crannell, Phys. Rev. , 1107 (1966).[31] I. Sick and J. McCarthy, Nucl. Phys. A , 631 (1970).[32] W. Reuter, G. Fricke, K. Merle, and H. Miska, Phys. Rev.C , 806 (1982).[33] F. J. Kline, H. Crannell, J. O’Brien, J. McCarthy, andR. R. Whitney, Nucl. Phys. A , 381 (1973).[34] J. A. Jansen, R. T. Peerdeman, and C. De Vries, Nucl.Phys. A , 337 (1972).[35] F. Hofmann, P. von Neumann-Cosel, F. Neumeyer,C. Rangacharyulu, B. Reitz, A. Richter, G. Schrieder,D. I. Sober, L. W. Fagg, and B. A. Brown, Phys. Rev. C , 024311 (2002). [36] J. Fregeau, Phys. Rev. , 225 (1956).[37] H. L. Crannell and T. A. Griffy, Phys. Rev. , B1580(1964).[38] J. H. Fregeau and R. Hofstadter, Phys. Rev. , 1503(1955).[39] M. Bernheim, T. Stovall, and D. Vinciguerra, Phys. Lett.B , 461 (1967).[40] H. Theissen, Springer Tracts in Modern Physics No. 65(Springer, Berlin, 1972) p. 145.[41] B. Pritychenko, M. Birch, B. Singh, and M. Horoi,Atomic Data and Nuclear Data Tables , 1 (2016).[42] P. Navr´atil, J. P. Vary, and B. R. Barrett, Phys. Rev. C , 054311 (2000).[43] P. Navr´atil, S. Quaglioni, I. Stetcu, and B. R. Barrett, J.Phys. G: Nucl. Part. Phys. , 083101 (2009).[44] H. Hergert, S. Binder, A. Calci, J. Langhammer, andR. Roth, Phys. Rev. Lett. , 242501 (2013).[45] H. Hergert, Phys. Scripta , 023002 (2016).[46] H. Hergert, J. M. Yao, T. D. Morris, N. M. Parzuchowski,S. K. Bogner, and J. Engel, J. Phys.: Conf. Series ,012007 (2018).[47] P. Maris, J. P. Vary, A. Calci, J. Langhammer, S. Binder,and R. Roth, Phys. Rev. C , 014314 (2014).[48] A. Tichai, J. M¨uller, K. Vobig, and R. Roth, Phys. Rev.C , 034321 (2019).[49] S. R. White, J. Chem. Phys. , 7472 (2002).[50] K. Vobig, Electromagnetic Observables and Open-ShellNuclei from the In-Medium No-Core Shell Model , Doc-toral thesis D17, Technical University Darmstadt (2019).[51] T. Hther, K. Vobig, K. Hebeler, R. Machleidt, andR. Roth, arXiv:1911.04955v1.[52] D. R. Entem, R. Machleidt, and Y. Nosyk, Phys. Rev. C , 024004 (2017).[53] R. Roth, J. Langhammer, A. Calci, S. Binder, andP. Navr´atil, Phys. Rev. Lett. , 072501 (2011).[54] R. Roth, A. Calci, J. Langhammer, and S. Binder, Phys.Rev. C , 024325 (2014).[55] E. Epelbaum, H. Krebs, and U. G. Meißner, Eur. Phys.J. A , 53 (2015).[56] S. Binder, A. Calci, E. Epelbaum, R. J. Furnstahl, J. Go-lak, K. Hebeler, H. Kamada, H. Krebs, J. Langham-mer, S. Liebig, P. Maris, U.-G. Meißner, D. Minossi,A. Nogga, H. Potter, R. Roth, R. Skibi´nski, K. Topol-nicki, J. P. Vary, and H. Wita(cid:32)la (LENPIC Collaboration),Phys. Rev. C , 044002 (2016).[57] S. Binder, A. Calci, E. Epelbaum, R. J. Furn-stahl, J. Golak, K. Hebeler, T. H¨uther, H. Kamada,H. Krebs, P. Maris, U.-G. Meißner, A. Nogga, R. Roth,R. Skibi´nski, K. Topolnicki, J. P. Vary, K. Vobig, andH. Wita(cid:32)la (LENPIC Collaboration), Phys. Rev. C98