Proton-neutron pairing correlations in the self-conjugate nucleus 42Sc

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Proton-neutron pairing correlations in the self-conjugate nucleus 42Sc © 2021 The Author(s). Published by Elsevier B.V. Funded by SCOAP3. Published version Koszorús, Á.; Vormawah, L.J.; Beerwerth, R.; Bissell, M.L.; Campbell, P.; Cheal, B.; Devlin, C.S.; Eronen, T.; Fritzsche, S.; Geldhof, S.; Heylen, H.; Holt, J.D.; Jokinen, A.; Kelly, S.; Moore, I.D.; Miyagi, T.; Rinta-Antila, S.; Voss, A.; Wraith, C. Koszorús, Á., Vormawah, L.J., Beerwerth, R., Bissell, M.L., Campbell, P., Cheal, B., Devlin, C.S., Eronen, T., Fritzsche, S., Geldhof, S., Heylen, H., Holt, J.D., Jokinen, A., Kelly, S., Moore, I.D., Miyagi, T., Rinta-Antila, S., Voss, A., & Wraith, C. (2021). Proton-neutron pairing correlations in the self-conjugate nucleus 42Sc. Physics Letters B, 819, Article 136439. https://doi.org/10.1016/j.physletb.2021.136439 2021 Physics Letters B 819 (2021) 136439 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Proton-neutron pairing correlations in the self-conjugate nucleus 42Sc Á. Koszorúsa,∗, L.J. Vormawaha, R. Beerwerthb, M.L. Bissellc, P. Campbell c, B. Cheala, C.S. Devlina, T. Eronen d, S. Fritzscheb, S. Geldhofd,e, H. Heylene,f, J.D. Holtg,h, A. Jokinend, S. Kellyc, I.D. Moored, T. Miyagig, S. Rinta-Antilad, A. Vossd, C. Wraitha aDepartment of Physics, University of Liverpool, Liverpool L69 7ZE, United Kingdom bHelmholtz Institute Jena, Fröbelstieg 3, 07743 Jena, Germany cSchool of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom dDepartment of Physics, University of Jyväskylä, PB 35(YFL) FIN-40351 Jyväskylä, Finland eKU Leuven, Instituut voor Kern-en Stralingsfysica, B-3001 Leuven, Belgium fCERN, Experimental Physics Department, CH-1211 Geneva 23, Switzerland gTRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada hDepartment of Physics, McGill University, 3600 Rue University, Montréal, QC H3A 2T8, Canada a r t i c l e i n f o a b s t r a c t Article history: Received 3 February 2021 Received in revised form 14 May 2021 Accepted 7 June 2021 Available online 10 June 2021 Editor: B. Blank Keywords: Collinear laser spectroscopy Hyperfine structure and isotope shift Proton-neutron pairing Charge radius Collinear laser spectroscopy of the N=Z=21 self-conjugate nucleus 42Sc has been performed at the JYFL IGISOL IV facility in order to determine the change in nuclear mean-square charge radius between the Iπ=0+ground state and the Iπ=7+isomer via the measurement of the 42g,42mSc isomer shift. New multi-configurational Dirac-Fock calculations for the atomic mass shift and field shift factors have enabled a recalibration of the charge radii of the 42−46Sc isotopes which were measured previously. While consistent with the treatment of proton-neutron, proton-proton and neutron-neutron pairing on an equal footing, the reduction in size for the isomer is observed to be of a significantly larger magnitude than that expected from both shell-model and ab-initio calculations. The measured nuclear magnetic dipole moment and electric quadruple moment, on the other hand, are in good agreement with simple empirical estimates and shell-model calculations. ©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. Odd-odd self-conjugate (N=Z) nuclei provide an ideal testing ground for proton-neutron pairing studies. More specifically, such nuclei are ideal for verifying if the same phenomena arise for an I=0 proton-neutron (πν) pair as for a proton-proton (ππ) or neutron-neutron (νν) pair with I=0[1]. The charge independence of the nucleon-nucleon interaction suggests symmetry in isospin between protons and neutrons, though recent findings elucidate the violation of the mirror symmetry [2]in the ground state of bound nuclei on the edge of the nuclear landscape. In odd-odd N=Znuclei, protons and neutrons occupy the same orbitals. Therefore, the interaction between the odd proton and neutron is enhanced, leading to a greater likelihood of the formation of a πν pair. For such a nucleus, it is expected that the charge radius will be greater for a state with I=0, T=1than for such with I= 0, T=0. This arises due to a similar orbital blocking picture to that accounting for the multi-quasiparticle isomer radii [3]. The I=0, T=1 (isovector) πν pair, can scatter into a *Corresponding author. E-mail address: [email protected] (Á. Koszorús). wide range of excited orbitals and still satisfy I=0; as these orbitals are less bound, they will have a greater spatial extent. The I= 0, T=0 (isoscalar) πν pair, on the other hand, is significantly restricted in the number of orbitals it can mix with. In such cases, orbitals with the same spin may well be energetically far apart (possibly even across a major shell closure), rendering such scattering unlikely [4]. A study of pairing effects on the relative mean-square charge radii of multi-quasiparticle isomers [3], such as 97m2Y, 176mYb and 178m1Hf, was motivated by the measurement of 178m2Hf by Boos et al. [5]. All of these isomers were found to have a smaller meansquare charge radius than their respective ground states irrespective of nuclear deformation. This offered an explanation into the observed odd-even staggering of nuclear charge radii [6]as arising from a combination of increasing rigidity (i.e. a reduction in the root mean square quadrupole deformation value towards the mean value) or decreasing surface diffuseness, due to the orbital (Pauli) blocking of the odd nucleon [3]. The multi-quasiparticle isomer studies looked into the effects of general nucleon pairing, not specifically πν pairs, the existence https://doi.org/10.1016/j.physletb.2021.136439 0370-2693/©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. Á. Koszorús, L.J. Vormawah, R. Beerwerth et al. Physics Letters B 819 (2021) 136439 of which has yet to be proved conclusively [7], although many studies have yielded results supporting the idea. The premise for using odd-odd N=Znuclei as a testbed for πν pairing correlations is supported by the investigation in 38K (N, Z=19) in which the relative difference in ground and isomeric state charge radii was probed via a direct measurement of the 38g,38mK isomer shift [4]. In this system, the I=3, T=0ground state was found to have a smaller charge radius than the I=0, T=1 isomer, thus validating the prediction of this intuitive πν pairing model. Shell-model calculations appear to reproduce the effect quantitatively for 38g,38mK[4]. Such a remarkable agreement motivates the study of other N=Znuclei. Above the N=Z=20 shell closures, an inversion of the ground and isomeric states is observed and the I=0+, T=1 becomes the ground state. The isomer shift has only been measured in one such system, 50g,50mMn [8], which shows an effect of similar magnitude as in 38K. In 42Sc a I=0+, T=1is the ground state, and a long-lived isomer is present with I=7+, T=0[9,10]. The measurement of this 42g,42mSc isomer shift, where the ground state is expected to be larger, will test if this simple trend continues and whether it can be quantitatively understood. 1. Experimental methodology Collinear laser spectroscopy [11,12]was performed at the IGISOL IV facility [13], located at the University of Jyväskylä JYFL Accelerator Laboratory, Finland, where beams of short-lived radioactive ions can be produced [14]. Singly-charged 42Sc ions were produced via a 30 MeV 40Ca(α,pn)42Sc fusion-evaporation reaction, taking place in an environment of a helium buffer gas at a pressure of ∼150 mbar inside the IGISOL chamber. The use of thin foil targets (thickness ∼2 mg/cm2) at IGISOL, coupled with the extraction of reaction products via a supersonic gas jet, enables fast release irrespective of physical or chemical properties. Once extracted from the target chamber, the reaction products with around 3500 ions per second rate, were then formed into a 30 keV beam of singlycharged ions which were subsequently mass separated using a 55◦ dipole magnet. Ions were cooled and bunched using a gas-filled radio-frequency quadrupole [15], from which bunches of temporal width 20 μs were released every 100 ms and directed to the laser spectroscopy station. Ions were overlapped in an anti-collinear geometry with 0.5 mW of light from a frequency doubled Spectra Physics 380D continuous-wave laser, running with Pyridine 2 dye. Optical spectra were taken using the same 363.1 nm 3d4s3D2→3d4p3F3 atomic transition as used in a previous study of radioactive scandium isotopes [16]. The fundamental frequency of the laser was locked to an I2absorption line and ions were Doppler tuned across the resonances by applying a voltage ramp to the photon-ion interaction region. For each voltage step an ion bunch was released and a gate was applied to the photon detection signal corresponding to the ion bunch transit time in front of the photomultiplier tube. The resulting voltage scans were transformed into frequency, f, via f=fL(1+α+2α+α2), (1) where α=eV/mc2, Vis the total accelerating voltage and fLis the laser frequency [11]. Several independent scans of 42g,42mSc were taken and summed. The frequency conversion was performed using the atomic mass, m, of 42gSc [17], and incorporating the 617 keV excitation of the isomer [9,10]as appropriate. Fig. 1. Measured and fitted hyperfine spectrum of 42Sc measured on the 363.1 nm line, together with the separate contributions from the ground and isomeric states (offset for clarity). Table 1 Values of the hyperfine magnetic dipole, Au(3F3), and electric quadrupole, Bu(3F3), parameters for 42mSc, measured on the 363.1 nm line. AAu(MHz) Bu(MHz) δν42g,m(MHz) 42mSc +82.6(3)−34(28)+74(5) 2. Data analysis and results A chi-squared minimisation routine was used in order to fit a series of Lorentzian peaks to the data, from which the hyperfine A(magnetic dipole) and B(electric quadrupole) parameters [11] were extracted for the upper electronic state (Au, Bu). Ratios of A and Bparameters for the upper and lower states were constrained to Al/Au=2.469 and Bl/Bu=0.552 [16]. Frequencies of the isomeric peaks were related according to: γ=ν+(αu−2.469αl)Au+(βu−0.552βl)Bu(2) where, for each state, α=K 2,(3) β=3K(K+1)−4I(I+1)J(J+1) 8I(2I−1)J(2J−1),(4) with K=F(F+1) −I(I+1) −J(J+1)and νis the frequency of the fine structure transition of the isomer. Each peak was assigned a free intensity parameter, but since one isomer peak is obscured by the single ground-state peak, its intensity was constrained with respect to the most intense isomer peak assuming the intensities given in Eq.(21) in [12]. Fig. 1shows the summation of the 42Sc measurements. Hyperfine Aand Bcoefficients and the isomer shift obtained from the fitting are shown in Table 1. Using the values in Table 1and the known hyperfine coefficients and nuclear moments of 45gSc [16], the magnetic dipole moment, μ, and spectroscopic electric quadrupole moment, Qs, were derived for 42mSc using: μ=μref AI Aref Iref ,Qs=Qsref B Bref .(5) The subscript ref refers to the known parameters of 45Sc. The obtained values are shown in Table 2. The change in mean-square charge radius between 42gSc and 42mSc was extracted from the 42g,42mSc isomer shift. The change 2 Á. Koszorús, L.J. Vormawah, R. Beerwerth et al. Physics Letters B 819 (2021) 136439 Table 2 Nuclear moments for 42mSc, calibrated using the published values for 45gSc [16]. Also shown is the change in mean-square charge radius between the ground and isomeric state, δr242g,42m, determined from the isomer shift using the revised calculation of F=−349(15)MHz/fm2, with the corresponding systematic error shown in square brackets. Aμ(μN)Qs(b) δr242g,m(fm2) 42mSc +3.820(14)−0.12(10)−0.212(14)[9] in nuclear mean-square charge radius, δr2A,A=r2A−r2A, is related to an isotope or isomer shift, δνA,A=νA−νA, by [18] δνA,A=MmA−mA mAmA +Fδr2A,A,(6) where mAand m Arepresent the mass of the isotopes A,A, respectively, Fand Mare the respective atomic factors for the field shift and mass shift. These are calculated using the multiconfiguration Dirac-Fock (MCDF) method for a specific atomic transition, but are independent of the isotopes under study [18]. Previous MCDF calculations yielded values of F=−355(50) MHz/fm2and M=+583(30)GHz·u for the 363.1 nm 3D2→3F3 transition in the Sc+ion [16]. Revised calculations were performed as part of this work, in which the full relativistic recoil Hamiltonian was applied [21,22]. Two sets of calculations were performed. The first utilised a model similar to that employed in [16], and a multi-reference set with a configuration of 3d4s, 3d2, 4s2, 4p2 for the ground state and 3d4pfor the excited state. These calculations provided an estimate of the effect of the full relativistic recoil Hamiltonian but failed to reproduce the experimental value of the transition energy, hinting at the use of an unbalanced multireference set. A second set of calculations was hence performed using an expanded configuration of 3d4p, 4s4pfor the excited state and including double excitations from the 3sshell to account for additional core effects. The result of this was a much reduced uncertainty on F, whilst the total mass shift is taken from the MCDF calculations, rather than using the scaling law for the normal mass shift. For the 363.1 nm transition used here and in [16], the newly calculated values of F=−349(15)MHz/fm2and M= +625(60)GHz·u are adopted. A recalculation of the previously measured mean-square charge radii [16]is shown in Table 3. While the field shift factor is similar to the previous value, an increase in the calculated mass shift factor produces a trend in the ground state radii which is consistent with the flattening trend after crossing over the middle of the f7/2towards N=20, as shown in Fig. 2(a). For changes in the mean-square charge radii in an isotopic chain δr2A,A, systematic errors are dominated by the mass shift factor. This error is identically zero for the reference isotope, 45Sc. On the other hand, the changes in the mean-square charge radii of different states in the same isotope δr2g,mare calculated with respect to the ground state, resulting in reduced systematic uncertainties. In addition, the mass shift (and its error) is negligible due to very small mass differences between ground and isomeric states. Therefore the value for δr242g,42m, shown in Table 3, was calculated solely from the Ffactor and is not affected by the error on M. When the nuclear charge radii Rch are calculated from the δr245,A, the charge radii of the stable 45Sc is used as a reference [19], thus the systematic uncertainties are propagated to all the other calculated values of the ground and isomeric states. 3. Discussion The measured electromagnetic moments of 42mSc presented in Table 2can be used to better understand the nuclear structure of this isomer. The magnetic dipole moment gives an insight into Fig. 2. (a) Experimental nuclear charge radii in the calcium region [19,20]. The full red squares show the charge radii of the Sc isotopes obtained using the newly calculated Fand M. For comparison, the literature values from [16]are presented by empty squares. The systematic uncertainties due to the atomic parameters are not shown. (b) The comparison of the measured charge radii to IMSRG calculations using two different interactions. The shaded area indicates the systematic uncertainty arising from the atomic parameters. (c) Measured and calculated changes of the mean-square charge radii of 42g,42mSc, 44g,44mSc and 45g,45mSc. the leading proton and neutron configuration of this state while the electric quadrupole moment provides insight on the deformation. The 42Sc self-conjugate isotope is expected to have a rather simple nuclear structure given that it is made up of one proton 3 Á. Koszorús, L.J. Vormawah, R. Beerwerth et al. Physics Letters B 819 (2021) 136439 Table 3 Changes in mean-square charge radius of the Sc isotopes, recalibrated using the newly calculated values of F=−349(15)MHz/fm2and M=+625(60)GHz·u calculated as part of this work. The systematic uncertainties due to the atomic parameters are added in square brackets. AAδr2A,A(fm2)Rch (fm) 42 45 −0.016(31)[273]3.544(5)[39] 42m 42 −0.212(14)[9]– 43 45 −0.042(14)[178]3.540(3)[25] 44 45 −0.081(11)[87]3.535(3)[12] 44m 44 −0.072(11)[3]– 45 45 0 3.5459(25) 45m 45 +0.189(6)[8]– 46 45 −0.097(9)[83]3.5322(28)[117] and one neutron outside the magic 40Ca core. The nuclear spin of I=7for 42mSc [9]results in an empirical estimate [23]of μ =+3.80 μNfor the magnetic dipole moment, from an average of the neighbouring 41Sc and 43Sc isotopes, coupled to an average of the 41Ca and 43Ti isotones [24]. This supports the stretched [πf7/2⊗νf7/2] 7+configuration. The effective quadrupole moment of 42mSc was also calculated using the measured quadrupole moments of 41Ca and 41Sc yielding in Q=-0.211(18) b, somewhat larger than the measured Q=-0.12(10) b, but consistent given the large uncertainty of the latter. The measured electromagnetic moments of 42Sc are thus reproduced by simple empirical calculations, confirming that the properties of this state are dominated by the coupling of a πand νin the f7/2orbital. The small value of Qand good agreement with the empirical estimate show that nuclear deformation is not expected to affect the size of this isomer. A simple shell-model approach had success in calculating the isomer shift between the T=0 and T=1states of 38K. In the shell model, changes in mean-square charge radius can be calculated from the difference in proton occupancies of the fp shell in the ground state and the isomer using equation [4] δr2A,A=1 Znπ fp(A,A)b2,(7) where bis the oscillator parameter, and nπ fp is the change in proton occupancy of the fp shell between two isotopes or nuclear states Aand A. To determine b2, the equation of Duflo and Zuker [25]was used: b2=1.07A1 3⎧ ⎪ ⎪ ⎩1−⎧ ⎪ ⎪ ⎩ 2T A ⎫ ⎪ ⎪ ⎭ 2⎫ ⎪ ⎪ ⎭e3.5 A,(8) where Tstands for the isospin, yielding a value of b2=4.043 fm2 for 42Sc. To determine the proton occupancy, shell-model calculations were performed in the model space consisting of the s1/2, d3/2, f7/2and p3/2orbitals for both protons and neutrons above an inert 28Si core, using the shell-model code NuShellX [27], for which full diagonalisation of this model space has been achieved. Two sets of calculations were performed; one using the original ZBM2 interaction, and one using the version with modified V0,1 d3/2d3/2matrix elements. The same methodology was used for the calculations of 38K[4], where the latter interaction was found to reproduce the properties of this self-conjugate isotope more accurately. Magnetic dipole moments (using free g-factors) and electric quadrupole moments calculated from the wave functions are presented in Table 4and show good agreement with the experimental values. Table 5shows a comparison between the experimental value of δr242g,42m and the theoretical calculation using the shell model via equation (7). Unlike the case of 38K where a close match is seen between the shell-model calculation and the experimental Table 4 Empirical estimates of the nuclear magnetic dipole and electric quadrupole moments for 42mSc together with those calculated in the shell model [26]with the ZBM2 and ZBM2M interactions [4]and the measured value. Empirical ZBM2 ZBM2M Experiment μ(μN)+3.8 +3.878 +3.878 +3.820(14) Qs(b) -0.211(18) -0.165 −0.178 -0.12(10) Table 5 Changes in the mean-square charge radii of the ground state and isomer in 42Sc expressed in fm2. The experimentally measured value is compared to calculations from the shell mode using the ZBM2 and ZBM2M interactions and to the results of the IMSRG method using the PWA and 1.8/2.0(EM) interactions. ZBM2 ZBM2M PWA 1.8/2.0(EM) Experiment -0.114 -0.099 -0.022 -0.027 -0.212(14)[9] value, for 42Sc the change in mean-square charge radius is underestimated by a factor of two. Ab initio calculations were performed using VS-IMSRG predictions for the charge radii of light scandium isotopes using two initial sets of NN+3N forces from chiral effective field theory [28,29]. The VS-IMSRG, developed over Refs. [30–37] decouples a valencespace Hamiltonian and consistent operators from the full Hilbert space via an approximate unitary transformation. Here, to provide some assessment of uncertainty from starting nuclear forces, we use both the 1.8/2.0(EM) and PWA from a well-established family of chiral interactions [38–40]. The former reproduces ground-state and excitation energies throughout the medium- to heavy-mass region, including nuclear driplines [41], but generally underpredicts absolute charge radii, while PWA gives larger radii but underbinds finite nuclei [40,42,43]. Recently, both were shown to reproduce the odd-even staggering in copper isotopes well [44]. To obtain charge radii, we first decouple the core and valence-space intrinsic proton mean-squared radius operator, then apply corrections arising from the mean-square charge radii of the proton and the neutron, as well as the relativistic Darwin-Foldy and spin-orbit corrections, as detailed in Ref. [42]. We use the IMSRG++ code [45], in the IMSRG(2) approximation where induced many-body operators are truncated at the twobody level. In addition, the capability to generate valence-space Hamiltonians across major harmonic-oscillator shells was developed in Ref. [46]. In the current work, we take 28Si as the core and decouple a valence-space Hamiltonian for the space spanned by both proton and neutron 1s1/2, 0d3/2, 0f7/2, and 1p3/2or- bitals unique for each isotope studied. The resulting valence-space Hamiltonians are diagonalized with the NuShellX@MSU shellmodel code [27] (and the KSHELL code in some cases [47]) to obtain ground-state energies and expectation values for the intrinsic proton mean-square radius operator. We start from a harmonic-oscillator basis of 15 major shells (i.e., e =2n +l ⩽emax =14) at ¯ hω=16 MeV then transform to the Hartree-Fock basis, capturing the effects of 3N forces among valence nucleons with the ensemble normal ordering described in Ref. [36]. Using the approximate unitary transformation from the Magnus framework we additionally decouple a valence-space radius operator consistent with the valence-space Hamiltonian. In addition, for storage requirements, we impose a cut of e1+e2+ e3⩽E3Max =16 for 3N matrix elements. Finally, spurious center- of-mass modes are separated by adding the center-of-mass Hamiltonian with the coefficient βat the beginning of the calculation as discussed in Ref. [46]. The β-dependence of the results is small around β=3, and thus the following discussion is based on the results with β=3. The comparison of the measured and calculated charge radii using the VS-MSRG is shown in Fig. 2(b). As expected, results us- 4 Á. Koszorús, L.J. Vormawah, R. Beerwerth et al. Physics Letters B 819 (2021) 136439 Fig. 3. Experimental isomer shifts of the T=0and T=1states in N=Zisotopes of K, Sc and Mn together with the theoretical values calculated with the IMSRG method. The error bars are smaller than the markers. ing the PWA interaction overestimate the charge radii, while the 1.8/2.0(EM) gives too-small values. While the flat gradient of the trend seen in the scandium chain is overall well reproduced, the details are not consistent with the experimental data. Next the changes in the mean-square charge radii of the ground state and isomer of 42g,42mSc, 44g,44mSc and 45g,45mSc are shown in 2(c). Note that the systematic error of the experimental values are reduced because the corresponding ground state is chosen as the reference state. The systematic errors are smaller than the statistical uncertainties for the ground-state isomer-pairs in 42,44Sc and are of similar magnitude for 45Sc, as shown in the third column of Table 3. The sign of the isomer shifts is correctly reproduced in all the measured cases in the Sc chain. However, the magnitude is underestimated. For 42mSc in particular, the change in mean-square charge radius with respect to the ground state is underestimated by a factor of 10, as shown in Table 5. This significant underestimation can be partially understood by looking into the occupancy in the ground and isomer states since the radius operator is dominated by the one-body piece. We observed that number of protons excited to the pf shell are ∼0.6 and ∼0.5in the ground and isomer states, respectively, and thus nπ fp ∼0.1, considerably smaller than the ∼0.5from the ZBM2 calculations. Less excitation to pf orbitals indicates the overestimation of the singleparticle gap between sd and pf shells, which would be due to the IMSRG(2) approximation as seen in comparison with the coupledcluster method [48]. Also, we observed that these fewer proton excitations from sd to pf could not solely explain the underestimation of the isomer shift. For example, even if we assume moderate proton one-particle-one-hole excitation from d3/2to f7/2in the ground state, the corresponding isomer shift is −0.165 fm2, which is still insufficient to explain the experimental isomer shift. We would need to include additional physics such as the radius operator renormalization effect, or the implementation of IMSRG(3)- level approximation, which is currently underway. Taking advantage of the universal applicability of the VS-IMSRG method, the difference in size between the T=0 and T=1states of the Z=Nisotopes for K, Sc and Mn were also calculated with the same interactions. These results are compared to experimental values in Fig. 3. The sign of these differences is correctly predicted, despite changing from the sd shell 38K isotope to the pf shell 42Sc and 50Mn. The magnitude, however, is clearly underestimated. Finally, Fig. 4shows the variation of absolute mean-square charge radii along N=Zfrom 36Ar to 50Mn. It is interesting to remark that the difference in size between the T=0 and T=1states is almost equal in 38K and 50Mn, while it is twice as large in 42Sc. In Fig. 4. Changes in mean-square charge radius of self-conjugate nuclei from 36Ar to 50Mn, relative to 40Ca. In each case, absolute values of r2for reference isotopes [19]are added to isotope shifts accordingly [4,16,19]. The shaded area corresponds to the systematic error due to the uncertainty of the atomic parameters. the πν pair blocking picture introduced earlier [4], the T=1state in an odd-odd nucleus should lie exactly on the N=Zline established by the T=1ground states in the even-even neighbours, while the T=0state is smaller due to the blocking of pair scattering, as illustrated by the 38K case. From the sign of the 42g,42mSc isomer shift, it is immediately clear that the charge radius of the 0+ground state is indeed greater than the charge radius of the 7+isomer, verifying qualitatively this prediction. However, an unambiguous comparison with the neighbouring T=1states is hindered due to a large systematic uncertainty on the determination of the absolute charge radii of 42gSc and 42mSc which arises from the uncertainty on r2relative to the reference isotope 45Sc [19] and therefore affects both states equally. Our result is thus fully consistent with the πν pairing picture. 4. Conclusions The results which we report here, together with those of the previous study [4], are qualitatively consistent with the intuitive picture of πν pairing correlations along the line of N=Z. The negative value obtained for δr242g,42m indicates a larger charge radius for the 0+ground state than for the 7+isomer, as expected from employing an orbital blocking picture. Quantitatively, on the other hand, the size of the phenomenon is not yet understood and requires further investigation both from the theoretical as from the experimental point of view. While the electromagnetic moments are well reproduced by the shell-model and simple empirical calculations, the mechanism determining the difference in the charge radius of the T=1and T=0states of the self-conjugate 42Sc is still missing. This is evidenced by both the shell-model and IMSRG calculations. On the experimental side, the difference in charge radius of these two states in odd-odd selfconjugate nuclei is only known in two other cases besides 42Sc. Amore complete picture of the evolution of such pairing correlations could be achieved by measuring the isomer shifts in other odd-odd self-conjugate nuclei over the nuclear chart. One ideal candidate for a future study would be 26Al, which has a 5+ground state and a 0+isomer. It is therefore the isomer in 26Al for which the charge radius is expected to be greater. Another such candidate would be 46V, in which the valence πν pair once again occurs in the f7/2shell, and further cases such as 52Fe, 54Co and 94Ag, providing fertile ground for this new avenue of research. 5 Á. Koszorús, L.J. Vormawah, R. Beerwerth et al. Physics Letters B 819 (2021) 136439 Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements We are grateful to J. Simonis for providing the 1.8/2.0 (EM) and PWA 3N matrix element files and S. R. Stroberg for the imsrg++ code [45]used to perform these calculations. TRIUMF receives funding via a contribution through the National Research Council of Canada. This work was further supported by NSERC, the Arthur B. McDonald Canadian Astroparticle Physics Research Institute, Canadian Institute for Nuclear Physics. 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