Proton-neutron pairing correlations in the self-conjugate nucleus 42 Sc
Agota Koszorus, Liam Vormawah, Randolf Beerwerth, Mark Bissell, Paul Campbell, Bradley Cheal, Charlie Devlin, Tommi Eronen, Stephan Fritzsche, Sarina Geldhof, Hanne Heylen, Jason Holt, Ari Jokinen, Sam Kelly, Iain Moore, Takayuki Miyagi, Sami Rinta-Antila, Annika Voss, Calvin Wraith
PProton-neutron pairing correlations in the self-conjugate nucleus Sc ´A. Koszor´us a , L.J. Vormawah a , R. Beerwerth b , M.L. Bissell c , P. Campbell c , B. Cheal a , C.S. Devlin a , T. Eronen d ,S. Fritzsche b , S. Geldhof d,e , H. Heylen e,f , J. D. Holt g,h , A. Jokinen d , S. Kelly c , I.D. Moore d , T. Miyagi g ,S. Rinta-Antila d , A. Voss d , C. Wraith a a Department of Physics, University of Liverpool, Liverpool L69 7ZE, United Kingdom b Helmholtz Institute Jena, Fr¨obelstieg 3, 07743 Jena, Germany c School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom d Department of Physics, University of Jyv¨askyl¨a, PB 35(YFL) FIN-40351 Jyv¨askyl¨a, Finland e KU Leuven, Instituut voor Kern-en Stralingsfysica, B-3001 Leuven, Belgium f CERN, Experimental Physics Department, CH-1211 Geneva 23, Switzerland g TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada h Department of Physics, McGill University, 3600 Rue University, Montr´eal, QC H3A 2T8, Canada
Abstract
Collinear laser spectroscopy of the N = Z =
21 self-conjugate nucleus Sc has been performed at the JYFL IGISOLIV facility in order to determine the change in nuclear mean-square charge radius between the I π = + ground stateand the I π = + isomer via the measurement of the , Sc isomer shift. New multi-configurational Dirac-Fockcalculations for the atomic mass shift and field shift factors have enabled a recalibration of the charge radii of the − Sc 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 asignificantly larger magnitude than that expected from both shell model and ab-initio calculations.
Keywords:
Collinear laser spectroscopy, Hyperfine structure and isotope shift, Proton-neutron pairing, Charge radiusOdd-odd self-conjugate ( N = Z ) nuclei provide anideal testing ground for proton-neutron pairing studies.More specifically, such nuclei are ideal for verifying ifthe same phenomena arise for an I = πν ) pair as for a proton-proton ( ππ ) or neutron-neutron( νν ) pair with I = N = Z nuclei, protons and neutrons oc-cupy the same orbitals. Therefore, the interaction be-tween the odd proton and neutron is enhanced, leadingto a greater likelihood of the formation of a πν pair.For such a nucleus, it is expected that the charge ra-dius will be greater for a state with I = T = I (cid:44) T =
0. This arises due to a sim-ilar orbital blocking picture to that accounting for the
Email address:
[email protected] ( ´A. Koszor´us) multi-quasiparticle isomer radii [3]. The I = T = πν pair, can scatter into a wide range of ex-cited orbitals and still satisfy I =
0; as these orbitalsare less bound, they will have a greater spatial extent.The I (cid:44) T = πν pair, on the other hand,is significantly restricted in the number of orbitals it canmix with. In such cases, orbitals with the same spin maywell be energetically far apart (possibly even across amajor shell closure), rendering such scattering unlikely[4].A study of pairing e ff ects on the relative mean-squarecharge radii of multi-quasiparticle isomers [3], suchas Y, Yb and
Hf, was motivated by themeasurement of
Hf by Boos et al. [5]. All ofthese isomers were found to have a smaller mean-squarecharge radius than their respective ground states irre-spective of nuclear deformation. This o ff ered an ex-planation into the observed odd-even staggering of nu-clear charge radii [6] as arising from a combination ofincreasing rigidity (i.e. a reduction in the root meansquare quadrupole deformation value towards the meanvalue) or decreasing surface di ff useness, due to the or- Preprint submitted to Physics Letters B February 4, 2021 a r X i v : . [ nu c l - e x ] F e b ital (Pauli) blocking of the odd nucleon [3].The multi-quasiparticle isomer studies looked intothe e ff ects of general nucleon pairing, not specifically πν pairs, the existence of which has yet to be provedconclusively [7], although many studies have yieldedresults supporting the idea. The premise for using odd-odd N = Z nuclei as a testbed for πν pairing correlationsis supported by the investigation in K ( N , Z = ff erence in ground and isomericstate charge radii was probed via a direct measurementof the , K isomer shift [4]. In this system, the I = T = I = T = πν pairing model. Shellmodel calculations appear to reproduce the e ff ect quan-titatively for , K [4]. Such a remarkable agreementmotivates the study of other N = Z nuclei. Above the N = Z =
20 shell closures, an inversion of the groundand isomeric states is observed and the I = + , T = , Mn [8], whichshows an e ff ect of similar magnitude as in K. In Sca I = + , T = I = + , T = , Sc isomer shift, where theground state is expected to be larger, will test if this sim-ple trend continues and whether it can be quantitativelyunderstood.
1. Experimental methodology
Collinear laser spectroscopy [11, 12] was performedat the IGISOL IV facility [13], located at the Univer-sity of Jyv¨askyl¨a JYFL Accelerator Laboratory, Fin-land, where beams of short-lived radioactive ions can beproduced [14]. Singly-charged Sc ions were producedvia a 30 MeV Ca( α ,pn) Sc fusion-evaporation reac-tion, taking place in an environment of a helium bu ff ergas at a pressure of ∼
150 mbar inside the IGISOL cham-ber. The use of thin foil targets (thickness ∼ / cm )at IGISOL, coupled with the extraction of reaction prod-ucts via a supersonic gas jet, enables fast release irre-spective of physical or chemical properties. Once ex-tracted from the target chamber, the reaction productswere then formed into a 30 keV beam of singly-chargedions which were subsequently mass separated using a55 ◦ dipole magnet. Ions were cooled and bunched us-ing a gas-filled radio-frequency quadrupole [15], fromwhich bunches of temporal width 20 µ s were releasedevery 100 ms and directed to the laser spectroscopy sta-tion. Ions were overlapped in an anti-collinear geome-try with 0.5 mW of light from a frequency-doubledcontinuous-wave narrow-linewidth dye laser, runningwith Pyridine 2 dye. Optical spectra were taken usingthe same 363.1 nm 3 d s D → d p F atomic tran-sition as used in a previous study of radioactive scan-dium isotopes [16]. The fundamental frequency of thelaser was locked to an I absorption line and ions wereDoppler tuned across the resonances by applying a volt-age ramp to the photon-ion interaction region. For eachvoltage step an ion bunch was released and a gate wasapplied to the photon detection signal corresponding tothe ion bunch transit time in front of the photomultipliertube. The resulting voltage scans were transformed intofrequency, f , via f = f L (1 + α + √ α + α ) , (1)where α = eV / mc , V is the total accelerating voltageand f L is the laser frequency [11]. Several indepen-dent scans of , Sc were taken and summed. Thefrequency conversion was performed using the atomicmass, m , of Sc [17], and incorporating the 617 keVexcitation of the isomer [9, 10] as appropriate.
2. Data analysis and results
A chi-squared minimisation routine was used in or-der to fit a series of Lorentzian peaks to the data, fromwhich the hyperfine A (magnetic dipole) and B (electricquadrupole) parameters [11] were extracted for the up-per electronic state ( A u , B u ). Ratios of A and B param-eters for the upper and lower states were constrained to A l / A u = .
469 and B l / B u = .
552 [16]. Frequencies ofthe isomeric peaks were therefore calculated accordingto γ = ν + ( α u − . α l ) A u + ( β u − . β l ) B u (2)where, for each state, α = K , (3) β = K ( K + − I ( I + J ( J + I (2 I − J (2 J − , (4)with K = F ( F + − I ( I + − J ( J +
1) and ν is the centroidof the isomer. Each peak was assigned a free intensityparameter, but since one isomer peak is obscured by thesingle ground-state peak, its intensity was constrainedwith respect to the most intense isomer peak assuminga model intensity distribution. Figure 1 shows the sum-mation of the Sc measurements. Hyperfine A and B o un t s Relative frequency (MHz)
Experiment Fitted structureGround state fi t Isomer fi t 9000800070006000100200300400500 Figure 1: Measured and fitted hyperfine spectrum of Sc measuredon the 363.1 nm line, together with the separate contributions from theground and isomeric states (o ff set for clarity).Table 1: Values of the hyperfine magnetic dipole, A u ( F ), andelectric quadrupole, B u ( F ), parameters for Sc, measured on the363.1 nm line. A A u (MHz) B u (MHz) δν , m (MHz) Sc + . − + coe ffi cients and the isomer shift obtained from the fit-ting are shown in Table 1.Using the values in Table 1 and the known hyper-fine coe ffi cients and nuclear moments of Sc [16], themagnetic dipole moment, µ , and spectroscopic electricquadrupole moment, Q s , were calculated for Sc andare shown in Table 2.The change in mean-square charge radius between
Sc and
Sc was extracted from the , Sc iso-mer shift. The change in nuclear mean-square chargeradius, δ (cid:104) r (cid:105) A , A (cid:48) = (cid:104) r (cid:105) A (cid:48) − (cid:104) r (cid:105) A , is related to an isotopeor isomer shift, δν A , A (cid:48) = ν A (cid:48) − ν A , by [18] δν A , A (cid:48) = M m A (cid:48) − m A m A m A (cid:48) + F δ (cid:104) r (cid:105) A , A (cid:48) , (5)where F and M are the respective atomic factors for the Table 2: Nuclear moments for
Sc, calibrated using the publishedvalues for
Sc [16]. Also shown is the change in mean-squarecharge radius between the ground and isomeric state, δ (cid:104) r (cid:105) , ,determined from the isomer shift using the revised calculation of F = − / fm , with the corresponding systematic errorshown in square brackets. A µ ( µ N ) Q s (b) δ (cid:104) r (cid:105) , m (fm ) m Sc + . − . − . field shift and mass shift. These are calculated usingthe multi-configuration Dirac-Fock (MCDF) method fora specific atomic transition, but are independent of theisotopes under study [18].Previous MCDF calculations yielded values of F = − / fm and M = + · u for the363.1 nm D → F transition in the Sc + ion [16]. Re-vised calculations were performed as part of this work,in which the full relativistic recoil Hamiltonian wasapplied [21, 22]. Two sets of calculations were per-formed. The first utilised a model similar to that em-ployed in [16], and a multi-reference set with a con-figuration of 3 d s , 3 d , 4 s , 4 p for the ground stateand 3 d p for the excited state. These calculations pro-vided an estimate of the e ff ect of the full relativistic re-coil Hamiltonian but failed to reproduce the experimen-tal value of the transition energy, hinting at the use of anunbalanced multi-reference set. A second set of calcu-lations was hence performed using an expanded config-uration of 3 d p , 4 s p for the excited state and includ-ing double excitations from the 3 s shell to account foradditional core e ff ects. The result of this was a muchreduced uncertainty on F , whilst the total mass shift istaken from the MCDF calculations, rather than using thescaling law for the normal mass shift.For the 363.1 nm transition used here and in [16], thenewly calculated values of F = − / fm and M = + · u are adopted. A recalculation ofthe previously measured mean-square charge radii [16]is shown in Table 3. While the field shift factor is similarto the previous value, an increase in the calculated massshift factor produces a trend in the ground state radiimore in keeping with regional systematics, as shownin Figure 2(a). For changes in the mean-square chargeradii in an isotopic chain δ (cid:104) r (cid:105) A , A (cid:48) , systematic errors aredominated by the mass shift factor. This error is identi-cally zero for the reference isotope, Sc. On the otherhand, the changes in the mean-square charge radii ofdi ff erent states in the same isotope δ (cid:104) r (cid:105) g , m are calcu-lated with respect to the ground state, resulting in re-duced systematic uncertainties. In addition, the massshift (and its error) is negligible due to very small massdi ff erences between ground and isomeric states. There-fore the value for δ (cid:104) r (cid:105) , , shown in Table 3, wascalculated solely from the F factor and is not a ff ected bythe error on M . When the nuclear charge radii R ch arecalculated from the δ (cid:104) r (cid:105) , A (cid:48) , the charge radii of the sta-ble Sc is used as a reference [19], thus the systematicuncertainties are propagated to all the other calculatedvalues of the ground and isomeric states.3
Neutron number
16 18 20 22 24 26 28 30 32 343.403.453.503.553.603.653.703.753.80
K ( Z =19)Ca ( Z =20)Sc ( Z =21)Ti ( Z =22) Cr ( Z =24)Mn ( Z =25) V ( Z =23)Co ( Z =27)Fe ( Z =26) Figure 2: (a) Experimental nuclear charge radii in the calcium re-gion [19, 20]. The full black squares show the charge radii of the Scisotopes obtained using the newly calculated F and M . For compar-ison, the literature values from [16] are presented by empty squares.The new results for Sc are shown in red squares. The systematicuncertainties due to the atomic parameters are not shown. (b) Thecomparison of the measured charge radii to IMSRG calculations us-ing two di ff erent interactions. The shaded area indicates the system-atic uncertainty arising from the atomic parameters. (c) Measuredand calculated changes of the mean-square charge radii of g , m Sc, g , m Sc and g , m Sc. Table 3: Changes in mean-square charge radius of the Scisotopes, recalibrated using the newly calculated values of F = − / fm and M = + · u calculated as partof this work. A (cid:48) A δ (cid:104) r (cid:105) A , A (cid:48) (fm ) R ch (fm)42 45 − . . − . − . − . . − . + . − . .
3. Discussion
The measured electromagnetic moments of m Scpresented in Table 2 can be used to better understand thenuclear structure of this isomer. The magnetic dipolemoment gives an insight into the leading proton andneutron configuration of this state while the electricquadruple moment provides insight on the deformation.The Sc self-conjugate isotope is expected to have arather simple nuclear structure given that it is made upof one proton and one neutron outside the magic Cacore. The nuclear spin of I = Sc [9] resultsin an empirical estimate [23] of µ = + . µ N for themagnetic dipole moment, from an average of the neigh-bouring Sc and Sc isotopes, coupled to an averageof the Ca and Ti isotones [24]. This supports thestretched [ π f / ⊗ ν f / ] 7 + configuration. The e ff ectivequadrupole moment of m Sc was also calculated us-ing the measured quadrupole moments of Ca and Scyielding in Q = -0.211(18) b, somewhat larger than themeasured Q = -0.12(10) b, but consistent given the largeuncertainty of the latter. The measured electromagneticmoments of Sc are thus reproduced by simple empir-ical calculations, confirming that the properties of thisstate are dominated by the coupling of a π and ν in the f / orbital. The small value of Q and good agreementwith the empirical estimate show that nuclear deforma-tion is not expected to a ff ect the size of this isomer.A simple shell model approach had success in calcu-lating the isomer shift between the T = T = K. In the shell model, changes in mean-square charge radius can be calculated from the di ff er-ence in proton occupancies of the f p shell in the groundstate and the isomer using equation [4] δ (cid:104) r (cid:105) A , A (cid:48) = Z ∆ n π f p ( A , A (cid:48) ) b , (6)4 able 4: Empirical estimates of the nuclear magnetic dipole andelectric quadrupole moments for Sc together with those calculatedin the shell model [26] with the ZBM2 and ZBM2M interactions [4]and the measured value.
Empirical ZBM2 ZBM2M Experiment µ ( µ N ) + + . + . + Q s (b) -0.211(18) -0.165 − .
178 -0.12(10)
Table 5: Changes in the mean-square charge radii of the ground stateand isomer in Sc expressed in fm . The experimentally measuredvalue is compared to calculations from the shell mode using theZBM2 and ZBM2M interactions and to the results of the IMSRGmethod using the PWA and 1.8 / ZBM2 ZBM2M PWA 1.8 / where b is the oscillator parameter, and ∆ n π f p is thechange in proton occupancy of the f p shell between twoisotopes or nuclear states A and A (cid:48) . To determine b , theequation of Duflo and Zuker [25] was used: b = . A − TA e . A , (7)where T stands for the isospin, yielding a value of b = .
043 fm for Sc. To determine the proton oc-cupancy, shell model calculations were performed inthe model space consisting of the s / , d / , f / and p / orbitals for both protons and neutrons above aninert Si core, using the shell model code NuShellX[27], for which full diagonalisation of this model spacehas been achieved. Two sets of calculations were per-formed; one using the original ZBM2 interaction, andone using the version with modified V , d / d / matrix ele-ments. The same methodology was used for the calcu-lations of K [4], where the latter interaction was foundto reproduce the properties of this self-conjugate isotopemore accurately. Magnetic dipole moments (using free g -factors) and electric quadrupole moments calculatedfrom the wave functions are presented in Table 4 andshow good agreement with the experimental values.Table 5 shows a comparison between the experimen-tal value of δ (cid:104) r (cid:105) , and the theoretical calculationusing the shell model via equation 6. Unlike the caseof K where a close match is seen between the shellmodel calculation and the experimental value, for Scthe change in mean-square charge radius is underesti-mated by a factor of two.Ab initio calculations were performed using VS-IMSRG predictions for the charge radii of light scan- dium isotopes using two initial sets of NN +
3N forcesfrom chiral e ff ective field theory [28, 29]. The VS-IMSRG, developed over Refs. [30–37] decouples avalence-space Hamiltonian and consistent operatorsfrom the full Hilbert space via an approximate uni-tary transformation. Here, to provide some assess-ment of uncertainty from starting nuclear forces, we useboth the 1.8 / ++ code [45], in the IMSRG(2)approximation where induced many-body operators aretruncated at the two-body level. In addition, the capabil-ity to generate valence-space Hamiltonians across majorharmonic-oscillator shells was developed in Ref. [46].In the current work, we take Si as the core and decou-ple a valence-space Hamiltonian for the space spannedby both proton and neutron 1 s / , 0 d / , 0 f / , and 1 p / orbitals unique for each isotope studied. The result-ing valence-space Hamiltonians are diagonalized withthe NuShellX@MSU shell-model code [27] (and theKSHELL code in some cases [47]) to obtain ground-state energies and expectation values for the intrinsicproton mean-square radius operator.We start from a harmonic-oscillator basis of 15 majorshells (i.e., e = n + l (cid:54) e max =
14) at (cid:126) ω =
16 MeVthen transform to the Hartree-Fock basis, capturing thee ff ects of 3N forces among valence nucleons with theensemble normal ordering described in Ref. [36]. Us-ing the approximate unitary transformation from theMagnus framework we additionally decouple a valence-space radius operator consistent with the valence-spaceHamiltonian. In addition, for storage requirements, weimpose a cut of e + e + e (cid:54) E =
16 for 3N ma-trix elements. Finally, spurious center-of-mass modesare separated by adding the center-of-mass Hamiltonianwith the coe ffi cient β at the beginning of the calcula-tion as descussed in Ref. [46]. The β -dependence of theresults is small around β =
3, and thus the followingdiscussion is based on the results with β = .200.1000.10 Mn Sc K δ < r > g , m ( f m ) PWA1.8/2.0 (EM)EXPERIMENT
Figure 3: Experimental isomer shifts of the T = T = N = Z isotopes of K, Sc and Mn together with the theoretical valuescalculated with the IMSRG method. The error bars are smaller thanthe markers. charge radii using the VS-MSRG is shown in Fig. 2(b).As expected, results using the PWA interaction over-estimate the charge radii, while the 1.8 / g , m Sc, g , m Sc and g , m Sc are shown in 2(c). Note that the systematic er-ror of the experimental values are reduced because thecorresponding ground state is chosen as the referencestate. The systematic errors are smaller than the statis-tical uncertainties for the ground-state isomer-pairs in , Sc and are of similar magnitude for Sc, as shownin the third column of Table 3. The sign of the iso-mer shifts is correctly reproduced in all the measuredcases in the Sc chain. However, the magnitude is un-derestimated. For m Sc in particular, the change inmean-square charge radius with respect to the groundstate is underestimated by a factor of 10, as shown inTable 5. This significant underestimation can be par-tially understood by looking into the occupancy in theground and isomer states since the radius operator isdominated by the one-body piece. We observed thatnumber of protons excited to the p f shell are ∼ . ∼ . ∆ n π f p ∼ .
1, considerably smaller than the ∼ . p f orbitals indicates the overestimation of the single-particle gap between sd and p f shells, which would bedue to the IMSRG(2) approximation as seen in com-parison with the coupled-cluster method [48]. Also, weobserved that these fewer proton excitations from sd to
18 19 20 21 22 23 24 25 N = Z r , A ( f m ) T = 1 state T = 0 state Figure 4: Changes in mean-square charge radius of self-conjugatenuclei from Ar to Mn, relative to Ca. In each case, ab-solute values of (cid:104) r (cid:105) for reference isotopes [19] are added toisotope shifts accordingly [4, 16, 19]. The shaded area correspondsto the systematic error due to the uncertainty of the atomic parameters. p f could not solely explain the underestimation of theisomer shift. For example, even if we assume moder-ate proton one-particle-one-hole excitation from d / to f / in the ground state, the corresponding isomer shiftis − .
165 fm , which is still insu ffi cient to explain theexperimental isomer shift. We would need to includeadditional physics such as the radius operator renormal-ization e ff ect, or the implementation of IMSRG(3)-levelapproximation, which is currently underway.Taking advantage of the universal applicability of theVS-IMSRG method, the di ff erence in size between the T = T = Z = N isotopes for K, Scand Mn were also calculated with the same interactions.These results are compared to experimental values inFig. 3. The sign of these di ff erences is correctly pre-dicted, despite changing from the sd shell K isotopeto the p f shell Sc and Mn. The magnitude, how-ever, is clearly underestimated. Finally, Fig. 4 showsthe variation of absolute mean-square charge radii along N = Z from Ar to Mn. It is interesting to remarkthat the di ff erence in size between the T = T = K and Mn, while it is twiceas large in Sc. In the πν pair blocking picture intro-duced earlier [4], the T = N = Z line establishedby the T = T = K case. From thesign of the , Sc isomer shift, it is immediately clearthat the charge radius of the 0 + ground state is indeedgreater than the charge radius of the 7 + isomer, veri-6ying qualitatively this prediction. However, an unam-biguous comparison with the neighbouring T = absolute charge radii of Sc and
Sc which arises from the uncertainty on (cid:104) r (cid:105) relativeto the reference isotope Sc [19] and therefore a ff ectsboth states equally. Despite this large systematic uncer-tainty, our result is fully consistent with the πν pairingpicture.
4. Conclusions
The results which we report here, together with thoseof the previous study [4], are qualitatively consistentwith the intuitive picture of πν pairing correlations alongthe line of N = Z . The negative value obtained for δ (cid:104) r (cid:105) , indicates a larger charge radius for the 0 + ground state than for the 7 + isomer, as expected fromemploying an orbital blocking picture.Quantitatively, on the other hand, the size of the phe-nomenon is not yet understood and requires further in-vestigation both from the theoretical as from the exper-imental point of view. While the electromagnetic mo-ments are well reproduced by the shell model and sim-ple empirical calculations, the mechanism determiningthe di ff erence in the charge radius of the T = T = Sc is still missing.This is evidenced by both the shell model and IMSRGcalculations. On the experimental side, the di ff erencein charge radius of these two states in odd-odd self-conjugate nuclei is only known in two other cases be-sides Sc. A more complete picture of the evolutionof such pairing correlations could be achieved by mea-suring the isomer shifts in other odd-odd self-conjugatenuclei over the nuclear chart. One ideal candidate fora future study would be Al, which has a 5 + groundstate and a 0 + isomer. It is therefore the isomer in Alfor which the charge radius is expected to be greater.Another such candidate would be V, in which the va-lence πν pair once again occurs in the f / shell, and fur-ther cases such as Fe, Co and Ag, providing fertileground for this new avenue of research.
5. Acknowledgments
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