Measurement and microscopic description of odd-even staggering of charge radii of exotic copper isotopes

This is a sel -a chi ed e sion o an o iginal a icle. This e sion may di e om he o iginal in pagina ion and ypog aphic de ails. Au ho (s): Ti le: Yea : Ve sion: Copy igh : Righ s: Righ s u l: Please ci e he o iginal e sion: CC BY 4.0 h ps://c ea i ecommons.o g/licenses/by/4.0/ Measu emen and mic oscopic desc ip ion o odd-e en s agge ing o cha ge adii o exo ic coppe iso opes © CERN 2020 Published e sion de G oo e, R. P.; Billowes, J.; Binne sley, C. L.; Bissell, M. L.; Cocolios, T. E.; Day Goodac e, T.; Fa ooq-Smi h, G. J.; Fedo o , D. V.; Flanagan, K. T.; F anchoo, S.; Ga cia Ruiz, R. F.; Gins, W.; Hol , J. D.; Koszo ús, Á.; Lynch, K. M.; Miyagi, T.; Naza ewicz, W.; Neyens, G.; Reinha d, P.-G.; Ro he, S.; S oke, H. H.; Ve non, A. R.; Wend , K. D. A.; Wilkins, S. G.; Xu, Z. Y.; Yang, X. F. de G oo e, R. P., Billowes, J., Binne sley, C. L., Bissell, M. L., Cocolios, T. E., Day Goodac e, T., Fa ooq-Smi h, G. J., Fedo o , D. V., Flanagan, K. T., F anchoo, S., Ga cia Ruiz, R. F., Gins, W., Hol , J. D., Koszo ús, Á., Lynch, K. M., Miyagi, T., Naza ewicz, W., Neyens, G., Reinha d, P.-G., . . . Yang, X. F. (2020). Measu emen and mic oscopic desc ip ion o odd-e en s agge ing o cha ge adii o exo ic coppe iso opes. Na u e Physics, 16(6), 620-624. h ps://doi.o g/10.1038/s41567- 020-0868-y 2020 Le e s h ps://doi.o g/10.1038/s41567-020-0868-y 1KU Leu en, Ins i uu oo Ke n- en S alings ysica, Leu en, Belgium. 2Depa men o Physics, Uni e si y o Jy äskylä, Jy äskylä, Finland. 3Pho on Science Ins i u e, Depa men o Physics and As onomy, The Uni e si y o Manches e , Manches e , UK. 4Enginee ing Depa men , CERN, Gene a, Swi ze land. 5TRIUMF, Vancou e , B i ish Columbia, Canada. 6Pe e sbu g Nuclea Physics Ins i u e, Ga china, Russia. 7Ins i u de Physique Nucléai e d’O say, O say, F ance. 8Massachuse s Ins i u e o Technology, Camb idge, MA, USA. 9Expe imen al Physics Depa men , CERN, Gene a, Swi ze land. 10Depa men o Physics, McGill Uni e si y, Mon eal, Quebec, Canada. 11Depa men o Physics and As onomy and FRIB Labo a o y, Michigan S a e Uni e si y, Eas Lansing, MI, USA. 12Ins i u ü Theo e ische Physik, Uni e si ä E langen, E langen, Ge many. 13Depa men o Physics, New Yo k Uni e si y, New Yo k, NY, USA. 14Ins i u ü Physik, Johannes Gu enbe g-Uni e si ä , Mainz, Ge many. 15School o Physics and S a e Key Labo a o y o Nuclea Physics and Technology, Peking Uni e si y, Beijing, China. ✉e-mail: deg [email p o ec ed] Nuclea cha ge adii globally scale wi h a omic mass num- be A as A1∕3, and iso opes wi h an odd numbe o neu ons a e usually sligh ly smalle in size han hei e en-neu on neighbou s. This odd–e en s agge ing, ubiqui ous h ough- ou he nuclea landscape1, a ies wi h he numbe o p o ons and neu ons, and poses a subs an ial challenge o nuclea heo y2–4. He e, we epo measu emen s o he cha ge adii o sho -li ed coppe iso opes up o he e y exo ic 78Cu (wi h p o on numbe Z = 29 and neu on numbe N = 49), p oduced a only 20 ions s–1, using he collinea esonance ioniza ion spec oscopy me hod a he Iso ope Mass Sepa a o On-Line De ice acili y (ISOLDE) a CERN. We obse e an unexpec ed educ ion in he odd–e en s agge ing o iso opes app oach- ing he N = 50 shell gap. To desc ibe he da a, we applied models based on nuclea densi y unc ional heo y5,6 and A-body alence-space in-medium simila i y eno maliza ion g oup heo y7,8. Th ough hese compa isons, we demons a e a ela ion be ween he global beha iou o cha ge adii and he sa u a ion densi y o nuclea ma e , and show ha he local cha ge adii a ia ions, which e lec he many-body pola iza- ion e ec s, na u ally eme ge om A-body calcula ions i ed o p ope ies o A ≤ 4 nuclei. The p ope ies o exo ic nuclei, in pa icula o hose close o (doubly) magic sys ems a om s abili y, ha e con inually p o en pi o al in deepening ou unde s anding o nuclea o ces and many-body dynamics. Owing o he p esence o he unpai ed p o on, odd-Z iso opes such as he coppe iso opes p o ide c u- cial insigh s in o he single-pa icle p o on s uc u e and how his a ec s he cha ge adii. Howe e , un il now, expe imen ally accessing cha ge adii o such iso opes close o exo ic doubly closed shells ( o example 78Ni and 100Sn) has been p ohibi i ely di icul . Ex ending he exis ing cha ge adius measu emen s9 beyond 75Cu has equi ed nea ly a decade o de elopmen s, culmina ing in he wo k p esen ed he e. The i s expe imen al challenge lies in he p oduc ion o a clean sample o hese sho -li ed species. We p oduced adioac i e ions a he ISOLDE labo a o y a CERN. This was done by impinging 1.4-GeV p o ons on o a neu on con e e , p oducing neu ons ha in u n induced ission o 238U a oms wi hin a hick a ge , hus minimizing o he unwan ed nuclea eac ions in he a ge . Se e al pu i ica ion s eps we e ne e heless equi ed o emo e con ami- nan s. Fi s , he coppe a oms ha di used ou o he a ge we e elemen -selec i ely lase -ionized by he ISOLDE esonance ioniza- ion lase ion sou ce (RILIS) in a ho ca i y. The ions we e hen accele a ed o 30 keV o mass sepa a ion using he ISOLDE high esolu ion sepa a o and p epa ed o high- esolu ion lase eso- nance ioniza ion spec oscopy. This equi ed sending he ions in o a gas- illed adio- equency linea Paul ap, ISCOOL, whe e hey we e cooled o up o 10 ms. The ions we e hen eleased in a sho bunch wi h a leng h o ~1 μs. The hype ine s uc u e o he coppe iso opes was measu ed in he inal s age o he expe imen using he collinea esonance ion- iza ion spec oscopy (CRIS)10 me hod. Fi s , he ions we e neu al- ized h ough a cha ge-exchange eac ion wi h a po assium apou . The non-neu alized ac ion o he beam was de lec ed, such ha only he neu alized a oms en e ed in o an ul ahigh- acuum egion. He e, hey in e ac ed wi h wo pulsed lase beams. The i s o hese lase sys ems, uned o he op ical ansi ion a 40,114.01 cm−1, esonan ly exci ed he a oms, while he second lase u he exci ed hese a oms o an au o-ionizing s a e, chosen o op imal ioniza ion e iciency. Owing o he acuum o 10−8 mba , he collisional ioniza- ion a e was less han 1 e e y 10 min o all excep he s able 63,65Cu, c ea ing a quasi-backg ound- ee measu emen . As illus a ed in he op panel o Fig. 1, by eco ding he numbe o ions as a unc- ion o he equency o he i s single-mode lase , he hype ine s uc u e o he coppe a oms could be measu ed. Changes o he cha ge adius o he nucleus esul in small changes in he cen oids o hese hype ine s uc u es o each iso ope, which is ypically a Measu emen and mic oscopic desc ip ion o odd–e en s agge ing o cha ge adii o exo ic coppe iso opes R. P. de G oo e   1,2 ✉ , J. Billowes3, C. L. Binne sley3, M. L. Bissell3, T. E. Cocolios   1, T. Day Goodac e   4,5, G. J. Fa ooq-Smi h   1, D. V. Fedo o    6, K. T. Flanagan3, S. F anchoo7, R. F. Ga cia Ruiz3,8,9, W. Gins1,2, J. D. Hol    5,10, Á. Koszo ús1, K. M. Lynch9, T. Miyagi5, W. Naza ewicz   11, G. Neyens1,9, P.-G. Reinha d12, S. Ro he   3,4, H. H. S oke13, A. R. Ve non1,3, K. D. A. Wend 14, S. G. Wilkins   3,4, Z. Y. Xu1 and X. F. Yang   1,15 NATuRE PHYSiCS | www.na u e.com/na u ephysics Le e s NaTuRE PHySICS 1:106 e ec . These iso ope shi s, while small, we e ex ac ed om he da a and we e used o de e mine he changes in he mean- squa ed cha ge adius (see Me hods o mo e de ails). Using his sequence o echniques, we could ex end ou knowl- edge o he cha ge adii o he coppe iso opes by ano he h ee neu- on numbe s, up o 78Cu (N = 49). Ou new da a hus ep esen an impo an s ep owa ds unde s anding he nuclea sizes in close p oximi y o he e y neu on- ich doubly magic sys em o 78Ni ( e . 11). The high e iciency and selec i i y o he CRIS echnique allows he obse a ion o signals wi h de ec ion a es o less han 0.05 ions s–1 on esonance (Fig. 1). Thanks o he ul a-low backg ound a es inhe en o he me hod, hese de ec ion a es we e su icien o a success ul measu emen o he cha ge adius o 78Cu in less han one day, while o he iso opes ypically only equi e a ew hou s o beam ime. The signal- o-backg ound a io ob ained is simila o hose achie ed in he s a e-o - he-a in-sou ce measu emen s12, bu wi h much na owe linewid hs o <100 MHz, ypical o con en ional as -beam collinea lase spec oscopy echniques, hus demons a - ing a bes -o -bo h-wo lds pe o mance. The changes in he mean-squa ed cha ge adii ex ac ed om he hype ine s uc u e spec a a e plo ed in he bo om panel o Fig. 1 (whi e do s), complemen ed by li e a u e alues o 58−62,67Cu ( e . 9) (whi e diamonds). The adii o he isome ic s a es a e shown wi h black ma ke s. The shaded a ea shows he unce ain y due o he a omic pa ame e s (see Me hods o mo e de ails). While hese a omic unce ain ies in luence he slope o he cha ge adii cu e, smalle -scale e ec s such as he odd–e en s agge ing (OES) o he cha ge adii a e la gely una ec ed. Values o he h ee-poin OES pa ame e o he adii Δ ð3Þ I , de ined as Δð3Þ ¼1 2 A þ 1 � 2 A þ A � 1 ðÞ ; ð 1 Þ Coun a e (Hz) F equency de uning om cen oid (MHz) 80 MHz 0.06 0.04 0.02 0 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 δ〈 2 〉( m 2 ) –0.8 –1.0 28 30 32 34 36 38 N N 15 10 5 0 –5 –10 –15 3028 32 34 36 38 40 42 44 46 48 50 40 42 44 46 48 50 –1,000 –750 –500 –250 250 500 Sys ema ic unce ain y Expe imen (g ound s a e) Expe imen (isome ) Li e a u e Δ (3)(10–3 m) 7500 Fig. 1 | O e iew o he expe imen al esul s. Top: ion a e ob ained du ing he lase scan o 78Cu as a unc ion o esonan lase equency ela i e o he hype ine s uc u e cen oid. E o ba s show he s a is ical unce ain y (one s anda d de ia ion). The solid line shows he bes - i ing se o Voig p o iles cen ed a he esonance loca ions (see Me hods). Bo om: alues o δ 2  I o 63−66,68−78Cu ( ela i e o 65Cu) om his wo k as a unc ion o neu on numbe N, alongside he li e a u e alues o 58−62,67Cu ( e . 9). The inse shows he OES o he adii de ined by equa ion (1). E o ba s a e s a is ical (one s anda d de ia ion). In he main plo , hese e o s a e oo small o be seen excep o 78Cu. The blue shaded a ea ep esen s he sys ema ic unce ain y due o he a omic pa ame e s. A a b c DFT: VS-IMSRG: Fy(s d) Fy(Δ ) EM1.8/2.0 PWA Expe imen 57 16 15 14 13 –8.4 –8.5 –8.6 –8.7 E/A (MeV) δ〈 2〉65,A( m 2) 〈 2〉( m 2) –8.8 0.5 0 –0.5 –1.0 28 30 32 34 36 38 40 N 42 44 46 48 50 59 61 63 65 67 69 71 73 75 77 79 Fig. 2 | Compa ison o selec ed expe imen al da a ( 2  I , E/A and δ 2  I ) and nuclea heo y p edic ions. a–c, The da a a e plo ed as a unc ion o neu on numbe N (bo om axis) and mass numbe A ( op axis). Expe imen al alues a e shown wi h whi e ci cles (e o ba s ep esen ing s a is ical unce ain ies a e oo small o be isible). Fo he DFT calcula ions, g een squa es a e used o he Fy(s d) unc ional and blue diamonds o he Fy(Δ ) unc ional, while he VS-IMSRG calcula ions a e displayed using ed squa es o he PWA in e ac ion and o ange diamonds o he EM1.8/2.0 in e ac ion. Panel a compa es o expe imen al squa ed cha ge adii, b o binding ene gies28,29 pe nucleon and c o di e en ial squa ed cha ge adii. Fo cla i y, only Fy(Δ ) and EM1.8/2.0 esul s a e shown in b and c. NATuRE PHYSiCS | www.na u e.com/na u ephysics Le e s NaTuRE PHySICS a e shown in he inse o Fig. 1. The OES o he adii is qui e p o- nounced nea N = 40, bu ou new da a poin s e eal a educ ion o he OES owa ds N = 50, s a ing a 74Cu. This is likely o be a ibu ed o he change in he g ound-s a e p o on con igu a ion. Indeed, as e lec ed in he g ound-s a e spins and momen s13,14, up o 73Cu, he odd p o on esides p edominan ly in he πp3∕2 o bi al, while om 74Cu onwa ds i occupies he π 5∕2 shell. We will now demons a e ha mode n densi y unc ional heo y (DFT) and he alence-space in-medium simila i y eno maliza- ion g oup (VS-IMSRG) amewo ks can bo h p o ide a sa is ac o y unde s anding o changes in he cha ge adii and binding ene gies o he coppe iso opic chain be ween neu on numbe s N = 29 and N = 49, down o he scale o he small OES. In he con ex o he ollowing discussion, i is impo an o emembe ha he global (bulk) beha iou o nuclea cha ge adii is go e ned by he Wigne – Sei z (o box-equi alen ) adius 0 ¼½ 3= ð 4πρ0 Þ 1= 3 I , which is gi en by he nuclea sa u a ion densi y ρ0. On he o he hand, he local luc ua ions in cha ge adii, including OES, a e p ima ily impac ed by he shell s uc u e and many-body co ela ions. The common in e p e a ion o OES in ol es a ious ypes o pola iza ion exe ed by an odd nucleon, occupying a speci ic shell-model (o one-quasi- pa icle) o bi al15. In pa icula , he sel -consis en coupling be ween he neu on pai ing ield and he p o on densi y p o ides a cohe en unde s anding o he OES o cha ge adii o sphe ical nuclei such as semi-magic iso opic chains5,16–18. Wi h measu emen s now spanning all iso opes be ween he wo exo ic doubly magic sys ems 56,78Ni, he coppe iso opes ep esen an ideal labo a o y o es ing no el heo e ical app oaches in he medium-mass egion. This egion o he nuclea cha ep esen s new e i o y o A-body heo ies based on wo-nucleon (NN) and h ee-nucleon (3N) o ces de i ed om chi al e ec i e ield he- o y19,20. In gene al, OES o masses has only been spa sely s udied wi hin he con ex o nuclea o ces and many-body me hods21,22. Howe e , he VS-IMSRG app oach7,8 has now su icien ly ad anced o s udy mos nuclea p ope ies in essen ially all open-shell sys- ems below A = 100, including masses, cha ge adii, spec oscopy and elec oweak ansi ions23. The p esence o a po en ial sub-shell closu e a N = 40 ( e . 9) and he well-e idenced s uc u al changes due o shell e olu ion as N = 50 is app oached13 all se e o es such calcula ions e en u he . F om he side o he DFT calcula ions, he ecen ly de eloped Fayans unc ional, success ul in desc ibing he global ends o cha ge adii in he Sn (Z = 50) and Ca (Z = 20) mass egions3–5, has no been es ed in his egion o he nuclea cha , no wi h da a on odd-Z iso opes in gene al. De ails on bo h he DFT and VS-IMSRG calcula ions can be ound in he Me hods, bu a ew key aspec s will be men ioned. The DFT calcula ions we e ca ied ou wi h he Fayans ene gy densi y unc ional24, which—impo an ly— ep oduces he mic oscopic equa ions o s a e o symme ic nuclea ma e and neu on ma e . The inclusion o su ace and pai ing e ms dependen on densi y g adien s has been shown o be c ucial o ep oducing ( he OES o ) he calcium cha ge adii5. The VS-IMSRG calcula ions we e pe - o med wi h wo se s o NN+3N o ces de i ed om chi al e ec- i e ield heo y, he PWA and 1.8/2.0(EM) in e ac ions o e . 25. Bo h a e cons ained by only wo-, h ee- and ou -body da a, wi h 3N- o ces speci ically i o ep oduce he 3H binding ene gy and 4He cha ge adius. The absolu e cha ge adii o he coppe iso opes a e compa ed o he heo e ical calcula ions in Fig. 2a. These o al cha ge adii a e ob ained using he e e ence adius1 65 = 3.9022(14) m. Excellen DFT: Fy(s d) Fy(Δ ) VS-IMSRG: EM1.8/2.0PWA Expe imen 3 2 1 0 –1 –2 –3 10 0 –10 Δ (3)(10–3 m) ΔE (3) (MeV) 28 30 32 34 36 38 40 N N 42 44 46 48 48464442403836343230 50 Fig. 3 | Compa ison o expe imen al and heo e ical h ee-poin s agge ing pa ame e s o binding ene gies ( Δð3Þ E I ) and adii ( Δð3Þ I ). Da a a e plo ed as a unc ion o neu on numbe N using he same colou scheme as in Fig. 2. Fo he DFT calcula ions, g een squa es a e used o he Fy(s d) unc ional and blue diamonds o he Fy(Δ ) unc ional, while he VS-IMSRG calcula ions a e displayed using ed squa es o he PWA in e ac ion and o ange diamonds o he EM1.8/2.0 in e ac ion. Expe imen al e o ba s a e s a is ical and ep esen one s anda d de ia ion. E o ba s on he DFT calcula ions ep esen only he s a is ical con ibu ion. The op panels compa e o he OES o he binding ene gies, while he bo om panels show cha ge adii. The calcula ions in he le panels we e pe o med wi h DFT, while he igh panels show he VS-IMSRG esul s. NATuRE PHYSiCS | www.na u e.com/na u ephysics Le e s NaTuRE PHySICS o e all ag eemen was ob ained wi h he wo se s o DFT calcula- ions, whe eas he VS-IMSRG calcula ions ei he gene a ed absolu e adii ha a e oo small (EM1.8/2.0) o oo la ge (PWA). This con- i ms ea lie indings2,26 ha he ep oduc ion o he absolu e adii equi es a co ec p edic ion o he nuclea sa u a ion densi y ρ0. While bo h Fayans unc ionals used he e mee his condi ion, his is no he case o he in e ac ions used in he VS-IMSRG calcula ions: he EM1.8/2.0 in e ac ion sa u a es a oo high o a densi y, and he PWA in e ac ion sa u a es a a lowe densi y. The misma ch be ween DFT and expe imen o he neu on- de icien iso opes is p ima ily ela ed o he pai ing in he π 5∕2p- shell egion: as discussed in he Me hods, he pai ing g adien e m ha was adjus ed using da a om he calcium egion is oo s ong in hea ie nuclei. We no e ha a simila educ ion a ound N = 30 was p edic ed by he Fayans DF3-a1 DFT calcula ions o he Ni chain27. Fo he binding ene gies pe nucleon28,29, shown in Fig. 2b, he DFT calcula ions ma ched pa icula ly well wi h expe imen , wi h bo h in e ac ions yielding p ac ically iden ical esul s. The VS-IMSRG binding ene gies appea o o e bind in he mid-shell egion bu ne e heless do well in e ms o he absolu e alue as well as he gene al end. Figu e 2c shows δ 2 hi 65;A I alongside EM1.8/2.0 VS-IMSRG and he Fy(Δ ) calcula ions. The ag eemen o he VS-IMSRG calcula ions wi h he da a is excellen o e all excep o a small disc epancy nea N = 40. Figu e 3 looks a he OES in mo e de ail, by compa ing expe i- men al alues o Δ ð3 Þ E I (de ined in analogy wi h Δ ð3Þ I ) and Δ ð3 Þ I o he- o e ical alues ob ained wi h DFT and VS-IMSRG me hods. No e how he educ ion in expe imen al Δ ð3 Þ I nea N = 50 is also isible in Δ ð3 Þ E I , al hough less p onounced. This demons a es ha cha ge adii a e mo e sensi i e o he unde lying nuclea s uc u e changes in his egion, such as o he spin change due o he in e sion o he p3∕2 and 5∕2 p o on o bi als om N = 45 onwa ds. The OES in binding ene - gies is ep oduced e y well wi h he VS-IMSRG calcula ions, while i is o e es ima ed wi h DFT. Fo he cha ge adii, he unc ional Fy(Δ ) ha includes he OES in calcium iso opes as i ing pa am- e e s, gene a es subs an ially mo e OES han Fy(s d), as expec ed. Nea N = 40, he ag eemen o Δ ð3 Þ I is pa icula ly good. Including he sys ema ic unce ain y o he uni o m blocking app oxima ion employed, he o al unce ain y o he DFT calcula ions on he OES is es ima ed o be less han 5 × 10−3 m. This unce ain y co e s he small quan i a i e misma ch o Fy(Δ ) a small N, hough he de ia- ion a la ge N emains no able and p o ides a help ul benchma k o u he de elopmen . The VS-IMSRG calcula ions p edic an OES o he igh o de o magni ude close o he neu on-shell closu es N = 28, 50, bu do no ep oduce he la ge OES in he mid-shell and close o N = 40. This is e y likely ela ed o he missing p o on exci a- ions om he π 7∕2 shell14, which we e shown o be impo an om N = 41 onwa ds, bu educed in 77,78Cu. The obse a ion ha he VS-IMSRG calcula ions pe o m as well o e en be e han DFT when i comes o p edic ing he local ea u es o he adii and ene gies illus a es ha he many-body co ela ion e ec s a e unde be e con ol in he VS-IMSRG app oach. This p o ides an impo an clue o he mic oscopic o igins o he OES. In pa icula , he na u al eme gence o he OES o bo h adii and binding ene - gies om an in e ac ion ha is only cons ained wi h ou -body da a is encou aging. In conclusion, we ha e epo ed new measu emen s o he changes in he mean-squa ed cha ge adii o e y neu on- ich cop- pe iso opes, which we e made possible hanks o he high esolu- ion and high sensi i i y o he newly de eloped CRIS me hod a ISOLDE/CERN. This echnique is now a ailable o s udy exo ic iso- opes, app oaching he nuclea d iplines, hus accessing o he i s ime he ancho poin s in he nuclea cha ha benchma k nuclea heo ies. We demons a ed good ag eemen be ween ou measu e- men s and esul s om DFT and VS-IMSRG me hods. Gi en he in insic complexi y o medium-mass sys ems wi h odd-Z, his ep esen s also a majo achie emen in nuclea heo y, and an impo an s ep o wa d in ou global unde s anding o he nuclea binding ene gy and cha ge adius o exo ic iso opes. The in e play be ween he bulk nuclea p ope ies (be e cap u ed by DFT) and local a ia ions (be e cap u ed by VS-IMSRG calcula ions) was shown o be c ucial in e ealing he mic oscopic desc ip ion o he OES e ec in adii and binding ene gies. The OES eme ges na u ally om NN+3N in e ac ions de i ed om chi al e ec i e ield heo y, cons ained o he p ope ies o iso opes wi h up o ou nucleons only, which p esen s an impo an s ep o wa d owa ds a p edic i e nuclea heo y. The compa ison wi h hea ie odd-Z sys ems nea he hea ies sel -conjuga e iso ope 100Sn, which can now be s udied expe imen ally o e a ange o mo e han 30 iso opes, is expec ed o p o ide he nex challenge o nuclea heo y. Online con en Any me hods, addi ional e e ences, Na u e Resea ch epo ing summa ies, sou ce da a, ex ended da a, supplemen a y in o ma- ion, acknowledgemen s, pee e iew in o ma ion; de ails o au ho con ibu ions and compe ing in e es s; and s a emen s o da a and code a ailabili y a e a ailable a h ps://doi.o g/10.1038/s41567- 020-0868-y. Recei ed: 13 Sep embe 2019; Accep ed: 4 Ma ch 2020; Published: xx xx xxxx Re e ences 1. Angeli, I. & Ma ino a, K. Table o expe imen al nuclea g ound s a e cha ge adii: an upda e. A om. Da a Nucl. Da a Tables 99, 69–95 (2013). 2. Reinha d, P.-G. & Naza ewicz, W. Nuclea cha ge and neu on adii and nuclea ma e : end analysis in Sky me densi y- unc ional- heo y app oach. Phys. Re . C 93, 051303 (2016). 3. Hammen, M. e al. F om calcium o cadmium: es ing he pai ing unc ional h ough cha ge adii measu emen s o 100−130Cd. Phys. Re . Le . 121, 102501 (2018). 4. Go ges, C. e al. 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A 95, 032502 (2017). 11. Taniuchi, R. e al. 78Ni e ealed as a doubly magic s onghold agains nuclea de o ma ion. Na u e 569, 53–58 (2019). 12. Ma sh, B. A. e al. Cha ac e iza ion o he shape-s agge ing e ec in me cu y nuclei. Na . Phys. 14, 1163–1167 (2018). 13. Flanagan, K. T. e al. Nuclea spins and magne ic momen s o 71,73,75Cu: in e sion o π2p3∕2 and π1 5∕2 le els in 75Cu. Phys. Re . Le . 103, 142501 (2009). 14. de G oo e, R. P. e al. Dipole and quad upole momen s o 73−78Cu as a es o he obus ness o he Z = 28 shell closu e nea 78Ni. Phys. Re . C 96, 041302 (2017). 15. Xie, L. e al. Nuclea cha ge adii o 62−80Zn and hei dependence on c oss-shell p o on exci a ions. Phys. Le . B 797, 134805 (2019). 16. Zawischa, D., Regge, U. & S apel, R. E ec i e in e ac ion and he s agge ing o nuclea cha ge adii. Phys. Le . B 185, 299–303 (1987). 17. Fayans, S. & Zawischa, D. Towa ds a be e pa ame iza ion o he nuclea pai ing o ce: densi y dependence wi h g adien e m. Phys. Le . 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Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use, sha ing, adap- a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons license, and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included in he a icle’s C ea i e Commons license, unless indica ed o he wise in a c edi line o he ma e ial. I ma e ial is no included in he a icle’s C ea i e Commons license and you in ended use is no pe mi ed by s a u- o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde . To iew a copy o his license, isi h p://c ea i ecommons. o g/licenses/by/4.0/. © CERN 2020 NATuRE PHYSiCS | www.na u e.com/na u ephysics Le e s NaTuRE PHySICS Me hods Lase sys ems. Coppe a oms we e lase -ionized using a wo-s ep lase ioniza ion scheme. Ligh o he i s s ep was p oduced using an injec ion-locked Ti:sapphi e lase sys em join ly de eloped by he Johannes Gu enbe g-Uni e si ä Mainz and he Uni e si y o Jy äskylä30. This lase ca i y is buil a ound a Ti:sapphi e c ys al ha is pumped wi h 532-nm ligh p oduced a a epe i ion a e o 1 kHz by a Lee LDP-100MQG pulsed Nd:YAG lase . Th ough pulsed ampli ica ion o con inuous-wa e seed ligh p oduced by an M-squa ed SolsTiS Ti:sapphi e lase , na owband (≈20 MHz) pulsed lase ligh was p oduced a a epe i ion a e o 1 kHz. This lase ligh was hen equency ipled using wo nonlinea c ys als o p oduce he equi ed 249-nm ligh o he esonan exci a ion s ep. A maximum o 0.5 μJ o 249-nm ligh was deli e ed in o he CRIS beamline, sa u a ing he esonan s ep. The wa eleng h o he scanning lase was eco ded by a HighFinesse WSU2 wa eme e e e y 10 ms, and used in a eedback loop o s abilize he wa eleng h o a a ge alue. The wa eleng h o a empe a u e-s abilized HeNe lase was simul aneously eco ded o e alua e he d i o he wa eme e du ing he measu emen s. Resonan ioniza ion o hese exci ed a oms was achie ed using a 314.2444-nm ansi ion o he 3d94s(3D)4d4P3∕2 au o-ionizing s a e a 71,927.28 cm−1, using ligh p oduced by a equency-doubled Spec olase 4000 pulsed dye lase pumped by a Li on LPY 601 50-100 PIV Nd:YAG lase , a a epe i ion a e o 100 Hz. Owing o he 344(20) ns li e ime o he exci ed s a e, he 314-nm lase pulse could be delayed by 50 ns, emo ing lineshape dis o ions and powe b oadening e ec s10 wi hou app eciable e iciency losses. A small pick-o o he undamen al lase ligh was sen o a High inesse WS6 wa eme e o moni o po en ial wa eleng h d i s. The maximum powe o his sys em was 125 μJ, which was no enough o ully sa u a e he ansi ion. Ex ac ion o cha ge adii om iso ope shi s. The hype ine pa ame e s and he cen oids νA a e ex ac ed om he hype ine spec a by i ing hem wi h co ela ed Voig p o iles cen ed a he esonance ansi ion equencies. This analysis was pe o med using he SATLAS analysis lib a y31. Mo e de ails on he analysis p ocedu e can be ound in e . 32. The iso ope shi , δν 65;A0 ¼νA�ν65 I , can be de e mined om he shi o he cen oid o he hype ine spec um o one iso ope ela i e o ha o a e e ence iso ope. Such e e ence measu emen s o 65Cu we e used o ake in o accoun possible changes in he beam ene gy o d i s in wa eme e calib a ion du ing he ou -day expe imen . Table 1 displays he iso ope shi s ob ained in his wo k. F om hese iso ope shi s, he mean-squa ed cha ge adius di e ence δ 2 hi 65;A0 I is ob ained om: δν65;A0 ¼ M mA 0 �m65 m65mA0 þ Fδ 2  65 ; A0 ; ð 2 Þ wi h m he mass o he iso ope, and M and F being he mass- and ield-shi ac o , espec i ely. The a omic ac o s o he 249-nm line we e de e mined h ough a King plo analysis32 using he da a o he 324.7540-nm line9. Using alues o M324 = 1,413(27) GHz u (whe e u is he uni ied a omic mass uni ) and F324 = −779(78) MHz m−2 ob ained om he same e e ence, we ind M249 = 2,284(24) GHz u and F249 = 439(80) MHz m−2. The ex ac ed adii a e in excellen ag eemen wi h hose measu ed in he 324-nm ansi ion (Table 1), al hough a ac o o 2-4 less p ecise due o he lowe alue o F249 as compa ed wi h F324. DFT calcula ions. In his wo k, we employed he ecen ly de eloped pa ame e iza ions Fy(s d)5 and Fy(Δ )4,6, which di e in he pool o da a used o he op imiza ion p o ocol. Fy(s d) uses he s anda d se o g ound-s a e da a (binding ene gies and cha ge/ma e adii o e en–e en semi-magic iso opes) om e . 33 complemen ed wi h he OES o ene gies, de ined in an analogous way as equa ion (1). Fy(Δ ), op imized a he Ha ee–Fock–Bogolyubo le el, addi ionally uses he cha ge adii o he e en–odd calcium iso opes. The inclusion o his in o ma ion subs an ially inc eases he s eng h o he g adien e m in he pai ing unc ional (which is absen in o he DFT o ms and p ac ically inac i e in Fy(s d)). This was ound o be c ucial in ep oducing he OES o cha ge adii in he calcium iso opes5. Howe e , his pa ame e iza ion o e es ima es he OES in he hea ie masses3 and he kink in cha ge adii a ound 132Sn ( e . 4). Fo all p ac ical de ails pe aining o ou Fayans-DFT calcula ions, we e e he eade o e . 6. VS-IMSRG calcula ions. Wo king in an ini ial ha monic-oscilla o basis o 13 majo shells, we i s ans o med o he Ha ee–Fock basis, hen used he Magnus p esc ip ion34 o he IMSRG o gene a e app oxima e uni a y ans o ma ions o decouple i s he gi en co e ene gy, hen as well as a speci ic alence-space Hamil onian om he ull A-body Hamil onian. In he IMSRG(2) app oxima ion used in his wo k he e, all ope a o s a e unca ed a he wo-body le el. In addi ion, we cap u e e ec s o 3N o ces be ween alence nucleons ia he ensemble no mal o de ing p ocedu e o e . 35. The same ans o ma ion is hen applied o he in insic p o on mean-squa ed adius ope a o , wi h app op ia e co ec ions o ob ain co e and alence-space (hence absolu e) cha ge adii. We ake 56Ni as ou co e e e ence s a e, wi h a alence space consis ing o he p o on and neu on p3∕2, p1∕2, 5∕2 and g9∕2 single-pa icle o bi s. Finally, using he NUSHELLX@MSU shell model code36, we diagonalize he alence-space Hamil onian o ob ain g ound- (and exci ed-) s a e ene gies, as well as expec a ion alues o he cha ge adius ope a o , whe e induced wo-body co ec ions a e included na u ally in he VS-IMSRG app oach. Da a a ailabili y The da a ep esen ed in Figs. 1–3 a e a ailable as Sou ce Da a. All o he da a ha suppo he plo s wi hin his pape and o he indings o his s udy a e a ailable om he co esponding au ho upon easonable eques . Re e ences 30. Sonnenschein, V. e al. Cha ac e iza ion o a pulsed injec ion-locked Ti:sapphi e lase and i s applica ion o high esolu ion esonance ioniza ion spec oscopy o coppe . Lase Phys. 27, 085701 (2017). 31. Gins, W. e al. Analysis o coun ing da a: de elopmen o he SATLAS Py hon package. Compu . Phys. Commun. 222, 286–294 (2018). 32. de G oo e, R. P. High Resolu ion Collinea Resonance Ioniza ion Spec oscopy o Neu on-Rich 76,77,78Cu Iso opes. PhD hesis, KU Leu en (2017); h ps://cds. ce n.ch/ eco d/2285661 33. Klüp el, P., Reinha d, P.-G., Bü enich, T. & Ma uhn, J. Va ia ions on a heme by Sky me: a sys ema ic s udy o adjus men s o model pa ame e s. Phys. Re . C 79, 034310 (2009). 34. Mo is, T. D., Pa zuchowski, N. M. & Bogne , S. K. Magnus expansion and in-medium simila i y eno maliza ion g oup. Phys. Re . C 92, 034331 (2015). 35. S obe g, S. R. e al. Nucleus-dependen alence-space app oach o nuclea s uc u e. Phys. Re . Le . 118, 032502 (2017). 36. B own, B. A. & Rae, W. D. M. The shell-model code NuShellX@MSU. Nucl. Da a Shee s 120, 115–118 (2014). Acknowledgemen s We acknowledge he suppo o he ISOLDE collabo a ion and echnical eams, and he Uni e si y o Jy äskylä o he use o he injec ion-locked ca i y. This wo k was suppo ed by he B iX Resea ch P og am no. P7/12 and FWO-Vlaande en (Belgium) and GOA 15/010 om KU Leu en, FNPMLS ERC Consolida o G an no. 648381, he Science and Technology Facili ies Council consolida ed g an ST/P004423/1 and con inua ion g an ST/L005794/1, he EU Se en h F amewo k h ough ENSAR2 (654002), he NSERC and he Na ional Resea ch Council o Canada, and by he O ice o Science, US Depa men o Ene gy unde awa d numbe s DE-SC0013365 and DE- SC0018083 (NUCLEI SciDAC-4 collabo a ion). We acknowledge he inancial aid o he Ed Schneide man Fund a New Yo k Uni e si y. Au ho con ibu ions R.P.d.G., J.B., C.L.B., M.L.B., T.E.C., T.D.G., G.J.F.-S., D.V.F., K.T.F., S.F., R.F.G.R., W.G., Á.K., K.M.L., G.N., S.R., H.H.S., A.R.V., K.D.A.W., S.G.W., Z.Y.X. and X.F.Y. pe o med Table 1 | Summa y o he iso ope shi s ob ained in his wo k A I iS (MHz) δ 2  65;A0 I ( m2) δ 2  65;A0 li I ( m2) 63 3/2 +1,055.4(2.6) −0.140(7)[20] −0.148(1)[17] 64 1+508(5) −0.09(1)[2] −0.118(3)[13] 65 3/2 0 0(0)[0] 0(0)[0] 66 1 −529(5) +0.01(1)[2] +0.033(5)[12] 67 3/2 − − +0.115(5)[18] 68 1 −1,498(5) +0.13(1)[3] +0.132(5)[31] 68 6 −1,479(4) +0.17(1)[3] +0.191(4)[31] 69 3/2 −1,937(5) +0.24(1)[4] +0.237(3)[34] 70 6 −2,375(7) +0.27(2)[5] +0.270(3)[44] 70 3 −2,390(7) +0.29(2)[5] +0.287(11)[44] 70 1 −2,397(6) +0.32(2)[5] +0.323(11)[44] 71 3/2 −2,787(4) +0.44(2)[7] +0.407(11)[44] 72 2 −3,240(7) +0.43(2)[7] +0.429(5)[55] 73 3/2 −3,634(5) +0.52(2)[8] +0.523(15)[58] 74 2 −4,057(5) +0.53(1)[9] +0.505(18)[72] 75 5/2 −4,455(4) +0.56(1)[9] +0.546(21)[80] 76 3 −4,848(5) +0.58(2)[10] − 77 5/2 −5,234(5) +0.59(2)[11] − 78 (6) −5,623(10) +0.58(3)[11] − The second column displays he nuclea spin I o he iso opes. Fo 78Cu, no i m spin assignmen has been made, which is why he alue is lis ed in b acke s. The iso ope shi s (IS) ob ained in his wo k a e compa ed o he a ailable li e a u e9. S a is ical and sys ema ic a omic unce ain ies a e enclosed wi hin pa en heses and squa e b acke s, espec i ely. NATuRE PHYSiCS | www.na u e.com/na u ephysics Le e s NaTuRE PHySICS he expe imen . R.P.d.G. and C.L.B. pe o med he da a analysis and R.P.d.G. p epa ed he igu es. J.D.H. and T.M. pe o med he VS-IMSRG calcula ions. W.N. and P.-G.R. pe o med he DFT calcula ions. R.P.d.G., W.N., P.-G.R. and J.H. p epa ed he manusc ip . All au ho s discussed he esul s and con ibu ed o he manusc ip a all s ages. Compe ing in e es s The au ho s decla e no compe ing in e es s. Addi ional in o ma ion Supplemen a y in o ma ion is a ailable o his pape a h ps://doi.o g/10.1038/ s41567-020-0868-y. Co espondence and eques s o ma e ials should be add essed o R.P.d.G. Pee e iew in o ma ion Na u e Physics hanks Da id Fo es , Bjö n Jonson and Pe e Ring o hei con ibu ion o he pee e iew o his wo k. Rep in s and pe missions in o ma ion is a ailable a www.na u e.com/ ep in s. NATuRE PHYSiCS | www.na u e.com/na u ephysics