1. Charge radii of medium-mass nuclei using the atomic physics input Magdalena Kowalska CERN, PH-Dept. kowalska@cern.ch below

2. Outline 2 Laser spectroscopy on radioactive nuclei Optical isotope shifts A long way from isotope shifts to charge radii Take-home message

3. Charge radii from optical transitions 3 Cocolios et al, Phys. Rev. Lett, 106, 052503 (2011) Atomic factors: (normal and specific mass shift, F-factor (main source of uncertainties)

4. Laser spectroscopy on radionuclides 4 Blue – since 1995

5. Collinear laser spectroscopy 5 Laser beam, Laser on fixed frequency Ion beam Electrostatic deflection Retardation zone Charge exchange region (if needed) Excitation / Observation region Detection method depends on chemical element or even isotope => optimised for best S/N ratio: - photon detection: all nuclei - beta asymmetry: short-lived nuclei - collision reionisation: noble gases Why collinear? Ions at RIB facilities come out as a beam, so laser-ion interaction is longest in (anti) collinear geometry Review: B Cheal and K T Flanagan 2010 J. Phys. G: Nucl. Part. Phys. 37 113101

6. Optical isotope shifts What is measured? (independently for each isotopes)  Laser frequency of resonance (including HFS) in laboratory frame (often: moving ions/atoms)  Or accelerating high voltage at which ion/atom beam is in resonance with laser What is needed to extract differences in charge radii?  Frequency difference (HFS centroid) for 2 isotopes at rest Conversion for ion beams (Collinear laser spectroscopy):  Uncertainty in all input values contributes  Statistical – collected statistics, quality of fit (especially for HFS)  Systematic: precision and accuracy of atomic masses and measuring devices (laser frequency and accelerating voltage) 6 set laser frequency (Wavemeter or frequency comb) atomic masses Accelerating voltage (voltage divider + voltmeter)

7. Optical isotope shifts - uncertainties Uncertainties:  Statistical – collected statistics, quality of fit (especially for HFS)  Systematic: precision and accuracy of atomic masses and measuring devices 7 set laser frequency (Wavemeter or frequency comb) atomic masses Accelerating voltage (usually voltage divider + voltmeter) Mg+, 2006 data Systematic uncertainties much larger than statistical

8. From isotope shifts to charge radii 8 Proportional to (change in) e- density at nucleus. Semi-empirical or ab initio methods. Due to reduced mass of nucleus-e- system and correlations between e-’s. Ab initio methods or “King plot”. A: HFS constant for ns e-  : correction for finite size of nucleus  – HFS anomaly g’ – reduced g factor of nucleus  From HFS: all inner e’s are paired off in alkali-like atom and make no contribution to magnetic HFS. HFS directly proportional to density of ns e- at point nucleus  Goudsmit-Fermi-Segre formula: ns electron density at nucleus is related to effective quantum number of ns level Screening effects:

9. K_MS: Modified King plot When data for at least 3 isotopes exists (i.e stable isotopes) partial help is there!  Combine absolute radii (transitions in muonic atoms or electron scattering) and isotope shifts in optical transition Usually resulting uncertainties too large -> so use F from semi-empirical or ab ignition approaches A good side effect: offset in accelerating voltage  Scales with mass like mass shift  -> can incorporate it in K_MS Result: Determine K_MS and F  usually too large uncertainties  So: use calculated F 9 Modified difference in charge radii Modified isotope shift A, A’ A, A’’ Slope: F 0 Offset: K_MS e.g. for Ne

10. From isotope shifts to charge radii It is not that easy: Many correlated systematic effects  In IS – accelerating voltage  In King-plot: Muonic data Inconsistent values from ab initio atomic theory and other approaches Large uncertainties in F and K_MS, whichever approach taken Which F and K_MS to use? Resulting radii trend must be the final judge:  Consistent with radii trends of neighbouring chains  Consistent with nuclear structure known from other observables 10

11. Some examples Below medium-mass nuclei The lucky cases with at least 3 stable nuclei + K with 2 stable isotopes 11

12. HFS structure of 21Mg observed in  -decay asymmetry Laser spectroscopy on Mg isotopes Mg+: Detection via:  Fluorescence photos  Or beta-decay asymmetry 12 (D1 line)

13. From Mg isotope shifts to charge radii 13 20% difference, seen also in other elements F [MHz/ fm2] Semi-empiricalAb initio GFSHFS[Tor85][Saf01][Ber03][Sah12] s->p1/2-158-148-117-125.81-127-126.221 s->p3/2-158-148-117-126.821-127-126.324 K_SMS [GHz/u] Ab initio [Saf01][Ber03][Sah12] s->p1/2-362-379-390.1-398.8 s->p3/2-361-373-386.1-389.9 [Tor85] G. Torbohm, B. Fricke, A. Rosen, PRA 31, 2038 (‘85) [Saf01] M.S. Safronova & W.R. Johnson, PRA 64, 052501 (‘01) [Ber03] J.C. Berengut, V.A. Dzuba, V.V. Flambaum, PRA 68, 022502 (‘03) [Sah12] B.K. Sahoo, J. Phys. B 43 231001 (’10) 10% error assumed F factor Specific mass shift Large differences or uncertainties -> let’s try a King plot (need Muonic data)

14. Mg charge radii from muonic transitions 14 Fricke et al, PRC 45, 81 (’92) Can they be determined more accurately nowadays? Large corrections and many systematic uncertainties

15. Mg modified King plot 15 Uncertainties: from transition energy + 10% of larger nucl. polarization uncert. Differences in radii from Muonic data vs electron scattering Too large uncertainties to be used d syst error 1: uncertainty in nuclear polarisation correction (30% of total nuclear polarization) Syst error 2: due to choice of skin thickness t of Fermi distribution Mg+, 24,25,26Mg Syst error in IS would lead to additional 5 GHz u uncertainty in K MS K_NMS = 365 GHz u

16. Resulting charge radii 16 [Ber03] [Sah10] [Saf01] King plot d (fm2) Uncertainties in atomic factors are equally important as central values Which atomic factors should I use? The resulting nuclear physics interpretation will tell me …

17. Mg charge radii and nuclear structure 17 D. Yordanov et al, Phys. Rev. Lett. 108, 042504 (2012) Atomic number, A Charge radius (fm) Uncertainty of the slope due to atomic F factor uncertainty not included N=20 N=14 Smallest radius at N=14, not N=20: Migration of the shell closure Best nuclear physics description from King plot: In general radii increase Minimum radii for 24,26Mg (new sub-shell closures) Radii around N=20 larger than around N=8

18. Laser spectroscopy of Ne isotopes 18 Ne atoms Special type of setup:  Excitation to a meta-stable level (neutralization)  Laser excitation from this state  Detection after reionization – ions or beta counts Acceleration-voltage calibration:  Use of (anti-) collinear laser

19. Charge radii of Ne isotopes F determined from semi-empirical approaches:  Goudsmit-Fermi-Segre: F=-40(4) MHz/fm2 (assuming 10% uncertainty)  HSF: F= -38 (4) MHz/fm2 (also, assuming 10% uncertainty)  Any ab initio calculations existing for this transition? Mass shift from modified King plot  Muonic radii for 20,21,22Ne  With fixed F 19 1 GHz 10 MHz

20. Ne charge radii and nuclear structure 20 Determined trend in radii can be explained by nuclear models Wealth of structures revealed:  Presence and disappearance of shells, clustering, onset of halo formation 20 Intrinsic density distributions of dominant proton FMD configurations Geithner et al, PRL 101, 252502 (‘08) Marinova et al, PRC 84, 034313 (‘11)

21. Laser spectroscopy of Ar isotopes Ar atom Technique: the same as later used for Ne (including energy calibration) Determination of F – semi-empirical approach: F= - 104(10) MHZ/fm2 Determination of mass shift constant: modified King plot  With fixed F Any ab initio calculations? Muonic data 21 Talmi – Zamick formula HF calculation, A. Klein et al, Nucl. Phys. A 607 (1996) 1. K. Blaum et al, NPA 799, 30 (2008)

22. Laser spectroscopy of K isotopes Collinear laser spectroscopy on atoms  Detection via fluorescence photons  Bunched ion beam (to lower laser background) K_SMS and F determined from ab initio calculations (only 2 data points from muonic atoms)  A.-M. Martensson-Pendrill, et al., JPB 23, 1749 (‘90) – reliable results for Ca  K_SMS = −15.4(3.8) GHz u  F =−110(3) MHz/fm2 22 N=28 shell closure well visible

23. Charge radii of Ar and K isotopes 23 K. Kreim et al, PLB 731, 97 (‘14) Resulting trends in radii make sense from nuclear structure point of view  N=28 shell closure visible (but not N=20)  Rather consistent with radii trends in neighbouring chains – although strong Z dependence is clearly visible K Ar Systematic errors in trends due to F uncertainty not shown

24. Take-away message Optical isotope shifts are sensitive to small changes in nuclear charge radii The art  lies in determining the proportionality (atomic) factors with both enough precision and enough accuracy to provide a quantitative relation and  to provide nuclear radii which make sense from the nuclear physics point of view Have to rely a lot on semi-empirical approaches and muonic data with large corrections calculated with limited computer power Hopefully ab initio approaches are catching up 24

25. Collinear-anticollinear 25

26. Modified King plot 26 0 Isotope pair A, A’ Isotope pair A,A’’ slope

27. King plot for 2 optical transitions In Mg+ s->p1/2 (D1) and s->p3/2 (D2) 27

28. From optical isotope shifts to radii Mass shift in spectra of many-electron atoms First order field shift  1 st order perturbation theory (finite nucleus as perturb to point-like nucleus)  1 st approach  Then corrections  For distribution 28 Specific mass shift, Due to e- correlations Normal mass shift nonrelativistic probability density of e- at point nucleus  -1 close to 1 for light isotopes, close to 0.1, 0.2 for heavy -> changes slowly with r, so can use

29. From optical isotope shifts to radii Electronic factor Determination of density of ns electron at point-like nucleus Ab initio atomic calculations Semi-empirical approaches:  From HFS: all inner e’s are paired off in alkali-like atom and make no contribution to magnetic HFS. HFS directly proportional to density of ns e- at point nucleus  Goudsmit-Fermi-Segre formula: ns electron density at nucleus is related to effective quantum number of ns level Screening effects: nonrelativistic probability density of e- at point nucleus, but it should be rather change in probability density A: HFS constant for ns e-  : correction for finite size of nucleus  – HFS anomaly g’ – reduced g factor of nucleus

30. F for Mg+ from GFS 30

31. F for Mg+ from HFS 31

32. Field shift according to Seltzer Seltzer 32

33. 2 nd order field shifts Due to 2nd-order contributions and from relativistic effects 33

34. Nuclear radii 34 Which radius? Depends on the used probe Charge radius – studied via Coulomb interaction of a charged particle with nucleus Matter radius – studied via strong interaction of nuclei and particles Nuclei don’t have abrupt boundaries - useful two parameters:  Mean radius – where density is 50% of maximum  Diffuseness/“skin thickness” – distance where density drops from near max to near min Mean square radius of charge distribution

35. Modified King plot When data for at least 3 isotopes exists (i.e stable isotopes) partial help is there!  Combine absolute radii (transitions in muonic atoms or electron scattering) and isotope shifts in optical transitions 35 0 A A A A A A Isotope pair A, A’ Isotope pair A,A’’ slope A A A A A A A A A A A A But if there are fewer stable isotopes … See Na, Mn, Cu, Ga …

36. Lasers + ion traps: n-def Hg & Au isotopes Several techniques combined RILIS lasers to probe the hyperfine structure of Hg & Au isotopes Detection:  Alpha spectroscopy with Windmill  Selective ion counting in MR-ToF 36 Bonn et al., PLB38 (1972) 308 Ulm et al., Z Physik A 325 (1986) 247 Au RILIS, Windmill, ISOLTRAP teams

37. Resulting charge radii 37 [Ber03] [Sah10] [Saf01] King plot d (fm2) King plot only Middle values K_MS + errorK_MS - error Too large uncertainties

38. Zamick 38

39. Electron scattering 39 Diffraction with radiation of wavelength smaller than the object – high-energy e- Lambda p> 100 MeV (e.g 100 MeV to 1 GeV e-) Result: rms radius: 1/2 Diffraction minima

40. Electron transitions Nuclei are not point-like particles -> atoms electrons can penetrate inside and probe the nuclear distribution (K X-ray transitions, especially 1s electrons) Biggest penetration into nucleus for 1s e- Much easier to compare distibutions of two neighbouring isotopes (than determine size of 1 radius absolutely) K X-ray isotope shifts – relatively large Optical isotope shifts in valence e-: much smaller penetration and thus smaller shift Both are still very small effects – due to a small size of nucleus compared to e- orbits 40 X-ray transition 1e-6 effect optical transition 1e-7 effect J. Bonn et al, Z. Phys. A 276, 203 (1976) P. Lee et al Phys Rev C 17, 1859 (1978)

41. Muonic atoms Muon’s mass = 200 e- mass -> Bohr (atomic) radius 200 smaller E.g. in 208 Pb: muon’s mean radius is inside the nucleus Isotope shifts – factor 1-2 (vs 1e-4 – 1e-6 for e-) Technique:  Muons produced at accelerators (in decay of pi mesons), e.g at PSI-Zurich  Bombard target made of isotope(s) of interest  Muons are captured in high-n orbits and cascade down to 1s orbits  Emitted photons (of MeV energy) are detected  Obtain directly charge-distribution parameters (although analysis is complex) 41