Magdalena Kowalska CERN, EP-Dept. kowalska@cern.ch Muonic atoms to measure charge radii of stable and soon unstable nuclei? Magdalena Kowalska CERN, EP-Dept. kowalska@cern.ch

Outline Charge radii from Muonic-atom transitions Input to optical isotope shifts Need for new Muonic-atom data and calculations

Study of charge radii Comparing and combining quantities from different techniques Model dependence

Nuclear charge radii & muonic atoms Muon’s mass = 200 e- mass -> Bohr (atomic) radius 200 smaller E.g. in 208Pb: 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 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)

Deriving radii from muonic transitions In 1970 Barrett showed that energies of muon transitions are best interpreted in terms of generalised moments of nuclear charge distribution k and α: smoothly varying functions of Z for a transition between given muonic levels α almost same for all transitions in particular element Barrett moments – almost independent of assumptions about actual shape of nucleus (so model independent) K(r) leads to Barrett equivalent radius Rka: radius of sphere with uniform charge distribution with same K(r) as actual nucleus (still model-independent) But optical isotope shifts measure root mean square radius, not Barrett moments, so similar quantity has to be derived for comparison => Model dependent radii For spherical nuclei: 2-parameter Fermi distribution For deformed nuclei: quadrupole deformation assumed

Uncertainties in radii from muonic atoms In absolute radius determination Statistical - measured energy Smaller than systematic errors, except light nuclei (for stable nuclei) Systematic Due to corrections to measured energies: quantum dynamical corrections, electron screening (small), and largest - nuclear polarization corrections (muon polarization of nucleus) For model-dependent rms radii: Related to different possible values of model parameters, e.g. skin thickness in Fermi distribution For difference in radii (for isotopes of the same element) Some part of systematic uncertainties are correlated, so total uncertainty is smaller But correlations are not well defined in literature and cannot be retraced based on published results

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

Combining results from muonic atoms & electron scattering, example of Ar When electron scattering data are precise enough to calculate V ratios: Use V to derive rms r2 from Barrett radius, instead nuclear models – closer to reality Radii from Muonic atoms Radii from electron scattering Radii from combined analysis 38Ar: interpolated

When can radii from muonic atoms help optical isotope-shift studies? Problem of deriving d<r2> of radioactive nuclei from optical isotope shifts: Atomic physics parameters F and K_MS for systems with > 4 e- are too uncertain to obtain small enough uncertainties in d<r2> F and K_MS from atomic theory for Mg+ d<r2> [Saf01] King plot A [Ber03] [Sah10] [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) [Sah10] B.K. Sahoo, J. Phys. B 43 231001 (’10)

=> Modified King plot When data for at least 3 isotopes exists (i.e stable isotopes): Combine absolute radii (transitions in muonic atoms and/or electron scattering) and isotope shifts in optical transitions to derive more precise F and K_MS values A A But if there are fewer stable isotopes … See Na, Mn, Cu, Ga … Isotope pair A, A’ A A slope A Isotope pair A,A’’ A A A

Elements with 2 stable isotopes, e.g. K 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 If 3 absolute radii were known, maybe the uncertainty in K radii was smaller

Other elements with 2 stable isotopes

Wish-list For stable isotopes where Muonic data exists: Better treatment of systematic uncertainties in radii differences New nuclear polarization calculations New measurements (also electron scattering) on long-lived isotopes for chains with 2 stable isotopes Will allow better calibration of atomic factors in optical isotope shifts

Mg modified King plot Differences in radii from Muonic data vs electron scattering Uncertainties: from transition energy + 10% of larger nucl. polarization uncert. d Too large uncertainties to be used Mg+, 24,25,26Mg Syst error in IS would lead to additional 5 GHz u uncertainty in KMS K_NMS = 365 GHz u 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

Comparison to nuclear theory 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 D. Yordanov et al, Phys. Rev. Lett. 108, 042504 (2012)

Resulting charge radii King plot King plot only Middle values d<r2> (fm2) K_MS + error K_MS - error Too large uncertainties [Saf01] King plot [Ber03] [Sah10]

F&H 2004

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

Deriving radii from muonic transitions Barrett moments Barrett radial moment can be derived model-independently from the transition energy Eif => (Barrett) equivalent radius (with dimention of length) can be derived Barrett radius

Radius of a sphere with constant charge density yielding the same radial moment as the actual radial distribution

Nuclear corrections Polarization of the nucleus by the muon Ratio of different radial moments

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

From Mg isotope shifts to charge radii F factor F [MHz/ fm2] Semi-empirical Ab initio GFS HFS [Tor85] [Saf01] [Ber03] [Sah12] s->p1/2 -158 -148 -117 -125.81 -127 -126.221 s->p3/2 -126.821 -126.324 20% difference, seen also in other elements 10% error assumed Specific mass shift 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) Large differences or uncertainties -> let’s try a King plot (need Muonic data)

Charge radii of Ne isotopes 1 GHz 10 MHz 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

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 Talmi – Zamick formula HF calculation, A. Klein et al, Nucl. Phys. A 607 (1996) 1. K. Blaum et al, NPA 799, 30 (2008)

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 N=28 shell closure well visible

Charge radii of Ar and K isotopes 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 Systematic errors in trends due to F uncertainty not shown K Ar K. Kreim et al, PLB 731, 97 (‘14)