1. 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.

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

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

4. 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)

5. 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

6. 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

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

8. 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

9. 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, (‘01)
[Ber03] J.C. Berengut, V.A. Dzuba, V.V. Flambaum, PRA 68, (‘03)
[Sah10] B.K. Sahoo, J. Phys. B (’10)

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

11. 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

12. Other elements with 2 stable isotopes

13. 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

14. 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

15. 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, (2012)

16. 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]

17. F&H 2004

18. 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

19. 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

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

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

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

23. 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
-127
s->p3/2
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, (‘01)
[Ber03] J.C. Berengut, V.A. Dzuba, V.V. Flambaum, PRA 68, (‘03)
[Sah12] B.K. Sahoo, J. Phys. B (’10)
Large differences or uncertainties -> let’s try a King plot (need Muonic data)

24. 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

25. 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)

26. 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

27. 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)