1. Alexandra Gade Professor of Physics NSCL and Michigan State University
Single-particle strengths, spectroscopic factors and effective single particle energies from experiment
Alexandra Gade
Professor of Physics
NSCL and Michigan State University

2. Outline Single-particle energies deduced from experiment
Simply strength-weighted … and one or two pit falls
Following Baranger
Experimental considerations and limitations
Spectroscopic factors from transfer reactions
Model dependences
Relative vs. absolute
Very elegant: Sum rules … or experimental self-consistency, sort of
Spectroscopic strength from Be or C-induced knockout reactions
A consistent approach… but absolute?
Perspective

3. Motivation Single-particle energies deduced from experiment
One of the foundations of the nuclear shell model
Estimate the size of shell gaps, track shell evolution
Measure: Energies (ground state and excited states, need spectroscopic factors though …)
Spectroscopic factors/strength from experiment
Relates to occupation numbers in the shell model scheme
Measure: Cross sections for the transfer to specific final states (transfer or knockout reactions)
However, both are non-observables in experiments and require varying degrees of theory input for their extraction from cross sections and energies
But, these non-observables turn out to be useful is many cases

4. Single-particle energies Example: Reduction of the Spin-Orbit Splittings at the N=28 Shell Closure
46Ar(d,p)47Ar transfer reaction at SPIRAL, spectroscopic factors deduced in comparison to DWBA with global optical potentials (Daehnick; Wales, Johnson; Varner; Perey, Perey, …)
Spin-orbit splitting for p orbits deduced from the energy difference of the first excited 1/2- states in 47Ar and 49Ca (spin-obit splitting reduced by ΔEso= -890keV in 47Ar relative to 49Ca)
Ex(1/2-)
Ex(1/2-)
~3.88
L.Gaudefroy et al., PRL (2006)

5. BUT … Single-particle energies Example: Comment on ‘‘Reduction of the Spin-Orbit Splittings at the N=28 Shell Closure’’
E0=E(g.s.) of A
Ef-=E(hole) in A-1
Ef+=E(particle) in A+1 D=
=-10(13) keV instead of -890 keV
Essentially no reduction of spin-orbit splitting when fragmentation of spectroscopic strength is taken into account
Problem: For rare isotopes, it is not necessarily possible to perform such complete spectroscopy of all particle and hole fragments
Calculated in the SM with 200 states
A. Signoracci and B. A. Brown, PRL 99, (2007)

6. Single-particle energies Example: Single-neutron energies near N = 28 and the absence of the N = 34 sub-shell closure in the Ti isotopes
P.D. Cottle and K.W. Kemper, PRC 78, (2008)

7. Single-particle energies Example: Single proton energies in the Si isotopes and the Z=14 sub-shell closure
Assess limitations of experimental setup (resolution and efficiency)
Specify the scheme used to extract quantity
Sometimes it is better not to extract certain non-observables if too uncertain
Often it is very valuable to compare excited state energies – testing how well theory describes fragmentation and the onset of collectivity rather than extracting “single-particle energies” from incomplete data
Rare isotope experiments are very hard (low rates) and we typically cannot extract the level of detail one is used to from stable nuclei (next generation facilities will be a step forward)
Example
P.D. Cottle, PRC 76, (2007)

8. Excited-state energies are useful, too
Often it is very valuable to compare excited state energies – testing how well theory describes fragmentation and the onset of configuration mixing rather than extracting “single-particle energies” from incomplete data
Towards the driplines, only few bound states and some of the most sensitive techniques will not be applicable to sample all spectroscopic strength
A. Gade, B. A. Brown et al., PRC 74, (2006) and A. Gade, Nuclear Physics News, in press

9. Experimental considerations Particle vs. γ-ray spectroscopy
d(46Ar,47Ar)p at GANIL – proton spectra
γ-tagging to measure excited states
Very good energy resolution, thick target can be used, high luminosity (lower beam rates sufficient)
Can only access bound states, germanium detectors have low detection efficiency at high energy, isomeric states are a problem, level scheme needs to be known
D.C. Radford et al., EPJ A15, 171 (2002)
43Cl
Particle tagging to measure excited states (classic)
Can access unbound states, high particle detection efficiency, no issues with feeding or isomers
Poor resolution, close-lying states can often not be resolved, comparably high exotic-beam beam rates needed
9Be(134Te,135Te+γ)8Be2α at

10. Spectroscopic factors from transfer reactions
SJL: Spectroscopic factor – scale factor between exp. and theory (must depend on the reaction theory used)
Natascha’s talk later today
FJL: Reaction theory – very model dependent, DWBA, Adiabatic Model, CCBA, CRC, … many assumptions, approximations, different optical model potentials
Differential cross section measured in the experiment
Observable: Excitation energy and cross section
Less model dependent: Shape of the angular distribution  ℓ-value of the transferred nucleon
J. P. Schiffer et al., PRL 92, (2004)

11. Transfer - Model dependence Example: Spectroscopic factors from the 51V(d,3He)50Ti reaction
Absolute spectroscopic factors are questionable due to the great model dependences (my opinion only – a fraction of the community thinks otherwise…)
Relative spectroscopic factors – relative within one nucleus or relative across an isotopic or isotonic chain and analyzed consistently are a very useful concept … although they are non-observables
However, there are traps, ℓ-dependent traps:
Kramer et al. (NIKHEF and KVI) did fantastic work on probing model dependences and benchmarking with (e,e’p) : Kramer et al., NPA 679, 267 (2001) and NPA 477, 55 (1988)
Often ignored or forgotten ... ?!
G. K. Kramer et al., NPA 477, 55 (1988)

12. Model Dependence – spatial extent Reduced neutron spectroscopic factors when using potential geometries constrained by Hartree-Fock calculations
Conventional: Traditional, fixed bound-state geometry a=0.65 fm and r0=1.25fm gives the naïve IPM value
HF: SFs deduced with the bound-state geometry constrained as well as possible with the result of mean field calculations (SkX Skyrme) agree with the magnitude in reduction observed in (e,e,’p)
CH89 global potential, a=0.65, r0=1.25fm
JLM folding potential, densities from SkX HF a=0.65, r adjusted to fit HF orbital radius
Three years later, back to standard parameters M. B. Tsang, Jenny Lee et al., PRL 102, (2009)
Jenny Lee, J. A. Tostevin et al., PRC 73, (2006)

13. Spectroscopic factors – self consistent Example: Test of Sum Rules in Nucleon Transfer Reactions
J. P. Schiffer et al., PRL 76, (2012)

14. Spectroscopic factors – self consistent Example: Test of Sum Rules in Nucleon Transfer Reactions
This analysis shows that careful experiments and the consistent use of reaction theory can yield self-consistent results for spectroscopic factors
Best of course are measurements with the same experimental setup
Often only possible for stable nuclei
Recently employed to benchmark theory (QRPA) in the description of nuclear structure around 76Ge (ββ decay matrix elements)
J. P. Schiffer et al., PRL 76, (2012)

15. Application: Spectroscopic Factors Example: Nuclear Structure Relevant to Neutrinoless Double β-Decay: 76Ge and 76Se
J. P. Schiffer, PRL 100, (2008)

16. Experimental considerations rare-isotope beams
For rare isotopes, all measurements in inverse kinematics
Modest resolution if light-ion-tagged, mandates thin targets and thus requires higher beam rates (104 – 106)
Clever approaches exist, like HELIOS, with improved resolution compared to conventional inverse kinematics transfer, but target thickness ultimately limits resolution there (give or take between resolution and required rate or yield)
Global potential are increasingly questionable for rare isotopes, folding models are preferred by some
K. L. Jones et al., Nature 465, 454 (2010)

17. One-nucleon knockout reactions P. G. Hansen and J. A. Tostevin, Annu
One-nucleon knockout reactions P. G. Hansen and J. A. Tostevin, Annu. Rev. Nucl. Part. Sci. 53, 219 (2003)
A nucleon is removed from a projectile upon collision with a C or Be target
In conjunction with reaction theory, spectroscopic strength can be assessed (eikonal and sudden approximations, folding potentials (HF distributions), bound-state wave function constrained with input from HF calculations)
The longitudinal momentum distribution (shape) is sensitive to the orbital angular momentum of the removed nucleon (like angular distributions in transfer)
Final-state identification with γ-ray spectroscopy  thick targets and high luminosities (measurements can be done at a few particles per second)
A. Gade et al., PRC 77, (2008)

18. Reduction close to stability
Shell M

19. Weakly-bound systems
Shell M

20. Strongly-bound systems
Shell M

21. Consistency with other probes?
For stable nuclei and near stability
Consistent with (e,e’p)
Consistent with transfer Jenny Lee, J.A. Tostevin et al., Reduced neutron spectroscopic factors when using potential geometries constrained by Hartree-Fock calculations, Phys. Rev. C 73, (2006).
In the regime of large asymmetry
Trend (not the magnitude) of increased reduction at larger asymmetry found consistent with conclusions from dispersive optical model analyses of elastic scattering data R. J. Charity et al., Phys. Rev. C 76, (2007), Phys. Rev. Lett. 97, (2006).

22. Consistent calculation of the single-particle cross section
Relative core-neutron wave function calculated in Woods-Saxon potential with a=0.7 fm and r0 adjusted to reproduce the core-nucleon rms separation in the ground state
The depth of the potential is chosen to reproduce the effective binding of the initial state
S-matrix from in double-folding optical limit of Glauber multiple scattering theory (Gaussian form factor for effective NN interaction)
Density distributions taken from HF

23. Input and sensitivity
From SKX Skyrme Hartree Fock B.A. Brown, Phys. Rev. C 58, 220 (1998)
rms radius of knockout residue R(r)
neutron and proton density distributions
root-mean-squared separation of the removed nucleon and the residue in the projectile Rsp
Sensitivity of the single-particle cross section to input parameters:
dσsp/σsp=1.1dRsp+1.2dR(r)
0.1 fm change in Rsp and R(r):
16% uncertainty for the single-particle cross section for the removal of a strongly bound nucleon

24. Single-particle cross sections vs. rsp 24Si: one-n and one-p removal
d5/2 proton removal
d5/2 neutron removal

25. Single-particle stripping cross section/ANC2 vs. rsp

26. Different Skyrmes

27. Experimental considerations rare-isotope beams
Experiments with a few particles per second are possible
Measured cross sections (observables) can be compared to theoretical cross sections (reaction theory x structure theory)
Complicated spectra and feeding relations can complicate determination of partial cross sections (deeply bound nuclei)
Gamma-ray detection efficiency is low at high energy
Cross sections to the ground state are often only upper limits (unobserved feeding)
A. Gade et al., PRC 71, (R) (2005)
L. A. Riley et al., PRC 78, (R) (2008)

28. Summary – Single-particle energies
The extraction of single-particle energies (certainly a non-observable) requires complete spectroscopy of particle and hole strength (Baranger)
Relies on the measurement of spectroscopic factors or strength – if particle and hole states are measured, this needs to be done consistently
This complete spectroscopy is difficult for stable nuclei, depending on the degree of fragmentation, and virtually impossible for very exotic systems (low beam intensities ...)
There is value in confronting measured excitation energies with shell model calculations without the extraction of “single-particle energies”  a more indirect and integral probe of the theory

29. Summary – Spectroscopic factors from transfer
Transfer reactions have been used for decades to deduce information about the single-particle degree of freedom
The extraction of spectroscopic factors is highly model dependent (optical model potential, bound-state geometry, …
Transfer reactions can be analyzed consistently with respect to the important spatial extent and agreement with the reduction observed in (e,e’p) can be achieved
Very elegant is the “self-consistent” extraction of occupancies and vacancies with the Mcfarlane and French sum rule. Indicates if the work was done consistently
Although spectroscopic factors are not observed in the experiment, their extraction is useful if all model-dependencies are in the open

30. Summary – Spectroscopic factors from knockout
Allows to quantify single-hole strength in the shortest lived nuclei (luminosity advantage)
Model independent information (cross sections and longitudinal momentum distributions) can be compared to theory for meaningful conclusions – observables and theoretical interpretation are clearly separate (needs to be analyzed consistently)
pn asymmetry dependent reduction factor Rs has generated much interest, trend but not the magnitude is in agreement with Dispersive Optical Model calculations
Disclaimer:
I have never deduced a single-particle energy from experiment
My research group has not quoted spectroscopic factors extracted from experiment in >7 years
Thanks to my theory collaborators: J.A. Tostevin, E. C. Simpson (reaction theory), B. A. Brown, T. Otsuka and Tokyo Group (structure)

31. Stripping
Stripping mechanism depends only on the absorptive content of the interaction – |S|2 and is highly constrained by the reaction cross section of the fragments and the nuclear sizes. These are calculated reliably using Glauber methods and by making use of Hartree-Fock for input that depends on nuclear sizes.
The description of the nucleon’s interaction with the target is common to all final states and different systems

32. Cross sections
Spectator-core approximation to many-body eikonal theory
(A-1) residue is at most elastically scattered
S matrices as function of impact parameter from Glauber theory (free nn np cross sections with Gaussian range parameters nn= np=0.5 fm. Real-to-imaginary ratios interpolated from tables in L. Ray, PRC 20, 1875 (1979)
nucleon-residue relative wave function calculated as eigenfunction of effective 2-body Hamiltonian containing local potential with the depth adjusted to reproduce the separation energy

33. Choice of Skyrmes
SkX has been shown to reproduce nuclear sizes will in Ca region
Skm* gives better surface diffuseness for charge density (difference in matter incompressibility)
Skxs15(20)(25) represent a reasonable variation in neutron skin thickness in 208Pb
Sly4 widely used …