Transcript Trento 2013

Single-particle strengths, spectroscopic
factors and effective single particle energies
from experiment
Alexandra Gade
Professor of Physics
NSCL and Michigan State University
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
– Experimental considerations and limitations
• Spectroscopic strength from Be or C-induced knockout reactions
– Model dependences
– A consistent approach… but absolute?
– Experimental considerations and limitations
• Perspective
A. Gade, 7/17/2015, Slide 2
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
– One of the foundations of the nuclear shell model
– 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
A. Gade, 7/17/2015, Slide 3
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 1/2states 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 97 092501 (2006)
A. Gade, 7/17/2015, Slide 4
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, 099201 (2007)
A. Gade, 7/17/2015, Slide 5
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, 037304 (2008)
A. Gade, 7/17/2015, Slide 6
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 “singleparticle 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)
P.D. Cottle, PRC 76, 027301 (2007)
A. Gade, 7/17/2015, Slide 7
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, 034322 (2006)
and A. Gade, Nuclear Physics News, in press
A. Gade, 7/17/2015, Slide 8
Experimental considerations
Particle vs. γ-ray spectroscopy
• 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(46Ar,47Ar)p at GANIL – proton spectra
43Cl
D.C. Radford et al., EPJ
A15, 171 (2002)
9Be(134Te,135Te+γ)8Be2α
at HRIBF@ORNL
• 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
A. Gade, 7/17/2015, Slide 9
Spectroscopic factors from transfer reactions
SJL: Spectroscopic factor – scale factor
between exp. and theory (must depend on
the reaction theory used)
Differential cross section
measured in the experiment
Observable: Excitation
energy and cross section
Natascha’s talk later today
FJL: Reaction theory – very model
dependent, DWBA, Adiabatic
Model, CCBA, CRC, … many
assumptions, approximations,
different optical model potentials
Less model dependent: Shape of the angular
distribution  ℓ-value of the transferred nucleon
J. P. Schiffer et al., PRL 92, 162501 (2004)
A. Gade, 7/17/2015, Slide 10
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)
A. Gade, 7/17/2015, Slide 11
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)
Three years later, back to
standardr =1.25fm
parameters ...
CH89 global potential, a=0.65,
0
M. B. Tsang, Jenny Lee et
JLM folding potential, densities from SkX HF a=0.65, r0
al., PRL 102, 062501 (2009)
adjusted to fit HF orbital radius
Jenny Lee, J. A. Tostevin et al., PRC 73, 044608 (2006)
A. Gade, 7/17/2015, Slide 12
Spectroscopic factors – self consistent
Example: Test of Sum Rules in Nucleon Transfer Reactions
J. P. Schiffer et al., PRL 76, 022501 (2012)
A. Gade, 7/17/2015, Slide 13
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, 022501 (2012)
A. Gade, 7/17/2015, Slide 14
Application: Spectroscopic Factors
Example: Nuclear Structure Relevant to Neutrinoless Double β-Decay: 76Ge
and 76Se
J. P. Schiffer, PRL 100, 112501 (2008)
A. Gade, 7/17/2015, Slide 15
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)
A. Gade, 7/17/2015, Slide 16
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, 044306 (2008)
A. Gade, 7/17/2015, Slide 17
Reduction close to stability
Shell M
A. Gade, 7/17/2015, Slide 18
Weakly-bound systems
Shell M
A. Gade, 7/17/2015, Slide 19
Strongly-bound systems
Shell M
A. Gade, 7/17/2015, Slide 20
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, 044608
(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, 044314 (2007), Phys. Rev. Lett. 97, 162503 (2006).
A. Gade, 7/17/2015, Slide 21
Consistent calculation of the singleparticle 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
A. Gade, 7/17/2015, Slide 22
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
A. Gade, 7/17/2015, Slide 23
Single-particle cross sections vs. rsp
24Si: one-n and one-p removal
d5/2 proton removal
d5/2 neutron removal
A. Gade, 7/17/2015, Slide 24
Single-particle stripping cross
section/ANC2 vs. rsp
A. Gade, 7/17/2015, Slide 25
Different Skyrmes
A. Gade, 7/17/2015, Slide 26
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, 051301(R) (2005)
L. A. Riley et al., PRC 78, 011303(R) (2008)
A. Gade, 7/17/2015, Slide 27
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
A. Gade, 7/17/2015, Slide 28
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
A. Gade, 7/17/2015, Slide 29
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)
A. Gade, 7/17/2015, Slide 30
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
A. Gade, 7/17/2015, Slide 31
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
1
2
 sp ( j , Bn ) 
2  b d b  jm (1  S N ) SC

2 j 1 m
2
jm 
A. Gade, 7/17/2015, Slide 32
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 …
A. Gade, 7/17/2015, Slide 33