Masses, Two Nucleon Transfer Reactions, and Collective Structural Evolution

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Masses, Two Nucleon Transfer Reactions, and
Collective Structural Evolution

• R. F. Casten
• WNSL, Yale
• Oak Ridge, Feb. 24, 2009

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Two-neutron separation energies
Binding Energies
S2n A BN S2n (Coll.)
Curvature in isotopic chains collecitve
effects deviations from linearity are a few
hundred keV
Normal behavior linear segments with drops
after closed shells
Discontinuities at first order phase transitions
Use any collective model to calculate the
collective contributions to S2n. We will use the
IBA.
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The IBA convenient model that spans the entire
triangle of colllective structures
Sph. Driving
Def. Driving
H e nd - ? Q ? Q
Parameters , c (within Q)
?/e
?/e
Competition 0 to infinity
Span triangle with z and c
c is overall energy scale factor fitted to first
2 state.
Many nuclei already mapped, hence parameters
already known. Inspect BEs.
z
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Which 0 level is collective and which is a
2-quasi-particle state?
Evolution of level energies in rare earth nuclei
But note
McCutchan et al
Do collective model fits, assuming one or the
other 0 state, at 1222 or 1422 keV, is the
collective one. Look at implications for
separation energies.
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4NB
?
H c
( 1 ? ) nd -
Q? Q?
Parameters
Now, lets look at the calculated collective
components of binding energies (S2n(Coll.)
values) with these two sets of previously known
parameters.
McCutchan et al
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!!!
Reminder These two levels differ by 200 keV
These collective binding energies added to A BN
cannot both be consistent with empirical S2n
trends
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Effects are largest for large numbers of valence
nucleons and for well-deformed nuclei. Previous
studies (e.g., Scholten et al., others) were in
regimes where the effects were 5-10 times smaller
and hence did not stand out.
Valence nucleon number
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Conclusions (strong version)
Conclusions (soft version) These results show a
link between masses and structure that is much
more sensitive than heretofore realized. Effect
is strongest in well-deformed nuclei near
mid-shell

• No one should do a collective structure
  calculation without looking at implications for
  masses.
• Anyone who measures a mass should check to see
  if there are possible implications for structure
  (e.g., can the BEs tell us about the structure
  of excited 0 states ?!).

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Two-Nucleon Transfer Reactions to 0 States

• A new interpretation

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The standard interpretation (since ca. 1960s)
of 2-nucleon transfer reactions to excited 0
states in collective nuclei

• Most nuclei Cross sections are small because
  the collective components add coherently for the
  ground state but cancel for the orthogonal
  excited states.
• Phase transition region Spherical and deformed
  states coexist and mix. Hence a reaction such as
  (p,t) on a deformed 156 Gd target populates both
  the quasi-deformed ground and quasi-spherical
  excited states of 154 Gd. Accepted signature of
  phase transitions.

Are these interpretations correct? Use IBA model
to calculate two-nucleon transfer cross sections
(rel. to g.s.)
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IBA well-suited to this embodies wide range of
collective structures and, being based on s and d
bosons, naturally contains an appropriate
transfer operator for L0 -- s-boson

• Parameters for initial, final nuclei known so
  calculations are parameter-free

Look at Hf isotopes as an example Exp all
excited state cross sections are small
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Gd Isotopes Undergo rapid shape transition at
N90. Excited state cross sections are comparable
to g.s.
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So, the model works well and can be used to look
at predictions for 2-nucleon transfer strengths
Expect from usual interpretation
Shape/phase trans. line
Def.
Small
Sph.
Big
Lets see what we get !
105 calculations
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Cross section ratios across triangle
Ratios as function of dR4/2 Monotonically grow
Huh !!??? Cross sections show nothing special
at shape transition. Rather, scale with
change of structure between target and final
nucleus.
dR4/2
dR4/2
dR4/2
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WHY?
H e nd - ? Q ? Q
The QQ term mixes the s and d boson basis states,
spreading the strength. The further apart the
two nuclei are, the greater the difference in the
distributions of s, d amplitudes, hence the
greater the spreading of cross section.
Example U(5) target ground state has (ns, nd)
(N,0). Therefore, only amplitude that
contributes to cross section, is (ns, nd)
(N-1,0).
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Why the earlier interpretation? Look at dR4/2
values.

• Of course, large changes occur in transitional
  regions. Can we find a case that does NOT
  involve a shape transition. Not easy but one
  case exists.

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Conclusions, Implications

• The cross sections are large in the transitional
  region but far larger in other cases. There is
  nothing special about the phase transition
  region.
• Rather, the cross sections depend on the change
  in structure between initial and final nuclei !
  This change can be measured by dR4/2
• For large dR4/2, cross sections may be spread
  over many 0 states

• A single framework now accounts for both the
  (usual) small cross sections (since most adjacent
  nuclei have small dR4/2 values), and for the
  large cross sections in regions of rapid change.
• The cross section distribution is a mixing effect
  but not of collective modes. Rather it is mixing
  at the shell model level (nucleon pairs coupled
  to spin 0) and therefore is general.
• Test in new nuclei by searching for large dR4/2
  values and doing 2-nucleon transfer in inverse
  kinematics.

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Collaborators

• Sensitivity of binding energies to structure
• Burcu Cakirli (Istanbul), Ryan Winkler (Yale),
  Klaus Blaum (Heidelberg), Magda Kowalska
  (CERN-ISOLDE)
• Two nucleon transfer cross sections
• Rod Clark (LBNL), Linus Bettermann (Koeln),
  Ryan Winkler (Yale)

This work could not have been done without the
prior mapping of nuclei into the triangle by
McCutchan et al, Phys. Rev C 69, 064306 (2004)
and subsequent mapping papers
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This unexpected result leads to a new
interpretation of these cross sections

• The cross sections are large in the transitional
  region but they are far larger in other cases.
  There is nothing special about the phase
  transition region.
• Rather, the cross sections depend only on the
  change in structure between initial and final
  nuclei ! This change can be measured by dR4/2
• For large dR4/2, cross sections may be spread
  over many 0 states

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The IBA a flexible, parameter-efficient
phenomenological collective model

• Enormous truncation of the Shell Model valence
  nucleons in
• pairs coupled to L 0 (s bosons) and L 2
  (d bosons), simple interactions
• Three dynamical symmetries, intermediate
  structures
• Two parameters (except scale)
• Symmetry triangle

Shape/phase trans.
Def.
Sph.
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Use the IBA to calculate the collective component
of the binding energy

• The same interactions in the IBA that give
  excitation spectra and E2 transition rates also
  give the collective component of the binding
  energy that is, those interactions depress the
  ground state due to s-d mixing which gives added
  binding compared to the vibrator U(5) limit.

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Empirical survey of (p,t) reaction strengths to 0
states
Nearly always cross sections to excited 0
states are a small percentage of the ground state
cross section. In the spherical deformed
transition region at N 90, excited state cross
sections are comparable to those of the ground
state.
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