DREB12, Pisa
March 26, 2012
LLNL-PRES-539552
This work was performed under the auspices of the U.S. Department
of Energy by Lawrence Livermore National Laboratory under contract
DE-AC52-07NA27344. Lawrence Livermore National Security, LLC
 Do transfer or knockout experiments measure
surface properties (ANC, reduced width), or
volume properties (norm of overlap function) ?
 Theory: only ‘asymptotic properties’ are observable:
invariant under off-shell (interior) unitary transformations.
 Reply: We have relied on local potentials for interior forms
 Conclusion: We must:
• pay attention to invariance if we derive effective potentials
(which may be local or non-local)
• separate the contributions from interior and exterior
• see if/how these contributions depend on higher-order couplings.
Lawrence Livermore National Laboratory
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LLNL-PRES-539552
Not clear what do we measure when we compare
(a) experimental magnitude to theory magnitude?
(b) experimental width to theory width?
Need a new general theory for resonant transfers!
• Preferably one easy to calculate!
• At present, to get convergence at large radii:
we use bins, or complex contour, or damping
• Should calculate actual shape of resonance peak
— Include wide / overlapping / multichannel resonances
— Ideally should fit using R-matrix resonance parameters
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LLNL-PRES-539552
 Look at dependence of transfer rate on rnA
= radius of neutron wave function fn(rnA) being probed
• Remember that fn(rnA) for rnA > rs (surface radius)
depends on the reduced width: g2 or the ANC: C
 Look at how post and prior transfers depend on maximum
value of rnA (cut wfn to zero outside).
 Later, try to express as much of the transfer as possible in
terms of the g2.
 This will help calculation of transfers to resonances
• Needed e.g. for Trojan Horse methods, and many expts.
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LLNL-PRES-539552
 Consider a deuteron d=n+p incident on target A,
and the A(d,p)B reaction, with B=A+n.
 Binding potentials Vnp for fd(r), VnA for fn (rnA)
• Entrance & exit optical potentials UdA(R), UpB(R)
• Also need ‘core-core’ potential UpA
Look at DWBA as first approximation:
 Tpost = <fp(-) fn | Vnp + UpA - UpB(R) | fd fd(+)>
(has ZR limit)
 As long-ranged in rnA as fn, as Vnp acts at all distances from target
 Tprior = <fp(-) fn | VnA + UpA - UdA(R) | fd fd(+)>
 Short-ranged in rnA than fn, as VnA , UpA , UdA all cut off away from target
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LLNL-PRES-539552
90
90
91
Zr(d,p) Zr
+
Ed = 11 MeV; stripping to 2f 7/2 resonance state
Ed = 11 MeV; stripping to 5/2 ground state
1.2
Normalized peak cross section
Normalized peak cross section
1.2
1
0.8
0.6
0.4
post
prior
0.2
0
0
2
4
6
8
10
12
Rmax [fm]
14
91
Zr(d,p) Zr
16
18
20
1
0.8
0.6
0.4
post
prior
0.2
0
0
2
4
6
8
10
12
Rmax [fm]
14
16
18
20
Thu Feb 9 13:15:28 2012
Wed Feb 8 13:54:08 2012
Bound state
Resonance bin at 1 MeV
Peak cross sections, calculated in the post and prior formalisms, are shown as a function of the cutoff radius,
(beyond which contributions from the neutron wave function are set to zero)
The cross sections are normalized relative to the peak cross sections obtained in the full calculation.
See that Post contributions are from large neutron radii.
Convergence to resonances is slow (especially for post form)
Very small post contributions from the interior
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LLNL-PRES-539552
 Define Tpost(a,b) & Tprior(a,b) with a < rnA< b limits
Mukhamedzhanov (PRC 84, 044616, 2011) showed recently:
T = Tpost(0,a) + Tsurf(a) + Tprior(a,∞)
where Tsurf(a) = <fp(-) fn |
| fd fd(+)>(in)
 Evaluate:
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LLNL-PRES-539552
Prove post-prior equivalence in DWBA:
 If a=0, then, since Tsurf(0) = 0, find T = Tprior(0,∞)
 If a=∞, then, since Tsurf(∞) = 0, find T = Tpost(0,∞)
Dependence on reduced width g2 of neutron wf:
 If a is outside radius of the potential, then
Tsurf(a) + Tprior(a,∞) depend on wfn only by g2
 Only dependence on interior is by
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(small) Tpost(0,a)
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LLNL-PRES-539552
90
91
Zr(d,p) Zr+
Ed = 11 MeV; stripping to 5/2 ground state
Normalized peak cross section
1
0.5
post
prior
surface
0
0
5
10
Rmax [fm]
15
Mon Mar 12 14:51:30 2012
20
Resonance
Bound state
Tsurf(a) = Tprior(0,a) - Tpost(0,a)
Now we see the surface term peaked at the surface (as expected).
But it does not produce all the cross section peak, or all the integral
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LLNL-PRES-539552
 The potentials in the prior matrix element
Vn + UpA - Ud(R)
are very similar to the
UnA + UpA - Ud(R)
used in CDCC breakup calculations.
Difference is that Vn = binding potl and UnA = optical potl.
 If we can ignore this difference, and calculate ΨCDCC,
then the ‘exterior prior’ term disappears:
T = TCDCCpost(0,a) + TCDCCsurf(a)
 For now:
regard the ‘exterior prior’ as indicator of breakup.
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LLNL-PRES-539552
Interior wfn
contribution
inside x-axis radius
Breakup
outside
radius
on x-axis
Surface
term at
x-axis radius
Plotting sqrt(cross-section) – to estimate amplitudes
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LLNL-PRES-539552
TCDCCsurf(a) = T - TCDCCpost(0,a)
≈ Tpost(a,∞)
Ratio of angle-integrated POST cross sections for
91
-- use this to estimate:
90
91
Zr(d,p) Zr
+
Ed = 11 MeV; stripping to 5/2 ground state
Zr(f7/2) resonance
1.5
Tpost(a,inf) = T
CDCC
surf
(a)
Normalized peak cross section
Tpost(0,a) = interior wf contribution
1
Proposed
R-matrix
radius a
0.5
0
1
0.8
0.6
0.4
interior wf term
Surface term
0.2
-0.5
Shape of neutron binding potential
-1
0
2
4
6
8
Radius a
10
12
14
0
2
4
6
8
Radius a
10
12
14
: Mar 20 10:53:27 2012
Black curve ratio of post cross sections spost(a,∞)/s = |Tpost(a,∞)/T|2Tue
Ratio of cross section peaks
Try to choose radius around 7.5 – 8 fm
outside potential, where the CDCC-surface
contribution is complete
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Larger interior contribution
for this bound-state transfer
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 See development of a model that separates
1. Interior contributions from shape of wave function
2. Breakup contributions from exterior tails
3. Dominant ‘surface contribution’ from exterior tails.
 ‘Surface Approximation’: if neglect other terms
 Good prospects for
• a new model of transfer reactions to resonances, that
• uses small-radius calculations (convergent!),
• to map R-matrix parameters onto resonance shapes.
 We are now developing the CDCC approach
 In future: fit neutron R-matrix parameters from expt.
Lawrence Livermore National Laboratory
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Define ‘spectroscopic factor’ S
= ratio of observed reduced width
to that of single-particle state
Maybe something for us to learn here?
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LLNL-PRES-539552
DOE Topical Collaboration
Charlotte Elster (OU)
Aim: develop new
methods that will
advance nuclear
reaction theory for
unstable isotopes by
using three-body
techniques to improve
direct-reaction
calculations
Goran Arbanas (ORNL)
Year 2 out of 5.
Ian Thompson, LLNL
Jutta Escher, LLNL
Filomena Nunes, MSU
Neelam Upadhyay (PD)
Akram Mukhamedzhanov
(TAMU)
V. Eremenko (PD)
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