neil rowley, ipn orsay
TRANSCRIPT
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Fusion oscillations in C+C systemsa comparison among 12C+12C, 13C+13C and 12C+13C
Kouichi Hagino, Tohoku UniversityNeil Rowley, IPN Orsay
1. Introduction: Wong formula2. Fusion oscillations
- 12C + 12C (spin zero bosons)- 13C + 13C (spin 1/2 fermions)- 12C + 13C (elastic transfer)
3. Summary
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Poffe, Rowley, Lindsay, NPA410(‘83) 498
two recent publications
alsoMUSIC (E. Rehm)
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IntroductionThe simplest approach to fusion: one-dimensional potential model
In this talk: potential model analyses for C+C fusion (above the barrier)cf. channel coupling effect Rowley’s talk
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C.Y. Wong, Phys. Rev. Lett. 31 (’73)766
i) Approximate the Coul. barrier by a parabola:
ii) l-independent barrier position and curvature:
iii) Replace the sum of l with an integral
Wong’s formula
(note) For
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Wong formula for light heavy-ion fusion
Wong formula:i) Approximate the Coul. barrier by a parabolaii) l-independent barrier position and curvatureiii) Replace the sum of l with an integral
small
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E-dependent Wong formulaN. Rowley, A. Kabir, and R. Lindsay, J. of Phys. G15(‘89)L269N. Rowley and K. Hagino, in preparation
use Vb, Rb, and Ω for the grazing angular momentum, lg
(note)
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grazing anuglar momentum:
for an exponential potential, VN(r) = -V0 e-r/a
or a better approximation:
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Exact:
Approximation-1:
Approximation-2:
a = 0.8 fm
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Continuum approximationWong formula:
the continuum approximation: appears very good but if you take a closer look…..
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Continuum approximationWong formula:
* Exact: a parabolic barrier with RE, VE, ΩE obtained with an exp. pot.
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the origin of the oscillations
1/k2 1/k2
* practically, the oscillations are invisible (∆σ ~ 1 mb in quantal calc.)
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effect of symmetrization: fusion oscillations in light symmetric systemsfusion of identical spin-zero bosons: wf has to be symmetric
the angular mom. is quantized in units of 2-hbara larger amplitude of fusion oscillations
1/k2
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exact:w/o parabola
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N. Poffe, N. Rowley, R. Lindsay, NPA410(‘83) 498
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Analytic formula for fusion oscillationsN. Poffe, N. Rowley, and R. Lindsay, Nucl. Phys. A410 (‘83) 498N. Rowley and K. Hagino, in preparation
Poisson sum rule
(still exact)
Approximation:
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(note)
in order for theosc. to be visible
light symmetric systems
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Comparison with experimental data
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i) 12C + 12C
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i) 12C + 12C12Cg.s. : 0+ the relative w.f. has to be spatially symmetric
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However, remember this:Fit with fixed Vb, Rb, Ω
Vb = 5.6 MeVRb = 6.3 fmΩ = 3.0 MeV
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ii) 13C + 13C13Cg.s. : 1/2- the relative w.f. has to be spatially symmetric for S = 0
spatially anti-symmetric for S = 1
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iii) 12C + 13C
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role of elastic transfer
θ
π−θ
elastic scattering
transfer
indistinguishable
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role of elastic transfer
θ
π−θ
if
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role of elastic transfer
θ
π−θ
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exponential potential with a = 0.9 fm
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exponential potential with a = 0.9 fm
parity-dependent potential
V = (1 + (-1)l ε) V0
W. von Oertzen and H.G. Bohlen, Phys. Rep. 19C(‘75) 1A. Vitturi and C.H. Dasso, Nucl. Phys. A458 (‘86) 157A. Kabir, M.W. Kermode and N. Rowley, Nucl. Phys. A481(‘88) 94
cf. 12C + 16O
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parity-dependent potential
V = (1 + (-1)l ε) V0
ε < 0
a smaller V
a higer barrier for even-l
Baye’s simple rule:D. Baye, J. Deenen, and Y. Salmon, Nucl. Phys. A289(‘77) 511D. Baye, Nucl. Phys. A460 (‘86)581
RGM with two-center HO shell model
(nuclear potential)
for 12C+13C(p1/2):
cf. sign(V+ - V-) = εV0 = - ε
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SummaryFusion oscillations: successive contribution of discrete centrifugal
barariers12C(0+) +12C(0+) 13C(1/2-) + 13C(1/2-)12C + 13C
symmetrization of relative wave function
--- elastic transfer
analytic formula for fusion oscillationsparabolic approximation
Next talk by Rowley: coupled-channels effects
cf. 14C + 14C: R.M. Freeman, C. Beck et al., PRC24 (‘81) 2390