Hindrance of heavy-ion fusion due to nuclear incompressibility

H. Esbensen; S. Misicu

REVIEW

Hindrance of heavy-ion fusion due to nuclear incompressibility

S. Misicu, H. Esbensen

Nuclear Theory · Submitted 22 Feb 2006 · 35 references

Pith's verdict: A physically motivated but not yet nailed-down mechanism for fusion hindrance; the fit is good, but the sudden-density assumption is the load-bearing and least stable piece.

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arXiv IDnucl-th/0602064 (version 1)

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CitationH. Esbensen, S. Misicu. "Hindrance of heavy-ion fusion due to nuclear incompressibility." arXiv:nucl-th/0602064 (2006). https://pith.science/paper/T6CMQZTD

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Editor's note

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Dear [colleague],

This paper proposes that the steep falloff in deep sub-barrier fusion comes from nuclear saturation: when the two nuclei overlap strongly, the local density would exceed saturation, costing energy, and that shows up as a repulsive core in the ion-ion potential. The specific transplant of that idea into a double-folding M3Y potential with a zero-range repulsive core is new in this context, and it does real work: for 64Ni+64Ni the coupled-channels calculation reproduces the excitation function with chi^2/N=0.86 and a tiny energy shift, and the S-factor maximum comes out in the right place. That is a genuine success, and the paper is honest about the limitations of the procedure.

The soft spot is the step from nuclear matter incompressibility to the ion-ion potential. Equation (2) calibrates the repulsive core by assuming that at R=0 the densities are frozen and simply add, so the local density doubles and the energy cost per nucleon is K/9. That is a sudden potential. But the fusion events in question are deep sub-barrier tunneling events, where the system has time to rearrange. An adiabatic potential would let the combined system relax toward normal density, which would make the pocket deeper and the barrier thinner. The shallow pocket with V_min around 85 MeV, and the resulting cutoff of the cross section at that energy, is a property of the sudden potential. The paper itself concedes that frozen densities are not reliable at strong overlap, which undercuts the calibration. So the mechanism is plausible, but the load-bearing identification is the weakest link.

The parameter situation is also softer than the abstract suggests. The strength V_core is set to reproduce K=228 MeV from a Thomas-Fermi calculation, but the shape of the pocket depends on the chosen diffuseness a=0.4 fm and on the neutron density parameters, which are adjusted to keep the barrier close to the Winther-Akyuz potential. One good fit to a single system with a few tuned parameters does not yet separate the incompressibility mechanism from a generic shallow-pocket fix.

For what it is worth, the reader's concern about Eq. (2) being incorrectly normalized does not hold up on inspection: K=9 rho0^2 d^2(e/rho)/drho^2 is the standard definition, so rho0^2 d^2 e/drho^2 = K/9 exactly. The sudden-versus-adiabatic issue is the real question.

Who gets value: anyone working on sub-barrier fusion or the astrophysical S-factor, because this is a concrete, testable proposal. It deserves a serious referee. The referee should push the authors to apply the potential to more of the systems in Ref. [5] and to address the adiabatic relaxation objection, perhaps with a density-constrained calculation. I would not desk reject it.

Recommendation: send to peer review.

Referee report

Significance

If correct, the mechanism would provide a physical explanation for the fusion hindrance phenomenon and for the S-factor maximum, with consequences for heavy-element nucleosynthesis. The paper's concrete achievement is a potential that fits the 64Ni+64Ni data well and correctly locates the S-factor maximum; the comparison with the Winther-Akyuz potential is instructive. The strengths are the clear statement of the idea, the good quality of the coupled-channels fit, and the explicit attempt to connect the repulsive core to a nuclear-matter property. The main weakness is that the central step from incompressibility to the ion-ion potential rests on a sudden, frozen-density identification that is not justified for slow sub-barrier fusion, and the quantitative evidence for the mechanism is still limited to one system with several adjusted ingredients.

Major objections (4)

[Eq. (2)] The relation Delta V/Ap = rho0^2 B''(rho0) = K/9 is not a direct consequence of the quadratic expansion of the energy per nucleon. For B the binding energy per nucleon, the energy increase at rho=2rho0 is (1/2) B'' rho0^2 = K/18; the K/9 result holds only if Delta V is interpreted as the total energy increase of all 2A overlapping nucleons divided by Ap. Because Vcore is calibrated through this relation, the manuscript should state this interpretation explicitly and derive Vcore from V(0) = Ap K/9; as written, the factor-of-two ambiguity makes the calibration unreproducible.
[Paragraph following Fig. 1] The repulsive core is calibrated at R=0 under the assumption that the two nuclei retain their ground-state densities and that the local density reaches 2rho0. The paper itself notes that the frozen-density assumption is 'less questionable' only in the tail region. Deep sub-barrier fusion proceeds slowly, and the dinuclear system has time to relax to the compound-nucleus configuration at normal saturation density. The relevant tunneling potential is therefore more nearly adiabatic, and the shallow pocket with Vmin approximate to 85 MeV, which is responsible for the hindrance, may be an artifact of the sudden potential. Please provide an argument based on the collision timescale, or repeat the coupled-channels calculation with an adiabatic (e.g., density-constrained) potential, to show that the hindrance is not an artifact of the sudden approximation.
[Potential parameters and fit] The potential contains several adjusted ingredients: Vcore (through K=228), the repulsive-core diffuseness a=0.4 fm, and the neutron-density parameters r0n and an, which are varied to match the Winther-Akyuz barrier. The success for 64Ni+64Ni (chi^2/N=0.86) is therefore not by itself evidence for the incompressibility mechanism. The sensitivity of the hindrance to a and to the adopted K value (e.g., K=180-260 MeV) should be quantified, and the same parameters should be applied, without refitting, to other systems with observed hindrance such as 58Ni+58Ni, 64Ni+100Mo, and 90Zr+90Zr. The single sentence on 58Ni+58Ni is insufficient for a general claim.
[Boundary conditions in the coupled-channels calculation] The calculation imposes ingoing-wave boundary conditions at the potential minimum and thus forces the cross section to vanish for E <= Vmin approximate to 85 MeV. The authors concede that 'one would expect a non-zero cross section even below Vmin.' For a shallow pocket, one should check whether the ingoing-wave approximation is valid, or whether reflection from the inner barrier (quasi-molecular resonances) changes the sub-barrier excitation function. At minimum, the sensitivity of the S-factor maximum to this boundary-condition choice should be discussed.

Minor objections (4)

[Eq. (2)] Please define B(rho) explicitly as the binding energy per nucleon and clarify the meaning of Delta V/Ap; the current wording conflates the per-nucleon energy increase with the total interaction energy divided by the projectile nucleon number.
[Fig. 3] The y-axis label is garbled in the reproduction; it should read sigma E exp[2 pi (eta - eta0)] (or the corresponding definition in the text).
[58Ni+58Ni discussion] The paper states that a similar study confirms the trend for 58Ni+58Ni but gives no numbers or figure. Please show the result quantitatively, since it is the only other system mentioned in support of the general mechanism.
[Related work] Reference [8] (Dasso and Pollarolo) is very close in spirit; the manuscript would benefit from a more explicit discussion of what the incompressibility mechanism adds beyond the earlier shallow-pocket hypothesis.

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The paper

The paper claims

Nuclear incompressibility adds a repulsive core to heavy-ion potentials, explaining the steep sub-barrier falloff of fusion cross sections.

The reading

This paper proposes that the unexpected, steep falloff of fusion cross sections at energies far below the Coulomb barrier is caused by nuclear incompressibility. When the colliding nuclei overlap strongly, saturated nuclear matter resists further compression, and that resistance shows up in the ion-ion interaction as a short-range repulsive core. The authors add such a core to a double-folding M3Y potential and calibrate its strength to the per-nucleon compression energy $K/9$ at full overlap. Coupled-channels calculations with the resulting M3Y+repulsion potential reproduce the $^{64}$Ni+$^{64}$Ni excitation function and the observed maximum in the astrophysical S-factor, which the standard empirical potential cannot. If correct, the mechanism locates a lower threshold for fusion and has consequences for heavy-element synthesis in stars.

What carries the argument

The central object is the M3Y+repulsion double-folding potential: the standard M3Y-Reid effective nucleon-nucleon interaction folded over the deformed proton and neutron densities, plus a short-range repulsive core modeled as a zero-range interaction of strength $V_{\rm core}=496$ MeV folded with densities of diffuseness $a\approx 0.4$ fm. The core is calibrated by the identity $\Delta V/A_p=K/9$, the per-nucleon compression-energy increase at full overlap, with incompressibility $K=228$ MeV from the Thomas-Fermi model for the compound nucleus. This added repulsion is what raises the inner minimum and thickens the barrier, producing both the disappearance of the classically allowed region below a threshold energy and the S-factor maximum.

Extended reading notes

Core claim

The central claim is that the saturation properties of nuclear matter are not just a bulk equation-of-state detail but directly determine the shape of the heavy-ion potential at small separations. At total overlap ($R=0$) the interaction energy per nucleon rises by $K/9$, where $K$ is the nuclear incompressibility of the fused compound system. The paper converts this compression energy into a zero-range repulsive nucleon-nucleon interaction, folded with densities of small diffuseness, and fixes its strength ($V_{\rm core}=496$ MeV) so the repulsion reproduces the $K/9$ shift for the compound nucleus $^{128}$Ba using $K=228$ MeV from a Thomas-Fermi model. The resulting M3Y+repulsion potential has a shallower inner pocket (minimum near 85 MeV) and a thicker classically forbidden barrier than the standard empirical potential. With the same coupled-channels treatment, this potential reproduces the $^{64}$Ni+$^{64}$Ni fusion excitation function ($\chi^2/N\approx 0.86$ versus $\approx 10.1$) and the S-factor maximum.

Load-bearing premise

The load-bearing premise is that the resistance of nuclear matter to compression at full overlap can be represented by a static, zero-range repulsive force added to the double-folding potential, with the incompressibility taken to be $K=228$ MeV and the repulsive densities given a 0.4 fm diffuseness.

Editorial extensions

If this is right

The fusion excitation function for $^{64}$Ni+$^{64}$Ni should vanish as the center-of-mass energy approaches the pocket minimum ($V_{\rm min}\approx 85$ MeV), rather than continuing as an exponential tail.
The astrophysical S-factor should exhibit a maximum at deep sub-barrier energies for medium-heavy systems, with the maximum position set by the pocket depth and barrier thickness.
Standard empirical potentials such as the Winther-Akyüz potential fail below the barrier because they lack the repulsive inner core; potentials with only a deeper pocket would predict quasimolecular resonances that are not observed.
The nuclear incompressibility of the compound system enters the fusion cross section, so the hindrance should be system-dependent through $K$ and also through the neutron-proton asymmetry of the colliding nuclei.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the core strength is set by $K$, fusion hindrance should be stronger for compound nuclei with larger incompressibility; systematic measurements across isotopic chains could map the nuclear equation of state.
The model's pocket shape could be probed independently by quasi-elastic scattering at deep sub-barrier energies, which is sensitive to the inner potential without relying on the absorption assumptions of fusion calculations.
Below $V_{\rm min}$ the calculation imposes ingoing-wave boundary conditions and predicts zero cross section, but physically the compound nucleus can form down to much lower energies; a residual non-tunneling contribution may appear and would test whether the sharp cutoff is real.

Original abstract

We propose a new mechanism to explain the unexpected steep falloff of fusion cross sections at energies far below the Coulomb barrier. The saturation properties of nuclear matter are causing a hindrance to large overlap of the reacting nuclei and consequently a sensitive change of the nuclear potential inside the barrier. We report in this letter a good agreement with the data of coupled-channels calculation for the {64}Ni+{64}Ni combination using the double-folding potential with M3Y-Reid effective N-N forces supplemented with a repulsive core that reproduces the nuclear incompressibility for total overlap.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.82 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We propose a new mechanism to explain the unexpected steep falloff of fusion cross sections at energies far below the Coulomb barrier. The saturation properties of nuclear matter are causing a hindrance to large overlap of the reacting nuclei and consequently a sensitive change of the nuclear potential inside the barrier. We report in this letter a good agreement with the data of coupled-channels calculation for the {64}Ni+{64}Ni combination using the double-folding potential with M3Y-Reid effective N-N forces supplemented with a repulsive core that reproduces the nuclear incompressibility for total overlap.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose a new mechanism to explain the unexpected steep falloff of fusion cross sections at energies far below the Coulomb barrier.	contribution 0.98	0.70	0.21	asserted 0.51
s1	The saturation properties of nuclear matter are causing a hindrance to large overlap of the reacting nuclei and consequently a sensitive change of the nuclear potential inside the barrier.	interpretation 0.75	0.20	0.56	asserted 1.00
s2	We report in this letter a good agreement with the data of coupled-channels calculation for the {64}Ni+{64}Ni combination using the double-folding potential with M3Y-Reid effective N-N forces supplemented with a repulsive core that reproduces the nuclear incompressibility for total overlap.	result 0.78	0.86	0.34	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.85 · evidence from simulation or computation 0.93.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
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T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 2.** FIG. 2: Experimental fusion excitation function for the sys

Showing 2 of the paper's 3 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:nucl-th/0602064 by the authors.

References

Works cited (35)

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C. L. Jiang et al. , Phys. Rev. Lett. 89, 052701 (2002)

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C. L. Jiang, H. Esbensen, B.B. Back, R.V.F. Janssens and K.E. Rehm, Phys. Rev. C 69, 014604 (2004)

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[3]

(dotted line) and with the M3Y+repulsion potential (solid line), in both cases using the same recipe for the c.c. calculations. The cross section for no couplings is 85 90 95 100 105 110 Center-of-Mass Energy E (MeV) 10 -5 10-4 10-3 10 -2 10 -1 1 10 10 2 103 f (mb) Experiment NOC CC (Winther-Akyuz) CC (M3Y+repulsion) 64 Ni+ 64 Ni FIG. 2: Experimental fusi...

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S ¸. Mi¸ sicu and W.Greiner, Phys. Rev. C 66, 044606 (2002)

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [5]

C. L. Jiang et al. , Phys. Rev. C 71, 044613 (2005)

View on Pith 2005

[2] [1]

C. L. Jiang et al. , Phys. Rev. Lett. 89, 052701 (2002)

View on Pith 2002

[3] [2]

C. L. Jiang, H. Esbensen, B.B. Back, R.V.F. Janssens and K.E. Rehm, Phys. Rev. C 69, 014604 (2004)

View on Pith 2004

[4] [3]

(dotted line) and with the M3Y+repulsion potential (solid line), in both cases using the same recipe for the c.c. calculations. The cross section for no couplings is 85 90 95 100 105 110 Center-of-Mass Energy E (MeV) 10 -5 10-4 10-3 10 -2 10 -1 1 10 10 2 103 f (mb) Experiment NOC CC (Winther-Akyuz) CC (M3Y+repulsion) 64 Ni+ 64 Ni FIG. 2: Experimental fusi...

View on Pith

[5] [4]

C. L. Jiang et al. , Phys. Rev. Lett. 93, 012701 (2004)

View on Pith 2004

[6] [6]

C. L. Jiang et al. , Phys. Rev. C 73, 014613 (2006)

View on Pith 2006

[7] [7]

K. Hagino, N. Rowley, and M. Dasgupta, Phys. Rev. C 67, 054603 (2003)

View on Pith 2003

[8] [8]

J. O. Newton, R.D. Butt, M. Dasgupta, D.J. Hinde, I.I. Gontchar, C.R. Morton, and K. Hagino,Phys. Rev. C 70, 024605 (2004)

View on Pith 2004

[9] [9]

C. H. Dasso and G. Pollarolo, Phys. Rev. C 68, 054604 (2003)

View on Pith 2003

[10] [10]

R. A. Broglia and A. Winther, Heavy Ion Reactions, Lecture Notes, Volume I: The Elementary Processes , (Addison-Wesley Pub. Co., CA, 1991), p. 114

View on Pith 1991

[11] [11]

E. M. Burbidge, G. R. Burbidge, W. A. Fowler and F. Hoyle, Rev. Mod. Phys. 29, 547 (1957)

View on Pith 1957

[12] [12]

I. I. Gontchar, D.J. Hinde, M. Dasgupta, and J.O. New- ton, Phys. Rev. C 69, 024610 (2004)

View on Pith 2004

[13] [13]

D.A. Bromley, J.A.Kuehner and E.Almqvist, Phys. Rev. Lett. 4, 365 (1960)

View on Pith 1960

[14] [14]

R. Zurm¨ uhleet al. , Phys. Lett. B129, 384 (1984)

View on Pith 1984

[15] [15]

R. R. Betts, B. B. Back, and B. G. Glagola, Phys. Rev. Lett. 47, 23 (1981)

View on Pith 1981

[16] [16]

H. J. Rose and G.A.Jones, Nature 307, 245 (1984)

View on Pith 1984

[17] [17]

A. Sandulescu, S ¸. Mi¸ sicu, F. Carstoiu and W. Greiner, Phys. Part. Nucl. 30, 386 (1999)

View on Pith 1999

[18] [18]

R. J. Munn, B. Block and F. B. Malik, Phys. Rev. Lett. 21, 159 (1968)

View on Pith 1968

[19] [19]

G. Michaud, Phys. Rev. C 8, 525 (1973)

View on Pith 1973

[20] [20]

B. Block and F.B. Malik, Phys. Rev. Lett. 19, 239 (1967)

View on Pith 1967

[21] [21]

W. Scheid, R. Ligensa and W. Greiner, Phys. Rev. Lett. 21, 1479 (1968)

View on Pith 1968

[22] [22]

J. Blocki, J. Randrup, W. J. Swiatecki and C. F. Tsang, Ann. Phys. N.Y. 105, 427 (1977)

View on Pith 1977

[23] [23]

W. D. Myers and W. J. Swiatecki, Phys. Rev. C 62, 044610 (2000)

View on Pith 2000

[24] [24]

G. R. Satchler and W. G. Love, Phys. Rep. 55 (1979) 183

View on Pith 1979

[25] [25]

M. E. Brandan and G. R. Satchler, Phys. Rep. 285 (1997) 143

View on Pith 1997

[26] [26]

S ¸. Mi¸ sicu and W.Greiner, Phys. Rev. C 66, 044606 (2002)

View on Pith 2002

[27] [27]

G. Bertsch, W. Borysowicz, H. McManus and W. G. Love, Nucl. Phys. A284, 399 (1977)

View on Pith 1977

[28] [28]

E. Uegaki and Y. Abe, Progr. Theor. Phys. 90, 615 (1993)

View on Pith 1993

[29] [29]

J. Eisenberg and W. Greiner, Nuclear Theory, vol.I, Phenomenological Models , (North-Holland, Amsterdam 1988)

View on Pith 1988

[30] [30]

S. Shlomo, Pramana Journal of Physics, 57, 557 (2001)

View on Pith 2001

[31] [31]

W. D. Myers and W. J. Swiatecki, Phys. Rev. C 57, 3020 (1998)

View on Pith 1998

[32] [32]

H. De Vries , C.W.Jaeger and C. De Vries, At. Data Nucl. Data Tables 36, 495 (1987)

View on Pith 1987

[33] [33]

H. Esbensen, Prog. Theor. Phys. Suppl. No. 154, 11 (2004)

View on Pith 2004

[34] [34]

J. R. Patterson, B. N. Nagorcka, G. D. Symons and W. M. Zuk, Nucl. Phys. A165, 545 (1971)

View on Pith 1971

[35] [35]

M. Beckerman, et al., Phys. Rev. C 25, 837 (1982)

View on Pith 1982