$\alpha$-$\alpha$ Double Folding Cluster Potential Description of the $^{12}$C+$^{24}$Mg System

I. Boztosun; M. Karakoc

REVIEW

$\alpha$-$\alpha$ Double Folding Cluster Potential Description of the $^{12}$C+$^{24}$Mg System

M. Karakoc, I. Boztosun

Nuclear Theory · Submitted 3 Mar 2006 · 21 references

Pith's verdict: A useful folding-model application whose central claim is undercut by comparing reaction, not fusion, cross sections for the microscopic potentials.

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arXiv IDnucl-th/0603010 (version 1)

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CitationI. Boztosun, M. Karakoc. "$\alpha$-$\alpha$ Double Folding Cluster Potential Description of the $^{12}$C+$^{24}$Mg System." arXiv:nucl-th/0603010 (2006). https://pith.science/paper/DYU4M2J4

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Editor's note

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Dear colleague,

This is a straightforward application of the alpha-alpha double folding cluster (DFC) method to 12C+24Mg, one of the first such analyses for this system. The authors compare four real potentials—NN-DF, alpha-alpha DFC, phenomenological deep and shallow—and show that the shallow potential fails to describe fusion even though it fits elastic scattering best. That contrast is a genuinely useful data point for potential-model systematics.

The paper has clear strengths: the folding calculation is standard, the grid search over the normalization N_R and imaginary depth W_0 is transparent, and the coupled-channels treatment of the 24Mg deformation for the phenomenological potentials is a sensible attempt to subtract inelastic flux. The conclusion that the shallow potential's long-range imaginary part is the culprit is plausible and well-argued.

The soft spot is the central comparison to fusion data. In Section IV the authors state that the optical model gives total reaction, not fusion, and they therefore do CC calculations for the deep and shallow potentials to subtract inelastic channels. But the NN-DF and DFC points in Table III and Figure 2 are the plain optical-model reaction cross sections, plotted as if they were fusion. At E_lab=21 MeV these are 320 and 354 mb against an experimental fusion value of 244 mb. Subtracting even 30 mb for the 2+ channel leaves a 20–40% overprediction. So the claim that the microscopic potentials describe fusion 'reasonably well' is not established. The authors could either run CC with the microscopic potentials or clearly relabel and discuss the difference.

There is also a secondary issue: the alpha-cluster distribution from Eq. (5) goes negative beyond r'≈0.75 fm for 12C, and the paper does not mention this. If it is an artifact of the Fourier deconvolution, the DFC potential loses its 'microscopic' status. That weakens the comparison but doesn't destroy it—it could still be an effective potential.

Overall, this is an honest, standard analysis with one easily fixable flaw. It deserves a serious referee, who should ask for the fusion comparison to be made properly. A revision addressing the reaction-versus-fusion issue and the density artifact could make the central claim credible. As it stands, it is a useful data point but not a demonstration.

Referee report

Significance

If the central claim were established, the paper would provide a useful discrimination between deep and shallow real potentials for 12C+24Mg and would support microscopic folding potentials as practical alternatives to phenomenological deep potentials. The manuscript contains a broad set of angular-distribution and cross-section comparisons with tabulated numerical values, which is convenient for readers. However, two load-bearing issues undermine the main conclusion: the folding-model 'fusion' cross sections are in fact total reaction cross sections, and the deconvolved alpha-cluster density used to build the alpha-alpha DFC potential is not positive definite. The paper's strengths are its comparative framework and the explicit tabulation of results, but these do not compensate for the identified gaps in the central argument.

Major objections (3)

[Section IV, Eq. (7), Table III, Fig. 2] The central comparison of the microscopic potentials with fusion data is not valid as presented. The coupled-channels subtraction sigma_F = sigma_R - sigma_in is explicitly applied only to the phenomenological deep and shallow potentials, while the NN-DF and alpha-alpha DFC entries in Table III and Fig. 2 are the optical-model total reaction cross sections - indeed the Table III heading itself says 'Theoretical reaction and experimental fusion cross-sections.' Because sigma_R includes inelastic and other non-fusion channels, the folding-model points are upper bounds on fusion, not predictions. For example, at E_lab = 21 MeV, Table III lists 320.64 and 354.45 mb for NN-DF and DFC against an experimental value of 243.56 mb; even subtracting the ~27 mb 2+ inelastic cross section reported in Table IV leaves a 20-34% overprediction. The claims in Section III that the deep potentials 'predict the fusion cross-section reasonably well' and that the folding potentials provide a consistent simultaneous description are therefore not supported by the evidence shown.
[Section II, Eq. (5), Table I] The deconvolved alpha-cluster density for 12C is not positive definite. Using the Table I parameters beta=0.3741 fm^-2 for 12C and the 4He density parameter 0.7024 fm^-2, the formulas in the text give mu approximately -1.785 fm^-2 and xi approximately 0.800 fm^-2, so that rho_c(r') = rho0c(1 - 1.785 r'^2) exp(-0.800 r'^2) changes sign at r' about 0.75 fm. The authors use this distribution as a physical cluster density in the folding integral of Eq. (1) without commenting on the sign change. If the deconvolved density is unphysical, the alpha-alpha DFC potential cannot be regarded as a genuine microscopic folding potential, and the fitted normalization N_R = 0.72 may be absorbing an artifact of the deconvolution. The authors should justify the use of a sign-changing density or replace it with a positive-definite cluster distribution.
[Section IV, Fig. 1] The conclusion that a shallow real potential cannot simultaneously describe elastic scattering and fusion is inferred from a single illustrative calculation at E_lab = 20 MeV (the solid line with plus in Fig. 1). No chi-square values or other quantitative measures are given for this short-ranged-imaginary variant, and the test is not repeated at the other energies studied. Because this conclusion is part of the paper's central message, the authors should either present a systematic scan over the measured energies or soften the claim to reflect the limited demonstration.

Minor objections (4)

[Table II] Several numerical entries in Table II are unreadable as typeset (for example '1. 8E + 1. 6' and '0 . 054E - 0. 47'); all potential parameters should be printed clearly and unambiguously.
[Section II, Eq. (5)] The symbols eta, lambda, and beta used in the definitions of xi and mu are not all defined in the text; in particular, lambda (presumably the 4He density parameter) is introduced only implicitly through Table I. Please define each symbol explicitly when it first appears.
[References] Reference [18] is cited together with [19] as a source of experimental fusion cross-section data, but the citation (Iwamoto, Moeller, Nix, Sagawa, Nucl. Phys. A506 (1996) 329) appears to be a theoretical paper rather than an experimental measurement; please verify and correct the reference.
[Section V] The conclusion states that the analysis covers energies from 16.0 to 24.0 MeV in the laboratory system, but Figure 1 and Table III show results only for 20.0 through 24.0 MeV; please clarify which data sets were used for the lower energies or cite the appropriate figures/tables.

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The paper

The paper claims

Only deep real potentials fit both 12C+24Mg scattering and fusion.

The reading

The paper reports a simultaneous optical-model and coupled-channels analysis of elastic scattering and fusion cross sections for 12C+24Mg from 16 to 24 MeV laboratory energy. It constructs a microscopic alpha-alpha double folding cluster potential by folding an effective alpha-alpha interaction with alpha-cluster densities of both nuclei, and compares it with a nucleon-nucleon double folding potential and two phenomenological potentials. The central claim is that only deep real potentials—phenomenological deep Woods-Saxon squared, nucleon-nucleon double folding, and alpha-alpha double folding cluster—can describe the angular distributions and fusion cross sections at the same time. The shallow real potential fits the elastic angular distributions but predicts fusion cross sections nearly twice the measured values. Including the 2+ and 4+ excited states of 24Mg improves the fusion predictions.

What carries the argument

The central object is the $\alpha$-$\alpha$ double folding cluster (DFC) potential, constructed by folding the effective $\alpha$-$\alpha$ interaction $v_{\alpha\alpha}(r) = -122.6225\,e^{-0.22 r^2}$ with $\alpha$-cluster distribution densities $\rho_{cP}$ and $\rho_{cT}$ of projectile and target. The cluster densities are obtained by Fourier deconvolution from nuclear matter densities using the relation $\rho_M = \rho_c \otimes \rho_\alpha$. This potential supplies the deep real part that, with a normalization of $N_R=0.72$, reproduces both elastic angular distributions and fusion cross sections; the comparison against shallow and deep phenomenological potentials is what carries the argument.

Extended reading notes

Core claim

The paper's central discovery is that the long-standing difficulty of describing the 12C+24Mg reaction with a single potential shape is resolved once the real potential is sufficiently deep. Using the optical model, the authors show that a shallow real Woods-Saxon potential, despite giving the lowest chi-squared for individual elastic angular distributions, overpredicts the fusion cross section by roughly a factor of two. In contrast, the deep phenomenological potential, the nucleon-nucleon double folding potential, and the new alpha-alpha double folding cluster potential all reproduce the elastic scattering and the fusion data consistently, with a normalization factor of about 0.72 for the alpha-alpha potential. Coupled-channels calculations that couple the 2+ and 4+ rotational states of 24Mg reduce the theoretical fusion cross section and improve agreement with experiment. The paper thereby argues that the real nuclear interaction for this system is deep, and that microscopic folding potentials built from known densities can replace phenomenological deep potentials.

Load-bearing premise

The load-bearing assumption is that the $\alpha$-cluster distribution $\rho_c(r')$ obtained by Fourier deconvolution is a valid physical density, even though the parameters listed for 12C make it negative beyond roughly 0.75 fm; if this distribution is not positive definite, the $\alpha$-$\alpha$ DFC potential is not a meaningful microscopic potential.

Editorial extensions

If this is right

Shallow real potentials, even when they fit elastic scattering extremely well, should not be trusted for fusion cross-section predictions in light heavy-ion systems.
The alpha-alpha DFC potential, with only two adjusted parameters, $N_R$ and the imaginary depth $W_0$, can serve as a predictive alternative to phenomenological deep potentials.
Coupled-channels coupling to rotational states of a deformed target like 24Mg removes enough flux to improve fusion predictions by tens of millibarns.
The energy-independent normalization $N_R=0.72$ supports a largely microscopic real potential for 12C+24Mg.

Reading between the lines

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

The deconvolved cluster density for carbon is negative for radii beyond about 0.75 fm; a positive-definite reconstruction would test whether the same data can still be described with a similarly simple folding potential.
The same deep-versus-shallow dichotomy could be tested systematically on other $4n$ projectile-target pairs, where alpha clustering is expected and folding potentials should be equally predictive.
An energy-dependent normalization factor $N_R(E)$ would reveal whether the fixed 0.72 is a genuine real-potential strength or a coarse average over the fitted energy range.

Original abstract

We present a simultaneous analysis of the elastic scattering and fusion cross-section data of the $^{12}$C+$^{24}$Mg system around the Coulomb barrier and over energies by using the microscopic $\alpha$-$\alpha$ double folding cluster potential within the framework of the optical model and the coupled-channels formalism. The $\alpha$-$\alpha$ double folding cluster potential is obtained by using the $\alpha$-cluster distribution densities of the nuclei in the usual double folding procedure. The microscopic potential results are compared with the findings of the phenomenological deep and shallow potentials. It is subsequently shown that only phenomenological deep real, microscopic nucleon-nucleon and $\alpha$-$\alpha$ double folding cluster potentials provide a consistent description of the angular distributions and fusion cross-section data simultaneously. The effect of the inclusion of the excited states of the target nucleus $^{24}$Mg on the fusion cross-section predictions is also determined by the coupled-channels calculations, which are shown to improve the agreement.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,449 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.33 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.97 · 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 present a simultaneous analysis of the elastic scattering and fusion cross-section data of the $^{12}$C+$^{24}$Mg system around the Coulomb barrier and over energies by using the microscopic $\alpha$-$\alpha$ double folding cluster potential within the framework of the optical model and the coupled-channels formalism. The $\alpha$-$\alpha$ double folding cluster potential is obtained by using the $\alpha$-cluster distribution densities of the nuclei in the usual double folding procedure. The microscopic potential results are compared with the findings of the phenomenological deep and shallow potentials. It is subsequently shown that only phenomenological deep real, microscopic nucleon-nucleon and $\alpha$-$\alpha$ double folding cluster potentials provide a consistent description of the angular distributions and fusion cross-section data simultaneously. The effect of the inclusion of the excited states of the target nucleus $^{24}$Mg on the fusion cross-section predictions is also determined by the coupled-channels calculations, which are shown to improve the agreement.

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 present a simultaneous analysis of the elastic scattering and fusion cross-section data of the $^{12}$C+$^{24}$Mg system around the Coulomb barrier and over energies by using the microscopic $\alpha$-$\alpha$ double folding cluster potential within the framework of the optical model and the coupled-channels formalism.	contribution 0.96	0.96	0.10	asserted 1.00
s1	The $\alpha$-$\alpha$ double folding cluster potential is obtained by using the $\alpha$-cluster distribution densities of the nuclei in the usual double folding procedure.	method 1.00	0.79	0.03	asserted 1.00
s2	The microscopic potential results are compared with the findings of the phenomenological deep and shallow potentials.	method 0.39	0.72	0.04	asserted 0.91
s3	It is subsequently shown that only phenomenological deep real, microscopic nucleon-nucleon and $\alpha$-$\alpha$ double folding cluster potentials provide a consistent description of the angular distributions and fusion cross-section data simultaneously.	result 0.94	0.84	0.74	asserted 1.00
s4	The effect of the inclusion of the excited states of the target nucleus $^{24}$Mg on the fusion cross-section predictions is also determined by the coupled-channels calculations, which are shown to improve the agreement.	result 0.93	0.94	0.08	asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.63 · evidence from analysis of existing data 0.66.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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 cross-sections (filled circles

Showing 1 of the paper's 2 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:nucl-th/0603010 by the authors.

References

Works cited (21)

[1]

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

View on Pith 1997
[2]

W. Sciani, A. Lepine-Szily, F.R. Lichtenthaeler, P. Fa- chini, L.C. Gomes, G.F. Lima, M.M. Obuti, J.M. Jr Oliveira and A.C.C. Villari, Nucl. Phys. A620, 91 (1997)

View on Pith 1997
[3]

A. Lepine-Szily, W. Sciani, Y.K. Watari, W. Mittig, F.R. Lichtenthaeler, M.M. Obuti, J.M. Jr Oliveira and A.C.C. Villari, Phys. Lett. B304, 45 (1993)

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[4]

R.L. Filho, A. Lpine-Szily, A.C.C. Villari and O.P. Filh o, Phys. Rev. C 39 (1989) 884

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[5]

J. Carter, R.G. Clarkson, V. Hnizdo, R.J. Keddy, D.W. Mingay, F. Osterfeld and J.P.F. Sellschop, Nucl. Phys. A273, 523 (1976)

View on Pith 1976
[6]

I. Boztosun, W.D.M. Rae, Phys. Rev. C64 054607 (2001)

View on Pith 2001
[7]

S. Erturk, I. Boztosun, Y. Kucuk, M. Karakoc and S. Aydin, J. Phys. G 31 (2005) S1837

View on Pith 2005
[8]

M. Karakoc, MSc Thesis, Erciyes University (2005)

View on Pith 2005
[9]

W.D.M. Rae, Int. J. of Mod. Phys. A. 3 (1988) 1343

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[10]

M. Freer, A.C. Merchant, J. Phys. G 23 (1997) 261

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[11]

M. El-Azab Farid, Z. M. M. Mahmoud and G. S. Hassan, Nucl. Phys. A691 (2001) 671

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M. El-Azab Farid, M. A. Hassanain, Nucl. Phys. A678 (2000) 39

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[13]

RIPL-2, Nuclear Matter Densities, IAEA, http://www-nds.iaea.org/RIPL-2/

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[14]

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

View on Pith 1979
[15]

Buck, H. Friedrich and C. Wheathly Nucl. Phys. A275 (1977) 246

View on Pith 1977
[16]

J. Cook, Comput. Phys. Commun., 25 (1982) 125

View on Pith 1982
[17]

I.J. Thompson, FRESCO, CC code, unpublished

View on Pith
[18]

A. Iwamato, P. Moeller, J.R. Nix and H. Sagawa, Nucl. 5 Phys. A506 (1996) 329

View on Pith 1996
[19]

S. Gary, C. Volant, Phys. Rev. C 25, 1877 (1982)

View on Pith 1982
[20]

J.T. Holdeman, R.M. Thaler, Phys. Rev. Lett. 14 (1965) 81; ibid, Phys. Rev. 139 B1186 (1965)

View on Pith 1965
[21]

T. Yamaya et al. , Phys. Lett. B417 (1998) 7

View on Pith 1998

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] [1]

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

View on Pith 1997

[2] [2]

W. Sciani, A. Lepine-Szily, F.R. Lichtenthaeler, P. Fa- chini, L.C. Gomes, G.F. Lima, M.M. Obuti, J.M. Jr Oliveira and A.C.C. Villari, Nucl. Phys. A620, 91 (1997)

View on Pith 1997

[3] [3]

A. Lepine-Szily, W. Sciani, Y.K. Watari, W. Mittig, F.R. Lichtenthaeler, M.M. Obuti, J.M. Jr Oliveira and A.C.C. Villari, Phys. Lett. B304, 45 (1993)

View on Pith 1993

[4] [4]

R.L. Filho, A. Lpine-Szily, A.C.C. Villari and O.P. Filh o, Phys. Rev. C 39 (1989) 884

View on Pith 1989

[5] [5]

J. Carter, R.G. Clarkson, V. Hnizdo, R.J. Keddy, D.W. Mingay, F. Osterfeld and J.P.F. Sellschop, Nucl. Phys. A273, 523 (1976)

View on Pith 1976

[6] [6]

I. Boztosun, W.D.M. Rae, Phys. Rev. C64 054607 (2001)

View on Pith 2001

[7] [7]

S. Erturk, I. Boztosun, Y. Kucuk, M. Karakoc and S. Aydin, J. Phys. G 31 (2005) S1837

View on Pith 2005

[8] [8]

M. Karakoc, MSc Thesis, Erciyes University (2005)

View on Pith 2005

[9] [9]

W.D.M. Rae, Int. J. of Mod. Phys. A. 3 (1988) 1343

View on Pith 1988

[10] [10]

M. Freer, A.C. Merchant, J. Phys. G 23 (1997) 261

View on Pith 1997

[11] [11]

M. El-Azab Farid, Z. M. M. Mahmoud and G. S. Hassan, Nucl. Phys. A691 (2001) 671

View on Pith 2001

[12] [12]

M. El-Azab Farid, M. A. Hassanain, Nucl. Phys. A678 (2000) 39

View on Pith 2000

[13] [13]

RIPL-2, Nuclear Matter Densities, IAEA, http://www-nds.iaea.org/RIPL-2/

View on Pith

[14] [14]

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

View on Pith 1979

[15] [15]

Buck, H. Friedrich and C. Wheathly Nucl. Phys. A275 (1977) 246

View on Pith 1977

[16] [16]

J. Cook, Comput. Phys. Commun., 25 (1982) 125

View on Pith 1982

[17] [17]

I.J. Thompson, FRESCO, CC code, unpublished

View on Pith

[18] [18]

A. Iwamato, P. Moeller, J.R. Nix and H. Sagawa, Nucl. 5 Phys. A506 (1996) 329

View on Pith 1996

[19] [19]

S. Gary, C. Volant, Phys. Rev. C 25, 1877 (1982)

View on Pith 1982

[20] [20]

J.T. Holdeman, R.M. Thaler, Phys. Rev. Lett. 14 (1965) 81; ibid, Phys. Rev. 139 B1186 (1965)

View on Pith 1965

[21] [21]

T. Yamaya et al. , Phys. Lett. B417 (1998) 7

View on Pith 1998