Pith. sign in

REVIEW 4 major objections 5 minor 76 references

Universal Wong formula for capture cross sections from light to super-heavy systems

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A single Wong-type formula now reproduces heavy-ion capture cross sections from carbon to super-heavy systems.

desk verdict Solid incremental upgrade with a real predictive success in Cr+U vs Ti+Pu, but the universal FDIS factor is only a four-point fit and the 'universal' label oversells it. read the letter →

arxiv 2411.17019 v2 pith:I4PMVSYM submitted 2024-11-26 nucl-th

Ning Wang , Jinming Chen , Yicheng Wang , Hong Yao This is my paper · ORCID
classification nucl-th
keywords capturecrosssectionheavy-ionfusionWongformulabarrierdistributionSkyrmeenergydensityfunctionaldeepinelasticscatteringsuper-heavynucleitime-dependentHartree-Fock
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a universal analytic formula for the capture cross section in heavy-ion fusion and argues that one fixed set of parameters can describe reactions from $^{12}$C+$^{14}$C up to $^{64}$Ni+$^{238}$U. Two refinements make this possible: a minimum width for the barrier-height distribution, taken from the quantum broadening of a parabolic barrier, and a suppression factor that shrinks the effective barrier radius when the capture pocket is shallow, encoding the loss to deep inelastic scattering. The authors validate the formula on thirty reactions that include spherical and deformed targets, neutron-shell-closed projectiles, and neutron-rich nuclei. If the claim holds, capture cross sections for unmeasured super-heavy synthesis reactions can be predicted quickly and without per-system tuning; the paper's prediction that $^{54}$Cr+$^{238}$U captures less than $^{50}$Ti+$^{242}$Pu is already consistent with measured evaporation-residue data.

What carries the argument

The machinery is the universal Wong formula $\sigma_{\rm cap}(E)=\int_0^\infty D(B)\sigma_{\rm Wong}(E,B)\,dB$ with a two-Gaussian barrier distribution whose centroid and width are set by the frozen Skyrme potential barrier height $B_0$. Two modifications carry the new results: the width constraint $w \geq \mathrm{FWHM} \approx 0.56\hbar\omega$, which supplies the quantum broadening missing for light systems, and the deep-inelastic-scattering factor $F_{\rm DIS}=\tfrac{1}{2}[1+\mathrm{erf}(\sqrt{B_{\rm cap}/c_1}-1)]$ with $c_1=2.0$ MeV, which shrinks the effective barrier radius $R_m=R_0F_{\rm DIS}$ when the capture pocket depth $B_{\rm cap}$ is small. This last factor is what converts the TDHF contact-time picture into a one-line analytic correction and is responsible for the super-heavy suppression.

What would settle it

Measure the capture excitation function for 54Cr+243Am at about 10% above the barrier: the deep-inelastic-suppressed formula gives values roughly a factor of two below estimates without that suppression, so the data would distinguish them. Alternatively, extract (Rm/R0)^2 from a measured capture-to-touching ratio for a shallow-pocket system not used in the fit and check it against 1/2[1+erf($\sqrt$(Bcap/2.0 MeV)-1)].

Watch

Extended reading notes

Core claim

The central claim is that the universal Wong formula, with barrier parameters from the Skyrme-energy-density-functional nucleus-nucleus potential, becomes genuinely universal once two corrections are added. The first is a constraint on the barrier-distribution width, $w \geq \mathrm{FWHM} \approx 0.56\hbar\omega$, which supplies the finite quantum width missing for light systems and restores the sub-barrier capture data for $^{14}$N+$^{16}$O, $^{16}$O+$^{16}$O, $^{12}$C+$^{14}$C, and $^{12}$C+$^{20}$Ne. The second is a capture-pocket-depth-dependent factor $F_{\rm DIS}=\tfrac{1}{2}[1+\mathrm{erf}(\sqrt{B_{\rm cap}/c_1}-1)]$ with $c_1=2.0$ MeV, which multiplies the barrier radius and the structure factor; TDHF simulations of heavy systems show the ratio of capture to touching cross section falling with decreasing pocket depth, and the formula tracks that ratio. With these changes, capture excitation functions for thirty systems from carbon to uranium are reproduced with fixed parameters, and the formula predicts that $^{54}$Cr+$^{238}$U capture lies below $^{50}$Ti+$^{242}$Pu capture because the Cr+U pocket is shallower, an ordering consistent with the measured evaporation residues.

Load-bearing premise

The whole super-heavy part of the argument rests on the assumption that the suppression of capture caused by deep inelastic scattering is the same universal error-function of the capture-pocket depth for every projectile-target combination, with a fixed scale c1=2.0 MeV and a TDHF contact-time cutoff near 600 fm/c.

Editorial extensions

If this is right

  • For unmeasured super-heavy reactions, capture cross sections can be obtained with the same fixed parameters used for light systems, removing the need to re-fit the potential per system.
  • The formula predicts that above-barrier capture for massive systems decreases as the capture pocket becomes shallower, so geometric radius alone is not a reliable scale for super-heavy capture.
  • The deep-inelastic-scattering suppression will reduce predicted evaporation-residue cross sections for super-heavy synthesis compared with estimates that ignore that suppression.
  • For light well-bound systems, the minimum-width constraint restores agreement with measured sub-barrier fusion data, notably for $^{14}$N+$^{16}$O and $^{16}$O+$^{16}$O.
  • The parabolic-barrier assumption limits the formula to energies not far below the barrier; deep sub-barrier fusion is outside its domain.

Reading between the lines

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

  • If the claimed universality is real, the same error-function suppression should appear in shallow-pocket systems outside the super-heavy mass region, and a few measured capture-to-touching ratios for such systems would test whether one scale, $c_1=2.0$ MeV, is sufficient.
  • The paper's comparison of $^{54}$Cr+$^{238}$U and $^{50}$Ti+$^{242}$Pu implies a strategy for searches for new super-heavy isotopes: among entrance channels forming the same compound nucleus, the more asymmetric combination with the deeper capture pocket should be favored in evaporation-residue experiments.
  • A natural test of the width constraint is to extract the experimental barrier distribution for a very light system such as $^{16}$O+$^{16}$O; the constrained formula effectively uses a single broadened Gaussian, whereas the unconstrained version uses a narrow two-Gaussian shape.
  • Extending the formula to radioactive or neutron-rich beams could be done without new parameters, since the structure factor already responds to $Q$-value and neutron-shell closures; data from such systems would provide a sharper check of the claimed universality.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper presents FUSION-v2, an extension of the universal Wong formula for capture cross sections based on a frozen Skyrme-density-functional nucleus-nucleus potential. Relative to FUSION-v1, two modifications are introduced: (i) a lower bound on the barrier-distribution width, w >= FWHM = 0.56 hbar omega, for light systems; and (ii) a pocket-depth-dependent reduction factor F_DIS, Eq. (8), calibrated to TDHF contact-time ratios and applied to the barrier radius and to the structure factor for heavy and super-heavy systems. The formula is compared with measured excitation functions for light systems and for a set of 30 systems induced by 12C, 16O, 32S, 48Ca and 64Ni, and it is used to predict that 54Cr+238U capture cross sections lie below those of 50Ti+242Pu, a trend consistent with the evaporation-residue data reported in Ref. [72].

Significance. If the claimed universality is established, the formula would be a practical single-parameter-set tool for estimating capture cross sections across the nuclear chart, which is particularly useful for planning super-heavy-element experiments. The paper has genuine strengths: the code is publicly available; the model ingredients are stated explicitly; the 54Cr+238U versus 50Ti+242Pu ordering is a falsifiable prediction consistent with later evaporation-residue data; and the parabolic-barrier limitation is acknowledged. However, the quantitative support for the central claim is not yet at the level of the claim: agreement is assessed visually without residuals or uncertainties, and the new F_DIS ingredient rests on only four TDHF points at a single energy. The empirical calibration is also not separated from the validation data, which weakens the word 'universal' as used in the title.

major comments (4)
  1. [Sec. III, Eq. (8), Fig. 2] The universal suppression factor F_DIS is calibrated with only four TDHF systems (86Kr+208Pb, 64Ni+208Pb, 58Fe+208Pb, 40Ca+96Zr), all at E = 1.05 EBass, using the criterion that contact time longer than 600 fm/c defines capture. The manuscript does not show that this TDHF criterion reproduces measured capture cross sections for these systems, does not test the sensitivity of the result to the contact-time cutoff or to the fitted value c1 = 2.0 MeV, and does not show that suppression depends only on Bcap rather than on mass asymmetry, deformation, or incident energy. Since the super-heavy predictions, including the 54Cr+238U versus 50Ti+242Pu ordering, use Eq. (8) at shallow pocket depths near or beyond the calibrated range, this is a load-bearing point that needs direct validation.
  2. [Sec. III, paragraph after Eq. (8)] The text states that 'the barrier radius R0 ... and the structure factor g are multiplied by FDIS in the calculations'. Only the relation Rm = R0 F_DIS is compared with the TDHF ratio in Fig. 2; the additional scaling of g by F_DIS is an independent assumption that changes the shape and normalization of the barrier distribution D1(B) in Eq. (3) and therefore affects sub-barrier cross sections. The effect of the g scaling should be shown separately, or g should be left unchanged until a specific justification is provided.
  3. [Sec. III, paragraph before Fig. 4] The statement that 'for all reactions under consideration the values of the model parameters are fixed and no additional adjustable parameter is introduced' is misleading as a claim of universality: f, c0 (with its two values), the neutron shell-closure flags delta_n, the cap 0 < g <= 2, the FWHM constraint, and c1 are all chosen constants. The paper does not identify which of the 30 reactions were used for calibration and which are genuine tests, and no numerical residual (average ratio, chi-square per point, or similar) is reported for the comparisons in Fig. 4. Visual agreement of a formula whose constants are tuned on the same class of data is not independent evidence of universality; a transparent training/test statement and a quantitative error measure are needed.
  4. [Sec. III, Fig. 6] The predicted capture cross sections in Fig. 6 are presented as point curves without any uncertainty estimate. Because the ordering of 54Cr+238U and 50Ti+242Pu depends on the difference between Bcap = 3.80 MeV and Bcap = 4.58 MeV through the rather flat erf function in Eq. (8), it would be helpful to show how stable the ordering is under reasonable variations of c1 and of the TDHF calibration. At present the reader cannot judge whether the predicted difference is significantly larger than the model uncertainty.
minor comments (5)
  1. [Title and abstract] The word 'ligh t' in the title and abstract is a typo and should read 'light'.
  2. [Sec. III, text discussing Fig. 6] The text refers to 'Fig. 5(a)' and 'Fig. 5(b)' when discussing the panels of the figure that is captioned as Fig. 6; these cross-references should be corrected.
  3. [Eq. (3)] The word 'Guassian' should be 'Gaussian'.
  4. [Sec. II, paragraph on reference systems] The rule for reference nuclei is stated as 'A0 - 1 < M_a.m. <= A0 (with a few exceptions which will be discussed later)', but the exceptions are never precisely specified; they should be listed or removed.
  5. [Sec. II, Eq. (5)] The min[ integral, integral ] rule in Eq. (5) is introduced without explanation; a short justification of why the smaller of the two integrals is chosen would improve the readability of the model definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the FDIS suppression is a calibrated extrapolation, and the final ordering is checked against external data.

full rationale

The paper's derivation chain is self-contained in the sense required by the circularity rules. The frozen Skyrme-EDF potential supplies B0, R0, hbar-omega, and Bcap without per-reaction adjustment; the two-Gaussian barrier distribution and f=0.926 come from FUSION-v1 [20] and the 443-system systematics [42]; the width constraint Eq. (7) is motivated by the standard quantum FWHM approx 0.56 hbar-omega; and the FDIS factor Eq. (8) is calibrated to four TDHF sigma_cap/sigma_T points at E=1.05 EBass. The highlighted prediction that 54Cr+238U capture lies below 50Ti+242Pu is then obtained by inserting computed pocket depths Bcap=3.80 MeV and Bcap=4.58 MeV into the fitted monotonic FDIS, and the measured evaporation-residue ordering [72] is used afterward as a check, not as a fitting input. The self-citations [20, 34, 42, 51] are load-bearing as prior model elements, but they are empirically or microscopically supported inputs rather than assumed uniqueness theorems, and the final validation is against external measured capture and evaporation-residue data. No equation in the paper makes a predicted quantity equal, by construction, to its fitting input, so no circular step is established.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The model is an empirically calibrated formula. Its predictive power rests on a chain of fitted constants and modeling choices: f, c0, c1, the shell-closure flags, the minimum-width constraint, the reference-system rules, and the TDHF contact-time criterion. The headline asymmetry for Cr+U versus Ti+Pu is an extrapolation that was later supported by measured evaporation residues, which provides an external check and keeps the circularity burden from being maximal.

free parameters (5)
  • f = 0.926
    Sets barrier-distribution centroids B1=fB0+w/2 and B2=fB0+w and width w=0.5(1-f)B0; taken from the SkM* parameter set, not derived.
  • c1 = 2.0 MeV
    Controls the pocket-depth dependence of the FDIS factor; tuned so Eq. (8) matches the TDHF ratio in Fig. 2.
  • c0 = 0.5 MeV^-1 for delta Q < 0, 0.1 MeV^-1 for delta Q > 0
    Determines the structure factor g in Eq. (6) from the Q-value difference to the reference system.
  • shell-closure flags delta_n = 1 or 0 per projectile and target
    Empirical corrections in Eq. (6) for neutron-shell-closed or neutron-rich nuclei; assignment is a modeling choice.
  • g cap = 2
    Truncation 0<g<=2 imposed to prevent the empirical structure factor from exceeding a reasonable range.
assumptions (6)
  • domain assumption Wong parabolic-barrier formula Eq. (1) is a valid representation of fusion penetration.
    Basis of the cross-section integral; the authors acknowledge in Sec. III that the formula breaks down at deep sub-barrier energies.
  • ad hoc to paper Capture is governed by a two-Gaussian barrier distribution with centroids B1=fB0+w/2 and B2=fB0+w and the min() rule in Eq. (5).
    The functional form and its parameters are empirical, inherited from FUSION-v1 (Ref. [20]).
  • domain assumption Entrance-channel nucleus-nucleus potential V(R) from the Skyrme energy density functional under the frozen-density approximation with ETF2 kinetic and spin-orbit densities.
    All barrier parameters B0, R0, and Bcap are computed from this potential; an inaccurate potential would shift every prediction.
  • ad hoc to paper A TDHF contact time of about 600 fm/c separates capture from deep inelastic scattering.
    This threshold is used to define the critical impact parameter bcap and to calibrate the pocket-depth trend in Fig. 2.
  • ad hoc to paper Constraint w >= FWHM = 0.56 hbar omega is imposed for light systems.
    Introduced to fix underprediction of light-system sub-barrier data; no derivation of the exact lower bound is given.
  • ad hoc to paper Reference-system mass numbers and the lanthanide rule A0=(A0+A0')/2 determine the structure factor baseline.
    A0 is set by relative atomic masses with exceptions, and A0' by the lightest stable isotope; these choices affect the g values.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Universal Wong formula for capture cross sections from light to super-heavy systems." pith.science (2026). https://pith.science/paper/I4PMVSYM

@misc{pith2026241117019,
  author       = {Pith},
  title        = {Pith review of: Universal Wong formula for capture cross sections from light to super-heavy systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I4PMVSYM}},
  note         = {Machine review of arXiv:2411.17019}
}
abstract

A universal Wong formula is proposed with refined model parameters for a systematic description of the capture cross sections for heavy-ion fusion reactions from C+C to Ni+U, in which the barrier parameters and the barrier distribution are determined by the entrance-channel nucleus-nucleus potential based on the Skyrme energy density functional. With introducing a constraint to the width of the barrier distribution and a pocket-depth dependent barrier radius, the capture excitation functions for a number of fusion reactions involving different nuclear structure effects are remarkably well reproduced, particularly for the reactions between light nuclei and those forming super-heavy nuclei. The systematic decreasing behavior of the geometric radii with the depth of capture pocket due to the influence of deep inelastic scattering is clearly observed in the TDHF calculations for super-heavy systems. The predicted capture cross sections for $^{54}$Cr + $^{238}$U at above barrier energies are evidently smaller than the corresponding results of more asymmetric projectile-target combination $^{50}$Ti + $^{242}$Pu due to the shallower capture pocket in Cr+U.

Figures

Figures reproduced from arXiv: 2411.17019 by the authors.

Figure 1
Figure 1. FIG. 1. Fusion excitation functions for reactions [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Ratio of capture cross section to touching cross sect [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) The same as Fig. 1, but for reactions [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Capture excitation functions for fusion reactions w [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Capture excitation functions for reactions [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Predicted capture excitation functions for fusion r [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

76 extracted references · 75 canonical work pages

  1. [72]

    Nhu Le, N

    N. Nhu Le, N. Ngoc Duy, N. Quang Hung, Eur. Phys. J. A 57, 187 (2021)

    work page 2021

  2. [1]

    Hofmann and G

    S. Hofmann and G. M¨ unzenberg, Rev. Mod. Phys. 72, 733 (2000). 14

    work page 2000

  3. [2]

    With the constraint being considered for 14N + 16O, one sees that the experimental data can be remarkably well reproduced

    61 MeV. With the constraint being considered for 14N + 16O, one sees that the experimental data can be remarkably well reproduced. The better reproductio n of the experimental data for these light systems indicate the constraint to the value of w is necessary and reasonable. For the fusion reactions leading to the synthesis of super-heavy n uclei, the dep...

    work page

  4. [3]

    Morita, K

    K. Morita, K. Morimoto, D. Kaji et al., J. Phys. Soci. Japa n 73, 2593 (2004)

    work page 2004

  5. [4]

    Hofmann, F.P

    S. Hofmann, F.P. Hessberger, D. Ackermanna et al., Nucl. Phys. A 734, 93 (2004)

    work page 2004

  6. [5]

    Yu. Ts. Oganessian, V.K. Utyonkov, Nucl. Phys. A 944, 62 (2015)

    work page 2015

  7. [6]

    Yu. Ts. Oganessian, F.Sh. Abdullin, P.D. Bailey et al., P hys. Rev. Lett. 104, 142502 (2010)

    work page 2010

  8. [7]

    Sobiczewski, Yu.A

    A. Sobiczewski, Yu.A. Litvinov, M. Palczewski, Atom. Da ta Nucl. Data Tables 119, 1 (2018)

    work page 2018

Show all 76 references
  1. [8]

    Yu. Ts. Oganessian, V.K. Utyonkov, N.D. Kovrizhnykh et a l., Phys. Rev. C 106, L031301 (2022)

    2022

  2. [9]

    Pomorski, B

    K. Pomorski, B. Nerlo-Pomorska, J. Bartel et al., Phys. R ev. C 97, 034319 (2018)

    2018

  3. [10]

    Tanaka, K

    T. Tanaka, K. Morita, K. Morimoto et al., Phys. Rev. Lett. 124, 052502 (2020)

    2020

  4. [11]

    Wang, S.J

    Y.Z. Wang, S.J. Wang, Z.Y. Hou et al., Phys. Rev. C 92, 064301 (2015)

    2015

  5. [12]

    Adamian, N.V

    G.G. Adamian, N.V. Antonenko, and W. Scheid, Phys. Rev. C 69, 011601(R) (2004)

    2004

  6. [13]

    T. A. Weaver, S.E. Woosley, Phys. Rep. 227, 65 (1993)

    1993

  7. [14]

    Guan, J.C

    D.W. Guan, J.C. Pei, Phys. Lett. B 851, 138578 (2024)

    2024

  8. [15]

    Dasgupta, D.J

    M. Dasgupta, D.J. Hinde, N. Rowley et al., Annu. Rev. Nuc l. Part. Sci. 48, 401 (1998)

    1998

  9. [16]

    X. Fang, W. P. Tan, M. Beard, et al., Phys. Rev. C 96, 045804 (2017)

    2017

  10. [17]

    Dasso, S

    C.H. Dasso, S. Landowne, Comput. Phys. Commun. 46, 187 (1987)

    1987

  11. [18]

    Hagino, N

    K. Hagino, N. Rowley, and A.T. Kruppa, Comput. Phys. Com mun. 123, 143 (1999)

    1999

  12. [19]

    Siwek-Wilczy´ nska and J

    K. Siwek-Wilczy´ nska and J. Wilczy´ nski, Phys. Rev. C 69, 024611 (2004)

    2004

  13. [20]

    Zagrebaev, Y

    V.I. Zagrebaev, Y. Aritomo, M.G. Itkis et al., Phys. Rev . C 65, 014607 (2001)

    2001

  14. [21]

    N. Wang, M. Liu, Y.X. Yang, Sci. China Ser. G - Phys. Mech. Astron. 52, 1554 (2009)

    2009

  15. [22]

    M. Liu, N. Wang, Z.X. Li et al., Nucl. Phys. A 768, 80 (2006)

    2006

  16. [23]

    Jiang and B.P

    C.L. Jiang and B.P. Kay, Phys. Rev. C 105, 064601 (2022)

    2022

  17. [24]

    B. Wang, K. Wen, W.J. Zhao et al., At. Data Nucl. Data Tabl es 114, 281 (2017)

    2017

  18. [25]

    Bass, Nucl

    R. Bass, Nucl. Phys. A 231, 45 (1974)

    1974

  19. [26]

    Wong, Phys

    C.Y. Wong, Phys. Rev. Lett. 31, 766 (1973)

    1973

  20. [27]

    Broglia, A

    R.A. Broglia, A. Winther, Heavy Ion Reactions , Frontiers in Physics, Vol. 84, (Addison- Wesley) (1991)

    1991

  21. [28]

    Bass, Lecture Notes in Physics , 117 (Berlin: Springer) 281 (1980)

    R. Bass, Lecture Notes in Physics , 117 (Berlin: Springer) 281 (1980)

    1980

  22. [29]

    Wen, C.J

    P.W. Wen, C.J. Lin, H.M. Jia et al., Phys. Rev. C 105, 034606 (2022)

    2022

  23. [30]

    Puri and Raj K

    Rajeev K. Puri and Raj K. Gupta, Phys. Rev. C 45, 1837 (1992)

    1992

  24. [31]

    Maruhn, P.G

    J.A. Maruhn, P.G. Reinhard, P.D. Stevenson et al., Phys . Rev. C 74, 027601 (2006)

    2006

  25. [32]

    N. Wang, W. Scheid, Phys. Rev. C 78, 014607 (2008). 15

    2008

  26. [33]

    Simenel, J

    C. Simenel, J. Phys. G: Nucl. Part. Phys. 41, 094007 (2014); arXiv:1403.3246v1

    2014 arXiv

  27. [34]

    Maruhn, and P.G

    Lu Guo, J.A. Maruhn, and P.G. Reinhard, Phys. Rev. C 76, 014601 (2007)

    2007

  28. [35]

    N. Wang, L. Ou, Y.X. Zhang and Z.X. Li, Phys. Rev. C 89, 064601 (2014)

    2014

  29. [36]

    H. Yao, H. Yang, N. Wang, Phys. Rev. C 110, 014602 (2024)

    2024

  30. [37]

    Vautherin, D.M

    D. Vautherin, D.M. Brink, Phys. Rev. C 5, 626 (1972)

    1972

  31. [38]

    N. Wang, K. Zhao and Z.X. Li, Phys. ReV. C 90, 054610 (2014)

    2014

  32. [39]

    V. Yu. Denisov and W. Noerenberg, Eur. Phys. J. A15, 375 (2002)

    2002

  33. [40]

    approach for the kinetic energy density and the spin-orbit densit y. The distribution function D(B) is expressed as a superposition of two Gaussian functions D1(B) and D2(B), D1(B) = 1√gπw exp [ −(B − B1)2 gw 2 ] (3) and D2(B) = 1 2√πw exp [ −(B − B2)2 (2w)2 ] . (4) In the rea...

  34. [41]

    Shorto, P.R.S

    J.M.B. Shorto, P.R.S. Gomes, J. Lubian, L.F. Canto, S. M ukherjee, L.C. Chamon, Phys. Lett. B 678, 77 (2009)

    2009

  35. [42]

    Bartel and K

    J. Bartel and K. Bencheikh, Eur. Phys. J, A14, 179 (2002)

    2002

  36. [43]

    Bartel, Ph

    J. Bartel, Ph. Quentin, M. Brack et al., Nucl. Phys. A 386, 79 (1982)

    1982

  37. [44]

    Y. Chen, H. Yao, M. Liu et al., Atom. Data Nucl. Data Table s 154, 101587 (2023)

    2023

  38. [45]

    Stokstad, Z

    R.G. Stokstad, Z. E. Switkowski, R.A. Dayras et al., Phy s. Rev. Lett. 37, 888 (1976)

    1976

  39. [46]

    Thomas, Y

    J. Thomas, Y. T. Chen, S. Hinds et al.,Phys. Rev. C 31, 1980(R) (1985)

    1985

  40. [47]

    Dasmahapatra, B

    B. Dasmahapatra, B. ˘Cujec, Nucl. Phys. A 565, 657 (1993)

    1993

  41. [48]

    Hulke, C

    G. Hulke, C. Rolfs, H. P. Trautvetter, Z. Phys. A 297, 161 (1980)

    1980

  42. [49]

    P. A. DeYoung, J. J. Kolata, R. C. Luhn et al., Phys. Rev. C 25, 1420 (1982)

    1982

  43. [50]

    Kozulin, G.N

    E.M. Kozulin, G.N. Knyazheva, I.M. Itkis et al., Phys. L ett. B 686, 227 (2010)

    2010

  44. [51]

    Itkis, G.N

    M.G. Itkis, G.N. Knyazheva, I.M. Itkisa et al., Eur. Phy s. J. A 58, 178 (2022)

    2022

  45. [52]

    H. Lee, P. McGlynn, and C. Simenel, Phys. Rev. C 110, 024606 (2024)

    2024

  46. [53]

    H. Yao, C. Li, H.B. Zhou, and N. Wang, Phys. Rev. C 109, 034608 (2024)

    2024

  47. [54]

    Newton, C.R

    J.O. Newton, C.R. Morton, M. Dasgupta et al., Phys. Rev. C 64, 064608 (2001)

    2001

  48. [55]

    Abriola, A.A

    D. Abriola, A.A. Sonzogni, M. di Tada et al., Phys. Rev. C 46, 244 (1992)

    1992

  49. [56]

    S. Gil, R. Vandenbosch, A. J. Lazzarini et al., Phys. Rev . C 31, 1752 (1985)

    1985

  50. [57]

    Mukherjee, D.J

    A. Mukherjee, D.J. Hinde, M. Dasgupta et al., Phys. Rev. C 75, 044608 (2007)

    2007

  51. [58]

    Viola, Torbjorn Sikkeland, Phys

    Victor E. Viola, Torbjorn Sikkeland, Phys. Rev. 128, 767 (1962)

    1962

  52. [59]

    Leigh, M

    J.R. Leigh, M. Dasgupta, D.J. Hinde et al., Phys. Rev. C 52, 3151 (1995)

    1995

  53. [60]

    H. Q. Zhang, Z. H. Liu, J. C. Xu et al., Phys. Rev. C 49, 926 (1994)

    1994

  54. [61]

    Gomes, I.C

    P.R.S. Gomes, I.C. Charret, R. Wanis et al., Phys. Re. C 49, 245 (1994)

    1994

  55. [62]

    D. J. Hinde, M. Dasgupta, N. Herrald et al., Phys. Rev. C 75, 054603 (2007). 16

    2007

  56. [63]

    Freifelder, P

    R. Freifelder, P. Braun–Munzinger, P. DeYoung et al., P hys. Rev. C 35, 2097 (1987)

    1987

  57. [64]

    Kolata, A

    J.J. Kolata, A. Roberts, A.M. Howard et al., Phys. Rev. C 85, 054603 (2012)

    2012

  58. [65]

    Trotta, A.M

    M. Trotta, A.M. Stefanini, S. Beghini et al., Euro. Phys . J. A 25, 615 (2005)

    2005

  59. [66]

    Prokhorova, A.A

    E.V. Prokhorova, A.A. Bogachev, M.G. Itkis et al., Nucl . Phys. A 802, 45 (2008)

    2008

  60. [67]

    Stefanini, L

    A.M. Stefanini, L. Corradi, H. Moreno et al., Phys. Lett . B 252, 43 (1990)

    1990

  61. [68]

    Wolfs, R.V.F

    F.L.H. Wolfs, R.V.F. Janssens, R. Holzmann et al., Phys . Rev. C 39, 865 (1989)

    1989

  62. [69]

    J. F. Liang, D. Shapira, J. R. Beene et al., Phys. Rev. C 75, 054607 (2007)

    2007

  63. [70]

    E. M. Kozulin, G. N. Knyazheva, K. V. Novikov, et al., Phy s. Rev. C 94, 054613 (2016)

    2016

  64. [71]

    Misicu and H

    S. Misicu and H. Esbensen, Phys. Rev. Lett. 96, 112701 (2006)

    2006

  65. [73]

    X.-J. Lv, Z. Y. Yue, W.-J. Zhao, B. Wang, Phys. Rev. C 103, 064616 (2021)

    2021

  66. [74]

    Oganessian, Eur

    Y. Oganessian, Eur. Phys. J. A 60, 227 (2024)

    2024

  67. [75]

    H. L¨ u, A. Marchix, Y. Abe, D. Boilley, Comp. Phys. Comm. 200, 381 (2016)

    2016

  68. [76]

    B. Mei, D. L. Balabanski, et al., Chin. Phys. C 45, 054001 (2021)

    2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.