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The total reaction cross section of heavy-ion reactions induced by stable and unstable exotic beams: the low-energy regime

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Abstract

In this review paper we present a detailed account of the extraction and the calculation of the total reaction cross section of strongly bound and weakly bound, stable and unstable, exotic, nuclei. We discuss the optical model and the more general coupled channels model of direct reactions, and how from fits to the data on elastic scattering supplies the elastic element of (partial wave) S-matrix and correspondingly the differential cross section and the total reaction cross section. The effect of long-range absorption due to the coupling to excited states in the target and to the breakup continuum in the projectile is also discussed. The semiclassical method is then analyzed and the Hill–Wheeler expression of the tunneling probability and the Wong formula for the fusion and the total reaction cross sections are discussed in details. The generalized optical theorem for charged particle scattering and the resulting sum-of differences method is then discussed. Also, the strong absorption model in its sharp cutoff form and its generalization, the smooth cutoff, is discussed. The so-called “quarter-point recipe” is discussed next, and the quarter-point angle is introduced as a simple and rapid mean to obtain the total reaction cross section. The last topic discussed is the reduction of the total reaction cross section that would allow a large body of data to sit on a single universal function. Such a universal function exists in the case of the fusion data, and the aim of this last topic of the review is to extend the fusion case to the total reaction, by adding the direct reaction contribution. Also discussed is the inclusive breakup cross section and how it can be used to extract the total reaction cross section of the interacting fragment with the target. This method is also known as the Surrogate method and represents a case of hybrid reactions. The sum of the integrated inclusive breakup cross section with the complete fusion cross section supplies the total fusion cross section. A review of the experimental methods and techniques used to measure elastic scattering and breakup reactions is also presented. Emphasis is given to recent experiments and new techniques developed to deal with reactions induced by unstable beams. We discuss also the experimental challenges in these measurements and highlight the new techniques being developed to optimize measurements with radioactive ion beams.

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Fig. 1
Fig. 2

The figure was taken from Ref. [31]

Fig. 3

Figure from Ref. [21]

Fig. 4

Figure taken from Ref. [41]

Fig. 5
Fig. 6
Fig. 7
Fig. 8

The figure, the data and the calculations are from Ref. [67]

Fig. 9

Figure extracted from Ref. [67]

Fig. 10

Figure taken from Ref. [58]

Fig. 11
Fig. 12

Figure taken from a private communication with A. Moro

Fig. 13
Fig. 14
Fig. 15

The figure was taken from Ref. [118] and the data are from Refs. [119, 121]

Fig. 16

Figure extracted from Ref. [151]

Fig. 17

Figure extracted from Ref. [159]

Fig. 18

Figure extracted from Ref. [185]

Fig. 19

Figure taken from Ref. [186]

Fig. 20

Figure extracted from Ref. [189]

Fig. 21

Figure extracted from Ref. [200]

Fig. 22

Figure extracted from Ref. [144]

Fig. 23

Figure taken from Ref. [243]

Fig. 24

Figure taken from Ref. [263]

Fig. 25

Figure taken from Ref. [264]

Fig. 26

Figure taken from Ref. [265]

Fig. 27

Figure taken from Ref. [279]

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There is no specific data to be deposited since it is a review paper.]

Notes

  1. For simplicity, we are neglecting spins of the collision partners.

  2. For a detailed discussion of this correction, we refer to Ref. [42].

  3. From now on, the collision collision vector will be denoted by \(\mathbf{R}\), instead of \(\mathbf{r}\). This change will prove convenient when we discuss collisions of a two-cluster projectile.

  4. Blackboard bold fonts are used to indicate operators acting on both the \(\mathbf{R}\) and the \(\xi \) spaces.

  5. For simplicity, we omit the spins of the collision partners. For non-zero spins, \({\mathcal {Y}}_\beta (\hat{\mathbf{r}})\) are linear combinations of spin-dependent terms, involving angular momentum coupling coefficients.

  6. Note that the angular part of the intrinsic wave functions guarantees the orthogonality for \(\beta \ne \beta ^\prime \).

  7. If \(c_1\) is not spherical, \(V_1\) depends both on the modulus and on the orientation of \(\mathbf{r}_1\).

  8. Alternatively, they can be expressed in terms of hyperspherical coordinates (see, e.g. Ref. [58]).

  9. Borromean nuclei are nuclei with a bound three cluster configuration, which becomes unbound when one of the clusters is removed.

  10. Note that Eq. (145) should be used with care. Owing to the real part of nuclear potential, this equation might not fully represent the elastic scattering probability [174, 205].

  11. The interaction cross section, \(\sigma _\mathrm{I}\), is defined as the sum of all nonelastic cross sections in which at least one nucleon is removed from the projectile. Then, we can write: \(\sigma _\mathrm{R} = \sigma _\mathrm{I} + \sigma _{\mathrm{inel}}\). Since in collisions of the halo nuclei \(\sigma _\mathrm{I} \gg \sigma _{\mathrm{inel}}\), one can approximate: \(\sigma _\mathrm{R} \simeq \sigma _\mathrm{I}\). Glauber theory has been widely used to analyze such data [14].

  12. A variant of the traditional method is to replace Eq. (148) by the transformation \(E \rightarrow E_{\mathrm{red}} = E - V_\mathrm{B}\).

  13. Although this has been a popular interpretation of the ICF process, the reaction mechanism involved is not fully stablished. Although the two-step picture leads to a good description of the data [253], recent calculations [254] using the IAV model have shown that the NEB mechanisms (which includes ICF) mostly consists on a one-step process. Experimentally, the two scenarios (one-step vs. two-step) can be hard or even impossible to disentangle.

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Acknowledgements

The authors acknowledge financial support from the Brazilian Funding Agencies: CNPq (Grant 304961/2017-5), FAPESP (Grants 2016/17612-7), INCT-FNA (Instituto Nacional de Ciência e Tecnologia-Física Nuclear e Aplicações) Proc. No. 464898/2014-5 and FAPERJ. The authors are indebted to Drs. Antonio Moro and Brett Carlson, for critically reading the manuscript.

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Author notes

  1. M. S. Hussein: Deceased

Authors and Affiliations

  1. Instituto de Física, Universidade Federal do Rio de Janeiro, CP 68528, Rio de Janeiro, Brazil

    L. F. Canto

  2. Instituto de Física, Universidade de São Paulo, Rua do Matão Nr. 1371, CEP 05508-090, São Paulo, SP, Brazil

    V. Guimarães & M. S. Hussein

  3. Instituto de Física, Universidade Federal Fluminense, Av. Litoranea s/n, Gragoatá, 24210-340, Niterói, RJ, Brazil

    J. Lubián

  4. Instituto de Estudos Avançados, Universidade de São Paulo C. P. 72012, 05508-970, São Paulo, SP, Brazil

    M. S. Hussein

  5. Departamento de Física, Instituto Tecnológico de Aeronáutica, CTA, São José dos Campos, São Paulo, SP, Brazil

    M. S. Hussein

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  1. L. F. Canto
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Canto, L.F., Guimarães, V., Lubián, J. et al. The total reaction cross section of heavy-ion reactions induced by stable and unstable exotic beams: the low-energy regime. Eur. Phys. J. A 56, 281 (2020). https://doi.org/10.1140/epja/s10050-020-00277-8

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