Phenomenological model for light-projectile breakup

Background: Projectile breakup can make a large contribution to reactions induced by projectiles with mass numbers 2, 3, and 4, yet there is no global model for it and no clear agreement on the details of the reaction mechanism.

Purpose: This project aims to develop a phenomenological model for light-projectile breakup that can guide the development of detailed theories and provide a useful tool for applied calculations.

Method: An extensive database of double-differential cross sections for the breakup of deuterons, ${}^3\mathrm{He}$ ions, and $\alpha$ particles was assembled from the literature and analyzed in a consistent way.

Results: Global systematics for the centroid energies, peak widths, and angular distributions of the breakup peaks have been extracted from the data. The dominant mechanism appears to be absorptive breakup, where the unobserved projectile fragment fuses with the target nucleus during the initial interaction. The global target-mass-number and incident-energy dependencies of the absorptive breakup cross section have also been determined, along with channel-specific normalization constants.

Conclusions: Results from the model generally agree with the original data after subtraction of a reasonable underlying continuum. Absorptive breakup can account for as much as 50%–60% of the total reaction cross section.

Figure 1

Effective radius parameter for projectile breakup. The points show the values deduced from the observed Coulomb shifts of the breakup peaks for the indicated projectiles and the heaviest available target. The curve is obtained from Eq. (4).

Figure 2

Peak energies for ($d,p$), (${}^3\mathrm{He},d$), and (${}^3\mathrm{He},p$) breakup. The points show the values extracted from the data at the indicated incident energies; the lines are from Eqs. (1, 2, 3, 4).

Figure 3

Peak energies for ($\alpha ,p$), ($\alpha ,d$), ($\alpha ,t$), and ($\alpha ,{}^3\mathrm{He}$) breakup at the indicated incident energies. The points and curves have the same significance as in Fig. 2.

Figure 4

FWHM for $d$ and ${}^3\mathrm{He}$ breakup peaks. The points show the experimental values, and the lines are obtained from Eq. (6). The incident energies for ($d,p$) breakup are (top to bottom) 80, 70, 56, 27.5, 25.5, and 14.8 MeV. At 56 MeV, the solid points include data from multiple angles, while the open points are from 30-deg spectra. The incident energies for (${}^3\mathrm{He},d$) breakup (open points) are 130, 110, 90, and 70 MeV. The (${}^3\mathrm{He},p$) results at 90 and 70 MeV are given by the solid points and the dashed curves.

Figure 5

FWHM for $\alpha$-particle breakup peaks. The points show the experimental values; the solid and dashed lines are obtained from Eqs. (6) and(7), respectively. The ($\alpha ,{}^3\mathrm{He}$) channel has an intermediate result for a bismuth target at 140 MeV. The data for ($\alpha ,t$) breakup at 80 MeV did not allow reliable widths to be extracted.

Figure 6

Normalized angular distributions for ($d,p$) breakup at 56 MeV. The points show the experimental breakup cross sections divided by ${\left(D_0\right)}^2$ as a function of laboratory angle, while the line shows the best-fit exponential.

Figure 7

Empirical values for $a_{\mathrm{bu}}$, the slope parameter of the breakup angular distributions, shown as a function of the average peak energy of the available targets. The lines show the systematics given by Eq. (15).

Figure 8

Angular distributions for the ($d,p$) breakup peaks at incident energies of 14.8 and 15.0 MeV. The points show the results obtained from the experimental spectra, the dashed curves show the results for a simple exponential in $\theta$ with the same normalization for all targets, and the solid curves are obtained when the angular barrier is included.

Figure 9

Normalized peak heights for (${}^3\mathrm{He},d$) breakup at 13 deg for 70 and 90 MeV ${}^3\mathrm{He}$ and 9 deg for 130 MeV ${}^3\mathrm{He}$. The 130-MeV points have been divided by a factor of two.

Figure 10

The critical angle ${\theta }_0$ for the forward-angle dip in absorptive breakup. The points show values inferred from the data; the error bars extending up from the $X$ axis are upper limits. The lines are given by Eq. (21). The ($d,p$) results are for incident energies of 15 MeV (circles and one upper limit) and 56 MeV. The ${}^3\mathrm{He}$ breakup results include data from 70 MeV (circles and three limits), 90 MeV (triangles and three limits), and 130 MeV (square and three limits. For $\alpha$-particle breakup, the incident energies are 80, 140, and 160 MeV and the ejectile $b$ can be any charged particle fragment.

Figure 11

Average normalization factors ${\sigma }_{\mathrm{ab}}\left(E_{\mathrm{inc}}\right)/{\left(D_0\right)}^2$ for the breakup peaks in the indicated breakup channels as a function of the projectile energy. The solid, dashed, and dotted curves show the model values for $A_a-A_b=1$, 2, and 3, respectively.

Figure 12

Comparison of measured double-differential spectra with calculated breakup peaks and an estimated continuum for ${}^{27}\mathrm{Al}\left(d,\mathrm{x}p\right)$ at 14.8 MeV. The points show the data from Ref. [6], the dashed curves show the calculated breakup peaks, the dotted curves show estimates of the underlying continuum, and the solid curves show the total estimated spectra.

Figure 13

Comparison between experimental results and the current breakup model for ${}^{\mathrm{nat}}\mathrm{Cu}(d,xp)$ at 14.8 MeV. The points and curves have the same significance as in Fig. 12.

Figure 14

Comparison between experimental results and the current breakup model for ${}^{\mathrm{nat}}\mathrm{Zr}(d,xp)$ at 14.8 MeV. The points and curves have the same significance as in Fig. 12.

Figure 15

Comparison between experimental results and the current breakup model for ${}^{\mathrm{181}}\mathrm{Ta}(d,xp)$ at 15.0 MeV. The points and curves have the same significance as in Fig. 12 except that the data are from Ref. [15] and the short-dash curves for angles greater than ${30}^{\mathrm{o}}$ show an extrapolation of the breakup-like peak observed at backward angles. The data are in the center-of-mass system, but for such a heavy target the differences between the laboratory and center-of-mass systems are small.

Figure 16

Comparison between experimental double-differential cross sections from Ref. [8] and the current breakup model for ${}^{27}\mathrm{Al}(d,xp)$ at 56.0 MeV. The points and curves have the same significance as in Fig. 12.

Figure 17

Comparison between experimental double-differential cross sections from Ref. [8] and the current breakup model for ${}^{118}\mathrm{Sn}(d,xp)$ at 56.0 MeV. The points and curves have the same significance as in Fig. 12.

Figure 18

Comparison between experimental double-differential cross sections from Ref. [8] and the current breakup model for ${}^{209}\mathrm{Bi}(d,xp)$ at 56.0 MeV. The points and curves have the same significance as in Fig. 12.

Figure 19

Comparison between experimental double-differential cross sections from Ref. [1] and the current breakup model for ${}^{90}\mathrm{Zr}\left({}^3\mathrm{He},d\right)$ at 70 MeV. The points and curves have the same significance as in Fig. 12.

Figure 20

Comparison between experimental double-differential cross sections from Ref. [1] and the current breakup model for ${}^{90}\mathrm{Zr}\left({}^3\mathrm{He},d\right)$ at 110 MeV. The points and curves have the same significance as in Fig. 12.

Figure 21

Comparison of experimental and calculated peak heights for (${}^3\mathrm{He},d$) absorptive breakup at ${13}^{\circ }$ and the indicated incident energies. The points show the experimental values; the curves connect the model points for the same targets.

Figure 22

Comparison between experimental double-differential cross sections from Ref. [2] and the current breakup model for ${}^{59}\mathrm{Co}\left({}^3\mathrm{He},d\right)$ at 130 MeV. The points and curves have the same significance as in Fig. 12.

Figure 23

Comparison between experimental double-differential cross sections from Ref. [2] and the current breakup model for ${}^{197}\mathrm{Au}\left({}^3\mathrm{He},d\right)$ at 130 MeV. The points and curves have the same significance as in Fig. 12.

Figure 24

Comparison between experimental double-differential cross sections from Ref. [1] and the current breakup model for ${}^{90}\mathrm{Zr}\left({}^3\mathrm{He},p\right)$ at 90 MeV. The points and curves have the same significance as in Fig. 12.

Figure 25

Comparison between experimental double-differential cross sections from Ref. [4] and the current breakup model for ${}^{90}\mathrm{Zr}\left(\alpha ,xd\right)$ at 80 MeV. The points and curves have the same significance as in Fig. 12 except that the lower energy calculated breakup peak in each spectrum is the ($\alpha ,p,d$) breakup peak.

Figure 26

The same comparison as in Fig. 25, except that the upper or main calculated breakup peak is shown shifted up by 4 MeV from the energy given by the systematics of the present model.

Figure 27

Comparison between experimental double-differential cross sections from Ref. [4] and the current breakup model for ${}^{58}\mathrm{Ni}\left(\alpha ,xd\right)$ at 160 MeV. The points and curves have the same significance as in Fig. 12 but with the inclusion of the calculated ($\alpha ,p,d$) breakup peak. The upper or main peak is shown shifted up by 4 MeV from the peak energy given by the present model.

Figure 28

Comparison between experimental double-differential cross sections from Ref. [4] and the current breakup model for ${}^{209}\mathrm{Bi}\left(\alpha ,xd\right)$ at 160 MeV. The points and curves have the same significance as in Fig. 12 but with the inclusion of the calculated ($\alpha ,p,d$) breakup peak. The upper or main peak is shown shifted up by 4 MeV from the peak energy given by the present model.

Figure 29

Comparison between experimental double-differential cross sections from Ref. [4] and the current breakup model for ${}^{90}\mathrm{Zr}\left(\alpha ,xp\right)$ at 80 MeV. The points and curves have the same significance as in Fig. 12.

Figure 30

Comparison between experimental double-differential cross sections from Ref. [4] and the current breakup model for ${}^{27}\mathrm{Al}\left(\alpha ,xp\right)$ at 160 MeV. The points and curves have the same significance as in Fig. 12.

Figure 31

Comparison between experimental double-differential cross sections from Ref. [4] and the current breakup model for ${}^{58}\mathrm{Ni}\left(\alpha ,xp\right)$ at 160 MeV. The points and curves have the same significance as in Fig. 12.

Figure 32

Comparison between experimental double-differential cross sections from Ref. [4] and the current breakup model for ${}^{27}\mathrm{Al}\left(\alpha ,xt\right)$ at 160 MeV. The points and curves have the same significance as in Fig. 12.

Figure 33

Comparison between experimental double-differential cross sections from Ref. [4] and the current breakup model for ${}^{27}\mathrm{Al}\left(\alpha ,x{}^3\mathrm{He}\right)$ at 160 MeV. The points and curves have the same significance as in Fig. 12.

Figure 34

Comparison between experimental double-differential cross sections from Ref. [3] and the current breakup model for ${}^{209}\mathrm{Bi}\left(\alpha ,x{}^3\mathrm{He}\right)$ at 140 MeV. The points and curves have the same significance as in Fig. 12.

Figure 35

Percentage of the total reaction cross section from absorptive breakup as calculated in the current model shown, (a) as a function of the projectile energy in the laboratory system and (b) as a function of the ratio $(E_{\mathrm{inc}}-B_{\mathrm{a}})/B_{\mathrm{a}}$. The points show the results for individual target-projectile combinations in the database.

Figure 36

Same as for Fig. 35 but with the cross section normalizations for the (${}^3\mathrm{He},n$) and (${}^3\mathrm{He},p$) breakup channels reduced as discussed in the text. The curve shows the overall trend as given by Eq. (33).

Figure 37

Comparison of the percent of breakup fusion from Avrigeanu et al. [26, 27] with the percent of absorptive breakup found in the present work. All breakup channels are included. The upper panel shows the percentages and the lower panel gives their ratios. The curve is given by Eq. (33).