Isospin conservation in preequilibrium reactions

Previous studies of the extent of isospin mixing in preequilibrium reactions in the continuum have concentrated on reactions with nucleons in both the entrance and exit channels and with incident energies of 30 MeV or less. Using an expanded database for light-particle reactions that includes complex particle channels and higher incident energies, a new investigation of preequilibrium and, at the lower incident energies, equilibrium mechanisms has been carried out. Although many of the measured inclusive energy spectra are insensitive to whether isospin is assumed to be mixed or conserved during the reaction, other spectra reveal evidence that isospin is conserved during the preequilibrium phase of a reaction when the excitation energy in the intermediate nucleus is less than four times the symmetry energy. When isospin is conserved during energy equilibration, good agreement with experiment is typically obtained when the corresponding amount of isospin conservation at equilibrium is taken to be around 40%, though this number is model dependent.

Figure 1

The effects of isospin conservation on sample ($p,\mathit{xp}$) reactions at 18 MeV. The circles give the experimental results, the dashed curves give the preequilibrium components calculated within the exciton model, and the short-dashed curves show the primary and secondary evaporation components. The primary evaporation peak is the one extending to higher emission energies. The solid curves give the total calculated spectrum, which also includes contributions from collective excitations of both spectroscopic and giant resonance states. In the left-hand panel, isospin is assumed to be fully mixed at all stages of the reaction. In the center panel, isospin is assumed to be conserved during the equilibration stage of the reaction, but mixed at equilibrium. Here the larger and harder preequilibrium component corresponds to emission from intermediate states with the ground state isospin, while the smaller, softer component gives emission from states with one additional unit of isospin. In the right hand panel, isospin is assumed to be conserved throughout the reaction. Elastic scattering is not included in the calculations. The data are from Refs. 8, 7.

Figure 2

The effects of isospin conservation on sample ($p,\mathit{xp}$) reactions at 25 MeV (${}^{159}\mathrm{Tb}$) and 28.8 MeV (${}^{54}\mathrm{Fe}$ and ${}^{197}\mathrm{Au}$). The points and curves have the same significance as in Fig. 1. Here in the left two panels, the secondary evaporation components for ${}^{159}\mathrm{Tb}$ and both evaporation components for ${}^{197}\mathrm{Au}$ are too small to be seen. This changes in the right hand panel, where isospin conservation at equilibrium makes these components larger. The iron and gold data are from [17] and are given in the laboratory system. The terbium spectrum is obtained from [6] by taking the backward hemisphere data and extrapolating it to forward angles as discussed in [9]. It is given in terms of the channel energy. All of the calculations are likewise given in terms of the channel energy.

Figure 3

The effects of isospin conservation in sample proton induced reactions on light targets. The solid curves show the results of calculations in which isospin is assumed to be a mixed (nonconserved) quantum number in all stages of the reaction; the dashed curves give the corresponding results when isospin is assumed to be a good quantum number during the equilibration phase of the reaction but to be mixed during the compound nucleus phase. The points show the experimental results. The 62 MeV data are from Ref. 17 and are given in the laboratory system, while the 90 MeV data are from Refs. 20, 21 and are given in terms of the channel energy. All of the calculations are given versus the channel energy.

Figure 4

The effects of isospin conservation in neutron-induced reactions on ${}^{28}\mathrm{Si}$ at 50 and 62.7 MeV. The curves and points have the same significance as in Fig. 3. The 50-MeV data are from Ref. 25 and are given in terms of the channel energy, whereas the 62.7-MeV data are from Ref. 26 and are given in the laboratory system. All calculations are given in terms of the channel energy. Although the calculated ${}^3\mathrm{He}$ spectra are shown separately, the experiments were unable to resolve ${}^3\mathrm{He}$ and α. Thus the contributions from ${}^3\mathrm{He}$ are included with the α spectra in both the data and the calculations.

Figure 5

The effects of isospin conservation in sample α-particle-induced reactions on light targets at 59 and 140 MeV. The points and curves have the same significance as in Fig. 3. The 59- and 140-MeV results are from Refs. 27 and [28], respectively. All of the results are given in terms of the channel energy.

Figure 6

Summary of experimental evidence on when isospin is conserved. The results are displayed as a function of the excitation energy and the symmetry energy in the intermediate nucleus. Solid points in the legend indicate stronger evidence for isospin mixing or conservation, whereas open symbols indicate weaker evidence. The dots represent systems for which the calculations are insensitive to isospin assumptions or where comparisons with the experimental results are inconclusive either because of the size of the error bars or because of discrepancies in spectral shape between calculation and experiment. The dots have been omitted from the panel displaying the equilibrium results. The dashed lines correspond to $E=4E_{\mathrm{sym}}$ and show the adopted criterion in this work for separating the isospin mixed and isospin conserved regimes. The dotted lines give an alternative boundary between these regimes, given by Eq. (4).

Figure 7

Summary of experimental evidence on the amount of isospin conservation at equilibrium shown as a function of the ratio $E/E_{\mathrm{sym}}$. The vertical lines give the ranges of reasonable values extracted from the data, whereas the curve represents the prescription given by Eq. (5).