Pion physics in the Liège intranuclear cascade model

The implementation of π production in the Liège intranuclear cascade model (INCL4) for spallation reactions is revisited to alleviate the overestimate of the π yield. Three modifications are proposed for this purpose: a better $\pi N$ cross section at high energy, the introduction of a π average potential, and the modification of the average mass of the Δ resonance. The π potential is determined from a global fit of a set of data bearing on π production in proton-induced reactions, on $\pi$-nucleus and absorption cross sections, and on proton production in π-induced reactions. The resulting π potential is poorly determined in the nuclear interior and agrees with the phenomenological optical-model potentials in the surface region. With these modifications, the predictions of the INCL4 model concerning π production cross sections in proton-induced reactions are considerably improved. Predictions of the improved version for $\pi$-nucleus reaction and absorption cross sections and for proton, residue, and fission cross sections in π-induced reactions are also presented and shown to give reasonably good agreement. Neutron production and some aspects of fission in π-induced reactions are also investigated and reasonably well predicted. Effects on the modifications on observables, which are not directly linked with π's, such as the neutron yield and the residue mass and charge spectra in proton-induced reactions are also investigated and shown to improve the description of these observable quantities. Several results on π production and the relative insentivity to the π potential in the nuclear interior are shown to be consistent with the fact that most π's are not produced in early collisions. Importance of rescattering in π absorption on nuclei is also pointed out. A comparison is made with the so-called Δ-hole model. Residual discrepancies are identified and are interpreted as due to the lack of π interaction with two nucleons at low energy, to the neglect of quantum motion effects, and to a possible underestimate of rescattering.

Figure 1

Influence of π production on neutron, proton, and γ multiplicities (per reaction) in proton-induced reactions on a Pb thin target, as functions of the incident proton kinetic energy $T_p$. The lines with the squares refer to the standard INCL4 calculations. The lines with the triangles are obtained by suppressing the inelastic $\mathit{NN}$ channel and the lines with circles correspond to keeping the same total $\mathit{NN}$ cross section but assuming purely elastic collisions. For neutrons, total multiplicities as well as multiplicities for three energy ranges are displayed.

Figure 2

Same as described in the legend to Fig. 1 for a thick cylindrical Pb target (diameter $=20$ cm, length $=60$ cm). Multiplicities of particles escaping from the target, per incident proton, are displayed. Same conventions as in Fig. 1.

Figure 3

Energy differential neutron multiplicity (per incident proton) induced by 800-MeV protons on a thin (upper panel) or thick (lower panel) Pb target. The full lines refer to the standard INCL4 calculation, whereas the dotted lines correspond to suppressing the $\mathit{NN}$ inelastic channel.

Figure 4

Phenomenology of the real part of the π optical-model potential based on local potentials, composed of a volume (left panels) plus a surface (right panels) components (the decomposition is either taken from the indicated references or made by us as a smooth separation around the half-density radius). The quantity $T_{\pi }$ is the π kinetic energy. The upper (lower) panels refer to positive (negative) π's. Experimental data are taken from Refs. [11, 13, 14, 15]. The convention corresponds to negative (positive) values for attractive (repulsive) potentials.

Figure 5

For each nucleus, the area of possible ($V_N^0,V_N^1$) values [see Eq. (3)] is delimited by similar line segments. They correspond roughly to to an increase of the χ squared by one unit. There is no dependence on $V_N^1$ for the (symmetric) ${}^{12}\mathrm{C}$ target. The triangle with heavy lines is the overlap of all areas of acceptable values.

Figure 6

Ratio of the calculated to the experimental ${\pi }^{+}$ (up) and ${\pi }^{-}$ (down) production cross section for reactions induced by 730-MeV protons on Pb, in function of the π potentials implemented in the INCL model. Based on the experimental data from Ref. [22]

Figure 7

Reaction (up) and absorption (down) cross section of positive pions on ${}^{12}\mathrm{C}$ as a function of the π kinetic energy $T_{{\pi }^{+}}$ for various values of $V_{{\pi }^{+}}$. The quantity $V_{{\pi }^{-}}$ is kept equal to 0. Experimental data are from Ref. [25].

Figure 8

Positive pion production cross-section in proton-induced reactions on various targets, as function of the incident proton kinetic energy $T_p$. The full lines are the standard INCL4 predictions. The dotted curves are obtained after introduction of the modified $\pi -N$ cross-section and of the pion potential (see text for detail). Experimental data are from Ref. [28]. Typical (statistical) uncertainties of the calculations are indicated by the error bars on the theoretical curves.

Figure 9

Production cross-section of ${\pi }^{+}$'s (left) and ${\pi }^{-}$'s (right) in proton-induced reactions on ${}^{208}\mathrm{Pb}$ (up) and ${}^{12}\mathrm{C}$ (down), as function of the incident proton kinetic energy $T_p$. Experimental data are from Refs. [21, 22]. Full lines are the standard INCL4 predictions. The dotted curves are obtained after introduction of the modified $\pi -N$ cross-section and of the pion potential (see text for detail). Typical (statistical) uncertainties of the calculations are indicated by the error bars on the theoretical curves.

Figure 10

Double differential production cross section of positive π's in proton-induced reactions on ${}^{208}\mathrm{Pb}$ at a kinetic energy of 730 MeV. Experimental data are from Ref. [22]. Full histograms correspond to standard INCL4 predictions. Dotted histograms are obtained after introduction of the modified $\pi N$ cross section and of the π potential (see text for detail). To ease the reading of the figure, the successive spectra have been multiplied by decreasing powers of 10, except for the smallest angle, in which case absolute values are given. The values of the detection angles are displayed close to the corresponding curves. Typical statistical uncertainties on the theoretical calculations are indicated by the error bars on the histograms.

Figure 11

Same as described in the legend to Fig. 10 for production of negative π's.

Figure 12

Same as described in the legend to Fig. 10 for a ${}^{12}\mathrm{C}$ target.

Figure 13

Same as described in the legend to Fig. 12 for production of negative πs.

Figure 14

Positive π reaction cross section with four different targets. Comparison between the standard INCL4 predictions (full lines), the predictions after modification of the $\pi N$ cross section and introduction of the π potential (see text for detail, dashed lines) and the experimental data (symbols) of Ref. [25]. Note that all cross sections for ${}^{12}\mathrm{C}$ have been multiplied by 2.

Figure 15

Positive π (true) absorption cross section on four different targets, indicated on the panels. Same convention as in Fig. 14. Data are from Refs. [25, 26]. Note that all cross sections for ${}^{63}\mathrm{Cu}$ have been multiplied by 2.

Figure 16

Negative π (true) absorption cross section on four different targets (indicated in the respective panels). Same convention as described in the legend to Fig. 15. Data are from Refs. [25, 26]. Note that all cross sections for ${}^{27}\mathrm{Al}$ have been multiplied by 2.

Figure 17

Double differential cross section for proton emission at ${30}^{°}$ in ${\pi }^{+}$-induced reactions on three different targets at 220 MeV. To ease the reading of the figure, the successive spectra have been multiplied by decreasing powers of 10, except for the lightest target, in which case absolute values are given. Data (dots) from Ref. [24] are compared with the standard predictions of INCL4 (full histograms) and with the predictions after the modifications described in the text (dotted histograms). The error bars give the typical uncertainty of the calculations.

Figure 18

Double differential cross section for proton emission in ${\pi }^{+}$-induced reactions on ${}^{12}\mathrm{C}$ at 220 MeV. To ease the reading of the figure, the successive spectra have been multiplied by decreasing powers of 10, except for the smallest angle, in which case absolute values are given. Data (dots) from Ref. [24] are compared with the standard predictions of INCL4 (full histograms) and with the predictions after the modifications described in the text (dotted histograms). The error bars give the typical uncertainty of the calculations.

Figure 19

Same as described in the legend to Fig. 17 for negative π's. Data are from Ref. [24].

Figure 20

Neutron double differential cross section in ${\pi }^{+}$-Pb collisions at 870 MeV. Data from Ref. [29] (dots) are compared with the calculations (histograms) using our modified version.

Figure 21

Mass yield of the fission fragments in ${\pi }^{+}$-induced reactions on ${}^{\mathrm{nat}}$U at 80 MeV. The yield is defined as the ratio between the production cross section for a given fragment and the fission cross section. The dots are the data from Ref. [30], reproduced as in the publication. However, they sum up to $~300\%$ instead of 200% for binary fission. The lines correspond to the yields calculated with the standard INCL4 (full line), with our modified version coupled to the ABLA evaporation-fission model [4, 5] (dotted line) or to the GEM model [31, 32].

Figure 22

Fission cross section in ${\pi }^{+}$-induced reactions on ${}^{235}\mathrm{U}$ and ${}^{209}\mathrm{Bi}$, as a function of the incident pion energy. The symbols represent the data from Ref. [33]. The full lines correspond to the yields calculated with the standard INCL4. The dotted lines correspond to our modified version, coupled to the ABLA evaporation-fission model [4, 5]. Note the U cross sections have been divided by 5.

Figure 23

Residue charge (upper panel) and mass (lower panel) production cross sections in $p$-Pb collisions at 1 GeV. Data (dots) from Ref. [35] are compared with our calculations using the standard (full lines) and the modified (dotted lines) versions. Only calculations for measured residues are displayed.

Figure 24

Distribution of the excitation energy $E^{\star }$ left in the remnant at the end of the cascade stage. Comparison between the results obtained with the standard version of INCL4 (solid histogram) and those obtained with the modified version (dashed histogram). The figure refers to the $p(800\mathrm{MeV})+{}^{208}$Pb case.

Figure 25

Reaction and true absorption cross sections for 165-MeV positive pions on various targets. Comparison between the predictions of standard INCL4 (dashed lines), of the modified version (dotted lines), of the Δ-hole model of Ref. [43] (full lines), and the experimental data (symbols) of Refs. [25, 26]

Figure 26

Reaction and true absorption cross sections for positive π's on a ${}^{93}\mathrm{Nb}$ target, as a function of the incident kinetic energy. Comparison between the predictions of standard INCL4 (dashed lines), of the modified version (dotted lines), of the Δ-hole model of Ref. [43] (full lines), and the experimental data (dots) of Ref [25].

Figure 27

Fits of ${\pi }^{+}\text{−}p$ and ${\pi }^{-}\text{−}p$ total cross sections. The crosses are the Particle Data Group world experimental data [48]. The full lines correspond to the parametrizations used in our modified version. The dotted lines [equivalent to the full lines in the (3,3) resonance region] give the parametrization used in the standard INCL4 model.