Three-nucleon interaction in heavy-ion collisions at intermediate energies

The effect of a three-nucleon ($3N$) interaction is studied for the production of high energy protons in heavy-ion collisions in the incident energy range of 44 to $400A\mathrm{MeV}$. The $3N$ interaction is incorporated into the antisymmetrized molecular dynamics transport model of Ono [A. Ono, Phys. Rev. C 59, 853 (1999)] as a $3N$ collision term, following a diagram of three consecutive binary collisions. For the theoretical ${}^{40}\mathrm{Ar}+{}^{51}\mathrm{V}$ reaction studies, no contribution from the $3N$ collisions is observed for high energy proton production at the incident energy of $44A\mathrm{MeV}$. However when the incident energy increases, the contribution increases gradually. At $200A\mathrm{MeV}$ and above, the contribution is observed as distinctly harder energy slopes in the proton energy spectra. The model is applied to the available Bevalac data for ${}^{40}\mathrm{Ar}+{}^{40}\mathrm{Ca}$ at $42,92$, and $137A\mathrm{MeV}$. The experimental proton energy spectra are reasonably well reproduced at angles $\theta \ge {70}^{\circ }$ for all three incident energies, showing negligible $3N$ contributions at $42A\mathrm{MeV}$ and significant contributions at $137A\mathrm{MeV}$ at the large laboratory angles. Good agreement at these large angles, where the $3N$ collision is a major mechanism to produce such protons, strongly indicates for the first time the importance of the $3N$ interaction in intermediate heavy ion reactions in a full transport calculation. The possible relation between the $3N$ collision term and the short range and the tensor interactions is suggested.

Figure 1

Diagrammatic representation of $3N$ collision. The lines indicate particle trajectories and the meeting points indicate the location of the particles at the time of collision. $P_1,P_2,P_3$ represent the initial states and $P_1^{\prime },P_2^{\prime },P_3^{\prime }$ are the final states. $\mathbf{a},\mathbf{b},\mathbf{c},$ are intermediate states treated as virtual states in the $3N$ collision process.

Figure 2

Simulated results of proton energy distributions for ${}^{40}\mathrm{Ar}+{}^{51}\mathrm{V}$ at different incident energies. Histograms represent the results of AMD-FM($3N$) and circles are from AMD-FM. The spectra in each column are plotted every ${20}^{\circ }$ from $\theta ={10}^{\circ }$ at the top to $\theta ={170}^{\circ }$ at the bottom. Each spectrum is multiplied by a factor of 10 from the bottom to avoid overlap each other. The incident energy is $100A\mathrm{MeV}$ on the left to $400A\mathrm{MeV}$ on the right, indicating on the top.

Figure 3

Summed number of two-body (circles) and three-body (squares) collisions are plotted as a function of incident energy. Insert: Number of $2N$ and $3N$ collisions are plotted as a function of reaction time. (a) ${}^{40}\mathrm{Ar}+{}^{51}\mathrm{V}$ at $100A\mathrm{MeV}$. The density in the overlap region becomes maximum at $\sim 30\mathrm{fm}/c$ at this energy. (b) ${}^{40}\mathrm{Ar}+{}^{51}\mathrm{V}$ at $300A\mathrm{MeV}$. The density becomes maximum at $\sim 20\mathrm{fm}/c$ at this energy. The sigma of the Gaussians are 21.7 (15.9) for $2N$ and 8.0 (6.8) for $3N$ attempted collisions at $100A\left(300A\right)\mathrm{MeV}$, respectively.

Figure 4

Simulated results of proton energy spectra for the ${}^{40}\mathrm{Ar}+{}^{40}\mathrm{Ca}$ reactions at 42 (a), 92 (b), and $137A\mathrm{MeV}$ (c), and for the ${}^{40}\mathrm{Ar}+{}^{51}\mathrm{V}$ reactions at $44A\mathrm{MeV}$ in (d). Data in (a) to (c) are taken from Ref. [57] and (d) are from Ref. [38]. Solid curves are for the results of AMD-FM($3N$) and dashed curves are for those of AMD-FM. Data and results are plotted in an absolute scale. All results are multiplied by a factor of ${10}^{-n}[n=0$ to $5)$ in (a) to (c) and 8 in (d)] from top to bottom for clarity. The data at $42A\mathrm{MeV}$ in (a) are multiplied by a factor of 0.2. See details in the text.