Frustrated fragmentation and re-aggregation in nuclei: A non-equilibrium description in spallation

Heavy nuclei bombarded with protons and deuterons in the 1 GeV range have a large probability of undergoing a process of evaporation and fission; less frequently, the prompt emission of a few intermediate-mass fragments (IMFs) can also be observed. We employ a recently developed microscopic approach, based on the Boltzmann-Langevin transport equation, to investigate the role of mean-field dynamics and phase-space fluctuations in these reactions. We find that the formation of few IMFs can be confused with asymmetric fission when relying on yield observables, but it cannot be ascribed to the statistical decay of a compound nucleus when analyzing the dynamics and kinematic observables; it can be described as a fragmentation process initiated by phase-space fluctuations, and successively frustrated by the mean-field resilience. As an extreme situation, which corresponds to non-negligible probability, the number of fragments in the exit channel reduces to two, so that fission-like events are obtained by re-aggregation processes. This interpretation, inspired by nuclear-spallation experiments, can be generalized to heavy-ion collisions from Fermi to relativistic energies, for situations when the system is closely approaching the fragmentation threshold.

Figure 1

(a) Isotopic cross sections $\sigma$ measured in the IMF region for the system ${}^{136}\mathrm{Xe}+p$ at $1 A\mathrm{GeV}$, from the experiment of Ref. [48]. (b) Zero-angle invariant velocity distributions ${\sigma }_{\mathrm{I}}\left(v_{\left|\right|}^{\mathrm{b}}\right)/\sigma$ extracted from the experimental data for some isotopes [50]. (c) Analysis detail for one isotope, ${}^{20}\mathrm{F}$: invariant velocity plot on a plane containing the longitudinal axis. Two contributions are present with a recoil mismatch: a Coulomb ring with radius $v_{\mathrm{peak}}$ and a Gaussian-like distribution. (d) Reduction of the two-dimensional plot to a distribution along the beam axis; two kinematic contributions to the invariant-velocity spectrum appear as a convex mode (Coulomb ring) and a concave mode (Gaussian-like), respectively.

Figure 2

Energy-deposition map with one bunch of spacial cascade trajectories in relief, calculated for central impact parameters in the reaction $p+{}^{208}\mathrm{Pb}$ at $1 A\mathrm{GeV}$. Each trajectory is associated with one test particle of the incoming projectile.

Figure 3

Effect of the cascade in modifying the momentum space from the initial state configuration (dashed line) to the excited configuration (full line), calculated for one event in the reaction $p+{}^{208}\mathrm{Pb}$ at $1 A\mathrm{GeV}$ (see text). The panel on the left represents the nucleon energy distribution, whereas on the right the integrated energy distribution is presented.

Figure 4

Upper panel: Evolution of the mean number of emitted nucleons per interval of time in the reactions $p+{}^{208}\mathrm{Pb}$ at 1 GeV. Lower panel: Evolution of the mean excitation energy calculated for the fraction of bound matter in the reaction $p+{}^{208}\mathrm{Pb}$. Central, intermediate, and peripheral impact parameters are tested. The widths of the bands give the standard deviation around the trajectories for the central collisions; other trajectories have a comparable standard deviation (not indicated).

Figure 5

Study of the fragment configuration for one event of the system ${}^{197}\mathrm{Au}+p$ at $750 A\mathrm{MeV}$, for reaction times which are compatible with the definition of the fragmentation pattern. The event shown in the figure is selected among those giving the largest fragment multiplicity.

Figure 6

Upper panel: As in Fig. 4, evolution of the mean number of emitted nucleons per interval of time for the spallation reactions described in the text. Middle panel: Probability of split as a function of time for $p+{}^{208}\mathrm{Pb}$ and $p+{}^{136}\mathrm{Xe}$ at 1 GeV. Lower panel: Saturation of the fragment multiplicity $M_{\mathrm{frag}}$ for $p+{}^{208}\mathrm{Pb}$ at 1GeV; mean and standard deviation are shown for the multiplicity of fragment with $Z>4$.

Figure 7

Evolution of the average excitation energy $\langle E^{*}/A\rangle$ for the fraction of bound matter during the reaction. The double arrow gives the average uncertainty in terms of standard deviation. The bunch of lines extending over the whole time evolution describes events where only a heavy residue is present; bunches of lines for systems which split into $M_{\mathrm{frag}}$ equal to two, three, or four fragments are divided by $M_{\mathrm{frag}}$ for better visibility, as they would all collapse on the line for $M_{\mathrm{frag}}=1$.

Figure 8

Top left: Evolution of the average isotopic content of bound matter constituting different hot systems as a function of time (calculated for central impact parameters). Top right, bottom right, bottom left: Distributions of the average isotopic content of single elements produced in the systems ${}^{124}\mathrm{Xe},{}^{136}\mathrm{Xe}$, and ${}^{208}\mathrm{Pb}$ (moving clockwise) bombarded by 1 GeV protons as a function of the element number at 200 fm/$c$ (before fragmentation), at 400 fm/$c$ (latest fragmentation probability), at 700 fm/$c$ (when the fragment multiplicity saturates in the dynamical calculations), and after secondary decay progressing from $t_{\mathrm{stop}}$. The $\beta$ stability and the residue corridor are indicated (see text). Residues and IMF regions are also indicated.

Figure 9

Normalized yields as a function of the multiplicity of fragments with $Z>4$, for central impact parameters, at various time steps of the dynamical process, and after secondary decay.

Figure 10

Correlation among the three heaviest IMFs of mass number $A_1,A_2$, and $A_3$ produced in the same event, for all events where at least two fragments are found in the range $5\le Z\le 11$. Combinations of relative sizes ${\mu }_1,{\mu }_2$, and ${\mu }_3$, where ${\mu }_i=A_i/(A_1+A_2+A_3)$, are studied in a Dalitz plot for the systems ${}^{136}\mathrm{Xe}$ and ${}^{208}\mathrm{Pb}$, for central impact parameters. Color maps refer to cold systems after secondary decay and the configurations at 700 fm/$c$ are indicated by black contour lines.

Figure 11

Production yields and kinetic energies for five spallation reactions, calculated as a function of the mass number, for central impact parameters, at 400 fm/$c$, at the time $t_{\mathrm{stop}}$, and after secondary-decay (cold).

Figure 12

Zero-angle invariant velocity distributions ${\sigma }_{\mathrm{I}}\left(v_{\left|\right|}^{\mathrm{b}}\right)/\sigma$ of carbon and fluorine isotopes calculated for the system ${}^{136}\mathrm{Xe}+p$ at $1 A\mathrm{GeV}$ and of IMFs with $7\le Z\le 11$ calculated for ${}^{208}\mathrm{Pb}+p$ at $1 A\mathrm{GeV}$. Left panels present distributions of hot fragments at 700 fm/$c$, while right panels present distributions of cold fragments after secondary decay. The integral of the distributions reflects the same number of events exploited for the two systems and, for better comparison, they are scaled by the factors indicated in the boxes. The spectra are shifted with respect to zero by the mean recoil velocity of ${}^{136}\mathrm{Xe}$ and ${}^{208}\mathrm{Pb}$ (indicated by arrows). The values of $M_{\mathrm{frag}}$ indicate the multiplicity of fragments of $Z>4$ (including heavy residues) associated with the events. The contribution for different fragment multiplicities is indicated.

Figure 13

Top: Time evolution of the density profile of the systems $p$(1 GeV)$+{}^{208}\mathrm{Pb}$ for one specific event selected among those giving multifragmentation. The system undergoes a spinodal behavior, visible from 100 to 200 fm/$c$, when developing inhomogeneities of comparable size. Later on, the fragmentation mechanism is frustrated by the mean-field resilience, resulting in a rather asymmetric fragmentation. Bottom: Time evolution of the size of potential concavities associated with the evolution of the density profile at the beginning of the process $(t<5$ fm/$c)$, during the phase of instability growth ($t=200$ fm/$c$), and when fragments appear ($t=400$ fm/$c$). See text for details.

Figure 14

A random selection of some reaction paths followed by the system ${}^{208}\mathrm{Pb}+p$ at $1 A\mathrm{GeV}$ on the time-dependent potential landscape, represented by the average size of potential concavities as a function of time.

Figure 15

Time evolution of the density profile for the systems ${}^{136}\mathrm{Xe}+p$ at $1 A\mathrm{GeV}$ for one event resulting in an asymmetric binary split.