High-resolution velocity measurements on fully identified light nuclides produced in ${}^{56}\mathrm{Fe}+\text{hydrogen}$ and ${}^{56}\mathrm{Fe}+\text{titanium}$ systems

New experimental results on the kinematics and the residue production are obtained for the interactions of ${}^{56}\mathrm{Fe}$ projectiles with protons and ${}^{\mathrm{nat}}\mathrm{Ti}$ target nuclei, respectively, at the incident energy of $1A\mathrm{GeV}$. The titanium-induced reaction serves as a reference case for multifragmentation. Already in the proton-induced reaction, the characteristics of the isotopic cross sections and the shapes of the velocity spectra of light residues indicate that high thermal energy is deposited in the system during the collision. In the ${}^{56}\mathrm{Fe}+p$ system the high excitation seems to favor the onset of fast break-up decays dominated by very asymmetric partitions of the disassembling system. This configuration leads to the simultaneous formation of one or more light fragments together with one heavy residue.

Figure 1

Idealistic plot of the phase diagram of nuclear matter, deduced from a Skyrme force ([27] parameterized according to [28]). Pressure is shown as a function of the average relative nucleon distance $r$ normalized to the distance $r_0$ at ground state. System configurations are drawn as possible final results of the expansion phase. When the thermalization path leads to the coexistence region, out of the spinodal region, damped density fluctuations occur. In the spinodal region density fluctuations are unstable and lead to cracking. At low density freeze-out is attained with different possible partition configurations: fragments are free to leave the system.

Figure 2

Experimental isotopic resolution. The isotopes are grouped in chains. One chain collects nuclei having the same value of the difference $N-Z$. The straight chain corresponds to $A∕Z=2$ or $N-Z=0$. On the right of this chain the isotopes are neutron rich, on the left they are proton rich.

Figure 3

Four steps of the analysis procedure to obtain the observed velocity spectrum of ${}^6\mathrm{Li}$ emitted in the reaction ${}^{56}\mathrm{Fe}+p$. (a) Raw spectra of counts as a function of $\beta \gamma$ in the laboratory frame. Each segment results from a different scaling of the magnetic fields of the FRS. One segment associated to the same magnetic scaling is marked with dark areas in this plot and in the two following ones. Arrows delimit the scanned $\beta \gamma$ range. (b) Yields normalized to the same beam dose. (c) Elimination of the angular-transmission distortion. Spectrum as a function of the longitudinal velocity in the beam frame $v_{\Vert }^b$. The broad Gaussian-type dark area indicates the contributions from nonhydrogen nuclei. (d) All components of the spectrum are composed together averaging overlapping points. Contributions from nonhydrogen nuclei were suppressed. The spectrum was divided by the number of nuclei per area of the liquid-hydrogen target. Statistical uncertainties and a fit to the data are shown.

Figure 4

Transmission of the FRS as a function of the positions in the dispersive and achromatic planes, relative to its maximum value. Numerical values are taken from Ref. 45.

Figure 5

(Top) Definition of the beam frame and of the center-of-mass frame of the emitting source with respect to the laboratory frame. The diagram corresponds to realistic conditions of the present experiment for ${}^6\mathrm{Li}$. The solid lines describe the slowing down of the beam and of the centroid of the velocity spectrum of ${}^6\mathrm{Li}$ in traversing the target. (Bottom) Mean longitudinal recoil velocities in the beam frame $⟨v_{\Vert }^b⟩$ of the reaction residues compared with the systematics of Morrissey [49] (solid line); only isotopes with sufficient statistics and entirely measured velocity spectra are considered.

Figure 6

Experimental isotopic production cross sections of some light elements for the reactions ${}^{56}\mathrm{Fe}+p$ and ${}^{56}\mathrm{Fe}+{}^{\mathrm{nat}}\mathrm{Ti}$ at $1A\mathrm{GeV}$. The cross sections related to the latter system are scaled of a factor 0.07. Numerical values are collected in Table .

Figure 7

Isotopic production cross sections shown on a chart of the nuclides for the reactions ${}^{56}\mathrm{Fe}+p$ and ${}^{56}\mathrm{Fe}+{}^{\mathrm{nat}}\mathrm{Ti}$ at $1A\mathrm{GeV}$. The values for ${}^{54}\mathrm{Mn}$ in ${}^{56}\mathrm{Fe}+p$ and for ${}^{53}\mathrm{Cr}$ in ${}^{56}\mathrm{Fe}+{}^{\mathrm{nat}}\mathrm{Ti}$ were obtained from systematics.

Figure 8

Experimental production cross sections as a function of the mass number. The statistical uncertainties are lower than 10%. The systematic uncertainties evolve from 10% for the heavy residues close to the projectile to 20% for the light fragments.

Figure 9

Reconstructed density plots in velocity space in the beam frame $(v_{\Vert }^b,v_{\perp }^b)$ representing the distribution on a plane containing the beam axis. The solid lines denote the angular acceptance of the spectrometer.

Figure 10

Velocity spectra of light residues produced in ${}^{56}\mathrm{Fe}+{}^{\mathrm{nat}}\mathrm{Ti}$ (upper diagram), and in ${}^{56}\mathrm{Fe}+p$ at $1A\mathrm{GeV}$ (lower diagram), ordered on a nuclear chart. They are represented as a function of the velocity in the beam direction in the beam frame $v_{\Vert }^b$. Crosses and points indicate measured spectra $\mathrm{d}{\mathcal{I}}^{\mathrm{Ti}}∕\mathrm{d}{v}_{\Vert }^b$ and $\mathrm{d}{\mathcal{I}}^p∕\mathrm{d}{v}_{\Vert }^b$, respectively, defined according to Eq. (3), and normalized to the unit. They represent all fragments transmitted through the FRS. Reconstructed velocity spectra ${\sigma }_r^{\mathrm{Ti}}$ and ${\sigma }_r^p$, defined according to Eq. (4) and normalized to the unit are marked with dashed and solid lines, respectively. In the lower diagram, the reconstructed spectra for ${}^6\mathrm{Li}$, ${}^{10}\mathrm{B}$, and ${}^{12}\mathrm{C}$ emitted from ${}^{56}\mathrm{Fe}+{}^{\mathrm{nat}}\mathrm{Ti}$ are superimposed as dashed lines for comparison.

Figure 11

(Left panel) Mean absolute velocities in the reference frame of the center of mass of the hot remnant, measured for residues of the ${}^{56}\mathrm{Fe}+p$ system (open circles) and deduced from the systematics of Tavares and Terranova [56] (dark bands). The width of the dark areas results from the range of the possible mother nucleus from ${}^{46}\mathrm{Ti}$ (lower values) to ${}^{56}\mathrm{Fe}$ (higher values). In the inset, data points on the total kinetic energy released in a symmetric split of nuclei close to iron, measured by Grotowski et al. [54] are compared to the systematics of Viola [55] (dotted-dashed line) and with the systematics of Tavares and Terranova [56] (solid line). (Right panel) Measured absolute-velocity spectra for the residues ${}^6\mathrm{Li}$, ${}^{10}\mathrm{B}$, and ${}^{12}\mathrm{C}$ produced in the ${}^{56}\mathrm{Fe}+p$ system. The arrows indicate the values obtained by the systematics of Tavares and Terranova.

Figure 12

Kinetic-energy spectra in the center of mass of the emitting source obtained from the reconstructed experimental velocity distributions in the case of emission of ${}^6\mathrm{Li}$ and ${}^{12}\mathrm{C}$, respectively. The spectra are compared for the ${}^{56}\mathrm{Fe}+p$ system (solid lines) and for the ${}^{56}\mathrm{Fe}+{}^{\mathrm{nat}}\mathrm{Ti}$ system (dashed lines). All spectra are normalized to the same area. The smooth distributions result from a spline fit procedure to the data.

Figure 13

Hot-fragment distributions generated in the intranuclear cascade [63], in the case of exclusion (left) and inclusion (right) of a pre-equilibrium stage [65], for the ${}^{56}\mathrm{Fe}+p$ system. Straight lines define constant values of excitation energy per nucleon, indicated in MeV. Five selected isotopes correspond to the most probable mass and nuclear charge for a given excitation energy per nucleon.

Figure 14

Calculated production of the hot fragments after the intranuclear cascade (modeled according to [63]) and a pre-equilibrium stage (simulated according to [65]), for the ${}^{56}\mathrm{Fe}+p$ system. The cross sections of the hot fragments are shown as a function of the excitation energy per nucleon $E*∕A$ of the source (bottom-left) and the mass number (top-right). The mass distribution of cross sections of the hot fragments is compared to the experimental final-residue production.

Figure 15

Comparison of the measured mass distributions as a function of the mass number for the system ${}^{56}\mathrm{Fe}+p$ with the results of GEMINI (upper part) and SMM (lower part). SMM is more sensitive than GEMINI to the effect of a pre-equilibrium phase.

Figure 16

(First row) Experimental velocity spectra (circles) and reconstructed velocity spectra (solid line). Each spectrum is drawn in the reference frame corresponding to the measured average velocity value of the fragment considered. (This frame corresponds to the “center of mass” frame of the reaction product drawn in Fig. 5). (Second row) Calculated velocity spectra obtained by GEMINI or SMM following INC and the pre-equilibrium stage, and from SMM following directly INC. Each spectrum is drawn in the reference frame corresponding to the calculated average velocity value of the fragment considered. (Third row) Velocity recoil introduced by the GEMINI or SMM phase alone (recoils by INC and pre-equilibrium stages not included). All spectra are normalized to the unit.

Figure 17

The dark areas represent portions of the residue production calculated with SMM, subdivided according to different multiplicities of intermediate-mass fragments (having $A>4$). The total production measured experimentally (dots) and calculated (solid line) is superimposed for comparison. The calculation disregards pre-equilibrium in the left diagrams and includes pre-equilibrium in the right diagrams.

Figure 18

Different portions (dark areas) of the residue production calculated with SMM are selected according to different ranges in the excitation energy per nucleon $E*∕A$ of the source. The calculation is performed only for the case of inclusion of pre-equilibrium. The total production measured experimentally (dots) and calculated (solid line) is superimposed for comparison.

Figure 19

(Upper part) Experimental data on the production cross-sections ratio as a function of the mass number for the reaction ${}^{56}\mathrm{Fe}+{}^{\mathrm{nat}}\mathrm{Ti}$ versus the reaction ${}^{56}\mathrm{Fe}+p$. The data are compared with the ratio of total nuclear cross sections for the two reactions, calculated according to the model of Karol [70]. (Lower part) SMM calculation of the probability for the formation of a residue as a function of the mass number and the multiplicity. The multiplicity is intended as the number of projectilelike residues heavier than an alpha particle produced in one collision.

Figure 20

Contribution of different multiplicity channels (A>4) to the velocity spectrum of ${}^6\mathrm{Li}$, as calculated by SMM. The representation is the same as in the second column of Fig. 16.

Figure 21

Integration domains of Eq. (3).