Isospin-dependent multifragmentation of relativistic projectiles

The $N/Z$ dependence of projectile fragmentation at relativistic energies has been studied with the ALADIN forward spectrometer at the GSI Schwerionen Synchrotron (SIS). Stable and radioactive Sn and La beams with an incident energy of 600 MeV per nucleon have been used in order to explore a wide range of isotopic compositions. For the interpretation of the data, calculations with the statistical multifragmentation model for a properly chosen ensemble of excited sources were performed. The parameters of the ensemble, representing the variety of excited spectator nuclei expected in a participant-spectator scenario, are determined empirically by searching for an optimum reproduction of the measured fragment-charge distributions and correlations. An overall very good agreement is obtained. The possible modification of the liquid-drop parameters of the fragment description in the hot freeze-out environment is studied, and a significant reduction of the symmetry-term coefficient is found necessary to reproduce the mean neutron-to-proton ratios $\langle N\rangle /Z$ and the isoscaling parameters of $Z⩽10$ fragments. The calculations are, furthermore, used to address open questions regarding the modification of the surface-term coefficient at freeze-out, the $N/Z$ dependence of the nuclear caloric curve, and the isotopic evolution of the spectator system between its formation during the initial cascade stage of the reaction and its subsequent breakup.

Figure 1

Measured cross section as a function of the fragment atomic number $Z$ determined with the TP-MUSIC IV detector for ${}^{124}\mathrm{Sn}$ (top panel) and ${}^{107}\mathrm{Sn}$ projectiles (bottom panel). Note that the experimental trigger conditions affected the yields for $Z>$ 40.

Figure 2

Inclusive mass distributions for fragments with $3⩽Z⩽6$ obtained for ${}^{107}\mathrm{Sn}$ projectiles (left panels) and ${}^{124}\mathrm{Sn}$ projectiles (right panels).

Figure 3

Reaction cross section $d\sigma /{\mathrm{dZ}}_{\mathrm{bound}}$ for ${}^{107}\mathrm{Sn}$ projectiles as recorded with the interaction trigger (thick solid line) in comparison with the expected value obtained with the subtraction technique (dashed line, see text). The solid thin line represents, on the same abscissa, the cross section $d\sigma /{\mathrm{dZ}}_{\mathrm{spect}}$ for the production of spectator nuclei with atomic number $Z_{\mathrm{spect}}$ according to the geometric participant-spectator model of Ref. [43].

Figure 4

Top panel: Triggered reaction cross section as a function of $Z_{\mathrm{bound}}$ for the three projectiles as indicated. Bottom panel: The corresponding distributions as obtained from ensemble calculations for 500 000 reaction events with standard parameters (see text).

Figure 5

Top panel: Mean reduced mass number $A/A_0$ of relativistic projectile residuals as a function of their excitation energy $E_x/A$ according to the fireball model of Gosset et al. [43], calculated for ${}^{107}\mathrm{Sn}$ projectiles, and INC calculations for ${}^{197}\mathrm{Au}+\mathrm{C}$ [44] (dashed lines), according to the experimental results for ${}^{197}\mathrm{Au}$ nuclei reported by Pochodzalla et al. [27] and including the widths in $E_x/A$ (dots) and according to the parametrization used in this work for $a_2=0.012$ and 0.015 MeV${}^{-2}$ (solid lines). Bottom panel: Ensemble of hot thermal sources represented in a scatter plot of reduced mass number $A/A_0$ versus excitation energy $E_x/A$, as used in the SMM calculations. The frequency of the individual sources is proportional to the area of the squares.

Figure 6

Relative yields of fragments for 10-unit-wide intervals of $Z_{\mathrm{bound}}$ as obtained with the SMM for the standard symmetry-term coefficient $\gamma =25$ MeV (solid lines) and with reduced coefficients $\gamma =14$ and 8 MeV (dashed and dotted lines, respectively) as a function of the fragment $Z$ for the case of ${}^{124}\mathrm{Sn}$ projectiles. The centers of the five $Z_{\mathrm{bound}}$ intervals relative to $Z_0$ and the scale factors used for displaying the yields are indicated in the figure.

Figure 7

Mean multiplicity of intermediate-mass fragments, $\langle M_{\mathrm{imf}}\rangle$, as a function of $Z_{\mathrm{bound}}$ as obtained with the SMM for the standard symmetry-term coefficient $\gamma =25$ MeV (solid lines) and with reduced coefficients $\gamma =14$ and 8 MeV (dashed and dotted lines, respectively), for reactions of ${}^{124}\mathrm{Sn}$ (top) and ${}^{124}\mathrm{La}$ (bottom) projectiles.

Figure 8

Relative yields of fragments for 10-unit-wide intervals of $Z_{\mathrm{bound}}$ as obtained with the SMM for the standard surface-term coefficient $B_0=18$ MeV (solid lines) and with the modified coefficient $B_0=20$ MeV (dotted line) as a function of the fragment $Z$ for the case of ${}^{124}\mathrm{Sn}$ projectiles. The centers of the five $Z_{\mathrm{bound}}$ intervals and the scale factors used for displaying the relative yields are indicated in the figure.

Figure 9

Mean multiplicity of intermediate-mass fragments, $\langle M_{\mathrm{imf}}\rangle$, as a function of $Z_{\mathrm{bound}}$ obtained with the SMM for the standard surface-term coefficient $B_0=18$ MeV (solid lines) and with the modified coefficient $B_0=20$ MeV (dotted lines), for reactions of ${}^{124}\mathrm{Sn}$ (top) and ${}^{124}\mathrm{La}$ (bottom) projectiles.

Figure 10

Experimental cross sections $d\sigma /\mathrm{dZ}$ for the fragment production following collisions of ${}^{107}\mathrm{Sn}$ projectiles sorted into five intervals of $Z_{\mathrm{bound}}/Z_0$ with centers as indicated and width 0.2 (symbols) in comparison with normalized SMM calculations (lines). Different scale factors were used for displaying the cross sections as indicated.

Figure 11

As Fig. 10 but for the case of ${}^{124}\mathrm{Sn}$ projectiles.

Figure 12

As Fig. 10 but for the case of ${}^{124}\mathrm{La}$ projectiles.

Figure 13

Experimental results (top panel) and results of SMM calculations (bottom panel) for the mean maximum charge $\langle Z_{\max }\rangle$ within an event versus $Z_{\mathrm{bound}}$, both normalized with respect to $Z_0$, for the three studied projectiles.

Figure 14

Comparison of SMM calculations (lines) with the experimental results (symbols) for the mean multiplicity $\langle M_{\mathrm{imf}}\rangle$ of intermediate-mass fragments versus $Z_{\mathrm{bound}}$ for the three studied projectiles.

Figure 15

Parameter $\tau$ obtained from power-law fits to the fragment yields $y(Z)$ in the range $5⩽Z⩽15$ according to $y(Z)\propto Z^{-\tau }$ as a function of $Z_{\mathrm{bound}}/Z_0$.

Figure 16

SMM calculations of the mean neutron-to-proton ratio $\langle N\rangle /Z$ of fragments with $2⩽Z⩽10$ produced in the fragmentation of ${}^{124}\mathrm{Sn}$ (top panel) and ${}^{124}\mathrm{La}$ (bottom panel) projectiles for the interval $0.4⩽Z_{\mathrm{bound}}/Z_0<0.6$ as a function of the fragment $Z$, performed with $\gamma =25$ MeV and with two surface energy coefficients $B_0=$ 18 and 20 MeV.

Figure 17

SMM calculations of the mean neutron-to-proton ratio $\langle N\rangle /Z$ as in Fig. 16, but for $B_0=18$ MeV and three different symmetry energy coefficients $\gamma =$ 8, 14, and 25 MeV.

Figure 18

Isotope distributions of fragments with $Z=$ 3–6 for ${}^{107}\mathrm{Sn}$ (left panels) and ${}^{124}\mathrm{Sn}$ (right panels), integrated over $0.2⩽Z_{\mathrm{bound}}/Z_0⩽0.8$. The dots represent the experimental data. Normalized results from calculations with the standard and modified parameters are shown by the dashed and solid lines, respectively.

Figure 19

Isotope distributions for ${}^{107}\mathrm{Sn}$ (left panels) and ${}^{124}\mathrm{Sn}$ (right panels) as in Fig. 18 but for fragments with $Z$ $=$7–10.

Figure 20

Measured mean neutron-to-proton ratios $\langle N\rangle /Z$ for fragments with $Z⩽10$ produced in the multifragmentation of ${}^{124}\mathrm{Sn}$ (top panel), ${}^{107}\mathrm{Sn}$ (middle panel), and ${}^{124}\mathrm{La}$ projectiles (bottom panel) and for three intervals of $Z_{\mathrm{bound}}/Z_0$ with centers as indicated and widths 0.2 (stars). In the calculations, performed for the range $0.2⩽Z_{\mathrm{bound}}/Z_0<0.8$, the surface-energy coefficients $B_0=$ 18 MeV and the symmetry-energy coefficient $\gamma =14$ MeV were used (open circles).

Figure 21

Mean neutron-to-proton ratio $\langle N\rangle /Z$ of fragments with $Z⩽10$ produced in the fragmentation of ${}^{124}\mathrm{La}$ projectiles in the range $0.4⩽Z_{\mathrm{bound}}/Z_0<0.6$. The experimental results (stars) are compared with SMM calculations using four combinations of symmetry-energy, ensemble, and surface parameters as indicated. Only the calculations with the symmetry-energy coefficient $\gamma =$ $14$ MeV (circles) are close to the data.

Figure 22

Mean neutron-to-proton ratio $\langle N\rangle /Z$ of fragments produced in the fragmentation of ${}^{124}\mathrm{Sn}$ (top panel) and ${}^{124}\mathrm{La}$ (bottom panel) projectiles in the range $0.4⩽Z_{\mathrm{bound}}/Z_0<0.6$. The experimental results (stars) are compared with SMM calculations using the three indicated values of the symmetry-term coefficient $\gamma$ and $E_x^{\mathrm{int}}=3$ MeV/nucleon as the start value of the interpolation interval for the secondary-decay stage of the calculations. With this choice, $\gamma =8$ MeV (triangles) gives the best agreement with the data.

Figure 23

Mean neutron-to-proton ratio $\langle N\rangle /Z$ of fragments with $Z⩽10$ produced in the fragmentation of ${}^{124}\mathrm{Sn}$ projectiles in the range $0.4⩽Z_{\mathrm{bound}}/Z_0<0.6$. The experimental results (stars) are compared with SMM calculations for the nominal $N/Z=$ 1.48 of that system (open circles) and for a reduced initial $N/Z=$ 1.33 ratio (dots). The standard liquid-drop parameters $B_0=$ 18 and $\gamma =$ 25 MeV were used.

Figure 24

Apparent temperatures $T_{\mathrm{HeLi}}$ (left panel) and $T_{\mathrm{BeLi}}$ (right panel) as a function of $Z_{\mathrm{bound}}/Z_0$ for the three reaction systems. For clarity, two of the three data sets are slightly shifted horizontally; statistical errors are displayed where they are larger than the symbol size (from Ref. [35]).

Figure 25

Mean microcanonical temperatures calculated for the three projectile ensembles with the $N/Z$ dependent surface coefficients $B_0=$ 17.5 and 19 MeV and with a symmetry coefficient $\gamma =14$ MeV. The experimental result is represented by the shaded band that extends from the lowest to the highest of the six apparent $T_{\mathrm{HeLi}}$ and $T_{\mathrm{BeLi}}$ temperatures shown in Fig. 24.

Figure 26

Experimental data (stars) and SMM ensemble calculations (open symbols) of isoscaling coefficients $\alpha$ extracted from fragment yield ratios ($3⩽Z⩽10$) for ${}^{124}\mathrm{Sn}$ and ${}^{107}\mathrm{Sn}$ projectiles (top panel) and for ${}^{124}\mathrm{Sn}$ and ${}^{124}\mathrm{La}$ projectiles (bottom panel) as a function of the reduced bound charge $Z_{\mathrm{bound}}/Z_0$. Four different symmetry-term coefficients $\gamma$ were used in the SMM calculations as indicated.

Figure 27

Theoretical isoscaling coefficients $\alpha$ calculated for three intervals in $Z_{\mathrm{bound}}/Z_0$ and for both hot primary fragments (open symbols, legend in upper panel) and cold fragments after secondary deexcitation (closed symbols, legend in lower panel).

Figure 28

Effective symmetry-term coefficient $\gamma$ as expected from the changing fragment-mass distributions using surface and volume symmetry-term coefficients from Refs. [65, 75, 93, 94] (open diamonds, circles, triangles, and squares, respectively) in comparison with the coefficient $\gamma$ for hot fragments obtained with the SMM from the $N/Z$ dependence of $B_0$ (solid triangles) and from the isoscaling analysis for the ${}^{107,124}\mathrm{Sn}$ pair of reactions (dots).

Figure 29

Mean neutron-to-proton ratio $\langle N\rangle /Z$ of fragments with $Z⩽10$ produced in the fragmentation of ${}^{124}\mathrm{Sn}$ (top panel) and ${}^{124}\mathrm{La}$ (bottom panel) projectiles in the range $0.4⩽Z_{\mathrm{bound}}/Z_0<0.6$. The experimental results (stars) are compared with SMM calculations with the Myers-Swiatecki parametrization [75] for the liquid-drop description of fragments and for two ensembles as indicated (open symbols).