Isoscaling of heavy projectile residues and $N/Z$ equilibration in peripheral heavy-ion collisions below the Fermi energy

The isoscaling of heavy projectile residues from peripheral heavy-ion reactions at 15–25 MeV/nucleon is employed to obtain information on the process of $N/Z$ equilibration. Recent mass spectrometric data of projectile residues from the reactions of ${}^{86}\mathrm{Kr}$ (15 MeV/nucleon) with ${}^{64,58}\mathrm{Ni}$ and ${}^{124,112}\mathrm{Sn}$ were first analyzed. The isotopically resolved yield distributions of the fragments in the range $Z=26–39$ were employed for the isoscaling analysis. The yield ratios $R_{21}(N,Z)$ of the fragments from each pair of systems exhibit isoscaling (i.e., an exponential dependence on the fragment neutron number $N$ for each atomic number $Z$) with the isoscaling parameter $\alpha$ increasing with decreasing (or increasing) $Z$ away from the projectile. This variation is related to the evolution toward $N/Z$ equilibration with increasing energy dissipation estimated from the residue velocities. In parallel to the new heavy-residue isoscaling data of ${}^{86}\mathrm{Kr}$ at 15 MeV/nucleon, our previous data at 25 MeV/nucleon for the reactions ${}^{86}\mathrm{Kr}+{}^{124,112}\mathrm{Sn}$ and ${}^{64}\mathrm{Ni}+{}^{64,58}\mathrm{Ni}$, ${}^{64}\mathrm{Ni}+{}^{124,112}\mathrm{Sn}$, as well as our data at 15 MeV/nucleon of the lighter system ${}^{40}\mathrm{Ar}+{}^{64,58}\mathrm{Ni}$, were analyzed in a similar way. Calculations with the stochastic nucleon-exchange model DIT (deep inelastic transfer) and the microscopic many-body model CoMD (constrained molecular dynamics) provided an overall fair description of data and valuable guidance for their interpretation. Interestingly, the data of the ${}^{86}\mathrm{Kr}+\mathrm{Ni},\mathrm{Sn}$ reactions at 15 MeV/nucleon show a retardation of the process of $N/Z$ equilibration which, as suggested by the CoMD calculations, is indicative of the collective character of the process. This retardation is not present in the investigated systems at 25 MeV/nucleon (and the light ${}^{40}\mathrm{Ar}+\mathrm{Ni}$ systems at 15 MeV/nucleon), whose behavior is found to be consistent with stochastic nucleon exchange.

Figure 1

(a) Yield ratios $R_{21}(N,Z)=Y_2(N,Z)/Y_1(N,Z)$ of projectile residues from the reactions of ${}^{86}\mathrm{Kr}$ (15 MeV/nucleon) with ${}^{64,58}\mathrm{Ni}$ with respect to $N$ for the $Z$'s indicated. (b) Yield ratios $R_{21}(N,Z)=Y_2(N,Z)/Y_1(N,Z)$ of projectile residues from the reactions of ${}^{86}\mathrm{Kr}$ (15 MeV/nucleon) with ${}^{124,112}\mathrm{Sn}$ with respect to $N$ for the $Z$'s indicated. The data are given by alternating filled and open circles, whereas the lines are exponential fits.

Figure 2

Isoscaling parameter $\alpha$ as a function of $Z$ for projectile residues from the reactions ${}^{86}\mathrm{Kr}$(15 MeV/nucleon)+${}^{64,58}\mathrm{Ni}$ (open circles) and ${}^{86}\mathrm{Kr}$(15 MeV/nucleon)+${}^{124,112}\mathrm{Sn}$ (closed circles).

Figure 3

(a) Average velocity vs $Z$ correlations for projectile residues from the reactions of ${}^{86}\mathrm{Kr}$ (15 MeV/nucleon) with ${}^{64}\mathrm{Ni}$ (open points) and ${}^{124}\mathrm{Sn}$ (closed points). The dashed line gives the projectile velocity. (b) Total kinetic energy loss (TKEL) vs $Z$ evaluated from residue velocities. The arrows indicate the maximum TKEL for Kr+Ni (lower arrow) and Kr+Sn (upper arrow). (c) Excitation energy per nucleon for the quasiprojectiles vs $Z$ evaluated from TKEL (see text).

Figure 4

(a) Isoscaling parameter $\alpha$ as a function of $Z$ for projectile residues from the reactions ${}^{86}\mathrm{Kr}$(25 MeV/nucleon)+${}^{124,112}\mathrm{Sn}$ [64]. (b) Average velocity vs $Z$ correlations for projectile residues from the reaction of ${}^{86}\mathrm{Kr}$ (25 MeV/nucleon) with ${}^{124}\mathrm{Sn}$. The dashed line gives the projectile velocity. (c) Total kinetic energy loss (TKEL) vs $Z$ evaluated from residue velocities. The maximum TKEL is $\sim 1040$ MeV (not shown). (d) Excitation energy per nucleon for the quasiprojectiles vs $Z$ evaluated from TKEL (see text).

Figure 5

Interaction time vs TKEL as calculated by DIT (dashed lines) and CoMD (full lines) for reactions at 15 MeV/nucleon (a) and 25 MeV/nucleon (b) as indicated in the figure. The arrows correspond to the maximum expected TKEL for the respective reactions.

Figure 6

(a) Calculated average $N/Z$ of quasiprojectiles as a function of TKEL for the 15-MeV/nucleon reactions: ${}^{86}\mathrm{Kr}+{}^{64}\mathrm{Ni}$ (upper set of curves) and ${}^{86}\mathrm{Kr}+{}^{58}\mathrm{Ni}$ (lower set of curves). DIT calculations: dashed (green) lines. CoMD calculations: solid lines (red) for asy-stiff and dotted lines (blue) for asy-soft. Upper and lower horizontal full lines: $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from ${}^{86}\mathrm{Kr}+{}^{64}\mathrm{Ni}$ and ${}^{86}\mathrm{Kr}+{}^{58}\mathrm{Ni}$ reactions, respectively. Horizontal dashed line: $N/Z$ of ${}^{86}\mathrm{Kr}$ projectile. (b) Difference in quasiprojectile average $N/Z$. Lines as in (a). Points: data from the present isoscaling analysis. Horizontal full line: difference of $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from ${}^{86}\mathrm{Kr}+{}^{64}\mathrm{Ni}$ and ${}^{86}\mathrm{Kr}+{}^{58}\mathrm{Ni}$ reactions. Arrow: maximum TKEL of the 15-MeV/nucleon Kr+Ni reactions.

Figure 7

(a) Calculated average $N/Z$ of quasiprojectiles as a function of TKEL for the 15-MeV/nucleon reactions: ${}^{86}\mathrm{Kr}+{}^{124}\mathrm{Sn}$ (upper set of curves) and ${}^{86}\mathrm{Kr}+{}^{112}\mathrm{Sn}$ (lower set of curves). DIT calculations: dashed (green) lines. CoMD calculations: solid lines (red) for asy-stiff and dotted lines (blue) for asy-soft. Upper and lower horizontal full lines: $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from ${}^{86}\mathrm{Kr}+{}^{124}\mathrm{Sn}$ and ${}^{86}\mathrm{Kr}+{}^{112}\mathrm{Sn}$ reactions, respectively. Horizontal dashed line: $N/Z$ of ${}^{86}\mathrm{Kr}$ projectile. (b) Difference in quasiprojectile average $N/Z$. Lines as in (a). Points: data from the present isoscaling analysis. Horizontal full line: difference of $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from ${}^{86}\mathrm{Kr}+{}^{124}\mathrm{Sn}$ and ${}^{86}\mathrm{Kr}+{}^{112}\mathrm{Sn}$ reactions. Arrow: maximum TKEL of the 15-MeV/nucleon Kr+Sn reactions.

Figure 8

(a) Calculated average $N/Z$ of quasiprojectiles as a function of TKEL for the 25-MeV/nucleon reactions: ${}^{86}\mathrm{Kr}+{}^{124}\mathrm{Sn}$ (upper set of curves) and ${}^{86}\mathrm{Kr}+{}^{112}\mathrm{Sn}$ (lower set of curves). DIT calculations: dashed (green) lines. CoMD calculations: solid lines (red) for asy-stiff and dotted lines (blue) for asy-soft. Upper and lower horizontal full lines give the $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from the ${}^{86}\mathrm{Kr}+{}^{124}\mathrm{Sn}$ and ${}^{86}\mathrm{Kr}+{}^{112}\mathrm{Sn}$ reactions, respectively. Horizontal dashed line: $N/Z$ of ${}^{86}\mathrm{Kr}$ projectile. (b) Difference in quasiprojectile $N/Z$. Lines as in (a). Points: the data from the present isoscaling analysis. Horizontal full line: difference of $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from the ${}^{86}\mathrm{Kr}+{}^{124}\mathrm{Sn}$ and ${}^{86}\mathrm{Kr}+{}^{112}\mathrm{Sn}$ reactions. Arrow: maximum TKEL of the 25-MeV/nucleon Kr+Sn reactions.

Figure 9

(a) Isoscaling parameter $\alpha$ as a function of $Z$ for projectile residues from the reactions ${}^{64}\mathrm{Ni}$(25 MeV/nucleon)+${}^{64,58}\mathrm{Ni}$ (open points) and ${}^{64}\mathrm{Ni}$(25 MeV/nucleon)+${}^{124,112}\mathrm{Sn}$ (closed points) [65]. (b) Average velocity vs $Z$ correlations for projectile residues from the reactions ${}^{64}\mathrm{Ni}$ (25 MeV/nucleon) + ${}^{64}\mathrm{Ni}$ (open points) and ${}^{64}\mathrm{Ni}$ (25 MeV/nucleon) + ${}^{124}\mathrm{Sn}$ (closed points). The dashed line gives the projectile velocity. (c) Total kinetic energy loss (TKEL) vs $Z$ evaluated from residue velocities. The symbols are as in (b). The maximum TKEL is $\sim 660$ MeV and $\sim 840$ MeV, respectively, for the above Ni+Ni and Ni+Sn reactions. (d) Excitation energy per nucleon for quasiprojectiles vs $Z$ evaluated from TKEL. The points are as in (b).

Figure 10

(a) Calculated average $N/Z$ of quasiprojectiles as a function of TKEL for the 25 MeV/nucleon reactions: ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ (upper set of curves) and ${}^{64}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ (lower set of curves). DIT calculations: dashed (green) lines. CoMD calculations (asy-stiff): solid (red) lines. Upper and lower horizontal full lines: $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ and ${}^{64}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ reactions, respectively. The upper line, of course, coincides with the $N/Z$ of the ${}^{64}\mathrm{Ni}$ projectile. (b) Difference in quasiprojectile average $N/Z$. Lines as in (a). Points: data from the present isoscaling analysis. Horizontal full line: difference of $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ and ${}^{64}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ reactions. Arrow: maximum TKEL of the 25-MeV/nucleon Ni+Ni reactions.

Figure 11

(a) Calculated average $N/Z$ of quasiprojectiles as a function of TKEL for the 25-MeV/nucleon reactions: ${}^{64}\mathrm{Ni}+{}^{124}\mathrm{Sn}$ (upper set of curves) and ${}^{64}\mathrm{Ni}+{}^{112}\mathrm{Sn}$ (lower set of curves). DIT calculations: dashed (green) lines. CoMD calculations (asy-stiff): solid (red) lines. Upper and lower horizontal full lines: $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from the ${}^{64}\mathrm{Ni}+{}^{124}\mathrm{Sn}$ and ${}^{64}\mathrm{Ni}+{}^{112}\mathrm{Sn}$ reactions, respectively. Horizontal dashed line: $N/Z$ of ${}^{64}\mathrm{Ni}$ projectile. (b) Difference in quasiprojectile average $N/Z$. Lines as in (a). Points: data from the present isoscaling analysis. Horizontal full line: difference of $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from the ${}^{64}\mathrm{Ni}+{}^{124}\mathrm{Sn}$ and ${}^{64}\mathrm{Ni}+{}^{112}\mathrm{Sn}$ reactions. Arrow: maximum TKEL of the 25 MeV/nucleon Ni+Sn reactions.

Figure 12

(a) Yield ratios $R_{21}(N,Z)=Y_2(N,Z)/Y_1(N,Z)$ of projectile residues from the reactions of ${}^{40}\mathrm{Ar}$(15 MeV/nucleon) with ${}^{64,58}\mathrm{Ni}$ with respect to $N$ for the $Z$'s indicated. The data are given by alternating filled and open circles, whereas the lines are exponential fits. (b) Isoscaling parameter $\alpha$ as a function of $Z$ for projectile residues from the reactions ${}^{40}\mathrm{Ar}$(15 MeV/nucleon)+${}^{64,58}\mathrm{Ni}$.

Figure 13

(a) Average velocity vs $Z$ correlations for projectile residues from the reaction of ${}^{40}\mathrm{Ar}$ (15 MeV/nucleon) with ${}^{64}\mathrm{Ni}$. The dashed line gives the projectile velocity. (b) Total kinetic energy loss (TKEL) vs $Z$ evaluated from residue velocities. The arrow indicates the maximum TKEL. (c) Excitation energy per nucleon for the quasiprojectiles vs $Z$ evaluated from TKEL (see text).

Figure 14

(a) Calculated average $N/Z$ of quasiprojectiles as a function of TKEL for the 15-MeV/nucleon reactions: ${}^{40}\mathrm{Ar}+{}^{64}\mathrm{Ni}$ (upper set of curves) and ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ (lower set of curves). DIT calculations: dashed (green) lines. CoMD calculations (asy-stiff): solid (red) lines. Upper and lower horizontal full lines: $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from ${}^{40}\mathrm{Ar}+{}^{64}\mathrm{Ni}$ and ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ reactions, respectively. Horizontal dashed line: $N/Z$ of ${}^{40}\mathrm{Ar}$ projectile. (b) Difference in quasiprojectile average $N/Z$. Lines as in (a). Points: data from the present isoscaling analysis. Horizontal full line: difference of $N/Z$ of fully $N/Z$ equilibrated quasiprojectiles from ${}^{40}\mathrm{Ar}+{}^{64}\mathrm{Ni}$ and ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ reactions. Arrow: maximum TKEL of the 15-MeV/nucleon Ar+Ni reactions.

Figure 15

Combined scaling of all studied systems: Horizontal axis: fraction of TKEL relative to the maximum allowed for binary collisions. Vertical axis: fraction of quasiprojectile $\Delta$ relative to that of the composite system ${\Delta }_{cs}$. Open circles (red): ${}^{86}\mathrm{Kr}$ (15 MeV/nucleon) + ${}^{64,58}\mathrm{Ni}$. Full circles (black): ${}^{86}\mathrm{Kr}$ (15 MeV/nucleon) + ${}^{124,112}\mathrm{Sn}$. Full squares (blue): ${}^{86}\mathrm{Kr}$ (25 MeV/nucleon) + ${}^{124,112}\mathrm{Sn}$. Upright triangles (magenta): ${}^{64}\mathrm{Ni}$ (25 MeV/nucleon) + ${}^{64,58}\mathrm{Ni}$. Diamonds (light blue): ${}^{64}\mathrm{Ni}$ (25 MeV/nucleon) + ${}^{124,112}\mathrm{Sn}$. Inverted triangles (green): ${}^{40}\mathrm{Ar}$ (15 MeV/nucleon) + ${}^{64,58}\mathrm{Ni}$. The horizontal line corresponds to full $N/Z$ equilibration. The full line that passes from the origin gives the course of $N/Z$ equilibration in the absence of retardation (see text).