Charge-changing cross sections for ${}^{42\text{–}51}\mathrm{Ca}$ and effect of charged-particle evaporation induced by neutron-removal reactions

Charge-changing cross sections ${\sigma }_{\mathrm{CC}}$ for ${}^{42\text{–}51}\mathrm{Ca}$ on a carbon target at around 280 MeV/nucleon have been measured. Though the existing point-proton radii $r_{\mathrm{p}}$ of Ca isotopes increase as the neutron number increases, the measured ${\sigma }_{\mathrm{CC}}$ data show a significant decrease, which is against the expectation from a simple Glauber-like model. We found that this observed phenomenon could be attributed to the charged-particle evaporation effect induced by the neutron-removal reaction. By taking the evaporation effect into account, various ${\sigma }_{\mathrm{CC}}$ data sets for nuclides from C to Fe isotopes on ${}^{12}\mathrm{C}$ measured at around 280 MeV/nucleon are reproduced with a standard deviation of 1.6%. It is also clarified that this evaporation effect becomes negligibly small in the neutron-rich region. The evaluated relation between ${\sigma }_{\mathrm{CC}}$ and $r_{\mathrm{p}}$ using the current model indicates that ${\sigma }_{\mathrm{CC}}$ data for neutron-rich Ca isotopes ($A\ge 51$) are highly sensitive to $r_{\mathrm{p}}$. This high sensitivity potentially allows one to determine the $r_{\mathrm{p}}$ of very neutron-rich nuclei.

Figure 1

Schematic view of the experimental setup.

Figure 2

Particle-identification plots of (a) the cocktail beam for the ${}^{42}\mathrm{Ca}$ setting before the reaction target and (b) particles transported to the F7 focal plane, with incoming ${}^{42}\mathrm{Ca}$ selection indicated by the ellipse in (a). In (b), $Z$ was determined from F7MUSIC $\mathrm{\Delta }E$. The appropriate beam-emittance constraint was adopted for both plots. The ${}^{48}\mathrm{Ca}$ plots are presented in Ref. [7].

Figure 3

(a) Identification of $Z$ after the reaction target with selection of incoming ${}^{42}\mathrm{Ca}$ in correlation between $Z_{\mathrm{F}5}$ and $Z_{\mathrm{F}7}$. The events inside the black lines were counted. (b) $Z_{\mathrm{F}5}$ histogram for $Z_{\mathrm{F}7}=19$ events indicated by the arrow in plot (a). The shaded region corresponds to the area enclosed by the black line in plot (a). (c,d) $Z_{\mathrm{F}5}$ histograms for events that were not transported to F7 in cases of ${}^{42}\mathrm{Ca}$ and ${}^{50}\mathrm{Ca}$, respectively. The dotted green line represents the fitting result. The red solid and dashed lines indicate the compositions.

Figure 4

(a) $A$ dependence of $P_{\mathrm{evap}}$ values [Eq. (18)] with $E_{\max }=20$ MeV (thin black dashed line), 45(8) MeV (red solid line), and 70 MeV (thin black dotted line). (b) $A$ dependence of ${\sigma }_{\mathrm{CC}}$ for Ca isotopes on ${}^{12}\mathrm{C}$. The present ${}^{42\text{–}51}\mathrm{Ca}$ results and existing data [20, 21] are indicated by closed circles and open triangles, respectively. The black solid line and green dashed line represent the ${\sigma }_{\mathrm{CC}}$ values from the Glauber(ZROLA) calculation [Eq. (12)] and the Glauber-like(ZROLA) calculation with the correction factor [18] (Eq. (13)), respectively. The thin black dashed line, red solid line, and thin black dotted line show the ${\sigma }_{\mathrm{CC}}$ calculations from the Glauber(CE) model [Eq. (17)] with $E_{\max }=20$, 45(8), and 70 MeV, respectively.

Figure 5

Energy dependence of ${\sigma }_{\mathrm{R}}$ for ${}^{12}\mathrm{C}$ on ${}^{12}\mathrm{C}$ [16, 38, 39]. The red solid and black dotted lines indicate the present ZROLA calculation with the best-fit parameter for the Fermi motion, $C_{\mathrm{FM}}=0.53$, and the conventional ZROLA calculation without the Fermi motion effect on ${\sigma }_{ij}$, respectively.

Figure 6

Ratio of experimental ${\sigma }_{\mathrm{CC}}$ value to the ${\tilde{\sigma }}_{\mathrm{CC}}$ value calculated using Eq. (12) as a function of one-proton separation energy $S_{\mathrm{p}}$. The present results of ${}^{42\text{–}51}\mathrm{Ca}$ are indicated by the red closed circles. The open triangles represent the experimental results [20, 21] of Ca (red), Ti (brown), Cr (green), and Fe (blue). The experimental ${\sigma }_{\mathrm{CC}}$ and calculated ${\tilde{\sigma }}_{\mathrm{CC}}$ values of Ti, Cr, and Fe isotopes are illustrated in Figs. 9 by the closed circles and the black solid lines, respectively.

Figure 7

(a,b) $f(E_{\mathrm{ex}},A_{\mathrm{P}}-x,Z_{\mathrm{P}})$ of $1n$ (black), $2n$ (red), and $3n$ (blue) channels for ${}^{42}\mathrm{Ca}$ and ${}^{48}\mathrm{Ca}$. (c,d) $E_{\mathrm{ex}}$ dependence of $r_{-x\mathrm{n}}{w}_{-x\mathrm{n}}\left(E_{\mathrm{ex}}\right)$ (dotted) and $r_{-x\mathrm{n}}{w}_{-x\mathrm{n}}\left(E_{\mathrm{ex}}\right)f(E_{\mathrm{ex}},A_{\mathrm{P}}-x,Z_{\mathrm{P}})$ (solid) of the respective channels, where $E_{\max }=45$ MeV. The inset of (d) shows a magnified view.

Figure 8

$r_{-x\mathrm{n}}$ (open) and $p_{-x\mathrm{n}}$ (closed) of (a) ${}^{42}\mathrm{Ca}$ (blue) and (b) ${}^{48}\mathrm{Ca}$ (red) as functions of the number of removed neutrons $x$.

Figure 9

Experimental ${\sigma }_{\mathrm{CC}}$ results at $E\simeq 280$ MeV/nucleon and Glauber(CE) ${\sigma }_{\mathrm{CC}}$ values for (a) Be, (b) C, (c) O, (d) Ti, (e) Cr, and (f) Fe isotopes as functions of $A$. The experimental results were taken from Refs. [19] (closed triangles), [20] (open triangles), [29] (open squares), [59, 60] (open diamonds), and [58] (open circles). Each line is defined as in Fig. 4. The ratios of the experimental values to the calculated ones using Eq. (12) (black solid lines) for Ti, Cr, and Fe isotopes are represented in Fig. 6. The ratios of the experimental values to the Glauber(CR) calculations with $E_{\max }=45$ MeV are represented in Fig. 10.

Figure 10

Ratios of experimental ${\sigma }_{\mathrm{CC}}$ results on ${}^{12}\mathrm{C}$ at around 280 MeV/nucleon to Glauber(CE) calculations for (a) Be, (b) C, (c) O, (d) Ca, (e) Ti, (f) Cr, and (g) Fe isotopes. Only the experimental data of nuclides, for which $r_{\mathrm{p}}$ were previously measured, are plotted. The marker symbols are the same as for Figs. 4 and 9. The shaded band shows the standard deviation ($\pm 0.016$) around $\mathrm{ratio}=1.004$ of the experimental data for nuclides from C to Fe isotopes, except for some ${}^{12}\mathrm{C}$ data (open diamond and open circle).

Figure 11

Relation between $r_{\mathrm{p}}$ and ${\sigma }_{\mathrm{CC}}$ for (a) ${}^{42}\mathrm{Ca}$ and (b) ${}^{51}\mathrm{Ca}$. The red solid lines indicate the calculations of ${\sigma }_{\mathrm{CC}}$ using the Glauber(CE) model with $E_{\max }=45$ MeV, while the red dashed lines indicate the uncertainty of the calculation owing to the $E_{\max }$ value ($\pm 8$ MeV).

Figure 12

Estimated relative uncertainty of the deduced $r_{\mathrm{p}}$ value from ${\sigma }_{\mathrm{CC}}$ of Ca isotopes as a function of $A$ of projectile Ca isotopes. The $P_{\mathrm{evap}}$ values using $E_{\max }=45$ MeV were adopted. The $\delta {\sigma }_{\mathrm{CC}}/{\sigma }_{\mathrm{CC}}$ value was assumed to be 1%. The black dotted, red solid, and blue dashed lines indicate the results under the assumptions of $\delta {\sigma }_{\mathrm{R}}/{\sigma }_{\mathrm{R}}=0\%$, 1%, and 3%, respectively. The estimation with the assumption of $\delta {\sigma }_{\mathrm{R}}/{\sigma }_{\mathrm{R}}=3\%$ simulates the case wherein the experimental ${\sigma }_{\mathrm{R}}$ data are not available.

Figure 13

Experimental ${\sigma }_{\mathrm{CC}}$ results at $E\simeq 900$ MeV/nucleon and Glauber(CE) values of ${\sigma }_{\mathrm{CC}}$ for (a) Be, (b) B, (c) C, (d) N, (e) O, and (f) F isotopes as functions of $A$. The systematic experimental results are represented by crosses [17] and closed triangles [24, 25, 26, 27]. The open triangles in (d) indicate ${\sigma }_{\mathrm{CC}}^{ex,\mathrm{noveto}}$ values from Ref. [27]. The open diamond and circle indicate experimental values from Refs. [24] and [58], respectively. Each line is defined as in Fig. 4.

Figure 14

Energy dependence of ${\sigma }_{\mathrm{CC}}$ for ${}^{28}\mathrm{Si}$ on ${}^{12}\mathrm{C}$. The experimental values are represented by closed triangles [18], open circles [58], open diamonds [80], and open squares [81]. Each line is defined as in Fig. 4.

Figure 15

Energy dependence of ${\sigma }_{\mathrm{R}}$ for ($\mathrm{a}){}^9\mathrm{Be}$ and ($\mathrm{b}){}^{27}\mathrm{Al}$ on ${}^{12}\mathrm{C}$ [16, 38, 39]. The red solid and black dotted lines indicate the present ZROLA calculation with the best-fit parameter for the Fermi motion, $C_{\mathrm{FM}}=0.53$, and the conventional ZROLA calculation without the Fermi motion effect on ${\sigma }_{ij}$, respectively. The experimental data are taken from Ref. [16] (triangle) and Ref. [38] (circles).

Figure 16

Same as Fig. 4, but with the Glauber-model calculations without the Fermi-motion effect under the ZROLA. The blue solid line and the corresponding shaded region indicate the Glauber(CE) model with $E_{\max }=55\left(6\right)$ MeV. The black solid line and green dashed line represent the ${\sigma }_{\mathrm{CC}}$ values from the Glauber(ZROLA) calculation [Eq. (12)] and the Glauber(ZROLA) calculation with the correction factor [18] [Eq. (13)], respectively. The present ${}^{42\text{–}51}\mathrm{Ca}$ results and existing data [20, 21] are indicated by closed circles and open triangles, respectively.

Figure 17

Same as Fig. 14, but with the Glauber-model calculations without the Fermi-motion effect under the ZROLA. The definitions of all the lines are the same as in Fig. 16. The experimental values are represented by closed triangles [18], open circles [58], open diamonds [80], and open squares [81].