Scaling phenomena of isobaric yields in projectile fragmentation, spallation, and fission reactions

Background: The isobaric ratio difference scaling phenomenon, which has been found for the fragments produced in projectile fragmentation reactions, is related to the nuclear density change in reaction systems.

Purpose: To verify whether the isobaric ratio difference scaling exists in the fragments produced in the spallation and fission reactions.

Methods: The isobaric ratio difference scaling, denoted by $S_{\mathrm{\Delta }ln{R}_{21}}$, is in theory deduced within the framework of the canonical ensemble theory at the grand-canonical limitation. The fragments measured in a series of projectile fragmentation, spallation, and fission reactions have been analyzed.

Results: A good $S_{\mathrm{\Delta }ln{R}_{21}}$ scaling phenomenon is shown for the fragments produced both in the projectile fragmentation reactions and in the spallation reactions, whereas the $S_{\mathrm{\Delta }ln{R}_{21}}$ scaling phenomenon for the fragments in the fission reaction is less obvious.

Conclusions: The $S_{\mathrm{\Delta }ln{R}_{21}}$ scaling is used to probe the properties of the equilibrium system at the time of fragment formation. The good scaling of $S_{\mathrm{\Delta }ln{R}_{21}}$ suggests that the equilibrium state can be achieved in the projectile fragmentation and spallation reactions. Whereas in the fission reaction, the result of $S_{\mathrm{\Delta }ln{R}_{21}}$ indicates that the equilibrium of the system is hard to achieve.

Figure 1

$S_{\mathrm{\Delta }ln{R}_{21}}$'s for the fragments produced in the $140A\mathrm{MeV}{}^{40}\mathrm{Ca}+{}^{181}\mathrm{Ta}/{}^9\mathrm{Be}$ [in (a)], ${}^{48}\mathrm{Ca}+{}^{181}\mathrm{Ta}/{}^9\mathrm{Be}$ [in (b)], ${}^{58}\mathrm{Ni}+{}^{181}\mathrm{Ta}/{}^9\mathrm{Be}$ [in (c)], and ${}^{64}\mathrm{Ni}+{}^{181}\mathrm{Ta}/{}^9\mathrm{Be}$ [in (d)] reactions measured by Mocko et al. [34]. $m$ denotes the difference between the $I$'s of the isobars.

Figure 2

$S_{\mathrm{\Delta }ln{R}_{21}}$'s of the fragments produced in the $140A\mathrm{MeV}{}^{58}\mathrm{Ni}/{}^{40}\mathrm{Ca}+{}^9\mathrm{Be}$ [in (a)], ${}^{48}\mathrm{Ca}/{}^{64}\mathrm{Ni}+{}^9\mathrm{Be}$ [in (b)], ${}^{58}\mathrm{Ni}/{}^{40}\mathrm{Ca}+{}^{181}\mathrm{Ta}$ [in (c)], and ${}^{48}\mathrm{Ca}/{}^{64}\mathrm{Ni}+{}^{181}\mathrm{Ta}$ [in (d)] reactions measured by Mocko et al. [34]. $m$ denotes the difference between the $I$'s of the isobars.

Figure 3

$S_{\mathrm{\Delta }ln{R}_{21}}$'s for the fragments produced in the $1A\mathrm{GeV}{}^{136}\mathrm{Xe}/{}^{124}\mathrm{Xe}+\mathrm{Pb}$ reactions measured by Henzlova et al. at the FRS-GSI [36]. $m$ denotes the difference between the $I$'s of the isobars.

Figure 4

$S_{\mathrm{\Delta }ln{R}_{21}}$'s for the fragments produced in the $500A\mathrm{MeV}{}^{136}\mathrm{Xe}+d/p$ spallation reactions. The cross sections of residue fragments measured in the ${}^{136}\mathrm{Xe}+p$ and ${}^{136}\mathrm{Xe}+d$ reactions are taken from Refs. [46, 47], respectively. $m$ denotes the difference between the $I$'s of the isobars.

Figure 5

$S_{\mathrm{\Delta }ln{R}_{21}}$'s for the fragments produced in the $1A\mathrm{GeV}{}^{238}\mathrm{U}/{}^{136}\mathrm{Xe}+p$ spallation reactions. The cross sections of the fragments measured in the ${}^{238}\mathrm{U}+p$ and ${}^{136}\mathrm{Xe}+p$ reactions are taken from Refs. [48, 49], respectively. $m$ denotes the difference between $I$'s of the isobars.

Figure 6

$S_{\mathrm{\Delta }ln{R}_{21}}$'s for the fragments produced in the $1A\mathrm{GeV}{}^{136}\mathrm{Xe}/{}^{56}\mathrm{Fe}+p$ spallation reactions. The cross sections of the fragments in the $1A\mathrm{GeV}{}^{136}\mathrm{Xe}+p$ and ${}^{56}\mathrm{Fe}+p$ reactions are taken from Refs. [49, 50], respectively. $m$ denotes the difference between the $I$'s of the isobars.

Figure 7

$S_{\mathrm{\Delta }ln{R}_{21}}$'s for the fragments produced in the $1A\mathrm{GeV}{}^{208}\mathrm{Pb}+d/p$ fission reactions. The cross section of fragments measured in the $1A\mathrm{GeV}{}^{208}\mathrm{Pb}+d$ and ${}^{208}\mathrm{Pb}+p$ reactions are taken from Refs. [52, 53], respectively. $m$ denotes the difference between the $I$'s of the isobars.

Figure 8

$S_{\mathrm{\Delta }ln{R}_{21}}$'s for the fragments produced in the $1A\mathrm{GeV}{}^{238}\mathrm{U}+d/p$ fission reactions. The cross sections of the fragments measured in the $1A\mathrm{GeV}{}^{238}\mathrm{U}+d$ and ${}^{238}\mathrm{U}+p$ reactions are taken from Refs. [51, 54], respectively. $m$ denotes the difference between the $I$'s of the isobars.

Figure 9

$S_{\mathrm{\Delta }ln{R}_{21}}$'s for the fragments produced in the $1A\mathrm{GeV}{}^{238}\mathrm{U}/{}^{208}\mathrm{Pb}+p$ fission reactions. The cross sections of the fragments measured in the $1A\mathrm{GeV}{}^{238}\mathrm{U}+p$ and ${}^{208}\mathrm{Pb}+p$ reactions are taken from Refs. [51, 53], respectively. $m$ denotes the difference between the $I$'s of the isobars.