Pairing-energy coefficients of neutron-rich fragments in spallation reactions

The ratio of pairing-energy coefficient to temperature ($a_{p}/T$) of neutron-rich fragments produced in spallation reactions has been investigated by adopting an isobaric yield ratio method deduced in the framework of a modified Fisher model. A series of spallation reactions, 0.5$A$ and 1$A$ GeV $^{208}$Pb + $p$, 1$A$ GeV $^{238}$U + $p$, 0.5$A$ GeV $^{136}$Xe + $d$, 0.2$A$, 0.5$A$ and 1$A$ GeV $^{136}$Xe + $p$, and $^{56}$Fe + $p$ with incident energy ranging from 0.3$A$ to 1.5$A$ GeV, has been analysed. An obvious odd-even staggering is shown in the fragments with small neutron excess ($I\equiv N - Z$), and in the relatively small-$A$ fragments which have large $I$. The values of $a_{p}/T$ for the fragments, with $I$ from 0 to 36, have been found to be in a range from -4 to 4, and most values of $a_{p}/T$ fall in the range from -1 to 1. It is suggested that a small pairing-energy coefficient should be considered in predicting the cross sections of fragments in spallation reactions. It is also concluded that the method proposed in this article is not good for fragments with $A/A_{s} >$ 85\% (where $A_{s}$ is the mass number of the spallation system).