Nucleon-Nucleon Scattering Parameters from Final-State Interactions

The energy spectra of $\alpha$ particles were measured at 5° lab for the reaction ${}_{}{}^3{}_{}{}^{}\mathrm{He}({}_{}{}^3{}_{}{}^{}\mathrm{He}, \alpha )pp$ using a 20-MeV ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ beam; at 5° and 8° lab for the reaction ${}_{}{}^3{}_{}{}^{}\mathrm{He}(t, \alpha )np$; and at 5°, 8°, and 10° lab for the reaction ${}_{}{}^3{}_{}{}^{}\mathrm{H}(t, \alpha )nn$ using a 22-MeV triton beam. The 5° lab spectra corresponding to low relative energy in the unobserved two-nucleon system were well fitted with Watson-Migdal final-state-interaction theory for the following scattering lengths: $a_{\mathrm{pp}}=-7.52\mathrm{}\pm 0.22$ F, $a_{\mathrm{np}}=-21.5\mathrm{}\pm 2.3$ F, and $a_{\mathrm{nn}}=-16.96\mathrm{}\pm 0.51$ F. Analyzing the change-symmetric reactions with the comparison method and using the accepted value for $a_{\mathrm{pp}}$, we find $a_{\mathrm{nn}}=-18.11\mathrm{}\pm 0.75$ F. When compared with Coulomb-corrected values of $a_{\mathrm{pp}}$, these values of $a_{\mathrm{nn}}$ imply that nucleon-nucleon forces are charge-symmetric to better than 1%. In obtaining the scattering lengths, the effective range was fixed at $r_0=2.84\mathrm{}$ F. Allowing both the effective range and scattering length to vary in the fitting procedure for the $n-n$ final-state data, we find $a_{\mathrm{nn}}=-17.4\mathrm{}\pm 1.8$ F and ${\left(r_0\right)}_{\mathrm{nn}}=2.4\mathrm{}\pm 1.5$ F. If the $n-n$ scattering length is fixed at the value - 17.0 F, our data determine the effective range to be ${\left(r_0\right)}_{\mathrm{nn}}=2.75\mathrm{}\pm 0.35$ F.