Elastic Coulomb breakup of ${}^{34}\mathrm{Na}$

Background: ${}^{34}\mathrm{Na}$ is conjectured to play an important role in the production of seed nuclei in the alternate $r$-process paths involving light neutron rich nuclei very near the $\beta$-stability line, and as such, it is important to know its ground state properties and structure to calculate rates of the reactions it might be involved in, in the stellar plasma. Found in the region of ‘island of inversion’, its ground state might not be in agreement with normal shell model predictions.

Purpose: The aim of this paper is to study the elastic Coulomb breakup of ${}^{34}\mathrm{Na}$ on ${}^{208}\mathrm{Pb}$ to give us a core of ${}^{33}\mathrm{Na}$ with a neutron and in the process we try and investigate the one neutron separation energy and the ground state configuration of ${}^{34}\mathrm{Na}$.

Method: A fully quantum mechanical Coulomb breakup theory within the architecture of post-form finite range distorted wave Born approximation extended to include the effects of deformation is used to research the elastic Coulomb breakup of ${}^{34}\mathrm{Na}$ on ${}^{208}\mathrm{Pb}$ at 100 MeV/u. The triple differential cross section calculated for the breakup is integrated over the desired components to find the total cross-section, momentum, and angular distributions as well as the average momenta, along with the energy-angular distributions.

Results: The total one neutron removal cross section is calculated to test the possible ground state configurations of ${}^{34}\mathrm{Na}$. The average momentum results along with energy-angular calculations indicate ${}^{34}\mathrm{Na}$ to have a halo structure. The parallel momentum distributions with narrow full widths at half-maxima signify the same.

Conclusion: We have attempted to analyze the possible ground state configurations of ${}^{34}\mathrm{Na}$ and in congruity with the patterns in the ‘island of inversion’ conclude that even without deformation, ${}^{34}\mathrm{Na}$ should be a neutron halo with a predominant contribution to its ground state most probably coming from ${}^{33}\mathrm{Na}(3/2^{+})\otimes$ $2p_{3/2}\nu$ configuration. We also surmise that it would certainly be useful and rewarding to test our predictions with an experiment to put stricter limits on its ground state configuration and binding energy.

Figure 1

The three-body Jacobi coordinate system with a deformed projectile.

Figure 2

The total cross section $\left({\sigma }_{-1n}\right)$ vs one neutron separation energy ($S_n$) for ${}^{34}\mathrm{Na}$ breaking on ${}^{208}\mathrm{Pb}$ at 100 MeV/u beam energy. The deformation parameter, ${\beta }_2$, has been set to 0. The solid line shows the cross section for $p$-wave neutron configuration while the dashed line shows the same from the $s$ wave. The dotted line is for the $f$-wave configuration whereas the dash-dotted line gives the effect of equal contribution from all the states taken together with spectroscopic factor ($C^{2}S$) for each being equal to 0.33.

Figure 3

The relative energy spectra of ${}^{34}\mathrm{Na}$ breaking elastically on ${}^{208}\mathrm{Pb}$ at 100 MeV/u beam energy due to Coulomb dissociation. For a neutron separation energy of 0.17 MeV, the deformation parameter, ${\beta }_2$ is varied from 0.0 to 0.3. (a) The g.s. for ${}^{34}\mathrm{Na}$ is assumed to be formed from contribution of $2p_{3/2}$ valence neutron. (b) The same when the g.s. is formed by $2s_{1/2}$ contribution.

Figure 4

Peak position of relative energy spectra as a function of $S_n$ in the breakup of ${}^{34}\mathrm{Na}$ on ${}^{208}\mathrm{Pb}$ at 100 MeV/u for deformation parameter, $-0.3\le {\beta }_2\le 0.0$ on the left, and $0.0\le {\beta }_2\le 0.3$ on the right. The average slope of all the curves is 0.9365. The g.s. of ${}^{34}\mathrm{Na}$ is supposed to be formed by contribution of $\nu$($2p_{3/2}$) state. The lines are a guide to the eye.

Figure 5

The parallel momentum distribution of ${}^{33}\mathrm{Na}$ for ${}^{34}\mathrm{Na}$ breaking elastically on ${}^{208}\mathrm{Pb}$ at 100 MeV/u beam energy. ${}^{33}\mathrm{Na}\otimes$ $2p_{3/2}\nu$ is assumed to form the ground state of ${}^{34}\mathrm{Na}$ for a neutron separation energy of 0.17 MeV. The solid, dashed, dash-dotted, and dotted lines correspond to a quadrupole deformation value of 0.0, 0.1, 0.2, and 0.3, respectively.

Figure 6

The neutron energy-angular distribution for ${}^{34}\mathrm{Na}$ breaking on ${}^{208}\mathrm{Pb}$ at 100 MeV/u for separation energy of 0.17 MeV. The ground state of ${}^{34}\mathrm{Na}$ is assumed to be given by ${}^{33}\mathrm{Na}\otimes$ $2p_{3/2}\nu$. The solid, dashed, dash-dotted, and dotted curves correspond to a quadrupole deformation (${\beta }_2$) value of 0.0, 0.1, 0.2, and 0.3, respectively. The neutron angle, ${\theta }_n=1^{\circ }$.

Figure 7

Average momentum of the ${}^{33}\mathrm{Na}$ fragment in the breakup of ${}^{34}\mathrm{Na}$ on ${}^{208}\mathrm{Pb}$ at 100 MeV/u as function of its detection angle.

Figure 8

The angular distribution for ${}^{34}\mathrm{Na}$ breaking on ${}^{208}\mathrm{Pb}$ at 100 MeV/u beam energy to form ${}^{33}\mathrm{Na}$ and a neutron. The deformation parameter, ${\beta }_2$ is varied from 0.0 to 0.3 for a neutron separation energy of 0.17 MeV. The g.s. of ${}^{34}\mathrm{Na}$ is supposed to be formed by $2p_{3/2}$ configuration of the valence neutron.