Final-state interactions and spin structure in $E1$ breakup of $^{11}\mathrm{Li}$ in halo effective field theory

Final-state interactions and spin structure in $E 1$ breakup of ${}^{11}{Li}$ in halo effective field theory

Matthias Göbel, Bijaya Acharya, Hans-Werner Hammer, and Daniel R. Phillips

Phys. Rev. C 107, 014617 – Published 25 January 2023

Abstract

We calculate the $E 1$ breakup of the $2 n$ halo nucleus ${}^{11}{Li}$ in halo effective field theory (Halo EFT) at leading order. In Halo EFT, ${}^{11}{Li}$ is treated as a three-body system of a ${}^{9}{Li}$ core and two neutrons. We present a detailed investigation of final-state interactions (FSIs) in the neutron-neutron $(n n)$ and neutron-core $(n c)$ channels. We employ Møller operators to formulate an expansion scheme that satisfies the non-energy-weighted cluster sum rule and successively includes higher-order terms in the multiple-scattering series for the FSI. Computing the $E 1$ strength up to third order in this scheme, we observe apparent convergence and good agreement with experiment. The neutron-neutron FSI is by far the most important contribution and largely determines the maximum value of the $E 1$ distribution. However, inclusion of $n c$ FSI does shift the peak position to slightly lower energies. Moreover, we investigate the sensitivity of the $E 1$ response to the spin structure of the neutron- ${}^{9}{Li}$ interaction. We contrast results for an interaction that is the same in the spin-1 and spin-2 channels with one that is only operative in the spin-2 channel, and find that good agreement with experimental data is only obtained if the interaction is present in both spin channels. The latter case is shown to be equivalent to a calculation in which the spin of ${}^{9}{Li}$ is neglected.

Received 5 August 2022
Accepted 9 November 2022

DOI:https://doi.org/10.1103/PhysRevC.107.014617

Physics Subject Headings (PhySH)

Breakup reactions Coulomb dissociation Effective field theory Few-body systems

6 ≤ A ≤ 19

Cluster models Nuclear reactions

Nuclear Physics

Authors & Affiliations

Matthias Göbel ^1,*, Bijaya Acharya ^2,3, Hans-Werner Hammer ^1,4, and Daniel R. Phillips ³

¹Technische Universität Darmstadt, Department of Physics, Institut für Kernphysik, 64289 Darmstadt, Germany
²Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
³Institute of Nuclear and Particle Physics and Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
⁴ExtreMe Matter Institute EMMI and Helmholtz Forschungsakademie Hessen für FAIR (HFHF), GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany

^*goebel@theorie.ikp.physik.tu-darmstadt.de

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Vol. 107, Iss. 1 — January 2023

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Figure 1
Diagrammatic representation of the $E 1$ matrix elements of distributions differing in the included FSIs. The neutrons are represented by blue solid lines and the ${}^{9}{Li}$ core is represented by an orange dashed line. The first row describes the matrix element without FSI, whereby the ellipse with the external line on the left side represents the complete matrix element resulting from the action of the $E 1$ operator on the ground state. On the right-hand side of the first row this is made more explicit: The $E 1$ photons are represented by wiggly lines and the ground state is composed from its Faddeev amplitudes represented by ellipses with corresponding labels. The $n n$ and $n c$ t-matrices are represented by circles. The second row shows the contributions for the matrix element that includes $n n$ FSI, while the third row describes the matrix element with $n c$ FSI.
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Figure 2
The left panels (a) shows $E 1$ strength distributions of ${}^{11}{Li}$ with different FSIs included. The right panel (b) shows the corresponding cumulative $E 1$ strength distributions. Numerical uncertainties are indicated by bands, which are very narrow here. They were obtained by comparing the calculations with ones having roughly two-thirds as many mesh points and a cutoff of three-fourths of the original one.
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Figure 3
$E 1$ strength distributions with $n n$ FSI included and different numbers of $n c$ interaction channels for the ground state. We show the result by Hongo and Son [51] (blue), which corresponds to no $n c$ interaction spin channels, in comparison with our results using one spin channel (orange) and two spin channels (green). The left panel (a) shows the theoretical curves. In the right panel (b) these distributions have been folded with the detector resolution and compared to the experimental data from Nakamura et al. [25] (adjusted to the current $S_{2 n}$ value).
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Figure 4
The left panel (a) shows $E 1$ strength distributions of ${}^{11}{Li}$ with different FSIs including higher-order schemes. The right panel (b) shows the corresponding cumulative $E 1$ strength distributions. The small horizontal band again shows the expected asymptotic value for the cumulative $E 1$ strength distribution, based on $〈 r_{c}^{2} 〉$ extracted from $F_{c} (k^{2})$ . Note that the results for $Ω_{n n}^{†} Ω_{n^{'} c}^{†} Ω_{n c}^{†}$ and for $\frac{1}{2} Ω_{n n}^{†} (Ω_{n^{'} c}^{†} Ω_{n c}^{†} + Ω_{n c}^{†} Ω_{n^{'} c}^{†})$ are on top of each other. The same is true for $Ω_{n^{'} c}^{†} Ω_{n c}^{†} Ω_{n n}^{†}$ and $\frac{1}{2} (Ω_{n^{'} c}^{†} Ω_{n c}^{†} + Ω_{n c}^{†} Ω_{n^{'} c}^{†}) Ω_{n n}^{†}$ .
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Figure 5
The left panel (a) shows our results for the $E 1$ strength distribution in comparison with the universal curve by Hongo and Son [51]. The right panel (b) shows our results and the result of Hongo and Son folded with the detector resolution in comparison with the experimental data from Nakamura et al. [25] (adjusted to the current $S_{2 n}$ value). The uncertainty bands show the estimated uncertainties of the leading-order EFT results. The uncertainty stemming from approximations of the multiple-scattering series by products of Møller operators can be estimated by comparing the curve using $Ω_{n n}^{†} Ω_{n^{'} c}^{†} Ω_{n c}^{†}$ with the one using $Ω_{n^{'} c}^{†} Ω_{n c}^{†} Ω_{n n}^{†}$ .
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Figure 6
Quotients of $E 1$ strength distributions with ${\bar{λ}}_{\max} = 5$ and with ${\bar{λ}}_{\max} = 3$ differing in the FSI treatment.
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Physical Review C

covering nuclear physics

Final-state interactions and spin structure in $E 1$ breakup of ${}^{11}{Li}$ in halo effective field theory

Matthias Göbel, Bijaya Acharya, Hans-Werner Hammer, and Daniel R. Phillips

Phys. Rev. C 107, 014617 – Published 25 January 2023

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Physical Review C

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Final-state interactions and spin structure in E1 breakup of Li11 in halo effective field theory

Matthias Göbel, Bijaya Acharya, Hans-Werner Hammer, and Daniel R. Phillips

Phys. Rev. C 107, 014617 – Published 25 January 2023

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Final-state interactions and spin structure in $E 1$ breakup of ${}^{11}{Li}$ in halo effective field theory