# Nuclear Matrix Elements for Heavy Ion Sequential Double Charge Exchange Reactions

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## Abstract

**:**

## 1. Introduction

## 2. Theory of Sequential Double Charge Exchange Reactions

## 3. The Heavy Ion DSCE Reaction Amplitude

## 4. Multipole Structure of the Transition Form Factors and Nuclear Matrix Elements

#### 4.1. Nuclear Structure Aspects

_{B}. Typically, such two particle–two hole states are rather stable against perturbations. Thus, good approximation admixtures of higher-order configurations, caused by residual interactions inducing core polarization, can be neglected.

#### 4.2. Brief on Spectral Properties of DCE Transitions

## 5. Approximations

#### 5.1. Nuclear NME in Closure Approximation

#### 5.2. Effective Form Factors

## 6. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Angular Momentum Couplings

## References

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**Figure 1.**Schematic graphical representation of a Double Single Charge Exchange (DSCE) reaction $a({N}_{a},{Z}_{a})+A({N}_{A},{Z}_{A})\to b({N}_{a}\pm 2,{Z}_{a}\mp 2)+A({N}_{A}\mp 2,{Z}_{A}\pm 2)$, proceeded by the sequential twofold action of the isovector NN T-matrix, indicated by wavy lines. Each of the interaction events acts similar to a one-body operator on the target and projectile, respectively. Note that the diagram on the right is related to left one by a change in time order. A striking formal similarity to a 2ν2β nuclear matrix element (NME) is apparent.

**Figure 2.**Second-order DSCE unit strength cross sections for the reaction ${}^{18}O+{}^{40}Ca\to {}^{18}N+{}^{40}Ar$ at ${T}_{lab}=270$ MeV. From top to bottom, results are shown for total angular momentum transfer in the first single charge exchange (SCE) interaction ${L}_{1}=0,2$ and the second SCE interaction ${L}_{2}=0,2,4$, respectively. The average excitation energy was chosen as ${\overline{\omega}}_{\gamma}=10$ MeV. The angular range corresponds to momentum transfers up to 1000 MeV/c. Optical potentials and transition potentials are calculated in a double folding approach by using the (newly derived) nucleon–nucleon (NN) T-matrix at ${T}_{lab}=15$ MeV, parameterized as in References [24,25]. Optical potentials are calculated with Hartree–Fock–Bogoliubov (HFB) ground state densities according to Reference [18]. The cross sections are calculated by using average form factors derived from QRPA transition densities as discussed in the text. For comparison, SCE unit strength cross sections are shown in Figure 3.

**Figure 3.**First-order SCE unit strength cross sections for the reaction ${}^{18}O+{}^{40}Ca\to {}^{18}N+{}^{40}Ar$ at ${T}_{lab}=270$ MeV. The angular range corresponds to momentum transfers up to 1000 MeV/c. Optical potentials and transition potentials are calculated in a double folding approach by using the (newly derived) nucleon–nucleon (NN) T-matrix at ${T}_{lab}=15$ MeV, parameterized as in References [24,25]. Optical potentials are calculated with Hartree–Fock–Bogoliubov (HFB) ground state densities according to Reference [18]. The cross sections are calculated by using average form factors derived from QRPA transition densities as discussed in the text.

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**MDPI and ACS Style**

Lenske, H.; Bellone, J.; Colonna, M.; Gambacurta, D.
Nuclear Matrix Elements for Heavy Ion Sequential Double Charge Exchange Reactions. *Universe* **2021**, *7*, 98.
https://doi.org/10.3390/universe7040098

**AMA Style**

Lenske H, Bellone J, Colonna M, Gambacurta D.
Nuclear Matrix Elements for Heavy Ion Sequential Double Charge Exchange Reactions. *Universe*. 2021; 7(4):98.
https://doi.org/10.3390/universe7040098

**Chicago/Turabian Style**

Lenske, Horst, Jessica Bellone, Maria Colonna, and Danilo Gambacurta.
2021. "Nuclear Matrix Elements for Heavy Ion Sequential Double Charge Exchange Reactions" *Universe* 7, no. 4: 98.
https://doi.org/10.3390/universe7040098