# Spectroscopic factor strengths using approaches

###### Abstract

We have calculated the spectroscopic factor strengths for the one-proton and one-neutron pick-up reactions Al(,He)Mg and Al(,)Al within the framework of the shell model. We employed two different approaches : an in-medium similarity renormalization targeted for a particular nucleus, and the coupled- cluster effective interaction. We also compared our results with recently determined experimental spectroscopic factors.

###### pacs:

21.60.Cs## I Introduction

The nature and occupancy of the single-particle orbits for a nucleus can be determined from the spectroscopic factors (SFs). Experimentally the SFs can be measured with single-particle transfer reactions. These reactions are of two types, first one is stripping in which one nucleon is stripped from the incoming projectile, while the second one is pick-up reaction in which one nucleon is picked up by the projectile. The examples of neutron transfer pick-up reactions are , and (He,), while stripping reactions are , and (, He) bru (). The SF is defined by a matrix element between initial and final state corresponding to entrance channel and exit channels, respectively. It is possible to describe the capture or emission of single nucleons in stellar burning processes by calculating the nuclear matrix elements for single-nucleon spectroscopic factors in the nuclear structure calculations.

Studies of SFs in different region of nuclear chart are reported in Refs. mh (); sc1 (); sc2 (); brown (); Tsang1 (); Franchoo (); Lee (). Survey of excited state neutron spectroscopic factors for nuclei are reported by Tsang et al Tsang2 (). In this work they extracted 565 neutron spectroscopic factors for and shell nuclei by analyzing angular distributions, they also compared the experimental results with shell-model results.

To study different excited states and their spectroscopic factors for Mg, many experimental results were reported in Refs. mg1 (); mg2 (); mg3 (); mg4 (); mg5 (); mg6 (); mg7 (). Recently, the spectroscopic factors for Mg is reported in Ref. vishal1 () using Al(d,He)Mg reaction. The structure of Al deduced from experiments which enlighten different reaction channels were reported in refs. al1 (); al2 (); al3 (); al4 (); al5 (); al6 (); al7 (). The experimental results for SFs of 14 excited states for Al using Al(d,)Al reaction are reported in Ref. vishal2 (). The studies of Al and Mg are important for astrophysics point of view. The massive stars throughout the Galaxy dominate in the production of Al nat (), and it decays by to Mg.

In the present work we have performed shell-model calculations using approaches for one-neutron and one-proton pick-up reaction on Al within the framework of the shell model.

## Ii Shell Model Analysis

We performed shell-model calculations using two modern approaches: an in-medium similarity renormalization targeted for a particular nucleus rag2 () and the coupled-cluster effective interaction (CCEI) ccei2 (). We also compared results with a phenomenological USDB interaction usdausdb (). For the diagonalization of matrices we used shell-model code NuShellX nushellx ().

Recently, Stroberg reported mass-dependent Hamiltonians for -shell nuclei using the in-medium similarity renormalization group (IM-SRG) based on chiral two- and three-nucleon interactions rag1 (). Further extension has been done for IM-SRG calculation based on ensemble reference states to consider residual forces among valence nucleons. This is a nucleus-dependent valence-space approach to study nuclear structure properties rag2 (). In the present work we performed calculations for the spectroscopic factor strengths using separate effective interactions for Mg, Al and Al based on nucleus-dependent valence space rag2 ().

The coupled-cluster effective interaction (CCEI) ccei2 (), uses A- dependent Hamiltonian,

(1) |

with an initial next-to-next-to-next-to-leading order () chiral interaction, and next-to-next-to-leading order () local chiral interaction, using similarity renormalization group transformation. From this the shell-model Hamiltonian in a valence-space obtained from coupled-cluster theory. Where, the CCEI Hamiltonian is

(2) |

For the shell-model space this Hamiltonian is limited for one- and two-body terms. The Two-body term is computed using Okubo-Lee-Suzuki similarity transformation. In the Eq. (2), A is the mass of the nucleus, is the mass of core. is the Hamiltonian for the core, is the valence one-body Hamiltonian, and is the additional two-body Hamiltonian. Coupled-cluster theory results for spectroscopic factor for proton and neutron removal from O is reported in Ref. nav ().

The Hamiltonian of the USDB is based on a renormalized matrix by fitting two-body matrix elements with experimental data for binding energies and excitation energies for the shell nuclei usdausdb (); brown2 (). The USDB interactions is fitted by varying 56 linear combinations of two-body matrix elements. The rms deviations of 130 keV were obtained between experimental and theoretical energies for USDB interaction. In the present work before calculating the SF, first we examined the wave functions of concerned nuclei using IM-SRG, CCEI and USDB effective interactions. The comparison between the calculated and experimental levels for Mg and Al is reported in Figs. 1 and 2. The results of the USDB interaction are much better than IM-SRG and CCEI interactions. In the case of Al we calculated only the g.s.().

We can define the spectroscopic amplitudes for pick-up and stripping reactions by taking the expectation values of the operators and between the states of nuclei with A-1 and A, and A+1 and A. The spectroscopic factor in terms of the reduced matrix elements of is given by:

(3) |

where the factor is by convention associated with heavier mass . Here, indices distinguish the various basis states with the same value poves (); brownbook ().

Experimentally vishal1 (); vishal2 (), the spectroscopic factors of different states were extracted using the relation between experimental cross section and theoretical cross sections;

(4) |

where, is the experimental differential cross-section and is the cross-section predicted by the DWUCK4 code. () is the total angular momentum of the orbital from where proton is picked up. is the isospin Clebsch-Gordon coefficient and S is the spectroscopic factor.

The uncertainty in the experimental SFs may come due to following: (i) the zero-range parameter may be uncertain, (ii) the optical potential may be uncertain, (iii) the zero-range distorted-wave Born-approximation is not sufficient rad1 (); rad2 (); rad3 (). In the Refs. vishal1 (); vishal2 (), it was reported that the replacement of the zero-range approximation with finite-range and nonlocal parameters reduces the SFs up to 45-50 %.

Furnstahl and Hameer, using effective field theory, tried to determine whether occupation numbers and momentum distributions of nucleons in nuclei are observables. They claimed that these quantities can only be defined if we take specific form of the Hamiltonian, regularization scheme etc. fur (). In the effective field theory, there is no definite form of the Hamiltonian, thus it is not possible to defined occupation numbers (or even momentum distribution). The “nonobservable” nuclear quantities such as momentum distribution and spectroscopic factor using parton distribution function ( PDFs) reported in Ref. fur_jpg (). The inclusion of long-range (low-momentum) pion-exchange tensor forces is important. But the recent study for the quenching of spectroscopic factors suggest that a long-range correlation is more dominating barbieri (). The uncertainty of the SFs coming from different sources was reported in the review article by Dickhoff and Barbieri barbieri1 (). In-elastic proton scattering is surface reaction, thus no detailed information is obtained related to the interior of the nucleus; this will gives rise to an error of 10%. Another uncertainty is due to the choice of the electron-proton cross-section; this will give a small uncertainty in the analysis of low- data.

Expt. | J | [USDB] | [IM-SRG] | [CCEI] | [Expt.] | [USDB] | [IM-SRG] | [CCEI] | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

keV | keV | keV | keV | |||||||||

0 | 0 | 0(0) | 0(0) | 0(0) | 0.170.05 | … | 0.276 | … | 0.232 | … | 0.339 | |

1806 | 2 | 1897(2) | 1631(2) | 1466(2) | 0.002 | 0.570.14 | 0.014 | 0.876 | 0.006 | 0.647 | 0.036 | 0.140 |

2935 | 2 | 3007(2) | 2374(2) | 2426(2) | 0.002 | 0.130.03 | 0.021 | 0.090 | 0.030 | 0.072 | 0.002 | 0.170 |

4317(3) | 3379(3) | 3972(3) | 0.072 | 0.005 | 0.0002 | 0.0001 | 0.055 | 0.000 | ||||

4365(4) | 4565(4) | 3960(4) | … | 1.620 | … | 1.038 | … | 0.795 | ||||

4449(2) | 3891(2) | 3970(2) | 0.044 | 0.074 | 0.0002 | 0.057 | 0.016 | 0.196 | ||||

2 | 4882(2) | 4209(2) | 4768(2) | 0.140 | 0.022 | 0.220 | 0.053 | 0.078 | 0.091 | |||

2 | 5386(2) | 5150(2) | 5766(2) | 0.009 | 0.004 | 0.103 | 0.024 | 0.205 | 0.122 | |||

4 | 5523(4) | 5608(4) | 5248(4) | … | 0.185 | … | 0.030 | … | 0.009 | |||

4 | 5892(4) | 6267(4) | 5988(4) | … | 0.001 | … | 0.030 | … | 0.025 | |||

3 | 6179(3) | 5881(3) | 6726(3) | 0.028 | 0.006 | 0.067 | 0.022 | 0.185 | 0.002 |

IM-SRG | USDB | |||
---|---|---|---|---|

J | Configuration | Configuration | ||

0 | 49 | 62 | ||

2 | 43 | 52 | ||

2 | 29 | 29 | ||

3 | 33 | 35 | ||

4 | 35 | 52 | ||

2 | 23 | 25 | ||

2 | 28 | 33 | ||

2 | 22 | 33 | ||

4 | 31 | 26 | ||

4 | 29 | 31 | ||

3 | 32 | 50 |

Expt. | J | [USDB] | [IM-SRG] | [CCEI] | [Expt.] | [USDB] | [IM-SRG] | [CCEI] | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

keV | keV | keV | keV | |||||||||

0 | 5 | 0(5) | 247(5) | 838(5) | … | 0.730.21 | … | 1.089 | … | 0.811 | … | 0.627 |

228.3 | 0 | 229(0) | 903(0) | 2563(0) | … | 0.090.03 | … | 0.139 | … | 0.117 | … | 0.004 |

416.8 | 3 | 520(3) | 1963(3) | 1873(3) | … | 0.320.07 | 0.201 | 0.0001 | 0.0003 | 0.321 | 0.028 | 0.004 |

1057.7 | 1 | 1034(1) | 1853(1) | 1224(1) | … | 0.170.05 | … | 0.004 | … | 0.0007 | … | 0.114 |

1759.0 | 2 | 1583(2) | 1691(2) | 1704(2) | … | 0.0380.006 | 0.031 | 0.004 | 0.001 | 0.0001 | 0.046 | 0.023 |

1850.6 | 1 | 1818(1) | 2399(1) | 3497(1) | … | 0.0190.004 | 0.007 | 0.438 | … | 0.001 | … | 0.027 |

2068.8 | 2 | 2126(2) | 2530(2) | 2757(2) | … | 0.260.06 | 0.007 | 0.438 | 0.0004 | 0.002 | 0.007 | 0.146 |

2365.1 | 3 | 2155(3) | 2825(3) | 2832(3) | … | 0.130.02 | 0.008 | 0.310 | 0.0005 | 0.005 | 0.022 | 0.071 |

2545.3 | 3 | 2402(3) | 3466(3) | 3260(3) | … | 0.160.03 | 0.002 | 0.120 | 0.001 | 0.008 | 0.049 | 0.068 |

3159.8 | 2 | 3236(2) | 3404(2) | 3853(2) | … | 0.060.01 | 0.010 | 0.045 | 0.016 | 0.051 | 0.000 | 0.068 |

3402.6 | 5 | 3398(5) | 3496(5) | 4223(5) | … | 0.060.01 | … | 0.077 | … | 0.133 | … | 0.0007 |

4705.3 | 4 | 4857(4) | 5282(4) | 4704(4) | … | 0.270.08 | … | 0.0003 | … | 0.007 | … | 0.008 |

IM-SRG | USDB | |||
---|---|---|---|---|

J | Configuration | Configuration | ||

5 | 48 | 62 | ||

0 | 54 | 65 | ||

3 | 23 | 18 | ||

1 | 15 | 57 | ||

2 | 13 | 16 | ||

1 | 16 | 19 | ||

2 | 14 | 42 | ||

3 | 17 | 30 | ||

3 | 14 | 21 | ||

2 | 18 | 13 | ||

5 | 16 | 17 | ||

4 | 24 | 17 |

### ii.1 Calculation of for pick-up reaction Al(d,He)Mg

Experimentally the states of Mg vishal1 () were studied by assuming pick-up from orbital only, and also a few states by assuming configuration mixing of two lower orbital of shell: and single particle orbitals. In Table 1, we compared the experimental values with shell-model results for IM-SRG, CCEI and USDB interactions (the corresponding wave functions are shown in Table II ). As extracted from experiment the calculated values were very large for () transfer as compared with () transfer. The shell-model results are larger for first two states, thus assigning larger single particle characteristics to these states. In the present work we have also predicted values for states up to 6 MeV. In the Fig. 3(a)-(d) we show the variation of for extracted experiment and calculated values. In all the three shell-model calculations, the spectroscopic factor for pickup from the orbital plays a major role for the higher excited states. In the Fig. 3(e), we have also plotted values for theory and extracted experimental value. The values calculated from IM-SRG and CCEI interactions are showing same trends as extracted values from the experiment.

### ii.2 Calculation of for pick-up reaction Al(d,)Al

The states of Al vishal2 () were experimentally studied by assuming pick-up from and single particle orbitals, while state at 3507 keV by pick-up from orbital. In the Table 3, we compared experimental values with shell-model results for IM-SRG, CCEI and USDB interactions. The experimentally extracted value for spectroscopic factors up to 4.7 MeV are reported in Ref. vishal2 (). In the present work we interpreted these extracted SFs in term of shell-model calculations (the corresponding wave functions are shown in Table 4). The experimental values for states are given in the Table 3. For the state the SF result with IM-SRG and USDB is slightly higher than extracted experimental value, while it is smaller with CCEI. For some states the SFs are very small this is because in these cases the wave functions are very fragmented. This is because large cancellations of contributions from different components of the wave functions. For at 1759 keV and at 4705 keV, the IM-SRG, CCEI and USDB interactions are predicting very small value of spectroscopic factors.

In the Figs. 4(a)-(d) we show the variation of for extracted experiment and calculated values. In the Fig. 4(e), we have also plotted values for theory and extracted experimental value. The extracted experimental values show a good trend with IM-SRG and CCEI.

## Iii Summary

We performed shell-model calculations for spectroscopic factors with two approaches: an in-medium similarity renormalization targeted for a particular nucleus and coupled-cluster effective interaction (CCEI). We also performed calculations with realistic USDB effective interaction. Along with the results, we present a comparison with recently determined experimental spectroscopic factors.

## Acknowledgments

P.C.S. thanks P. Navrátil and S. R. Stroberg for useful discussions during this work. We also thank R. Shyam and V. K. B. Kota for useful comments to improve this manuscript. P.C.S. acknowledges the hospitality extended to him during his stay at TRIUMF, Vancouver City, Canada. P.C.S. acknowledges financial support from faculty initiation grants. V.K.âs work was supported in part by the CSIR GrantNo.09/143(0844)/2013-EMR-1-India Ph.D.fellowship program

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