^{7}Be(n,α)^{4}He Reaction and the Cosmological Lithium Problem: Measurement of the Cross Section in a Wide Energy Range at n_TOF at CERN

M. Barbagallo, *et al.* (The n_TOF Collaboration)

Physical Review Letters **117**, 152701 (2016)

Open Access, Editors' Suggestion

DOI: 10.1103/physrevlett.117.152701

- Q: Which are the
^{8}Be states allowed (forbidden) to decay into two α particles? - Q: Why?
- Q: Why the J
^{π}=4^{+}state at 11.35 MeV in^{8}Be is not included in the (n,γα) reaction modeling? - Q: Since you mentioned, what are the selection rules for EM transitions?
- Q: What sort of states can be formed by incident neutrons on
^{7}Be? - Q: Is there a level scheme of
^{8}Be with the related γ and α decay channels? - Q: Can you provide some additional infos on the shell model calculations?
- Q: How about the DRC calculations?
- Q: Do you have a high-resolution version of Figure 3 of the paper?
- Q: Could you show a comparison of the n_TOF results with the data of some ENDF library?
- Q: How did you estimate the energy range of interest for the big bang nucleosynthesis?
- Q: Can you provide with a new evaluation of the
^{7}Be(n,α)^{4}He reaction rate, based on the present experimental data? - Q: What are the implications of these new results on the CLIP?

**A:** States allowed and forbidden to decay into two α particles in ^{8}Be are:

allowed | forbidden |
---|---|

0^{+}, 2^{+}, 4^{+}, ... |
0^{-}, 1^{+}, 1^{-}, 2^{-}, 3^{+}, 3^{-}, 4^{-}, ... |

**A:** The two α particles emitted in the decay will have a J^{π}=0^{+} total angular momentum and parity. The orbital angular momentum of the relative motion between the two α's in the center of mass system is L and parity (-)^{L}. Therefore, the total angular momentum of the final state is L and parity is (+) (-)^{L} (+) = (-)^{L}. Only states for which the total angular momentum and parity are conserved, such as 0^{+}, 1^{-}, 2^{+}, 3^{-}, ..., are allowed to decay.

In addition, since the two α's (in their ground state) are bosons, the total wave function must be symmetric under interchange and this excludes odd values of the relative orbital angular momentum. Hence, only 0^{+}, 2^{+}, 4^{+}, ... states are allowed to decay.

**Attention:** for excitation energies above the first excited state in ^{4}He (the 0^{+} at 20.21 MeV), one of the two α could be emitted excited. Therefore, the two α's would not be anymore identical particles and decay is allowed again for states such as a 1^{-}. This is the case, for example, of the (1, 2)^{-} state at 24 MeV in ^{8}Be.

**A:** s-wave neutrons on ^{7}Be (g.s. with J^{π}=3/2^{-}) can form only states with J^{π}=1^{-} and 2^{-}. These states can decay to final states shown in the following table:

1^{-} |
2^{-} |
EM transition |
---|---|---|

0^{+} 1^{+} 2^{+} |
1^{+} 2^{+} 3^{+} |
E1 |

0^{-} 1^{-} 2^{-} |
1^{-} 2^{-} 3^{-} |
M1 |

0^{-} 1^{-} 2^{-} 3^{-} |
0^{-} 1^{-} 2^{-} 3^{-} 4^{-} |
E2 |

0^{+} 1^{+} 2^{+} 3^{+} |
0^{+} 1^{+} 2^{+} 3^{+} 4^{+} |
M2 |

The J^{π}=4^{+} state at 11.35 MeV in ^{8}Be can only be populated by an M2 transition from the 2^{-} state. An M2 transition is strongly suppressed in comparison to other multipolarities and the corresponding (n,γ) cross section leading to this state is negligible.

In addition, the 4^{+} state cannot have a simple |^{7}Be ⊗ 1p3/2> configuration. It is, therefore, outside the model space adopted in the present calculation.

In general, these are the selection rules for EM transitions between states J_{i} → J_{f} with ΔJ = |J_{f} - J_{i}|:

EM | ΔJ | π_{f} • π_{i} |
excluded |
---|---|---|---|

E1 | 0, 1 | -1 | 0 ↔ 0 |

M1 | 0, 1 | +1 | 0 ↔ 0 |

E2 | 0, 1, 2 | +1 | 0 ↔ 0, 1, 1/2 ↔ 1/2 |

M2 | 0, 1, 2 | -1 | 0 ↔ 0, 1, 1/2 ↔ 1/2 |

**A:** Considering that ^{7}Be ground state is a 3/2^{-} state, the following ^{8}Be states with total angular momentum and parity can be formed by incident s- and p-wave neutrons:

s-waves | p-waves |
---|---|

1^{-}, 2^{-} |
0^{+}, 1^{+}, 2^{+}, 3^{+} |

**A:** Yes. Here below is a plot of the levels of interest for the (n,α) and (n,γα) reaction channels. The levels allowed to decay into the 2α channel and the α emission energy are indicated. The γ-ray lines (solid) are connecting states which can be populated by E1 transitions, initiated by 1^{-} and 2^{-} capturing states, formed by s-wave neutrons on ^{7}Be.

A plot of the full level scheme in the Nuclear Data Sheets style is Here >>, while a list of the adopted levels is available from the NuDat program. See also Tiley et al..

**A:** Sure. Calculations were performed with the OXBASH shell model code (hpc version 2005-12) [2], using the ppn model space (1p1/2, 1p3/2 orbits) and the Kumar interaction [3]. The resulting energy spectrum, in comparison with the experimental data is show here

The level sequence up to 20 MeV is reproduced well by the calculation and the lower part of the spectrum is sufficiently accurate. Of course, in the DRC calculations reported in the paper the experimental data for the energy levels are used, in combination with the spectroscopic factors calculated with OXBASH. We did the calculations using other interactions as well (e.g. Cohen and Kurath, in Nuclear Physics **A101** (1967) 1), obtaining very similar results for the spectroscopic factors:

Ex [MeV] | J^{π} |
Cohen and Kurath | Kumar (adopted) |
---|---|---|---|

0.0 | 0^{+} |
1.48 | 1.51 |

3.03 | 2^{+} |
0.58 | 0.57 |

16.626 | 2^{+} |
0.41 | 0.30 |

16.922 | 2^{+} |
0.50 | 0.47 |

**A:** The DRC model calculations [4] have been performed using the bound state wave functions obtained from Wood-Saxon potential with the parameters reported in the paper. The well depth has been adjusted to reproduce the experimental binding energy of the bound states in ^{8}Be. Continuum state wave functions have been calculated with the same parameters as for the bound states, with the well depth adjusted to reproduce the experimental thermal cross section (see below).

As mentioned in the paper, both J^{π} = 1^{-} and 2^{-} incident channels have been included in the DRC cross section calculations. Here is a table with the resulting DRC partial cross sections at thermal neutron energy (E_{n} = 0.0253 eV), splitted for the two initial channel total angular momenta:

1^{-} |
2^{-} |
All | |||||
---|---|---|---|---|---|---|---|

Ex [MeV] | J^{π} |
σ(n,γα) [b] | % | σ(n,γα) [b] | % | σ(n,γα) [b] | % |

0.0 | 0^{+} |
1.278 | 54.7 | 0.000 | 0.0 | 1.278 | 37.6 |

3.03 | 2^{+} |
0.988 | 42.2 | 0.988 | 93.2 | 1.977 | 58.2 |

16.626 | 2^{+} |
0.032 | 1.4 | 0.032 | 3.0 | 0.063 | 1.9 |

16.992 | 2^{+} |
0.040 | 1.7 | 0.040 | 3.8 | 0.080 | 2.3 |

total | 2.338 | 100 | 1.059 | 100 | 3.398 | 100 |

**Note** that the contribution of the two J^{π} = 1^{+} states at 17.64 and 18.15 MeV has been omitted because, while they can be populated by E1 transitions from 1^{-} and 2^{-} initial states, these two states cannot decay into the α emission channel. In other words, they are a component (of the oder of 0.5% of the total) in the (n,γ), but not in the (n,γα) reaction channels.

**A:** Certainly. Click on the figure to enlarge it and

to the hyperlinks for the ps and pdf versions.

The tabulated cross sections shown in Figure 3 (green line), which can be used to make a new estimate of the reaction rate, can be found here.

**A:** Sure. Here is the plot of Figure 3 which includes the (n,α) cross section of the ENDF/B-VII.1 and TENDL-2014 evaluated nuclear data files:

The **ENDF/B-VII.1** evaluation includes the contribution of the two 2^{+} states at 20.1 and 22.24 MeV (both can be populated by p-wave neutrons), via R-matrix (see Page and Hale, 2004). It has to be noted that, though a direct breakup of these two states into two α is allowed, our data do not show any increase in the cross section, at least up to ~ 10 keV. The adopted widths produce a cross section which is consistent with our experimental results up to ~1 keV incident neutron energy.

The **TENDL-2014** evaluation is based on Hauser-Feshbach statistical model calculations, complemented by a 1/v component. Both components seem to be off by large factors in comparison to the experimental data. The HFS model is not supposed to work for light nuclei, with very low level densities.

**A:** Here below is a plot of the BBN yields in a standard cosmology. You can see that the relevant energy range for the production/destruction of the Li and Be isotopes is between 20 keV and 120 keV, corresponding to temperature between 0.23 and 1.4 in units of T_{9}

All the yields are mass fractions relative to p, except for p and ^{4}He that are absolute mass fractions.

**A:** Here it is. The reaction rate of Wagoner (1969), used in most of the big bang nucleosynthesis reaction codes, reads

in units of cm^{3} s^{-1} mole^{-1}. The first term in this rate is due to the 1/v component of the cross section, with a thermal value of 145 mb. The second term, linear in T_{9}, is normalized to the upper limit (< 0.1 mb) of the (n,α) cross section, also at thermal energy and corresponds to a cross section proportional to the velocity, as one would expect for incident p-wave neutrons. The related cross section terms, can be expressed analytically in the form

and

where the cross section is in b when E is in MeV.

Because our predicted thermal cross section (based on the DRC calculations) is 3.4 b, the first term of the reaction rate, valid at low temperatures, should be instead

in units of cm^{3} s^{-1} mole^{-1}.

As far as the second term (linear in T_{9}) is concerned, in addition to the n_TOF data, we can consider the data of Hou et al. (2015) [5], where indirect/time-reversal reactions have been considered to derive the (n,α) cross section. If we combine the two experimental sets, we can fix an upper and a lower limit for the p-wave component as shown in this figure

In this first simple evaluation (others can be made), we have assumed a p-wave component which could be consistent with the combined experimental data sets. Note that the last (first) few points of both experimental sets have a 100% uncertainty.

We can then evaluate the upper and lower limits as

Even though our estimate for the (n,α) cross section is a factor of over 20 larger than previously evaluated in the energy range up to ~ 1 keV, we predict a reaction rate a factor of 10 to 100 lower in the energy/temperature range of interest for the big bang nucleosynthesis (shaded area shown in the figure above).

If we assume the cross section indicated in Figure 3 of the paper with the green curve, formed by the presently calculated (n,γα) cross section, summed to the data from the ENDF/B-VII.1 evaluation, we can provide an analytical expression for the total reaction rate of the ^{7}Be(n,α)^{4}He reaction as

valid in the temperature range from 1 MK to 5 GK, in units of cm^{3} s^{-1} mole^{-1}.

In the figure below, this rate is shown in comparison with the previously adopted rate of Wagoner (1967) and the recently proposed rate of Hou et al. [5]

**Note:** the evaluation of a new reaction rate requires some additional work. What is presented here is a preliminary track for working on the issue.

**A:** From the analysis shown above, we cannot expect a strong effect of the new reaction rate on the BBN Li yields. In fact, in a standard cosmology calculation performed with the **standard rate** by Wagoner (1967) and with the **new** reaction rate given above, the relevant yields (mass fractions relative to H) are

Li6 | Li7 | Be7 | Li8 | Li7 + Be7 | |
---|---|---|---|---|---|

standard rate | 1.114E-14 | 2.818E-11 | 4.069E-10 | 1.325E-25 | 4.3151E-10 |

new rate | 1.114E-14 | 2.836E-11 | 4.120E-10 | 1.325E-25 | 4.4036E-10 |

with a difference of the order of only 2% (increasing) in the summed ^{7}Li + ^{7}Be yield obtained using the new rate.

A plot of the mass fraction as a function of the baryon-to-photon ratio, η, is given here below. The results obtained with the new reaction rate are indistinguishable from those obtained with the standard rate, in this plot.

- D.R. Tilley, J.H. Kelley, J.L. Godwin, D.J. Millener, J.E. Purcell, C.G. Sheu, H.R. Weller,
*Energy levels of light nuclei A=8,9,10*, Nuclear Physics**A745**(2004) 155-362, ISSN 0375-9474, (pdf) - B. A. Brown, A. Etchegoyen, N. S. Godwin, W. D. M. Rae, W. A. Richter, Ormand, E. K. Warburton, J. S. Winfield, L. Zhao and C. H. Zimmerman, MSU-NSCL report, Volume 1289 (2004).
- N. Kumar,
*Effective interaction calculations for nuclei of mass 6 to 9*, Nuclear Physics**A225**(1974), 221. - A. Mengoni, T. Otsuka and M. Ishihara,
*Direct Radiative Capture of p-wave Neutrons*, Phys. Rev. C**52**(1995) R2334 (pdf). - S. Q. Hou, J. J. He, S. Kubono, and Y. S. Chen,
*Revised thermonuclear rate of*, Physical Review C^{7}Be(n,α)^{4}He relevant to Big-Bang nucleosynthesis**91**(2015) 055802.

-- Alberto Mengoni - last update: 2016-07-07

Topic revision: r23 - 2018-11-16 - AlbertoMengoni

- ABATBEA
- ACPP
- ADCgroup
- AEGIS
- AfricaMap
- AgileInfrastructure
- ALICE
- AliceEbyE
- AliceSPD
- AliceSSD
- AliceTOF
- AliFemto
- ALPHA
- Altair
- ArdaGrid
- ASACUSA
- AthenaFCalTBAna
- Atlas
- AtlasLBNL
- AXIALPET
- CAE
- CALICE
- CDS
- CENF
- CERNSearch
- CLIC
- Cloud
- CloudServices
- CMS
- Controls
- CTA
- CvmFS
- DB
- DefaultWeb
- DESgroup
- DPHEP
- DM-LHC
- DSSGroup
- EGEE
- EgeePtf
- ELFms
- EMI
- ETICS
- FIOgroup
- FlukaTeam
- Frontier
- Gaudi
- GeneratorServices
- GuidesInfo
- HardwareLabs
- HCC
- HEPIX
- ILCBDSColl
- ILCTPC
- IMWG
- Inspire
- IPv6
- IT
- ItCommTeam
- ITCoord
- ITdeptTechForum
- ITDRP
- ITGT
- ITSDC
- LAr
- LCG
- LCGAAWorkbook
- Leade
- LHCAccess
- LHCAtHome
- LHCb
- LHCgas
- LHCONE
- LHCOPN
- LinuxSupport
- Main
- Medipix
- Messaging
- MPGD
- NA49
- NA61
- NA62
- NTOF
- Openlab
- PDBService
- Persistency
- PESgroup
- Plugins
- PSAccess
- PSBUpgrade
- R2Eproject
- RCTF
- RD42
- RFCond12
- RFLowLevel
- ROXIE
- Sandbox
- SocialActivities
- SPI
- SRMDev
- SSM
- Student
- SuperComputing
- Support
- SwfCatalogue
- TMVA
- TOTEM
- TWiki
- UNOSAT
- Virtualization
- VOBox
- WITCH
- XTCA

Copyright &© 2008-2023 by the contributing authors. All material on this collaboration platform is the property of the contributing authors.

or Ideas, requests, problems regarding TWiki? use Discourse or Send feedback

or Ideas, requests, problems regarding TWiki? use Discourse or Send feedback