Q&A on the ⁷Be(n,γα)⁴He paper (Supplemental Material)

⁷Be(n,α)⁴He 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 ⁸Be states allowed (forbidden) to decay into two α particles?

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

allowed	forbidden
0+, 2+, 4+, ...	0-, 1+, 1-, 2-, 3+, 3-, 4-, ...

Q: Why?

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 ⁴He (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 ⁸Be.

Q: Why the J^π=4⁺ state at 11.35 MeV in ⁸Be is not included in the (n,γα) reaction modeling?

A: s-wave neutrons on ⁷Be (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 ⁸Be 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 |⁷Be ⊗ 1p3/2> configuration. It is, therefore, outside the model space adopted in the present calculation.

Q: Since you mentioned, what are the selection rules for EM transitions?

In general, these are the selection rules for EM transitions between states J_i → J_f with ΔJ = |J_f - J_i|:

EM	ΔJ	πf • πi	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

Q: What sort of states can be formed by incident neutrons on ⁷Be?

A: Considering that ⁷Be ground state is a 3/2^- state, the following ⁸Be 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+

Of these, only the 0⁺, the 1^- and the 2⁺ states can do a direct breakup into 2α particles (see above). The 1⁺, 2^- and 3⁺ states must do a γ transition before decaying into 2α.

Q: Is there a level scheme of ⁸Be with the related γ and α decay channels?

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 ⁷Be.

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..

Q: Can you provide some additional infos on the shell model calculations?

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]	Jπ	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

Q: How about the DRC calculations?

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 ⁸Be. 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]	Jπ	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.

Q: Do you have a high-resolution version of Figure 3 of the paper?

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.

Q: Could you show a comparison of the n_TOF results with the data of some ENDF library?

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.

Q: How did you estimate the energy range of interest for the big bang nucleosynthesis?

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 T₉

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

Q: Can you provide with a new evaluation of the ⁷Be(n,α)⁴He reaction rate, based on the present experimental data?

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

in units of cm³ 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 T₉, 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 cm³ s^-1 mole^-1.

As far as the second term (linear in T₉) 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 ⁷Be(n,α)⁴He reaction as

valid in the temperature range from 1 MK to 5 GK, in units of cm³ 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.

Q: What are the implications of these new results on the CLIP?

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 ⁷Li + ⁷Be 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.

References

1. 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)
2. 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).
3. N. Kumar, Effective interaction calculations for nuclei of mass 6 to 9 , Nuclear Physics A225 (1974), 221.
4. A. Mengoni, T. Otsuka and M. Ishihara, Direct Radiative Capture of p-wave Neutrons , Phys. Rev. C 52 (1995) R2334 (pdf).
5. S. Q. Hou, J. J. He, S. Kubono, and Y. S. Chen, Revised thermonuclear rate of ⁷Be(n,α)⁴He relevant to Big-Bang nucleosynthesis, Physical Review C 91 (2015) 055802.

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