Supplemental Material for the ⁷Be(n,p)⁷Li paper

⁷Be(n,p)⁷Li Reaction and the Cosmological Lithium Problem: Measurement of the Cross Section in a Wide Energy Range at n_TOF at CERN
L. Damone, M. Barbagallo, M. Mastromarco, A. Mengoni, L. Cosentino, E. Maugeri, S. Heinitz, D. Schumann, R. Dressler, F. Käppeler, N. Colonna, P. Finocchiaro, J. Andrzejewski, J. Perkowski, A. Gawlik, et al. (The n_TOF Collaboration)
Physical Review Letters 121, 042701 (2018)     Open Access  
10.1103/PhysRevLett.121.042701

Adopted cross section data for R-matrix fitting

As mentioned in the paper draft, the n_TOF measurement provides cross section data for the ⁷Be(n,p) reaction in the energy range 25 meV E_n 325 keV. In order to perform a reliable fit of the cross section, hence to derive the corresponding reaction rate in the wide temperature range required by Big-Bang Nucleosynthesis (BBN) network calculations, we must complement the n_TOF data with data derived from the time-reversal reaction ⁷Li(p,n)⁷Be.

To this end, the detailed balance relation, from time-reversal invariance of strong interaction (see for example [1]),

can be used. Here, the direct and inverse reactions 1 + 2 3 + 4, are characterized by the center-of-mass momenta k_1,2 and k_3,4 and by the total angular momenta J_i of the reacting particles, while the δ's avoid double counting in the case particles 1 and 2 and/or 3 and 4 are identical. Note that this general expression simplifies to the Eq. (2) of the draft paper in the specific case of p + ⁷Li n + ⁷Be(g.s.) since J₁ = J₃ = 1/2 and J₂ = J₄ = 3/2.

The data of the ⁷Li(p,n)⁷Be reaction from Ref. [2], transformed into ⁷Be(n,p)⁷Li cross section, are available here >>.

Here is a plot of the adopted data

Experimental data from the ⁷Be(n,p)⁷Li cross section measured at n_TOF, combined with the data obtained from the time-reversal ⁷Li(p,n)⁷Be reaction.

A table with the numerical data, which includes the n_TOF experimental results and the data from the time-reversal reaction, is available here.

R-matrix fit of the adopted cross section

The adopted (n,p) cross section has been fitted with a single-level Breit-Wigner expression

where

is the reduced mass in the entrance channel and the spin of the target nucleus ( in the present case). Both, the neutron and charged-particle widths are taken to be energy dependent in the form

where is the energy-independent reduced width and can be calculated at the resonance energy, once a given width is known. The penetrability factor is given by

Here, F and G are, respectively, the regular and irregular Coulomb functions for positive energies, the usual Sommerfeld parameter and the reduced mass for the entrance or exit channels, respectively in case of neutron or charged-particle widths.

A total of N_r = 9 levels (resonances) have been included in the fit of the adopted cross section data. For each resonance, the parameters available from the ENSDF library [3] have been adopted as starting values (see the table here below).

The energies of each resonance has been kept constant while the widths have been allowed to vary. Here below is a table with the final values.

The SLBW (reduced) cross section is shown in comparison with the adopted experimental data in the figures below. Note that, most of the resonance parameters are purely fit parameters, obtained to reproduce the best representation of the adopted cross section. They are not to be considered accurate physical properties of the ⁸Be excited states.

Adopted reduced (n,p) cross section shown in comparison with the SLBW fit obtained with the parameters of Table 2

Cross section in the energy range above 10 keV

A table with the pointwise cross section calculated using the SLBW formula is provided here.

Uncertainties

Here is a table of the estimated uncertainties in the measured (n,p) cross section, all in %, with the systematics added in quadrature.

(*) Including: sample mass, flux normalization and detector efficiency estimation. This component has been estimated and confirmed with the data of the measurement performed with a ⁶Li sample (see paper text for more information).

The 5% uncertainty above E_n = 50 keV is assumed considering that the time-reversal ⁷Li(p,n)⁷Be cross section is know with this leve of accuracy.

New reaction rate for the ⁷Be(n,p)⁷Li

After a reliable fit of the (n,p) cross section σ(E) has been obtained as described in the previous section, the maxwellian averaged cross section (MACS)

can be readily calculated numerically for a full set of thermal energies kT. In turn, the MACS can be promptly converted into a reaction rate

in units of cm³/s/mole when <σ> is in barn and the temperature T₉ in 10⁹ degrees. The numerical results can be accurately described by the following analytical expression [4], which includes a power expansion in T₉ (coefficients to ) plus an exponential term (), as well as a resonance term ()

in units of cm³/s/mole when = 6.805e+09, = -1.971e+00, = 2.042e+00, = -1.069e+00, = 2.717e-01, = -2.670e-02, = 1.963e+08, = 2.889e+07, and = 2.811e-01.

A code segment (in c) with the expression above is given here

    1/* Be7 + n -> p + Li7 */
    2/* rate from n_TOF  - central value */
    3        a0=   6.8048e+09; a1= -1.9706e+00; a2=  2.0419e+00; a3= -1.0687e+00; a4=  2.7172e-01; a5= -2.6699e-02; a6=  1.9610e+08; a7=  2.8899e+07; b0=  2.8114e-01;
    4        rrate = a0*(1. + a1*pow(T9,1./2.) + a2*T9 + a3*pow(T9,3./2.) + a4*T9*T9 + a5*pow(T9,5./2.)) + a6*pow(T9/(1.+13.076*T9),3./2.)/pow(T9,3./2.) + a7/pow(T9,3./2.)*exp(-b0/T9);

Considering now that the reaction rate is linear with the maxwellian averaged cross section, the reaction rate with its uncertainty can be estimated. The result is shown in the figure below, where a comparison of the reaction rates for the ⁷Be(n,p)⁷Li reaction of the present work with some of the commonly adopted rates ([4], [5], [6]) with respect to the rate of Cyburt (2004) [7]. The uncertainty associated with the presently determined rate is shown by the corresponding grey band. The temperature range of interest for BBN is indicated by the vertical band.

This figure is equivalent to Figure 3 of the draft paper, with the addition of the error bars in the present data, estimated as described in the text.

The reaction rate in tabular form can be found here.

Implications on the BBN

The calculations of the BBN yields have been performed using an updated version of the AlterBBN code of Alexander Arbey et al. [8] where the 12 most important rates have been updated.

Details of the reaction rates used are given here >>.

As mentioned in the paper draft, the BBN calculations have been performed adopting a neutron average life-time of s and neutrino species. The baryon-to-photon number density ratio (in units of 10^-10) has been allowed to vary within the uncertainty quoted in the CMB analysis and in the wider range established by the concordance of observation of primordial ⁴He and deuterium as evaluated in the review of the most recent Particle Data Group publication [9]. The results of the BBN calculation for the main observables are shown in the following table.

The uncertainty or range of variation of the Lithium production is associated only to the corresponding variation of the ⁷Be(n,p)⁷Li reaction rate uncertainty.

A typical plot of the ⁷Li and ⁷Be yields, plotted vs are shown here. Similar plots for Y_p, D and ³He are here.

Calculations performed with an updated version of L Kawano's code NUC123 [10] produced results of all the yields practically identical to those obtained with the AlterBBN code, once the reaction rates (and the Cosmology parameters) have been set the same.

(n,p) and (p,n)

Time-reversal invariance, mentioned above, can be invoked to derive the ⁷Li(p,n)⁷Be cross section from the present ⁷Be(n,p)⁷Li measured cross section data. Here below is a plot of the derived (p,n) cross section.

7Li(p,n)7Be near threshold

7Li(p,n)7Be in the full energy range up to Ep = 2 MeV

A comparison is shown here with the data of Gibbons and Macklin [11] and Sekharan et al. [2], in the left panel, while a more complete set, available from the EXFOR database are shown on the right panel. The ENDF/B-VII.1 [12] data are plotted as well. The SLBW cross section, calculated with the resonance parameters of Table 2 above, is plotted as well.

Both, direct and time-reversal data, are provided here >> in tabular form.

Q&A

Q: The ⁷Be(n,p)⁷Li is actually producing ⁷Li. How come that a higher rate for this reaction results in a lower ⁷Li yield?

A: The ⁷Be(n,p)⁷Li reaction is producing ⁷Li, while destroying ⁷Be. However, during BBN, the ⁷Li is readily consumed by the ⁷Li(p,α) reaction. Therefore, the net result of the ⁷Be(n,p)⁷Li process is a reduction of the ⁷Be yield. In the end, the ⁷Be that survives BBN, will undergo electron-capture decay

More than 95% of final cosmic ⁷Li abundance is, in fact, due to the survival of ⁷Be during BBN. Therefore, an increase in the rate for the ⁷Be(n,p)⁷Li reaction results in a reduction of the ⁷Li abundance.

See also the Q&A below.

Q: Could you provide a time/temperature evolution plot of the BBN yields obtained with the adopted parameters and reaction rates?

A: Certainly. Here it is (click on the image or reload the page to see the time evolution):

Q: Have you considered the contribution of the (n,p₁) component to the ⁷Be(n,p)⁷Li cross section?

A: Yes. The first exited state in ⁷Li at 478 keV has a spin and parity J^π = 1/2^- (see figure below).

Energy levels for ⁷Be, ⁸Be, and ⁷Li relevant for the (n,p) and (p,n) reactions.

Therefore, if a J^π = 2^- is formed by s-wave neutron on ⁷Be, the state cannot decay into ⁷Li(1st) by emitting l=0 (no sufficient angular momentum), nor l=1 (no parity conservation) protons. Because the 2^- state just above threshold in ⁸Be dominates the reaction mechanisms in a wide energy range, it is possible to estimate the (n,p₁) contribution to the ⁷Be(n,p)⁷Li cross section by calculating the ration of penetrabilities of l=2 to l=0 protons in the p + ⁷Li(1s) exit channel. The result of this estimate is shown in the figure below.

Estimate of the (n,p₁) contribution to the ⁷Be(n,p)⁷Li cross section.

Considering the simplicity of the assumption, this estimate is compatible with the experimental value quoted by Koehler et al. [13] of (1.2 ± 0.5)%.

Q: Why the (n,p) reaction taking place in the ¹⁴N contained in the backing is not producing background at low neutron energies?

A: In order to identify the protons in the ΔE-E telescope, the protons must have an energy > 1.2 MeV. In the case of the ¹⁴N(n,p) reaction, at least E_n > 500 keV must be added to the the Q-value of 625 keV to generate protons with enough kinetic energy to go through the ΔE detector.

Q: You did not cite our paper on a CLiP solution or attempted solution. Why?

A: A Scholarly Article google search for the Cosmological Lithium Problem (CLiP) produced 9,920 items, as of today (13 March 2018). Here is a table with a very few (arbitrarily) selected examples.

Category	Description	CLiP solved	Reference
nuclear/non-standard physics	non-maxwellian velocity distribution during BBN	yes	S. Q. Hou et al., ApJ 834 (2017) 165. pap
nuclear physics	7Be beta-decay rate in hot plasma	no	S. Simonucci et al., ApJ 764 (2013) 118. doi
non-standard physics	massive gravitino particle decay	maybe	R. H. Cyburt et al., JCAP 10 (2010) 32. pdf
non-standard physics	sterile neutrino, decaying after BBN	yes	L. Salvati et al., JCAP (2016) 2. pdf
astronomy	interstellar lithium observations	no	J. C. Howk et al., Nature 489 (2012) 121. doi
nuclear physics	trojan horse method for reaction rate determination	no	R. G. Pizzone et al., ApJ 786 (2014) 112. pap
non-standard physics	exotic late-decaying particles with lifetimes exceeding ∼1 sec	yes	D. Cumberbatch et al. Phys. Rev. D 76 (2007) 123005. pap
particle/nuclear physics	dynamic screening	maybe	X. Yao, T. Mehen, and B. Müller, Phys. Rev. D 95 (2017) 116002 doi
particle physics	time-dependent quark mass	yes	K. Mori and M. Kusakabe, Phys. Rev. D 99 (2019) 083013. doi
...

If you send in a reference, we can add it to this table.

References

1. Claus E. Rolfs, William S. Rodney, Couldrons in the Cosmos, University of Chicago Press, 1988
2. K.K.Sekharan, H.Laumer, B.D.Kern, F.Gabbard, A neutron detector for measurement of total neutron production cross sections, Nuclear Instruments and Methods in Physics Res. 133 (1976) 253. doi.
3. D.R. Tilley and J.H. Kelley and J.L. Godwin and D.J. Millener and J.E. Purcell and C.G. Sheu and H.R. Weller, Energy levels of light nuclei A=8,9,10, Nuclear Physics A745 (2004) 155. doi, url.
4. M.S. Smith, L. Kawano, and L.H. Malaney, Experimental, computational, and observational analysis of primordial nucleosynthesis, The Astrophysical Journal 85 (1993) 219. url.
5. P. Descouvemont, A. Adahchour, C. Angulo, A. Coc, and E. Vangioni-Flam, Compilation and R-matrix analysis of Big Bang nuclear reaction rates, Atomic Data and Nuclear Data Tables 88 (2004) 203. doi
6. P.D. Serpico, S. Esposito, F. Iocco, G. Mangano, G. Miele, and O. Pisanti, Nuclear reaction network for primordial nucleosynthesis: a detailed analysis of rates, uncertainties and light nuclei yields, Journal of Cosmology and Astroparticle Physics 12 (2004) 10. doi
7. Cyburt, Richard H, Primordial nucleosynthesis for the new cosmology: Determining uncertainties and examining concordance, Phys. Rev. D 70 (2004) 023505. doi
8. A. Arbey, AlterBBN: A program for calculating the BBN abundances of the elements in alternative cosmologies, Computer Physics Communications 183 (2012) 1822. url, doi.
9. C. Patrignani et al. (Particle Data Group), Big-Bang Nucleosynthesis (review), Chin. Phys. C 40 (2016) 100001. url pdf
10. L. Kawano, Primordial Nucleosynthesis - The Computer Way, FERMILAB-Pub-92/04-A (1992). pdf.
11. J.H. Gibbons, and R.L. Macklin, Total Neutron Yields from Light Elements under Proton and Alpha Bombardment, Phys. Rev. 114 (1959) 571. doi, url
12. M. Chadwick et al., ENDF/B-VII.1 Nuclear Data for Science and Technology: Cross Sections, Covariances, Fission Product Yields and Decay Data, Nuclear Data Sheets 112 (2011) 2887. doi, url.
13. P.E. Koehler, C.D. Bowman, F.J. Steinkruger, D.C. Moody, G.M. Hale, J.W. Starner, S.A. Wender, R.C. Haight, P.W. Lisowski, and W.L. Talbert, Phys. Rev. C 37 (1988) 917. url.

-- AlbertoMengoni for the n_TOF Collaboration - 2018-02-13