^{7}Be(n,p)^{7}Li 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
As mentioned in the paper draft, the n_TOF measurement provides cross section data for the ^{7}Be(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 ^{7}Li(p,n)^{7}Be.
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 + ^{7}Li n + ^{7}Be(g.s.) since J_{1} = J_{3} = 1/2 and J_{2} = J_{4} = 3/2.
The data of the ^{7}Li(p,n)^{7}Be reaction from Ref. [2], transformed into ^{7}Be(n,p)^{7}Li cross section, are available here >>.
Here is a plot of the adopted data
A table with the numerical data, which includes the n_TOF experimental results and the data from the time-reversal reaction, is available here.
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).
# | E[MeV] | Ex[MeV] | l | J | Gn[MeV] | Gp[MeV] | Ga[MeV] | Gtot[MeV] |
---|---|---|---|---|---|---|---|---|
1 | 0.013 | 18.910 | 0 | 2.0 | 6.100e-02 | 6.100e-02 | 0.000e+00 | 1.220e-01 |
2 | 0.194 | 19.069 | 1 | 3.0 | 1.000e-03 | 2.710e-01 | 0.000e+00 | 2.720e-01 |
3 | 0.384 | 19.235 | 1 | 3.0 | 1.140e-01 | 1.140e-01 | 0.000e+00 | 2.280e-01 |
4 | 0.573 | 19.400 | 0 | 1.0 | 3.200e-01 | 3.200e-01 | 0.000e+00 | 6.400e-01 |
5 | 1.098 | 19.860 | 3 | 4.0 | 1.000e-03 | 2.100e-01 | 4.900e-01 | 7.010e-01 |
6 | 1.373 | 20.100 | 1 | 2.0 | 1.000e-01 | 1.270e-01 | 5.730e-01 | 8.000e-01 |
7 | 1.486 | 20.199 | 1 | 0.0 | 1.500e-01 | 1.500e-01 | 3.600e-01 | 6.600e-01 |
8 | 2.287 | 20.900 | 2 | 4.0 | 8.000e-01 | 8.000e-01 | 0.000e+00 | 1.600e+00 |
9 | 3.544 | 22.000 | 0 | 1.0 | 2.000e+00 | 2.000e+00 | 0.000e+00 | 4.000e+00 |
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.
# | E[MeV] | Ex[MeV] | l | J | Gn[MeV] | Gp[MeV] | Ga[MeV] | Gtot[MeV] |
---|---|---|---|---|---|---|---|---|
1 | 0.013 | 18.910 | 0 | 2.0 | 3.500e-02 | 1.190e-01 | 0.000e+00 | 1.540e-01 |
2 | 0.194 | 19.069 | 1 | 3.0 | 4.800e-02 | 4.150e-01 | 0.000e+00 | 4.630e-01 |
3 | 0.384 | 19.235 | 1 | 3.0 | 1.220e-01 | 7.500e-02 | 0.000e+00 | 1.970e-01 |
4 | 0.573 | 19.400 | 0 | 1.0 | 9.355e+00 | 3.390e-01 | 0.000e+00 | 9.694e+00 |
5 | 1.098 | 19.860 | 3 | 4.0 | 4.000e-03 | 4.400e-01 | 5.230e-01 | 9.670e-01 |
6 | 1.373 | 20.100 | 1 | 2.0 | 1.529e+00 | 1.258e+00 | 1.480e-01 | 2.935e+00 |
7 | 1.486 | 20.199 | 1 | 0.0 | 1.000e-01 | 1.000e-01 | 3.250e-01 | 5.250e-01 |
8 | 2.287 | 20.900 | 2 | 4.0 | 1.460e-01 | 1.243e+00 | 0.000e+00 | 1.389e+00 |
9 | 3.544 | 22.000 | 0 | 1.0 | 2.000e+00 | 2.000e+00 | 0.000e+00 | 4.000e+00 |
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 ^{8}Be excited states.
A table with the pointwise cross section calculated using the SLBW formula is provided here.
Here is a table of the estimated uncertainties in the measured (n,p) cross section, all in %, with the systematics added in quadrature.
statistical [%] | systematics [%] | ||||
---|---|---|---|---|---|
f_{C} | ang. dist. | others(*) | Total | ||
0.01 eV ≤ E ≤ 1 keV | 1 - 5 | 8 | 0 | 5 | 10 |
1 keV ≤ E ≤ 50 keV | 5 - 10 | 8 | 5 - 10 | 5 | 10 - 15 |
E > 50 keV | 5 |
(*) 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 ^{6}Li sample (see paper text for more information).
The 5% uncertainty above E_{n} = 50 keV is assumed considering that the time-reversal ^{7}Li(p,n)^{7}Be cross section is know with this leve of accuracy.
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 cm^{3}/s/mole when <σ> is in barn and the temperature T_{9} in 10^{9} degrees. The numerical results can be accurately described by the following analytical expression [4], which includes a power expansion in T_{9} (coefficients to ) plus an exponential term (), as well as a resonance term ()
in units of cm^{3}/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 ^{7}Be(n,p)^{7}Li 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.
The reaction rate in tabular form can be found here.
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 ^{4}He 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.
Y_{p} | D/H [ 10^{-5} ] | ^{3}He/H [ 10^{-5} ] | ^{7}Li/H [ 10^{-10} ] | |
present with standard rates | 0.246 | 2.43 | 1.08 | 5.46 |
present with new rate (η_{10} = 6.09) | 0.246 | 2.43 | 1.08 | 5.26 ± 0.40 |
present with new rate (5.8 ≤ η_{10} ≤ 6.6) | 0.246 | 2.43 | 1.08 | 4.73 - 6.23 |
observations | 0.245 ± 0.003 | 2.569 ± 0.027 | - | 1.6 ± 0.3 |
The uncertainty or range of variation of the Lithium production is associated only to the corresponding variation of the ^{7}Be(n,p)^{7}Li reaction rate uncertainty.
A typical plot of the ^{7}Li and ^{7}Be yields, plotted vs are shown here. Similar plots for Y_{p}, D and ^{3}He 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.
Time-reversal invariance, mentioned above, can be invoked to derive the ^{7}Li(p,n)^{7}Be cross section from the present ^{7}Be(n,p)^{7}Li measured cross section data. Here below is a plot of the derived (p,n) cross section.
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.
A: The ^{7}Be(n,p)^{7}Li reaction is producing ^{7}Li, while destroying ^{7}Be. However, during BBN, the ^{7}Li is readily consumed by the ^{7}Li(p,α) reaction. Therefore, the net result of the ^{7}Be(n,p)^{7}Li process is a reduction of the ^{7}Be yield. In the end, the ^{7}Be that survives BBN, will undergo electron-capture decay
More than 95% of final cosmic ^{7}Li abundance is, in fact, due to the survival of ^{7}Be during BBN. Therefore, an increase in the rate for the ^{7}Be(n,p)^{7}Li reaction results in a reduction of the ^{7}Li abundance.
See also the Q&A below.
A: Certainly. Here it is (click on the image or reload the page to see the time evolution):
A: Yes. The first exited state in ^{7}Li at 478 keV has a spin and parity J^{π} = 1/2^{-} (see figure below).
Therefore, if a J^{π} = 2^{-} is formed by s-wave neutron on ^{7}Be, the state cannot decay into ^{7}Li(1st) by emitting l=0 (no sufficient angular momentum), nor l=1 (no parity conservation) protons. Because the 2^{-} state just above threshold in ^{8}Be dominates the reaction mechanisms in a wide energy range, it is possible to estimate the (n,p_{1}) contribution to the ^{7}Be(n,p)^{7}Li cross section by calculating the ration of penetrabilities of l=2 to l=0 protons in the p + ^{7}Li(1s) exit channel. The result of this estimate is shown in the figure below.
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)%.
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 ^{14}N(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.
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 | ^{7}Be 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.
-- AlbertoMengoni for the n_TOF Collaboration - 2018-02-13