# Current status and desired accuracy of the isotopic production cross-
sections relevant to astrophysics of cosmic rays II. Fluorine to Silicon (and
updated LiBeB)

Yoann Génolini<EMAIL_ADDRESS>LAPTh, Université Savoie Mont
Blanc & CNRS, Chemin de Bellevue, 74941 Annecy Cedex, France  David Maurin
<EMAIL_ADDRESS>LPSC, Université Grenoble-Alpes, CNRS/IN2P3, 53 avenue
des Martyrs, 38026 Grenoble, France  Igor V. Moskalenko<EMAIL_ADDRESS>W.
W. Hansen Experimental Physics Laboratory and Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA
Michael Unger<EMAIL_ADDRESS>Institute for Astroparticle Physics
(IAP), Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany  Institutt
for fysikk, Norwegian University of Science and Technology (NTNU), Trondheim,
Norway

###### Abstract

High-precision cosmic-ray data from ongoing and recent past experiments
(Voyager, ACE-CRIS, PAMELA, ATIC, CREAM, NUCLEON, AMS-02, CALET, DAMPE) are
being released in the tens of MeV/n to multi-TeV/n energy range. Astrophysical
and dark matter interpretations of these data are limited by the precision of
nuclear production cross-sections. In Paper I, Phys. Rev. C 98, 034611 (2018),
we set up a procedure to rank nuclear reactions whose desired measurements
will enable us to fully exploit currently available data on CR Li to N
($Z=3-7$) species. Here we extend these rankings to O up to Si nuclei
($Z=8-14$), also updating our results on the LiBeB species. We also highlight
how comprehensive new high precision nuclear data, that could e.g. be obtained
at the SPS at CERN, would be a game-changer for the determination of key
astrophysical quantities (diffusion coefficient, halo size of the Galaxy) and
indirect searches for dark matter signatures.

###### Contents

  1. I Introduction
  2. II Calculation setup and flux origin
    1. II.1 Propagation setup
    2. II.2 Nuclear cross-section models
    3. II.3 Primary vs. secondary origin of F to Si
  3. III Definitions and formulae
    1. III.1 Ghost nuclei in cumulative cross-sections
    2. III.2 Ranking of cross-sections: $f_{abc}$ coefficients
    3. III.3 Impact of nuclear uncertainties on fluxes
    4. III.4 Beam time calculation: ${\cal C}_{ab}$ coefficients
    5. III.5 Other rankings of astrophysical interest
      1. III.5.1 Direct contributions from measured fluxes: $D_{i\to j}$ coefficients
      2. III.5.2 Channels from primary fluxes: $P^{\rm 1-step}_{ij}$ and $P^{\rm 2-step}_{ikj}$ coefficients
      3. III.5.3 Progenitors from cumulative of channels: $P_{i\to j}$ coefficients
  4. IV Results at 10 GeV/n (on TOA fluxes)
    1. IV.1 Graphical views of the various rankings
    2. IV.2 Desired reactions
    3. IV.3 Desired number of events (beamtime)
  5. V Forecasts from improved cross-sections
    1. V.1 Methodology
      1. V.1.1 Set-up of the mock cross-sections
      2. V.1.2 Fit to secondary/primary ratios
    2. V.2 Improvements on propagation parameters
    3. V.3 Reduced uncertainty for secondary $\bar{p}$
    4. V.4 Reduced uncertainty for $L$ and primary $\bar{p}$
  6. VI Conclusions
  7. A Energy-dependent view of the fragmentation contributions, channels, and progenitors
    1. A.1 Origin of contributions
    2. A.2 Ranking of channels
    3. A.3 Ranking of progenitors
  8. B Figures for $D_{i\to j}$ and $P_{i\to j}$ at 10 GeV/n
  9. C Tables of ranked $f_{abc}$ and plots at 10 GeV/n

##  I Introduction

In the last two decades, Galactic cosmic-ray (GCR) physics has entered a
precision era with the PAMELA [1, 2] and AMS-02 [3] experiments. The energy
domain of GCR direct high-precision measurements now starts from a few MeV/n
with Voyager 1, 2 [4] and ACE-CRIS [5], and goes up to hundreds of TeV/n with
CALET [6], DAMPE [7], CREAM [8], and NUCLEON [9]. Many GCR data sets are now
dominated by systematic uncertainties, with AMS-02 data reaching an
unprecedented few percent precision. On the other hand, the modeling of the
GCR nuclear component (compared to these data) involves a large network of
production reactions whose precision only reaches $10-20\%$ level on average
(see for instance [10], hereafter Paper I). This is a severe limitation to
provide answers to important questions in this field of research, such as the
origin of the CRs, transport in the Galaxy and unveiling astropysical dark
matter (DM).

The aim of this series of paper is to rank the most crucial nuclear isotopic
production cross-sections, whose new measurements will enable us to fully
exploit the present and future GCR data. We refer the reader to Paper I for a
more extensive introduction, motivations, and references, in particular for
the historical context of nuclear data and models relevant for GCR physics.
While Paper I dealt with isotopic cross-section reactions to produce Li, Be,
B, C, and N elements (i.e. $Z=3-7$), this second paper extends the study to
O–Si ($Z=8-14$) species. In particular, overwhelmingly secondary F and mostly
primary Si and the F/Si ratio can be used as a heavier analog [11, 12, 13] to
the B/C ratio, which is widely used to deduce the CR propagation parameters.
Given that various CR species can have different origin, a set of such
secondary to primary ratios with different atomic mass numbers and properties
can be utilized to probe interstellar propagation on different spatial scales
[14, 15].

Our results allow the experimental teams to concentrate on a small number of
crucial reactions rather than blindly measure thousands of relevant reactions.
The implications are broad and can potentially lead to major breakthroughs.

Remarkable is the number and breadth of the newly discovered features.
Perhaps, the first to mention is the discovery of a rise of the positron
fraction $e^{+}/(e^{+}$$+$$e^{-})$ up to 100 GeV by PAMELA [16], contrary to
the expectations of a monotonic decrease with energy [17, 18]. The rise was
confirmed by Fermi-LAT [19], and with higher precision and up to $\sim$500 GeV
by AMS-02 [20]. The latest AMS-02 data indicate a cutoff in the $e^{+}$
spectrum at $\sim$350 GeV [3] that may hint at the origin of the excess. The
all-electron ($e^{-}$$+$$e^{+}$) spectrum up to 1 TeV was measured by Fermi-
LAT [21, 22]. It appears to be too flat, contrary to the expectations of a
steep decrease. A sharp cutoff above $\sim$1 TeV was reported by H.E.S.S. [23,
24] and confirmed by VERITAS [25]. Subsequent measurements by PAMELA [26],
AMS-02 [27, 28, 3], CALET [29], DAMPE [30], and Fermi-LAT [31], though do not
fully agree in detail, confirmed these results and reveal some features in the
100 GeV–TeV region. The all-electron spectrum includes the $e^{+}$ excess (a
so-called “signal”) and may include the identical $e^{-}$ “signal” if the
source is charge-sign symmetric (e.g., pulsars or DM) [32, 28]. Fast energy
losses due to the inverse Compton and synchrotron emission at TeV energies
ensure that TeV electrons could only come from nearby DM clumps or local
sources [33, 34, 35, 36] that would appear as spectral bumps (yet to be
detected).

Analysis of the Fermi-LAT observations yielded a weak extended residual
component at a few GeV peaked at the Galactic Center (GC) [37, 38], consistent
with minimal supersymmetric model (MSSM) and limits from direct detection
experiments [39, 40]. A recent detection of the $\gamma$-ray emission from the
extensive 400-kpc-across halo of M31 at 3-20 GeV [41] hints at the similarity
with the GC and $\bar{p}$ excesses [42, 43]. The excess of 10-20 GeV $\bar{p}$
[44, 45], which is clearly seen in a high precision spectrum by AMS-02 [46,
3], may be an artifact due to uncertainties in the $\bar{p}$ production cross
section, but surprisingly this excess appears in the same energy range as the
excesses in $\gamma$-ray emission from the GC and the halo of the M31 galaxy.
Meanwhile, outside of the excess range the $\gamma$-ray emission and $\bar{p}$
data agree well with conventional predictions [47, 41]. The absence of a
corresponding signal in direct detection experiments may point to the hidden
sector DM [45].

The most striking is the AMS-02 preliminary claim of the detection of six
${}^{3}\overline{\rm He}$ and two ${}^{4}\overline{\rm He}$ events [48]. The
number of detected events (8) and the ratio ${}^{4}\overline{\rm
He}/^{3}\overline{\rm He}$ $=$$1/3$ is too high for them to be produced
through coalescence [49, 50, 51, 52].

Besides, new data yield a deeper understanding of the internal works of our
Galaxy. Observations of the breaks (hardening) in the spectra of (mostly)
primary ($p$, He, C, O, Ne, Mg, Si, Fe), secondary (Li, Be, B, F), and
intermediate (N, Na, Al) nuclei at the same magnetic rigidity $R$$\approx$300
GV by PAMELA [53], Fermi-LAT [54], AMS-02 [55, 56, 57, 58, 59, 60, 3, 61, 62,
63], while the spectral slopes of these groups of nuclei are different. This
may imply a change in the spectrum of the interstellar turbulence [64, 65, 66]
or an influence of a passing star (such as $\epsilon$ Eri) [67, 68]; for a
summary of current models see [69]. There is also an evidence of some fraction
of primary Li in CRs [70], the unexpected low-energy excess in CR Fe [71] and
Al [72], and a hint at the presence of primary F in CRs [11], all discovered
using the combined GalProp–HelMod framework; alternative interpretations of
the Li and F data invoke spatially-dependent diffusion model [13] or, what
concerns us particularly in this paper, uncertainties on the production cross-
sections [73, 74, 75, 76, 12].

Anomalously high CR isotopic ratios 12C/16O, 22Ne/20Ne, and 58Fe/56Fe [77, 78]
and a recent detection of radioactive 60Fe in CRs by the ACE-CRIS [79]
indicate a relatively recent supernovae (SNe) activity in the solar
neighborhood and align well with the discovered low-energy excess in CR Fe
[71]. Discovery of the Fermi Bubbles in the Fermi-LAT data [80, 81], huge
$\sim$10-kpc-across structures emanating from the GC, and even larger bubbles
by eROSITA [82] add to understanding of the activity of the central
supermassive black hole and its ability to accelerate particles.

The listed discoveries imply that new physics may be involved, however, the
significance of these findings critically depends on the accuracy of the
underlying datasets, especially isotopic production cross sections. The latter
are used for evaluation of the CR propagation parameters, which, in turn, is
the cornerstone of all calculations involving CR particles, their emissions,
and related backgrounds.

The paper is organized as follows: in Section II, we present the propagation
setup and reference isotopic production cross-section parametrizations used
for our calculations. We also highlight the contributing fractions to CR
fluxes (CR source, fragmentation on the interstellar medium, or radioactive
decay of parent nucleus). In Section III, we gather all relevant definitions
and formulae used for the ranking of the individual isotopic production cross-
sections (in terms of their contribution to the production of a given CR
isotope or element), and provide formulae to propagate the cross-section
uncertainties to the calculated GCR fluxes. We also provide additional
coefficients (built from the ranking production cross section coefficients)
enabling the calculation of the beam time and number of event estimates to
improve GCR flux modeling to a user-desired precision. Of interest for the
broader CR community, we also define additional rankings in terms of the
dominant reactions for the direct production (from GCR fluxes as measured),
the most important production channels (defined to be the ensemble of 1-step
or multi-step production paths starting from primary fluxes), and the most
important progenitors (defined to be the sum of all contributions starting
from a primary flux). In Section IV, we show a subset of our results,
illustrating the various ranking coefficients at 10 GeV/n in different
graphical views, the desired reactions to improve GCR flux predictions, and
the number of events necessary to measure these reactions at a precision that
would be on par to model GCR fluxes at an accuracy below the $3\%$ precision
reached by current GCR data. In Section V, we consider mock nuclear data
obtained from the above beam-time properties to make forecasts on the
improvement brought on key CR quantities for different possible scenarios of
new nuclear measurements (for instance at NA61/SHINE).

For readability, we report a lot of our figures and tables in the Appendix:
Appendix A shows the energy-dependence of (i) the primary/secondary content of
GCR elemental fluxes and the relative importance of 1-step vs. multi-step
production, (ii) the most important production channels, (iii) the most
important progenitors. Appendix B provides a graphical view at 10 GeV/n of the
most important direct production reactions and most important progenitors at
the isotopic level. Appendix C provides tables of the $f_{abc}$ coefficients
(also at 10 GeV/n), i.e. the ranking of the most important production
reactions $a+b\to c$ for Li to Si GCR fluxes; we also show graphical views of
the $f_{aHc}$ (i.e. restricting our consideration by reactions on H, the
dominant component of the ISM).

##  II Calculation setup and flux origin

The transport of CRs in the Galaxy is described by a system of second-order
differential equations (e.g., [83]), which include CR sources and transport
(diffusion, convection), energy gains and losses, and production of secondary
isotopes and particles and destruction of nuclei species in the interstellar
medium (ISM), assumed to be made of $90\%$ of H and $10\%$ of He in number. To
describe the nuclear component, we assume steady-state. In order to produce
the observed amount of secondary species, GCRs must cross a specific amount of
matter in the ISM, called the grammage. In this respect, the discussion and
the results presented here are largely independent of the specific GCR model
implementation, as long as the model reproduces standard observables of GCR
physics [84].

###  II.1 Propagation setup

As in Paper I, we rely on the 1D semi-analytical propagation model implemented
in the USINE package111https://dmaurin.gitlab.io/USINE/ [85]. Since Paper I,
several high-precision elemental fluxes have been released by the AMS-
collaboration [58, 59, 60, 61, 62, 63], in particular F to Si fluxes and also
the Fe flux. These data have been used to update the values of the transport
parameters employed in different models [66, 71, 11, 69], including studies
done with the USINE code [86, 73, 87, 76]. We use here the so-called SLIM
configuration introduced in [86], that is a purely diffusive configuration
without convection and distributed reacceleration. The diffusion coefficient
$D(R)$ depends on the rigidity $R=pc/Ze$ and is taken to be a broken-power law
with breaks both at low (around a few GV) and high (at a few hundreds of GV)
rigidities:

$\\!\\!\\!D(R)\\!=\\!{\beta}D_{0}\\!\\!\left[\frac{R}{\\!1\,{\rm
GV}\\!}\right]^{\delta}\\!{\left[1\\!+\\!\left(\\!\frac{R_{l}}{R}\right)^{\\!\\!\\!\frac{\delta-\delta_{l}}{s_{l}}}\\!\right]^{\\!s_{l}}}\\!{\left[1\\!+\\!\left(\\!\frac{R}{R_{h}}\right)^{\\!\\!\\!\frac{\delta-\delta_{h}}{s_{h}}}\\!\right]^{\\!-s_{h}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!.\\!\\!$
(1)

For practical purpose, the values of the parameters in the above equation are
taken from [12]. It is important to stress that for these parameters, $D$ is a
decreasing function of the energy below the low-energy break, and a growing
function above this break.

At variance with Paper I, we assume species-dependent broken power-laws for
the CR source spectra instead of a universal power-law for all species: this
ensures a better match of the existing GCR data, but it has in practice a very
minor impact on the results [see, e.g., 12]. The isotopic fractions and
relative elemental abundances at source are initialized to their Solar system
values [88] and iteratively adjusted until they match both the elemental
fluxes and isotopic ratios: for elemental fluxes, we use AMS-02 data at 50 GV
for H to Si and Fe, and HEAO-3 CR data [89] at 10.6 GeV/n for other elements
up to Zn; for isotopic ratios, we use low-energy data points mostly from ACE-
CRIS [90, 91, 92] but also from ULYSSES [93].

We stress that other hypotheses for the transport setup or the source spectra
would lead to slightly different energy dependences of the various rankings
presented in this study, respectively at low and high-energy. First,
considering reacceleration and convection (instead of, or in addition to, the
low-energy diffusion break) would change the residence time in the Galaxy
below a few GeV/n, hence the grammage crossed and the amount of secondary CR
produced. However, in this non-relativistic regime, the interactions rate also
decreases by a $\beta$ factor, and eventually goes to zero when the threshold
for the cross-section is reached. Furthermore, below a few hundreds of MeV/n,
energy losses in the ISM become the dominant process that shapes the fluxes,
so that all these details become somewhat irrelevant. Actually, our results
are shown for solar-modulated fluxes (using the Force-Field approximation [94,
95]), and the so-called top-of-atmosphere (TOA) fluxes considered here
correspond to calculation made for interstellar energies always above a few
hundreds of MeV/n. All in all, changing the transport setup should only
slightly change the relative importance of secondary production with regard to
the total flux (for mixed species) in the low energy regime; more importantly,
the relative importance of the various cross-sections (and hence the rankings)
should not. Second, assuming a universal spectral index for all species or
allowing for different low- and high-energy spectral indices for different
elements [66, 71, 11, 12] would change the ranking of the most-important
cross-sections above several tens of TeV, where high-energy data are not
constraining enough, and in particular AMS-02 data are dominated by
statistical uncertainties. To mitigate this potential effect, we provide the
ranking of the production cross-sections at 10 GeV/n: this is an energy around
which the overall production level (or grammage) is perhaps best-constrained
(thanks to the best precision reached for GCR data there).

###  II.2 Nuclear cross-section models

Both types of the cross sections, the total fragmentation and isotopic
production cross sections, are necessary for the GCR modeling, the former
being measured and modeled at a better precision than the latter. For flux
calculations, uncertainties from production cross-sections dominate compared
to uncertainties from inelastic cross-sections, and we fix the
parameterization of the latter to that of [96]; we refer readers to [97] for a
recent discussion of the inelastic cross-sections models222Note that there are
many typos in the original paper [96], which are fixed in [97]. used in GCR
physics.

The existing models for the isotopic production cross-sections were discussed
at length in Paper I. To encompass the typical uncertainties on these models,
as in Paper I, we consider two parameterizations available from the GALPROP
code333http://galprop.stanford.edu [97], namely its options OPT12 and OPT22.
Both are based on a careful inspection of the quality and systematics of all
available nuclear datasets for each reaction in MeV/n–GeV/n range, and the use
of specifically designed approximations, semi-empirical formulae, and nuclear
codes normalized to data when available [98, 99, 100, 101, 102, 103]: _for
marginally important reactions,_ OPT12 uses the WNEW code [104, 105, 106, 107]
and OPT22–the YIELDX code [108, 109], providing a representative dispersion
for cross-sections where no data are available. For more details on the
nuclear package in GALPROP see [66] and references therein. For most of our
rankings, we actually use a third parameterization, based on a mixture of
GALPROP-OPT12 and GALPROP-OPT22 [76, 12]. This parameterization is based on
GALPROP-OPT12 and renormalizes, for the most important reaction involved in
the production of LiBeB [76] and F [12], GALPROP-OPT22 cross-sections to
nuclear data, which became available after the GALPROP nuclear cross-section
package was developed. Indeed, with these new nuclear data, the fragmentation
of Fe appears to be very important [76], compared to how it was ranked in
Paper I. This parameterization can be seen as a partial update of the original
GALPROP routines, which is used here to update the ranking for the LiBeB
elements provided in Paper I.

A full update of the GALPROP parameterizations, based on a comprehensive
survey of the literature for nuclear data relevant for the production of all
elements considered here would be ideal. However, this is an extremely time-
consuming task yet to be completed. While working to achieve this goal, we
provide the best rankings available at this time. As new data becomes
available, our results will be updated in the follow-up papers.

###  II.3 Primary vs. secondary origin of F to Si

Before defining which rankings are performed and what they are, it is
important to briefly introduce the most salient properties, production-wise,
of the CR elements under scrutiny. For an advanced discussion on the origin of
these elements in GCRs, see [110].

GCRs are broadly separated in two categories, namely primary species,
accelerated in sources (e.g., 1H, 4He, O, Si, Fe), and secondary species,
resulting from the nuclear fragmentation of heavier species in the ISM (e.g.,
2H, 3He, LiBeB, F, Sc, Ti, and V). Most species are formed from an energy-
dependent mixture of these two categories. Schematically, for each secondary
component, its spectrum appears to be steeper than the spectrum of its
progenitor by the value of the index of the diffusion coefficient. Given the
shape of $D(R)$, see Eq. (1), which is minimum at a few GeV/n and increases
otherwise, the maximum of secondary content in CRs peaks at a few GeV/n.
Moreover, in case of consequent nuclear interactions, the energy dependence of
the spectrum of the daughter nucleus will suffer such steepening each time
resulting in even steeper spectrum (see also Paper I). Obviously, secondary-
dominated species will be much more sensitive to uncertainties in the cross
sections than primary-dominated ones, but the maximum of uncertainties will
arise at this peak, where the direct and multi-step contributions maximized.
Showing calculations at a few GeV/n thus highlights the role of nuclear
uncertainties on CR fluxes for the most unfavorable case.

We show in Table 1 the different origins of the elemental fluxes (and their
isotopes) calculated at 10 GeV/n. To avoid any possible ambiguity, we denote
the contributing fractions shown as $f_{\rm prim}$ (for primary), $f_{\rm
sec}$ (for secondary, including direct and multi-step production), and $f_{\rm
rad}$ (for radioactive, a rare case of a CR species receiving a contribution
from the $\beta$ or EC capture decay of another CR progenitor444Relevant
unstable species for GCR studies are those whose decay lifetime is not too
small or too large compared to the propagation time in the Galaxy, see Section
III.1.), whose sum makes up $100\%$ of the total flux. We further separate
$f_{\rm sec}$ into direct, two-step, and more than two-step contributions
(whose total makes up $100\%$ of the fragmentation flux).

Table 1: Fractions of primary, fragmentation, and radioactive origin (with regard to total flux), and contributions of 1-step, 2-step, and more-than-2-step channels (with regard to total fragmentation production) for fluxes at 10 GeV/n modulated with orce-field modulation potential $\phi_{\rm FF}=700$ MV. CR element |  | % of |  | % of multi-step
---|---|---|---|---
or isotope (%) |  | total flux |  | secondaries
|  | $f_{\rm prim}$ | $f_{\rm sec}$ | $f_{\rm rad}$ |  |  1 |  2 | $>\\!2$
Li |  |  | 0 | 100 | 0 |  | 74 | 20 | 6
6Li | (61%) |  | 0 | 100 | 0 |  | 75 | 19 | 6
7Li | (39%) |  | 0 | 100 | 0 |  | 72 | 22 | 6
Be |  |  | 0 | 100 | 0 |  | 75 | 19 | 6
7Be | (58%) |  | 0 | 100 | 0 |  | 80 | 15 | 5
9Be | (33%) |  | 0 | 100 | 0 |  | 69 | 24 | 7
10Be | (9%) |  | 0 | 100 | 0 |  | 70 | 24 | 6
B |  |  | 0 | 96 | 4 |  | 77 | 18 | 5
10B | (34%) |  | 0 | 90 | 10 |  | 71 | 23 | 6
11B | (66%) |  | 0 | 100 | 0 |  | 80 | 15 | 4
C |  |  | 80 | 20 | 0 |  | 78 | 17 | 5
12C | (90%) |  | 88 | 12 | 0 |  | 72 | 22 | 6
13C | (10%) |  | 7 | 93 | 0 |  | 84 | 12 | 3
14C | (0.03%) |  | 0 | 100 | 0 |  | 60 | 33 | 6
N |  |  | 22 | 76 | 2 |  | 88 | 8 | 3
14N | (51%) |  | 43 | 52 | 5 |  | 86 | 11 | 3
15N | (49%) |  | 0 | 100 | 0 |  | 90 | 7 | 3
O |  |  | 94 | 6 | 0 |  | 65 | 24 | 10
16O | (97%) |  | 97 | 3 | 0 |  | 64 | 25 | 11
17O | (1%) |  | 2 | 98 | 0 |  | 65 | 24 | 10
18O | (1%) |  | 9 | 91 | 0 |  | 69 | 22 | 9
F |  |  | 0 | 100 | 0 |  | 74 | 18 | 8
19F | (100%) |  | 0 | 100 | 0 |  | 74 | 18 | 8
Ne |  |  | 69 | 31 | 0 |  | 71 | 20 | 9
20Ne | (61%) |  | 85 | 15 | 0 |  | 66 | 24 | 10
21Ne | (10%) |  | 0 | 100 | 0 |  | 70 | 21 | 9
22Ne | (29%) |  | 62 | 38 | 0 |  | 76 | 17 | 7
Na |  |  | 13 | 87 | 0 |  | 80 | 13 | 7
23Na | (100%) |  | 13 | 87 | 0 |  | 80 | 13 | 7
Mg |  |  | 83 | 16 | 1 |  | 70 | 19 | 10
24Mg | (74%) |  | 90 | 10 | 0 |  | 72 | 18 | 10
25Mg | (13%) |  | 53 | 47 | 0 |  | 70 | 20 | 11
26Mg | (13%) |  | 69 | 22 | 9 |  | 68 | 22 | 10
Al |  |  | 35 | 65 | 0 |  | 85 | 9 | 7
26Al | (9%) |  | 0 | 100 | 0 |  | 77 | 15 | 8
27Al | (91%) |  | 38 | 62 | 0 |  | 86 | 8 | 6
Si |  |  | 91 | 9 | 0 |  | 50 | 29 | 21
28Si | (90%) |  | 96 | 4 | 0 |  | 52 | 26 | 21
29Si | (6%) |  | 50 | 50 | 0 |  | 48 | 31 | 22
30Si | (4%) |  | 40 | 60 | 0 |  | 50 | 29 | 20

In Table 1, we first see that between O and Si, Mg is the only element with a
radioactive contribution (Be, C, N were already discussed in Paper I), owing
to a non-negligible amount of 26Al decaying into 26Mg, weighted down in Mg by
the subdominant isotopic content of 26Mg. We then see that F is a quasi-pure
secondary species, as discussed for instance in [12]. We then have Ne, Mg, and
Si predominantly of primary origin, and finally Na and Al as prototypes of
mixed species. The last three columns show that direct production (1-step) is
always the dominant fraction with regard to the total fragmentation
production, but with a significant amount of 2-step production (up to $20\%$),
and a non-negligible amount of $>2$-step production (up to $10\%$); these
numbers must be compared to the $\lesssim 3\%$ precision we have nowadays on
CR data, a precision we wish to match from the modeling side. We stress that
the odds of having multi-step reactions decrease with the number of steps $n$:
the timescale for $n$ successive interactions to happen is $n$ times that of a
single interaction, which makes it $n$ times less likely than escape from the
Galaxy to happen. As a consequence, as we will see later on, this significant
$10\%$ level of multi-step production seen in Table 1 is actually made up of a
huge number of tiny contributions.

##  III Definitions and formulae

In Paper I, definitions and formulae used in our ranking analyses were
scattered in different sections. Here, we gather them all in one section to
facilitate reading. We also expand the formulae to new cases and
configurations not considered in Paper I.

We start with the definition of ghost nuclei, highlighting the scope of
cumulative cross-sections in a GCR context in Section III.1. We then define
the $f_{abc}$ coefficients to rank the production cross-sections to measure
with high priority in Section III.2. From these coefficients, we define in
Section III.3 formulae for the error propagation of nuclear cross-section
uncertainties to the GCR fluxes (under several plausible assumptions on
nuclear data). In Section III.4, we consider the case where all fragments from
many progenitors can be measured at once, for which we can provide formulae to
calculate the beam time properties (reactions and associated number of events)
necessary to reach a desired precision on the predicted GCR fluxes. To do so,
we introduce the ${\cal C}_{ab}$ coefficients, formed from the $f_{abc}$
coefficients and the reaction cross sections.

In order to rank other quantities of interest for CR physicists, other
families of coefficients are introduced in Section III.5: the $D_{i\to j}$
coefficients to rank the most important direct production channels calculated
from GCR fluxes as measured (not in Paper I); the $P^{\rm 1-step}_{ij}$,
$P^{\rm 2-step}_{ikj}$… to rank the production channels, i.e. reaction paths
linking a primary flux to a fragment, accounting for direct and multi-step
production; the $P_{i\to j}$ and $P_{Z_{i}\to j}$ coefficients (not in Paper
I) to rank the most important progenitors of a given CR species.

###  III.1 Ghost nuclei in cumulative cross-sections

Unstable nuclei created by interactions of CRs (on the ISM) whose half-life
are $\lesssim$ kyr play a specific role in the GCR modeling. These short-lived
nuclei decay before having another interaction and are thus dubbed ghost
nuclei: they are not detected in CR experiments, but they increase the
production of their stable daughter nuclei (detected in CR experiments), which
is reflected in the cumulative cross sections $\sigma^{\rm cumul}$. In nuclear
physics experiments, the production cross-section of these ghosts can be
measured if their half-life is longer than the time of flight between the
target and the detector. If this is not the case, this implies that the
measured cross-sections are themselves cumulative of some sort. Hence, going
from measured or modeled production cross-sections to cumulative cross-
sections for CR applications cannot be ignored for most CR species.

For a CR projectile $X$ into a stable CR nucleus $Y$ (for any given target
$T$), we define $\sigma^{\rm cumul}_{X\rightarrow Y}$ to be the direct
production of $Y$, plus the production of all related ghost nuclei $g$
weighted by their branching-ratio decay ${\cal B}r_{g}$ in $Y$:

$\sigma^{\rm cumul}_{X+T\rightarrow{\rm Y}}=\sigma_{X+T\rightarrow{\rm
Y}}+\sum_{g}{\cal B}r_{g}\cdot\sigma_{X+T\rightarrow g}\;.$ (2)

We use in the following the ghost and branching ratios555Note that GALPROP
nuc_package also includes the nuclear reaction network with branching ratios.
These include naked and H-like atoms, which may have different decay channels.
gathered in [111] and expanded in [112]. We list in Table 2 the properties of
the ghost nuclei whose production for F to Si fluxes is large enough to make
them appear in the ranking tables shown in App. III.2.

For the rankings presented in the next section, it is also useful to define
$\sigma^{\rm cumul}_{i\rightarrow j}$, the cumulative cross-section of CR
species $Y$ from $X$ over all ISM targets $T$:

$\sigma^{\rm cumul}_{X\rightarrow Y}\equiv\frac{\sum_{T}n_{\rm
ISM}^{T}\sigma^{\rm cumul}_{X+T\rightarrow Y}}{\sum_{T}n_{\rm ISM}^{T}}\,,$
(3)

with $n_{\rm ISM}^{T}$ the density of the ISM target $T$.

Table 2: List of ghost nuclei with significant contributions to CR fluxes $Z=3-14$, i.e. appearing in Tables 6 to 16. The different columns show their half-life, decay channel, and branching ratio, taken from [113]. Nucleus | $T_{1/2}$ | Daughter | Decay mode
---|---|---|---
6He | 806.92 ms |  6Li |  $\beta^{-}$ ($100\%$)
9Li | 178.3 ms |  9Be |  $\beta^{-}$ ($49.2\%$)
11Be | 13.76 s |  11B |  $\beta^{+}$ ($97.1\%$)
10C | 19.3009 s |  10B |  $\beta^{+}$ ($100\%$)
11C | 20.364 m |  11B |  $\beta^{+}$ ($100\%$)
12B | 20.20 ms |  12C |  $\beta^{-}$ ($98.4\%$)
13N | 9.965 m |  13C |  $\beta^{+}$ ($100\%$)
16N | 7.13 s |  16O |  $\beta^{-}$ ($99.99855\%$)
17N | 4.173 s |  16O |  $\beta^{-}n$ ($95\%$)
13O |  8.58 ms {23mm |  13C |  $\beta^{+}$ ($89.1\%$)
12C |  $\beta^{+}p$ ($10.9\%$)
14O | 70.620 s |  14N |  $\beta^{+}$ ($100\%$)
15O | 122.24 s |  15N |  $\beta^{+}$ ($100\%$)
19O | 26.470 s |  19F |  $\beta^{-}$ ($100\%$)
17F | 64.370 s |  17O |  $\beta^{+}$ ($100\%$)
18F | 109.739 m |  18O |  $\beta^{+}$ ($100\%$)
20F | 11.163 s |  20Ne |  $\beta^{-}$ ($100\%$)
21F | 4.158 s |  21Ne |  $\beta^{-}$ ($100\%$)
19Ne | 17.274 s |  19F |  $\beta^{+}$ ($100\%$)
23Ne | 37.140 s |  23Na |  $\beta^{-}$ ($100\%$)
21Na | 22.422 s |  21Ne |  $\beta^{+}$ ($100\%$)
22Na | 2.60 yr |  22Ne |  $\beta^{+}$ ($100\%$)
24Na | 14.957 h |  24Mg |  $\beta^{-}$ ($100\%$)
25Na | 59.1 s |  25Mg |  $\beta^{-}$ ($100\%$)
22Mg | 3.8755 s |  22Ne |  $\beta^{+}$ ($100\%$)
23Mg | 11.317 s |  23Na |  $\beta^{+}$ ($100\%$)
27Mg | 9.435 m |  27Al |  $\beta^{-}$ ($100\%$)
25Al | 7.183 s |  25Mg |  $\beta^{+}$ ($100\%$)
28Al | 2.245 m |  28Si |  $\beta^{-}$ ($100\%$)
29Al | 6.56 m |  29Si |  $\beta^{-}$ ($100\%$)
27Si | 4.15 s |  27Al |  $\beta^{+}$ ($100\%$)
30P | 2.498 m |  30Si |  $\beta^{+}$ ($100\%$)
32P | 14.268 d |  32S |  $\beta^{-}$ ($100\%$)
44Sc | 4.0420 h |  44Ca |  $\beta^{+}$ ($100\%$)

###  III.2 Ranking of cross-sections: $f_{abc}$ coefficients

In the context of GCRs, we wish to establish a priority list of nuclear
production cross-sections to measure and improve, in order to better model F
to Si CR fluxes. Each element has its own ranking (of the most important
nuclear reaction), and both the production cross-sections and CR fluxes
(progenitors) have different energy dependences, so that the various
contributions are also energy dependent. Nevertheless, for legibility,
throughout Section III, these dependences on the specific element considered
and energy is left implicit in the various definitions and formula.

Let us consider the production cross-section $\sigma_{{a+b\rightarrow c}}$,
where $a$ is a CR projectile, $b$ an ISM target (H or He), and $c$ a fragment
(CR or ghost). The impact of this cross-section on the total secondary flux
$\psi_{\rm sec}$ of a given CR isotope or element is obtained from the
relative difference between the standard (or reference) flux calculation,
$\psi_{\rm sec}^{\rm ref}$, and the calculation in which we set this cross-
section to zero, $\psi_{\rm sec}^{\sigma_{{a+b\rightarrow c}}=0}$. In Paper I,
we defined the $f_{abc}$ coefficients to be exactly this difference (i.e.
impact on the flux):

$f_{abc}=\frac{\psi_{\rm sec}^{\rm ref}-\psi_{\rm
sec}^{\sigma_{{a+b\rightarrow c}}=\,0}}{\psi_{\rm sec}^{\rm ref}}\;.$ (4)

With this definition, the ranking of the most important production cross-
sections amounts to the ranking of the $f_{abc}$ coefficients; we stress that
this ranking remains unchanged if we consider the impact on the total flux
$\psi_{\rm tot}$ instead of $\psi_{\rm sec}$ in Eq. (4). As underlined in
Paper I, the $f_{abc}$ can be roughly interpreted as fractional contributions,
although

$\sum_{\forall(a,b,c)}f_{abc}\gtrsim 1\,,$ (5)

i.e. the sum of all cross-section contributions (for a CR species) is usually
larger than one. Indeed, most $\sigma_{a+b\to c}$ are involved in both direct
and multi-step production, leading to a double counting of some sort666In a
multi-step reaction, we kill the associated production when we set to zero the
cross-section of any of the steps involved: when we sum the associated
$f_{abc}$, we add multiple times the same production, hence the double (or
multiple) counting.: the more important these multi-step contributions are
(which peaks at a few GeV and then decrease with energy, see Paper I), the
larger (than one) the sum of the $f_{abc}$ is.

###  III.3 Impact of nuclear uncertainties on fluxes

The uncertainty on any calculated CR flux $\psi_{\rm tot}$ depends on the
individual cross-section uncertainties $\Delta\sigma_{{a+b\rightarrow c}}$. As
discussed in Paper I, this uncertainty can be related to the $f_{abc}$
coefficients (introduced in Section III.2) and $f_{\rm sec}$ fraction (shown
in Table 1).

We consider here the case of nuclear data coming from patchy measurements of
subsets of reactions at different facilities; this is the current status of
nuclear cross-section data. Different plausible assumptions on the presence or
absence of correlations between these datasets (and in practice, on the
modeling of the cross-sections based on these data) lead to different error
propagation formulae. We discussed three cases in Paper I, namely:

  * •

fully correlated uncertainties (or ‘corr’ for short)

$\displaystyle\left(\frac{\Delta\psi_{\rm tot}}{\psi_{\rm tot}}\right)^{\rm
corr}\\!\\!\approx f_{\rm
sec}\,\sum_{a,b,c}f_{abc}\frac{\Delta\sigma_{{a+b\rightarrow
c}}}{\sigma_{{a+b\rightarrow c}}}\,;$ (6)

  * •

fully uncorrelated uncertainties (‘uncorr’)

$\displaystyle\left(\frac{\Delta\psi_{\rm tot}}{\psi_{\rm tot}}\right)^{\rm
uncorr}\\!\\!\\!\\!\\!\\!\\!$ $\displaystyle\approx$ $\displaystyle f_{\rm
sec}\,\sqrt{\sum_{a,b,c}\left(f_{abc}\frac{\Delta\sigma_{{a+b\rightarrow
c}}}{\sigma_{{a+b\rightarrow c}}}\right)^{2}}\,;$ (7)

  * •

mixed case (hereafter ‘mix’), with uncorrelated uncertainties for fragments of
the same projectile, but correlated uncertainties for different projectiles

$\displaystyle\left(\frac{\Delta\psi_{\rm tot}}{\psi_{\rm tot}}\right)^{\rm
mix}\\!\\!\\!\\!$ $\displaystyle\approx$ $\displaystyle\\!\\!f_{\rm
sec}\sum_{a}\sqrt{\sum_{b,c}\left(f_{abc}\frac{\Delta\sigma_{{a+b\rightarrow
c}}}{\sigma_{{a+b\rightarrow c}}}\right)^{2}}\\!.$ (8)

The first and second case correspond, respectively, to an optimistic and
pessimistic view on the existing status of nuclear data, and consequently on
the estimated uncertainties on the CR fluxes. The third case is probably the
most realistic, though this is difficult to assess in practice (see discussion
in Paper I). These formulae were used in Paper I to illustrate how the flux
uncertainties decrease (below the target precision of AMS-02 data), when a
growing number of important reactions are perfectly measured. We use these
formulae for the same purpose here in Section IV.2.

###  III.4 Beam time calculation: ${\cal C}_{ab}$ coefficients

In our previous paper, we introduced a simple method to use the ranked
production reactions as a guidance for laboratory measurements on how to
estimate the required number of interactions to achieve a specific flux
uncertainty. Since in an experiment dedicated to the measurements of
fragmentation cross sections, many fragments associated with a single
projectile are measured at once, we can estimate the statistical uncertainty
that such an experiment can achieve depending on the amount of recorded
interactions.

##### Flux uncertainty from one measured reaction.

In the reaction $a+b$, fragments of type $c$ are produced with probability

$p_{c}=\frac{\sigma_{{a+b\rightarrow c}}}{\sigma_{a+b}},$ (9)

where $\sigma_{a+b}$ is the total inelastic cross section for $a+b$ reaction
and $\sigma_{{a+b\rightarrow c}}$ is the fragmentation cross section
${a+b\rightarrow c}$ to produce fragment $c$. If $N$ interactions are
recorded, then the measured number of fragments, $n_{c}=p_{c}\,N$, is
multinomially distributed. If the number of interactions itself is a
Poissonian random variable, then the number of fragments is independent and
follow a Poissonian distribution with variance $n_{c}$, see e.g. [114],
leading to an experimental variance of the partial cross section of

$V(\sigma_{a+b\rightarrow
c})=\frac{1}{N}\,\sigma_{a+b}^{2}\,p_{c}=\frac{1}{N}\,\sigma_{a+b}\,\sigma_{a+b\rightarrow
c}.$ (10)

The combined relative uncertainty on the flux (total flux $\psi_{\rm tot}$
denoted $\psi$ in the rest of the section) from all reactions in $a+b$
interactions follows from error propagation,

$\left(\frac{\Delta\psi}{\psi}\right)_{a+b}^{2}=\frac{f_{\rm
sec}^{2}}{\psi^{2}}\sum_{c}\left(\frac{\partial\psi}{\partial\sigma_{{a+b\rightarrow
c}}}\right)^{2}V(\sigma_{a+b\rightarrow c})\,,$ (11)

where $f_{\rm sec}$ denotes the fraction of secondaries contributing to the
flux (cf. Table 1). Using $\partial\psi/\partial\sigma_{a+b\rightarrow
c}=(\psi/\sigma_{a+b\rightarrow c})f_{abc}$ and Eq. (10), this can be written
as

$\left(\frac{\Delta\psi}{\psi}\right)_{a+b}^{2}=\frac{f_{\rm sec}^{2}}{N}{\cal
C}_{ab}^{2},$ (12)

where we introduced the ${\cal C}_{ab}$ coefficients

${\cal
C}_{ab}^{2}\equiv\sum_{c}f_{abc}^{2}\frac{\sigma_{a+b}}{\sigma_{a+b\rightarrow
c}}.$ (13)

Note that the ${\cal C}_{ab}$ defined here is mathematically equivalent to our
previous definition after expanding the squared sum in Eq. (18) of Paper I:

$\left(\,\sum
x_{i}\right)^{2}=\sum_{i}x_{i}^{2}+2\sum_{i=1}^{m-1}\sum_{j=i+1}^{m}x_{i}x_{j},$

and cancelling the common terms.

##### Flux uncertainty from all contributing reactions.

The above uncertainty is only for one contributing reaction. The total flux
uncertainty is obtained from the quadratic sum over all contributing reactions
(i.e. reactions $a+b$, $d+e$, $f+g$, etc.). Labelling these $n_{r}$ reactions
with the index $k$, we can write $\\{{\cal C}_{ab},{\cal C}_{de},{\cal
C}_{fg},\dots\\}$ as $\\{{\cal C}_{k}\\}_{k=1\dots n_{r}}$, and assuming
$N_{k}$ interactions recorded for each reaction $k$, we get

$\left(\frac{\Delta\psi}{\psi}\right)^{2}=f_{\rm
sec}^{2}\sum_{k=1}^{n_{r}}\frac{1}{N_{k}}\,{\cal C}_{k}^{2}.$ (14)

##### Optimizing the total number of reactions to measure.

The requirement of having the total relative flux uncertainty to be smaller
than a certain value $\xi$,

$f_{\rm sec}^{2}\sum_{k=1}^{n_{r}}\frac{1}{N_{k}}\,{\cal
C}_{k}^{2}\equiv\sum_{k=1}^{n_{r}}{V}_{k}\leq\xi^{2}\,,$ (15)

has no unique solution for the $N_{k=1\dots n_{r}}$ number of interactions
(for each reaction $k$) to be measured. We try the ansatz

${V}_{k}=\xi^{2}\frac{{\cal C}_{k}^{\beta}}{\sum_{k=1}^{n_{r}}{\cal
C}_{k}^{\beta}}$ (16)

for the partial contribution ${V}_{k}$ of reaction $k$ to the overall
variance, which corresponds to

$N_{k}=\frac{f_{\rm sec}^{2}}{\xi^{2}}\,\,{\cal
C}_{k}^{2-\beta}\sum_{k=1}^{n_{r}}{\cal C}_{k}^{\beta}$ (17)

interactions to be measured for reaction $k$, and in total

$\displaystyle N_{\text{tot}}=$
$\displaystyle\sum_{k=1}^{n_{r}}N_{k}=\frac{f_{\rm
sec}^{2}}{\xi^{2}}\,\left(\sum_{k=1}^{n_{r}}{\cal
C}_{k}^{2-\beta}\right)\left(\sum_{k=1}^{n_{r}}{\cal C}_{k}^{\beta}\right)\,.$
(18)

In Paper I we investigated equal partial variances, $\beta=0$ and
${V}_{k}=\xi^{2}/n_{r}$, which leads to $N_{k}=n_{r}\,(f_{\rm sec}{\cal
C}_{k}/\xi)^{2}$. This scheme has the undesired effect that adding reactions
with low flux impact (small ${\cal C}_{k}^{2}$) will artificially increase
$N_{k}$ of all the other reactions via the number of considered reactions
$n_{r}$.

The scheme that _minimizes_ the total number of interactions to be recorded is
given by $\beta=1$, since for this value the derivative of Eq. (18),

$\displaystyle\frac{\partial N_{\text{tot}}}{\partial\beta}=\frac{f_{\rm
sec}^{2}}{\xi^{2}}\Biggl{[}$
$\displaystyle\;\quad\left(\sum_{k=1}^{n_{r}}{\cal C}_{k}^{\beta}\log{\cal
C}_{k}\right)\left(\sum_{k=1}^{n_{r}}{\cal C}_{k}^{2-\beta}\right)$
$\displaystyle-\left(\sum_{k=1}^{n_{r}}{\cal C}_{k}^{2-\beta}\log{\cal
C}_{k}\right)\left(\sum_{k=1}^{n_{r}}{\cal C}_{k}^{\beta}\right)\Biggr{]},$
(19)

is zero. Such an optimization allows significant work effort to be conserved
and yet fulfills our primary goal.

We show in Section IV.3 the ${\cal C}_{ab}$ coefficients we obtain for Li to
Si, and the desired number of events to reach a specified accuracy in the
modeled GCR fluxes. These numbers are then be used to provide forecasts in
Section V, in order to highlight how such measurements would be the game
changer for GCR physics.

###  III.5 Other rankings of astrophysical interest

We stress that nuclear or particle physicists, whose goal is to make new
measurements, should focus on the ranking of reactions obtained via the
$f_{abc}$ (and associated ${\cal C}_{ab}$) coefficients above. However, other
coefficients, corresponding to different groupings of the production cross-
sections, can be formed. These coefficients enable the ranking of other
quantities of interest for CR physicists, as described below.

####  III.5.1 Direct contributions from measured fluxes: $D_{i\to j}$
coefficients

At first order, the most important reactions for the production of CR species
$j$ are those responsible from the direct production of $j$ from the measured
CR isotopic fluxes $i$, as provided for instance in [115]. Indeed, as some of
the measured fluxes are partly or fully of secondary origin, accounting for
the direct production only misses cross sections involved in two-step
reactions (reactions allowing to go from a primary species into this secondary
flux).

Denoting $\sigma^{\rm cumul}_{i\rightarrow j}$ the cumulative cross-section of
CR species $j$ from $i$ over all ISM targets $t$, see Eq. (3), these direct
contributions are calculated from

$D_{i\to j}=\frac{\psi_{\rm sec}^{j,\,\rm ref}-\psi_{\rm sec}^{j,\,\sigma^{\rm
cumul}_{i\to j}=\,0}}{\psi_{\rm sec}^{j,\,\rm ref}}\,,$ (20)

and by construction we have

$\sum_{i}D_{i\to j}=1.$ (21)

The ranking of the above coefficients is directly the ranking of most
important production reactions (not separating the various targets and ghost
contributions), as opposed to the $f_{abc}$ corresponding to the ranking of
all involved cross-sections (Section III.2).

We did not consider these coefficients in Paper I, but we do in this paper
(see next section). We also consider the case of an element $Z_{i}$ ($n_{i}$
isotopes) into $Z_{j}$ ($n_{j}$ isotopes), given by the average

$\displaystyle D_{Z_{i}\to Z_{j}}=\frac{1}{n_{Z_{i}}n_{Z_{j}}}\times\sum_{i\in
Z_{i}}\sum_{j\in Z_{j}}D_{i\to j}\,;$ (22)

these coefficients also satisfy $\sum_{Z_{i}}D_{Z_{i}\to Z_{j}}=1$.

####  III.5.2 Channels from primary fluxes: $P^{\rm 1-step}_{ij}$ and $P^{\rm
2-step}_{ikj}$ coefficients

In Paper I, a channel was defined to be a unique production path linking a CR
progenitor $i$ to a CR fragment $j$. This production path can be of length one
(direct production) or larger than one (multi-step production), associated in
this paper to the $P^{\rm 1-step}_{ij}$, $P^{\rm 2-step}_{ikj}$, etc.
coefficients defined below. What we did not stress enough in Paper I is that a
channel can only start from a CR having a primary source term, hence the
notation $P$ here (at variance with the notation chosen in Paper I). Compared
to the $D_{ij}$ coefficients above (Section III.5.1), the $P^{\rm
1-step}_{ij}$ coefficients only start from species $i$ having a non-zero
primary component. However, the $P^{\rm 2-step}_{ij}$ coefficients describe
the 2-step contributions not accounted for in the $D_{ij}$. In that respect,
the full list of the production cross-sections that matter most is directly
the reactions appearing in the ranked channels.

These coefficients are obtained from the relative difference between the
secondary flux (of a CR species $j$) calculated from the case in which only a
specific channel is open (i.e. all cross-sections set to zero except for those
involved in this channel) and the reference calculation $\psi^{j,\,\rm
ref}_{\rm sec}$ (where all cross-sections are set to their fiducial values)
define the $P^{\rm 1-step}_{ij}$ (and $P^{\rm 2-step}_{ikj}$, …) coefficients
associated with 1-step (2-step, …) channels:

$\begin{cases}\displaystyle P^{\rm 1-step}_{ij}=\frac{\psi_{\rm
sec}^{j,\,\sigma^{\rm cumul}_{m\rightarrow
n}=\,0\;\,\forall(m,n)\neq(i,j)}}{\psi_{\rm sec}^{j,\,\rm ref}}\;,\\\
\displaystyle P^{\rm 2-step}_{ikj}=\frac{\psi_{\rm sec}^{j,\,\sigma^{\rm
cumul}_{m\rightarrow n}=\,0\;\,\forall(m,n)\neq\\{(i,k),(k,j)\\}}}{\psi_{\rm
sec}^{j,\,\rm ref}}\,,\\\\[11.38092pt] \displaystyle P^{\rm
3-step}_{ikpj}=\dots\\\\[5.69046pt] \dots\end{cases}$ (23)

As underlined in Paper I, the list of channels for all possible progenitors
$i$ of a CR fragment $j$ (under scrutiny) defines all its possible and
independent contributions. Hence, contrarily to the $f_{abc}$ coefficients
(see Eq. [5]), but similarly to the $D_{ij}$ coefficients, we do have, for the
sum of these coefficients,

$\sum_{i}^{i>j}\Bigg{(}P^{\rm 1-step}_{ij}+\sum_{k}^{i>k>j}\bigg{(}P^{\rm
2-step}_{ikj}+\sum_{k}^{i>k>j}\big{(}\dots\big{)}\bigg{)}\Bigg{)}=1.$ (24)

In practice, we only calculate the $P^{\rm 1-step}_{ij}$ and $P^{\rm
2-step}_{ikj}$ coefficients, and regroup the remaining coefficients under

$P_{j}^{\rm>2-step}=1-\sum_{i}^{i>j}\Bigg{(}P^{\rm
1-step}_{ij}+\sum_{k}^{i>k>j}P^{\rm 2-step}_{ikj}\Bigg{)}.$ (25)

This is sufficient to rank the most important direct and 2-step channels, and
to estimate the relative importance of the remaining multi-step contributions.
Although individual multi-step reactions are generally subdominant, their sum
can contribute to up to $25\%$ at 10 GeV/n (see Table 1). Actually, multi-step
contributions decrease with energy, and the ratio of contributing fractions
$P^{(n+1)-\rm step}/P^{n-\rm step}\propto R^{-\delta}$ (with $\delta\approx
0.5$ the slope of the diffusion coefficient), as discussed in detail in Paper
I.

Finally, if one is interested in the ranking of the production channels of a
CR element $Z_{j}$ instead of a CR isotope $j$, the above formulae can still
be used by substituting $\psi_{\rm sec}^{j\,,\rm ref}\to\psi_{\rm
sec}^{Z_{j}\,,\rm ref}$ in Eq. (23), and summing also over all $j\in Z_{j}$ in
Eqs. (24)-(25), with the meaning that $Z_{j}$ is the element under scrutiny
and $j$ is any isotope of the element $Z_{j}$.

####  III.5.3 Progenitors from cumulative of channels: $P_{i\to j}$
coefficients

As introduced in [76], we can further group the above coefficients in order to
specifically highlight the relative importance of different CR primary
progenitors $i$ in the production of a CR isotope $j$. By summing over all
possible intermediate fragments between $i$ and $j$, we form

$P_{i\to j}\\!=\\!P^{\rm
1-step}_{ij}+\\!\\!\\!\sum_{k}^{i>k>j}\\!\\!\\!\Bigg{(}P^{\rm
2-step}_{ikj}+\\!\\!\\!\sum_{p}^{k>p>j}\\!\\!\\!\bigg{(}P^{\rm
3-step}_{ikpj}+\dots\bigg{)}\\!\Bigg{)}.$ (26)

As for the channel coefficient, see Eq. (24), the sum of these new
coefficients is also one, i.e.

$\sum_{i}P_{i\to j}=1\,.$ (27)

We stress that the $P_{i\to j}$ and $D_{i\to j}$ (see Section III.5.1)
correspond to different quantities: while the $P_{i\to j}$ coefficients
quantify the importance of the primary species from which a CR $j$ under
scrutiny comes from (accounting from multi-step fragmentation starting from
this primary progenitor), the $D_{i\to j}$ coefficients quantify the
importance of the direct production of CR $j$ from the measured CR fluxes
(independently of their primary and secondary origin).

In practice, because it is time consuming to calculate all orders of the
$P_{i\dots j}^{n-\rm step}$ coefficients, we directly calculate the $P_{i\to
j}$ via

$P_{i\to j}=\frac{\psi_{\rm sec}^{j,\;\sigma_{k\to p}=\,0\,\;\forall
k>i,\,\forall p}\;-\;\psi_{\rm sec}^{j,\;\sigma_{i\to k}=\,0\,\;\forall
k}}{\psi_{\rm sec}^{j,\,\rm ref}}\;,$ (28)

that is the ratio of the difference between the calculation ‘all contributions
from CRs heavier than $i$ set to zero’ and ‘all contributions starting from
$i$ set to zero’, to the reference calculation $\psi^{j,\,\rm ref}_{\rm sec}$;
by construction and as also checked directly from the numerical calculation of
these coefficients, their sum is one.

Instead of the production of an isotope $j$, we can extend these coefficients
to the production of an element $Z_{j}$, i.e. $P_{i\to Z_{j}}$, by
substituting $\psi_{\rm sec}^{j}\to\psi_{\rm sec}^{Z_{j}}$ in Eq. (28) and
summing over all isotopes $j$ (of this element $Z_{j}$) in the right-hand side
of Eqs. (26) and (27). We can also extend the formula to the production from
an element $Z_{i}$ (instead of an isotope $i$), with $n_{i}$ the number of
isotopes of element $Z_{i}$,

$\begin{cases}\displaystyle P_{Z_{i}\to j}=\frac{1}{n_{Z_{i}}}\times\sum_{i\in
Z_{i}}P_{i\to j}\,,\\\ \displaystyle P_{Z_{i}\to
Z_{j}}=\frac{1}{n_{Z_{j}}}\times\sum_{i\in Z_{i}}P_{i\to Z_{j}}\,,\end{cases}$
(29)

which also satisfies $\sum_{Z_{i}}P_{Z_{i}\to j}=1$ and
$\sum_{Z_{i}}P_{Z_{i}\to Z_{j}}=1$. The $P_{Z_{i}\to Z_{j}}$ coefficients are
shown in Sect. IV.1 and their isotopic counterparts, the $P_{i\to j}$
coefficients, are shown in App. B.

The coefficients introduced in this section are directly the contributing
fractions of CR $i$ (or element $Z_{i}$) into CR $j$ (or element $Z_{j}$),
they weight the importance of both the various CR fluxes and production cross-
sections in a single energy-dependent number. As a result, ranking these
coefficients amounts to ranking the various contributors (or progenitors) of
$j$. We stress that in some cases, a progenitor may rank quite high while the
associated channels (i.e. $P_{i\dots j}$ coefficient discussed in Section
III.5.2) all rank low: the contribution of numerous sub-percent multi-step
production channels sometimes combine into a significant overall contribution.
Finally, because these coefficients include direct and multi-step
contributions in variable amounts, their energy dependence is a weighted-
combination of the single and multi-step associated energy dependence, and
thus less trivial than their $P_{i\dots j}$ counterparts.

##  IV Results at 10 GeV/n (on TOA fluxes)

We discuss here our main results in a few summary figures. We provide more
information and details in the Appendix, where additional plots and tables are
gathered.

###  IV.1 Graphical views of the various rankings

In Paper I, we provided tables for the rankings of the $f_{abc}$ coefficients
(III.2) and the channels (III.5.2). Here, we highlight via graphics the
different information contained in the $D_{Z_{i}\to Z_{j}}$ (III.5.1),
$P_{Z_{i}\to Z_{j}}$, and $f_{abc}$ coefficients, and what they tell us about
GCRs and more importantly, about the production cross sections.

Figure 1 shows the contributing production fractions (larger than $0.1\%$) of
elements $Z_{i}$ into $Z_{j}$ at 10 GeV/n; for each $Z_{j}$, the sum of the
coefficients on a row, by definition, adds up to $100\%$ of the secondary
production. The top panel shows the $D_{Z_{i}\to Z_{j}}$ coefficients (not
considered in Paper I), defined in Eq. (22), corresponding to the direct
production of the elements under scrutiny ($y$-axis on the right) from GCR
elemental fluxes ($x$-axis on top) as measured. This plot highlights the
standard results, for instance, that most of LiBeB comes from the
fragmentation of C and O, with some contributions from other primary species
(eg., Si, Fe). For the mostly secondary species F, the main progenitors are
Ne, Mg, and Al. Moving to heavier elements, the main progenitors shift to Mg,
Si, and Fe. Going back to Li, this plot also shows that some of its fraction
comes from the fragmentation of Be and B (and even its own isotopes, like
7Li), i.e. secondary species. What this plot does not show is the origin and
importance of the reactions contributing to the production of these secondary
species.

Figure 1: Secondary production of GCR elements $Z_{j}$ (right-hand side
labels) from GCR elements $Z_{i}$ at 10 GeV/n. The color bar encodes
$\log_{10}$ of the contributing fractions for this production, from 2.0
($100\%$) down to -1.0 ($0.1\%$); only contributions larger than $0.1\%$ are
shown. The top panel corresponds to the $D_{Z_{i}\to Z_{j}}$ coefficients
(III.5.1) and the bottom panel to the $P_{Z_{i}\to Z_{j}}$ ones (III.5.3). For
both cases, the sum of coefficients on one row is 100%, i.e. all the secondary
production, but we recall that the latter ($f_{\rm sec}$) is sometimes only a
small fraction of the total flux (see Table 1). See text for discussion.

This information is actually encoded into the ranking of channels, via the
associated coefficients defined in Eq. (23). A full discussion of the ranking
and importance of these channels for Li to Si elements, and their energy
dependences, are shown in Appendix A. Here, we only show in the bottom panel
of Fig. 1 the $P_{Z_{i}\to Z_{j}}$ coefficients defined in Eq. (29): these
coefficients are built from regrouping the 1-step and multi-step channels, and
correspond to the cumulated production of $Z_{j}$ from the primary fluxes
$Z_{i}$. Compared to the top panel, as illustrated on the Li case, we stress
that: (i) there is no production of Li from B, Be, and Li, because the latter
have no primary component; (ii) the production of Li or F from Fe is more
important in the bottom panel, because not only does it include direct
fragmentation of Fe into Li (as seen in the top panel) but it also includes
multi-step fragmentation of Fe into Li. In other words, these coefficients
provide a direct view of which primary elements contribute to the production
of al GCR fluxes. We stress that we focused the discussion on elements, but
the behavior and conclusions we just drew are similar if isotopes are
considered instead; the associated $D_{i\to j}$ and $P_{i\to j}$ coefficients
are shown in Appendix B.

Figure 2: Graphical view of the ranked reactions to measure for the production
of GCR Li (left) and F (right) at 10 GeV/n; similar plots for other elements
are gathered in Appendix C. The reactions shown are those making up $90\%$
(respectively $85\%$) of the total Li flux (respectively F flux),
corresponding to $f_{aHc}$ coefficients larger than 0.05 (respectively 0.10).
The color bar encodes $\log_{10}(f_{aHc})$ from 2.0 (100) down to the minimal
$f_{aHc}$ value shown; we recall that $\sum_{\forall a,b,c}f_{abc}>1$, so that
the $f_{aHc}$ coefficients cannot be interpreted as percentage fractions. The
top labels show the list of CR nuclei $a$ involved, while the right labels
show the list of $c$ fragments produced from $a$ interacting on H. These
fragments are GCR isotopes themselves but also short-lived, i.e. ghost,
nuclei: for instance 6He in the left panel decays into 6Li (see Table 2). The
blue rectangles highlight the direct production of Li (left panel) and F
(right panel); the reactions outside these rectangles involve intermediate
steps in the production of the element under scrutiny.

As stressed in Section III.2, the full information on the importance of the
various production cross-sections (involved in direct or multi-step
production) are encoded in the $f_{abc}$ coefficients. Tables of these ranked
coefficients at 10 GeV/n are provided in Appendix C for Li to Si species. We
consider below the $f_{aHc}$ coefficients, i.e. where the ISM target $b$ is H,
discarding interactions on He (but keep in mind that some of the latter rank
high, see Appendix C). We show these coefficients in Fig. 2 for the production
of Li (respectively F) on the left (respectively right) panel. As in Fig. 1,
we identify the dominant progenitors (here 56Fe, 28Si, 24Mg, etc.) for the
production of Li via 6Li and 7Li (respectively F via 19F and the ghost nucleus
19Ne). At variance with Fig. 1, the $y$-axis on the right-hand side now
includes all isotopes up to 55Fe. The latter can be intermediary steps to the
production of Li (respectively F). As a result, we see a lot of smaller
contributions associated to multi-step fragmentation, which include
fragmentation of 56Fe into Sc to Fe isotopes (respectively S to Fe isotopes).
The sparse matrix of the $f_{aHc}$ coefficients directly tells us what
reactions are involved to reach a given fraction of the total flux. We provide
in Appendix C similar figures for Be, B, N, Na, and Al.

We finally stress that for Li and F shown in Fig. 2, we limited ourselves to
the $f_{abc}$ coefficients leading to $90\%$ and $85\%$ of the total flux
respectively (in order to have legible and matching in size figures). For
instance, to build up $97\%$ of the total flux, we need to consider 767
reaction for Li, 823 for Be, 550 for B, 85 for C, 452 for N, 33 for O, 1030
for F, 417 for Ne, 840 for Na, 189 for Mg, 651 for Al, and 139 for Si. Hence,
even for mostly primary species, the number of reactions to consider can be
large; the number of reactions needed to make up different fractions of the
secondary or total flux are indicated in Tables 5 to 16. As a result, reaching
higher precision in the GCR flux modeling will always be at the price of
increasing efforts to measure a growing number of reactions at high precision.
We can also stress that fragmentation of Fe has a growing importance when
moving to heavier GCR elements.

###  IV.2 Desired reactions

We discussed in Section III.3 different assumptions made on the status of
current nuclear data uncertainties to propagate them to GCR fluxes
predictions. These assumptions are fully uncorrelated (‘uncorr.’), fully
correlated (‘corr’), or a mixture of both (‘mix’), and are expected to bracket
the reality of patchy data obtained over the years from very different
facilities and measuring techniques. If future nuclear data are taken
following the same patchiness, we can use the above assumptions to make rough
estimates on how new measurements will incrementally improve GCR modeling.

Figure 3: Improvement on the CR flux modeling precision when fragments of more
and more reactions ($x$-axis labels) are ideally measured (i.e. no
uncertainty). The first bin on the left-hand side, ‘Current’, gives the error
on the calculated elemental fluxes given the assumption made on the nuclear
cross-section uncertainties without new cross-section measurements. From left
to right, this error decreases when a growing number of reaction are measured.
We consider three different hypotheses for the propagation of cross-section
uncertainties (see Section III.3): correlated as given by Eq. (6),
uncorrelated given by Eq. (7), or a mixture of these two given by Eq. (8). The
shaded areas encompasses the more or less optimistic assumption that current
production cross-section uncertainties are in the $15-25\%$ range.

Following the approach used in Paper I, let us consider a scenario where we do
incremental nuclear measurements over time, measuring first the fragments of
the most highly ranked CR projectile (on H or He), then the second-highly
ranked, etc. In an idealized situation where these measurements deliver near
perfect nuclear data, Fig. 3 shows, in the above propagation uncertainties
scenarios, which reactions to measure and how many of them need to be measured
in order to get a better than $3\%$ uncertainty on the GCR flux prediction (on
par with the GCR data uncertainties). We see that for primary species (O and
Si) or mostly primary species (C and Mg), the current estimated $\sim 20\%$
uncertainty on production cross-sections already ensures that the GCR fluxes
of these elements already have a precision on par with the data. On the other
end, purely secondary species (Li, Be, B, and F), and to a lesser extent mixed
species (N, Ne, Na, and Al), need a significant number of reactions to be
better measured. This number strongly depends on the propagation error
scenario, moving from more up to 20 (‘corr’ scenario in blue) down to two
(‘uncorr’ scenario in orange). We believe that the most realistic scenario is
given by ‘mix’ (green), which typically requires five to ten reactions to be
measured.

Having a closer look at the reactions to measure, we recover the usual
suspects (that depends on the CR element considered), i.e. mostly abundant GCR
isotopes (12C, 16O, 24Mg, 30Si, 56Fe) interacting on H. But we also have
similar reactions on He that pop up for purely secondary GCR species.

###  IV.3 Desired number of events (beamtime)

In the context of a recent pilot run at NA61/SHINE [116, 117], and possible
future measurements at the SPS, it is interesting to consider the case in
which every desired cross-section is measured at once. By everything, we mean
that, thanks to this facility, many progenitors and fragments of interest for
CR physics can be measured in a ‘single’ run. In such a case, we do not have
to rely on approximations as above for error propagation, but we can directly
calculate the desired number of events necessary for each reaction to reach
the desired precision on GCR fluxes, as described in Section III.4.

Table 3: Table of $C_{ab}$ coefficients calculated from Eq. (13). Only
$C_{ab}>0.05$ are shown.

Li
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{16}\text{O}$+ | H | 0.841
${}^{12}\text{C}$+ | H | 0.684
${}^{16}\text{O}$+ | He | 0.176
${}^{56}\text{Fe}$+ | H | 0.165
${}^{28}\text{Si}$+ | H | 0.143
${}^{12}\text{C}$+ | He | 0.140
${}^{24}\text{Mg}$+ | H | 0.135
${}^{11}\text{B}$+ | H | 0.114
${}^{15}\text{N}$+ | H | 0.105
${}^{13}\text{C}$+ | H | 0.098
${}^{14}\text{N}$+ | H | 0.091
${}^{20}\text{Ne}$+ | H | 0.086
${}^{10}\text{B}$+ | H | 0.057
${}^{7}\text{Li}$+ | H | 0.057
${}^{56}\text{Fe}$+ | He | 0.053

Be
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{16}\text{O}$+ | H | 1.079
${}^{12}\text{C}$+ | H | 0.928
${}^{16}\text{O}$+ | He | 0.216
${}^{28}\text{Si}$+ | H | 0.200
${}^{24}\text{Mg}$+ | H | 0.200
${}^{56}\text{Fe}$+ | H | 0.188
${}^{12}\text{C}$+ | He | 0.181
${}^{11}\text{B}$+ | H | 0.147
${}^{14}\text{N}$+ | H | 0.120
${}^{20}\text{Ne}$+ | H | 0.117
${}^{15}\text{N}$+ | H | 0.115
${}^{13}\text{C}$+ | H | 0.092
${}^{10}\text{B}$+ | H | 0.078
${}^{56}\text{Fe}$+ | He | 0.058
${}^{22}\text{Ne}$+ | H | 0.054

B
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{12}\text{C}$+ | H | 0.802
${}^{16}\text{O}$+ | H | 0.690
${}^{12}\text{C}$+ | He | 0.147
${}^{16}\text{O}$+ | He | 0.130
${}^{11}\text{B}$+ | H | 0.103
${}^{28}\text{Si}$+ | H | 0.098
${}^{24}\text{Mg}$+ | H | 0.098
${}^{15}\text{N}$+ | H | 0.093
${}^{56}\text{Fe}$+ | H | 0.085
${}^{14}\text{N}$+ | H | 0.082
${}^{13}\text{C}$+ | H | 0.074
${}^{20}\text{Ne}$+ | H | 0.066

C
---
$(a$+ | $c)$ | ${\cal C}_{ab}$
${}^{16}\text{O}$+ | H | 1.054
${}^{16}\text{O}$+ | He | 0.185
${}^{24}\text{Mg}$+ | H | 0.125
${}^{15}\text{N}$+ | H | 0.116
${}^{28}\text{Si}$+ | H | 0.110
${}^{14}\text{N}$+ | H | 0.104
${}^{20}\text{Ne}$+ | H | 0.092
${}^{13}\text{C}$+ | H | 0.083
${}^{56}\text{Fe}$+ | H | 0.068

N
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{16}\text{O}$+ | H | 1.245
${}^{16}\text{O}$+ | He | 0.213
${}^{24}\text{Mg}$+ | H | 0.138
${}^{28}\text{Si}$+ | H | 0.125
${}^{20}\text{Ne}$+ | H | 0.113
${}^{15}\text{N}$+ | H | 0.087
${}^{56}\text{Fe}$+ | H | 0.065
${}^{22}\text{Ne}$+ | H | 0.054

O
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{24}\text{Mg}$+ | H | 0.499
${}^{28}\text{Si}$+ | H | 0.459
${}^{20}\text{Ne}$+ | H | 0.354
${}^{56}\text{Fe}$+ | H | 0.224
${}^{22}\text{Ne}$+ | H | 0.175
${}^{25}\text{Mg}$+ | H | 0.092
${}^{27}\text{Al}$+ | H | 0.091
${}^{23}\text{Na}$+ | H | 0.089
${}^{24}\text{Mg}$+ | He | 0.086
${}^{28}\text{Si}$+ | He | 0.083
${}^{19}\text{F}$+ | H | 0.079
${}^{26}\text{Mg}$+ | H | 0.067
${}^{21}\text{Ne}$+ | H | 0.066
${}^{18}\text{O}$+ | H | 0.062
${}^{20}\text{Ne}$+ | He | 0.060
${}^{56}\text{Fe}$+ | He | 0.057
${}^{32}\text{S}$+ | H | 0.056

F
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{20}\text{Ne}$+ | H | 0.827
${}^{24}\text{Mg}$+ | H | 0.703
${}^{28}\text{Si}$+ | H | 0.702
${}^{56}\text{Fe}$+ | H | 0.433
${}^{22}\text{Ne}$+ | H | 0.306
${}^{23}\text{Na}$+ | H | 0.139
${}^{20}\text{Ne}$+ | He | 0.138
${}^{21}\text{Ne}$+ | H | 0.137
${}^{25}\text{Mg}$+ | H | 0.137
${}^{27}\text{Al}$+ | H | 0.133
${}^{26}\text{Mg}$+ | H | 0.128
${}^{28}\text{Si}$+ | He | 0.127
${}^{24}\text{Mg}$+ | He | 0.122
${}^{56}\text{Fe}$+ | He | 0.112
${}^{32}\text{S}$+ | H | 0.079
${}^{22}\text{Ne}$+ | He | 0.051

Ne
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{24}\text{Mg}$+ | H | 0.662
${}^{28}\text{Si}$+ | H | 0.509
${}^{56}\text{Fe}$+ | H | 0.243
${}^{22}\text{Ne}$+ | H | 0.213
${}^{23}\text{Na}$+ | H | 0.119
${}^{25}\text{Mg}$+ | H | 0.112
${}^{24}\text{Mg}$+ | He | 0.108
${}^{27}\text{Al}$+ | H | 0.100
${}^{26}\text{Mg}$+ | H | 0.088
${}^{28}\text{Si}$+ | He | 0.085
${}^{32}\text{S}$+ | H | 0.064
${}^{56}\text{Fe}$+ | He | 0.058
${}^{21}\text{Ne}$+ | H | 0.056

Na
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{24}\text{Mg}$+ | H | 1.161
${}^{28}\text{Si}$+ | H | 0.709
${}^{56}\text{Fe}$+ | H | 0.320
${}^{24}\text{Mg}$+ | He | 0.185
${}^{25}\text{Mg}$+ | H | 0.172
${}^{27}\text{Al}$+ | H | 0.155
${}^{26}\text{Mg}$+ | H | 0.124
${}^{28}\text{Si}$+ | He | 0.116
${}^{32}\text{S}$+ | H | 0.091
${}^{56}\text{Fe}$+ | He | 0.074
${}^{29}\text{Si}$+ | H | 0.058

Mg
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{28}\text{Si}$+ | H | 0.977
${}^{56}\text{Fe}$+ | H | 0.484
${}^{27}\text{Al}$+ | H | 0.225
${}^{28}\text{Si}$+ | He | 0.156
${}^{26}\text{Mg}$+ | H | 0.147
${}^{25}\text{Mg}$+ | H | 0.134
${}^{32}\text{S}$+ | H | 0.121
${}^{56}\text{Fe}$+ | He | 0.107
${}^{29}\text{Si}$+ | H | 0.080
${}^{30}\text{Si}$+ | H | 0.054

Al
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{28}\text{Si}$+ | H | 1.441
${}^{56}\text{Fe}$+ | H | 0.426
${}^{28}\text{Si}$+ | He | 0.229
${}^{32}\text{S}$+ | H | 0.125
${}^{56}\text{Fe}$+ | He | 0.091
${}^{29}\text{Si}$+ | H | 0.091
${}^{27}\text{Al}$+ | H | 0.070
${}^{30}\text{Si}$+ | H | 0.052

Si
---
$(a$+ | $b)$ | ${\cal C}_{ab}$
${}^{56}\text{Fe}$+ | H | 1.112
${}^{32}\text{S}$+ | H | 0.336
${}^{56}\text{Fe}$+ | He | 0.223
${}^{30}\text{Si}$+ | H | 0.142
${}^{29}\text{Si}$+ | H | 0.139
${}^{54}\text{Fe}$+ | H | 0.112
${}^{31}\text{P}$+ | H | 0.085
${}^{34}\text{S}$+ | H | 0.069
${}^{33}\text{S}$+ | H | 0.064
${}^{40}\text{Ca}$+ | H | 0.063
${}^{38}\text{Ar}$+ | H | 0.056
${}^{42}\text{Ca}$+ | H | 0.054
${}^{32}\text{S}$+ | He | 0.053
${}^{52}\text{Cr}$+ | H | 0.051

The constants ${\cal C}_{ab}$ are listed in Table 3 to aid the optimization of
the beam requests for future measurements. These constants also allow to
estimate the uncertainty of a particular secondary flux due to the statistical
uncertainty of the cross section measurements in $a+b$ interactions to be
predicted for the given number of recorded interactions and they provide
guidelines on which combinations of projectile and target are the most
important ones to measure. For instance, it can be seen that the dominating
${\cal C}_{ab}$ coefficients for Li are ${\cal C}_{\rm Cp}^{\rm Li}$ and
${\cal C}_{\rm Op}^{\rm Li}$, but also Si and Fe interactions with protons are
important reactions. The constants derived here differ from the values quoted
in Paper I due to updated cross-section values, in particular for the
fragmentation of Fe (see Sect. II.2). Furthermore, the importance of a few
reactions was overestimated previously, due to an error in the decoding of the
data tables during calculation. In addition, Paper I did not include the
factor $f_{\rm sec}$ in Eq. (11) leading to a too large weight of reactions
contributing to the flux of nuclei that are mostly of primary origin. In
summary, the ${\cal C}_{ab}$ coefficients and related calculations listed here
supersede our previous results.

We aim at the desired model uncertainty to be smaller than the uncertainty of
the current and near future CR experiments. The AMS-02 experiment claims
$\sim$3% uncertainty for most of its data. Therefore, since the contribution
from cross-section uncertainties should be a sub-dominant of the overall
uncertainty, we investigate how keep this contribution at the $1\%$ level. If
in addition an experimental systematic uncertainty of typically $0.5\%$ can be
achieved (e.g. Abgrall  _et al._ 118), we arrive to the required statistical
accuracy of $\xi=\sqrt{0.01^{2}-0.005^{2}}=0.0087$ as in [119]. Adopting the
optimal power-law exponent $\beta=1$ derived in Sec. III.3 results in the
required number of interactions listed in Table 4. It is worthwhile noting
that a scaling with $\beta=0$, as investigated in Paper I, would require about
a factor of 2 more interactions to be recorded to obtain the same accuracy,
but it involves fewer interaction channels.

Table 4: Required number of interactions to be recorded per reaction, as calculated from Eq. (17) with $\beta=1$. The reactions are given in three groups of increasing projectile mass (up to O, Si, or Fe). The cumulative number of required interactions is quoted at the end of each group. reaction | $N_{\text{int}}$
---|---
${}^{16}\text{O}$+ | H | 60k
${}^{12}\text{C}$+ | H | 50k
${}^{16}\text{O}$+ | He | 20k
${}^{11}\text{B}$+ | H | 10k
${}^{15}\text{N}$+ | H | 10k
${}^{14}\text{N}$+ | H | 10k
${}^{12}\text{C}$+ | He | 10k
${}^{10}\text{B}$+ | H | 5k
${}^{13}\text{C}$+ | H | 5k
${}^{7}\text{Li}$+ | H | 5k
| $N(\leq\text{O})=1.9\times 10^{5}$
${}^{28}\text{Si}$+ | H | 50k
${}^{24}\text{Mg}$+ | H | 50k
${}^{20}\text{Ne}$+ | H | 50k
${}^{22}\text{Ne}$+ | H | 20k
${}^{28}\text{Si}$+ | He | 10k
${}^{27}\text{Al}$+ | H | 10k
${}^{26}\text{Mg}$+ | H | 10k
${}^{24}\text{Mg}$+ | He | 10k
${}^{23}\text{Na}$+ | H | 10k
${}^{25}\text{Mg}$+ | H | 10k
${}^{21}\text{Ne}$+ | H | 10k
${}^{20}\text{Ne}$+ | He | 10k
${}^{32}\text{S}$+ | H | 5k
${}^{29}\text{Si}$+ | H | 5k
${}^{22}\text{Ne}$+ | He | 5k
| $N(\leq\text{Si})=3.8\times 10^{5}$
${}^{56}\text{Fe}$+ | H | 30k
${}^{56}\text{Fe}$+ | He | 10k
| $N(\leq\text{Fe})=4.2\times 10^{5}$

##  V Forecasts from improved cross-sections

Here we investigate different scenarios for new nuclear cross-section
measurements and the impact of these measurements on key questions addressed
by the CR community, namely: the derivation of CR transport parameters and
their impact on indirect DM searches. To assess the improvement brought by new
nuclear cross-section measurements, we generate mock cross-section models
which, within a given propagation scenario, are used to fit the transport
parameters on the secondary to primary ratios. We start by explaining the
framework used before coming to the results.

###  V.1 Methodology

####  V.1.1 Set-up of the mock cross-sections

We define several scenarios from which we draw a thousand of mock cross-
section models.

##### Current uncertainty (Scenario I):

In this scenario, for each cross-section of our benchmark cross-section model
(see Section II.2), we apply a default energy-independent random Gaussian bias
with variance equals to 20% of the cross section value. Moreover, we apply
another random Gaussian bias, chosen to be identical for reactions involving
the same progenitor, with a variance again of 20% the cross section value.
This way, we try to encompass experimental biases present in cross-section
measurements, and so in cross-section models.

##### Updated cross sections up to O/Si/Fe (Scenario II):

By default, the mock cross-section models of this kind are generated in the
same way as in Scenario I. However, they do include improvements of some of
the cross-section uncertainties, according to our proposition (Table 4). For
each reaction ($a+b\to c$) in this list, we estimate the Poisson parameter
$\mu_{abc}=N_{a+b}\times\sigma_{a+b\to c}/\sigma_{a+b}$, with $N_{a+b}$ being
the number of $a+b$ interactions proposed in Table 4. Here $\sigma_{a+b\to c}$
is the production cross-section of $c$, and $\sigma_{a+b}$ is the $a+b$ total
inelastic cross-section. Hence $\mu_{abc}$ corresponds to the number of events
including the multiplicity, hence the following updates are given for a
detector able to measure all fragments of a given fragmentation event. When
the relative Poissonian error $1/\sqrt{\mu_{abc}}$ is smaller than 20% for a
given reaction, we draw a random value $Y$ from a Poissonian distribution of
parameter $\mu_{abc}$, and apply the rescaling factor $Y/\mu_{abc}$ to the
corresponding cross section. By drawing a thousand of mock cross-section
models, we mimic the improvement of the statistical uncertainty achieved with
new fragmentation data. We consider three sub-scenarios, taking O, Si, and Fe
consequently as the heaviest progenitors up to which our new experiment was
able to take fragmentation data.

####  V.1.2 Fit to secondary/primary ratios

For each scenario we fit alternatively the secondary/primary ratios: Li/C,
Be/C, B/C and F/Si, to compare improvements achieved for each observable. For
the transport model, we use the minimalist, purely diffusive, SLIM propagation
benchmark presented above in the Section II.1. For each mock cross-section
model (a thousand for each scenario) we fit only the four parameters
($\delta_{\rm l},R_{\rm l},\delta,D_{0}$) which impact the low and
intermediate rigidity dependence of the diffusion coefficient.

The results of the corresponding fits are shown as contour plots in Fig. 4
with solid colored lines defining the $1\sigma$ contour in the
$(\delta,D_{0})$ relative error plane. This way we propagate the uncertainty
of the cross section onto the propagation parameters values. To gauge if the
improvements of the scenarios described in Section V.1.1 are satisfactory, we
have to compare the irreducible/intrinsic error on the propagation parameters
coming from the uncertainty of the secondary/primary data. We do so by fitting
a thousand mock AMS-02 datasets generated from a given known model (i.e. with
fixed $\delta_{\rm l},R_{\rm l},\delta,D_{0}$) from which each data point is
scattered according to a Gaussian law, whose variance is
$\sigma=\sqrt{\sigma_{stat}^{2}+\sigma_{\rm syst}^{2}}$, with $\sigma_{stat}$
and $\sigma_{\rm syst}$ being the statistical and systematic uncertainties
provided by the AMS-02 collaboration. Note that the fragmentation cross
sections are fixed to their benchmark value (see Section II.2). The results of
these fits are shown in Fig. 4 with a solid black line defining the $1\sigma$
contour in the $(\delta,D_{0})$ relative error plane. The central point of
that region is highlighted in red.

###  V.2 Improvements on propagation parameters

Figure 4: Forecast of transport parameters determination from new cross-
section measurement campaigns. Each figure shows $1\sigma$ contours in the
($D_{0}$, $\delta$) relative error plane in different scenarios. The left
panel shows the estimated current uncertainty (solid red line) and three cases
were a subsets of cross sections have been updated according to our
proposition Table 4, increasing the mass of the heavier progenitor from O to
Fe. Finally, for comparison, we show the irreducible/intrinsic data
uncertainty (solid black line). The right panel is a zoom of the left one, and
compare subcases where we would not measure the fragmentations of Table 4 on
Helium target. More details on how these bounds were computed can be found in
the text.

Several important remarks can be drawn from the left panel of Fig. 4: (i) For
each secondary/primary ratio considered, the current uncertainty on
$(D_{0},\delta)$ (solid red line) is much larger than the data would allow if
the uncertainties on the fragmentation cross sections (solid black line) were
vanishing. Note the difference in correlation in the ($D_{0}$,$\delta$) plane,
where the relative range of $\delta$ is much smaller than that of $D_{0}$.
This implies that consequent improvements in the accuracy of the cross
sections would result in tighter relative constrains for $\delta$ than for
$D_{0}$. On the other hand, such measurements will bring down the uncertainty
in $D_{0}$ four-fold from 40% to better than 10%. (ii) Our proposition (Table
4) leads to significant improvements in the accuracy of ($D_{0}$,$\delta$).
The three sub-scenarios, taking different heaviest progenitors for the new
fragmentation data (O, Si, or Fe), give similar improvements derived from the
Li/C, Be/C, and B/C ratios, with the slightly reduced uncertainties as we move
from O to Si and to Fe. (iii) For the F/Si ratio, significant improvements in
the accuracy of ($D_{0}$,$\delta$) are observed when the new fragmentation
data involve progenitors heavier than F, as anticipated. We anticipate that
this will also be the case for all heavier ratios, hence the importance to get
new cross-section dataset including heavy nuclei fragmentation.

While the left panel of Fig. 4 includes the fragmentations on H and He targets
proposed in Table 4, the right panel shows a comparison when fragmentation on
He target is not measured. We remark that including the He fragmentation
always lead to reduced uncertainties for ($D_{0}$,$\delta$), however the
improvement is mild for the heavier nuclei ratios (B/C and F/Si) while it is
significant for the lightest ones (Li/C and Be/C), decreasing the uncertainty
in $D_{0}$ by a few percent.

###  V.3 Reduced uncertainty for secondary $\bar{p}$

GCR antiprotons is one of the most powerful probe for the indirect detection
of dark matter [120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131,
44, 132, 133]. However, as highlighted in many studies [120, 122, 124, 129,
130, 132] and in particular in [133], the modeling uncertainties of the
$\bar{p}$ secondary astrophysical background are significantly larger than
AMS-02 data uncertainties [46]. The two most important modeling uncertainties
are (i) the nuclear production cross sections of $\bar{p}$ and $\bar{n}$ from
GCRs and (ii) the transport parameter uncertainties. As underlined just above,
the latter are related to the nuclear production uncertainties.

Figure 5: Reduction of the uncertainties on secondary $\bar{p}$ flux brought
by new nuclear data. The red envelope shows the current ‘transport’
uncertainty, i.e. uncertainty related to the transport parameter uncertainties
(linked to the nuclear cross-section uncertainties). The magenta envelopes
show the envelopes with new cross section measurements (including reactions up
to Fe), for the case of new measurements both on H and He as targets (thick
lines) or H target only (thin lines). With this improvement, the uncertainty
is smaller than the current AMS-02 data uncertainties [46]; the symbols show
the residuals of the data with regards to the best-fit $\bar{p}$ prediction of
[133], [Bo20] in the legend. See text for discussion.

Considering the same scenarios as above, we calculate from the best-fit
coefficients the associated $\bar{p}$ flux (for each mock cross-section
model). Doing so, we can calculate at each energy the distribution of
$\bar{p}$ values and its quantiles, forming an envelope of uncertainties over
all energies, as shown in Fig. 5. In this plot, the red envelope shows the
current ‘transport’ uncertainties777This band does not exactly match the one
shown in [133] because we do not use exactly the same approach. In particular,
here we fix the solar modulation instead of propagating its uncertainty as in
[133], where we were also making use of the BIG benchmark scenario instead of
SLIM here. Both these effects enlarge the uncertainty band at low rigidities.,
which is much larger than the $\bar{p}$ data uncertainties. The black symbols
show the residuals between the AMS-02 data and the best-fit model of [133],
featuring a ‘bump’ seen at $\sim 10$ GV (also from other studies) which has
triggered a lot of excitement in the community, possibly hinting at a new
physics signal. This feature can actually be related to some correlations in
the data, as advocated in [133]. In any case, the current precision on the
transport parameters, related to our current knowledge of nuclear cross
sections, is not on par with the AMS-02 data precision.

The magenta envelopes show the configurations where we fit the transport
parameters using the B/C ratio only (see Section V.2), in a scenario where all
nuclear cross sections have been improved for reactions both on H and He
targets (thick lines) or only on H targets (thin lines). We do not consider
transport parameters extracted from the fits of Li/C, B/C or F/Si, because
they only need to be calibrated with the best-measured ratio: B/C, or ideally,
a combined fit of these ratios. Actually, using Li/C, Be/C or B/C only leads
to minor differences as these ratios all lead to similar transport parameter
uncertainties. Using transport parameter uncertainties from F/Si would lead to
minor or similar improvements depending on the new nuclear data taken, as
discussed above. The improved nuclear data, and so the improved transport
parameter uncertainties, would be the game changers for indirect dark matter
searches, as AMS-02 data could be fully exploited then. Actually, as advocated
in Fig. 2 of [133], both the nuclear data (highlighted in this paper) and
$\bar{p}$ and $\bar{n}$ production (see also [50]) should be improved to
overcome this challenge.

###  V.4 Reduced uncertainty for $L$ and primary $\bar{p}$

Primary dark matter signals also suffer from transport parameter
uncertainties. However, as highlighted in several studies [134, 135], these
uncertainties are dominated by the uncertainty on the halo size of the Galaxy,
$L$ (the primary flux is actually roughly proportional to $L$).

Figure 6: Forecast for the relative precision of the halo size $L$ of the
Galaxy. The red envelope shows the current uncertainty on $L$, and the magenta
envelopes shows the improvement achieved with by new cross section
measurements (including reactions up to Fe), for the case of measurements both
on H and He as targets (thick line) or H target alone (thin line). With this
improvement, the uncertainty becomes smaller than the uncertainties from
current data (as estimated in [136]). See text for discussion.

This halo size cannot be directly extracted from the above secondary-to-
primary nuclei ratios because it largely constraints the $L/D_{0}$ ratio, but
this degeneracy can be broken using the radioactive clocks measurements, such
as 10Be (e.g., [137, 138, 139, 140, 141]). Recent studies emphasized the role
of the accuracy of the nuclear cross-sections in the determination of $L$ via
the 10Be/9Be ratio [87, 142, 143, 136, 144]. We take advantage of a simplified
calculation of this ratio proposed in [136] to fit the halo size $L$ from
10Be/9Be ratio for each mock cross-section scenarios discussed above (using
the best-fit transport parameters derived for each of these mocks). This
allows us to estimate the uncertainties on the determination of $L$, as shown
in Fig. 6. The red histogram shows the probability distribution function of
$L$ for current nuclear data. The standard deviation of this distribution
($\approx$30%) is much larger than that of the black histogram $\approx$12%,
estimated from the data uncertainties only assuming perfectly known nuclear
data [136]. New nuclear data (magenta histogram) allow to decrease this number
down to $\approx$5%, with slightly better precision if these nuclear data are
taken on both H and He targets (thick line) compared to the case of H target
alone (thin line).

The black histogram is actually estimated from pre-AMS-02 data, with 10Be/9Be
data precision at the level of ten to twenty percent. Preliminary data from
the AMS-02 and the HELIX project [145] are expected to reach similar precision
$\sim$15%), but to cover a much larger energy range. The black line estimate
is thus expected to improve (see e.g. [146]), although these challenging
forthcoming measurements may still not go as far to high energies as desired
to provide significant improvement [136]. In any case, the message is that the
new cross section data will allow the uncertainty in the halo size $L$ to be
reduced by more than a factor of two. This is already a huge step forwards as
it means that, for instance, dark matter exclusion plots based on $\bar{p}$
studies [147] will benefit from a similar gain. Additionally, such an
improvement will help constraining the underlying physics at the origin of the
diffusive halo.

##  VI Conclusions

This paper follows up on our previous work (Paper I) to prioritize the list of
cross-sections of interest for Galactic CR studies that have to be measured
with a higher precision. The current generation CR experiments (AMS-02, CALET,
DAMPE, Fermi-LAT) has been bringing a revolution in astrophysics of CRs, and
to fully exploit these data, we need a combined effort of the astrophysics,
nuclear, and particle physics communities and their facilities to meet the
demand for high precision nuclear fragmentation data.

In this paper, we have reviewed the production of F to Si and also updated our
previous predictions for Li to O GCR elements. We have provided several plots
to highlight the importance of direct production of Li to Si elements, but
also to highlight the respective importance of the primary progenitors. The
raking of these quantities is mostly of interest for the CR community.

As in Paper I, the main result of our paper are the so-called $f_{abc}$
coefficients, which directly rank the most important reactions involved in the
production of GCRs. This allows us to select the most critical reactions for
production of the so-called secondary Li, Be, B, Fe, and Na elements, used to
calibrate the interstellar transport of GCRs. Their accurate prediction
significantly impacts our ability to make progress and fully benefit from
recent GCR data on a variety of important questions, such as understanding the
origin of CRs, properties of the interstellar matter, our local galactic
environment, indirect dark matter searches, and many others.

From these $f_{abc}$ coefficients, we have highlighted the way towards
actionable measurements, by providing the exact reactions to measure and the
desired number of reactions necessary to render GCR model accuracy on par with
existing GCR data precision. From our numbers, the beam time can be easily
estimated thus facilitating the experiment planning. We have shown that new
nuclear data would dramatically improve the precision on the derived transport
parameters. They would (i) reduce the uncertainties of some transport
parameters by a factor of ten, allowing many disputed interpretations of the
observed features in spectra of CR species to be addressed; (ii) reduce the
uncertainties in production of secondary antiprotons by a factor of five,
allowing the dark matter searches and exploration of discrepancies between the
data and the models with exquisite precision; (iii) reduce the uncertainty on
the halo size of the Galaxy by a factor of a few, allowing the constraints on
dark matter candidates to be automatically improved by the same factor. While
we focused in this paper on these three examples, the benefit of these new
nuclear data would not end here, as multi-messengers studies of the Galaxy
(radio- to $\gamma$-rays, neutrinos, CRs) are to some level all calibrated on
the transport coefficients that can only be improved with these new nuclear
data.

In this second paper of the series, we have covered GCR elements from Li to
Si, and similar studies for heavier species are highly desirable. AMS-02 has
already provided measurements of the Fe flux [61] and will soon fill the gap
of the elements between Si and Fe. Of particular importance are the so-called
sub-Fe elements: Sc, Ti, and V. Because of their mostly secondary origin, they
can be used to probe GCR transport of heavy species, and comparing with light
secondary nuclei results, will provide new insights on GCR transport. It is
already known that fragmentation of Fe is a key ingredient for this modeling.
Our follow-up effort should quantify how much so.

###### Acknowledgements.

This work has been supported by the “Investissements d’avenir, Labex ENIGMASS”
and by the Programme National des Hautes Energies of CNRS/INSU with INP and
IN2P3, co-funded by CEA and CNES. IVM acknowledges partial support through
NASA grants Nos. 80NSSC23K0169, 80NSSC22K0718. We thank the NA61/SHINE
collaboration for their very useful feedback during the NA61++/SHINE:
NA61++/SHINE: Physics opportunities from ions to pions meeting, which
motivated the calculations made for Section V.

##  Appendix A Energy-dependent view of the fragmentation contributions,
channels, and progenitors

In this Appendix, we present an energy-dependent view of the various
fragmentation contributions (shown at 10 GeV/n only in Table 1), alongside an
energy-dependent view of the rankings of the progenitors and channels.

Figure 7:  Contributions and rankings for Li (left) and Be (right) fluxes at
$\phi_{\rm FF}=700$ MV. Top panel: primary, radioactive, and fragmentation
fractions also broken down into the sum of all 1-step, 2-step, or $>2$-step
contributions, see Eqs. (24-25); gray dotted lines highlight the importance of
sub-percent 1-step and 2-step contributions, when grouped according to their
value at 10 GeV/n (in $\sum_{N}$, $N$ is their number). Middle panel: relative
to $f_{\rm sec}$, 1-step and 2-step channels (see Eq. 24) shown individually
when larger than $1\%$ at 10 GeV/n (right legend), or grouped by contributing
fractions (gray-dotted lines in bottom legend). The sum of all these curves,
plus all $>\\!2$-step channels, $f^{>\\!2\rm-step}$ (blue-dashed line in
bottom legend), makes up $100\%$ of $f_{\rm sec}$. The black solid line (with
vertical arrows) separates contributions larger or smaller than $1\%$ of the
total flux (non-trivial shape if $f_{\rm sec}<100\%$). Bottom panel: relative
to $f_{\rm sec}$, CR progenitors whose contribution $f_{Z\to\rm Li}$ (see Eq.
26) is larger than $0.1\%$ at 10 GeV/n. Their sum, plus all smaller
contributions (thick gray solid line) makes up $100\%$ of $f_{\rm sec}$. As in
the middle panel, the black solid line separates contributions above or below
$1\%$ of the total flux.

Figure 8: Same as in Fig. 7, but for B (left) and C (right).

Figure 9: Same as in Fig. 7, but for N (left) and O (right).

Figure 10: Same as in Fig. 7, but for F (left) and Ne (right).

Figure 11: Same as in Fig. 7, but for Na (left) and Mg (right).

Figure 12: Same as in Fig. 7, but for Al (left) and Si (right).

Figures 7 to 12 provide, as a function of $E_{k/n}$, a graphical view of the
CR origin, and the ranking of channels and progenitors (in three different
panels, from top to bottom respectively). We choose to show a TOA energy
domain ranging from hundreds of MeV/n to hundreds of TeV/n, which corresponds
to the range covered by recent and current CR experiments, Voyager 1, ACE-
CRIS, AMS-02, and TRACER, ATIC, CREAM, NUCLEON, CALET, DAMPE. We consider here
all CR elements from Li to Si, updating the results shown in Paper I for Li to
N.

###  A.1 Origin of contributions

The top panels of Figs. 7-12 extend the conclusions already drawn (at one
energy) from Table 1, relative to the primary ($f_{\rm prim}$), fragmentation
($f_{\rm sec}$), and radioactive ($f_{\rm rad}$) nature of the various CR
elements. Concerning $f_{\rm rad}$, only B (top-left panel of Fig. 8) and Mg
(top-right panel of Fig. 11) have non-zero contributions (orange dotted
lines), which quickly drop to zero as energy increases owing to a boosted
lifetime of the associated $\beta$-unstable parent CR. Concerning the energy
dependence of $f_{\rm sec}$ with respect to $f_{\rm prim}$ (for mixed
species), it peaks at a few GeV/n, as already discussed in Section II.3: the
interaction times is constant with energy (cross-sections are assumed to be
constant above a few GeV/n) while the escape time (from the Galaxy) decreases.
As a result, as illustrated for instance in the N plot in the top-left panel
of Fig. 9, the fragmentation contribution (blue solid line) decreases with
energy while the primary component grows (green dash-dotted line). A similar
energy dependence is observed when moving from 1-step to 2-step reactions, and
then to $>$2-step contributions with respect to the total flux (thin to
thicker blue dashed lines) reflecting the decrease in the probability of
multiple interactions. For pure secondary species like F (top left panel of
Fig. 10), the steepening of the energy dependence of such a contribution with
the increase of the number of steps involved in its production is seen even
more clearly.

In addition, these plots show how a larger number of individually negligible
contributions adds up to a significant amount. To do so, we focus on
individual 1-step and 2-step contributions whose $P^{\rm 1-step}_{ij}$ and
$P^{\rm 2-step}_{ikj}$ coefficients (see Eq. 23) fall in the range $[x\%,y\%]$
with $(x,y)<0.1$: there are $N$ such coefficients, and we sum over all of them
weighted by the secondary fraction $f^{\rm frac}$ to get the desired result.
The thin to thick gray dotted lines and their caption show, for instance for
the pure secondary F (top-left panel of Fig. 10), that moving from
$[0.1\%,1\%]$ to $[0.01\%,0.1\%]$, and then to $[0.001\%,0.01\%]$, a growing
number of channels is involved (respectively $N=20$, 75, and 119), adding up
to a sizable amount of the total flux (respectively $\sim$10%, 3%, 0.4%);
because these contributions combine 1- and 2-step channels, the different
grouping intervals share the same energy dependence. Roughly, to get a $3\%$
percent precision on the modeled fluxes, all reactions making up at least
$97\%$ of the total flux must be accounted for, which can implicate up to
hundreds of sub-percent reactions.

###  A.2 Ranking of channels

The middle panels of Figs. 7-12 are very busy and difficult to read at a
single glance. Nevertheless, we decided to keep them this way, because they
give a comprehensive view of all channels (see definitions in Section
III.5.2), in the sense that the sum of all the curves shown corresponds to
$100\%$ of the fragmentation flux. They are separated into 1-step or 2-step
channels whose individual contributions are larger than $1\%$ at 10 GeV/n
(color-coded legend), and those below this level which are further grouped by
range (gray dotted lines). We also group together all $>\\!2$-step
contributions (dashed-blue curve). These groups were already shown in the top
panels with respect to the total flux, while they are shown here with respect
to the secondary component888For pure secondary species for which $f_{\rm
sec}=100\%$ (Li, Be, B, and F), the curves are by construction the same.
$f_{\rm sec}$. We do not repeat the discussion of the energy dependence of
these contributions, as they were amply commented in the previous paragraph.

As expected and as already stressed in Paper I, most of the LiBeB (Figs. 7 and
8) comes from the direct and 2-step fragmentation of 12C and 16O. However, the
ranking of the reactions at 10 GeV/n is slightly different from those shown in
Paper I (as gathered in Tables V, VI, and VII there): reactions with 56Fe
progenitor are ranked significantly higher for Li and Be here. As another
illustration of the impact of some cross-section updates, the 16O$\to^{6}$Li
reaction is now contributing about twice the 12C$\to^{6}$Li reaction (left
panel of Fig. 7), whereas it was reported as only a few percent larger in
Paper I (see Table V there): this is related to the renormalization of the of
16O+p$\to^{6}$Li cross-section to a higher value as measured in [148].

Looking globally at the dominant channels for the various elements from F up
to Si, we see that: F production is dominated by fragmentation of 20,22Ne,
24Mg, 28Si, and 56Fe; Ne and Na by 24Mg and 28Si; Mg and Al by 28Si; Si by
56Fe and 32S. However the detail of the number of contributions larger than
$1\%$ at 10 GeV/n, or whether one single channel dominates vs several ones
contribute equally, is strongly dependent on the CR species under scrutiny.
Interestingly, in almost all elements shown, the sum of all $>$2-step
contributions (blue-dashed line) ranks second at 10 GeV/n, though it rapidly
decreases below and above this energy; we will come back to this in the next
paragraph. For the primary species (O, Ne, and Si), the dominant contribution
appears to be the sum of $\sim$50 1-step and 2-step channels in the
$[0.1\%-1\%]$ range (thinnest dotted line), rather than a certain reaction.

It is important to question the relevance of these various channels (or groups
of channels) with respect to the total flux, especially for fluxes with a
significant primary fraction. To do so, the solid black lines (with upward
arrows) separate the region above which a channel (or group of channels)
accounts for more than $1\%$ of the total flux; this black line is a straight
line at $1\%$ for secondary species (Li, Be, B, and F), but it has a more
complicated shape for mixed (combining primary and secondary components)
species. For instance, for Mg (right-middle panel of Fig. 11), only the direct
production from 28Si is above this line for the individual channels. The
grouped 1-step and 2-step channels in the $[0.1\%-1\%]$ range (thinnest dotted
line) and the sum of all $>\\!2$-step contributions (blue-dashed line) also
are. However, as the energy increases and the element becomes predominantly
primary, the number of relevant channels (and groups of channels) further
decreases.

###  A.3 Ranking of progenitors

The bottom panels of Figs. 7-12 show yet another complementary perspective on
the reactions to consider. Compared to the channel ranking above, the
progenitor ranking is based on the sum of the direct and all multi-step
fragmentation channels starting from this progenitor (see Section III.5.3).
The colored curves show CR progenitors whose contribution is larger than
$0.1\%$ at 10 GeV/n, and the thick gray line–the sum of all remaining
progenitors contributions. As discussed in [12], the different energy
dependences between the contributions of various elements reflects the
different energy dependences of the elemental fluxes themselves. Indeed, the
ratios of heavier-to-lighter primary CR fluxes decrease at low energy [149,
61] owing to a larger fragmentation cross-section of heavier species and their
faster ionization energy losses. For instance, for the production of secondary
Si (right panel of Fig. 3), Fe is the dominant progenitor (pink dotted line).
Similarly, elements with similar atomic numbers share the same energy
dependence, i.e. contribution of heavier elements, e.g., Mn and Cr, decreases
while contribution of lighter elements, e.g., S, Si, Ca, etc., increases at
low energy.

One can also compare contributions of various progenitors to the black solid
line (with upward arrows) corresponding to 1% of the total flux. We account
for small individual contributions of different channels tagging them to the
original element they came from. Interestingly, in the case of Mg (right panel
of Fig. 11), Fe progenitor appears above the 1% line in the bottom panel,
whereas none of the individual channels originating from 56Fe is contributing
above the black solid line in the middle panel.

##  Appendix B Figures for $D_{i\to j}$ and $P_{i\to j}$ at 10 GeV/n

In this Appendix, we show the $D_{i\to j}$ and $P_{i\to j}$ coefficients at 10
GeV/n (Fig. 13). These coefficients are introduced and defined in Section
III.5.1 and Eq. (20), and Section III.5.3 and Eq. (26) respectively. They
describe the contributing fraction of a progenitor $i$ into the isotope $j$
for elements Li through Si studied in this paper.

Similarly to the discussion of Section IV.1 for the $D$ and $P$ coefficients
for elements, the $D_{i\to j}$ coefficients in the left panel of Fig. 13 show
the direct production of the various CR isotopes of Li to Si from the GCR
isotopic fluxes as measured. The $P_{i\to j}$ coefficients on the right panel
correspond to the cumulated production from GCR primary fluxes only. In that
respect, the values for species of pure secondary origin (LiBeB isotopes),
mostly secondary origin (Sc, Ti, V isotopes), and subdominant isotopes in
primary species (for instance 41-44Ca) are null or below $0.1\%$.

##  Appendix C Tables of ranked $f_{abc}$ and plots at 10 GeV/n

In this Appendix, we provide the ranking of the most impacting production
cross sections based on the ranking of the $f_{abc}$ coefficients introduced
in Section III.2. In order to mitigate possible cross-section uncertainties
for those channels where no data is available, our calculations are based on
the average $f_{abc}$ coefficients over the three cross-section models
discussed in Section II.2. The full list of $f_{abc}$ coefficients can be
obtained from the original ASCII files, available to download from ZENODO
[150].

In the first column, Tables 6 to 16 list the reactions ranked at 10 GeV/n for
the production of secondary Li through Si species in CRs. The next 3-column
block shows the flux impact (minimum, mean, and maximum value calculated from
our three cross-section parameterizations), i.e. how much this reaction
contributes, in fraction, to the calculated GCR fluxes. As discussed in
Section III.2 and contrarily to the $D$ and $P$ coefficients, we recall that
the sum of the $f_{abc}$ coefficients (for a given $c$) is larger than one.
The fifth column presents the minimal and maximal cross-section values for
each reaction. The next-to-last column (check symbol) indicates when at least
one data point exists for the listed cross section. While there can be several
data points available for some reactions, for some others only 1–2 points may
exist, and yet in a limited energy range. This is for instance the case for
the reactions involved in the production of F (see Table 11): at the first
glance, the cross sections are well-constrained as the most important
reactions have at least one data point, but as discussed in [12], it appears
that all these data points are located in the rising portion of the cross
sections, leaving the exact asymptotic values at high energy quite uncertain.

The last column shows the ratio of the reaction cross-section to the
cumulative one, $\sigma/\sigma^{\rm cumul}$ (see Section III.1); in Paper I,
we showed instead $\sigma^{\rm cumul}/\sigma$, but we feel the inverse of this
ratio is a natural choice. Reactions with $\sigma/\sigma^{\rm cumul}<1$
indicate that the isotope produced comes from its direct production and the
production of ghosts (highlighted in boldface); see also Section III.1 and Eq.
2). For instance, for F (see Table 11), the contribution from the short-lived
nucleus 19Ne (see Table 2) can reach half the cumulative production at most.
We stress that these coefficients and tables are relative to the secondary
production. As shown in Table 1, some of these species are mostly primary
(e.g., O and Si). Hence, the relative importance of the reactions for the
total flux is reduced. To illustrate this, we intercalate lines (in the
tables) highlighting that coefficients above these lines make up some given
fractions ($25\%$, $50\%$, $75\%$; etc.) of the secondary or total fluxes. We
also provide graphical views in Figs. 14 to 17 of the $f_{aHc}$ coefficients
not shown in Section IV; we refer the reader to this section for detailed
information on how to read and interpret these plots.

To conclude this Appendix, we underline that tables for Li to N supersede
those presented in Paper I, mostly differing from the ranking position of
reactions involving 56Fe (as already discussed in n Section III.2).

Figure 13: Same as Fig. 1 for for isotopes instead of elements. The top panel corresponds to the $D_{i\to j}$ coefficients (III.5.1) and the bottom panel to the $P_{i\to j}$ ones (III.5.3). For both cases, the sum of coefficients on one row is 100%, i.e. all the secondary production. We nevertheless recall that the latter ($f_{\rm sec}$) is sometimes only a small fraction of the total flux (see Table 1). See text and Section IV.1 for discussion.  Table 5: Reactions sorted according to their flux impact $f_{abc}$ (with $\sum f_{abc}=1.39$) on Li at 10 GeV/n ($f_{\rm sec}=100\%$, see Table 1). The first 50 reactions shown provide 76.4% of the total flux (767 reactions to reach 97%); reactions in bold highlight short-lived fragments (see Sect. III.1 and Table 2). Reaction | Flux impact $f_{abc}$ |  $\sigma$ [mb] | Data | $\displaystyle\frac{\sigma}{\sigma^{\rm cumul}}$
---|---|---|---|---
| Min | Mean | Max | [Min,Max] |  |
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 11.4 | 15.2 | 17.1 | [13.0, 18.4] | ✓ | 1.0
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 11.3 | 12.5 | 13.1 | [14.0, 15.4] | ✓ | 0.9
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 9.91 | 9.93 | 9.97 | [11.9, 12.6] | ✓ | 1.0
($>$25% of $f_{\rm sec,tot}$)
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 9.67 | 9.74 | 9.77 | [10.7, 11.2] | ✓ | 1.0
$\rm{}^{11}B\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 2.89 | 2.92 | 2.99 | [21.5, 21.5] | ✓ | 1.0
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{6}$$\rm Li$ | 2.01 | 2.86 | 3.28 | [20.6, 31.8] |  | 1.0
$\rm{}^{12}C\\!+\\!He\\!\\!\to^{6}$$\rm Li$ | 1.94 | 2.14 | 2.25 | [21.6, 23.7] |  | 0.9
$\rm{}^{7}Li\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 2.03 | 2.11 | 2.27 | [31.5, 31.5] | ✓ | 0.7
$\rm{}^{13}C\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 1.98 | 2.05 | 2.09 | 22.1 |  | 1.0
$\rm{}^{56}Fe\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 1.97 | 2.03 | 2.06 | [23.0, 23.0] | ✓ | 1.0
$\rm{}^{15}N\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 1.92 | 1.95 | 1.97 | 18.6 | ✓ | 1.0
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{15}$$\rm N$ | 1.77 | 1.88 | 1.94 | 34.3 | ✓ | 0.5
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{7}$$\rm Li$ | 1.70 | 1.82 | 1.88 | [17.8, 18.6] |  | 1.0
$\rm{}^{56}Fe\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 1.63 | 1.74 | 1.96 | [17.8, 22.5] | ✓ | 0.8
$\rm{}^{12}C\\!+\\!He\\!\\!\to^{7}$$\rm Li$ | 1.70 | 1.71 | 1.71 | [18.4, 19.4] |  | 1.0
($>$50% of $f_{\rm sec,tot}$)
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{13}}$${\rm\mathbf{O}}$ | 1.67 | 1.70 | 1.72 | 30.5 | ✓ | 0.5
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 1.64 | 1.69 | 1.72 | 12.6 |  | 0.9
$\rm{}^{28}Si\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 1.63 | 1.69 | 1.72 | 12.6 |  | 0.9
$\rm{}^{13}C\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 1.62 | 1.68 | 1.71 | 17.8 |  | 1.0
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{15}}$${\rm\mathbf{O}}$ | 1.58 | 1.68 | 1.73 | 30.5 | ✓ | 0.5
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 1.41 | 1.51 | 1.56 | 32.3 | ✓ | 0.9
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 1.45 | 1.50 | 1.52 | 11.4 |  | 1.0
$\rm{}^{28}Si\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 1.45 | 1.50 | 1.52 | 11.4 |  | 1.0
$\rm{}^{10}B\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 1.37 | 1.41 | 1.43 | 20.0 |  | 1.0
$\rm{}^{14}N\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 1.32 | 1.39 | 1.43 | [13.0, 13.0] | ✓ | 0.7
$\rm{}^{15}N\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 1.34 | 1.37 | 1.38 | 12.8 | ✓ | 0.9
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{14}$$\rm N$ | 0.86 | 1.27 | 1.47 | [22.9, 29.2] | ✓ | 1.0
$\rm{}^{20}Ne\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 1.02 | 1.16 | 1.22 | 12.6 |  | 0.9
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 1.13 | 1.15 | 1.18 | [27.7, 30.0] | ✓ | 0.5
$\rm{}^{7}Be\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 1.13 | 1.14 | 1.16 | 21.0 |  | 1.0
$\rm\mathbf{{}^{12}C\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 1.06 | 1.11 | 1.14 | [26.9, 27.9] | ✓ | 0.5
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 1.07 | 1.10 | 1.11 | 17.5 | ✓ | 0.4
$\rm{}^{20}Ne\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 0.91 | 1.02 | 1.08 | 11.4 |  | 1.0
$\rm{}^{56}Fe\\!+\\!He\\!\\!\to^{7}$$\rm Li$ | 0.92 | 0.99 | 1.03 | [97.0, 103.3] |  | 1.0
$\rm{}^{14}N\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 0.93 | 0.98 | 1.01 | [9.3, 9.4] | ✓ | 1.0
$\rm{}^{56}Fe\\!+\\!He\\!\\!\to^{6}$$\rm Li$ | 0.81 | 0.85 | 0.92 | [79.9, 94.6] |  | 0.8
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 0.50 | 0.72 | 1.17 | [11.2, 27.3] | ✓ | 0.5
$\rm\mathbf{{}^{7}Li\\!+\\!H\\!\\!\to^{6}}$${\rm\mathbf{H}e}$ | 0.68 | 0.70 | 0.76 | 10.5 |  | 0.3
$\rm{}^{10}B\\!+\\!H\\!\\!\to^{7}$$\rm Li$ | 0.67 | 0.69 | 0.70 | 10.0 |  | 1.0
$\rm{}^{11}B\\!+\\!H\\!\\!\to^{6}$$\rm Li$ | 0.68 | 0.69 | 0.71 | 5.0 | ✓ | 0.8
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.53 | 0.57 | 0.59 | [12.3, 13.3] | ✓ | 0.9
$\rm\mathbf{{}^{12}C\\!+\\!H\\!\\!\to^{6}}$${\rm\mathbf{H}e}$ | 0.00 | 0.55 | 0.82 | [0.0, 1.0] | ✓ | 0.1
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{6}}$${\rm\mathbf{H}e}$ | 0.00 | 0.53 | 0.80 | [0.0, 0.9] | ✓ | 0.0
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.51 | 0.51 | 0.51 | [10.5, 11.0] | ✓ | 0.9
$\rm{}^{11}B\\!+\\!He\\!\\!\to^{7}$$\rm Li$ | 0.47 | 0.48 | 0.51 | [31.3, 33.2] |  | 1.0
$\rm\mathbf{{}^{14}N\\!+\\!H\\!\\!\to^{6}}$${\rm\mathbf{H}e}$ | 0.00 | 0.47 | 0.70 | [0.0, 6.4] | ✓ | 0.3
($>$75% of $f_{\rm sec,tot}$)
$\rm\mathbf{{}^{56}Fe\\!+\\!H\\!\\!\to^{6}}$${\rm\mathbf{H}e}$ | 0.40 | 0.46 | 0.60 | [4.3, 6.9] | ✓ | 0.2
$\rm{}^{28}Si\\!+\\!He\\!\\!\to^{6}$$\rm Li$ | 0.35 | 0.42 | 0.45 | [24.5, 29.7] |  | 0.9
$\rm{}^{24}Mg\\!+\\!He\\!\\!\to^{6}$$\rm Li$ | 0.33 | 0.38 | 0.41 | [22.5, 26.8] |  | 0.9
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 0.37 | 0.38 | 0.38 | [9.7, 10.0] | ✓ | 1.0
Table 6: Reactions sorted according to their flux impact $f_{abc}$ (with $\sum f_{abc}=1.35$) on Be at 10 GeV/n ($f_{\rm sec}=100\%$, see Table 1). The first 50 reactions shown provide 74.7% of the total flux (823 reactions to reach 97%); reactions in bold highlight short-lived fragments (see Sect. III.1 and Table 2). Reaction | Flux impact $f_{abc}$ |  $\sigma$ [mb] | Data | $\displaystyle\frac{\sigma}{\sigma^{\rm cumul}}$
---|---|---|---|---
| Min | Mean | Max | [Min,Max] |  |
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 16.5 | 16.8 | 16.9 | [9.7, 10.0] | ✓ | 1.0
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 14.4 | 14.5 | 14.6 | [9.0, 9.7] | ✓ | 1.0
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 8.12 | 8.25 | 8.51 | [6.1, 6.8] | ✓ | 1.0
($>$25% of $f_{\rm sec,tot}$)
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 4.89 | 6.09 | 6.68 | [3.6, 4.6] | ✓ | 1.0
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{7}$$\rm Be$ | 2.68 | 2.87 | 2.97 | [14.7, 15.4] |  | 1.0
$\rm{}^{28}Si\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 2.62 | 2.72 | 2.77 | [10.8, 10.8] | ✓ | 1.0
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{10}$$\rm Be$ | 2.52 | 2.63 | 2.69 | [4.0, 4.0] | ✓ | 1.0
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 2.43 | 2.58 | 2.65 | [10.0, 10.3] | ✓ | 1.0
$\rm{}^{14}N\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 1.92 | 2.26 | 2.43 | [10.1, 11.8] | ✓ | 1.0
$\rm{}^{12}C\\!+\\!He\\!\\!\to^{7}$$\rm Be$ | 2.25 | 2.26 | 2.27 | [12.7, 13.6] |  | 1.0
$\rm{}^{11}B\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 2.20 | 2.22 | 2.23 | [10.0, 10.4] | ✓ | 0.9
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{10}$$\rm Be$ | 1.50 | 1.69 | 1.79 | [2.2, 2.4] | ✓ | 1.0
$\rm{}^{20}Ne\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 1.47 | 1.67 | 1.76 | [9.7, 9.7] |  | 1.0
$\rm{}^{56}Fe\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 1.61 | 1.66 | 1.78 | [9.4, 10.9] | ✓ | 1.0
($>$50% of $f_{\rm sec,tot}$)
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 1.61 | 1.61 | 1.62 | 32.3 | ✓ | 0.9
$\rm{}^{10}B\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 1.49 | 1.53 | 1.55 | [13.9, 13.9] |  | 1.0
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 0.86 | 1.52 | 1.85 | [4.3, 8.7] | ✓ | 1.0
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{15}$$\rm N$ | 1.34 | 1.41 | 1.45 | 34.3 | ✓ | 0.5
$\rm{}^{56}Fe\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 1.02 | 1.33 | 1.49 | [7.5, 10.4] | ✓ | 1.0
$\rm{}^{12}C\\!+\\!He\\!\\!\to^{9}$$\rm Be$ | 1.27 | 1.29 | 1.33 | [8.6, 9.5] |  | 1.0
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 1.25 | 1.28 | 1.34 | [27.7, 30.0] | ✓ | 0.5
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{15}}$${\rm\mathbf{O}}$ | 1.20 | 1.26 | 1.29 | 30.5 | ✓ | 0.5
$\rm\mathbf{{}^{12}C\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 1.21 | 1.24 | 1.26 | [26.9, 27.9] | ✓ | 0.5
$\rm{}^{15}N\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 1.20 | 1.22 | 1.24 | [7.3, 7.3] | ✓ | 1.0
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{14}$$\rm N$ | 0.89 | 1.20 | 1.35 | [22.9, 29.2] | ✓ | 1.0
$\rm{}^{11}B\\!+\\!H\\!\\!\to^{10}$$\rm Be$ | 0.93 | 1.10 | 1.45 | [8.6, 12.9] | ✓ | 1.0
$\rm{}^{15}N\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 1.06 | 1.08 | 1.09 | [5.4, 5.4] | ✓ | 1.0
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{13}}$${\rm\mathbf{O}}$ | 1.03 | 1.08 | 1.10 | 30.5 | ✓ | 0.5
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{9}$$\rm Be$ | 0.80 | 1.05 | 1.17 | [5.3, 7.3] |  | 1.0
$\rm{}^{11}B\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 0.96 | 1.04 | 1.08 | [3.6, 4.2] | ✓ | 1.0
$\rm{}^{13}C\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 0.94 | 0.97 | 0.99 | 6.6 | ✓ | 0.9
$\rm{}^{28}Si\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 0.91 | 0.95 | 0.97 | [4.5, 4.6] | ✓ | 1.0
$\rm{}^{10}B\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 0.86 | 0.87 | 0.88 | [6.5, 6.9] | ✓ | 1.0
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 0.55 | 0.81 | 1.34 | [11.2, 27.3] | ✓ | 0.5
$\rm{}^{56}Fe\\!+\\!He\\!\\!\to^{7}$$\rm Be$ | 0.75 | 0.76 | 0.78 | [39.4, 43.3] |  | 1.0
$\rm{}^{13}C\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 0.69 | 0.71 | 0.73 | 4.0 | ✓ | 1.0
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 0.66 | 0.69 | 0.71 | 17.5 | ✓ | 0.4
$\rm{}^{28}Si\\!+\\!He\\!\\!\to^{7}$$\rm Be$ | 0.53 | 0.63 | 0.67 | [19.8, 23.7] |  | 1.0
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.58 | 0.62 | 0.64 | [12.3, 13.3] | ✓ | 0.9
$\rm{}^{56}Fe\\!+\\!He\\!\\!\to^{9}$$\rm Be$ | 0.45 | 0.61 | 0.70 | [29.8, 43.9] |  | 1.0
$\rm{}^{20}Ne\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 0.54 | 0.61 | 0.64 | 4.3 |  | 1.0
$\rm{}^{9}Be\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 0.57 | 0.60 | 0.61 | 10.6 | ✓ | 1.0
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.55 | 0.56 | 0.56 | [10.5, 11.0] | ✓ | 0.9
$\rm{}^{24}Mg\\!+\\!He\\!\\!\to^{7}$$\rm Be$ | 0.45 | 0.54 | 0.58 | [16.8, 20.4] |  | 1.0
$\rm{}^{22}Ne\\!+\\!H\\!\\!\to^{7}$$\rm Be$ | 0.38 | 0.45 | 0.60 | [4.7, 6.4] |  | 1.0
$\rm{}^{13}C\\!+\\!H\\!\\!\to^{10}$$\rm Be$ | 0.43 | 0.44 | 0.45 | 5.9 | ✓ | 1.0
$\rm{}^{12}C\\!+\\!He\\!\\!\to^{10}$$\rm Be$ | 0.39 | 0.41 | 0.42 | [5.6, 5.6] |  | 1.0
$\rm{}^{22}Ne\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 0.29 | 0.38 | 0.55 | [4.4, 7.1] |  | 1.0
$\rm{}^{14}N\\!+\\!He\\!\\!\to^{7}$$\rm Be$ | 0.30 | 0.37 | 0.40 | [14.4, 17.6] |  | 1.0
$\rm{}^{14}N\\!+\\!H\\!\\!\to^{9}$$\rm Be$ | 0.34 | 0.34 | 0.34 | [2.0, 2.1] | ✓ | 0.9
Table 7: Reactions sorted according to their flux impact $f_{abc}$ (with $\sum f_{abc}=1.36$) on B at 10 GeV/n ($f_{\rm sec}=96\%$, see Table 1). The first 50 reactions shown provide 81.9% of the total flux (550 reactions to reach 97%); reactions in bold highlight short-lived fragments (see Sect. III.1 and Table 2). Reaction | Flux impact $f_{abc}$ |  $\sigma$ [mb] | Data | $\displaystyle\frac{\sigma}{\sigma^{\rm cumul}}$
---|---|---|---|---
| Min | Mean | Max | [Min,Max] |  |
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 17.1 | 17.2 | 17.4 | [27.7, 30.0] | ✓ | 0.5
$\rm\mathbf{{}^{12}C\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 15.6 | 16.6 | 17.2 | [26.9, 27.9] | ✓ | 0.5
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 7.5 | 10.8 | 17.3 | [11.2, 27.3] | ✓ | 0.5
($>$25% of $f_{\rm sec}$ $\Rightarrow$ 28% of $f_{\rm tot}$)
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 7.15 | 7.85 | 8.19 | [12.3, 13.3] | ✓ | 0.9
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 6.98 | 7.05 | 7.09 | [10.5, 11.0] | ✓ | 0.9
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 0.00 | 4.42 | 6.62 | [0.0, 9.9] | ✓ | 0.5
$\rm{}^{11}B\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 3.98 | 4.12 | 4.19 | [38.9, 42.2] | ✓ | 1.0
($>$50% of $f_{\rm sec}$ $\Rightarrow$ 52% of $f_{\rm tot}$)
$\rm{}^{12}C\\!+\\!He\\!\\!\to^{11}$$\rm B$ | 2.44 | 2.46 | 2.49 | [35.7, 38.6] |  | 0.5
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 2.38 | 2.45 | 2.49 | 32.3 | ✓ | 0.9
$\rm{}^{15}N\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 1.96 | 2.25 | 2.39 | [26.1, 30.6] | ✓ | 0.8
$\rm\mathbf{{}^{12}C\\!+\\!He\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 2.06 | 2.20 | 2.27 | [32.0, 33.2] |  | 0.5
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{15}$$\rm N$ | 1.58 | 1.84 | 1.98 | 34.3 | ✓ | 0.5
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{11}$$\rm B$ | 1.20 | 1.66 | 2.57 | [16.1, 36.6] |  | 0.5
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{15}}$${\rm\mathbf{O}}$ | 1.40 | 1.64 | 1.76 | 30.5 | ✓ | 0.5
$\rm{}^{14}N\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 1.39 | 1.64 | 2.13 | [17.5, 29.2] | ✓ | 0.6
$\rm{}^{13}C\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 1.45 | 1.52 | 1.55 | 22.2 |  | 0.7
$\rm\mathbf{{}^{12}C\\!+\\!H\\!\\!\to^{10}}$${\rm\mathbf{C}}$ | 1.28 | 1.45 | 1.80 | [2.1, 3.1] | ✓ | 0.1
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{13}}$${\rm\mathbf{O}}$ | 1.25 | 1.32 | 1.35 | 30.5 | ✓ | 0.5
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{14}$$\rm N$ | 1.02 | 1.31 | 1.45 | [22.9, 29.2] | ✓ | 1.0
$\rm{}^{12}C\\!+\\!H\\!\\!\to^{10}$$\rm Be$ | 1.10 | 1.16 | 1.18 | [4.0, 4.0] | ✓ | 1.0
$\rm{}^{12}C\\!+\\!He\\!\\!\to^{10}$$\rm B$ | 1.02 | 1.12 | 1.17 | [15.8, 17.1] |  | 0.9
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{10}$$\rm B$ | 1.04 | 1.10 | 1.13 | [14.7, 15.1] |  | 0.9
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 0.97 | 1.01 | 1.03 | 10.4 |  | 0.6
$\rm{}^{28}Si\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 0.89 | 0.93 | 0.94 | [9.5, 9.5] |  | 0.6
$\rm{}^{15}N\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.72 | 0.91 | 1.00 | [9.6, 12.8] | ✓ | 1.0
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 0.80 | 0.85 | 0.87 | 17.5 | ✓ | 0.4
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{10}}$${\rm\mathbf{C}}$ | 0.00 | 0.82 | 1.23 | [0.0, 1.8] | ✓ | 0.1
$\rm{}^{20}Ne\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 0.70 | 0.80 | 0.84 | 12.0 |  | 0.6
$\rm{}^{14}N\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.76 | 0.76 | 0.76 | [9.5, 10.4] | ✓ | 0.9
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{10}$$\rm Be$ | 0.66 | 0.74 | 0.79 | [2.2, 2.4] | ✓ | 1.0
$\rm{}^{56}Fe\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 0.70 | 0.73 | 0.75 | [11.2, 11.2] | ✓ | 0.7
$\rm\mathbf{{}^{14}N\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 0.00 | 0.69 | 1.03 | [0.0, 13.0] | ✓ | 0.4
$\rm\mathbf{{}^{16}O\\!+\\!He\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 0.00 | 0.64 | 0.97 | [0.0, 13.0] |  | 0.5
($>$75% of $f_{\rm sec}$ $\Rightarrow$ 76% of $f_{\rm tot}$)
$\rm{}^{13}C\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.59 | 0.62 | 0.63 | 9.0 |  | 1.0
$\rm\mathbf{{}^{24}Mg\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 0.55 | 0.60 | 0.63 | [5.9, 6.3] | ✓ | 0.4
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.58 | 0.60 | 0.61 | 6.2 |  | 0.9
$\rm{}^{11}B\\!+\\!He\\!\\!\to^{10}$$\rm B$ | 0.57 | 0.57 | 0.57 | [50.0, 51.7] |  | 1.0
$\rm\mathbf{{}^{13}C\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 0.54 | 0.56 | 0.57 | 8.2 |  | 0.3
$\rm{}^{28}Si\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.53 | 0.55 | 0.56 | 5.7 |  | 0.9
$\rm{}^{22}Ne\\!+\\!H\\!\\!\to^{11}$$\rm B$ | 0.49 | 0.54 | 0.64 | [15.9, 17.8] | ✓ | 0.9
$\rm\mathbf{{}^{28}Si\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 0.46 | 0.54 | 0.58 | [5.0, 5.9] | ✓ | 0.4
$\rm{}^{11}B\\!+\\!H\\!\\!\to^{10}$$\rm Be$ | 0.41 | 0.48 | 0.63 | [8.6, 12.9] | ✓ | 1.0
$\rm{}^{20}Ne\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.42 | 0.47 | 0.50 | 7.1 |  | 0.9
$\rm\mathbf{{}^{20}Ne\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 0.42 | 0.46 | 0.48 | [6.8, 7.2] | ✓ | 0.4
$\rm{}^{56}Fe\\!+\\!H\\!\\!\to^{10}$$\rm B$ | 0.40 | 0.43 | 0.49 | [6.0, 7.8] | ✓ | 0.9
$\rm{}^{14}N\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 0.34 | 0.43 | 0.47 | [40.3, 51.5] | ✓ | 0.9
($>$80% of $f_{\rm sec}$ $\Rightarrow$ 81% of $f_{\rm tot}$)
$\rm\mathbf{{}^{15}N\\!+\\!H\\!\\!\to^{11}}$${\rm\mathbf{C}}$ | 0.40 | 0.41 | 0.41 | [5.3, 5.3] | ✓ | 0.1
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{12}$$\rm C$ | 0.33 | 0.34 | 0.34 | 39.8 |  | 0.9
$\rm{}^{15}N\\!+\\!He\\!\\!\to^{11}$$\rm B$ | 0.29 | 0.34 | 0.36 | [34.1, 41.6] |  | 0.8
$\rm{}^{56}Fe\\!+\\!He\\!\\!\to^{11}$$\rm B$ | 0.29 | 0.31 | 0.33 | [41.6, 44.2] |  | 0.7
Table 8: Reactions sorted according to their flux impact $f_{abc}$ (with $\sum f_{abc}=1.29$) on C at 10 GeV/n ($f_{\rm sec}=20\%$, see Table 1). The first 50 reactions shown provide 95.9% of the total flux (85 reactions to reach 97%); reactions in bold highlight short-lived fragments (see Sect. III.1 and Table 2). Reaction | Flux impact $f_{abc}$ |  $\sigma$ [mb] | Data | $\displaystyle\frac{\sigma}{\sigma^{\rm cumul}}$
---|---|---|---|---
| Min | Mean | Max | [Min,Max] |  |
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 21.2 | 21.5 | 22.2 | 32.3 | ✓ | 0.9
($>$80% of $f_{\rm tot}$ $\Rightarrow$ 2% of $f_{\rm sec}$)
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{13}}$${\rm\mathbf{O}}$ | 18.2 | 18.4 | 18.8 | 30.5 | ✓ | 0.5
($>$25% of $f_{\rm sec}$ $\Rightarrow$ 85% of $f_{\rm tot}$)
($>$85% of $f_{\rm tot}$ $\Rightarrow$ 26% of $f_{\rm sec}$)
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 11.7 | 11.9 | 12.1 | 17.5 | ✓ | 0.4
$\rm{}^{14}N\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 3.19 | 3.73 | 4.00 | [40.3, 51.5] | ✓ | 0.9
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{13}}$${\rm\mathbf{N}}$ | 3.42 | 3.46 | 3.53 | 5.1 | ✓ | 0.1
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{12}$$\rm C$ | 2.91 | 2.95 | 3.04 | 39.8 |  | 0.9
$\rm{}^{15}N\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 2.61 | 2.65 | 2.75 | 33.4 | ✓ | 0.9
($>$50% of $f_{\rm sec}$ $\Rightarrow$ 90% of $f_{\rm tot}$)
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{15}$$\rm N$ | 2.32 | 2.46 | 2.53 | 34.3 | ✓ | 0.5
($>$90% of $f_{\rm tot}$ $\Rightarrow$ 51% of $f_{\rm sec}$)
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 2.06 | 2.31 | 2.44 | [20.2, 25.2] |  | 0.9
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{13}}$${\rm\mathbf{O}}$ | 2.19 | 2.22 | 2.28 | 30.5 | ✓ | 0.5
$\rm\mathbf{{}^{16}O\\!+\\!He\\!\\!\to^{13}}$${\rm\mathbf{O}}$ | 2.18 | 2.20 | 2.24 | 32.8 |  | 0.5
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{15}}$${\rm\mathbf{O}}$ | 2.06 | 2.19 | 2.25 | 30.5 | ✓ | 0.5
$\rm{}^{20}Ne\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 1.91 | 2.08 | 2.16 | [30.1, 31.4] | ✓ | 1.0
$\rm{}^{15}N\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 1.86 | 2.07 | 2.17 | [22.8, 28.4] | ✓ | 0.8
$\rm{}^{13}C\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 1.93 | 1.94 | 1.95 | [27.1, 28.7] |  | 0.5
$\rm{}^{16}O\\!+\\!H\\!\\!\to^{14}$$\rm N$ | 1.35 | 1.76 | 1.97 | [22.9, 29.2] | ✓ | 1.0
$\rm\mathbf{{}^{13}C\\!+\\!H\\!\\!\to^{12}}$${\rm\mathbf{B}}$ | 1.47 | 1.76 | 1.91 | [21.0, 28.5] |  | 0.5
$\rm{}^{28}Si\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 0.86 | 1.68 | 2.08 | [8.5, 21.5] |  | 1.0
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{13}$$\rm C$ | 1.61 | 1.63 | 1.66 | 21.6 |  | 0.4
$\rm{}^{20}Ne\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 0.96 | 1.02 | 1.05 | [14.9, 15.0] | ✓ | 0.8
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 0.82 | 1.01 | 1.11 | [8.0, 11.2] |  | 0.8
$\rm\mathbf{{}^{16}O\\!+\\!H\\!\\!\to^{12}}$${\rm\mathbf{B}}$ | 0.87 | 0.90 | 0.97 | [1.3, 1.4] | ✓ | 0.0
$\rm{}^{28}Si\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 0.67 | 0.67 | 0.69 | 6.8 |  | 0.9
$\rm{}^{14}N\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 0.63 | 0.63 | 0.63 | 7.9 | ✓ | 0.6
$\rm{}^{22}Ne\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 0.51 | 0.55 | 0.64 | [16.2, 16.4] | ✓ | 1.0
$\rm{}^{22}Ne\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 0.47 | 0.54 | 0.67 | [15.6, 17.1] | ✓ | 0.9
$\rm{}^{14}N\\!+\\!He\\!\\!\to^{12}$$\rm C$ | 0.43 | 0.50 | 0.54 | [48.6, 62.1] |  | 0.9
$\rm\mathbf{{}^{14}N\\!+\\!H\\!\\!\to^{13}}$${\rm\mathbf{N}}$ | 0.49 | 0.49 | 0.50 | 6.2 | ✓ | 0.4
$\rm\mathbf{{}^{16}O\\!+\\!He\\!\\!\to^{13}}$${\rm\mathbf{N}}$ | 0.44 | 0.44 | 0.45 | 5.8 |  | 0.1
$\rm\mathbf{{}^{15}N\\!+\\!H\\!\\!\to^{12}}$${\rm\mathbf{B}}$ | 0.39 | 0.44 | 0.46 | [4.8, 6.2] |  | 0.2
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{16}$$\rm O$ | 0.41 | 0.42 | 0.42 | [30.2, 30.4] | ✓ | 0.9
($>$75% of $f_{\rm sec}$ $\Rightarrow$ 95% of $f_{\rm tot}$)
$\rm{}^{56}Fe\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 0.40 | 0.41 | 0.42 | [5.9, 6.5] | ✓ | 1.0
($>$95% of $f_{\rm tot}$ $\Rightarrow$ 75% of $f_{\rm sec}$)
$\rm{}^{23}Na\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 0.37 | 0.37 | 0.37 | [21.0, 22.6] |  | 0.9
$\rm{}^{24}Mg\\!+\\!He\\!\\!\to^{12}$$\rm C$ | 0.33 | 0.37 | 0.39 | [29.3, 36.5] |  | 0.9
$\rm{}^{27}Al\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 0.31 | 0.36 | 0.38 | [15.9, 20.4] |  | 1.0
$\rm{}^{25}Mg\\!+\\!H\\!\\!\to^{12}$$\rm C$ | 0.33 | 0.36 | 0.37 | [18.3, 21.1] |  | 0.9
$\rm{}^{15}N\\!+\\!He\\!\\!\to^{13}$$\rm C$ | 0.35 | 0.36 | 0.37 | 40.3 |  | 0.9
$\rm{}^{20}Ne\\!+\\!H\\!\\!\to^{16}$$\rm O$ | 0.29 | 0.35 | 0.38 | [33.4, 39.0] | ✓ | 1.0
$\rm{}^{28}Si\\!+\\!H\\!\\!\to^{27}$$\rm Al$ | 0.31 | 0.33 | 0.34 | 50.4 | ✓ | 0.6
$\rm\mathbf{{}^{15}N\\!+\\!H\\!\\!\to^{13}}$${\rm\mathbf{N}}$ | 0.32 | 0.32 | 0.33 | 4.0 | ✓ | 0.1
$\rm{}^{16}O\\!+\\!He\\!\\!\to^{15}$$\rm N$ | 0.30 | 0.31 | 0.32 | 39.3 |  | 0.5
$\rm{}^{56}Fe\\!+\\!H\\!\\!\to^{13}$$\rm C$ | 0.26 | 0.31 | 0.41 | [3.9, 5.9] | ✓ | 1.0
$\rm{}^{20}Ne\\!+\\!He\\!\\!\to^{12}$$\rm C$ | 0.28 | 0.30 | 0.32 | [39.7, 41.5] |  | 1.0
$\rm{}^{28}Si\\!+\\!He\\!\\!\to^{12}$$\rm C$ | 0.15 | 0.30 | 0.37 | [13.6, 34.5] |  | 1.0
$\rm{}^{24}Mg\\!+\\!H\\!\\!\to^{23}$$\rm Na$ | 0.29 | 0.29 | 0.30 | 39.2 | ✓ | 0.6