#  Spectra of Cosmic Ray Sodium and Aluminum and Unexpected Aluminum Excess

M. J. Boschini INFN, Milano-Bicocca, Milano, Italy  CINECA, Segrate, Milano,
Italy  S. Della Torre INFN, Milano-Bicocca, Milano, Italy  M. Gervasi INFN,
Milano-Bicocca, Milano, Italy  Physics Department, University of Milano-
Bicocca, Milano, Italy  D. Grandi INFN, Milano-Bicocca, Milano, Italy  Physics
Department, University of Milano-Bicocca, Milano, Italy  G. Jóhannesson
Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
NORDITA, Roslagstullsbacken 23, 106 91 Stockholm, Sweden  G. La Vacca INFN,
Milano-Bicocca, Milano, Italy  Physics Department, University of Milano-
Bicocca, Milano, Italy  N. Masi INFN, Bologna, Italy  Physics Department,
University of Bologna, Bologna, Italy  I. V. Moskalenko Hansen Experimental
Physics Laboratory, Stanford University, Stanford, CA 94305  Kavli Institute
for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA
94305  S. Pensotti  INFN, Milano-Bicocca, Milano, Italy  Physics Department,
University of Milano-Bicocca, Milano, Italy  T. A. Porter Hansen Experimental
Physics Laboratory, Stanford University, Stanford, CA 94305  Kavli Institute
for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA
94305  L. Quadrani  INFN, Bologna, Italy  Physics Department, University of
Bologna, Bologna, Italy  P. G. Rancoita INFN, Milano-Bicocca, Milano, Italy
D. Rozza INFN, Milano-Bicocca, Milano, Italy  M. Tacconi INFN, Milano-Bicocca,
Milano, Italy  Physics Department, University of Milano-Bicocca, Milano, Italy

(Accepted May 15, 2022)

###### Abstract

Since its launch, the Alpha Magnetic Spectrometer–02 (AMS-02) has delivered
outstanding quality measurements of the spectra of cosmic-ray (CR) species,
$\bar{p}$, $e^{\pm}$, and nuclei (H–Si, Fe), which resulted in a number of
breakthroughs. The most recent AMS-02 result is the measurement of the spectra
of CR sodium and aluminum up to $\sim$2 TV. Given their low solar system
abundances, a significant fraction of each element is produced in
fragmentations of heavier species, predominantly Ne, Mg, and Si. In this
paper, we use precise measurements of the sodium and aluminum spectra by
AMS-02 together with ACE-CRIS and Voyager 1 data to test their origin. We show
that the sodium spectrum agrees well with the predictions made with the
GalProp-HelMod framework, while aluminum spectrum shows a significant excess
in the rigidity range from 2–7 GV. In this context, we discuss the origin of
other low-energy excesses in Li, F, and Fe found earlier. The observed
excesses in Li, F, and Al appear to be consistent with the local Wolf-Rayet
(WR) stars hypothesis, invoked to reproduce anomalous 22Ne/20Ne, 12C/16O, and
58Fe/56Fe ratios in CRs, while excess in Fe is likely connected with a past SN
activity in the solar neighborhood. We also provide updated local interstellar
spectra (LIS) of sodium and aluminum in the rigidity range from few MV to
$\sim$2 TV. Our calculations employ the self-consistent GalProp–HelMod
framework that has proved to be a reliable tool in deriving the LIS of CR
$\bar{p}$, $e^{-}$, and nuclei $Z\leq 28$.

cosmic rays — diffusion — interplanetary medium — ISM: general — Sun:
heliosphere

##  1 Introduction

Precise data at low and high energies delivered by the modern space
instrumentation have triggered a number of discoveries of new features in
spectra of CR species enabling unprecedented probes of the stellar
nucleosynthesis, properties of the interstellar medium (ISM), and the origin
of CRs. A combination of data from individual spacecraft, can be combined to
cover the enormous range of rigidities, from few MV to tens of TV, where the
individual spectra of CR species are shaped by many different processes.

With their latest paper on CR sodium and aluminum, the AMS-02 collaboration
(Aguilar et al., 2021a) has now completed their series of papers on CR species
H–Si, and Fe (Aguilar et al., 2014, 2015a, 2015b, 2016a, 2016b, 2017, 2018a,
2018b, 2019a, 2019b, 2020, 2021b, 2021a, 2021c). Fluorine, sodium, and
aluminum in CRs share some common features: all three have odd atomic numbers,
and are mostly represented by a single stable isotope with odd number of
nucleons, ${}^{19}_{\phantom{0}9}$F, ${}^{23}_{11}$Na, ${}^{27}_{13}$Al, while
long-lived radioactive 26Al is present in CRs only fractionally. Because of
the properties of stellar nucleosynthesis, the source and CR abundances of
these elements are considerably lower than their even-$Z$ neighbors, 8O, 10Ne,
12Mg, 14Si. A significant fraction of each element, F, Na, Al, is produced
through fragmentation of their more abundant neighbors, Ne, Mg, Si. A
comparison of the spectra of F, Na, and Al, their common and distinct
features, may provide new insights into the origin of these elements.

In this paper we analyze the new CR measurements of the spectra of Na and Al
and test their consistency with measurements of other species. We also provide
updated Na and Al LIS in the rigidity range from few MV to $\sim$2 TV. Our
calculations and interpretation employ the GalProp111Available from
http://galprop.stanford.edu –HelMod222http://www.helmod.org/  framework that
is proved to be a reliable tool in deriving the LIS of CR species (Boschini et
al., 2019a, 2020a).

##  2 Calculations

In this work we are using the same CR propagation model with distributed
reacceleration and convection that was used in our previous analyses (for more
details see Boschini et al., 2017, 2018a, 2018b, 2020a, 2020b, 2021, 2022).
The latest version 57 of the GalProp code for Galactic propagation of CRs and
diffuse emissions is described in detail in Porter et al. (2021), see also
Boschini et al. (2020a) and references therein.

Full details of the latest HelMod code version 4 for heliospheric propagation
are provided in Boschini et al. (2019b). It solves the Fokker-Planck equation
for heliospheric propagation in Kolmogorov formulation backward in time (Bobik
et al., 2016). The accuracy of the solution was tested using the Crank-
Nicholson technique and found to be better than 0.5% at low rigidities. The
large number of simulated events ensures that the statistical errors are
negligible compared to the other modeling uncertainties.

When comparing our calculations with data collected over extended period of
time, variations in the solar activity are addressed in the following way. The
propagation equation is solved for each Carrington rotation, and the numerical
results are then combined accordingly to the instrument exposure and the time
period. This approach is equivalent to application of a weighted average that
accounts for both exposure time and absolute counting rate variations.

The values of propagation parameters in the ISM along with their confidence
limits are derived from the best available CR data using the Markov Chain
Monte Carlo (MCMC) routine. Here we use the same method as described in
Boschini et al. (2017). Five main propagation parameters, that affect the
overall shape of CR spectra, were left free in the scan using GalProp running
in 2D mode: the Galactic halo half-width $z_{h}$, the normalization of the
diffusion coefficient $D_{0}$ at the reference rigidity $R=4$ GV and the index
of its rigidity dependence $\delta$, the Alfvén velocity $V_{\rm Alf}$, and
the gradient of the convection velocity $dV_{\rm conv}/dz$ ($V_{\rm conv}=0$
in the plane, $z=0$). Their best-fit values tuned to the AMS-02 data are
listed in Table 1 and are the same as obtained in Boschini et al. (2020a). The
radial size of the Galaxy does not significantly affect the values of
propagation parameters and was set to 20 kpc. We also introduced a factor
$\beta^{\eta}$ in the diffusion coefficient, where $\beta=v/c$, and $\eta$ was
left free. The best fit value of $\eta=0.70$ improves the agreement at low
energies, and slightly affects the choice of injection indices ${\gamma}_{0}$
an ${\gamma}_{1}$. A detailed discussion of the injection (I) and propagation
(P) scenarios of the 350 GV break can be found in the works by Vladimirov et
al. (2012) and Boschini et al. (2020a).

The corresponding B/C ratio also remains the same (see Fig. 4 of Boschini et
al., 2020a), and compares well with all available measurements: Voyager 1
(Cummings et al., 2016), ACE-
CRIS333http://www.srl.caltech.edu/ACE/ASC/level2/cris_l2desc.html (Lave et
al., 2013), AMS-02 (Aguilar et al., 2018a), ATIC-2 (Panov et al., 2009), CREAM
(Ahn et al., 2008, 2009), and NUCLEON (Grebenyuk et al., 2019).

In the calculations, we use our standard formalism. The convection velocity is
assumed to increase linearly with distance $z$ from the plane, $V_{\rm
conv}(z)=V_{0}+z\cdot dV_{\rm conv}/dz$. The spatial diffusion coefficient is
parametrized as $D_{xx}=\beta D_{0}R^{\delta}$, where $\beta=v/c$ is the
particle velocity. If reacceleration is included, the momentum-space diffusion
coefficient $D_{pp}$ is related to $D_{xx}$ as $D_{pp}=p^{2}V_{\rm
Alf}^{2}/\left(9D_{xx}\right)$ (Berezinskii et al., 1990; Seo & Ptuskin,
1994). See also a review by Strong et al. (2007) and our papers cited in the
beginning of this section. The injection spectra of CR species are
parametrized by the rigidity-dependent function:

$q(R)\propto(R/R_{0})^{-\gamma_{0}}\prod_{i=0}^{2}\bigg{[}1+(R/R_{i})^{\frac{\gamma_{i}-\gamma_{i+1}}{s_{i}}}\bigg{]}^{s_{i}},$
(1)

where $\gamma_{i=0,1,2,3}$ are the spectral indices, $R_{i=0,1,2}$ are the
break rigidities, $s_{i}$ are the smoothing parameters ($s_{i}$ is
negative/positive for $|\gamma_{i}|\lessgtr|\gamma_{i+1}|$).

Table 1: Best-fit propagation parameters for I\- and P-scenarios Parameter | Units | Best Value | Error
---|---|---|---
$z_{h}$ | kpc | 4.0 | 0.6
$D_{0}(R=4\ {\rm GV})$ | cm2 s-1 | $4.3\times 10^{28}$ | 0.7
$\delta$aaThe P-scenario assumes a break in the diffusion coefficient with index $\delta_{1}=\delta$ below the break and index $\delta_{2}=0.15\pm 0.03$ above the break at $R=370\pm 25$ GV (for details see Boschini et al., 2020b). |  | 0.415 | 0.025
$V_{\rm Alf}$ | km s-1 | 30 | 3
$dV_{\rm conv}/dz$ | km s-1 kpc-1 | 9.8 | 0.8

##  3 Results and Discussion

Fig. 1 shows a comparison of the calculated spectrum with Voyager 1 (Cummings
et al., 2016), ACE-CRIS (Lave et al., 2013), HEAO-3-C2 (Engelmann et al.,
1990), and AMS-02 sodium data (Aguilar et al., 2021c). The calculated spectrum
reproduces the data in the whole energy range from 100 MV–2 TV quite well,
although there is insignificant 1$\sigma$ excess below 7 GV. The injection
spectrum of primary sodium is adjusted to the AMS-02 data (Table 2), but
remains similar to other species (Boschini et al., 2020a).

Table 2: The injection spectra of primary sodium and aluminum and aluminum
excess

| Source | Spectral parameters
---|---|---
| abundance | $\gamma_{0}$ |  $R_{0}$ (GV) | $s_{0}$ | $\gamma_{1}$ |  $R_{1}$ (GV) | $s_{1}$ | $\gamma_{2}$ |  $R_{2}$ (GV) | $s_{2}$ | $\gamma_{3}$
23Na | default | 50 | 0.24 | 0.92 | 0.22 | 2.33 | 8.2 | 0.20 | 2.49 | 350 | 0.16 | 2.15
27Al | default | 94 | 0.20 | 0.60 | 0.17 | 2.04 | 7.0 | 0.20 | 2.46 | 355 | 0.17 | 1.93
27Al | excess | 11.5 | $-5.00$ | 4.0 | 0.30 | 5.00 |  $\cdots$ | $\cdots$ |  $\cdots$ |  $\cdots$ | $\cdots$ |  $\cdots$

Note. — The source abundances are relative, see Table 3 in Boschini et al.
(2020a). For parameter definitions see Eq. (1). Shown are $|s_{i}|$ values,
note that $s_{i}$ is negative/positive for
$|\gamma_{i}|\lessgtr|\gamma_{i+1}|$.

Figure 1:  A comparison of the calculated spectrum of sodium with Voyager 1
(Cummings et al., 2016), ACE-CRIS (Lave et al., 2013), HEAO-3-C2 (Engelmann et
al., 1990) (left panel), and AMS-02 data (Aguilar et al., 2021c) (right
panel). In the left panel, to match the units of the published data, the
spectra are plotted vs. kinetic energy per nucleon, $E_{\rm kin}$. The dashed
gray lines show the LIS, and the solid black lines are the corresponding
modulated spectra. The lower panels show the relative difference between our
calculations and the data sets. The gray shaded areas indicate 1$\sigma$
confidence limits for the calculated modulated spectra.

Figure 2:  The calculated Na/F and Na/Si ratios as compared with Voyager 1
(Cummings et al., 2016), ACE-CRIS (Lave et al., 2013), and AMS-02 data
(Aguilar et al., 2020, 2021a, 2021c). In the left panel, the dashed gray line
shows the default LIS Na/F ratio, while the solid gray line shows the ratio
for the renormalized ($\times$0.896) fluorine spectrum (for details see
Boschini et al., 2022). The solid gray line in the right panel shows the LIS
Na/Si ratio. In both panels, the modulated ratios are shown with solid black
lines. The gray shaded areas indicate 1$\sigma$ confidence limits for the
calculated modulated ratios. The Voyager 1 and ACE-CRIS data are converted
from kinetic energy per nucleon to rigidity (assuming $A/Z=2$ for Si). These
data are shown as shaded areas with the width corresponding to 1$\sigma$
error.

Figure 3:  A comparison of the calculated aluminum spectrum with Voyager 1
(Cummings et al., 2016), ACE-CRIS (Lave et al., 2013), HEAO-3-C2 (Engelmann et
al., 1990), and AMS-02 data (Aguilar et al., 2021c). Top panels show the
calculations made with the default aluminum injection spectrum (Table 2)
similar to other Si-group species (Table 2, Boschini et al., 2020a). The
bottom panels show the total LIS (default+excess), with the injection spectrum
of the aluminum excess (Table 2) tuned to the AMS-02 data. The excess is shown
separately with the dash-dotted gray lines. In the left panels, to match the
units of the published data, the spectra are plotted vs. kinetic energy per
nucleon, $E_{\rm kin}$. In all panels, the dashed gray line shows the LIS, and
the solid black line is the corresponding modulated spectrum. The gray shaded
areas indicate 1$\sigma$ confidence limits for the calculated modulated
spectrum. The lower panels show the relative difference between our
calculations and the data sets.

Figure 4:  The calculated Al/F and Al/Si ratios as compared with Voyager 1
(Cummings et al., 2016), ACE-CRIS (Lave et al., 2013), HEAO-3-C2 (Engelmann et
al., 1990), and AMS-02 data (Aguilar et al., 2020, 2021a, 2021c). In all
panels, the solid gray line shows the corresponding LIS ratio, while the
modulated ratios are shown with solid black lines. The top panels show
calculations for a default Al injection spectrum (Table 2); in the left panel
shown is the ratio with the renormalized ($\times$0.896) fluorine spectrum
(for details see Boschini et al., 2022). The bottom panels show calculations
with an additional low-energy primary excess component on top of the default
injection spectrum. For a comparison, the dashed gray lines show the default
LIS ratios. The corresponding modulated ratios are shown with solid black
lines. The gray shaded areas indicate 1$\sigma$ confidence limits for the
calculated modulated ratios. The Voyager 1 and ACE-CRIS data are converted
from kinetic energy per nucleon to rigidity (assuming $A/Z=2$ for Si). These
data are shown as shaded areas with the width corresponding to 1$\sigma$
error.

Figure 5: A comparison of the calculated isotopic ratios 26Al/27Al (left
panels) and 10Be/9Be (right panels) with available data: Voyager 1, 2 data
(Lukasiak et al., 1994) collected over a period of 16 years (1977–1993) in the
heliosphere, Ulysses (Simpson & Connell, 1998), ACE-CRIS (Yanasak et al.,
2001), Balloon (Webber, 1982), and ISEE3-HKH (Wiedenbeck, 1983). Calculations
for the halo size $z_{h}=4$ kpc are shown in the top panels, and for $z_{h}=6$
kpc in the bottom (with the adjusted normalization of the spatial diffusion
coefficient $D_{0}(R=4\ {\rm GV})=7.5\times 10^{28}$ cm2 s-1). The dashed gray
lines in all plots show the default LIS ratios calculated for $z_{h}=4$ kpc,
while the solid gray line in the top left panel shows the LIS ratio with the
additional low-energy primary aluminum component. The dash-dotted lines show
the default LIS ratios calculated for $z_{h}=6$ kpc. The colored symbols show
the original published data from different instruments, while the same symbols
plotted in black show our predictions based on the calculated LIS ratios for
$z_{h}=4$ (top panels) or 6 kpc (bottom panels). The black line in the gray
area shows the modulated ratio, and the gray area marks 1$\sigma$ error for
our predictions.

Fig. 2 shows the calculated Na/F and Na/Si ratios as compared with Voyager 1
(Cummings et al., 2016), ACE-CRIS (Lave et al., 2013), and AMS-02 data
(Aguilar et al., 2020, 2021a, 2021c). In the left panel, the dashed gray line
shows the default LIS Na/F ratio (with non-renormalized fluorine spectrum),
while the solid gray line shows the ratio for the renormalized ($\times$0.896)
fluorine spectrum (for details see Boschini et al., 2022). The solid gray line
in the right panel shows the LIS Na/Si ratio. In both panels, the modulated
ratios are shown with solid black lines. The Voyager 1 and ACE-CRIS data are
converted from kinetic energy per nucleon to rigidity (assuming $A/Z=2$ for
Si). These data are shown as shaded areas with the width corresponding to
1$\sigma$ error. Note that the ACE-CRIS data taken during consequent solar
minima overlap.

The ratios, Na/F for renormalized fluorine and Na/Si, exhibit a good overall
agreement with data. There is some discrepancy of the calculated Na/Si ratio
with AMS-02 data at 2–5 GV, which corresponds to $\approx$1$\sigma$ (5%) if we
take into account the uncertainties in the modulated ratio shown by the gray
band. This can be due to the errors in the production cross sections, which we
discuss in Sect. 4. At lower energies, there is a discrepancy with the Voyager
1 data where the Na/Si ratio shows a concave shape, while the calculated ratio
shows a monotonic decrease toward the lower energies. This discrepancy is
related to large fluctuations (well above the experimental error bars) of
Voyager 1 data points for sodium (see Fig. 8 in Boschini et al., 2020a).

Fig. 3 shows a comparison of the calculated Al spectra with Voyager 1
(Cummings et al., 2016), ACE-CRIS (Lave et al., 2013), HEAO-3-C2 (Engelmann et
al., 1990), and AMS-02 data (Aguilar et al., 2021c). The top panels show
calculations for a default Al injection spectrum (Table 2, Boschini et al.,
2020a), which is slightly adjusted to reproduce AMS-02 data at the highest
energies (Table 2). In the top left panel, the Al spectrum is tuned to Voyager
1, ACE-CRIS, and the middle range of the HEAO-3-C2 instrument. The lower and
higher energy HEAO-3-C2 data points are biased as was shown in Boschini et al.
(2020a), and therefore we did not not use them in our fits. To match the units
of the published data, the spectra in the left panels are plotted vs. kinetic
energy per nucleon, $E_{\rm kin}$. Obvious is a significant excess in the
rigidity range 2–7 GV when compared with AMS-02 data, but outside of this
range the agreement is good (Fig. 3 top right).

The bottom panels show calculations with a low-energy primary excess component
added to the default injection spectrum (see Table 2); its shape was tuned to
match the AMS-02 data after propagation. The modified Al spectrum has now a
clear bump, but the agreement with AMS-02 data is significantly improved at
the cost of fine tuning of the injection spectrum. Possible reasons for the
aluminum excess and other excesses are discussed in Sect. 4.

Fig. 4 shows the calculated Al/F and Al/Si ratios as compared with Voyager 1
(Cummings et al., 2016) ACE-CRIS (Lave et al., 2013), and AMS-02 data (Aguilar
et al., 2020, 2021a, 2021c). The Voyager 1 and ACE-CRIS data are converted
from kinetic energy per nucleon to rigidity (assuming $A/Z=2$ for Si). These
data are shown as shaded areas with the width corresponding to 1$\sigma$
error. The ACE-CRIS data taken during consequent solar minima are overlapping
in the plots. In all panels, the solid gray line shows the corresponding LIS
ratio, while the modulated ratios are shown with solid black lines. The top
panels show calculations for a default Al injection spectrum (Table 2); in the
left panel shown is the ratio with the renormalized ($\times$0.896) fluorine
spectrum (for details see Boschini et al., 2022). Again, one can see a
significant excess in the rigidity range 2–7 GV, while outside of this range
the agreement with data is good. The bottom panels show calculations with a
low-energy primary excess component (see Table 2 and Fig. 3 bottom) added to
the default injection spectrum. For a comparison, the dashed gray lines show
the default LIS ratios. Again, an additional low-energy component
significantly improves the agreement with data.

The radioactive 26Al isotope in CR is entirely secondary. Therefore, it
provides an independent test of the propagation model and our tuned Al
spectrum. The calculated isotopic ratio 26Al/27Al is compared with available
CR data in Fig. 5. In the top right panel, we also show the 10Be/9Be ratio
calculated for the same propagation parameters (Table 1). Bottom panels show
both ratios calculated for a larger halo size $z_{h}=6$ kpc with the adjusted
normalization of the spatial diffusion coefficient $D_{0}(R=4\ {\rm
GV})=7.5\times 10^{28}$ cm2 s-1.

The calculated LIS and modulated ratios are compared with the most precise CR
isotopic data available: Voyager 1, 2 data (Lukasiak et al., 1994) collected
over a period of 16 years (1977–1993) as the spacecraft moved from 1 au to 53
au and 41 au, correspondingly, Ulysses (Simpson & Connell, 1998), and ACE-CRIS
(Yanasak et al., 2001). In the plots the colored symbols show the original
published data from different instruments, while the same symbols plotted in
black show our predictions based on the calculated LIS. The gray area marks
1$\sigma$ error for our predictions.

One can see that the agreement with Voyager 1, 2 and Ulysses data in the top
left panel (26Al/27Al) is good, while the discrepancy with ACE-CRIS data
reaches 3$\sigma$–4$\sigma$ (green squares vs. the shaded area with black
squares). The increase in the halo size to $z_{h}=6$ kpc improves the
agreement to $\sim$1$\sigma$ for all data sets (bottom left). The 10Be/9Be
ratio agrees well with available data in both cases $z_{h}=4,6$ kpc.

The difference in the halo size as derived from different “radioactive clocks”
is a well-known issue (Moskalenko et al., 2001, 2005), where the halo size
derived from the 10Be/9Be ratio is systematically lower ($z_{h}=1.5-6$ kpc)
than that derived from other ratios, 26Al/27Al, 36Cl/Cl, 54Mn/Mn; meanwhile,
$z_{h}=4-6$ kpc range was found to be consistent will all ratios. We note that
calculations in Moskalenko et al. (2001, 2005) were done using the force-field
approximation and CR data available at that time.

Overall, the agreement of the calculated 10Be/9Be and 26Al/27Al ratio with
data is good (Fig. 5), but indicates that halo size limits perhaps have to be
reassessed based on new CR data and advances in treatment of the heliospheric
modulation. On the other hand, we emphasize that the production cross sections
for Be isotopes are known better than that for isotopes of Al, Cl, and Mn, and
this has to be taken into account in a new assessment.

In the Appendix, we provide analytical parameterizations of the Na and default
Al LIS, Eqs. (A1), (A2) and the total Al LIS, Eq. (A3), that includes also the
additional primary component. The parameters are provided in Tables 3–5,
correspondingly. We also provide numerical tables for all three cases, which
tabulate the Na and Al LIS in rigidity $R$ (Tables 6, 8) and in kinetic energy
$E_{\rm kin}$ per nucleon (Tables 7, 9), and corresponding Tables 10, 11 for
the total Al LIS, that includes also the additional primary component.

##  4 Low-energy Excesses in Lithium, Fluorine, Aluminum, and Iron

In this Section we are speculating on the physical reasons for the observed
low-energy excess in the aluminum spectrum in context of other excesses that
were found in recent CR data. We do not discuss here a possibility of
instrumental errors as they are rather unlikely.

The AMS-02 detector, with its multiple redundancies, provides the state-of-
the-art measurements of the CR species (Aguilar et al., 2021d). Before its
launch, AMS-02 was tested extensively at the CERN test beam with electrons,
positrons, protons, and pions. During more than 10 years of AMS-02
construction, a large international group of physicists have developed a
comprehensive Monte Carlo simulation program for AMS-02. Importantly, the
fragmentation cross sections of CR species inside the instrument are measured
by the AMS-02 itself using the silicon tracker layers. The procedure and
survival probabilities for different elements are described in Aguilar et al.
(2021d). The data analysis is performed in parallel by several groups, which
employ different methods. A good agreement between their results is required
before a paper is submitted for a publication.

Let us first discuss the default injection spectrum of Al. It is usually
assumed that the majority of CR particles are accelerated in supernova remnant
(SNR) shocks. The diffusive shock acceleration operating in the shocks as well
as the processes in the ISM are insensitive to the nature of particles, while
the relevant variable is rigidity. The standard SNR shock acceleration (Malkov
& Drury, 2001) results in almost identical spectra of injected particles in
terms of rigidity. Therefore, the default injection spectrum of Al should be
very similar to the spectra of other elements in the Si group, except perhaps
at very low energies which could be sensitive to the local distributions of
the ISM gas and CR sources. Table 2 shows the default Al injection spectrum,
which is similar to the injection spectra of its neighbors indeed (see Table 2
in Boschini et al., 2020a). There is no viable reason of why the Al injection
spectrum in the 2–10 GV range should be different from its neighbors. This
forces us to the conclusion that the observed excess must have a different
origin, which is discussed below.

There are four possible physical reasons of the observed excess in the Al
spectrum: (i) an incorrect spectrum of 28Si, the major progenitor of secondary
26,27Al, (ii) errors in the total inelastic cross sections of Al, (iii) errors
in the isotopic production cross sections of 26,27Al, and (iv) an additional
local component of primary Al. Below we are discussing them one by one. For
completeness, in point (iii) we are also discussing the accuracy of 23Na cross
sections. The calculated spectra of Mg and Si, the major progenitors of Na,
are tuned to the recent AMS-02 data.

(i) The Si spectrum is tuned to Voyager 1, ACE-CRIS, and AMS-02 data (Boschini
et al., 2020a). Importantly, AMS-02 data are available above 2 GV, the same
rigidity range for all CR species. This is exactly the energy range where the
Al excess is observed. Contributions of other CR species are very minor and
cannot be responsible for the observed excess.

(ii) Significant errors in the total inelastic cross section of Al as a
primary reason for the excess can be excluded by consideration of the
accelerator data. Bobchenko et al. (1979) provide a table of measured proton-
nucleus cross sections for the proton momenta from 5 GeV/c to 9 GeV/c that
correspond to the ambient 27Al rigidity range 10–19 GV in the inverse
kinematics, while the measurement accuracy is stated as 1-2%. The average
value of the total inelastic cross section in this interval is 456.1 mb and
typical error bars are $\pm$4 mb. Most of the data points are consistent with
a constant value within the error bars with just four out of total seventeen
points being clear outliers at 443$\pm$4 mb, 445$\pm$4 mb, and 467$\pm$4 mb
(two points). This is the energy range where the total inelastic cross
sections are flat (e.g., see Fig. 1 in Wellisch & Axen, 1996) and they are
easier to measure than the isotopic production. There are also abundant data
on the total inelastic cross section of 27Al below $\sim$1 GeV nucleon-1 ($<$3
GV) (Bauhoff, 1986). The parameterizations of the total inelastic cross
sections used in our calculations (e.g., Barashenkov & Polanski, 1994) are
tuned to the available data. That makes a significant error below 7 GV, where
the excess is observed, rather unlikely. We note the absence of a similar
excess in the spectra of neighboring nuclei, such as Ne, Mg, Si (Boschini et
al., 2020a).

(iii) Almost 100% of secondary 27Al is produced through fragmentation of 28Si
with some minor contributions from 29Si, 32S, and 56Fe (Moskalenko et al.,
2013). Unfortunately, the isotopic production cross section
$p+^{28}$Si$\to$27Al is a major source of uncertainty. There are only a couple
of measurements available for this reaction: $(E_{\rm kin},\sigma)$ = (218 MeV
nucleon-1, 25 mb) (Artun et al., 1975), and (506 MeV nucleon-1, 48.5 mb)
(Webber et al., 1998). These energies correspond to 1.4 GV and 2.3 GV (27Al)
in the inverse kinematics, respectively, and constrain the production of 27Al
in the lower range of rigidities where the excess is observed. Clearly, the
full energy dependence of the cross section can only be evaluated using semi-
phenomenological calculations or nuclear codes (see the description of GalProp
in Boschini et al., 2020a). However, measurements of 28Si fragmentation
28Si$+p\to$Al made by a Siegen group (Hirzebruch et al., 1991) indicate that
the cross section remains flat between 1 and 14.5 GeV nucleon-1. This energy
range corresponds to the rigidity range from 3.5–31 GV (27Al), which covers
the excess range and extends to significantly higher rigidities. Therefore,
even though the cross section between $\approx$500 MeV nucleon-1 and
$\approx$1000 MeV nucleon-1 remains unconstrained, it has no effect on the
excess, which is observed in the rigidity range 2–7 GV with the maximum at
$\approx$4 GV. The latter corresponds to $E_{\rm kin}\approx$ 1.2 GeV
nucleon-1, i.e. where the cross section is flat.

To further investigate the issue, we look at another aluminum isotope, 26Al,
which is mostly produced through fragmentation of 28Si and 27Al (Moskalenko et
al., 2013). Since we compare our calculations with the observed CR isotopic
ratio 26Al/27Al below 1 GeV nucleon-1 (Fig. 5), our concern is the accuracy of
calculations of production of 26Al in this energy range with a correction for
heliospheric modulation. Fortunately, production of 26Al is well-constrained
below 1 GeV nucleon-1. The experimental data points with references can be
found in the EXFOR444https://www-nds.iaea.org/exfor/ data base; the data
points from different experiments are consistent within the error bars.

The calculated 26Al/27Al ratio (Fig. 5, bottom left, $z_{h}=6$ kpc) agrees
well with available CR data corrected for the corresponding level of
heliospheric modulation. To summarize, it is unlikely that the uncertainties
in the production cross sections of 26,27Al can result in the observed excess.

We also comment here on the accuracy of sodium cross sections. The total
inelastic cross section is calculated using the same parameterizations that
are used for other species (Barashenkov & Polanski, 1994). It should be
accurate enough given that total inelastic cross sections of its neighbors, Ne
and Mg, are well-measured. The production cross sections of 23Na are less
certain. The secondary sodium 23Na is produced through fragmentation of 24Mg
(dominant channel) and 28Si with minor contributions of 25Mg, 26Mg, and 27Al
(Moskalenko et al., 2013). Unfortunately, there is only a single measurement
for each parent nucleus, 24,25,26Mg, 28Si, 27Al, made by Webber et al. (1990)
at the LBL Bevalac using a $\sim$600 MeV nucleon-1 beams. Therefore, we are
using the updated semi-phenomenological parameterizations by Webber et al.
(2003) renormalized to the data. The beam energy of $\sim$600 MeV nucleon-1
(corresponds to 23Na rigidity of $\sim$2.5 GV) above which the expected cross
section behavior is quite smooth. It is reasonable to expect that Na
production cross sections should be flat above 2.5 GV, i.e. extending into the
rigidity range 2–7 GV of the observed Al excess.

(iv) The observed low-energy excess in aluminum spectrum is most likely due to
the stable isotope 27Al. However, because it is stable, its yield and Galactic
distribution are difficult to measure. Meanwhile, abundant literature exists
on the observations of the distribution of the diffuse $\gamma$-ray 1.809 MeV
line emission from the decay of the radioactive 26Al isotope and on its
origin. Therefore, we are discussing the origin of 26Al assuming that the same
sources are producing 27Al as well.

Studies of the primitive meteoritic grains indicate that the early solar
system material was enriched with short-lived radionuclides, such as 10Be,
26Al, 36Cl, 41Ca, 53Mn, 60Fe, and others (Goswami et al., 2005). In
particular, the presolar meteoritic ratio 26Al/27Al$\sim$$4.5\times 10^{-5}$
derived from the excess of 26Mg is surprisingly well determined despite the
fact that the half-life of 26Al of $\sim$700 kyr is much shorter than the
formation time of the solar system. Therefore, a local source of 26Al, such as
nearby SN or an AGB star, which injected fresh 26Al into the forming solar
system, is usually invoked.

Observations of the distribution of the diffuse Galactic 1.809 MeV
$\gamma$-ray emission line from 26Al decay by COMPTEL (Diehl et al., 1995;
Oberlack et al., 1996; Knödlseder et al., 1999) and INTEGRAL (Bouchet et al.,
2015) have shown that 26Al nucleosynthesis is ongoing in the present Galaxy.
Potential sources include AGB stars, novae, core collapse supernovae (SNe),
and Wolf-Rayet (WR) star winds (Prantzos & Diehl, 1996; Diehl et al., 2006).
Very recent review of the nucleosynthesis reactions involved in the production
and destruction of 26Al and 60Fe and characteristics of the stellar sites of
their nucleosynthesis can be found in Diehl et al. (2021).

Apparently, the sources of cosmic Al are numerous, and are simultaneously also
the sources of other rare isotopes, such as 7Li, 19F, 60Fe and others (see
discussions in our papers: Boschini et al., 2020b, 2021, 2022, and references
therein). Particularly interesting is a possible contribution of the massive
WR stars proposed to explain the observed anomalous 22Ne/20Ne ratio and other
observed ratios, 12C/16O, and 58Fe/56Fe, in CRs (Binns et al., 2008). It was
shown that all isotopic ratios measured with ACE-CRIS are consistent with CR
source composition consisting of $\sim$20% of WR material mixed with $\sim$80%
material with solar-system composition. Since most of WR stars are found in
the OB associations and are the evolutionary stages of OB stars, the OB
associations (through multiple SN explosions) are the likely sources of a
substantial fraction of CRs. WR stars are also most likely local sources of
7Li, and 19F, while 60Fe may come from a SN or a series of SN explosions 1–2
Myr ago in the solar neighborhood (Binns et al., 2016).

Regarding the spectrum of sodium, we note the absence of a similar low-energy
excess. An interesting analysis using the Gaia-ESO Survey to study sodium and
aluminum abundances in giants and dwarfs and its implications for stellar and
Galactic chemical evolution can be found in Smiljanic et al. (2016). WR stars
do not seem to be a significant source of sodium. Absence of the excess in the
sodium spectrum apparently supports the hypothesis of the origin of the
observed excesses in the local OB associations. Meanwhile, studies of sodium
nucleosynthesis in different stellar environments are desirable as they may
help to discriminate between possible scenarios of the origin of the observed
excesses.

##  5 Conclusion

Using the combined data of AMS-02 (Aguilar et al., 2021b), ACE-CRIS (Lave et
al., 2013), and Voyager 1 (Cummings et al., 2016) we analyzed the spectra of
sodium and aluminum over a wide rigidity range from 100 MV to 2 TV. We found
that the sodium spectrum agrees well with the predictions made with the
GalProp-HelMod framework, while aluminum spectrum shows a significant excess
in the rigidity range from 2–7 GV.

Our analysis shows that among four possible reasons of the observed excess, a
contribution of local sources is most likely. In this context, we discuss the
origin of other low-energy excesses in Li, F, and Fe found earlier (Boschini
et al., 2020b, 2021, 2022). It seems that the WR hypothesis that was invoked
to reproduce the observed isotopic ratios, 22Ne/20Ne, 12C/16O, 58Fe/56Fe, and
other observed CR abundances (Binns et al., 2008) is also consistent with the
observed excesses in Li, F, and Al, while excess in Fe may be connected with a
past SN activity in the solar neighborhood. Absence of a corresponding excess
in sodium supports this hypothesis, as the WR winds are not a significant
source of sodium.

Finally, we note that the exploration of the low-energy features in the
spectra of CR species has just begun, thanks to the data from the interstellar
probes Voyager 1, 2, and precise measurements by AMS-02, ACE-CRIS, and other
instruments. These features harbor the keys to understanding our local
Galactic environment and the history of formation of the solar system. The
increase in the collected statistics and reduction in the systematic errors
will help to establish the precise spectral shapes of observed features and to
facilitate their interpretation.

Special thanks to Pavol Bobik, Giuliano Boella, Marian Putis, and Mario
Zannoni for their continuous support of the HelMod project and many useful
suggestions. This work was carried out using HelMod tool, which is currently
supported within ASIF–ASI (Agenzia Spaziale Italiana) Supported Irradiation
Facilities–framework for space radiation environment activities, e.g., ASIF
implementation agreements 2017-22-HD.0 ASI-ENEA, 2017-15-HD.0 ASI-INFN,
2021-39-HH.0 ASI-ENEA and ASIF implementation agreement 2021-36-HH.0 involving
ASI and Milano-Bicocca University. This work is also supported by ASI contract
ASI-INFN 2019-19-HH.0 and ESA (European Space Agency) contract
4000116146/16/NL/HK. Igor V. Moskalenko and Troy A. Porter acknowledge support
from NASA Grants No. NNX17AB48G, 80NSSC22K0718, 80NSSC22K0477. This research
has made use of the SSDC Cosmic rays database (Di Felice et al., 2017) and
LPSC Database of Charged Cosmic Rays (Maurin et al., 2014).

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##  Appendix A Analytical parameterization and numerical tables of the sodium
and aluminum LIS

Here we provide an analytical parameterization of the sodium LIS (m-2 sr-1 s-1
GV-1):

$\displaystyle F_{\rm Na}(R)=$ $\displaystyle\begin{cases}\displaystyle
a-bR\tilde{R}-G(R)\left(c+dR-f\,R^{3}\right)-g\left(h\,R^{4}\right)^{G(R)},&R\leq
2.8\,{\rm GV},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \displaystyle
R^{-2.7}\left[-i-lR+m\sqrt{n-\frac{o}{R}+p\sqrt{R}}+qR\tanh\left(-\frac{rR}{sR+R^{2}-t\sqrt{R}}\right)\right],&R>2.8\,{\rm
GV},\end{cases}$ (A1)

where $R$ is the particle rigidity in GV, expressions for $\tilde{R}$ and
$G(x)$ are given in Eq. (A), and the values of the fitting parameters from $a$
to $z$ are summarized in Table 3.

Table 3: Parameters of the analytical fit to the sodium LIS, Eq. (A1)  Param | Value | Param | Value | Param | Value | Param | Value
---|---|---|---|---|---|---|---
$a$ | 2.8359e-1 | $g$ | 9.2254e-2 | $n$ | 3.3852e+0 | $s$ | 4.100e+1
$b$ | 3.1660e-2 | $h$ | 1.0122e-1 | $o$ | 4.0058e+0 | $t$ | 4.8223e+1
$c$ | 2.8544e-1 | $i$ | 2.9718e+0 | $p$ | 9.9286e-4 |  $\cdots$ | $\cdots$
$d$ | 7.4636e-2 | $l$ | 3.3525e-7 | $q$ | 2.8346e-2 |  $\cdots$ | $\cdots$
$f$ | 1.2551e-1 | $m$ | 3.3510e+0 | $r$ | 7.3991e+1 |  $\cdots$ | $\cdots$

We also provide analytical parameterizations of the “default” aluminum LIS and
total aluminum LIS (with the excess), units are m-2 sr-1 s-1 GV-1:

$\displaystyle F_{\rm Al}^{\rm default}(R)=$
$\displaystyle\begin{cases}\displaystyle
a-\frac{b}{R}+cR^{3}+dR^{2}G(R)-fR^{2}-g\,R\,G(R)-hG^{4}(R),&R\leq 3.6\,{\rm
GV},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \displaystyle
R^{-2.7}\left[i\tilde{R}+l\sqrt{R}+\frac{m}{\log(n+oR)}+p\log^{2}(n+oR)-q-r\,R\tilde{R}-s\tilde{R}^{2})\right],&R>3.6\,{\rm
GV},\end{cases}$ (A2) $\displaystyle F_{\rm Al}^{\rm total}(R)=$
$\displaystyle\begin{cases}\displaystyle
aR+\frac{b}{R}+cR^{3}+d\,R\tilde{R}^{4}G^{2}(R)-f-gR^{2}-h\tilde{R}^{2},&R\leq
3.2\,{\rm GV},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \displaystyle
R^{-2.7}\left\\{iR-l+m\tanh{(R)}+\tilde{R}\left(n-oR\right)+p\log(q+R\tilde{R})-r\,L\left[s\log(q+R\tilde{R})\right]\right\\},&R>3.2\,{\rm
GV},\end{cases}$ (A3)

where $R$ is the particle rigidity in GV,

$\displaystyle\tilde{R}$ $\displaystyle=\log{R},$ $\displaystyle G(x)$
$\displaystyle=e^{-x^{2}},$ (A4) $\displaystyle L(x)$
$\displaystyle=1/(1+e^{-x}),$

and the values of the fitting parameters from $a$ to $z$ are given in Tables 4
and 5.

Table 4: Parameters of the analytical fit to the “default” aluminum LIS, Eq. (A2)  Param | Value | Param | Value | Param | Value | Param | Value
---|---|---|---|---|---|---|---
$a$ | 2.1141e-1 | $g$ | 2.0344e-1 | $n$ | 2.7104e+0 | $s$ | 1.4698e-1
$b$ | 9.7044e-4 | $h$ | 1.8791e-1 | $o$ | 3.0751e-2 |  $\cdots$ | $\cdots$
$c$ | 4.0776e-3 | $i$ | 1.8105e+0 | $p$ | 1.4367e-1 |  $\cdots$ | $\cdots$
$d$ | 1.0513e-1 | $l$ | 2.2979e-2 | $q$ | 7.6313e+0 |  $\cdots$ | $\cdots$
$f$ | 2.6449e-2 | $m$ | 7.4947e+0 | $r$ | 3.1067e-7 |  $\cdots$ | $\cdots$
Table 5: Parameters of the analytical fit to the total aluminum LIS, Eq. (A3)  Param | Value | Param | Value | Param | Value | Param | Value
---|---|---|---|---|---|---|---
$a$ | 2.3066e-1 | $g$ | 8.9912e-2 | $n$ | 6.3989e-1 | $s$ | 2.3391e-1
$b$ | 2.4724e-2 | $h$ | 9.5355e-2 | $o$ | 8.5007e-6 |  $\cdots$ | $\cdots$
$c$ | 1.2692e-2 | $i$ | 1.2796e-4 | $p$ | 1.6380e+0 |  $\cdots$ | $\cdots$
$d$ | 9.6944e-2 | $l$ | 9.5849e+1 | $q$ | 1.2755e+2 |  $\cdots$ | $\cdots$
$f$ | 2.9140e-2 | $m$ | 1.4909e+2 | $r$ | 7.8534e+1 |  $\cdots$ | $\cdots$

The analytical representations, Eqs. (A1)–(A3), are also complemented by
numerical tables calculated for the I-scenario, which tabulate the LIS in
rigidity $R$ (Tables 6, 8, 10), and in kinetic energy $E_{\rm kin}$ per
nucleon (Tables 7, 9, 11).

Table 6: $Z=11$ – Sodium LIS

Rigidity | Differential | Rigidity | Differential | Rigidity | Differential | Rigidity | Differential | Rigidity | Differential
---|---|---|---|---|---|---|---|---|---
GV | intensity | GV | intensity | GV | intensity | GV | intensity | GV | intensity
9.027e-02 | 7.982e-04 | 7.289e-01 | 1.083e-01 | 1.008e+01 | 4.958e-03 | 5.271e+02 | 5.598e-08 | 3.313e+04 | 7.803e-13
9.473e-02 | 9.152e-04 | 7.661e-01 | 1.167e-01 | 1.094e+01 | 3.994e-03 | 5.802e+02 | 4.277e-08 | 3.649e+04 | 6.047e-13
9.941e-02 | 1.035e-03 | 8.054e-01 | 1.251e-01 | 1.188e+01 | 3.206e-03 | 6.387e+02 | 3.271e-08 | 4.018e+04 | 4.687e-13
1.043e-01 | 1.171e-03 | 8.469e-01 | 1.334e-01 | 1.292e+01 | 2.563e-03 | 7.032e+02 | 2.504e-08 | 4.424e+04 | 3.633e-13
1.095e-01 | 1.324e-03 | 8.906e-01 | 1.413e-01 | 1.405e+01 | 2.040e-03 | 7.741e+02 | 1.919e-08 | 4.872e+04 | 2.817e-13
1.149e-01 | 1.498e-03 | 9.369e-01 | 1.488e-01 | 1.530e+01 | 1.617e-03 | 8.523e+02 | 1.471e-08 | 5.365e+04 | 2.185e-13
1.206e-01 | 1.694e-03 | 9.857e-01 | 1.556e-01 | 1.668e+01 | 1.277e-03 | 9.383e+02 | 1.129e-08 | 5.908e+04 | 1.695e-13
1.265e-01 | 1.916e-03 | 1.037e+00 | 1.619e-01 | 1.819e+01 | 1.004e-03 | 1.033e+03 | 8.668e-09 | 6.506e+04 | 1.315e-13
1.328e-01 | 2.167e-03 | 1.092e+00 | 1.674e-01 | 1.985e+01 | 7.862e-04 | 1.137e+03 | 6.661e-09 | 7.164e+04 | 1.020e-13
1.393e-01 | 2.451e-03 | 1.150e+00 | 1.721e-01 | 2.168e+01 | 6.136e-04 | 1.252e+03 | 5.122e-09 | 7.889e+04 | 7.918e-14
1.462e-01 | 2.772e-03 | 1.211e+00 | 1.760e-01 | 2.369e+01 | 4.771e-04 | 1.379e+03 | 3.939e-09 | 8.687e+04 | 6.146e-14
1.535e-01 | 3.134e-03 | 1.276e+00 | 1.789e-01 | 2.590e+01 | 3.696e-04 | 1.518e+03 | 3.031e-09 | 9.566e+04 | 4.772e-14
1.611e-01 | 3.543e-03 | 1.345e+00 | 1.809e-01 | 2.834e+01 | 2.854e-04 | 1.671e+03 | 2.333e-09 | 1.053e+05 | 3.705e-14
1.690e-01 | 4.005e-03 | 1.419e+00 | 1.816e-01 | 3.103e+01 | 2.199e-04 | 1.840e+03 | 1.796e-09 | 1.160e+05 | 2.878e-14
1.774e-01 | 4.526e-03 | 1.497e+00 | 1.812e-01 | 3.398e+01 | 1.690e-04 | 2.026e+03 | 1.383e-09 | 1.277e+05 | 2.235e-14
1.862e-01 | 5.114e-03 | 1.580e+00 | 1.795e-01 | 3.723e+01 | 1.295e-04 | 2.231e+03 | 1.066e-09 | 1.407e+05 | 1.736e-14
1.954e-01 | 5.776e-03 | 1.668e+00 | 1.764e-01 | 4.081e+01 | 9.914e-05 | 2.457e+03 | 8.214e-10 | 1.549e+05 | 1.349e-14
2.050e-01 | 6.522e-03 | 1.763e+00 | 1.720e-01 | 4.475e+01 | 7.572e-05 | 2.705e+03 | 6.332e-10 | 1.706e+05 | 1.049e-14
2.152e-01 | 7.362e-03 | 1.863e+00 | 1.662e-01 | 4.909e+01 | 5.773e-05 | 2.979e+03 | 4.883e-10 | 1.878e+05 | 8.150e-15
2.259e-01 | 8.307e-03 | 1.971e+00 | 1.592e-01 | 5.387e+01 | 4.394e-05 | 3.280e+03 | 3.766e-10 | 2.068e+05 | 6.335e-15
2.371e-01 | 9.369e-03 | 2.087e+00 | 1.510e-01 | 5.913e+01 | 3.340e-05 | 3.612e+03 | 2.905e-10 |  $\cdots$ |  $\cdots$
2.488e-01 | 1.056e-02 | 2.211e+00 | 1.419e-01 | 6.492e+01 | 2.536e-05 | 3.977e+03 | 2.241e-10 |  $\cdots$ |  $\cdots$
2.611e-01 | 1.190e-02 | 2.344e+00 | 1.321e-01 | 7.130e+01 | 1.923e-05 | 4.379e+03 | 1.729e-10 |  $\cdots$ |  $\cdots$
2.741e-01 | 1.340e-02 | 2.487e+00 | 1.220e-01 | 7.832e+01 | 1.457e-05 | 4.822e+03 | 1.334e-10 |  $\cdots$ |  $\cdots$
2.877e-01 | 1.508e-02 | 2.640e+00 | 1.115e-01 | 8.605e+01 | 1.103e-05 | 5.310e+03 | 1.030e-10 |  $\cdots$ |  $\cdots$
3.020e-01 | 1.695e-02 | 2.806e+00 | 1.010e-01 | 9.457e+01 | 8.347e-06 | 5.847e+03 | 7.952e-11 |  $\cdots$ |  $\cdots$
3.170e-01 | 1.905e-02 | 2.985e+00 | 9.058e-02 | 1.039e+02 | 6.310e-06 | 6.438e+03 | 6.140e-11 |  $\cdots$ |  $\cdots$
3.328e-01 | 2.138e-02 | 3.179e+00 | 8.052e-02 | 1.143e+02 | 4.766e-06 | 7.090e+03 | 4.742e-11 |  $\cdots$ |  $\cdots$
3.493e-01 | 2.397e-02 | 3.388e+00 | 7.097e-02 | 1.256e+02 | 3.598e-06 | 7.807e+03 | 3.663e-11 |  $\cdots$ |  $\cdots$
3.667e-01 | 2.686e-02 | 3.615e+00 | 6.205e-02 | 1.382e+02 | 2.716e-06 | 8.596e+03 | 2.830e-11 |  $\cdots$ |  $\cdots$
3.850e-01 | 3.005e-02 | 3.862e+00 | 5.379e-02 | 1.519e+02 | 2.050e-06 | 9.466e+03 | 2.187e-11 |  $\cdots$ |  $\cdots$
4.042e-01 | 3.358e-02 | 4.129e+00 | 4.627e-02 | 1.671e+02 | 1.547e-06 | 1.042e+04 | 1.690e-11 |  $\cdots$ |  $\cdots$
4.244e-01 | 3.748e-02 | 4.421e+00 | 3.951e-02 | 1.838e+02 | 1.168e-06 | 1.148e+04 | 1.307e-11 |  $\cdots$ |  $\cdots$
4.456e-01 | 4.178e-02 | 4.738e+00 | 3.352e-02 | 2.022e+02 | 8.820e-07 | 1.264e+04 | 1.010e-11 |  $\cdots$ |  $\cdots$
4.679e-01 | 4.649e-02 | 5.083e+00 | 2.831e-02 | 2.225e+02 | 6.664e-07 | 1.392e+04 | 7.814e-12 |  $\cdots$ |  $\cdots$
4.913e-01 | 5.163e-02 | 5.460e+00 | 2.377e-02 | 2.448e+02 | 5.039e-07 | 1.533e+04 | 6.045e-12 |  $\cdots$ |  $\cdots$
5.160e-01 | 5.724e-02 | 5.872e+00 | 1.987e-02 | 2.694e+02 | 3.813e-07 | 1.688e+04 | 4.676e-12 |  $\cdots$ |  $\cdots$
5.419e-01 | 6.330e-02 | 6.323e+00 | 1.654e-02 | 2.965e+02 | 2.888e-07 | 1.858e+04 | 3.619e-12 |  $\cdots$ |  $\cdots$

Note. — Differential Intensity units: (m2 s sr GV)-1.

Table 7: $Z=11$ – Sodium LIS

$E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential
---|---|---|---|---|---|---|---|---|---
GeV/n | intensity | GeV/n | intensity | GeV/n | intensity | GeV/n | intensity | GeV/n | intensity
1.000e-03 | 3.605e-02 | 6.309e-02 | 6.463e-01 | 3.981e+00 | 1.056e-02 | 2.512e+02 | 1.170e-07 | 1.585e+04 | 1.632e-12
1.101e-03 | 3.939e-02 | 6.948e-02 | 6.668e-01 | 4.384e+00 | 8.482e-03 | 2.766e+02 | 8.942e-08 | 1.745e+04 | 1.264e-12
1.213e-03 | 4.246e-02 | 7.651e-02 | 6.846e-01 | 4.827e+00 | 6.793e-03 | 3.046e+02 | 6.839e-08 | 1.922e+04 | 9.800e-13
1.335e-03 | 4.577e-02 | 8.425e-02 | 6.993e-01 | 5.315e+00 | 5.419e-03 | 3.354e+02 | 5.236e-08 | 2.116e+04 | 7.597e-13
1.470e-03 | 4.934e-02 | 9.277e-02 | 7.104e-01 | 5.853e+00 | 4.306e-03 | 3.693e+02 | 4.012e-08 | 2.330e+04 | 5.891e-13
1.619e-03 | 5.318e-02 | 1.022e-01 | 7.175e-01 | 6.446e+00 | 3.408e-03 | 4.067e+02 | 3.076e-08 | 2.566e+04 | 4.568e-13
1.783e-03 | 5.733e-02 | 1.125e-01 | 7.206e-01 | 7.098e+00 | 2.688e-03 | 4.478e+02 | 2.361e-08 | 2.825e+04 | 3.543e-13
1.963e-03 | 6.181e-02 | 1.239e-01 | 7.199e-01 | 7.816e+00 | 2.111e-03 | 4.931e+02 | 1.812e-08 | 3.111e+04 | 2.749e-13
2.162e-03 | 6.663e-02 | 1.364e-01 | 7.156e-01 | 8.607e+00 | 1.652e-03 | 5.430e+02 | 1.393e-08 | 3.426e+04 | 2.133e-13
2.381e-03 | 7.182e-02 | 1.502e-01 | 7.078e-01 | 9.478e+00 | 1.288e-03 | 5.980e+02 | 1.071e-08 | 3.773e+04 | 1.656e-13
2.622e-03 | 7.741e-02 | 1.654e-01 | 6.968e-01 | 1.044e+01 | 1.001e-03 | 6.585e+02 | 8.237e-09 | 4.155e+04 | 1.285e-13
2.887e-03 | 8.343e-02 | 1.822e-01 | 6.826e-01 | 1.149e+01 | 7.750e-04 | 7.251e+02 | 6.338e-09 | 4.575e+04 | 9.977e-14
3.179e-03 | 8.990e-02 | 2.006e-01 | 6.654e-01 | 1.266e+01 | 5.982e-04 | 7.985e+02 | 4.878e-09 | 5.038e+04 | 7.747e-14
3.501e-03 | 9.686e-02 | 2.209e-01 | 6.450e-01 | 1.394e+01 | 4.606e-04 | 8.793e+02 | 3.755e-09 | 5.548e+04 | 6.017e-14
3.855e-03 | 1.043e-01 | 2.432e-01 | 6.218e-01 | 1.535e+01 | 3.538e-04 | 9.682e+02 | 2.892e-09 | 6.109e+04 | 4.674e-14
4.245e-03 | 1.124e-01 | 2.678e-01 | 5.957e-01 | 1.690e+01 | 2.712e-04 | 1.066e+03 | 2.228e-09 | 6.727e+04 | 3.631e-14
4.675e-03 | 1.210e-01 | 2.949e-01 | 5.671e-01 | 1.861e+01 | 2.075e-04 | 1.174e+03 | 1.717e-09 | 7.408e+04 | 2.821e-14
5.148e-03 | 1.303e-01 | 3.248e-01 | 5.360e-01 | 2.049e+01 | 1.585e-04 | 1.293e+03 | 1.324e-09 | 8.157e+04 | 2.192e-14
5.669e-03 | 1.402e-01 | 3.577e-01 | 5.027e-01 | 2.257e+01 | 1.208e-04 | 1.424e+03 | 1.021e-09 | 8.983e+04 | 1.704e-14
6.242e-03 | 1.508e-01 | 3.938e-01 | 4.679e-01 | 2.485e+01 | 9.194e-05 | 1.568e+03 | 7.873e-10 | 9.892e+04 | 1.325e-14
6.874e-03 | 1.621e-01 | 4.337e-01 | 4.318e-01 | 2.736e+01 | 6.988e-05 | 1.726e+03 | 6.073e-10 |  $\cdots$ |  $\cdots$
7.569e-03 | 1.743e-01 | 4.776e-01 | 3.953e-01 | 3.013e+01 | 5.305e-05 | 1.901e+03 | 4.685e-10 |  $\cdots$ |  $\cdots$
8.335e-03 | 1.872e-01 | 5.259e-01 | 3.592e-01 | 3.318e+01 | 4.023e-05 | 2.093e+03 | 3.615e-10 |  $\cdots$ |  $\cdots$
9.179e-03 | 2.010e-01 | 5.791e-01 | 3.239e-01 | 3.654e+01 | 3.048e-05 | 2.305e+03 | 2.790e-10 |  $\cdots$ |  $\cdots$
1.011e-02 | 2.157e-01 | 6.377e-01 | 2.897e-01 | 4.023e+01 | 2.308e-05 | 2.539e+03 | 2.154e-10 |  $\cdots$ |  $\cdots$
1.113e-02 | 2.313e-01 | 7.022e-01 | 2.570e-01 | 4.431e+01 | 1.746e-05 | 2.795e+03 | 1.663e-10 |  $\cdots$ |  $\cdots$
1.226e-02 | 2.479e-01 | 7.733e-01 | 2.261e-01 | 4.879e+01 | 1.320e-05 | 3.078e+03 | 1.284e-10 |  $\cdots$ |  $\cdots$
1.350e-02 | 2.654e-01 | 8.515e-01 | 1.974e-01 | 5.373e+01 | 9.968e-06 | 3.390e+03 | 9.916e-11 |  $\cdots$ |  $\cdots$
1.486e-02 | 2.840e-01 | 9.377e-01 | 1.712e-01 | 5.916e+01 | 7.525e-06 | 3.733e+03 | 7.660e-11 |  $\cdots$ |  $\cdots$
1.637e-02 | 3.035e-01 | 1.033e+00 | 1.474e-01 | 6.515e+01 | 5.679e-06 | 4.110e+03 | 5.918e-11 |  $\cdots$ |  $\cdots$
1.802e-02 | 3.240e-01 | 1.137e+00 | 1.260e-01 | 7.174e+01 | 4.286e-06 | 4.526e+03 | 4.573e-11 |  $\cdots$ |  $\cdots$
1.985e-02 | 3.456e-01 | 1.252e+00 | 1.070e-01 | 7.900e+01 | 3.235e-06 | 4.984e+03 | 3.535e-11 |  $\cdots$ |  $\cdots$
2.185e-02 | 3.682e-01 | 1.379e+00 | 9.027e-02 | 8.699e+01 | 2.442e-06 | 5.489e+03 | 2.732e-11 |  $\cdots$ |  $\cdots$
2.406e-02 | 3.917e-01 | 1.518e+00 | 7.579e-02 | 9.580e+01 | 1.844e-06 | 6.044e+03 | 2.113e-11 |  $\cdots$ |  $\cdots$
2.650e-02 | 4.161e-01 | 1.672e+00 | 6.338e-02 | 1.055e+02 | 1.394e-06 | 6.656e+03 | 1.634e-11 |  $\cdots$ |  $\cdots$
2.918e-02 | 4.414e-01 | 1.841e+00 | 5.278e-02 | 1.162e+02 | 1.054e-06 | 7.329e+03 | 1.264e-11 |  $\cdots$ |  $\cdots$
3.213e-02 | 4.673e-01 | 2.027e+00 | 4.378e-02 | 1.279e+02 | 7.972e-07 | 8.071e+03 | 9.778e-12 |  $\cdots$ |  $\cdots$
3.539e-02 | 4.938e-01 | 2.233e+00 | 3.620e-02 | 1.409e+02 | 6.039e-07 | 8.887e+03 | 7.567e-12 |  $\cdots$ |  $\cdots$

Note. — Differential Intensity units: (m2 s sr GeV/n)-1.

Table 8: $Z=13$ – Default aluminum LIS

Rigidity | Differential | Rigidity | Differential | Rigidity | Differential | Rigidity | Differential | Rigidity | Differential
---|---|---|---|---|---|---|---|---|---
GV | intensity | GV | intensity | GV | intensity | GV | intensity | GV | intensity
8.967e-02 | 1.397e-03 | 7.240e-01 | 1.196e-01 | 1.002e+01 | 5.553e-03 | 5.236e+02 | 1.016e-07 | 3.291e+04 | 4.657e-12
9.410e-02 | 1.595e-03 | 7.610e-01 | 1.249e-01 | 1.087e+01 | 4.530e-03 | 5.763e+02 | 7.925e-08 | 3.624e+04 | 3.713e-12
9.875e-02 | 1.801e-03 | 8.000e-01 | 1.299e-01 | 1.180e+01 | 3.680e-03 | 6.345e+02 | 6.195e-08 | 3.991e+04 | 2.961e-12
1.036e-01 | 2.034e-03 | 8.412e-01 | 1.348e-01 | 1.283e+01 | 2.975e-03 | 6.985e+02 | 4.852e-08 | 4.395e+04 | 2.361e-12
1.087e-01 | 2.296e-03 | 8.847e-01 | 1.393e-01 | 1.396e+01 | 2.394e-03 | 7.689e+02 | 3.807e-08 | 4.840e+04 | 1.882e-12
1.141e-01 | 2.592e-03 | 9.306e-01 | 1.435e-01 | 1.520e+01 | 1.918e-03 | 8.466e+02 | 2.990e-08 | 5.329e+04 | 1.501e-12
1.198e-01 | 2.926e-03 | 9.791e-01 | 1.473e-01 | 1.656e+01 | 1.530e-03 | 9.320e+02 | 2.352e-08 | 5.868e+04 | 1.197e-12
1.257e-01 | 3.302e-03 | 1.030e+00 | 1.506e-01 | 1.806e+01 | 1.216e-03 | 1.026e+03 | 1.852e-08 | 6.462e+04 | 9.548e-13
1.319e-01 | 3.726e-03 | 1.085e+00 | 1.534e-01 | 1.972e+01 | 9.622e-04 | 1.130e+03 | 1.461e-08 | 7.116e+04 | 7.615e-13
1.384e-01 | 4.204e-03 | 1.142e+00 | 1.556e-01 | 2.153e+01 | 7.589e-04 | 1.244e+03 | 1.153e-08 | 7.836e+04 | 6.074e-13
1.453e-01 | 4.742e-03 | 1.203e+00 | 1.572e-01 | 2.353e+01 | 5.964e-04 | 1.370e+03 | 9.105e-09 | 8.629e+04 | 4.845e-13
1.524e-01 | 5.348e-03 | 1.268e+00 | 1.582e-01 | 2.573e+01 | 4.670e-04 | 1.508e+03 | 7.196e-09 | 9.502e+04 | 3.865e-13
1.600e-01 | 6.029e-03 | 1.336e+00 | 1.586e-01 | 2.815e+01 | 3.646e-04 | 1.660e+03 | 5.691e-09 | 1.046e+05 | 3.083e-13
1.679e-01 | 6.795e-03 | 1.409e+00 | 1.584e-01 | 3.082e+01 | 2.839e-04 | 1.828e+03 | 4.503e-09 | 1.152e+05 | 2.459e-13
1.762e-01 | 7.655e-03 | 1.487e+00 | 1.578e-01 | 3.375e+01 | 2.205e-04 | 2.013e+03 | 3.565e-09 | 1.269e+05 | 1.962e-13
1.849e-01 | 8.620e-03 | 1.569e+00 | 1.562e-01 | 3.698e+01 | 1.709e-04 | 2.216e+03 | 2.824e-09 | 1.397e+05 | 1.565e-13
1.941e-01 | 9.703e-03 | 1.657e+00 | 1.538e-01 | 4.054e+01 | 1.322e-04 | 2.440e+03 | 2.239e-09 | 1.539e+05 | 1.249e-13
2.037e-01 | 1.092e-02 | 1.751e+00 | 1.504e-01 | 4.445e+01 | 1.020e-04 | 2.687e+03 | 1.775e-09 | 1.694e+05 | 9.962e-14
2.138e-01 | 1.227e-02 | 1.851e+00 | 1.460e-01 | 4.876e+01 | 7.861e-05 | 2.959e+03 | 1.408e-09 | 1.866e+05 | 7.947e-14
2.243e-01 | 1.379e-02 | 1.958e+00 | 1.407e-01 | 5.351e+01 | 6.045e-05 | 3.258e+03 | 1.117e-09 | 2.054e+05 | 6.341e-14
2.355e-01 | 1.549e-02 | 2.073e+00 | 1.344e-01 | 5.873e+01 | 4.642e-05 | 3.588e+03 | 8.870e-10 |  $\cdots$ |  $\cdots$
2.471e-01 | 1.738e-02 | 2.196e+00 | 1.273e-01 | 6.449e+01 | 3.560e-05 | 3.950e+03 | 7.043e-10 |  $\cdots$ |  $\cdots$
2.594e-01 | 1.948e-02 | 2.328e+00 | 1.195e-01 | 7.082e+01 | 2.727e-05 | 4.350e+03 | 5.594e-10 |  $\cdots$ |  $\cdots$
2.723e-01 | 2.182e-02 | 2.470e+00 | 1.111e-01 | 7.780e+01 | 2.086e-05 | 4.790e+03 | 4.445e-10 |  $\cdots$ |  $\cdots$
2.858e-01 | 2.442e-02 | 2.623e+00 | 1.022e-01 | 8.548e+01 | 1.595e-05 | 5.274e+03 | 3.533e-10 |  $\cdots$ |  $\cdots$
3.000e-01 | 2.729e-02 | 2.788e+00 | 9.325e-02 | 9.394e+01 | 1.218e-05 | 5.808e+03 | 2.808e-10 |  $\cdots$ |  $\cdots$
3.149e-01 | 3.047e-02 | 2.965e+00 | 8.441e-02 | 1.032e+02 | 9.299e-06 | 6.395e+03 | 2.233e-10 |  $\cdots$ |  $\cdots$
3.305e-01 | 3.397e-02 | 3.158e+00 | 7.574e-02 | 1.135e+02 | 7.091e-06 | 7.042e+03 | 1.776e-10 |  $\cdots$ |  $\cdots$
3.470e-01 | 3.782e-02 | 3.366e+00 | 6.736e-02 | 1.248e+02 | 5.404e-06 | 7.754e+03 | 1.413e-10 |  $\cdots$ |  $\cdots$
3.642e-01 | 4.202e-02 | 3.591e+00 | 5.940e-02 | 1.372e+02 | 4.117e-06 | 8.539e+03 | 1.124e-10 |  $\cdots$ |  $\cdots$
3.824e-01 | 4.657e-02 | 3.836e+00 | 5.194e-02 | 1.509e+02 | 3.136e-06 | 9.403e+03 | 8.946e-11 |  $\cdots$ |  $\cdots$
4.015e-01 | 5.143e-02 | 4.102e+00 | 4.505e-02 | 1.660e+02 | 2.389e-06 | 1.035e+04 | 7.121e-11 |  $\cdots$ |  $\cdots$
4.215e-01 | 5.654e-02 | 4.391e+00 | 3.880e-02 | 1.826e+02 | 1.821e-06 | 1.140e+04 | 5.669e-11 |  $\cdots$ |  $\cdots$
4.426e-01 | 6.191e-02 | 4.706e+00 | 3.322e-02 | 2.009e+02 | 1.389e-06 | 1.255e+04 | 4.514e-11 |  $\cdots$ |  $\cdots$
4.647e-01 | 6.747e-02 | 5.049e+00 | 2.830e-02 | 2.210e+02 | 1.060e-06 | 1.383e+04 | 3.595e-11 |  $\cdots$ |  $\cdots$
4.880e-01 | 7.321e-02 | 5.424e+00 | 2.400e-02 | 2.432e+02 | 8.097e-07 | 1.522e+04 | 2.863e-11 |  $\cdots$ |  $\cdots$
5.125e-01 | 7.907e-02 | 5.833e+00 | 2.027e-02 | 2.676e+02 | 6.196e-07 | 1.676e+04 | 2.281e-11 |  $\cdots$ |  $\cdots$
5.383e-01 | 8.500e-02 | 6.280e+00 | 1.707e-02 | 2.945e+02 | 4.750e-07 | 1.846e+04 | 1.817e-11 |  $\cdots$ |  $\cdots$

Note. — Differential Intensity units: (m2 s sr GV)-1.

Table 9: $Z=13$ – Default aluminum LIS

$E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential
---|---|---|---|---|---|---|---|---|---
GeV/n | intensity | GeV/n | intensity | GeV/n | intensity | GeV/n | intensity | GeV/n | intensity
1.000e-03 | 6.256e-02 | 6.309e-02 | 7.060e-01 | 3.981e+00 | 1.179e-02 | 2.512e+02 | 2.122e-07 | 1.585e+04 | 9.680e-12
1.101e-03 | 6.811e-02 | 6.948e-02 | 7.058e-01 | 4.384e+00 | 9.592e-03 | 2.766e+02 | 1.655e-07 | 1.745e+04 | 7.717e-12
1.213e-03 | 7.329e-02 | 7.651e-02 | 7.037e-01 | 4.827e+00 | 7.776e-03 | 3.046e+02 | 1.294e-07 | 1.922e+04 | 6.153e-12
1.335e-03 | 7.886e-02 | 8.425e-02 | 6.996e-01 | 5.315e+00 | 6.275e-03 | 3.354e+02 | 1.013e-07 | 2.116e+04 | 4.906e-12
1.470e-03 | 8.485e-02 | 9.277e-02 | 6.935e-01 | 5.853e+00 | 5.042e-03 | 3.693e+02 | 7.946e-08 | 2.330e+04 | 3.912e-12
1.619e-03 | 9.128e-02 | 1.022e-01 | 6.855e-01 | 6.446e+00 | 4.034e-03 | 4.067e+02 | 6.241e-08 | 2.566e+04 | 3.120e-12
1.783e-03 | 9.820e-02 | 1.125e-01 | 6.756e-01 | 7.098e+00 | 3.215e-03 | 4.478e+02 | 4.908e-08 | 2.825e+04 | 2.488e-12
1.963e-03 | 1.056e-01 | 1.239e-01 | 6.636e-01 | 7.816e+00 | 2.552e-03 | 4.931e+02 | 3.864e-08 | 3.111e+04 | 1.984e-12
2.162e-03 | 1.136e-01 | 1.364e-01 | 6.497e-01 | 8.607e+00 | 2.019e-03 | 5.430e+02 | 3.047e-08 | 3.426e+04 | 1.582e-12
2.381e-03 | 1.222e-01 | 1.502e-01 | 6.340e-01 | 9.478e+00 | 1.591e-03 | 5.980e+02 | 2.404e-08 | 3.773e+04 | 1.262e-12
2.622e-03 | 1.313e-01 | 1.654e-01 | 6.168e-01 | 1.044e+01 | 1.250e-03 | 6.585e+02 | 1.899e-08 | 4.155e+04 | 1.007e-12
2.887e-03 | 1.412e-01 | 1.822e-01 | 5.982e-01 | 1.149e+01 | 9.785e-04 | 7.251e+02 | 1.500e-08 | 4.575e+04 | 8.030e-13
3.179e-03 | 1.517e-01 | 2.006e-01 | 5.785e-01 | 1.266e+01 | 7.638e-04 | 7.985e+02 | 1.186e-08 | 5.038e+04 | 6.405e-13
3.501e-03 | 1.629e-01 | 2.209e-01 | 5.578e-01 | 1.394e+01 | 5.946e-04 | 8.793e+02 | 9.385e-09 | 5.548e+04 | 5.110e-13
3.855e-03 | 1.750e-01 | 2.432e-01 | 5.366e-01 | 1.535e+01 | 4.618e-04 | 9.682e+02 | 7.429e-09 | 6.109e+04 | 4.076e-13
4.245e-03 | 1.878e-01 | 2.678e-01 | 5.141e-01 | 1.690e+01 | 3.579e-04 | 1.066e+03 | 5.884e-09 | 6.727e+04 | 3.252e-13
4.675e-03 | 2.015e-01 | 2.949e-01 | 4.901e-01 | 1.861e+01 | 2.768e-04 | 1.174e+03 | 4.664e-09 | 7.408e+04 | 2.594e-13
5.148e-03 | 2.161e-01 | 3.248e-01 | 4.647e-01 | 2.049e+01 | 2.137e-04 | 1.293e+03 | 3.697e-09 | 8.157e+04 | 2.070e-13
5.669e-03 | 2.316e-01 | 3.577e-01 | 4.380e-01 | 2.257e+01 | 1.646e-04 | 1.424e+03 | 2.933e-09 | 8.983e+04 | 1.651e-13
6.242e-03 | 2.481e-01 | 3.938e-01 | 4.102e-01 | 2.485e+01 | 1.266e-04 | 1.568e+03 | 2.327e-09 | 9.892e+04 | 1.317e-13
6.874e-03 | 2.656e-01 | 4.337e-01 | 3.816e-01 | 2.736e+01 | 9.719e-05 | 1.726e+03 | 1.847e-09 |  $\cdots$ |  $\cdots$
7.569e-03 | 2.841e-01 | 4.776e-01 | 3.522e-01 | 3.013e+01 | 7.454e-05 | 1.901e+03 | 1.466e-09 |  $\cdots$ |  $\cdots$
8.335e-03 | 3.036e-01 | 5.259e-01 | 3.226e-01 | 3.318e+01 | 5.709e-05 | 2.093e+03 | 1.165e-09 |  $\cdots$ |  $\cdots$
9.179e-03 | 3.243e-01 | 5.791e-01 | 2.931e-01 | 3.654e+01 | 4.369e-05 | 2.305e+03 | 9.252e-10 |  $\cdots$ |  $\cdots$
1.011e-02 | 3.460e-01 | 6.377e-01 | 2.640e-01 | 4.023e+01 | 3.340e-05 | 2.539e+03 | 7.352e-10 |  $\cdots$ |  $\cdots$
1.113e-02 | 3.688e-01 | 7.022e-01 | 2.359e-01 | 4.431e+01 | 2.551e-05 | 2.795e+03 | 5.844e-10 |  $\cdots$ |  $\cdots$
1.226e-02 | 3.926e-01 | 7.733e-01 | 2.096e-01 | 4.879e+01 | 1.947e-05 | 3.078e+03 | 4.646e-10 |  $\cdots$ |  $\cdots$
1.350e-02 | 4.175e-01 | 8.515e-01 | 1.848e-01 | 5.373e+01 | 1.485e-05 | 3.390e+03 | 3.695e-10 |  $\cdots$ |  $\cdots$
1.486e-02 | 4.433e-01 | 9.377e-01 | 1.617e-01 | 5.916e+01 | 1.131e-05 | 3.733e+03 | 2.939e-10 |  $\cdots$ |  $\cdots$
1.637e-02 | 4.698e-01 | 1.033e+00 | 1.405e-01 | 6.515e+01 | 8.618e-06 | 4.110e+03 | 2.338e-10 |  $\cdots$ |  $\cdots$
1.802e-02 | 4.968e-01 | 1.137e+00 | 1.211e-01 | 7.174e+01 | 6.564e-06 | 4.526e+03 | 1.861e-10 |  $\cdots$ |  $\cdots$
1.985e-02 | 5.235e-01 | 1.252e+00 | 1.037e-01 | 7.900e+01 | 5.001e-06 | 4.984e+03 | 1.481e-10 |  $\cdots$ |  $\cdots$
2.185e-02 | 5.492e-01 | 1.379e+00 | 8.831e-02 | 8.699e+01 | 3.811e-06 | 5.489e+03 | 1.179e-10 |  $\cdots$ |  $\cdots$
2.406e-02 | 5.738e-01 | 1.518e+00 | 7.479e-02 | 9.580e+01 | 2.906e-06 | 6.044e+03 | 9.387e-11 |  $\cdots$ |  $\cdots$
2.650e-02 | 5.969e-01 | 1.672e+00 | 6.311e-02 | 1.055e+02 | 2.217e-06 | 6.656e+03 | 7.475e-11 |  $\cdots$ |  $\cdots$
2.918e-02 | 6.182e-01 | 1.841e+00 | 5.306e-02 | 1.162e+02 | 1.694e-06 | 7.329e+03 | 5.953e-11 |  $\cdots$ |  $\cdots$
3.213e-02 | 6.376e-01 | 2.027e+00 | 4.446e-02 | 1.279e+02 | 1.296e-06 | 8.071e+03 | 4.742e-11 |  $\cdots$ |  $\cdots$
3.539e-02 | 6.548e-01 | 2.233e+00 | 3.721e-02 | 1.409e+02 | 9.936e-07 | 8.887e+03 | 3.778e-11 |  $\cdots$ |  $\cdots$

Note. — Differential Intensity units: (m2 s sr GeV/n)-1.

Table 10: $Z=13$ – Total aluminum LIS

Rigidity | Differential | Rigidity | Differential | Rigidity | Differential | Rigidity | Differential | Rigidity | Differential
---|---|---|---|---|---|---|---|---|---
GV | intensity | GV | intensity | GV | intensity | GV | intensity | GV | intensity
8.967e-02 | 1.397e-03 | 7.240e-01 | 1.197e-01 | 1.002e+01 | 5.812e-03 | 5.236e+02 | 1.016e-07 | 3.291e+04 | 4.657e-12
9.410e-02 | 1.596e-03 | 7.610e-01 | 1.250e-01 | 1.087e+01 | 4.701e-03 | 5.763e+02 | 7.925e-08 | 3.624e+04 | 3.713e-12
9.875e-02 | 1.802e-03 | 8.000e-01 | 1.301e-01 | 1.180e+01 | 3.791e-03 | 6.345e+02 | 6.195e-08 | 3.991e+04 | 2.961e-12
1.036e-01 | 2.034e-03 | 8.412e-01 | 1.350e-01 | 1.283e+01 | 3.047e-03 | 6.985e+02 | 4.852e-08 | 4.395e+04 | 2.361e-12
1.087e-01 | 2.297e-03 | 8.847e-01 | 1.395e-01 | 1.396e+01 | 2.440e-03 | 7.689e+02 | 3.807e-08 | 4.840e+04 | 1.882e-12
1.141e-01 | 2.593e-03 | 9.306e-01 | 1.438e-01 | 1.520e+01 | 1.947e-03 | 8.466e+02 | 2.990e-08 | 5.329e+04 | 1.501e-12
1.198e-01 | 2.927e-03 | 9.791e-01 | 1.476e-01 | 1.656e+01 | 1.549e-03 | 9.320e+02 | 2.352e-08 | 5.868e+04 | 1.197e-12
1.257e-01 | 3.303e-03 | 1.030e+00 | 1.509e-01 | 1.806e+01 | 1.227e-03 | 1.026e+03 | 1.852e-08 | 6.462e+04 | 9.548e-13
1.319e-01 | 3.727e-03 | 1.085e+00 | 1.537e-01 | 1.972e+01 | 9.696e-04 | 1.130e+03 | 1.461e-08 | 7.116e+04 | 7.615e-13
1.384e-01 | 4.206e-03 | 1.142e+00 | 1.560e-01 | 2.153e+01 | 7.635e-04 | 1.244e+03 | 1.153e-08 | 7.836e+04 | 6.074e-13
1.453e-01 | 4.744e-03 | 1.203e+00 | 1.577e-01 | 2.353e+01 | 5.993e-04 | 1.370e+03 | 9.105e-09 | 8.629e+04 | 4.845e-13
1.524e-01 | 5.350e-03 | 1.268e+00 | 1.588e-01 | 2.573e+01 | 4.688e-04 | 1.508e+03 | 7.196e-09 | 9.502e+04 | 3.865e-13
1.600e-01 | 6.031e-03 | 1.336e+00 | 1.593e-01 | 2.815e+01 | 3.657e-04 | 1.660e+03 | 5.691e-09 | 1.046e+05 | 3.083e-13
1.679e-01 | 6.797e-03 | 1.409e+00 | 1.592e-01 | 3.082e+01 | 2.846e-04 | 1.828e+03 | 4.503e-09 | 1.152e+05 | 2.459e-13
1.762e-01 | 7.657e-03 | 1.487e+00 | 1.586e-01 | 3.375e+01 | 2.210e-04 | 2.013e+03 | 3.565e-09 | 1.269e+05 | 1.962e-13
1.849e-01 | 8.623e-03 | 1.569e+00 | 1.572e-01 | 3.698e+01 | 1.712e-04 | 2.216e+03 | 2.824e-09 | 1.397e+05 | 1.565e-13
1.941e-01 | 9.706e-03 | 1.657e+00 | 1.549e-01 | 4.054e+01 | 1.324e-04 | 2.440e+03 | 2.239e-09 | 1.539e+05 | 1.249e-13
2.037e-01 | 1.092e-02 | 1.751e+00 | 1.517e-01 | 4.445e+01 | 1.021e-04 | 2.687e+03 | 1.775e-09 | 1.694e+05 | 9.962e-14
2.138e-01 | 1.228e-02 | 1.851e+00 | 1.476e-01 | 4.876e+01 | 7.867e-05 | 2.959e+03 | 1.408e-09 | 1.866e+05 | 7.947e-14
2.243e-01 | 1.380e-02 | 1.958e+00 | 1.426e-01 | 5.351e+01 | 6.049e-05 | 3.258e+03 | 1.117e-09 | 2.054e+05 | 6.341e-14
2.355e-01 | 1.549e-02 | 2.073e+00 | 1.367e-01 | 5.873e+01 | 4.644e-05 | 3.588e+03 | 8.870e-10 |  $\cdots$ |  $\cdots$
2.471e-01 | 1.738e-02 | 2.196e+00 | 1.300e-01 | 6.449e+01 | 3.561e-05 | 3.950e+03 | 7.043e-10 |  $\cdots$ |  $\cdots$
2.594e-01 | 1.949e-02 | 2.328e+00 | 1.227e-01 | 7.082e+01 | 2.728e-05 | 4.350e+03 | 5.594e-10 |  $\cdots$ |  $\cdots$
2.723e-01 | 2.183e-02 | 2.470e+00 | 1.150e-01 | 7.780e+01 | 2.087e-05 | 4.790e+03 | 4.445e-10 |  $\cdots$ |  $\cdots$
2.858e-01 | 2.443e-02 | 2.623e+00 | 1.070e-01 | 8.548e+01 | 1.595e-05 | 5.274e+03 | 3.533e-10 |  $\cdots$ |  $\cdots$
3.000e-01 | 2.730e-02 | 2.788e+00 | 9.897e-02 | 9.394e+01 | 1.219e-05 | 5.808e+03 | 2.808e-10 |  $\cdots$ |  $\cdots$
3.149e-01 | 3.048e-02 | 2.965e+00 | 9.132e-02 | 1.032e+02 | 9.300e-06 | 6.395e+03 | 2.233e-10 |  $\cdots$ |  $\cdots$
3.305e-01 | 3.398e-02 | 3.158e+00 | 8.404e-02 | 1.135e+02 | 7.092e-06 | 7.042e+03 | 1.776e-10 |  $\cdots$ |  $\cdots$
3.470e-01 | 3.783e-02 | 3.366e+00 | 7.722e-02 | 1.248e+02 | 5.404e-06 | 7.754e+03 | 1.413e-10 |  $\cdots$ |  $\cdots$
3.642e-01 | 4.204e-02 | 3.591e+00 | 7.083e-02 | 1.372e+02 | 4.117e-06 | 8.539e+03 | 1.124e-10 |  $\cdots$ |  $\cdots$
3.824e-01 | 4.659e-02 | 3.836e+00 | 6.448e-02 | 1.509e+02 | 3.136e-06 | 9.403e+03 | 8.946e-11 |  $\cdots$ |  $\cdots$
4.015e-01 | 5.146e-02 | 4.102e+00 | 5.738e-02 | 1.660e+02 | 2.389e-06 | 1.035e+04 | 7.121e-11 |  $\cdots$ |  $\cdots$
4.215e-01 | 5.657e-02 | 4.391e+00 | 4.941e-02 | 1.826e+02 | 1.821e-06 | 1.140e+04 | 5.669e-11 |  $\cdots$ |  $\cdots$
4.426e-01 | 6.194e-02 | 4.706e+00 | 4.167e-02 | 2.009e+02 | 1.389e-06 | 1.255e+04 | 4.514e-11 |  $\cdots$ |  $\cdots$
4.647e-01 | 6.750e-02 | 5.049e+00 | 3.476e-02 | 2.210e+02 | 1.060e-06 | 1.383e+04 | 3.595e-11 |  $\cdots$ |  $\cdots$
4.880e-01 | 7.325e-02 | 5.424e+00 | 2.881e-02 | 2.432e+02 | 8.097e-07 | 1.522e+04 | 2.863e-11 |  $\cdots$ |  $\cdots$
5.125e-01 | 7.911e-02 | 5.833e+00 | 2.378e-02 | 2.676e+02 | 6.196e-07 | 1.676e+04 | 2.281e-11 |  $\cdots$ |  $\cdots$
5.383e-01 | 8.505e-02 | 6.280e+00 | 1.959e-02 | 2.945e+02 | 4.750e-07 | 1.846e+04 | 1.817e-11 |  $\cdots$ |  $\cdots$

Note. — Differential Intensity units: (m2 s sr GV)-1.

Table 11: $Z=13$ – Total aluminum LIS

$E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential | $E_{\rm kin}$ | Differential
---|---|---|---|---|---|---|---|---|---
GeV/n | intensity | GeV/n | intensity | GeV/n | intensity | GeV/n | intensity | GeV/n | intensity
1.000e-03 | 6.258e-02 | 6.309e-02 | 7.068e-01 | 3.981e+00 | 1.234e-02 | 2.512e+02 | 2.122e-07 | 1.585e+04 | 9.680e-12
1.101e-03 | 6.813e-02 | 6.948e-02 | 7.066e-01 | 4.384e+00 | 9.952e-03 | 2.766e+02 | 1.655e-07 | 1.745e+04 | 7.717e-12
1.213e-03 | 7.331e-02 | 7.651e-02 | 7.045e-01 | 4.827e+00 | 8.010e-03 | 3.046e+02 | 1.294e-07 | 1.922e+04 | 6.153e-12
1.335e-03 | 7.888e-02 | 8.425e-02 | 7.005e-01 | 5.315e+00 | 6.426e-03 | 3.354e+02 | 1.013e-07 | 2.116e+04 | 4.906e-12
1.470e-03 | 8.487e-02 | 9.277e-02 | 6.946e-01 | 5.853e+00 | 5.138e-03 | 3.693e+02 | 7.946e-08 | 2.330e+04 | 3.912e-12
1.619e-03 | 9.131e-02 | 1.022e-01 | 6.867e-01 | 6.446e+00 | 4.095e-03 | 4.067e+02 | 6.241e-08 | 2.566e+04 | 3.120e-12
1.783e-03 | 9.823e-02 | 1.125e-01 | 6.768e-01 | 7.098e+00 | 3.254e-03 | 4.478e+02 | 4.908e-08 | 2.825e+04 | 2.488e-12
1.963e-03 | 1.057e-01 | 1.239e-01 | 6.650e-01 | 7.816e+00 | 2.577e-03 | 4.931e+02 | 3.864e-08 | 3.111e+04 | 1.984e-12
2.162e-03 | 1.136e-01 | 1.364e-01 | 6.512e-01 | 8.607e+00 | 2.034e-03 | 5.430e+02 | 3.047e-08 | 3.426e+04 | 1.582e-12
2.381e-03 | 1.222e-01 | 1.502e-01 | 6.357e-01 | 9.478e+00 | 1.601e-03 | 5.980e+02 | 2.404e-08 | 3.773e+04 | 1.262e-12
2.622e-03 | 1.314e-01 | 1.654e-01 | 6.187e-01 | 1.044e+01 | 1.256e-03 | 6.585e+02 | 1.899e-08 | 4.155e+04 | 1.007e-12
2.887e-03 | 1.412e-01 | 1.822e-01 | 6.003e-01 | 1.149e+01 | 9.823e-04 | 7.251e+02 | 1.500e-08 | 4.575e+04 | 8.030e-13
3.179e-03 | 1.517e-01 | 2.006e-01 | 5.808e-01 | 1.266e+01 | 7.661e-04 | 7.985e+02 | 1.186e-08 | 5.038e+04 | 6.405e-13
3.501e-03 | 1.630e-01 | 2.209e-01 | 5.604e-01 | 1.394e+01 | 5.961e-04 | 8.793e+02 | 9.385e-09 | 5.548e+04 | 5.110e-13
3.855e-03 | 1.750e-01 | 2.432e-01 | 5.395e-01 | 1.535e+01 | 4.627e-04 | 9.682e+02 | 7.429e-09 | 6.109e+04 | 4.076e-13
4.245e-03 | 1.879e-01 | 2.678e-01 | 5.174e-01 | 1.690e+01 | 3.585e-04 | 1.066e+03 | 5.884e-09 | 6.727e+04 | 3.252e-13
4.675e-03 | 2.016e-01 | 2.949e-01 | 4.938e-01 | 1.861e+01 | 2.772e-04 | 1.174e+03 | 4.664e-09 | 7.408e+04 | 2.594e-13
5.148e-03 | 2.162e-01 | 3.248e-01 | 4.689e-01 | 2.049e+01 | 2.139e-04 | 1.293e+03 | 3.697e-09 | 8.157e+04 | 2.070e-13
5.669e-03 | 2.317e-01 | 3.577e-01 | 4.429e-01 | 2.257e+01 | 1.647e-04 | 1.424e+03 | 2.933e-09 | 8.983e+04 | 1.651e-13
6.242e-03 | 2.482e-01 | 3.938e-01 | 4.159e-01 | 2.485e+01 | 1.267e-04 | 1.568e+03 | 2.327e-09 | 9.892e+04 | 1.317e-13
6.874e-03 | 2.657e-01 | 4.337e-01 | 3.881e-01 | 2.736e+01 | 9.724e-05 | 1.726e+03 | 1.847e-09 |  $\cdots$ |  $\cdots$
7.569e-03 | 2.842e-01 | 4.776e-01 | 3.598e-01 | 3.013e+01 | 7.456e-05 | 1.901e+03 | 1.466e-09 |  $\cdots$ |  $\cdots$
8.335e-03 | 3.038e-01 | 5.259e-01 | 3.314e-01 | 3.318e+01 | 5.711e-05 | 2.093e+03 | 1.165e-09 |  $\cdots$ |  $\cdots$
9.179e-03 | 3.244e-01 | 5.791e-01 | 3.034e-01 | 3.654e+01 | 4.370e-05 | 2.305e+03 | 9.252e-10 |  $\cdots$ |  $\cdots$
1.011e-02 | 3.461e-01 | 6.377e-01 | 2.762e-01 | 4.023e+01 | 3.340e-05 | 2.539e+03 | 7.352e-10 |  $\cdots$ |  $\cdots$
1.113e-02 | 3.689e-01 | 7.022e-01 | 2.504e-01 | 4.431e+01 | 2.551e-05 | 2.795e+03 | 5.844e-10 |  $\cdots$ |  $\cdots$
1.226e-02 | 3.928e-01 | 7.733e-01 | 2.267e-01 | 4.879e+01 | 1.947e-05 | 3.078e+03 | 4.646e-10 |  $\cdots$ |  $\cdots$
1.350e-02 | 4.176e-01 | 8.515e-01 | 2.050e-01 | 5.373e+01 | 1.485e-05 | 3.390e+03 | 3.695e-10 |  $\cdots$ |  $\cdots$
1.486e-02 | 4.435e-01 | 9.377e-01 | 1.853e-01 | 5.916e+01 | 1.131e-05 | 3.733e+03 | 2.939e-10 |  $\cdots$ |  $\cdots$
1.637e-02 | 4.701e-01 | 1.033e+00 | 1.674e-01 | 6.515e+01 | 8.619e-06 | 4.110e+03 | 2.338e-10 |  $\cdots$ |  $\cdots$
1.802e-02 | 4.970e-01 | 1.137e+00 | 1.503e-01 | 7.174e+01 | 6.565e-06 | 4.526e+03 | 1.861e-10 |  $\cdots$ |  $\cdots$
1.985e-02 | 5.238e-01 | 1.252e+00 | 1.321e-01 | 7.900e+01 | 5.001e-06 | 4.984e+03 | 1.481e-10 |  $\cdots$ |  $\cdots$
2.185e-02 | 5.495e-01 | 1.379e+00 | 1.124e-01 | 8.699e+01 | 3.811e-06 | 5.489e+03 | 1.179e-10 |  $\cdots$ |  $\cdots$
2.406e-02 | 5.741e-01 | 1.518e+00 | 9.378e-02 | 9.580e+01 | 2.906e-06 | 6.044e+03 | 9.387e-11 |  $\cdots$ |  $\cdots$
2.650e-02 | 5.972e-01 | 1.672e+00 | 7.750e-02 | 1.055e+02 | 2.217e-06 | 6.656e+03 | 7.475e-11 |  $\cdots$ |  $\cdots$
2.918e-02 | 6.186e-01 | 1.841e+00 | 6.368e-02 | 1.162e+02 | 1.694e-06 | 7.329e+03 | 5.953e-11 |  $\cdots$ |  $\cdots$
3.213e-02 | 6.380e-01 | 2.027e+00 | 5.215e-02 | 1.279e+02 | 1.296e-06 | 8.071e+03 | 4.742e-11 |  $\cdots$ |  $\cdots$
3.539e-02 | 6.553e-01 | 2.233e+00 | 4.269e-02 | 1.409e+02 | 9.936e-07 | 8.887e+03 | 3.778e-11 |  $\cdots$ |  $\cdots$

Note. — Differential Intensity units: (m2 s sr GeV/n)-1.