with the level of the interstellar turbulence parameter w being 1. The spatial diffusion
coefficient is usually written in this form
D
xx
= βD
0
(R/R
0
)
δ
, (2.3)
with R and β being the rigidity and particle velocity divided by light speed respectively.
This transport equation is numerically solved based on given boundary conditions, that
is, the cosmic ray density ψ vanishes at the radius R
h
and the height z
0
of the cylindrical
diffusion halo.
The above key propagation parameters can be constrained by fitting the latest ratios
of nuclei, that is the Boron-to-Carbon ratio (B/C) and the Beryllium ratio (
10
Be/
9
Be).
We adopt the values of propagation parameters shown in table 1, determined by the B/C
and
10
Be/
9
Be data [6]. The benchmark model of the propagation we use is diffusion
reacceleration model. The
10
Be/
9
Be ratio data are sensitive to the diffusion halo size
z
0
. For a given halo size the diffusion coefficient as a function of momentum and the
reacceleration parameter are determined by B/C ratio data. The values in table 1 are
the best fit results. The diffusion reacceleration scenario is found to be best consistent
with the data compared with other propagation scenarios, such as the diffusion convection
model [27].
In eq. (2.1), the source term of cosmic ray species i can be described by the product
of the spatial distribution and the injection spectrum function
Q
i
(~r, p) = f(r, z)q
i
(p). (2.4)
For the spatial distribution of the primary cosmic rays we use the following supernova
remnants distribution
f(r, z) = f
0
r
r
a
exp
−b
r − r
r
exp
−
|z|
z
s
, (2.5)
where r
= 8.5 kpc is the distance between the Sun and the Galactic center, the height of
the Galactic disk is z
s
= 0.2 kpc, and the two parameters a and b are chosen to be 1.25 and
3.56, respectively [27]. We assume the following power law with one break for the injection
spectrum of various nuclei
q
i
∝
R/R
p
br
−ν
1
, R ≤ R
p
br
R/R
p
br
−ν
2
, R > R
p
br
. (2.6)
The corresponding injection parameters in eq. (2.6), i.e. rigidity break R
p
br
and power law
indexes ν
1
, ν
2
, can be determined by fitting the latest AMS-02 proton data [1]. We adopt
injection parameters obtained by performing such a fit in ref. [6]. The values of these
injection parameters are shown in table 1, together with the Fisk potential φ
i
(i = p, ¯p)
for solar modulation effect. The values of Fisk potential come from ref. [6]. They are the
approximate constants of the time-varying modulation form employed in the reference.
– 3 –