content_image/432124.jpg content_image/432134.jpg content_image/432133.jpg content_image/432130.jpg content_image/432129.jpg content_image/432123.jpg content_image/432131.jpg
where the dimensionless variables are defined as x 2 ≡ u $^{2}$/v2 0 and η 2 ≡ v 2 $_{⊙}$/v 2 $_{0}$, v$_{0}$ =220 km/s is the average local circular velocity which is related to DM velocity dispersion ¯ v as v 2 $_{0}$/ ¯ v 2 = 2 / 3. 

Taking into account the form factor suppression, the scattering function Ω( w ) is given by 

<img src='content_image/432124.jpg'>

where and we will omit it below, n is the number den- sity of the nuclei, Q is the energy transfer in the scattering, Q$_{max}$ ( Q$_{min}$ ) denotes the possible max- imum(minimum) energy transfer and Q$_{cap}$ is the minimal energy transfer which requires DM could be captured only with a velocity less than v , and we define lower limit of integration as Q ′ min = max ( Q$_{cap}$, Q$_{min}$ ). These energy transfer func- tions are given by [14] 

<img src='content_image/432134.jpg'>

<img src='content_image/432133.jpg'>

and one has to impose the condition 

for the capture to happen. σ SI ( SD ) is the inelastic SI (SD) cross section and F ( Q ) is the SI form fac- tor. The term σ $^{SD}$S ( Q ) will be decomposed into SD cross sections and SD form factors for different spin J later. The inelastic SI (SD) cross section is given by 

<img src='content_image/432130.jpg'>

<img src='content_image/432129.jpg'>

where A is the atomic mass number, Z is the charge of the nucleus, µ$_{χn}$ ≡ m$_{χ}$m$_{n}$/ ( m$_{χ}$ + m$_{n}$ ) is the DM-nucleon reduced mass, f$_{p}$ ( f$_{n}$ ) is the SI DM coupling to protons (neutrons) and we normalize it to f$_{p}$ = f$_{n}$ , σ SI n is the DM-nucleon SI scatter- ing cross section. We can see that if f$_{p}$ ≃ f$_{n}$ , the DM-nuclei SI cross section is proportional to A $^{2}$. Moreover, if m$_{χ}$ >> m$_{N}$ , the reduced mass µ m$_{N}$ will provide another enhancement factor of A $^{2}$. We can expect heavy nuclei with large abundance in the Sun, such as Fe, to play a more important role in the capture process. For the SI form factor, we use the Helm form factor as, 

<img src='content_image/432123.jpg'>

with nuclear skin thickness s = 1 fm, effective nu- clear radius r = √$_{R}$ 2 − 5 s $^{2}$, R = 1 . 2 A 1 / 3 fm, and momentum transfer k = √$_{2}$$_{m}$$_{N}$$_{Q}$$_{.}$ 

The SD cross section with form factor for spin- non-zero nuclei is 

<img src='content_image/432131.jpg'>

where − spin and isovector spin couplings, a$_{p}$ and a$_{n}$ are the effective spin DM-proton and DM-neutron cou- plings which we assume a$_{p}$ =1 and a$_{n}$ =0 in the below, S$_{00}$ ( k ) , S$_{00}$ ( k ) , S$_{01}$ ( k ) are nuclear structure functions. In many nuclei models, the nuclear spin is carried by the unpaired nucleons. Therefore we focus on the nuclei with an odd nucleon number in the SD scattering process. In the calculation we use the nuclear structure functions provided by Ref. [21]. 

The capture rate of 50 GeV DM due to differ- ent species of nuclei in the Sun as a function of the mass splitting δ is calculated for both iSI and iSD DM. The results depend on some astrophysi- cal parameters and as a reference point, we choose ρ$_{χ}$ = 0 . 3 GeV cm − $^{3}$, v$_{⊙}$ = 220 km/s and v$_{0}$ = 220 km/s in the Maxwell-Boltzmann distribution. The element abundances are given by Ref. [22] in the form of ϵ$_{i}$ = log( n$_{i}$/n$_{H}$ ) + 12 and the mass frac- tion of hydrogen (helium) is 0.7381 (0.2485). In the calculation for SI capture, we pick out the species with ϵ$_{i}$ > 6. For SD capture, we calculate $^{1}$H, $^{19}$F, $^{23}$Na, $^{27}$Al, $^{29}$Si and $^{39}$K. As we can see in FIG. 1 and 2, Fe and Al provide most contributions to the iSI and iSD scattering processes respectively and 

