dipole moment F$_{2}$ contribution. We point out that this sizable F$_{2}$ contribution is possible in the non-relativistic scattering of WIMP and nucleus, leading to a several factor improvement in the exclusion plot. In the future refined direct DM search experiments, the possibility of DM magnetic moment of N can be probed with another independent information on its mass. For the dipole-dipole interaction contribution, the cross- section in the leading order becomes We thank J. H. Yoo for useful discussions. WSC, JHH, JEK and BK are supported in part by the Ko- rea Research Foundation, Grant No. KRF-2005-084- C00001. B.K. is also supported in addition by the FPRD of the BK21 program and the KICOS Grant No. K20732000011-07A0700-01110 of Ministry of Education and Science. IWK is supported by the U.S. Department of Energy under grant No. DE-FG02-95ER40896. where the dot product is over the SO(3) space and Tr is over the spin multiplicities. Here, if we use g$_{nuclei}$ = 2 Mµ/e √ s$_{N}$ ( s$_{N}$ + 1), we can see that the term propor- tional to µ 2 is independent of the nucleus spin. Here, if we use g$_{nuclei}$ = 2 Mµ/es$_{N}$ we see that the term involving µ 2 is proportional to ( s$_{N}$ + 1) / 3 s$_{N}$ . It confirms that our parametrization of F$_{2}$ in the text is appropriate. In the calculation of scattering cross-section of DM with target nuclei, we treated the nuclei as a spin 1 2 particle. However, many nuclei used in the direct de- tection of DM have spin different from 1 $_{2}$. Moreover, our parametrization of the magnetic moment using F$_{2}$ , in which spin 1 2 for nuclei has been assumed, should be matched to produce a realistic value. For that, we con- sider two body wave function, in non-relativistic quan- tum mechanics with non-local interaction, which is ob- tained after integrating out the photon field, where N $^{α}$( ⃗ x ) is the nuclear wave function with S ( N ) z = α and χ $^{i}$( ⃗ y ) is the DM wave function with S ( χ ) z = i . The Scr¨ odinger equation for this two body system reads where H$_{mo-di}$ and H$_{di-di}$ are the monopole–dipole and dipole–dipole interactions, respectively. Here, we ne- glected unimportant normalization for each terms be- cause we only need to match the dependency on the nu- cleon spin. Since the initial state of the problem is unpolar- ized, its density matrix is proportional to the iden- tity in (2 s$_{N}$ + 1)(2 s$_{χ}$ + 1)-dimensional space, that is, ρ = 1 / (2 s$_{N}$ + 1)(2 s$_{χ}$ + 1), where s$_{N}$ and s$_{χ}$ are the spin of nuclei and DM, respectively. Therefore, after averag- ing the polarizations, the cross-section is proportional to tr[ ρM $^{†}$M ], where M is M = H$_{int}$ + H$_{int}$ 1 E − H + iϵ H$_{int}$. (14) H Ψ = ( − ∇ 2 x 2 M − ∇ 2 y 2 m$_{N}$ ) Ψ + g$_{DM}$Ze 2 2 m$_{N}$ | ⃗ x − ⃗ y | ∇ x · ⃗ S ( N $^{)}$Ψ + g$_{DM}$g$_{nuclei}$e 2 2 m$_{N}$ 2 M | ⃗ x − ⃗ y | 3 ⃗ S ( N ) · ⃗ S ( χ $^{)}$Ψ = ( H$_{0}$ + H$_{mo-di}$ + H$_{di-di}$ )Ψ Ψ $^{αi}$( ⃗ x, ⃗ y ) = N $^{α}$( ⃗ x ) ⊗ χ $^{i}$( ⃗ y ) (13) Acknowledgments Appendix 6 σ ∝ tr[ ρM $^{†}$M ] ∝ | g$_{DM}$g$_{nuclei}$ | 2 (2 s$_{N}$ + 1)(2 s$_{χ}$ + 1) tr [ ⃗ S ( N ) · ⃗ S ( χ ) ⃗ S ( N ) · ⃗ S ( χ ) ] = | g$_{DM}$g$_{nuclei}$ | 2 9 s$_{N}$ ( s$_{N}$ + 1) s$_{χ}$ ( s$_{χ}$ + 1) (15)