sections by factors of $A^2$, making it advantageous to use targets consisting of heavy nuclei. This enhancement is due to the fact that the WIMP wavelength is of order the size of the nucleus, thus the scattering amplitudes on individual nucleons add coherently. The spin-dependent contribution, in contrast, couples to the spin of the target nuclei and scales with $J (J+1)$. Naively, this could be considered a coherent subtraction of amplitudes of opposite signs of pairs of nucleons. As the current spin-dependent scattering constraints are not strong enough to test many dark matter models, we devote our attention primarily to the process of spin-independent scattering.

The spin-independent WIMP-nucleus elastic scattering cross section is given by: $$\label{sig}
\sigma \approx \frac{4 m^2_{X} m^2_{T}}{\pi (m_{X}+m_T)^2} [Z f_p + (A-Z) f_n]^2,$$ where $m_T$ is the mass of the target nucleus, $m_X$ is the WIMP’s mass and $Z$ and $A$ are the atomic number and atomic mass of the nucleus. $f_p$ and $f_n$ are the WIMP’s couplings to protons and neutrons, given by: $$f_{p,n}=\sum_{q=u,d,s} f^{(p,n)}_{T_q} a_q \frac{m_{p,n}}{m_q} + \frac{2}{27} f^{(p,n)}_{TG} \sum_{q=c,b,t} a_q  \frac{m_{p,n}}{m_q},
\label{feqn}$$ where $a_q$ are the WIMP-quark couplings and $f^{(p,n)}_{T_q}$ denote the quark content of the nucleon.

The first term in Eq. \[feqn\] 