Antiferromagnetism in Simple Metals

The antiferromagnetic or spin-density-wave (SDW) state is studied for the case of a simple metal with the aid of a simplified electron-electron exchange interaction of the form $v\left(\mathbf{q}\right)=\frac{B}{({\lambda }^2+q^2)}$. As was found by Overhauser, the SDW state has lower energy than the normal state for sufficiently large values of $B$ and $\frac{1}{\lambda }$. A detailed analysis of the finite-amplitude SDW state is carried out. We find that the wave number $Q$ of the SDW state may be significantly less than $2k_F$ even for an interaction just strong enough for antiferromagnetism. The Fermi surface has an appreciable area of contact with the magnetic zone boundary. The phase transition from the paramagnetic to the antiferromagnetic state is first-order as a function of the coupling constant.