Spin susceptibility in superconductors

The wave-vector-dependent spin susceptibility $\chi \mathrm{}\left(q\right)\mathrm{}$ of a superconductor in one, two, and three dimensions is calculated numerically. These calculations show that for large $q\sim k_F$, where $k_F$ is the Fermi momentum, $\chi \mathrm{}\left(q\right)\mathrm{}$ is essentially the same as that in the normal state, indicating that for distances $r\ll {\xi }_0$ (${\xi }_0$ the coherence length), Ruderman-Kittel-Kasuya-Yosida coupling between magnetic atoms is essentially unchanged by superconductivity of the system. In one and two dimensions we find a reduction in $\chi \mathrm{}\left(q\right)\mathrm{}$ at very small $q$, whose structure affects any possible cryptomagnetic order of the type suggested by Anderson and Suhl for the case of three dimensions. In three dimensions our calculation verifies the validity of the approximate $\chi \mathrm{}\left(q\right)\mathrm{}$ of Anderson and Suhl in the appropriate $q$ region. For small $q$, $\chi \mathrm{}\left(q\right)\mathrm{}$ shows a much stronger dependence on $q$ for lower dimensions than in the three-dimensional case, which has a bearing on the range of magnetic ordering, and may be relevant to rare-earth materials with Fermi-surface "nesting."