Optimization and improvement of Halbach cylinder design
  In this paper we describe the results of a parameter survey of a 16 segmented Halbach cylinder in three dimensions in which the parameters internal radius, $r_{\mathrm{\scriptsize{in}}}$, external radius, $r_{\mathrm{\scriptsize{ex}}}$, and length, $L$, have been varied. Optimal values of $r_{\mathrm{\scriptsize{ex}}}$ and $L$ were found for a Halbach cylinder with the least possible volume of magnets with a given mean flux density in the cylinder bore. The volume of the cylinder bore could also be significantly increase by only slightly increasing the volume of the magnets, for a fixed mean flux density. Placing additional blocks of magnets on the end faces of the Halbach cylinder also improved the mean flux density in the cylinder bore, especially so for short Halbach cylinders with large $r_{\mathrm{\scriptsize{ex}}}$. Moreover magnetic cooling as an application for Halbach cylinders was considered. A magnetic cooling quality parameter, $\Lambda_{\mathrm{cool}}$, was introduced and results showed that this parameter was optimal for long Halbach cylinders with small $r_{\mathrm{\scriptsize{ex}}}$. Using the previously mentioned additional blocks of magnets can improve the parameter by as much as 15% as well as improve the homogeneity of the field in the cylinder bore. 
R. Bj{\o}rk, C. R. H. Bahl, A. Smith, N. Pryds
physics.ins-det