Conceptual design of a compact high gradient quadrupole magnet of varying strength using permanent magnets

A concept is presented to design magnets using cylindrical-shaped permanent-magnet blocks, where various types of magnetic fields can be produced by either rotating or varying the size of the magnetic blocks within a given mechanical structure. A general method is introduced to calculate the 3D magnetic field produced by a set of permanent magnets. An analytical expression of the 2D field and the condition to generate various magnetic fields like dipole, quadrupole, and sextupole are derived. Using the 2D result as a starting point, a computer code is developed to get the optimum orientation of the magnets to obtain the user-specific target field profile over a given volume in 3D. Designs of two quadrupole magnets are presented, one using 12 and the other using 24 permanent-magnet blocks. Variation of the quadrupole strength is achieved using tuning coils of a suitable current density and specially designed end tubes. A new concept is introduced to reduce the integrated quadrupole field strength by inserting two hollow cylindrical tubes made of iron, one at each end. This will not affect the field gradient at the center but reduce the integrated field strength by shielding the magnetic field near the ends where the tubes are inserted. The advantages of this scheme are that it is easy to implement, the magnetic axis will not shift, and it will prevent interference with nearby devices. Around 40% integrated field variation is achieved using this method in the present example. To get a realistic estimation of the field quality, a complete 3D model using a nonlinear $B\text{\ensuremath{-}}H$ curve is also studied using a finite-element-based computer code. An example to generate around an $80\text{ }\text{ }\mathrm{T}/\mathrm{m}$ quadrupole field gradient is also presented.