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The multi-bend achromat (MBA), which often serves as a building block formodern low-emittance storage rings, is composed of a repetition of unit cellswith optimized optical functions for low emittance in the achromat center, aswell as end cells for dispersion and optics matching to insertion devices.  In this work, we describe the simplest stable class of unit cells that arebased on a longitudinal Fourier expansion, transforming Hill equations toMathieu equations. The resulting cell class exhibits continuously changingdipolar and quadrupolar moments along the beam path. Although this elementarymodel is defined by only three parameters, it captures a significant amount ofnotions that are applied in the design of MBAs. This is especially interestingas Mathieu cells can be viewed as an elementary extension of Christofilos'original model of alternating-gradient focusing, while their sinusoidal bendingand focusing functions lend themselves to future applications in undulator-likestructures.  Mathieu cells can be used to estimate the range of reasonable cell tunes andput an emphasis on the combination of longitudinal gradient bending and reversebending, as well as on strong horizontal focusing to reach emittances lowerthan the classic theoretical minimum emittance cell. Furthermore, the lowestemittances in this model are accompanied by small absolute momentum compactionfactors.

The Mathieu unit cell as a template for low emittance lattices