The poloidal angle a magnetic field line advances per toroidal circuit -- the stellarator community's measure of field-line twist and the reciprocal of the tokamak safety factor.
The rotational transform, denoted by the Greek letter ι (iota) or sometimes written as ι̸ ("iota-bar," equal to ι/2π), measures how much a magnetic field line rotates in the poloidal direction for each complete toroidal transit around a torus. It is the fundamental geometric quantity describing the helical structure of confining magnetic fields and is used predominantly in the stellarator community, while tokamak physicists more often use its reciprocal, the safety factor q = 1/ι̸.
The concept was introduced by Lyman Spitzer in his foundational work on the stellarator in the early 1950s. Spitzer recognized that a purely toroidal field cannot confine a plasma because vertical drifts cause charge separation and an outward E × B drift. By twisting the flux tube—either mechanically, as in the original figure-8 stellarator, or magnetically, with helical windings—the field lines acquire a poloidal component, and particles sample both the top and bottom of the torus, averaging out the vertical drift.1
Surfaces where ι̸ = n/m (a rational number, with n and m integers) are called rational surfaces. On these surfaces, field lines close after m toroidal circuits, making them especially susceptible to resonant perturbations. Even small field errors can open magnetic islands at rational surfaces, degrading confinement. Stellarator optimization codes like VMEC and ROSE carefully position the ι profile to avoid low-order rationals in the core while sometimes deliberately placing them at the edge for island-divertor operation.3
In a stellarator, the rotational transform is produced entirely by the three-dimensional shaping of the external magnetic coils—no net toroidal plasma current is needed. This is the defining advantage of the stellarator concept: the plasma is intrinsically steady-state and immune to current-driven disruptions. In a tokamak, most of the rotational transform comes from the plasma current, with a smaller contribution from the Shafranov shift and shaping effects.
Modern stellarator design treats the ι profile as a central optimization target. The goal is to achieve sufficient rotational transform for good confinement while maintaining a low-shear profile that avoids dangerous rational surfaces. Quasi-symmetry concepts—quasi-axisymmetry (QA), quasi-helical symmetry (QH), and quasi-isodynamic (QI) configurations—all impose constraints on how ι varies across flux surfaces, ensuring that particle orbits remain well confined despite the three-dimensional geometry.