# Rotational transform

The rotational transform (or field line pitch) ι/2π is defined as the number of poloidal transits per single toroidal transit of a field line on a toroidal flux surface. The definition can be relaxed somewhat to include field lines moving in a spatial volume between two nested toroidal surfaces (e.g., a stochastic field region).

Assuming the existence of toroidally nested magnetic flux surfaces, the rotational transform on such a surface may also be defined as [1]

$\frac{\iota}{2 \pi} = \frac{d \psi}{d \Phi}$

where ψ is the poloidal magnetic flux, and Φ the toroidal magnetic flux.

## Safety factor

In tokamak research, the quantity q = 2π/ι is preferred (called the "safety factor"). In a circular tokamak, the equations of a field line on the flux surface are, approximately: [2]

$\frac{r d\theta}{B_\theta} = \frac{Rd\varphi}{B_\varphi}$

where $\phi$ and θ are the toroidal and poloidal angles, respectively. Thus $q = m/n = \left \langle d\varphi /d\theta \right \rangle$ can be approximated by

$q \simeq \frac{r B_\varphi}{R B_\theta}$

Where the poloidal magnetic field ${B_\theta}$ is mostly produced by a toroidal plasma current. The principal significance of the safety factor q is that if $q \leq 2$ at the last closed flux surface (the edge), the plasma is magnetohydrodynamically unstable.[3]

In tokamaks with a divertor, q approaches infinity at the separatrix, so it is more useful to consider q just inside the separatrix. It is customary to use q at the 95% flux surface (the flux surface that encloses 95% of the toroidal flux), q95.