Safety Factor

The safety factor (q) is a dimensionless quantity that describes how loosely or tightly the magnetic field lines are twisted in a plasma confined in a torus (doughnut shape). It is used as a key indicator that determines plasma stability.

Simple Definition (High School)

Inside a doughnut-shaped vessel, the magnetic field lines are not straight; they wind around in a helical, spiraling path. The safety factor is a number that expresses how many times a field line goes around the whole doughnut (the large loop) while it makes one trip around the cross section of the doughnut (the small loop).

For example, if the safety factor is 3, the field line goes around the whole doughnut 3 times while making one trip around the cross section, which is a loose twist. If this value becomes small and the twist becomes too tight, the plasma wriggles like a jump rope and starts to thrash about, breaking the confinement. This is why it is called the “safety” factor.

Precise Definition (Undergraduate and Above)

The safety factor q is defined as the ratio of the number of toroidal turns to poloidal turns that a magnetic field line makes as it goes once around the torus. For a simple tokamak with a circular cross section, it is expressed as follows.

$$q = \frac{r B_\phi}{R B_\theta}$$

Here r is the minor radius (the radius of the cross section), R is the major radius (the distance to the center of the doughnut), $B_\phi$ is the toroidal magnetic field, and $B_\theta$ is the poloidal magnetic field. In other words, the safety factor can be read as the ratio of the strength of the toroidal magnetic field to the poloidal magnetic field, multiplied by the shape of the device (the ratio of r to R).

The safety factor varies with location, and the way it changes from the center of the plasma toward the edge is called the q profile. The value at the edge $q_a$ is especially important, and to maintain stable operation, $q_a > 2$ is generally taken as a rule of thumb.

Role in Fusion Research

The safety factor governs the MHD stability of the plasma, in particular the kink instability (an instability in which the field lines twist and kink). The smaller q is, the tighter the twist, and the more likely the kink instability is to occur. If there is a region where q falls below 1, sawtooth oscillations arise, and on surfaces where q becomes an integer or a simple fraction (rational surfaces), magnetic islands tend to form. Controlling the q profile well is central to fusion reactor design in order to achieve high performance while maintaining stable confinement.

Note that this is an entirely different quantity from the energy gain factor Q (the ratio of fusion output to input energy), whose symbol is similar. The safety factor q represents the twist of the field lines, while the gain factor Q represents the performance of the reactor, so be careful not to confuse them.

Learn More

• Magnetic Configuration
• Plasma Instabilities

Related Terms

• Tokamak - The device where the safety factor is especially important
• Plasma - What is confined by the twisted field lines
• Confinement - The performance that the safety factor governs
• Stellarator - A method that twists field lines without relying on plasma current