# Safety factor (plasma physics)
The safety factor, denoted q, is a dimensionless parameter in magnetic confinement fusion that measures the helical pitch of magnetic field lines on a flux surface: it is the number of toroidal transits a field line makes for each complete poloidal circuit around the torus[1](https://www.fusionenergynews.com/kb/safety-factor.html). Higher values of q generally correspond to greater stability against magnetohydrodynamic (MHD) perturbations, which is the origin of the name[1](https://www.fusionenergynews.com/kb/safety-factor.html). The q profile, its variation across the plasma radius, is treated as the dominant parameter for classifying tokamak operating scenarios[2](https://people.physics.anu.edu.au/~bdb112/ITER_Physics_2007/nf7_6_S01.pdf).
| Key fact | Detail |
|---|---|
| Definition | Number of toroidal turns a field line makes per poloidal turn on a given flux surface[1](https://www.fusionenergynews.com/kb/safety-factor.html) |
| Cylindrical approximation | \( q(r) \approx r B_{T} / (R_{0} B_{p}(r)) \) for a large-aspect-ratio tokamak with circular cross-section[1](https://www.fusionenergynews.com/kb/safety-factor.html) |
| Exact toroidal form | \( q(\psi) = \frac{1}{2\pi} \oint \frac{B_\phi}{R B_p}\, d\ell \), integrated around the poloidal cross-section of one flux surface[3](https://magnetohydrodynamics.physics.wisc.edu/lecture12.html) |
| Kink stability | Kruskal-Shafranov criterion \( q_a > 1 \) at the plasma edge; in practice tokamaks operate with \( q_a \gtrsim 3 \)[4](https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node13.html) |
| Typical profile | Conventional tokamak: monotonically increasing \( q(r) \) with \( q(0) \simeq 0.8 \)[4](https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node13.html) |
| Rational surfaces | Where \( q = m/n \) with integers m and n, field lines close on themselves; these surfaces are prone to tearing modes and magnetic islands[5](https://www.osti.gov/etdeweb/servlets/purl/20238431) |
| Stellarator relation | \( q = 1/\iota \), the reciprocal of the rotational transform \( \iota \) used in [stellarator](https://www.edgepedia.org/stellarator) physics[1](https://www.fusionenergynews.com/kb/safety-factor.html) |
## Definition and formulas
On each closed flux surface of an axisymmetric toroidal plasma, the magnetic field is the sum of a toroidal component \( B_\phi \) (around the long way) and a poloidal component \( B_p \) (around the short way). The safety factor characterizes the average pitch angle of the resulting helical field lines[6](https://youjunhu.github.io/research_notes/tokamak_equilibrium/node10.html). The exact toroidal definition integrates the field-line trajectory around a full poloidal circuit on each flux surface[1](https://www.fusionenergynews.com/kb/safety-factor.html):
\[ q(\psi) = \frac{1}{2\pi} \oint \frac{B_\phi}{R B_p}\, d\ell, \]
where the line integral is taken around the poloidal cross-section of one flux surface[3](https://magnetohydrodynamics.physics.wisc.edu/lecture12.html).
For a large-aspect-ratio tokamak with circular cross-section, this reduces to the cylindrical approximation
\[ q(r) \approx \frac{r B_{T}}{R_{0} B_{p}(r)}, \]
Here r denotes the minor radial coordinate, R_{0} is the major radius, B_{T} stands for the toroidal field, and B_{p} denotes the poloidal field evaluated at radius r.[1](https://www.fusionenergynews.com/kb/safety-factor.html) Because the toroidal field is generated by external coils and is largely independent of plasma behavior, changes to q are driven mainly by the poloidal field, which in a tokamak is produced by the plasma current itself[7](https://pure.mpg.de/rest/items/item_3572066_2/component/file_3572067/content).
The radial variation of the safety factor, \( dq/dr \), is called the magnetic shear[8](https://pure.tue.nl/ws/portalfiles/portal/3287209/200910788.pdf). For typical tokamak conditions the profile is well represented by a parabolic form, \( q(R) = q_0 + (q_a - q_0)((R - R_0)/a)^2 \), with the on-axis value approximately \( q_0 = q_a/(1 + q_a) \) and the edge value \( q_a \) set by the total plasma current[8](https://pure.tue.nl/ws/portalfiles/portal/3287209/200910788.pdf).
## Rational surfaces and instabilities
When q is a rational number \( q = m/n \), the field line joins up on itself after \( m \) poloidal and \( n \) toroidal rotations around the torus[5](https://www.osti.gov/etdeweb/servlets/purl/20238431). When q is irrational, a field line never closes and densely fills the surface; rational surfaces are where magnetic islands arise most naturally once axisymmetry is weakly broken[3](https://magnetohydrodynamics.physics.wisc.edu/lecture12.html). This geometric resonance makes rational flux surfaces particularly prone to magnetic instabilities[8](https://pure.tue.nl/ws/portalfiles/portal/3287209/200910788.pdf): internally, the appearance of surfaces where \( q = m/n \) can seed tearing modes and magnetic islands[1](https://www.fusionenergynews.com/kb/safety-factor.html).
**Kink stability.** A tokamak plasma remains stable to ideal kink modes only if the edge safety factor satisfies the Kruskal-Shafranov criterion \( q_a > 1 \), which sets an upper limit on the toroidal plasma current at fixed toroidal field strength; in practice a tokamak can only safely operate when \( q_a \gtrsim 3 \)[4](https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node13.html). More broadly, most unstable kink modes with poloidal mode numbers \( m = 1, 2, 3 \) can be stabilized if \( q_{95} > 3 \), which is the ITER standard operation condition[9](https://www.mdpi.com/1996-1073/3/11/1741).
**The q = 1 surface and sawteeth.** In a conventional tokamak the central safety factor sits slightly below unity, typically \( q(0) \simeq 0.8 \)[4](https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node13.html). The repetitive disruptive behavior of the plasma core within the \( q \sim 1 \) layer, the sawtooth oscillation, is the \( m/n = 1/1 \) internal kink mode[10](https://bp-pub.pppl.gov/pub_report/2008/PPPL-4295.pdf). Sawtooth control techniques act on this surface: localized ion cyclotron resonance heating in JET can destabilize sawteeth by reducing the critical magnetic shear needed to trigger a crash and increasing the local shear near the \( q = 1 \) surface[11](https://doi.org/10.1088/0741-3335%2F49%2F12B%2FS35).
## The q95 convention and the separatrix
Because the poloidal magnetic field vanishes at the X-point of a diverted plasma, the safety factor diverges there; the most significant deviation from the simple formula happens at the separatrix for this reason[7](https://pure.mpg.de/rest/items/item_3572066_2/component/file_3572067/content). The edge value of q is therefore quoted at a nearby surface: \( q_{95} \) is the safety factor of the flux surface that encloses 95% of the poloidal magnetic flux enclosed by the last closed flux surface[12](https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node167.html). In a KSTAR H-mode discharge example, \( B_0 = 1.79 \) T, \( R_0 = 1.80 \) m, and \( q_{95} = 4.04 \)[13](https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node177.html).
## Measuring the q profile
The q profile is inferred rather than directly measured. Because \( q(\psi) \sim B_\phi / B_p \), reconstructing it requires the poloidal field, and therefore the toroidal current density, throughout the plasma[3](https://magnetohydrodynamics.physics.wisc.edu/lecture12.html). External magnetic data determine the total plasma current and boundary shape well but do not uniquely determine how current is distributed on flux surfaces, so internal measurements are essential. [Motional Stark effect](https://www.edgepedia.org/motional-stark-effect) (MSE) polarimetry measures the local magnetic pitch angle, and those constraints are folded into EFIT and related Grad-Shafranov solvers to determine the current profile in a way external magnetics alone cannot[3](https://magnetohydrodynamics.physics.wisc.edu/lecture12.html). Deep-learning surrogate models have also been developed to reconstruct the q profile from measurements without MSE while mimicking traditional equilibrium reconstruction[14](https://arxiv.org/abs/2208.08730).
## Profile design and current research
MHD stability theory places critical constraints on the q profile[1](https://www.fusionenergynews.com/kb/safety-factor.html). In reactor scenario design, the profile is chosen to be monotonic with maximized shear on rational surfaces, with an elevated \( q_{\min} \) that avoids low-order resonances where potentially disruptive neoclassical tearing modes (NTMs) may occur and is consistent with direct access to second ballooning mode stability[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC11423683/). [Advanced tokamak](https://www.edgepedia.org/advanced-tokamak) operation, by contrast, seeks q profiles that are not monotonic and feature reversed shear, placing an internal transport barrier where q reaches its minimum; this yields improved confinement along with a larger share of bootstrap current.[1](https://www.fusionenergynews.com/kb/safety-factor.html) Control algorithms using spatially variable electron cyclotron current drive can regulate the local slope of the q profile at a given rational surface, useful because rational values such as \( q = 1.5 \) and \( q = 2 \) are impossible to avoid in certain scenarios[16](https://www.sciencedirect.com/science/article/pii/S0920379623004969).
**Turbulence feedback on q.** Nonlinear gyrokinetic simulations, both local and global, demonstrate that currents driven by turbulence can flatten the q profile around low-order rational surfaces in weakly sheared magnetic fields, and that this occurs even at small plasma beta[17](https://journals.aps.org/pre/abstract/10.1103/958f-qzjf). The resulting stepped profiles, with extended regions of zero magnetic shear, dramatically reduce heat transport and appear robust across linear and nonlinear q profiles and reversed-shear cases; this mechanism may play a key role in triggering internal transport barriers[17](https://journals.aps.org/pre/abstract/10.1103/958f-qzjf).
## Relation to rotational transform
The safety factor is the reciprocal of the [rotational transform](https://www.edgepedia.org/rotational-transform) \( \iota \), \( q = 1/\iota \)[1](https://www.fusionenergynews.com/kb/safety-factor.html). Stellarator physicists prefer \( \iota \), the number of poloidal transits per toroidal transit, while tokamak physicists use its reciprocal[18](https://www.fusionenergynews.com/kb/rotational-transform.html). The source of the transform differs: in tokamaks it is produced by coupling the symmetric toroidal field with the poloidal field of the plasma current, whereas in stellarators it is produced entirely by the three-dimensional shaping of external coils, so no net toroidal plasma current is needed[18](https://www.fusionenergynews.com/kb/rotational-transform.html). As in a tokamak, magnetic islands in stellarators tend to form near rational values of the transform[19](https://www.fusionnica.com/confinement/stellarator/).
## References
1. [Safety Factor q, The Fusion Record, Fusion Energy News](https://www.fusionenergynews.com/kb/safety-factor.html)
2. [Chapter 1: Overview and summary, ITER Physics (2007)](https://people.physics.anu.edu.au/~bdb112/ITER_Physics_2007/nf7_6_S01.pdf)
3. [Equilibrium Reconstruction and Properties, Classic Problems in MHD, University of Wisconsin](https://magnetohydrodynamics.physics.wisc.edu/lecture12.html)
4. [Tokamaks, Fusion Plasma Physics course notes, University of Texas at Austin](https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node13.html)
5. [Chapter 1. Introduction, OSTI](https://www.osti.gov/etdeweb/servlets/purl/20238431)
6. [Expression of safety factor in terms of magnetic field, Youjun Hu research notes](https://youjunhu.github.io/research_notes/tokamak_equilibrium/node10.html)
7. [Physics background: Tokamak equilibrium, Max Planck Institute for Plasma Physics](https://pure.mpg.de/rest/items/item_3572066_2/component/file_3572067/content)
8. [Tokamak physics, Eindhoven University of Technology](https://pure.tue.nl/ws/portalfiles/portal/3287209/200910788.pdf)
9. [A Review of Fusion and Tokamak Research Towards Steady-State Operation: A JAEA Contribution, Energies](https://www.mdpi.com/1996-1073/3/11/1741)
10. [Paradigm Changes in High Temperature Plasma Physics Research and Implications for ITER, PPPL-4295](https://bp-pub.pppl.gov/pub_report/2008/PPPL-4295.pdf)
11. [The physics of sawtooth stabilization, Plasma Physics and Controlled Fusion](https://doi.org/10.1088/0741-3335%2F49%2F12B%2FS35)
12. [Equilibrium Magnetic Field, Fusion Plasma Physics course notes, University of Texas at Austin](https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node167.html)
13. [Example Tokamak Discharge, Fusion Plasma Physics course notes, University of Texas at Austin](https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node177.html)
14. [Reconstruction of tokamak plasma safety factor profile using deep learning, arXiv](https://arxiv.org/abs/2208.08730)
15. [Plasma burn, mind the gap, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC11423683/)
16. [Nonlinear control of safety factor gradient in tokamaks using spatially variable electron cyclotron current drives, Fusion Engineering and Design](https://www.sciencedirect.com/science/article/pii/S0920379623004969)
17. [Turbulence-generated stepped safety factor profiles in tokamaks with low magnetic shear, Phys. Rev. E 112, L043201 (2025)](https://journals.aps.org/pre/abstract/10.1103/958f-qzjf)
18. [Rotational Transform / Iota, The Fusion Record, Fusion Energy News](https://www.fusionenergynews.com/kb/rotational-transform.html)
19. [Stellarator / Helical Approach, Fusionnica](https://www.fusionnica.com/confinement/stellarator/)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Plasma physics › Magnetized plasmas and confinement › MHD equilibria and safety factor*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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