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Concepts & Physics

Plasma Beta

The ratio of plasma kinetic pressure to magnetic pressure — a dimensionless figure of merit that governs the economic and physical viability of magnetically confined fusion devices.

Reviewed Last reviewed: 9 Aug 2026 · Category: Concepts & Physics

Definition

β = 2μ0 n kB T / B2

where μ0 is the permeability of free space, n is the particle number density, kB is the Boltzmann constant, T is the plasma temperature, and B is the magnetic field strength. Beta is dimensionless, typically expressed as a percentage.[1]

Physical Significance

Fusion power density scales as β2·B4, so a device at higher beta can achieve the same fusion power with a weaker magnet system. However, above a critical value, pressure-driven MHD instabilities disrupt confinement.[1]

The Troyon Limit

βmax (%) = βN × Ip / (a · BT)

where βN is typically 2.5–3.5 for conventional tokamaks.

In practice, conventional tokamaks operate at 3–8%. NSTX achieved βT > 35% transiently.[2]

Beta in Different Configurations

Conventional tokamaks: 3–8%. Spherical tokamaks: 15–40%. Stellarators: 3–5%. Field-reversed configurations: β approaching unity (~0.9), the highest of any confined plasma.[1]

Engineering and Economic Implications

Since magnet cost scales with B2·V, a high-beta device can be physically smaller for the same fusion power. However, the advent of HTS magnets (fields >20 T) has altered the calculus: moderate beta at high B may be preferable to high beta at low B.[1]

Sources

  1. Freidberg, J.P. Plasma Physics and Fusion Energy. Cambridge University Press, 2007.
  2. Troyon, F. et al. "MHD-Limits to Plasma Confinement." Plasma Physics and Controlled Fusion, vol. 26, no. 1A, 1984, pp. 209–215.
  3. Wesson, J. Tokamaks. 4th ed., Oxford University Press, 2011.
  4. Peng, Y.-K.M. and Strickler, D.J. "Features of Spherical Torus Plasmas." Nuclear Fusion, vol. 26, no. 6, 1986, pp. 769–777.

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