Plasma Beta

The ratio of plasma pressure to magnetic pressure, the key measure of how efficiently a field confines plasma.

Definition

Beta is the ratio of the plasma thermal pressure to the magnetic pressure:

beta = p / (B^2 / 2 mu0) = 2 mu0 n k_B T / B^2

It measures how much plasma pressure a given magnetic field can hold. Higher beta means the field is used more efficiently, which is economically and physically desirable, but stability limits how high beta can go before the plasma tears itself apart.

Toroidal, poloidal, and normalized beta

Several betas are used: toroidal beta (against the toroidal field), poloidal beta (against the poloidal field, related to bootstrap current), and total beta. For comparing machines the normalized beta, beta_N = beta a B / I_p, is standard because the stability limit is nearly constant in beta_N.

Why high beta is hard

• Pressure-driven ballooning and kink modes set an upper limit on beta
• The limit is captured empirically by the Troyon scaling beta_max ~ beta_N I_p / (a B)
• Exceeding the limit typically triggers a disruption or a resistive mode

Spherical tokamaks and beta

Low-aspect-ratio spherical tokamaks are notable for reaching high toroidal beta, because their strong shaping and favorable field configuration allow more pressure per unit toroidal field. The Hyperion breeder is a spherical tokamak precisely to exploit this high-beta capability, with equilibria and stability computed from Grad-Shafranov and ballooning analysis.

How it is evaluated

Beta is a direct output of the equilibrium solve: integrate the pressure profile and divide by the magnetic energy. Stability codes then check the equilibrium against the Troyon and ballooning limits to confirm the operating beta is achievable.