Plasma Beta

Plasma beta is a dimensionless quantity defined as the plasma’s thermal pressure divided by the pressure of the magnetic field that confines it, denoted by the symbol $\beta$. It measures how efficiently an expensive resource, the magnetic field, is being used to confine the plasma.

Simple Definition (High School)

In a fusion reactor, a strong magnetic field is created and hot plasma is confined inside it. The coils that produce this magnetic field are very expensive. Even if you prepare a strong magnetic field, it would be wasteful if the plasma confined inside it were thin and cold. Plasma beta is like a report card that shows how dense and hot the plasma you can confine is, relative to the magnetic field you prepared. The larger this value, the more energy you can generate with the same magnetic field, making the reactor more economical.

Precise Definition (Undergraduate and Above)

Plasma beta is defined as the ratio of the plasma pressure $p$ to the magnetic pressure $B^2 / (2\mu_0)$, as follows.

$$\beta = \frac{p}{B^2 / (2\mu_0)}$$

Here $p$ is the plasma’s thermal pressure, $B$ is the magnetic flux density, and $\mu_0$ is the permeability of free space. In other words, beta is the plasma pressure divided by the magnetic pressure, representing what fraction of the magnetic field energy can be converted into plasma confinement.

In a tokamak, the achievable upper limit of beta is determined by the device geometry and the plasma current. This empirical upper limit is called the Troyon beta limit, and it is organized using the normalized beta $\beta_N$ as follows.

$$\beta_N = \frac{\beta \, [\%] \, a \, B}{I_p}$$

Here $a$ is the minor radius, $B$ is the toroidal magnetic field, and $I_p$ is the plasma current. When $\beta_N$ exceeds roughly 3, magnetohydrodynamic (MHD) instabilities tend to arise, and this is taken as a rule of thumb for the Troyon beta limit.

Role in Fusion Research

Fusion power is roughly proportional to the square of the plasma pressure, and most of the construction cost is taken up by the magnetic field coils. Therefore, if a higher beta can be achieved with the same magnetic field, the same power output can be obtained with a smaller and cheaper reactor, making beta an indicator that directly governs the economics of the reactor. On the other hand, raising beta too much invites MHD instabilities and disruptions, so how to stably maintain a high beta within the range of the Troyon beta limit is an important challenge in design and operation.

Learn More

• Magnetohydrodynamics (MHD)
• Tokamak

Related Terms

• Plasma - the object whose pressure forms the numerator of beta
• Magnetohydrodynamics - the theory that deals with the instabilities setting the beta limit
• Tokamak - the device to which the Troyon beta limit applies
• Safety Factor - another indicator that governs MHD stability