Debye Length

The Debye length is the characteristic distance over which the electric field of a charge placed in a plasma is screened by the surrounding charged particles, so that its influence barely reaches beyond it. It is the most fundamental length scale for measuring whether an ionized gas actually behaves as a plasma.

Simple Definition (High School)

If you place a small ball carrying positive charge inside a plasma, the electrons flying around it are drawn in and form a “cloud of electrons” that surrounds the ball. This cloud makes the ball’s charge hard to see from the outside, so from a little distance away the ball’s presence becomes almost undetectable.

This “distance over which the electric influence reaches” is the Debye length. If the particles were cold and sitting still, the cloud would cling tightly to the ball and cancel its charge completely, but real particles are constantly moving because of their heat, so the cloud becomes blurry. The Debye length is the length set by the balance between the electric force that pulls particles in and the thermal motion that tends to scatter them apart. It grows longer when the temperature is higher and becomes shorter when the particles are denser.

Precise Definition (Undergraduate and Above)

The electric potential around a charge in a plasma does not follow the vacuum Coulomb potential but instead a Yukawa-type (screened Coulomb) potential

$$\phi(r) = \frac{q}{4\pi\varepsilon_0 r}\, e^{-r/\lambda_D}$$

Because of the exponential factor $e^{-r/\lambda_D}$, once the distance $r$ exceeds the Debye length $\lambda_D$ the potential rapidly approaches zero. Here $q$ is the charge and $\varepsilon_0$ is the vacuum permittivity.

Using the electron temperature $T_e$ and the electron density $n_e$, the Debye length can be written as

$$\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}$$

where $k_B$ is the Boltzmann constant and $e$ is the elementary charge. Temperature appears in the numerator and density in the denominator, matching the intuition that a higher temperature makes it longer and a higher density makes it shorter. In a fusion plasma (temperature around 10 keV, density around $10^{20}\ \text{m}^{-3}$), the Debye length is roughly 70 micrometers.

Role in Fusion Research

The Debye length is directly tied to the very condition that defines an ionized gas as a plasma. To be a plasma, the Debye length must be sufficiently smaller than the device size $L$ ($\lambda_D \ll L$), which ensures that charge is screened throughout most of the system and quasineutrality (a state where the electron and ion densities are nearly equal) holds. In addition, a large number of particles must lie within the Debye sphere (a sphere of radius $\lambda_D$) ($N_D \gg 1$), which is the condition for screening to work collectively and for collective effects to dominate over individual two-body collisions. In fusion plasmas the Debye length is more than four orders of magnitude smaller than the device, so these conditions are satisfied with ample margin.

Learn More

• Debye Shielding
• Plasma Physics (Learning Roadmap)
• MHD (Magnetohydrodynamics)

Related Terms

• Plasma - The state of matter defined by the Debye length
• Confinement - Methods for holding a plasma
• Tokamak - A magnetic confinement device
• Magnetohydrodynamics - Plasma theory that starts from quasineutrality