Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part I, Chapter 1

Basic Properties & Definition

What is plasma and when does matter behave collectively?

1.1 Definition of Plasma

A plasma is a quasi-neutral gas of charged and neutral particles that exhibits collective behavior. This definition contains three key criteria:

Three Criteria for Plasma

  1. Sufficient ionization: Enough charged particles for collective effects
  2. Quasi-neutrality: Equal positive and negative charge densities on macroscopic scales
  3. Collective behavior: Long-range Coulomb interactions dominate over binary collisions

The ionized gas must be sufficiently dense that the Coulomb interaction between charged particles extends over many particle separations, creating collective electromagnetic fields that affect all particles simultaneously.

1.2 Debye Shielding

Consider a test charge q placed in a plasma. Mobile charges will rearrange to screen the electric field. For an electron-ion plasma with equal temperatures Te = Ti = T, the potential around the test charge is:

$$\phi(r) = \frac{q}{4\pi\epsilon_0 r} e^{-r/\lambda_D}$$

where λD is the Debye length, the characteristic shielding distance:

$$\lambda_D = \sqrt{\frac{\epsilon_0 k_B T}{n_e e^2}} = \sqrt{\frac{\epsilon_0 k_B T_e}{n_e e^2} + \frac{\epsilon_0 k_B T_i}{n_i Z^2 e^2}}$$

Derivation from Poisson-Boltzmann

Starting with Poisson's equation and assuming Boltzmann distributions for electrons and ions:

$$\nabla^2 \phi = -\frac{\rho}{\epsilon_0} = -\frac{e}{\epsilon_0}(n_i - n_e)$$

With Boltzmann distributions ne = n0 exp(eφ/kBTe) and ni = n0 exp(−eφ/kBTi), linearizing for small φ:

$$\nabla^2 \phi = \frac{1}{\lambda_D^2} \phi$$

For spherical symmetry, this yields the shielded Coulomb potential above.

Physical Interpretation

The Debye length is the distance over which electric fields are shielded in a plasma. For r >> λD, the test charge is completely screened. This is a fundamental characteristic of plasma behavior—unlike a neutral gas, plasmas actively shield out electric fields.

1.3 Plasma Frequency

If charge neutrality is perturbed, the plasma responds by oscillating at the plasma frequency. Consider displacing electrons by a small distance δx from a uniform ion background:

$$E = \frac{n_e e \delta x}{\epsilon_0}$$

The restoring force on electrons gives the equation of motion:

$$m_e \frac{d^2 \delta x}{dt^2} = -e E = -\frac{n_e e^2}{\epsilon_0} \delta x$$

This is simple harmonic motion with frequency:

$$\boxed{\omega_{pe} = \sqrt{\frac{n_e e^2}{\epsilon_0 m_e}} \approx 56.4 \sqrt{n_e} \text{ rad/s}}$$

where ne is in m−3. The ion plasma frequency is:

$$\omega_{pi} = \sqrt{\frac{n_i Z^2 e^2}{\epsilon_0 m_i}} = \sqrt{\frac{m_e}{m_i}} \omega_{pe} \ll \omega_{pe}$$

Typical Values

  • • Laboratory plasmas (ne ∼ 1018 m−3): fpe ∼ 9 GHz
  • • Ionosphere (ne ∼ 1012 m−3): fpe ∼ 9 MHz
  • • Solar corona (ne ∼ 1014 m−3): fpe ∼ 90 MHz

1.4 Plasma Parameter

A crucial parameter is the number of particles in a Debye sphere (volume ∼ λD3):

$$N_D = n_e \frac{4\pi}{3} \lambda_D^3 = \frac{4\pi}{3} n_e \left(\frac{\epsilon_0 k_B T_e}{n_e e^2}\right)^{3/2}$$

Simplifying:

$$N_D = \frac{4\pi}{3} \frac{(\epsilon_0 k_B T_e)^{3/2}}{e^3 \sqrt{n_e}} \propto \sqrt{T_e^3/n_e}$$

Criterion for Plasma Behavior

For collective behavior to dominate, we require:

$$\boxed{N_D \gg 1}$$

This ensures that many-particle collective effects dominate over binary collisions. The inverse is the plasma coupling parameter:

$$\Gamma = \frac{1}{N_D} = \frac{(e^2/4\pi\epsilon_0 \lambda_D)}{k_B T} = \frac{U_{\text{potential}}}{U_{\text{kinetic}}}$$

When Γ << 1 (weakly coupled), the plasma behaves classically. When Γ ∼ 1 or greater (strongly coupled), correlations become important, leading to phenomena like crystallization in dusty plasmas.

1.5 Quasi-Neutrality

On length scales L >> λD, a plasma is quasi-neutral:

$$n_e \approx Z n_i \quad \text{for } L \gg \lambda_D$$

This doesn't mean the plasma is perfectly neutral—small deviations create the electric fields that govern plasma dynamics. The charge imbalance is typically:

$$\left|\frac{n_e - n_i}{n_e}\right| \sim \frac{\lambda_D}{L}$$

Example: Laboratory Plasma

Consider a plasma with ne = 1018 m−3, Te = 10 eV ≈ 1.16 × 105 K:

$$\lambda_D \approx 7.4 \times 10^3 \sqrt{\frac{T_e[\text{eV}]}{n_e[\text{m}^{-3}]}} \approx 74 \mu\text{m}$$

For a system size L = 1 m, the charge imbalance is only ∼10−4, yet this tiny imbalance creates electric fields strong enough to enforce quasi-neutrality.

1.6 Summary: Three Plasma Criteria

For a gas to behave as a plasma, it must satisfy:

1. Many particles in Debye sphere

$$N_D = n_e \lambda_D^3 \gg 1$$

Ensures collective effects dominate over binary collisions.

2. System size exceeds Debye length

$$L \gg \lambda_D$$

Ensures quasi-neutrality and allows collective oscillations.

3. Plasma period shorter than observation time

$$\omega_{pe} \tau \gg 1$$

Ensures the plasma can respond collectively on the timescale of interest.

These three criteria define plasma behavior across 20 orders of magnitude in density and temperature—from fusion reactors to the interstellar medium!

← Part I OverviewNext: Single Particle Motion →