Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part I, Chapter 4

Plasma Parameters

Characterizing plasmas through dimensionless quantities

4.1 Temperature and Density

Plasma state is characterized by two fundamental quantities: temperature T anddensity n. These span enormous ranges across different plasmas.

Temperature

In plasma physics, temperature is often expressed in electron volts (eV):

$$k_B T = 1 \text{ eV} \equiv 11,600 \text{ K}$$

Electrons and ions can have different temperatures: Te ≠ Ti. Equilibration occurs through collisions on timescale:

$$\tau_{eq} \sim \frac{m_i}{m_e} \tau_{ei} \approx 1836 \tau_{ei}$$

Density Ranges

  • • Interstellar medium: n ∼ 106 m−3, T ∼ 1 eV
  • • Solar wind: n ∼ 107 m−3, T ∼ 10 eV
  • • Ionosphere: n ∼ 1012 m−3, T ∼ 0.1 eV
  • • Tokamak core: n ∼ 1020 m−3, T ∼ 10 keV
  • • Laser fusion: n ∼ 1031 m−3, T ∼ 10 keV

4.2 Beta Parameter

The plasma beta is the ratio of thermal to magnetic pressure:

$$\beta = \frac{P_{thermal}}{P_{magnetic}} = \frac{n(k_B T_e + k_B T_i)}{B^2/2\mu_0} = \frac{2\mu_0 nk_B T}{B^2}$$

Beta determines the relative importance of thermal vs. magnetic forces:

  • • β << 1: Magnetic field dominates, particles gyrate freely
  • • β ∼ 1: Thermal and magnetic pressures comparable
  • • β >> 1: Thermal pressure dominates, field can be expelled

Beta in Different Plasmas

  • • Solar corona: β ∼ 10−4 – 10−2 (magnetically dominated)
  • • Solar wind at 1 AU: β ∼ 1 (pressure balance)
  • • Tokamak: β ∼ 0.05 (must keep low for MHD stability)
  • • Reverse field pinch: β ∼ 0.1 – 0.2

In fusion, achieving high β is crucial for reactor economics (higher β means less magnetic field needed for confinement).

4.3 Coupling Parameter

The coupling parameter Γ compares potential to kinetic energy:

$$\Gamma = \frac{e^2/4\pi\epsilon_0 a}{k_B T} = \frac{U_{Coulomb}}{U_{thermal}}$$

where a = (3/4πn)1/3 is the mean inter-particle spacing (Wigner-Seitz radius).

Plasma Regimes

  • • Γ << 1 (weakly coupled): Ideal gas, kinetic theory applies
  • • Γ ∼ 1 (intermediate): Correlations important
  • • Γ >> 1 (strongly coupled): Liquid-like or crystalline

Examples

  • • Fusion plasmas: Γ ∼ 10−3 (weakly coupled)
  • • White dwarf interior: Γ ∼ 1 – 100 (strongly coupled)
  • • Dusty plasma crystals: Γ ∼ 100 – 1000 (Coulomb crystal)
  • • Warm dense matter: Γ ∼ 1 (emerging field)

Relation to plasma parameter:

$$\Gamma \sim \frac{1}{N_D^{2/3}}$$

4.4 Magnetization Parameter

In magnetized plasmas, the key parameter is the ratio of gyroradius to system size:

$$\delta = \frac{r_L}{L}$$

Equivalently, compare cyclotron frequency to collision frequency:

$$\frac{\omega_c}{\nu_{ei}} = \frac{|q|B/m}{\nu_{ei}}$$
  • • ωc/ν >> 1: Magnetized plasma (particles complete many gyro-orbits between collisions)
  • • ωc/ν ∼ 1: Weakly magnetized
  • • ωc/ν << 1: Unmagnetized (collisions dominate)

For electrons in a tokamak: ωce/νei ∼ 105, highly magnetized!

4.5 Classification of Plasmas

Plasmas can be classified along several axes:

By Temperature

  • • Cold: T < 1 eV (fluorescent lamps, ionosphere)
  • • Thermal: T ∼ 1 – 100 eV (arcs, processing plasmas)
  • • Hot: T > 1 keV (fusion, astrophysical)

By Density

  • • Low density: n < 1016 m−3 (space, laboratory)
  • • Medium: n ∼ 1018 – 1020 m−3 (fusion)
  • • High density: n > 1026 m−3 (ICF, stellar)

By Generation Method

  • • DC discharge: Glow discharge, arc
  • • RF discharge: Inductively coupled, capacitively coupled
  • • Magnetic confinement: Tokamak, stellarator
  • • Inertial confinement: Laser-driven, Z-pinch
  • • Natural: Solar, magnetospheric, astrophysical

4.6 The n-T Diagram

Plasmas span 20+ orders of magnitude in density and temperature. Key boundaries:

$$\text{Plasma parameter: } N_D = n\lambda_D^3 \propto n^{-1/2} T^{3/2} > 1$$
$$\text{Quantum effects: } \Lambda_{thermal} = \frac{h}{\sqrt{2\pi m k_B T}} < n^{-1/3}$$
$$\text{Relativistic: } k_B T \sim m_e c^2 = 511 \text{ keV}$$

The fusion triple product criterion:

$$n T \tau_E > 3 \times 10^{21} \text{ m}^{-3} \cdot \text{keV} \cdot \text{s}$$

defines the region where fusion power exceeds input power (ignition).

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