Part I, Chapter 3

Collisions & Mean Free Path

Coulomb collisions and their role in plasma dynamics

3.1 Coulomb Collisions

In a plasma, particles interact via the Coulomb force. Unlike neutral gases where collisions are hard-sphere interactions, plasma collisions are long-range and cumulative. The fundamental difference is that the Coulomb force decays as 1/r², leading to qualitatively different scattering behavior.

The Coulomb potential energy between two charges q₁ and q₂ separated by distance r is:

$$V(r) = \frac{q_1 q_2}{4\pi\epsilon_0 r} = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 r}$$

where Z₁ and Z₂ are the charge numbers. The corresponding Coulomb force is:

$$\vec{F} = -\nabla V = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 r^2}\hat{r}$$

Scattering Geometry

Consider a test particle of mass m and velocity v approaching a field particle at rest. The trajectory is a hyperbola characterized by the impact parameter b - the perpendicular distance between the initial trajectory and the field particle.

The scattering angle θ is derived from conservation of energy and angular momentum.

Derivation of Small-Angle Scattering

Step 1. A test particle (mass m, charge q₁ = Z₁e, velocity v) passes a field particle (charge q₂ = Z₂e) at impact parameter b. For b ≫ b₉₀, the trajectory is nearly straight. The transverse impulse is:

$$\Delta p_\perp = \int_{-\infty}^{\infty} F_\perp\, dt = \int_{-\infty}^{\infty} \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 (b^2 + v^2 t^2)} \frac{b}{\sqrt{b^2 + v^2 t^2}}\, dt$$

Step 2. Substituting u = vt/b and using ∫(1 + u²)^−3/2du = 2:

$$\Delta p_\perp = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 b v} \int_{-\infty}^{\infty}\frac{du}{(1+u^2)^{3/2}} = \frac{2Z_1 Z_2 e^2}{4\pi\epsilon_0 b v}$$

Step 3. The deflection angle is θ ≈ Δp_⊥/p = Δp_⊥/(mv):

$$\theta \approx \frac{\Delta p_\perp}{mv} = \frac{2Z_1 Z_2 e^2}{4\pi\epsilon_0 m v^2 b} = \frac{2b_{90}}{b}$$

where b₉₀ is defined as the impact parameter giving θ = 1 radian (approximately 90°).

$$\boxed{\theta \approx \frac{2b_{90}}{b} = \frac{Z_1 Z_2 e^2}{2\pi\epsilon_0 m v^2 b}}$$

where the characteristic impact parameter for 90° scattering is:

$$b_{90} = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 m v^2} = \frac{Z_1 Z_2 e^2}{8\pi\epsilon_0 E_{\text{kin}}}$$

Small-Angle Scattering Dominates

The key insight is that for large impact parameters b >> b₉₀, we get small deflections θ << 1. Since the cross-sectional area grows as πb², most collisions occur at large b, makingsmall-angle scattering far more frequent than large-angle deflections.

However, the cumulative effect of many small-angle scatterings can be significant. The change in perpendicular velocity after N collisions grows as:

$$\Delta v_\perp \sim \sqrt{N} \langle\theta\rangle v \sim \sqrt{N\sigma n v t} \langle\theta\rangle v$$

This random walk in velocity space is described by the Fokker-Planck equation, which we'll explore in detail later.

Why Coulomb Collisions Are Different

• Long range: Force extends to infinity (modified by Debye shielding)
• 1/r² force: No characteristic length scale (except λ_D)
• Small-angle dominant: Most collisions are glancing, not head-on
• Cumulative: Many weak deflections add up to large angular changes
• Non-additive: Cannot simply sum cross sections - need statistical treatment

3.2 Rutherford Scattering & Cross Sections

The Rutherford Formula

The exact scattering angle for Coulomb scattering is given by the Rutherford formula.

Derivation of Exact Scattering Angle

Step 1. In the center-of-mass frame, the orbit equation from conservation of energy E and angular momentum L = μvb (μ = reduced mass) is:

$$\frac{1}{r} = \frac{\mu}{L^2}\left(\frac{Z_1 Z_2 e^2}{4\pi\epsilon_0}\right)\left(\cosh\xi - 1\right) + \frac{1}{b\sinh\xi}$$

For a 1/r potential, the orbit is a hyperbola. The scattering angle relates to the eccentricity ε of the hyperbola and the distance of closest approach r_min.

Step 2. The turning point occurs where the radial kinetic energy vanishes. Conservation of energy and angular momentum give:

$$\frac{1}{2}\mu v^2 = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 r_{\min}} + \frac{L^2}{2\mu r_{\min}^2}$$

Step 3. The deflection angle is found from the orbit integral. For a repulsive Coulomb potential, using the orbit equation for a hyperbola:

$$\tan\frac{\theta}{2} = \frac{b_{90}}{b} \quad \Rightarrow \quad \theta = 2\arctan\!\left(\frac{b_{90}}{b}\right)$$

This reduces to θ ≈ 2b₉₀/b for b ≫ b₉₀ (the small-angle limit from before).

Step 4. Inverting: b = b₉₀ cot(θ/2). The differential cross section follows from the geometric relation dσ = |b db| = |b (db/dθ) dθ|, and dΩ = 2π sin θ dθ:

$$\frac{d\sigma}{d\Omega} = \frac{b}{\sin\theta}\left|\frac{db}{d\theta}\right| = \frac{b_{90}\cot(\theta/2)}{2\sin(\theta/2)\cos(\theta/2)} \cdot \frac{b_{90}}{2\sin^2(\theta/2)}$$

Simplifying with sin θ = 2 sin(θ/2)cos(θ/2) and cot(θ/2) = cos(θ/2)/sin(θ/2):

$$\boxed{\frac{d\sigma}{d\Omega} = \left(\frac{b_{90}}{2}\right)^2 \frac{1}{\sin^4(\theta/2)} = \left(\frac{Z_1 Z_2 e^2}{8\pi\epsilon_0 \mu v^2}\right)^2 \frac{1}{\sin^4(\theta/2)}}$$

This is the famous Rutherford scattering formula. The sin⁻⁴(θ/2) divergence as θ → 0 reflects the dominance of small-angle scattering from the long-range Coulomb force. Unlike hard-sphere scattering where dσ/dΩ is isotropic, Coulomb scattering is strongly forward-peaked.

90° Scattering Cross Section

Define b₉₀ as the impact parameter that gives exactly 90° scattering (θ = π/2):

$$b_{90} = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 m v^2} = \frac{Z_1 Z_2 e^2}{8\pi\epsilon_0 T}$$

where we used E_kin = ½mv² = T (kinetic energy). The cross section for 90° scattering is:

$$\sigma_{90} = \pi b_{90}^2 = \frac{\pi (Z_1 Z_2 e^2)^2}{(4\pi\epsilon_0)^2 m^2 v^4}$$

The Coulomb Logarithm

The total Coulomb cross section formally diverges due to long-range interactions. We need two cutoffs:

• Minimum b: b_min ≈ b₉₀ (quantum effects or closest approach)
• Maximum b: b_max ≈ λ_D (Debye shielding cuts off potential)

Derivation of the Coulomb Logarithm

Step 1. The total cross section for 90° deflection is the cumulative effect of many small-angle scatterings. Consider the mean-square deflection angle:

$$\langle\theta^2\rangle = \int \theta^2 \frac{d\sigma}{d\Omega}\, d\Omega = \int_{b_{\min}}^{b_{\max}} \theta^2(b) \cdot 2\pi b\, db \cdot n_i \Delta x$$

Step 2. Using θ ≈ 2b₉₀/b for small angles (valid for b ≫ b₉₀):

$$\langle\theta^2\rangle = 2\pi n_i \Delta x \int_{b_{\min}}^{b_{\max}} \left(\frac{2b_{90}}{b}\right)^2 b\, db = 8\pi n_i \Delta x\, b_{90}^2 \int_{b_{\min}}^{b_{\max}} \frac{db}{b}$$

Step 3. The integral evaluates to ln(b_max/b_min), defining the Coulomb logarithm:

$$\langle\theta^2\rangle = 8\pi n_i b_{90}^2 \ln\Lambda \cdot \Delta x, \qquad \ln\Lambda \equiv \ln\!\left(\frac{b_{\max}}{b_{\min}}\right)$$

Step 4. Setting b_max = λ_D (Debye shielding cutoff) and b_min = b₉₀ (for classical plasmas) or b_min = ℏ/(2μv) (for quantum limit, whichever is larger):

$$\ln\Lambda = \ln\!\left(\frac{\lambda_D}{b_{90}}\right) = \ln\!\left(\frac{4\pi\epsilon_0 \lambda_D k_B T}{Z_1 Z_2 e^2}\right) \approx \ln(12\pi N_D)$$

The last form uses N_D = n_eλ_D³. Since N_D ≫ 1 for any valid plasma, ln Λ is always a well-defined, positive number (typically 10-20).

$$\boxed{\ln\Lambda = \ln\left(\frac{b_{\max}}{b_{\min}}\right) = \ln\left(\frac{\lambda_D}{b_{90}}\right) \approx \ln\left(1.6\times10^4 \frac{T_e^{3/2}}{n_e^{1/2}}\right)}$$

Using the Debye length λ_D = √(ε₀k_BT/(ne²)) and thermal velocity, we can write:

$$\boxed{\ln\Lambda \approx \ln\left(\frac{12\pi n_e \lambda_D^3}{Z}\right) = \ln\left(1.6\times10^4 \frac{T_e^{3/2}}{n_e^{1/2}}\right)}$$

where T_e is in eV and n_e in cm⁻³. Typically ln Λ ≈ 10-20 for most plasmas.

Physical Meaning of ln Λ

• Measures the ratio of maximum to minimum impact parameters
• Represents the number of particles in a "Debye sphere": n_D ~ λ_D³n ~ exp(2 ln Λ)
• Weak (logarithmic) dependence on plasma parameters
• Validates plasma approximation when ln Λ >> 1
• Typically 10-20, varies slowly: doubling temperature only adds ~0.5 to ln Λ

Momentum Transfer Cross Section

For transport calculations, we need the momentum transfer cross section.

Derivation

Step 1. The fractional momentum loss per collision is (1 − cos θ). The momentum transfer cross section integrates this over all impact parameters:

$$\sigma_m = \int (1-\cos\theta)\frac{d\sigma}{d\Omega}\, d\Omega = 2\pi \int_0^\pi (1-\cos\theta)\left(\frac{b_{90}}{2}\right)^2 \frac{\sin\theta}{\sin^4(\theta/2)}\, d\theta$$

Step 2. Using 1 − cos θ = 2 sin²(θ/2) and sin θ = 2 sin(θ/2)cos(θ/2):

$$\sigma_m = 2\pi b_{90}^2 \int_{\theta_{\min}}^{\pi} \frac{\cos(\theta/2)}{\sin(\theta/2)}\, d\theta = 4\pi b_{90}^2 \ln\!\left[\sin(\pi/2)/\sin(\theta_{\min}/2)\right]$$

Step 3. For small θ_min ≈ 2b₉₀/b_max = 2b₉₀/λ_D:

$$\sigma_m = 4\pi b_{90}^2 \ln\!\left(\frac{\lambda_D}{b_{90}}\right) = 4\pi b_{90}^2 \ln\Lambda$$

$$\boxed{\sigma_m = 4\pi b_{90}^2 \ln\Lambda}$$

The factor (1 − cos θ) weights large-angle collisions more heavily since they transfer more momentum. The logarithmic factor ln Λ captures the cumulative contribution of small-angle collisions.

3.3 Collision Frequencies

Electron-Ion Collisions

The collision frequency is the rate at which a particle experiences 90° deflections. For electrons colliding with ions, starting from the cross section:

$$\nu_{ei} = n_i \sigma_m v = n_i (4\pi b_{90}^2 \ln\Lambda) v$$

Substituting b₉₀ and using quasineutrality (n_i ≈ n_e/Z):

$$\nu_{ei}(v) = \frac{Z n_e e^4 \ln\Lambda}{4\pi\epsilon_0^2 m_e^2 v^3}$$

This is the collision frequency for a single particle with speed v. Note the v⁻³ dependence: slow particles collide more frequently!

Thermal Average

For a Maxwellian distribution, we average ν_ei(v) over the velocity distribution.

Derivation of Thermally-Averaged Collision Frequency

Step 1. The velocity-dependent collision frequency is ν_ei(v) = n_iσ_mv. Substituting σ_m = 4πb₉₀²ln Λ and b₉₀ = Ze²/(4πε₀m_ev²):

$$\nu_{ei}(v) = n_i \cdot 4\pi\left(\frac{Ze^2}{4\pi\epsilon_0 m_e v^2}\right)^2 \ln\Lambda \cdot v = \frac{Z^2 n_i e^4 \ln\Lambda}{4\pi\epsilon_0^2 m_e^2 v^3}$$

Step 2. Average over the Maxwellian f(v) = (m_e/(2πk_BT_e))^3/2 exp(−m_ev²/(2k_BT_e)):

$$\langle\nu_{ei}\rangle = \int_0^\infty \nu_{ei}(v)\, f(v)\, 4\pi v^2\, dv \propto \int_0^\infty v^{-3} \cdot v^2 e^{-m_e v^2/(2k_B T_e)}\, dv$$

Step 3. The integral reduces to ∫₀^∞ v⁻¹ e^−αv² dv with α = m_e/(2k_BT_e). Substituting u = αv²:

$$\int_0^\infty v^{-1} e^{-\alpha v^2}\, dv = \frac{1}{2}\int_0^\infty u^{-1} e^{-u}\, du$$

This diverges logarithmically at small v (slow particles collide infinitely often!). In practice, the thermal average is defined using the inverse cube average ⟨v⁻³⟩ evaluated over the bulk distribution, giving:

$$\langle v^{-3}\rangle_{\text{eff}} = \frac{4}{3\sqrt{\pi}} \left(\frac{m_e}{2k_BT_e}\right)^{3/2} = \frac{4}{3\sqrt{\pi}\, v_{th,e}^3}$$

where v_th,e = √(2k_BT_e/m_e). Combining all factors (using Zn_i = n_e):

$$\boxed{\nu_{ei} = \frac{Z n_e e^4 \ln\Lambda}{12\pi^{3/2}\epsilon_0^2 m_e^{1/2} (k_B T_e)^{3/2}}}$$

In practical units (SI):

$$\nu_{ei}[\text{s}^{-1}] = 2.9 \times 10^{-12} Z \frac{n_e[\text{m}^{-3}] \ln\Lambda}{T_e[\text{eV}]^{3/2}}$$

Or in CGS units:

$$\nu_{ei}[\text{s}^{-1}] = 3.0 \times 10^{-6} Z \frac{n_e[\text{cm}^{-3}] \ln\Lambda}{T_e[\text{eV}]^{3/2}}$$

Other Collision Frequencies

Electron-electron collisions:

$$\nu_{ee} \approx \nu_{ei} \quad \text{(within factor of } \sqrt{2})$$

Ion-ion collisions (same mass, slower):

$$\nu_{ii} = \frac{Z^4 n_i e^4 \ln\Lambda}{12\pi^{3/2}\epsilon_0^2 m_i^{1/2} (k_B T_i)^{3/2}} \approx \frac{\nu_{ei}}{\sqrt{m_i/m_e}} \sim \frac{\nu_{ei}}{43}$$

For hydrogen plasma. Ions collide ~40 times less frequently than electrons.

Ion-electron collisions (momentum exchange):

$$\nu_{ie} = \frac{m_e}{m_i}\nu_{ei} \approx 10^{-3}\nu_{ei}$$

Very small due to mass ratio - electrons barely affect ion momentum.

Scaling Laws

• Density: ν ∝ n_e (linear - more targets)
• Temperature: ν ∝ T_e^-1.5 (fast particles see smaller cross section)
• Mass: ν ∝ m^-0.5 (heavier particles move slower, collide less)
• Charge: ν ∝ Z² (stronger force, larger cross section)
• Coulomb log: ν ∝ ln Λ (weak logarithmic dependence)

Numerical Examples

Fusion plasma (n_e = 10²⁰ m⁻³, T_e = 10 keV, ln Λ = 17):

$$\nu_{ei} = 2.9\times10^{-12} \frac{10^{20} \times 17}{(10^4)^{3/2}} \approx 5\times10^4 \text{ s}^{-1}$$

Solar corona (n_e = 10¹⁵ m⁻³, T_e = 100 eV, ln Λ = 20):

$$\nu_{ei} \approx 6\times10^{-3} \text{ s}^{-1}$$

Ionosphere (n_e = 10¹¹ m⁻³, T_e = 0.1 eV, ln Λ = 15):

$$\nu_{ei} \approx 1.4\times10^{4} \text{ s}^{-1}$$

3.4 Collision Timescales

Various physical processes occur on different collision timescales. Understanding these hierarchies is crucial for plasma modeling.

Collision Time

The collision time is simply the inverse of the collision frequency:

$$\tau_{ei} = \frac{1}{\nu_{ei}} = \frac{12\pi^{3/2}\epsilon_0^2 m_e^{1/2} (k_B T_e)^{3/2}}{Z n_e e^4 \ln\Lambda}$$

This is the time for an electron to experience a 90° deflection.

Energy Equilibration Time

The time for electron and ion temperatures to equilibrate through collisions:

$$\tau_{eq} = \frac{3m_i}{m_e}\tau_{ei} \approx 1836 A \tau_{ei}$$

where A is the ion mass number. For hydrogen, τ_eq ~ 60 ms in a fusion plasma.

Slowing-Down Time

A fast particle (e.g., fusion alpha, beam ion) slows down via collisions. The slowing-down time is:

$$\tau_s = \frac{3m}{m_e}\tau_{ei}\frac{T_e}{E_0}$$

where E₀ is the initial particle energy. A 3.5 MeV alpha particle in a fusion plasma takes τ_s ~ 0.5-1 second to thermalize.

Hierarchy of Timescales

For a typical plasma, the timescale hierarchy is:

$$\omega_{pe}^{-1} \ll \omega_{ce}^{-1} \ll \tau_{ei} \ll \tau_{ii} \ll \tau_{eq} \ll \tau_{\text{conf}}$$

• ω_pe^-1 ~ 10^-11 s: Plasma oscillation period
• ω_ce^-1 ~ 10^-8 s: Electron cyclotron period
• τ_ei ~ 10^-5 s: Electron collision time
• τ_ii ~ 10^-3 s: Ion collision time
• τ_eq ~ 10^-2 s: Energy equilibration
• τ_conf ~ 1 s: Confinement time (tokamak)

3.5 Mean Free Path & Collisionality

The mean free path is the average distance a particle travels between collisions:

$$\lambda_{mfp} = v_{th} \tau_{ei} = \frac{v_{th}}{\nu_{ei}}$$

Using v_th = √(k_BT_e/m_e) and substituting the collision frequency:

$$\lambda_{mfp} = \frac{12\pi^{3/2}\epsilon_0^2 (k_B T_e)^2}{Z n_e e^4 \ln\Lambda}$$

In practical units:

$$\lambda_{mfp}[\text{m}] \approx 3.4 \times 10^{11} \frac{T_e^2[\text{eV}]}{Z n_e[\text{m}^{-3}] \ln\Lambda}$$

Relationship to Debye Length

The ratio of mean free path to Debye length is:

$$\frac{\lambda_{mfp}}{\lambda_D} \sim \frac{(n_D)^2}{\ln\Lambda} \sim n_D$$

where n_D = n_eλ_D³ is the number of particles in a Debye sphere. For a valid plasma approximation, λ_mfp >> λ_D, which requires n_D >> 1 - the plasma parameter condition!

The Knudsen Number

The Knudsen number Kn = λ_mfp/L compares the mean free path to the system size L, determining the appropriate description:

Kn << 1

Collisional/Fluid Regime
• Many collisions across system
• Local thermodynamic equilibrium
• MHD equations valid
• Transport by diffusion

Kn ~ 1

Transitional Regime
• Comparable timescales
• Neither limit valid
• Need collision integrals
• Kinetic-fluid hybrid models

Kn >> 1

Collisionless/Kinetic Regime
• Few collisions
• Non-Maxwellian distributions
• Vlasov/Boltzmann equation
• Wave-particle interactions

Examples Across Regimes

ITER tokamak core:
n_e = 10²⁰ m⁻³, T_e = 10 keV, L = 2 m
λ_mfp ~ 10 km → Kn ~ 5000 (highly collisionless!)
Solar wind (1 AU):
n_e = 10⁷ m⁻³, T_e = 10 eV, L = 1.5×10¹¹ m
λ_mfp ~ 10¹⁰ m → Kn ~ 0.07 (weakly collisional)
Z-pinch:
n_e = 10²⁶ m⁻³, T_e = 1 keV, L = 1 mm
λ_mfp ~ 0.03 mm → Kn ~ 0.03 (collisional)
Ionosphere E-layer:
n_e = 10¹¹ m⁻³, T_e = 0.1 eV, L = 10 km
λ_mfp ~ 7 m → Kn ~ 7×10⁻⁴ (fluid limit)

3.6 Collision Operators in Kinetic Theory

In kinetic theory, the evolution of the distribution function f(r, v, t) includes a collision term:

$$\frac{\partial f}{\partial t} + \vec{v} \cdot \nabla f + \frac{\vec{F}}{m} \cdot \nabla_v f = \left(\frac{\partial f}{\partial t}\right)_{\text{coll}} = C(f)$$

This is the Boltzmann equation. The collision operator C(f) must conserve particles, momentum, and energy:

$$\int C(f) d^3v = 0, \quad \int m\vec{v} C(f) d^3v = 0, \quad \int \frac{1}{2}mv^2 C(f) d^3v = 0$$

BGK Collision Operator

The simplest model is the Bhatnagar-Gross-Krook (BGK) operator:

$$C_{\text{BGK}}(f) = -\nu(f - f_M)$$

where f_M is the local Maxwellian with the same density, velocity, and temperature as f. This relaxes f toward equilibrium at rate ν. Simple but non-physical for plasma (doesn't preserve momentum in multi-species collisions).

Fokker-Planck Collision Operator

For Coulomb collisions, the proper operator is a Fokker-Planck equation describingdiffusion in velocity space:

$$C(f) = \frac{\partial}{\partial v_i}\left[-A_i(v) f + \frac{\partial}{\partial v_j}\left(D_{ij}(v) f\right)\right]$$

This has two parts:

• Dynamical friction A_i: Average drag force (systematic slowing)
• Diffusion D_ij: Random walk in velocity (heating/scattering)

The dynamical friction coefficient is:

$$A_i = -\frac{4\pi Z^2 e^4 \ln\Lambda}{m^2} \frac{\partial}{\partial v_j}\int \frac{v_j - v_j'}{|\vec{v}-\vec{v}'|^3} f'(\vec{v}') d^3v'$$

For a test particle in a Maxwellian field plasma:

$$\vec{A}(v) = -\nu(v) \left[1 + \frac{m}{M}\right]\vec{v} \quad \text{where} \quad \nu(v) \propto \frac{n\ln\Lambda}{v^3}$$

Rosenbluth Potentials

The Fokker-Planck coefficients can be written using Rosenbluth potentials:

$$A_i = -\Gamma \frac{\partial h}{\partial v_i}, \quad D_{ij} = \frac{\Gamma}{2}\frac{\partial^2 g}{\partial v_i \partial v_j}$$

where Γ = 4πZ²e⁴ln Λ/m² and:

$$g(\vec{v}) = \int \frac{f'(\vec{v}')}{|\vec{v}-\vec{v}'|} d^3v', \quad h(\vec{v}) = \int |\vec{v}-\vec{v}'| f'(\vec{v}') d^3v'$$

These satisfy Poisson-like equations that can be solved efficiently numerically.

Landau Collision Operator

The complete form derived by Landau (1936) for species α colliding with species β:

$$C_{\alpha\beta}(f_\alpha) = \Gamma_{\alpha\beta} \frac{\partial}{\partial v_i}\int \frac{U_{ij}(v-v')}{|v-v'|^3}\left[f_\beta' \frac{\partial f_\alpha}{\partial v_j} - f_\alpha \frac{\partial f_\beta'}{\partial v_j'}\right] d^3v'$$

where U_ij = δ_ij - (v_i-v_i')(v_j-v_j')/|v-v'|² projects perpendicular to the relative velocity.

3.7 Energy and Momentum Exchange

Momentum Exchange Rate

When electrons and ions have different drift velocities u_e and u_i, collisions exchange momentum at rate:

$$\frac{dP_{ei}}{dt} = -m_e n_e \nu_{ei}(u_e - u_i) = -m_i n_i \nu_{ie}(u_i - u_e)$$

This friction force appears in the momentum equations for each species. Note that m_eν_ei = m_iν_ie ensures momentum conservation.

Energy Exchange Rate

When T_e ≠ T_i, energy flows from hot to cold species.

Derivation of Energy Exchange Rate

Step 1. In a single electron-ion collision, the energy transferred from the electron to the (nearly stationary) ion is of order:

$$\Delta E = \frac{2m_e m_i}{(m_e + m_i)^2}E_e(1-\cos\theta) \approx \frac{2m_e}{m_i}E_e(1-\cos\theta)$$

since m_e ≪ m_i. The fraction of energy transferred per collision is ∼ 2m_e/m_i.

Step 2. Averaging over all scattering angles (using the momentum transfer cross section) and over the Maxwellian, the energy loss rate per electron is:

$$\frac{dE_e}{dt}\bigg|_{\text{coll}} = -\frac{2m_e}{m_i}\nu_{ei}\frac{3}{2}k_B(T_e - T_i)$$

Step 3. The volumetric energy exchange rate is Q_ei = n_e · dE_e/dt:

$$Q_{ei} = \frac{3m_e n_e \nu_{ei}}{m_i}k_B(T_e - T_i)$$

The equilibration time follows from dT_e/dt = −Q_ei/(3n_ek_B/2): τ_eq ≈ (m_i/(3m_e))τ_ei.

$$\boxed{Q_{ei} = \frac{3m_e n_e}{m_i}\nu_{ei}\, k_B(T_e - T_i)}$$

The factor 3m_e/m_i ∼ 1/600 for hydrogen means energy exchange is slow! The equilibration time is:

$$\tau_{eq} \approx \frac{m_i}{3m_e}\tau_{ei} \approx 1836 A \cdot \tau_{ei}$$

Physical Picture

In each collision, the relative energy change is:

$$\frac{\Delta E}{E} \sim \frac{m_e}{m_i}$$

Many collisions (~m_i/m_e) are needed to transfer significant energy between electrons and ions.

3.8 Classical Transport Coefficients

Spitzer Resistivity

The classical electrical resistivity arises from electron-ion momentum exchange.

Derivation of Spitzer Resistivity

Step 1. The electron momentum equation in steady state with an applied electric field and collisional drag from ions:

$$0 = -n_e e \mathbf{E} - m_e n_e \nu_{ei}(\mathbf{u}_e - \mathbf{u}_i) \quad \Rightarrow \quad \mathbf{E} = -\frac{m_e \nu_{ei}}{e}(\mathbf{u}_e - \mathbf{u}_i)$$

Step 2. The current density is j = −n_ee(u_e − u_i), so u_e − u_i = −j/(n_ee). Substituting:

$$\mathbf{E} = \frac{m_e \nu_{ei}}{n_e e^2}\mathbf{j} = \eta\,\mathbf{j}$$

Step 3. Inserting the thermal-averaged ν_ei:

$$\eta = \frac{m_e}{n_e e^2}\cdot\frac{Z n_e e^4 \ln\Lambda}{12\pi^{3/2}\epsilon_0^2 m_e^{1/2}(k_BT_e)^{3/2}} = \frac{Z m_e^{1/2} e^2 \ln\Lambda}{12\pi^{3/2}\epsilon_0^2 (k_BT_e)^{3/2}}$$

Note: the exact Spitzer calculation includes an additional correction factor ≈ 0.51 from solving the Fokker-Planck equation (fast electrons carry disproportionately more current because ν_ei ∝ v⁻³), giving the tabulated Spitzer resistivity.

$$\boxed{\eta_{\text{Spitzer}} = \frac{Z m_e^{1/2} e^2 \ln\Lambda}{12\pi^{3/2}\epsilon_0^2 (k_B T_e)^{3/2}} \approx 5.2\times10^{-5} \frac{Z\ln\Lambda}{T_e^{3/2}[\text{eV}]} \quad \Omega\cdot\text{m}}$$

The T_e^−3/2 scaling means hotter plasmas are better conductors! This is opposite to ordinary metals where resistivity increases with temperature. At T_e = 10 keV (fusion plasma), η ∼ 10⁻⁹ Ω·m — making a tokamak plasma a better conductor than copper!

Thermal Conductivity

Heat flows down temperature gradients via Fourier's law: q = −κ∇T.

Derivation of Electron Thermal Conductivity

Step 1. From kinetic theory, the heat flux carried by particles with mean free path λ_mfp and thermal velocity v_th is:

$$q = -\frac{1}{3} n v_{th} \lambda_{mfp} \frac{d(\frac{3}{2}k_B T)}{dx} = -n v_{th} \lambda_{mfp} \frac{k_B}{2}\frac{dT}{dx}$$

Step 2. Using λ_mfp = v_th/ν_ei and the thermal velocity v_th = √(k_BT_e/m_e):

$$\kappa_e = \frac{1}{2}\frac{n_e k_B v_{th}^2}{\nu_{ei}} = \frac{n_e k_B^2 T_e}{2m_e \nu_{ei}}$$

Step 3. The exact Braginskii transport calculation (solving the linearized Fokker-Planck equation with Chapman-Enskog expansion) gives a numerical prefactor of 5/2 instead of 1/2, accounting for the v-dependence of ν_ei ∝ v⁻³:

$$\kappa_e = \frac{5}{2}\frac{n_e k_B^2 T_e}{m_e \nu_{ei}} = \frac{5}{2}\frac{n_e k_B v_{th}^2}{\nu_{ei}}$$

Since ν_ei ∝ T_e^−3/2, the thermal conductivity scales as κ_e ∝ T_e · T_e^3/2 = T_e^5/2. Hot plasmas are extremely good thermal conductors — this rapid temperature equilibration along magnetic field lines is why tokamak temperatures are nearly uniform on flux surfaces.

$$\boxed{\kappa_e = \frac{5}{2}\frac{n_e k_B^2 T_e}{m_e \nu_{ei}} \propto T_e^{5/2}}$$

The thermal diffusivity χ = κ/(n_ek_B) is:

$$\chi_e = \frac{5}{2}\frac{k_B T_e}{m_e \nu_{ei}} = \frac{5}{2}\frac{v_{th}^2}{\nu_{ei}} \sim v_{th} \lambda_{mfp}$$

Particle Diffusion

Density gradients drive particle diffusion:

$$\vec{\Gamma} = -D \nabla n \quad \text{where} \quad D = \frac{v_{th}^2}{\nu_{ei}} \sim \lambda_{mfp} v_{th}$$

Viscosity

Velocity gradients induce viscous stress with kinematic viscosity:

$$\mu = \rho \frac{v_{th}^2}{\nu} \sim \rho v_{th} \lambda_{mfp}$$

Key Scalings

All classical transport coefficients scale as v_th²/ν ~ λ_mfp·v_th:

• Resistivity: η ∝ T_e^-1.5 (decreases with temperature)
• Thermal conductivity: κ ∝ T_e^2.5 (increases rapidly)
• Diffusion: D ∝ T_e^2/n (increases with temp, decreases with density)
• Viscosity: μ ∝ T_e^2.5/n (similar to thermal conductivity)

Classical vs Anomalous Transport

In magnetized plasmas (tokamaks, stellarators), transport perpendicular to B is strongly reduced:

$$\chi_\perp^{\text{classical}} = \frac{\chi_\parallel}{\omega_{ce}^2 \tau_{ei}^2} \ll \chi_\parallel$$

However, turbulence-driven anomalous transport typically dominates in experiments:

$$\chi_\perp^{\text{anomalous}} \sim \frac{\rho_s^2 c_s}{L_n} \gg \chi_\perp^{\text{classical}}$$

where ρ_s is the ion gyroradius at electron temperature and L_n is the density scale length. Anomalous transport can exceed classical by factors of 10²-10⁴!

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