Part 5, Chapter 3

Runaway Electrons

Dreicer field, critical velocity, avalanche multiplication, and mitigation

3.1 The Runaway Mechanism

In a plasma with an applied electric field, electrons are accelerated while simultaneously experiencing frictional drag from Coulomb collisions. A remarkable feature of Coulomb collisions is that the drag force decreases with increasing velocity:

$$F_{\text{drag}} = -m_e\,\nu_{ee}(v)\,v \propto v^{-2}$$

because the collision frequency scales as $$\nu_{ee} \propto v^{-3}$$. This means that if an electron is fast enough, the electric force exceeds the collisional drag and the electron continuously accelerates -- it “runs away.” These runaway electrons can reach relativistic energies (tens of MeV) and carry megaampere-level currents, posing a severe threat to tokamak first walls during disruptions.

Key Danger: A single runaway electron beam in ITER could carry ~10 MA of current at ~20 MeV, depositing its energy on a localized area of the first wall. Runaway mitigation is a critical ITER design requirement.

3.2 The Dreicer Field

The Dreicer field $$E_D$$ is the electric field at which all electrons run away (the drag at the thermal velocity equals the electric force). It is defined as:

$$E_D = \frac{n_e\,e\,\ln\Lambda}{4\pi\epsilon_0\,T_e}$$

For typical tokamak parameters ($$n_e = 10^{20}$$ m^-3, $$T_e = 10$$ keV,$$\ln\Lambda = 15$$), the Dreicer field is $$E_D \sim 30$$ V/m. In practice, the loop voltage in tokamaks produces fields of order 0.01-1 V/m, well below the Dreicer field. However, during disruptions, the sudden loss of plasma current inductively generates electric fields that can approach or exceed the critical field.

Critical Field and Critical Velocity

At electric fields below the Dreicer field, only electrons in the tail of the distribution (above a critical velocity) can run away. The critical velocity is found by balancing the electric force against the collisional drag:

$$v_c = \sqrt{\frac{eE}{m_e\,\nu_{ee}(v_c)}} = v_{th}\sqrt{\frac{E_D}{E}}$$

The Connor-Hastie critical field includes relativistic effects and is the minimum field for which any runaway generation can occur:

$$E_c = \frac{n_e\,e^3\,\ln\Lambda}{4\pi\epsilon_0^2\,m_e\,c^2} = \frac{E_D}{(c/v_{th})^2}$$

For relativistic electrons (v approaching c), the drag increases again due to synchrotron radiation and bremsstrahlung losses. The Connor-Hastie field is typically 100-1000 times smaller than the Dreicer field, meaning runaway generation can occur at relatively modest electric fields.

3.3 Avalanche Multiplication

Beyond primary (Dreicer) generation, runaway electrons can multiply exponentially through the avalanche mechanism. A relativistic runaway electron can knock a thermal electron above the critical velocity through a close Coulomb collision, creating a secondary runaway. This leads to exponential growth:

$$\frac{dn_r}{dt} = n_r\,\frac{1}{c\,\tau_c\,\ln\Lambda}\left(\frac{E}{E_c} - 1\right)$$

where $$n_r$$ is the runaway density and $$\tau_c = 4\pi\epsilon_0^2 m_e^2 c^3 / (n_e e^4 \ln\Lambda)$$ is the relativistic collision time. The avalanche growth rate is:

$$\gamma_{\text{aval}} = \frac{1}{c\,\tau_c\,\ln\Lambda}\left(\frac{E}{E_c} - 1\right)$$

The total number of e-foldings during a disruption of duration $$\Delta t$$ is:

$$n_r(t) = n_{r,0}\,\exp\!\left(\gamma_{\text{aval}}\,\Delta t\right)$$

In ITER-scale devices, the avalanche can amplify the seed population by factors of $$10^{10}$$or more, converting a large fraction of the plasma current to runaway current. This makes the avalanche the dominant concern for runaway generation in large tokamaks.

Mitigation Strategies: Massive gas injection (MGI), shattered pellet injection (SPI), and resonant magnetic perturbations (RMPs) are used to dissipate the runaway beam or prevent its formation by raising the critical energy through impurity radiation and collisional drag.

3.4 Computational Examples

The following code computes the runaway generation rate vs electric field and plots the electron distribution function showing the runaway tail.

Runaway Electron Generation Rate and Distribution

Python

script.py103 lines

import numpy as np
import matplotlib.pyplot as plt

# Constants
e = 1.602e-19
me = 9.109e-31
eps0 = 8.854e-12
c = 3e8
kB = 1.381e-23

# Plasma parameters
ne = 1e20       # m^-3
Te_eV = 1000    # eV
Te_J = Te_eV * e
ln_Lambda = 15
vth = np.sqrt(2*Te_J/me)

# Dreicer field
E_D = ne * e * ln_Lambda / (4*np.pi*eps0 * Te_J / e)  # Note: Te in J -> divide by e for V

# More careful: E_D = ne * e^3 * lnLambda / (4*pi*eps0^2 * kB*Te)
# but Te_J = kB*Te, so:
E_D = ne * e**3 * ln_Lambda / (4*np.pi*eps0**2 * Te_J)

# Connor-Hastie critical field
E_c = ne * e**3 * ln_Lambda / (4*np.pi*eps0**2 * me * c**2)

# Relativistic collision time
tau_c = 4*np.pi*eps0**2 * me**2 * c**3 / (ne * e**4 * ln_Lambda)

fig, axes = plt.subplots(1, 2, figsize=(13, 5))

# --- Plot 1: Avalanche growth rate vs E/E_c ---
ax1 = axes[0]
E_ratio = np.linspace(1, 50, 200)  # E/E_c
gamma_aval = (1.0/(c * tau_c * ln_Lambda)) * (E_ratio - 1)

ax1.semilogy(E_ratio, gamma_aval, color='#f59e0b', linewidth=2.5)
ax1.axvline(x=E_D/E_c, color='#f87171', linestyle='--', alpha=0.7,
            label=f'Dreicer field E_D/E_c = {E_D/E_c:.0f}')
ax1.set_xlabel('E / E_c', fontsize=12)
ax1.set_ylabel('Avalanche Growth Rate (s$^{-1}$)', fontsize=12)
ax1.set_title('Runaway Avalanche Growth Rate', fontsize=13, fontweight='bold')
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)

# Add number of e-foldings for 10ms disruption
ax1_twin = ax1.twinx()
dt = 10e-3  # 10 ms disruption
n_efold = gamma_aval * dt
ax1_twin.semilogy(E_ratio, n_efold, color='#38bdf8', linewidth=0, alpha=0)
ax1_twin.set_ylabel('e-foldings in 10 ms', fontsize=11, color='#38bdf8')
# Mark where n_efold = 10, 20, 30
for nef in [10, 20, 30]:
    idx = np.argmin(np.abs(n_efold - nef))
    if idx > 0 and idx < len(E_ratio)-1:
        ax1.annotate(f'{nef} e-folds', xy=(E_ratio[idx], gamma_aval[idx]),
                    fontsize=8, color='#38bdf8', ha='left',
                    xytext=(E_ratio[idx]+1, gamma_aval[idx]*1.5),
                    arrowprops=dict(arrowstyle='->', color='#38bdf8', lw=0.8))

# --- Plot 2: Distribution function with runaway tail ---
ax2 = axes[1]
v = np.linspace(0, 8*vth, 1000)

# Maxwellian
f_maxwellian = (me/(2*np.pi*Te_J))**1.5 * 4*np.pi*v**2 * np.exp(-me*v**2/(2*Te_J))
f_maxwellian = f_maxwellian / np.max(f_maxwellian)

# Critical velocity for different E fields
E_fields = [0.01*E_D, 0.05*E_D, 0.1*E_D]
colors_E = ['#34d399', '#fbbf24', '#f87171']

ax2.semilogy(v/vth, f_maxwellian, color='#94a3b8', linewidth=2.5, label='Maxwellian')

for E_val, col in zip(E_fields, colors_E):
    vc = vth * np.sqrt(E_D / E_val)
    if vc/vth < 8:
        ax2.axvline(x=vc/vth, color=col, linestyle='--', linewidth=1.5,
                   label=f'v_c (E/E_D={E_val/E_D:.2f})')
        # Shade runaway region
        mask = v/vth > vc/vth
        ax2.fill_between(v[mask]/vth, 1e-10, f_maxwellian[mask],
                        alpha=0.15, color=col)

ax2.set_xlabel('v / v_th', fontsize=12)
ax2.set_ylabel('f(v) (normalized)', fontsize=12)
ax2.set_title('Distribution Function & Runaway Threshold', fontsize=13, fontweight='bold')
ax2.legend(fontsize=9, loc='upper right')
ax2.grid(True, alpha=0.3)
ax2.set_ylim(1e-10, 2)
ax2.set_xlim(0, 8)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0f172a')
plt.show()

print(f"Dreicer field E_D = {E_D:.2f} V/m")
print(f"Connor-Hastie field E_c = {E_c:.4f} V/m")
print(f"Ratio E_D/E_c = {E_D/E_c:.0f}")
print(f"Relativistic collision time tau_c = {tau_c:.3e} s")
print(f"Growth rate at E=5*E_c: {(1/(c*tau_c*ln_Lambda))*4:.2e} s^-1")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

3.5 Key Scales Summary

Electric Field Hierarchy

$$E_c \ll E_{\text{loop}} \ll E_D$$ during normal operation. During disruptions, E can exceed E_c by factors of 10-100, triggering avalanche growth.

ITER Concern

With $$I_p \sim 15$$ MA and $$L/R \sim 1$$ s, a fast disruption can generate$$E \sim 50\,E_c$$, producing runaway beams carrying multi-MA currents at MeV energies.

← Previous Next →