Part 5, Chapter 5

Atomic Processes

Ionization, recombination, charge exchange, and the Saha equation

5.1 Atomic Physics in Plasmas

Even in “fully ionized” plasmas, atomic processes play a critical role at the edge, in the divertor, and wherever neutral particles interact with the plasma. The three fundamental processes are ionization (creating ions from neutrals), recombination (ions capturing electrons), and charge exchange (ions swapping electrons with neutrals). These processes determine the ionization balance, neutral penetration depth, impurity transport, and energy loss in the plasma boundary.

In thermodynamic equilibrium, the ionization state is determined by the Saha equation. In laboratory plasmas, deviations from equilibrium require solving rate equations with collisional-radiative models, but the Saha equation provides the essential framework.

5.2 Electron Impact Ionization

Electron impact ionization is the dominant ionization mechanism in most plasmas. An electron with kinetic energy exceeding the ionization potential $$\chi_i$$ can liberate a bound electron:

$$e + A^{z+} \rightarrow 2e + A^{(z+1)+}$$

The ionization rate coefficient, averaged over a Maxwellian electron distribution, is approximately:

$$\langle\sigma v\rangle_{\text{ion}} \approx S_0\,\sqrt{\frac{T_e}{\chi_i}}\,\exp\!\left(-\frac{\chi_i}{T_e}\right)$$

where $$S_0 \sim 10^{-13}$$ m^3/s is a characteristic rate. The exponential dependence on$$\chi_i/T_e$$ means ionization is strongly temperature-sensitive. For hydrogen ($$\chi_i = 13.6$$ eV), significant ionization begins above ~3 eV and is essentially complete by ~30 eV.

The Saha Equation

In local thermodynamic equilibrium (LTE), the ratio of ionization states is given by the Saha equation:

$$\frac{n_{z+1}\,n_e}{n_z} = \frac{2\,g_{z+1}}{g_z}\left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2}\exp\!\left(-\frac{\chi_z}{k_B T}\right)$$

where $$g_z$$ are the statistical weights (partition functions) of the charge states, and$$\chi_z$$ is the ionization energy from state z to z+1. The Saha equation shows that at high temperatures and low densities, matter is more highly ionized. The density dependence arises from the three-body nature of recombination.

Validity: The Saha equation requires LTE, meaning collisional rates dominate over radiative rates. This holds when $$n_e \gtrsim 10^{23}$$ m^-3 for hydrogen at ~1 eV. At lower densities (most fusion and space plasmas), coronal equilibrium or collisional-radiative models are needed instead.

5.3 Recombination Processes

Recombination is the inverse of ionization. Three mechanisms exist:

Radiative Recombination

$$e + A^{z+} \rightarrow A^{(z-1)+} + h\nu$$

A free electron is captured into a bound state with photon emission. Rate scales as:

$$\alpha_{\text{rad}} \propto Z^2\,T_e^{-1/2}$$

Three-Body Recombination

$$2e + A^{z+} \rightarrow e + A^{(z-1)+}$$

A second electron carries away the excess energy. Dominant at high density. Rate scales as:

$$\alpha_{3b} \propto n_e\,T_e^{-9/2}$$

Dielectronic Recombination

$$e + A^{z+} \rightarrow A^{(z-1)+**} \rightarrow A^{(z-1)+} + h\nu$$

Resonant capture into a doubly-excited state, followed by radiative stabilization. Often the dominant recombination mechanism for multi-electron ions.

The total recombination rate coefficient for hydrogen-like ions is approximately:

$$\alpha_{\text{rec}} \approx 2.6 \times 10^{-19}\,Z^2\,T_e^{-1/2}\;\text{m}^3/\text{s}$$

where T_e is in eV. In coronal equilibrium (low density), the ionization balance is set by equating the ionization and recombination rates: $$n_e n_z \langle\sigma v\rangle_{\text{ion}} = n_e n_{z+1}\,\alpha_{\text{rec}}$$.

5.4 Charge Exchange

Charge exchange (CX) is the transfer of an electron from a neutral atom to an ion during a close collision:

$$A^{z+} + B \rightarrow A^{(z-1)+} + B^+$$

For the most important case of proton-hydrogen charge exchange:

$$H^+ + H \rightarrow H + H^+$$

The cross section is remarkably large and nearly energy-independent at low energies:

$$\sigma_{\text{CX}} \approx 3 \times 10^{-19}\;\text{m}^2 \quad (E \lesssim 10\;\text{keV})$$

The resulting rate coefficient is $$\langle\sigma v\rangle_{\text{CX}} \approx 10^{-14}$$ m^3/s, which exceeds the ionization rate below ~100 eV. Charge exchange has several important consequences:

Energy loss: Hot ions exchange with cold neutrals, replacing a confined energetic ion with a fast neutral that escapes the magnetic field
Neutral penetration: CX allows neutrals to hop through the plasma edge as a random walk of fast neutrals
CX recombination spectroscopy (CXRS): Injected neutral beams undergo CX with impurity ions, producing characteristic line emission that reveals ion temperature, rotation, and impurity density profiles
Neutral beam heating: CX is the mechanism by which injected fast neutral atoms become trapped ions that heat the plasma

5.5 Computational Example

The following code computes the hydrogen ionization fraction from the Saha equation and plots ionization, recombination, and charge exchange rate coefficients versus temperature.

Ionization Balance and Atomic Rate Coefficients

Python

script.py90 lines

import numpy as np
import matplotlib.pyplot as plt

# Constants
e = 1.602e-19
me = 9.109e-31
kB = 1.381e-23
h = 6.626e-34
chi_H = 13.6  # Ionization potential of hydrogen (eV)

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

# --- Plot 1: Saha ionization fraction vs temperature ---
ax1 = axes[0]
Te_eV = np.logspace(-0.5, 2, 300)  # 0.3 to 100 eV
Te_K = Te_eV * e / kB

ne_values = [1e18, 1e19, 1e20, 1e21]  # m^-3
colors_ne = ['#34d399', '#38bdf8', '#f59e0b', '#f87171']

for ne, col in zip(ne_values, colors_ne):
    # Saha: n_i * n_e / n_0 = (2 * g_i/g_0) * (2*pi*me*kT/h^2)^{3/2} * exp(-chi/kT)
    # g_i/g_0 = 1/2 for hydrogen (proton has g=1, ground state H has g=2)
    saha_rhs = (2*np.pi*me*kB*Te_K / h**2)**1.5 * np.exp(-chi_H*e/(kB*Te_K))
    # If n_e = n_i (pure hydrogen), and n_total = n_0 + n_i:
    # n_i^2 / (n_total - n_i) = saha_rhs
    # Let x = n_i/n_total (ionization fraction)
    # x^2 * n_total / (1-x) = saha_rhs
    n_total = ne  # Approximate total density
    # Quadratic: x^2 + (saha_rhs/n_total)*x - saha_rhs/n_total = 0
    a_coeff = n_total
    b_coeff = saha_rhs
    c_coeff = -saha_rhs
    # x = (-b + sqrt(b^2 - 4ac)) / (2a)  with a=n_total, b=saha_rhs, c=-saha_rhs
    discriminant = b_coeff**2 + 4*a_coeff*saha_rhs
    x_ion = (-b_coeff + np.sqrt(discriminant)) / (2*a_coeff)
    x_ion = np.clip(x_ion, 0, 1)

ax1.semilogx(Te_eV, x_ion, color=col, linewidth=2.5, label=f'ne = {ne:.0e} m$^{{-3}}$')

ax1.set_xlabel('Temperature (eV)', fontsize=12)
ax1.set_ylabel('Ionization Fraction', fontsize=12)
ax1.set_title('Hydrogen Ionization (Saha Equation)', fontsize=13, fontweight='bold')
ax1.legend(fontsize=9)
ax1.grid(True, alpha=0.3)
ax1.set_ylim(0, 1.05)
ax1.axvline(x=chi_H, color='white', linestyle=':', alpha=0.3)
ax1.text(chi_H*1.1, 0.5, 'chi_H = 13.6 eV', fontsize=9, color='white', alpha=0.5, rotation=90, va='center')

# --- Plot 2: Rate coefficients vs temperature ---
ax2 = axes[1]
Te_eV2 = np.logspace(0, 4, 300)  # 1 eV to 10 keV

# Ionization rate (approximate Lotz formula for hydrogen)
sigma_v_ion = 1.1e-14 * np.sqrt(Te_eV2/chi_H) * np.exp(-chi_H/Te_eV2) / (chi_H/Te_eV2 + 0.5)

# Radiative recombination rate
alpha_rec = 2.6e-19 * Te_eV2**(-0.5)

# Charge exchange rate (H+ + H -> H + H+), approximately constant below 10 keV
sigma_v_cx = 1.0e-14 * np.ones_like(Te_eV2)
# Slight decrease at high energy
sigma_v_cx *= np.exp(-0.1*(np.log10(Te_eV2/100))**2)
sigma_v_cx = np.clip(sigma_v_cx, 1e-16, 3e-14)

ax2.loglog(Te_eV2, sigma_v_ion, color='#f59e0b', linewidth=2.5, label='Ionization <sigma*v>')
ax2.loglog(Te_eV2, alpha_rec, color='#38bdf8', linewidth=2.5, label='Recombination alpha')
ax2.loglog(Te_eV2, sigma_v_cx, color='#a78bfa', linewidth=2.5, label='Charge exchange <sigma*v>')

ax2.set_xlabel('Electron Temperature (eV)', fontsize=12)
ax2.set_ylabel('Rate Coefficient (m$^3$/s)', fontsize=12)
ax2.set_title('Atomic Rate Coefficients for Hydrogen', fontsize=13, fontweight='bold')
ax2.legend(fontsize=10)
ax2.grid(True, alpha=0.3)
ax2.set_xlim(1, 1e4)
ax2.set_ylim(1e-21, 1e-13)

# Mark where ionization exceeds recombination
cross_idx = np.argmin(np.abs(sigma_v_ion - alpha_rec))
ax2.axvline(x=Te_eV2[cross_idx], color='white', linestyle=':', alpha=0.3)

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

print(f"Ionization rate at 50 eV: {sigma_v_ion[np.argmin(np.abs(Te_eV2-50))]:.2e} m^3/s")
print(f"Recombination rate at 50 eV: {alpha_rec[np.argmin(np.abs(Te_eV2-50))]:.2e} m^3/s")
print(f"CX rate at 50 eV: {sigma_v_cx[np.argmin(np.abs(Te_eV2-50))]:.2e} m^3/s")
print(f"Ionization-recombination crossover at ~{Te_eV2[cross_idx]:.0f} eV")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

MIT: Principles of Plasma Diagnostics

Lectures on line broadening and neutral particle diagnostics from MIT's Principles of Plasma Diagnostics course.

Lecture 16: Line Broadening

Lecture 18: Neutral Particles

← Previous Next →