Part 8, Chapter 7

Plasma Diagnostics

Langmuir probes, Thomson scattering, interferometry, and spectroscopy

7.1 Langmuir Probes

The Langmuir probe is the simplest and most widely used plasma diagnostic. A small conducting electrode is inserted into the plasma and biased at different voltages. The resulting current-voltage (I-V) characteristic reveals the electron temperature, electron density, and plasma potential.

The I-V characteristic has three distinct regions:

Ion saturation region ($V \ll V_p$): All electrons are repelled; only ions reach the probe.
Transition (electron retardation) region: The electron current increases exponentially with voltage as faster electrons overcome the retarding potential.
Electron saturation region ($V > V_p$): All electrons reaching the probe sheath are collected.

In the transition region, the electron current for a Maxwellian plasma is:

$$I_e = I_{e,\text{sat}} \exp\!\left(\frac{e(V - V_p)}{T_e}\right), \quad V < V_p$$

where the electron saturation current is:

$$I_{e,\text{sat}} = \frac{1}{4} n_e e A_p \sqrt{\frac{8T_e}{\pi m_e}}$$

and $A_p$ is the probe surface area and$V_p$ is the plasma potential. The electron temperature is extracted from the slope of $\ln(I_e)$ vs$V$:

$$T_e = e\left(\frac{d\ln I_e}{dV}\right)^{-1}$$

The ion saturation current provides the density:

$$I_{i,\text{sat}} \approx 0.6\, n_e e A_p \sqrt{\frac{T_e}{m_i}}$$

The factor 0.6 comes from the Bohm sheath criterion, which requires ions to enter the sheath at the ion sound speed $c_s = \sqrt{T_e/m_i}$.

7.2 Thomson Scattering

Thomson scattering -- the scattering of electromagnetic radiation by free electrons -- is the gold-standard diagnostic for measuring electron temperature and density simultaneously without perturbing the plasma. A high-power laser pulse is fired through the plasma, and the spectrum of scattered light is analyzed.

The differential cross section for Thomson scattering by a single electron is:

$$\frac{d\sigma}{d\Omega} = \frac{r_e^2}{2}(1 + \cos^2\theta)$$

where $r_e = e^2/(4\pi\epsilon_0 m_e c^2) = 2.82 \times 10^{-15}$ m is the classical electron radius and $\theta$ is the scattering angle. The total cross section is $\sigma_T = 8\pi r_e^2/3 = 6.65 \times 10^{-29}$ m².

The scattering regime depends on the Salpeter parameter:

$$\alpha = \frac{1}{k\lambda_D}$$

where $k = 2k_0\sin(\theta/2)$ is the scattering wavevector.

Non-collective ($\alpha \ll 1$): Scattering from individual electrons. The spectrum is a Gaussian whose width gives $T_e$ and whose integral gives $n_e$.
Collective ($\alpha \gg 1$): Scattering from electron density fluctuations (plasma waves). The spectrum shows distinct electron and ion features, giving $T_e$, $T_i$, and $n_e$.

7.3 Interferometry

Interferometry measures the line-integrated electron density by detecting the phase shift that a probing electromagnetic wave accumulates while passing through the plasma. The refractive index of a plasma is:

$$\eta = \sqrt{1 - \frac{n_e}{n_c}} \approx 1 - \frac{n_e}{2n_c}$$

The accumulated phase shift relative to propagation in vacuum is:

$$\Delta\phi = \frac{\omega}{c}\int (\eta - 1)\, dl \approx -\frac{e^2}{2m_e c \omega \epsilon_0}\int n_e\, dl = -\frac{r_e \lambda_0}{\phantom{0}} \int n_e\, dl$$

More precisely:

$$\Delta\phi = -\frac{\omega}{2cn_c}\int n_e\, dl$$

A single-chord interferometer gives the line-integrated density. Multiple chords or two-dimensional imaging (holographic interferometry) can be used with Abel inversion to reconstruct the density profile. Common configurations include Mach-Zehnder and Michelson interferometers using microwave, far-infrared, or visible laser sources.

7.4 Spectroscopy

Optical emission spectroscopy analyzes light emitted by the plasma (from atomic/ionic transitions) to determine temperature, density, composition, and flow velocities.

Doppler broadening gives the ion temperature. Thermal motion of emitting ions broadens spectral lines with a Gaussian profile of width:

$$\Delta\lambda_D = \lambda_0 \sqrt{\frac{8\ln 2\, T_i}{m_i c^2}}$$

Stark broadening gives the electron density. The electric microfields from surrounding electrons and ions split and broaden spectral lines (especially hydrogen Balmer series). The half-width at half-maximum scales as:

$$\Delta\lambda_S \propto n_e^{2/3}$$

For the hydrogen H-beta line (486.1 nm), the empirical Stark broadening relation is well tabulated and widely used for density measurements in the range$10^{14} - 10^{18}$ cm^-3.

Line ratios between different transitions of the same species give the electron temperature through the Boltzmann relation:

$$\frac{I_1}{I_2} = \frac{A_1 g_1 \lambda_2}{A_2 g_2 \lambda_1} \exp\!\left(-\frac{E_1 - E_2}{T_e}\right)$$

MIT: Principles of Plasma Diagnostics

Lectures from MIT's Principles of Plasma Diagnostics course covering introduction, Langmuir probes, imaging, proton imaging, and Thomson scattering.

Lecture 1: Introduction to Principles of Plasma Diagnostics

Lecture 4: Langmuir Probe

Lecture 15: Equilibria and Imaging

Lecture 17: Proton Imaging

Lecture 20: Thomson Scattering: Basics

Lecture 21: Thomson Scattering: Non-Collective

Lecture 22: Thomson Scattering: Collective

Lecture 23: Thomson Scattering: Advanced

Interactive Simulation

Langmuir Probe I-V Curve: Simulate and Fit for T_e and n_e

Python

script.py121 lines

import numpy as np

# Constants
e = 1.602e-19
m_e = 9.109e-31
m_i = 40 * 1.672e-27   # Argon ions
k_B = 1.381e-23
eps0 = 8.854e-12

# True plasma parameters (to be recovered from the I-V curve)
T_e_true = 3.0          # eV
n_e_true = 1e17         # m^-3
V_p_true = 15.0         # Plasma potential (V)
A_probe = 1e-6          # Probe area (m^2) - 1 mm^2

T_e_J = T_e_true * e
v_e_th = np.sqrt(8 * T_e_J / (np.pi * m_e))
c_s = np.sqrt(T_e_J / m_i)

# Currents
I_e_sat = 0.25 * n_e_true * e * A_probe * v_e_th
I_i_sat = 0.6 * n_e_true * e * A_probe * c_s

print("=== Langmuir Probe I-V Curve Simulation ===")
print()
print(f"True parameters:")
print(f"  T_e = {T_e_true:.1f} eV")
print(f"  n_e = {n_e_true:.1e} m^-3")
print(f"  V_p = {V_p_true:.1f} V")
print(f"  Probe area = {A_probe*1e6:.1f} mm^2")
print(f"  I_e,sat = {I_e_sat*1e3:.4f} mA")
print(f"  I_i,sat = {I_i_sat*1e3:.4f} mA")
print()

# Generate I-V curve
V = np.linspace(-30, 30, 200)
I_probe = np.zeros_like(V)

for j in range(len(V)):
    # Ion current (roughly constant for V < V_p, slight increase for very negative V)
    I_ion = -I_i_sat * (1.0 + 0.01 * (V_p_true - V[j]))

# Electron current
    if V[j] < V_p_true:
        I_elec = I_e_sat * np.exp(e * (V[j] - V_p_true) / T_e_J)
    else:
        I_elec = I_e_sat  # Saturation

I_probe[j] = I_ion + I_elec

# Add realistic noise
np.random.seed(42)
noise_level = 0.02 * I_e_sat
I_noisy = I_probe + noise_level * np.random.randn(len(V))

# === Fit to extract T_e and n_e ===
# Step 1: Find floating potential (V where I = 0)
idx_float = np.argmin(np.abs(I_noisy))
V_float = V[idx_float]

# Step 2: Use the transition region to fit T_e
# Select region: V_float - 5 < V < V_float + 10 (well in transition region)
mask = (V > V_float - 2) & (V < V_float + 8)
V_fit = V[mask]
I_fit = I_noisy[mask]

# Subtract approximate ion current
I_ion_approx = -I_i_sat
I_e_fit = I_fit - I_ion_approx

# Only use positive electron current
pos_mask = I_e_fit > 0
V_pos = V_fit[pos_mask]
I_e_pos = I_e_fit[pos_mask]

# Linear fit to ln(I_e) vs V
ln_Ie = np.log(I_e_pos)
coeffs = np.polyfit(V_pos, ln_Ie, 1)
slope = coeffs[0]

T_e_fit = 1.0 / slope  # in eV (since slope = e/T_e with T_e in eV... but we used e*V/T_e_J)
T_e_fit_eV = e / (slope * e)  # = 1/slope in eV

# Step 3: Find plasma potential (knee in I-V curve)
# dI/dV has a maximum at V_p
dIdV = np.gradient(I_noisy, V)
idx_Vp = np.argmax(dIdV)
V_p_fit = V[idx_Vp]

# Step 4: Extract density from I_e,sat
I_e_sat_fit = np.max(I_noisy) + I_i_sat
v_e_th_fit = np.sqrt(8 * T_e_fit_eV * e / (np.pi * m_e))
n_e_fit = I_e_sat_fit / (0.25 * e * A_probe * v_e_th_fit)

print("=== Fit Results ===")
print()
print(f"{'Parameter':>15} {'True':>12} {'Fitted':>12} {'Error':>10}")
print("-" * 52)
print(f"{'T_e (eV)':>15} {T_e_true:12.2f} {T_e_fit_eV:12.2f} {abs(T_e_fit_eV-T_e_true)/T_e_true*100:9.1f}%")
print(f"{'V_p (V)':>15} {V_p_true:12.2f} {V_p_fit:12.2f} {abs(V_p_fit-V_p_true)/V_p_true*100:9.1f}%")
print(f"{'V_f (V)':>15} {'---':>12} {V_float:12.2f} {'':>10}")
print(f"{'n_e (m^-3)':>15} {n_e_true:12.2e} {n_e_fit:12.2e} {abs(n_e_fit-n_e_true)/n_e_true*100:9.1f}%")
print()

# Print I-V curve data
print("=== I-V Characteristic ===")
print(f"{'V (V)':>8} {'I (mA)':>10} {'I_noisy (mA)':>14} {'Region':>15}")
print("-" * 50)
for i in range(0, len(V), 10):
    if V[i] < V_float - 3:
        region = "ion saturation"
    elif V[i] < V_p_true:
        region = "transition"
    else:
        region = "e- saturation"
    print(f"{V[i]:8.1f} {I_probe[i]*1e3:10.4f} {I_noisy[i]*1e3:14.4f} {region:>15}")

print()
print(f"Floating potential: V_f = {V_float:.1f} V")
print(f"Theoretical: V_f = V_p - (T_e/2)*ln(m_i/(2*pi*m_e)) = {V_p_true - T_e_true/2*np.log(m_i/(2*np.pi*m_e)):.1f} V")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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