Part 5, Chapter 6

Opacity and Radiative Transfer

Bremsstrahlung opacity, Rosseland mean, and optically thin vs thick plasmas

6.1 Introduction to Plasma Opacity

Opacity quantifies how effectively a plasma absorbs or scatters radiation at a given frequency. It determines whether photons can escape freely (optically thin) or are trapped and thermalized (optically thick). The opacity controls the radiative energy balance, spectral emission, and temperature structure of astrophysical and laboratory plasmas.

The absorption coefficient $$\kappa_\nu$$ (units of m^-1) appears in the radiative transfer equation:

$$\frac{dI_\nu}{ds} = -\kappa_\nu I_\nu + j_\nu$$

where $$I_\nu$$ is the specific intensity, s is the path length, and $$j_\nu$$ is the emission coefficient. The optical depth along a ray is:

$$\tau_\nu = \int_0^L \kappa_\nu\,ds$$

A plasma is optically thin when $$\tau_\nu \ll 1$$ (photons escape freely) and optically thick when $$\tau_\nu \gg 1$$ (radiation is trapped and approaches a blackbody spectrum).

6.2 Bremsstrahlung (Free-Free) Opacity

The dominant opacity mechanism in fully ionized plasmas is inverse bremsstrahlung(free-free absorption), where a photon is absorbed by an electron in the Coulomb field of an ion. By Kirchhoff's law, the absorption coefficient is related to the emission coefficient by:

$$\kappa_\nu^{ff} = \frac{j_\nu^{ff}}{B_\nu(T)}$$

where $$B_\nu(T)$$ is the Planck function. The resulting opacity scales as:

$$\kappa_{ff} \propto n_e\,n_i\,Z^2\,T^{-7/2}\,\nu^{-2}\,\bar{g}_{ff}$$

where $$\bar{g}_{ff}$$ is the thermally averaged Gaunt factor (of order unity). The strong temperature dependence T^(-7/2) means that opacity increases dramatically as the plasma cools. The frequency dependence $$\nu^{-2}$$ shows that low-frequency (long-wavelength) radiation is absorbed much more strongly.

Kramers' Opacity Formula

In CGS units, the frequency-integrated free-free opacity (Kramers approximation) is:

$$\kappa_{ff}^{\text{Kramers}} = 3.68 \times 10^{22}\,\bar{g}_{ff}\,(1 - e^{-h\nu/k_BT})\,Z^2\,n_i\,n_e\,T^{-1/2}\,\nu^{-3}\;\text{cm}^{-1}$$

6.3 Rosseland Mean Opacity

In optically thick regions, radiation transport is diffusive and can be described by a single frequency-averaged opacity. The appropriate average is the Rosseland mean, which weights by the temperature derivative of the Planck function:

$$\frac{1}{\kappa_R} = \frac{\displaystyle\int_0^\infty \frac{1}{\kappa_\nu}\,\frac{\partial B_\nu}{\partial T}\,d\nu}{\displaystyle\int_0^\infty \frac{\partial B_\nu}{\partial T}\,d\nu}$$

The Rosseland mean is a harmonic average, meaning it is dominated by frequency windows where the opacity is lowest (the most transparent regions). This is physically sensible because radiation preferentially escapes through the most transparent spectral windows.

For pure free-free opacity, the Rosseland mean evaluates to:

$$\kappa_R^{ff} \propto n_e\,n_i\,Z^2\,T^{-7/2}$$

Planck Mean Opacity

In contrast, for optically thin emission, the relevant average is the Planck mean:

$$\kappa_P = \frac{\displaystyle\int_0^\infty \kappa_\nu\,B_\nu\,d\nu}{\displaystyle\int_0^\infty B_\nu\,d\nu}$$

This is a direct average weighted by the Planck spectrum, emphasizing frequencies where the opacity is highest. The Planck mean always exceeds the Rosseland mean.

6.4 Optically Thin vs Thick Regimes

Optically Thin ($$\tau \ll 1$$)

Every emitted photon escapes the plasma
Radiated power depends on volume: $$P \propto n_e^2 \sqrt{T}\,V$$
Spectrum reflects local emission processes
Use Planck mean for energy balance
Most fusion and space plasmas

Optically Thick ($$\tau \gg 1$$)

Radiation is reabsorbed and thermalized
Radiative flux depends on surface area: $$F = \sigma T^4$$
Interior approaches LTE (blackbody spectrum)
Use Rosseland mean for diffusive transport
Stellar interiors, dense Z-pinches

The transition between regimes depends on density, temperature, and plasma size. Radiative cooling is most effective in the optically thin limit, where increasing density rapidly increases radiation losses. In the optically thick limit, radiation is trapped and the cooling rate saturates at the blackbody surface flux.

6.5 Computational Example

The following code computes and plots the bremsstrahlung emission spectrum and evaluates the Rosseland mean opacity as a function of temperature.

Bremsstrahlung Spectrum and Rosseland Mean Opacity

Python

script.py91 lines

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import trapezoid

# Constants
h = 6.626e-34    # Planck constant (J s)
kB = 1.381e-23   # Boltzmann constant (J/K)
c = 3e8          # Speed of light (m/s)
e = 1.602e-19    # Elementary charge (C)
me = 9.109e-31   # Electron mass (kg)

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

# --- Plot 1: Bremsstrahlung spectrum at different temperatures ---
ax1 = axes[0]
temperatures_eV = [10, 100, 1000, 10000]
colors = ['#f87171', '#fb923c', '#fbbf24', '#34d399']

for Te_eV, color in zip(temperatures_eV, colors):
    Te_K = Te_eV * e / kB
    # Frequency range
    nu = np.logspace(12, 19, 500)
    # Bremsstrahlung emissivity j_nu ~ exp(-h*nu / kT) * T^{-1/2} * g_ff
    # (normalized shape)
    j_nu = np.exp(-h * nu / (kB * Te_K)) / np.sqrt(Te_K)
    j_nu = j_nu / np.max(j_nu)  # Normalize for shape comparison
    ax1.loglog(nu, j_nu, color=color, linewidth=2, label=f'T = {Te_eV} eV')

ax1.set_xlabel('Frequency (Hz)', fontsize=12)
ax1.set_ylabel('Normalized Emissivity', fontsize=12)
ax1.set_title('Bremsstrahlung Spectrum Shape', fontsize=13, fontweight='bold')
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)
ax1.set_xlim(1e12, 1e19)
ax1.set_ylim(1e-6, 2)

# --- Plot 2: Rosseland mean opacity vs temperature ---
ax2 = axes[1]
Te_range_eV = np.logspace(0, 4, 100)
ne = 1e20  # m^-3
ni = ne
Z = 1

kappa_R = []
for Te_eV in Te_range_eV:
    Te_K = Te_eV * e / kB
    # Frequency grid
    nu = np.logspace(10, 20, 2000)
    x = h * nu / (kB * Te_K)

# Planck function derivative dB/dT
    # dB/dT = (2h*nu^3/c^2) * (h*nu/kT^2) * exp(x)/(exp(x)-1)^2
    exp_x = np.exp(np.clip(x, 0, 500))
    dBdT = (2*h*nu**3/c**2) * (h*nu/(kB*Te_K**2)) * exp_x / (exp_x - 1)**2
    dBdT = np.nan_to_num(dBdT)

# Free-free opacity: kappa_nu ~ ne*ni*Z^2 * T^{-1/2} * nu^{-3} * (1-exp(-x))
    kappa_nu = ne * ni * Z**2 * (kB*Te_K)**(-0.5) * nu**(-3) * (1 - np.exp(-np.clip(x, 0, 500)))
    kappa_nu *= 3.7e8  # Normalization constant (CGS -> SI approximate)

# Rosseland mean: 1/kappa_R = integral (1/kappa)*dB/dT dnu / integral dB/dT dnu
    integrand_num = dBdT / np.where(kappa_nu > 0, kappa_nu, 1e-99)
    integrand_den = dBdT

inv_kR = trapezoid(integrand_num, nu) / max(trapezoid(integrand_den, nu), 1e-99)
    kappa_R.append(1.0 / max(inv_kR, 1e-99))

kappa_R = np.array(kappa_R)

ax2.loglog(Te_range_eV, kappa_R, color='#f59e0b', linewidth=2.5)
# Overplot T^{-7/2} scaling
T_ref = 10
kR_ref = kappa_R[np.argmin(np.abs(Te_range_eV - T_ref))]
scaling = kR_ref * (Te_range_eV / T_ref)**(-3.5)
ax2.loglog(Te_range_eV, scaling, '--', color='#94a3b8', linewidth=1.5,
           label=r'$T^{-7/2}$ scaling')

ax2.set_xlabel('Electron Temperature (eV)', fontsize=12)
ax2.set_ylabel('Rosseland Mean Opacity (m$^{-1}$)', fontsize=12)
ax2.set_title(f'Rosseland Mean vs Temperature (ne={ne:.0e})', fontsize=13, fontweight='bold')
ax2.legend(fontsize=11)
ax2.grid(True, alpha=0.3)

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

print(f"Rosseland opacity at 10 eV: {kappa_R[np.argmin(np.abs(Te_range_eV-10))]:.2e} m^-1")
print(f"Rosseland opacity at 1 keV: {kappa_R[np.argmin(np.abs(Te_range_eV-1000))]:.2e} m^-1")
print(f"Opacity ratio (10 eV / 1 keV): {kappa_R[np.argmin(np.abs(Te_range_eV-10))]/kappa_R[np.argmin(np.abs(Te_range_eV-1000))]:.1f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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