Radiation Processes

4.1 Radiation in Plasmas

Radiation is one of the primary energy loss mechanisms in plasmas and simultaneously provides the most important diagnostic tool. Charged particles radiate whenever they are accelerated. In a plasma, three main processes produce radiation: bremsstrahlung (free-free), cyclotron/synchrotron emission (gyromotion), and line radiation(bound-bound transitions). The relative importance of each depends on the plasma temperature, density, magnetic field, and composition.

Understanding radiation is essential for: (1) estimating power balance in fusion devices, (2) interpreting spectroscopic diagnostics, (3) modeling astrophysical sources, and (4) designing radiative divertor concepts that spread exhaust power over large areas.

4.2 Bremsstrahlung Radiation

Bremsstrahlung (“braking radiation”) is emitted when an electron is deflected in the Coulomb field of an ion. It is the dominant radiation mechanism in fully ionized, high-temperature plasmas. The total bremsstrahlung power density radiated by a thermal plasma is:

where $$T_e$$ is in eV. Key features:

• Scales as $$n_e^2$$ (binary collision process)
• Scales as $$Z^2$$ (strongly enhanced by impurities)
• Scales as $$T_e^{1/2}$$ (increases slowly with temperature)
• Spectrum is flat up to the cutoff $$h\nu \sim k_B T_e$$, then falls exponentially

For a D-T fusion plasma at 10 keV with $$n_e = 10^{20}$$ m^-3, the bremsstrahlung power density is approximately $$P_{\text{br}} \approx 1.7 \times 10^4$$ W/m^3, which must be offset by fusion heating for ignition.

4.3 Cyclotron and Synchrotron Radiation

Electrons gyrating around magnetic field lines emit electron cyclotron emission (ECE) at the cyclotron frequency and its harmonics:

The total cyclotron radiation power from a single non-relativistic electron is given by the Larmor formula applied to circular motion:

Averaging over a Maxwellian distribution, the total cyclotron power density is:

where T_e is in eV. At fusion-relevant temperatures, the low harmonics of ECE become optically thick, meaning the plasma reabsorbs most of its own cyclotron radiation. The net cyclotron loss is typically 1-5% of the total emitted power, making it a smaller concern than bremsstrahlung in most scenarios.

For relativistic electrons, radiation is emitted in a broad synchrotron spectrum extending to high harmonics, with the critical frequency:

4.4 Line Radiation and Radiation Cooling

Partially ionized species emit line radiation through bound-bound transitions. The power radiated per unit volume by impurity species of charge state Z is characterized by a cooling rate function $$L_z(T_e)$$:

The cooling rate peaks at temperatures where the impurity is partially ionized (typically 10-1000 eV for medium-Z elements like carbon, nitrogen, and neon). Common impurities used for radiative cooling in divertors include:

Radiative Cooling and Thermal Instability

The total radiation cooling function (bremsstrahlung + line + recombination) determines the plasma energy balance. In regions where the cooling function increases faster than the heating, a radiative thermal instability (MARFE) can develop, leading to localized dense, cold, strongly radiating regions. This is both a concern (MARFEs can trigger disruptions) and a tool (controlled radiative divertors use this to spread exhaust power).

4.5 Computational Example

The following code compares bremsstrahlung, cyclotron, and approximate line radiation power densities across a range of temperatures for fusion-relevant conditions.