The emission of radiation is one of the most important tools for diagnosing conditions in plasmas and can play a significant role in moving energy. For plasmas at all but the highest densities, we can assume that electrons are either in bound quantum states with energies dominated by the central potential of ionic nuclei, or they are unbound, occupying a continuum of free-electron states. We have seen that free electrons do not really occupy a true energy continuum (see Section 1.3), but a free electron means that the density per unit energy of quantum states is high, so that we can often consider the free-electron energies as continuous.

Radiation arising from free electrons dominates in low atomic number plasmas. Radiation transition probabilities scale rapidly with increasing atomic number: for example, as Z 4 for hydrogen-like ions, while radiation for free-electron transitions scales at a lesser rate proportional to Z 2 (see Section 5.2). In complete thermal equilibrium with the radiation field in equilibrium with particle temperatures, the emission of radiation is given by the Planck black-body formulas derived in Chapter 4. However, complete thermal equilibrium is rare in laboratory plasmas and the more tenuous astrophysical plasmas as radiation absorption within the dimensions of the plasmas is small.

We consider the radiation processes involving free electrons in this chapter. A full quantum mechanics understanding is generally not required to model emission from free electrons, so we discuss the emission of radiation in this chapter using largely classical non-quantum treatments. The Bohr model for bound energy states is utilised in considering emission from an electron making a transition from a free to bound state.

Cyclotron Radiation

Astrophysical plasmas such as those near a neutron star have strong embedded magnetic fields and even in interstellar space there is a weak magnetic field (≈ 10−10 Tesla). The plasmas studied in magnetic fusion research have magnetic fields in the range 0.1–10 Tesla confining the plasma. Confinement works as charged particles orbit with helical-shaped trajectories around an imposed magnetic field B0.

An important type of emission from plasmas consists of spectral lines originating from transitions between bound quantum states. Quantum mechanics gives information on the energies of the quantum states and can give information on the intensity of emission. The absorption of radiation resulting in transitions between bound quantum states can dominate the calculation of absorption coefficients, particularly for higher atomic number ions where the spectral density of absorption and emission lines is large and the radiative-transition probabilities are high (for example, hydrogen-like ion transition probabilities scale proportionally to Z 4).

The Bohr model of the atom where electrons are said to orbit the nucleus like planets orbiting the Sun was introduced in Section 1.5. The Bohr model gives a good approximation of the energies of hydrogen and hydrogen-like ions. It is also a reasonable model for the energies of excited states of higher-atomic-number atoms and ions. Excited electrons of multi-electron atoms and ions have orbits some distance from the nucleus and other electrons, so their kinetic and potential energy is determined by the near point-like net charge near the nucleus as occurs with hydrogen-like ions. The Bohr model, however, fails to predict the correct angular momentum of orbiting electrons and fails to predict the fine structure of the energy levels. Fine structure causes, for example, the energy states to split and produce two or more closely spaced spectral lines, rather than a single line. The Bohr model also does not enable a satisfactory method of evaluating the rates of radiative absorption and emission (which can be done with time-dependent quantum mechanics; see Section 10.1).

We consider the quantum mechanics of hydrogen and hydrogen-like ions in this chapter. Hydrogen and hydrogen-like ions only differ in the charge Z of the nucleus, so it is possible to treat both almost identically. We commence with solutions of the Schrodinger equation, treat the effects leading to the fine structure and mention hyperfine energy splitting arising from electron interaction with the nuclear spin. There are many books dealing with the details of the quantum mechanics of hydrogen. We refer the reader to Haken and Wolf [43] for further reading.

A Quantum Mechanical Treatment of Atoms and Ions

A quantum mechanical treatment of atoms and ions can start with the timedependent Schrodinger equation.

This book provides an introduction to the physics of emission, absorption and interaction of light in astrophysics and in laboratory plasmas. Such study necessarily requires a wide range of modern physics understanding involving electricity and magnetism, relativity, atomic structure, quantum mechanics, particle collision theory, statistical physics and more. Indeed, the analysis of light emission and collisional processes relevant to plasmas has provided much of the experimental evidence for quantum mechanics. The atomic and radiation physics of plasmas is, consequently, an ideal subject for study as an extension to material taught to physics undergraduates. The book combines undergraduate-level studies of the quantum mechanics of ions/atoms with the atomic and radiation physics of plasmas, though non-quantum models are used extensively. Atomic and radiation physics is presented at a level aimed at undergraduates in their final two years through to graduate students and researchers. Material needed for research in plasma physics and astrophysics is derived.

Plasma physicists working in a range of areas from astrophysics, magnetic fusion and inertial fusion to low-temperature plasmas of technological significance utilise atomic and radiation physics to interpret measurements. Plasma physics is a growing research area with the construction of the ITER tokamak, new laser-plasma facilities and the development of new methods of creating plasma, such as with free-electron lasers. Atomic and radiation physics is also an essential component in the theoretical development and simulation of astrophysical and laboratory plasmas. One aim of this book is to emphasise the overlap of atomic/radiation physics between astrophysical and laboratory plasmas, an imbrication exploited in the expanding field of laboratory astrophysics where physical scenarios relevant to astrophysics are simulated in the laboratory.

Due to the range of understanding required for research in the atomic and radiation physics of plasmas, the underlying physics is often not developed in research publications in astrophysics and plasma spectroscopy. An aim of this book has been to start with the knowledge obtained by physics graduates before they begin to specialise and to develop formulae and explain techniques used in plasma spectroscopy. The areas of plasma research utilising aspects of atomic and radiation physics are briefly introduced before spectroscopic applications are covered, but this book concentrates on the underlying atomic and radiation physics.

In a plasma in complete thermodynamic equilibrium, the radiation field is given by the Planck black-body expression (see Section 4.1), the ionisation is determined by the Saha-Boltzmann equation (see Section 1.4.1) and the population ratios of bound quantum states are determined by the Boltzmann ratio. In local thermodynamic equilibrium (LTE), quantum state populations are given by the Saha-Boltzmann equation and Boltzmann ratio, but the radiation field is not in equilibrium with the particles. We discuss the plasma conditions needed to establish equilibrium later in this chapter, but it is worthwhile to note that LTE often occurs when collisional processes dominate the populating and de-populating of the quantum state populations and radiative processes are not significant.

Radiative rates of decays for bound quantum states were determined in Section 10.1 and between free and bound states in Section 5.4. The cross-sections for collisional processes were discussed in Chapter 11. The cross-sections depend on the energy of the incident colliding electron, but in a plasma we have a Maxwellian distribution of the energies of the free electrons. The cross-section values need to be averaged over the Maxwellian distribution to produce a rate coefficient which when multiplied by the density of free electrons and the initial quantum state density yields the rate of change of the quantum state. The radiative reactions involved were listed in Table 11.1. A list of collisional reactions affecting quantum state populations has been given in Table 11.2. Models calculating plasma quantum state densities and consequent radiation emission and absorption properties using rates of radiative and collisional processes are known as collisional radiative models [89].

Collisional Excitation and De-Excitation

Our investigation of cross-sections for excitation by inelastic electron collisions has shown a variation with the energy E of the incident electron approximately proportional to 1/E (see Section 11.4). The cross-section for collisional excitation can be written in terms of a collision strength Ωpq(E) such that

where πa 2 0 is a cross-section for the ground state of the hydrogen atom (taken as the area associated with the Bohr radius a 0) and gp is the degeneracy of the initial quantum state. ‘Effective collision strengths’ γpq are tabulated for different temperatures where the collision strength has been averaged over the electron distribution.

Free and bound electrons in a plasma are accelerated by electromagnetic radiation. The interaction with the electrons affects the propagation of the radiation by altering the phase of the oscillating electric and magnetic field of the electromagnetic wave and by absorption of the electromagnetic wave energy as discussed in Chapter 2. As well as affecting a propagating electromagnetic wave, the acceleration of the free and bound electrons in a medium also gives rise to radiation emission: a process referred to as ‘scattering’.

As the acceleration of electrons affects the propagation of electromagnetic waves while producing emission of radiation, scattering of light by electrons in a medium can be regarded as determining the optical properties of the medium. Resonances in the responses of free and bound electrons to oscillations from electromagnetic waves tend to have a dominant effect on light propagation. We determined the refractive index arising in plasmas from free electrons and the resonance at the plasma frequency (see Section 2.1). Other resonances associated, for example, with bound electrons also produce refractive index effects.

By determining the refractive index of the medium in which light propagates, scattering processes ultimately govern the reflection and refraction behaviour of light at the junction between materials with different refractive indices. For example, macroscopic particles such as dust in plasmas or water droplets in clouds in the atmosphere reflect light from surfaces (known as Mie scattering). Gradients of refractive index lead to refractive bending of the direction of light propagation.

The fraction of electromagnetic radiation scattered by free electrons is typically a small loss mechanism for radiation of frequency much greater than the plasma frequency, but it is useful for diagnosing conditions in plasmas. For diagnostic measurements of plasmas at optical (ultra-violet to infra-red) frequencies, a laser radiation source is usually employed so that light can be spatially located and the high laser power per unit area ensures that the scattered light is greater than the emission associated with the thermal energy of the plasma. With radio waves and ionospheric scattering, high-power radar systems are employed. In dense plasmas of relevance to inertial fusion, incoherent X-ray sources or free-electron laser sources are used [99].