To calculate quantum state densities when a plasma is not in thermodynamic equilibrium, it is necessary to examine the individual processes populating and de-populating the quantum states. We have already considered the radiative processes (spontaneous radiative decay, photo-excitation and stimulated emission in Section 4.2 and free-bound radiative recombination in Section 5.4). Typical radiative reactions involved are listed in Table 11.1.

Other processes which need to be taken into account to evaluate quantum state population densities are collision induced. In colliding with an ion, electrons (and to a lesser extent other ions) can cause transitions between quantum states and between discrete quantum states and free electrons. A list of collisional reactions affecting quantum state populations is 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].

Collisions in Plasmas

When the electrons, ions and atoms in a plasma undergo collisions they share energy and momentum so that a thermal distribution (Equation 1.22) characterised by a temperature and number density is produced. We considered the idea of a collision of an electron with an ion in Section 5.2 where the acceleration of the electron produces bremsstrahlung emission. Only a small loss of energy to radiation occurs with each collision.

In collisions between similar mass particles such as electron–electron and ion– ion collisions, the charged particle momentum and energy can be readily exchanged between the colliding particles. Collision between electrons and ions also transfer energy, but at a rate which is slower by the relative mass of the ions and electrons. Where the kinetic energy of colliding particles is conserved (apart from the small bremsstrahlung emission), the collision is referred to as being elastic.

Inelastic collisions involve changes in the bound quantum state of an ion or atom as a result of the collision. Energy is transferred from an electron kinetic energy to the potential energy of a bound electron (or vice versa).

An important measure with all collisions is the value of the cross-section for the collision.

The opacity of a medium is its impenetrability to radiation. For electromagnetic radiation in plasmas, the opacity arises due to absorption by free and bound electrons or due to scattering (where some radiation is re-emitted). We discussed scattering in Chapter 3, absorption of radiation in Chapter 4 and the particular physics of absorption by free electrons in Chapter 5.

Radiation opacity in plasma is important at higher densities or in larger plasmas. However, even low-density plasma may have some frequencies of radiation, say in the microwave or radiofrequency spectral range where radiation absorption is significant. Plasma opacity can limit the ability to diagnose conditions within the interior of a plasma and plasma opacity can slow the outflow of energy within a plasma and lead to non-local radiation heating.

Radiation transport calculations use a measure of dimensionless distance known as the optical depth. The optical depth of a thickness of material is the natural logarithm of the ratio of the incident to transmitted radiation power through the material. In the absence of any source of radiation, the radiation power falls exponentially with optical depth. When the optical depth approaches zero, a medium is said to be optically thin for the specified radiation and the opacity can be neglected. Large values of optical depth #x226B; 1 for a plasma or other medium suggest that opacity and radiation transport is significant: a condition known as ‘optically thick’.

We explore in this chapter the issue of opacity and the movement of radiation energy in plasmas. After some general treatment of opacity, we consider plasmas which are in, or close to, thermal equilibrium, so that the radiation field can be described by the Planck black-body distribution. Opacity modeling and radiation transport in plasmas close to thermal equilibrium is important in solar and stellar radiation modelling and in laser-produced plasmas.

The Equation of Radiative Transfer

The radiation intensity within a plasma exchanges energy with the bound and free electrons by emission and absorption processes. Electron transitions between bound discrete quantum states associated with the energy states of an ion or atom produce spectral-line emission, while transitions from free electrons to other free quantum states or to bound quantum states produce continuum emission.

A plasma is created by adding energy to a gas so that electrons are removed from atoms, producing free electrons and ions. Electric and magnetic fields interact strongly with the charged electrons and ions in plasmas (unlike solids, liquids and gases) and, consequently, plasmas behave differently to imposed electric and magnetic fields and modify electromagnetic waves in different ways to solids, liquids and gases. The different behaviour of plasmas has caused them to be regarded as a fourth fundamental state of matter in addition to solids, liquids and gases.

More than 99% of the observable universe is plasma. For example, the Sun is a plasma and has mass comprising 99.85% of the solar system, so the fraction of plasma in the solar system is slightly higher once interplanetary plasma is included. Present understanding of the universe has been enabled by the detection of electromagnetic radiation emitted by or passing through plasma material. To understand the universe, we need to understand plasmas, and, in particular, we need to understand the processes of light emission and propagation in plasmas.

Plasmas have many realised and potential applications. The fusion of isotopes of hydrogen in plasmas confined using magnetic fields or confined by inertia before a dense plasma can expand should provide a new source of energy production to replace the burning of fossil fuels, though the exact physics and many technical issues are not yet resolved [35]. The fuel for a fusion reactor (the deuterium isotope of hydrogen) is abundant in seawater (at concentration 33mg/litre). Large-scale experiments are under way to make fusion reactors because of the enormous potential impact of the development of a fusion power plant [79, 67].

Plasmas are used in many technological applications, including semiconductor etching and thin-film coating [15]. Plasma is created during the welding of solid material and is under study for biological and medical applications such as bacterial sterilisation. The emission of light from plasmas has many applications, ranging from fluorescent tubes to the use of extreme ultra-violet light emitted from laser plasmas for the lithography of semiconductors [105]. Many different lasers utilising plasmas have been developed, including argon ion lasers and an extensive array of plasma lasers designed to operate at short wavelengths [108, 91], with the record for saturated lasing achieved at wavelength 5.9 nm [125, 100].

Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved).

The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function.

Quantum Theory of the Atom–Radiation Interaction

An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.

In this chapter we initially concentrate on radiation in thermal equilibrium. We derive the Planck radiation law for an equilibrium radiation field by considering the density of photons in a black-body cavity. As well as being applicable to many astrophysical and some laboratory plasmas, the concept of a radiation field in thermal equilibrium is useful in deducing the relationships between inverse processes interacting with a radiation field. The three possible radiative processes (spontaneous emission, photo-excitation and stimulated emission) between two quantum states in an atom or ion in the presence of an equilibrium radiation field are examined. It is shown that a simple thought experiment, where the rates of the three radiative processes are in balance with an equilibrium radiation field, leads to universal relationships between the radiative rates.

The Planck Radiation Law

If the radiation and electrons in a medium have many interactions through emission and absorption, the radiation field becomes thermalised and can be regarded as having a temperature equal to that of the electrons. In this section, we calculate the form of the thermalised radiation known as the Planck radiation intensity.

An equilibrium radiation field is a collection of photons which follow the rules of statistical mechanics for a collection of bosons. We need to calculate the density of modes for light in a steady-state system. We imagine a cube in space with perfectly conducting (and hence perfectly reflecting) walls and sides of length L. Such a volume with an assumed radiation energy in equilibrium with the walls is known as a black-body cavity (though in inertial fusion work the German word hohlraum is often used).

The Density of Modes

The wavelengths ƛ of light that can exist in a steady state inside a conducting cube of sides L are quantised by the condition that a half-integer number of wavelengths only can exist in directions perpendicular to the walls. Otherwise, the oscillating electric field for the light causes transient interference effects and a steady-state solution for the radiation field does not exist.

A molecule can be defined as a group of two or more atoms held together by sharing one or more electrons. Approximately equal electron sharing between atoms is said to produce a covalent bond between the atoms. The shared electron(s) in a covalent bond attract the positively charged atomic nuclei sufficiently to overcome the mutual electrostatic repulsion between the nuclei. When the atoms in a molecule are not the same element, the electron(s) may not be shared equally around the nuclei. Depending on the level of electron sharing, the bond or attraction between atoms can have a contribution due to a net negative charge near one atom (associated with that atom attracting more of the electron wavefunction) and a net positive charge near another, leading to an ionic bond.

We discussed the energy levels of atoms and ions comprising a single element in Chapters 7 and 8. In discussing the periodic table of the pure elements, we saw that, after placing electrons into sub-shells of increasing energy, the last un-filled subshell largely controls the chemical behaviour of the element as the wavefunction(s) of the electrons(s) in this sub-shell usually extend farther away from the nucleus so that these electrons are more likely to interact with neighbouring atoms.

Neutral and ionised molecules occur in plasmas if the particle temperatures are sufficiently low that vibrations and collisions do not have sufficient energy to cause the molecules to dissociate. In ‘plasma chemistry’, plasma ions react with surfaces or other plasma constituents to form molecules [36]. Temporary molecules are sometimes formed in plasmas. Charge exchange occurs when a neutral atom such as hydrogen interacts with a highly stripped ion (with no or few electrons) to form a temporary molecule. It has been shown that to achieve good accuracy the Rosseland mean opacity (discussed in Chapter 6) should include molecular transitions at temperatures below 5,000K [2].

We consider a simple molecular ion comprising two protons and an electron: the hydrogen ion molecule. The physics associated with this simple H2+ molecule illustrates the behaviour of more complex molecules and is relevant to plasma physics. Ionisation of neutral molecular hydrogen gas creates the hydrogen ion molecule.

Much of the physics determined for hydrogen and hydrogen-like ions is relevant to atoms and ions with many electrons. In a multi-electron atom or ion, the quantum state energies and wavefunctions are dominated by the central potential arising from the nuclear charge as is the case for hydrogen and hydrogen-like ions. Consequently, quantum wavefunctions similar in form to those found for hydrogen or hydrogen-like ions are produced. There are some corrections needed to the exact electron energies associated with the nuclear electric field experienced by each electron due to shielding effects by other electrons – and some perturbing energy effects due to electron–electron Coulomb repulsion. The electron wavefunctions have similar orbital or angular shapes as for hydrogen and hydrogen-like ions (see Figures 7.2 and 7.3). The same electron-configuration system (i.e. 1s, 2s, 2p, …) can be used and the same degeneracies are associated with different values of principal quantum number n (degeneracy 2n 2) and orbital angular momentum quantum number l (degeneracy 2×2l(l+1)) as found for hydrogen and hydrogen-like ions. As multi-electron atoms and ions have quantum states following the ‘rules’ established for hydrogen and hydrogen-like ions, the quantum state energy values are primarily determined by a principal quantum number n with values n=1, 2, 3, …. Angular quantum numbers l with l=0, 1, 2, … n − 1 are found to have a small effect on energy values due to shielding of the nuclear charge. As for hydrogen, the number of individual quantum states at a particular energy involves the different magnetic quantum numbers m with values m= − l,−l + 1, …, 0, 1 … + l and the two possible spin orientations (relative to the orbital angular momentum). The states which the electrons occupy are called the electronic configuration for the atom or ion.

An important feature of multi-electron atoms is established by the Pauli exclusion principle which requires that only one electron can occupy an individual quantum state. In the ground state of an atom or ion, the required number of electrons effectively fills the lowest energy quantum states designated by the n, l, m and s values according to the hydrogen degeneracies.

The definition of the necessary density to have a high-density plasma probably depends on the research area of the person seeking the definition. Many regard departure from coronal equilbrium (see Section 12.5) a suitable definition of high density. However, we will consider plasma material at sufficiently high densities to be ‘high density’ if LTE or near-LTE occurs between the ground states of different ionisation stages (see Section 12.7). With LTE conditions, new concepts (not, for example, considered in Chapter 12) often need to be considered. Even with comparatively low temperatures (e.g. a few eV), plasmas formed at high density have a high energy content per unit volume and form a subset of the research field of high-energy density physics [20, 28]. A high-energy density is defined as an energy exceeding 1011 Jm−3 by most authors working in the field (e.g. [20]). Such energy densities are found in materials at solid density and above when temperatures exceed a few eV.

The physics of plasmas at high density requires an understanding of equilibrium relationships. Equilibrium relations are valid at sufficiently high densities which often makes for simpler calculations of plasma ionisation and radiation emission. In plasmas at high density, many interactions can lead to statistical distributions, so that equilibrium ionisation populations and equilibrium radiation distributions as introduced in Sections 1.4.1 and 4.1, respectively, are present. The equilibrium relationships for ionisation and radiation distribution were used to deduce rates of inverse processes by invoking detailed balance (see Chapter 4 for radiative processes and Chapter 12 for collisional processes).

At very high densities and low temperatures, we need to modify the equilibrium ionisation relation (the Saha-Boltzmann equation) as it is necessary to allow for free-electron quantum states becoming fully occupied so that Fermi-Dirac rather than Maxwellian electron energy distribution is required (see Section 13.4). At high density, photons are more likely to interact with particles. However, the Planck black-body radiation distribution does not require modification at high-photon or particle density as photons are bosons and any number can occupy an excitation state.

If the density is high and the temperature of the plasma is not high, the chemical potential and Fermi energy associated with the near-full occupancy of free-electron quantum states are important.

To achieve a plasma with free electrons requires elevated temperatures and hence light emission, propagation and absorption can be important. The propagation of light, unlike many other familiar waves, does not need a medium in which to oscillate. Light propagates in plasma and the free electrons are driven to oscillate, but the electrons generally impede the wave oscillation rather than aid the process. It becomes impossible for light to travel through an unmagnetised plasma if the frequency of the radiation is less than the plasma's natural oscillation frequency: the plasma frequency is discussed in Section 1.1. At low frequencies, the electron oscillations relative to the ions dampen the electromagnetic oscillation of light. Light of all frequencies passes with no absorption or alteration in phase in vacuum, though the intensity from any finite-sized source ultimately falls proportionally to the inverse of the square of distance from the source.

In this chapter, we show how Maxwell's equations describing the relationships between electric and magnetic fields (and electric current and electric charge) are consistent with oscillating electric and magnetic fields propagating in vacuum at the velocity of light c. The oscillating fields are solutions of Maxwell's equations. We treat the electric currents generated in a plasma by light to show how the currents affect electromagnetic waves. The acceleration of any charge is shown to produce transverse electric field oscillations, thus providing a mechanism for the production of electromagnetic waves.

The electromagnetic spectrum of interest in plasma physics ranges from radio waves to X-rays and gamma rays (see Figure 2.1). The propagation of the different components of the electromagnetic spectrum involves identical physics with variations only occurring when the radiation interacts with matter. The highestfrequency, highest-energy gamma rays at photon energies above, say, 100 keV are not created typically by thermal processes, but can be important when fast ‘superthermal’ particles are created in plasmas. Radio-wave propagation can be important in low-density plasmas, such as the ionosphere.

Electromagnetic Waves in Plasmas

The propagation of radiation through a medium can be examined using Maxwell's equations.