Phenomenological introduction

Model atoms

In the preceding section, we discussed the interaction between light and two-level atoms – probably one of the most popular physical models. The basis of this popularity is its intuitively simple interpretation combined with the posibility of explaining a wide range of physical phenomena. An interesting aspect of the two-level system is that, although its dynamics are formally equivalent to those of a classical angular momentum, it can explain many aspects of quantum mechanics. Once these aspects are understood, it is tempting to look further into the behaviour of real systems, trying to find patterns inconsistent with the predictions of the two-level model.

This section discusses aspects of the interaction between matter and radiation that are incompatible with the two-level model. We do not consider specific atomic systems or attempt a complete analysis of the dynamics that a three-level system can exhibit. Instead we select a number of phenomena that play an important role in the discussion of those physical systems that form the subject of the subsequent sections.

In view of the complexity of the various kinetic models discussed near the beginning of the previous chapter and of the vast array of cross sections and other atomic data required for their implementation, there is a great need for a more basic theory. Such a theory will have a restricted region of validity but can be used with confidence for the interpretation of plasma spectroscopic experiments performed under appropriate conditions. This basic theory is provided by equilibrium statistical mechanics or thermodynamics. It, together with the various conditions for its validity, is the subject of the present chapter. Before introducing this new topic, a second role for the more basic theory should be emphasized, namely its service in testing kinetic models under conditions for which thermodynamic equilibrium should hold.

The practical goal of any of the kinetic models and of the local thermodynamic equilibrium (LTE) relations introduced here is the calculation of populations of the various states of atoms or ions and of the free electron density for specified temperature(s), pressure or mass density, and assumed chemical composition. As already discussed in the preceding chapter, one deals with local, and usually also transient, equilibria, because strict thermodynamic equilibrium would either require an unbounded, spatially and temporally homogeneous plasma or a plasma enclosed in an ideal blackbody hohlraum (cavity).

In the preceding chapters, the basic radiative and collisional processes governing local radiative properties of a plasma were introduced; and their quantitative evaluation or acquisition was discussed to help the reader in the selection of data needed for analysis or prediction of a spectrum. We also learned about various kinetic or thermodynamic models designed for comprehensive descriptions of level populations.

There are two reasons for this seemingly all-encompassing body of theory and basic data to be insufficient, nevertheless, for both analysis and predictions. First, one generally cannot measure local values of the plasma emission, but must infer them from some distance and averaged over the various contributing volume elements. Second, and even more fundamentally, radiative processes also influence level populations so that the emission or absorption in one location depends on the radiation flux coming from the rest of the plasma.

Therefore a self-consistent treatment of radiation transport and level populations is necessary, requiring in general a nonlocal and nonlinear theory. Such theory has been developed over many years, mainly by astronomers and astrophysicists. Much progress has been made after the two basic treatises (Chandrasekhar 1950 and Sobolev 1963) were written, mostly by computational methods (see, e.g., Athay 1972, Kalkofen 1984, Crivellari, Hubeny and Hummer 1991), but also through more or less analytic models (see, e.g., Thomas and Athay 1961, Jefferies 1968, Ivanov 1973).

Depending on various plasma conditions, such as size, composition, densities and temperatures, the electron density and related quantities can be measured using a number of techniques. Of these, Langmuir probes (Tonks and Langmuir 1929, Hutchinson 1987) provided the first means to infer local values of the electron density, mostly at relatively low densities. Much more recently, Thomson scattering of laser light has become a method of choice for localized electron density measurements (Kunze 1968, Evans and Katzenstein 1969, DeSilva and Goldenbaum 1970, Sheffield 1975) over a range of about 1011 to 1021 cm−3. Then there are the usually inherently most accurate interferometric techniques, mostly laser-based as well (Jahoda and Sawyer 1971, Hauer and Baldis 1988), which span a similar range, recently extended to Ne ≈ 3 × 1021 cm−3 (DaSilva et al. 1995), but provide localized values of the electron density only indirectly. Using two or more wavelengths, one can, however, separate the free-electron contribution (1.37) to the refractive index from any boundstate contribution (2.103). In single-species, partially ionized plasmas, this boundstate contribution is a direct measure of the neutral atom density. Interferometric local values of the density can, in principle, be determined by methods analogous to those discussed for emission or absorption measurements in section 8.5.

In a fully ionized plasma, in which all bound electrons have been removed from their original atoms or molecules, there is besides the continuous spectrum no line emission or absorption, except possibly for features related to plasma resonances or waves (Bekefi 1966, Stix 1992, Swanson 1989). These often nonthermal features are usually at such low frequencies that they do not obscure or interfere with atomic radiation, the subject of principal interest here. Since atoms and incompletely stripped ions possess a continuous spectrum, besides the discrete spectrum providing the pairs of states involved in line radiation, continuous emission and absorption spectra underly and accompany the line spectra discussed in the preceding chapters. These processes are not only important as background to line emission, but also because continuum intensities can provide relatively direct measures of electron density and temperature (see sections 10.2 and 11.4, respectively).

With the usual convention for the energies of bound states as being negative relative to those of zero kinetic energy electrons at large distances from the nucleus of any isolated atom or ion, one might infer that all positive energy states belong to the continuous spectrum. In practice, this is an oversimplification, because there are states corresponding to the excitation of two or more bound electrons which are almost discrete.

Observable intensities of spectral lines and continua depend just as strongly on the appropriate level populations as on the transition probabilities and related quantities discussed in chapters 2 and 3, and on the radiative transfer processes to be discussed in chapter 8. The level population and transfer problems are not really separable. One must therefore strive for internal consistency, following the example of astrophysicists (see, e.g., Mihalas 1978). Nevertheless, it is reasonable to discuss first the various population and depopulation processes as if these interconnections were not very important. Often only processes affecting discrete states that are not very close to the corresponding continuum limits need to be evaluated explicitly. This important simplification arises because the rates of collisional processes controlling the relative populations of highly excited states tend to be very large, thus facilitating the establishment of so-called partial local thermodynamic equilibrium (PLTE, see sections 7.1 and 7.6). However, the lower excited level populations often need explicit evaluation, especially because these levels tend to give rise to the strongest lines. Note also that electron collisional rates usually, but not always, dominate because of their high velocities and of their energy-dependent cross sections that are similar to those for ions.

The quantitative description of atomic radiation reviewed in the previous chapter requires knowledge of energy levels and of level populations, which will be the principal subjects of chapters 6 and 7, and of oscillator strengths or of the closely related line strengths. Their calculation and measurement are the main topics of the present chapter. As to energy levels and wavelengths, a large body of high quality empirical data is available (Moore 1949-1958, Martin, Zalubas and Hagen 1978, Cowan 1981, Kelly 1987a and b, Bashkin and Stoner 1975, 1978, 1981), and atomic structure calculations (Cowan 1981, Sobel'man 1992) are usually of an accuracy that is sufficient for most plasma spectroscopy applications. Our knowledge of oscillator strengths, and therefore also of transition probabilities for spontaneous transitions and line strengths, has also greatly improved since the predecessor of this monograph (Griem 1964) was written. These advances are the result of improved experiments and computations and of critical evaluations of data (Wiese, Smith and Glennon 1966, Wiese, Smith and Miles 1969, Fuhr and Wiese 1995, Martin, Fuhr and Wiese 1988, Fuhr, Martin and Wiese 1988). Especially complete and accurate data are now available for atoms and ions of carbon, nitrogen and oxygen (Wiese, Fuhr and Deters 1995).

Although most of the electromagnetic radiation from many natural and laboratory plasmas is atomic in origin and therefore subject to quantum effects, it remains useful to introduce some of the basic radiative processes via classical theory. Other important foundations of plasma spectroscopy are atomic physics and plasma physics, especially the statistical mechanics of ionized gases. Generally, the basic theory is well established in these parent disciplines. However, the large variety of processes contributing to emission or absorption spectra often requires more or less drastic simplifications in their theoretical description or computer modeling. Critical experiments are playing an essential role in checking the reliability of various models and in delineating the region of their applicability.

Plasma spectroscopy, although being a highly specialized subfield, is at the same time a very interdisciplinary science. It not only owes its origins largely to astronomy, but also returns to astronomy and astrophysics methods of analysis of spectra and a multitude of basic data, which have both been subjected to experimental scrutiny. The state of stellar plasmas is significantly influenced by radiation, and the latter is more or less controlled by radiative energy transfer. Internally consistent treatments of the states of matter and radiation first developed by astronomers are now also becoming important for the description of plasma experiments.

In addition to the most general applications of spectroscopic methods to density and temperature measurements, which were discussed in the two preceding chapters, there are numerous special applications. Some of these will be discussed in this concluding chapter, without any prejudice against any well-established or recently developed methods which are omitted. If there is any common thread, it is in the strong role played by atomic collision theory and by the physics of atoms and ions in electric and magnetic fields. As in the other chapters, no attempt will be made to describe the often very sophisticated instrumentation or other experimental details, which the interested reader should be able to find with the help of the references in the original papers.

The first two special applications to be discussed, namely charge exchange recombination and beam emission spectroscopy, obviate the essential difficulty in emission or absorption spectroscopy in obtaining spatial resolution along the line of sight. This is accomplished by injecting heating or diagnostic neutral beams into magnetically confined plasmas either to preferentially populate some excited states of plasma ions by charge exchange recombination, as discussed in section 6.5, or by having the atoms in the beam ionized and excited by the plasma electrons, as discussed in sections 6.2 and 6.3, or even by protons and other plasma ions (Mandl et al. 1993), see section 6.5.

In this and the following chapters, various applications of plasma spectroscopy will be discussed. Their selection is necessarily somewhat arbitrary, but they will hopefully serve as useful demonstrations of the general methods and principles described in the preceding chapters. A very broad class of applications is concerned with the energy loss or gain of plasmas because of emission or absorption of electromagnetic radiation. As usual, the need for comprehensive calculations of these processes is shared with astrophysics. Here the requirement of energy conservation within a stellar atmosphere not subject to any significant nonradiative energy transport must be imposed by having zero divergence of the spectrally integrated radiative flux which, in turn, is obtained from the radiative energy transfer equations of the preceding chapter. In many laboratory plasmas such a general approach is not necessary, because most of the emission normally comes from optically thin layers and because radiative heating, except for radio-frequency (Golant and Fedorov 1989) and microwave heating (Bekefi 1966), is not involved.

Very notable exceptions to the last point are laser-produced plasmas, in which the absorption of the, typically, visible laser light is indeed essential (Kruer 1988). Other exceptions are x-ray heated plasmas produced, e.g., for the measurement of absorption coefficients of hot and dense low, medium and high Z materials (Davidson et al. 1988, Foster et al. 1991, Perry et al. 1991, Springer et al. 1992, 1994, Schwanda and Eidmann 1992, DaSilva et al. 1992, Eidmann et al. 1994, Winhart et al. 1995).

Next to the qualitative determination of the chemical composition of a plasma from emission and absorption line identifications, the measurement of electron and ion or atom temperatures is the oldest application of spectroscopic methods to plasma and gaseous electrical discharge physics, not to mention astronomy. It continues to play an important role, e.g., in fusion research (DeMichelis and Mattioli 1981, 1984 and Kauffman 1991). In the laboratory, independent methods based on laser light scattering and Langmuir probes are available, as already mentioned in the introduction to the preceding chapter. However, in astronomy spectroscopic methods normally must stand alone. Another important distinction is the usually dominant role of radiative transfer (see chapter 8) in astronomical applications, compared with the relatively small optical depth in some useful portion of the spectrum of most laboratory plasmas.

In many cases, it is necessary to distinguish between kinetic temperatures of electrons, ions and atoms, say, Te, Tz, and Ta. These temperatures may differ from each other even if the individual velocity distributions are close to Maxwellians, because, e.g., electron-electron energy transfer rates are much larger than electron-ion collision rates, as are ion-ion rates (Spitzer 1962). In most applications, at least the electrons do have a Maxwellian distribution, and we will assume this here.

This book was written for the benefit of young researchers in diverse disciplines ranging from experimental plasma physics to astrophysics, and graduate students wanting to enter the interdisciplinary area of research now generally called plasma spectroscopy. The author has attempted to develop the theoretical foundations of the numerous applications of plasma spectroscopy from first principles. However, some familiarity with atomic structure and collision calculations, with quantum-mechanical perturbation theory and with statistical mechanics of plasmas is assumed. The emphasis is on the quantitative mission spectroscopy of atoms and ions immersed in high-temperature plasmas and in weak radiation fields, where multi-photon processes are not important.

As in the author's previous books on plasma spectroscopy and spectral line broadening written, respectively, over three and two decades ago, various applications are discussed in considerable detail, as are the underlying critical experiments. Hopefully, the reader will find the numerous references useful and current up to the latter part of 1995. They provide advice concerning access to basic data, which are needed for the implementation of many of the experimental methods, and to descriptions of instrumentation.

The author has once more benefited from his experience in teaching special lecture courses at the University of Maryland and recently also at the Ruhr University in Bochum and some of its neighboring institutions.

Atoms and ions containing residual bound electrons do not quite resemble the simple harmonic oscillator model used so successfully in the classical theory of radiation. However, replacing the atoms or ions with sets of harmonic oscillators of a great number of discrete resonance frequencies and having various amplitudes, together with the results of classical radiation theory, go a long way toward a quantitative description of emission or absorption spectra. The set of resonance frequencies is obtained from measured or calculated energy levels using Ritz's combination principle. The amplitudes are associated with matrix elements of appropriate quantum mechanical operators between wave functions of the two energy eigenstates involved at a given frequency. In other words, quantities of the emitters, absorbers, or scatterers are described quantum-mechanically, whereas the electromagnetic field is treated classically.

Such semi-classical description of matter-electromagnetic field interactions became unnecessary very early in the development of quantum theory. It will therefore not be discussed in any detail. Instead, we will begin immediately with the combined theory of matter and radiation (Heitler 1954, Dirac 1958, Loudon 1983).

Quantum theory of particles and fields

There are various ways to also quantize the electromagnetic fields (Cohen-Tannoudji, DuPont-Roc and Grynberg 1989), of which that performed on the combined Hamiltonian equations of motion for the field-matter system is followed here.