stark broadening models for plasma diagnostics
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Stark broadening models for plasma diagnostics
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2014 J. Phys. D: Appl. Phys. 47 343001
(http://iopscience.iop.org/0022-3727/47/34/343001)
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Journal of Physics D: Applied Physics
J. Phys. D: Appl. Phys. 47 (2014) 343001 (33pp) doi:10.1088/0022-3727/47/34/343001
Topical Review
Stark broadening models for plasmadiagnostics
Marco Antonio Gigosos
Departamento de Optica, Universidad de Valladolid, 47071 Valladolid, Spain
E-mail: [email protected]
Received 5 March 2014, revised 19 May 2014Accepted for publication 9 June 2014Published 8 August 2014
AbstractA review of the models used for the calculation of the Stark broadening spectrum for plasmasis presented. The aim is to provide an overview of the related theories using a simpleformalism, useful for an introduction to the study of this phenomenon, which is encountered inplasma diagnostics. A historical development is followed and some of the problems that arecurrently under investigation are mentioned.
Keywords: absolute gravimeter, speed-of-light correction, parasitic perturbations
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1. Plasma spectroscopy
The analysis of emission spectra has been, for a long time, theonly means available for studying the properties of bodies atextremely high temperatures—not to mention stellar objects,which are completely inaccessible. Even now, while othertechniques are available for the study of industrial or laboratoryplasmas, spectroscopy—passive or active—is still the mainsource of information on the internal physical processes.
Since Langmuir (1928) minted the term plasma, therehas been much progress as regards knowledge on ionic andatomic collision processes, and the study of plasma kineticsand the interaction of plasma with radiation. Together, allthese developments have helped to focus research attention onplasma physics.
There are several research avenues giving rise to thegrowing interest in the study of plasmas. Among them wemust include the efforts to achieve controlled thermonuclearreactions, the interest in natural plasmas present in the upperatmosphere and elsewhere in the solar system—almost thewhole of the universe is plasma—as well as the study ofthe Sun, of the behaviour of industrial plasmas—in welding,cutting, etc—and of systems involved in high-performancelighting, and, more recently, the high energy plasmas producedby lasers at very high density.
Broadly, we can say that a plasma is a gas formedessentially from charged particles—ions and electrons that
are not bound—but approximately neutral on a macroscopicscale. To maintain this state of matter, a large amount ofenergy is required, with the aim of dissociating moleculesand ionizing atoms, and to give ions and electrons enoughkinetic energy to avoid immediate recombination. Thereforeit is an unbalanced state which requires, for its maintenance,a permanent supply of energy, which can be internal—a resultof chemical or nuclear reactions—or external, as happens indischarge plasmas, shock plasmas, etc.
This essentially energetic character of the plasma meansthat the radiative processes have a fundamental relevance.Fortunately, the atoms and ions of the plasma are subject tostrong interactions with their environment, so the details ofthe processes of emission and absorption of radiation dependsignificantly on that environment, and thus, from the analysisof the radiation, we can obtain information about the physicalstate of the plasma.
The dominant interactions between particles in a plasmaare electromagnetic in nature. The particles have charge, sothe local electric field produced by them provides the dominantmeans of interaction. Emitting ions and atoms are under theinfluence of strong electric fields which alter their naturalemission frequencies—the Stark effect. The analysis of theStark shift and the broadening of the spectra is one of the mostimportant techniques of plasma diagnostics for plasma that isoptically thin, and it constitutes the subject of this work.
0022-3727/14/343001+33$33.00 1 © 2014 IOP Publishing Ltd Printed in the UK
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
The interpretation of spectroscopic measures as adiagnostic tool for investigating the physical state of a plasmastarts from a model of the plasma and a model of the emittingatom. The quantities more commonly measured for use asdiagnostic tools are the wavelengths of the radiation emittedby the plasma and the widths and shifts of the spectral lines.The former show the presence of elements or ionization statesinside the plasma. The kinetic temperatures of the emittersand the density of perturbers can be obtained from widths andshifts. Also, the ratios between the intensities of differentspectral lines emitted by atoms or ions of the same speciesallow us to find out whether the state populations of that specieshave a distribution in thermodynamic equilibrium, and, if theydo, what the occupation temperatures of the levels are.
The plasma model has to properly take into account theinteractions of the emitter with its environment of charged orneutral particles, the collective phenomena, etc, for differentphysical conditions of the charged perturber density andtemperature of the plasma. On the other hand, the model whichrepresents the emitter must consider, with enough accuracy, thephysical structure of states and the transition probabilities.
The combination of the two models will allow us tointerpret the widths, shifts and intensities of the spectrallines as a function of the parameters which characterize theplasma. In view of this, it is extremely important to maintain aproper balance between the inherent complexity of the physicsof the phenomena and the tractability of the accompanyingequations, in order to provide useful diagnostic tools andfeasible computational models.
2. The broadening of the spectral lines
The ideal situation of a spectral line corresponding to radiationof a perfectly defined frequency ω0 cannot be observed innature, not even in isolated atoms. A spectrum of this typewould be described by a function I0δ(ω − ω0) of zero width.The existence of the natural width, a consequence of the limitedduration of the emission process of the radiation due to theinteraction of the emitting atom with the radiation itself (Betheand Salpeter 1957), determines that the isolated spectrum has aLorentz profile shape, besides what is introduced unavoidablyby the instrument used to measure it (the instrumental functionof the spectrometer). However the most important causes ofthe spectral line broadening rest on the process of interactionof the ions and emitting atoms with their environment and ontheir own dynamic behaviour.
Two important mechanisms are able to producebroadenings and shifts of the spectral lines in a plasma.The first of these, the statistical Doppler effect, appears asa consequence of the fact that the radiation registered hassuffered the Doppler effect due to the relative movement ofthe emitter and the observer. The radiation results in anincoherent statistical mixture of individual emissions of eachemitting atom with a frequency shift caused by the Dopplereffect associated with its velocity. The shape and width of theobserved line are determined by the velocity distribution ofthose emitters. Then, the spectra give us information aboutthe temperature of the emitters. This broadening mechanism
depends very weakly on the nature or magnitude of theinteractions between particles and only takes into account theemitter velocities1. This is the phenomenon which lets us findthe kinetic temperature of low density plasmas, in which theinteractions between particles are unimportant.
The second broadening mechanism is what is knownas pressure broadening. Its origin lies in the interactionsthat affect the emitter arising from the surrounding particles.Depending on which interaction affects the emitters, we talk ofStark broadening, caused by the electric field generated by thefree charged particles of the plasma, or a van der Waals effect,when the interaction is due to the dipole–dipole interactionbetween the emitter and the perturbers, etc. These phenomenadepend strongly on the magnitude of the perturbations, which,at the same time, depend on the number of particles capableof producing them. Thus this is a broadening mechanism thatdepends on the pressure in the plasma.
2.1. The Stark broadening
In the case of Stark broadening, the local electric fields fromthe free charges which surround the emitters are capableof modifying the emission process, giving rise to shifts inthe emission frequencies or changes in the phase of theradiation, which are observed, as a whole, as a phenomenon ofbroadening and shift of the lines. This is an effect determinedfundamentally by the intensity of the local electric fields,thereby enabling us to find the density of charged particlesin the plasma.
Since the start of the studies of this sort of line broadening,the theories have adopted two different mathematicaltreatments according to whether the effect of the electric fieldsis produced by electrons—highly mobile—or ions—slower,due to the large mass. In the first case—very fast interactions—one can talk of two sorts of phenomena. On the one hand,there are the perturbations caused by particles which generatevery intense fields by colliding with the emitters at very shortdistances. They are the strong fast collisions. In those casesthe broadening of the lines appears as a consequence of asuccession of cuts in the emission phase, which alters itscoherence and leads to a set of shorter independent wave trains.In this sense, line broadening is ruled by the collision rate andnot by the detailed effect of each individual collision.
In contrast, when the collisions are fast and weak, theformal treatment considers that the broadening and the shift ofthe line are caused by the accumulation of small changes ofthe emission phase, changes that, in practice, are independentof each other, which leads also to a Lorentzian broadening.
At the other extreme, when the electric fields experiencedby the emitter vary slowly—on the timescale of the atomemission—the effect produced is the typical Stark shift ofthe spectral line. In fact, the broadening shows the statisticaldistribution of the quasi-static electric field experienced by theemitter. In this sense, this kind of broadening is a non-coherentsuperposition of spectral lines individually shifted by the Starkeffect, resembling the way in which Doppler broadening arises
1 Except with respect to the phenomenon of Dicke narrowing (Dicke 1953,Firstenberg et al 2007), but this is not relevant to what we are discussing.
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as a superposition of individual lines shifted by the Dopplereffect.
In 1959, a special circumstance arose in the history ofthe knowledge of spectral line broadening processes. The firstcomprehensive review of the studies in this field appeared,published by Margenau and Lewis (1959). This review coversall the knowledge developed from 1906, with the worksof Lorentz (1906) concerning impact broadening, up to themodern formulation of this model constructed by Anderson(1949), and covers the range from extremely important paperby Holtsmark concerning the statistical distribution of theelectric microfield, which was published in 1919 (Holtsmark1919), to the work of Ecker (1957), which extends thecalculations of this distribution, including the Debye shielding.
It is between 1958 and 1969 that the fundamental worksconcerning Stark broadening of the spectral lines appear.These give the first models that permit performing realisticcalculations. They are the works by Baranger (Baranger1958a, 1958b, 1958c, 1958d), Griem et al (Griem 1962a,1962b, 1966, 1968, Griem and Shen 1961, Griem et al 1959,1962a, 1962b, Kolb and Griem 1958), Lewis (1961), Fano(1963), Sahal-Brechot (1969a, 1969b), Cooper and Oertel(1967), Smith and Hooper (1967), Smith et al (1969a, 1969b),Vidal et al (1970a, 1970b), Smith et al 1969c, Voslamber(1969, 1970, 1972a, 1972b, 1976) and so on. The first resultsused a semiclassical treatment—the atom is treated accordingto quantum mechanics and the radiation is considered aclassical object—in which the separation between the effectsof electrons and ions was admitted. On the one hand, theelectrons cause a sequence of almost instantaneous changesin the phase of the emitter (the impact effect) giving riseto a Lorentzian broadening of the line. On the other hand,the ions are considered quasi-static and their effect is ruledby the statistical distribution of the local electric microfield(Baranger and Mozer 1959, Mozer and Baranger 1960, Hooper1966). From that moment, and until the end of the 1970s,a breakthrough in the development of models that accountfor the Stark broadening occurred. Griem himself publishedin 1964 a book which collects together all of the knowledgeabout the subject and provides spectra tables for use in plasmadiagnostics (Griem 1964). In these calculations the case ofhydrogen lines was discussed in detail.
Hydrogen is the ideal candidate for use as a means ofconducting plasma diagnostics for two reasons. First, it is thesimplest atom and, therefore, the simplest to model. Second,due to the accidental degeneracy of its energy levels, the Starkeffect is linear, so its spectrum is very sensitive to local electricfields. It is also the testbench for any theoretical developmentthat seeks to explain the line broadening.
In the 1960s a large number of works were published withthe aim of improving the models and providing experimentalinformation about the broadening of spectral lines. Note that inmany cases, this technique is the only one available for plasmadiagnostics. Most of those works have hydrogen as the objectof study, in part because of the magnitude of the effect in thatelement, which make it very useful for the purpose. This is thecase for the works by Mclean and Ramsden (1965), Hill andFellerhoff (1966), Gerardo and Hill (1966), Hill and Gerardo
(1967), Morris and Krey (1968), Shumaker and Popenoe(1968), Bengtson et al (1970), Konjevic et al (1970a, 1970b,1970c, 1970d, 1971), Puric et al (1971a, 1971b), Fussmann(1972), Wiese et al (1972) and so on.
However, just because of the enormous effect observed,the lines of the hydrogen or hydrogen-like ions have someinconveniences: at very high densities their lines overlapwith lines of other elements and the analysis is very difficult;and when the temperatures are very high, the hydrogen iscompletely ionized and no spectrum can be recorded. Inaddition, it is not always possible to find hydrogen in theplasma, so the need to extend the studies to other emittersquickly emerged.
The first semiclassical calculations of the Starkbroadening parameters for isolated lines2 were also carriedout by Griem and his collaborators (Griem 1962a, Griemand Shen 1961, Griem and Shen 1962, Griem et al 1962b),Cooper and Oertel (1969), and in particular by Sahal-Brechot(1969a), Sahal-Brechot (1969b), who presented a treatmentwith remarkable mathematical improvements.
In 1974, Griem published an extensive work compilingall the information available up to that moment and providingvalues of the broadening parameters—the width and shift—fora huge number of lines from neutral emitters—from heliumto calcium and caesium—and singly ionized emitters—fromhelium to calcium (Griem 1974).
2.2. The effects of the ion dynamics
In 1973 the first relevant discrepancy between the theoreticaland experimental results was observed. In an impeccable work,Kelleher and Wiese (1973) showed that the ion movement hasremarkable consequences for the shape of the spectral lines.As usual, the study was carried out on hydrogen. The completework appeared in Wiese et al (1975). This experimental facthad been predicted some time before by Dufty (1970). Theresults of Wiese et al (1975) allow us to extrapolate the data tothe case where the ion dynamics vanishes (which is equivalentto considering the reduced mass of the emitter–perturber pairto be infinite). The authors (Wiese et al 1975) found that theextrapolation fits relatively well to the values obtained usingthe models which consider static ions and emitters. This lastresult suggested that the quasi-static field approximation forthe ions was the source of the discrepancies.
In 1977, Grutzmacher and Wende (1977) managed torecord precisely the Stark broadening spectra of the Lymanalpha line (which requires vacuum ultraviolet spectroscopy),and they observed a strong discrepancy between the resultsobtained and the data provided by the models published upto that date. This was a very relevant result since the Lymanalpha line is the simplest line of hydrogen, and at the same timethe hydrogen atom is the simplest one. This work was furtherextended one year later, including experimental recordings of
2 In speaking about spectra, the term isolated line refers to a spectral linegenerated by transitions between energy levels for which the perturbationsare not able to induce any transition (isolated energy levels). On this basis,line broadening is studied as a process of spontaneous emission in which theenergy between the states involved is altered because of the collisions, but noother surrounding states are involved.
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the Lyman beta line (Grutzmacher and Wende 1978), withidentical conclusions. The discrepancies were not negligible,since, for example, in the case of the Lyman alpha line thedifferences between theory and experiment as regards thewidth of the line are of about a factor of 4. These authorsfound, as had already been done in the experiment by Wieseet al (1975), that, in the central zone of the spectral lines,the theory–experiment discrepancies were considerable. Inthe central zone of the spectra, the shape of the line profileis determined by the effect of the weak and fast collisions;therefore it is for the dynamic aspects of the line broadeningprocess that the theoretical treatment needs to be improved.
Around that time, new models were developed withthe aim of including the effect of the movement of ionperturbers on the spectra calculations. This aim was hardto achieve, since the already established model assumed, asan accepted simplification, a complete separation of the twoextreme mechanisms—static fields on one side, and nearlyinstantaneous fields on the other—which would mean a radicalchange in orientation with respect to the state of the art models.
In 1971, the works Frisch and Brissaud (1971) andBrissaud and Frisch (1971) proposed an entirely newcalculation technique, which was named the model microfieldmethod and remained almost unknown to the Stark broadeningcommunity up to the publication of the Lyman alpha profilesand the work of Seidel (1977a), which will be discussed later.Like the previous ones, this treatment considers the emitter tobe subjected to an electric microfield which alters its emissionprocess and thereby gives rise to line broadening. The physicalmodel of the plasma was not revised. In addition, somesimplifications with a basis in the physics of the problem wereaccepted: perturber particles move along classical trajectories;there is no correlation between them; and the action of theperturbers on the emitter is described as a charge–dipoleinteraction. The authors who proposed the MMM started froma result which was already assumed in the previous models: thestatistical distribution of the local field and the time correlationdetermine the shape of the spectral lines. The differences fromthe models developed previously lie in the statistical treatmentnecessary for performing the calculation. The publishedmodels assumed that the perturber field can be treated as asequence of collisions of ions and electrons with the emitter,and take the parameters of these collisions (the velocity of theperturber, impact parameter and orientation of the collisiontrajectory) as the stochastic variables for which the statisticalcalculations are done. On the other hand, the MMM considersthe time sequence of the microfield experienced by the emitteras a sample of a stochastic process and its calculation methodis based on statistical considerations of that time sequence,treated as a time random variable. The authors suggested that,with an appropriate stochastic model for that sequence, thedipole moment autocorrelation functions of the emitter couldbe obtained, and from this, the spectrum.
In short, these authors propose making the mathematicalapproximations in the step where the field statistics and itscorrelation are calculated, in such a way that the equationswhich allow us to obtain the spectrum can be solved exactlyonce those two functions are known. Previous models
approached this in a different way: once the statistics of thefield and its time correlation had been exactly obtained, themathematical approximations were applied in the spectrumcalculation. Frisch and Brissaud preferred to perform an exactcalculation of the spectrum with approximant field statistics.No matter which microfield model was chosen, the onlyphysics based requirement is that both its field distributionand its time correlation should match pre-established ones (forexample, the field distribution proposed by Hooper (1966) andthe time correlation by Taylor (1960), or any others). In theirwork, Frisch and Brissaud illustrate this using, as a test, a verysimple Markovian model for the perturber electric field. Thisnew treatment can be called definitely unified, namely, it cantreat equally the perturber fields caused by ions and electrons,so it can include perfectly the ion movement in the spectralcalculation.
Shortly after the experimental results concerning theeffects of ionic movements were published, Seidel (1977a,1977b) applied the idea of Frisch and Brissaud to computesome hydrogen spectra including the dynamics of both theionic and electronic fields, although not under a truly unifiedscheme since the timescales of the two species are toodifferent. Seidel uses two different stochastic processes forthe ions and electrons without any correlation between them,although the method treats them in the same way. Thistechnique deviates drastically from the standard proceduresused for plasma line broadening (Lee 1973) and, in someway, dispenses with some physical information related tothis phenomenon (for example, the collision cross sections ofions and electrons); therefore the possibilities of comparingits results with experimentally measurable quantities are lost.Some authors (Smith et al 1981) found that this technique is insome ways based on mathematical approximation more thanon physical approximation, only seeking to find an analyticalexpression for the spectral shape, at the expense of losingphysical information that can be extracted from them. Inaddition, with certain values of the perturber mass the spectraobtained have no physical sense (Sorge et al 1999). However,this procedure gives extremely useful and good quality results.Following this technique, tables of spectra for the hydrogenlines have been constructed (Stehle and Hutcheon 1999, Stehleand Fouquet 2010).
The publication of the MMM had another remarkableconsequence: the appearance of computer simulation appliedto spectral calculations. Note that the main problem of theMMM is that it needs to set a function which establishes thestatistics of the time changes of the perturber electric field—the source of the Stark broadening. It is at that point wherethis method is forced to make approximations, sometimespoorly justified. The computer simulations allow us to obtainthose time sequences of the local microfield without anymathematical approximation and with the correspondence withphysical reality easily controllable.
2.3. Computer simulation
The first paper with a computer simulation applied in spectralline shape calculations appeared in 1979. In that work,
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Stamm and Voslamber (1979) studied the effect of the ionicmovements on the shape of the Lyman beta line from hydrogen.The simulation was proposed as an alternative to the Brissaudand Frisch (1971) treatment, recently commented on. After all,the computer simulation methods are intended to generate thetime sequence for the electric microfield so that their statisticalmoments can be considered in a natural way. This is a spectralcalculation method with the same scheme as the MMM butwith the electric perturber field ‘calculated well’, or, to bemore precise, obtained from an ‘ideal experiment’.
Essentially, the simulations consider numerically usinga computer a very simple model of the plasma formed fromcharged particles (perturbers) and emitters (neutral or charged).Through the calculations, the movement of the perturbers overtime is reproduced and the electric field experienced by theemitter due to the perturbers is calculated at every moment.This microfield sequence determines the time behaviour ofthe emitter; this evolution is obtained by solving numericallythe corresponding Schrodinger equation. Once the evolutionoperator of the atom or ion under that perturbing field isknown, the emission or absorption process of the radiationcan be calculated. Then, the spectral profile is obtainedfrom an average taken over a large number of individualprocesses which repeat the same procedure with differentplasma configurations, that is, when there is a representativesampling of the electric field affecting the emitters in a plasma.
Stamm’s formalism and simulation technique did notlead to the widespread use of these methods for calculatingspectral lines because the computational cost was prohibitivewith the means available at that time. Indeed, in workfrom 1979, Stamm and Voslamber only simulate the ionicmovement and they use a theoretical approximation to includethe effect of the electronic collisions. This simplificationmakes the calculation possible, since it drastically reducesthe number of numerical operations required to solve thedifferential equations: the timescale of the ions is much biggerthan the electron timescales, so fewer simulation steps arerequired. In addition, they use a simulation technique with freeand independent particles moving with constant speed alongstraight paths inside a cubic box with periodic conditions. Thisre-injection procedure was modified in a later work (Stammand Smith 1984). To integrate the equations they employed aRunge–Kutta technique which does not guarantee the evolutionoperator’s unitarity, so they were forced to truncate the verystrong ion-emitter collisions. In spite of these practicallimitations, the idea could be useful for studying in detailsome aspects of the analytical models, so from the beginningthis technique was proposed as a testing laboratory for theapproximations adopted by theoreticians. The first works—between 1979 and 1985—were directed in that sense. Oneof those, which is particularly relevant, was used to providethe basis of one of the hypotheses used in the model microfieldmethod—the approximation known as the µ-ion model (Seideland Stamm 1982)—that has been adopted since then, notonly in that treatment, but in all simulations. Stamm’s workallowed us to check that the discrepancies between theory andexperiments as regards the Lyman lines decreased notably ifthe ionic movement was included (Stamm et al 1984a).
In 1985, Gigosos et al (1985) published an algorithmcapable of notably reducing the computational cost of solvingthe differential equation for emitter evolution in the caseof hydrogen, which permitted calculating the spectra ofBalmer lines using this technique. Thereafter the systematiccalculation of Stark broadening spectra using simulationswhich include ions and electrons in the plasma could beconsidered (Gigosos and Cardenoso 1987). At the same time,several numerical methods for treating the simulated plasmawere developed (essentially as regards the re-injection particletechnique (Gigosos and Cardenoso 1987)), using sphericalenclosures, avoiding the cycles that appear in simulations withthe cubic box and periodic conditions and even nullifyingthe limitations of the finite simulation enclosure by usingan ingenious mechanism of particle generation (Hegerfeldtand Kesting 1988). These methods were used and improvedin the following years (see: Calisti et al (1988), Frerichs(1989), Alexiou et al (1999), Poquerusse et al (1996),Alexiou and Leboucher-Dalimier (1999), Sorge et al (2000),Halenka and Olchawa (1996), Olchawa (2002), Wujec et al(2002), Halenka et al (2002), Wujec et al (2003), Olchawaet al (2004))—all of them with simulation techniques usingindependent particles, without interaction). The calculationshave permitted the construction of tables of spectra to beused in plasma diagnostics, both for hydrogen (Gigosos andCardenoso 1996, Gigosos et al 2003) and for neutral helium(Gigosos and Gonzalez 2009, Lara et al 2012, Gigosos et al2014), and even in studying the spectra obtained through two-photon polarization spectroscopy techniques (Gigosos andGonzalez 1998, Gonzalez and Gigosos 2000).
The simulation methods have been employed to obtaininformation about some details of the plasma behaviour and itsconsequences for the spectral line broadening, starting with thedynamics of the local electric field (Smith et al 1984, Stammet al 1984b), or the range of the approximations of binary andcomplete collision used in the standard model (Gigosos et al1986). Frerichs (1989) analyse the approximations adopted inthe calculations based on the MMM in which the interactionsof ions and electrons are split into two processes that arestatistically independent; Godbert-Mouret et al (1998) usethe simulations to validate the effects included in the analyticcalculations of spectra from ionic emitters at high electrondensity; Barbes et al (2001) study the coupling between thebroadening mechanism of impact and effects due to the quasi-static field; Alexiou (2005) uses the simulations to contrasthypotheses (which are rejected) used in some models ofanalytical calculations; Gigosos et al (2006) studies the effectof strong collisions produced by the electrons in evaluatingthe dependence of the so-called Wiesskopf radius on thetemperature; Stambulchik et al (2007) analyse the effects of thecorrelations in the movements of the charged particles and theirconsequences for the broadening of Balmer alpha line; Gigososand Gonzalez (2009) use the effects of the ion dynamics toquantify the distance from balance in a plasma discharge; andso on.
In general, the simulations have accomplished two goals:they permit analysing the analytical model and studying it indetail since they work as an ideal experiment in which the
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conditions can be not physically realistic, while also showingthe reach of the mathematical approximations employed in themodels; and, on the other hand, they give directly a practicalresult which allows us to link the parameters of the plasmawith the shape of the spectral lines—the main purpose ofthis diagnostic technique. This last goal can be achievedat the expense of a huge amount of computation; however,current computer systems allow this way of working withoutprohibitive cost, so simulations have been gaining relevanceand reliability.
2.4. The opacity project
In 1984 an international collaboration was formed with the aimof establishing reliable methods of calculation for obtaining theatomic data required for estimating stellar envelope opacitiesand other related quantities. The team was coordinatedby Professor M J Seaton, from the Department of Physicsand Astronomy, University College, and there were groupsinvolved in it from France, Germany, the United Kingdom, theUnited States and Venezuela. It was intended to give an answerto the Simon (1982) conjecture: the opacities usually used donot agree with the models of the observed stellar properties.Then, it was proposed that it should collect informationconcerning state equations, atomic physics and spectral linebroadening. The project was not intended to develop theoriesof high precision. Reliable and fast methods of calculationof transition probabilities and widths of spectral lines wererequired. And simulations cannot compete in this sense. Theseefforts led to the development of programmes of calculationsof transition probability and microfield distributions, for a veryshort time—the project was terminated in 1995. The resultswere especially useful in astrophysics, but also in laboratoryplasma diagnostics; the institution which was the most activein spectral calculation was the Lawrence Livermore NationalLaboratory, with the aim of using the data in diagnostics forplasma produced by lasers.
Nowadays, this group have a database availableat http://cdsweb.u-strasbg.fr/topbase/home.html (see Seaton(2005)).
2.5. The frequency fluctuation model
With the aim of supplying a fast method of calculation ofspectral lines, the group Diagnosis dans les gaz et les plasmas,formed by researchers from CNRS and professors from theUniversity of Marseille, developed a technique called thefrequency fluctuation model (FFM) (Calisti et al 1994, Talin etal 1995).
The analytical methods used until that moment obtainedspectra as superpositions of Stark components, each withhomogeneous broadening (Lorentzian curves) due to the effectof electronic collisions. The shift and intensity for each of thesecomponents are given by the statistics of the ionic field, whichis considered static. The homogeneous part of the broadeningis due to the electrons (fast) and the non-homogeneous part tothe ions (static). The new treatment follows the same scheme,but avoiding the quasi-static approximation for the ions, so itcorrects the causes of the main observed discrepancies between
theory and experiment. For that purpose, the calculationsare organized into two steps. In the first one, as in thealready known models, a set of components with homogeneousbroadening are established, components which arise from theStark splitting and whose sum, properly weighted, leads tothe quasi-static Stark spectrum. These components includethe broadening due to the fast collisions of the electrons, andwhose magnitude is known through a conventional impactmodel. In the second step, these components are mixedfollowing a Markov process, in a similar way to that adoptedfor the phenomenon produced in saturation spectroscopy (Talinet al 1983).
The aim is to calculate the spectra through a frequencyfluctuation model of the radiation emitted by the averageemitter. Talin and Klein (1982) published a study proposalconcerning the redistribution of radiation in a system subjectedto random perturbations based on the same idea. The frequencyfluctuation is caused by the perturbations (very complex)experienced by the emitter (Talin et al 1995). It is not necessaryto put forward any hypothesis concerning the nature of thestochastic process related to the perturber field. Instead, it isconsidered that the process of frequency mixing is a Markovprocess with a certain exchange rate. This parameter is chosenas the average rate of the fluctuations caused by the ionic fieldand it is obtained from simulations or from a simple model ofthe field–field correlations.
Then, the process is established through two functions,similarly to what happens with the MMM: the statistical dis-tribution of the static ionic field and the change rate whichrules the frequency fluctuations. The ionic field distributionalso determines the transition channels which are going to beconsidered in the calculations, and their transition probabili-ties. The model is considered valid since its results are verysimilar to the ones from the simulations (Calisti et al 1995).
The formal description of the average emitter is similar towhat is used for a molecule with a very large set of states linkedthrough the thermal bath, which is what plasma represents(Talin et al 1997). The calculation method has a physical(and formal) justification very similar to that for describingthe Dicke narrowing, which uses a kinetic process equationto explain the Doppler broadening in a gas in which theemitters change their velocities due to the collisions with thesurrounding particles (Bureyeva et al 2010, Calisti et al 2010).
This calculation technique turns out to be very efficient, asit provides data from the very complex spectra in a simple andfast way, and with a quality similar to that offered by computersimulations, which are taken as the accuracy reference.
This calculation method was extended to the study ofother phenomena in plasma—for example, the radiationredistribution in a dense and hot plasma (Mosse et al 1999).
2.6. Other calculation methods
The aim proposed for The Opacity Project remains inforce as regards the calculations of Stark broadeningspectra. The analytical calculation methods which mix theimpact approximation adopted for considering the electroniccollisions with the quasi-static field approximation used for
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the treatment of the ionic field effects give results of pooraccuracy, since the ion dynamics effects, which are veryimportant under some circumstances, are not included. Thecomputer simulations cover those cases perfectly; however, thecomputational cost makes them inappropriate for the purpose.The FFM provides a very convenient solution, but it requiressome verifications, for which, ultimately, calculations usingcomputer simulations are required.
There are other calculation methods which, in somecircumstances, give appropriate results. Some of thesemethods have been developed in searching to improve thecomputational efficiency, at the cost of loss of the highaccuracy. For example, the so-called frequency separationtechnique (FST), developed by Alexiou (1996), separatesthe correlation function of the emitter electric dipole (whoseFourier transform is the spectra) into the product of twofunctions, one with the information of the correlation functionsdue to the perturbations of high frequency and another whichtakes into account the effects of low frequency fields. Thisseparation is justified in an empirical way from analysing theresults obtained in computer simulations. The author admitsthat this method restricts the accuracy of the calculationsbut, in recompense, it simplifies them numerically in a veryimportant way.
The so-called BID method (standing for the Boercker–Iglesias–Duffy method (Boercker et al 1987)), which followsan analytical treatment, studies the line broadening processas one further example of a physical process for which itis necessary to determine the correlation function of twomagnitudes which depend on the degrees of freedom of theatoms and on the pseudorandom effects of the plasma in whichthey are immersed. Boercker et al (1987) presents very generalequations for setting the dynamics of this correlation function,based on the method of the projector operator of Zwanzig(1961) and linearizing the integrodifferential equation obtainedby them. This procedure is quite general, and, in fact, it allowsthe authors to determine transport and radiative properties ofions in highly coupled plasmas following the same formalism.As also happens with other treatments, their results are in goodagreement with molecular dynamics simulations.
Other models have progressed, finding high accuracy forthe approximation used in the standard theory (ST), whichsplits the effects of the ions and of the electrons into twoextreme treatments (impact and quasi-static). In particular,there have been attempts to resolve the weak points ofthat model, as it is the fixing of the cut-off radius that isrequired to avoid the divergences appearing as a consequenceof the linearizations carried out in the calculations (see thereview by Touma et al (2000)). Other authors have tried toimprove the calculation of the electronic collisions, retainingthe impact approximation, by performing a more detailedquantum calculation of the scattering matrix at different levelsof approximation, depending of the degree of coupling betweenstates and including fine structure effects (Elabidi et al 2004).
Recently two workshops have been held concerningspectral line shapes for plasmas (the first and second SLSP codecomparisons, held in Vienna in 2012 and 2013), where severalresearch groups have contrasted their methods of spectral
calculation with examples for extreme situations, with the aimof improving their own methods and determining the physicalor numerical basis of the differences between the results.One of the most commonly discussed aspects is the precisenumerical complexity of this problem and the need for methodsthat are well physically substantiated but with a realistic chanceof being applicable in plasma diagnostics. Their results willbe published soon.
At this moment, it is accepted that computer simulationsprovide the best quality results. However, their use is stillvery computationally intensive and it is not possible to usethis approach for real time diagnosis in plasma spectroscopylaboratories. For this reason, there is a concern to reducecomputation times which is generating continuous researchinterest in simplified calculation methods.
3. Formal treatment of the Stark broadening
In this section the formalism of the calculations of the Starkbroadening spectra which is common to all the treatmentswhich can be consulted in the bibliography concerning thistopic will be shown. In the development, we will explain theformal aspects or the mathematical or physical approximationwhich distinguish between models. In general, all thetreatments share a physical model: the Stark broadeningphenomenon is caused by the local electric fields producedby the plasma over the atoms and emitting ions. There aretwo subsystems involved in the process: on the one hand, theemitting atoms, subjected to the plasma action and, on theother, the plasma itself or the set of perturbers. The latter isconsidered as a thermal bath which alters the emission processbut whose behaviour is not affected by this process. In thissense, the plasma is always considered optically thin and theequilibrium of the radiation is not taken into account.
Since the early works by Anderson (1949), andaccording with the Wiener–Khintchine theorem (Wiener 1930,Khintchine 1934), the shape of a spectral line from a dipoletransition can be obtained from the power spectrum of thedipole moment of the atomic emitter. Formally,
I (ω) = Re1
π
∫ ∞
0dt eiωt {C(t)}, (1)
where C(t) is the autocorrelation function of the atomic dipolemoment, which, in the quantum treatment of the emitter, hasthe expression
C(t) = tr [ D(0) · D(t)ρ] , (2)
where ρ is the density matrix which accounts for thepopulations of the states involved, and D(t) is the operatordipole moment in the Heisenberg picture:
D(t) = U+(t)D(0)U(t). (3)
Here U(t) is the evolution operator of the emitter which obeysthe Schrodinger equation:
ihd
dtU(t) = H(t)U(t) = (
H0 + qE(t) · R)U(t), (4)
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
in which the emitter Hamiltonian, H(t), includes the structureof the unperturbed states, H0, and the effects of the chargedperturbers through the dipole interaction qE(t)·R. The vectorE(t) describes the time sequence of the electric microfieldwhich acts over the emitter, caused by all surrounding chargedparticles.
Then, this formalism considers that the recorded spectrumis an incoherent superposition of the dipole emissions of theatoms or ions from the plasma. In this way, the average that wehave denoted with the symbols { } in expression (1) is, actually,an average over all possible microfield time sequences whichcan be experienced by the emitter: it is an average over theplasma, which can be written as { }E . The other averaging,related to the possible initial states of the emitter, is performedby taking the trace of the operator product which appears inexpression (2), in which the density matrix accounts for thestatistical distribution of those states.
4. The quasi-static field model
If the electric field appearing in (4) is static, the evolutiondifferential equation has an immediate solution:
U(t) = exp
[− i
hH t
]
= exp
[− i
h
(H0 + qE · R
)t
], (if E is constant).
(5)
In this situation, the autocorrelation function of the dipoleemitter is a sum of harmonic oscillations. In fact, QE isthe unitary matrix which converts the eigenstates of H0 toeigenstates of the complete Hamiltonian H . So, QEHQ+
E
is diagonal and for that state basis (determined by the value ofthe field E) one can write
D(E)ji (t) = e
ihEj t D
(E)ji (0)e− i
hEi t , (6)
so we have
C(E)(t) =∑ij
e− ih(Ei−Ej )tD
(E)ij · D
(E)ji ρii . (7)
The ρii set the population of the ith state. This sum is extendedto the eigenstates of H whose eigenvalues are the quantitiesEi . These energies, naturally, depend on the static perturberfield and can be written as a sum of the contribution of thenon-perturbed atom Hamiltonian H0 and the contribution ofthe interaction between the emitter and the perturber field:
E (E)k = E (0)
k + �E (E)k . (8)
(We have used a superscript (E) to indicate quantities thatdepend on the perturber field.) The quantities �E (E)
k accountfor the spectral line shifts due to the electric field—the Starkeffect. Proceeding in this way, the autocorrelation function ofthe emitter dipole is written
C(E)(t) =∑ij
e−i(ω(0)ij +�ω
(E)ij )t D
(E)ij · D
(E)ji ρii
≡∑ij
I(E)ij e−iω(E)
ij t , (9)
ω0 ω0
ns
n sn p
E
S
S
P
(a) (b)
= 0 E = 0
Figure 1. Stark effect in a transition between n′p P and ns S states.The initial state n′p P is coupled to the n′s S state by the externalelectric field. The spectral lines have been represented with anatural width that has nothing to do with the Stark effect.
with ωij = (Ei − Ej )/h and Iij the intensity of thecorresponding spectral component.
Figure 1 presents an example of this situation. In side(a) of the figure the spectrum of a typical transition betweena P state and an S state without an external field is shown.For the same atom—side (b) of the figure—in the presenceof a stationary electric field a splitting of the spectral line canbe observed and, for the example shown, the appearance of aforbidden transition3.
For a plasma, the resulting spectrum is a statistical mixtureof configurations of this sort (see expression (1)), each onecorresponding to a value of the perturbing field:
I (ω) =∑ij
{I
(E)ij Re
1
π
∫ ∞
0dt exp
[i(ω − ω
(E)ji
)t] }
E
=∑ij
{I
(E)ij δ
(ω − ω
(E)ji
) }E
. (10)
This average can be expressed as
I (ω) =∑ij
∫d3E W(E) Iij (E) δ
(ω − ωji(E)
), (11)
where W(E) is the probability density for the perturbing fieldhaving a value of E. To calculate the spectrum it is necessaryto know the eigenstates and eigenvalues of the Hamiltonianof the expression (4) and the probability distribution of theelectric field inside the plasma.
Figure 2 gives a picture of the meaning of theexpression (11). It represents a simplified example of atransition with a unique spectral component. The resultingspectrum is a superposition of spectra, each one shifted by the
3 The appearance of these lines is caused by the new stationary states—eigenstates of Hamiltonian H with a perturbing electric field—being a mixtureof the eigenstates of H0, and, thus, they do not obey the selection rules whichimpede the transition between the states n′s S and ns S. The position andintensity of these forbidden lines strongly depend on the value of the perturbingfield, which depends on the charged particle density. Therefore, these linesare very useful for identifying the electron density.
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
ω0(E1) ω0(E2)
ω0(E1) ω0(E2) E1
E2
ω0(E1) ω0(E2)
ω
P (ω(E))I(ω(E))
++
++
+
++ +
++
+
+
+
+
+
Figure 2. Stark broadening quasi-static model. Two diagrams showelemental spectral transitions corresponding to two differentconfigurations of the emitter subjected to the field created by thesurrounding particles. The frequency of the transition depends onthe perturber electric field E. The resulting spectrum is theincoherent superposition of these heavy transitions with theprobability distribution for the perturbing field taking thecorresponding value E.
Stark effect and weighted by the probability distribution W(E)
for the perturbing field taking the value corresponding to thatshift and having the line intensity Iij (E) in that configuration.The appearance of an isolated component in this model hasthe shape shown in the figure. This is one of the causes of theasymmetry of a Stark broadened spectral line.
4.1. The Holtsmark model
The first complete calculation of a statistical distribution ofthe field inside a plasma was performed by Holtsmark, whoin 1919 (Holtsmark 1919) obtained such a distribution for thecase where the plasma can be considered formed of free andindependent particles.
Holtsmark starts from the following expression:
W(F ) =∫
d3r1 d3r2 . . . , d3rn P (r1, r2, . . . , rn)
×δ (F − E(r1, r2, . . . , rn)) , (12)
which accounts for the fact that the probability density W(F )
for the field taking the value F is given by the weightof the configurations of a set of n particles located at thepoints (r1, r2, . . . , rn), producing a field E(r1, r2, . . . , rn)
which matches with F . P(r1, r2, . . . , rn) accounts for theprobability density of the configurations of the aforementionedparticles.
This calculation becomes clearer if one handles thecharacteristic function of the probability distribution, namely,that in the Fourier space. So we can write
A(k) ≡∫
dF eik·F W(F )
=∫
d3r1 d3r2 . . . , d3rn P (r1, r2, . . . , rn)
× exp [ik · E(r1, r2, . . . , rn)]
= ⟨eik·E ⟩. (13)
Thus, the characteristic function of our distribution is the meanvalue of the complex exponential exp(ik · E) in which theelectric field appears. This field can be expressed as the sum ofthe fields produced by each of the n particles of the plasma, so
A(k) =⟨e∑n
j=1 ik·Ej
⟩=⟨
n∏j=1
eik·Ej
⟩. (14)
Holtsmark (1919) considers here the cases in which theparticles of the plasma can be considered statisticallyindependent. This situation will occur when the energy oftheir interaction is negligible compared with the energy of theplasma. This is the case for a high temperature plasma (infact, it is the limit for T → ∞). Under such conditions, theaverage of the product indicated in (14) can be written as aproduct of the individual averages which, in this situation, areall the same:
A(k) = ⟨eik·E ⟩n. (15)
This is equivalent to having adopted in the expression(13) P(r1, r2, . . . , rn) = (P (r))n. If the plasma ishomogeneous and isotropic, this probability distribution issimply P(r) d3r = d3r/V , with V being the volume of theplasma considered. Then,
A(k) =[
1
V
∫V
d3r eik·E(r)
]n
=[
1
V
∫V
r2 dr sin θ dθ dϕ eikE(r) cos θ
]n
=[
4π
V
∫V
dr r2 sin(kE(r))
kE(r)
]n
. (16)
Holtsmark (1919) uses Coulombian fields, so E(r) =q/(4πε0r
2), with q being the value of the charge of eachparticle considered. The integral which appears in (16) is moreeasily solved if we take
A(k) =[
4π
V
∫V
dr r2 − 4π
V
∫V
dr r2
(1 − sin(kE(r))
kE(r)
)]n
=[
1 − 2π
V
∣∣∣∣ q
4πε0k
∣∣∣∣3/2 ∫ ∞
0
dx
x5/2
(1 − sin(x)
x
)]n
(17)
where we have made the substitution x ≡ kE(r) =kq/(4πε0r
2). This variable change allows us to give theintegral as the simple numerical value 4
15
√2π . In addition,
we have extended the integral to the extremes (0, ∞), takingit that in the limit V → ∞ we have n → ∞, but keeping theplasma density N ≡ n/V constant. Thus, putting V = n/N
we have
A(k) = limn→∞
1 − 1
n
(2π
(4
15
)2/3q
4πε0N2/3 k
)3/2
n
= exp[−(kF0)
3/2]
(18)
where we have defined the so-called Holtsmark normal field
F0 ≡ 2π
(4
15
)2/3q
4πε0N2/3 =
(2
5
√2π
)2/3q
4πε0r20
= 1.001768 E0. (19)
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
H
0.4
0.3
0.2
0.1
0.00 5 10 15 20
Holtsmark�distribution
(β)
β = E/F0
Figure 3. The Holtsmark distribution. The abscissa axis gives thefield using the scale F0, which depends on the electron density asN2/3.
E0 is the Coulombian field generated by a charge located atthe typical distance in the plasma, r0 ≡ (3/(4πN))1/3.
Finally, the distribution function of the field is obtainedby inverting the Fourier transform in (13):
W(E) = 1
8π3
∫d3k e−iE·k A(k). (20)
The function A(k) depends only on the modulus of k, so wecan follow the same steps as in (16):
W(E) = 1
2π2
∫ ∞
0dk k2 A(k)
sin(kE)
kE. (21)
As is expected, this distribution depends only on the modulusof the field E, so we can use the probability density P(E) ofthe said modulus:
P(E) = 4πE2 W(E) = 2
πE
∫ ∞
0dk k e−(kF0)
3/2sin(kE).
(22)
We introduce a normalized field
β ≡ E
F0(23)
and we put x ≡ kF0 in (22), so the function that we are lookingfor is expressed in terms of the Holtsmark distribution:
H(β) = F0 P(E) = 2
πβ
∫ ∞
0dx exp
[−x3/2]
x sin(βx).
(24)
Figure 3 shows the graph of this function (calculated bynumerical integration of (24)). If we had a spectral line witha single component and its shift due to Stark effect was linearwith the field, then the quasi-static Stark spectrum would havethe shape indicated in the figure. Both the width of the lineand its shift would depend on the electron density as N2/3, asthis is the scale of the abscissa axis in the figure. In a real case,naturally, there are always several Stark components whoserelative intensities depend on the perturber field, so a spectralline is a superposition of curves similar to the one shown.
4.2. Models of non-independent particles
The Holtsmark calculation oversimplifies the problem, as itdoes not consider any effect of the correlation between plasmacharges. Ecker (1957) and Ecker and Muller (1958) slightlymodified the Holtsmark calculations to include, at least in asimplified way, the effect of the coupling between charges.For this, they considered the effective field produced by theheavy particles to be Debye screened. To be more precise,Ecker (1957) starts from the same calculation as Holtsmark,but alters the integration over r , with a change from coveringthe interval (0, ∞) to covering the interval (0, rD), withrD being the Debye radius. This calculation allows us toobtain a statistical distribution which now depends on theplasma temperature, since rD =
√ε0kT /(q2N) depends on
the temperature. In the limit T → ∞, the Ecker distributionmatches the Holtsmark one.
Barely two years after the publication of Ecker, Barangerand Mozer (1959) presented a new calculation of the statisticaldistribution of the local microfield in a plasma with theaim of applying it in the calculations of Stark broadeningspectra. Their treatment employs a ‘cluster-type expansion’using the Mayer functions (Mayer and Mayer 1940) and theyextend the Holtsmark calculations considered in the first twofunctions of the development. Moreover, Baranger and Mozer(1959) introduces the concepts of a high frequency field anda low frequency field, which is essential for understandingthe physical process which gives rise to the broadening ofthe spectral lines. However, these authors separate the twokinds of fields following a criterion that is not sensitive to theeffect produced on a spectral line. Thus, they consider thatthe high frequency field is essentially the field produced bythe electrons, to which there has to be added a homogeneousdistribution of neutralizing charges. The low frequency fieldis the superposition of the fields generated by the ions, towhich is added the screening effect of the electrons—yielding,therefore, a low frequency field coupled to the ionic field. Thisseparation has led to many misunderstandings in the analysisof the Stark broadening lines.
Baranger and Mozer (1959) obtained the statisticaldistribution of the high frequency field starting from the sameexpression (20) with A(k) given by (14). Following Mayer,they set each one of the factors of (14) as
exp(ik · Ej
) = 1 +[
exp(ik · Ej
)− 1] ≡ 1 + φj . (25)
In this way, the exponential of (13) can be developed in theform
exp [ik · E] =∏j
(1 + φj
) = 1 +∑
j
φj +∑
j,k
j<k
φjφk + . . .
(26)
and, then,
A(k) = 1 +
⟨∑j
φj
⟩+
⟨∑j,k
j<k
φjφk
⟩+ . . . ≡ 1 +
∑n=1
An(k).
(27)
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
Each one of the terms An(k) is the average of products of thefunctions φj in the configurations of n particles:
An(k) =∑
j1 ,...,jnj1<j2<...<jn
∫d3rj1 d3rj2 . . . , d3rjn
×Pn(rj1 , rj2 , . . . , rjn)∏
m=j1,...,jn
φm(k, rm) (28)
with Pn(rj1 , rj2 , . . . , rjn) the probability density of the
configuration of n entities in the positions (rj1 , rj2 , . . . , rjn)
when there is a test particle at r = 0. Moreover, A(k) admitsan expansion in terms of the Ursell functions (Ursell 1927) Un
such thatA(k) = exp [H(k)] (29)
with
H(k) =∞∑
n=1
nnp
n!hn(k), (30)
hn(k) ≡∫
d3r1 d3r2 . . . , d3rn Un(r1, r2, . . . , rn)
×n∏
m=1
φm(k, rm) (31)
where we have removed the restriction (j1 < j2 < . . . < jn)
and grouped the equal terms using np!/(np −n)! ≈ nnp, which
is valid when the number of particles np tends to infinity at thesame time as V → ∞. The Ursell development allows us toexpress the functions Un in terms of Pn. Specifically, we have(Munster 1969)
U1(r1) = P1(r1) ,
U2(r1, r2) = P2(r1, r2) − P1(r1)P1(r2),
U3(r1, r2, r3) = P3(r1, r2, r3) − P2(r1, r2)P1(r3)
−P2(r2, r3)P1(r1)
−P2(r3, r1)P1(r2) + 2P1(r1)P1(r2)P1(r3) ,
. . . (32)
The form (29) is more convenient, since the function Un
converges very fast as n increases. In fact, Baranger and Mozer(1959) limited the development to the first two elements, sothey have
A(k) ≈ exp [H1(k) + H2(k)] (33)
where the values of the functions Hn(k) can be written in termsof the functions gn(r
(n)) ≡ V n Un(r(n)) according to (31).
For the case of a distribution of a high frequency field overa neutral point, the authors used the Coulomb field withoutscreening to set the functions φm which appear in (31). For thedistributions gn(r
(n)), in the first case g1(r) = 1 is obtainedand the result of (31) matches that obtained for the Holtsmarkdistribution (expression (18)):
h1(k) = −(kF0)3/2. (34)
For the second-order term, Baranger and Mozer (1959)uses the distribution function g2(r1, r2) corresponding to theDebye–Huckel potential which describes the screening effect
experienced by the ions as a consequence of the presence ofthe electrons, tending to surround them:
g2(r1, r2) = exp
[−VD(|r1 − r2|)
kT
]− 1 (35)
VD(r) = q2
4πε0
e−r/rD
r, rD =
(ε0kT
q2N
)1/2
. (36)
The authors performed the calculation of h2(k) through theintegral appearing in (31), in an approximate way. For this,first, they limit the distribution to the linear term, so they adoptg2(r1, r2) ≈ VD(|r1 − r2|)/kT , and also, they carry out adevelopment in spherical harmonics of the functions φm andVD . This development is truncated from the third order, whichleads to a more manageable equation. The result is presentedin the form of tables of the field distribution normalized forseveral cases of the ratio between the electron density N andthe temperature T (which appears in the expression for theDebye radius).
For the case of a low frequency distribution over neutralpoint, Mozer and Baranger (1960) consider that it is necessaryto take a time average in the fast fluctuations produced by thefields generated by the electrons. In this case, the electric fieldconsidered in the expression for the functions φm has to be theDebye screened field:
E = q
4πε0
r
r3
(1 +
r
rD
)e−r/rD , (37)
where the value rD is as defined before in (36). Concerning tosecond-order term h2(k), to the effect of the electron screeningthere has to be added the effect of the ion–ion correlation in theinteraction potential expression, resulting in the density whichhas to be used in the expression (36) for the Debye radius forthe calculation of g2(r1, r2) having to be 2N , to include thepresence of both species.
The same numerical treatment has been employed toevaluate the probability distribution of the field over a chargedpoint. In that case, the distribution expression g1(r) = 1 isno longer valid. When the observation point of the field isoccupied by an electron, the high frequency field is obtainedby following the same steps, by putting
g1(r) = exp
[− q2
4πε0
1
r kTe−r/rD
](high frequency on an electron) (38)
and using the Coulomb field in the expressions for φm. Inthe case of a point occupied by an ion, the low frequencydistribution is obtained in the same way, but through the Debyescreened field for the functions φm and putting the Debye radiuscorresponding to the density 2N in the distribution functionsg1 and g2. The effects which can originate from the presenceof a charge at the origin for the function g2 are considerednegligible—of the same magnitude as the approximationsmade to carry out the numerical calculation4.
A few years after the publication of Baranger and Mozer,Hooper (1966) presented an alternative method for obtaining
4 Pfennig and Trefftz (1966) reviewed the work of Baranger and Mozer, anddetected some numerical mistakes and corrected them. These authors providedthe corrected tables.
11
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
the field distributions of high and low frequency. Hiscalculation method is based on the same idea: using Mayer–Ursell functions, but with terms of another sort. He startsfrom the expression (21) (in which one goes from W(E) toP(E) = 4πE2W(E)) and writes
P(E) = 2
πE
∫ ∞
0dk A(k) k sin(kE) (39)
where
A(k) = 1
Z
∫d3r1
∫. . .
∫d3rnp
exp[−qV (r1, . . . , rnp)/kT ]
× exp[ik · E(r1, . . . , rnp)] (40)
in which Z represents the partition function of the set, V is theinteraction potential of the charges (which in the calculationsof the high frequency distribution is the Coulomb potential andin which the effect of the neutralizing background is excluded),T is the temperature of the plasma and E is the field evaluatedat the coordinate origin. As Broyles (1955, 1958) did, Hoopersplits the potential V into two terms: V = V0 +
∑i wj0 where
wj0 ≡ q
4πε0
e−αrj /rD
rj
(41)
with α being a parameter to be determined later and rD beingthe Debye radius of the plasma. This approach defines apotential V0 ≡ V − ∑
i wj0 which contains informationconcerning a Debye-type mean field. At the same time wereplace the vector E of (40) by its value as a function ofthe potential E = −∇V = −∇V0 − ∑
j ∇wj0. Someway to separate the central interactions from non-centralinteractions is sought. The central part includes an averagewhich, logically, can be discounted for the non-central part.This development is carried over into the expression (40):
A(k) = 1
Z
∫d3r1
∫. . .
∫d3rnp
exp[−qV0/kT + k · ∇V0]
×np∏
j=1
exp[−qwj0/kT + k · ∇wj0]. (42)
Then, Hooper applies the same Mayer method as was used byBaranger, but with the function
χ(k, rj ) ≡ exp[−qwj0/kT + k · ∇wj0] − 1 (43)
playing the role of the φj .This formal change does not alter the physics of the
statistical study. It simply concerns performing a seriesexpansion (which will have to be truncated, since the exactcalculation is unknown) which converges in a different way.And here Hooper introduces a new element in his process.The arbitrary parameter α introduced in (41) should be fixedsuch that it minimizes the error produced by the truncationof the series and by the simplification introduced throughthe Jacobian transformation to collective coordinates (Broyles1958) which appears in the calculation (see the details in(Hooper 1966)). However, it is more convenient to distributethe truncation errors so that an area of α values is searchedsuch that the function A(k) (and, therefore, P(E)) results
as stationary in α. Thereby, α is set at the centre of thisarea. This is justified because the truncation errors and theones from the criterion of separation between the central fieldsand non-central fields are compensated. In addition, Hooperverifies his results ‘empirically’ by comparing them with theones obtained in a computer simulation using the Monte Carlomethod. This verification procedure is maintained in all thesubsequent developments in this study.
The Hooper treatment (Hooper 1966, 1968a, 1968b),although formally different from the one employed byBaranger and Mozer (1959), actually leads to the sameresults. The data from the calculations of Baranger andMozer (1959) and Mozer and Baranger (1960), with themathematical corrections provided by Pfennig and Trefftz(1966), are essentially the same (see the Demura (2010) reviewin which the history of this development is described).
The idea of introducing an adjustable parameter tooptimize the calculation was taken up again a few years laterby Iglesias et al (1983), who developed the method known asAPEX (the adjustable parameter exponential approximation).This is a method of calculation that is an alternative to thosedeveloped by Baranger and by Hooper; however, it has behindit the same physics, and we can take advantage of the twoexact results. Iglesias calculated the statistical distributionof the field in a plasma formed by ions with a distributedcharge of Zq in thermodynamic equilibrium, together witha neutralizing continuous medium (background) with a chargedensity of Z0q. First, he made use of the knowledge of themean square value of the field
{ E · E } = kT N
ε0
Z
Z0. (44)
In addition, he considered that the interactions between allparticles can be averaged in advance except those acting overthe test charge. Namely, he considered pseudoparticles ina one-component plasma (OCP) acting on the test particlethrough an effective field which, arbitrarily (but with physicalmeaning (Dufty et al 1985)), he writes as
E∗[j ] = Zq
4πε0
rj0
r3j0
(1 + αrj0)e−αrj0 , (45)
where α is the adjustable parameter in this model. The factor(1 +αr)e−αr transforms the real charge of the ion, Zq, into theeffective charge, which already includes an average. Whenall the interactions between particles except those with thetest one are omitted, all the Ursell functions of the Mayer–Ursell development are zero except the first one, U1. So, in thedevelopment equivalent to the expression (30), we limit to
h(k) =∫
d3r U1(r)φ(k, r) =∫
d3r g(r)[eik·E(r) − 1
].
(46)
In this expression, both the radial distribution function g(r) andthe field E have to contain already the preset average. That is,we must use a certain effective density g∗(r) and the effectivefield E∗(r). To achieve this task and determine the adjustableparameter α, Iglesias et al (1983) establish that the product
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
of the effective density with the effective field must be equalto the product of the exact density with the exact Coulombianfield, E(r), that is
g∗(r)E∗(r) = g(r)E(r) ⇒ g∗(r) = g(r)eαr
1 + αr, (47)
and also, we impose that the mean square value of the fieldobtained with the distribution PAPEX(E) sought matches withthe exact value∫
dE PAPEX(E) E2 = kT N
ε0
Z
Z0. (48)
This last relation allows us to determine the adjustableparameter α. As regards the value of the radial distributiong(r), the authors calculate it using the HNC approximation ofthe Ornstein–Zernike equation (Munster 1969). The resultsobtained show an excellent agreement with the ones obtainedthrough computer simulation.
It is not the aim of this review to discuss the detailsof the methods of calculation of the field distribution insidethe plasma. This is a statistical physics problem whoseapplications go beyond the Stark broadening calculations.After all, since the Baranger and Mozer (1959) publications,the physical model considered in all treatments has remainedthe same. Most of the development work in this field has beenfocused on finding mathematical solutions that are as accurateas possible. Even nowadays, a concern of researchers isimproving the calculation algorithms to obtain the distributionfunction accurately and, above all, quickly, using a computer(Laulan et al 2008, Iglesias et al 2000). For a quick referenceon the details of the calculations, see the recent Demura (2010)review in which the unsolved questions are commented on.As regards the spectral calculations, we simply use the fielddistribution—with accuracy that is appropriate for the formaltreatment used—where the fields can be considered static.
4.3. Limitations of the quasi-static model
The quasi-static field model used in the spectral calculation isbased on an assumption which is unrealistic in most situations:the perturber electric field must be static. This, naturally, isnot true. The plasma particles and the emitter itself move,so the local electric microfield ‘seen’ by a typical emitter isnecessarily dynamic. However, when we talk about a quasi-static field we are considering that the characteristic timescaleof the typical field variations is much larger than the timescaleof the emitter dipole correlation. To fix ideas, we are going toconsider an example. Take the Balmer beta line of hydrogenin a plasma with an electron density of Ne ≈ 8×1022 m−3 andan electron temperature of T ≈ 12 000 K. We know that, withthese conditions, the full-width at half-maximum (FWHM) ofthe line is �λ1/2 ≈ 4 nm (Wiese et al 1975). According tothis, the characteristic time of loss correlation for the emitterdipole is
τd =[
2πc
λ2Hβ
�λ1/2
]−1
≈ 3 × 10−14 s. (49)
Moreover, the characteristic time of the time variation of thefield ‘seen’ by an emitting atom is given by
τc = r0
v0(50)
r0 =(
3
4πN
) 13
(51)
v0 =√
2kT
m, (52)
where r0 denotes the typical distance between plasma chargeswhose electron density is N , and v0 is the typical velocityof the particles with mass m in a plasma with temperatureT . In the example that concerns us, τc is of the order of2 × 10−14 s for the electronic field and 10−12 s for the ionicfield (H+ ions). During the time needed for the emitter dipoleto lose the correlation, the ions have barely moved 1% ofthe typical distance between them, so the field generated bythem can be considered as constant. However, the electronhas a much greater mobility, and cannot be even remotelyconsidered static, even in a configuration like this, with sucha wide spectral line as the Balmer beta one. In a case with anarrower line the situation is worse: the field produced by theelectrons has dynamics with a characteristic time much smallerthan those for the atomic dipole correlation, so the broadeningof the lines caused by these fields has a nature very differentto what is deduced from the quasi-static model.
5. Characteristic timescales in the Stark broadening
In the study of line broadening by the Stark effect, fourcharacteristic timescales are handled. We have talked aboutthree of them: the characteristic times of the fields caused by theelectrons (τcE
) and by the ions (τcI) (determined by expression
(50) with v0 the typical velocity for each species), and thecharacteristic time of the correlation loss of the emitter dipole(τd , which is the inverse of the spectral linewidth, the unknownof our problem). The other timescale is ruled by the frequencyshift of the spectral line produced by the typical Stark effectin the plasma, that is, the typical magnitude �E (E) which hasbeen used in (8):
τH = h
�E (E0). (53)
This quantity depends on the magnitude of the typical localelectric field E0 in the plasma and on the state structure of theemitting atom or ion. For hydrogen, the relation between theperturbing field and the displacement of the energy levels bythe Stark effect is linear, and has the form
�E (E0) ≈ qea0E0S = qea0S
(q
4πε0
1
r20
,
)(the linear Stark effect) (54)
where qea0S is the atomic dipole moment in the starting state ofthe optical transition under study (qe is the electron charge, a0 isthe Bohr radius and S is the reduced element of the matrix R forthe state concerned, which depends on the principal quantumnumber of the state as∼n2). Hydrogen is a special case because
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
Figure 4. Calculation example for the autocorrelation function ofthe atomic emitter dipole moment (upper figure) subjected to aperturbation whose electric field (modulation) evolves according tothe lower figure. The example shown corresponds to one of thespectral components of the Balmer beta line for a high densityplasma (see the details in the text).
the local electric field links degenerate states of the atom, whichleads to a linear Stark effect. For other elements, this situationoccurs in some special cases or when the energy differencebetween levels perturbed by the local field is small comparedwith the interaction energy of the emitter with the perturberplasma. In other situations, the Stark effect is quadratic withthe field and has a form which, in general, can be expressed as
�E (E0) ≈ q2e a2
0S2
hωij
E20 , (the quadratic Stark effect) (55)
where hωij is the energy difference between the states i andj linked by the perturber electric field which gives rise to theStark effect.
The relation between τH and τc fixes the relevant physicalphenomenon of the spectral line broadening. When τc � τH
we are in the ‘quasi-static’ broadening domain. In that case,the shape and width of the spectral lines are determined by thestatistical distribution of the perturber electric fields. Then wehave τd ∼ τH .
In the example mentioned before, corresponding to theclassical experiment of Wiese et al (1975) for the Balmer betacase, we have �E (E0) ≈ 0.01 eV and τH ≈ 7 × 10−14 s. Inthis case, τcI
� τH , so there is no doubt that the dominantbroadening effect for the spectral line is the effect of the quasi-static ionic field.
In figure 4 an example of a situation like what has beendescribed here is shown. This is the autocorrelation function ofthe dipole moment corresponding to a spectral component ofthe Balmer beta line for a plasma with the same conditionsas in the Wiese et al (1975) experiment (this spectral linehas 36 components, like what has been represented, with 18different Stark shifts). The calculation reproduces the resultsof one computer simulation—see the details below. In thelower picture, part (b), the modulation of the perturbing fieldis represented as a function of time. The scale E0 is usedfor the field (see expression (19)) and t0 = τcE
for the time(see expression (50)). In the upper picture, part (a), the
Figure 5. Calculation example for the autocorrelation function ofthe atomic emitter dipole moment in a very narrow spectral linesituation, where the correlation loss of the dipole is much slowerthan for the perturbing fields. Note the range of ordinates in side (a)of the figure (see the details in the text).
function C(t) is shown according to expression (2), wherea fixed oscillating term cos(ω0t) corresponding to the mainfrequency ω0 of the Balmer beta transition has been eliminated.What is shown, then, is the modulation of the autocorrelationfunction. Using a dashed line, the calculation result obtainedwhen considering only the ionic field is presented (see thedashed line in part (b) of the figure). Using a solid line,the result from the calculation including the total field, ionicplus electronic, is shown. With the ionic field, one can seea modulation which adapts to the field—the frequency of themodulation increases when the field increases. This curve fitsvery well to expression (7), which includes a sum of terms likethe one shown. The total resulting spectra, averaged over manysituations like this, will reproduce the statistics of the perturberfield. When the effect of the electronic field is added, with fastfluctuations, the correlation loss of the emitter is accentuated—this taking the form of relatively abrupt changes in the phase—but its net contribution is relatively weak, comparing with theeffect of the ionic field, and has slower effects and, therefore,is more effective. With the electronic field, the autocorrelationfunction of the emitter is no longer similar to a frequencymodulated signal, but now includes a modulation in amplitudewhich accounts for the fast peaks of the electronic field.
When τc τH , the dipole moment D(t) evolves muchmore slowly than the perturbing fields because they can nolonger be considered static. This has a very remarkableconsequence: the perturbed emitting atom has time to averagethe perturber field, so it becomes weaker and inefficient. Thistime-averaged field may have a value much lower than thetypical statistical one. Only occasionally are very strongfields, produced by very slow particles, effective. In this casethe treatment developed using the quasi-static model is notsuitable.
Figure 5 shows another calculation obtained throughsimulation for another extreme situation. In this case, weconsider the He I 1083 nm line for a pure helium plasma withan electron density of Ne = 2 × 1021 m−3 and a temperatureof 5000 K. The example in the figure shows the general aspectof the emitter dipole autocorrelation function (as before, the
14
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
modulation is shown since the fixed part of cos(ω0t) has beenomitted). Note that the ordinate axis only covers a smallinterval from the beginning; a time interval larger than the oneshown is required. In the figure, the effect of the perturbingfield is clearly shown. Only the very strong collisions cause anappreciable change. The rest of them give rise to a slow lossof correlation. The aspect of the modulation of C(t) is oneof a sequence of steps caused by the collisions of the particlesapproaching the emitter. When a strong collision occurs, C(t)
starts to oscillate, as we saw in figure 4, but C(t) just starts tobe modulated; the perturber particle disappears and its effectvanishes. A very intense ionic collision has been included toallow better appreciation. Even in this case the correlation lossis very small, because the ion moves away relatively quickly.
This example shows us that when the broadening is causedby very rapidly varying fields, it is necessary to distinguish twocases: the effect of most collisions, which are weak, and theeffect produced by the perturbers, which generates the intensepeaks in the local field. These latter collisions, in spite of beingvery fast, may cause abrupt changes in the phase of the emitterdipole with magnitude larger than that of the one caused by thetypical field, so it is convenient to treat them separately: theyare the strong collisions, or penetrating collisions as they havebeen called in some cases. We will explain the models for bothcases in what follows, starting with the case of fast and weakfields.
6. The impact broadening model
6.1. Fast and weak collisions
The treatment known as impact approximation is applicablewhen the scales of time characteristics are in this order: τc τH τd . The result that we are going to find is completelypredictable. The average of the autocorrelation function of theemitter dipole moment (see expression (2)) is obtained from theaverage { D(t) }, which can be written, formally, as (see (3))
{ D(t) } = { U(t, 0) }D(0). (56)
In this expression the operator U(t, t ′) marks the evolution ofthe dipole moment between the times t ′ and t and is givencompletely by the time sequence of the perturbing microfieldE(t) between the instants t ′ and t . If t τd , then {U(t, 0)} ≈11. Let us take t � τd and write
{ D(t) } = { U(t, t1)U(t1, 0) }D(0) (57)
with t1 being any intermediate time between 0 and t such thatt1 � τc and (t − t1) � τc. The evolution of the dipole betweenthe times t1 and t is marked by the perturbing field betweenthose instants and, since that interval is very large comparedwith τc, this field is uncorrelated with the previous field int1. In consequence, the operators U(t, t1) and U(t1, 0) can beconsidered statistically independent:
{ D(t) } ≈ { U(t, t1) } { U(t1, 0) }D(0) . (58)
This argument can be continued by splitting the interval (0, t)
into pieces of size �t such that τd � �t � τc:
{ D(n �t) } ≈ { U(n �t, (n − 1) �t) }{ U((n − 1) �t, (n − 2) �t) } . . .
. . . { U(�t, 0) } D(0). (59)
If the interval �t is large enough compared with τc, then it cancover a representative sample of the electric field experiencedby the emitter, so all those terms are equal and we can put
{ D(t) } ≈ { U(�t, 0) }t/�t D(0). (60)
Since �t τd , we will have that {U(�t, 0)} ≈ 11 − W �t ,with W being a certain operator which contains the informationof the mean effect of the field over the emitter. The condition of�t being very small compared with the characteristic time ofthe dipole evolution allows us to write
{ D(t) } ≈ exp [ −Wt ] D(0). (61)
In the following, we will see how to evaluate the operator W .However, what is plain is that the impact broadening is ofhomogeneous type with Lorentzian shape spectra.
In most of the cases, the dipolar transition between a setof upper levels and a set of lower levels whose distance inenergy is within the optical domain or ultraviolet is studied.The processes of collision between emitters and perturbersgive rise to exchanges of energy which are much lower thanthat energy gap, so we can consider that they are not capableof inducing transitions between the two groups of states (theno-quenching approximation) in the characteristic time of theemission process. In consequence, the evolution operatorwhich has been considered has the form
U(t) =(
Uu(t) 00 Ul(t)
)(62)
where Uu(t) and Ul(t) are two matrices which mark thetime evolutions of the states of the upper and lower groups,respectively. This is also the structure of the matrix ρ, whichaccounts for the population of the states at the beginning of theemission process: the matrix has no coherences. In fact, in thecases which will be treated in the following, the states of eachof the groups—upper and lower—have, in practice, the samepopulation, so, hereafter, the density matrix will be omitted.
On the other hand, when the emission spectrum wasconsidered, we focused only on the surroundings of the spectralline centre if our frequency shifts were small enough so as notto have to consider dipole transitions outside of the two groupsconsidered, and even less between states of the same group,whose distance in energy is far below the corresponding opticaltransition. Thereby, the dipole moment matrix appearing in (2)has the form
D =(
0 d
d+ 0
). (63)
Proceeding in this way, the trace which appears in (2) can bewritten as
C(t) = tr(U+u d Ul · d+) + tr(U+
l d+Uu · d). (64)
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
The matrices Uu and Ul are, respectively, solutions of theequations
ihd
dtUu = HuUu , ih
d
dtUl = HlUl (65)
with
Hu = Eu + H0u + qE(t) · Ru
Hl = El + H0l + qE(t) · Rl , (66)
where Eu+H0u and El +H0l are the total Hamiltonian projectorsof the unperturbed emitter over the upper and lower statessubspaces, respectively, and analogously for the projectionsof the operator R over those same subspaces. The matrices Eu
and El , each one proportional to the identity matrix, account forthe energy gap between the two groups of states. We can omitthose constant quantities, writing in (64) Uu exp(−iEut/h)
and Ul exp(−iEl t/h) instead of Uu and Ul respectively. And,similarly, hereafter, when we write Hu or Hl , we will considerthat we have dispensed with the values of Eu and El . We willthen write the autocorrelation function in the form
C(t) = eiω0t tr(U+u d Ul · d+) + e−iω0t/h tr(U+
l d+Uu · d) ,
(67)
with ω0 ≡ (Eu−El)/h, the central frequency of the line studied.Let us write
tr(U+
l (t)d+Uu(t) · d) ≡ CR(t) + i CI (t) (68)
with CR(t) and CI (t) real functions of time. This way, weobtain
C(t) = 2CR(t) cos(ω0t) + 2CI (t) sin(ω0t). (69)
With the expression written in this way, we have separated theoptical frequency part (ω0) from the low frequency ones (thefunctions CR(t) and CI (t))5. The emission profile is then
I (ω)
= 1
π
∫ ∞
0dt cos(ωt) [2CR(t) cos(ω0t) + 2CI (t) sin(ω0t)]
= 1
π
∫ ∞
0dt {[cos((ω + ω0)t) + cos((ω − ω0)t)] CR(t)
+ [sin((ω + ω0)t) − sin((ω − ω0)t)] CI (t)} . (70)
We put ω = ω0+�ω, and we calculate the spectrum around thefrequency ω0. In addition, we will omit the contribution to thespectra of the quantities such as
∫∞0 dt cos((ω + ω0)t)CR(t),
which would make sense if the functions CR(t) or CI (t) hadtime variation in that scale of frequencies. We have then
I (�ω) = 1
π
∫ ∞
0dt [cos(�ω t) CR(t) − sin(�ω t) CI (t)] .
(71)
We use the trace invariance under cyclic permutations towrite
CR(t) + i CI (t) = tr(d+ · Uud U+l ). (72)
5 The function which has been represented in figures 4 and 5 is preciselyCR(t), the term of the cosine of the optical transition.
We defined(t) ≡ Uu(t)d U+
l (t) , (73)
and carry the evolution equations (65) into the matrix d(t):
ihd
dtd(t) = Hs(t)d(t) − d(t)Hl(t). (74)
Each element dij of the matrix links one state |ψ(u)i 〉 of
the upper state set with a state |ψ(l)j 〉 of the lower state set. In
the following, it will be practical to work in the state space ofthe form |i, j〉 ≡ |ψ(u)
i 〉⊗|ψ(l)j 〉. This is the so-called Liouville
space. In this new space, the matrix d has the structure of avector, and the differential equation (74) can be written as
ihd
dt||d(t)〉〉 = L||d(t)〉〉, (75)
where each element of the matrix(Ld)ij
is obtained from(Ld)ij
=∑
k
H(u)ik dkj −
∑k
dikH(l)kj
=∑n,m
(H
(u)in δjm − δinH
(l)mj
)dnm. (76)
In compact form, it can be written as
L = Hu ⊗ 11 − 11 ⊗ H tl (77)
where the superscript t indicates the transpose.With this formalism, the average of the autocorrelation
function is written as
{ CR(t) + i CI (t) } = 〈〈d(0)|| {d(t)} 〉〉 (78)
on the understanding that the ‘scalar product’ of two matricesA and B is
〈〈A||B〉〉 ≡ tr(A+B
). (79)
The evolution of the matrix d(t) is marked by the‘Liouvillian’ L, which can be written as a sum of twocontributions:
L(t) = L0 + V(t) (80)
L0 ≡ H0u ⊗ 11 − 11 ⊗ H t0l (81)
V(t) ≡ E(t) · (Ru ⊗ 11 − 11 ⊗ Rtl
) ≡ E(t) · RRR. (82)
It is convenient to separate, in the atomic dipole evolution,the parts of fixed oscillation—the carrier signal, determinedby the energy differences of the unperturbed states—and themodulations, caused by the perturbations6. Thus, we write (forwriting convenience we omit the || 〉〉 symbols)
d(t) ≡ e−iL0t/hd(t), V(t) ≡ eiL0t/h V(t)e−iL0t/h,
(83)
ihd
dtd(t) = V(t)d(t). (84)
6 Do not forget that we have already removed the ω0 value, the centre of theoptical transition under study. The characteristic values of L0 are of the formof ωul −ω0, with ωul being the frequency of an unperturbed transition betweena certain state of the upper group and another from the lower group, and ω0the centre of the spectrum.
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
The differential equation (84) is equivalent to the integralequation
d(t) = d(t0) +1
ih
∫ t
t0
dt ′ V(t ′)d(t ′). (85)
We seek the average { d(t) } = e−iL0t/h{ d(t) } so we take
{ d(t) } = { d(t0) } +1
ih
∫ t
t0
dt ′ { V(t ′)d(t0) }
+
(1
ih
)2 ∫ t
t0
dt ′∫ t ′
t0
dt ′′ { V(t ′)V(t ′′)d(t ′′) } (86)
(we have developed the eigenvalue of { d(t) } in the integrandof (85)). In the first integral of this expression, the average{V(t ′) d(t0)} = {V(t ′)}{d(t0)}, since V(t ′) is subsequent to,and therefore independent of, d(t0). Now, the average of V(t ′)is zero, because it is an average over the field itself, which is anisotropic vector in our plasma. So the first non-zero term in ourdevelopment of integrals (86) is the one of second order. Atthis point we are going to use the impact approximation. Wehave said that when τc τH , the dipole moment d(t) evolvesmuch more slowly than the perturbing fields, so the correlationbetween V(t ′′) and d(t ′′) is lost very quickly (t ′ is the futurefor d(t ′′), so this is also independent of it); then in the averageof the product of the three magnitudes appearing in the secondintegrand of (86), we can separate the part for d(t ′′). Thus
{ d(t) } ≈ { d(t0) } +
(1
ih
)2
×∫ t
t0
dt ′∫ t ′
t0
dt ′′ { V(t ′)V(t ′′) } { d(t ′′) }. (87)
The perturbation V(t) experienced by the emitter is a stationaryrandom process, so the average {V(t ′)V(t ′′)} depends only onthe modulation of the difference τ = t ′ − t ′′. This average is,precisely, the autocorrelation function of the perturbing field(see expressions (82) and (83)):
{V(t ′)V(t ′′)} = q2∑
i,j=x,y,z
{Ei(t′)Ej (t
′′)} × Ri (t′)Rj (t
′′)
= q2
3{E(t ′) · E(t ′′)} RRR(t ′) · RRR(t ′′)
≡ q2
3�(|t ′ − t ′′|) RRR(t ′) · RRR(t ′′), (88)
where we have considered an isotropic system, so{Ei(t
′)Ej (t′′)} = 1
3δij {E(t ′) · E(t ′′)}, a function only of
|t ′ − t ′′|. We develop the matrix RRR(t) to write
{V(t ′)V(t ′′)}= q2
3�(|t ′ − t ′′|) eiL0t
′/h RRR e−iL0(t′−t ′′)/h · RRRe−iL0t
′′/h
≡ eiL0t′/h M(t ′ − t ′′) e−iL0t
′/h , (89)
where
M(t ′ − t ′′) ≡ q2
3�(|t ′ − t ′′|) RRRe−iL0(t
′−t ′′)/h · RRReiL0(t′−t ′′)/h.
(90)
Figure 6. Illustration of the magnitudes involved in the integral(91). The characteristic timescales of the field correlation (�(|t |))and of the emitter dipole evolution (d(t)) are very different.
Figure 7. Illustration of the application of the so-called impactapproximation. The integral of (91) has a typical shape, as shown inthe figure.
Let us carry this result into (87):
{ d(t) } ≈ { d(t0) } +
(1
ih
)2
×∫ t
t0
dt ′ eiL0t′/h∫ t ′
t0
dt ′′ M(t ′ − t ′′) e−iL0t′/h { d(t ′′)}.
(91)
Figure 6 shows the typical shape of the function involvedin the integral of this last expression. The field correlation islost in a characteristic time τc such that τc τH τd , withτH being the characteristic scale of the Stark effect magnitudeand τd the typical time of the emitter dipole evolution. Figure 7shows the typical shape of the integral appearing in (91)(considering that d(t ′′) is constant).
Let us now use the impact approximation7. This consistsof assuming that the evolution of the dipole moment is muchslower than those of the other functions, so we have taken atime interval �t = (t − t0) such that τc �t τd . Inequation (91) it can be considered that the integral acquiresits extreme value very quickly and d(t ′′) is replaced by d(t ′)
7 As has been pointed out, it would be more correct to call it the fast and weakinteraction approximation.
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
(the function �(|t ′ − t ′′|) acts as a Dirac δ); therefore we have
{ d(t) } ≈ { d(t0) } +
(1
ih
)∫ t
t0
dt ′ eiL0t′/hKe−iL0t
′/h { d(t ′) }.(92)
K ≡(
1
ih
)∫ ∞
0dτ M(τ ), (93)
with M(τ ) given by (90). We have taken the limitτ = t ′ − t ′′ → ∞ precisely because in the scale handled,�t � τc and the integrals already have their limit value. Theexpression (92) is, formally, similar to (85), and is equivalent,then, to the differential equation
ihd
dt{ d(t) } = K(t) { d(t) }, K(t) ≡ eiL0t/h K e−iL0t/h.
(94)
If the change introduced in (83) is reversed, we have
ihd
dt{ d(t) } = (L0 + K) { d(t) } (95)
with the solution
{ d(t) } = exp
[− i
h(L0 + K) t
]d. (96)
With this solution, the autocorrelation function of the emitterhas the form
{ CR(t) + i CI (t) } = tr
(d+ exp
[− i
h(L0 + K) t
]d
)(97)
whose Fourier transform, the Stark spectrum, is, effectively, asuperposition of Lorentzian curves:
I (�ω) = Re1
πtr
(d+
[i
(1
h(L0 + K) − �ω
)]−1
d
).
(98)
The operator in the Liouville space K contains all theinformation related to the broadening and shifting of the lines.In the literature concerning Stark broadening, the so-calledwidth operator, � ≡ iK/h, is used. Let us see the physicalmeaning of its matrix elements.
The expression (95), in terms of matrix elements, can bewritten as
ihd
dt{ diα(t) } =
∑j,β
(L0 + K)jβ
iα { djβ(t) }. (99)
Figure 8 illustrates the notation used. We label with italicindices the states of the upper set and with Greek indices thoseof the lower set. The operator which appears in parentheses inthe expression (99) accounts for the rate of jumps between thestate |ψi〉 and the state |ψj 〉 of the same group.
The first of the operators of (99), which is diagonal, hasas matrix elements (see expression (76))
L0jβ
iα = H(u)0 ij
δαβ − δijH(l)0 βα
= hωiαδjβ
iα (100)
ik
Figure 8. Scheme of transitions in the broadening line process. Theaction of the perturbing field produces transitions between states ofthe same group (upper or lower) which in the Liouville space areobserved as jumps between the emission modes.
and describes the natural oscillations of the lines studied, andthe operator � ≡ iK/h, where
�jβ
iα =∑k,γ
q2
3h2
∫ ∞
0dτ �(τ) ei(ωjβ−ωkγ )τ RRRkγ
iα · RRRjβ
kγ , (101)
accounts for the effect of the perturbing field. The magnitudeof this effect is given by the collisional cross section ofthe emitter, through the quadratic values of the operator R,measuring the mean quadratic value of the emitter dipolarmoment without perturbation, and by a Fourier transform ofthe autocorrelation function of the perturbing field. This lastfactor is a measure of the power spectra of the said fieldevaluated in the frequency (ωjβ −ωkγ ), which is what couplesthe intermediate states in the transitions caused by the action ofthe perturbing field. Those matrix elements—and, therefore,the matrix—are complex numbers, since they are the resultof a complex Fourier transform. Their real part describes thecorrelation loss of the dipolar moment, and therefore, the Starkwidth of the line, and their imaginary part is added to the termcorresponding to L0 and accounts for the shift of the line. Thecharacteristic scale of this operator is given by
� ∼ q2
h2 E20τcn
4a20 ∼ N√
T(102)
(E0 is the characteristic field, τc is the characteristic durationof the field correlation, and n is the principal quantum numberof the starting state of the transition). The impact width(caused by fast and weak collisions) is linear with the electrondensity and decreases with the temperature as 1/
√T . To this
functional dependence, there has to be added the effect causedby that Fourier transform which couples the differences inthe frequencies of the intermediate transitions. This factordepends on the dynamics of the perturbing field and, therefore,depends on the density and on the temperature in a way that isnot easy to establish.
At this point, we can check easily where the developmentthat we are describing is applicable up to. As in the firsttreatment of the quasi-static field model, let us consider aplasma of free and independent particles. In this model, the
18
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
charged particles of the plasma move along straight lines withconstant speed whose modulation fits to a Maxwell–Boltzmanndistribution. In that case, the correlation function of the fieldcan be obtained analytically without difficulty. Effectively,
�(t) = { E(0) · E(t) } with
E(t) =nP∑k=1
Ek(t) =nP∑k=1
q
4πε0
rk + vkt
|rk + vkt |3(103)
(we have considered a plasma with nP particles which occupiesa volume V ). The average over the particle configurations,indicated by { }, causes the appearance of a double sum∑
k
∑k′ {Ek(0) · Ek′(t)} whose elements are zero when
k �= k′, because different particles are considered statisticallyindependent. Then �(t) = np{Ek(0) · Ek(t)}; namely,
�(t) = nP
∫d3v
π3/2
1
v30
e−(v/v0)2∫
V
d3r
VE(0)E(t) cos(EEt )
(104)
where v0 = (2kT /me)1/2 and
E(0) = α
r2= − d
drφ(r), E(t) = α
r ′2 = − d
dr ′ φ(r ′),
with φ(r) ≡ α
r,
cos(EEt ) = r + vt cos θ
r ′ = dr ′
dr,
with r ′(r) ≡ (r2 + v2t2 + 2rvt cos θ)1/2
(we have put α ≡ qe/(4πε0) and θ is the angle between r andv), so
�(t) = 8√
π
v30
Ne
∫ ∞
0dv v2e−(v/v0)
2∫ π
0sin θdθ
×∫ ∞
0r2dr
dφ(r)
dr
dφ(r ′)dr ′
dr ′
dr. (105)
The integration over dr is immediate:∫ ∞
0r2dr
(−α
r2
)dφ(r ′)
dr= α
[φ(r ′(0)) − φ(r ′(∞))
]= α2
|vt | . (106)
The integration over dθ give us a factor 2 and the completeresult is, actually, the average over the speeds of 4πNeα
2/|vt |:
�(t) = 8Neα2√π
v0
1
|t | . (107)
It is practical to write Ne = 1/( 43πr3
0 ), E0 = α/r20 and
τc = r0/v0, so
�(t) = 6√π
E20τc
1
|t | . (108)
To evaluate the magnitude of the Stark width and shiftdue to the impact effect, it is necessary to carry this result intothe expression (101). In the case of hydrogen, in which allthe quantities ωkγ are equal, the impact width of the lines isdirectly proportional to the time integral of the correlation ofthe perturbing electric field. However, in a plasma with freeand independent particles or, to be more precise, in a plasma
that is weakly coupled, this time integral, evaluated at a neutralpoint, diverges logarithmically, both for short times and forlong times. A similar situation can be appreciated in the casesin which the states are not degenerate.
The physical meaning of this result is plain. When t → 0,the function �(0) accounts for the quadratic average of theelectric field in the plasma which, in the case of weakly coupledparticles, diverges. In other words: the local electric fieldscan become arbitrarily intense, which invalidates one of ourhypotheses adopted in the development of the impact model:that the action of the perturbers is fast and weak. The treatmentof the strong collisions, caused by very close particles, requiresanother formalism.
On the other hand, the divergence of the correlation fieldintegral at long times is a consequence of the long rangecharacter of the Coulombian interaction. However, in a realplasma, these long range effects disappear as a consequenceof the charge coupling. The Debye–Huckel model describesthis phenomenon very well. The screening between chargesavoids the long range effects; thus, the mean field is, actually,a Debye field instead of a Coulomb field.
To continue the development of the impact model and, atthe same time, consider these effects, we are going to modifysome of the expressions used.
From the point of view of the �(t) calculation, thedivergence at the origin of times has to be avoided and alsowe have to consider that for very long times, the correlationhas to tend to zero more quickly than 1/t . It is necessary tosomehow omit the particles which for t = 0 produce veryintense fields (above a certain value Emax). On the other hand,the particles which for t = 0 are located outside of a certainsphere with a radius of rmax should be considered as the causesof an initial field which vanished due to the screening producedby the particle correlation, which is not considered in thissimple calculation. Thereby, rmax will be related to the Debyeradius.
With this approach, we rewrite the expression (106):∫ rmax
rmin
r2dr
(−α
r2
)dφ(r ′)
dr= α
[φ(r ′(rmin)) − φ(r ′(rmax))
].
(109)
Now we need to evaluate the integration in dz ≡ − sin θ dθ .Specifically,∫ +1
−1
α2 dz(r2k + v2t2 + 2rkvtz
)1/2 = α2
rkvt[|rk + vt | − |rk − vt |]
={
2α2/rk if rk > |vt |,2α2/|vt | if rk < |vt |,
(110)
where rk is rmin or rmax depending on the case. Averaging thisfunction over the modulation of the velocity gives us
1
v30
[ ∫ rk/|t |
0dv v2e−(v/v0)
2 2
rk
+∫ ∞
rk/|t |dv v2e−(v/v0)
2 2
v|t |]
=√
π
2rk
erf
(rk
v0|t |)
, (111)
19
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
so, finally,
�(t) = 4πNeα2
[erf(rmin/|v0t |)
rmin− erf(rmax/|v0t |)
rmax
]. (112)
As before, we write this expression in terms of the typicalelectric field of the plasma:
�(t) = 3E20
[erf(rmin/|v0t |)
rmin/r0− erf(rmax/|v0t |)
rmax/r0
], (113)
which gives us a very compact expression for theautocorrelation function of the perturbing electric fieldproduced by the charges of the plasma excluding the oneswhich produced fast and strong collisions and taking intoaccount, at least approximately, the limitation on the rangeof the fields due to the screening of the charges. To that end,we set rmax = rD , the Debye radius in the plasma.
The function obtained avoids the divergences whichappear in the calculation of the elements of the width operator.To see this, let us come back to hydrogen. Let us evaluate thetime integral of the function �(t):∫ ∞
0�(t) dt
= 3E20r0
∫ ∞
0
dt
v0t
[erf(rmin/v0t)
rmin/v0t− erf(rmax/v0t)
rmax/v0t
]. (114)
We carry out the replacement of the factor r0/v0 = τc and thevariable change x = 1/v0t , so dx/x = −dt/t :∫ ∞
0�(t) dt = 3E2
0τc
∫ ∞
0
dx
x
[erf(x rmin)
x rmin− erf(x rmax)
x rmax
].
(115)
This integral is of the general form∫ x2
x1
dx
x[F(xy2) − F(xy1)] =
∫ y2
y1
dy
y[F(x2y) − F(x1y)]
(116)
with F(xy) = erf(xy)/xy, y1 = rmax and y2 = rmin. Inthe limit x2 → ∞ the function erf(xy) tends to 1, so F(xy)
tends to zero. In the other limit, when x1 is 0, we havelimx→0erf(xy)/xy = 2/
√π . In conclusion, we have that the
integral (115) is reduced to
− 2√π
∫ rmin
rmax
dy
y= 2√
πln
(rmax
rmin
), (117)
and, then, ∫ ∞
0�(t) dt = 6√
πE2
0τc ln
(rmax
rmin
), (118)
and thus, for hydrogen, the width operator has the form(see (101))
� = 2√π
q2
h2 E20τc ln
(rmax
rmin
)R2 (119)
For spectra of non-hydrogen-like elements, it is necessaryto calculate the matrix elements (101) numerically from amodel of the correlation function of the perturbing field, �(t),such as the one which was set up in this section or somethingsimilar. There will be more on this after we treat the strongfast collisions.
6.2. The effect of the strong fast collisions
When an electron–emitter collision can alter the phase of thedipole oscillator to a considerable extent (comparing with π ),then the model developed in the previous subsection is nolonger valid. After all, the truncation done in the development(86) is equivalent to the replacement of the phase exponentialvalue by the value. Under such circumstances, the dipolemoment correlation function C(t) experiences a change of theorder of unity, which means that the coherence of the radiationemitted before and after the collision is completely lost. Thenwe can consider, to all intents and purposes, that the radiationis formed by two wave trains that are totally independent.This situation is shown in the example drawn in figure 4, inwhich those changes are observed when there is an electroniccollision with very intense fields. The classical treatment byLorentz (1906) describes that situation perfectly. The quantumtreatment does not give a different result. After all, when acollision of these characteristics happens, the emitter statesbefore and after the collision are uncorrelated, and if thatprocess takes a very short time (comparing with the meancoherence time of the emission process), then the details ofhow the emitter evolves in those cases are irrelevant. So thequantum treatment considers, as the classical ones did, thatin those cases the correlation function suddenly falls to zeroand no longer recovers. The key aspect is setting a criterionfor detecting when these full correlation loss events are goingto happen and what the probability for them happening is.The homogeneous width of the spectral line can be readilycomputed from the frequency of these events.
To fix ideas, let us consider the phase change ofthe evolution operator produced by an individual collision(complete) between the emitter and an electron with impactparameter b and speed v (b ⊥ v):
�ϕ ≈ 1
h
∫ +∞
−∞dt 〈V(t)〉
=∫ +∞
−∞dt
(q2
4πε0h
)⟨b + vt(
b2 + v2t2)3/2 · RRR
⟩
=(
q2
4πε0h
)2
bv〈ub · RRR〉. (120)
To set this relation, we have dispensed with the feature thatthe perturbation operator V(t) does not commute with V(t ′)at different times, so the expression (120) only allows us toevaluate the magnitude of that phase (we have not consideredthe ‘time ordering’ in the calculation of the evolution operator).In any case, we consider that a fast collision is strong when thatphase change is of the order of unity (a value of π/2 changes thevalue of C(t) from 1 to 0). This gives us an impact parametervalue b below which the phase change will be even higher. It isset then as something which is known as the Weisskopf radius(Weisskopf 1933):
rw ≡ 1
ϕw
(q2
4πε0h
)2
v〈ub · RRR〉, (121)
with ϕw ≈ 1. The collisions with an impact parameter belowthat value break the coherence completely and give rise to
20
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
a homogeneous broadening of the spectral line whose widthis determined by the mean frequency of this collisions. Themodel considers that when a collision of this sort happens, itcan be assumed that the effects of the rest of the perturbers arenegligible and that a situation in which two collisions of thissort occur at the same time never happens. Therefore we canaccept that a collision frequency with impact parameter belowthat value can be obtained through a simple relation:
�w = N {vσw} = N{v πr2
w
}(122)
where N is the electron density and {·} denotes the average overthe particle velocities. We carry the result (121) into (122) andobtain the width of each Stark component originating from thefast strong collision:
�w = N
∫ ∞
0dv
f (v)
v
4π
ϕ2w
(q2
4πε0h
)2 ⟨(ub · RRR)2⟩, (123)
where f (v) = (4/√
π)v2 e−(v/v0)2/v3
0,v0 = (2kT /me)1/2.
We make the following replacements in this expression:N = (1/( 4
3πr30 )) and 〈(ub · RRR)2〉 = R2/3, and, as before,
E0 = (q/(4πε0r20 )), τc = r0/v0, so
�w = 2√π
q2
h2 E20τc
1
ϕ2w
R2. (124)
This term is added to the width operator (119) which takesthe weak collisions (with impact parameter over rmin = rw)into consideration, so, for hydrogen and for transitions withdegenerate states, we have
�w = 2√π
q2
h2 E20τc
[1
ϕ2w
+ ln
(rD
〈rw〉)]
R2, (125)
with ϕw ≈ 1. In his approximation, Griem (1974) adoptsthe value ϕw = 2, taking as its basis an analysis of theevolution process of the evolution operator phase with moreaccuracy than has been shown in this summary and using, forthe hydrogen case, the exact analytical solution of the evolutionoperator, which can be solved for that element (see such asolution in Lisitsa and Sholin (1972), or in Gigosos et al (1986),where a compact expression for the exact phase change for thehydrogen is provided).
In (125) we have adopted rmin = 〈rw〉, an averageover the velocities and the different states in the transitions.Specifically,
〈rw〉 =(
q2
4πε0h
) ⟨1
vR⟩
= 2√π
h
mev0
〈R〉a0
, (126)
where a0 = q2/(4πε0meh2) is the Bohr radius. This
calculation is nothing more than an approximation of themagnitude of the strong collision effect, and it is precisely thedetermination of the cut-off parameters rmin and rmax that hasgiven rise to several formal developments. Recent works havereturned to this topic with the aim of improving the precisionof the calculations and avoiding the divergences appearing inthis treatment (Touma et al 2000) which, we stress, are simplythe consequence of the approximation which dispenses withthe unitary character of the evolution operator.
In the spectral line calculation for non-hydrogen-liketransitions, the calculation becomes more complicated dueto the appearance of the factors exp(i(ωjβ − ωkγ )t) in thetime integral of the field coherence �(t) (see (101)). Thenit is convenient to evaluate the said correlation {E(0) · E(t)},parametrizing the electronic trajectories through the vectorsb and v and the time tb at which the particle is found at theminimum distance from the emitter, since for each value ofb and v the power spectrum of the perturbing field can bedetermined, which will facilitate the evaluation of the Fouriertransform appearing in (101). The spectrum is given in terms ofthe modified Bessel functions K0(z) and K1(z). However (seethe development in Griem (1974)), the divergence problemsat short distances appear again and, as happens with thedevelopment presented, it is indispensable to set a cut-offparameter rmin to avoid them.
7. The standard theory
The treatment of the so-called ST includes the ion effects(static) and the electron effects (of fast action) together and,from some aspect, independent. The formalism followed inthe previous section, where we split the Hamiltonian whichdescribes the plasma action over the emitter into two parts, oneof them (H0) constant in time, and which fixes the structure ofthe emitter without perturbation, and another dynamical part(V (t)) which includes the action of the electron over the emitterthrough the dipole potential, can be generalized easily if weconsider that the stable part includes the action of the ionsthrough a potential, also dipolar, in the form
HE0 = H0 + qE · R, (127)
in which the ionic field E is static. Everything developed inthe impact model is unaffected by this addition, as we see if weconsider that the electronic collisions suffered by the emitterfollow the same statistics, with the ion’s static field value beingindependent of what it is subjected to at that moment. The STassumes that the whole spectra can be obtained through anaveraging of the different configurations of the ionic field ofthe impact spectrum in each case. Then we have
I (ω) = Re1
π
∫d3E W(E) tr
×(
d+ ·[
i
(1
h
(LE
0 + KE)− ω
)]−1
d
), (128)
where the Liouvillian LE0 includes the structure of the emitter
states subjected to a static electric field E,
LE0 = L0 + qE · RRR, (129)
and the integration is done with the field statistic, as was done inthe Stark broadening quasi-static model. Naturally, the impactwidth of each Stark component (whose values are set in theoperator KE) is affected by the ionic field, since this fielddetermines the stationary values of the emitter state energy, andthe differences between those values appears in the elementsωjβ which modulates the efficiency of the electronic collisions(see the expression for the width operator in (101)).
21
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
8. The model microfield method
The Stark broadening spectrum calculation includes one stepin which it is necessary to know the statistical behaviour of thelocal electric field, which alters the emission. In the extremecase in which the emission process is much faster than thedynamics of the field (the quasi-static approximation), the fieldcan be considered time independent and it will be enough toknow just the stationary statistical distribution of the localfield. In the opposite case, when the fields change with acharacteristic time much lower than the correlation time of theemitter dipole, it will be enough to know the time correlationfunction of the electric microfield to determine the spectrumshape. It seems reasonable to think that in intermediatesituations, we would have to calculate the spectrum usingproperly the information of those two statistical functions.In the treatment described so far, the averaging needed hasbeen performed considering that the dynamical field is asuperposition of collisions, and we have taken the collisionparameters as random variables. In the so-called modelmicrofield method (Frisch and Brissaud 1971, Brissaud andFrisch 1971, Seidel 1977a, 1977b) we will embark on thiscalculation dealing directly with the field as a time randomprocess. To illustrate the procedure followed in this calculationmethod, we are going to summarize here the model employedby Brissaud and Frisch (1971) to determine the spectra.
We maintain our physical model of the line broadeningprocess and we start from the expression (56) which determinesthe time evolution of the emitter dipole from the operatorU(t, 0). As we did with expression (57), we are going tosplit the evolution operator into a sequence of operators, eachone covering a very short time interval compared with thecharacteristic scale of the field evolution. This is differentfrom what we saw earlier: in our development of the impactapproximation (expressions (56) to (60)) we took time intervalsthat were short on the dipole evolution scale, but largecompared with the scale of the field evolution. Here we taketime intervals that are arbitrarily small. In some way, we arereplacing the time evolution of the perturbing field E(t) with astep function in which the field is considered constant during acertain very short time. Then our field sequence is consideredto be defined by the relations
E(t) = Ek , tk � t < tk+1, (130)
where the times tk are the jumping times at which the fieldchanges its value, and the values Ek describe the history ofthe perturbing field. Our treatment must include a statisticalmodel which determines the distribution of those jumpingtimes and the statistical laws of the field values at those times.In their exposition, Brissaud and Frisch (1971 use a kangarooprocess, which means that the jumping times follow a Poissondistribution with a density ν(E) which depends on the fieldvalue. This model is determined by the probability transition(assumed as stationary) given by
Ptr(E, �t |E′, 0) = [1 − ν(E′)�t
]δ(E′ − E)
+ν(E′)�t Q(E). (131)
In this expression, Ptr d3E accounts for the probability thatthe electric field takes the value E (with a tolerance of d3E)in an infinitesimal time �t after the previous value of the fieldwas E′. This is a conditional probability of the transition fromE′ to E in the interval �t . The function Q(E) representsa normalized statistical distribution which will be determinebelow. ν(E′) realizes the jump frequency corresponding to theinitial value E′ and its inverse provides the typical duration ofa field of magnitude E′.
The physical meaning of the expression (131) is clear: thefirst term, (1 − ν(E′)�t), accounts for the probability that inthat time the field value does not change (there is no jump).In the second term, ν(E′)�t accounts for the probabilitythat the field value changes. The factor Q(E) weights theprobability that the new value is, effectively, E. Therefore,this last distribution is not the same as the stationary statisticaldistribution of the field (which will be called W(E), and whichhas been studied before) since it is a conditional distribution.Note that the distribution W(E) measures the probability thatthe field has a value of E at any time independently of itsprevious value. On the other hand, Q(E) accounts for theprobability that the field has the value E as a consequence ofa change. So the field values with small jumping frequenciesν(E′) have values of W(E) larger than those set by Q(E).The Fokker–Planck equation for the statistical distribution ofthe field in this process has the form
∂W(E, t)
∂t
= lim�t→0
1
�t
[ ∫d3E′ Ptr(E, �t |E′, 0) − W(E, t)
]
= −ν(E)W(E, t) + Q(E)
∫d3E W(E, t)ν(E).
(132)
If W(E) is the stationary distribution of the field, then
Q(E) = ν(E)W(E)∫d3E W(E)ν(E)
= ν(E)W(E)
〈 ν(E) 〉P . (133)
(We have denoted by 〈·〉P the averaging done with thestationary statistical distribution W(E), which corresponds towhat is used in the quasi-static model.)
The knowledge of the jumping probabilities allows us toobtain the average of the evolution operator. If {U [n](0, t)} isthe average operator for the situations in which the field hasexperimented n jumps in the interval (0, t), the total averagecan be obtained as
{ U(t, 0) } =∞∑
n=0
{ U [n](t, 0) }. (134)
Here, the symbol {·} has been used to denote the average overthe possible field sequences. It is clear that
{ U [0](t, t ′) } = { exp[− i
hLE(t − t ′)] exp[−ν(E)(t − t ′)] }E
=∫
d3E W(E) e−ν(E)(t−t ′) US(E; t − t ′),
(135)
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
where LE = L0 + qE · RRR, the Liouville operator whichdescribes the evolution of the emitter subjected to the singlestatic field E. We have set US(E; t, 0) as the solution of thisevolution in this case. In general, for n > 0, according to ourdefinition (131), we have{ U [n](t, 0) }=∫ t
0dtn
∫d3En Q(En)ν(En) e−ν(En)(t−tn) US(En; t − tn)
×∫ tn
0dtn−1
∫d3En−1 Q(En−1)ν(En−1) e−ν(En−1)(tn−tn−1)
×US(En−1; tn − tn−1) × . . .
. . . ×∫ t1
0dt ′∫
d3E W(E)ν(E) e−ν(E)t ′US(E; t ′). (136)
We must use the conditional field probability Q(E) in everyinterval except the first one, which lacks a previous condition.This expression can be written in the compact formfor n > 1 :
{ U [n](t, 0) } =∫ t
0dt ′∫
d3E Q(E)ν(E) e−ν(E)(t−t ′)
×US(E; t − t ′) { U [n−1](t ′, 0) } , (137)
for n = 1 :
{ U [1](t, 0) } =∫ t
0dt ′∫
d3E Q(E)ν(E) e−ν(E)(t−t ′)
×US(E; t − t ′)
×∫
d3E W(E)ν(E) e−ν(E)t ′ US(E; t ′), (138)
and the case n = 0 is as in (135). These recurrence relationscan be written as convolution products:
{U [n](t, 0)} =∫ t
0dt ′TQ(t − t ′){U [n−1](t ′, 0)}, for n > 1,
(139)
{ U [1](t, 0) } =∫ t
0dt ′ TQ(t − t ′) TW(t ′), (140)
where
TQ(t) ≡∫
d3E ν(E)Q(E) e−ν(E)(t−t ′) US(E; t, 0) (141)
TW(t) ≡∫
d3E ν(E)W(E) e−ν(E)(t−t ′) US(E; t, 0). (142)
The Laplace transformation allows us to transform theconvolution products into ordinary products and the recurrencerelation (139) shows that the sum of the terms from n = 1 isthe sum of a geometric series. Then{U(s)} = 〈US(s + ν) 〉P + 〈νUS(s + ν)〉P
× [〈ν11 − ν2US(s + ν)〉P]−1 〈νUS(s + ν)〉P . (143)
We have replaced Q(E) according to its value given in (133),so all the averaging is done with the set field distribution W(E).Therefore, in (143), for instance (L [·] symbolizes the Laplacetransformation),
TQ(s) = L[TQ(t)
] = 1
〈ν〉P
∫d3E ν(E)2W(E)
×∫ ∞
0dt e−st e−ν(E)(t−t ′) US(E; t, 0)
= 1
〈ν〉P 〈 ν2US(s + ν) 〉P , (144)
and (we set s = −iω)
US(E; s + ν) = [ (s + ν(E))11 + iLE ]−1
= [ ν(E)11 − i(ω11 − LE) ]−1 . (145)
This is our microfield model. The development has madeit possible to carry out exactly the calculation of the evolutionoperator average in the Fourier space (which is what we need toobtain the spectrum). To apply this to the spectral calculationwe have to set two functions: the stationary distribution of thefield W(E) (we can take, for instance, that of Hooper (1966))and the jumping frequency ν(E) depending on the electricfield value. However, in this case, this latter function can beobtained from the correlation function of the field. Effectively,our model assumes that the field sequence follows a Poissonprocess with a jumping frequency which depends on the fieldbut without memory, that is, the fields before and after the jumpare uncorrelated. In this way, we have
�(t) = { E(t) · E(0) } =∫
d3E W(E) E2 e−ν(E)t
=∫ ∞
0dE E2P(E) e−ν(E)t . (146)
We are dealing with an isotropic plasma, so the two functionsdepend only on the modulation of the field. We have replaced4πE2 W(E) by P(E) and ν(E) by ν(E). The distributionP(E) is normalized, so
∫∞0 dE P(E) = 1.
The jumping frequency ν(E) is expected to be anincreasing (in fact, monotone increasing) function of the field.The duration of the field becomes shorter as its value becomesbigger. Therefore, the integral (146) converges (except for thecase t = 0, as the covariance of the field, {E(0) · E((0)},measured at a neutral point diverges). If we consider that, infact, ν(E) is a monotonic function of E with ν(∞) = ∞,we can invert the calculation operation in (146) and determineν(E) from �(t). We perform a variable change in the integral,taking ν as the new variable of the integration:
�(t) =∫ ∞
ν(0)
dν e−νt E2(ν)P (E(ν))dE
dν≡∫ ∞
ν(0)
dν e−νt g(ν).
(147)
This is a Laplace transformation which can be inverted, so,knowing �(t), we determine ν(E). For instance, if we takethe expression (108), we have that g(ν) = 6E2
0τc/√
π , so
ν(E) =√
π
6τc
∫ E
0dE P(E)
(E
E0
)2
. (148)
This step is crucial in the model development: the setting ofthe jumping frequency ν(E) to enable us to reproduce thecorrelation function of the field which we consider valid. Inhis work, Seidel (1977b) uses an expression for �(t) that ismore sophisticated than (108). To do this, he considers thatthe plasma is formed from independent pseudoparticles whichproduce Debye fields. To obtain the correlation function ofthe field in this case, we have to use the Debye potentialφ(r) = α exp(−r/rD)/r in the expression correspondingto (105). In that case, the integration is equally simple andleads to the result (Brissaud et al 1976)
�(t) = 4πNeα2
rD
⟨(rD
|vt | − 1
2
)exp(−v|t |/rD)
⟩v
, (149)
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
where rD is the Debye radius and 〈〉v denotes the average overthe velocities:
�(t) = 8√
π
v30rD
Neα2∫ ∞
0dv v2e−(v/v0)
2
(rD
|vt | − 1
2
)e−vt/rD .
(150)
The expression (150) itself is a Laplace transform in thevariable ν = v/rD . Specifically,
�(t) = 8√
π
ν3DrD
Neα2
(L
[ ∫ ν
0dν νe−(ν/νD)2
]
−1
2L[ν2 e−(ν/νD)2
]), (151)
so, in this case,
g(ν) = 4√
π
νDrD
Neα2(
1 − [1 + (ν/νD)2]e−(ν/νD)2)
, (152)
where we have set νD ≡ v0/rD = ωp/√
2 (ωp is the plasmafrequency). From the definition of g(ν), we can determine thejumping frequencies corresponding to this microfield model:
ν(E)
νD
+1
2
(ν(E)
νD
)2
e−(ν/νD)2 − 3
4
√πerf(ν/νD)
=√
π
3
rD
r0
∫ E
0dE P(E)
(E
E0
)2
, (153)
which can be numerically solved without difficulty once thestationary field distribution P(E) is set.
This treatment concentrates on the Stark spectracalculation problem in the determination of the jumpingfrequency ν(E) from a microfield model which sets the rightfunction �(t). From there, the calculation is exact. Howeverits weakness came from the fact that such a microfield modelassumes that the jumps produced at the times tk are Markovianand allows that, in these jumps, the fields change arbitrarily. Infact, when establishing the relation (131), one is consideringthat the probability of going from a field E′ to another, E,P(E′ → E), is proportional to a product of two functions,ν(E′) and Q(E), each one independent of one of the fields—and thus, independence between them. It is not reasonableto think that that situation is real, since one would expect adependence of the type f (E − E′)ν(E′) with f (x) a certainfunction with a peak around the value x = 0 (Smith et al 1981).It is necessary to introduce some kind of statistical correlationbetween the two values in each jump, and this eliminates thepossibility of continuing with the exact analytical developmentas had been done previously.
9. Computer simulations
The formal treatment of the Stark effect in computer simulationis, actually, the simplest of all. The aim is to obtain thepower spectrum of the dipole moment correlation of a typicalemitter of the plasma through the expression (1). For this it isnecessary to know an average of the correlation function C(t),defined in expression (2); each one of these is obtained throughthe matrix operation indicated in (3) once the differentialequation (4) has been solved. The only requirement, asidefrom the numerical calculation tools for solving this differential
equation, is a time sequence of perturbing electric microfieldsE(t) whose statistical pattern fits with the correspondingplasma whose physical characteristics (composition, densityand temperature) we want to reproduce. That is all.
Naturally, a simulation can be considered an idealexperiment carried out through numerical calculation; it cannotreproduce with accuracy a real physical situation. It isnecessary to make approximations. The first of the limits ison the size of the plasma. Only a small set of particles isinvolved in the calculation. In some cases, a cubic enclosurewith periodic conditions (the particles which escape througha wall of the cube enter through the opposite wall) hasbeen used, and in other cases, a spherical enclosure. Inthis latter scheme, it is necessary to set up an algorithm toreinject the particles, ensuring the stability of the position andvelocity statistics in the plasma enclosure (see the details ofthe procedure in Gigosos and Cardenoso (1996)). On theother hand, the simulation can include or ignore the interactionbetween particles. In most of the published works which giveresults for Stark spectra, particles without interaction havebeen used, namely, the simulated perturbers travel followingstraight paths with constant speed with no correlation betweentheir movements. Simply, a sampling of positions andvelocities which satisfy the expected statistical distribution(homogeneous and isotropic plasma with velocities distributedaccording to Maxwell–Boltzmann statistics for the equilibriumtemperature) is established. In other cases, a mutual interactionis included, which requires one then to solve the Newtonequation for their movements. In these cases, the volume ofcomputation is enormous. Note that it is necessary to have aplasma enclosure with the number of particles of the order ofthousands (at least), since it is necessary to guarantee that suchenclosures have dimension clearly larger than the reach of themean field—the Debye radius.
Once the movement of the particles of the plasma (ions,electrons and neutral emitters, as the case may be) hasbeen established, the field over the emitter is evaluated.All the particles of the enclosure contribute to this field.If the interactions between particles are not considered—an independent particle simulation—then it is necessary tocorrect the calculation by introducing artificially some effectof correlation between the particles. To do that, a Debye fieldis used instead of the Coulomb field. In this way, at leastapproximately, the behaviour of a real plasma is reproduced.Conversely, if the interaction has been included, the field actingover the emitters and which is introduced in the evolutionequation (4) is directly the Coulomb field. The effects ofcoupling between charges and screening of the interactionsemerges naturally if the interactions have been taken intoaccount in the plasma dynamics.
In a simulation with a spherical enclosure, only the centreof the sphere is a useful point for calculating the field at. Atthe centre, then, is where the emitter whose evolution is goingto be determined is placed. In such cases, it is necessary toadd a physical approximation: it is considered that such anemitter is at rest (this is in its own inertial reference system) and,then, any dynamic effect of the emitter itself is not taken intoaccount. It is necessary, then, to modify the velocity statistics
24
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
of all the other particles, since we are considering statisticsof relative movements. This is what is known as the µ-ionmodel (Seidel and Stamm 1982): the velocity distribution ofthe perturbers fits to a Maxwell–Boltzmann distribution inwhich the mass of the particles is replaced by the reducedmass µ of the emitter–perturber pair. This model permitsus, also, to consider plasmas in which the ion temperatureand that of the electrons are different, that is non-equilibriumsituations, which are very frequently encountered in the areaof industrial plasmas: the µ parameter of the emitter–perturberpair appears in the formal equations of calculations as a termµ/T in the velocity statistics. So through that parameter, boththe perturber mass and its temperature can be regulated.
Simulation with a cubic enclosure and periodic conditionsallows us to obtain the field at several points (with manyemitters included in the calculation). However, this approachcannot be used with independent particles, since the entrancesand exits of the particles in the enclosure are periodic andthis is reflected in the spectrum calculation. Naturally, withinteracting particles, the enclosure must have dimensionsclearly larger than the Debye radius (which must also holdfor the spherical enclosure in the other case). This simulationmodel does not need the µ-ion approximation, but the non-equilibrium configurations are not easily reproducible.
Once the particle dynamics of the plasma is obtained andthe electric field acting over the emitter has been calculated, itis necessary to solve the Schrodinger equation (4) numerically.At this point, the simulations assume two approximations: theemitter–plasma interaction is limited to the dipole interaction(or to the sum of the dipole and quadrupole interactions, forwhich it is necessary to evaluate also the field gradients),and the matrices H0 and R are limited to including a finitenumber of states. Moreover, in this stage of the calculationsit is convenient to avoid the problems appearing in theanalytical calculations and which force one to introduce cut-off parameters to avoid the divergences. Let us examine thisin more detail.
In a practical calculation, the emitter evolution is carriedout through time steps of size �t , so the time sequence of theperturbing field is defined as described in the model microfieldmethod (see expression (130)) with tk = k �t . The step sizehas to be chosen such that the time evolution of the fieldis obtained accurately. Therefore �t τc, where τc isthe characteristic time of the electron field evolution. Thus,in the short time interval between tk and tk+1 = tk + �t
the electric field can be considered constant. Thereby, theevolution operator can be written as
U(tk+1) = UL(tk+1, tk) U(tk), U(0) = 11 (154)
and taken as the solution of the Schrodinger equation in theinterval (tk, tk+1):
UL(tk+1, tk) = exp
[− i
hHk �t
]
= exp
[− i
h(H0 + qEk · R) �t
]. (155)
The operator constructed in this way is unitary and remainsunitary at any time. Thereby, some mathematical problems
which appear when the perturbing electric field Ek takes verylarge values are avoided. Not all simulations which can befound in the literature use this procedure. In some cases,Runge–Kutta–Fehlberg techniques which do not guaranteethe unitarity of the operator U(t) are used. The method ofexpression (155) is the most computationally expensive, whichis the price that you have to pay if you require that the verystrong collisions do not present any problem.
To calculate the exponential of a matrix, the most efficientmethod requires one to diagonalize this matrix, and thiscalculation is very expensive computationally, especially ifthe matrix includes a large number of states. In the caseof hydrogen lines, the objective for the first simulations,this calculation stage is considerably simplified, because thesymmetry of the structure of the states for this element and thefact that they are degenerate allow us to reduce the dimensionof the matrix used considerably: the calculation can be donewith real 4 × 4 matrices independently of the main quantumnumber of the states involved (Gigosos et al 1985). Also,this matrix can be parametrized as a function of only eightreal numbers (two quaternions, which describe the Euler–Rodrigues parameters of a rotation). However, for other atoms,this simplicity cannot be maintained and, if one wants toguarantee the unitarity of the operator U at every time, it isnecessary to diagonalize this matrix in every time step. For this,it is particularly useful to have an algorithm based on successiveapproximations like the Jacobi method (Press et al 1992),since, in each time step of the simulation, the diagonalizationdone in the previous step is a good approximation to that forthe current step, because the perturbing electric field changesby a very small amount from one step to the next one.
This procedure ensures that the very strong collisions areproperly considered. The matrix elements of the operator U
are sums of sine and cosine functions of the H eigenvalues, so,regardless of how large these values are, the matrix U remainsbounded (unitary, in fact). When a strong collision occurscaused by a particle which is very close to the emitter, theoperator U is the exponential of a very large phase and, inpractice, that means a total breaking of the emitter coherence,but this does not cause any problems in the calculation. Then,the broadening phenomenon due to very strong collisions istreated naturally without numerical divergences. It is notnecessary to introduce cut-off parameters in the calculation.
Of course, effects such as the ion dynamics are consideredin the calculation in a natural way. Note that the simulationis a treatment that is truly unified, namely, the same methodis used to consider the ionic and electronic fields when thesimulations include both fields at the same time. Of course, thestatistical laws which rule the time sequence of the microfieldsare induced naturally according to the physical model ofthe plasma used in the simulation. When the simulation isof free and independent particles, the time sequence of themicrofields will have a correlation function, as expressed in(108) if the fields are Coulombian, or as expressed in (149)if the Debye field is used. However, unlike the MMM, thesimulation includes naturally all degrees of correlation ofthe field statistics and not only the first two. Moreover, whenthe plasma simulation includes the particle interaction, both the
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J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
field distributions and the statistical parameters of its dynamicsare considered naturally.
10. The frequency fluctuation model
The so-called FFM is used in a technique of calculation forline broadening that is an alternative to the expensive computersimulation methods. Its physical basis is similar to those forother procedures and, in fact, it does not modify any of thehypotheses concerning the causes of line broadening.
To understand its formalism, let us start from theexpression (128) which provides the spectra in the standardmodel. In a practical calculation, the integral appearing in thisexpression is dealt with numerically and the operator whichhas to be inverted is represented by a matrix in a certain basis(usually the spherical basis is adopted, in which the matrixelements of the operator RRR are constructed with Clebsch–Gordan coefficients). The simplest approach would be to adoptthe basis in which the matrix is diagonal, but this basis dependson the static electric field E appearing in the expression, andthen, it would be necessary to diagonalize for each value of E.In any case, QE is the matrix which diagonalizes that operator,namely[
i
(1
h
(LE
0 + KE)− ω
)]−1
= QE
1
γ E1 − i(ω − ωE
1 ). . .
1
γ En − i(ω − ωE
n )
Q−1
E ,
(156)
where h(ωEk − iγ E
k ) is the eigenvalue of that operatorcorresponding to the kth state. Let us not forget that, in theLiouville space, each state represents a transition betweena level of the upper group and a level of the lower group(see figure 8). Therefore, ωE
k represents the frequencyof that transition (which includes the Stark effect of thestatic field) and γ E
k represents the impact width due tothe fast electronic collisions. Proceeding in this way, thespectral line in the standard model is calculated through theexpression
IS(ω) = Re1
π
∫d3E W(E)
∑k
[d+QE
]k
· 1
γ Ek − i(ω − ωE
k )
[Q−1
E d]k, (157)
in which the sum is extended over the dipolar transitionsconsidered. The statistical distribution of the field does notdepend on the orientation of the vector E. Neither the set ofmatrix eigenvectors nor the complete spectrum (which includesa scalar product of the vectors d) depends on the orientationof E, so the angular integration can be done analytically.Ultimately, this development can be achieved numericallythrough a summation over all values of the electric field and
the transitions involved:
IS(ω) = Re1
π
∑�
P�
∑k
[d+Q�
]k
× 1
γ �k − i(ω − ω�
k)
[Q−1
� d]k, (158)
with P� the statistical weight of the field configurations E�.The frequency fluctuation method starts from this formalexpression. This is equivalent to considering a ‘molecule’whose transitions fit to the frequencies ω�
k and to the widths ofthe homogeneous lines γ �
k , with all possible values of the indexk and �. The corresponding state space can be assimilated intothe space of the tensor product of the emitting atom states spaceby adopting a state space corresponding to the values of theelectric field in the development, namely, one for which eachstate is of the form |ψ(u)
i 〉 ⊗ |ψ(l)j 〉 ⊗ |E�〉. Here i and j cover
the states of the upper and lower energy groups, respectively,of the emitter (both define the index k of (158)) and � covers thevalues of the static electric field. We have, in compact form,grouping terms,
IS(ω) = Re1
π
∑j
Ij
γj − i(ω − ωj)(159)
where the sum is extended over all possible values of theindex k and � of the expression (158). The coefficientsIj (in general, complex numbers) are the results of takingthe products between the quantities [d+ Q�]k and [Q−1
� d]k ,together with the statistical weighting P� of the field values.Each transition is determined by the ωj and γj values. Ofcourse, we consider the spectra to be area normalized, so∑
j Re Ij = 1. This allows us to adopt the quantity pj ≡ Re Ij
as the statistical weight of the j th channel.The first stage of the FFM consists precisely in preparing
this development, for which the range of static electric fieldvalues needs to be set. The refinement of this choice sets thenumber of transitions (or channels) considered and, thus, thecomputational cost of the model.
The spectrum obtained in (159) includes the effects of thequasi-static ionic field broadening plus the impact effect ofthe electronic collisions. The calculation model is required toinclude the effects of the ion dynamics, that is, it is necessaryto abandon the quasi-static field simplification.
Let us consider now a ‘molecule’ model whose spectrum,without perturbations, is described by an expression like(159). The dipole autocorrelation function corresponding tothis situation is given by
C(t) =∑
j
Ij e−(iωj +γj )t , (160)
where Ij accounts for the intensities of each of the transitionchannels. The frequency fluctuation model considers that themost general expression of the spectrum has the form
C(t) =∑
j
Aj (t) e−(iωj +γj )t , (161)
in which the intensities Aj(t) follow a Markov process with nomemory caused by the ‘jumps’ between channels of different
26
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
frequencies which is induced by the ionic field dynamics.The fluctuation of the plasma–emitter interaction causesfluctuations in the emission frequencies of that ‘molecule’.This treatment does not assume a microfield model, but simplyassumes that the dynamics of the emission process is whatreally is random and that the characteristics of that randomprocess must be obtained from the emission process itself,instead of speculating about the stochastic process of the field‘seen’ by an emitter (Talin et al 1995). Naturally, the meanvalue {Aj(t)} has to be the stationary value Ij . Here thesymbol {·} is used to denote the average which has beenobtained according to the statistics of that Markov process.The complete spectrum which includes the effects of thosefluctuations has to be obtained through a Fourier transform ofthe average: ∑
j
{Aj(t) e−(iωj +γj )t
}. (162)
The quantities Aj(t) will follow a stationary Markovprocess without memory characterized by a certain transitionmatrix S, so for a specific embodiment of that process,
Aj(t + τ) =∑
k
Sjk(t + τ, t)Ak(t). (163)
The transition matrix will be stationary, and its average will beexpressed as
{ S(t + τ, t) } = eWτ . (164)
Also, since the averages of the quantities Aj(t) set thestatistical weights of the channels j , we have to ensure thatthe said weight remains stable in time, which requires∑
j
Wijpj = 0. (165)
The quantities Wijpj account for the jumps occurring inthe time unit between the channels j and i. In the FFM,a microreversibility of those processes is considered, soWijpj = Wjipi for i �= j . This allows us to write
Wij = νpi, i �= j (166)
and the condition (165), together with the normalizationrelation,
∑j pj = 1, sets the diagonal elements of the
matrix W :Wii = −ν(1 − pi) . (167)
Here ν accounts for the jumping frequency of the Markovprocess.
Once the Markov matrix which regulates the jumpprobability is set, the (162) average can be calculated:{Aj(t) e−(iωj +γj )t
} = {Aj(0)
}eWjj t e−i�j t
+∫ t
0dt ′ eWjj (t−t ′) e−i�j (t−t ′)
∑k �=j
Wjk
{Ak(t
′) e−i�kt′ }(168)
(for convenience, we have put �j = ωj − iγj ). The first termof the right side of this equality accounts for the cases in whichthere is no jump in the interval (0, t). Naturally, {Aj(0)} = Ij .
The integral of the second term, which deals with the averagingof the last jump instant previous to t and the summationover k, in which there appear the jumping probabilities Wjk ,accounts for the possible transitions. This expression is themaster equation of the Markov process considered here. It isconvenient to take the Laplace transform, since the integralappearing in (167) is a convolution, which leads to the simplealgebraic relation
Gj(ω) = Ij
i(�j − ω) − Wjj
+1
i(�j − ω) − Wjj
×∑k �=j
Wjk Gk(ω) (169)
where we have written G(ω) = L { Aj(t) e−i�j t }. This resultcan be written in matrix form:
[ i(� − ω11) − W ] |G(ω)〉 = |I 〉 (170)
or|G(ω)〉 = [ i(� − ω11) − W ]−1 |I 〉 (171)
where the matrix � is diagonal in the basis used, with �j
values. The vectors |G(ω)〉 and |I 〉 have as components Gj(ω)
and Ij respectively. The spectrum including the frequencyfluctuations is, then,
I (ω) = Re1
π
∑j,k
[[ i(� − ω11) − W ]−1 ]
jkIk. (172)
It is practical to use the matrix notation which has been usedin (170) and (171), so
I (ω) = Re1
π〈s | [ i(� − ω11) − W ]−1 |I 〉 (173)
where we have used that the vector |s〉, defined in all of itscomponents, in the basis used, equals 1.
The FFM spectrum requires the inversion of a matrixwhich can reach a considerable dimension if many states areused and the interval of the static field values is very fine.Note that W is not diagonal in the basis used; therefore thecalculation of its inverse (or its diagonalization) is a veryexpensive operation. However, the structure of the matrixwhich has to be inverted facilitates this considerably.
In fact, let us separate the matrix W into two matrices, oneof them a scalar (see (166) and (167)):
W = −ν 11 + ν W, Wij = pi, (174)
or in compact form,
W = −ν 11 + ν |p 〉〈s | (175)
(the matrix W transforms any vector to the |p 〉 vector).Therefore we have
I (ω) = Re1
π〈s | [ ν 11 − i(ω11 − �) − νW
]−1 |I 〉. (176)
Let us use the matrix identity
[ A − B ]−1 = A−1 + A−1B [ A − B ]−1 (177)
27
J. Phys. D: Appl. Phys. 47 (2014) 343001 Topical Review
Figure 9. Illustration of the ion dynamics effect in the FFM treatment. The left figure shows the case in which the jumping frequency ν isvery small, comparing with the total width of the spectrum. In the right figure there appears the same spectrum with a larger ν value, whichis translated into a broadening of the line and a greater mix of the spectral components. The spectrum corresponding to the ST and the singleStark components with the broadening caused by the electronic impact (SSC) distributed according to the static field statistics are shown.
with A = ν 11− i(ω11−�) and B = νW = ν|p 〉〈s | so having
〈s | [ ν 11 − i(ω11 − �) − νW]−1 |I 〉
= 〈s | [ ν 11 − i(ω11 − �) ]−1 |I 〉+ ν 〈s | [ ν 11 − i(ω11 − �) ]−1 |p 〉×〈s | [ ν 11 − i(ω11 − �) − νW
]−1 |I 〉. (178)
Therefore,
〈s | [ ν 11 − i(ω11 − �) − νW]−1 |I 〉
= 〈s | [ ν 11 − i(ω11 − �) ]−1 |I 〉1 − ν 〈s | [ ν 11 − i(ω11 − �) ]−1 |p 〉 , (179)
and, then,
I (ω) = Re1
π
〈s | [ ν 11 − i(ω11 − �) ]−1 |I 〉1 − ν 〈s | [ ν 11 − i(ω11 − �) ]−1 |p 〉 . (180)
This expression only requires sums of terms which have beenprepared in the first stage of the process. The numerator ofthis formula is, actually, a variant of the standard model (see(159)):
〈s | [ ν 11 − i(ω11 − �) ]−1 |I 〉 =∑
j
Ij
ν + γj − i(ω − ωj);
(181)
the width of each Stark component has been increased bya quantity ν, which rules the jumps frequency; and in thedenominator of (180) a similar sum appears in which theintensities Ij (generally complex) has been replaced by theirreal parts pj .
The very simple form of the final expression for thespectrum allows us to analyse the consequences of the iondynamics, in the light of this model, easily. If we limit toobserving the numerator of the expression (180), we couldthink that such effects add an additional broadening ν toeach Stark component, which must be translated into a globalbroadening of the line. This interpretation is, in part, correct;however it can lead to a misunderstanding. Note that thedenominator of (180) plays the opposite role. In fact, let usremember that in all of our development we have worked with
area normalized spectra (the sum of all terms pj or Ij equals1), so if we call �ω the width of the spectrum, its peak heightwill be of the order of 1/�ω. Therefore, in the denominator of(180) we will have a function with an approximated minimumvalue of 1−ν/�ω. If ν �ω, the denominator is practicallyequal to 1 in all of the ω range, and then, effectively, theion dynamics causes a broadening which is added to that ofthe electronic impact. However if ν takes relevant values,comparing with �ω, then the opposite effect arises: the quasi-static spectral curve (the numerator of (180)) is divided by afunction which ‘sharpens’ this curve and, then, it is narrowed.
In figure 9 an example in which a few Stark components(DSC) are considered is shown. When ν �ω, the effect ofthe ion dynamics smooths the spectrum and removes the profiledetails (this is the effect which can be observed in the historicalmeasures of Wiese et al (1975) relating to the central depth ofthe Balmer beta case). When the jumping frequency increases,this mixing effect is greater and, in consequence, the spectralline narrows. In fact, the quasi-static spectrum corresponds tothe FFM spectrum when ν = 0 and, in the other limit, whenν → ∞, any effects of the ionic field (including the the quasi-static effect) vanish and the spectrum is reduced to that of asingle Stark component with the width caused by the electronicimpact.
The frequency fluctuation method, as was said, wasinspired from the mode mixing phenomenon (line mixing) inemission spectroscopy (Talin et al 1983). Figure 10 shows acalculation of the FFM applied to the case of a transition withonly two Stark components. For such small values of ν, themixing dynamics of the components produces, effectively, anindividual broadening which is added to the impact one. Asν increases, the aspect of the spectrum changes and reachesthe shape of a single peak whose width, contrary to what mayseem to be the case, starts to decrease as ν increases, until theimpact width value for each Stark component is reached.
The model sets the parameter ν according to the dynamicsof the ionic field—to what is necessary for an analysis similarto that for the MMM. In practice, the setting of the ν value isdecided by comparing the FFM results with the results obtained
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Figure 10. Illustration of the mode mixing phenomenon accordingto the FFM treatment. The spectrum of two Stark components cangive rise to a profile which looks like it is arising from a singlecentral transition if the jumping frequency takes very large values.
in simulations in the case of a simple spectrum or from a modelof the field–field correlations (Talin et al 1995). Keep in mindthat the simulations are prohibitive when the spectrum studiedis from transitions with many states involved. For those cases,the FFM has no special computational problems.
11. Applications
The spectral line broadening caused by the Stark effect is thebasis of one of the most efficient and, at the same time, simplestplasma diagnostics methods. Passive spectroscopy whichrecords a line spectrum is a non-invasive technique whichpermits determining the electron density in a plasma with goodaccuracy. On the other hand, active spectroscopy methodswith lasers can be used for diagnosis in limited and localizedenclosures in a plasma. In all of these cases, the quality of thediagnostics technique is grounded on a subsequent analysisof the recorded spectral line, and this analysis is groundedon models such as the ones summarized in this review. Thisindirect measurement technique is strongly dependent on themodel that we apply. And to be sure of our results, it is essentialto confront such models with laboratory measurements carriedout on well controlled plasmas and with their well establishedcomposition parameters, densities and temperatures. Thistask has been carried out by many laboratories (a recentcritical review can be consulted in (Konjevic et al 2002),as can the extensive bibliography cited there, as well as themonograph published by NIST (Fuhr and Lesage 1993) whichincludes a comprehensive list of the bibliography relevant tothis field—or consult the theoretical and experimental data inhttp://physics.nist.gov/Linebrbib).
In general, a calibration measure for the width and shiftparameter can be considered valid when the plasma source iswell characterized, that is, it is homogeneous in the observationregion, has a quasisteady state, and is highly reproducible(Konjevic et al 2002). This aspect must be addressed carefully,especially in those cases in which the experiments have beendone on pulsed plasmas and the spectrum measures have
been obtained point by point at different times. Moreover, itis necessary that the electron density is determined througha measure alternative to the spectral lines’ own widths.Interferometry with one or two wavelengths, registration ofthe absolute intensity of the line referred to the continuouslevel, Thomson scattering studies, measurements of the electricconductivity, etc, are accepted techniques (Griem 1997).
At the same time, the plasma temperature should beadequately characterized. It should be taken into account thatin many applications, plasmas are non-equilibrium or partialequilibrium plasmas (PLTE—partial local thermodynamicequilibrium—cases). For non-equilibrium plasmas we cantalk about the kinetic temperature of electrons or of heavyspecies (which are usually different, since the temperature willbe higher for electrons than neutrals or ions), the temperatureof filled excited states (which parametrizes the statistics of thepopulation of internal energy levels of ions and neutrals, andis related to the kinetic temperature of electrons), and, finally,the temperature of the continuous radiation (which does notusually match any of the previous ones, since plasmas areoptically thin in general and do not reach thermal equilibriumwith the radiation). Thus, calibration experiments should beable to characterize all these temperatures while they keepthe state of equilibrium of the plasma under control. Wellrecognized techniques (Griem 1997) include: the Boltzmannplot method for one or several species, measurements of theabsolute or relative intensity emission or the ratio of lineintensities with respect to the continuum, Thomson scatteringstudies, electrical conductivity measurements, etc.
For plasmas very far away from kinetic equilibrium,models should take into account that the electron temperatureis much higher than those of ions and neutrals. Wheneverwe can talk about electronic and ionic or gas temperatures,meaning that electrons or heavy species are in equilibriumwith others of the same species, and in spite of the lack ofequilibrium among species, the models can be extended andstill provide sensible results, since the effects of electronic andionic kinetic temperatures can be separated in the calculations(some computer simulations, for instance, have been carriedout with unequilibrated species using an artificial alterationof the mass of the particles, which regulates their mobility(Gigosos et al 2003)). On the other hand, for plasmas with verylow electron densities, where the Stark effect is very weak, theappearance of other line broadening mechanisms may renderuseless for line spectra analysis all the models which havebeen reported in this review. For high pressure plasmas witha low ionization level, the line broadening associated withcollisions of the emitter with neutral species (van der Waalsbroadening) dominates in most cases over the Stark effect, andspectroscopic diagnosis has, as a consequence, to look for otherapproaches. This phenomenon has received a treatment similarto the one presented here, but there is not such a completeliterature, and the control experiments performed have not beencomparable in number to those performed in investigating theStark effect (see the book by Griem (1997) for a summary).
The results of these analysis and comparisons between theexperimental data and the theoretical models are, in general,in good agreement (discrepancies below 20% and in many
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cases, below 10%) for the Stark width for emission lines ofneutral atoms. However, in many cases for lines of chargedemitters (even for ions with charge +1), the results are notconclusive and the quality of the models cannot be proven. Thediscrepancies are significant in many cases and it is necessaryto have more theoretical and experimental information. Tocite an example, Uzelac et al (1993), after recording spectraof Ne and F for different ionization states, conclude that itis necessary to review the models, because they consider itnecessary to characterize spectra both for diagnostic purposesand to achieve progress in the development of laser sources.
It is known that the models are less accurate when theemitter is an ion. In those cases, the emitter itself conditions,with its electric action, the plasma which surrounds it, andmany of the treatments described in this summary dispensewith the plasma reaction against the emitter. This is the case forthe perturbing field correlation, which naturally has dynamicsconditioned by the correlation of the charged emitter itself (seein Dufour et al (2005) a study of the effects of the emitter–plasma correlation in a charged emitter).
To illustrate the kinds of theory–experiment discrepanciesfound, it is enough to cite just two examples. The first onerelates to the regularities of the Stark broadening spectra inisoelectronic series. Note that the ST, particularly, establishesthat the widths of the isolated lines are determined by theplasma parameters and by the structure of the emitter states.This state structure fixes the cross section for collisionsbetween the emitter and the charged perturbers, and this crosssection, in a simplified way, depends on the emitter charge Z
as 1/Z2. It is expected, then, that the widths of the spectrallines corresponding to emitters with similar state structures willreproduce those regularities. The search for those regularitieshas been the subject of many works, with the aim of verifyingthe model results (Wiese and Konjevic 1982, Puric et al 1987,Bottcher et al 1988, Kobilarov and Konjevic 1990, Wiese andKonjevic 1992, Glenzer et al 1992, Moreno et al 1993, Glenzeret al 1994, Blagojevic et al 1996, Wrubel et al 1998, Hegazyet al 2003, Puric et al 2008).
If we look at the expression (101), we can appreciatethat the widths of the lines are proportional to the quadraticvalues of the operator R and, thus, to 1/Z2, with Z beingthe charge of the emitter nucleus. However, the experimentalresults show a trend for the linewidth with Z more similar to1/Z (Bottcher et al 1988, Wrubel et al 1998, Hegazy et al2003). It is important to note that the model does not exactlyfix a width dependence as 1/Z2, since in the expression (101)other elements which depend also on the nucleus charge andon the state structure (particularly the coupling frequency ωjβ)are included; however, a dependence with Z more markedthan what is observed in the experiments is expected. Ofcourse, this has led to some reviews of the models, in whichthe authors try to avoid all of the approximations adopted (forinstance, including effects of the inelastic collisions (Alexiou1994) or quadrupole effects (Alexiou and Ralchenko 1994),or in the frame of the impact approximation in quantumcalculations which include intermediate coupling (Elabidiet al 2008, Elabidi et al 2009)), which have not solved theproblem definitively. In fact, Griem and Ralchenko (2000)
suggests that there might be other mechanisms of electronicbroadening which have not been considered. Also, a pendingtask related to the ST is properly setting the values of the cut-off parameters rmax and rmin which the model requires to avoidthe divergences appearing in calculations—a consequenceof using a perturbation theory which involves a non-unitaryoperator (Alexiou 1995).
The other example concerns the broadening of the radio-recombination lines observed in stellar atmospheres. Theserelate to transitions in the radiofrequency spectrum whichare produced in Rydberg atoms between states with adjacentprincipal quantum numbers or nearly adjacent ones (�n ≈ 1).They are transitions which start from very high quantumnumbers (n ≈ 200) and continue until the ground stateis reached. They are observed in H II regions. They arespectral lines that are very useful since they do not sufferoptical attenuation along the light line and they can be usedto diagnose the conditions of the observed atmospheres. Theywas recorded for the first time in 1965 (see the proceedings ofthe symposium on the first 25 years of research on the subjectin Gordon and Sorochenko (1990)). From the beginning itbecame clear that the Stark width observed was completelynegligible compared with the predictions of the models, tosuch an extent that one might consider this width non-existent (see chapter 2 of the Gordon and Sorochenko (2002)monograph, whose first section talks about the ‘absence ofStark broadening’).
From the beginning, Griem (1967) considered that theimpact theory was valid in those situations. The standardmodel of Stark broadening—the Griem one—establishes afunctional dependence of the impact width on the principalquantum number as n4.4, approximately. Some measures seemto confirm that trend (Smirnov et al 1984). However, the widthvalues observed are much lower than the ones expected fromthis model. The theory predicts a considerable increase ofthe width as the rise to the ionization level proceeds. Theastronomers found that it is by no means the case that thisvery pronounced growth occurs. Moreover, according to theirobservations, the linewidth tends to stabilize. The work didnot generate much reaction, since from the beginning it wasconsidered that other effects—mainly Doppler ones—wouldmask the results, to which the poor quality was attributed.
However, the issue re-emerged with force when Bell et al(2000), with a new technique, obtained new data covering awide interval of �n values with high levels of the principalquantum number (between 102 and 274 and, later, 289 (Bell2012)). The results not only do not fit to the model, but show theopposite trend: the linewidths seem to decrease as the startingprincipal quantum number increases.
This issue, which has been described as ‘mysterious’(Griem 2005), has led to some discussions (Oks 2004, Griem2005, Alexander and Gulyaev 2012) which have not led on toa definitive explanation of such discrepancies, although theyhave provided an accurate formal expression for the impactwidth in relation to the principal quantum number in thehydrogen spectrum (which does not fit to a ∼n4.4 function,and which is more similar to an ∼n2�n2 form (Gigosos et al2007)). As Lisitsa (2006) has pointed out, this is still an openproblem, remaining unsolved.
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Acknowledgments
The author thanks Bernard Talin for advice concerning theFFM. The research activity of the author was supported by theSpanish Ministerio de Ciencia e Innovacion under contract No.ENE2010-19542/FTN.
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