Spectroscopy and Atomic Physics

CHAPTER

Atmospheric structure, Non-Equilibrium Thermodynamics and Magnetism

5 SUBCHAPTER

Spectroscopy and Atomic Physics

5.1 Philip G. Judge

National Center for Atmospheric Research, High Altitude Observatory, Boulder, CO, United States

CHAPTER OUTLINE 1. 2. 3. 4. 5.

Overview ........................................................................................................... 128 Regimes of Solar Plasmas .................................................................................. 128 Origin and Types of Atomic Transitions ................................................................ 132 Atomic Structure ................................................................................................ 133 Spectrum Formation in a Nutshell........................................................................ 138 5.1 Optically Thick Formation .................................................................... 140 5.2 Optically Thin Formation...................................................................... 140 5.3 NoneLocal Thermodynamical Equilibrium and Further Complications...... 143 6. Plasma Spectroscopy ......................................................................................... 145 7. Closing Remarks ................................................................................................ 152 References ............................................................................................................. 153

The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00005-4 Copyright © 2019 Elsevier Inc. All rights reserved.

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

1. OVERVIEW We review the spectroscopy of atoms embedded in plasmas, specifically in solar plasmas. The general goal of remote spectroscopic work is to discover properties of the emitting plasma, whether they are in small laboratory plasmas or clusters of galaxies. Fujimoto (2004) calls such studies “plasma spectroscopy,” as opposed to “atomic spectroscopy,” whose focus is atomic structure. In the Sun we can use spectroscopy to measure elemental abundances, plasma motions, densities, temperatures, and magnetic and electric fields. First, we place the Sun into the context of other plasmas. Then we review the origins of spectral lines and continua in unperturbed atoms, and present scalings for atomic parameters. Next, we sketch the formation of solar spectral features, relating the emergent spectrum to elementary ideas from radiative transfer, under local thermodynamical equilibrium (LTE), non-LTE, coronal, and collisional-radiative (CR) conditions. We examine the interactions of individual atoms and atomic ions with solar plasmas, stressing scaling laws and highlighting common threads. The reader can refer to reviews on plasma spectroscopy by Cooper (1966), on the interpretation of spectral intensities from laboratory and astrophysical plasmas by Gabriel and Jordan (1971), and on atomic processes in the Sun by Dufton and Kingston (1981).

2. REGIMES OF SOLAR PLASMAS The solar spectrum originates from plasma in a variety of regimes. A low-resolution UV spectrum showing emission and absorption lines and various continua is shown in Fig. 5.1.1. Based on more than a century of research using data such as these, Fig. 5.1.2 shows the range of solar plasmas in the context of astrophysical and terrestrial plasmas. In astronomy it is usual to term the visible surfaces of stars as “atmospheres,” those regions from which the bulk of the luminosity of the star is emitted. However, the photosphere and chromosphere are also partially ionized plasmas. A plasma is characterized and defined through three criteria: 1. the Debye screening length (using Gaussian units) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi rD ¼ kTe =4pe2 ne ¼ 6:92 Te =ne cm

(5.1.1)

must be far smaller than the macroscopic size of the plasma. The plasma is quasineutral and its dynamics are controlled mostly through its own bulk selfinteractions and not with boundaries; 2. the plasma parameter L ¼ 4pne r3D is [1; and 3. the electron plasma frequency qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi (5.1.2) upe ¼ ne e2 =pme ¼ 8:98  103 ne rad=s

FIGURE 5.1.1 A “classic” figure of the solar spectrum, highlighting emission and absorption features across the solar atmosphere. Some strong lines are deep in absorption yet have emission in the core (Mg II is the case shown here; emission is not visible at this modest spectral resolution). Notice the Dn ¼ 1 transitions of H, He I, and He II, marked with asterisks, at comparable wavelengths with Dn ¼ 0 transitions of much more highly ionized species. Adapted from Scheffler, H., Elsa¨sser, H., 1974. Physik der Sterne und der Sonne.

Solar core

1025

ne cm-3

1020

LTE

metals F-D M-B

Λ

lightning arc photosphere chromosphere

1015 1010

Ionosphere

105

EBIT--> Active coronaFlares Quiet corona

PN

Magnetosphere interplanetary 1AU

100 interstellar

10-5 10-2

10-1

pinch fusion tokamak

AGN BLR

flame

nLTE

lasers

=1

15

10

Λ=

Galaxy halos

100

101

102

103

104

105

Te eV

FIGURE 5.1.2 Plasmas are identified as a function of electron temperature (1 eV h 11,605K) and electron density. Loci for plasma parameters L ¼ 1 and L ¼ 1015 are shown, in which L is the number of free electrons inside a Debye sphere. Plasma behavior requires L [ 1. Solar plasmas are identified in red; the plus sign shows weakly ionized plasmas. The dotdashed lines separate regions in which local thermodynamical equilibrium (LTE) versus non-LTE conditions prevail; the upper line is for ionization and the lower line is for a typical E1 line. The dashed line separates Fermi-Dirac from MaxwelleBoltzmann statistics for electrons. EBIT, electron beam ion trap device; PN, planetary nebulae.

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

exceeds collision frequencies with neutral particles. Electrostatic interactions domipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nate the electron dynamics. (Note that rD upe ¼ kTe =4p2 me ). All observable solar plasmas easily satisfy these definitions. They clump in the center of Fig. 5.1.2. They have lower densities than terrestrial plasmas and higher densities than photoionized astronomical plasmas. The spectra in Fig. 5.1.1 form in the corona (x-rays), chromosphere (Ly a), transition region (C IV), and photosphere (Mg II line wings). The mean times between particle collisions within plasmas are a matter of some delicacy, because collisions involve solutions to the Boltzmann transport equation with its associated closure problem. Assuming that departures from thermal distributions are small (Braginskii, 1965; Mihalas and Mihalas, 1984), approximate times needed for electrons and ions with charge Z  1 and ionization potential IZ1 to relax to the same temperatures are of the order  3=2 1 6 kTe see w 1:6  10 s (5.1.3) ne R rffiffiffiffiffiffi 3=2 mi kTi 1 4 7 sii w7  10 ðZ  1Þ s (5.1.4) ni mp R   1 2 mi kTe 9 4 kTi mp þ 5:4  10 sie w1:4  10 ðZ  1Þ s. (5.1.5) mp R R m i ni For neutrals (Z ¼ 1), collision times are gas-kinetic (cross-sections are z pa2a , a0 ¼ -2/2mee2 is the Bohr radius) or sometimes are dominated by resonant processes such as charge transfer. In the case of charge transfer, one must make distinctions between times for the transfer of momentum and energy. Z is the net core charge “seen” by the outermost “optical electron.” For H I Z ¼ 1; for He II Z ¼ 2, etc. Temperatures  are normalized by the Boltzmann constant k and Rydberg energy unit R ¼ e2 2a0 . In anticipation of later sections, here are time scales for spontaneous radiative transitions and “inelastic” collisions involving changes-of-state of the internal structures of atomic ions by electron impact: sE1 w108 Z 4 s; spontaneous decay of excited levels through electric dipole ðE1Þ transitions sm w10þ1 Z 8 s; metastable levels     IZ1 2 IZ1 1 9 1 exp s; ionization sion w9  10 pffiffiffiffiffi R kTe ne Te srr w2  1010 Z 2 Te0:7

1 s; radiative recombination ne

(5.1.6) (5.1.7) (5.1.8) (5.1.9)

2. Regimes of Solar Plasmas

In the Sun’s atmosphere, srr can be small or, for chromospheric ions with small Z, they can exceed 60 s, longer than dynamical times of interest (Carlsson and Stein, 2002; Judge, 2005). In any case, sion wsrr [sie [sii [see ;

(5.1.10)

and sion wsm [see [sE1 . (5.1.11)  ffiffiffiffiffi p h Another parameter is LTe ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 7  106 Te , the “thermal” de Broglie 2pme kTe wavelength of electrons. When ne L3Te  1; or ne  3  1015 Te3=2 ;

(5.1.12)

the electrons are nondegenerate and obey MaxwelleBoltzmann statistics; otherwise Fermi-Dirac statistics must be used. Fig. 5.1.2 shows a dashed line demarking regimes of classical and quantum statistics. With some notable exceptions (e.g., charge transfer) (Butler and Dalgarno, 1980), free electrons dominate the plasmaeatom interactions. The Maxwelle Boltzmann distribution for electron speed v is   3=2  m mv2 f ðvÞdv ¼ exp  (5.1.13) 4pv2 dv 2pkTe 2kTe which is normalized to unity. We use the notation Z ∞ vsðvÞf ðvÞdv hvsi ¼

(5.1.14)

0

to describe the probability that per each impacting electron, a process described by the cross-section s(v) (such as for electron impact ionization) takes place, its units are cm3/s. When atomic ions are in thermal equilibrium with a bath of electrons, two level populations q2 and q1 obey (* denotes LTE):   q2 g2 E2  E1 ¼ exp  . (5.1.15) q1 g1 kTe The equation describing the ionization state of stages of the element of interest in equilibrium is the Saha equation (which allows for the extra degrees of freedom of the free electron):   qZ 2 UZ ðTe Þ EZ  EZ1 exp  . (5.1.16) ne  ¼ 3 qZ1 LTe UZ1 ðTe Þ kTe Here, U are partition functions and IZ1 ¼ EZ  EZ1 is the ionization potential of ion Z  1. Truly isolated atoms have U ¼ ∞, a problem resolved in real plasmas via collisions and plasma microfields (Section 6). The lower dot-dashed line in Fig. 5.1.2 separates LTE from non-LTE conditions between bound levels, in which ne(Te) ¼ A z 108Z2 s, divided by

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

Cz104Z 2T 1/2 cm3/s, assuming that ions of charge Z  1 form near Te z 104Z2 (1 < Z < 92). This locus is meaningful only on a log-log plot. Also, we will see that for plasmas close to 2-body ionization equilibrium (equality of Eqs. 5.1.8 and 5.1.9), IZ1> and (often [) at temperatures where ion Z  1 is abundant.

3. ORIGIN AND TYPES OF ATOMIC TRANSITIONS The full intricacy of atomic structure and associated transitions is highly specialized. It is not surprising that atomic physics is complicated. One must solve “many-” and “few-” body problems that have no closed-form solutions, even in the classical limit. It draws upon mathematics of vector spaces, symmetries, special relativity, nuclear physics, and quantum electrodynamics. The purpose here is to provide some physical insight. There are many textbooks on atomic structure and spectra. A fine reference is Cowan (1981). From where do “spectral lines” originate? One of the greatest triumphs in 20th-century physics was the development of wave mechanics, based on clear experimental evidence of the need to abandon or build on classical mechanics. In mathematical language, the wavefunctions j for isolated atoms are solutions to H0 j ¼ Ej;

(5.1.17)

where H0 is the Hamiltonian. The fundamental postulates of quantum mechanics are equivalent to writing particle momenta p and energy E in the form of operators p/  iZvi ; E/iZvt ;

(5.1.18)

which operate on complex wavefunctions j(xi,t). According to Born, we are to interpret j*(xi,t)j(xi,t) as the probability of finding a particle at the given position and time. Importantly, these operators are fully compatible with Einstein and de Broglie’s arguments that E ¼ -u and p ¼ -k from experiments, implying that particles behave like waves, with f w exp i(k.r  ut). With these substitutions for p and E, Eq. (5.1.17) becomes an eigenfunction equation, the Schro¨dinger equation, the solution of which yields a set of eigenvalues and functions. The eigenstates have a definite energy E, but there is nothing about the Schro¨dinger equation that demands that the eigenvalue spectrum be discrete. It is only when we impose boundary conditions compatible with Born’s interpretation that we find discrete eigenvalues. If we demand that the wavefunction be localized by a negative potential (i.e., that it has zero amplitude at infinity), or if cyclic coordinates (such as angular variables) exist, we find that the eigenvalues are discrete. The “lines” of Fraunhofer, Bunsen, and Kirchoff are identified with the interaction between electromagnetic radiation and two of the discrete states, assuming that the Hamiltonian H0 is a good “zeroth order” approximation to the atom. The induced transition probabilities between levels (Einstein “B” coefficients) are most simply evaluated treating the radiation field classically via a small perturbation term HR added to H0. Spontaneous emission (through the “A” coefficient) was invoked by

4. Atomic Structure

Einstein to account for the phenomenon of unprovoked emission, a process also needed for compatibility with MaxwelleBoltzmann populations and Planck’s thermal equilibrium function for radiation (Eq. 5.1.27). This yields 2hn3 B21 ; (5.1.19) c2 relations obtained from the principle of detailed balance (microreversibility) for a thermal system. In other words, the two levels are coupled through a thermal radiation field (a black body). The spontaneous emission is understood physically from a treatment of the coupled and quantized electromagnetic field and atoms, in which fluctuations in the vacuum field induce spontaneous emission. Here we assume that the atomic lines have finite widths owing to natural broadening (quantum uncertainty in energy resulting from the finite lifetimes), thermal plasma motions, and other processes that contain further information on plasma conditions, but without further discussion. This summarizes the physical origin of the “boundebound” (b-b) spectral transitions. Transitions between two unbound states are called “freeefree” ( f-f ) transitions, or “Bremsstrahlung”. Finally, “boundefree” (b-f ) radiative transitions occur between bound states with n electrons to free states with n  1 bound electrons and a free electron. These processes are called photoionization, and in reverse, radiative recombination. Only b-b transitions are considered to be “line” radiation. Sharp (“autoionization”) resonances in b-f transitions are common when the free states are mixed with levels in which two electron spin-orbitals are excited. g1 B12 ¼ g2 B21 ;

A21 ¼

4. ATOMIC STRUCTURE To understand spectra, naturally one must first understand the structure of atoms and atomic ions. In general, the atomic structure and transitions are solutions to Schro¨dinger’s (or Dirac’s) equation for atoms, which are many body systems (many electrons clumped around the oppositely charged nucleus). As such, the solutions are generally complicated. We provide only a brief guide to understanding some elementary properties of atomic levels, eigenfunctions, and transitions. The case of hydrogenic ions provides valuable information because the wavefunctions are analytical. It is only useful in multielectron atomic ions to the extent that an optical electron is well-separated from the nucleus and other electrons, and in highly charged ions, ideas that will be used subsequently. The correspondence principle, namely that classical behavior be recovered as a limiting case of quantum mechanics, is also useful. States with large quantum numbers can be surprisingly well-described using classical concepts. For example, cross-sections tend to increase geometrically with the classical orbit areas, which are proportional simply to the principal quantum number n4/Z2. Electron spin, however, is an entirely quantum phenomenon. The wavefunctions are intrinsically built

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upon manifolds in both real and spin space. The implications simply have to be accepted with only limited classical analogies. Full wavefunctions for complex atoms and atomic ions can be expanded in terms of products of single electron “spin-orbitals” to form a hierarchy of configurations, terms, levels, and states, and the “good” quantum numbers (parity and total angular momentum) are readily obtained from the straightforward symmetries of these single electron wavefunctions. Thus, the reflection of the wavefunction in real space (the parity of a given state, another constant of the motion) has an eigenvalue of 1; it depends only on the orbital angular momenta li of the individual electrons through PN P ¼ ð1Þ i li ; (5.1.20) for a state described by an N-electron configuration or combination of configurations when they are mixed. P for a complex state built upon say, a 1s22p62s23p53d electron configuration, has a parity determined only by the partly filled 3p5 shell. (The s,d orbitals and even occupation numbers of p, f,. orbitals contribute only unit multipliers to P.) This is therefore an odd-parity state P ¼ 1. This picture is useful because we can then identify the kinds of transitions that are going to be strong electric dipole (E1) transitions, such as H La, because the E1 operator eri is of odd parity. Thus, strong E1 transitions connect levels with opposite parity only. Similarly, the magnetic dipole M1 (m) and E2 quadrupole (erirj) operators can lead to transitions only between levels of the same parity. Examples in solar physics include M1 transitions seen during eclipse among the ground levels of ions of iron, in the fourth row ˚ , which is the tranof the periodic table, such as the “green line” of Fe XIV at 5304 A 2 0 2 0 sition 2p P3=2 /2p P1=2 of the Al-like isoelectronic sequence (ions with the same number of electrons as Al). Other conservation laws guide our understanding of which lines might be strong and which weak. As well as constraints on parity, the “triangle” selection rules, coming from the need for photoneatom interactions to conserve angular momentum, allow one to identify “permitted” and “forbidden” transitions by inspection of quantum number J for the total angular momentum. Even the “not-so-good” quantum numbers S and L obey approximate rules, such as DS ¼ DL ¼ 0 when relativistic terms in the Hamiltonian (e.g., spin-orbit interactions) are small. This is readily understood because the E1 operator that induces the strongest line transitions operates only on the wavefunction in real space, not the component in spin space. Another important class of transitions is the “intercombination” (“IC”) transitions, E1 transitions for which DS s 0. These lines can be bright yet remain optically thin and sensitive to plasma densities in a manner different from that of E1 transitions, which permits one to explore plasma densities through joint observations of IC (or M1, E2) and E1 lines. There are other limiting cases in which further simplifying ideas can be used. Single electrons orbiting closed shells occur along the first column of the periodic table. Even in open shell cases, single electron excitations often occur between states

4. Atomic Structure

that are far from the strongly coupled electrons buried deeper in the net electrostatic potentials in the core atomic configurations. Wavefunctions that correspond almost to single electron excitations are amenable to simpler treatments. Hydrogen itself also serves as a pedagogical tool for at least two reasons. First, analytical solutions for eigenstates are available. Second, solutions for large quantum numbers, in which electrons spend time far from the nucleus, are (closely) hydrogenic in nature. For a nucleus of charge Z, hydrogenic (one-electron) eigenfunctions satisfying 1 Z H ¼  V2  (5.1.21) 2 r    yield the following scalings: EZ,n ¼ Z2/2n2 and rZ;n /n2 Z, both of which have important consequences. Electrons are Fermions. Thus, for N-electron atomic ions, antisymmetric combinations of one-electron orbitals are needed to find physically meaningful eigenfunctions of  N  X 1 2 Z 1X 1 H0 ¼  Vi  þ . (5.1.22) 2 r 2  rj j jr i i i; j i¼1 These “orbitals” comprise solutions to the purely radial part of Eq. (5.1.22), each characterized by quantum numbers n and l in the same way as for hydrogen (the Schro¨dinger equation separates into radial and angular components in both cases). The orbitals and their combination to yield atomic wavefunctions cannot be closed-form functions because the unperturbed potential, although of the form V(r), is no longer of the Coulomb form Z/r. The residual noncentral part is treated as a perturbation. Solutions are called atomic “terms” with well-defined antisymmetric products of orbitals. The solutions commute with total orbital and spin angular momentum operators, yielding well-defined orbital and spin angular momentum quantum numbers L and S, respectively. In this case, new energy levels are produced, lifting the central field’s degeneracy. For a given n, several levels exist, which can be of the same or different parity. Finally, by including the various relativistic effects (spin-orbit, etc.), the terms are split into levels through the additional terms in the Hamiltonian. Only parity and total angular momentum operators commute with the Hamiltonian so that only parity and J are good quantum numbers. This hierarchy of splitting of levels is essential to understanding real spectra that occur via transitions between atomic levels and, if external magnetic or electric fields are applied, between magnetic substates. The central-field approximation clearly works well when Z  (N  1) [ 1, i.e., the outer electron experiences a strong central field owing to the nuclear charge and incomplete screening by core electrons. On this basis, it is straightforward to find the scalings listed in Table 5.1.1. Early x-ray spectroscopy of atomic ions by Moseley helped to complete the period table of elements by allowing Z to have noninteger values as core electrons shield nuclear charge. There is a powerful theory based on “quantum defects” (Seaton, 1983), in which energy eigenvalues in complex atoms and ions are used to determine the “defect” d through EZ,n ¼ Z2/2(n  d)2.

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Table 5.1.1 Scalings of Parameters of Isolated Atomic Ions With Z, n Parameter

Example

Scaling With Z, n

Term in Hamiltonian

hri DEji (Ionization) DEji(Dn s 0) DEji(Dn ¼ 0) DEji (Fine structure, spinorbit)a

. 2s2 1 S0 /2s2 S1=2 þ e 0 2s2 1 S0 / 2s3p1 P1 0 2s2 1 S0 /2s2p1 P1 0 2s2p 3 P2 /2s2p3 P10

a0n2/Z Z2 Z2 Z/n2 Z4/n3

. Ze2/r Ze2/r e2/jrj  rij m$Bf12a2 Z 4 R

The fine structure induced by spin-orbit interactions arises from the term m$B ¼ ZmB /r3, which is z m2B Z 4 =a30 h12Z 4 a2 R in the Hamiltonian, noting that RZ 2 is the central field energy. Here, a ¼ e2/Zc is the fine structure constant; mB ¼ eZ/2mec ¼ a(ea0)/2 is the Bohr magneton. The magnetic field strengths B ¼ cv Ze2 =r experienced by electrons orbiting the nucleus are of the order 105e108 G. In  contrast, observable solar fields lie below 4000 G. R ¼ me e4 2Z2 is the Rydberg unit of energy (h13.6 eV, the ionization potential of neutral hydrogen for an infinitely heavy proton).

a

The E1 operator does not operate in spin space. If the wavefunctions used to evaluate matrix elements are pure LS-coupled states, IC (DS s 0) transitions are forbidden. Such transitions arise naturally because of mixing of LS-coupled states by relativistic effects. They are present in solar and stellar photoionized objects and their spectra, generally weaker than E1 collisions owing to smaller crosssections for excitation by electron impact (Table 5.1.1). Inspection of Tables 5.1.1 and 5.1.2 reveals several important properties of atomic ions of relevance to plasma spectroscopy, and that are reflected in spectra such as shown in Fig. 5.1.1: •

•

•

In H-like and noble gas spectra, to excite any electron requires a change in principle quantum number n and z RZ 2 of energy. Thus, the lowest excited levels of H, He, Ne and isoelectronic ions have strong lines principally at extreme UV radiation (EUV) wavelengths. La lines of H and He II are obvious in Fig. 5.1.1 at 1215 ˚ . Lines of highly ionized H and He-like ions for abundant elements (C, and 304 A N, O, Si, Fe, etc.) collect in the softex-ray region as a result (a few angstroms to a ˚ [see Fig. 5.1.3]). few tens of angstroms [see Fig. 5.1.1]) and below 100 A In ions with incomplete subshells, the lowest excited levels have the same principal quantum number as the ground level, requiring energies of only z RZ. As a result, such lines (having Dn ¼ 0) lie at much longer wavelengths and can be intense. In the Sun, lines of the Li- and Na-isoelectronic ˚, sequences are prominent: resonance lines of Ca II are close to 3933 and 3969 A ˚ ˚ ˚ ˚ Mg II 2800 A, Na I 5890 A, C IV 1550 A, O VI 1026 A, and so on (Fig. 5.1.1). ˚. In contrast, the 2s  3p transitions of O VI lie at 150 A As noted before, when close to ionization equilibrium, ions form when kTe or z RZ 2. Thus, these Dn ¼ 0 lines contribute more strongly to total radiation losses than Dn ¼ 1 lines, which is of importance in laboratory as well as astrophysical plasmas.

4. Atomic Structure

Table 5.1.2 Scalings of Transition Probabilities Parameter Type Aji/s

Y21(Te)

S cm3/s

brr cm3/s

Magnitude and Z,n,Te Dependence

Type

Example

E1, Dn s 0 E1, Dn ¼ 0 IC, Dn ¼ 0, D Ss 0 M1, Dn ¼ Dl ¼ 0 E2, Dn ¼ 0

2s2 1 S0 / 2s3p1 P1 0 2s2 1 S0 / 2s2p 1 P1 1 3 0 2 2s S0 / 2s2p P1 2p2 3 P0 / 2p2 3 P1 2p2 3 P0 / 2p2 3 P2

Two-photon

2s2 1 S0 / 2s2p 1 P0

E1 E1 (small gf ) Intercombination Forbidden e-Impact ionization

Rad recombination

0

0

2s2 1 S0 / 2s3p

1 0 P1

0 Si II 3s2 3p 2 P3=2 / 3s2p2 2 D5=2 0 2s2 1 S0 / 2s2p 3 P1 3 3 2 2 2p P0 / 2p P1 2s2 1 S0 þ e / 2s 2 S1=2 þ 2e

2s 2 S1=2 þ e / 2s2 1 S0 þ hn

109 Z4 108 Z1 100 Za, a ¼ 5  7 10 Za, a ¼ 3  12 100 Z1 10 Z 6 n2 lnTe/Z2 0.1 lnTe/Z2 1/Z2 1000/(Te Z 2)  2 pffiffiffiffiffi 1010 Te ℛ IZ   IZ1 exp  kTe 5  1011 Z 2 Te0:7

Data in this table are intended largely for transitions with modest principal quantum numbers n ( nG þ 4, say, where nG is the largest principal quantum number of the ion’s ground state. Hydrogenic values can be used for higher-n levels (refer to Fujimoto, 2004). Dependences on principal quantum number n are needed only for E1 transitions that dominate the collisions and decays of high-n levels. Values of a for radiative decays of intercombination and forbidden transitions depend on the coupling schemes. Refer to Cowan (1981). Scalings for Maxwellian-averaged collision strengths Y are from Burgess and Tully (1992). “Small gf” cases are those in which accidental cancellation occurs in E1 radial integrals.

•

•

IC and forbidden lines often have conditions in which collisional depopulation is comparable in magnitude to radiative deexcitation. The emission from such lines can become a linear function of density (compare with Eq. [5.1.31], for permitted E1 lines). The comparison of E1 and the IC and forbidden lines are thus “density-sensitive line ratios.” As Z increases along an isoelectronic sequence, the IC and F transitions increase dramatically. Indeed, by the time one reaches H- and He-like spectra, such lines are of comparable strength in plasmas, yielding singularly rich information of the plasma temperature and density in the soft x-ray region (Gabriel and Jordan, 1971; Jordan and Veck, 1982).

To highlight the rich variety of lines in the x-ray region, Fig. 5.1.3 shows lines of calcium observed during a solar flare. To see the regularities in spectra represented by the scalings in Table 5.1.2, refer to Gabriel and Jordan (1971) and Dufton and Kingston (1981).

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FIGURE 5.1.3 Ca XIX and Ca XVIII (Z ¼ 19 and 18) lines from a solar flare observed with the bent crystal spectrometer on the Solar Maximum Mission satellite. The broad variety of permitted, intercombination, and forbidden transitions includes Ca XIX lines (upper arrows): w is the 0 0 /1s 2 1 S respectively; and z resonance line 1s2p 1 P 1 /1s 2 1 S 0 ; x,y are from 1s2p 3 P2;1 0 1 3 2 is the fully forbidden 1s2s S 1 /1s S 0 . The other lines (a  t) are from Ca XVIII from doubly excited 1s2p2s and 1s2p2 configurations that decay to 1s22s and 1s22p configurations. Adapted from Jordan, C., Veck, N.J., May 1982. Comparison of observed CA XIX and CA XVIII relative line intensities with current theory. Sol. Phys. 78, 125e135.

5. SPECTRUM FORMATION IN A NUTSHELL Unlike laboratory plasmas, the Sun and astronomical sources are vast. Radiation transport, with notable exceptions, must therefore be dealt with. The reader should refer to a standard radiative transfer text (e.g., Mihalas, 1978), or to Chapter 5.2, because here we only review some essential ideas. In this section, we look at sources

5. Spectrum Formation in a Nutshell

and sinks of radiative energy given by source functions S and opacity k in terms of atomic-level populations and appropriate cross-sections (which in turn determine transition probabilities), leaving until later the ingredients needed for these quantities. Conservation of radiant energy along a certain ray yields the equation of radiative transfer in terms of three macroscopic quantities: the optical depth s along the ray, the specific intensity I, and the source function S (initially dropping the subscript for frequency n): dI ¼IS (5.1.23) ds in which ds ¼ kds along an element of the ray ds. I and S depend on the ray geometry (angles through an atmosphere) and frequency. The “source function” S ¼ h/k ¼ hl in which l ¼ k1 cm is the photon mean free path and h is the emission coefficient in Hz1. S is easily understood to be the contribution to the radiant energy integrated along one photon mean free path. Given a source function S, the solution to Eq. (5.1.23) is Z 0 IðtÞ ¼ SðsÞejtsj ds (5.1.24) smax

where, in particular, the emergent (observed) intensity (at s ¼ 0): Z 0 Ið0Þ ¼ SðsÞes ds smax

(5.1.25)

When mean free paths exceed the size of the emitting plasma (optically thin plasma), it is more meaningful to work with h and z (see subsequent discussion). I(0) represents the observed intensity of radiation at each frequency n. Unlike many astronomical objects, on the Sun we can resolve the surface. I(0) is a function of position r, f on the solar disk projected onto the plane of the sky. Historically, solutions to radiation transport generally assume rotational symmetry; I depends only on the cosine m of the angle q between the local gravity vector (the direction of atmospheric stratification) and the line of sight. Thus, we can write I(0) h Inm. Next, we examine plasma at high and low densities, leading to LTE (the detailed balancing of thermal processes) and the “coronal approximation” (balancing only between binary, or two-body, collisions), respectively. The intermediate case, the CR model, is reviewed. A discussion of the collisional processes responsible for setting, along with any radiation, the state (i.e., level populations q) of the atomic ions in plasma is delayed until the next section. Along with basic atomic crosssections (oscillator strengths, photoionization cross-sections, and other quantities), these populations q determine the emergent radiation along with the transport equation.

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5.1 OPTICALLY THICK FORMATION When thick, we can approximate modest depth variations of S (such as occur in radiative equilibrium) as Sn ¼ an þ bnsnmþ., which, with lim SðsÞes /0 and keeping s/∞ only the linear term: Inm ð0Þ z an þ bn m ¼ Sn ðs ¼ mÞ;

(5.1.26)

which is the “EddingtoneBarbier relation.” If thermal processes dominate the emission and absorption of photons, such as collisions with a bath of particles at temperature T and density n, and/or the radiation field is a black body at temperature T, then Sn ¼ Sn ¼ Bn ðTÞ in which Bn is the Planck function Bn ðTÞ ¼

2hn3 1 c2 expðhn=kTÞ  1

(5.1.27)

and k ¼ k*(n,T) is the absorption coefficient in LTE (denoted by *). In LTE, calculations of k consume most of the effort needed to compute the spectrum. In a typical stellar atmosphere, one has a mix of elements, each satisfying thermal statistical distributions, through the Saha equation. LTE is sometimes valid for visible and infrared spectral lines formed in the solar photosphere, and often for submillimeter to centimeter wavelength radiation formed in the upper photosphere and chromosphere where thermal (free-free) processes dominate both S and k. Whereas the photosphere is the densest visible plasma on the Sun (Fig. 5.1.2), transport of radiation in boundfree transitions often leads to significant non-LTE effects in k, if not also in S (Bruls et al., 1992). This should be unsurprising, when we acknowledge that thinner radiation in continua, freely escaping to space, implies no thermal equilibrium. A spectral line is characterized by having more opacity at core frequencies than in the line wings. Therefore, in the line core, we see source functions from regions higher in the atmosphere than in the wings and neighboring continuum. If the temperature decreases with height, when close to LTE, we thus see an absorption line spectrum. This explains the nature of Fraunhofer’s photospheric absorption spectrum when the photospheric temperature decreases with height in radiative equilibrium. Such observations allow the photospheric temperature structure to be mapped as a function of optical depth, and height if the relationship between line opacity and height is known, through a model atmosphere. Line depths yield information on abundances, line profiles on plasma densities, motions, electric and magnetic fields, and other parameters beyond this chapter’s scope (Griem, 1964).

5.2 OPTICALLY THIN FORMATION Solar plasma above the chromosphere can often be treated as optically thin. In plasmas where s < 1, with s measured from the observer to the source, Z 0 hðsÞds; (5.1.28) Ið0Þ  Iðs1 Þ ¼ s1

5. Spectrum Formation in a Nutshell

The intensity owing to the emitting plasma is just an integral of h along the line of sight minus any “incident radiation” behind the thin layer I(s1). In the Sun, the hot, thin corona emits on top of emission from the underlying photosphere I(s1) z S(s ¼ 1) or space (I[s1] ¼ 0 when observed above the visible limb). S(s ¼ 1) is large at visible or infrared wavelengths, so that I(0) z I(s1) and the thin plasma’s radiation is negligible. However, at EUV or x-ray wavelengths, I(0) [ S(s ¼ 1) and the corona is readily visible in emission above a much darker background. Again, a spectral line is characterized by having more opacity and emissivity at core frequencies than in the wings. Therefore, in the line core, more atoms emit in the hot optically thin layer between s1 and the observer than in the wings. Eq. (5.1.28) therefore predicts an emission line spectrum on top of any background (I(s1)). This explains the nature of the coronal spectrum at EUV and x-ray wavelengths (Fig. 5.1.1). Unlike the photospheric case, we cannot probe the temperature as a function of height; instead, we can derive a particular distribution of emitting material as a function of temperature from many spectral lines, as follows. The simplest pedagogical case is where Eq. (5.1.28) applies to emission lines between two atomic levels labeled 2 and 1, integrating over frequencies where line emission is significant. Using the rate of spontaneous emission of photons given by the Einstein Ae coefficient A21, we have h ¼ hn4p21 q2 ðsÞA21 and Z 0 hn21 q2 ðsÞA21 ds. (5.1.29) Ið0Þ  Iðs1 Þ ¼ s1 4p where q2(s) is the population density of level 2. In the particular case of a two-level atomic ion in which all photon emissions originate from collisions with plasma electrons from level 1 to level 2, we find the first part of the familiar “coronal approximation,” q2 A21 ¼ q1 ne hvs12 ihq1 ne C12 ðTe Þ. those electrons with 12me v2 > hn21 contribute, so that relaxed to temperature Te, C12(Te) f exp(hn21/kTe).

(5.1.30)

Only when the electrons are fully The second part of the coronal approximation is that the fraction of all ions of the element containing levels 1 and 2 is also controlled by two-body electron collisions (impacts between ions and single electrons). The latter dominates at low densities over three-body collisions (see the upper dot-dashed line in Fig. 5.1.2), such as in the solar corona (Woolley and Allen, 1948). The relevant rates are the inverse of the times sion and srr provided earlier. Then, in a fully ionized plasma in statistical equilibrium (dynamical time scales exceeding both sion and srr), q1 fAne F1 ðTe Þ (in which A is the elemental abundance relative to hydrogen) and then Z 0 Z 2 ne G12 ðTe Þdsh xðTe ÞG12 ðTe ÞdTe (5.1.31) Ið0Þ  Iðs1 Þ ¼ s1

DTe

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

in which the distribution function x(Te) measures a purely solar quantity, the emission measure x(Te)dTe. x(Te)dTe is visualized by seeking all plasma between Te and Te þ dTe along the line of sight and forming the sum of the product of n2e and dsk for all matching segments k found. x(Te) is derived by inverting the integral equation from a set of emission lines, “regularized” to produce a minimally structure solution compatible with the data, as to deal with nonuniqueness characteristic of such inversions (Craig and Brown, 1976). Several lines can used to determine plasma elemental abundances following early work by Woolley and Allen (1948). The CR model of Bates et al. (1962) extends the coronal approximation to optically thin plasmas at higher densities of laboratory and some astrophysical sources. Their treatment includes the large number of bound levels of high principal quantum number, n [ 1. Bates et al. (1962) developed a heuristic argument to treat the atomic populations by identifying and tracking populations of long-lived (“metastable”) levels. Two sets of equations are developed and solved independently. One is for ionization and recombination involving only long-lived or “metastable” levels and the other is for the fully relaxed set of number densities of the many short-lived excited levels (108Z4 s), given the populations of slowly evolving (metastable) levels from the first equation. The model compresses the general problem of time evolution by using just two modes corresponding to “ionizing” and “recombining-plasma” conditions. For a modern review, see Fujimoto (2004), who provides a qualitative justification for this method. A mathematical justification is given by Judge (2005). Fujimoto (2004) examines the ionizing and recombining components, allowing one to understand the role of the vast number of high-n levels in the atomic rate equations. Above a certain principal quantum number, the “quasithermal” value nqt, collisions dominate and connect higher levels to the ground state of the ion. For all levels with n > nqt, the populations approach LTE wrt the ground state of the ion above. For levels with n < nqt the coronal approximation is valid, with the levels being fed by collisions from lower-lying metastable levels. Given the Saha and Boltzmann equations, the populations of excited levels for most plasmas (neLTe  1) (see Eq. 5.1.16) are far smaller than the number densities of electrons and ions in the metastable (e.g., ground) state. The reader should refer to Cooper (1966), Section 7, and to Fujimoto (2004), Chapters 4 and 5, for details. Using hydrogenic approximations to levels with high principal quantum numbers n[1, the critical value is roughly  n 2=17 e nqt z 95 7 (5.1.32) Z which is Eq. (4.29b) of Fujimoto (2004) with ne in cm3. For the solar atmosphere with lgne ¼ 14, 11, 8 for the photosphere, chromosphere, and corona, nqtz3, 5, 14 respectively. These values agree well with detailed calculations for the Sun’s photosphere and chromosphere (Table 17 of Vernazza et al., 1981).

5. Spectrum Formation in a Nutshell

In astrophysics, the CR model helps us to identify limits of the coronal approximation. We can often make the following simplifications: 1. Above the chromosphere, ionization balance (b-f processes) can be computed separately from the excitation (b-b). The b-f processes are slowest, roughly 1 s1 ion þ srr per second. Ionization and statistical equilibrium are valid when dynamical processes are slower; 2. As found in the coronal ionization equilibrium approximation, each ion tends to be abundant at electron temperatures such that kTe  IZ; 3. We can compute ionization balance reasonably accurately using the coronal approximation (two-body processes) up to perhaps 1011 cm3, with care (see, e.g., Fig. 7 of Cooper, 1966); and 4. We can interpret the spectra of low-lying levels (with n ( nqt) using collisional excitation only from lower metastable levels. Point 3 is not valid in the cases where dielectronic recombination (DR) is important because of the role of high-n doubly excited states whose “spectator electron” has n [ nqt (Summers, 1974).

5.3 NONeLOCAL THERMODYNAMICAL EQUILIBRIUM AND FURTHER COMPLICATIONS These cases are limits of the more general case in which particle collisions are not frequent enough to maintain LTE, yet the plasma remains optically thick in some atomic transitions of interest. These conditions plague the chromosphere and prominences, and strong lines in the transition region. In this case, one must solve equations for radiative transfer simultaneously with kinetic equations for the populations q of atomic levels belonging to several neighboring ionization stages. This coupled set of nonlinear and nonlocal equations requires iterative procedures to bring the radiation and population densities to consistency. The nonlocality is illustrated clearly in the “two-level atom,” in which the non-LTE population equations



 dq2 dq1 ¼ q2 A21 þ B21 J þ qe C21 þ q1 B12 J þ qe C12 ¼  dt dt

(5.1.33)

i become, assuming statistical equilibrium (dq dt ¼ 0, which can include derivatives for advection and diffusion) (Delache, 1967; Fontenla et al., 1990; Pietarila and Judge, 2004),

S¼

h 2hn3 1  ¼ 2  hεB þ ð1  εÞJ; q1 g2 k c 1 q2 g1

e C21 where ε ¼ A21nþn , and C21 ¼ C21(1  exp  hn/kT). e C21

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The emergent spectra reflect the nonlocal nature of the formation of the spectrum. The source function (the term in J) includes J, J ¼ L½S.

(5.1.34)

in which L[.] is an operator extending over the entire atmosphere and over all frequencies across the line profile. When more than two levels are considered, the spectra also become nonlinearly coupled because transitions become implicitly dependent on one another through the multilevel population equations (Scharmer and Carlsson, 1985). Thus, the relatively “simple” interpretation of spectra that applies in LTE or the coronal limit becomes modified by the effects of photon scattering. For the strong Fraunhofer lines, source functions depend on the conditions across almost the entire chromosphere (which can exceed 104 local photon mean free paths). The ratio of the range of influence to the (much smaller) mean free path, the “thermalization length” (Mihalas, 1978), is fundamentally important in stellar atmospheres. In general, the emergent intensity of an optically thick plasma depends on the geometry and the boundary conditions, as well as the plasma itself. As such, numerical solutions are required. Algorithms to solve the coupled nonlinear and nonlocal equations of radiative transfer and atomic populations in modern astrophysics usually fall into two categories: linearization (Scharmer and Carlsson, 1985) and preconditioning (Rybicki and Hummer, 1991, 1992). Three-dimensional codes are readily available for the statistical equilibrium problem, including the linearization code MULTI3D (Botnen and Carlsson, 1999) and the preconditioning code RH (Uitenbroek, 2000; Uitenbroek, 2001). Further departures from the non-LTE case are encountered in solar physics. For example, statistical equilibrium becomes a poor assumption when relaxation times for the populations exceed dynamical times, typically the slowest of the ionization and recombination times. These effects occur in both “quiet” as well as active and flaring regions of the chromosphere. These are best studied on a case-by-case basis, although there are some general ideas related to the “ionizing” or “recombining” conditions from the CR model (Judge, 2005; Laming and Feldman, 1992). Another example is the unambiguous evidence of high-energy tails in electron distribution functions, which are found in the form of type III radio bursts and hard x-ray flare emission (Fletcher et al., 2011). Coulomb collisions are infrequent at high energies (see(ε) f ε2), so that once they are generated, such electrons thermalize only when they penetrate to the deep chromosphere or photosphere. Table 5.1.3 summarizes these various regimes.

6. Plasma Spectroscopy

Table 5.1.3 Regimes of Spectrum Formation Regime

Conditions

Solar Example

Thermal equilibrium LTE

Detailed balance High densities

Coronal

Hot, tenuous plasma (<108/ cm3) Optically thin, two-body processes Denser plasma (>108/cm3) optically thin, including nonequilibrium Solution of coupled RTa and SEa equations Populations evolve in time with or without RT

. Weak lines, deep photosphere Quiet solar corona

approximation Collisional-radiative

Non-LTE Nonequilibrium

Active solar corona

Strong lines chromosphere, prominences Chromosphere, transition region corona, flares

a LTE, local thermodynamical equilibrium; RT, radiative transfer; SE, statistical equilibrium. All of these regimes assume that electrons are fully relaxed to an equilibrium temperature Te.

6. PLASMA SPECTROSCOPY In contrast to isolated atoms, the wavefunctions of atoms and ions embedded in a tenuous plasma are solutions to equations such as v j (5.1.35) vt in which HP(t) is the additional term in the atomic Hamiltonian owing to the collective effect of particles and fields in the plasma. HP(t) is a function of time t because warm gases and plasmas contain vast numbers of particles, each undergoing complicated motions in the presence of external and self-generated fields. When the particles are thermalized, these motions are stochastic, and so they must be the wavefunctions of individual atoms. However, Eq. (5.1.35) is not generally solved in this fashion (nevertheless, see DeWitt and Nakayama, 1964), for it is only in a statistical sense that we need to know the evolution of the atomic wavefunctions from macroscopic objects. Instead, the quantized atoms are examined in a plasma described classically. Then we can speak of various processes that are treated as perturbations of H0, in which ensembles of atoms and atomic ions evolve within a plasma: ðH0 þ HP ðtÞÞj ¼ iZ

1. 2. 3. 4.

Lowering of ionization potential by the plasma electric microfields Stimulated emission and absorption of radiation Inelastic particle collisions with charged and neutral particles Elastic particle collisions

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The ionization potential is lowered because half of the time, thermally varying electric fields in each Debye sphere supply small amounts of energy to aid ionization processes. The infinite sums in partition functions are removed physically; the amount of the reduction is DIZ z

Ze2 2 kTe ; rD z rD 4pe2 ne

(5.1.36)

Now IZ z (Ze2/2a0), so that DIZ/IZ z 2a0/rD. The maximum quantum number pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi corresponding to this energy is rD =2a0 . This is not to be confused with the quasithermal (collisional) limit nqt or with the IngliseTeller estimate arising from line-broadening, which makes the discrete spectrum appear as a continuum at high n. Table 5.1.4 lists processes that are important in changing the states of atomic ions in plasmas, and which therefore contribute to the radiation emerging from the plasmas. Changes in the electron wavefunctions are most readily induced by radiation (processes 2, 3, 5, and 6) through E1 interactions, and by collisions with plasma electrons. Electron collisions are far more efficient than ion collisions because of a classical result: namely, that the energy exchanged between colliding particles is maximized when the particles have the same mass. Collisions with protons and ions can have significant effects under specific conditions, such as between fine structure levels or when Ti [ Te (Seaton, 1964a). One more general result is clear: that three-body recombination (process 8) is rare in most plasmas except for the densest conditions, because it requires the simultaneous presence of three particles (the atomic ion and two electrons) within a volume of order (a0/Z)3. In Fig. 5.1.2, we see that for most plasmas of interest, three-body recombination is negligible. In contrast, the inverse process, collisional ionization (5.1.7) is often dominant. The balance between ionization (5.1.7) and two-body recombinations (5.1.4) and (5.1.13) constitutes part of the coronal approximation. Surprisingly, classical arguments yield the correct asymptotic dependence on Te (kTe  IZ) and order of magnitude for (process 7) (Seaton, 1962b). The formula listed in Table 5.1.4 is in fact a semiempirical fit to cross-sections based on the classical behavior for kTe  IZ. Updates to such fits are to be found in Arnaud and Rothenflug (1985) and Arnaud and Raymond (1992) and later citations to these two publications. Collisions between bound levels with n < nqt are fundamentally different depending on whether the target atom is neutral or ionized. In the neutral case, the cross-section at threshold (12 mv2 ¼ Ej  Ei ) is zero and increases algebraically with energy up to a maximum close to 2(Ej  Ei), when it declines. In contrast, because of the Coulomb attraction of the projectile electron by the target charge, cross-sections at threshold are finite for charged targets. Yji(Te) is the Maxwellianaveraged collision strength. Yji(Te), a measure of probability, approaches zero for

Table 5.1.4 Atomic Processes in Plasmas a

Name

Processb

Example

1

Spontaneous decay

XZ,k / XZ,j þ hn

2s2p1 P1 / 2s2 1 S0 þ hn

2

Stimulated emission

XZ,k þ hn / XZ,j þ 2hn

2s2p1 P1 þ hn / 2s2 1 S0 þ 2hn

3(1)

Absorption

XZ,j þ hn / XZ,k

4(5)

Radiative recombination

XZ,k þ e / XZ1,j þ hn

2s2 1 S0 þ hn / 2s2p1 P1

5(4)

Photo-ionization

6

Stimulated recombination

0 0

0

2s2 S1=2 þ e / 2s2 1 S0 þ hn 

XZ,j þ hn / XZ þ 1,k þ e

2s2 1 S0 þ hn / 2s2 S1=2 þ e



XZ,k þ hn þ e / XZ1,j þ 2hn 



2s2 S1=2 þ e þ hn / 2s2 1 S0 þ 2hn



7(8)

Direct ionization

XZ,k þ e / XZ þ 1,j þ e þ e

2s2 1 S0 þ e / 2s2 S1=2 þ e þ e

8(7)

Three-body recombination

XZ,k þ e þ e / XZ1,j þ e

2s2 S1=2 þ e þ e / 2s2 1 S0 þ e

9(10)

Excitation (direct)

10(9)

12(11)

c

13

c

14

c

a

XZ,j þ e / XZ,k

0

2s2 1 S0 þ e / 2s2p1 P1 þ e



Deexcitation (direct)

XZ,k þ e / XZ,j

Dielectronic capture

XZ;j þ e / XZ1;k 

Autoionization Dielectronic recombination Excitation via autoionization

XZ1;k  / XZ;j þ

0

2s2p1 P1 þ e / 2s2 1 S0 þ e 2s2 S1=2 þ e / 2 pnp 1 S1

e

2pnp 1 S1 / 2s2 S1=2 þ e

XZ1;k  / XZ1;k  þ hn / XZ1;j 11 then XZ1;k  / XZ;k

þ e

þ hn0

0

2pnp 1 S1 / 2snp1 P1 þ hn / 2s2 1 S0 þ hn0 0

0 2s2 S1=2 þ e / 2pns1 P1 / 2p2 P1=2 þ e0

The column lists P(I) in which P represents the process written and I is the inverse process, where applicable. XZ,k refers to atomic species “X,” of ion stage Z (core charge) and energy level k. The energies are in alphabetical order: the energy of level XZ,j is lower than that of XZ,k. c These processes involve “two-electron” excitations. “One” and “two” electron processes mean that the atomic wavefunctions have just one spin-orbital and two spin-orbitals that have been excited, leaving behind one and two holes in the “core,” respectively. b

6. Plasma Spectroscopy

11(12)

c



147

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

Te / 0 for neutral targets and a finite value for ions. The collision rate per target atom is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Yji ðTe Þ Yji ðTe Þ 2 2 Cji ¼ pa0 2R ne ; ¼ 8:63  106 Te1=2 ne . (5.1.37) pme kTe gj gj The upward collision rate is simply given by qi Cij ¼ qj Cji and Y is symmetric in ij. Given Eqs. (5.1.15), (5.1.29), and (5.1.30), this means that lines with different values of E2  E1 have ratios that are sensitive to Te. For neutrals, unlike the ions, the collision strengths are zero at zero temperature. The different kinds of transitions listed in Table 5.1.2, which apply to ions, are thoroughly discussed in a penetrating publication by Burgess and Tully (1992). The different temperature (i.e., energy) dependences listed in the table arise from conceptually simple physics. The E1 transitions are dominated by long-range interactions at relatively low energies; as a result, the cross-sections are very large, and the effects of doubly excited states (“autoionization resonances”) on the cross-sections are modest. These crosssections are proportional to the E1 oscillator strengths, which is the origin of the well-known “Van Regemorter” and related approximations (Van Regemorter, 1962). Generally speaking, the cross-sections for ions are much better known than for neutrals, with the exceptions of neutral H and He, which have naturally received much attention because of their astrophysical abundances. Packages such as CHIANTI (Young et al., 2016) contain the results of decades of efforts to compute, measure, and compile such data. For levels with principal quantum numbers above nqt, cross-sections become large owing to the classical orbit size and the correspondence principle. Hydrogenic results can be used. For IC transitions, a change of spin is needed that requires penetration of the ion “target,” i.e., short-range interactions. There is no E1 component; the cross-sections are therefore smaller and exchange of electrons can be important. As a result, the cross-sections drop rapidly with increasing energy as the interaction time is reduced. More seriously, the cross-sections and the Maxwellian averaged Yji(Te) become sensitive to the doubly excited states through autoionization (process 14). The resonances are particularly difficult to compute in energy, so that the IC collisional transition probabilities entail substantial uncertainties unless (rare) measurements are made. Forbidden transitions behave somewhat like IC transitions, and as noted earlier, they often have contributions from proton collisions. Radiative recombination is computed employing the principle of detailed balance, using populations from the Saha equation and with photoionizing radiation given by the Planck function. It is dominated by the spontaneous recombination component. Most of the recombination occurs to levels with low principal quantum number n. Recombination involves collisions with electrons that, in the attractive Coulomb field of the target, can excite doubly excited states. Use of the term “radiative recombination” explicitly excludes resonance structures associated with doubly excited states. Hence, the rate depends monotonically on Te, as indicated in the Table 5.1.2. Except for recombination onto a bare nucleus, separation between

6. Plasma Spectroscopy

“radiative” recombination and DR, which involves the resonances, is artificial (Nahar and Pradhan, 1992). Recombinations directly to levels with n > nqt are effectively not recombinations at all because collisions upward from n to neighboring n þ 1 dominate in determining the flux of populations from nqt up to the continuum (see Chapter 3 of Fujimoto, 2004). This leads us to discuss the last process: DR (process 13). Following a suggestion by Unso¨ld in the late 1950s, this was first studied by Seaton’s group. Burgess (1964, 1965) was able to show its importance and develop a general formula based on the Rydberg series of high-n states converging on to a doubly excited ionization limit (Fig. 5.1.4). Burgess’s work showed that discrepancies in coronal temperatures using a variety of techniques were largely resolved by inclusion of this process (Seaton, 1962a, 1964b). The process is illustrated by Fig. 5.1.4 taken from Cooper (1966). To understand the processes involved, we look at the impact of an electron with energy E on the state marked “First ionization limit.” To fix ideas, let this be the ground level of Be-like ions 1s22s2S10 at E1, and let the second limit (“first excited state of the 0 ion”) be 1s2 2s2p1 P1 at E2. Now consider the possible outcomes of varying E: •

•

• •

• •

If E > (E2  E1), the impacting electron can excite level 2 with an outgoing electron with lower energy (process 9). The rapid radiative decay of this level yields emission of a photon in the resonance lines of the Be-like ion. If E < (E2  E1), lying between the doubly excited levels, the electron may emit a photon owing to direct recombination at UV wavelengths (forming a localized recombined atomic state) or its acceleration in the charge of the Be-like ion (Bremsstrahlung, leaving the ion unperturbed and a free electron). • If E < (E2  E1) and we are at energies coincident with a doubly excited level energy (there is in principle an infinity of these levels converging onto E2), the incoming electron’s wavefunction may change from that of a continuum (Coulomb wavefunction extending to infinity) of the form 1s22s2pkl in which -k2/2m ¼ E  E1 to a local wavefunction characterized by a two-electron excitation that, in terms of spin-orbitals, will have the form 1s22s2pnl. The wavefunctions can be seen as a superposition of these two types of “single electron” wavefunctions. The wavefunction collapses to either 1s22s2pkl (autoionization: process 12) or 1s22s2pnl (dielectronic capture, process 11). If collapse occurs to 1s22s2pnl, this state will most rapidly decay (for large nl values) via the transition 1s22s2pnl / 1s22s2nl (see the line marked “radiative transition probability” in the Figure 5.1.4). This line will have a longer wavelength than the resonance line of ion Z because it occurs in the presence of the core plus the “spectator” nl electron. Such lines are called “dielectronic satellites.” Radiative decay of the nl electron to a lower orbital is termed “stabilization.” The net effect is that the electron has been captured by ion Z, leading to an excited level of ion Z  1. This is DR.

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FIGURE 5.1.4 (Left) A series of energy levels for single-electron excitations such as those found for hydrogen-like ions. They converge onto the shaded continuum (“First ionization limit”). (Right) Two electrons are excited and the energy levels shown converge onto a higher continuum. Those doubly excited levels lying between these continua are the cause of autoionization and dielectronic recombination. Adapted from Cooper, J., January 1966. Plasma spectroscopy. Rep. Prog. Phys. 29, 35e130.

The sum over all the nl terms was examined by Burgess (1965) to derive a general formula applicable to low-density plasmas. It should be clear that at finite densities, lowering of the ionization potential by plasma microfields and the important effects of upward collisions in the series of high nl levels will suppress DR. Summers (1974) addressed this problem, solving the large matrix equations needed to account for these effects in detail.

6. Plasma Spectroscopy

Using these results and the scalings of nqt with density, we can make general statements about the importance of DR in the solar atmosphere and interior. DR is unimportant below the photosphere of the Sun because there, nqt  3. In other words, all levels with nl > 3 are strongly collisionally coupled to the next ionization stage and so LTE prevails for these levels. In the chromosphere (ne z 1011 cm3, Te z 7000K), DR may be important but only for one or two levels, because nqt z 5. The corona is the optimal region of the Sun’s atmosphere for Burgess’ DR mechanism to operate because nqt > 10. The effects of DR are most important when E2  E1 is smallest, because then impacting electrons with modest energies can contribute to the DR process. Impacts of electrons on H- and He-like ions tend to have lower DR rates than those of Li-like ions for this reason. Fig. 5.1.5 shows recombination rates split into radiative and DR components, as well as a self-consistent calculation from Nahar and Pradhan (1992) for two ions with prominent lines in the solar transition region. Several lessons are immediately learned from this Fig. 5.1.5. DR is a phenomenon that peaks in energy and electron temperature, unlike radiative recombination, which decreases monotonically with temperature. This is because the states involved converge on to the first excited levels of ion Z. In the example shown, the O IV to O III DR rate peaks near 1.6  10 K, corresponding to 13.6 eV. The first excited levels

FIGURE 5.1.5 Recombination rate coefficients computed using standard approaches and a full treatment of the electron-ion collision problem (Nahar and Pradhan, 1992). All calculations apply to the zero-density limit. Radiative recombination is characterized by the monotonic drop of the coefficient with Te, dielectronic recombination peaks close to the energy of the first excited levels of the doubly excited state.

151

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connected by E1 transitions to the ground level of O IV lie near 15 eV. The second important lesson is that care is needed in using atomic data. It is seen that factors of two exist between different calculations, even without the effects of collisions in levels with n > nqt. Finally, when DR is present, it can exceed radiative recombination by orders of magnitude. Storey (1981) pointed out the importance of a type of DR efficient at low temperatures, when the first excited levels E2 are metastable levels lying close to the ground level of ion Z. These occur in Be-, C-, N-, and O-like ions in the second row of the periodic table (also, of course, in third and higher rows), which are abundant in many astrophysical plasmas. Stabilization occurs via excitation and decay of a doubly excited electron, which, in contrast to Burgess’ high temperature case, has n  5.

7. CLOSING REMARKS This chapter is necessarily just a short overview. It is hoped that the information provided a little insight and physical understanding about how spectra are formed in the solar atmosphere. The reader should refer to the reviews and textbooks referenced for a more complete understanding. Spectroscopy, with radiation transfer and polarimetry (Chapters 5.2 and 5.3 in this volume), lie at the heart of all quantitative knowledge of the Sun and stars. Essentially all interesting problems in solar physics must be addressed at some level through the building blocks outlined in this chapter, including the most important remaining unsolved problem: How does the Sun regenerate its global magnetic field every 22 years? For completeness, here is a list of useful spectroscopic resources, plus a brief assessment of the goals of the various repositories and projects: •

•

•

The National Institute of Standards and Technology provides critically assessed atomic energy levels and spectral lines. These are necessarily incomplete as critical evaluations take time to ingest. See https://physics.nist.gov/ PhysRefData/ASD/levels_form.html CHIANTI has become a workhorse for solar physicists. It endeavors to capture the latest and best energy levels, transition probabilities, collision strengths, and various tabulations of ionization equilibrium for use primarily in low-density, coronal, and transition-region plasma. The user should be aware of the need to allow for finite-density effects in ionization equilibria. CHIANTI targets UV, EUV, and x-ray spectra. See http://www.chiantidatabase.org/ HAOS-DIPER is an ingestion of CHIANTI and OPACITY and IRON PROJECT data with some allowance for finite density effects and nonequilibrium conditions. The database adopts quantum numbers for energy levels and as such can be used to approximate various parameters and modify atomic models when needed. See http://www.hao.ucar.edu/modeling/haos-diper/

References

•

•

•

• •

The OPACITY and IRON projects used state-of-the-art calculations of opacities. The data must be understood by the user as applicable to opacity calculations, and their use for spectroscopy must be handled with appropriate care. See http:// cdsweb.u-strasbg.fr/topbase/home.html The Atomic Data and Analysis Structure project is de facto a primary source for plasma spectroscopy under optically thin conditions. It is a major provider for the plasma machine and nuclear fusion communities. See http://www.adas.ac. uk/about.php For radiative transfer, LTE or otherwise, perhaps the best starting place is the code RH (“Rybicki-Hummer”) written by Uitenbroek (2000) because of its ready adaptation to one-, two-, or three-dimensional cases. Uitenbroek can be contacted for the code itself. R. Kurucz maintains a large database of atomic transition probabilities for opacity calculations. See http://kurucz.harvard.edu/linelists.html There are many other packages that are more specialized: for instance, for application to molecules and/or soft s-ray astronomy. See, e.g., https://heasarc. gsfc.nasa.gov/xanadu/xspec/or http://www.atomdb.org/

Finally, the reader is recommended not to use packages as black boxes for reasons that should be clear.

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