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Electron–Ion and Ion–Ion Recombination General References 825

54.7 J. N. Bardsley: J. Phys. A Proc. Phys. Soc. 1,365

(1968)

54.8 M. R. Flannery: Atomic Collisions: A Sympo-

sium in Honor of Christopher Bottcher,ed.

by D. R. Schultz, M. R. Strayer, J. H. Macek

(American Institute of Physics, New York 1995)

p. 53

54.9 A. Giusti: J. Phys. B 13, 3867 (1980)

54.10 P. van der Donk, F. B. Yousif, J. B. A. Mitchell,

A. P. Hickman: Phys. Rev. Lett. 68, 2252 (1992)

54.11 S. L. Guberman: Phys. Rev. A 49, R4277 (1994)

54.12 M. R. Flannery: Int. J. Mass Spectrom. Ion Process

149/150, 597 (1995)

54.13 M. R. Flannery: J. Chem. Phys. 95, 8205 (1991)

54.14 A. Burgess: Mon. Not. R. Astron. Soc. 118,477(1958)

54.15 J. A. Gaunt: Philos. Trans. Roy. Soc. A 229, 163 (1930)

54.16 M. J. Seaton: Mon. Not. R. Astron. Soc. 119, 81 (1959)

54.17 B. F. Rozsnyai, V. L. Jacobs: Astrophys. J. 327,485

(1988)

54.18 D. R. Flower, M. J. Seaton: Comp. Phys. Commun. 1,

31 (1969)

54.19 P. G. Martin: Astrophys. J. Supp. Ser. 66, 125 (1988)

General References (General Recombination)

G.1 M. R. Flannery: Adv. At. Mol. Phys. 32, 117 (1994)

G.2 M. R. Flannery: Recombination Processes.In:Mo-

lecular Processes in Space, ed. by T. Watanabe,

I. Shimamura, M. Shimiza, Y. Itikawa (Plenum, New

York 1990), Chapt. 7, p. 145

G.3 D. R. Bates: Recombination.In:Electronic and

Atomic Collisions, ed. by H. B. Gilbody, W. R. Newell,

F. H. Read, A. C. H. Smith (North-Holland, Amster-

dam 1988), p. 3

G.4 D. R. Bates: Recombination.In:Case Studies in

Atomic Physics, ed. by E. W. McDaniel, M. R. C. Mc-

Dowell (North-Holland, Amsterdam 1974), Vol. 4,

p. 57

G.5 D. R. Bates, A. Dalgarno: Electronic Recom-

bination.In:Atomic and Molecular Processes,

ed. by D. R. Bates (Academic Press, New York

1962)

G.6 D. R. Bates: Adv. At. Mol. Phys. 15, 235 (1979)

G.7 W. G. Graham (Ed.): Recombination of Atomic Ions

(Plenum Press, New York 1992)

G.8 J. N. Bardsley: Recombination Processes in Atomic

and Molecular Physics.In:Atomic and Molecular

Collision Theory, ed. by F. A. Gianturco (Plenum

Press, New York 1980), p. 123

G.9 J. Dubau, S. Volonte: Rep. Prog. Phys. 43,199

(1980)

G.10 Y. Hahn: Adv. At. Mol. Phys. 21,123(1985)

G.11 Y. Hahn, K. J. LaGattuta: Phys. Rep. 116, 195 (1988)

G.12 E. W. McDaniel, E. J. Mansky: Adv. At. Mol. Opt. Phys.

33, 389 (1994)

(Three-Body Electron–Ion Collisional-Radiative Recombination: Theory)

G.13 D. R. Bates, A. E. Kingston, R. W. P. McWhirter: Re-

combination between Electrons and Atomic Ions:

I. Optically Thin Plasmas; II. Optically Thick Plas-

mas; Proc. R. Soc. (London) Ser. A 267, 297 (1962);

270,155(1962)

G.14 D. R. Bates, S. P. Khare: Proc. Phys. Soc. 85,231

(1965)

G.15 D. R. Bates, V. Malaviya, N. A. Young: Proc. R. Soc.

(London) Ser. A 320,437(1971)

G.16 A. Burgess, H. P. Summers: Mon. Not. R. Astron. Soc.

174,345(1976)

G.17 H. P. Summers: Mon. Not. R. Astron. Soc. 178,101

(1977)

G.18 N. N. Ljepojevic, R. J. Hutcheon, J. Payne: Comp.

Phys. Commun. 44,157(1987)

(Electron–Ion Recombination: Molecular Dynamics Simulations)

G.19 W. L. Morgan: J. Chem. Phys. 80, 4564 (1984)

G.20 W. L. Morgan: Phys. Rev. A 30,979(1984)

G.21 W. L. Morgan, J. N. Bardsley: Chem. Phys. Lett. 96,

93 (1983).

G.22 W. L. Morgan: Recent Studies in Atomic and Molecu-

lar Processes, ed. by A. E. Kingston (Plenum Press,

New York 1987), p. 149

Part D G

826 Part D Scattering Theory

(Ion–Ion Recombination: Review Articles)

G.23 M. R. Flannery: Three-Body Recombination between

Positive and Negative Ions.In:Case Studies in

Atomic Collision Physics, ed. by E. W. McDaniel,

M. R. C. McDowell (North-Holland, Amsterdam 1972),

Vol. 2, 1, p. 1

G.24 M. R. Flannery: Ionic Recombination.In:Atomic

Processes and Applications, ed. by P. G. Burke,

B. L. Moiseiwitsch (North-Holland, Amsterdam

1976), 12, p. 407

G.25 M. R. Flannery: Ion–Ion Recombination in High

Pressure Plasmas.In:Applied Atomic Collision

Physics, ed. by E. W. McDaniel and W. L. Nighan

(Academic Press, New York 1983), Vol. 3 Gas Lasers,5,

p. 393

G.26 M. R. Flannery: Microscopic and Macroscopic Per-

spectives of Termolecular Association of Atomic

Reactants in a Gas.In:Recent Studies in Atomic and

Molecular Processes, ed. by A. E. Kingston (Plenum

Press, London 1987), p. 167

G.27 D. R. Bates: Adv. At. Mol. Phys. 20,1(1985)

G.28 B. H. Mahan: Adv. Chem. Phys. 23, 1 (1973)

G.29 J. T. Moseley, R. E. Olson, J. R. Peterson: Case Studies

in Atomic Collision Physics, ed. by M. R. C. McDowell,

E. W. McDaniel (North-Holland, Amsterdam 1972),

p. 1

G.30 D. Smith, N. G. Adams: Studies of Ion–Ion Re-

combination using Flowing Afterglow Plasmas.In:

Physics of Ion–Ion and Electron–Ion Collisions,ed.

by F. Brouillard, J. W. McGowan (Plenum Press, New

York 1982), p. 501

(Termolecular Ion–Ion Recombination: Theory)(A) Low Gas Densities: Linear Region

G.31 M. R. Flannery, E. J. Mansky: J. Chem. Phys. 88, 4228

(1988)

G.32 M. R. Flannery, E. J. Mansky: J. Chem. Phys. 89,4086

(1988)

G.33 M. R. Flannery: J. Chem. Phys. 89, 214 (1988)

G.34 M. R. Flannery: J. Chem. Phys. 87, 6947 (1987)

G.35 M. R. Flannery: J. Phys. B 13, 3649 (1980)

G.36 M. R. Flannery: J. Phys. B 14, 915 (1981)

G.37 D. R. Bates, I. Menda

s: J. Phys. B 15,1949

(1982)

G.38 D. R. Bates, P. B. Hays, D. Sprevak: J. Phys. B 4,962

(1971)

G.39 D. R. Bates, M. R. Flannery: Proc. R. Soc. (London)

Ser. A 302,367(1968)

G.40 D. R. Bates, R. J. Moffett: Proc. R. Soc. (London) Ser.

A 291, 1 (1966)

(B) All Gas Densities: Non-Linear Region

G.41 M. R. Flannery: Microscopic and Macroscopic

Theories of Termolecular Recombination be-

tween Atomic Ions.In:Dissociative Recom-

bination, ed. by B. R. Rowe, J. B. A. Mitchell,

A. Canosa (Plenum Press, New York 1993),

p. 205

G.42 M. R. Flannery: J. Chem. Phys. 95, 8205 (1991)

G.43 M. R. Flannery: Phil. Trans. R. Soc. (London) Ser. A

304, 447 (1982)

G.44 D. R. Bates, I. Menda

s: Proc. R. Soc. (London) Ser. A

359,275(1978)

G.45 J. J. Thomson: Phil. Mag. 47, 337 (1924)

Ion–Ion Recombination: Monte-Carlo Simulations

G.46 P. J. Feibelman: J. Chem. Phys. 42, 2462 (1965)

G.47 A. Jones, J. L. J. Rosenfeld: Proc. R. Soc. (London)

Ser. A 333, 419 (1973)

G.48 D. R. Bates, I. Menda

s: Proc. R. Soc. (London) Ser. A

359, 287 (1978)

G.49 D. R. Bates: Chem. Phys. Lett. 75, 409 (1980)

G.50 J. N. Bardsley, J. M. Wadehra: Chem. Phys. Lett. 72,

477 (1980)

G.51 D. R. Bates: J. Phys. B 14, 4207 (1981)

G.52 D. R. Bates: J. Phys. B 14, 2853 (1981)

G.53 D. R. Bates, I. Menda

s: Chem. Phys. Lett. 88,528

(1982)

G.54 W. L. Morgan, J. N. Bardsley, J. Lin, B. L. Whitten:

Phys. Rev. A 26, 1696 (1982)

G.55 B. L. Whitten, W. L. Morgan, J. N. Bardsley: J. Phys.

B 15, 319 (1982)

Part D G

Electron–Ion and Ion–Ion Recombination General References 827

Ion–Ion Tidal Recombination: Molecular Dynamics Simulations

G.56 D. R. Bates, W. L. Morgan: Phys. Rev. Lett. 64, 2258

(1990)

G.57 W. L. Morgan, D. R. Bates: J. Phys. B 25,5421

(1992)

Radiative Recombination: Theory

G.58 M. J. Seaton: Mon. Not. R. Astron. Soc. 119, 81 (1959)

G.59 D. R. Flower, M. J. Seaton: Comp. Phys. Commun. 1,

31 (1969)

G.60 A. Burgess, H. P. Summers: Mon. Not. R. Astron. Soc.

226,257(1987)

G.61 Y. S. Kim, R. H. Pratt: Phys. Rev. A 27, 2913 (1983)

G.62 D. J. McLaughlin, Y. Hahn: Phys. Rev. A 43,1313

(1991)

G.63 F. D. Aaron, A. Costescu, C. Dinu, J. Phys. II (Paris) 3,

1227 (1993)

Dissociative Recombination: Theory and Experiment

G.64 D. Zajfman, J. B. A. Mitchell, B. R. Rowe, D. Schwalin

(Eds.): Dissociative Recombination: Theory, Ex-

periment and Applications III (World Scientiﬁc,

Singapore 1996)

G.65 D. R. Bates: Adv. At. Mol. Phys. 34, 427 (1994)

G.66 B. R. Rowe, J. B. A. Mitchell, A. Canosa (Eds.): Dis-

sociative Recombination: Theory, Experiment and

Applications II (Plenum, New York 1993)

G.67 J. B. A. Mitchell, S. L. Guberman (Eds.): Dissociative

Recombination: Theory, Experiment and Applica-

tions I (World Scientiﬁc, Singapore 1989)

G.68 J. B. A. Mitchell: Phys. Rep. 186, 215 (1990)

G.69 A. Giusti-Suzor: Recent Developments in the Theory

of Dissociative Recombination and Related Pro-

cesses.In:Atomic Processes in Electron-Ion and

Ion-Ion Collisions, ed. by F. Brouillard (Plenum

Press, New York 1986)

G.70 J. N. Bardsley, M. A. Biondi: Adv. At. Mol. Phys. 6,

1 (1970)

Part D G

829

Dielectronic Re

55. Dielectronic Recombination

Dielectronic recombination (DR)isatwo-step

process that greatly increases the effciency for

electrons and ions to recombine in a plasma. The

process therefore plays an important role in the

theoretical modeling of plasmas, whether in the

laboratory or in astrophysical sources such as the

solar corona. The purpose of this chapter is to

present the theoretical formulation for DR,andthe

principal methods for calculating rate coefﬁcients.

The results are compared with experiment over a

broad range of low-Z ions and high-Z ions where

relativistic effects become important.

55.1 Theoretical Formulation ....................... 830

55.2 Comparisons with Experiment .............. 831

55.2.1 Low-Z Ions............................... 831

55.2.2 High-Z Ions

and Relativistic Effects ............... 831

55.3 Radiative-Dielectronic Recombination

Interference........................................ 832

55.4 Dielectronic Recombination

in Plasmas .......................................... 833

References .................................................. 833

Electron–ion recombination into a particular ﬁnal re-

combined state may be schematically represented as

−

+ A

→ A

(q−1)+

+ ω, (55.1)

and

−

+ A

→



(q−1)+



→ A

(q−1)+

+ ω, (55.2)

where q is the charge on the atomic ion A, ω is

the frequency of the emitted light, and the brackets

in (55.2) indicate a doubly-excited resonance state. The

ﬁrst process is called radiative recombination (RR),

while the second is called dielectronic recombination

(DR). Both recombination mechanisms are the inverse

of photoionization. At sufﬁciently high electron dens-

ity, three-body recombination becomes possible. The

three-body mechanism is the inverse of electron impact

ionization.

The review article by Seaton and Storey [55.1]in-

cludes an interesting history of the theoretical work

on dielectronic recombination. The process was ﬁrst

referred to as dielectronic recombination by Massey

and Bates [55.2], after a suggestion of its possible

importance in the ionosphere by Sayers in 1939. How-

ever, estimates of the rate coefﬁcient for this process

indicated that DR is not an important process in

the ionosphere, where the temperatures are too low

to excite anything but the lower energy resonance

states.

In 1961, Unsold, in a letter to Seaton, suggested

that DR might account for a well-known tempera-

ture discrepancy in the solar corona. Seaton initially

concluded that DR would not signiﬁcantly increase re-

combination in the solar corona. However, he had only

included the lower energy resonance states in his analy-

sis; Burgess [55.3] showed that when one includes the

higher members of the Rydberg series of resonance

states that are populated at coronal temperatures, DR

can indeed explain this discrepancy.

Dielectronic recombination has since received much

theoretical attention due, in part, to its importance

in modeling high temperature plasmas. Various ap-

proaches to the theory are discussed in a review by

Hahn [55.4] and in [55.5]. Recently, there have been

various projects aimed at the generation of large quan-

tities of DR data for use in astrophysical and fusion

plasma modeling. One such project is based on the re-

sults of the AUTOSTRUCTURE code, with both the

total and partial (i. e., resolved by recombined level)

DR rate coefﬁcients being archived. The methodology

is outlined in Badnell et al. [55.6]. Data are calcu-

lated for all members of an isoelectronic sequence

from H to Ar, along with various ions relevant to

astrophysics or fusion, namely Ca, Ti, Cr, Fe, Ni,

Zn, Kr, Mo, and Xe. Work has been completed for

the oxygen [55.7], beryllium [55.8], carbon [55.9],

lithium [55.10], boron [55.11], neon [55.12] and nitro-

gen [55.13] isoelectronic sequences. There has also been

Part D 55

830 Part D Scattering Theory

a large quantity of data generated using a fully relativistic

Dirac–Fock code [55.14]. This data includes calcula-

tions of Na-like ions [55.15] and H-like through to

Ne-like [55.16] ions of certain astrophysically important

elements.

Interest in dielectronic recombination has in-

creased dramatically in the last twenty years. Mitchell

et al. [55.17] published the DR cross section for C

us-

ing a merged electron–ion beams apparatus, and Belic

et al. [55.18] reported on a crossed beams measure-

ment of the DR cross section for Mg

.Also,Dittner

et al. [55.19] published merged beams measurements

of the DR cross section for the multiply charged ions

and C

. Since that time, atomic physics exper-

iments carried out using heavy-ion traps, accelerators,

and storage-cooler rings have produced high-resolution

mappings of the resonance structures associated with

electron-ion recombination. The experiments have been

carried out using a wide range of facilities and technolo-

gies, such as the test storage ring (TSR) at Heidelberg,

the experimental storage ring (ESR) at Darmstadt, the

accelerator-cooler ring facility at Aarhus, the electron

beamiontrap(EBIT) at Livermore, and the electron

beam ion source (EBIS) at Kansas State. A good review

of the dramatic experimental progress in DR measure-

ments is again found in the NATO proceedings [55.5].

55.1 Theoretical Formulation

In the independent-processes approximation, the two

paths for recombination are summed incoherently. The

radiative recombination cross section for (55.1), in

lowest-order of perturbation theory, is given by

8π



 j









 jJM





(55.3)

The set (α

) represents the quantum numbers for

the N-electron target ion state, (α

) represents the

12.0

10.0

8.0

6.0

4.0

2.0

0.0

0.0 2.5 5.0 7.5 10.0 12.5

(10

–10

/s)

Energy (eV)

具σν典

Fig. 55.1 Dielectronic Recombination for O

. Calcula-

tions were performed for ﬁelds of 0 V/cm (dotted curve),

3V/cm (dashed curve), 5 V/cm (chain curve), and 7 V/cm

(solid curve)

quantum numbers for the (N + 1)-electron recombined

ion state, (k j) represents the quantum numbers for the

continuum electron state, (JM) represents the quantum

numbers for the (N + 1)-electron system of target plus

free electron state, and g

is the statistical weight of

a J

level. The dipole radiation ﬁeld operator is given

D =



2ω

3πc

N+1



s=1

. (55.4)

Continuum normalization is chosen as one times a sine

function, and atomic units (e =

= m = 1) are used. In

the isolated-resonance approximation, the dielectronic

recombination cross section for (55.2), in lowest-order

perturbation theory, is given by

8π



 j





α

|D|α

α

|V|α

 jJM

− E

+ iΓ



(55.5)

where the set (α

) represents the quantum numbers

for a resonance state with energy E

and total width Γ

and the electrostatic interaction between electrons is

given by

V =



s=1

− r

N+1

−1

. (55.6)

By the principle of detailed balance, σ

of (55.3)

is proportional to the photoionization cross section from

Part D 55.1

Dielectronic Recombination 55.2 Comparisons with Experiment 831

the bound state, while the energy-averaged σ

may be

written as





2π

∆k



, (55.7)

where the autoionization decay rate A

is given by



 j



|

 jJM|V|α

|

, (55.8)

the radiative decay rate A

is given by

= 2π



|

|D|α

|

, (55.9)

and ∆ is the energy bin width. Each resonance level

in (55.7) makes a contribution at a ﬁxed continuum en-

ergy k

/2; thus σ

 plotted as a function of energy is

a histogram.

55.2 Comparisons with Experiment

55.2.1 Low-Z Ions

For the most part, the agreement between the re-

cent high-resolution measurements and theoretical

calculations based on the independent-processes and

isolated-resonance (IPIR) approximations is quite

good [55.20–38]. We illustrate the agreement with ex-

periment obtained by calculations employing the IPIR

approximations with three examples from the Li isoelec-

tronic sequence. The pathways for dielectronic capture,

in terms of speciﬁc levels, are given by

−

+ A

(2s

1/2

) →



(q−1)+

(2p

1/2

nlj)







(q−1)+

(2p

3/2

nlj)



, (55.10)

wherea1s

core is assumed to be present. Both Rydberg

series autoionize by the reverse of the paths in (55.10),

while for sufﬁciently high n, the 2p

3/2

nlj levels may au-

toionize to the 2p

1/2

continuum. Both series radiatively

stabilize by either a 2p → 2s core orbital transition, or by

a nlj → n



valence orbital transition, where 2p



with j =

is a bound level.

Dielectronic recombination cross section calcula-

tions [55.28]intheIPIR approximation for O

are

compared with experiment in Fig. 55.1. Fine-structure

splitting of the two series of (55.10) is minimal for

this light ion, so there appears only one Rydberg se-

ries. The 2p 6 resonances are located at 2.5eV, the

2p 7 at 5.0 eV, and so on; accumulating at the se-

ries limit around 11.3 eV. Electric ﬁeld effects on the

high-n resonances are strong in O

, so that calcula-

tions were done for ﬁelds of 0, 3, 5, and 7 V/cm. Since

the precise electric ﬁeld strength in the experiment is

not known, the accuracy of an electric ﬁeld dependent

theory, in this case, has yet to be determined. However,

the effects of state mixing by extrinsic ﬁelds in the col-

lision region for the dielectronic recombination of Mg

have been investigated both experimentally [55.39]and

theoretically [55.40].

There have been a signiﬁcant number of recent

experiments on low-Z ions. These experiments, in

general, show good agreement with theory, see Fogle

et al. [55.37]andSchnell et al. [55.38]. However, it is

also clear that discrepancies remain between theory and

experiment for certain low energy resonances, due to the

difﬁculty in calculating the energy positions of such res-

onances. As has been pointed out in Savin et al. [55.41]

and Schippers et al. [55.42], this can lead to signiﬁcant

uncertainties in low temperature DR rate coefﬁcients.

Calculating such low energy resonances to sufﬁcient ac-

curacy for low temperature DR rate coefﬁcients remains

a signiﬁcant challenge for theory. Some success in this

area has been achieved using relativistic many-body per-

turbation theory, obtaining very good agreement with

low energy resonance positions for a range of sys-

tems, see, for example, Lindroth et al. [55.43], Fogle

et al. [55.44], and Tokman et al. [55.45].

55.2.2 High-Z Ions and Relativistic Effects

Dielectronic recombination cross section calcula-

tions [55.31]intheIPIR approximation for Cu

26+

are

compared with experiment in Fig. 55.2. The two ﬁne-

structure Rydberg series are now clearly resolved; the

ﬁne-structure splitting is about 27 eV for Cu

26+

.The

1/2

13 resonances are just above threshold, while the

3/2

11 resonances are found around 5.0 eV. Electric

ﬁelds in the range 0–50 V/cm have little effect on the

26+

spectrum. Overall, the agreement between theory

and experiment is excellent.

Electron-ion recombination cross section calcula-

tions [55.32]intheIPIR approximation for low-lying

resonances in Au

76+

are compared with experiment in

Fig. 55.3. The 2p

1/2

nlj series limit is at 217 eV, while

the 2p

3/2

nlj series limit is at 2.24 keV; yielding a ﬁne

Part D 55.2

832 Part D Scattering Theory

150

120

0 20406080

150

120

0 20406080

具σν典(10

–10

/s)

Experiment

3/2

1911

1/2

⬁

具σν典(10

–10

/s)

Energy (eV)

Theory

×10

Fig. 55.2 Dielectronic Recombination for Cu

26+

structure splitting of 2.03 keV. QED effects alone shift

the 2p

3/2

nlj series limit by 22.0 eV. Thus accurate

atomic structure calculations must be made to locate

the 2p

3/2

6ljresonances in the 0–50 eV energy range of

the experiment. The ﬁgure shows that the perturbative

relativistic, semirelativistic, and fully relativistic calcu-

lations for the dielectronic recombination cross section

ride on top of a strong radiative recombination back-

ground. In principle, the fully relativistic theory contains

the most physics, and thus it is comforting that on the

whole it is in good agreement with the experiment. It is

instructive, however, to see how well the computation-

ally simpler perturbative relativistic and semirelativistic

theories do for such a highly charged ion.

5.6

5.2

4.8

4.4

4.0

3.6

0 10203040

4.8

4.4

4.0

3.6

0 10203040

4.8

4.4

4.0

3.6

0 10203040

Energy (eV)

Recombination rate (10

–9

/s)

1/2

n = 20 n = 21

3/2

j = 3/2 3/2 5/2 7/2

Recombination rate (10

–9

/s)

Energy (eV)

Recombination rate (10

–9

/s)

Energy (eV)

Fig. 55.3a–c Dielectronic recombination for Au

76+

.The

curves show

(a) perturbative relativistic, (b) semirelativis-

tic, and

recombination cross section

There have been several recent experimental

measurements on high-Z ions, in particular, for astro-

physically abundant species. In general, there is good

agreement between theory and experiment. Examples

of high-Z element DR studies include those done on

Fe XXI and Fe XXII by Savin et al. [55.41], and on

Fe XX by Savin et al. [55.46].

55.3 Radiative-Dielectronic Recombination Interference

There has been a great deal of effort in recent years to

develop a more general theory of electron-ion recombi-

nation which would go beyond the IPIR approximation

to include radiative–dielectronic recombination inter-

Part D 55.3

Dielectronic Recombination References 833

ference and overlapping (and interacting) resonance

structures [55.47–54]. In almost all cases, the interfer-

ence between a dielectronic recombination resonance

and the radiative recombination background is quite

small and difﬁcult to observe. The best possibility for ob-

servation of RR-DR interference appears to be in highly

charged atomic ions. In the cases studied to date, the

combination of electron and photon continuum coupling

selection rules and the requirement of near energy degen-

eracy make the overlapping (and interacting) resonance

effects small and difﬁcult to observe. Heavy ions in rela-

tively low stages of ionization are the best place to look,

since there are resonance series attached to the large

numbers of LSterms or ﬁne structure levels.

The distorted wave approximation (Chapt. 52)has

been so successful in describing dielectronic recombi-

nation cross sections for most atomic ions because, for

low charged ions, the DR cross section is proportional to

the radiative rate, while for highly charged ions the DR

cross section is proportional to the autoionization rate.

Thus the weakness of the distorted wave method in cal-

culating accurate autoionization rates for low charged

ions is masked by a DR cross section that is highly de-

pendent on radiative atomic structure. As one moves to

more highly charged ions, the DR cross section becomes

more sensitive to the autoionization rates, but at the same

time, the distorted-wave method becomes increasingly

more accurate.

55.4 Dielectronic Recombination

Dielectronic recombination is an important atomic pro-

cess that is included in the theoretical modeling of the

ionization state and emission of radiating ions, which

is fundamental to the interpretation of spectral emission

from both fusion and astrophysical plasmas (Chapts. 82,

86,and87). The dielectronic recombination rate coef-

ﬁcient, into a particular ﬁnal recombined state, is given



πT

∆



(

i → f

)



exp



−k



(55.11)

where T is the electron temperature. Dielectronic re-

combination rate coefﬁcients, from the ground and

metastable states of a target ion into fully resolved

low-lying states and bundled high-lying states of a re-

combined ion, are required for a generalized collisional

radiative treatment [55.55, 56] of highly populated

metastable states, the inﬂuence of ﬁnite plasma dens-

ity on excited state populations, and of ionization in

dynamic plasmas.

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Part D 55

835

Rydberg Collis

56. Rydberg Collisions: Binary Encounter,

Born and Impulse Approximations

Rydberg collisions are collisions of electrons, ions

and neutral particles with atomic or molecular

targets which are in highly excited Rydberg states

characterized by large principal quantum numbers

(n  1). Rydberg collisions of atoms and molecules

with neutral and charged particles include the

study of collision-induced transitions both to

and from Rydberg states and transitions among

Rydberg levels. The basic quantum mechanical

structural properties of Rydberg states are given

in Chapt. 14. This Chapter collects together many

of the equations used to study theoretically the

collisional properties of both charged and neutral

particles with atoms and molecules in Rydberg

states or orbitals. The primary theoretical scattering

approximations enumerated in this Chapter are

the impulse approximation, binary encounter

approximation and the Born approximation. The

theoretical techniques used to study Rydberg

collisions complement and supplement the

eigenfunction expansion approximations used for

collisions with target atoms and molecules in their

ground (n = 1) or ﬁrst few excited states (n>1),

as discussed in Chapt. 47. Direct application of

eigenfunction expansion techniques to Rydberg

collisions, wherein the target particle can be in

a Rydberg orbital with principal quantum number

in the range n ≥ 100, is prohibitively difﬁcult

due to the need to compute numerically and

storewavefunctionswithn

nodes. For n =100

this amounts to ∼ 10

nodes for each of the

wave functions represented in the eigenfunction

expansion. Therefore, a variety of approximate

scattering theories have been developed to deal

speciﬁcally with the pecularities of Rydberg

collisions.

56.1 Rydberg Collision Processes .................. 836

56.2 General Properties of Rydberg States..... 836

56.2.1 Dipole Moments........................ 836

56.2.2 Radial Integrals ........................ 836

56.2.3 Line Strengths........................... 837

56.2.4 Form Factors............................. 838

56.2.5 Impact Broadening.................... 838

56.3 Correspondence Principles.................... 839

56.3.1 Bohr–Sommerfeld Quantization .. 839

56.3.2 Bohr Correspondence Principle ... 839

56.3.3 Heisenberg Correspondence

Principle .................................. 839

56.3.4 Strong Coupling Correspondence

Principle .................................. 840

56.3.5 Equivalent Oscillator Theorem..... 840

56.4 Distribution Functions ......................... 840

56.4.1 Spatial Distributions .................. 840

56.4.2 Momentum Distributions ........... 840

56.5 Classical Theory ................................... 841

56.6 Working Formulae for Rydberg Collisions 842

56.6.1 Inelastic n,-Changing

Transitions ............................... 842

56.6.2 Inelastic n → n



Transitions........ 843

56.6.3 Quasi-Elastic -Mixing

Transitions ............................... 844

56.6.4 Elastic n  → n 



Transitions ...... 844

56.6.5 Fine Structure n  J → n  J



Transitions ............................... 844

56.7 Impulse Approximation........................ 845

56.7.1 Quantal Impulse Approximation.. 845

56.7.2 Classical Impulse Approximation . 849

56.7.3 Semiquantal Impulse

Approximation.......................... 851

56.8 Binary Encounter Approximation .......... 852

56.8.1 Differential Cross Sections .......... 852

56.8.2 Integral Cross Sections ............... 853

56.8.3 Classical Ionization Cross Section . 855

56.8.4 Classical Charge Transfer Cross

Section .................................... 855

56.9 Born Approximation ............................ 856

56.9.1 Form Factors............................. 856

56.9.2 Hydrogenic Form Factors ............ 856

56.9.3 Excitation Cross Sections ............ 858

56.9.4 Ionization Cross Sections ............ 859

56.9.5 Capture Cross Sections................ 859

References .................................................. 860

Part D 56