Excitation of atoms and broadening of spectral lines

(4.2.42)

(J

4n Z2 A.2 = 27 v: R~
(4.2.43)

v and for quadrupole transitions (n=3),

Xo

= Rowlv.

The functions
4.3 Charge Exchange 4.3.1

Special Features of Charge Exchange Processes

Charge exchange in atomic collisions, i.e., the process A

+ B+ --; A + + B

(4.3.1 )

in which an electron goes over from the atom A to the ion B, is an example of a process with redistribution of particles. The electron wave functions of initial state 'P(rA) and final state
102

4 Collisions Between Heavy Particles

In the limiting case of high velocities, perturbation theory can be used. In the framework of perturbation theory, the velocity dependence of the charge exchange cross section is given by the Brinkman-Kramers formula [4.l9]. a(v) ex v- 12 ,

(v~vo).

(4.3.2)

4.3.2 Resonance Charge Exchange We shall consider the collision of the atom A in the state y with the ion of the same atom A+ (4.3.3 ) We shall assume that the state y is the state ns and that the ion A+ contains only completely filled electronic shells. At low relative velocities, the system A( ns) + A+ can be considered as molecular ion At. As R ~ 00, the ground term of the system is doubly degenerate because the electron can be located near either of the two ions A +. We shall denote the corresponding wave functions by 'l'1 and 'l'2. When R decreases, the ground term splits into two terms, an even term Ve 1:g ) = Vg , and an odd term Ve 1:u ) = VU. In the adiabatic approximation the wave functions of these terms are expressed in terms of wave functions 'l'1 and 'l'2 in the following way

(4.3.4 )

The wave function satisfying the initial condition 'l'(t form 1 'l'(t) = J2['l'g(t) + 'l'u(t)] . By substituting (4.3.4) into (4.3.5) we have in the limit

~

-00)

=

'l'1 has the

( 4.3.5) t ~

00

(4.3.6)

(4.3.7) The probability of the transition I W = sin2 '1 .

~

2 is (4.3.8)

The corresponding cross section can be obtained by integrating W over the impact

4.3 Charge Exchange

103

parameter, (J

= 2n Jpdp sin2'1(p) .

(4.3.9)

o

If '1 ~ 1, the function sin2'1 can be replaced by its mean value 112. Therefore it is enough to know the function at comparatively large p when '1 < 1. In the case of the molecular ion Hi, the exchange splitting at large distances has the form [4.8, 20]

1

2 [Vu(R) -

Vg(R)]

= 2R exp (-1

(4.3.10)

- R) .

Here atomic units are used. In the rectilinear-trajectory approximation, R

=

J p2 + v2t2,

'1

= 2~ vOx3/2exp(_I_x), x = plao.

(4.3.11)

(J

x 2 ....Q. = na02'

(4.3.12)

v Using (4.3.9) we have 2

where Xo is determined using the equation sin '1(xo) 0.28(C = 0.5772 is Euler's constant):

3/2 Xo exp(-xo)

~

'1(xo)

=

exp (-C)/2

=

0.15v (4.3.13 ) --. Vo A detailed discussion of the resonance and quasi-resonance charge exchange is given in [4.6, 20, 21]. The expression (4.3.12) for the charge exchange cross section is valid if v < Vo = e2 /;". In the limiting case vivo ~ 1, see [4.6,21],

(J

4.3.3

=

~

8 na~ -2 (VO)2 15 v

[1-4 (V)2 21518 (VO)12 + 1]-5 ---. na~. Vo v

(4.3.14)

Contribution of Inner Shells

If the atom A in (4.3.1) is a multielectron atom there exists a range of velocities v for which the condition v > vov'i for the valence electrons and the condition v < Vo v'i for the electrons of the inner shell are valid. Here, I is the ionization potential expressed in atomic units, i.e., in units of 2 Ry. In this range of velocities, inner electronic shells give the main contribution to the total charge exchange cross section. Now we shall consider the cross section of the charge exchange process ( 4.3.15)

104

4 Collisions Between Heavy Particles

in which an electron from the shell na/~ is captured by a proton, and as a result the hydrogen atom in the state with principal quantum number n is formed. If the relative velocity of the colliding particles is not too low, the cross section can be described by a simple generalization of the Brinkman-Kramers formula [4.22,23]

( IN _ ) _ 2 28 " N[2I(na l a)]5/2 (VO)2 n3 f(v) v' u na a n - 1tao 5

(4.3.16)

where (4.3.17)

The ionization potential I of the shell na I~ is expressed in atomic units. In the original Brinkman-Kramers formula the factor" is equal to unity. The accurate first-order perturbation theory [4.24] gives in the limit of high velocity, " ~ 1/3. In the estimates given below, we shall assume" = 1/3. At low velocities, the cross section decreases more rapidly than follows from (4.3.16); in fact exponential decrease occurs: 2 n (nLJI Vo ) . u(nalN -n)=nao-exp --a 8y2 2y V

(4.3.18)

This result can be obtained by the method described in Sect. 4.1.2. We shall denote the velocity for which the curves u(v) defined by (4.3.16) and (4.3.18) intersect by Vc. When V > vc , the cross section can be estimated by means of (4.3.16), and when v < Vc. one has to use (4.3.18). The cross section for capture from the inner shells is usually given by (4.3.16). In this case, I ~ LJI ~ 1/2n2, and the maximum value of the cross section of capture from the shell nal~ is achieved at the velocity Vm = voy'4I/3: 25N n

u(vrn) ~ na~-3-(2l)-7/2 .

(4.3.19)

When v > Vrn , the cross section decreases rapidly (ex: V- 12 ). In the range of velocities v:n < v < v~, where v:n corresponds to the outer shell and v~ corresponds to the inner shell nala, the dependence of the total cross section on energy tf (summed over all shells nala) has the form (4.3.20) The limiting value of p = 2.5 corresponds to the heaviest atoms. The charge-exchange cross sections for the Ne and Ar atoms are shown in Figs. 4.1,2. At energies tf ~ 100keV, the outer shell gives the main contribution. At higher energies, the capture of electrons from inner shells becomes significant.

4.3 Charge Exchange

105

In the case of alkali elements, inner shells give the main contribution to the cross section of charge exchange beginning from energies tff ;<: 20 keY [4.27]. Experimental data on the electron capture cross sections in collisions of protons with neutral targets were given in [4.28, 29]. 10- 15

cm l 10- 17

t 10-

19

b

10- /1 10-23 10

c-

Fig. 4.1. Effective cross section of the charge transfer H+ +Ne-+H+Ne+. Solid curves show the capture from different shells. Dotted line is the total theoretical cross section. Points correspond to experiment [4.25] .. Crosses correspond to experiment [4.26] I

I

H+ +Ar(1s / 2s 1 2p6 3s 1 3p 6) -

H+ Ar+

4 0

o~

6

~

8

b

Ly/

0

\,

/~,,~) \

/

2

\.

.-1/

4

K / /

6

'/

/

~~.~

~ \~

10

Fig. 4.2. Electron capture cross section in collision H+ + Ar: broken curves show the contribution from K - and L- shells, the solid curve shows the total capture cross section calculated in a modified Brinkman-Kramers approximation. Circles and crosses are various experimental data (for details see [4.23]).

106

4 Collisions Between Heavy Particles

4.3.4 Charge Exchange in the Case of Multicharged Ions

The Brinkman-Kramers formula can be easily generalized to the case of charge transfer from an arbitrary ion to a neutral atom [4.23): A + Bz+1

-->

A + + Bz , (4.3.21)

The function I( v) is given by (4.3.17) when AI = I - In, In = z2/2n 2. Here the levels n of an ion Bz, are assumed to be hydrogenlike. The cross section of quasi-resonance charge exchange to the level of an ion Bz with the principal quantum number

no = z../ii;.

(4.3.22)

has evidently the maximum magnitude. However, applicability of (4.3.21) for the resonance charge exchange is limited to the range of high velocities v ~ vm . The Coulomb field induces a strong perturbation of the terms in the final channel. Therefore, the exponential decrease of the cross section with decrease of velocity occurs at velocities considerably larger than Vc defined by (4.3 .16, 18). The Coulomb perturbation of terms is responsible for another mechanism of charge exchange in multicharged ions. If the ionization potential In of the ion Bz is larger than the ionization potential of the atom A, then electronic terms of the system A + + Bz ( n) intersect the ground term of the system A + Bz+ I at the points Xn determined by the condition z2

2n 2

-

h -

z - I

Xn

=0.

(4.3.23)

In the vicinity of the point of intersection of the terms, nonadiabatic LandauZener transitions can take place. At small velocities such transitions give the principal contribution to the transition probability. At higher velocities, transitions at other values of R also must be taken into consideration. The calculations [4.30, 31] show that the most effective transitions at high velocities are those in the states of ion Bz with principal quantum number no for which the quasi-resonance condition holds at high X [4.5,25]. When v decreases, capture to other levels with n < no becomes efficient. For n < no the cross section can have two maxima corresponding to the two above-mentioned mechanisms. The total cross section of electron transfer to all excited levels of an ion Bz has relatively weak dependence on velocity up to those velocities at which an ion Bz in the state no is mainly formed. The magnitude of the total cross section O"tot increases with z. Although the different models give somewhat different dependences on z, the order of magnitude of O"tot can be estimated by means of

4.3 Charge Exchange

107

formula

A more detailed discussion of charge exchange in the case of multiply charged ions is given in [4.5,32-35]. The available experimental data on electron capture by multicharged ions from simple H- and He- targets are given in [4.7,36,37]. An extensive bibliography on electron transfer processes was recently published by Tawara [4.38].

5 Some Problems of Excitation Kinetics

This chapter may be considered as an introduction to some problems of plasma spectroscopy. In the beginning we give a summary of the analytic formulas which can approximate the dependence of the rate coefficients (va) on temperature and atomic characteristics. These formulas contain two fitting parameters which should be determined from the results of numerical calculations. Tables of these parameters will be described in Chap. 6. Some useful semiempirical formulas are also given below. Special attention is paid to the process of dielectronic recombination and formation of the dielectronic satellites. These processes are very important for the spectroscopy of high-temperature laboratory and astrophysical plasmas. The last section of the chapter is devoted to the simplest approaches to the kinetics of level populations in a plasma.

5.1

Rate Coefficients for Elementary Processes in a Plasma. Approximation of Cross Sections and Rate Coefficients by Analytic Formulas

5.1.1 Excitation of Atoms and Ions In this section we summarize the formulas expressing cross sections a and rate coefficients (va) in terms of radial integrals and angular factors Q. It is convenient for applications to plasma kinetics problems to use the approximate analytic formulas. In this section, simple fitting formulas are proposed which describe the results of numerical calculations by means of two or three adjustable parameters and provide the correct asymptotic behavior of the quantities a and (va). Some semiempirical formulas, which are often used, are also presented. When constructing the approximate formulas we are mainly concerned with the energy region 8 ;S Ez (Ez is the ionization energy of the ion Xz ) which is of. the most importance for plasma diagonostics. In particular, the asymptotic behavior of the cross sections for optically allowed transition is taken in the form 8- 1, instead of the form 8- 1 In 8 (Sect. 3.1.3). Therefore one has to be cautious when applying the formulas given here to the problems of excitation and ionization in beam experiments. In these cases, the factor In 8 can be substantial. In this section the initial state is denoted by index 0, and the final state by index 1. The final results are given in cas units.

I. I. Sobel'man et al., Excitation of Atoms and Broadening of Spectral Lines © Springer-Verlag Berlin Heidelberg 1995

5.1 Rate Coefficients for Elementary Processes in a Plasma

109

We shall assume everywhere that the electron energy distribution function JF(8) is the Maxwellian, see (1.2.1). For excitation 0 -+ 1 by electron impact the rate coefficient (VO"OI) averaged over the electron distribution function is ex>

(VO"OI)

=

J vO"oIJF(G)dG ,

(5.1.1 )

tiE

where G is the energy of the incident electron, and AE is the threshold excitation energy. On substituting the Maxwellian distribution function (1.2.1) we obtain VO"

_Kex> ~exp(-G/T)8d8 na~ (Ry)I/2T3/2 '

L

( 01) -

(5.1.2)

217 x 10- 8cm3-1 - . K -- 2.Jili.ao s . m It is convenient to express the energy and temperature (Sect. 3.1.3). On setting u

= (8 -

In

P= DE/T.

AE)/DE,

the scaled units

(5.1.3)

Eq. (5.1.2) can be written as DE) 1/2 ex> ( ) (VO"OI) = K ( -R p3/2e - pp J 0" u2 (u y 0 nao

+ p)e -

pudu

(5.1.4)

with p = AE/DE. The rate of the inverse process (deexcitation) can be expressed through the excitation rate by (1.2.7). As was shown in Sects. 2.3 and 3.1,2 the cross section for the transition ao - al can be written, in the general case, in the form O"aOal

= E [Q~(ao,adO"~(lo, ld + Q~(ao,adO"~(lo, IdJ ,

(5.1.5)

I(

where the quantities O"~ and O"~ depend only on the quantum numbers nolo and nlll of the optical electron. The dependence on the total angular momenta of an atom (J,S,L, ... ) is described by the factors Q~ and Q~. For the cross section summed with respect to J,S,L, ... ) in the case of configurations no/~-nol~-Inl/h we have

O"(no/~,no/~-Inl/t>

= m E[O"~(lo, Id + O"~(lo, II)] .

(5.1.6)

I(

Thus the quantities O"~ and O"~ correspond to the single-electron transition cross sections. The quantity O"~ includes the direct and interference terms ,.1

VI(

=,.

VI(

+ ,.int

VI('

Q'(ao,al ) ex 0SoSI ~ .' I(

(5.1.7)

110

5. Some Problems of Excitation Kinetics

the quantity a~ representing purely the exchange term. The expressions of a~ and a~ in terms of radial integrals are given in Sect. 2.3. These quantities, evidently, can be obtained only by means of numerical calculations. The index K varies in the interval of Kmin to K max , where Kmin

= 1/0 -

Id,

Kmax

= 10

+ II

(5.1.8)

.

In accordance with (2.3.9, 10), a~ -=1= 0 only for K with the same parity as Kmin. We shall use the analytical fitting formulas to represent the results of the numerical calculations. In computer codes one can explore formulas with many adjusted parameters providing very accurate fitting to numerically calculated or measured cross sections. However, good accuracy is based on mutual compensations of essential contributions from nearby terms of opposite signs. Therefore, a small change of cross section results in a much larger change of parameters. In particular, the extrapolation of parameters along an isoelectronic sequence becomes very difficult if possible at all. We use comparatively simple formulas with true asymptotic and 2 or 3 adjusted parameters. The accuracy of these formulas is usually a few percents, but an interpolation and extrapolation of parameters for similar transitions, in particular in isoelectronic sequence, is very simple. Generally we use (5.1.5) and the fitting formulas for a~, a~. For cross sections, summed over J, the factors Q' and Q" differ only in the spin parts: Q~ = QK(Lo,LI)Ao, Ao

= b(So, SI ),

Q~

A2

= QK(Lo,Ld A 2 ,

=

(2S1 + l)j2(2Sp

(5.1.9)

+ 1).

In this case the fitting formula may be applied to the total one-electron cross section (5.1.10) This provides often the better accuracy of fitting since for a direct transition a~ is energy function smoother than a~, which may be negative at small and medium energies. It should be noted, however, that for transitions between fine-structure components the J o - JI orbital parts of Q' and Q" are different and the form of (5.1.10) is not valid. Taking into account properties described in Sects. 3.1-2 the one-electron excitation cross sections are written in the form 2 a~(lo, II) = ~

2/0

+I

(R) --L DE

2 (

EI ) Eo

3/2

. C(]I, (u) u + q> u=---

a"(1 1)= K

0, I

1ta~ 2/0

+1

(Ry)2 (EI)3/2. DE Eo

DE

(5.1.11)

C(]I"(u)

u + q>

where Eo,E I are atomic level energies (from the ionization limit). For a~ the function (]II = (]I' is used. C, q> are the fitting parameters, DE is a scaling factor.

5.1 Rate Coefficients for Elementary Processes in a Plasma

111

The functions depend on the type of transition, and generally can include a third fitting parameter D. If one prefers to use a two-parameter formula D = 0 should be assumed. For the excitation cross sections, we have '(u)

= [1 -

D/(u + Ii] '11(U)

"(u)

= [u +

0.4 +Dr2 '11(U)

(5.1.12)

where '11(U)

= {(U+ p)I/2, Z= 1

(neutrals) ,z>l (ions)

1

p = ,1E/DE

(5.1.13)

The excitation rate coefficient can be written in a similar way:

"

,

(5.1.14)

10- (RY 10- 8 (RY 8

EI)3/2 AG'(f3) . - - e - f3p [cm 3 S-I] , f3 + X

(vu (10 Id) = - - _. 2/0 + 1 DE Eo '" _ (vu"(1 0 I») I - --2/0 + 1 '" f3

EI )3/2 AG"(f3) n [ 3 -I] (5.1.15) e - I'P cm s DE Eo f3 + X '

--. - .

= DE/T.

Here A and X are adjusted parameters, the functions G(f3) can include the third adjusted parameter

= f31/2(f3 + 1 + D)'11 (p-I) , G"(f3) = f31/2(f3 +D)'1I(P-I). G'(f3)

(5.1.16)

For the total one-electron rate coefficient (vu~) =Ao(vu~) +A2(VU~)

(5.1.17)

the function Gt = G' is used. Eqs. (5.1.11-16) can be employed in any of the scales DE given by (3.1.30a, b, c). But in the case of closely spaced levels only DE = lEo I or z2 Ry are appropriate. The functions and G are symmetric with respect to the initial and final states. Therefore, in the case of deexcitation collisions 1 ~ 0, we have

(vu(aJ, ao») =

(vu'(1 I») " I, 0

L [Q~(al' ao)(vu~(ll' 10») + Q~(al' ao)(vu~(lJ, 10»)]

" = ~ (RY . EI)3/2. AG'(f3) 2/1 + 1 DE Eo P+ X

_ (vu""(1 I, I») 0 -

10- 8

--2/0 1

+

(RY

[ 3 -I] cm s ,

EI) 3/2 A G"(P) [cm3 s -I] . DE Eo f3 + X

--. - .

, (5.1.18)

(5.1.19)

112

5. Some Problems of Excitation Kinetics

5.1.2 Ionization Cross sections and rate coefficient of ionization can be described in the way similar to that outlined in Sect. 5.1.1. In this book we do not consider exchange effects in the case of ionization. The ionization cross section summed with respect to the quantum numbers J),L),SI is usually of interest. In this case the angular factor Qj does not depend on K. For the transition (5.1.20) We have (5.1.21) Thus we can write fitting formula for the total one-electron ionization cross section summed over K and II : CTj(lO)=

1ta~ (Ry)2CtP(U) 210 + 1 DE U + cp

10- (RD~ ) 8

(VCTj(lo») = 210 + 1

3/2

.

8-Ez DE

(5.1.22)

U=--

A G(P)

P+ X e - Pp [cm3 S-I]

,

(5.1.23)

P=DE

T'

where Ez is the ionization energy of the level ao, the scale DE is Ez or zZRy. The functions

=

U

1

u+ +

D"I(U) ,

(5.1.24)

(5.1.25) It should be noted that in the case of neutral atoms (z = 1), the formulas for

(

2 11,2)3/2 1t T exp (P p )(VCTj) .

m

(5.1.26)

5.1 Rate Coefficients for Elementary Processes in a Plasma 5.1.3

113

Recombination

In high-density plasmas the principal recombination process is three-body recombination. The number of these recombination events per second is Nz+IN;ICr, and the rate coefficient of recombination is related to the ionization rate coefficient by (5.1.26). In plasmas with moderate and low densities, two-body recombination prevails. The two-body processes are the radiative recombination and dielectronic recombination, see (1.1.4,5). The number of two-body recombination events per second is Nz+INe", where" = "v + "d. In two-body recombination a photon is involved, and therefore it is a weaker process than three-body recombination. When the density is not high, the probability of triple collisions is substantially smaller than the probability of binary collisions. This compensates the weakness of the interaction. Dielectronic recombination will be considered in detail in Sect. 5.2. For the rate coefficient of dielectronic recombination

Xz+I(CXo) + e

--t

Xz(cxlnl)

--t

Xz(cxonl)

+ liw,

(5.1.27)

the fitting formula has the form (Sect. 5.2.5)

"d = 10-13 QdAdP3/ 2 exp (-PXd) [cm3

P=

S-I],

(5.1.28)

(z + liRy/T ,

where Ad and Xd are the adjusted parameters. The parameters Ad, Xd for several types of transitions txo - cx\, are given in Sect. 6.1.7. The factor Qd depends on the type of the transition txo - CXI. For transitions CXo = 1'0 --t CXI = 1'0-111,

Qd = m/(2/0 + I), m Ik and for transitions 1o I

(5.1.29a) --t

/m-I/k+ 0 I I,

(5.1.29b) In most cases, recombination occurs onto the highly excited levels nl. We shall consider now the radiative recombination process,

Xz+I(aj)

+ e --t Xz(a) + liw

(5.1.30)

[Ref. 1.1, Sect. 9.5]. Taking into account the known properties of the radiative recombination cross section [1.1], we adopt the following fitting formula for the rate coefficient of recombination to the level a:

P=

Ea/T

(5.1.31)

where A,X and D are the adjusted parameters [5.1.], and Ea is the ionization energy of the level a, which corresponds to the term LjSj of an ion Xz+I. (We

114

5. Some Problems of Excitation Kinetics

note that the numerical factor adopted here is different from the factor used in [Ref. 5.1, Sect. 9.7.4]. The factor Qv depends on the angular momenta of the level a. When a = ajnlLS, aj = no/~-ILjSj,

(2S + 1)(2L + I) Qv = 2(21 + 1) (2Sj + I) (2Lj

(5.1.32)

+ 1)

For the cross section summed with respect to L, S,

= Qv(no/~-Inl) = and aj = no/~-ILjSj,

Qv(no/;;'"""" I [LjSj]nl) When a

= no/~LS

(2S + 1)(2L + 1) Qv = 2(21 + 1)(2Sj + 1)(2Lj

+ l)m

I .

(LS G LiSi

(5.1.33)

)2

(5.1.34)

For the cross section summed with respect to LjSj and LS m-I

Qv(no/~) = I - 2( 2/0 + I)

(5.1.35)

For the total rate coefficient of radiative recombination Xz+1 a formula similar to that of (5.1.31):

"v = ~ "v(a) = 10- 13 (:~) 1/2

A/Jlli:: D)

[cm3

-+ Xz

one can adopt

S-I] ,

(5.1.36) where Ez is the ionization potential of an ion Xz • The factor Qv should be taken into account in the summation with respect to a. It can be shown (see below) that radiative recombinations occurs mainly to those levels a for which lEal> T. Here and anywhere in this book the energy of a level Ea < 0 is measured from the ionization limit. For these levels "v(a) decreases slowly with increasing n, being proportional to n- I . The higher levels with lEa I < T produce only small contributions to "v, since for these levels "v(a) ex n- 3 • In the case of high temperature T> IEzl, radiative recombination to the states with minimal value of n provides the major contribution to the total recombination rate. We shall now give for reference the formulas which allow the calculation of the cross section and the rate coefficient of radiative recombination (for details see [5.1]). The rate coefficient of radiative recombination is given by expression

"v(a)

E 11/2 00 () p3 /2 J an ~ u exp (-pu) du, y 0 7tao

= K I Ra

u

= &/IEal ,

(5.1.37)

5.1 Rate Coefficients for Elementary Processes in a Plasma where

P= IEal/T, _

Urv -

li,w

and

2

2

1ta03 (137)3

= lEal + $,

Urv

Qv(a)

1m

115

is the cross section of recombination to the level a:

(lEal +$)3 $

E

2

ImP (kA.,nl),

).=1±1

(5.1.38)

= max (/, A.) .

The radial integral p(kA., nl) is equal to 00

p(kA.,nl) =

J Pnl(r)Pk).(r)rdr,

(5.1.39)

°

where Pnl and Pk). are the radial wave functions of the optical electron in the discrete and continuous spectra of an atom X z ' The function Pnl is normalized, as usually, to unity, and the function Pk). is normalized in accordance with (3.2.2):

. ( kr - -A.n - 1 sm Vk 2

~

Pk).

r-+oo

z + -lnkr + '1 )

k

.

(5.1.40)

(5.1.3~0) atomic units with the Ry unit for the energy are used. Recombination to highly excited levels n ~ 1 is well described by the quasiclassical formula [5.2]

In

2 16n:z2 LlnRy ~ urv(nl) = nao3(137)3 . ----;;&V 1 + 117 e (LIn, Ill) ,

e(Lln, Ill)

d

= ( dx JAn (X)

LI - Ii,wn 2 n - 2z2Ry'

_

III -

)2

-

(Llnf _x2 2 X2 [JAn(X)] , X = IlILln,

(5.1.42)

/1 _ (/ + 1/2)2

V

(5.1.41)

n2

On summing (5.1.41) with respect to I we get the known Kramers formula for the radiative recombination to the hydrogenic levels: uKr'(n) = na~ rv

16z2 Ry 3V3(137)3 LIn $ .

(5.1.43)

On substituting (5.1.43) into (5.1.37) and using (5.1.2) we find "v(n) Pn

=

64y'n aoli,

3/2

.

V3 Z Pn [-exp (Pn) El (-Pn)] , 33m 1373

= z2Ry/n2T

(5.1.44) ,

where Ei(-P) is the exponential integral. With the use of (5.1.44) it is not difficult to show that (5.1.45)

116

5. Some Problems of Excitation Kinetics

Using (5.1.44) for the levels with n ~ nl and the more accurate method for n < nl, one can write the radiative recombination rate coefficient summed over all levels in the form lev

E

=

lev(a) + ~'(n ~ nl) ,

(5.1.46)

a(n
Kr.(

lev

X

5.1.4

~ nl

n

)

=

32J1t aoh r::; z 3v3 • m • 137 3

nl

[In 1.78PI - exp (Pd Ei( -Pd

pI/2 I

(I + !: )l

PI = z 2Ry/niT .

(5.1.47)

Semiempirical Formulas for the Rates of Excitation, Ionization and Dielectronic Recombination

a) Excitation

In the range of high energies of the electron, the excitation cross section for optically allowed transition (L11 = ± I, L1S = 0) is expressed in terms of the oscillator strength by the formula (3.1.24). There are a number of semiempirical formulas for estimating cross sections which are based on the Bethe formula. Here we shall give the Van Regemorter formula [5.3] which is currently the most frequently used: 2

O'aOa(

= nao

8n y'3faoa(

(Ry)2 L1E u+

y(u) 1 '

(5.1.48)

where f aoa( is the oscillator strength for the transition ao -+ a I. The factor y( u) is determined from the experimental data and the results of numerical calculations of excitation cross sections. Its values for neutral atoms and ions are given in Table 5.1. When u -+ 00 , y(u)~

y'3

2nIn(l+u),

and (5.1.48) becomes the Bethe formula.

Table 5.1. Factor y( u) for atoms (z = 1) and ions (z > 1) y, y,

y'u z=1 z > 1

0.0 0.000 0.200

0.2 0.Q15 0.200

0.4 0.034 0.200

0.6 0.057 0.200

0.8 0.084 0.200

y, y,

y'u z=1 z > 1

2.0 0.328 0.328

3.0 0.561 0.561

4.0 0.775 0.775

5.0 0.922 0.922

6.0 1.040 1.040

1.0 0.124 0.200

5.1 Rate Coefficients for Elementary Processes in a Plasma

117

On using (5.1.48) for the cross section the excitation rate coefficient can be written in the fonn

(VO"aoa,) = 10- 8 x 32jaoa,

When

(~~r/2 pl/2exp(-p). p(PHcm3 s-I].(5.1.49)

P~l,

p(P)

~ - ~ Ei(-P)·

The values of the factor p(P) are given in Table 5.2. In various applications also the fonnu1as by Drawin [5.4], Mewe [5.5], and Gryzinsky [5.6, 7] are often used. In fact, all these fonnulas practically do not differ. The analytic fonnulas for the excitation cross sections can be obtained in the Born approximation using the model interaction potential VK(R) ()( RK / (R~+R2)K+l/2 [5.8]. For transitions between the levels with small energy spacing (with the same principal quantum number) such a potential is often fairly close to the real potential. In case of optically allowed transitions (AI = 1, K = 1) (5.1.50) where Zp is the projectile charge, M is the reduced mass of the colliding pair, iaoa, is the oscillator strength for transition ao -+ al (5.1.51) Kn(x) are the modified Bessel functions, Xmin = max

Ro ao

fM( V"RY Is

Y-;;

=f

~) VRY

,

(5.1.52)

(5.1.53)

Table 5.2. Factor p(fJ) for atoms (z fJ

z=1

p, p,

z > 1

p, p,

z=1 z > 1

= 1)

and ions (z > 1)

O.oI 1.160 1.160

0.02 0.956 0.977

0.04 0.758 0.788

0.1 0.493 0.554

0.2 0.331 0.403

0.100 0.214

2 0.063 0.201

4 0.040 0.200

10 0.023 0.200

>10 0.066p-'/2 0.200

fJ

0.4 0.209 0.290

118

5. Some Problems of Excitation Kinetics

In the case of quadrupole transitions 4

Uaoal = 135

(I(;

= 2)

22M Ry , -;cQ2(ao,a l )

7ta oZ p -;;;

x (21

I) (10 2/1)2 (l0Ir21/1)2 1+ 000 R~a~ x [4>2(Xmin) -4>2(Xmax )] ,

(5.1.54) (5.1.55)

00

(l0Ir21/1)

= J Pno/o(r)Pnl/l(r)r2 dr

(5.1.56)

o

iff~AE

When both Xmin

~~

Ro 2 ao

and Xmax > I, 4>1(Xmax ) and 4>2(Xmax ) can be neglected, and

(M) m

1/2 AE Ry

(RY) 1/2 iff

(5.1.57)

This approximation corresponds to the first-order impact parameter approximation with rectilinear trajectories. In this case the functions 4>1 (Xmin) and 4>2 (Xmax ) can be approximated by 4>1(X)

~

exp (-2x) . In (2.25 + 0.681/x) ,

4>2(X) ~ exp (-2x) . (2 + xJ3n/2)2 .

(5.1.58) (5.1.59)

The rate coefficients, averaged over the Maxwellian distribution, can be written in the form 16n l / 2 e 2 2 Q~(aoad (VU,,) = 21(; + 1 h ao [(21(; - 1)!!]2 x(21 I

+

I) (/ 0 I(; 11)2 (/olr"I/I)2 000 R2,,-2a2

o 0 xexp (-AE/T) . (Ry/T)1/2I,,(AE/T, Yo) ,

(5.1.60)

00

I,,(x, y) = exp (x/2)

J dt t2"-IKr(t) exp

(5.1.61)

o

Yo

= Ro (AE) 1/2 aO

RY

For the dipole transitions (AI

=

I) (vu) can be written in the following form

< vu >= 1.7410- 7 faoal (Ry/AE)(Ry/T)I/2 (5.1.62)

5.1 Rate Coefficients for Elementary Processes in a Plasma

119

When the argument of Ko does not exceed 3.0 the integral II (x, y) is approximated by the following asymptotic formula (5.1.63) For transitions with small energy spacing in plasmas usually rff ~ AE. Therefore, the Coulomb attraction of the electrons is not important. However, in cases of neutral atoms and ions with low z the effect of normalization can be very substantial. In such cases the formulas given above can be used only for rff~Eo, EI, i.e., when the Born approximation is valid. For multiply-charged ions (z > 3) and rff > AE they give fairly reasonable values of cross sections for the dipole transitions, and allow to estimate the order of magnitude of the cross section for quadrupole transitions. b) Ionization The well-known classical Thomson formula for the cross section of ionization from the shell nolo corresponds to (5.1.22 and 24) when C = 4m(2/0 + 1), ({J = I, DE = Ez and D = 0: aj

2

(Ry)2

= 1tao • 4m Ez

(u

+u 1)2

(5.1.64)

.

To estimate the rate coefficient of ionization for atoms and ions from the ground state, the Seaton formula [5.9] is often used:

(vai)

=

R )3/2 ( 10- 8 x 4.3m; p-1/2exp(-{3) [cm3 S-I], (5.1.65)

{3

= Ez/T, {3

~ 1,

where E z is the ionization potential of an ion X z • This formula corresponds to (5.1.23,24) when z > 1, X = 0 and A = 4.3· (2/0 + 1). Expression (5.1.65) is valid only for {3 2: 1. Sufficiently universal semiempirical formula was suggested by Lotz [5.10]:

(vai) = 10- 8 • 6m

1~~13/2 {3-1/2exp(-{3)f({3)

[cm3 s-I], (5.1.66)

f({3)

=

-{3exp({3)Ei(-{3).

This formula corresponds approximately to (5.1.23,24) when z > I,X = 0.4, and A = 6(210 + 1). The values of the factor f({3) are given in Table 5.3. Table 5.3. Factor f(fJ) fJ f(fJ)

= 1/4 = 0.34

1

4

8

0.59

0.83

0.90

120

5. Some Problems of Excitation Kinetics

The compact semiempirical fonnula which is also often used is given in [5.4]. Classical fonnulas for the ionization cross sections are given in [5.7, II]. c) Dielectronic Recombination A detailed treatment of dielectronic recombination is given in Sect. 5.2. To exhaust the list of analytic fonnulas for bound-bound and free-bound electronic collisional processes we present here a semiempirical fonnula for the rate coefficient of dielectronic recombination proposed by Burgess [5.12]. This fonnula can be written in a fonn similar to (5.1.28) "d(a)

= 1O- 13 BdP3/ 2exp(-PXd)

[cm 3 S-I],

P= (z + 1)2Ry/T ,

(5.1.67)

with Bd = 480jcxocx

Xd

5.2

(

z2

:~3.4 )

1/2

[I

+ O.105(z + l)X + O.015(z + 1ilrl ,

3 )-1 ,

(

= X I + O.Ol\z ~ 1)2

X = (z + 1)2Ry .

(5.1.68)

Dielectronic Recombination

In this section and the following one we discuss some problems related to dielectronic recombination I and fonnation of dielectronic satellites. We have used cgs units here. We recall also that for ions which are members of the isoelectronic sequence of an atom A the designation [A] is used. For example, the designation [H] is used for a set He+, Li 2 + and so on. 5.2.1

Electron Capture and Under threshold Resonances (Simplified Model)

As noted above, the excitation cross section for positive ions has a nonzero value at threshold due to the long-range Coulomb attraction. This attraction also allows the excitation of an ion Xz+ I at an energy below threshold, the electron being captured on some level nl ofthe ion Xz • For example, at an energy lower than the excitation threshold for the resonance level of the He-like ion 06+, the following process is possible: 06+(li) + e

--t

05+(1s2p nl) .

The doubly excited state which is the result of electron capture is unstable, and may decay either through autoionization or spontaneous emission of the resonance 1 Dielectronic recomb~tion is widely discussed in the literature, see e.g. the review articles [5.13-15]. In [5.15] one can find an excellent historical review and numerous references to original articles.

5.2 Dielectronic Recombination

121

photon 2p - Is. In the latter case, the atom X z is transferred to the stationary state, i.e., recombination occurs. This process is called dielectronic recombination (abbreviated to DR below). Generally, the process of dielectronic recombination of an ion Xz+! via the intermediate doubly excited state of an ion Xz is written in the form (5.2.1)

y = anlLSJ,

y' = a'niL'S'J' .

Below, the LS-coupling scheme is adopted. Besides it is assumed that photon emission occurs due to transition of the "inner" electron a - a', and the state nl of the "outer" electron does not change. Radiative transitions of the "outer" electron cause the complementary satellites (Sect. 5.3), but they do not play an essential role in the total balance of dielectronic recombination. The three most important effects of a DR process are: A. Dielectronic recombination for all ions other than bare nuclei is an additional recombination process. In many actual cases, as shown by Burgess and Seaton [5.16] the rate of DR can considerably exceed the rate of radiative recombination. Therefore, in low density plasmas dielectronic recombination should be necessarily taken into account. B. Satellites to the resonance and other lines of an ion Xz+1 originating from radiative transitions in reaction (5.2.1). C. Complementary excitation of levels a' when the autoionization occurs in reaction (5.2.1) with a' =I- ao. The latter has been considered in detail in Sect. 3.4, the satellites will be discussed in Sect. 5.3. Here we shall consider the intrinsic dielectronic recombination. For simplicity we discuss here only the process (5.2.1) neglecting the secondary ionization of the excited ion Xiy'). Both collisional and radiative secondary ionization in a plasma are considered in [5.17]. In some cases the secondary autoionization of ion Xz(y') is also possible. (For a discussion of the secondary processes see also [5.15].) In Sect. 5.2.1 we confine ourselves to the description of the simplified model making the following assumptions: a) the state of ion Xz is described by quantum numbers ani without specifying the terms LS; b) the value of n is large enough, so that the influence of electron nl on the state a of the core can be neglected; the levels nl can be considered as hydrogenic, and capture cross section can be expressed in terms of excitation cross section for transition ao - a using the correspondence principle; and c) in the process of photon emission, the electron returns to the initial state r:t.o. In this case the formulas for calculation of dielectronic recombination cross

5. Some Problems of Excitation Kinetics

122

sections prove to be sufficiently simple. Discussion of these assumptions will be given in Sect. 5.2.3. The general case will be discussed in Sect. 5.2.2. Within the frame of our simplified model, DR process is written in the form /' X;(aonl)

XZ+I (lXo) + e ---- X;*(a nl) '\.

+ hw (5.2.2)

Xz+1(IXO) + e , the lower branch of the reaction (autoionization) being the competing process. Therefore the cross section for dielectronic recombination via the state IXnl is A(a, IXQ) (I) , Wa IXn

I

ad(anl)

= ad(lXQ, anI) A ( IX) +

(5.2.3 )

where A( a, IXQ) is the probability of a radiative transition IX - lXo in an ion Xz+ I, Wa is the autoionization probability for the level anI of an atom X z , A(IX) = LiXJ A(IX, IXd is the total probability of radiative decay of the levela,a~(IXQ,anl) is the cross section for electron capture to the level nl when the transition IXQ - IX is excited. This cross section is represented by a set of resonances at the energies .to (0

~ LIAE _ z2 R2Y < LIAE,

AE

n

= Ea.a.o = Ea. - Ea.o .

(5.2.4)

It is convenient in this section to use again eGS units. The resonance width equals F = hWa • The cross section averaged over the resonances can be obtained

with the aid of the correspondence principle by extrapolating below the threshold the partial cross section for the excitation ao - IX: 2z2Ry n

a~(IXQ, IXnl)F = a(lXQ, 1X1)--3- .

(5.2.5)

Here a(lXQ, IX/) = EAo a(IXQAo, 1X1), where a(IXQAo, 1X1) is the partial cross section for the transition IXQ - IX in the threshold C = AE; Ao, 1 are the orbital momenta of the outer electron. In accordance with condition (b) of the model we should sum the cross section over total angular momenta LTST. The corresponding formulas are given in Sects. 2.3 and 3.2, in which the sum over A. is to be replaced by one definite value of A. = I. The values of Wa and a~(or a) are related to each other as characteristics of direct and reverse processes. To derive this relation it should be noted that at A = 0, the ratio of the populations of Xz+I(IXQ) and Xz(lXnl) is given by the Saha formula. Using this formula, we obtain (21

+ 1)ga. W,(a IXn 1)

= z2C .I:.

1tnn

goa(lXo,1X1)

3

2

1tao

'

(5.2.6)

where ga. and go are the statistical weights of the states a and ao. The rate coefficient of dielectronic recombination is Kd

= E Kd(a), a.

Kd

(a)

= E vad(aO, IX nl)F:#,( C) , ~

(5.2.7)

5.2 Dielectronic Recombination

123

where r is the resonance width, and ff(C) is the Maxwellian distribution for the energies of the electrons. The value of C is given by (5.2.4). Substituting (5.2.3) and (5.2.5) in (5.2.7) we obtain (5.2.8) where ns and

nl

are determined by the relations

( ns)3 n

= Wa(rxnl, rxo) .

(5.2.9)

A(rx, rxo)

One can see that nl is in fact the minimum value of n, i.e., it determines the lowest level at which the capture of an electron is possible in accordance with (5.2.4 ). We now transfer (5.2.8) into a form more convenient for applications. We substitute in (5.2.8) the explicit expression for ff(C) and use the relation (5.2.6), and the relation between A and the oscillator strength /:

g!J.A(rx, rxo)

=

1 (RY) T (.1E)2 Ry go/!J.o!J. .

(5.2.10)

1373

We write the result in the form

Kd(rx)

=

(z

+ 1)2Ry

=

{3

1O-13Bd(rx){33/2exp(-{3x) ,X =

T

[cm3 s-I], (5.2.11)

.1E (z + 1)2Ry ,

where {33/2exp (-{3X) provides the main temperature dependence of Kd, and Bd(rx) only slightly depends on temperature and is equal to

, z 21 + 1 Bd(rx)=C ·4·/"0!J. I: exp(J{3)I: 1+( / )3 nl n>nl l
= ns(l) = 137 ( 2 ~i(1( rxo, rxl) n ao(21

C'

=

+ 1)/!J.0!J.

1013 x 4n 3/2aoli (_z_)3 1373 m z+1

) 1/3,

= 0.53

J{3

=

I: Bn exp(J{3) , n>nl

Zn2~TY ,

(_Z_)3 [cm3 z+1

(5.2.12)

S-I] .

The value of ns separates all the levels into two parts. For n > ns(l), Wa < A and after the capture of an electron the ion Xz transfers to a stable state, i.e., recombination occurs. The contribution of these levels to Kd is ,

Kd(rx) ~

LIE

f

I: V(1( rxo,rxl)ff(C)dC, Es =

LlE-IEsl I

z2Ry --2 .

(5.2.13)

ns

For n < ns(l), Wa > A and most of captures are followed by autoionization; only a small part rv (n/ns)3 provides the DR. Nevertheless these levels (n < ns)

124

5. Some Problems of Excitation Kinetics

contribute mainly to the total rate "d. If we temporarily omit the factor exp (c5P) in (5.2.12), then all levels with n < ns(l) will almost equally contribute to "d. If, as usually is the case, ns ~ 1 the ratio of contributions of the levels with n < ns and n > ns is

"

,-

led '" led

2ns -

ns

n)

,....,

-

2

.

If now we take into account the factor exp(c5p), this ratio will be even larger since c5p '" n- 2 • The levels n < ns( I) of the ion Xz are evidently in thermodynamical equilibrium with the ion Xz+) and hence the rate of DR is proportional to the probability of radiative transition, A( IX, ao) oc tJE2 j fY.{ja. oc ni 4 j fY.{ja., in accordance with (5.2.12). As seen above DR proceeds mainly in the levels n) - ns , and since in most cases ns ~ 1 the condition (b) of our model (n ~ 1) is justified. The exception is the case of the Is - 2p transition at z ~ 20 when ns < n) and n > ns for all possible values of n. We consider now as an illustrative example the recombination of a [Li] ion with excitation of 2s - 2 p transition and recombination of [H] ion with excitation of Is - 2 p transition. In the first case, the energy level distance tJE is small, so that X = tJE/(z + I)Ry '" l/z~l, and u(ao, IXI) is large. That means the factor exp( -PX) ~ 1 in (5.2.11), ns ~n) ~ 1, and a great number oflevels concentrated in very narrow energy band ('" tJE) contribute to "d. Because of the small value of A '" X2 j, the value of led is comparatively small in spite of a great number of levels. In the case of recombination of [H] ion or [He] ion X ~ 3/4, and due to the factor exp (-PX), the rate of DR at small temperatures is negligible. The value of ns isn't large and for z ~ 20, ns is even smaller than n) ~ 2. For this reason, a comparatively small number of levels contribute to Bd in (5.2.12), but the contribution of each one is great because of the high value of A. According to numerical calculations the total values of Bd for the Is - 2 P transition usually exceed those for the 2s - 2 p transition. The value of led, however, is greatly dependent on temperature in the case of the Is - 2 p transition.

5.2.2

General Case

The formulas (5.2.11, 12) obtained above provide a useful method of calculation of the DR rate coefficient within the simplified model. In this section we derive a general expression for the DR rate coefficient without the assumptions of the simplified model. We shall consider the process (5.2.1) again using the detailed balance principle (Sect. 1.2) to derive the general formula for recombination rate coefficient. The total DR rate is (5.2.14)

5.2 Dielectronic Recombination

125

where R(y) is the probability of electron capture into the state y of Xz; and A(y) = E y' A(y, y') and Wa(Y) = E y ' Wa(y, y') are the probabilities of radiative and autoionization decays. Since the latter decay is associated with internal electrostatic interaction, it cannot change the total momenta LSJ. If we suppose that A = 0, and hence the system is in thermodynamical equilibrium, then according to detailed balance principle, we can write

NeNz+lR(y) = Nz(y)Wa(y,tXo) ,

Wa(y,tXo) =

E Wa(y, Yo) , ~

where the ratio Nz+dNz is determined by Saba equation (Sect. 1.2). The value of R, of course, is independent of any assumption concerning the radiative decay associated with an electromagnetic interaction. Therefore we can use the last equation to determine the value of R in the general case, and substituting it in (5.2.14), we obtain

I 8n3/ 2 a3 Kd(Y) = 2go . (z + 1~ p3/2exp (-PX

+ bP)

x gyA(y) Wa(y, tXo) ; bP = AE - EylXo , T A(y) + Wa(Y)

(5.2.15)

the values of P and X being determined by (5.2.11); go and gy are the statistical weights of the states tXo(Xz+1) and y(Xz ), and AE = EalXo and EylXO are the excitation energies of the states IX and y. The difference of these energies bE is equal to the bound energy of the captured electron, (5.2.16) The radiative decay probability A(y) summed over all final states y' does not depend on LSJ and is denoted below by A(IX). The values of P and bP are in fact also independent of LSJ. The autoionization probability in the LS coupling scheme does not depend on J, but essentially depends on LS. Therefore we shall write the rate coefficient of DR in the form (5.2.11), the factor Bd being equal to Bd(lX) = C

E exp(bP)q(y),

y = IXnILS,

niLS

C = 1013

4 3/2 3 n ao 3[c m 3 -1] s. go (z+ 1)

(5.2.17)

Due to the nonlinear dependence of Bd on Wa(y) we cannot explicitly sum over LS in (5.2.17). For this reason, the use of this equation requires a great deal of computation. In most applications an approximate formula is used, in which an averaged value of Wa(y),

Wa(lXnl) =

it

(2L + 1)(2S + 1) 2(21 + 1)( 2L a + 1)(2Sa + 1) Wa(Y) ,

(5.2.18)

126

5. Some Problems of Excitation Kinetics

is substituted in the denominator of (5.2.17). After this substitution we can write (5.2.17) in the form Bd(a.) = CEexp(bP)q(rxn/) , nl

(5.2.19)

(a.nl) = 2(21 + l)gIXA(a.) Wa(a.nl,lXo) . A(a.) + Wa(a.n/) q

Calculations with these formulas are much simpler than (5.2.17). Summing of linear expressions of the type (5.2.18) can be accomplished analytically, and only the sum over nl has to be done numerically. Besides, the expression for Wa(rxn/) is much simpler than that for Wa(Y) (Sect. 5.2.3). Approximation (5.2.19) corresponds to the assumption (a) of the simplified model. If n ~ 1 [assumption (b)], we can use the relation (5.2.6) and substitute the threshold excitation cross section in place of Wa in (5.2.19). Thus we obtain Bd(a.)

, z

(21+1)B'

= C . 4" . I IXfJIX E exp (bP) E B B'( I nl

n>nl

l
+

n ns

)3'

(5.2.20)

where B' =

E g1X1I1X/1X 1X' go/lXfJlX

B=

(EIXIX/)2 , EIXIXfJ

Wa(a.n/) Wa(rxnl,lXo) ,

(5.2.21 )

C',nJ,ns , being defined by (5.2.9 and 12).

The simplified model (5.2.11,12) is readily obtained from (5.2.20,21) if we assume that B = B' = 1. This corresponds to the assumption (c) with the single final state IXo for all decays. 5.2.3 Formulas for Autoionization Probability In this subsection we shall give the formulas for Wa without using the approximation (5.2.6). First of all we consider the probability Wa(Y) for decay of the state Y = rxnILS.!. The equation for probability of autoionization was derived in Sect. 3.4 on the base of the general theory of the excitation of highly charged atoms. We can obtain the same result more directly as the transition probability Y - Yo in firstorder perturbation theory. The latter way is similar to deriving the formula for the partial Y - Yo excitation cross section of the ion Xz+l. In this case, however, we deal with transitions at given values of LS.! which correspond to total angular momenta LTSTJT in the problem of excitation of the ion Xz+l. Therefore, we cannot use the simple formulas from Sect. 2.3.2 where the cross section was summed over LTSTh. In the case of LS coupling, Wa(Y) does not depend on J,

5.2 Dielectronic Recombination

127

but depends on LS. Using (2.3.27,28) we obtain

2Ry Wa(Y,IXo) = ~

E 1EA"(bsIX()SaR,,d Ao

n

"

" e R"",,) 12 , BSaosa. 'L.J ,,"

(5.2.22)

(5.2.23)

In the radial integrals Rd and Re , the functions F;.(r) for the electron in continuum are replaced by the functions PI(r) for a discrete spectrum, PI being normalized to unity similarily to Plo and Pia. The functions PI and FAo in the exchange integral ~ are orthogonalized functions related to PI and Flo according to (3.2.5). For the averaged quantity Wa(IXnl), we can use simpler formulas, similar to those given in Sect. 2.3.2 for the cross section summed over LTST. Thus we obtain [cf. (2.3.8,9)] Wa(IXnl,IXo) =

2Ry", T ~[Q,,(IX, IXo) Wi1a. nl, 10) I

I

+Q~(IX,IXo) W~/(la.nl, 10)],

W~(la.nl, 10) =

W~/(la. nl, 10) =

( 21 a.

( 2/ a.

+ 1;(21 + 1) ~R~(~ -

(5.2.24)

~R~",,),

+ 1; (21 + 1) ~(~R~""i .

(5.2.25)

Formulas for Q factors are given in Sect. 2.3.2, but the direction of transition has to be reversed to IX - t IXo instead of IXo - t IX. In the limit n~ 1, the expression (5.2.24) is proportional to the partial excitation cross section for the transition 1Xo-<X in the ion Xz+I. To derive an explicit relation we should use the following relation of the discrete and continuous spectra functions: Pnl(r)

-t n--+oo

(

2z2) 1/2 Fkl (k nn

-3

-t

0) .

(5.2.26)

Thus we obtain (5.2.8).

5.2.4 Some Inaccuracies of the Simplified Model The error introduced in the results of calculation of "d due to substitution of the average value Wa(lXnl) in place of Wa(Y) depends on the relation between the

128

5. Some Problems of Excitation Kinetics

probabilities Wa and A. In the case Wa ~A (large z or n), we have

= 2(21 + 1)gl1 Wa ( rx.nl, rx.o) , Lq(y) = L9 y Wa(y, (1.0) = q(rx.nl) , q( rx.nl)

LS

(5.2.27)

LS

the last equality resulting from (5.2.18). Thus the replacement of Wa(Y) by its average value does not introduce any additional errors in this case. The alternative case, Wa ~ A, is more complicated. If Wa ~ A for every possible values of L,S, then q(y) = gyA(rx.) and L q(y) = q(rx.nl), and using the average value Wa(rx.nl) does not introduce any additional error. It is possible, however, that for some values of L,S Wa(Y) = 0 in accordance with selection rules, or Wa(r) is very small, and the terms with these Wa(y) do not contribute to Lyq(y). As an illustration we consider an s- p transition (1.o-rx. in an ion Xz+l with one electron outside the closed shells. At first we consider the orbital momentum only. If Yo = sAo LS, y = pILS, then from orbital momentum and parity conservation it follows that L = Ao = I ± 1 and Wa(y) = 0 for L = I. Therefore we have Lq(y)

=L

L

q(rx.nl)

gyA(rx.)

= (2S + 1)(41 + 2)A(rx.) ,

L-,il

(5.2.28)

= g8 ·3(21 + l)A(rx.) ,

where g8 is the spin part of the statistical weight glXllt. We see that q(rx.nl) overestimates the result by a factor of 1.5. We emphasize that to produce an accurate result isn't a simple problem since LS coupling is replaced by jl coupling with increasing values of n. In jl coupling, the selection rule L =1= I fails. The quantitative treatment of the coupling transfer is difficult. We note however that this inaccuracy is essential only for s - p transitions. In other cases, the inaccuracy introduced by substitution of Wa(rx.nl) in place of Wa(Y) is small with the possible exception of the lowest values of n. If we neglect the above-mentioned change of a coupling scheme, we can derive a sufficiently simple and accurate expression for the DR rate coefficient in an s - p transition. In this case L = Ao, and therefore it is sufficient to average Wu(y) over S and to substitute in the denominator of (5.2.17) the quantity Wa(rx.n/)

2S + 1

3(21 + 1)

= ~ 2(2S + 1) Wa(Y) = 2Ao + 1 I1

Wa(rx.nl).

In the way, which is similar to that used for the derivation of (5.2.20), we obtain for both cases (1.0 = ns and rx.o = ns 2 : (5.2.29)

This formula can be used in place of (5.2.12) n~ differing from ns by the substitution 3(21 + 1) u«(1.o, rx./) = L U«(1.oAo, tx/) -+ 2Ao U(txoAo, rx./) . (5.2.30) Ao +1

5.2 Dielectronic Recombination

129

We consider now the dependence on spin momentum S. For a single electron outside the closed shells, in accordance with (2.3.27,28),

where f and g are the direct and exchange scattering amplitudes. For a ls2p transition, f ~ g (cf. Sect. 3.2.3) and WaeL) is very small. If WaeL)~A, we have

L:q(y) s

= (2L + l)A(oc),

q(ocnl)

= 4gLA(oc) ,

where gL is the orbital part of the statistical weight grxnl. Therefore using q(ocnl) in this case, one can overestimate the result up to a factor of 4, if we assume LS coupling and f ~ g. The second assumption of the simplified model is the use of an extrapolation formula (5.2.16). This formula is applicable for n ;::: 4, but for n = 2 or 3 it may not be sufficiently accurate. In most cases terms with large n make the main contribution to the DR rate coefficient. However at z ;::: 15 for recombination of [H] and [He] ions, the levels with n = 2 and 3 are the most important. Numerical calculations show that in this case the extrapolation formula underestimates the result. The third assumption of the simplified model implies that only transitions to the ground state OCo take place. For the resonance level oc any other transitions are, evidently, impossible. For higher levels the additional radiative transitions oc - oc' lead to some increase of Kd through the factor B' in (5.2.20). As a rule, this effect is small because A <X E;rxl' and Erxrxl ~Erxrxo. On the other hand, the additional channel of autoionization y - oc' can affect the magnitude of the factor B' in (5.2.20) substantially, because Wa(ocnl,oc') increases rapidly with decrease of Erxrx/. As a result, the contribution of the levels nl satisfying the condition

decreases considerably. Accurate treatment of this effect makes the computations considerably more cumbersome (see [5.10]). It can be estimated approximately assuming Wa ~ A in the cases when autoionization to the such level oc' is possible. In other words, summation over n in (5.2.12) can be restricted to 2

n < (z RyjErxrx'

) 1/2

,Erxrx' f ~Erxrxo

An example of such a process is the dielectronic recombination

130

5. Some Problems of Excitation Kinetics

5.2.5 Numerical Calculations and Analytical Approximation Formulas Equation (5.2.11) describes a general temperature dependence of the DR rate coefficient. The temperature dependence of the factors exp (c5P) in Bd (5.2.12), although small, can nevertheless be important in some cases. To include this effect in a simple analytical formula, we replace the quantity X = ,1E/(z + 1fRy by an adjustable parameter Xd, and rewrite (5.2.11,12) in the form Kd(a)

= 1O-13Bd/P/2exp( -PXd) [cm3 S-I] ,

P = (z + ~)2RY,

Bd

=

z= B

n,

Xd

= X -,1

(5.2.31)

.

n:o:::nl

According to (5.2.31) the value of Bd is equal to its limit at high T( c5p = 0). The value of Xd can be determined by the condition that Kd is equal to its true value for some value of T = T l • The value of Bd can be calculated numerically using the simplified model, i.e., via threshold values of the partial cross sections for the transition ao - a. Burgess [5.7] proposed a simple empirical formula for the DR rate coefficient which is given in (5.1.55,56). In spite of the absence of any adjustable parameter the Burgess formula in most cases gives results in a good agreement with the results of numerical calculations. This is due to the general structure of (5.2.12). If we replace exp(c5p) by its average value, the DR rate coefficient is proportional to ,1n(l)

+ = ns - max ( I nl

I)

. (5.2.32)

We can see from (5.2.12) that ns <X (a/f)I/3 and is almost independent of the type of ion and transition. Nevertheless in some cases the error of the Burgess formula isn't negligible.

5.3 Satellites of Resonance Lines in Spectra of Highly Charged Atoms 5.3.1

Excitation by Means of DR

The photon Y-YI(anILSJ-1XQni l L l S I Jz) emitted during DR has an energy slightly different from that of the transition a - 1XQ in the ion X z + l • This corresponds to additional spectral lines in the vicinity of the line a - ao. These lines are called satellites or dielectronic satellites [5.18]. In this section we consider the simplest and most important case of the satellites of [He]- and [H]-ion l resonance lines. I We recall that the designation [H) is used for the hydrogenlike ion, [He] for the heliumlike ion and so on.

5.3 Satellites of Resonance Lines in Spectra of Highly Charged Atoms

131

Satellites of the resonance line 2 p - Is in [H] ions Xz+l correspond to transitions (5.3.1 )

2pnl - lsn/l

in [He] ions X Z • Satellites of the line Is2p I P -ls2 I S in [He] ions X;+ I correspond to transitions ls2pnl - ls2 nil

(5.3.2)

in [Li] ions, and also to numerous transitions in ions of lower charge: ls2s2pnl -

Ii 2sn/l

[Be]

Is2i2pknl-li2s22pk-In/l

(k

=

(5.3.3 ) 1 ... 6)

[B] ... [Ne].

The satellites (5.3.1-3) were observed in the spectra of ions with z ~ 10 in laboratory and astrophysical plasmas. They proved to be an effective method of plasma diagonostics, since the satellite-to-resonance-line intensity ratio is essentially dependent on temperature and in some cases on electron density. At the same time the satellites are situated very close to the corresponding resonance line, and are excited by means of DR from the same initial state as the resonance line, see (5.2.1). These are important features for plasma diagnostics. In a plasma in the state of ionization equilibrium, the satellites (5.3.3) are usually very weak. Their intensity relative to the resonance line becomes considerable at very low temperature when the absolute intensities are negligible. In transient plasmas with small characteristic time the satellites due to ions in the isoelectronic sequences [Be], [B], ... ,[Ne] can be sufficiently strong. They provide a convenient method of the estimation of the ionization state in a plasma. The energy distance between satellites and the resonance line decreases rapidly (as n- 3 ) with increasing n. In fact only satellites with n = 2 or 3 have been resolved. from the resonance line up to now. Satellites with n ~ 4 can however change the shape of the line. Apart from the transition 2 p - Is of an internal electron, the transitions of the outer one (nl - Is) are possible, although less probable. The satellites of the members of principal spectral series np - Is, lsnp - ls2 are due to such transitions. Below we confine ourselves to the most important cases of n = 2 and 3 satellites of the types (5.3.1,2). It was mentioned above that the principal way of satellite excitation is DR of the ion XZ + I . Direct excitation of an inner-shell electron Is of the ion Xz can also provide satellites of the type (5.3.2). We consider now the relative intensity of the satellite .(

I

s yd ) = I./( y, YI ) + I.II( y, YI ) , y, YI ) = TI (''/, ( ires IX, IXo

(5.3.4)

where Is and Ires are the absolute intensities of the satellite due to the transition Y - Yl. and of the resonance line IX - 1Xo. We suppose that y = IXnlLSJ and

132

5. Some Problems of Excitation Kinetics

YI = rx.onItLISIJ\, where II can be unequal to I due to configuration mixing (corresponding satellites usually being weak). The terms i' and i" in (5.3.4) correspond to two mechanisms of excitation, namely DR and direct excitation of the Is electron; for satellites of [H]-ion resonance line, i" = O. We begin with the DR excitation mechanism. In this case, (5.3.5) where Kd(Y, yd is defined by (5.2.15) with substitution of the probability of radiative transition Y - YI in place of A(y) in the numerator. (In this section we shall use the notation W for the radiative transition probabilities instead of A in order to avoid mixing the radiative probabilities and parameters of approximation of collisional rate coefficients.) In our case, rx.o = Is or Isl, IX = 2 p or Is2 p, and only one autoionization channel is possible; hence Wa(y, rx.o) = Wa(Y). Taking this into account and using the approximation from Sect. 5.1 for (vulZQat), we obtain i'(Y,YI) = a'~fJexp(~fJ)q(y,yd, q(y Y ) = 1O- 13 g 'W(Y,YI)Wa(y) , I y W(y) + Wa(Y) ,

~fJ = EIXIZ() - E ylZQ '" ..!!.. T

- n2

(5.3.6)

fJ = (z + I iRy . '

T

Here the factor 10- 13 is introduced for convenience: for z - 20, q is of the order of unity. The factor a' is

a

'=

2.

4

X

I

~2

3/2 3 7r:

2 fJ" ----±.!... _n_. + I goA'Q"

aoe fJ'

4

E (E4E )

e-__3 ~. n2~E

_0

3/2

,(5.3.7)

at

where A', "I, Q' are approximation parameters (Sect. 5.1) for (vulZQat); at large z, l. In the most interesting case fJ' > 1, the dependence of a' on temperature is very slight. Using the results of numerical calculation in the Bom-Coulomb approximation with exchange, we obtain for fJ' = 3, = 1,

e-

e

a' = 0.75 x 10-3 for satellites of [He] ions, a' = 0.47 x 10-3 for satellites of [H] ions . For applications it is very important that for all satellites with given n the dependence of i' on T is practically the same. The value of i'(y, yd is determined by a factor of q(y, yd. For small z the value of Wa exceeds the transition probability W by a factor of the order of (li,c/e 2 )3 '" 106 : (5.3.8)

5.3 Satellites of Resonance Lines in Spectra of Highly Charged Atoms

133

We see that for small z, if ~ 1 but rapidly increases with z. At large z, (5.3.9) and if is almost independent of z. In this case if becomes ~ 1 for lJ{J ~ 1. The transfer from (5.3.8) to (5.3.9) occurs at z "" 20. As an illustration, in Table 5.4 the calculated and experimental results for the satellite structure of the [He]-ion Fe XXV are shown. The value of lJ{J was determined according to the observed value of i for y = Is 2 p2 2D5/2 (the brightest and best separated satellite). The calculations of wavelengths and spontaneous transition probabilities were made using the perturbation theory expansion in the parameter 1/:?E. We note that for :?E = 26 one should take into account all relativistic interactions, even the Lamb shift of the electron Is, which is equal to 0.08 A.

Table 5.4. Wavelengths l and relative intensities for satellites of Fe XXV resonance line in X-ray spectra of the solar corona. The temperature of the plasma was determined from the intensity of l = 1.8662 A satellite Experiment

[5.19]

l[A]

i[%]

l[A]

Transition

1.8510 1.8564 1.8571

100 21 15

1.8579 1.8594 1.8608 1.8618 1.8634

23

1.8660 1.8674 1.8679 1.8700 1.8729

47 38

1.8504 1.8555 1.8566 1.8571 1.8578 1.8595 1.8610 1.8622 1.8630 1.8635 1.8659 1.8674 1.8685 1.8699 1.8727

Is 2p IpI -> Ii ISO Is 2 p 3P2 -> Ii I So a 2SI /2 -> m 2P3/2 b l 2P I/2 -> n 2SI/2 a 2P3/2 -> m 2P I/2 Is 2p 3PI -> Ii ISO b2 2P3/2 -> n 2 SI/2 a 2p3/2 -> m 2P3/2 a 2D3/2 -> m 2P I/2 b2 2P I/2 -> n 2SI / 2 a 2D5/2 -> m 2P3/2 a 2D3/2 -> m 2P3/2 Is 2s 3S1 -> Is2 ISO Is 2p 3 3P I,2 -> Is 22p2 3PI a 4P5/2 -> m 2P3/2

12 22 47

21

Theory

l: a = Is 2p2, m = Is 22p,

b l = Is (2s2ppl), n = li2s.

i'[%}

i"[%]

100 29 4.7 10.2 0.18 0.18 13 32 4.9 47 3.8

2.6 17 19.7

6

14 7.7 127 b2 = Is(2s2pp3),

134

5. Some Problems of Excitation Kinetics

5.3.2

Direct Inner-SheD Excitation

We consider now a direct inner-shell excitation of satellites of the [He]-ion resonance line (5.3.10) In the case of DR, both the resonance line and satellites are excited from the same state, namely the ground state (Xo of the ion Xz+I. In contrast, the initial state for direct excitation is the ground state of the ion X z ; hence the relative intensity i" is proportional to the ratio Nz/Nz+ I of the populations of [Li] to [He] ions. In the case of low-density (e.g., astrophysical) plasma, almost all the X z ions are in the state Is22s. In high-temperature laboratory plasma, the electron density is as high as Ne '" 1020 cm- 3 , and the levels 2s and 2p are populated in accordance with the Boltzmann formula. Since AE~T, N2p/N2s = g2p/g2s = 3. It is evident that the direct excitation of satellite levels isn't effective in the case of [H]-ion resonance line satellites since the population of the corresponding initial state Is 21 of [He] is too small. The relative intensity i" of the [He] resonance line satellite due to the direct inner-shell excitation is i"

= L:Nz("),o) . (vO"YOY). Yo Nz+I

(vO"<X()<x)

W(")',")'I) W(1') + Wi1') .

(5.3.11)

For a low-density plasma,

"v

(vO"i(2s») [I

+ D(T)] ,

D(T)

=

"d ,

"v

(5.3.12)

"v

and "d are the total rate coefficients for radiative and dielectronic where recombination, and (VO"i) is the ionization rate coefficient for the ion X z . Using approximations from Sect. 5.1 for these quantities and the approximate equality, see (5.3.6), (5.3.13) we obtain z NN

z+1

= (JE)2 AA [1 + D(T)]JP exp (JP);; + Xi V I

+ Xv

The relative intensity i" will be written then in the form i" (")" 1'd = a"[1 + D( T)] JP exp (JP) q" ("),,1'1) , "( ) _ A(1'o, 1') • W(1', 1'1) q 1',1'1 W(1') + Wa(1') ,

(5.3.14)

5.3 Satellites of Resonance Lines in Spectra of Highly Charged Atoms

{3'

= E~~o b{3 bE

135

(5.3.15)

.

Here A v, Xv, Ai(yo), Xi(YO), A', X', A(yo, y), and X(Yo, y) are parameters of the approximations (Sect. 5.1) for Kv, (vo"j(yo») , (vO"~o~)' and (va yOY ); the value of ~" --+ 1 at z ~ 1. It is clear that in the low-density case, only the satellites with y = Is 2s 21 are excited, of which only satellites beginning from the levels of electronic configuration Is 2s 2 p can be strong enough. In the case of a high-density plasma both the Is2 2s and Is2 2 p levels are populated. Therefore i"

= Nz Eo (vO"yOY)

W(y, Yl) NZ+l Yo Yo (vO"~o~) W(y) + Wa(y) (5.3.16)

Kv ,,L- gyO (VO"i ( Yo »)' gyO

=

gyO ~ L-gyO Yo

,

and all satellites can be excited. We obtain an equation similar to (5.3.15) with the change

A(yo, y)

--+

Ai(2s) b{3 + Xi(2s)

EOYoA(yo, y) , Yo

--+

Ai(2/o) ~4[b{3 + Xi(2/o)]

(5.3.17)

Substitution of numerical values in the expression for a" in (5.3 .15) shows that the values of a" in both cases of low and high densities are practically the same, and for b{3 = 1, {3' = 3, ~" = 1 are equal to

a"

=

1.4 x 10-3

(z -IOn0.6)4

.

(5.3.18)

By comparing (5.3.15) and (5.3.6), one can see that except for the value of D(T) the temperature dependences i'(T) and i,,(T) are the same. The factor D(T)

corresponds to the influence of DR on the ionization equilibrium. For temperature T '" Tm , at which Ires has the maximum value, D(T) significantly exceeds unity, and hence influences i"(T). However, at lower temperatures, D(T) rapidly decreases as exp (-3b{3) and the dependences i/(T) and i"(T) become similar. If the plasma is in the state of ionization equilibrium, satellite excitation due to DR prevails, especially at z ;:S 15 according to (5.3.18). In the case of transient

136

5. Some Problems of Excitation Kinetics

(ionizing) plasma, the role of direct excitation can be very important at z > 15. It should be noted that the characteristic ionization time increases with increasing z. In Table 5.4, values of i" for the ion Fe XXV in ionization equilibrium are given.

5.4

Populations of Excited Levels in a Plasma

By the intensity of a spectral line is usually understood the energy emitted per second by a unit volume of a plasma as a result of spontaneous transitions. For the transition k -+ i of an atom, this quantity is Iki = "'WkiAkiN(k) [ergcm- 3 s- l ]

,

(5.4.1)

where Aki is the spontaneous emission probability, and N(k) is the population density of atoms excited to the level k. We shall consider the case of an optically thin plasma in which the line radiation may freely escape and does not affect the level populations. The transition probabilities Aki are the atomic characteristics. The excited level populations are, in general case, dependent upon the variety of collisional and radiative processes. In this section we shall consider some approaches to the calculation of the level populations and also the ionization and recombination coefficients S and IX. These coefficients determine the rates of ion production and loss and are expressed in units [cm3 S-I]. Using the example of the hydrogen atom (and hydrogenlike ions) we shall describe below the approach to the numerical solution of this problem. A survey of the analytic methods of calculati~g the populations and recombination coefficients is given in [5.20]. For a detailed discussion of the problem of line intensities and populations, see [5.21-24]. Article [5.24] contains an extensive list of references. Three types of models of a plasma are usually considered: the coronal model in the limit of low density, the model of local thermodynamic equilibrium (LTE) at high density, and the models of a plasma in the intermediate region which are often referred to as the collisional-radiative models. (For the conditions of applicability of the coronal and LTE approximations see Sect. 1.2.) Simple relations between the atomic level populations exist in the high-density limit when relaxation for any atomic level is determined by collisional processes. In LTE plasmas populations are completely determined by the temperature, total density of atoms and chemical composition of a plasma, and are independent of the cross sections of elementary processes, see (1.2.2, 3). A comprehensive discussion of the validity conditions for LTE can be found in [5.25]. When LTE conditions are violated, the populations of atomic levels are determined by all collisional and radiative processes. The processes to be included in calculations are the following: collisional excitation and deexcitation, ionization and the inverse process of three-body recombination, radiative and dielectronic recombination, spontaneous radiative decay, and collisional and radiative cas-

5.4 Populations of Excited Levels in a Plasma

137

cades. When the degree of ionization is not too low, only the electronic collisions are of importance. 5.4.1

Populations of the Hydrogen Levels at Low Plasma Densityl

In the low-density limit (coronal litnit, see Sect. 1.2), decay of the excited

levels is provided exclusively by spontaneous radiative transitions2 • Hence it follows that the population densities for all the levels except the ground state are extremely low. An excited level n is populated by direct processes which are the excitation from the ground state by electron impact and radiative recombination, and by cascade from the higher levels. For nonhydrogenic ions, dielectronic recombination has to be taken into account as a direct process. The steady-state rate equations may be written in the form

2: N(n')An'n + qn . (5.4.2) n'>n In this equation, N(n) is the density of atoms in the level with the principal quantum number n, An = 2:nll
(5.4.3) where N(l) is the population density of the ground level, (VO'ln) is the rate coefficient [cm3 S-I] of excitation by electron impact averaged over the Maxwellian distribution, N(H+) is the density of bare nuclei, and "v(n) is the rate coefficient of radiative recombination to form an atom in level n. It is assumed that all levels with the different orbital quantum numbers are populated proportionally to their statistical weights:

N(nl)

= 21 -; 1N(n)

. (5.4.4) n The solution of the equilibrium equations (5.4.2) may be written in the form N(n) =

" C(n,, n)qn' A1L..J nn'?,n

,

(5.4.5)

where C( n, n) = 1. The element of the cascade matrix C( n', n) is the probability that excitation of n' is followed by all possible cascade radiative transitions to level n. Let P(n', n) be the probability that the direct radiative transition from n' to n takes place:

P(n', n)

= An'n/An'

.

The content of this section is based on [5.26]. The relaxation of metastable levels or of levels with large principal quantum numbers n ~ 1 is determined by collisional processes even at very low density. 1

2

5. Some Problems of Excitation Kinetics

138

For C(n', n), we then have

+ 1, n) =

C(n

P(n + 1, n) , (5.4.6)

C(n +2, n) = P(n + 2, n + 1)C(n + 1, n) +P(n +2, n) ,

C(n + m, n) = P(n + m, n + m - 1) C(n + m - 1, n) +P(n + m, n + m - 2)C(n + m - 2, n) + ... + P(n + m, n) . With C(n, n) = 1 we obtain generally n+m-I

C(n

+ m, n) = E

+ m, n') C(n', n)

P(n

(5.4.7)

.

n'=n

Using (5.4.7) one may calculate similarly the radiative cascade matrix for any other atom once the matrix elements P( n', n) (or radiative transition probabilities) are known. The number n in this case may be considered as an index labelling the atomic levels in the order of increase of the energy. Table 5.5 gives the cascade matrix elements C(n + m, n) for n ~ 10, m ~ 10 calculated by Seaton [5.26]. For m 2:: 5, the calculated values can be fitted to

C(n + m, n) = C(oo, n) + An/m . The quantities C( 00, n) are also given in Table 5.5. For application to the excited levels with n > 10, one may use an analytical formula obtained in [5.27] using the Kramers approximation:

2 ln (n3) C( n + m, n) = n3 2"

x

(1 1) n2 - (n + m)2

-I [

0.5772 -In

(

r

1 1)

n2 - (n + m)2

(5.4.8)

2

Table 5.5. Cascade matrix C( n + m, n) for hydrogen in the case of optically thin plasma m

n

1 2 3 4 5 6 7 8 9 10 C(oo,n)

2

3

4

5

6

7

8

9

10

0.4418 0.4105 0.3991 0.3934 0.3903 0.3883 0.3870 0.3860 0.3853 0.3848 0.3796

0.2978 0.2601 0.2454 0.2380 0.2336 0.2308 0.2289 0.2274 0.2264 0.2255 0.2174

0.2336 0.1947 0.1791 0.1710 0.1661 0.1627 0.1605 0.1589 0.1575 0.1564 0.1468

0.1975 0.1587 0.1430 0.1347 0.1294 0.1260 0.1236 0.1217 0.1203 0.1191 0.1088

0.1744 0.1361 0.1206 0.1120 0.1068 0.1033 0.1008 0.0990 0.0975 0.0964 0.0860

0.1581 0:1203 0.1052 0.0968 0.0916 0.0881 0.0854 0.0835 0.0821 0.0809 0.0703

0.1464 0.1093 0.0942 0.0858 0.0806 0.0771 0.0745 0.0725 0.0710 0.0698 0.0590

0.1369 0.1006 0.0858 0.0775 0.0723 0.0688 0.0662 0.0642 0.0627 0.0615 0.0505

0.1295 0.0939 0.0791 0.0711 0.0660 0.0624 0.0598 0.0578 0.0563 0.0551 0.0441

5.4 Populations of Excited Levels in a Plasma

139

The error of formula (5.4.8) does not exceed 15 percent. When m is large, the error decreases. This is in accordance with the accuracy of the Kramers approximation. The formula (5.4.5) for the population density includes the total spontaneous decay probability An. Let An be written in the form

(5.4.9)

where oc = e2 /fic = 1/137 is the fine-structure constant, c is the velocity of light, and ao = fi2/me 2 = 0.529 x 10- 8 cm is the Bohr radius. The values of r(n) are given in Table 5.6. For n ~ 20, r(n) may be fitted to r(n)c::'3lnn-0.247. For applications, the analytical formulas of the Kramers approximation are often useful: 2AOZ4

Ann' = n3n'( n2 - n'2)'

= Aoz4ln

A

n5

n

(n3 2- n)

According to (5.4.3,5) in the limit of Ne -+ 0 the population density for any excited level is negligibly small compared to the electron density, densities of the ground level atoms and bare nuclei. The total number density of the excited levels is also much smaller than these quantities. Therefore at low density, the ionization rate is provided by the direct collisional ionization from the ground state. The only process of recombination for hydrogen and hydrogenlike ions is radiative recombination. The rate coefficient of recombination is determined by the sum over principal quantum numbers, 00

oc

=

Kv

=

LKv(n) . n=!

Table 5.6. Factor r(n) n

2

3

r(n)

1.911

3.084

n r(n)

10 6.657

11 6.943

4

5

3.929 12 7.205

13 7.442

4.589

8

9

6

7

5.132 14 7.668

5.590

5.989

15 7.876

20 8.740

6.389 25 9.410

140

5. Some Problems of Excitation Kifietics

5.4.2 Intermediate Density. Collisional-Radiative Model of a Plasma At intermediate densities one has to take into account both radiative and collisional cascade transitions. Here we shall also consider hydrogenlike ions. It is assumed that the relationship (5.4.4) is satisfied, the free electrons have a Maxwellian distribution, and that the plasma is optically thin for the line radiation. The rate equations which determine the population densities N(z)(n) of the bound levels of an ion Xz may be written in the form (5.4.10) where rn is the total decay probability of the level n[s-I] including the radiative decay probability An = L:nl
r

_ {Wmn +Amn mn -

Wmn

when m < n , when m < n .

The recombination frequency Rn includes the radiative recombination term NeKv(n) and three-body recombination term Kr(n)N;. N(z+I) is the bare-nuclei number density. The calculation of the instantaneous population densities requires the solution of a set of equations (5.4.10). Due to the electric microfield of a plasma, the higher levels merge into a continuum and the number of bound levels is finite (this problem is beyond the scope of this text, see, e.g. [5.21,28-30]). Nevertheless this number is large enough, especially at low densities. In actual numerical calculations one can confine oneself to considerably smaller principal quantum number no. Here we give a simple explicit estimate of no (for a detailed analysis see [5.31]). The values of no can be chosen taking into account the fact that for n large enough, the collisional processes are- much more important than the radiative processes. The radiative decay probability decreases rapidly with n: An ()( n- 5. The collisional frequency for the transitions n -4 n ± 1 increase ()( n4, and the collisional ionization frequency W~ ()( n2. Therefore the population and decay of highly excited levels are determined by exclusively electronic collisions. When n ~ 1, Wn, n± I ~ W~, and the electron stepwise motion between the highly excited states is like a slow diffusive motion [5.32-34]. Hence for n greater than some value no, the populations are determined by the balance of three-body recombination and ionization by electron impact, and therefore satisfy

5.4 Populations of Excited Levels in a Plasma

141

the Saha-Boltzmann equations, see (1.2.3,4),

N(z)(n)

= NE(n) = N(z+l)

Bn(T) =

. NeBn(T) ,

~~ exp (z:~;),

s = 2 (2~;1t y/2 Ne

(5.4.11)

.

The value of no may be roughly estimated from the condition

W~o ~Ano .

(5.4.12)

The frequency of ionization W~ = Ne (VO"j (n )) may be estimated using the classical Thomson formula for the cross section (Sect. 5.1). In this case (VO"i(n)) x[1

IR

= 8Vn~2a~ E~ 13/2 f33/2exp (-13)

+ f3exp(f3)Ei(-f3)],

For highly excited levels j

Wn~

13 ~ I,

13 = IEnl/T = z 2Ry/n2T

.

(5.4.13)

and it follows from (5.4.13)

8.7 x 10- 8 2 (z2RY) 1/2 3 n -Ne z T

.

(5.4.14)

Using (5.4.9) for estimation of the radiative decay probability An [assuming for simplicity -r(n) = 10] we obtain from the condition (5.4.12) no

~ z (1018 )

1/7

Ne

(~) 1/14

.

z2Ry

(5.4.15)

For the range of electron temperature of 0.01-100 eV and an electron density of 108-IO I5 cm- 3, it follows from (5.4.15) that 2 ~ no/z ~ 30 . To find the level populations and the ionization state of a plasma one has to solve the several sets of equations (5.4.10) for ions of the different stages of ionization z under the conditions that the density of the nuclei of each element is conserved and the plasma is electrically neutral. If temperature and density change not too rapidly, the computational problem can be simplified. 5.4.3

Quasi-Stationary Approach for Hydrogen

The small perturbations from the steady state relax in a time of order 1/ r n. The relaxation time of any of the excited levels is always much shorter than that of the ground level l . Hence the time in which the population densities ofthe excited 1 The estimate for the relaxation time given above is rather rough. In actual cases the relaxation time for one level depends on that for all other levels and on initial conditions, and may be an order of magnitude larger than 1/ Tn [5.35]. For metastable levels of atoms and ions other than hydrogen and hydrogenlike ions the relaxation time can be great.

142

5. Some Problems of Excitation Kinetics

levels come into correspondence with the instantaneous population density of the ground level, free electrons and bare nuclei H+, is very short. Thus the quasisteady approximation is obtained by setting dN(z)(n)/dt = 0 for all the levels except the ground level. The population density of the ground level should change comparatively slow, on a time scale much longer than the longest relaxation time for the excited levels, but in general will not be necessarily in equilibrium with the electron density and the number density of bare nuclei. The quasi-stationary approach given in the works of Bates et al. [5.36,37] enables population densities and coefficients of recombination and ionization to be tabulated in a wide range of electron temperature and density. These quantities are tabulated for hydrogen and hydrogenlike ions (see, e.g. [5.37-41]. More recent values are given in [5.40]. New results for helium and additional references may be found in [5.42]. It is convenient to express the populations in terms of Saba-Boltzmann equilibrium values NE(n), see (5.4.11), b(n)

= N(z)(n)/NE(n)

.

For hydrogen N(z+l) in (5.4.11) is equal to N(H+). Using this notation the rate equations for the levels with 2 :s; n :s; no can be rewritten b(n)rn - L:b(m)NE(m) rmn - N(H+) Rn min NE(n) NE(n)

=0 .

(5.4.16)

Taking into account (5.4.11) and the relationship NE(1)Wln = NE(n)Wnl ,

(5.4.16) may be reduced to Bm(T)

b(n)rn-

Rn

~b(m)Bn(T) • rmn= NeBn(T)

N(l)

+ Ndl)Wnl.

(5.4.17)

m¥1

The first term in the right-hand side of (5.4.17) corresponds to the direct recombination to the level n and depends on Ne and Te only. The second term corresponds to the excitation from the ground state and is proportional to N(1). The solution of this set of no - 1 equations may be expressed by the sum of two terms N(n) N(1) b(n) = NE(n) = ro(n) + rl(n) NE(1) .

(5.4.18)

The first term determines both direct and cascade populating from the continuum and the second term corresponds to the direct and stepwise excitation from the ground state. With the use of (5.4.18) the differential equation for N(1) may be written in the form (5.4.19)

5.4 Populations of Excited Levels in a Plasma

143

The quantity IX =

1 [ Ne Rl

NE(n)

+ n~lN(H+{o(n)rnl

]

,

following the works by Bates et at. [5.36,37] is referred to as the collisionalradiative recombination coefficient, and the quantity S

=

1 [ Ne r

l -

NE(n)

n~lNE(1{l(n)rnl

]

is referred to as collisional-radiative ionization coefficient. The steady-state value of N(1) is given by the relation N(1)=N(H+):§ .

(5.4.20)

As was noted in Sect. 1.2, the total population of all the excited levels is usually considerably smaller than the ground level population densityl LN(n)~N(1) , Ne , N(H+) .

(5.4.21 )

n>l

Therefore the coefficients IX and S may be regarded as the total rate coefficients of recombination and ionization2 • When one of the inequalities (5.4.21) is not satisfied the steady state solution may not be valid. In the low-density limit the net ionization rate is equal to the rate coefficient of ionization from the ground state (VO"i (1»), and IX is equal to the radiative recombination rate. At low densities, Wnl

rl(n) = -A ex: Ne , nl

and ro(n) depends on T only. In the high-density limit, collisional cascades are of major importance and the quantities ro and rl become independent of Ne . Some numerical data for ro, rl, IX, and S from the work by Johnson and Hinnov [5.39] are given in Table 5.7 (for more recent results see [5.40]). The quasi-steady level population densities at a given temperature T are determined by (5.4.18) on substituting the values of electron density N e, the density of protons N(H+) and the population density of the ground level N (I). In a plasma composed of hydrogen only Ne = N(H+). The steady-state population densities are to be found with the condition (5.4.20) taken into account. The steady-state populations for the ground level are given in [5.43].

I The problem of an upper limit of n for the sum over excited levels in (5.4.21) is outside the scope of this text. 2 S and 0( do not correspond to coefficients giving the rate at which electrons leave the ground level and come to it. They are smaller than such coefficients would be.

109

4.9-6 2.5-17Ne 1.4-3 1.0-17Ne 1.1-2 7.2-18Ne 3.2-2

6.1-18N.

5.8-2 5.5-18Ne 1.2-26 7.9-13

rl(3)

n(5)

ro(6)

rl(6) S

IX

rl(6) S

ro(6)

rl(5)

ro(5)

rl(4)

ro(4)

n(3)

7.5-18 4.9"':13

4.5-18Ne

2.4-1 4.4-7 4.5-1 3.0-7 1.3-17 7.3-13

2.0-2 8.2-18Ne 6.1-2 6.0-18Ne l.l-I 5.0-18Ne 1.5-1

ro(3)

1.5-1 4.8-8 2.4-1 4.1-8 1.0-17 6.1-13

1.9-17N.

rl(2)

2.6-2 8.2-8 8.2-2 5.9-8

1.2-3

ro(2)

1.8-3 1.9-6 3.3-2 8.1-7 l.l-I 5.7-7

1.5-3 1.9-7

0

Ne

2.0-17 1.0-12

7.9-1 1.2-{j

5.5-1 2.7-{j

5.0-2 8.0-6 2.2-1 5.1-{j

2.5-3 1.9-5

10

lOll

1010

IX

ro(5)

rl(4)

ro(4)

12

3.1-3 1.0-6 2.8-2 7.1-7 1.0-1 5.7-7 2.7-1 4.3-7 3.2-26 1.7-12

2.2-3 1.0-7 1.8-2 7.2-8 5.5-2 6.0-8 1.1-1 5.2-8 2.1-26 1.2-12

ro(3)

rl(2)

ro(2)

1.1-5 2.5-{j

1011

7.6-6 2.5-7

1010

6.2-6 2.5-8 1.8-3 1.0-8 1.5-2 7.2-9 4.2-2 6.0-9 8.1-2 5.4-9 1.6-26 1.0-12

Ne

0

Table 5.7. Parameters ro(n), rl(n) and coefficients S and IX

1013

10 14

8.6-1 1.0-5 2.1-25 7.1-12

T = 8 x 103K. 4.5-3 1.3-2 1.9-3 1.9-4 1.2-1 4.3-1 7.7-5 6.1-4 5.6-1 8.3-1 3.1-5 1.7-4 9.5-1 8.4-1 1.1-5 5.0-5 9.4-1 9.8-1 1.7-5 4.0-6 4.3-17 1.5-16 1.7-12 3.9-12

1013

6.4-1 2.3-{j 6.5-26 2.9-12

T = 4 x 103K. \.9-5 4.9-5 2.5-5 2.5-4 2.2-2 6.0-3 1.0-5 1.0-4 3.1-1 7.3-2 5.7-5 6.9-6 3.3-1 6.8-1 4.4-{) 2.5-5

10 12

7.1-2 1.8-2 7.2-1 4.5-3 9.3-1 l.l-3 9.8-1 3.2-4 9.9-1 1.1-4 9.4-16 1.4-11

10 15

9.4-1 7.2-5 1.3-24 2.7-11

2.4-4 2.5-3 \.3-1 1.1-3 6.0-1 4.8-4 8.5-1 1.8-4

10 14

3.7-1 1.0-1 8.5-1 2.4-2 9.6-1 5.9-3 9.9-1 1.7-3 1.0 5.9-4 5.0-15 7.1-11

10

16

9.6-1 7.7-4 1.4-23 1.6-10

9.0-1 2.0-3

2.2-3 2.4-2 3.5-1 1.3-2 7.4-1 5.3-3

10 15

1.0 3.3-3 1.0 1.1-3 9.5-15 3.2-10

7.0-1 2.0-1 9.3-1 4.7-2 9.8-1 1.1-2

10 17

\.8-2 2.0-1 4.2-1 1.1-1 7.7-1 4.5-2 9.2-1 1.6-2 9.7-1 6.5-3 1.2-22 1.4-9

10 16

1.0-14 2.4-9

1.0 3.6-3 1.0 1.2-3

7.7-1 2.2-1 9.5-1 5.1-2 9.9-1 1.2-2

10

18

6.2-2 6.7-1 4.5-1 3.9-1 7.9-1 1.5-1 9.2-1 5.6-2 9.7-1 2.2-2 4.0-22 1.3-8

10 17

2.3-27Ne

1.0 3.6-3 1.0 \.3-3 1.1-14

7.1-1 2.2-1 9.5-1 5.2-2 9.9-1 1.3-2

00

9.7-1 3.0-2 5.4-22 1.3-25Ne

m ~

9.2-1 7.7-2

0

on

(")

:t.

[

0 ::I

~.

...o.

....0

I'D

a"

a'"C

(II

8

til

8on

y.

-t

8.5-2 9.2-1 4.7-1 5.3-1 7.9-1 2.1-1

00

for optically thin hydrogen plasma [5.39]. T=103 K. 4.9 -6 denotes 4.9x 10- 6 •

S ex

ro(6) rl(6)

ro(5) rl(5)

ro(4) rl(4)

n(3)

ro(3)

ro(2) rl(2)

S ex

ro(6) rl(6)

n(5)

ro(5)

8.6-11 2.1-13

7.3-11

6.7-11

1.8-13

1.8-13

5.3-1 3.1-8

1.9-13

5.7-1 3.4--6 8.0-1 1.5--6 9.1-1 6.0-7

3.4-1 6.3--6

4.1-1 4.5-7 5.5-1 3.2-7 7.2-1 1.9-7

2.7-1 6.8-8

3.7-1 4.8-8 4.5-1 3.8-8

1.5-1 1.5-5

4.4-13 4.3-13

1.4-1 1.5--6 2.9-1 6.7-7

3.6-13

6.0-1 2.2-7 3.4-13

1.6-1 6.8--6 3.9-1 4.0--6 7,0-1 1.8--6 8.7-1 7.7-7

3.5-2 1.6-5

10 12

1.3-1 1.5-7

6.2-11

3.6-18Ne

1.3-1 1.5-17Ne 2.5-1 6.9-18Ne 3.4-1 5.0-18Ne 4.0-1 4.1-18Ne 4.5-1

3.0-13

3.8-1 3.4-8 3.0-13 3.3-13

4.0-1 3.5-7

2.4-1 4.9-7

2.0-1 5.1-8 2.9-1 4.1-8

ro(4) rl(4)

ro(3) rl(3)

n(2)

2.9-2 1.6-6 1.3-1 7.0-7

2.6-2 1.6-7 1.1-1 7.1-8

2.3-2 1.6-17Ne 9.8-2 7.2-18Ne 1.8-1 5.2-18Ne 2.4-1 4.4-18Ne 2.9-1 3.8-18Ne 2.6-13

ro(2)

1011

1010

0

Ne

Table 5.7. (continued) 10 14

9.9-1 6.6-6 1.7-12 9.2-13

9.8-1 1.9-5

9.2-1 7.1-5

2.4-13

8.4-1 1.4-5 9.5-1 4.2--6 9.8-1 1.5--6 1.1-10

5.0-1 4.9-5

3.1-13

2.0-10

1.0 3.8--6

9.6-1 4.2-5 9.9-1 1.1-5

8.2-1 2.1-4

T = 3.2 x 104 K. 1.9-1 2.8-1 1.5-4 1.3-3

7.4-1 1.9-5 9.1-1 5.9--6 9.7-1 2.1--6 7.1-13 5.7-13

T = 1.6 x 104 K. 4.9-2 9.6-2 1.5-3 1.6-4 3.0-1 6.8-1 5.9-5 3.3-4

lOB

4.9-10 4.8-13

7.0-13

7.6-10

1.0 1.8-5

1.0 5.5-5

1.0 3.4-5 1.0 1.1-5

1.0 2.1-4

9.9-1 1.2-3

9.2-1 1.3-2

1.0 2.0-5 8.2-10 9.0-12

1.5-12

1.0 6.0-5

1.0 2.3-4

1.0 1.3-3

9.8-1 1.4-2

8.0-11

1.0 8.4-5 1.9-11

1.0 9.6-4 1.0 2.5-4

1.0 5.1-3

9.6-1 4.0-2

lOIS

1.0 2.0-5 8.2-10

1.0 5.9--6

1.0 2.3-4

1.0 1.3-3

9.8-1 1.4-2

1.0 8.3-5 1.8-11 1.2-11

1.0 6.8-5 1.5-11 4.8-12

1.0 9.4-4 1.0 2.5-4

9.9-1 4.9-3

9.4-1 3.9-2

1O l7

9.9-1 7.8-4 1.0 2.0-4

9.7-1 4.1-3

7.8-1 3.2-2

10 16

9.9-1 1.3-4

9.5-1 7.3-4

6.1-1 7.2-3

2.0-12

1.0 2.7-5 6.1-12

9.8-1 3.0-4 9.9-1 8.1-5

8.9-1 1.6-3

3.2-1 1.1-2

10 15

8.3-30Ne

1.0 2.0-5 8.2-10

1.0 5.0-5

1.0 2.3-4

1.0 1.3-3

9.9-1 1.4-2

7.5-29Ne

1.0 8.5-5 1.9-11

1.0 2.5-4

1.0 9.6-4

1.0 5.1-3

9.6-1 4.0-2

00

t""'

~

VI

-

I»

'"3

5i

'tI

I»

'"S·

< g.

C1l

0..

C1l

a·

ttl ><

'"0 ....,

::s

...5io·

~

>c:I

0

:,:.. 'tI

VI

5.6--14

6.7-9

5.6--14

1.5 2.3-17Ne

S

IX

ro(2) n(2) ro(3) rl(3)

IX

S

ro(4) n(4) ro(5) rl(5) ro(6) rl(6)

1.1 4.5-7

1.2 5.3-8

1.1 4.4-8 2.8-8 1.5-14

1.1 5.0-18Ne 2.8-8 1.5-14

2.9-8 1.5-14

1.1 2.9-7

1.2 6.3-7

1.5 2.3-6 1.3 9.5-7

5.7-14

7.2-9

9.0-1 2.1-7

7.9-1 5.2-7 8.4-1 3.6--7

1.2 6.7-8

1.5 2.3-7 1.3 9.7-8

7.7-1 5.6--8 8.0-1 4.4-8 8.2-1 3.6--8

7.4-1 7.8-7

6.9-1 1.8-6

1011

1.2 7.0-18Ne 1.2 5.7-18Ne

1.3 1.0017Ne

6.9-9

7.6--1 5.9-18Ne 7.8-1 4.8-18Ne 7.9-1 4.2-18Ne

ro(4) rl(4) ro(5) rl(5) ro(6) rl(6)

7.3-1 8.1-8

6.9-1 1.8-7

6.8-1 1.8-17Ne 7.2-1 8.2-18Ne

ro(2) rl(2)

ro(3) rl(3)

1010

0

Ne

Table 5.7. (continued)

3.0-8 1.5-14

1.0 9.8-7

1.1 2.2-6

1.1 4.6-6

1.5 2.3-5 1.2 8.8-6

5.7-14

7.7-9

7.0-1 1.8-5 7.7-1 7.2-6 8.5-1 3.7-6 9.3-1 1.6-6 9.7-1 6.7-7

10 12

1.2-8

1.0 7.5-6

1.0 3.5-6

3.3-8 1.5-14

1.0 6.2-6 1.0 2.4-6

1.1 1.9-5

1.0 5.3-6 4.1-8 1.5-14

1.0 1.5-5

1.0 5.2-5

5.8-8 1.5-14

1.0 1.1-5

1.0 1.1--4 1.0 3.1-5

1.1 6.3-3 1.0 6.0--4

1.9-8 6.5-14

1.0 8.1-5

9.9-1 4.5--4

9.2-1 5.1-3

1.0 2.2-5

6.1--41 T = 5.12 x IO sK. 1.4 1.5 1.7-3 2.2--4 1.0 1.1 6.5-5 2.5--4

8.9-9 5.9-14

9.5-1 1.4-5 9.8-1 4.3-6 9.9-1 1.6-6

x IO sK. 7.9-1 1.4-3

T = 1.28 7.3-1 1.7--4 8.5-1 5.1-5

lOIS

9.5-1 1.9--4 9.9-1 3.6--5 1.0 1.0-5

10 14

1013

6.7-8 1.5-14

1.0 1.4-5

1.0 3.9-5

1.0 1.4--4

1.0 8.7-3 1.0 7.6--4

7.2-14

2.2-8

1.0 9.5-6

1.0 1.0--4 1.0 2.8-5

1.0 5.8--4

9.8-1 7.0-3

10 16

1.0 1.4-5 6.8-8 2.4-14

1.0 4.0-5

1.0 1.5--4

1.0 9.0-3 1.0 7.9--4

1.4-13

2.2-8

1.0 9.8-6

1.0 1.1--4 1.0 2.8-5

1.0 6.0--4

9.9-1 7.3-3

10 17

1.0 1.4-5 6.8-8 1.2-13

1.0 4.0-5

1.0 1.5--4

1.0 9.0-3 1.0 7.9--4

7.7-13

2.2-8

1.0 9.8-6

1.0 1.1--4 1.0 2.9-5

1.0 6.0--4

9.9-1 7.3-3

10 18

6.8-8 1.0-31Ne

1.0 1.5--4 1.0 4.0-5 1.0 1.4-5

1.0 9.0-3 1.0 7.9--4

7.0-31Ne

2.2-8

1.0 9.8-6

1.0 1.1--4 1.0 2.9-5

1.0 6.0--4

9.9-1 7.3-3

00

0

'"

(1)

e. 0

~

::I

e. 0

O.

><

-'"

tTl

0

'"....,

!3

0

0-

...'"ti 0

(1)

!3

en

VI

0'1

. j>.

3.4 1.6-6 2.4 6.4-7

1.5 4.S-S

3.4-S

3.4-15

3.4 1.6-7 2.5 6.5-8

1.9 3.5-8

I.S 3.O-S 2.5-S 6.5-16

1.6 5.1-ISN.

1.5 4.4-ISN.

3.3-8

3.4-15

3.4 1.6-17N.

2.5 6.6-ISN.

2.2 4.5-1SN.

2.0 3.7-ISN.

1.9 3.2-ISN. 2.5-8 6.5-16

ro(5) n(5)

ro(6) n(6)

S

ex

ro(2) rl(2)

ro(3) n(3)

ro(4) rl(4)

ro(5) rl(5) ro(6) n(6)

ex

S

3.4-15

1.7 6.1-8

1.7 6.3-1SN.

ro(4) rl(4)

2.1 4.4-8

1.5 4.O-S

1.9 S.9-8

1.9 0.9-ISN.

2.6-8 6.5-16

1.7 2.4-7

1.9 3.2-7

2.1 4.2-7

3.4-S

1.3 3.0-7

1.5 4.3-7

1.6 5.S-7

1.9 S.7-7

2.4 2.2-6

2.4 2.2-7

2.5 2.2-17N.

1011

ro(2) n(2) ro(3) rl(3)

1010

0

N.

Table 5.7. (continued)

2.6-8 6.4-16

1.3 1.1-6

1.6 2.1-6

1.9 3.7-6

3.3 1.6-5 2.4 6.1-6

3.3-15

3.2-15

4.7-8

1.0 7.0-6

1.0 1.9-5

4.6-8 5.8-16

1.0 I.S-5

1.0 4.S-5

5.8-S 5.7-16

1.0 2.6-5

1.0 7.2-5

1.0 2.5-4

1.0 1.2-3

l.l S.0-4 1.0 1.6-4

1.2 1.3-2

3.0-15

7.7-8

1.0 2.1-5

1.0 5.7-5

2.0 7.2-3

3.0-15

6.5-S

1.0 1.5-5

1.0 4.3-5

1.0 1.5-4

l.l 6.3-5 1.0 2.0-4

1.0 1.0-3

l.l 7.6-4

1.2 2.8-4

1016 l.l 1.1-2

1015 1.4 7.4-3

T = S.l92 x 106 K. 3.2 2.9 1.4-3 1.6-4 2.1 1.5 5.2-5 2.7-4 1.5 l.l 2.0-5 6.4-5 1.2 1.0 7.5-6 2.0-5 l.l 1.0 3.1-6 7.5-6 3.3-S 2.S-8 6.4-16 6.2-16

3.9-8 3.3-15

1.0 3.0-6

l.l 1.2-6

1.2 2.2-5

1.6 6.5-5

1.1 7.6-6

3.6-8

1014

T = 2.04S x 106 K. 2.1 1.4 2.1-4 1.7-3

1013

1.3 2.4-6

1.5 4.7-6

I.S S.2-6

2.3 2.1-5

1012

6.O-S 6.6-16

1.0 2.8-5

1.0 7.6-5

1.0 2.6-4

1.0 1.3-3

1.0 1.4-2

4.1-15

7.S-8

1.0 2.1-5

1.0 5.9-5

1.0 2.1-4

1.0 1.1-3

1.0 1.2-2

10 17

6.0-8 1.6-15

1.0 2.8-5

1.0 7.7-5

1.0 2.7-4

1.0 1.3-3

9.9-1 1.4-2

1.5-14

7.9-8

1.0 2.1-5

1.0 5.9-6

1.0 2.1-4

1.0 1.1-3

9.9-1 1.2-2

1018

6.6-S 1.1-33N.

1.0 2.8-5

1.0 7.7-5

1.0 2.7-4

1.0 1.3-3

9.9-1 1.4-2

1.2-32N.

7.9-8

1.0 2.1-5

1.0 5.9-5

1.0 2.1-4

1.0 1.1-3

9.9-1 1.2-2

00

.".

-I

-

1>0'>

!3

1>0'> fI.I

:g

1>0'>

S·

fI.I

a.

<

~

8-

!3.

>e

-

trI

0

fI.I

= ....

1>0'>

g.

E.

.g

~

'"C

VI

5. Some Problems of Excitation Kinetics

148

5.4.4 Hydrogen6k.e Ions Both in the coronal limit and at high density, the temperature at which the ions Xz exist is proportional to the ionization potential Ez. For hydrogenlike ions, Ez = z 2Ry. When T <X z2, the quantities (va) <X z-3. The spontaneous radiative transition probabilities Ann' <X z4. Therefore the reduced density and temperature fie

= Ne/z7, f = T/z2

are convenient when considering the ions. At given fie and f, the quantities ro, rio rx./z, and z 3S do not depend on z. The Saba-Boltzman equation (5.4.11) may be rewritten in the form

NE(n) = where Q density

~ n28~/2 :~ (z2:y) 3/2 exp ( ; )

,

(5.4.22)

= Ne/N(z+I). From (5.4.18), it follows that the reduced population

fI(n) = QN(z)(n)/zll

(5.4.23)

will not depend on z. Table 5.8 gives some values of ro, rio rx./z, and z 3S from data of the work by McWhirter and Hearn [5.38). The population densities are obtained by substituting NE(n) from (5.4.22) into (5.4.18) with N(z+l) instead of N(H+). The more recent quantitative results can be found in [5.40,43). The difference between two sets of coefficients ro, rl given by Tables 5.7 and 5.8 is great at low temperature T ~z2Ry. It can be explained by different threshold behaviour of the cross sections for neutral atom and for ion (the excitation cross Table s.s. Parameters ro(n),rl(n) and coefficients S and thin plasma [5.38]. 4.7--6 denotes 4.7 x 10-6 1/e

0

108

1010

1012

ro(2)

4.7--6

5.9-6 1.2-8

8.8--6 1.2--6

3.6-5 1.2-4

1.6-4 1.2-3

1.3-3

1.6-3 5.5-9

2.5-3 5.5-7

1.5-2 5.5-5

1.0-2

1.4-2 5.1-9

2.2-2 5.0-7

2.9-2

3.9-2 4.3-9

7.9-2

1013

ro(3)

rl(3) ro(4)

rl(4) ro(5)

rl(5) ro(7)

rl(7)

for hydrogenlike ions in the optically lOIS

1016

1018

00

103z 2K. 1.4-3 1.2-2

1.3-2 1.1-1

6.1-2 5.1-1

1.1-1 8.9-1

1.1-1 8.9-1

9.6-2 5.7-4

3.2-1 6.4-3

4.3-1 6.0-2

4.7-1 2.9-1

5.0-1 5.1-1

5.0-1 5.1-1

2.1-1 4.2-5

5.2-1 2.9-4

6.9-1 2.9-3

7.5-1 2.7-2

7.7-1 1.3-1

7.8-1 2.2-1

7.8-1 2.2-1

7.0-2 4.1-7

5.5-1 2.3-5

7.8-1 1.3-4

8.7-1 1.3-3

8.9-1 1.2-2

9.0-1 5.8-2

9.0-1 1.0-1 I

9.0-1 1.0-1

1.1-1 3.4-9

3.2-1 2.7-7

8.8-1 6.3--6

9.5-1 3.0-5

9.7-1 3.0-4

9.8-1 2.8-3

9.8-1 1.4-2

9.8-1 2.4-2

9.8-1 2.4-2

T

rl(2)

IX

=4 x

1014

S ·z3

9.1-26

1.1-25

1.9-25

1.0-24

6.1-24

4.6-23

3.7-22

2.0-21

3.5-21

3.8-21

IXlz

7.9-13

9.2-13

1.4-12

5.2-12

2.0-11

1.1-10

8.6-10

8.0-9

7.6-7

7.6-251/e

5.4 Populations of Excited Levels in a Plasma

149

Table 5.8. (continued) fIe

0

108

1010

1012

ro(2) n(2)

1.1-3

1.3-3 8.9-9

1.5-3 8.9-7

ro(3) rl(3)

1.9-2

2.2-2 4.2-9

2.7-2 4.1-7

3.1-3 8.8-5 6.9-2 4.0-5

ro(4) n(4)

5.9-2

6.7-2 3.5-9

7.5-2 3.4-7

3.3-1 2.6-5

ro(5) rl(5)

1.0-1

1.2-1 3.2-9

6.5-1 1.3-5

ro(7) rl(7)

1.8-1

2.2-1 2.5-9

1.6-1 3.0-7 4.0-1 2.0-7

S ·z3

4.9-17

5.3-17

(lIz

4.8-13

ro(2) n(2)

1013

1014

10 15

1016

1018

2.1-1 6.4-2 7.7-1 1.8-2

6.0-1 1.9-1

7.6-1 2.4-1

8.9-1 5.1-2

9.4-1 6.5-2

7.6-1 2.4-1 9.4-1 6.5-2

9.3-1 5.5-3

9.7-1 1.6-2

9.8-1 2.0-2

9.8-1 2.0-2

00

T=8xI03~K.

7.4-3 8.8-4 2.5-1 3.6-4 6.9-1 1.4-4

3.4-2 8.5-3

9.5-1 3.1-4 9.9-1 5.9-5

9.8-1 2.0-3

9.9-1 5.7-3

9.9-1 7.3-3

9.9-1 7.3-3

9.2-1 3.3--6

8.8-1 5.4-5 9.8-1 9.9--6

1.0 3.8-4

1.0 1.1-3

1.0 1.4-3

1.0 1.4-3

6.8-17

1.7-16

4.1-16

2.2-15

1.6-14

4.5-14

4.5-14

5.1-13

6.1-13

1.2-12

2.5-12

7.6-12

3.7-11

3.8-14 1.7-10

1.0-8

1.0-26,,_

2.2-2

2.4-2 6.8-9

8.6-1 5.7-2

9.4-1 6.3-2

9.4-1 6.3-2

9.3-2

1.0-1 3.2-9

9.7-1 5.8-3

9.8-1 9.9-3

9.9-1 1.1-2

9.9-1 1.1-2

ro(4) n(4)

1.7-1

1.8-1 2.6-9

2.0-1 2.5-7

9.8-1 1.5-3

9.9-1 2.5-3

1.0 2.8-3

1.0 2.8-3

ro(5) rl(5)

2.3-1

2.5-1 2.3-9

2.9-1 2.2-7

T = 1.6 x 104~K. 5.7-2 1.5-1 3.6-2 6.7-5 6.6-4 6.0-3 3.8-1 1.8-1 7.4-1 3.0-5 2.4-4 1.3-3 4.3-1 9.3-1 7.6-1 1.8-5 8.6-5 3.6-4 7.1-1 9.1-1 9.8-1 3.2-5 9.3--6 1.2-4

5.2-1 3.2-2

ro(3) rl(3)

2.6-2 6.8-7 1.1-1 3.2-7

9.9-1 4.9-4

1.0 8.4-4

1.0 9.1-4

1.0 9.1-4

ro(7) rl(7)

3.1-1

3.5-1 1.8-9

4.8-1 1.5-7

9.3-1 2.1--6

9.8-1 5.7--6

1.0 2.1-5

1.0 8.5-5

1.0 1.5-4

1.0 1.6-4

1.0 1.6-4

S

1.3-12

1.4-12

1.5-12

2.4-12

4.2-12

1.2-11

4.3-11

7.2-11

7.8-11

7.8-11

(lIz

2.9-13

3.0-13

3.2-13

4.3-13

6.3-13

3.0-12

7.9-12

3.1-10

3.1-28".

ro(2) rl(2)

1.2-1

1.3-1 5.4-9

6.8-1 2.0-2

9.3-1 3.0-2

9.7-1 3.1-2

9.7-1 3.1-2

ro(3) n(3)

2.5-1

2.6-1 2.6-9

1.3-1 5.4-7 2.7-1 2.5-7

1.2-12 4 = 3.2 x 10 z2K. 1.9-1 3.1-1 4.5-3 5.2-4 5.1-1 8.1-1 1.8-4 8.8-4

9.5-1 2.9-3

9.9-1 4.2-3

1.0 4.4-3

1.0 4.4-3

ro(4) n(4) ro(5) rl(5)

3.3-1

3.5-1 1.9-9 4.1-1 1.7-9

ro(7) rl(7)

4.6-1

5.0-1 1.3-9

S ·z3

2.4-10 1.7-13

2.4-10 1.8-13

'Z3

(lIz

3.9-1

T 1.5-1 5.4-5 3.3-1 2.4-5

6.1-1 2.6-3 8.7-1 8.4-4

3.7-1 1.9-7 4.4-1 1.6-7 5.9-1 1.1-7

5.4-1 1.4-5 7.5-1 7.6--6

8.1-1 6.4-5 9.3-1 2.3-5

9.5-1 2.3-4 9.8-1 7.4-5

9.9-1 6.8-4 1.0 2.2-4

1.0 9.8-4 1.0 3.1-4

1.0 1.0-3 1.0 3.3-4

1.0 1.0-3 1.0 3.3-4

9.4-1 2.0--6

9.9-1 4.1--6

1.0 1.2-5

1.0 3.6-5

1.0 5.1-5

1.0 5.4-5

1.0 5.4-5

2.5-10

3.2-10

4.4-10

8.5-10

2.8-9

2.9-9

2.9-9

1.8-13

1.8-13

2.0-13

2.4-13

2.0-9 3.3-13

5.0-13

2.9-11

2.9-29".

5. Some Problems of Excitation Kinetics

150

Table 5.8. (continued) 1015

1016

10 18

00

5.2-1 3.6--3

7.8-1 1.5-2

9.5-1 2.3-2

9.8-1 2.4-2

9.8-1 2.4-2

6.5-1 1.5--4

8.6--1 7.1--4

9.7-1 2.1-3

9.9-1 2.9-3

1.0 3.1-3

1.0 3.1-3

6.7-1 1.2-5

8.6--1 5.5-5

9.6--1 1.8--4

9.9-1 4.8--4

1.0 6.7--4

1.0 7.0--4

1.0 7.0--4

6.2-1 1.1-7

8.1-1 5.9-6

9.4-1 2.0-5

9.9-1 5.8-5

1.0 1.5--4

1.0 2.1--4

1.0 2.2--4

1.0 2.2--4

6.6--1 9.2-10

7.1-1 7.9-8

9.6--1 1.3-6

9.9-1 3.3-6

1.0 9.7-6

1.0 2.4-5

1.0 3.4-5

1.0 3.5-5

1.0 3.5-5

3.4-9

3.5-9

4.0-9

4.8-9

7.3-9

1.4-8

1.8-8

1.8-8

1.8-8

1.0-13

1.1-13

1.2-13

1.6--13

2.1-13

5.6--12

5.6--30'1e

= 2.56 x

105z2 K.

'1e

0

108

1010

10 12

= 6.4 x

104z2 K.

ro(2) rl(2)

3.1-1

3.5-1 4.5-9

3.6--1 4.5-7

3.8-1 4.4-5

4.2-1 4.3--4

ro(3) rl(3)

4.6--1

4.8-1 2.1-9

4.9-1 2.1-7

5.3-1 1.9-5

ro(4) rl(4)

5.3-1

5.5-1 1.5-9

5.7-1 1.5-7

ro(5) rl(5)

5.7-1

6.0-1 1.2-9

ro(7) rl(7)

6.2-1

s. Z3

3.4-9

1013 T

rx/z

1.0-13

1.0-13

1.0-13

T

10 14

ro(2) n(2)

1.1

1.1 3.5-9

1.1 3.5-7

1.1 3.5-5

1.1 3.4--4

1.1 2.9-8

1.0 1.4-2

9.8-1 2.4-2

9.7-1 2.6--2

9.7-1 2.6--2

ro(3) rl(3) ro(4) rl(4)

1.0

1.0 1.6--9 1.0 1.1-9

1.0 1.6--7 1.0 1.1-7

1.0 1.5-5 1.0 9.2-6

1.0 1.3--4 1.0 5.2-5

1.0 6.7--4 1.0 1.8--4

1.0 2.0-3 1.0 4.5--4

1.0 3.0-3 1.0 6.8--4

1.0 3.2-3 1.0 7.2--4

1.0 3.2-3 1.0 7.2--4

ro(5) rl(5)

9.9-1

1.0 6.0-10

1.0 5.8-8

1.0 3.9-6

1.0 1.8-5

1.0 5.4-5

1.0 1.4--4

1.0 2.1--4

1.0 2.2--4

1.0 2.2--4

ro(7) rl(7)

9.7-1

1.0 4.6--10

1.0 4.1-8

1.0 9.0-7

1.0 3.0-6

1.0 8.5-6

1.0 2.2-5

1.0 3.2-5

1.0 3.2-5

1.0 3.2-5

1.0

section for a neutral atom at the threshold is equal to zero, but, for ions, the threshold value is not zero: see Sect. 3.2). In the range T rv z2Ry, the values of ro, rl from the Tables 5.7 and 5.8 are found to be in reasonable agreement. 5.4.5 Population Densities of Highly Excited Levels at High Density; Steady-Flow Regime The highly excited bound levels with n ~ no are populated and evacuated exclusively by collisions. After an electron has been transferred to a level through three-body recombination it may be either reionized through electron impact, or transferred to another bound level through inelastic or super-elastic (quenching) collision. If n > no, the collisional transition frequency between the bound levels is n2 times larger than the frequency of reionization. The most probable are

5.4 Populations of Excited Levels in a Plasma

151

collisional transitions between adjacent levels, followed by transfer of a small energy amount. The electron wandering between the highly excited levels can be treated as diffusion in the space of quantum numbers which can be described by Focker-Planck equation [5.32-34,44]. Neglecting the radiative processes the electron flux j(n) [cm- 3 s- l ] in the space of quantum numbers can be determined by 00

j(n) =

L:

L:[N(n + k' + l)Wn+k'+I,n-k - N(n - k)Wn-k,n+k,+d, (5.4.24)

k'~Ok~O

the first term in the square brackets representing the electron flux via n directed to the ground state, the second term representing the flux directed to the continuum. Using the ratio N(n)/NE(n) = b(n) which shows the departure of the level population from the Saba-Boltzmann population density, (5.4.24) may be rewritten in the form

j(n) = N(Z+I)-S2 L:[(n + k' k,k'

+ 1iexp (En+k,+dT)b(n + k' + l)Wn+k'+I,n-k

-(n - kiexp (En-k/T) b(n - k)Wn-k,n+k,+d ,

(5.4.25)

where S = z383/2/4n3/2aijNe, 8 = T/z 2Ry. Assuming b(n + k) = b(n) + kob/on, retaining the terms of the order and neglecting the weak functions of n, one can obtain from (5.4.25)

j(n) =

N(Z+I)Ne~exp

(ETn)

~b n2 L: k 2 (vun,n+k)

un

Ikl~1

.

rv

k 2,

(5.4.26)

In n ~ 1 the sum with respect to k in (5.4.26) may be extended to infinity. Let the basic dependence of (VUn,n+k) on temperature and the numbers n and k, which determine the rate coefficient order of magnitude, be written in explicit form (Sect. 3.5) 2

2

4

2 e nao -1/2n (VUn,n+k ) -_ .jn'h--;'38 k 3 cp(n, k, 8) ,

(5.4.27)

where cp(n, k, 8) is a weak function of its arguments. It is convenient to introduce the continuous variable e = 1/n2. Using (5.4.27) we can write the formula for the flux as

'() __ N(Z+I)N232n2e2 58-2-3/2 (~)LOb ] e e z6 Ii, ao e exp 8 oe'

(5428) . .

The factor L = L:k k-1qJ(n = e- I/2 , k,8) is a weak function of e and 8. (In reference [5.20] it is estimated to be ~ 0.2). Using the variable e the diffusion equation can be written

2e3/2 d~~) = q(e) - N(e)fB .

(5.4.29)

Here q(e) [cm- 3 s-I] is the rate for the direct population of the level e, and

152

5. Some Problems of Excitation Kinetics

f e[s -I]

is the total frequency of electron transfer to the continuum and to the lower levels n' with n' < no. The collisional transfer to the highly excited levels is already taken into account in the expressions (5.4.26,28) for the electron flux j(e) and should be excluded from fe. In order to obtain b(e) and j(e), the equation (5.4.29) should be supplemented with appropriate boundary conditions. The boundary condition at I'. = 0 is found easily. In the limit of I'. ~ 0 (n ~ 00), the function b( e) should correspond to the continuum distribution. Wlien Maxwellian velocity distribution is valid, limb(e)

e..... O

=

I .

(5.4.30)

The second boundary condition in general must be chosen by fitting the solution of (5.4.29) to the solution of the set of rate equations (5.4.10) for lower levels where the discrete structure and radiative transitions are of importance. In some cases it is possible to obtain the explicit form of this boundary condition. Somewhat later we shall consider low-temperature recombining plasmas when the condition of total absorption at some value 1'.1 can be treated as the second boundary condition. Without any particular pumping or evacuation of the highly excited levels, q( e) is equal to the rate of three-body recombination to the levell'., and f B is the probability of the inverse process, ionization by electron impact. Assuming the departure of b(e) from the Saha-Boltzmann equilibrium value (equal to unity) to be small, the right-hand part of (5.4.29) may be put equal to zero, because q(e) ~ N(e)fe due to detailed balance. Thus we obtain the constant-flow approximation j(e)

= const. = j

Then solving the equation (5.4.28) together with the boundary condition (5.4.30) yields the relationship b(e)

=

z 6Le 2 e I - j N(Z+llNic[(e')3/2 exp

(

-

1'.') e

de'

(5.4.31)

C = 32n2 aije2 jh .

If

e~e

. z 6Le2

b(e) ~ 1- ] N(z+I>N2Ce

5/2

(5.4.32)

e

The flux value should be determined using the second boundary condition. Now we shall consider low-temperature recombining plasmas. The flux j( e) is positive definite, and b( e) decreases with e. As the second boundary condition for equation (5.4.29), one can use the condition of total absorption of the flux at some value 1'.1, b(eJ)

=0

.

(5.4.33)

5.4 Populations of Excited Levels in a Plasma

153

If we deal only with low temperature e ~ 81, the value of j determined from (5.4.31) and (5.4.33) does not depend on 8t, and the integration with respect to 8' may be extended to infinity. Thus j = N(z+I)Ne a. ,

(5.4.34)

where

N. 27

3/2

2

a. = ~ _7t_ ~a~e-9/2 z6 3 Ii. =

N ez3

4V27t3/ 2 e 10 3 ml/2T9/2L

(5.4.35)

is the recombination coefficient. The order of magnitude and the dependence of the coefficient a. on temperature appear to be in agreement with the results of numerical calculations by Johnson and Hinnov, and Bates et al. given in Tables 5.7 and 5.8.

6 Tables and Formulas for the Estimation of Effective Cross Sections

In this chapter, tables of cross sections and rate coefficients of excitation and

ionization by electron impact and rate coefficients of dielectronic recombination are given. The cross sections and rate coefficients are presented in the form of products of angular and radial factors, the latter being expressed in analytical form containing two or three adjusted parameters. The tabulated parameters are obtained from the results of numerical calculations. The first section contains a description of the contents of the tables and relevant fitting formulas. In the second and third sectins of the chapter the formulas for angular factors are given which are necessary for applying the tables.

6.1 6.1.1

Tables of Numerical Results Methods of Calculations and Survey of the Tables

In this section the results of numerical calculations for the cross sections a, collision rate coefficients (va), and dielectronic recombination rate coefficient /Cd are given. The calculations are made using the Born method (see Sects. 2.3 and 3.1) and its modifications which are described in Sect. 3.2. The atomic wave functions are assumed to be constructed from single electron wave functions in accordance with a specific scheme of angular momenta coupling. A singleconfiguration approximation is used. The radial wave functions for all levels of a given electronic configuration are assumed to be the same. Under these assumptions the cross sections may be expressed in the form (2.3.4,8). In this chapter, however, we use the formulas somewhat different from (2.3.8). For the excitation cross sections of the transitions ao ---+ al we write in the general case a(ao,

ad = a'(ao, ad + a"(ao, ad = I: Q~(ao,ada~(lo, Id K

+ I: Q~(ao,ada~(lo, Id

(6.1.1 )

K

where a' consists of the direct plus interference terms, and a" is the purely exchange contribution to the cross section (Sects. 2.3,3.1,2). A summary of formulas for the Q-factors is given in Sect. 6.2. In order to simplify the use of the tables given below the subsequent subsections comprise the specific formulas for analytical approximation and Q-factors which can be used for specific tables. The exchange cross sections are given either summed over /C or for those cases in which index /C has a single value. Therefore, in most cases the exchange

I. I. Sobel'man et al., Excitation of Atoms and Broadening of Spectral Lines © Springer-Verlag Berlin Heidelberg 1995

6.1 Tables of Numerical Results

155

cross sections can be determined only for transitions between the atomic terms as a whole, so (6.1.2) where index K is not included, and a"(lo, 1\) = ~Ka~(lo, 1\). The calculated quantities a~, a~ have been approximated by means of simple analytic formulas which contain the two or three fitting parameters: C, cp, D for a and A, X, D for (va). The fitting parameters have been found from the results of numerical calculations by the method of least squares. The errors of analytic approximation R are also given in the tables. We pass on to a brief description of the tables and of the approximate methods which have been used for calculations. The Born approximation with normalization has been used for calculation of the cross sections for hydrogen atom summed over 10 and 1\ (Table 6.1). Tables 6.2 and 6.3 contain the fitting parameters for cross sections and rate coefficients calculated in the Born approximation with the Bates-Damgaard approximation for the atomic wave functions. The Bates-Damgaard approximation used for calculations of Tables 6.2 and 6.3 is most valid in cases when the maxima of both radial functions lie outside the atomic core. This condition is usually formulated explicitly as

no = no

Vz2Ryi lEo I >

ni

> nc ,

10 + 1/2,

nr = Vz 2Ry/lEd > 1\ + 1/2,

(6.1.3) (6.1.4)

> nc ,

where nc is the largest of the principal quantum numbers of the electrons of the atomic core. The condition (6.1.4) is sometimes stricter than (6.1.3). However, in many cases when conditions (6.1.3) are fulfilled, but (6.1.4) are not fulfilled, the error does not exceed the factor of 2. Such errors are inherent in the Born method itself. The excitation cross sections for specific atoms and ions (Tables 6.1,4-10), have been normalized with the use of the K matrix. For calculation of radial BOO

t 600 N<§'400

-!:: b

200

o

....

3d-4f

- .J

'

I

I I,

Y

1

2

.........

,

r----,_

-

---- -eV

B

Fig. 6.1. Effective excitation cross section for the transition 3d-4f of the H atom: (1) Normalized cross section; (2) Born approximation

156

6. Tables and Formulas for the Estimation of Effective Cross Sections

integrals, the semiempirical numerical wave functions [6.1] have been used. These functions provide considerably better accuracy in all cases, especially for transitions from the shells of equivalent electrons, than the Bates-Damgaard radial functions. In the case of neutral atoms, the normalized Born cross sections are tabulated for transitions without spin change. For intercombination transitions the method of orthogonalized functions (Sect. 3.2) with normalization has been used. In case of ions for transitions with no change of spin the normalized CoulombBorn cross sections have been calculated for Table 6.4. The Coulomb-Bomexchange method (Sect. 3.2) has been used for the cross sections tabulated in Tables 6.5-10. The exchange interaction has been taken into account within the framework of the method of orthogonalized functions. The ionization cross sections have been calculated in the Coulomb-Born approximation. We shall give a brief description of corrections to the Born approximation which are induced by treating the Coulomb attraction and the exchange interaction, and by normalization of the cross sections. The difference between the Born and Coulomb-Born cross sections is great only near the threshold 8::6 11E, T::6 11E; see Figs. 3.5, 6.2, 6.3. The treatment of exchange affects the value of the cross section for transitions with no change of spin also not far from the threshold (see Fig. 3.6). Therefore the treatment of exchange and Coulomb attraction are necessary for the cross sections of excitation from the ground state of ions. In the LS-coupling approximation, the intercombination-transition excitation is solely due to exchange. The effect of normalization upon the cross sections of excitation from the ground state is not substantial.

2s 3p

z=lO

30r-=:::::::::~~:;"';';;'_J------I-'~-~---H

~C>

t>

~

~~10~------~----~~------~~----~ Bom opproximotion

A,

O~----~~----~-------L------~

10-2

10-1

1

10

102

u=(lo-M)/M-

Fig. 6.2. Effective cross section for the transition 23-3 p of the hydrogen and hydrogenlike ions. (- • - • -) Born approximation. The effect of nonnalization is very small even for neutral hydrogen

6.1 Tables of Numerical Results

157

2s-3p

1 10 TlIlE=l/P_

Fig. 6.3. Excitation rate coefficient (vu) for the transition 2s-3 p of the hydrogen and

hydrogenlike ions

For transitions between the excited levels of neutral atoms, first order perturbation theory can give considerably overestimated results at low energies 8 < Ry (Figs. 3.7, 6.1). Normalization removes this deficiency. In the case of multiply charged ions, the matrix element of interaction of the outer electron with the bound one is z times smaller than the interaction of the bound electron with the atomic core. In Coulomb units, this matrix element is proportional to the parameter 1/z. Therefore at z ~ I (actually at z > 3) the normalization effect is negligible even for transitions between closely spaced levels. The Coulomb attraction for such transitions is also not important. Hence when 8 > AE, T > AE, the Born cross sections may be used for multiply charged ions. In cases not covered by the tables which are given in this section, one can use the semiempirical formulas given in Sect. 5.1. 6.1.2

Excitation Cross Sections for Neutral Hydrogen. Transitions no - nl (Table 6.1)

The tabulated cross sections are the sums of normalized Born cross sections for nolo - t nlll transitions. They are fitted by means of the following formula

1ta~ (no i

C 4)'(u) ,

(1

=

U

= (c - AE)/Ry .

nl) n~ U + qJ

(6.1.5)

The rate coefficients are given by A G'(P) exp(-pP) [cm3 /s], (V(1) = 10- 8 ( -no) 2-pnl no + X

P=

Ry/T,

p

= AE/Ry.

(6.1.6)

158

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.1. Normalized Born cross sections for neutral hydrogen Transition

C

qJ

D

R

A

3.26 2.70 2.58 2.53 2.50

0.00 0.00 0.00 0.00 0.00

0.07 0.08 0.08 0.08 0.08

X

D

R

0.26 0.38 0.42 0.44 0.45

0.00 0.00 0.00 0.00 0.00

0.01 0.01 0.02 0.02 0.02

no-nl

1-2 1-3 1-4 1-5 1-6

68.28 39.61 33.62 31.27 30.10

32.17 25.66 23.82 23.03 22.61

2-3 2-4 2-5 2-6

1275.0 454.0 318.0 268.0

1.23 0.62 0.53 0.50

0.00 0.00 0.00 0.00

0.10 0.16 0.16 0.15

332.0 316.0 281.0 264.0

0.70 0.59 0.52 0.43

3.80 0.80 0.30 0.00

0.02 0.02 0.02 0.02

3-4 3-5 3-6

8928.0 2298.0 1435.0

0.88 0.08 0.09

0.00 0.70 0.60

0.08 0.17 0.14

1437.0 1337.0 1202.0

0.96 1.19 1.02

9.90 3.30 1.60

0.02 0.02 0.01

4-5 4-6

35494.0 8492.0

0.19 0.06

0.70 0.70

0.13 0.12

7505.0 4201.0

1.14 1.59

9.90 5.50

0.04 0.02

5-6

118366.0

0.42

0.00

0.19

30951.0

1.49

9.90

0.04

The formulas for 1P'(u) and G'(P) are expressed by (5.1.12, 13 and 16). The set of parameters C, qJ, D is adjusted for the range 0.02 < u < 16, and the set A, X, D for 0.25 < P < 8. For the transitions no -+ no + 1 with no > 5 the quasiclassical cross sections given in Tables 3.3 and 3.4 may be used. 6.1.3 Born Cross Sections Calculated in the Bates--Damgaard Approximation for Atomic Wave Functions (Tables 6.2, 6.3) The cross sections for transitions with no change of spin of the atom, L1S = 0, are tabulated in a form analogous to the tables of oscillator strengths in BatesDamgaard approximation given in [6.2]. The quantities a and (va) are expressed in the form 2 (

a(ao,ad = 7tao

/2

Ry ) ( E ll, ) Eo 2/0 + 1 ~ Q,,(ao,ad 1P,,(u), u

L1E

=

If - L1E

L1E

'

(6.1.7)

(va(ao, ad) = 10

-8

(RY i/2 (EI i/2 exp( -P) , L1E) Eo) 2/ + 1 ~ Q,,(ao, ad

xG,,(P) [cm3 s- l ]

,,= 110 -

0

,p = L1E/T

111, 1/0 -/11 + 2,···1/0 + 111 .

(6.1.8)

6.1 Tables of Numerical Results

159

Table 6.2. Transitions with no change of spin AS = 0; Born cross sections in the Bates-Damgaard approximation for atomic wave functions. Asterisks indicate cases where the errors exceed 10%. Parameters C and q>. The effective principal quantum number An

0.5

1.0

1.5

2.0 Transition s - s, parameter C. a )

0.6 0.7 0.8 0.9 1.0

67 + 1 72+ I 69+ 1 58 + 1 42+ 1

55 + 1 55 + 1 49+ 1 39+ 1 26+ 1

57 + 55 + 48 + 37 + 25 +

1.1

16 + 1 67+0 60+0* 13 + 1* 27 + 1 41 + 1 52+ 1 57 + 1 54+ 1 43 + 1

15 + 1 81 +0 65 +0* 10+ 1 17 + 1 25 + 1 30+ 1 31 + 1 28 + 1 21 + 1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

1 1 1 1 1

14+ 1 84+0 68+0 97+0 15 + 1 21 + 1 25+ 1 26+ 1 23 + 1 18 + 1 15 + 1 13 + 1 11 + I 71 +0 78+0 73+0 57 +0* 60+0* 58+0 12 + i* 84+0 75+0 24+ 1 14 + 1 11 + 1 38 + 1 20+ 1 16+ 1 50+ 1 26+ 1 20+ 1 55 + 1 27 + 1 21 + 1 54+ 1 25 + 1 20+ 1 45 + 1 19 + 1 16 + 1 15 + 1 12 + 1 10+ 1 72+0 76+0 69+0 56+0* 58 +0* 54+0 11 + 1* 79+0 68+0 23 + 1 13 + 1 10 + 1 37 + 1 19 + 1 14 + 1 49 + 1 24+ 1 18 + 1 55 + 1 26+ 1 19+ 1 54+ 1 24+ 1 18 + 1 46+ 1 19 + 1 15 + 1 15 + 1 12 + 1 10+ 1 72+0 76+0 66+0 56+0* 57 +0* 52+0 11 + 1* 76+0 65+0

61 + 57 + 49+ 37 + 24+

no

2.5 I I 1 1 1

66+ 61 + 51 + 38 + 25 +

3.0 1 1 1 1 1

14 + 1 85 +0 69+0 94+0 14 + 1 19 + 1 22 + 1 22 + 1 20+ 1 15 + 1

15 + 1 87+0 70+0 93 +0 13 + 1 18 + 1 21 + 1 21 + 1 18 + 1 14 + 1

10 + 1 68+0 54+0 68+0 10 + 1 13 + 1 16 + 1 17 + 1 16 + 1 12 + 1

99+0 64+0 52+0 63+0 91 +0 12 + 1 14 + 1 15 + 1 14 + 1 11 + 1

91 +0 61 +0 49 -0 59+0 87 +0 12 + 1 14 + 1 15 + 1 14 + 1 11 + 1 84+0 57+0 46 -0 55 +0

82+0 55 +0 45 -0 53 +0 76+0 10 + 1 12 + 1 13 + 1 12 + 1 10+ 1 74+0 51 + 0 42 -0 48 -0

71 + 65 + 54+ 40+ 26+

4.0

3.5 1 1 1 1 1

77+ 69+ 57+ 42+ 27+

1 I 1 1 1

15 + 1 90+0 72+0 94+0 13 + 1 18 + 1 20+ 1 20+ 1 18 + 1 13 + 1 95+0 63 +0 50+0 60+0 86+0 11 + 1 13 + 1 14 + 1 13 + 1 10 + 1

16+ 1 94+0 75+0 96+0 13 + 1 18 + 1 20+ 1 20+ 1 17 + 1 13 + 1 93+0 62+0 49-0 58+0 82+0 11 + 1 13 + 1 13 + 1 12 + 1 10+ 1

76+0 52+0 42 -0 49-0 69+0 93+0 11 + 1 11 + 1 11 + 1 91 +0

72+0 49-0 40-0 46-0 64+0 86+0 10+ 1 11 + 1 10+ 1 84+0 62+0 43 -0 35 - 0 40-0

67+0 47 -0 38 -0 44-0

83 + 1 74+ 1 60+ 1 44+ 1 28 + 1 16 + 1* 97+0 77+0 98+0 14 + 1 18 + 1 20+ 1 20+ 1 17 + 1 13 + 1 93 +0* 61 +0 49-0 58+0 80+0 10 + 1 12 + 1 13 + 1 12 + 1 96+0 69+0* 48-0 39 -0 44-0 61 +0 81 +0 98+0 10 + 1 96+0 79+0 58 +0* 41-0 ·33 - 0 38 -0

4.5

5.0

89+ 1 79+ 1 64+ 1 46+ 1 30 + 1 17 + 1* 10 + 1* 80+0 10 + 1 14 + 1 18 + 1 20+ 1 20+ 1 17 + 1 13 + 1 93 +0* 61 + 0* 49-0 57+0 79+0 10+ 1 12 + 1 12 + 1 11 + 1 94+0 67 +0* 47 -0* 38-0 43 -0 59+0 78+0 93 +0 98+0 92+0 75 +0 55 + 0* 39 - 0* 32 -0 36-0

95 + 1 84+ 1 67+ 1 49+ 1 31 + 1 18 + 1* 10+ 1 83+0 10+ 1 14 + 1 18 + 1 21 + 1 20+ 1 18 + 1 13 + 1 94+0* 62+0* 49-0 57+0 79+0 10+ 1 12 + 1 12 + 1 11 + 1 92+0* 66+0* 46+0* 37 -0 42-0 57+0 76+0 90+0 95+0 88+0 72+0* 54+0* 38 - 0* 31- 0* 35 - 0

This table can be applied to transitions between the levels of two different electronic configurations ns-n's with no change of any of the angular quantum numbers

a)

160

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no Lin

0.5

0.6 0.7 0.8 0.9 1.0

92+0 88+0 85 +0 84+0 84+0

80+0 78+0 78+0 81 + 0 89+0

76+0 78+0 81 +0 89+0 10 + 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

10 + 1 15 + 1 15 + I" 10 + 1 78+0 70+0 68+0 69+0 72+0 81 + 0

12 + 1 15 + 1 12 + 1 77+0 62+0 60+0 62+0 68+0 76+0 89+0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

93 +0 15 + 1 20 + I" 13 + I" 10 + 1 91 +0 85 +0 82+0 81 + 0 81 +0 89+0 13 + 1 20 + I" 13 + I· 10+ 1 91 +0 85 +0 82+0 81 + 0 81 + 0

99+0 13 + 1 15 + I" 10 + 1 78+0 68+0 66+0 67+0 70+0 78+0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

88+0 13 + 1 20 + I· 13 + I· 10 + 1 91 +0 85+0 82+0 81 + 0 81 + 0

4.1 4.2 4.3 4.4

88+0 13 + 1 20 + I· 14 + I'

1.0

1.5

2.5

3.0

3.5

78 +0 82 +0 89+0 10 + 1 11 + 1

81 + 0 88 + 0 97+0 11 + 1 13 + 1

86+0 94+0 10 + 1 12 + 1 15 + 1

91 + 10 + 11 + 13 + 17 +

16 + 1 18 + 1 12 + 1 67 +0 55 +0 57 +0 65 +0 75 +0 90+0 11 + 1

19 + 1 20 + 1 12 + 1 66+0 53 +0 57 +0 67+0 80+0 98+0 12 + 1

11 + 1 13 + 1 12 + 1 81 + 0 61 +0 57+0 59+0 65 +0 73+0 85 +0

14 + 1 16 + 1 11 + 1 69+0 56+0 57 +0 63 +0 71 + 0 83 +0 10+ 1 12 + 1 14 + 1 12 + 1 74+0 56+0 55 +0 59+0 67 +0 78+0 95 +0

14 + 1 16 + 1 12 + 1 74+0 55 +0 54+0 60+0 70+0 84+0 10 + 1

15 + 1 18 + 1 13 + 1 74+0 53 +0 53 +0 61 +0 73 +0 90+0 11 + 1

21 + 1 23 + 1 13 + 1 64+0 52+0 58+0 69+0 85+0 10 + 1 13 + 1 17 + 1 20+ 1 14 + 1 73 +0 52+0 53+0 63 +0 77+0 97+0 12 + 1

96+0 13 + 1 15 + I· 10 + 1 78+0 68+0 65+0 66+0 69+0 78+0

10+ 1 13 + 1 12 + 1 82+0 61 + 0 56+0 58+0 63+0 71 +0 84+0

11 + 1 14 + 1 12 + 1 76+0 57 +0 54+0 58+0 65 +0 76+0 92+0

13 + 1 15 + 1 12 + 1 77+0 55 +0 53 + 0 58+0 68+0 81 + 0 10 + 1

14 + 1 17 + 1 13 + 1 77+0 54+0 52+0 59+0 70+0 87+0 11 + 1

16+ 1 18 + 1 14 + 1 77+0 53+0 52+0 60+0 73+0 93 +0 11 + 1

95 +0 13 + 1 15 + I" 10 + 1

10 + 1 12 + 1 12 + 1 82+0

11 + 1 13 + 1 12 + 1 77+0

12 + 1 15 + 1 12 + 1 79+0

14 + 1 16 + 1 13 + 1 79+0

15 + 1 18 + 1 14 + 1 79+0

Transition s - s, parameter

2.0

4.0

4.5

96+0 10 + 1 12 + 1 15 + 1 18 + 1

10 + 11 + 13 + 16 + 20+

5.0

qJ

0 1 1 1 1

1 1 1 1 1

24 + I· 26 + I" 25 + 1 27 + I" 13 + 1 14 + 1 64+0 63 +0 52+0 52+0 59+0 60+0 72+0 75 +0 90+0 96+0 11 + 1 12 + 1 15 + 1 16 + 1 19 + I" 21 + I" 21 + 1 23 + I" 15 + 1 15 + 1 73 +0 73 +0 51 + 0 51 + 0 53 +0 54+0 64+0 67+0 81 + 0 86+0 10+ 1 11 + 1 13 + 1 14 + 1 17 + I" 19 + I" 20 + 1 22 + I" 15 + 1 15 + 1 78+0 78 + 0 52+0 52+0 52+0 52+0 61 +0 63 +0 77+0 81 + 0 99+0 10 + 1 13 + 1 14 + 1 16 + I· 18 + I· 19 + 1 21 + I· 15 + 1 15 + 1 80+0 81 + 0

10 + 12 + 14 + 17 + 22+

1 1 1 1 1

29 + I· 30+ I" 14 + 1 62+0 52+0 62+0 79+0 10 + 1 13 + 1 17 + 1 23 + I" 25 + I· 16 + 1 73 +0 51 + 0 55+0 69+0 90+0 12 + 1 16 + I· 21 + I" 24+ I" 16 + 1 79+0 52+0 53+0 65+0 85+0 11 + 1 15 + I· 19 + I· 22 + I' 16 + 1 82+0

161

6.1 Tables of Numerical Results Table 6.2. (continued) The effective principal quantum number no Lin

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Transition s - p, parameter C 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

11 +2 14+2 15 + 2 15 +2 13 +2

II + 1 27 + 1 49 + 1 73 + 1 96+ 1 11 +2 11 +2 11 +2 93 + I 69 + 1

16 + 1 37 + 1 59 + 1 80 + 1 97 + 1 10 + 2 10+2 94 + 1 75 + 1 52 + l'

21 + 1 44 + 1 67 + 1 86 + 1 10+2 10+2 98 + 1 84 + 1 63 + I" 41 + l'

25 + 1 50+ 1 73 + 1 92 + 1 10 +2 10+2 96+ 1 80 + l' 59 + l' 37 + I"

29+ 1 56 + 1 80+ 1 98 + 1 10+2 10+2 95 + 1 77 + l' 55 + I" 34 + I"

32 + 1 61 + 1 86 + 1 10+2 11 +2 10+2 96+ l' 77 + l' 54 + I' 32 + l'

36 + 1 67 + 1 92 + 1 10+2 11 +2 11 +2 97 + l' 77 + l' 53 + I' 31 + l'

39 + 1 72 + 1 98 + 1 II +2 12 +2 11 +2 99 + l' 77 + l' 53 + I" 31 + l'

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

30+2 25 +2 18 + 2 10+2 51 + 1 14 + 1 10- 0 76+0 26+ 1 48 + 1

10+2 61 + 1 27 + 1 71 +0 53 + O' 19 + I" 42+ 1 68 + 1 87 + 1 95 + I

46 + 1 23 + 1 74+0" 43 -0 11 + 1 24+ 1 38 + 1 47 + 1 49 + 1 44+ 1

30 + l' 12 + 1 42 -0 55 + O' 13 + l' 24+ 1 34 + 1 39 + 1 38 + 1 31 + 1

22 + I" 86 +0" 37 - 0 63 + 0" 13 + I" 22 + 1 29 + 1 32 + 1 29 + 1 23 + l'

19 + 1 68 +0" 37 - 0 70+ 0" 14 + l' 21 + 1 27 + 1 29 + 1 26 + l' 20 + l'

16 + 58 + 38 75 + 14 + 21 + 25 + 26 + 23 + l7 +

1 O' 0" O' l' 1 1 l' l' I'

15 + l' 53 + 0" 40 - 0" 80+0' 14 + l' 21 + l' 25 + l' 25 + I' 21 + l' 15 + I"

14 + l' 50 +0" 43 - 0" 84 +0' 14 + I" 20 + I" 24 + l' 24 + l' 20 + I' 14 + I"

13 + I" 48 - 0" 45 - 0" 88 +0" 15 + l' 21 + l' 24 + l' 23 + I" 19 + l' 13 + I"

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

19+2 18 + 2 14+2 10+2 54 + 1 20 + 1 31 - 0 28 -0 15 + I 33 + I

80 + 1 55 + 1 29+ I 10 + 1 37 -0 11 + 1 30 + 1 52 + I 73 + 1 84+ 1

36 + 1 21 + 1 87+0 34-0 72+0 16 + l' 27 + 1 36 + 1 40 + 1 37 + 1

21 + l' 11 + 1 42 -0 34 - O· 77 +0 15 + 1 23 + 1 29 + 1 30 + 1 26+ I

15 + I" 74+0 30 - O· 35 - O· 77 + O· 13 + 1 19 + 1 22 + 1 22 + 1 18 + I"

12 + 1 55 + O' 26 -0 37 - O· 76+0' 12 + I l7 + 1 19 + 1 18 + l' 15 + l'

10 + 1 45 - O· 24-0 38 - O· 75 + 0" 12 + l' 15 + 1 17 + l' 16 + l' 12 + I"

91 +0 39 - 0" 24-0 39-0 74+0' II + l' 14 + l' 16 + I" 14 + I' 11 + l'

82+0 35 - 0" 24-0 40 -0 74 +0' 11 + l' 14 + l' 14 + I" 13 + I' 10 + I"

75 +0 32 - O· 24- 0 41 - 0 74+0' 11 + I" 13 + I" 14 + l' 12 + l' 93 +0'

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

16+2 16 +2 13 +2 96 + 1 55 + 1 32 + I 46-0 16 - O· 11 + 1 27 + 1

73 + 1 53 + 1 30 + I 11 + 1 35 - 0 93 +0 25 + 1 47 + 1 67 + 1 80+ 1

32 + 1 20 + 1 90+0 34-0 59+0 13 + l' 23 + 1 32 + 1 36 + 1 34 + 1

18 + l' 10 + I 41 - O· 28 - O· 62+0 12 + I" 20+ 1 25 + 1 26 + I 23 + 1

12 + I" 67+0 28 -0 28 - O· 59 + O· 10 + l' 15 + 1 18 + 1 18 + l' 15 + l'

10 + 1 49 - 0" 23 -0 28 - 0" 57 +0' 99 +0' 13 + 1 16 + 1 15 + I" 13 + I"

81 + O· 39 - O· 20-0 28 - O· 55 +0" 90+0' 12 + 1 13 + I 13 + I' 10 + I·

70+0' 32 - 0" 19 - 0 27 - O· 53 + 0" 85 +0' 11 + I 12 + I" 11 + l' 93 +0·

61 +0 28 - O· 18 - 0 27 - O· 51 + O· 80+0· 10 + 1 11 + l' 10 + 1· 81 +0'

55+0 25 - O· l7 -0 28 - O· 50+0' 77 +0' 98+0 10 + I" 97 +0· 74+0·

4.1 4.2 4.3 4.4

15 + 2 15 + 2 13 +2 94+ 1

71 + 52 + 30 + 11 +

31 + 1 17 + 1 20+ 1 99+0 92+0 41 - O· 34- 0 26 - O·

11 + I" 62+0 27.-0 25 -0

89+0 45 - O· 21 - 0 24 - O·

71 +0 35 - 0" 18 - 0 23 - O·

59+0 29 - 0" 16- 0 22 - O·

51 + 25 15 22 -

46-0 22 - O· 14 - 0 22 - O·

1.1

1 1 1 1

0 O· 0 O·

162

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no Lin

0.5

1.0

1.5

2.0

Transition s - p, parameter 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2.5

3.0

3.5

4.0

4.5

5.0

qJ

22+ 1 25 + 1 28+ 1 32+ 1 37 + 1

lO+ 1 14 + 1 17 + 1 21 + 1 25 + 1 29+ 1 34+ 1 40+ 1 47 + 1 58 + 1

11 + 1 16 + 1 21 + 1 25 + 1 30+ 1 36+ 1 42+ 1 50 + 1 61 + 1 75 + 1*

12 + 18 + 24+ 29+ 36 + 43 + 51 + 62 + 76 + 93 +

1 1 1 1 1 1 1 1 1* 1-

13 + 1 20 + 1 27 + 1 34+ 1 41 + 1 49+ 1 60+ 1 73 + 1* 89 + 111+2*

15 + 1 22 + 1 30+ 1 38 + 1 46+ 1 56+ 1 68 + 1 84 + 1* 10+2* 12 +2*

16+ 1 24+ 1 33 + 1 42+ 1 52+ 1 63 + 1 77 + 194+ 111 +2* 14+2*

17 + 1 26+ 1 36+ 1 46+ 1 57 + 1 70+ 1 85 + l' lO+212+2* 15 + 2*

1.4 1.5 1.6 1.7 1.8 1.9 2.0

26+ 1 29+ 1 33 + 1 38 + 1 47 + 1 70+ 1 57+0 45 -0 16 + 1 23 + 1

44+ 1 56 + 1 73 + 1 37 + 1 10- 015 - 0* 10 + 1 17 + 1 23 + 1 29+ 1

70+ 1 85 + 1 38 + 1* 10-0 10-0 74+0 16 + 1 25 + 1 34+ 1 45 + 1

II +2* 61 + 1* lO-O lO - 0* 37 - 0* 14 + 1 26+ 1 38 + 1 53 + 1 69 + 1*

12 +2 40 + 110-0 10 - 0* 59+018 + 1 31 + 1 45 + 1 62 + 1* 81 + 1*

12+2 27 + 1* 10 -0* 10 - 0* 79+0* 21 + 1 40 + 1* 52 + 1* 71 + 1* 92 + 1*

13 +2* 19 + 110-010 - 098+024+ 1* 40+ 1* 58 + 1* 79+ 1* 10+2*

12 +2* 14+ 1* lO-O* lO-O* 11 + 127 + 1* 45 + 1* 65 + 1* 88 + 1* 11 + 2*

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

24+ 1 28 + 1 32+ 1 57 + 1 46+ 1 63 + 1 73 + 1 lO-O 11 + 1 20+ 1

37 + 1 48 + 1 64+ 1 59+ 1 10 - 0 10 - 0 69+0 14 + 1 21 + 1 27 + 1

86 + 1* 68 + 1* 44 -0* 10 - 0 10 - O' 91 +0 20+ 1 32 + 1 46+ 1 62 + 1*

lO+2 19 + 1* 10 - 0 10-0 63 -0* 21 + 1* 38 + I' 59 + I' 82 + I' 10 + 2*

35 + 1 46+ 1 61 + 1 63 + 1 10-0 10-0 54+0 13 + 1 27 + 1 27 + 1

78 + I' 67 + 1 80+0 10 - O' lO -0* 69 +0* 17 + 1 29+ 1 42 + 1* 58 + 1*

10+2 39 + I' 10-0 10 - O· 23 - O· 14 + I' 27 + 1 43 + I' 61 + 1* 80 + I' 92 + 1* 44 + I' 10-0 10 - 0* 10-0* 11 + 1* 24+ 1 39+ 1 56 + I' 74 + I'

lO+2 23 + 1* lO-O lO-O 50 - O· 18 + I' 35 + 1* 53 + 1* 75 + 1* 97 + 1*

23 + 1 27 + 1 32 + 1 37 + 1 45 + 1 61 + 1 88+ 1 10 - O' 91 + O· 19 + 1

96+ 1 53 + 1* 10-0 10 -0' 10 -0* 11 + 1 23 + 1 37 + 1 53 + I' 7l + I' 86 + 1* 56 + 1* 21 - 0 10 - 0* 10 - O' 90+0' 20 + 1 34+ 1 49 + 1* 66 + 1*

10+2 30 + 1* 10-0 lO-O 37 - 0* 16 + 1* 31 + 1* 48 + I' 68 + 1* 89 + 1*

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

23 + 27 + 31 + 37 +

34+ 45 + 60+ 64+

63 + 71 + 25 + 10 -

73 + I' 66+ 1 10+ 1 10-0

81 + 1 56 + I' 70-0 10 -0*

87 + 1 45 + I' 10-0 10 - O'

96+ 1 n + I' 10-0 10 - 0* 24-0* 15 + 1* 30+ 1 48 + 1* 68 + 1* 89+ I' 90+ 1 91 + 1* 36 + I' 29 + 1* lO-O 10-0 10 - O' lO-O*

95 + 1 22 + 1* 10- 0 10- 0* 35 - 0* 17 + 1* 33 + 1 52 + 1* 74+ 1* 96 + 1*

4.1 4.2 4.3 4.4

56 + 1 71 + 1 56+ 1 10 -0 lO-O 33 - 0* 12 + 1 21 + 1 30+ 1 41 + 1 52 + 1 67 + 1 59 + 1 10-0 10-0 18 - 0* lO+ 1 19 + 1 28 + 1 38 + 1 57 + 1 65 + 1 61 + 1 20-0

93 + I84+ 1 49-0 10 -0* 13 - 0* 11 + 1 21 + 1 31 + 1 43 + 1 57 + 1 73 + 1* 78 + 1 17 + 1 10 - O' 10-0 67+0 16 + 1 26+ 1 38 + 1 51 + 1 67 + 1* 74+ 1 23 + 1* 10 - 0* lO-O 49 -0* 14 + 1 24+ 1 35 + 1 48 + 1

18 + 1 28+ 1 39 + 1 50+ 1 62 + 1 76+ 1 94+ 111 + 2* 14 + 2* 17 +2* 11 +2* 10 + 1* lO - 0* 10 - 0* 13 + 1* 30 + 1* 50 + 1* 71 + 1* 96 + 1* 12 +2*

1.1

1.2 1.3

1 1 1 1

1 1 1 1

1 1 1* 0*

95 + 1 34 + 1* lO-O lO- 0* 14 - 0* 13 + 1* 27 + 1 43 + 1* 62 + I' 82 + 1*

91 + 1 24+ 1* lO-O lO- 0*

6.1 Tables of Numerical Results

163

Table 6.2. (continued) The effective principal quantum number LIn

0.5

1.0

1.5

2.0

no

2.5

3.0

3.5

4.0

4.5

5.0

1 1 1 1 1

28 - 0 79+0 15 + 1 25 + 1 37 + 1 49+ 1 59 + 1 66 + 1 67 + 1 65 + 1

47 -0 11 + 1 21 + 1 32 + 1 43 + 1 53 + 1 60 + 1 63 + 1 61 + 1 54+ 1

63 +0 15 + 1 26 + 1 37 + 1 48 + 1 57 + 1 62 + 1 62+ 1 57 + 1 51 + 1

79+0 18 + 1 30 + 1 42 + 1 52 + 1 60+ 1 63 + 1 61 + 1 55 + 1 45 + 1

93+0 21 + 1 33 + 1 46+ 1 56 + 1 62 + 1 64+ 1 61 + 1 53 + 1 48 + 1

10 + 23 + 37 + 50 + 60+ 65 + 66 + 61 + 53 + 41 +

99 + 1 10+2 95 + 1 80 + 1 60 + 1 34 + 1 16 + 1 91 +0 12 + 1 23 + 1

74 + 1 63 + 1 49 + 1 33 + 1 19 + 1 11 + 1 89+0 13 + 1 21 + 1 31 + 1

54 + 1 42 + 1 29 + 1 18 + 1 11 + 1 90+0 11 + 1 17 + 1 24+ 1 29+ 1

43 + 1 31 + 1 21 + I" 13 + 1 93 +0 96+0 13 + 1 18 + 1 22 + 1 25 + 1

37 + 1 26 + I" 17 + 1 11 + 1 90+0 10 + 1 14 + 1 18 + 1 21 + 1 23 + 1

34 + I" 23 + 1 15 + 1 10 + 1 92+0 11 + 1 14+ 1 18 + 1 20 + 1 20+ 1

31 + I" 21 + I" 13 + 1 99 +0" 95 +0 11 + 1 15 + 1 18 + 1 20 + 1 20+ 1

30 + I" 20 + I" 13 + 1 98 + 0" 99+0 12 + 1 15 + 1 18 + 1 19 + 1 18 + I"

33 + 1 46 + 1 55 + 1 57 + 1 51 + 1 35 + 1 20 + 1 10 + 1 90+0 15 + 1

38 + 1 40+ 1 37 + 1 29 + 1 20+ 1 12 + 1 84+0 90+0 14 + 1 22 + 1

30 + 1 28+ 1 23 + 1 16 + 1 11 + 1 80+0 83+0 11 + 1 16 + 1 21 + 1

24+ 1 21 + 1 16 + 1 11 + 1 83 +0 71 + 0 85 + 0 11 + 1 15 + 1 18 + 1

21 + 1 17 + I" 13 + 1 95 +0 72+0 70+0 88 +0 11 + 1 14 + 1 16 + 1

18 + 1 15 + I" 11 + I" 83 +0 67+0 71 + 0 90+0 11 + 1 13 + 1 14 + 1

17 + I" 13 + I" 10 + I" 75 +0 65+0 72+0 91 +0 11 + 1 13 + 1 13 + 1

16 + I' 12 + I" 94+0" 71 +0 64+0 73+0 92+0 11 + 1 12 + 1 12 + I"

Transition s - d, parameter C 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

39 + 53 + 66 + 75 + 78 +

Ll

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

1 1 1 1 1 1 1 1 I" I"

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

26+2 14+2 60 + 1 15 + 1 83 - 1 29 - 0 11 + 1 19 + 1 23 + 1 23 + 1

14+2 11 +2 75 + 1 40 + 1 16 + 1 40 -0 14 + 1 38 + 1 62 + 1 79 + 1 83 + 1 75 + 1 58 + 1 37 + 1 19 + 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

12 + 2 78 + 1 37 + 1 11 + 1 11- 0 13 -0 66+0 12 + 1 16 + 1 17 + 1

50+0 85 +0 24+ 1 43 + 1 59 + 1 66+ 1 63 + 1 52 + 1 36 + 1 20+ 1

22 + 1 34 + 1 44 + 1 49 + 1 47 + 1 34+ 1 21 + 1 11 + 1 83 +0 12 + 1

29 + 1 33 + 1 32 + 1 27 + 1 20 + 1 13 + 1 85 +0 79+0 11 + 1 19 + 1

23 + 1 23 + 1 20+ 1 15 + 1 10 + 1 77+0 73 +0 96+0 13 + 1 18 + 1

18 + 1 17 + 1 14 + 1 10 + 1 77+0 63 +0 70+0 94+0 12 + 1 15 + 1

16 + 1 14 + 1 11 + 1 85 +0 64+0 59+0 70+0 93 +0 11 + 1 13 + 1

14 + 1 12 + I" 96+0' 72+0 57+0 57+0 69+0 89+0 10 + 1 12 + 1

12 + I" 10 + I' 84+0' 64+0 64+0 56+0 69+0 87+0 10 + 1 11 + 1

11 + I" 98 + 0' 76+0" 59+0 51 +0 55+0 68+0 84+0 97+0 10+ 1

4.1 4.2 4.3 4.4

97 + 1 61 + 1 30+ 1 10 + 1

58+0 66+0 18 + 1 35 + 1

17 + 28 + 39 + 45 +

25 + 29 + 30 + 26 +

20+ 21 + 19 + 15 +

16 + 15 + 13 + 10 +

13 + 1 12 + 1 10 + 1 79+0

11 + 1 10 + I' 86+0 66+0

10 + 1 92+0' 74+0' 58+0

96+0' 83 + 0' 66+0' 52+0

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

164

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no Lin

0.5

1.0

2.5

3.0

3.5

4.0

72+0 78+0 84+0 91 + 0 10 + I

81 + 0 62+0 63 +0 68+0 74+0 82+0 90+0 10 + 1 11 + 1 14+ 1

70+0 61 +0 66+0 74+0 83 +0 92+0 10 + 1 11 + 1 13 + 1 15 + 1

64+0 62+0 70+0 80+0 91 + 0 10 + 1 11 + 1 13 + 1 15 + 1 22+ I

88+0 94+0 10 + 1 11 + 1 14 + I

85 +0 91 + 0 99+0 10 + I 12 + 1 15 + 1 18 + I 13 + I 56+0 50+0

II + I 12 + 1 14 + 1 17 + I 20+ I 19 + I 89+0 46 -0 48 -0 61 +0

14+ I 16 + 1 19 + 1 22 + I 20 + I 98+0 49-0 47 - 0 60+0 77 +0

17 + 1 21 + 1 24 + I· 24 + 1 15 + I 64+0 45 -0 54+0 71 + 0 93 +0

21 + I 25 + I· 28 + 1 23 + 1 11 + I 53+0 48-0 62+0 83+0 11 + I

1.5

2.0

4.5

5.0

Transition s - d, parameter cp 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

84+0 86+0 90+0 98+0 16 + I 66+0 78+0 85 +0 90+0 96+0

14 + 1 59+0 65+0 71 +0 77+0 84+0 92+0 10 + 1 11 + 1 13 + 1

57 +0 68+0 80+0 92 +0 10 + I 12 + I 16 + I 16 + 1 83 +0 48 -0

75+0 92+0 11 + 1 13 + 1 16 + 1 18 + I 14 + I 62+0 44-0 52+0

96+0 11 + I 14 + 1 17 + 1 19 + 1 14 + 1 67+0 45 - 0 50+0 65+0

11 + 1 14 + I 18 + 1 21 + I 18 + I 10 + I 52 +0 46-0 58+0 76+0

14 + 1 17 + I· 21 + I 22+ 1 15 + 1 76+0 47 -0 50+0 66+0 92+0

62+0 60+0 60+0 64+0 66+0 69+0 74+0 79+0 83 +0 86+0 93 +0 99+0 99+0 10 + 1 11 + 1 11 + I 12 + 1 13 + 1 13 + 1 14 + 1 16 + 1 15 + 1 17 + 1 19 + 1 17 + 1 20 + 1 22 + I· 21 + 1 34+ 1 27 + I· 25 + I· 28 + I· 32 + I· 29 + I 33 + I· 36 + I· 30 + 1 31 + I 31 + 1 20+ I 17 + I· 15 + I' 87+0 74+0 66+0 49 - 0 48 - 0 49-0 52+0 57+0 63 +0 70+0 80+0 90+0 96+0 11 + 1 12 + 1 12 + 1 15 + 1 16 + I· 16 + 1 19 + I· 21 + I· 20 + I" 23 + I" 26 + I" 24 + I· 26 + I" 27 + I· 22 + 1 21 + 1 20+ 1 13 + 1 II + 1 97+0 63 +0 56+0 53 +0 46 -0 48 -0 51 + 0 55 +0 62+0 68+0 76+0 86+0 97 +0 10 + 1 12 + 1 13 + I"

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

86+0 89+0 94+0 10 + 1 16 + 1 57 +0 75+0 83 +0 90+0 96+0

18 + I 62+0 59+0 67+0 74+0 82+0 90+0 10 + I 11 + 1 13 + 1

49 -0 61 + 0 73 +0 86+0 10 + I 12 + I 15 + I 17 + I 10 + I 49-0

65 +0 81 + 0 99+0 12 + I 14 + 1 17 + I 15 + 1 76+0 45 -0 49-0

82+0 10 + 1 13 + I 16 + 1 18 + 1 15 + I 80+0 47 -0 47 -0 60+0

99+0 12 + I 16 + I 19 + 1 18 + 1 11 + 1 59+0 45 -0 53 +0 69+0

II + 1 15 + I 18 + 1 20+ 1 17 + 1 91 +0 51 + 0 47-0 59+0 82+0

13 + I 17 + I· 21 + I" 21 + I 14 + 1 75 +0 48 -0 50+0 67+0 92+0

4.1 4.2 4.3 4.4

87+0 90+0 96+0 10+ 1

19 + 1 68+0 57+0 64+0

47 -0 57+0 70+0 83 + 0

61 +0 76+0 94+0 11 + 1

76+0 97+0 12 + I 15 + 1

90+0 11 + 1 14 + 1 17 + 1

10 + 13 + 17 + 19 +

12 + 16 + 19 + 20+

1.1

1.2 1.3

1.4 1.5 1.6 1.7 1.8 1.9 2.0

I I I 1

I I" 1 1

15 + I· 17 + I" 20 + I· 22 + I" 23 + I" 24 + I' 21 + I 20+ 1 13 + 1 11 + I 65 +0 59+0 47 -0 48 -0 55 +0 60+0 76+0 85 + 0 10 + 1 II + 1 14 + 1 15 + I" 18 + I" 20 + I" 21 + I" 23 + I" 21 + 1 20+ 1

6.1 Tables of Numerical Results

165

Table 6.2. (continued) The effective principal quantum number LIn

1.5

2.0

2.5

3.0

n~

4.0

4.5

5.0

1 1 1 1 1 1 I' I' 1 O·

29 + 1 51 + 1 66+ 1 72 + 1 70 + 1 62 + I' 48 + I' 33 + I' 19 + 1 89 +0'

33 + 1 57 + 1 73 + 1 79+ 1 77 + 1 67 + I' 53 + I' 36 + I' 21 + I' 97 +0'

36+ 1 62 + 1 79+ 1 86 + 1 84+ 1 73 + I' 57 + I' 39 + I' 23 + I' 10 + l'

3.5

Transition p - s, parameter C 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

83 +0 15 + 1 21 + 1 24+ 1 24+ 1 21 + 1 16 + 1 11 + 1 58 +0 24-0

13 + 25 + 33 + 37 + 37 + 33 + 26 + 18 + 10 + 47 -

Ll 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

15 - 0 21 - O· 41 - O· 64+0' 81 + 0 85 +0 75 +0 53 +0 30-0 17 - 0

21 - 0 25 - O· 47 - O· 77 + O· 10 + l' 11 + 1 10+ 1 83 +0 55 +0 29 - O'

26- 0 27 - O· 51 + O· 83 + O· 11 + 1 12 + 1 11 + 1 91 + 0 60+0 31 - O·

30 -0 29 - O' 53 + O· 87 + O· 11 + 1· 13 + 1 12 + l' 10 + I· 68+0 36 - O·

33 - 0 31 - O· 55 + O· 90 +0' 12 + l' 13 + l' 13 + l' 10 + I' 71 +0 38 - O·

36-0 32 - O· 56 + O· 93 + O· 12 + 1· 14 + l' 13 + l' 11 + l' 76+0 41 - O·

39- 0 33 - O· 58 + O· 96+0· 13 + I' 14 + I' 14 + I' 11 + l' 79+0 42 - O·

42 -0 35 - O· 60+0' 99 +0· 13 + I· 15 + I' 14 + I' 12 + I· 83 +0 45 - O·

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

16 - 0 18 - O· 33 - O· 53 - O· 71 + O· 81 + 0 78+0 63+0 43 -0 24 - O·

18 - 0 19 - O· 33 - O· 54+0' 73 +0' 83+0 80+0 65+0 43 -0 24 - O·

19 - 0 19 - O· 32 - O· 52 + O· 72 +0' 83 +0 82+0' 69 +0' 48 - 0 27 - O·

20-0 19 - O· 32 - O· 52 + O· 71 + O· 83 +0 82 +0· 69 +0' 48 -0 27 - O·

21 - 0 19 - O· 31 - O· 51 + O· 71 + O· 83 +0' 83 + O· 70 +0· 50 + O· 28 - O·

22- 0 19 - O· 31- O' 51 + O· 70+0' 83 + O· 83 +0· 70+0' 50+0 28 - O·

22- 0 19 - O· 31 - O· 51 + O· 70+0' 83 +0· 83 + O· 72 +0' 51 + 0 29 - O'

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

12 - O· 17 - O· 32 - O· 49 - O· 63 +0 66+0 58+0 41 -0 23 -0 15 - 0 12 - O· 16 - O· 29 - O· 44 - O· 56+0 59+0 51 + 0 36-0 20- 0 14 - 0

14 - 0 16 - O· 28 - O· 45 - O· 61 + 0 70+0 68+0 56+0 38-0 21 - O·

15 - 0 16 - O· 27 - O· 44- O· 60+ O· 68+0 66+0 54+0 37 - 0 21- 0

15 - 0 15 - O· 25 - O· 41 - O· 57 + O· 67 +0 66+0 56+0 40-0 23 - O·

15 - 0 15 - O· 24 - O· 40 - O· 55 + O· 64+0 64+0 55 + O· 39 - 0 22 - O·

16 - 0 15 - O· 23 - O· 38 - O· 53 + O· 63 +0 63 +0· 55 + O· 39- 0 23 - O·

16 - 0 14 - O· 23 - O· 37 - O· 51 + O· 61 + 0 62+0' 53 + O· 39-0 23 - O·

16 - 0 14- O· 22 - O· 36 - O· 50+0' 60+0 61 + O· 53 + O· 39-0 23 - O·

4.1 4.2 4.3 4.4

11 15 27 42

13 -0 15 - O· 26 - O· 42 - O·

14- 0 15 - O· 25 - O· 40 - O·

14 14 22 36 -

13 13 21 34 -

13 -0 12 - O' 20 - O· 32 - O·

13 -0 12 - O· 19 - O· 30 - O·

13 -0 12 - O· 18 - O· 29 - O·

-

O· O· O· O·

1 1 1 1 1 1 1 1 1 O·

18 + 32 + 43 + 48 + 47 + 42 + 33 + 22 + 13 + 58 +

1 1 1 1 1 1 1 I' l' O·

22 + 39 + 51 + 57 + 56 + 49 + 39 + 27 + 15 + 71 +

1 1 1 1 1 1 I' I' 1 O·

0 O· O· O·

26 + 45 + 59 + 65 + 63 + 56 + 44 + 30 + 17 + 80 +

0 O' O· O·

166

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number An

1.5

3.0

3.5

4.0

4.5

5.0

1 1 1 1 1 1 1 I· 1* I·

14 + 1 21 + 1 29 + 1 37 + 1 46+ 1 58 + 1 72 + 1* 88 + I· 10+2 66 + 1*

15 + 1 23 + 1 32 + 1 41 + 1 51 + 1 64+ 1 80 + I· 99 + 1* 11 +2 72 + 1*

16 + 1 25 + 1 34+ 1 45 + 1 57 + 1 71 + I· 89 + I· 11 +2· 12 +2 84+ 1*

17 + 1 27 + 1 37 + 1 49+ 1 62+ 1 78 + I· 97 + I· 12 +2· 14+ 2· 90 + 1*

18 + 1 29+ 1 40+ 1 53 + 1 67 + 1 84 + I· 10+2· 13 +2· 15 +2* 98 + 1*

10-0 10- O· 10-0* 66+0· 15 + 1· 25 + 1 36+ 1 47 + 1 51 + 1 27 + 1·

29-0 10 - 0* 10 - 0* 69+0* 17 + 1 28 + 1 41 + 1 54+ 1 57 + 1 27 + 1·

52+0 10 - 0* 10 - 0* 72+0· 18 + 1· 31 + 1 46 + I· 61 + 1· 68 + 1 40 + 1·

70+0 10-0· 10 - O· 77+0· 20 + 1* 34 + 1* 50 + 1· 67 + 1· 75 + 1 42 + 1·

86+0 10 - 0* 10 -0* 82 +0* 21 + 1* 37 + 1* 55 + 1· 74 + 1· 85 + 1 51 + 1*

99+0 10- 0* 10- 0* 88+0· 23 + 1* 40 + 1· 59 + I· 80 + I· 92 + 1 54 + 1·

11 + 1 10 - 0* 10-0* 94+0· 25 + 1* 43 + 1· 64+ 1* 87 + I· 10+2 60 + 1*

10-0 10- O· 10 - O· 43 - 0* 12 + 1* 22+2 32+ 1 42+ 1 44+ 1 22 + 1*

10 - 0 10 - O· 10 - O· 41 - O· 13 + 1· 24+ 1 35 + 1 46+ 1 47 + 1 19 + 1·

28-0 10 - O· 10 - O· 41 - O· 14 + I· 26+ 1 39 + 1· 52 + I· 57 + 1 33 + I·

42 -0 10 - O· 10 - O· 42 - 0* 15 + 1· 28 + 1 43 + 1* 57 + I· 62 + 1 33 + I·

57+0 10 - O· 10 - O· 44 - 0* 16 + I· 20 + 1* 46 + 1* 63 + 1· 70+ 1 41 + 1·

68+0 10 - 0* 10 - 0* 46 - 0* 17 + 1· 32 + 1* 50 + I· 68 + I· 76+ 1 42 + I·

79+0 10 - O· 10 - 0* 49 - 0* 18 + I· 34 + 1* 53 + 1· 73 + I· 83 + 1 48 + I·

10-0 10 - O· 10 - O· 34- O· 11 + 1 20+ 1 30+ 1 39+ 1 41 + 1 20+ 1·

10 - 0 10 - 0* 10 - O· 32 - O· 12 + 1* 22+ 1 33 + 1 43 + 1 42 + 1 16 + 1 10 - 0 10 - 0* 10 - O· 27 - O·

17 - 0 10 - O· 10 -0· 30 - 0* 12 + 1· 23 + 1 36+ 1 48 + 1 52+ 1 29 + 1· 12 - 0 10 - O· 10 - O· 25 - O·

30-0 10 -0· 10 - O· 30 - O· 13 + I· 25 + 1 39+ 1 53 + 1 56+ 1 28 + 1· 23 -0 10 -0· 10-0· 24-0·

43 -0 10 - O· 10 - O· 31- 0* 14 + I· 27 + 1 42+ I· 57 + I· 64+ 1 36 + 1· 35 - 0 10 -0· 10 -0· 24-0·

53+0 10 - O· 10 - O· 32 - O· 14+ I· 29+ 1 45 + 1· 62 + I· 68 + 1 37 + 1· 45 -0 10 - O· 10 -0· 24-0·

63+0 10- O· 10 - O· 33 - O· 15 + 1· 30+ 1 48 + I· 66+ I· 74+ 1 43 + 1·

2.0

2.5

Transition p - s, parameter 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ll

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

10+ 1 14 + 1 19 + 1 24+ 1 30+ 1 37 + 1 46+ 1 55 + 1 52 + 1 11 + 1 10-0 10- O· 10- O· 67 +0· 14 + 1 23 + 1 32 + 1 38 + 1 27 + 1 10-0 10- O· 10 - O· 10- O· 49-0* 12 + 1 20+ 1 28 + 1 33 + 1 19 + 1 10-0 10 - O· 10- 0* 10-0· 42 -0· 11 + 1 19 + 1 27 + 1 30 + 1 14 + 1 10-0 10-0* 10 - O· 10 - O· 39- O·

no

11 + 17 + 22+ 28+ 35 + 44+ 54+ 66+ 74+ 45 +

1 1 1 1 1 1 1 1 1 1·

10 - 0 10- 0* 10- O· 30- O·

12 + 19 + 25 + 33 + 41 + 51 + 63 + 78 + 87 + 49 +

qJ

54+0 10 - O· 10 - O· 25 - O·

167

6.1 Tables of Numerical Results Table 6.2. (continued) The effective principal quantum number no LIn

2.0

1.5

3.0

3.5

4.0

4.5

5.0

16+2 15 +2 13 +2 10+2 73 + 1 44+ 1 26+ 1 19 + 1 22+ 1 33 + 1 45 + 1 54+ 1 56+ 1 51 + 1 41 + 1

19+2 17 + 2 14+2 11 +2 77 + 1 46+ 1 27 + 1 20+ 1 24+ 1 35 + 1 47 + 1 55 + 1 56 + 1 51 + 1 40+ 1

21 +2 19 +2 16+2 12 +2 81 + 1

23 +2 21 +2 17+2 12 +2 86+ 1

27+2 24+2 19 +2 14+2 96+ 1

49+ 1 28 + 1 21 + 1 25 + 1 36+ 1 48 + 1 56+ 1 57 + 1 51 + 1 40+ 1

51 + 1 30+ 1 22+ 1 27 + 1 38+ 1 50+ 1 58 + 1 58 + 1 51 + 1 40+ 1

25+2 22+2 18+2 13+2 91 + 1 54+ 1 31 + 1 23 + 1 28+ 1 40+ 1 52+ 1 59+ 1 59+ 1 52+ 1 41 + 1

28 + 19 + 14 + 16 + 22 + 30 + 36 + 39 + 36 + 30 +

28 + 19 + 14 + 15 + 22 + 29 + 35 + 38 + 35 + 29 +

1 1 1 1 1 1 1 1 1 1

28+ 1 19 + 1 14 + 1 16 + 1 22 + 1 29+ 1 35 + 1 37 + 1 34+ 1 28+ 1

28 + 1 19 + 1 14 + 1 16 + 1 22+ 1 29+ 1 35 + 1 37 + 1 34+ 1 28 + 1

29+ 1 19 + 1 14+ 1 16 + 1 22+ 1 29+ 1 35 + 1 36+ 1 34+ 1 27 + 1

21 + 1 14 + 1 11 + 1 12 + 1 16+ 1 22 + 1 27 + 1 29+ 1 27 + 1 23 + 1

20+ 1 14 + 1 11 + 1 12 + 1 16 + 1 21 + 1 26+ 1 28 + 1 27 + 1 22 + 1

20+ 14 + 11 + 11 +

21 + 1 15 + 1 11 + 1 12 + 1 17 + 1 23 + 1 28 + 1 30+ 1 29+ 1 25 + 1 18 + 1 13 + 1 10 + 1 11 + 1

17 + 1 12 + 1 10+ 1 10+ 1

17 + 1 12 + 1 96+0 10+ 1

20+ 1 14+ 1 11 + 1 11 + 1 16 + 1 21 + 1 25 + 1 27 + 1 26+ 1 22+ 1 16 + 1 12 + 1 93+0 99+0

2.5

Transition p - p, parameter C ,,=0 0.6 0.7 0.8 0.9 1.0

10+2 11 +2 10+2 87 + 1 65 + 1

14+2 13 +2 12+2 96+ 1 67 + 1

1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

39 + 22+ 14+ 16 + 26+ 39 + 51 + 58 + 57+ 49+

1 1 1 1 1 1 1 1 1 1

42 + 25 + 18 + 21 + 31 + 43 + 52+ 55 + 51 + 41 +

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

31 + 20+ 13 + 13 + 20+ 31 + 42+ 49+ 50+ 45 +

1 1 1 1 1 1 1 1 1 1

30+ 1 20+ 1 15 + 1 16 + 1 22 + 1 31 + 1 39+ 1 43 + 1 41 + 1 34+ 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

29+ 1 19 + 1 13 + 1 12 + 1 18 + 1 28+ 1 39+ 1 46+ 1 48+ 1 44+ 1 28+ 1 19 + 1 13 + 1 12 + 1

26+ 1 18 + 1 14 + 1 14 + 1 19 + 1 27 + 1 34+ 1 38 + 1 37 + 1 31 + 1

29+ 1 19 + 1 14 + 1 15 + 1 22+ 1 30+ 1 38 + 1 41 + 1 39 + 1 33 + 1 24+ 1 16 + 1 12 + 1 13 + 1 18 + 1 25 + 1 32 + 1 35 + 1 34+ 1 29 + 1

24+ 17 + 13 + 13 +

21 + 15 + 11 + 12 +

1 1 1 1 1 1 1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1 1 1 1 1 1 1

22 + 1 16 + 1 12 + 1 13 + 1 17 + 1 24+ 1 29+ 1 32+ 1 31 + 1 26+ 1 1 1 1 1

57 + 32 + 24+ 29+ 41 + 54+ 61 + 61 + 53 + 41 +

1 1 1 1 1 1 1 1 1 1

168

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no LIn

1.5

2.0

2.5

Transition p - P. parameter K=O

3.0

3.5

4.0

4.5

5.0

qJ

0.6 0.7 0.8 0.9 1.0

90+0 84+0 81 +0 80+0 82+0

77+0 77+0 80+0 87+0 10 + 1

78+0 82+0 90+0 10 + 1 12 + 1

83+0 90+0 10 + 1 11 + 1 14 + 1

88+0 97+0 11 + 1 13 + 1 16 + 1

94+0 10+ 1 12 + 1 14 + 1 19 + 1

99+0 11 + 1 13 + 1 16+ 1 22+ 1

10 + 12 + 14 + 18 + 24+

1.1

94+0 13 + 1 20+ 1 17 + 1 11 + 1 85+0 74+0 70+0 69+0 72+0 82+0 11 + 1 17 + 1 17 + 1 11 + 1 86+0 73+0 68+0 67+0 69+0

12 + 1 16 + 1 15 + 1 92+0 64+0 57+0 58+0 62+0 71 +0 84+0

15 + 1 18 + 1 14+ 1 74+0 56+0 55+0 60+0 69+0 81 +0 99+0 12 + 1 15 + 1 14 + 1 83 +0 57+0 53 +0 56+0 64+0 75+0 92+0

18 + 1 21 + 1 14 + 1 70+0 54+0 56+0 63 +0 75 +0 91 +0 11 + 1

22 + 1 25 + 1 14 + 1 67+0 53 +0 57+0 67+0 81 +0 10 + 1 13 + 1

26+ 1 28+ 1 14+ 1 64+0 52+0 58+0 71 +0 88+0 11 + 1 15 + 1

29+ 1 31 + 1 13 + 1 61 +0 51 +0 60+0 75 +0 96+0 12 + 1 17 + 1

34+ 1 34+ 1 13 + 1 59+0 52+0 62+0 80+0 10+ 1 13 + 1 19 + 1

15 + 1 18 + 1 15 + 1 81 +0 56+0 52+0 58+0 68+0 83 +0 10 + 1

17 + 1 21 + 1 15 + 1 77+0 53 +0 52+0 60+0 73+0 92+0 12 + 1

20+ 1 24+ 1 16+ 1 74+0 51 +0 52+0 63 +0 79+0 10 + 1 13 + 1

23 + 1 27 + 1 16 + 1 72+0 50+0 53 +0 66+0 85+0 11 + 1 15 + 1

26+ 1 30+ 1 16 + 1 70+0 50+0 55 +0 70+0 92+0 12 + 1 17 + 1

78+0 10+ 1 16+ 1 17 + 1 12 + 1 87+0 73+0 67+0 66+0 68+0 76+0 10+ 1 12 + 1 17 + 1

10 + 1 13 + 1 13 + 1 97+0 65+0 54+0 53 +0 57+0 64+0 76+0

11 + 1 14+ 1 13 + 1 87+0 59+0 52+0 55+0 62+0 73 +0 89+0 11 + 1 14 + 1 13 + 1 88+0

13 + 1 17 + 1 15 + 1 86+0 57+0 51 + 0 56+0 65+0 80+0 10 + 1

16 + 1 19 + 1 15 + 1 83 +0 54+0 51 + 0 57+0 69+0 87+0 11 + 1

21 + 1 25 + 1 17 + 1 78+0 51 +0 51 +0 62+0 80+0 10+ 1 14 + 1

23 + 1 27 + 1 17 + 1 76+0 50+0 52+0 65+0 86+0 11 + 1 16+ 1

13 + 1 16 + 1 15 + 1 89+0

15 + 1 19 + 1 15 + 1 85+0

18 + 1 22+ 1 16 + 1 80+0 52+0 51 +0 59+0 74+0 96+0 12 + 1 17 + 1 21 + 1 16 + 1 83 +0

19 + 1 23 + 1 17 + 1 81 +0

22+ 1 26+ 1 17 + 1 80+0

1.2 1.3

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 ~.2

3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4·

10 + 1 13 + 1 14 + 1 96+0 64+0 55+0 54+0 58+0 66+0 79+0

97+0 12 + 1 13 + 1 97+0

1 1 1 1 1

169

6.1 Tables of Numerical Results Table 6.2. (continued) The effective principal quantum number no LIn

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Transition p - p, parameter C

K=2 O.l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

19 - 0 47 - 0 83+0 12 + 1 15 + 1 18 + 1 19 + 1 18 + 1 15 + 1 12 + 1

42 -0 93 +0 14 + 1 19 + 1 23 + 1 25 + 1 25 + 1 22 + 1 19 + 1 14 + 1

59+0 12 + 1 19 + 1 25 + 1 29 + 1 30 + 1 29 + 1 26 + 1 21 + 1 16 + 1

76+0 15 + 1 23 + 1 30 + 1 34 + 1 35 + 1 33 + 1 29 + 1 23 + 1 17 + 1

91 +0 18 + 1 27 +11 34 + 1 38 + 1 39 + 1 37 + 1 31 + 1 25 + 1 18 + 1

10+ 1 21 + 1 31 + 1 38 + 1 43 + 1 43 + 1 40 + 1 34+ 1 27 + 1 19 + 1

12 + 1 24+ 1 35 + 1 43 + 1 47 + 1 47 + 1 43 + 1 37 + 1 29 + 1 21 + 1

13 + 26+ 38 + 47 + 51 + 51 + 47 + 39 + 31 + 22 +

1.1

1.4 1.5 1.6 1.7 1.8 1.9 2.0

84+0 57+0 43 -0 44-0 59+0 79+0 99+0 11 + 1 10 + 1 96+0

10 + 1 73 +0 60+0 64+0 79+0 99+0 11 + 1 12 + 1 11 + 1 97 +0

11 + 1 80+0 67 +0 72+0 88+0 10 + 1 12 + 1 12 + 1 11 + 1 10 + 1

12 + 1 86+0 72 +0 78 +0 94+0 11 + 1 12 + 1 12 + 1 11 + 1 10 + 1

12 + 1 91 +0 76+0 83 +0 10 + 1 11 + 1 13 + 1 13 + 1 12 + 1 10 + 1

13 + 1 95 +0 81 + 0 87 +0 10 + 1 12 + 1 13 + 1 13 + 1 12 + 1 10+ 1

14 + 1 10 + 1 84+0 92+0 11 + 1 12 + 1 14 + 1 14 + 1 12 + 1 10 + 1

15 + 1 10 + 1 88+0 96+0 11 + 1 13 + 1 14 + 1 14 + 1 13 + 1 11 + 1

2.l 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

70+0 52+0 40- 0 39-0 50 + 0 67+0 84+0 96+0 98+0 90+0

78+0 62 +0 52 +0 53 +0 64+0 78+0 91 +0 98+0 95 +0 84+0

80+0 63 +0 55 +0 56+0 66+0 80+0 91 + 0 97 +0 94+0 83 +0

82+0 65 + 0 56 + 0 58+0 67 +0 80+0 91 +0 95 +0 91 +0 81 + 0

83 +0 65 +0 57+0 59+0 68+0 81 +0 91 + 0 94+0 90+0 80+0

84+0 66+0 57 +0 59+0 69+0 81 +0 90+0 94+0 89+0 79+0

86+0 67 +0 58 + 0 60+0 70+0 82+0 91 +0 94+0 89+0 78+0

87+0 68+0 59+0 61 +0 71 +0 83 +0 91 +0 94+0 89+0 78+0

3.l 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

66+0 58 + 0 39 - 0 38 - 0 47 -0 63+0 79+0 92+0 95+0 87+0

70+0 57 +0 49 -0 50+0 58+0 71 + 0 83 + 0 89+0 87+0 78+0

69+0 56+0 49 -0 50+0 58 + 0 69+0 80+0 85 +0 84+0 76+0

68+0 56+0 49 -0 50+0 57 +0 68 + 0 77 +0 81 + 0 79+0 72+0

67 +0 55 +0 48 -0 49 - 0 56+0 66+0 75 +0 79+0 77+0 69+0

66+0 54+0 48 -0 49 - 0 56+0 65 +0 73 +0 76+0 74+0 67+0

65 +0 53 + 0 47 - 0 48 -0 55 +0 64+0 72+0 75+0 72+0 65 +0

65 +0 53 +0 47 -0 48 - 0 55+0 63 +0 71 +0 74+0 71 +0 64+0

4.l 4.2 4.3 4.4

64+0 49-0 39-0 37 - 0

66+0 54+0 47 -0 48 - 0

63 +0 53 + 0 47 - 0 47 -0

61 51 45 46

59+0 49 -0 44-0 45 - 0

57 +0 48 -0 42 -0 43 -0

55 +0 46-0 41 - 0 42 -0

54+0 45 -0 40 -0 41 - 0

1.2 1.3

+0 +0 -0 - 0

1 1 1 1 1 1 1 1 1 1

170

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no LIn.

1.5

2.0

2.5

Transition p - p, parameter ,,=2

3.0

3.5

4.0

4.5

5.0

qJ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

18 + 1 10 + 1 79+0 70+0 68+0 70+0 75+0 84+0 96+0 11 + 1

13 + 1 75 +0 64+0 65 +0 71 + 0 79+0 91 + 0 10 + 1 12 + 1 14 + 1

98 +0 64+0 63 + 0 70+0 79+0 91 +0 10 + 1 12 + 1 14 + 1 17 + 1

81 + 0 61 + 0 66+0 76+0 88 + 0 10 + 1 12 + 1 14 + I 17 + I 20 + 1

71 + 0 61 + 0 70+0 83 +0 98 +0 11 + 1 13 + 1 16 + 1 20 + 1 24 + 1

66+0 63 +0 75 + 0 89+0 10 + 1 12 + 1 15 + 1 19 + 1 23 + 1 27 + 1

62+0 65 +0 79+0 97+0 11 + 1 14 + 1 17 + 1 22 + 1 27 + 1 32 + 1

60+0 68 +0 84+0 10 + 1 12 + 1 15 + 1 19 + 1 24+ 1 30 + 1 35 + 1

1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

14 + I 16 + 1 12 + 1 72+0 49-0 46-0 51 + 0 61 + 0 73 +0 90+0

16 + 1 14 + 1 87+0 51 + 0 42 -0 47 -0 58+0 72+0 90+0 10 + 1

18 + I 14 + 1 84+0 49 -0 43 - 0 49 -0 63 +0 80+0 10 + 1 12 + 1

21 + 1 15 + 1 83 +0 48 - 0 43 -0 53 +0 69+0 91 + 0 11 + I 14 + 1

24 + 1 16 + 1 82+0 47 - 0 45 - 0 57+0 77+0 10 + I 13 + 1 16 + 1

26 + 1 17 + 1 80+0 47 -0 47 -0 61 +0 85 +0 11 + 1 15 + 1 18 + 1

29 + I 17 + 1 79+0 47 -0 49 - 0 66+0 94+0 13 + 1 17 + 1 21 + 1

32 + 1 17 + 1 78+0 48 -0 52+0 72+0 10 + 1 14 + 1 19 + 1 23 + 1

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

11 + 1 13 + 1 12 + 1 78+0 58+0 44-0 48 - 0 56+0 68+0 83 +0

12 + 1 12 + 1 88+0 56+0 43 -0 44 -0 52 +0 65 +0 81 + 0 98 +0

14 + 1 12 + 1 88 +0 55 + 0 43 -0 45 - 0 55 +0 70+0 89+0 11 + 1

16 + 1 13 + 1 89+0 54+0 43 -0 47 - 0 60+0 78 +0 10 + 1 12 + 1

18 + 1 14 + 1 90+0 53 +0 43 - 0 49 -0 65 +0 87+0 11 + 1 14 + 1

20+ 1 15 + 1 89+0 52+0 44 - 0 52 +0 71 + 0 97 +0 12 + 1 16 + 1

22 + 1 16 + 1 89+0 52+0 45 -0 56+0 77+0 10 + 1 14 + 1 18 + 1

24 + 1 16 + 1 88+0 51 + 0 46-0 59+0 84+0 11 + 1 16 + 1 20 + 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

10 + 1 12 + 1 12 + 1 79+0 51 +0 44-0 47 - 0 54+0 66+0 80+0

11 + 1 11 + 1 87+0 58+0 44-0 43 -0 50+0 62+0 77+0 94+0

12 + 1 12 + 1 88 +0 57+0 43 -0 44-0 52 + 0 66+0 84+0 10 + 1

14 + 1 13 + 1 90+0 56+0 43 -0 45 -0 56+0 73 +0 95 +0 11 + 1

15 + 1 13 + 1 91 + 0 55 +0 43 - 0 47 -0 60+0 81 + 0 10 + 1 13 + 1

17 + I 14 + 1 91 + 0 55 +0 44 - 0 49 -0 65 +0 89+0 11 + 1 14 + 1

19 + I 15 + 1 92+0 54+0 44-0 52 +0 70+0 98 +0 13 + 1 16 + 1

21 + 1 16 + 1 91 +0 54+0 45 -0 55 +0 76+0 10 + 1 14 + 1 18 + 1

4.1 4.2 4.3 4.4

10 + 1 12 + 1 11 + 1 80+0

10 + 1 10 + 1 86+0 58+0

11 + 1 11 + 1 87 + 0 58+0

13 + 1 12 + 1 90+0 58 +0

14 + 1 13 + 1 92+0 57 +0

16 + 1 14 + 1 92+0 56+0

17 + 1 14 + 1 92+0 55 +0

19 + 1 15 + 1 93 +0 55 +0

6.1 Tables of Numerical Results

171

Table 6.2. (continued) The effective principal quantum number no ,1n

3.0

3.5

4.0

4.5

5.0

22+2 27+2 30+2 30+2 28+2

24+ 1 58 + 1 10 + 2 15 + 2 20+2 23 +2 25 +2 25 +2 23 +2 19 + 2

37 + 1 80 + 1 13 +2 17 +2 22+2 24+2 25 +2 24+2 20 +2' 15 + 2'

47 + 1 98 + 1 15 + 2 19 + 2 23 +2 25 +2 25 +2 22 +2' 18 + 2' 13 + 2'

57 + 1 11 +2 16 + 2 21 +2 24+2 26+2 25 + 2' 21 + 2' 17 + 2' 11 + 2'

65 + 1 12 +2 18 + 2 23 +2 26+2 26+2 25+2* 21 + 2' 15 + 2' 99 + I'

73 + 1 13 +2 19 + 2 24+2 27 +2 27 +2' 25 + 2' 20 +2' 14 + 2' 89 + I'

1.4 1.5 1.6 1.7 1.8 1.9 2.0

34+2 31 + 2 26+2 19+2 12 +2 50 + I' 99+0 76 +0' 31 + 1 68 + 1

23+2 16+2 94 + I' 40+ 1 10 + 1 14 + l' 38 + I' 73 + 1 11 +2 13+2

14 + 2' 86 + I' 40+ 1 10 + I' 10 + I' 25 + I' 49+ 1 73 + 1 89 + 1 93 + 1

10 + 55 + 17 + 83 + 16 + 34 + 55 + 74 + 82 + 78 +

2' I' I' 0' I' I' 1 1 1 1

79 + I' 39 + 1 10 + I' 96 +0' 20 + I' 37 + 1 55 + 1 68 + 1 71 + I' 63 + I'

67 + I' 26 + I' 91 + 0' 11 + I' 23 + I' 40+ 1 56 + 1 66 + I' 66 + I' 55 + I'

61 + I' 18 + I' 87 +0 13 + I' 25 + I' 41 + 1 55 + I' 62 + I' 59 + I' 47 + I'

56 + I' 15 + I' 89+0 14 + I' 27 + I' 43 + I' 55 + I' 61 + I' 55 + I' 42 + I'

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

13 +2 16+2 16+2 14+2 10+2 58 + 1 20 + I' 41- 0 13 + I' 39 + 1

13+2 11 +2 80 + I' 44 + I' 14 + 1 79 +0' 19 + I' 43 + 1 74+ 1 10+2

86 + I' 64 + I' 37 + I' 13 + I' 65 +0' 12 + I' 27 + 1 45 + 1 61 + 1 69 + 1

63 + 41 + 19 + 67 + 79 + 17 + 31 + 46 + 56 + 57 +

1 1 1 0' 0' I' 1 1 1 I'

48 + I' 29 + I' 10 + I' 57 +0 94+0' 18 + I' 30 + 1 41 + 1 47 + I' 45 + I'

39 + I' 22 + 1 76 +0' 58 + 0' 10 + I' 20 + I' 31 + 1 39 + I' 43 + I' 39 + I'

33 + I' 16 + I' 62 +0' 62 +0' 11 + I' 20 + I' 30 + I' 37 + I' 38 + I' 33 + I'

30 + I' 12 + I' 56 + 0' 66+0' 12 + I' 21 + I' 30 + I' 35 + I' 35 + I' 29 + I'

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

94+ 1 12 + 2 13 +2 12+2 10+2 61 + 1 24 + I' 47 -0 88 + 0' 28 + 1

10+2 98 + 1 74+ 1 45 + 1 18 + 1 70+0 13 + I' 33 + 1 61 + 1 89 + 1

69 + 1 55 + I' 35 + I' 15 + 1 57+0 92+ 0' 20 + I' 36 + 1 50 + 1 59 + 1

49 + I' 35 + I' 18 + 1 65 +0' 59 +0' 12 + I' 24 + 1 36 + 1 46 + 1 48 + 1

37 + I' 24 + I' 10 + I' 48 - 0 66 +0' 13 + I' 22 + 1 31 + 1 37 + I' 37 + I'

30 + 19 + 71 + 45 73 + 14 + 22 + 30 + 34 + 32 +

I' 1 0' 0' 0' I' 1 1 I' I'

25 + I' 14 + 1 54+0' 44 - 0' 78 + O· 14 + I' 21 + I' 27 + I' 29 + I' 27 + I'

22 + I' 10 + I' 46 - O· 45 - 0' 82 +0' 14 + I' 21 + I' 26 + I' 27 + I' 23 + I'

4.1 4.2 4.3 4.4

77 + 1 10+2 12 + 2 11 +2

97 + 91 + 71 + 45 +

62 + 51 + 34 + 15 +

43 + I' 31 + I' 18 + 1 64 +0'

32 + 22 + 10 + 45 -

25 + I' 17 + I' 68 +0' 39-0

21 + I' 12 + 1 50+0' 37 - 0'

18 + I' 99+0' 41 - 0' 37 - 0'

1.5

2.0

2.5

Transition p - d, parameter C K = 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

1.2 1.3

1 1 1 1

1 I' I' 1

I' I' 1 0

172

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no

An

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

14+ 1 19 + 1 24+ 1 29 + 1 35 + 1 41 + 1 50+ 1 60+ 1 74 + 1* 94 + 1* 11 +2* 14+ 2* 49 + 1* 12 - 0* 24 - 0* 10 + 1* 20+ 1 31 + 1 45 + 1 62+ 1 85 + 1* 11 +2 91 + 1 90+0* 10 - 0* 55 +0* 14 + 1 24+ 1 37 + 1 52 + 1*

15 + 1 21 + 1 27 + 1 34+ 1 41 + 1 50 + 1 62 + 1 76 + 1* 96 + 1* 11+2* 14 +2* 14+2 16 + 1* 10 - 0* 52+0* 15 + 1 27 + 1 41 + 1 59 + 1* 81 + 1*

16 + 1 23 + 1 30+ 1 38 + 1 48 + 1 59+ 1 73 + 1* 92 + 1* 11+2* 13+2* 17 +2* 93 + 1* 70+0* 10 - 0* 81 + 0* 19 + 1 33 + 1 51 + 1* 72 + 1* 97 + 1*

17 + 1 25 + 1 34+ 1 43 + 1 55 + 1 68+ 1 86 + 1* 10 +2* 12+2* 15 +2*

18 + 1 27 + 1 37 + 1 48+ 1 62+ 1 78 + 1* 98 + 1* 12 +2* 13 +2* 16 +2*

20+2* 54+ 1* 35 - 0* 22 - 0* 11 + 1* 24+ 1 40+ 1* 61 + 1* 85 + 1* 11 + 2*

10+2* 13 +2* 42 + 1* 21- 0 14 - 0* 87 +0* 19 + 1 31 + 1 48 + 1* 68 + 1*

12 +2* 12 +2* 20 + 1* 10 - 0* 33 - 0* 12 + 1* 24+ 1 39 + 1* 59 + 1* 82 + 1*

14 + 2* 92 + 1* 11 + 1* 10 - 0* 53 +0* 15 + 1* 29 + 1* 47 + 1* 70 + 1* 95 + 1*

22+2* 35 + 1* 22-0 36 -0* 14 + 1* 28 + 1* 47 + 1* 70 + 1* 96 + 1* 12 + 2* 16 + 2* 63 + I· 67+0* 10 - 0* 73 +0* 19 + 1* 35 + 1* 56 + 1*' 80 + 1* 10 + 2*

73 + 1* 98 + 1* 98 + 1 15 + 1* 10 - 0* 36 - 0* 11 + 1 21 + 1 33 + 1 48 + 1 67 + 1* 91 + 1* 98 + 1 20+ 1*

92 + 1* 11+2* 57 + 1* 48 -0 10 - 0* 63 +0* 15 + 1 27 + 1 42 + 1* 62 + 1*

10+2* 12 +2 30 + 1* 15 - 0* 17 - 0* 92+0* 20+ 1 34+ 1 52 + 1* 75 + 1*

12 + 2* 10+2 16 + 1* 10 - O· 32 - 0* 12 + 1* 24 + 1* 41 + 1* 62 + 1* 87 + 1*

14+2* 79 + 1* 10 + 1* 10 - 0* 49-0* 15 + 1* 29 + 1* 48 + 1* 7l + 1* 96 + 1*

85 + 1* 10+2* 66+ 1 68+0

10+2* 11 + 2* 36 + 1* 25 -0

11 + 2* 10+2 21 + 1* 10 - 0*

12 + 2* 87 + 1* 13 + 1* 10-0*

Transition p - d, parameter (() ,,=1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

28+ 1 32+ 1 37 + 1 43 + 1 54+ 1 83 + 1* 57+ 1 10 - 0* 85+0 15 + 1

52 + 1 66+ 1 88 + 1* 98+ 1 10+ 1 10-0* 67+0* 14+ 1 21 + 1 29+ 1

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

21 + 1 26+ 1 32+ 1 39+ 1 49+ 1 69+ 1 96+ 1* 28-0 34- 0* 11 + 1

38+ 1 50+ 1 68 + 1* 88 + 1* 40+ 1 10 - 0* 25 - 0* 98+0 17 + 1 25 + 1

13 + 1 16 + 1 20+ 1 24+ 1 28 + 1 33 + 1 39 + 1 46+ 1 56+ 1 69 + 1 87 + 1* 11+2* 12 +2 15 + 1* 10 - 0* 56+0* 14 + 1 23 + 1 33 + 1 46+ 1 61 + 1* 83 + 1* 10 + 2* 47 + 1* 11 - 0* 17 - 0* 90+0 17 + 1 27 + 1 39+ 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

18 + 1 24+ 1 30+ 1 37 + 1 46+ 1 65 + 1 93 + 1* 12 + 1 13 - 0* 94+0

34+ 1 45 + 1 61 + 1 81 + 1 58+ 1 25 -0 10 - 0* 78+0 15 + 1 23 + 1

53 + 1 73 + 1* 95 + 1* 64+ 1 35 -0 10 - 0* 69+0* 15 + 1 24+ 1 36+ 1

4.1 4.2 4.3 4.4

17 + 1 23 + 1 29+ 1 36+ 1

32+ 1 43 + 1 58 + 1 77+ 1

49 + 68 + 90 + 71 +

1.1

24+ 28+ 32 + 36 + 42+

1 1 1 1 1

1 1* 1* 1

173

6.1 Tables of Numerical Results Table 6.2. (continued) The effective principal quantum number no Lin

2.0

3.0

3.5

4.0

4.5

5.0

95 +0 12 + 1 14 + 1 15 + 1 15 + 1

10 - 0 26- 0 47 -0 73 +0 10 + 1 12 + 1 14 + 1 15 + 1 14 + 1 13 + 1

16 - 0 38 - 0 64+0 93 +0 11 + 1 14 + 1 15 + 1 15 + 1 14 + 1 12 + 1

22- 0 48 - 0 78+0 10 + 1 13 + 1 15 + 1 16 + 1 15 + 1 14 + 1 12 + 1

27 -0 57 +0 91 +0 12 + 1 15 + 1 16 + 1 17 + 1 16 + 1 14 + 1 12 + 1

31 - 0 66+0 10 + 1 31 + 1 16 + 1 17 + 1 18 + 1 16 + 1 14 + 1 12 + 1

35 - 0 74+0 11 + 1 14 + 1 17 + 1 18 + 1 18 + 1 17 + 1 15 + 1 12 + 1

1.6 1.7 1.8 1.9 2.0

13 + 1 12 + 1 11 + 1 90+0 63 +0 37 -0 20-0 15 - 0 21 - 0 34-0

13 + 1 10 + 1 82+0 58 + 0 42 -0 36 -0 41- 0 54+0 70+0 84+0

11 + 1 88 +0 66+0 50+0 44-0 46 - 0 54+0 65 +0 74+0 79+0

10 + 1 79+0 60+0 49 -0 46-0 50+0 59 + 0 68+0 73 +0 73 +0

98+0 76+0 59+0 50+0 49 -0 54+0 61 +0 68+0 72+0 70+0

97+0 75 +0 60+0 53 +0 54+0 59+0 66+0 70+0 71 +0 68+0

98 +0 76+0 62+0 55 + 0 57+0 62+0 68 +0 72+0 72+0 67+0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

59+0 71 + 0 76+0 71 +0 60+0 40-0 24-0 16 - 0 16- 0 24-0

88+0 83 + 0 71 + 0 56+0 42 -0 35 - 0 35 - 0 43 -0 56+0 70+0

77 +0 68+0 57 +0 46- 0 40-0 39 - 0 43 - 0 51 + 0 59+0 65 +0

68+0 60+0 50 + 0 42 -0 39-0 40-0 45 -0 52+0 57+0 59+0

64+0 56+0 47 -0 41- 0 39-0 41- 0 45 -0 51 + 0 54+0 55 +0

97+0 75 +0 59+0 51 + 0 51 + 0 57 +0 64+0 7(}+0 71 + 0 69+0 62+0· 54+0 46-0 41 - 0 40-0 42 -0 46-0 51 + 0 53 +0 53 +0

61 +0 52+0 45 -0 40-0 39-0 42 -0 46-0 49-0 51 + 0 51 + 0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

44-0 57+0 64+0 65+0 57+0 41 - 0 26-0 17 -0 15 - 0 20-0

75+0 74+0 66+0 54+0 42 -0 34 -0 33 - 0 38 - 0 50+0 63 +0

66+0 61 + 0 52+0 44 - 0 38 - 0 36-0 39 -0 46- 0 53 +0 59+0

57 + 0 52+0 45 -0 39 - 0 35 - 0 36 - 0 40-0 45 -0 50+0 52+0

53 +0 47 -0 41 - 0 37 - 0 34-0 35 - 0 39-0 43 -0 47 -0 48 -0

50+0 45 -0 40-0 36-0 35 - 0 36 -0 39-0 44-0 46-0 46-0

48-0 42 -0 38 - 0 34-0 33 - 0 33 - 0 38 -0 41- 0 41 - 0 43 -0

60+0 52+0 45 - 0 41- 0 41- 0 44-0 47 - 0 50+0 52+0 50+0 46 -0 42-0 38 - 0 33 - 0 34-0 37 -0 40-0 40-0 43 -0 43 -0

4.1 4.2 4.3 4.4

38-0 50+0 59+0 61 +0

70+0 70+0 64+0 54+0

61 +0 57+0 50+0 42 - 0

51 + 0 47 -0 42 -0 37 - 0

47 -0 43 -0 38 -0 34-0

44-0 41 - 0 36-0 34-0

41 -0 37 -0 33 -0 30-0

39 -0 37 -0 33 - 0 28-0

1.5

2.5

Transition p - d, parameter C K=3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

1.2 1.3 1.4 1.5

174

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no 3.0

3.5

4.0

4.5

5.0

64+0 56+0 51 + 0 49 -0 48 -0

35 + 1 13 + 1 81 + 0 60+0 51 + 0 47 -0 46- 0 46-0 48 - 0 51 + 0

23 + 1 90+0 58+0 48 -0 45 -0 45 -0 47 - 0 49 -0 52+0 56+0

17 + I 68+0 49-0 45 - 0 45 - 0 47 - 0 49- 0 52+0 56+0 62+0

13 + 1 57+0 45 -0 45 - 0 46-0 49 -0 52+0 56+0 61 +0 68 +0

11 + 1 51 + 0 44-0 45 - 0 48 - 0 51 + 0 55 +0 60+0 67+0 76+0

92 +0 47 - 0 44-0 46- 0 49- 0 53 +0 58 + 0 64+0 73+0 84+0

1.6 1.7 1.8 1.9 2.0

62+0 56+0 53 +0 51 + 0 51 + 0 56+0 76+0 10 + 1 89+0 67+0

48 -0 51 + 0 57 +0 67 +0 78 +0 78+0 63 +0 51 + 0 45 -0 43 -0

55 + 0 60+0 66+0 69+0 63 +0 53 + 0 45 - 0 42 - 0 42 - 0 43 - 0

61 + 0 67 +0 71 + 0 70+0 60+0 50+0 44-0 42 -0 42 -0 45 -0

68+0 75 +0 79+0 73 +0 60+0 49 - 0 43 -0 42 -0 43 - 0 47 -0

77 +0 84+0 86+0 75 +0 60+0 48 -0 43 -0 42 - 0 45 -0 49 -0

86+0 94+0 92+0 77 +0 59+0 47 -0 42 -0 43 -0 46-0 53 +0

95+0 10 + 1 98+0 78+0 57 +0 46-0 42 -0 44 -0 49-0 56+0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

60+0 53 + 0 49 - 0 48 -0 48 - 0 51 + 0 63 +0 86+0 90+0 78+0

43 - 0 45 - 0 49 - 0 56+0 65 +0 70+0 64+0 53 + 0 45 - 0 42 -0

46-0 49 -0 54+0 58 +0 59+0 54+0 48 - 0 43 -0 42 - 0 42 -0

48 -0 53 +0 58 +0 62+0 60+0 54+0 47 -0 43 -0 42 -0 43 -0

52+0 58 + 0 65 +0 67+0 62+0 54+0 47 - 0 43 -0 42 -0 44- 0

56+0 64+0 70+0 71 + 0 63 +0 53 +0 46 -0 42 -0 42 -0 45 -0

61 + 0 70+0 77 +0 75 +0 64+0 54+0 45 - 0 42 -0 43 - 0 48 - 0

66+0 77+0 85 +0 78 +0 65 +0 54+0 43 - 0 42 -0 44- 0 49- 0

3.l 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

60+0 52+0 48-0 46-0 47 - 0 49-0 59 + 0 78+0 88+0 71 + 0

42 -0 44-0 47 -0 53 +0 60+0 66+0 63 +0 53 +0 46 -0 42 -0

44-0 47 - 0 50+0 55 +0 56+0 54+0 49 -0 44 - 0 42 - 0 42 - 0

45 -0 49-0 54+0 58 +0 59+0 55 +0 48 -0 44-0 42 -0 42 -0

47 - 0 53 +0 59+0 63 +0 62+0 55 + 0 48 - 0 44-0 42 -0 43 -0

50+0 57+0 64+0 67 +0 62+0 52+0 46-0 43 -0 43 -0 45 -0

53 +0 62+0 69+0 69+0 59+0 48 -0 48 -0 44-0 40-0 44-0

59+0 68+0 86+0 77 +0 77 +0 73 +0 56+0 39 -0 37 -0 46-0

4.1 4.2 4.3 4.4

60+0 52+0 48 - 0 46-0

42 43 46 51

43 - 0 45 -0 49-0 53 +0

44 -0 48 -0 52+0 56+0

46-0 51 + 0 57+0 61 +0

48 - 0 55 +0 61 +0 64+0

50+0 59+0 67+0 71 + 0

47 -0 60+0 63 +0 68+0

LIn

1.5

2.0

2.5

Transition p - d, parameter cp

K=3 O.l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

1.2 1.3 1.4 1.5

- 0 -0 -0 +0

6.1 Tables of Numerical Results

175

Table 6.2. (continued) The effective principal quantum number nO' LIn

2.5

3.0

3.5

4.0

4.5

5.0

Transition d - s, parameter C 0.1 0.2 30.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

17 - 0 31 - 0 41 - 0 45 -0 45 -0 41 - 0 34 - 0 27 -0 21 - 0 30 - 0'

33 - 0 60+0 79+0 88+0 87 +0 79+0 66+0 52 +0 40 -0 31 - 0

48 -0 88 + 0 11 + 1 12 + 1 12 + 1 11 + 1 97 +0 76+0 57 +0 13 + I'

62 +0 11 + 1 15 + 1 16 + 1 16 + 1 15 + 1 12 + 1 99+0 74+0 55 +0

76+0 14 + 1 18 + 1 20+ 1 20 + 1 18 + 1 15 + 1 12 + 1 90+0 16 + I'

89+0 16 + 1 21 + 1 24 + 1 24+ 1 22 + 1 18 + 1 14 + 1 10 + 1 76+0

1.4 1.5 1.6 1.7 1.8 1.9 2.0

16 16 17 19 19 19 18 16 14 25 -

0 0 0 0 0 0 0 0 0 O·

26 -0 26 -0 28 -0 31 - 0 33 - 0 33 - 0 31 - 0 28 - 0 24-0 21 - 0

36 - 0 34-0 36 - 0 40-0 43 - 0 43 - 0 41 - 0 37 - 0 32 - 0 33 - O'

44-0 41 - 0 43 -0 48 - 0 51 + 0 52+0 50+0 44-0 38 - 0 33 - 0

51 + 0 47 - 0 49- 0 54+0 59+0 60+0 57+0 51 + 0 44- 0 44 - O·

58 + 0 52+0 54+0 60+0 66+0 67 +0 65 +0 58 +0 49 -0 41 - 0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

13 13 14 15 16 16 15 14 13 14 -

0 0 0 0 0 0 0 0 0 0'

20- 0 20 - 0 22 - 0 23 - 0 24-0 25 - 0 24-0 22 - 0 20- 0 18 - 0

25 - 0 25 - 0 26 -0 29 - 0 31 - 0 31 - 0 30-0 28 - 0 25 - 0 23 - 0

29 -0 29 -0 30 -0 33 - 0 35 - 0 36 - 0 35 - 0 32 - 0 29 -0 26 -0

33 - 0 31- 0 33 - 0 36-0 39 - 0 40 -0 39 - 0 36 -0 32 -0 29- 0

36 -0 34-0 35 - 0 38 - 0 42 -0 43 -0 42 -0 39 - 0 35 - 0 30 - 0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

12 12 13 14 14 14 14 13 12 13 -

0 0 0 0 0 0 0 0 0 0'

18 - 0 18 - 0 19 - 0 20 - 0 21 - 0 21 - 0 21 - 0 20- 0 18 - 0 17 - 0

21 - 0 21 - 0 23 - 0 24 -0 26 - 0 26 -0 26 -0 24 - 0 22 - 0 21 - 0

24-0 24-0 25 - 0 27 -0 29 -0 29 -0 29 -0 27 -0 24 -0 22 -0

26-0 25 - 0 27 -0 29- 0 31- 0 32 - 0 31 - 0 29 - 0 26 -0 25 -0

27 -0 27 - 0 28 -0 30 -0 32 - 0 34 - 0 33 - 0 31 - 0 28 -0 25 - 0

4.1 4.2 4.3 4.4

11 12 12 13 -

0 0 0 0

16 17 18 19 -

19 - 0 19 - 0 20 - 0 22 -0

21 - 0 21 - 0 22 -0 24 -0

22 -0 22 -0 23 - 0 25 - 0

23 -0 23 -0 24 -0 26 -0

1.1

1.2 1.3

0 0 0 0

176

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no An

2.5

Transition d - s, parameter

({J

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

76+0 63+0 70+0 80+0 92+0 10 + 1 11 + 1 12 + 1 13 + 1 10 -0*

68+0 62+0 72+0 84+0 98+0 11 + 1 13 + 1 14 + 1 14 + 1 12 + 1

1.1

1.4 1.5 1.6 1.7 1.8 1.9 2.0

81 +0 61 +0 52+0 53 +0 59+0 68+0 79+0 92+0 10 + 1 58 + 1*

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

1.2 1.3

4.0

4.5

5.0

63 +0 63 +0 75+0 89+0 10 + 1 12 + 1 14 + 1 16 + 1 17 + 1 93 + 1*

61 +0 66+0 79+0 95 +0 11 + 1 13 + 1 16 + 1 18 + 1 19 + 1 17 + 1

59+0 68+0 84+0 10+ 1 12 + 1 14+ 1 17 + 1 20+ 1 22+ 1 87 + 1*

59+0 71 +0 88+0 10 + 1 13 + 1 16 + 1 19 + 1 23 + 1 25 + 1 22+ 1

90+0 59+0 47 -0 47-0 56+0 69+0 84+0 96+0 10 + 1 92+0

10 + 1 63+0 47 -0 47 -0 57+0 73 +0 91 +0 10+ 1 11 + 1 22 + 1*

11 + 1 69+0 48 -0 48-0 59+0 77+0 99+0 12 + 1 13 + 1 12 + 1

13 + 1 75+0 49-0 48-0 61 +0 82+0 10+ 1 13 + 1 14+ 1 24 + 1*

15 + 1 81 +0 51 +0 49-0 63 +0 86+0 11 + 1 14 + 1 16 + 1 15 + 1

71 +0 59+0 52+0 51 +0 54+0 62+0 72+0 85+0 97+0 21 + 1*

73+0 55+0 46-0 46-0 52+0 62+0 74+0 85+0 89+0 82+0

82+0 59+0 47 -0 46-0 53 +0 65 -0 80+0 95 +0 10 + 1 12 + 1

10+ 1 68+0 49-0 47-0 55+0 71 +0 92+0 11 + 1 12 + 1 14+ 1

11 + 1 73+0 51 +0 47-0 57+0 75+0 98+0 12 + 1 13 + 1 13 + 1

68+0 58+0 51 + 0 50+0 53+0 60+0 70+0 82+0 96+0 25 + 1*

68+0 53+0 46-0 45 -0 50+0 60+0 71 +0 80+0 84+0 78+0 65 +0 52+0 45 -0 45 -0

75 +0 57+0 46-0 45 -0 51 + 0 62+0 76+0 89+0 96+0 11 + 1 72+0 56+0 46 -0 45 -0

93+0 64+0 48 -0 46-0 54+0 68+0 86+0 10 + 1 11 + 1 10+ 1 84+0 61 +0 48 -0 46-0 52+0 65+0 81 +0 96+0 10+ 1 98+0 79+0 59+0 47 -0 45 -0

92+0 65+0 49-0 46-0 53+0 67+0 86+0 10+ 1 11 + 1 13 + 1 87+0 63+0 49-0 46-0

10+ 1 69+0 50+0 47 -0 54+0 70+0 91 +0 11 + 1 12 + 1 11 + 1 95 +0 67+0 50+0 46-0

66+0 57+0 51 + 049-0

3.0

3.5

6.1 Tables of Numerical Results

177

Table 6.2. (continued) The effective principal quantum number no

An

2.5

3.0

3.5

4.0

4.5

5.0

46+ 1 76+ 1 94+ 1 99+ 1 93 + 1 78 + I· 59 + I· 38 + I· 19 + 1 74+0 49-0 66+0· 10 + I· 14 + 1 17 + I" 17 + 1* 15 + 1* 11 + 1 68+0 37 - 0*

55 + 1 90+ 1 11 +2 11 +2 11 +2 93 + I· 69 + I· 46 + I· 22 +'1 85 +0·

62+ 1 10+2 12 +2 13 +2 12 + 2· 10+2· 79 + I· 53 + I· 26 + I· 97 +0·

55+0 73 +0· 11 + I· 16 + 1 19 + 1* 20 + 1* 18 + 1* 13 + 1 77+0 41 - O·

60+0 78 -0· 12 + I· 17 + I· 21 + I" 22 + I· 20 + I· 15 +1· 88 +0· 45 - 0*-

23 -0 30- O· 44-0* 60+0· 73 +0· 78+0 71 +0 55+0 35 -0 22 - O·

32 - O· 42 - 0* 63 +0· 87+0 10+ 1 11 + 1· 10 + 1· 79+0 48 -0 29-0 24 - O· 31 - O· 46-0· 63+0 77+0 82+0 76+0 59+0 37 -0 23 -0

34 -0· 43 -0· 65 +0· 92+0· 11 + 1· 12 + 1· 11 + I· 89+0· 54+0· 31 - o· 25 - O· 31 - O· 47 - O· 65 +0· 81 + O· 87 +0· 82 +0· 65 +0· 40-0· 24-0·

20 - O· 26 -0· 37 - O· 51 +0·

21- 0 26- O· 38 - O· 52+0

21 - O· 26 - O· 38 -0· 53 +0·

Transition d - p, parameter C ,,=1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

16 + 1 25 + 1 38 + 1 31 + 1 27 + 1 22 + 1 15 + 1 87+0 44-0 30-0 27 - O· 37 -0· 50+0· 59+0 61 +0 56+0 43 -0 31- 0 24-0 24-0 21 - O· 27 - O· 34- O· 39 - O· 39-0 35 - 0 28-0 22 -0 20-0 23 -0 19-0 24-0 29-0 32-0 31-0 28-0 23 -0 19 - O· 19 - O· 24-0·

4.1 4.2 4.3 4.4

18 - O· 23 -0· 27 -0 29-0

1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

27 + 1 44+ 1 54+ 1 57 + 1 53 + 1 44+ 1 33 + 1 21 + 1 10 + 1 45 - O· 34-0· 47 - O· 73 +0· 98+0 11 + 1 11 + 1 99+0 72+0 43 -0 26-0 23 - O· 31 - 0* 45 - O· 60+0 70+0 72+0 63+0 48-0 31 - 0 20-0 19 - O· 25 - 0* 36 - O· 47 -0 55+0 57+0 50+0 38-0 26-0 18 - 0 17 - O· 22 -0· 31 - O· 41-0

37 + 1 62 + 1 75 + 1 79+ 1 74+ 1 62 + 1 46 + I· 29 + I· 14 + 1 59+0 43 -0 60+0 92+0 12 + 1 14 + 1 14 + 1 12 + 1 92+0 53 +0 32 -0 28 -0· 37 - 0* 54+0· 73 +0 86+0 88+0 77+0 57+0 36-0 24-0 22- O· 29 -0· 42 -0· 56+0· 65+0 67+0 59+0 44-0 29-0 21 - 0 20 -0· 26 - O· 36- O· 48-0

30- O· 39 -0· 59+0· 81 +0 98+0 10 + 1 94+0 73 +0 45 -0 27 -0

178

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no LIn

2.5

3.0

Transition d - p, parameter ,,=1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ll

1.2 1.3

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

13 + 1 18 + I 24+ I 31 + 1 40+ I 50+ I 56+ I 43 + 1 14 + I 13 - 0 10 - 0* 15 - O· 61 +0* 11 + 1 18 + I 22 + I 20+ I 11 + I 25 -0 10-0 10-0* 13 - 0* 48 -0* 92+0* 13 + 1 15 + 1 12 + 1 52+0 10-0 10-0 10-0 14 -0 45 -0 83 +0 11 + 1 12 + 1 87+0 30 -0· 10 - O· 10 - O· 10 - 0* 14 - O· 44-0 78+0

14 + 1 20+ I 27 + I 36+ 1 46+ I 60+ I 76+ I 85 + I 55 + I 12 + I· 10 - 0* 10- 0* 61 + O· 13 + I 22 + I 33 + I 44+ I 45 + I 26+ 1 67+0 10 - O· 10 - O· 46 -0* 10 + I 18 + 1 27 + 1 35 + 1 34+ I 19 + 1 51 +0 10 -0· 10 - O· 40 - 0* 96+0 16 + I 24+ I 31 + I 29+ I 16 + I 43 -0 10 - O· 10 - O· 37 -0 90+0

3.5

4.0

4.5

5.0

16 + 1 25 + 1 34+ 1 46+ 1 61 + 1 80+ 1· 10+2· 12 + 2* 88+ 1 18 + 1 11- 0 12 - 0* 74+0· 16 + 1 29+ 1 45 + 1* 62 + 1· 70+ 1 42 + 1 10 + I· 10 - O· 10 - O· 54+0* 13 + 1 24+ I 37 + 1 50+ 1 54+ 1 31 + 1 83 +0 10-0 10- O· 48 -0* 11 + 1* 21 + 1* 33 + 1 45 + 1 46+ 1 26+ 1 72+0 10- O· 10- 0* 44-0* 11 + I·

17 + I 27 + I 37 + I 51 + 1 68+ I 90 + I· II +2* 14 +2· 10+2 19 + 1·

19 + 1 29+ 1 41 + 1 56+ 1 75 + I· 10 +2· 12 +2* 15 +2* 11+2* 23 + 1* 21 - 0 17 - O· 87 +0* 20 + 1* 36 + I· 56 + I· 79 + 1· 94 + 1· 56 + I· 14 + I·

qJ

15 + I 23 + I 31 + 1 41 + 1 53 + 1 70 + 1 90 + I· 10 + 2* 64+ 1 12 + I 10 - 0 11- 0 68+0 15 + 1 26 + 1 39 + 1 52+ 1 53 + 1 27 + I 61 +0 10 - 0* 10 - O· 51 + O· 12 + 1 21 + I 32 + 1 41 + I 38+ 1 19 + I 41-0 10 - 0* 10-0* 45 -0* 10 + 1* 19 + I 28 + 1 36+ 1 32 + 1 15 + 1 31 - 0 10 - O· 10 - O· 42 - 0* 10 + 1

15 - 0 15 - O· 80+0* 18 + 1 32 + 1* 51 + 1· 70+ 1· 80+ I 46+ 1 11 + 1* 10 - O· 10- 0* 59+0· 14+ I 26+ 1 41 + 1· 57 + 1* 61 + I 33 + 1 83+0 10- O· 10-0* 52+0· 13 + 1* 23 + 1 37 + I 50+ 1 51 + 1 27+ 1 69+0 10-0 10- 0* 48 - O· 12 + I

13 -0· 10 -0· 64+0* 15 + 1· 29 + 1* 46+ 1* 65 + 1* 72 + I· 41 + 1· 11 + 1* 10 - O· 10 - 0* 56+0· 14 + 1· 26 + 1* 41 + 1· 58 + 1* 62 + 1· 35 + 1* 96+0· 10 - O· 10 -0· 51 + O· 13 + 1*

179

6.1 Tables of Numerical Results Table 6.2. (continued) The effective principal quantum number no

Lin

2.5

3.0

3.5

4.0

4.5

5.0

Transition d - p, parameter C ,,=3 13 -0 24-0 33 -0 39-0 42-0 42-0 39-0 34-0 29 -0 25 -0 22-0 20-0 20-0 21- 0 22-0 23 -0 23 -0 22-0 20-0 19 -0

18 - 0 34-0 46-0 55 +0 59+0 58+0 53 +0 46-0 39-0 33 -'- 0

23 -0 43 -0 59+0 70+0 74+0 72+0 66+0 57+0 48 -0 40-0

27-0 51 +0 70+0 83+0 88+0 86+0 78+0 68+0 56+0 46-0

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

74 -1 13 - 0 18 - 0 22-0 23 -0 23 -0 22-0 20-0 17 - 0 15 - 0 13 -0 13 -0 13 -0 13 -0 14 - 0 15 -0 15 -0 14 -0 14 -0 13 -0

28 26 26 27 28 29 29 27 25 23

39-0 35 -0 34-0 35 -0 37 -0 37 -0 37 -0 36-0 33 -0 30-0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

12 -0 12 -0 12 - 0 12 -0 12 -0 13 -0 13 -0 13 -0 13 -0 12 -0

18 -0 17 -0 17 -0 18 -0 19 -0 19 -0 19 -0 19 -0 18 -0 17 -0

22 -0 21 - 0 21 - 0 21 - 0 22 -0 23 -0 23 -0 22 -0 21 - 0 20-0

34-0 31- 0 30-0 31- 0 33 -0 34-0 33 -0 32-0 29-0 27 -0 25 -0 24-0 24-0 24-0 25 -0 26-0 26-0 25 -0 24-0 23 -0

32-0 59+0 82+0 96+0 10+ 1 99+0 90+0 77+0 64+0 52+0 44-0 39-0 38 -0 39-0 40-0 41- 0 41- 0 39-0 36-0 32-0

27-0 26-0 26-0 26-0 27-0 28-0 28-0 27-0 26-0 24-0

29-0 28-0 28 -0 28-0 29-0 29-0 29-0 29-0 27 -0 24-0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

11-0 11-0 11-0 12 -0 12 -0 12 -0 12 -0 12 -0 12 -0 12 -0

16 -0 16 -0 16 -0 16 -0 17 -0 17 -0 17 -0 17 - 0 16 -0 16 -0

19 -0 19 - 0 18 - 0 19 - 0 20-0 20 -0 20 -0 20-0 19 - 0 18 - 0

21- 0 21- 0 21-0 22-0 22-0 23 -0 23 -0 23 -0 21- 0 20-0

22-0 21- 0 21- 0 22-0 23 -0 23 -0 24-0 24-0 22-0 22-0

23 -0 23 -0 23 -0 23 -0 24-0 24-0 25 -0 23 -0 24-0 22 -0

4.1 4.2 4.3 4.4

11-0 11-0 11-0 11-0

15 - 0 15 -0 15 -0 16 - 0

18 17 17 17 -

20-0 20-0 20-0 20-0

21-0 20-0 19 - 0 20-0

20-0 19 - 0 18 - 0 18 - 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-0 -0 -0 -0 -0 -0 -0 -0 -0 -0

0 0 0 0

180

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.2. (continued) The effective principal quantum number no

An

2.5

3.0

Transition d - p, parameter ,,=3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

22+ 1 81 +0 52+0 47 -0 49-0 53 +0 58+0 60+0 62+0 61 +0

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

33 + 1 11 + 1 67+0 52+0 51 +0 57+0 66+0 76+0 88+0 77+0 68+0 58+0 53+0 53+0 57+0 62+0 66+0 68+0 66+0 61 +0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

3.5

4.0

4.5

5.0

89+0 46-0 45 -0 47 -0 51 +0 55 +0 61 +0 69+0 77+0 83+0 82+0 72+0 59+0 48 -0 43 -0 43 -0 44-0 51 +0 57+0 61 +0

qJ

12 + 1 54+0 45 -0 46-0 49-0 52+0 55 +0 60+0 65+0 69+0

10 + 1 49-0 44-0 47-0 49-0 53+0 58+0 64+0 71 +0 76+0

59+0 56+0 52+0 50+0 48 -0 47-0 47 -0 47-0 48-0 49-0

16 + 1 63 +0 46-0 46-0 48 -0 52+0 55+0 58+0 61 +0 63 +0 62+0 58+0 52+0 47 -0 44-0 42 -0 43 -0 45 -0 48 -0 51 +0

68+0 62+0 54+0 47-0 43 -0 42-0 43 -0 47 -0 51 + 0 55+0

75+0 67+0 57+0 48-0 43 -0 42-0 44-0 48-0 54+0 59+0

55 +0 52+0 52+0 54+0 59+0 63+0 66+0 65+0 62+0 57+0

51 +0 52+0 51 +0 50+0 48 -0 46-0 45 -0 44-0 45 -0 47 -0

53 +0 53 +0 50+0 47 -0 43 -0 42 -0 42 -0 44-0 46-0 49-0

57+0 56+0 52+0 48 -0 43 -0 42-0 43 -0 45 -0 47 -0 51 +0

63 +0 64+0 60+0 51 +0 45 -0 42-0 43 -0 48-0 49-0 52+0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

52+0 50+0 51 +0 55+0 68+0 64+0 65+0 64+0 68+0 55 +0

49-0 51 +0 50+0 49-0 47-0 45 -0 44-0 43 -0 44-0 46-0

51 +0 51 +0 49-0 46-0 44-0 42 -0 42 -0 43 -0 45 -0 48 -0

53 +0 53 +0 50+0 48-0 44-0 41-0 42-0 44-0 47-0 50+0

61 +0 59+0 55+0 49-0 44-0 43 -0 42-0 46-0 51 +0 53+0 54+0 54+0 53+0 48-0 41-0 38-0 45-0 45-0 48-0 54+0

4.1 4.2 4.3 4.4

50+0 49-0 51 +0 55+0

48-0 50+0 50+0 49-0

50+0 50+0 50+0 47 -0

54+0 55+0 53+0 47 -0

59+0 58+0 53+0 52+0

54+0 39-0 41-0 41-0

1.1

57+0 62 +0 71 +0 54+0 53+0 52+0 50+0 44-0 53+0 60+0

181

6.1 Tables of Numerical Results

Table 6.3. Transitions with no change of spin LIS = 0; excitations rate coefficients (va) in the Born approximation. Parameters A and x The effective principal quantum number

LIn 0.5

1.0

1.5

2.0

no

2.5

3.0

3.5

4.0

4.5

5.0

Transition s - s, parameter A" 0.6 0.7 0.8 0.9 1.0

10+2 12 + 2 12 + 2 10 +2 80 + 1

10 + 2 11 +2 10+2 80 + I 51 + I

11 +2 11 +2 10+2 72+ I 41 + I

13 +2 12 + 2 94+ I 62 + I 32 + I

13 +2 11 +2 88 + I 54 + I 26 + I

14 +2 11 +2 81 + I 47 + I 22 + I

14+ 2 11 +2 75 + I 42 + 1 18 + I

14+2 10+2 70 + I 37 + 1 61 + I

14+2 10+2 65 + I 34+ 1 14 + I

14 +2 10+2 61 + I 31 + 1 12 + I

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

29 + I 67+0 21 - 0 11 + I 35 + I 64 + I 91 + I 10+2 10+2 85 + 1

23 + I 65 +0 32 -0 11 + I 29 + I 52 + I 69 + I 74+ I 65 + I 45 + 1

17 + I 54+0 54+0 17 + I 37 + I 57 + I 68 + I 66+ I 53 + I 34 + 1

12 + I 50+0 80+0 23 + I 44 + I 60+ I 64+ I 56 + I 41 + I 23 + 1

10 + I 49 -0 10 + I 28 + I 48 + I 59 + I 57 + I 47 + I 32 + I 17 + 1

83 + 0 50+0 12 + I 32 + I 49+ I 55 + I 51 + I 39 + I 25 + 1 13 + 1

70+0 51 + 0 14 + I 35 + I 49 + I 51 + I 45 + I 34 + I 21 + 1 10 + 1

61 +0 53 +0 15 + I 36 + I 47 + I 47 + I 40 + I 29+ I 17 + 1 88 +0

54+0 54+0 16 + 0 36 + I 45 + I 44+ I 36 + I 25 + I 15 + 1 74+0

49-0 55 + 0 17 + I 36 + I 43 + I 40 + I 33 + I 23 + I 13 + I 63 +0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

30 + 1 80+0 19 -0 94+0 30 + 1 59 + 1 86 + 1 10 +2 10+2 88 + I

22 + 1 74+0 30 -0 84+0 23 + 1 44 + 1 61 + 1 67 + 1 61 + I 43 + I

15 + 1 56+0 45 -0 12 + 1 29+ 1 47 + 1 58 + 1 59 + 1 48 + I 32 + I

10 + 1 46 -0 60+0 16 + 1 33 + 1 47 + 1 52 + 1 47 + 1 35 + I 20 + I

78+0 41 - 0 70+0 19 + 1 35 + 1 44 + 1 44 + 1 37 + 1 26 + I 15 + I

60+0 38 - 0 77 +0 20 + 1 34 + 1 39 + 1 37 + 1 29+ 1 20+ I 10 + I

48 -0 35 - 0 80+0 20+ 1 31 + 1 35 + 1 31 + 1 24+ 1 15 + I 84+0

40- 0 33 -0 82+0 20+ 1 29+ 1 30 + 1 27 + 1 20+ 1 12 + I 66+0

34-0 32 -0 82+0 19 + 1 26+ 1 27 + 1 23 + I 17 + 1 10 + I 54+0

29- 0 30- 0 81 + 0 18 + 1 24 + 1 24+ 1 20+ 1 14 + 1 89+0 45 -0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

30 + I 84+0 19 - 0 87 +0 28 + I 57 + I 84+ I 10+2 10+2 90+ I

22 + I 76+0 30 - 0 76+0 21 + I 41 + I 58 + I 65 + I 59 + I 42 + I

14 + I 56+0 42-0 11 + I 26 + I 43 + I 55 + I 55 + I 46 + I 31 + I

97+0 44-0 53 +0 14 + I 29 + I 42 + I 47 + I 43 + I 32 + I 19 + I

69+0 38 - 0 60+0 16 + I 30 + I 38 + I 39 + I 33 + I 23 + I 13 + I

51 + 0 33 - 0 62+0 16 + I 28 + I 33 + I 31 + I 25 + I 17 + I 97+0

40-0 29-0 62+0 16 + I 25 + I 28 + I 26+ I 20+ I 13 + I 73+0

32 -0 26-0 60+0 15 + I 22 + I 24+ I 21 + I 16 + I 10 + I 56+0

26-0 24-0 58+0 14 + I 20+ I 21 + I 18 + I 13 + I 86+0 45 -0

22+0 22 -0 56+0 13 + I 18 + I 18 + I 15 + I II + I 71 +0 37 - 0

4.1 30+ I 22 + I 14 + I 93 +0 64+0 47 -0 4.2 85 +0 77+0 56+0 43 - 0 36- 0 30- 0 4.3 19 - 0 30-0 41- 0 51 + 0 55 +0 55 +0 4.4 84+0 72+0 10 + I 13 + I 14 + I 14 + I a This table can be applied to transitions between the configurations ns-n's

35 - 0 28 -0 22 -0 19 - 0 26- 0 23 -0 20 -0 19 - 0 53 +0 50+0 47 -0 44-0 13 + 1 12 + I 11 + I 10 + I levels of two different electronic

182

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.3. (continued) The effective principal quantum number no Lin

0.5 1.0 1.5 2.0 Transition s - s, parameter X

0.6 0.7 0.8 0.9 1.0

13 + 14 + 15 + 15 + 16 +

1 I 1 1 1

15 + 16 + 17 + 17 + 16 +

1 1 1 1 1

17 + 1 17 + 1 17 + 1 16+ 1 14+ 1

17 + 17 + 16 + 14 + 11 +

I 1 1 1 1

2.5

3.0

3.5

4.0

4.5

5.0

17 + I 16 + 1 14 + I 12 + I 90+0

16 + I 15 + 1 13 + 1 10 + 1 72+0

15 + I 13 + 1 11 + 1 87+0 58+0

14+ I 12 + 1 10 + 1 74+0 47-0

13 + I 11 + 1 88+0 63 +0 39-0

12 + I 10+ 1 78+0 54+0 32-0

1.1 15 + 1 13 + 1 99+0 74+0 57+0 44-0 35 -0 29-0 24-0 20-0 1.2 88+0 68+0 53 +0 48 -0 47 -0 48-0 49-0 50+0 50+0 50+0 1.3 26-0 39 -0 64+0 98+0 13 + 1 15 + 1 17+ 1 18 + 1 19 + 1 19 + 1

1.4 1.5 1.6 1.7 1.8 1.9 2.0

72+0 10+ 1 13 + 1 14 + 1 15 + 1 16 + 1 16 + 1

91 +0 14 + 1 17 + 1 18 + 1 19 + 1 19 + 1 18 + 1

14 + 20+ 22 + 22 + 21 + 19 + 16 +

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

16 + 1 99+0 26-0 65 +0 10 + 1 13 + 1 14 + 1 15 + 1 16 + 1 16 + 1

14 + 1 81 +0 40-0 82+0 13 + 1 17 + 1 19 + 1 20+ 1 19 + 1 18 + 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

16 + 1 10 + 1 26-0 63 +0 10+ 1 12 + 1 14 + 1 15 + 1 16 + 1 16 + 1

4.1 4.2 4.3 4.4

16 + I 10 + 1 26-0 62+0

I I I 1 1 I I

25 + 1 28 + I 26 + 1 22+ 1 18 + 1 14 + 1 10+ 1

28 + 1 29+ 1 25 + 1 20+ 1 16 + 1 12 + 1 82+0

30+ 1 28+ 1 23 + 1 18 + 1 13 + 1 10+ 1 66+0

30+ 1 27 + 1 21 + 1 16 + 1 11 + 1 83+0 53+0

29 + 1 25 + 1 19 + I 14+ 1 10 + 1 70+0 43 -0

28 + 1 23 + 1 17 + 1 12 + 1 90+0 60+0 36-0

11 + 1 63 +0 62+0 13 + 1 20+ 1 23 + 1 23 + 1 22+ 1 20+ 1 16 + 1

86+0 56+0 93+0 20+ 1 27 + I 27 + I 25 + 1 22 + 1 18 + 1 13 + 1

66+0 54+0 11 + 1 25 + 1 31 + 1 28 + 1 24+ 1 19 + 1 15 + 1 10+ 1

52+0 52+0 13 + 1 28 + 1 32 + 1 27 + 1 22 + 1 17 + 1 12 + 1 86+0

42-0 50+0 14+ 1 30+ 1 31 + 1 25+ 1 19+ 1 14+ 1 10+ 1 69+0

34-0 48 -0 15 + 1 29+ 1 29+ 1 23 + 1 17 + 1 12 + 1 87+0 56+0

28 -0 46-0 15 + 1 28+ 1 27 + 1 20+ 1 15 + 1 10+ 1 74+0 46-0

23 -0 45 -0 15 + 1 27 + 1 24+ 1 18 + 1 13 + 1 95 +0 63 +0 38 -0

15 + 1 86+0 41- 0 79+0 13 + 1 17 + 1 19 + 1 20+ 1 20+ 1 18 + 1

11 + 1 68+0 63 +0 13 + 1 20+ 1 23 + I 24+ 1 23 + 1 20+ 1 17 + 1

90+0 60+0 93 +0 20+1 28 + I 28 + 1 26+ 1 22 + 1 18 + 1 14 + 1

70+0 57+0 11 + 1 25 + 1 32+ 1 30+ 1 25 + 1 20+ 1 15 + 1 11 + 1

55 +0 54+0 13 + 1 28 + 1 33 + 1 28 + 1 23 + 1 17 + 1 13 + 1 88+0

44-0 51 +0 14+ 1 29+ 1 32 + 1 26+ 1 20+ 1 15 + 1 10+ 1 71 +0

36-0 48-0 14 + 1 29+ 1 30+ 1 23 + 1 17 + 1 13 + 1 90+0 57+0

30-0 46-0 14 + 1 27 + 1 27 + 1 21 + 1 15 + 1 11 + 1 76+0 47-0

25 -0 43 -0 14+ 1 26+ 1 25 + 1 19 + 1 14 + 1 98+0 64+0 39-0

15 + 1 88+0 41- 0 78+0

12 + 1 70+0 63+0 13 + 1

93 +0 62+0 93 +0 20+ 1

72+0 58+0 11 + 1 25 + 1

57+0 55 +0 12 + 1 28 + 1

46-0 52+0 13 + 1 28+ 1

37 -0 48-0 13 + 1 28+ 1

31-0 46-0 13 + 1 27 + 1

26-0 43 -0 13 + 1 25 + 1

1 1 1 1 1 1 1

20 + 25 + 25 + 23 + 20+ 17 + 13 +

Table 6.3. (continued) The effective principal.quantum number Lin

2.5

3.0

3.5

4.0

4.5

5.0

14 + 1 20 + 1 30 + 1 37 + 1 46+148+1 58 + 1 52 + 1 64 + 1 51 + 1 62+144+1 53 + 1 34 + 1 40+123+1 25+113+1 13 + 1 66 + 0*

25 + 1 40+ 1 46+ 1 46+ 1 41 + 1 33 + 1 23 + 1 14 + 1 79 + O· 41 - 0

28+ 1 41 + 1 44+ 1 41 + 1 34+ 1 26+ 1 17 + 1 10 + 1* 57 + 0* 33 -0

31 + 1 41 + 1 42 + 1 37 + 1 30+ 1 21 + 1 14 + 1 81 + 0* 44 - 0* 30-0

33 + 1 41 + 1 40+ 1 34+ 1 26+ 1 18 + 1 11 + 1* 66 + 0* 37 - 0* 28-0

35 +1 41 + 1 37 + 1 31 + 1 23 + 1 15 + 1 97 + 0* 55 + 0* 33 - 0* 28 -0

36+ 1 40+ 1 35 + 1 28 + 1 20+ 1 13 + 1* 84 + 0* 48 - 0* 30 - 0* 27 - 0*

0.5 1.0 1.5 2.0 Transition s - p, parameter A

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

89+ 95 + 91 + 77+ 57 +

1 1 1 1 1

no

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

19 + 2 32 + 1 14 + 2 13 + 1 89 + 1 37 - 0* 44 + 1 52 + 0 15 + 1 29 + 1 21 - 0 45 + 1 34 - 0 56 + 1 14 + 1 61 + 1 25+158+1 34 + 1 49 + 1

66 + 0 34 - 0 32 - 0 43 - 0* 73 + 0 21 + 1 29 + 1 28 + 1 37 + 1 27 + 1 36 + 1 24 + 1 33 + 1 20 + 1 27 + 1 15 + 1 19+199+0 11 + 1 55 + 0

31 - 0 64 + 0* 22 + 1 22 + 1 19 + 1 16 + 1 12 + 1 88 + 0 54+0 32 - 0

33 - O· 86 + 0 18 + 1 17 + 1 14 + 1 12 + 1 89 + 0 60 + 0 37-0* 24 - 0

36 - 0* 10 + 1 15 + 1 13 + 1 11 + 1 90 + 0 65 + 0 43 - 0* 27-0* 20 - 0

38 - 0* 10 + 1 13 + 1 11 + 1 92 + 0 71 + 0 51 + O· 43 - O· 22-0* 18 - 0

40 - 0* 10 + 1 11 + 1 94 + 0 76 + 0 58 + 0 41 - O· 27 - 0* 19-0* 17 - 0

41 - O· 97 + 0 94 + 0 80 + 0 64 + 0 49 - O· 34 - O· 23 - 0* 17-0* 16 - 0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

13 +2 30 + 1 10 +2 14 + 1 73+146-0 41 + 1 32 - 0 16 + 1 22 + 1 37 - 0 39 + 1 75 -1 49 + 1 10+154+1 18 + 1 54 + 1 26+146+1

66+031-0 32-031-0 44-011+1 23+122+1 32 + 1 23 + 1 32 + 1 20 + 1 29 + 1 17 + 1 24+113+1 17 + 1 88 + 0 10+151+0

24-0 39 - O· 15 + 1 17 + 1 15 + 1 13 + 1 10 + 1 72+0 46-0 28- 0

23 - O· 46 - O· 13 + 1 12 + 1 11 + 1 91 +0 69+0 48-0 31 - O· 21- 0

23 - O· 52 + 0 10 + 1 97+0 82+0 65+0 49-0 33 - O· 22 - O· 17 - 0

23 - O· 53 +0 84+0 75 +0 63 +0 50+0 37 - O· 25 - O· 17 - O· 14 - 0

23 - O· 52+0 67+0 60+0 50+0 39 -0 29 - O· 20 - O· 14 - O· 13 -0

22 - 0* 48 -0 55+0 49-0 41- 0 32 - O· 23 - O· 16 - 0* 12 - O· 12 -0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

11 + 2 29 + 1 96 + 1 14 + 1 68 + 1 50 + 0 39 + 1 29 - 0 17 + 1 19 + 1 44 - 0 36+ 1 65-146+1 82 +0 51 + 1 15 + 1 51 + 1 23 + 1 45 + 1

66 + 32 38 21 + 30 + 30 + 27 + 22 + 16 + 10 +

0 0 0 1 1 1 1 1 0 0

22-0 32 -0· 12 + 1 15 + 1 14 + 1 11 + 1 91 +0 65+0 42 -0 26-0

20-0 36 - O· 11 + 1 11 + 1 96+0 79+0 61 +0 43 -0 28 - O· 19 - 0

19 - 0* 38 - 0 86+0 81 +0 69+0 55+0 42-0 29 - O· 20 - O· 15 - 0

18 - O· 38-0 66+0 61 +0 52+0 41- 0 31-0· 22 - O· 15 - 0* 12 - 0

17 - 0* 36-0 52+0 47 -0 40-0 32 -0 24-0· 17 - O· 12 - 0 11-0

16 - 0* 34-0 42-0 38-0 32-0 25 -0· 19-0· 14 - O· 10-0 10-0

4.1 4.2 4.3 4.4

10 +2 91 + 1 65 + 1 38 + 1

66+028-0 33 - 0 27 - 0 35 - 0 76 + 0 19 + 1 19 + 1

20-0 29-0· 10 + 1 14 + 1

18 - 0 31-0 97+0 10 + 1

16 - O· 32-0 76+0 72+0

15 - 0* 31- 0 57+0 53+0

14 - 0* 30-0 45 -0 41-0

13 - O· 27 -0 35 -0 32-0

29 + 1 14 + 1 52 +0 28 - 0

0 0 0 1 1 1 1 1 1 1

29 28 87 + 20 + 21 + 19 + 16 + 12 + 84 + 49 -

Table 6.3. (continued) The effective principal quantum number no L1n 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Transition s - p, parameter X 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

71 + 0 63 +0 56 + 0 48 - 0 41 - 0

l.l 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

10+ 1 99+0 87+0 75 +0 64+0 54 + 0 44 - 0 35 - 0 26 - 0 18 - 0

10 + 1 92+0 76+0 62+0 51 +0 40-0 31 - 0 23 -0 16 - 0 10 - O·

10+ 1 84+0 66+0 52+0 40-0 30-0 22-0 15 - 0 10- O· 10-0

10 + 1 77+0 58+0 44-0 32-0 23 -0 16 - 0 10 - O· 10 -0' 10-0

97+0 70+0 51 + 0 37 -0 26-0 18 - 0 12 - 0 10 - O· 10 - O· 10 - 0

93 +0 64+0 45 -0 31 - 0 22 -0 14 - 0 10 - O· 10 -0' 10 - O· 10-0

89+0 59+0 40-0 27 -0 18 - 0 ll- 0 10 - O· 10 - O· 10 -0' 10-0

85 +0 54+0 35 - 0 23 -0 15 - 0 10 - O· 10 - O· 10 - O· 10- O· 10 - O·

61+031-0 55 + 0 20 - 0 48 - 0 11 - 0 40 - 0 81 + 0 29 - 0 31 + 1 12 - 0 18 + 1 26 + 1 11 + 1 14 + 1 82 + 0 87 + 0 63 + 0 67 + 0 50 + 0

12-0 13 - 0 11 + 1* 38 + 1 21 + 1 12 + 1 78 + 0 53 + 0 36 - 0 24 - 0

10- 0 44 - 0* 42 + 1 30 + 1 15 + 1 86+0 54+0 35 - 0 23 -0 14 - 0

15 - 0 96 + 0* 39 + 1 21 + 1 10 + 1 62+0 38-0 23 -0 15 - 0 II - 0

23 - O· 15 + 1 30 + 1 16 + 1 81 +0 46-0 27 - 0 17 - 0 II - 0* 10-0

30 - O· 19 + 1 23 + 1 12 + 1 62+0 35 - 0 20-0 12 - 0* 10 - O· II - 0

37 - O· 21 + 1 18 + 1 93 +0 48 -0 27 -0 15 - O· 10 - O· 10 - 0* 11 - o·

43 - O· 20 + 1 14 + 1 73 +0 38 -0 21-0 12 - O· 10 - O· 10 - O· 12 -0

47 - O· 17 + 1 II + 1 59+0 30-0 16 - 0* 10 - O· 10 - O· 10 - O· 12 - 0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

65 + 57 + 49 40 30 16 27 23 + 10 + 73 +

0 0 0 0 0 0 0 1 1 0

36 24 13 36 35 + 23 + 13 + 92 + 69 + 53 +

0 0 0 0 1 1 1 0 0 0

16 13 61 + 45 + 28 + 15 + 92 + 60 + 40 26 -

0 0 1 1 1 1 0 0 0 0

12 - 0 34-0 27 + 1 37 + 1 20+ 1 10 + 1 64+0 40-0 26 - 0 16 - 0

16 - 0 68 +0* 41 + 1 27 + 1 14 + 1 77+0 45 -0 27 -0 17 - 0 13 -0

22 - O· 10+1* 35 + 1 20+ 1 10 + 1 56+0 33 - 0 20-0 13 - 0* II - 0

29 - O· 13+1 27 + 1 15 + 1 78+0 42 -0 24-0 15 - 0* 10 - 0* 12 - 0

34 -0* 15 + 1 21 + 1 II + 1 60+0 33 -0 19 - O· 11 - 0* 10 - 0* 12 -0

38 - O· 15 + 1 16 + 1 90+0 47 -0 26 -0 14 - O· 10 - O· 10 - 0* 13 -0

41 - O· 15 + 1 13 + 1 72+0 37 - 0 20 - O· 11 - O· 10 - O· 10 - O· 13 - 0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

66 + 57 + 49 40 30 17 13 28 + 11 + 78 +

0 0 0 0 0 0 0 1 1 0

39 25 14 28 35 + 25 + 14 + 97 + 71 + 54 +

0 0 0 0 1 1 1 0 0 0

17 14 50 + 45 + 31 + 17 + 10 + 64 + 42 27 -

0 0 0 1 1 1 1 0 0 0

13 - 0 32 - 0 22 + 1 40 + 1 22 + 1 11 + 1 69+0 43 -0 27 -0 17 - 0

17 - 0 61 + O· 37 + 1 30+ 1 16 + 1 85 +0 49-0 30-0 18 - 0 14- 0

23 -0 92 +0' 36 + 1 22+ 1 II + 1 62+0 36-0 22 -0 14 - O· 12 - 0

28 - O· II + 1 28 + 1 16 + 1 87+0 47 -0 27 -0 16 - 0* 11 - 0* 13 - 0

33 - O· 13 + 1 22+ 1 12 + 1 67+0 36-0 21 - O· 12 - 0* 10 - O· 13 - 0

37 - O· 14 + 1 17 + 1 99+0 52+0 28 -0 16 - O· 10 - O· 10-0 14 -0

40 - O· 13 + 1 14 + 1 78+0 42 -0 23 - 0* 13 - O· 10 - 0* 10-0 14 - 0

32 12 + 23 + 13 +

36 -0' 13 + 1 18 + 1 10 + 1

38 - O· 13 + 1 14 + 1 83 +0

4.167+040-0 4.2 57 + 0 26 - 0 4.3 49 - 0 15 - 0 4.4 40 - 0 25 - 0

18-0 14 - 0 17 - 0 14 - 0 31 - 0 59 +0' 45 - 0 19 + 1 35 + 1 43 + 1 42 + 1 32 + 1

23 -0 28 - O· 86+0 11 + 1 35 + 1 29 + 1 23 + 1 17 + 1

O· 1 1 1

Table 6.3. (continued) The effective principal quantum number no LIn 0.5

2.5

3.0

3.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

93 + 1 12 +2 13 +2 14+2 13 +2

36-0 16 + 1 38+ I 64+ 1 89+ 1 10+2 11 +2 11 +2 97 + 1 74+ 1

73+0 28+ 1 55 + 1 79+ 1 96+ 1 10+2 96+ 1 82+ 1 63 + 1 43 + 1

11 + 38 + 66+ 86+ 94+ 91 + 80+ 63 + 45 + 28 +

27+2 19+2 11 +2 55 + 1 15 + 1

19+2 18+2 16+2 11 +2 74+ 1 30+ 1 89+0 86+0 36+ 1 79+ 1

10+2 76+ 1 45 + 1 22+ 1 97+0 84+0 26+ 1 61 + 1 84+ 1 92+ 1

50 + 1 29+ 1 14 + 1 84+0 10 + 1 30 + 1 60+ 1 70+ 1 68 + 1 58 + 1

1.0 1.5 2.0 Transition s - d, parameter A

1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

4.0

4.5

5.0

I I I 1 1 1 1 1 1 1

15 + 1 47+ 1 74+ I 88 + I 89+ 1 80+ 1 66+ 1 49+ 1 33 + 1 20+1

20+ I 55 + 1 79+ 1 87 + 1 83 + 1 71 + 1 56+ 1 39+ 1 25 + 1 15 + 1

24+ I 61.+ 1 81 + 1 . 84+ 1 77 + 1 63 + 1 47+ 1 32+ 1 20+ 1 12 + 1

26+ 1 14 + 1 86+0 91 +0 20+ 1 46+ 1 56+ 1 53 + 1 44+ 1 34+ 1

16 + 1 96+0 80+0 12 + 1 30+ 1 46+ 1 46+ 1 39+ 1 31 + 1 22+ 1

11 + 1 80+0 88+0 17 + 1 34+ 1 40+ 1 36+ 1 29+ 1 21 + 1 15 + 1

94+0 75+0 10+ 1 20+ 1 33 + 1 34+ 1 29+ 1 22 + 1 16+ 1 17 + I"

81 +0 76+0 11.+ 1 22+ 1 30+ 1 28+ 1 23 + 1 17 + 1 12 + 1 86+0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

49+2 26+2 10+2 25 + 1 66 -1 69+0 24+ 1 38 + 1 43 + 1 40+ 1

31- 0 36 + 1 91 + 1 14+2 16+2 16+2 13+2 93 + 1 51 + 1 19 + 1

10+2 12+2 11 +2 10+2 75 + 1 37 + 1 13+ 1 65 +0 19 + 1 55 + 1

85 + 1 68 + 1 47 + 1 27 + 1 13 + 1 75+0 12 + 1 39+ 1 68 + 1 81 + 1

43 + 1 29+ 1 16 + 1 93 +0 77+0 15 + 1 39+ 1 58 + 1 59 + 1 53 + 1

23 + 1 14+ 1 87+0 70+0 10 -i- 1 25 + 1 42+ 1 43 + 1 38 + 1 30+ 1

14 + 1 89+0 65+0 76+0 14 + 1 29+ 1 35 + 1 32 + 1 26+ 1 19 + 1

97+0 66+0 61 +0 89+0 17 + 1 27 + 1 27 + 1 23 + 1 18 + 1 12 + 1

73+0 56+0 61 +0 10 + 1 18 + 1 23 + 1 21 + 1 17 + 1 13 + 1 93+0

60+0 51 +0 63 +0 10 + 1 17 + 1 19 + 1 17 + 1 13 + 1 10 + 1 70+0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

23+2 14+2 63 + 1 17 + 1 91-1 35 -0 14 + 1 24+ 1 30+ 1 29+ 1

24-0 20+ 1 62+ 1 10+2 13+2 13 +2 11 +2 85 + 1 49+ 1 21 + 1

79 + 1 10+2 10+2 96+ 1 75 + 1 40+ 1 16 + 1 67+0 13 + 1 44+ 1

78 + 1 65 + 1 47 + 1 29+ 1 14 + 1 79+0 97+0 29+ 1 60+ 1 75 + 1

40+ 1 28 + 1 17 + 1 98+0 71 +0 11 + 1 31 + 1 51 + 1 55 + 1 50+ 1

21 + 1 13 + 1 86+0 64+0 84+0 18 + 1 35 + 1 39+ 1 35+ 1 28+ 1

13+ 1 84+0 60+0 62+0 10+ 1 22 + 1 30+ 1 28 + 1 23 + 1 17 + 1

87+0 60+0 51 +0 67+0 12 + 1 21 + 1 23 + 1 20+ 1 16 + 1 11 + 1

64+0 48-0 49-0 72+0 13 + 1 18 + 1 18 + 1 15 + 1 11 + 1 84+0

50+0 42-0 48-0 75+0 12 + 1 15 + 1 14 + 1 11 + 1 87+0 62+0

4.1 4.2 4.3 4.4

17 +2 10+2 50+ 1 14+ 1

28-0 14+ 1 51 + 1 90+ 1

69+ 90+ 98 + 92 +

74+ 63 + 47 + 29+

39 + 1 27 + 1 17 + 1 10+ 1

20+ 1 13 + 1 84+0 61 +0

12 + 1 80+0 57+0 56+0

81 +0 56+0 47 -0 57+0

58+0 44-0 42-0 59+0

45 -0 37 -0 40-0 60+0

1 1 1 1

1 1 1 1

186

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.3. (continued) The effective principal quantum number no LIn 0.5

2.5

3.0

3.5

4.0

4.5

5.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

14+ 1 20+ 1 21 + 1 19 + 1 16 + 1 13 + 1 11 + 1 88+0 67+0 49-0 36-0 29-0 42-0 11 + 1 30 + 1 36+ 1 25 + 1 17 + 1 11 + 1 79+0

15 + 1 21 + 1 20+ 1 17 + 1 14 + 1 11 + 1 91 +0 69+0 51 +0 36-0 27 -0 29-0 58+0 16 + 1 32 + 1 28+ 1 19 + 1 12 + 1 85+0 58+0

18 + 1 21 + 1 18 + 1 14 + 1 11 + 1 83+0 62+0 44-0 31-0 23 -0

15 + 1 14+ 1 13 + 1 11 + 1 85+0

12 + I 19 + 1 21 + 1 20+ 1 18 + 1 16+ 1 13 + 1 11 + 1 91 +0 70+0 51 +0 37-0 34-0 66+0 21 + 1 40+ 1 34+ 1 23 + 1 16 + 1 11 + 1

17 + 1 21 + 1 19 + 1 16 + 1 12 + 1 98+0 74+0 55+0 39-0 28 -0

16+ 1 15 + 1 14+ 1 12 + 1 10 + 1 77+0 44-0 82+0 24+ 1 27+ 1

19 + 1 18 + 1 17 + 1 16 + 1 14 + 1 12 + 1 10+ 1 81 +0 57+0 41- 0 68+0 25 + 1 37 + 1 31 + 1 24+ 1

10 + 1 17 + 1 20+ 1 20+ 1 20+ 1 18 + 1 16 + 1 14 + 1 12 + 1 10 + 1 80+0 59+0 42 -0 39 -0 90+0 29+ 1 41 + 1 32 + 1 23 + 1 16 + 1

23 -0 31 - 0 74+0 19 + 1 29+ 1 22+ 1 14 + 1 97+0 65 +0 10 - O·

21 -0 36-0 89+0 21 + 1 25 + 1 18 + 1 11 + 1 76+0 50+0 35 - 0

1.0 1.5 2.0 Transition s - d, parameter X

1.1

1.2 1.3

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

15 + 1 15 + 1 15 + 1 13 + 1 70+0 19 + 1 17 + 1 16 + 1 15 + 1 14 + 1

64+0 19 + 1 19 + 1 18 + 1 17 + 1 16 + 1 15 + 1 13 + 1 11 + 1 90+0

24+ 1 21 + 1 18+ 1 15 + 1 12 + 1 93+0 58+0 51 +0 17 + 1 29+ 1

18 + 1 14 + 1 10 + 1 78+0 53 +0 50+0 13 + 1 36+ 1 37 + 1 29+ 1

12 + 1 86+0 60+0 46-0 61 +0 17 + 1 40 + 1 39+ 1 28 + 1 20 + 1

79+0 55+0 42-0 52+0 12 + 1 32 + 1 40+ 1 29+ 1 20+ 1 13 + 1

54+0 40-0 41- 0 76+0 19 + 1 36+ 1 31 + 1 21 + 1 14 + 1 96+0

41-0 35 -0 48-0 10+ 1 24+ 1 32+ 1 24+ 1 15 + 1 10+ 1 70+0

33 -0 34-0 57+0 13 + 1 25 + 1 26+ 1 18 + 1 12 + 1 79+0 53+0

28 -0 35 -0 66+0 14 + 1 24+ 1 21 + 1 14+ 1 93 +0 61 +0 42-0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

15 + 1 15 + 1 14+ 1 13 + 1 71 +0 21 + 1 18 + 1 16 + 1 15 + 1 14 + 1

40-0 19 + 1 21 + 1 19 + 1 18 + 1 17 + 1 15 + 1 13 + 1 11 + 1 92+0

28+ 1 23 + 1 19 + 1 16+ 1 13 + 1 10+ 1 64+0 48-0 14 + 1 29+ 1

22 + 1 16 + 1 12 + 1 88+0 60+0 49-0 10 + 1 31 + 1 40+ 1 31 + 1

14 + 1 10 + 1 70+0 51 +0 57+0 14 + 1 36 + 1 42+ 1 31 + 1 22+ 1

94+0 65+0 48-0 51 +0 10+ 1 26+ 1 41 + 1 33 + 1 22+ 1 15 + 1

65+0 47-0 43 -0 68+0 15 + 1 32+ 1 34+ 1 24+ 1 16 + 1 10 + 1

49-0 39-0 47+0 89+0 20+ 1 31 + 1 26+ 1 17 + 1 11 + 1 78+0

39-0 36-0 53+0 10+ 1 22+ 1 26+ 1 20+ 1 13 + 1 88+0 59+0

33 -0 36-0 59+0 12 + 1 21 + 1 22 + 1 15 + 1 10+ 1 68+0 46-0

4.1 4.2 4.3 4.4

15 + 1 15 + 1 14+ 1 12 + 1

39-0 17 + 1 21 + 1 20+ 1

30+ 1 25 + 1 20+ 1 17 + 1

23 + 1 17 + 1 13 + 1 94+0

15 + 1 10+ 1 76+0 54+0

10+ 1 71 +0 52+0 51 +0

72+0 51 + 0 45 -0 64+0

54+0 42-0 46-0 82+0

42-0 38-0 51 +0 99+0

36 -0 36-0 56+0 11 + 1

6.1 Tables of Numerical Results

187

Table 6.3. (continued) The effective principal quantum number Lin

1.5

2.0

2.5

no

3.0

3.5

4.0

4.5

5.0

Transition p - s, parameter A 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10 + 1 16 + 1 18 + 1 16 + 1 12 + 1 84+0 45 -0 21 - 0 16 - 0 51 + 0

17 + 1 24+ 1 24+ 1 21 + 1 15 + 1 99+0 55 +0 29 -0 21 - 0 37 -0

21 + 1 28 + 1 27 + 1 22 + 1 15 + 1 99+0 55 + 0 30- 0 26-0 49 -0

25 + 1 31 + 1 28 + 1 22 + 1 15 + 1 95 +0 53 +0 31 - 0 27 -0 45 - O·

28 + 1 32 + 1 28 + 1 21 + 1 14+ 1 89+0 50+0· 31 - 0 28 - O· 48 - O·

30 + 1 33 + 1 28 + 1 21 + 1 14+ 1 84 +0' 48 - O· 30 -0 28 - O· 45 - O·

32 + 1 33 + 1 27 + 1 20 + 1 13 + 1 79+ O· 45 - O· 30-0 28 - O· 45 - O·

33 + 1 33 + 1 27 + 1 19 + 1 12 + 1 74+0· 43 - O· 29 - O· 28 - O· 43 - O·

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 l.9 2.0

92+0 12 + 1 12 + 1 10+ 1 75 +0 48 - 0 27 -0 15 - 0 19 - 0 71 + 0

13 + 1 15 + 1 13 + 1 10 + I 76+0 51 + 0 31 - 0 20 -0 20-0 39-0

15 + 1 15 + 1 12 + 1 96+0 69+0 46 -0 29- 0 21 - 0 23 - 0 51 + 0

14 + 1 14 + 1 11 + 1 84+0 60+0 40 -0 27 - 0 20-0 22 - o· 39-0

13 + 1 12 + 1 99+0 74+0 53 +0 36 - 0 24-0 19 - 0 21 - O· 39 - O·

11 + 1 11 + 1 87 +0 65 +0 46-0 32 -0 22 -0 18 - 0 20 - O· 33 - O·

10 + 1 98 +0 77 +0 58+0 41- 0 29 - O· 20-0 17 - 0 19 - O· 31 - O·

94+0 86+0 68+0 51 + 0 37 - O· 26 - O' 19 - 0 16 - 0 17 - O· 28 - 0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

85+0 11 + 1 11 + 1 89+0 65+0 41 - 0 23 -0 14 - 0 22 -0 79+0

11 + 1 12 + 1 11 + 1 85 +0 61 +0 41 - 0 26 -0 18 - 0 19 - 0 39 -0

12 + 1 11 + 1 97 +0 73 +0 52+0 35 - 0 24- 0 18 - 0 23 -0 53 +0

10 + 1 99+0 80+0 60+0 43 -0 30- 0 21 - 0 17 - 0 19 - O· 36-0

89+0 83 +0 67+0 50+0 36-0 26-0 18 - 0 16 - 0 19 - O· 36-0

74+0 69+0 56+0 42 -0 31 - 0 22 -0 16 - 0 14 - 0 16 - O' 28 -0

62+0 58+0 47 - 0 36-0 27 -0 19 - O· 14 - 0 13 - 0 15 - O· 26-0

53 +0 49 -0 40-0 31 - 0 23 - O· 17 - O· 13 - 0 12 - 0 14 - O· 22 -0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

83 +0 10 + 1 10 + 1 85 + 0 61 +0 38 - 0 21 - 0 14 - 0 24-0 83 +0

10 + 1 11 + 1 10 + 1 77+0 56+0 38 - 0 24- 0 17 - 0 19 - 0 39 - 0

10 + 1 10 + 1 85 +0 64+0 46 -0 31 - 0 21 - 0 17 - 0 22 - 0 55 +0

89+0 83 +0 68+0 51 + 0 37 -0 26-0 19 - 0 16 - 0 18 - O· 35 - 0

72+0 67+0 55+0 41 - 0 30-0 21- 0 16 - 0 14 - 0 17 - O· 34-0

58+0 54+0 44 - 0 34-0 25 -0 18 - 0 14 - 0 12 - 0 15 - O· 25 -0

47 - 0 44 - 0 36-0 28 -0 21- 0 15 - 0 12 - 0 11 - 0 13 - O· 23 -0

39- 0 37 - 0 30- 0 24-0 18 - O· 13 - O' 11- 0 10 - 0 12 - O· 19 - 0

4.1 4.2 4.3 4.4

81 + 0 10 + 1 10 + 1 82+0

10 + 1 10 + 1 96+0 73 +0

10 + I 95 +0 79+0 59+0

82+0 75 +0 62+0 46 -0

64+0 59+0 49-0 37 -0

50+0 47 - 0 38 - 0 30 -0

40 38 31 24

33 -0 30- 0 25 -0 20-0

-0 - 0 - 0 -0

188

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.3. (continued) The effective principal quantum number no Lin

1.5

2.0

2.5 3.0 Transition p - s, parameter X

3.5

4.0

4.5

5.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10 + 1 97 +0 81 +0 66+0 51 + 0 37 -0 25 -0 17 - 0 29-0 18 + 1

10 + 1 89+0 71 +0 54+0 40+0 28 -0 19 - 0 13 -0 21 - 0 93 +0

10 + 1 82+0 62+0 45 -0 32-0 22-0 14 - 0 11- 0 23 -0 10+ 1

10 + 1 75+0 54+0 38 - 0 26-0 17 - 0 ll- 0 10- 0 20-0 84 +0'

97+0 68+0 47 -0 32 -0 21 - 0 13 - 0 10- O· 10-0 19 - O· 81 + O·

93 +0 62+0 41 - 0 27 -0 18 - 0 11 - o· 10 - O· 10-0 18 - O· 71 + O·

89+0 57+0 37 -0 24-0 15 - 0 10 - O· 10 -0· 10- 0 17 - O· 66+0'

85 +0 52+0 33 -0 20-0 12 - 0 10 - O' 10 - O· 10 - O· 15 - O· 60+0'

1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

39 + 1 33 + 1 21 + 1 12 + 1 81 +0 51 + 0 32 -0 26-0 64+0 30+ 1

46 + 1 34 + 1 19 + 1 11 + 1 65 +0 40-0 26-0 22 -0 39 -0 14 + 1

45 + 1 31 + 1 17 + 1 91 +0 53 +0 32- 0 22 -0 22 -0 45 -0 17 + 1

39 + 1 27 + 1 14 + 1 76+0 43 -0 26- 0 18 - 0 19 - 0 38 - O· 12 + 1

33 + 1 23 + 1 12 + 1 63 +0 35 - 0 21 - 0 15 - 0 17 - 0 37 - O· 12 + I·

28 + 1 19 + 1 10 + 1 52+0 29-0 17 -0 13 -0 15 - 0 32 - O· 10 + 1·

24+ 1 16 + 1 85 +0 44-0 24-0 14 - O· 11- 0 14 - 0 30 - O· 93 +0'

20+ 1 13 + 1 71 +0 37 -0 20 - O· 12 - O· 10-0 12 - 0 27 - O· 80+0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

42 + 1 36+ 1 24+ 1 14 + 1 89+0 55+0 35 - 0 31 - 0 89+0 34+ 1 43 + 1 37 + 1 25 + 1 15 + 1 92+0 57+0 36-0 34-0 10 + 1 35 + 1 43 + 1 38 + 1 25 + 1 15 + 1

50+ 1 37 + 1 22 + 1 12 + 1 72+0 45 -0 30-0 27 -0 48 -0 16 + 1 52 + 1 39 + 1 23 + 1 13 + 1 76+0 47 -0 32 -0 29 -0 53 +0 18 + 1 53 + 1 39 + 1 24+ 1 13 + 1

49+ 1 34 + 1 19 + 1 10 + 1 59+0 36-0 26-0 28-0 59+0 22+ 1 50 + 1 35 + 1 20+ 1 10+ 1 63 +0 39-0 28 -0 32 -0 68+0 25 + 1 51 + 1 35 + 1 20+ 1 11 + 1

42 + 1 29 + 1 16 + 1 86+0 49-0 30-0 22-0 24-0 47 - O· 14 + 1 43 + 1 29 + 1 16 + 1 90+0 52+0 32 -0 24-0 27 -0 53 + O· 16 + 1 43 + 1 30+ 1 17 + 1 93 +0

35 + 1 24+ 1 13 + 1 71 +0 40-0 25 -0 19 - 0 23 -0 47 - O· 14 + 1 35 + 1 24+ 1 13 + 1 75+0 43 -0 27 -0 21 - 0 26-0 53 + O· 16 + 1 36 + 1 24+ 1 14 + 1 77+0

28 + 1 20+ 1 11 + 1 59+0 33 - 0 21- 0 16 - 0 20-0 40 -0' 11 + 1 29+ 1 20+ 1 11 + 1 62+0 36-0 23 -0 18 - 0 23 -0 44 - O· 12 + 1 29 + 1 20+ 1 11 + 1 64+0

23 + 1 16 + 1 92+0 49 -0 28 -0 17 - O· 14 - 0 18 -0 37 - O· 10 + 1 23 + 1 17 + 1 95+0 52+0 30 -0 19 - 0 16 - 0 21 - 0 42 - O· 11 + 1 24 + 1 17 + 1 97+0 54+0

19 + 1 14 + 1 77+0 41 - 0 23 - O· 15 - O· 12 - 0 16 - 0 33 - O· 90+0 19 + 1 14 + 1 80+0 44- 0 25 - O· 16 - O· 14 - 0 18 - 0 36 - O· 97+0 19 + 1 14 + 1 82+0 45 - 0

189

6.1 Tables of Numerical Results Table 6.3. (continued) The effective principal quantum number Lin

1.5

2.0

2.5

no

3.0

3.5

4.0

4.~

5.0

Transition p - p; parameter A ,,=0 0.5 0.6 0.7 0.8 0.9 1.0

14+2 18+2 20+2 20+2 18+2 13 +2

26+2 29+2 29+2 25+2 18+2 11 +2

34+2 36+2 33+2 26+2 17 +2 92+ 1

40+2 39+2 33 +2 24+2 15 +2 75 + 1

44+2 41 +2 33+2 23 +2 13 +2 62 + 1

47+2 42+2 32+2 21 +2 11 +2 53 + 1

49+2 42+2 31 +2 20+2 10+2 46+ 1

50+2 41 +2 30+2 18+2 97 + 1 40+ 1

1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

71 + 1 27 + 1 79+0 97+0 31 + 1 68 + 1 10+2 13+2 13+2 11 +2

50+ 1 17 + 1 12 + 1 32 + 1 77 + 1 12+2 15 +2 15 +2 12+2 83 + 1

37 + 1 15 + 1 19 + 1 56+ 1 11 +2 15 + 2 16+2 14+2 10+2 64+ 1

29+ 1 14+ 1 26+ 1 75 + 1 12+2 15 +2 15 +2 12+2 85 + 1 47 + 1

23 + 1 14 + 1 33 + 1 89 + 1 13 +2 15 +2 13 +2 10+2 69+ 1 37 + 1

20+ 1 14+ 1 39+ 1 97+ 1 13 +2 14+2 12 +2 92+ 1 57 + 1 29+ 1

17 + 1 15 + 1 43 + 1 10+2 13 +2 13 +2 11 +2 81 + 1 49+ 1 24+ 1

15 + 1 15 + 1 46+ 1 10+2 12+2 12+2 10+2 71 + 1 42+ 1 20+ 1

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

67+ 1 30+ 1 97+0 74+0 22+ 1 53 + 1 98+ 1 11 +2 12+2 11 +2

43 + 1 17 + 1 10+ 1 22+ 1 56+ 1 98+ 1 12+2 13+2 11 +2 74+ 1

30 + 1 13 + 1 14 + 1 36+ 1 77 + 1 11 +2 12+2 11 +2 88 + 1 55 + 1

21 + 1 11 + 1 17 + 1 46+ 1 87 + 1 11 +2 11 +2 94+ 1 66+ 1 38 + 1

16 + 1 10+ 1 19 + 1 51 + 1 88+ 1 10+2 97 + 1 77 + 1 52+ 1 29+ 1

13 + 1 97+0 20+ 1 53 + 1 84+ 1 92+ 1 83 + 1 64+ 1 41 + 1 22+ 1

11 + 1 91 +0 21 + 1 54+ 1 78+ 1 82+ 1 72+ 1 54+ 1 34+ 1 18 + 1

95+0 88+0 21 + 1 52+ 1 72+ 1 74+ 1 63 + 1 46+ 1 28+ 1 14+ 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

66+ 1 31 + 1 10 + 1 69+0 19+ 1 48+ 1 83 + 1 11 +2 12+2 11 +2

39+ 1 17 + 1 10+ 1 20+ 1 49+ 1 88+ 1 11 +2 12+2 10+2 70+ 1

26+ 1 12 + 1 12 + 1 30 + 1 65 + 1 98 + 1 11 +2 10+2 80+ 1 50+ 1

18 + 1 10+ 1 14+ 1 36+ 1 71 + 1 93 + 1 95 + 1 81 + 1 58 + 1 34+ 1

14 + 1 88+0 14 + 1 39+ 1 69+ 1 83 + 1 88 + 1 65 + 1 45 + 1 25 + 1

10+ 1 78+0 15 + 1 39+ 1 64+ 1 73,+ 1 67 + 1 52+ 1 34+ 1 19 + 1

88+0 71 +0 15 + 1 38+ 1 59+ 1 63 + 1 57 + 1 43 + 1 28+ 1 15 + 1

73+0 65+0 14+ 1 36+ 1 53 + 1 56+ 1 48+ 1 36+ 1 23 + 1 12 + 1

4.1 4.2 4.3 4.4

65 + 1 32+ 1 11 + 1 67+0

38+ 1 16+ 1 10+ 1 18 + 1

25 + 12 + 11 + 27 +

17 + 1 96+0 12 + 1 31 + 1

12 + 1 80+0 12 + 1 33 + 1

96+0 69+0 12 + 1 32+ 1

76+0 60+0 12 + 1 31 + 1

62+0 54+0 11 + 1 29+ 1

1 1 1 1

190

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.3. (continued) The effective principal quantum number no

An

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Transition p - p; parameter X

K=O 0.5 0.6 0.7 0.8 0.9 1.0

12 + 14 + 15 + 16 + 17 + 17 +

1 1 1 I 1 I

16+ 1 17 + 1 17 + I 17 + I 16 + I 14+ I

17 + 18 + 17 + 16 + 14 + II +

1 I 1 I 1 I

18 + 1 17 + I 16 + I 14 + I II + I 85+0

17 + 1 16 + I 14 + 1 12 + I 96+0 67+0

17 + 1 15 + I 13 + 1 10 + I 80+0 53+0

16 + 1 14+ I II + I 94+0 68+0 43 -0

15 + 1 13 + I 10+ 1 82+0 58+0 35 -0

l.l

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

15 + I 10 + 1 45 -0 49-0 10+ 1 14+ 1 17 + 1 19 + 1 19 + 1 19 + 1

10 + I 59+0 55+0 12 + 1 20+ 1 23 + 1 24+ 1 23 + 1 20+ 1 17 + 1

73 +0 48 -0 88+0 20+ I 27 + 1 27 + 1 24+ 1 21 + 1 17 + 1 13 + 1

54-0 44-0 II + I 25 + I 29+ 1 26+ 1 22 + 1 18 + 1 14 + 1 10 + 1

41-0 44-0 14 + I 28 + I 29+ 1 25 + 1 20+ I 15 + 1 II + 1 78+0

32-0 45 -0 16 + 1 30+ I 28 + 1 22 + 1 17 + I 13 + I 94+0 61 +0

26-0 45-0 17 + I 29+ 1 26+ I 20+ 1 15 + 1 11 + 1 78+0 49-0

21 - 0 46-0 18 + 1 28 + 1 24+ 1 18 + 1 13 + 1 97+0 65+0 40-0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

17 + 1 12+ 1 61 +0 44-0 93+0 14+ 1 17 + 1 19 + 1 20+ 1 20+ 1

12 + 1 73+0 58+0 II + I 20+ 1 25 + 1 26+ 1 24+ 1 22+ 1 18 + 1

87+0 58+0 83 +0 19 + I 28 + I 29+ 1 27 + 1 23 + 1 18 + 1 14 + 1

64+0 51 + 0 10 + 1 24+ 1 32 + 1 29 + 1 25 + 1 19 + 1 15 + 1 10 + 1

49-0 7-0 12 + 1 27 + 1 32 + 1 27 + 1 22 + I 16 + 1 12 + 1 83 +0

38 -0 45 -0 13 + 1 28+ 1 30+ 1 25 + 1 19 + 1 14 + 1 10+ 1 65 +0

31 - 0 43 -0 13 + 1 28 + 1 28 + 1 22+ I 16 + 1 12 + 1 82+0 52+0

25 -0 41-0 14 + 1 27 + 1 26+ 1 20+ 1 14+ 1 10+ 1 69+0 42-0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

18 + 1 13 + 1 68+0 44-0 89+0 14 + 1 17 + I 19 + 1 20+ I 20+ I

13 + 1 79+0 61 +0 II + 1 20+ 1 25+ I 27 + 1 25 + 1 23 + 1 18 + I

94+0 63 +0 84+0 19 + 1 29 + 1 31 + 1 28 + I 24+ I 19 + I 14 + I

69+0 54+0 10 + 1 23 + I 32 + I 31 + I 26+ 1 20+ I 15 + I 11 + I

53 +0 49-0 11 + 1 26+ 1 33 + 1 29+ 1 23 + I 17 + 1 12 + I 86+0

41 - 0 45 -0 12 + 1 27 + I 31 + 1 26+ 1 20+ I 14 + 1 10 + I 67+0

33 -0 43 -0 12 + 1 27 + 1 29+ I 23 + I 17 + 1 12 + I 85+0 54+0

27-0 40-0 12 + 1 26+ 1 26+ 1 20+ 1 15 + I 10 + 1 71 +0 44-0

4.1 4.2 4.3 4.4

19 + 1 14 + 1 72+0 44-0

13 + I 82+0 62+0 II + 1

97+0 66+0 85 +0 18 + I

72+0 56+0 99+0 23 + I

55+0 50+0 II + 1 25 + I

43 -0 46-0 II + 1 26+ I

35 -0 43 -0 11 + 1 26+ 1

28 -0 40-0 12 + 1 25 + I

191

6.1 Tables of Numerical Results Table 6.3. (continued) The effective principal quantum number .dn

l.5

2.0

2.5

no

3.0

3.5

4.0

4.5

5.0

Transition p - p, parameter A

K=2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l.0

13 -0 63+0 15 + 1 25 + 1 36 + I 42 + 1 42 + 1 37 + 1 28 + 1 17 + 1

42-0 16 + 1 33 + 1 48 + 1 57 + I 56 + I 48 + 1 35 + 1 22 + 1 12 + I

77+0 27 + I 49+ 1 63 + 1 65 + 1 58 + 1 45 + 1 30 + 1 18 + 1 10 + 1

11 + 1 38 + I 61 + 1 71 + 1 68 + 1 56 + 1 40+ 1 26+ 1 15 + 1 95 +0

16 + I 47 + 1 69 + 1 75 + 1 67 + 1 52 + 1 36 + 1 22 + 1 13 + I 91 +0

21 + 1 56 + 1 75 + 1 76+ 1 64 + I 48 + I 32 + 1 20 + 1 12 + 1 90+0

25 + 1 63 + 1 79+ I 75 + 1 61 + I 45 + 1 29+ I 18 + 1 11 + I 89+0

30 + 1 69 + 1 81 + 1 74+ 1 58 + 1 41 + 1 27 + 1 16 + 1 10 + 1 88+0

1.1 l.2

l.6 l.7 l.8 l.9 2.0

78+0 36-0 34-0 82+0 18 + 1 28 + 1 33 + 1 32 + 1 26+ I 17 + 1

67+0 61 +0 11 + 1 24+ 1 37 + 1 41 + 1 37 + 1 28+ 1 18 + I 11 + I

73 +0 95 +0 19 + 1 36 + 1 43 + 1 40+ 1 31 + 1 22 + I 14 + 1 92+0

83 + 0 12 + 1 26+ 1 41 + 1 42 + 1 35 + 1 26+ 1 17 + 1 11 + I 82+0

92+0 15 + 1 30 + 1 41 + 1 37 + 1 29 + 1 21 + 1 14 + I 99+0 75 +0

99+0 16 + 1 31 + 1 38 + 1 33 + 1 25 + 1 18 + 1 12 + 1 87 +0 70+0

10 + 1 17 + 1 31 + 1 34 + 1 29+ 1 21 + 1 15 + 1 10 + 1 77 +0 66+0

10 + 1 18 + 1 29+ 1 31 + 1 25 + 1 19 + 1 13 + 1 94+0 70+0 63 +0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.8 3.0

83 +0 41 - 0 33 - 0 66+0 15 + 1 25 + 1 31 + 1 31 + 1 26+ 1 18 + 1

68+0 60+0 98+0 20 + 1 32 + 1 37 + 1 33 + 1 26+ 1 17 + 1 10 + 1

70+0 85 + 0 15 + 1 28 + I 36 + 1 34 + I 27 + 1 19 + 1 13 + 1 87+0

74+0 10 + 1 19 + 1 31 + 1 33 + 1 28 + 1 21 + 1 15 + 1 10 + 1 75 +0

77 +0 11 + 1 20 + I 29 + 1 28 + 1 23 + 1 17 + 1 12 + 1 86+0 67 +0

78+0 11 + I 20+ I 26 + 1 24+ 1 19 + 1 14 + 1 10 + 1 73 +0 61 +0

77+0 11 + 1 19 + 1 23 + 1 20 + 1 15 + 1 11 + 1 84+0 63 +0 56+0

76+0 11 + 1 17 + 1 19 + 1 17 + 1 13 + 1 98+0 72+0 56+0 52+0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

86+0 43 -0 32 -0 60+0 14 + 1 24+ 1 30 + 1 31 + 1 26 + 1 18 + 1

67 +0 60-0 93+0 18 + 1 30 + 1 35 + 1 32 + 1 25 + 1 16 + 1 10 + 1

68+0 81 + 0 14 + 1 25 + 1 33 + 1 31 + 1 25 + 1 18 + 1 12 + 1 84+0

70+0 94+0 16 + 1 26 + 1 30 + 1 25 + 1 19 + 1 14 + 1 97 +0 72+0

69+0 99+0 17 + 1 24+ 1 24+ 1 20+ 1 15 + 1 11 + 1 79+0 62+0

67+0 98+0 16 + 1 21 + 1 20 + 1 16 + 1 12 + 1 88+0 66+0 56+0

65 +0 95 +0 14 + 1 18 + 1 16 + 1 13 + 1 99+0 73 +0 56+0 50+0

61 +0 90+0 13 + 1 15 + 1 13 + 1 10 + 1 82+0 61 +0 49- 0 46 - 0

4.1 4.2 4.3 4.4

87 +0 44 -0 32 -0 58+0

67 +0 59+0 90+0 17 + 1

66+0 78+0 13 + 1 23 + 1

67 +0 88 +0 15 + 1 24 + 1

65 + 0 90+0 15 + 1 22 + 1

61 +0 88+0 14 + 1 18 + 1

58+0 83 +0 12 + 1 15 + 1

54+0 77 +0 11 + 1 13 + 1

1.3

1.4 1.5

192

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.3. (continued) The effective principal quantum number

no 3.5

4.0

4.5

5.0

12 + 1 19 + 1 21 + 1 19 + 1 16 + 1 13 + 1 10 + 1 78+0 56+0 47 -0

14 + 1 20 + 1 21 + 1 18 + 1 14 + 1 11 + 1 85+0 61 +0 46 -0 42 -0

16 + 1 21 + 1 20+ 1 16 + 1 12 + 1 96+0 70+0 50+0 38 -0 40-0

17 + 1 21 + 1 19 + 1 15 + 1 11 + 1 82+0 58+0 41 - 0 33 - 0 38 - 0

18 + 1 21 + 1 17 + 1 13 + 1 98+0 70+0 49-0 34-0 29 -0 37 - 0

56+0 10+ 1 25 + 1 40+ 1 39 + 1 30 + 1 21 + 1 14+ 1 10 + 1 76+0

62+0 13 + 1 31 + 1 42 + 1 35 + 1 24+ 1 16 + 1 11 + 1 81 + 0 67+0

67+0 15 + 1 33 + 1 39 + 1 30 + 1 20 + 1 13 + 1 91 +0 67+0 62+0

71 +0 16 + 1 33 + 1 35 + 1 25 + 1 16 + 1 10 + 1 74+0 57+0 58+0

73 +0 17 + 1 31 + 1 30 + 1 20 + 1 13 + 1 89+0 62+0 50+0 48 -0

73 +0 16 + 1 28 + 1 25 + 1 17 + 1 11 + 1 74+0 52+0 44-0 52+0

71 +0 82+0 15 + 1 30 + I 40+ 1 37 + 1 29 + 1 21 + 1 15 + 1 10 + 1

74+0 11 + 1 24+ 1 41 + I 43 + 1 34 + 1 24+ 1 16 + 1 11 + 1 86+0

80+0 14 + 1 29+ 1 43 + I 39 + 1 28 + I 19 + 1 13 + 1 93+0 77+0

85+0 16 + 1 31 + 1 40+ 1 33 + 1 23 + 1 15 + 1 10 + 1 78+0 71 + 0

86+0 16 + 1 30+ 1 35 + 1 27 + 1 18 + 1 12 + 1 86+0 67 +0 67+0

86+0 16 + 1 28 + 1 30 + 1 22 + 1 15 + 1 10+ 1 71 +0 58+0 62+0

84+0 16 + 1 26+ 1 26+ 1 18 + 1 12 + 1 84+0 60+0 51 + 0 58+0

11 + 1 71 +0 67+0 13 + 1 23 + 1 30+ 1 30 + I 27 + 1 22 + 1 17 + 1

78+0 87 +0 15 + 1 30 + 1 40+ 1 39 + 1 31 + 1 23 + 1 16 + 1 10 + 1

83 +0 12 + 1 24+ 1 41 + 1 45 + 1 36 + 1 25 + 1 17 + 1 12 + 1 91 +0

89+0 15 + 1 29 + 1 43 + 1 41 + 1 30 + 1 20+ 1 14 + 1 10 + 1 83 +0

93 +0 16 + 1 30 + 1 40+ 1 34 + 1 24+ 1 16 + 1 11 + 1 84+0 76+0

94+0 16 + 1 29 + 1 35 + 1 28 + 1 19 + 1 13 + 1 92+0 72+0 72 +0

92+0 16 + 1 27 + 1 30 + 1 23 + 1 16 + 1 10 + 1 77+0 63 +0 66+0

89+0 15 + 1 24+ 1 26+ 1 19 + 1 13 + 1 90+0 65 +0 56+0 62+0

11 + 1 74+0 67+0 12 + I

82+0 90+0 16 + I 30+ 1

88+0 12 + 1 24+ 1 41 + 1

94+0 15 + 1 28 + 1 43 + 1

98+0 16 + 1 30 + 1 40+ 1

98+0 16 + 1 29+ 1 35 + 1

96+0 16 + 1 26+ 1 30 + 1

92+0 15 + 1 24+ 1 25 + 1

Lin

1.5 2.0 2.5 3.0 Transition p - p, parameter X

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

63+0 11 + 1 14 + 1 17 + 1 18 + 1 19 + 1 18 + 1 17 + I 15 + 1 12 + 1

84+0 14 + 1 18 + 1 20+ 1 20+ 1 18 + 1 16 + 1 13 + 1 10 + 1 71 + 0

10 + 1 17 + 1 20+ 1 20+ 1 18 + 1 16 + 1 13 + 1 10+ 1 73+0 54+0

1.1

1.4 1.5 1.6 1.7 1.8 1.9 2.0

88+0 52+0 66+0 15 + 1 24+ 1 28 + 1 27 + 1 24+ 1 20+ 1 15 + 1

54+0 71 +0 16 + 1 31 + 1 37 + 1 33 + 1 26+ 1 19 + 1 13 + 1 94+0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

10 + 1 65+0 66+0 13 + 1 24+ 1 29 + I 29+ 1 26+ 1 22 + 1 17 + 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

K=2

1.2 1.3

Table 6.3. (continued) The effective principal quantum number no An

1.5

2.0 2.5 3.0 Transition p - d, parameter A

4.0

4.5

5.0

1 I 1 1 1 1 1 1 I· I·

59 + 1 82 + 1 86+ 1 81 + 1 69 + 1 54+ 1 38 + 1 24 + 1· 13 + 1· 74+0·

65 + 1 82 + 1 82 + 1 74+ 1 60+ 1 45 + 1 30 + 1· 18 + 1· 10 + I· 61 +0

68 + I 82 + 1 78 + 1 67 + 1 53 + 1 38 + 1· 25 + 1· 15 + I· 84+0· 54+0

3.5

K=I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

15 +2 16 + 2 15 + 2 12 +2 97 + 1

30 + 1 60+ 1 86 + 1 10+2 11 +2 10+2 96+ 1 76 + 1 53 + 1 32 + 1

43 + 73 + 91 + 99+ 96 + 85 + 68 + 49+ 31 + 16 +

1 1 1 1 1 I 1 1 1 I·

53 + 79+ 90 + 89 + 81 + 66+ 49 + 32 + 19 + 10 +

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

19+2 15 + 2 10+2 64+ 1 29 + 1 66 +0· 38 -0 34+ 1 56 + 1 72 + 1

59 + 1 29+ 1 11 + I· 65+0 29+ 1 68 + 1 79+ 1 83 + 1 80+ 1 69+ 1

17 + 1· 79+0· 64+0 23 + 1 52 + 1 53 + 1 49 + 1 43 + 1 34+ 1 24+ 1

83 + O· 57+0 11 + 1 41 + 1 42 + 1 39 + 1 34+ 1 28 + 1 20 + 1 12 + 1·

60+0 65 +0· 19 + 1 34 + 1 32 + 1 28 + 1 23 + 1 17 + 1 12 + 1· 73 +0·

54+0 78 + O· 25 + 1 28 + 1 25 + 1 21 + 1 17 + 1 12 + 1· 83 + O· 51 + O·

54+0 93 +0· 24+ 1 23 + 1 20+ 1 17 + 1 13 + 1· 93 +0· 60+0· 39 - O·

55 +0 10 + 1 21 + 1 19 + 1 17 + 1 13 + 1 10 + I· 73 +0· 48 - O~ 33 -0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

10+2 98 + 1 79+ 1 54+ 1 30 + 1 97+0 26-0 20+ 1 41 + 1 54 + 1

48 + 1 27 + 1 12 + 1 61 +0 10 + 1 50+ 1 65 + 1 68 + 1 67 + 1 61 + 1

15 + 1 79+0 50+0 87 + O· 39 + 1 42 + 1 39 + 1 35 + 1 29+ 1 21 + 1

71 + 45 57 + 21 + 31 + 30 + 27 + 22 + 17 + 11 +

O· 0 O· 1 1 1 1 1 1 1·

46 -0 41 - 0 78 +0· 23 + 1 23 + 1 21 + 1 17 + 1 13 + 1 98 + O· 62 + O·

36-0 42 -0 10 + 1 18 + 1 17 + 1 15 + 1 12 + 1 97 +0· 67 +0· 43 - O·

33 - 0 45 -0 11 + 1 14 + 1 13 + 1 11 + 1 94+0· 69 +0· 47 - O· 32 - O·

31 - 0 48 -0 11 + 1 12 + 1 10 + 1 93 +0 73 +0· 53 + O· 36 - O· 26 -0·

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

84+ 1 81 + 1 68 + 1 50 + 1 30 + 1 11 + 1 29-0 10 + 1 35 + 1 47 + 1

44+ 1 27 + 1 13 + 1 63 +0 73 +0 42 + 1 58 + 1 62+ 1 61 + 1 57 + 1

14+ 1 79+0· 47 - 0 65+0 32 + 1 37 + 1 35 + 1 31 + 1 26+ 1 19 + 1

66+0· 41- 0 44 - O· 13 + 1 27 + 1 26 + 1 24 + 1 20+ 1 15 + 1 10 + 1

40 -0· 33 - 0 54+0 17 + 1 19 + 1 17 + 1 15 + 1 12 + 1 87 +0· 57 + O·

30-0 32 -0 66+0 14 + 1 14 + 1 13 + 1 10 + 1 84+0· 59+0· 39 - O·

26-0 33 - O· 75 +0 11 + 1 11 + 1 96+0 78+0 59 +9· 41 - O· 28 - O·

23 -0 33 - 0 75 +0 92+0 86+0 74+0 60+0· 45 - O· 31 - O· 22 - O·

4.1 4.2 4.3 4.4

74+ 73 + 63 + 48 +

42 + 1 26+ 1 13 + 1 65+0

13 + 1 78 + O· 46-0 56+0

63 +0· 38 - 0 39-0 10 + 1

37 -0 30 -0 45 -0 14 + 1

27 -0 28 -0 52+0 12 + 1

22 -0 27 -0 58+0 99+0

20-0 26-0 58+0 78+0

1 1 1 1

Table 6.3. (continued) The effective principal quantum number no An

1.5 2.0 2.5 3.0 Transition p - d, parameter X K

3.5

4.0

4.5

5.0

= I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

65+0 57+0 49 -0 41 - 0 33 - 0

10 + 1 94+0 79+0 66+0 55 + 0 45 -0 36 -0 28 -0 21 - 0 15 - 0

10 + 1 84+0 67+0 53 + 0 42 -0 33 - 0 25 -0 19 - 0 13 - 0 10 - O·

10 + 1 76+0 58+0 44 -0 33 - 0 25 -0 18 - 0 12 - 0 10 - O· 10 - O·

96+0 69+0 50+0 37 -0 27 -0 19 - 0 13 -0 10 - O· 10 - O· 10 - O·

92+0 63 +0 44 - 0 31 - 0 22 -0 15 - 0 10 - O· 10 - O· 10 - O· 10-0

88+0 57+0 38 -0 26 -0 18 - 0 12 - o· 10 - O· 10 - O· 10 - O· 10 -0

1.6 1.7 1.8 1.9 2.0

56+0 48 - 0 41 - 0 33 - 0 24-0 11 - O· 44-0 30+ 1 15 + 1 95 +0

25 - 0 16 - 0 10 - O· 18 - 0 26+ 1 32 + 1 16 + 1 10 + 1 67 - 0 48 -0

10 - O· 10 - O· 19 - 0 21 + 1 32 + 1 16 + 1 87+0 54+0 35 - 0 23 - 0

10 - O· 11 - 0 76+0 36 + 1 19 + 1 95 +0 55 +0 34- 0 22 - O· 13 - O·

10-0 23 - O· 18 + 1 24+ 1 12 + 1 63 +0 37 -0 22 -0 13 - O· 10 - O·

10-0 40- 0 24 + 1· 16 + 1 84+0 45 -0 26-0 15 -0· 10 - O· 10 - O·

11 - 0 59+0· 21 + 1 12 + 1 61 +0 33 - 0 19 - O· 11 - O· 10 - O· 10 - O·

14 - 0 77 +0 17 + 1 92 +0 46-0 25 -0 14 - O· 10 - O· 10 - O· 10-0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

73 +0 58+0 47 -0 36-0 27 -0 14 - 0 12 -0 37 + 1 22 + 1 12 + 1

34-0 22 -0 13 - 0 12 - 0 76+0 43 + 1 25 + 1 13 + 1 81 +0 55 +0

15 - 0 10 - O· 13 - 0 76+ O· 43 + 1 23 + 1 11 + 1 68+0 42 -0 27 -0

10 - O· 10-0 37 - O· 29+ 1 26+ 1 13 + 1 72+0 43 -0 26-0 16 - O·

10 - 0 16 - 0 82 +0· 30 + 1 17 + 1 87+0 48 -0 28 -0 17 - O· 10 - O·

10-0 25 - 0 13 + 1 21 + 1 11 + 1 60+0 33 - 0 19 - O· 11 - O· 10 - O·

10-0 36-0 17 + 1 15 + 1 83 +0 44-0 24 - O· 14 - O· 10 - O· 10 - O·

12 -0 48 -0 16 + 1 11 + 1 62 +0 33 -0 18 - O· 10 - O· 10 - O· 10 - O·

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

82+0 63 +0 49-0 38 - 0 28 -0 16 - 0 10-0 20+ 1 28 + 1 14 + 1

38 - 0 25 -0 15 - 0 12 - 0 47 -0 45 + 1 30 + 1 15 + 1 90+0 59+0

18 - 0 11 - O· 12 - 0 53+0 44+ 1 28 + 1 13 + 1 76+0 47 -0 30-0

10 - O· 10-0 30 - O· 20+ 1 30 + 1 15 + 1 82+0 48 -0 29-0 18 - 0

10 - O· 14 - 0 61 +0 29 + 1 19 + 1 10 + 1 55 +0 32 -0 19 - O· 11 - O·

10-0 21 - 0 10 + 1 23 + 1 13 + 1 70+0 38 - 0 22 - O· 13 - O· 10 - O·

10- 0 30 - O· 13 + 1 17 + 1 96+0 51 +0 28-0 16 - O· 10 - O· 10 - O·

11 - 0 38 -0 14 + 1 13 + 1 71 +0 38 -0 21 - O· 12 - O· 10 - O· 10 - O·

4.1 4.2 4.3 4.4

88+0 66+0 51 +0 39-0

41 - 0 27-0 17 - 0 12 -0

19 - 0 12 - O· 12 - 0 44-0

11 - O· 10-0 26-0 16 + 1

10 - O· 14 - 0 52+0 27 + 1

10-0 20 -0 85+0 24+ 1

10-0 27 -0 11 + 1 18 + 1

10-0 34-0 13 + 1 14 + 1

1.1

1.2 1.3

1.4 1.5

6.1 Tables of Numerical Results

195

Table 6.3. (continued) The effective principal quantum number no

.dn

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

27 - 1 19 - 0 63+0 14 + 1 25 + 1 38 + 1 49+ 1 56 + 1 55 + 1 47 + 1 35 + 1 22+ 1 12 + 1 69+0 65 +0 10 + 1 20+ 1 31 + 1 39 + 1 39 + 1 33 + 1 23 + 1 14 + 1 84+0 65 +0 85 +0 15 + 1 24+ 1 33 + 1 36 + 1

66 - 1 42 -0 12 + 1 25 + 1 40 + 1 53 + 1 59 + 1 58 + 1 49 + 1 36 + 1 23 + 1 14 + 1 86+0 77 +0 II + 1 21 + 1 32 + 1 39 + 1 39 + 1 32 + 1 23 + 1 14 + 1 96+0 76+0 92+0 15 + 1 24+ 1 32 + 1 34 + 1 29 + 1

ll- 0 72+0 19 + 1 36 + 1 51 + 1 61 + 1 62 + 1 54+ 1 41 + 1 28 + 1 17 + 1 10 + 1 83 +0 10 + 1 17 + 1 29 + 1 38 + 1 38 + 1 32 + 1 24+ 1 16 + 1 11 + 1 83 +0 85 +0 12 + 1 20+ 1 28 + 1 31 + 1 28 + 1 22 + 1

18 - 0 10 + 1 27 + 1 46+ 1 60+ 1 65 + 1 59 + 1 48 + 1 34 + 1 22 + 1 13 + 1 95 +0 92+0 13 + 1 23 + 1 34 + 1 37 + 1 33 + 1 26 + 1 18 + 1 12 + 1 92+0 81 + 0 10 + 1 15 + 1 23 + 1 28 + 1 27 + 1 22+ 1 16 + 1

26-0 14 + 1 34+ 1 54+ 1 65 + 1 65 + 1 56 + 1 42 + 1 29 + 1 18 + 1 11 + 1 91 +0 10 + 1 16 + 1 27 + 1 34+ 1 34 + 1 28 + 1 21 + 1 15 + 1 10 + 1 80+0 81 + 0 10 + 1 16 + 1 23 + 1 25 + 1 22 + 1 17 + 1 13 + 1

36- 0 19 + 1 42 + 1 61 + 1 68 + 1 63 + 1 51 + 1 37 + 1 25 + 1 15 + 1 10+ 1 92+0 11 + 1 19 + 1 29 + 1 33 + 1 30 + 1 24+ 1 17 + 1 12 + 1 88+0 74+0 83 +0 II + 1 17 + 1 22+ 1 22 + 1 19 + 1 15 + 1 10 + 1

Transition p - d, parameter A

K=3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

31 + 1 34+ 1 33 + 1 28 + 1 21 + 1 11 + 1 40-0 13 - 0 22 -0 62+0 13 + 1 20+ 1 24+ 1 25 + 1 21 + 1 14 + 1 65 +0 21 - 0 15 - 0 37 -0

19 + 1 29+ 1 40+ 1 48 + 1 51 + 1 48 + 1 38 + 1 25 + 1 13 + 1 59+0 38 - 0 61 + 0 13 + 1 23 + 1 33 + 1 38 + 1 36 + 1 27 + 1 17 + 1 86+0 46-0 48-0 92+0 17 + 1 28 + 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

99+0 16 + 1 21 + 1 23 + 1 21 + 1 15 + 1 78+0 28 -0 14 - 0 29-0

34+ 1 34 + 1 28+ 1 18 + 1 99+0 52+0 45 -0 79+0 15 + 1 26+ 1

32 + 1 23 + 1 15 + 1 91 +0 66+0 78+0 13 + 1 22 + 1 30 + 1 34 + 1

22 + 1 14 + 1 98+0 75 +0 84+0 12 + 1 21 + 1 28 + 1 31 + 1 27 + 1

15 + 1 10+ 1 80+0 78+0 10 + 1 16 + 1 24+ 1 27 + 1 26+ 1 20+ 1

11 + 1 88 +0 76+0 87 +0 12 + 1 19 + 1 24+ 1 24 + 1 20+ 1 15 + 1

95 +0 71 +0 71 +0 89+0 13 + 1 19 + 1 21 + 1 19 + 1 15 + 1 11 + 1

78+0 68 + 1 69+0 93 +0 14 + 1 19 + 1 19 + 1 16 + 1 15 + 1 10+ 1

4.1 4.2 4.3 4.4

83 +0 14+ 1 20+ 1 22+ 1

33 + 34+ 28 + 19 +

31 + 1 23 + 1 15 + 1 94+0

21 + 1 14 + 1 98+0 75 +0

15 + 1 10 + 1 79+0 74+0

11 + 1 86+0 75 +0 85 +0

87+0 66+0 62+0 76+0

79+0 69+0 63 +0 73 +0

1 1 1 1

196

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.3. (continued) Lin

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 l.l

1.2 1.3

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

The effective principal quantum number 1.5 2.0 2.5 3.0 Transition p - d, parameter X ,,=3 19 -0 30-0 59+0 88+0 15 + I 10 + I 15 + I 21 + I 20+ I 26+ I 16+ I 24+ I 29+ I 19+ I 27 + I 31 + I 22+ I 29+ I 30 + I 25 + I 30+ I 27 + I 27+ I 28 + I 23 + I 19 + I 28+ I 25 + I 18 + I 21 + I 28+ I 20+ I 14 + I 24+ 1 25 + I 14 + I II + I 25 + I 18 + 1 12 + I 10 + 1 26+ 1 II + I 11 + I 20+ I 24+ 1 82+0 18 + 1 33 + 1 16 + I II + 1 29+ 1 44+ I 68+0 19 + I 38 + I 46+ I 84+0 26+ 1 41 + I 42 + I 14+ 1 31 + I 40+ I 34+ I 18 + 1 22+ 1 26+ 1 28 + 1 29+ 1 28+ I 21 + 1 10 + I 72+0 12 + I 17 + I 22+ I 26+ 1 29+ I 30+ I 29+ 1 23 + I 13 + I 75+0 II + 1 17 + 1 22 + I 27 + 1 29+ 1

34+ I 34+ I 31 + I 24+ I 16+ I 10+ I 10+ I 17 + 1 25 + I 32+ I 36+ I 37+ I 34+ I 27 + I 18 + I II + 1 10+ I 16+ I 24+ I 32+ I 37+ 38+ 35 + 28+

I I I I

34+ I 27 + I 20+ I 14 + I 12 + I 17 + I 27 + I 38 + I 44+ I 44+ I 38+ I 31 + I 22 + I 16 + I 13 + I 16 + I 26+ I 37 + I 45 + I 45 + I 41 + I 33 + I 24+ I 17 + I

no 3.5

4.0

4.5

5.0

41-0 11 + I 19 + I 26+ I 30+ I 31 + I 30+ I 27 + I 23 + I 18 + I

54+0 14 + I 23 + I 29+ I 32+ I 31 + I 28 + I 23 + I 19 + I 14+ I

66+0 17 + I 26+ I 31 + I 32+ I 29+ I 25 + I 20+ I 15 + I 12 + I

79+0 20+ I 29+ I 32 + I 31 + I 27 + I 22 + I 17 + I 13 + I 10 + I

14 + I 11 + I II + I 17 + 1 29+ I 44+ 1 40+ I 44+ I 35 + I 27 + 1

11 + 10 + 12 + 22+ 37 + 48 + 46+ 38 + 29+ 21 +

I I 1 I I 1 1 1 I 1

97+0 99+0 14+ I 26+ 1 41 + I 47 + I 41 + 1 31 + I 23 + I 17 + 1

86+0 10+ 1 16 + I 28 + I 42+ 1 43 + I 35 + I 26+ 1 19 + I 14 + I

26+ I 19 + I 15 + I 14+ I 18 + I 30+ 1 43 + I 49+ I 47 + I 39+ I 30+ I 22+ I 17 + I 15 + I 18 + 1 28 + I 42+ 1 50+ I 49+ I 42+ I

20+ I 15 + I 13 + 1 16 + I 26+ I 40+ 1 50+ I 49+ I 41 + I 31 + I 23 + 1 17 + I 15 + I 17 + I 24+ 1 37 + I 49+ I 51 + I 43 + I 34+ 1

16 + I 13 + I 14+ I 20+ I 32+ I 45 + I 49+ I 43 + I 33 + I 25 + I 18 + 1 15 + 1 15 + I 19 + I 30+ I 42+ I 49+ 1 45+ 1 35 + I 26+ 1

11 + I 11 + I 15 + I 24+ I 35 + I 42+ I 37 + I 29 + 1 22 + I 16+ 1 13 + I 13 + 1 15 + I 23 + 1 35 + I 43 + I 41 + I 31 + 1 24+ I 18 + 1

32+ I 24+ I 18 + I 16 + 1

25 + 19 + 16 + 17 +

20+ 1 16+ 1 16 + 1 20+ 1

13 + 1 12 + I 14 + 1 22 + I 34+ I 45 + 1 44+ I 35 + 1 26+ I 20+ 1 15 + 1 13 + I 15 + 1 21 + I 32 + 1 45 + 1 45 + 1 37 + 1 27+ 1 21 + 1 16 + 1 13 + 1 14+ I 20+ 1

1 I 1 1

15 + 14 + 15 + 21 +

1 1 1 1

6.1 Tables of Numerical Results

197

Table 6.3. (continued)

An

The effective principal quantum number no 2.5 3.0 3.5 4.0 Transition d - s, parameter A

4.5

5.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

24-0 71 +0 10+ 1 10 + 1 83 +0 56+0 34-0 21- 0 17 -0 11 - O·

54+0 15 + 1 20+ 1 19 + 1 14 + 1 96+0 59+0 38 -0 35 -0 49-0

90+0 23 + 1 28+ 1 25 + 1 18 + 1 12 + 1 75+0 51 +0 47 -0 57+0

12 + 1 30+ 1 35 + 1 30+ 1 21 + 1 13 + 1 86+0 59+0 56+0 78+0

16 + 1 36 + 1 40+ 1 33 + 1 23 + 1 14+ 1 92+0 65+0 62+0 78+0

20+ 1 42+ 1 44+ 1 35 + 1 24+ 1 15 + 1 96+0 68+0 66+0 89+0

1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

44-0 67+0 74+0 65+0 49-0 34-0 24-0 19 - 0 18 - 0 18 -0

85 +0 12 + 1 13 + 1 11 + 1 80+0 56+0 41- 0 35 -0 38-0 53 +0

11 + 1 17 + 1 17 + 1 13 + 1 97+0 68+0 50+0 43 -0 47 -0 62+0

13 + 1 19 + 1 18 + 1 14+ 1 10+ 1 73 +0 55+0 48 -0 53+0 74+0

14+ 1 20+ 1 18 + 1 14 + 1 10+ 1 74+0 56+0 49-0 54+0

14 + 1 20+ 1 18 + 1 14+ 1 10+ 1 73+0 55+0 49-0 55+0 73 +0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

45 -0 59+0 61 +0 52+0 40-0 29-0 22 -0 18 -0 18 - 0 18 -0

81 +0 10+ 1 10+ 1 86+0 64+0 47 -0 37 -0 34-0 39-0 54+0

10 + 1 13 + 1 12 + 1 10 + 1 75 +0 55+0 44-0 40-0 46-0 59+0

11 + 1 13 + 1 13 + 1 10 + 1 76+0 57+0 46-0 44-0 50+0 69+0

11 + 1 13 + 1 12 + 1 99+0 74+0 56+0 45 -0 43 -0 49 -0 64+0

10+ 1 12 + 1 11 + 1 93+0 70+0 53+0 44-0 42-0 48-0 63+0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

44-0 55+0 55+0 47-0 36-0 27 -0 21-0 18 - 0 17 -0 17 -0

77+0 95+0 92+0 76+0 57+0 43 -0 35 -;- 0 33 -0 39-0 53 +0

94+0 11 + 1 10 + 1 86+0 65+0 50+0 41- 0 39-0 44-0 57+0

97+0 11 + 1 10+ 1 86+0 65+0 50+0 42-0 41- 0 48-0 65 +0

93+0 11 + 1 10 + 1 81 +0 62+0 48 -0 40-0 39-0 45 -0 5~+0

87+0 10+ 1 92+0 74+0 57+0 45 -0 38-0 37 -0 43 -0 57+0

4.1 4.2 4.3 4.4

43 -0 52+0 52+0 44-0

75+0 90+0 86+0 70+0

89+0 10+ 1 98+0 79+0

90+0 10 + 1 96+0 77+0

84+0 96+0 88+0 71 +0

76+0 86+0 79+0 64+0

198

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.3. (continued)

An

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

The effective principal quantum number 2.5 3.0 3.5 Transition d - s, parameter X ,,=2 11 + 1 13 + 1 14 + 1 18 + 1 20 + 1 21 + 1 21 + 1 21 + 1 20+ 1 19 + 1 18 + 1 16 + 1 15 + 1 14 + 1 12 + 1 11 + 1 10 + 1 90+0 84+0 74+0 65+0 64+0 61 +0 56+0 68+0 76+0 73 +0 27 -0· 14 + 1 11 + 1

no 4.0

4.5

5.0

16 + 1 21 + 1 19 + 1 15 + 1 11 + 1 78+0 57+0 51 +0 70+0 13 + 1

17 + 1 21 + 1 18 + 1 13 + 1 97+0 67+0 49-0 45 -0 64+0 10+ 1

18 + 1 21 + 1 17 + 1 12 + 1 86+0 59+0 43 -0 41-0 59+0 11 + 1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

23 + 1 34+ 1 34+ 1 27 + 1 20+ 1 14 + 1 10+ 1 93+0 10+ 1 10+ 1

27 + 1 39 + 1 37 + 1 28 + 1 19 + 1 13 + 1 10+ 1 10+ 1 13 + 1 21 + 1

28 + 1 41 + 1 36 + 1 26+ 1 18 + 1 12 + 1 99+0 98+0 12 + 1 19 + 1

27 + 1 39+ 1 34+ 1 23 + 1 15 + 1 11 + 1 90+0 91 +0 12 + 1 20+ 1

25 + 1 36+ 1 30+ 1 21 + 1 13 + 1 98+0 79+0 82+0 11 + 1

23 + 1 32+ 1 27 + 1 18 + 1 12 + 1 85+0 70+0 73+0 10+ 1 16 + 1

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

28 + 1 35 + 1 34+ 1 27 + 1 20+ 1 14 + 1 11 + 1 10+ 1 11 + 1 12 + 1

33 + 41 + 38 + 28 + 20+ 15 + 12 + 12 + 16 + 24+

34+ 1 42 + 1 37 + 1 27 + 1 19 + 1 14 + 1 11 + 1 12 + 1 15 + 1 22 + 1

32+ 1 39 + 1 34+ 1 24+ 1 17 + 1 12 + 1 10+ 1 11 + 1 15 + 1 23 + 1

29+ 1 35 + 1 30+ 1 21 + 1 15 + 1 11 + 1 95+0 10+ 1 13 + 1 20+ 1

26+ 1 31 + 1 26+ 1 18 + 1 13 + 1 97+0 84+0 91 +0 12 + 1 18 + 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

30+ 1 35 + 1 33 + 1 27 + 1 20+ 1 15 + 1 12 + 1 11 + 1 12 + 1 13 + 1

36+ 1 42 + 1 38 + 1 29+ 1 21 + 1 15 + 1 13 + 1 13 + 1 18 + 1 26+ 1

37 + 43 + 37 + 27 + 19 + 15 + 12 + 13 + 17 + 24+

1 1 1 1 1 1 1 1 1 1

34+ 1 40+ 1 34+ 1 24+ 1 17 + 1 13 + 1 11 + 1 12 + 1 17 + 1 25 + 1

31 + 1 35 + 1 30+ 1 21 + 1 15 + 1 11 + 1 10+ 1 11 + 1 15 + 1 21 + 1

27+ 1 31 + 1 26+ 1 18 + 1 13 + 1 10 + 1 93+0 10+ 1 13 + 1 20+ 1

4.1 4.2 4.3 4.4

30+ 1 35 + 1 33 + 1 26+ 1

37 + 42+ 38 + 29 +

38 + 43 + 37 + 27 +

1 1 1 1

35 + 1 40+ 1 34+ 1 25 + 1

32 + 35 + 29+ 22 +

28+ 1 30+ 1 25 + 1 19 + 1

1.1

1 1 1 1 1 1 1 1 1 1

1 1 1 1

1 1 1 1

6.1 Tables of Numerical Results

199

Table 6.3. (continued) The effective principal quantum number no LIn

2.5

3.0

3.5

4.0

4.5

5.0

Transition d - p, parameter A K=1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

19 + 1 24 + 1 21 + 1 15 + 1 94+0 52+0 32-0 35 - 0 77+0 18 + 1

31 + 1 37 + 1 32 + 1 23 + 1 15 + 1 87+0 50+0 38 - 0 50+0' 12 + 1

41 + 1 46 + 1 38 + 1 27 + 1 17 + 1 99 +0 58+0 47 - 0 63 + 0" 15 + 1

48 + 1 51 + 1 41 + 1 29 + 1 18 + 1 10 + I' 63 +0 49- 0 60+0' 13 + 1

53 + 1 55 + 1 43 + 1 29 + 1 18 + 1 10 + l' 65+0 52 +0 63 +0" 13 + 1

58 + 1 57 + 1 43 + 1 29 + 1 18+ 1 10 + I" 66+0 52 + 0 62 +0' 11 + 1

1.1

1.4 1.5 1.6 1.7 1.8 1.9 2.0

16 + 1 14 + 1 99+0 63 +0 39-0 28 -0 31- 0 57 +0 13 + 1 17 + 1

20+ 1 17 + 1 12 + 1 87+0 59 - 0 40-0 31 - 0 33 - 0 51 +0 12 + 1

23 + 1 18 + 1 13 + 1 92+0 62+0 43 - 0 35 - 0 39 - 0 64+0 15 + 1

21 + 1 17 + 1 12 + 1 89+0 62+0 43 -0 35 - 0 35 - 0 50+0 10 + 1

20+ 1 16 + 1 11 + 1 85 +0 59 + 0" 42 -0 34-0 35 - 0 50+0 10 + 1

18 + 1 14 + 1 11 + 1 79+0 56 + O' 40-0 32 - 0 33 - 0 44 - 0" 81 + 0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

13 + 1 11 + 1 75 +0 47 -0 31 - 0 26 -0 36 - 0 75 +0 14 + 1 17 + 1

14 + 1 12 + 1 89+0 63 +0 44-0 32-0 28-0 32-0 52+0 11 + 1

15 + 1 12 + 1 89+0 63 +0 44- 0 33 - 0 30-0 37 - 0 65 +0 14 + 1

13 + 1 10 + 1 81 + 0 58+0 42 -0 32 - 0 28 -0 30 -0 45 -0 92+0

12 + 1 96+0 73 +0 53 +0 39 -0 30 -0 26 -0 29 -0 44-0 86+0

10 + 1 84+0 65 +0 48 -0 36 - 0 27 - 0 24 -0 26 -0 36 -0 65 +0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

12 + 1 10 + 1 67+0 42 -0 28 -0 26-0 40 -0 86+0 14 + 1 17 + 1

12 + 1 10 + 1 75 +0 53 + 0 38 - 0 29-0 26-0 31 - 0 52+0 10 + 1

12 + 1 10 + 1 73 + 0 52 +0 38 - 0 29 - 0 28 - 0 36 -0 66+0 13 + 1

10 + 1 85 +0 64+0 47 -0 35 - 0 27 - 0 25 - 0 28 -0 42 -0 84+0

91 +0 74+0 56+0 42 -0 31 - 0 25 -0 23 - 0 27 -0 41 - 0 77+0

77+0 63 +0 49-0 37 -0 28 -0 22 - 0 20 - 0 23 - 0 32 -0 57 +0

4.1 4.2 4.3 4.4

12 + 1 95 +0 63 +0 39 - 0

11 + 1 92+0 68+0 49-0

11 + 1 88+0 65 +0 47 - 0

91 + 0 74+0 56+0 41 - 0

77 +0 63 +0 48 -0 36-0

64+0 52+0 41 - 0 31 - 0

1.2 1.3

200

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.3. (continued) The effective principal quantum number 2.5 3.0 3.5 Transition d - p, parameter l ,,=1 10 + 1 10 + 1 99+0 88+0 79+0 71 +0 49-0 67+0 57+0 48-0 40-0 33 -0 32-0 27 -0 21- 0 21-0 17 - 0 14-0 19-0 I3 -0 ll-O 42-0 19 - 0 18 -0 16+ 1 57 = 0 55+0 44+ 1 26+ 1 25 + 1

no 4.0

4.5

5.0

95+0 64+0 43 -0 28 -0 18 - 0 ll-O 10 -0 14 -0 42 -0 18 + 1

90+0 59+0 38 -0 24-0 15 -0 10 -0 10-0 12 -0 38 -0 15 + 1

86+0 53 +0 33 -0 20-0 12 - 0 10-0 10-0 10-0 32-0 12 + 1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

40+ 1 27 + 1 15 + 1 88+0 54+0 44-0 65+0 16+ 1 41 + 1 46+ 1

41 + 1 25 + 1 I3 + 1 74+0 45 -0 31- 0 30-0 49-0 12 + 1 40+ 1

36+ 1 21 + 1 II + 1 61 +0 36-0 26-0 27 -0 48-0 12 + 1 39+ 1

31 + 1 17 + 1 93+0 50+0 30-0 21-0 21-0 34-0 84+0 26+ 1

26+ 1 15 + 1 78+0 42 -0 25 -0 17 -0 18 - 0 31-0 75 +0 23 + 1

22 + 1 12 + 1 66+0 35 -0 20-0 14-0 15 -0 i5 -0 60+0 17 + 1

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

42+ 1 28 + 1 16 + 1 98+0 65 +0 63+0 II + 1 27 + 1 47 + 1 45 + 1

42+ 1 26+ 1 14 + 1 85+0 53+0 40-0 42-0 70+0 16 + 1 45 + 1

36+ 1 22+ 1 12 + 1 70+0 44-0 34-0 39-0 70+0 18 + 1 45 + 1

31 + 1 18 + 1 10 + 1 57+0 36-0 27-0 29-0 47-0 II + 1 29+ 1

26+ 1 15 + 1 85+0 47 -0 29-0 23 -0 25 -0 43 -0 10 + 1 26+ 1

21 + 1 12 + 1 71 +0 40-0 24-0 19 -0 21-0 34-0 76+0 19 + 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

42+ 1 29+ 1 17 + 1 10+ 1 72+0 77+0 14+ 1 33 + 1 47 + 1 45 + 1

42+ 1 27 + 1 15 + 1 91 +0 58+0 46-0 50 + 1 84+0 19 + 1 48+ 1

36+ 1 22+ 1 12 + 1 74+0 48-0 39-0 47 -0 86+ 1 21 + 1 47 + 1

30+ 1 18 + 1 10+ 1 61 +0 39-0 31-0 34-0 56+0 12 + 1 31 + 1

25 + 1 15 + 1 88+0 51 +0 33 -0 26-0 30-0 51 +0 II + 1 27 + 1

21 + 1 I3 + 1 73+0 42-0 27-0 22 -0 24-0 40-0 87+0 20+ 1

4.1 4.2 4.3 4.4

42+ 1 29+ 1 17 + 1 10+ 1

42+ 1 27 + 1 16 + 1 94+0

36+ 1 22+ 1 I3 + 1 77+0

30+ 1 19 + 1 10+ 1 64+0

25 + 1 15 + 1 90+0 53 +0

21 + 1 I3 + 1 75+0 44-0

An

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

6.1 Tables of Numerical Results Table 6.3. (continued) The effective principal quantum number no

An

2.5 3.0 3.5 Transition d - p, parameter A

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

20 - 1 II - 0 29-0 53 +0 73 +0 80+0 70+0 51 + 0 32 -0 20-0

55 - 1 30 - 0 74+0 12 + 1 15 + 1 15 + 1 12 + I 88 +0 55 +0 37 -0

1.1

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

15 - 0 17 - 0 26-0 40-0 53+0 58+0 51 + 0 39-0 27 -0 20-0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

4.0

4.5

5.0

10 - 0 55 +0 12 + 1 20+ 1 23 + 1 21 + 1 16 + 1 11 + 1 71 + 0 51 + 0

16 - 0 85 +0 19 + 1 27 + 1 29 + 1 25 + 1 18 + 1 12 + 1 80+0 62+0

24- 0 12 + 1 25 + 1 34 + 1 34 + 1 27 + 1 19 + 1 12 + 1 87 +0 70+0

33 - 0 15 + 1 31 + 1 39 + 1 37 + 1 29 + 1 20+ 1 13 + 1 90+0 76+0

32 - 0 41 - 0 62+0 91 +0 II + 1 II + 1 93 +0 69+0 49 -0 38 - 0

49 -0 67 +0 10 + 1 14 + 1 16 + 1 15 + 1 11 + 1 85 + 0 62+0 52+0

65 +0 92+0 14 + 1 18 + 1 19 + 1 16 + 1 12 + 1 93 +0 70+0 61 +0

78 +0 11 + I 16 + 1 21 + 1 20+ 1 17 + 1 12 + 1 94+0 73 +0 66+0

89+0 13 + 1 19 + 1 22 + 1 21 + 1 17 + 1 12 + 1 92+0 73 +0 67 +0

17 -0 20-0 27 -0 38 - 0 48 -0 51 + 0 45 -0 34- 0 25 -0 19 - 0

36 -0 44-0 60+0 82+0 97 +0 96+0 81 + 0 62+0 46-0 38 - 0

53 +0 67 +0 93 +0 12 + 1 13 + 1 12 + 1 10 + 1 76+0 58+0 51 + 0

66+0 87 +0 11 + 1 15 + 1 15 + 1 13 + 1 10 + 1 82 +0 64+0 58+0

74+0 96+0 13 + 1 15 + 1 15 + 1 13 + 1 10 + 1 81 + 0 65+0 62+0

76+0 10 + 1 13 + 1 16 + 1 15 + 1 12 + 1 99+0 76+0 63 +0 59+0

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

18 - 0 28 -0 28 -0 38 -0 46-0 48 -0 42 - 0 32 -0 24-0 19 -0

37 - 0 44 -0 59+0 77 +0 90+0 88 +0 75 +0 58 +0 45 -0 37 - 0

52+0 65 +0 87+0 11 + 1 12 + 1 11 + 1 92+0 71 + 0 56+0 50+0

64+0 83 +0 11 + 1 13 + 1 13 + 1 12 + 1 98 +0 76+0 62+0 56+0

69+0 84+0 11 + 1 13 + 1 13 + 1 II + 1 93 +0 74+0 62+0 61 +0

65 +0 92+0 11 + 1 13 + 1 13 + 1 II + 1 85 +0 69+0 58+0 50+0

4.1 4.2 4.3 4.4

18 - 0 21- 0 28 -0 37 -0

37 - 0 44-0 58+0 75 +0

52+0 63 +0 83 +0 10 + 1

64+0 83 +0 II + 1 12 + 1

68+0 75 +0 98+0 12 + 1

57 +0 82+0 98+0 II + 1

K=3

201

202

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.3. (continued) The effective principal quantum number no

LIn

2.5

3.0

3.5 Transition d - p, parameter X 1(=3

4.0

4.5

5.0

O.l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

21 - 0 68+0 12 + 1 19 + 1 24+ 1 27 + 1 26+ 1 21 + 1 14 + 1 10 + 1

32 -0 97 +0 17 + 1 24 + 1 29 + 1 29 + 1 26 + 1 20+ 1 15 + 1 11 + 1

43 -0 12 + 1 21 + 1 28 + 1 31 + 1 29 + 1 24 + 1 19 + 1 14 + 1 12 + 1

56+0 15 + 1 25 + 1 31 + 1 31 + 1 28 + 1 22 + 1 17 + 1 13 + 1 12 + 1

69+0 18 + 1 28 + 1 32 + 1 31 + 1 26+ 1 20+ 1 15 + 1 12 + 1 12 + 1

82 +0 21 + 1 30 + 1 32 + 1 29 + 1 23 + 1 18 + 1 13 + 1 11 + 1 11 + 1

1.1 1.2

1.6 1.7 1.8 1.9 2.0

86+0 10 + 1 15 + 1 23 + 1 30 + 1 32 + 1 28 + 1 21 + 1 15 + 1 11 + 1

11 + 15 + 24+ 33 + 39 + 38 + 32 + 24+ 18 + 15 +

1 1 1 1 1 1 1 1 1 1

13 + 20 + 31 + 42 + 45 + 40 + 32 + 24 + 19 + 17 +

1 1 1 1 1 1 1 1 1 1

15 + 24 + 37 + 47 + 46+ 39 + 30 + 22 + 18 + 17 +

1 1 1 1 1 1 1 1 1 1

2.l 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

10 + 12 + 17 + 24+ 30 + 32 + 28 + 21 + 15 + 12 +

1 1 1 1 1 1 1 1 1 1

15 + 19 + 27 + 36 + 41 + 39 + 33 + 25 + 20 + 17 +

1 1 1 1 1 1 1 1 1 1

21 + 29 + 40 + 48 + 48 + 41 + 32 + 25 + 20+ 20 +

1 1 1 1 1 1 1 1 1 1

16 + 27 + 41 + 46+ 41 + 32 + 24+ 18 + 16 + 16 + 20+ 30 + 41 + 46+ 41 + 33 + 25 + 20+ 18 + 18 +

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

11 + 13 + 18 + 25 + 31 + 31 + 27 + 21 + 15 + 12 +

1 1 1 1 1 1 1 1 1 1

17 + 21 + 28 + 37 + 41 + 40+ 33 + 26 + 20 + 18 +

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

18 + 25 + 35 + 44+ 47 + 42 + 34 + 26 + 21 + 19 + 21 + 27 + 37 + 46 + 48 + 43 + 35 + 27 + 22 + 20 +

16 + 26 + 40+ 48 + 45 + 36 + 27 + 20+ 17 + 17 + 21 + 30 + 41 + 48 + 45 + 37 + 29+ 23 + 19 + 20+

23 + 31 + 41 + 49 + 48 + 42 + 33 + 26 + 22 + 21 +

1 1 1 1 1 1 1 1 1 1

24+ 31 + 42 + 48 + 46 + 39 + 30 + 24+ 21 + 22 +

1 1 1 1 1 1 1 1 1 1

22 + 33 + 44 + 47 + 43 + 34+ 27 + 22 + 20+ 19 +

1 1 1 1 1 1 1 1 1 1

4.l 4.2 4.3 4.4

12 + 14 + 19 + 26 +

1 1 1 1

18 + 22 + 29 + 37 +

1 1 1 1

22 + 29 + 38 + 46 +

1 1 1 1

25 + 33 + 43 + 49+

1 1 1 1

26 + 31 + 39+ 47 +

1 1 1 1

22 + 31 + 43 + 49 +

1 1 1 1

1.3

1.4 1.5

6.1 Tables of Numerical Results

203

A summary of formulas defining the angular factors QK for the actual cases most frequently met is given in Sect. 6.2. The functions cPK(u) and GK«(3) are approximated by means of the following formulas: for K = 1:

cPI(u)

= C(_U_) 1/21n(16 + u) , u+l

G I «(3)

for

K

(6.1.9)

u+q>

= A J ~«(3 + 3) In( 16 + 1/(3) . +X

(6.1.10)

= C (_u_) 1/2 _ _ ,

(6.1.11 )

¥- 1 :

cPK(u)

u+l GK «(3)

= A J~«(3 + 3) +X

u+q> .

(6.1.12)

The transition under consideration is characterized by the assignment of the effective principal quantum numbers of the lower level no and of the upper level nj no

= VZ2RY/IEol, nj = Vz2Ry/IE I I

The quantities Eo and EI are the ionization energies corresponding to a specified state of the atomic core; z is the spectroscopic symbol of an ion. For a neutral atom, z = 1, for a singly charged ion, Z = 2, and so on. The parameters C, q> and A, X for transitions s -+ S, S -+ p, S -+ d, p -+ s, p -+ p, p -+ d, d -+ s, d -+ P are given in Tables 6.2 and 6.3 as functions of no and LIn = nj - no' The spacing with respect to no and LIn adopted in the tables ensures the possibility of linear interpolation almost everywhere. The tables give the order and the mantissa of the number; for example, 24 - 1, 47 - 0, 59+0, 42+ 1, and 12+2 denote respectively 0.024, 0.47, 0.59, 4.2, and 12. The range of approximation for cross sections is 1 :S u :S 128. The cases where the errors of approximation exceed 10 percent are indicated in Table 6.2 by asterisks. These errors, however, do not exceed a factor of 2. The rate coefficients are approximated in the range 1/32 :S (3 :S 4. The asterisks in Tables 6.3 indicate the cases in which errors of approximation are greater than 25 percent. 6.1.4 Normalized Cross Sections for Specific Atoms and Ions (Tables 6.4--8) (i) LIS = 0 For transitions with no change of spin (LIS = 0) the cross sections (J and rate coefficients (v(J) are fitted by

(J' = naij [EI] 3/2 Q~(aoad CcP'(u) z4 Eo 2/0 + 1 u + q> ,

(6.1.13)

204

6. Tables and Formulas for the Estimation of Effective Cross Sections u = (8 - AE)/z'l-Ry ,

(vn) =

1O-8~ [EI] 3/2 Q~(aoad z3 Eo 210 + 1

P=

Ry/T,

z2

AE/Z2

p=

• AG'(P)ex (_ P) P+ X P P ,

cm3 s- 1

(6.1.14)

Ry.

Here, z is the charge of the parent ion (spectroscopic symbol of ion, Z = 1 for a neutral atom, Z = 2 for a singly charged ion, and so on), and Q~(aoad is the angular factor. In fact, in all tables only the states with s core electrons (besides closed shells) are considered. In this case

Lp

Gfo~o p p

= 0,

= 1,

Q'

=m

for excitation from the shells of equivalent electrons 1'0, and Q' = 1 for one electron outside closed electron shells. Since Q' are independent of K we give the values, summed over K, of n and (vn). The formulas for tP'(u) and G'(P) are given by (5.1.12, 13 and 16). (ii) AS = 1 In the case of intercombination transitions n" = naij

z4

[EI]3/2 Q~(aoat>CtP"(u) Eo

210 + 1

(6.1.15)

u + cp ,

u = (8 - AE)/z2 Ry,

(vn") =

1O-8~ [EI]3 /2Q~(aoad.AG"(P) exp(-pp) [cm3 S-I], z3

P=

z2 Ry/T,

Eo

210

+I

P+ X

(6.1.16)

p = AE/z2 Ry.

In the tables for intercombination transitions, as in those for AS = 0 only the states with s core electrons (besides closed shells) are considered. In this case

L

p = 0,

"LnSo" lTL;Sp

= 1,

Q

= mA 2 = m

2S1 + 1 2(2Sp + 1)

(6. l.l 7)

for excitation from the shells of m equivalent electrons. Since Q" are independent of K we give summed over K values of n" and (vn"). The formulas for tP"(u) and G"(P) are given by (5.1.12, 13 and 16). The energy dependence of exchange cross sections varies from one transition to another, and the errors of fitting are rather large. For this reason only the rate coefficients are tabulated in most cases. The set of parameters C, cp,D is adjusted for the range 0.02 < u < 16, and the set A,X,D, for 0.25 < P < 16.

6.1 Tables of Numerical Results

205

Table 6.4. Normalized Born and Coulomb-Born excitation cross sections. Transitions with no change of spin (LIS = 0) Atom

HI

He I

C 3.46 70.32 2.33 38.11 1.81 2.06 31.58 2.07 0.03

0.67 4.27 0.68 3.64 1.13 0.69 3.49 1.29 0.61

D 0.00 0.00 0.00 0.00 0.30 0.00 0.00 0.20 0.40

R 0.04 0.03 0.01 0.05 0.01 0.01 0.05 0.Q1 0.01

A 7.52 24.12 5.72 18.59 2.03 5.24 16.77 2.40 0.06

X 12.13 0.34 2.94 0.28 0.68 2.53 0.32 0.65 0.87

D 9.90 0.50 1.40 0.00 0.20 1.00 0.00 0.10 0.00

R 0.03 0.02 0.02 0.02 0.02 0.02 0.Q1 0.02 0.02

2s-3s 2s-3p 2s-3d 2s-4s 2s-4p 2s-4d 2s-4f 2p-3s 2p-3p 2p-3d 2p-4s 2p-4p 2p-4d 2p-4f

18.27 139.93 57.52 8.34 59.82 16.73 8.94 10.51 60.96 1014.92 4.89 27.75 313.54 38.55

0.26 0.60 0.18 0.22 0.60 0.12 0.13 0.02 0.41 1.59 0.Q1 0.24 1.05 0.35

0.00 0.90 0.00 0.00 0.90 0.00 0.00 0.90 0.00 0.20 0.90 0.00 0.00 0.00

0.02 0.53 0.02 0.02 0.58 0.01 0.03 0.57 0.06 0.06 0.44 0.02 0.11 0.08

20.15 15.20 89.18 13.57 7.72 46.84 21.24 8.73 42.94 147.18 5.80 42.45 122.06 41.11

1.64 0.53 1.14 0.74 0.68 10.17 4.10 0.26 1.75 0.67 0.54 0.84 0.80 1.74

2.10 9.90 0.60 0.00 9.90 6.70 2.60 0.00 3.90 6.70 0.00 0.20 2.70 2.30

0.02 0.16 0.02 0.02 0.12 0.02 0.02 0.08 0.02 0.02 0.17 0.02 0.02 0.02

3s-4s 3s-4p 3s-4d 3s-4f 3p-4s 3p-4p 3p-4d 3p-4f 3d-4s 3d-4p 3d-4d 3d-4f 11S-2 1S 11S-ip 11S-3 1S 11S_3 1P 11S-3 1D 11S-4 1S 11S-4 1p 11S-41D 11S-41F

55.53 0.16 359.31 0.26 114.84 0.11 56.65 0.03 61.83 -0.01 197.65 0.27 1644.09 0.51 394.47 0.13 0.05 6.43 52.52 0.02 251.01 0.17 5485.46 0.73 1.26 1.85 8.22 26.71 1.43 1.32 21.53 7.78 2.90 0.36 1.34 1.32 21.77 8.99 0.44 2.91 0.00 1.48

0.00 0.90 0.00 0.00 1.00 0.00 0.60 0.00 0.10 0.70 0.50 0.70 0.00 0.20 0.00 0.20 0.20 0.00 0.10 0.20 0.30

0.07 0.25 0.12 0.01 0.44 0.07 0.16 0.10 0.05 0.25 0.04 0.14 0.Q1 0.02 0.01 0.03 0.02 0.01 0.02 0.02 0.02

40.45 44.43 111.48 121.39 25.01 88.72 228.43 332.30 10.84 49.05 84.22 502.57 4.13 8.09 3.43 7.55 0.33 3.25 7.35 0.42 0.00

3.07 0.57 4.54 0.99 0.63 2.46 0.74 4.44 1.29 0.50 1.83 0.38 1.61 0.25 1.23 0.25 0.52 1.25 0.27 0.54 0.97

7.70 9.90 8.70 0.00 2.90 9.90 9.90 9.90 0.70 0.30 9.90 9.90 0.40 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.20

0.02 0.22 0.02 0.02 0.02 0.02 0.13 0.02 0.Q1 0.02 0.02 0.31 0.02 0.02 0.02 0.01 0.02 0.Q1 0.01 0.02 0.02

1.00 0.00

0.15 0.02

72.85 17.95

1.32 1.32

9.90 1.30

0.07 0.02

Transition Is-2s Is-2p Is-3s ls-3p Is-3d ls--4s Is-4p Is-4d Is-4f

2 1S-21p is-3 1S

287.51 14.84

qJ

0.00 0.28

206

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.4. (continued) Atom Transition He I

Li I

LiII

D

R

0.46 0.79 0.79 0.53 12.25 2.77

7.20 0.10 0.00 7.60 9.90 1.50

0.02 0.02 0.00 D.03 0.02 0.02

5.55 42.80 147.66 3.59 42.42 121.92 40.64

0.73 1.75 0.67 0.67 0.84 0.81 1.74

9.60 3.90 6.60 3.90 0.20 2.70 2.30

0.04 0.02 0.02 0.03 0.02 0.02 0.02

0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.07 53.34 0.03 15.61 0.65 8.05 0.01 48.95 0.01 10.51 0.53 5.23 0.01 34.76 0.04 5.97

1.13 0.94 0.22 0.71 0.85 0.16 0.93 1.51

8.60 0.50 0.00 0.00 0.00 0.10 0.00 0.40

0.03 0.02 0.13 0.01 0.01 0.09 0.02 0.02

0.34 0.43 1.81 0.34 0.24 1.21

0.90 0.00 0.10 0.90 0.00 0.00

0.31 0.04 0.05 0.42 0.03 0.11

9.16 41.32 137.32 3.45 40.94 112.13 30.57

0.80 1.60 0.66 0.93 0.72 0.77 1.70

9.90 3.30 5.70 7.80 0.00 2.70 2.30

0.07 0.02 0.02 0.06 0.01 0.02 0.02

242.74 9.92 6.02 24.23 129.92 57.91 917.36 1261.32 75.59 2994.65

1.14 0.26 0.12 0.37 1.30 0.42 1.79 0.13 0.06 0.10

0.10 0.20 0.10 0.00 0.00 0.00 0.10 0.90 0.00 0.20

0.06 58.31 0.04 14.43 0.06 20.23 0.02 36.32 0.30 26.90 0.04 46.49 0.05 148.53 0.15 232.48 0.04 115.65 0.231495.79

0.86 0.66 7.03 0.70 0.86 1.22 0.63 0.93 2.17 2.48

5.20 0.00 3.40 0.00 6.00 2.10 5.40 9.90 2.10 9.10

0.02 0.01 0.02 0.01 0.03 0.02 0.02 0.07 0.02 0.02

1.87 43.32 1.37 31.48 0.79

0.78 7.77 0.80 7.63 3.46

0.00 0.20 0.00 0.20 0.40

0.06 0.02 0.08 0.02 0.02

1.30 0.21 1.41 0.22 0.51

0.00 0.20 0.00 0.10 0.80

0.01 0.02 0.01 0.02 0.02

({J

D

R

62.58 45.04 7.12 34.69 18.89 5.27

3.01 0.23 0.25 2.21 0.18 0.17

0.20 0.00 0.00 0.50 0.00 0.00

0.11 5.35 0.01 70.67 0.01 12.06 0.09 3.39 0.01 43.96 0.03 12.09

ip-3 l S 21P-3 1P 21p-3 1D 21p-4 IS 2lp-4lp iP-4 ID iP-4 IF

44.98 60.81 1010.04 16.10 27.69 313.07 38.15

0.97 0.41 1.59 1.73 0.24 1.05 0.36

0.60 0.00 0.20 0.00 0.00 0.00 0.00

0.21 0.06 0.06 0.09 0.02 0.11 0.08

23S_23P 23S-3 3S 23S_3 3P 23S-3 3D 23S-43S 23S-43P 23 S-4 3 D 23S-43F

255.67 11.64 11.93 31.92 5.91 10.22 18.13 2.72

0.83 0.31 0.60 0.30 0.28 1.02 0.27 0.23

23P_3 3 S 23 p_3 3 p 23p_3 3 D 23 P-4 3 S 23 P-4 3 P 23 P-4 3 D 23 P-4 3 F

60.27 55.97 865.03 16.22 25.80 309.41

2 1S-3 1P is-3 ID 2 IS-4 IS 2 1S-4lp 2 IS-4ID is-4 IF

2s-2p 2s-3s 2s-3p 2s-3d 2p-3s 2p-3p 2p-3d 3s-3p ·3s-3d 3p-3d lis-is I Is-ip 11S-3 1S 11S-3 1P 11S-3 1D

C

A

5.62 11.31 4.64 9.38 0.39

X

207

6.1 Tables of Numerical Results Table 6.4. (continued) Atom Transition

C

cp

D

R

A

X

D

R

2 l s-ip is-3 IS 2 1S-3 1P is-3 ID ip-3 l S 21P-3 1P 21p-3 1D

309.65 14.84 93.49 50.10 32.36 59.47 892.85

0.03 0.19 1.66 0.32 2.31 0.21 1.12

0.70 0.00 0.60 0.00 0.00 0.00 0.00

0.15 211.50 0.02 27.82 8.41 0.15 0.03 61.23 0.07 4.48 0.01 93.50 0.16292.64

1.33 1.02 0.53 1.18 0.54 1.21 0.62

3.10 0.20 9.10 1.10 5.20 0.70 2.60

0.02 0.02 0.02 0.02 0.02 0.02 0.02

23S-23p 23S-33S 23S-3 3p 23S-3 3D 23P_3 3S 23p_3 3p 23p-3 3D

249.92 13.55 57.24 41.93 43.37 54.68 780.14

0.03 0.23 1.87 0.41 2.15 0.22 1.21

0.80 0.00 0.70 0.00 0.10 0.00 0.00

0.18 164.06 0.01 23.92 0.15 4.15 0.02 48.10 0.17 5.70 0.01 87.89 0.16268.62

1.11 1.12 0.45 1.12 0.60 1.09 0.51

2.60 0.40 9.90 1.10 6.20 0.50 1.90

0.02 0.02 0.02 0.02 0.03 0.02 0.02

Be I

2S-2P

151.61

2.11

0.00

222.45 12.24 29.67 35.05 70.98 54.88 790.48 1517.48 68.55

0.02 0.26 4.26 0.44 1.72 0.21 1.20 0.03 0.03

0.90 0.00 0.00 0.10 0.00 0.00 0.00 0.80 0.10

0.92 1.11 0.30 1.08 0.65 1.10 0.51 1.51 1.11

0.90 2.20 0.40 3.00 1.10 4.80 0.50 1.90 6.60 0.30

0.02

2s-2p 2s-3s 2s-3p 2s-3d 2p--3s 2p--3p 2p--3d 3s-3p 3s-3d

58.21 141.37 21.28 3.48 38.34 13.17 89.02 273.34 615.05 129.97

0.40

Be II

0.09 0.24 0.01 0.03 0.02 0.10 0.01 0.16 0.15 0.02

0.02 0.02 0.02 0.02 0.03 0.02 0.02 0.02 0.02

CI

2p2 3p-2p3s 3p 18.47 2p21D-2p3s Ip 20.37 2p2 IS-2p3s IP 30.78 2 1s-ip 209.19

2.92 2.36 2.09

0.50 0.50 0.20

0.08 0.10 0.08

4.14 4.97 9.70

0.41 0.43 0.46

2.00 2.00 1.50

0.03 0.03 0.02

0.04

0.70

0.25 193.41

0.53

0.40

0.02

223.10 303.97 27.94 10.46 19.82 1392.49 254.78 70.07 85.49

0.03 1.37 0.34 0.33 0.11 1.44 0.92 0.38 0.45

0.70 0.10 0.00 0.00 0.60 0.30 0.00 0.00 0.00

0.23 0.06 0.01 0.04 0.09 0.05 0.11 0.10 0.07

71.46 1286.57

0.05 0.25

0.00 0.80

209.07 67.57 42.88 14.91 31.17 159.32 59.15 53.67 55.35 676.14 0.03 109.74 0.14215.54

0.54 0.76 0.72 0.65 0.74 0.68 0.97 1.39 1.36 1.62 2.24 0.88

0.40 4.80 0.00 0.00 0.00 9.10 6.40 2.70 3.10 9.90 2.20 9.90

0.02 0.02 0.01 0.02 0.02 0.02 0.03 0.02 0.01 0.04 0.02 0.06

Lill

OV F VI Na I

is-2IP 3s-3p 3s-3d 3s-4s 3s-4p 3p--3d 3p-4s 3p-4p 3p-4p 3d-4p 4s-3d 4s-4p

Na VIII

is-ip

244.83

0.02

0.70

0.20 225.78

0.62

0.60

0.02

MgI

3 1S-3 1P 3 IS-4 I S

214.17 5.29

2.32 0.54

0.00 0.10

0.06 62.81 0.03 6.87

0.45 -0.61

1.60 0.00

0.020.01

208

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.4. (continued) Atom Transition MgI

C

qJ

D

R

A

X

D

R

26.04 25.34 281.84 1140.87 84.65

0.44 0.79 0.84 1.06 0.18

0.00 0.00 0.00 0.50 0.00

om

3 1S-4lp 3 I P-4 I S 3 1P-3 1D 3 1P-4lp

45.35 27.74 68.56 130.28 127.20

0.85 0.55 1.02 0.68 1.18

0.00 0.00 6.50 9.90 0.70

0.01 0.02 0.02 0.01 0.02

33 P-4 3 S 33 P-43 P 33 p-3 3 D

103.99 298.92 706.60

1.56 2.62 2.22

0.10 0.00 0.10

0.18 23.65 0.51 68.46 0.05 141.69

0.73 0.15 0.53

4.30 0.60 3.30

0.03 0.03 0.02

406.66 15.14 51.07 208.84 1604.38 48.25

0.49 0.21 0.32 0.74 0.79 0.00

0.00 0.00 0.00 0.00 0.00 0.00

0.22 198.56 0.02 25.99 0.03 63.40 0.21 66.65 0.17475.20 0.01 104.53

0.87 1.33 1.20 0.76 0.81 0.99

2.70 0.70 1.10 3.70 4.30 0.00

0.02 0.02 0.02 0.02 0.02 0.00

3 1S-3 1D

0.13 0.10 0.07 0.02

Mg II

3s-3p 3s-4s 3s-3d 3p-4s 3p-3d 4s-3d

Mg IX

2S-2P

253.98

0.02

0.70

0.19 235.97

0.63

0.60

0.02

MgX

2s-2p 2s-3s 2s-3p 2s-3d 2p-3s 2p-3p 2p-3d

319.36 16.14 119.68 49.81 23.35 56.80 745.42

om 0.16 2.26 0.10 1.69 0.15 0.68

0.70 0.00 0.00 0.60 0.00 0.00 0.00

0.14278.24 0.02 34.07 0.11 22.37 0.06 68.92 0.17 5.40 0.01 119.17 0.29439.66

0.90 0.96 0.38 0.65 0.39 0.95 0.40

1.30 0.00 2.70 0.10 2.30 0.00 0.60

0.02 0.01 0.02 0.01 0.02 0.01 0.02

Al I

3p-4s

77.72

1.68

0.20

0.13

18.76

0.66

3.50

0.02

329.94

0.81

0.00

0.19 157.76

0.57

1.50

0.02

390.91 12.79 40.96 23.27 294.73 1522.11 80.13 33.26 1586.55 858.63

1.02 0.30 0.22 0.13 0.95 0.98 0.36 0.02 0.14 0.53

0.30 0.00 0.00 0.50 0.00 0.50 0.00 0.00 0.90 0.00

0.08 0.02 0.01 0.08 0.13 0.06 0.13 0.06 0.16 0.11

73.65 16.82 70.01 35.02 61.19 171.45 59.54 45.52 281.97 196.56

0.83 0.92 0.79 0.70 0.99 0.64 1.50 3.70 0.90 1.34

6.80 0.50 0.00 0.00 7.30 9.90 3.10 5.00 9.90 9.90

0.02 0.02 0.00 0.01 0.03 0.05 0.02 0.02 0.07 0.03

AlII

3 1S-3 1P

KI

4s-4p 4s-5s 4s-3d 4s-5p 4p-5s 4p-3d 4p-5p 5s-3d 5s-5p 3d-5p

Ca I

4 1S-3 1D 4 1S-4lp 3 1O-4 l p 43 p-3 3 D

12.19 302.55 304.09 283.09

0.16 1.82 0.19 0.59

0.00 0.10 0.20 0.00

0.02 0.06 0.10 0.09

25.12 68.44 116.87 69.84

0.99 0.62 1.95 1.40

0.00 3.50 9.10 9.40

0.02 0.02 0.02 0.02

Ca II

4s-3d 4s-4p 4s-5s 3d-4p 3d-5s 4p-5s

10.89 596.95 20.11 369.95 1.75 294.60

0.03 0.03 0.17 0.02 0.06 0.68

0.00 0.90 0.00 0.80 0.40 0.00

0.03 24.65 0.27 178.03 om 33.95 0.15 236.50 2.88 0.03 0.22 85.36

1.05 0.99 1.54 1.23 0.73 0.82

0.00 3.70 1.00 3.00 0.00 4.60

0.00 0.02 0.02 0.02 0.01 0.02

Table 6.4. (continued) Atom Transition

C

cp

D

R

A

X

D

R

Cu I

4s--4p

167.62

1.71

0.00

0.08

58.81

0.51

1.60

0.02

Zn I

4 1S--41P 4 1S-5 1S 4 1S--4 1D 4 1S_5 1P 41P_5 1S 41p--41D 41p_5 1p

163.76 3.66 15.12 30.46 240.58 1082.46 66.48

2.58 0.77 0.67 1.22 0.83 1.73 0.24

0.00 0.00 0.00 0.00 0.00 0.20 0.00

0.06 0.04 0.00 0.13 0.10 0.04 0.03

56.38 4.67 23.90 27.88 62.92 152.82 88.16

0.38 0.62 0.79 0.48 1.02 0.63 1.24

0.90 0.00 0.00 0.00 6.00 6.40 1.00

0.02 0.00 0.Ql 0.02 0.02 0.02 0.02

4 3P_5 3S 43p_5 3p 43P--4 3D

88.10 753.48 470.12

1.62 5.43 2.25

0.00 0.00 0.10

0.10 22.88 0.47 85.75 0.07 114.73

0.71 0.08 0.44

3.50 1.00 2.10

0.02 0.02 0.02

315.14 10.72 35.49 13.28 4.17 177.20 1253.98 243.28 71.52

0.62 0.24 0.39 0.19 0.38 0.80 0.96 0.69 0.15

0.00 0.00 0.20 0.30 0.00 0.00 0.00 0.00 0.60

0.21 0.02 0.02 0.05 0.05 0.20 0.17 0.67 0.03

156.02 20.05 38.25 24.07 6.70 57.89 388.52 133.56 69.26

0.78 0.84 1.05 0.83 1.41 0.74 0.70 0.26 1.19

2.20 0.00 1.10 0.00 0.90 3.40 3.30 0.20 1.70

0.02 0.02 0.02 0.00 0.02 0.02 0.02 0.03 0.02

Zn II

4s--4p 4s-5s 4s--4d 4s-5p 4s--4f 4p-5s 4p--4d 4p-5p 4p--4f

Ga I

4p-5s 4p-5p 4p--4d 4p-6s 5s-5p 5s--4d 5s--6s 4 1S--4 1p 4 1S-5 1S 4 1S--4 1D 41P_5 1S 41p--41D 43P_5 3S 4 3P--4 3D

78.66 22.83 479.96 21.00 700.64 57.69 21.33

1.83 0.00 2.64 1.47 0.69 0.11 0.24

0.00 1.00 0.00 0.30 0.50 0.00 0.00

0.09 0.04 0.05 0.12 0.11 0.04 0.06

20.93 30.47 123.45 8.21 105.03 96.59 23.53

0.66 0.70 0.42 0.54 0.88 0.99 1.52

3.00 0.20 1.70 1.50 9.90 0.30 1.90

0.02 0.02 0.02 0.03 0.02 0.02 0.02

299.06 7.91 40.57 230.70 1285.02 124.23 986.67

0.94 0.26 0.48 0.63 0.74 1.23 1.20

0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.18 138.94 0.04 16.22 0.02 48.84 0.23 90.02 0.17417.23 0.15 33.28 0.15 336.24

0.54 0.93 1.07 0.76 0.83 0.67 0.52

1.40 0.00 0.90 3.00 4.00 3.50 2.00

0.02 0.Ql 0.02 0.02 0.02 0.02 0.02

5s-5p 5s--4d 5s--6s 5s--6p 5s-5d 5p--4d 5p-6s 5p-6p 5p-5d 4d--6s 4d--6p 4d-5d

421.39 39.60 13.38 29.20 8.52 1330.95 338.26 85.10 42.03 22.90 395.52 194.77

1.03 0.19 0.30 0.12 0.11 0.66 0.92 0.34 0.03 0.02 0.55 0.03

0.30 0.00 0.00 0.60 0.00 0.60 0.00 0.00 0.80 0.00 0.00 0.90

0.08 0.01 0.02 0.11 0.01 0.07 0.11 0.13 0.37 0.04 0.15 0.17

0.83 0.82 1.03 0.60 5.07 0.71 1.01 1.52 0.41 2.89 1.20 0.79

7.00 0.00 0.70 0.00 2.30 9.90 7.70 3.20 0.00 2.90 7.60 3.40

0.02 0.00 0.02 0.00 0.01 0.06 0.03 0.02 0.04 0.02 0.03 0.02

Ga II

RbI

76.70 70.04 17.21 38.59 27.98 174.03 68.96 62.76 42.62 37.65 103.37 83.95

210

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.4. (continued) Atom Transition Sr I

SI8-4 1D SiS-Sip SIS-6 IS SIS-6 l p 4 10-S1P 41D-(j IS 4 1D-(jl P S3P-43D S3P-63S S3P-63P 4 3D-(j3S 43D-(j3p

A

D

R

0.93 0.63 1.01 3.58 1.81 0.82 0.46

0.00 4.10 0.80 1.90 9.90 0.00 1.60

0.01 0.02 0.02 0.01 0.03 0.01 0.02

0.13 112.4S 0.24 33.94 62.69 0.03 11.S9 12.21

1.32 0.79 1.38 0.86 0.31

9.90 6.60 1.70 0.00 0.40

0.03 0.03 0.02 0.00 0.01

cp

D

R

21.03 338.S3

0.16 1.66

0.00 0.20

8.23 SS2.67 6.46 31.88

0.20 0.29 0.10 0.73

0.00 0.00 0.00 0.00

0.02 41.28 0.07 70.01 11.6S 0.01 21.27 0.08 181.86 0.02 11.79 0.32 13.04

448.12 181.20

0.00 0.89

1.00 O.SO

6.11

0.09

0.00

C

X

Sr II

Ss-4d Ss-Sp Ss-6s Ss-Sd Ss-6p Ss-4f Sp-6s Sp-Sd Sp-6p Sp-4f 4d-Sp 4d-6s 4d-Sd 4d-Sd 4d-Sd 4d-6p 4d-4f

18.13 726.97 21.9S 33.20 2S.6S 20.16 S03.13 2168.30 128.12 281.14 489.30 2.98 108.S2 SO.72 SO.72 22.17 978.39

0.03 0.S1 O.1S 0.16 0.11 0.16 0.03 0.69 0.11 0.12 0.03 0.06 0.13 0.01 0.01 0.07 1.02

0.00 0.00 0.00 0.50 0.30 0.00 0.90 0.00 0.10 0.30 0.80 0.40 0.10 0.70 0.70 0.40 0.00

0.01 39.07 0.22 288.11 0.01 36.83 O.OS 29.87 0.04 43.48 0.03 38.S2 0.32 199.26 0.19 SS8.87 0.02 183.3S 0.04 308.03 0.19277.21 0.04 4.S6 0.02 162.86 0.8S 82.76 0.8S 82.76 O.OS 3S.77 0.18280.00

0.99 0.90 1.61 1.32 0.77 0.86 0.93 0.88 1.70 1.88 1.21 0.96 1.36 0.81 0.81 0.74 0.66

0.00 3.70 1.10 2.20 0.00 0.00 4.30 S.70 1.60 2.70 3.S0 0.40 1.00 0.00 0.00 0.00 3.40

0.01 0.02 0.02 0.02 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.02 0.23. 0.23 0.00 0.02

Ag I

Ss-Sp Ss-6s Ss-6p Ss-Sd Sp-6s Sp-6p Sp-Sd 6s-6p 6s-Sd

189.97 6.10 20.24 13.98 162.11 48.7S 820.00 986.91 77.S6

1.82 0.47 0.62 0.56 1.06 0.38 1.89 0.39 0.08

0.00 0.00 0.00 0.10 0.00 0.10 0.10 0.70 0.00

0.07 0.03 0.12 0.02 0.10 0.04 O.OS 0.13 0.06

62.12 9.29 26.49 19.52 42.21 43.26 140.91 lS6.80 119.28

0.50 0.73 0.6S 0.67 0.87 1.21 0.61 0.86 1.84

1.70 0.00 0.00 0.00 4.80 1.80 4.80 9.90 1.60

0.02 0.00 0.03 0.01 0.03 0.02 0.02 0.04 0.02

Cd I

SiS-Sip S IS-6 IS SiS-SiD S IS-6 l p Slp-6 IS Sip_SiD Slp-6 l p

194.S9 4.44 17.94 29.17 269.74 1162.16 76.03

2.81 0.66 0.69 1.20 0.87 I.S4 0.19

0.00 0.10 0.00 0.00 0.00 0.30 0.00

O.OS 0.02 0.01 0.12 0.09 0.06 0.03

S7.54 S.78 26.70 26.20 66.26 lS4.64 114.68

0.42 0.63 0.74 0.46 1.01 0.63 1.12

1.30 0.00 0.00 0.00 6.30 7.10 0.60

0.01 0.01 0.01 0.01 0.02 0.02 0.02

6.1 Tables of Numerical Results Table 6.4. (continued) Atom Transition

C

D

R

0.71 0.17 0.45

3.90 0.50 2.40

0.02 0.03 0.02

0.67 0.85 0.42

4.90 0.50 2.40

0.03 0.02 0.03

0.18 162.58 0.02 17.44

0.54 1.08

1.50 0.30

0.02 0.02

0.00

0.16 42.80

0.68

3.70

0.02

1.03 0.13 0.29 0.12 0.19 0.15 0.00 0.92 0.33 0.87 0.21 0.03 0.02 0.03 1.12

0.30 0.00 0.00 0.60 0.30 0.00 1.00 0.00 0.00 0.60 0.00 0.00 0.90 0.90 0.30

0,07 0.02 0.02 0.11 0.02 0.02 0.15 0.14 0.13 0.16 0.04 0.02 0.35 0.11 0.09

80.69 55.13 18.03 41.66 6.68 15.99 191.12 72.16 66.24 57.17 170.92 20.70 21.00 63.48 169.13

0.85 0.87 1.11 0.55 0.66 2.13 1.19 1.03 1.57 0.74 1.60 0.90 0.77 0.90 0.80

7.90 0.00 0.90 0.00 0.00 0.90 9.90 8.20 3.40 9.90 1.70 0.00 2.40 3.20 9.20

0.02 0.00 0.02 0,0} 0.01 0.02 0.04 0.03 0.02 0.03 0.02 0.01 0.02 0.02 0.02


D

R

53p_53D

109.64 233.46 587.17

1.71 2.37 2.33

0.00 0.00 0.10

Inl

5p-6s 5p-6p 5p-5d

66.44 25.49 340.34

1.18 0.30 2.38

0.60 0.40 0.30

0.14 25.88 0.50 61.24 0.06 130.10 0.17 13.28 0.03 30.80 0.08 71.91

In II

5 1S-5 1P 5 1S-6 I S 53P-63S

362.32 9.61

0.90 0.28

0.00 0.00

162.86

1.16

482.05 29.52 14.48 33.30 4.59 6.84 812.31 370.22 90.97 466.25 136.19 10.35 39.98 128.76 1253.22

Cd I

53P-63S 53p~p

211

A

X

Cs I

6s-6p 6s-5d 6s-7s 6s-7p 6s-6d 6s-4f 6p-5d 6p-7s 6p-7p 6p-6d 6p-4f 5<1-7s 5d-7p 5d-6d 5d-4f

Ba I

6 1S-5 1D 6 1S-6 l p

6.24 319.72

0.11 1.32

0.00 0.20

0.04 11.63 0.07 69.95

0.86 0.72

0.00 4.70

0.01 0.02

5 1D-6 l p 5 10-7 1S 53 D-63P 530-73S 63P-73S

194.93

0.57

0.00

0.09 60.67 0.81

1.31 3.84

6.80 6.70

0.02 0.02

191.96

0.37

0.00

124.76

0.78

0.50

0.09 67.23 1.24 0.27 25.11

1.63 2.31 0.84

8.00 3.00 6.60

0.02 0.02 0.03

18.71 836.47 24.63 41.98 597.78 2.60 102.46 439.45 2135.33

0.01 0.03 0.14 0.17 0.04 0,07 0.13 0.54 0.70

0.00 0.90 0.00 0.40 0.80 0.30 0.10 0.00 0.00

0.04 42.83 0.27 355.84 0,0} 41.03 0,07 38.18 0.21 308.59 4.53 0.03 0,0} 158.29 0.24 134.20 0.20538.64

1.08 1.02 1.60 1.42 1.16 0.78 1.41 0.88 0.86

0.00 4.30 1.10 2.40 3.80 0.00 1.00 4.90 5.70

0.00 0.02 0.02 0.02 0.02 0.01 0.02 0.02 0.02

Ba II

6s-5d 6s-6p 6s-7s 6s-6d 5d-6p 5d-7s 5d-6d 6p-7s 6p-6d

Hg I

6 1S-6 l p 6 18-7 1S 6 1P-7 1S

195.00 10.02 309.30

2.95 0.29 1.01

0.40 0.00 0.00

0.08 44.18 0.02 14.91 0.08 68.78

0.39 0.82 0.96

1.80 0.20 6.50

0.02 0.02 0.02

63P-73S

168.16

1.70

0.10

0.14 36.45

0.69

4.20

0.02

212

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.5. Excitation of multiply charged ions. Normalized Coulomb-Bom-exchange cross sections for hydrogenlike ion Ne X. Parameters C, lp, D and A, X, D given below can be used for arty hydrogenlike ions with z > 3. Transition

C

ls-2s ls-2p Is-3s Is-3p Is-3d ls-4s Is-4p Is-4d Is-4f 2s-3s 2s-3p 2s-3d 2s-4s 2s-4p 2s-4d 2s-4f 2p-3s 2p-3p 2p-3d 2p-4s 2p-4p 2p-4d 2p-4f 3s-4s 3s-4p 3s-4d 3s-4f 3p-4s 3p-4p 3p-4d 3p-4f 3d-4s 3d-4p 3d-4d 3d-4f

3.42 65.74 2.25 34.74 1.33 1.97 28.24 1.46 0.02 17.64 168.98 55.08 8.12 71.75 16.06 8.70 11.25 57.98 789.52 4.23 26.75 261.84 32.88 52.74 405.74 122.48 57.08 70.33 190.45 1274.21 396.20 6.47 50.47 250.15 4204.00

lp

D

R

0.94 4.63 1.02 4.07 0.89 1.05 3.86 0.84 0.14 0.17 1.87 0.07 0.23 1.87 0.07 0.03 0.65 0.10 0.72 0.03 0.09 0.64 0.02 0.06 1.15 0.02 0.05 0.79 0.05 0.00 0.04 0.02 0.00 0.04 0.02

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.00 0.00 0.70 0.00 0.00 0.70 0.80 0.00 0.20 0.00 0.90 0.30 0.00 0.90 0.00 0.00 0.80 0.30 0.00 0.10 1.00 0.60 0.50 0.90 0.10 0.90

0.02 0.08 0.03 0.12 0.38 0.04 0.14 0.40 0.39 0.01 0.13 0.08 0.01 0.13 0.09 0.09 0.63 0.01 0.29 0.66 0.02 0.29 0.21 0.03 0.17 0.22 0.14 0.56 0.03 0.45 0.15 0.11 0.30 0.03 0.35

A 5.90 23.92 4.20 16.34 2.65 3.79 14.60 3.55 0.29 32.45 34.82 64.69 15.15 19.11 24.72 14.46 5.90 122.67 431.23 4.10 66.44 200.66 39.02 105.23 49.19 80.69 85.15 10.50 364.70 453.05 393.76 10.54 42.37 531.28 2319.00

X 0.82 0.20 0.89 0.28 5.52 0.91 0.31 8.45 4.56 1.08 0.41 0.75 0.84 0.39 0.68 0.74 0.23 0.98 0.39 0.34 2.20 0.29 0.67 0.91 0.72 2.81 1.77 0.72 0.87 0.72 2.22 0.73 0.41 0.99 0.62

D

R

0.00 0.00 0.00 0.00 4.40 0.00 0.00 6.00 0.00 0.30 2.70 0.50 0.00 1.80 0.00 0.00 0.40 0.00 0.70 0.00 0.90 0.00 0.40 0.00 9.90 7.50 1.60 9.40 0.00 4.10 3.70 0.00 0.30 0.00 1.80

0.00 0.01 0.01 0.02 0.09 0.01 0.03 0.08 0.01 0.02 0.02 0.01 0.02 0.02 0.01 0.01 0.02 0.01 0.02 0.10 0.02 0.02 0.02 0.01 0.02 0.02 0.02 0.02 0.01 0.02 0.02 0.01 0.01 0.00 0.02

Table 6.6. Excitation of multiply charged ions. Normalized Coulomb-Born-exchange

cross sections for heliumlike ion Na x. Transitions with no change of spin (6S = 0) Parameters C, lp, D and A, X, D can be used for any heliumlike ion with z > 3 Transition 11 S-21 S 11 S-21 P 11 S-3 1 S

C 3.31 62.45 2.25

qJ

1.43 5.56 1.64

D 0.10 0.50 0.20

R 0.02 0.04 0.01

A

X

D

R

3.77 10.57 2.38

0.60 0.31 0.56

0.10 1.70 0.10

0.02 0.01 0.02

6.1 Tables of Numerical Results Table 6.6. (continued) Transition

C

qJ

D

R

R

41.37 1.70 2.01 36.13 1.96 0.02

6.90 3.76 1.78 7.33 3.99 1.69

0.50 0.70 0.20 0.50 0.70 0.00

0.04 0.07 0.Ql 0.06 0.08 0.33

6.37 0.31 2.05 5.46 0.35 0.02

X 0.32 0.44 0.54 0.32 0.43 0.55

D

IIS-3 IP IIS-3 ID IIS-4 IS IIS-4IP IIS-4ID IIS-4IF

1.80 3.20 0.10 1.80 3.10 0.00

0.Ql 0.02 0.02 0.02 0.02 0.08

zIS-2lp 21S-3 1S 2 1S-3 1P 2 1S-3 1D zIS-4 IS 2 1S-41P 2 1S-41D zIS-4IF 21P-3 1S 21P-3 1P 21p-3 1D 21P-4 1S 21p-41p 21p-41D zIP-4 IF

409.42 17.37 166.30 55.29 8.14 70.82 17.46 8.20 21.23 58.14 836.51 8.47 27.18 291.36 38.25

0.03 0.20 2.46 0.20 0.28 2.16 0.32 0.00 2.84 0.18 0.89 2.56 0.24 0.95 0.51

0.50 0.00 0.00 0.50 0.00 0.20 0.40 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.30

0.10 0.02 0.09 0.07 0.03 0.12 0.04 0.08 0.04 0.02 0.23 0.08 0.02 0.19 0.07

290.52 29.40 23.32 47.55 12.86 11.51 15.60 6.44 2.59 103.91 373.36 1.54 48.39 155.65 23.51

1.99 1.63 0.50 1.08 1.76 0.55 1.03 1.83 0.38 0.87 0.48 0.36 0.79 0.48 0.95

4.80 1.10 4.70 1.80 1.40 4.50 1.50 4.20 4.30 0.10 1.30 2.60 0.00 0.90 2.30

0.02 0.02 0.02 0.02 0.02 0.03 0.02 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02

3 1S-4 1S 3 1S-41P 31S-41D 3 1S-41F 31P-4 1S 3 1P-41P 3 1P-41D 31P-41F 3 1D-4 1S 3 1D-41p 3 1D-41D 3 1D-41F

5l.98 387.56 116.56 56.59 99.72 190.96 1463.19 380.73 6.80 50.17 247.32 4214.

0.06 1.41 0.03 0.10 1.41 0.06 0.56 0.90 0.13 0.02 0.05 0.02

0.00 0.00 0.80 0.10 0.00 0.10 0.00 0.30 0.20 0.90 0.00 0.90

0.04 0.11 0.22 0.16 0.27 0.04 0.31 0.19 0.27 0.32 0.02 0.35

102.46 41.23 68.13 64.90 10.43 354.90 427.74 341.30 5.48 25.18 505.88 2237.

0.89 0.65 2.91 2.85 0.61 0.92 0.74 2.51 2.54 0.90 0.93 0.64

0.00 9.90 8.90 4.40 9.90 0.10 4.50 5.10 5.80 3.20 0.00 2.00

0.01 0.02 0.02 0.02 0.05 0.02 0.02 0.01 0.02 0.02 0.Ql 0.02

23S_23P 23S-3 3S 23S_3 3P 23S-3 3D 23S-43S 23S_43P 23S-43D 23S-43F 23P_3 3S

361.31 16.59 146.76 51.27 7.82 68.41 16.82 7.36 16.66

0.Ql 0.19 2.41 0.09 0.26 2.43 0.10 0.05 1.27

0.70 0.00 0.00 0.70 0.00 0.00 0.70 0.80 0.00

0.10 0.02 0.08 0.09 0.01 0.08 0.10 0.11 0.37

270.49 29.62 22.56 54.73 13.75 14.29 22.59 10.68 4.91

1.48 1.22 0.46 0.82 1.12 0.39 0.56 0.81 0.27

3.10 0.50 3.90 0.80 0.40 2.30 0.00 0.30 1.40

0.Ql 0.02 0.02 0.01 0.02 0.03 0.Ql 0.02 0.01

A

213

214

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.6. (continued) Transition 23p_3 3p 23p_3 3D 23P-43S 23p_4 3p 23p_43D 23p-43F 33S-43S 33S_43P 33S-43D 33S-43F 33P-43S 33p_43p 33p_4 3D 33p_4 3F 33D-43S 33D_4 3p 33D-43D 33D-43F

C

D

qJ

A

R

X

D

R

0.00 0.02 0.03 0.01 0.02 0.02 0.01 0.02 0.02 0.02 0.03 0.02 0.02 0.02 0.02 0.02 0.00 0.02

56.23 772.12 5.64 26.31 269.86 30.56

0.13 0.79 0.73 0.11 0.75 0.03

0.10 0.00 0.00 0.30 0.00 0.90

0.02 0.27 0.54 0.02 0.26 0.22

114.22 390.74 3.40 58.49 181.43 33.07

0.92 0.42 0.21 1.04 0.37 0.58

0.00 0.90 0.10 0.00 0.30 0.40

50.54 350.20 106.33 56.00 98.32 187.59 1351.28 391.17 7.65 53.06 250.60 4199.

0.06 1.41 0.02 0.06 1.15 0.05 0.55 0.04 0.01 0.00 0.04 0.02

0.00 0.00 0.80 0.30 0.00 0.10 0.00 0.60 0.70 0.90 0.10 0.90

0.04 0.13 0.22 0.14 0.37 0.03 0.34 0.15 0.20 0.35 0.03 0.35

100.15 37.00 66.88 76.69 11.50 356.10 403.69 371.17 9.49 37.24 522.72 2302.

0.90 0.64 2.72 2.15 0.65 0.86 0.72 2.26 1.44 0.50 0.97 0.61

0.00 9.90 7.70 2.40 9.90 0.00 4.30 4.00 1.40 0.90 0.00 1.80

Table 6.7. Normalized cross sections for intercombination transitions (AS = 1) Atom

Transition

He I

lIs-is IIS-2IP IIS-3 IS IIS-3 IP IIS-3 ID IIS-4 IS IIS-4IP IIS-4ID IIS-4IF 23S-2 1S

23 S-ip 23S-3 1S 23S-3 1P 23S-23D

is-23p is-3 3S

is-3 3p is-3 3 D 23p-2 1 p 23P-3 1S

C

qJ

D

R

22.66

0.69

2.60

0.31

27.63

0.37

3.70

0.38

28.54

0.34

3.90

0.38

X

D

R

1.10 9.92 1.91 12.18 0.75 2.14 12.94 0.98 0.Q3

2.25 1.88 3.68 2.37 0.73 4.06 2.52 0.74 0.93

0.20 0.20 0.20 0.20 0.00 0.20 0.20 0.00 0.00

0.03 0.04 0.Q3 0.04 0.07 0.Q3 0.04 0.07 0.14

9.38 22.20 13.66 15.76 88.17 13.13 4.93 19.10 108.73 19.58 17.22

1.77 1.78 3.89 2.16 4.80 1.69 2.59 4.26 5.50 1.04 3.70

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.04 0.02 0.02 0.02 0.01 0.04 0.04 0.03 0.01 0.05 0.02

A

6.1 Tables of Numerical Results

215

Table 6.7. (continued) Atom

Lill

Transition 23p_3 1p 23p_3 1D 21P_3 3S 21p_3 3p 21p_3 3D 11S-23S 11S-23P 11S-3 3S 11S_3 3P 11S-3 3D 23S-2 1S 23S-21P 23S-3 1S 23S_3 1P 23S-3 1D is-23p 2 1S-3 3S is-3 3p is-3 3D 23p_2 1p 23P-3 1S 23p_3 1p 23p_3 1D 21P-3 3S 21p_3 3p 21p_3 3D

Mg IX

2 1S-23P is-23p 2 1S-23P 2 1S_23P is-23p 2 1S-23P 3 1S-3 3P 31S-43S 3 1S_43P 31S-3 3D 33p_3 1p 33P-4 1S 33p_3 1D 33p_4 1p 2 1S_23P

AlII

3 1S-3 3P

Be I BII CIII OV F VI Na VIII MgI

C

D

R

1l.43

l.20

l.20

0.42

1l.64

l.23

1.40

0.48

X

D

R

36.07 166.02 14.15 40.87 159.60

3.89 5.17 3.22 4.57 4.81

0.10 0.00 0.00 0.10 0.00

0.05 0.04 0.01 0.06 0.03

0.04 7.20 0.04 6.35 l.53

0.87 0.45 0.85 0.40 0.90

9.90 0.00 9.90 0.00 0.00

0.33 0.05 0.34 0.06 0.02

12.95 25.83 8.42 9.80 52.71 14.93 6.10 6.54 50.03 41.35 9.70 22.67 107.21 14.98 19.92 123.15

3.64 3.60 5.78 3.20 4.79 3.27 4.59 2.50 4.77 l.87 3.28 l.64 3.63 4.25 l.69 3.94

0.00 0.00 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.04 0.03 0.08 0.04 0.03 0.03 0.04 0.02 0.02 0.02 0.03 0.05 0.01 0.02 0.05 0.01

10.55 14.67 10.16 6.83 7.55 7.05

0.80 l.76 2.37 2.05 2.09 2.15

0.00 0.00 0.00 0.00 0.00 0.00

0.04 0.01 0.04 0.02 0.02 0.02

15.38 2.36 11.54 64.55 38.63 5.74 185.98 37.61

0.95 0.92 1.31 2.36 l.66 l.59 4.44 3.46

0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.10

0.05 0.08 0.02 0.02 0.03 0.04 0.06 0.05

7.01

2.21

0.00

0.02

35.75

3.87

0.00

0.02

A

216

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.7. (continued) Atom Ca I

Zn I

Ga II

Sr I

Cd I

In II

Ba I

Transition 4 1S_43P 4 1S-3 3D 4 3p-4 1p 43p-3 1D 4 1S_43P 4 1S-5 3S 4 1S_53P 4 1S-43D 43p-4 1p 43P-5 1S 43p-41D 43p-5 1D 4 1S_43P 4 1S-53S 4 1S-43D 43p-4 1p 43P-5 1S 43p-41D 51S_5 3P 5 1S-43D 51S-63S 51S_63P 53p-41D 53P-5 1P 53P-6 1S 53p-6 1p 43D-5 1P 51S-5 3P 51S-63S 51S_63P 51S-5 3D 53P-5 1P 53P-6 1S 53p-5 1D 53P-6 1P 51S-53P 51S-63S 53P-5 1P 53P-6 1S 6 1S-5 3D 6 1S_63P 6 1S-73S

C

D

R

X

D

R

20.73 34.57 37.l9 31.84

1.54 2.12 1.78 1.42

0.00 0.00 0.00 0.00

0.04 0.02 0.00 0.Q1

13.08 1.64 10.63 36.60 32.85 8.37 222.55 35.37

0.68 0.73 1.04 1.90 1.40 3.75 6.75 3.27

0.00 0.00 0.00 0.00 0.00 0.10 0.10 0.10

0.06 0.03 0.02 0.02 0.04 0.08 0.06 0.04

37.95 7.66 62.19 72.04 16.86 127.73

3.53 4.37 4.24 3.02 4.45 2.81

0.00 0.00 0.00 0.00 0.00 0.00

0.02 0.04 0.03 0.Q1 0.Q1 0.Q1

25.27 43.30 8.71 23.66 36.50 39.25 8.87 29.75 22.87

1.99 2.38 3.08 3.l3 1.47 2.07 2.14 2.89 1.40

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.03 0.Q1 0.03 0.02 0.02 0.Q1 0.05 0.02 0.07

15.11 2.l3 11.86 45.80 35.l7 8.47 241.55 33.72

0.80 0.91 1.27 2.32 1.49 3.70 7.05 3.41

0.00 0.00 0.00 0.00 0.00 0.l0 0.l0 0.10

0.08 0.09 0.02 0.02 0.03 0.06 0.08 0.05

40.08 6.08 89.74 20.63

4.32 4.49 4.11 5.93

0.00 0.00 0.00 0.00

0.03 0.04 0.Q1 0.02

12.56 22.01 5.62

1.30 1.71 2.48

0.00 0.00 0.00

0.04 0.03 0.03

A

6.1 Tables of Numerical Results

217

Table 6.7. (continued) Atom

Hg I

A

X

D

R

53 D-6 1P 53 D-7 1 S 5 1D_63 p

16.62 6.85 11.02

1.12 1.90 0.65

0.00 0.00 0.00

0.08 0.05 0.13

61 S_63 p 6 1 S-73 S 63 p-6 1P 63 P-7 1 S 6 1P_7 3 S

19.83 3.84 37.16 12.30 21.52 13.63

1.17 1.34 1.61 4.18 4.36 3.94

0.00 0.00 0.00 0.10 0.00 0.00

0.08 0.12 0.01 0.06 0.04 0.01

Transition

C

D

R

73 S-is

Table 6.S. Excitation of multipy charged ions. Normalized exchange cross section for heliumlike ion Na X. Intercombination transitions (LIS = 1). Parameters C, q>, D and A, X, D can be used for any heliumlike ion with z > 3. Transition

C

q>

D

R

A

X

D

R

IS-2S IS-2P IS-3S IS-3P IS-3D IS-4S IS-4P IS-4D IS-4F 2S-3S 2S-3P 2S-3D 2S-4S 2S-4P 2S-4D 2S-4F 2P-3S 2P-3P 2P-3D 2P-4S 2P-4P 2P-4D 2P-4F 3S-4S 3S-4S 3S-4P 3S-4D 3S-4F 3P-4S

2.45 4.61 2.27 4.50 0.36 2.30 4.35 0.46 0.01

1.31 0.81 1.52 0.84 0.52 1.37 0.90 0.57 0.36

1.00 0.50 1.00 0.60 0.30 1.10 0.60 0.30 0.10

0.09 0.11 0.08 0.10 0.21 0.08 0.10 0.21 0.35

1.59 10.64 1.49 9.57 2.29 1.53 9.13 2.83 0.17 2.10 5.42 16.26 1.92 4.90 11.33 8.76 4.02 26.37 65.05 3.65 23.65 54.02 22.88 2.01 2.01 4.54 8.38 20.05 3.87

0.69 1.12 0.69 1.09 1.60 0.70 1.08 1.57 2.14 3.33 3.46 3.47 3.35 3.44 3.23 4.51 3.06 3.64 4.42 3.05 3.65 4.25 5.53 6.90 6.90 7.02 7.17 6.76 6.44

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.02 0.03 0.02 0.03 0.02 0.02 0.03 0.02 0.02 0.03 0.03 0.03 0.04 0.03 0.03 0.03 0.03 0.03 0.02 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03

218

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.8. (continued) Transition 3P-4P 3P-4D 3P-4F 3D-4S 3D-4P 3D-4D 3D-4F

C

D

qJ

R

A

17.28 32.97 58.74 6.48 24.22 50.84 121.00

X

6.79 8.34 7.55 6.77 7.53 7.86 8.86

D 0.00 0.00 0.00 0.00 0.00 0.00 0.00

R

0.02 0.01 0.02 0.02 0.02 0.02 0.02

6.1.5 Transitions between Closely Spaced Levels (Tables 6.9-10)

In the case of transitions between closely spaced levels under the conditions AE ~ Eo, Eland AE ~ Iff, the dependence of multipole and exchange cross sections calculated by means of the first-order methods on AE is almost absent. For the optically allowed transitions (AI = 1) a weak logarithmic dependence exists. The calculations for multiply-charged ions have been made using the Coulomb-Born approximation for transitions with no change of spin (AS = 0) and using the orthogonalized functions method for intercombination transitions. The data of Tables 6.9 and 6.10 were obtained for a set of values of AE and can be applied to arbitrary multiply-charged ions with z > 3. For quadrupole and intercombination transitions the value of AE is not important. For dipole transitions one has to interpolate data for particular values of AE. The fitting formulas and the range of analytic approximation are quite the same as in Sect. 6.1.4. 6.1.6 Ionization Cross Sections (Table 6.11 and 6.12)

The ionization cross sections have been calculated in accordance with (3.1.38) in the partial wave representation. In cases of ions the Coulomb-Born approximation has been used:

O'i(aO)

= QiO'(lO),

(vO'i(ao»)

u

= Qi(VO'i(lO»)

= (Iff - EO)/z2

P= z2

DE

Ry/T,

= z2

Ry ,

.

(6.1.18) (6.1.19)

Ry,

p = EO/z2 Ry.

The fitting formulas and the angular factors Qi are given by (5.1.21-25). For the total cross section of ionization from a shell 1'0,

Qi=m,

(6.1.20)

where m is a number of equivalent electrons. The set of parameters C, cp,D is adjusted for the range 0.0625 < u < 64, and the set A,X,D, for 0.125 < P < 8.

Table 6.9. Transitions between the closely spaced levels with no change of spin (LIS Coulomb-Born-exchange cross sections for multiply charged ions Transition

1:l.E/z2

C

cp

R

D

A

[cm- I ]

= 0)

The

X

D

R

0.96 1.72 2.06 2.16 1.96 1.72 1.52 1.42 1.66 2.10 2.24 2.19 1.89 1.55 1.47 1.68 2.11 2.24 2.15 1.84 1.55 1.46

1.60 4.00 4.80 4.30 3.20 2.30 1.70 1.40 4.10 5.20 4.60 3.70 2.60 1.70 1.40 4.10 5.20 4.60 3.60 2.50 1.70 1.40

0.02 0.02 0.02 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.02 0.02 0.02 0.Q2 0.02 0.02 0.01 0.02 0.Q2 0.02

1.15 1.14 1.69 0.85 0.84 0.87 0.87 0.85 0.93

0.70 0.70 1.20 0.00 0.00 0.00 0.00 0.00 0.10

0.02 0.02 0.02 0.Q2 0.02 0.Q1 0.02 0.02 0.02

Dipole transitions 2s-2p 2s-2p 2s-2p 2s-2p 2s-2p 2s-2p 2s-2p 2s-2p 3s-3p 3s-3p 3s-3p 3s-3p 3s-3p 3s-3p 3s-3p 3p-3d 3p-3d 3p-3d 3p-3d 3p-3d 3p-3d 3p-3d

1480. 740. 370. 185. 93. 46. 23. 12. 856. 428. 214. 107. 54. 27. 13. 856. 428. 214. 107. 54. 27. 13.

2p-2p 2p-2p 3s-3d 3s-3d 3s-3d 3p-3p 3p-3p 3d-3d 3d-3d

100. 10. 1000. 100. 10. 100. 10. 100. 10.

321. 389. 433. 491. 531. 589. 647. 704. 1849. 2163. 2439. 2762. 3092. 3436. 3677. 2356. 2738. 3080. 3482. 3892. 4319. 4617. 13.53 13.51 51.26 51.64 51.53 91.07 91.17 75.06 75.09

0.Q1 0.01 0.02 0.Q1 0.01 0.01 0.01 0.00 0.Q1 0.03 0.02 0.Q1 0.Q1 0.01 0.01 0.01 0.03 0.02 0.Q1 0.01 0.01 0.Q1

0.70 0.15 0.08 0.70 0.50 0.09 0.50 0.06 0.08 0.40 0.06 0.40 0.40 0.05 0.40 0.04 0.12 0.70 0.50 0.10 0.40 0.10 0.08 0.40 0.40 0.06 0.05 0.40 0.30 0.07 0.70 0.11 0.50 0.10 0.40 0.10 0.40 0.08 0.40 0.06 0.40 0.05 0.07 0.30 Quadrupole transitions 0.00 0.30 0.03 0.00 0.04 0.30 0.Q1 0.20 0.05 0.00 0.20 0.04 0.00 0.20 0.04 0.00 0.10 0.03 0.00 0.10 0.03 0.00 0.03 0.10 0.00 0.10 0.03

265. 269. 315. 406. 524. 650. 773. 883. 1232. 1492. 2024. 2671. 3403. 4173. 4890. 1588. 1904. 2565. 3387. 4304. 5242. 6132. 20.46 20.37 86.06 96.24 95.54 178.25 178.31 144.24 143.78

Table 6.10. Intercombination transitions between the closely spaced levels (LIS = I). The summed over K exchange rate coefficients for multiply charged ions Transition 2s-2s 2s-2p 2p-2p 3s-3s 3s-3p 3s-3d 3p-3p 3p-3d

1:l.E/z2 200. 747. 200. 20. 210. 252. 20. 42.

[cm- I ]

A 6.00 7.84 62.32 6.08 9.00 15.36 41.08 47.72

X 3.07 2.83 3.51 6.62 7.80 8.35 7.15 8.59

D

R

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.03 0.02 0.03 0.02 0.02 0.01 0.02 0.01

220

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.11. Ionization cross sections for atoms and ions in Coulomb-Born approximation Atom Level A D R C D R cp X Is 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 5g

9.581 32.6 86.9 66.9 175.0 294.0 106.0 283.0 483.0 680.0 145.0 396.0 674.0 956.0 1296.0

2.37 0.13 0.33 0.03 0.11 0.08 0.00 0.01 0.00 -0.01 -0.01 -0.02 -0.03 -0.03 -0.03

0.60 0.00 -0.60 -0.50 -0.80 -0.80 -0.70 -0.80 -0.80 -0.80 -0.80 -0.80 -0.80 -0.80 -0.80

0.07 0.11 0.08 0.11 0.10 0.12 0.13 0.11 0.12 0.21 0.16 0.12 0.13 0.19 0.28

7.371 198.5 430.0 746.3 1961.0 3462.0 2084.0 5919.0 10416.0 17566.0 4445.0 13595.0 24815.0 35980.0 57240.0

0.12 -0.70 0.01 21.45 5.10 0.05 5.25 1.00 0.02 15.23 1.60 0.03 9.00 0.60 0.02 7.11 0.20 0.01 19.66 1.00 0.02 15.61 0.50 0.02 14.02 0.30 0.02 14.32 0.10 0.00 0.80 0.02 26.75 25.52 0.50 0.02 25.57 0.40 0.02 22.74 0.20 0.01 23.95 0.00 0.01

lIS 23S 23p 21S

5.986 19.67 88.1 25.3 88.0

4.50 0.21 0.44 0.16 0.34

3.50 0.20 -0.60 0.10 -0.60

0.05 0.11 0.07 0.12 0.08

1.583 86.14 366.8 128.4 426.8

0.16 -0.10 0.02 17.47 5.60 0.06 5.65 1.50 0.03 18.37 5.10 0.06 5.42 1.10 0.02

Is

2s 2p 3s 3p 3d

9.570 32.45 85.1 66.2 176.0 287.0

2.41 0.22 0.40 0.07 0.09 0.11

1.10 0.00 -0.60 -0.50 -0.70 -0.80

0.07 0.12 0.05 0.12 0.09 0.03

4.500 89.60 295.6 406.4 1388.0 2548.0

0.25 -0.10 0.04 13.12 7.20 0.06 3.81 1.10 0.02 1.60 0.03 8.30 6.37 0.60 0.02 5.65 0.30 O.oI

Li I

2s 2p 3s 3p 3d

13.80 88.7 34.1 179.0 294.0

0.24 0.40 0.08 0.13 0.08

0.00 -0.60 -0.70 -0.80 -0.80

0.06 0.08 0.03 0.12 0.12

77.41 389.9 386.5 1913.0 3458.0

16.35 5.45 9.60 9.65 7.12

4.10 1.30 0.70 0.80 0.20

0.04 0.02 0.02 0.02 0.01

Li II

lIS 23S 21S 23p 21p

8.046 25.00 29.41 85.3 85.7

2.68 0.30 1.33 0.49 0.41

4.00 0.20 -0.80 -0.60 -0.60

0.07 0.11 0.09 0.07 0.05

1.872 44.27 60.58 259.3 294.2

0.20 7.56 9.83 3.89 3.95

0.30 6.00 6.80 1.40 1.20

0.02 0.07 0.07 0.03 0.02

Bel

2 1S 2~P 21p

BI

2p

8.56 64.51 88.5 53.08

0.73 1.05 0.56 1.70

0.60 0.10 -0.50 0.60

0.08 0.09 0.08 0.08

11.87 77.73 285.4 36.56

0.17 -0.80 0.09 0.13 -0.80 O.oI 7.85 3.30 0.04 0.29 -0.30 0.04

CI

2s 2p3p 2plD 2p l S

4.529 36.88 41.11 47.58

3.25 2.60 2.30 1.88

1.10 1.70 1.30 0.90

0.05 0.10 0.08 0.07

HI

He I

ip He II

2.349 12.47 16.79 25.70

0.18 -0.40 0.01 0.36 0.50 0.02 0.36 0.30 0.02 0.30 -0.10 0.02

6.1 Tables of Numerical Results

221

Table 6.11. (continued) Atom

cn CIll

C IV CV

NI 01

on o III o IV OV

o VI FI Ne I Na I

MgI

MgIX

Level 2p 2 1S 23 p 21p 2s 2p lIS 23 S 23 p 2p 28 2p 2s 2p 2s 2p 2s 2p 21S 23 p 21p 2S 2p 2p 38 3p 3d 31S 33 p 31P

is 23 p

ip MgX

2s 2p

Mg XI ArI KI

lIS

Ca I

3p 48 1 4 S 43 p 33 D 3 1D

C

A

D

R

ffJ

D

R

67.48

1.01

0.00

0.11

74.04

0.23 -0.60 0.04

21.63 76.3 84.6 25.86 84.4

0.31 0.40 0.34 0.23 0.42

0.50 -0.10 -0.30 0.30 -0.60

0.10 0.09 0.09 0.11 0.08

31.72 165.9 239.8 53.42 302.0

0.17 -0.80 0.05 11.48 8.20 0.05 6.21 3.00 0.04 9.73 6.70 0.07 4.89 1.60 0.03

9.203 29.04 83.4 26.43 2.644 24.92 8.321 50.62 13.67 66.66 19.41 75.1

1.42 0.19 0.40 3.78 3.66 3.64 2.56 1.88 0.92 0.74 0.40 0.41

3.80 0.20 -0.60 3.00 8.00 2.70 1.50 0.80 0.80 0.20 0.70 0.00

0.05 0.11 0.07 0.10 0.04 0.06 0.12 0.08 0.11 0.11 0.11 0.10

3.519 0.21 -0.10 0.02 76.64 12.60 7.00 0.06 4.89 312.3 1.50 0.03 0.25 0.70 0.02 5.18 0.485 -0.11 -0.10 0.01 0.19 5.45 0.30 0.01 0.45 2.540 1.10 0.03 0.54 0.90 0.03 21.90 0.23 -0.50 0.04 11.63 0.11 -0.80 0.03 80.48 24.52 0.13 -0.80 0.02 9.42 7.40 0.05 143.3

23.95 79.1 83.9 28.03 17.07 13.17 13.06 85.2 296.0

0.26 0.31 0.33 0.19 5.88 5.82 0.42 0.26 0.09

0.50 -0.20 -0.40 0.30 5.00 6.50 -0.60 -0.70 -0.80

0.11 0.08 0.09 0.10 0.06 0.03 0.02 0.11 0.11

37.80 226.1 273.7 66.22 1.98 1.39 83.04 553.7 3362.0

5.43 7.11 5.95 11.34 0.10 0.09 8.29 5.03 6.89

9.21 68.4 77.3

0.68 0.57 0.32

-0.40 -0.40 -0.70

0.03 0.08 0.10

39.92 209.8 464.8

9.37 6.26 4.99

27.47 80.4 83.1 29.94 83.7

0.20 0.32 0.35 0.17 0.29

0.40 -0.40 -0.50 0.30 -0.50

0.10 0.08 0.08 0.10 0.07

57.88 272.0 304.3 80.34 331.4

9.80 5.79 5.79 12.65 5.37

9.477 3Q.62 16.36

1.23 1.95 0.36

3.20 1.70 -0.60

0.06 0.41 0.05

0.71 0.53 0.39 0.39

-0.40 -0.50 0.60 0.40

0.11 0.14 0.14 0.15

12.31 88.3 174.1 188.4

4.417 18.03 92.98 34.93 285.1 200.9 234.6

X

4.40 3.50 2.40 6.90 0.40 0.30 1.50 0.50 0.20

0.08 0.04 0.03 0.07 0.01 0.01 0.02 0.02 0.02

2.90 0.03 2.60 0.03 0.60 0.02 6.50 2.20 2.00 6.80 1.60

0.07 0.03 0.03 0.06 0.03

0.23 -0.20 0.02 0.73 0.70 0.06 6.23 1.10 0.02 2.40 0.03 5.37 4.35 1.40 0.03 0.12 -0.80 0.02 0.13 -0.80 0.03

222

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.11. (continued) Atom

Cu I Krl Rb I

Sr I

Sr II Ag I Xe I Cs I

Ba I

Ba II

Hg I

Level

C

41p

99.4

0.42

-0.60

0.09

405.2

4s 4p 5s 5p 4d SiS 53 p 43D 41D Sip

10.27 39.98 16.32 95.7 306.0 13.22 99.7 236.3 39.77 117.9 25.0 325.5 11.28 46.68 19.2 125.3 297.0

0.75 2.47 0.42 0.23 0.30 0.59 0.54 0.46 1.59 0.36 0.18 0.22 0.71 1.87 0.38 0.22 0.23

-0.30 1.70 -0.60 -0.70 -0.80 -0.50 -0.50 0.70 1.40 0.20 0.20 0.60 0.60 1.60 0.10 0.10 0.40

0.04 0.31 0.05 0.05 0.13 0.04 0.11 0.14 0.09 0.13 0.03 0.16 0.04 0.18 0.05 0.08 0.16

37.25 16.19 77.12 624.6 1926.0 56.38 331.0 393.7 36.03 585.7 144.2 785.6 40.13 35.78 122.0 869.6 1128.0

13.17 5.40 0.04 0.60 1.00 0.05 5.69 1.30 0.02 6.07 0.80 0.02 7.56 1.30 0.03 8.94 2.90 0.03 7.97 3.20 0.04 6.67 5.40 0.07 0.20 -0.60 0.03 9.49 2.50 0.04 5.44 0.90 0.02 8.18 4.90 0.05 18.90 8.30 0.04 0.34 -0.30 0.03 6.53 1.00 0.02 6.49 0.80 0.02 4.20 0.05 1l.l0

14.52 153.3 163.3 96.6 27.3 352.1 127.0

0.51 0.45 0.38 0.43 0.14 0.21 0.16

0.20 2.70 2.60 0.20 0.20 0.60 0.00

0.05 0.09 0.09 l.l0 0.02 0.16 0.04

73.18 100.5 118.8 399.1 171.0 883.3 1018.0

6.39 1.40 0.02 0.22 -0.40 0.01 0.20 -0.50 0.01 4.96 1.20 0.03 5.58 0.80 0.02 6.90 3.80 0.05 5.52 0.40 0.02

9.67 89.2 114.9

0.24 0.22 0.85

2.60 1.80 0.10

0.12 0.13 0.12

18.60 129.9 413.2

12.23 8.50 0.16 0.18 -0.80 0.12 12.41 4.90 0.05

5s 4d 5s 5p 6s 6p 5d 6 1S 53 D SiD 63 p 6s 5d 6p 6 1S 63 p 6 1P

D

qJ

A

R

Table 6.12. Rate coefficients of dielectronic recombination in Coulomb-Born-exchange approximation. Parameters and Xd.

He II BeN

C VI

o VIII NeX Mg XII SiXN

S XVI

IXo (XI H-like ions Is - 2p Is - 2p Is - 2p Is - 2p Is - 2p Is - 2p Is - 2p Is - 2p

31.18 36.69 31.34 25.78 21.06 17.20 14.08 11.56

0.74 0.73 0.71 0.69 0.67 0.66 0.64 0.63

A.I

X

6.91

D

R

2.10 0.03

6.1 Tables of Numerical Results

223

Table 6.12. (continued) Xz+1

IXo

(XI

A.i

CaXX Fe XVI

Is Is He-like lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS -

2p 2p ions 21p

7.828 4.420

0.61 0.60

17.43 22.21 0.024 18.21 0.078 14.37 0.170 11.16 0.309 8.659 0.495 6.738 0.714 4.144 1.151 2.138 1.500

1.13 0.85 0.90 0.79 0.85 0.75 0.82 0.72 0.80 0.69 0.78 0.68 0.75 0.66 0.71 0.64 0.65

10.81 6.405 3.706 2.400 1.690 1.182 .9387 .6229 .5012

0.07 0.04 0.02 0.02 0.01 0.01 0.01 0.01 0.01

21.07 11.62 7.010 5.665 4.663 1.003

0.10 0.06 0.04 0.03 0.03 0.01

Li II CV CV o VII o VII NeIX Ne IX Mg XI Mg XI Si XIII Si XIII S XV SXV CaXIX Ca XIX Fe XXV Fe XXV Be II C IV o VI Ne VIII MgX Si XII S XIV Ca XVIII Fe XXIV CIII

ov

Ne VII Na VIII Mg IX Fe XXIII

ip 23 p

ip 23p

ip 23p

ip 23 p

ip 23p

ip 23 p

ip 23 p

ip

23 p Li-like ions 2s - 2p 2s - 2p 2s 2p 2s - 2p 2s - 2p 2s 2p 2s - 2p 2s - 2p 2s - 2p Be-like ions 21p 2 1S

is - ip 21S - ip is ip 21S - ip ip 2 1S

Xd

6.1.7 Dielectronic Recombination Rate Coefficients (Table 6.12) The methods of calculations of the rate coefficients for the dielectronic recombination process,

Xz+\(IXo) + e

--+

Xz(lX\nl)

--+

Xz(lXonl)

+ fut),

224

6. Tables and Formulas for the Estimation of Effective Cross Sections

are described in Sect. 5.2. The simplified model (5.2.12) with its modification (5.2.29) for s - p transitions was used. The excitation cross section for the transition IXo - 0(1 of an ion XZ+I has been calculated in the Coulomb-Born approximation with exchange whenever it has been substantial. The rate coefficient for dielectronic recombination connected with the transition IXo - IXI is expressed in the form

f3

=

(z

+ 1)2Ry T

.

(6.1.21) Parameters Ad and Xd for the most important actual cases are given in Table 6.12 and the angular factors Qd for these cases are given by

Qd(nolO', nol:;'-lnI1d = m, Qd(noltnllf', nol:-lnllf'+I)

=N

(1- 2(21~+ 1»)·

(6.1.22)

In the case of heliumlike ions the total rate coefficient for dielectronic recombination is the sum of contributions from excitation of both singlet and triplet P levels.

6.2 Formulas Defining the Angular Factors 6.2.1

Rules for the Addition of Cross Sections

In various applications, cross sections are required for transitions between separate levels, between two groups of closely spaced levels, for transitions from a given level to a group of levels, and for transitions from the whole group of levels to a given level. For example, one may be interested in transitions between separate fine structure components LoSoJo - LISIJI of two terms or in the transition between the terms LoSo - LISI as a whole. The cross section for transition from a given level a of the group A to the group B of levels b is, clearly, a(Aa, B)

= L a(Aa, Bb) ,

(6.2.1 )

b

where a(Aa, Bb) is the cross section for the transition a-b. If every level a of the group A is populated proportionally to its statistical weight, then the cross section for the transition A - Bb is defined by a(A, Bb)

=

1 g(A) ~g(a)a(Aa, Bb),

(6.2.2)

and the cross section for the transition A - B, by 1 a(A, B) = g(A) ~g(a)a(Aa, Bb) .

(6.2.3)

6.2 Formulas Defining the Angular Factors

225

La

Here g(a) is the statistical weight of level a, and g(A) = g(a) is the statistical weight of the group of levels A. The tabulated cross sections are given by formulas (6.1.1,2) where the dependence of effective cross sections on angular momenta is determined by the factors Q~ and Q~. Therefore the summation of the cross sections over the finestructure components of terms and over the terms belonging to a single electronic configuration is equivalent to the summation of these angular factors. The next subsections give a summary of formulas defining the factors Q~ and Q~ for the cases which can be met when using the tables of cross sections given in Sect. 6.1. 6.2.2

LS-Coupling; Q" for transitions between levels LSI

In this and the following subsections we give general formulas for Q-factors in the LS-coupling. The derivation for transitions not involving the shells of equivalent electrons was given in Sect. 2.3. We consider also some most important particular cases. Q~P) means Q~ or Q~. To simplify the notation, we denote by y the whole set of quantum numbers defining the term, specifying if necessary the spin S P and orbital angular momentum L p of the atomic core, the orbital momentum of an electron I, the total spin S, and the total orbital momentum of an atom L. The unnecessary quantum numbers will be omitted in formulas. The multipole order " in general can vary between "min

= 1/0 -

Id,

"max

= 10 + II

,

and Q~ is not zero only if " = "min, "min + 2, ... ,10 + II ,

For transitions between LS.! levels Q-factor can be written as follows, compare (2.3.3-5) 2/0 + 1

Q~p)(LoSoJo, LI SIJd = 2Jo + 1 FvB~lCv(J)C(P)(q), C'(q)

= 2b(q),

C/(q)

= ~[qf ,

Bq1u(J) = BqlC(SL)MqlCv(SLJ)

(6.2.4)

(6.2.5) (6.2.6)

The factor M according to (2.2.23) is equal to

(6.2.7)

BqlC(SL) == BqiSoLo, SILd does not depend on J, but depends on the type of transition. It is discussed in the next subsection.

226

6. Tables and Formulas for the Estimation of Effective Cross Sections

The sum over fine-structure components J) is independent of J o:

Q£p)(LoSoJo. L)S)

= Q£p)(LoSo• L)S) (6.2.8)

= ~ " B2 (SL) C(p) (q) [LoSoF Lq' qK • QK averaged over Jo of the initial levels is Q£p)(LoSo. L)S)JI)

[J ]2

= [L)~)F Q£p)(LoSo• L)SI) •

(6.2.9)

i.e .• it is proportional to the upper level statistical weight.

6.2.3 LS-Coupling; Q" for transitions between Terms LS Q-factors for transition LoSo-LIS) are defined by (6.2.8). On substitution of C(p) we obtain I 2[/0]2 2 QK(LoSo• L) S) = [LoSoF BOK (SoLo. S)L) •

(6.2.10)

2[/of " 2 2 QK(LoSo. L) SI) = [LoSoF Lq' BqK (SoLo. SI LI) [q] . /I

a) Transitions not involving the shells of equivalent electrons ao = [LpSp]loLoSo. BqK(SL)

a) = [LpSp]I)L)SI.

(6.2.11)

= MOqq(SpsS)MoKK(LpIL)[SpLp] .

Therefore Q~(LoSo. L)S)

= QK(L o• L I ) • Ao. Q~(LoSo. LISI ) = QK(L o• L I ) • A2 •

Ao = !5(So. SI) • A2

= (2S1 + l)j2(2Sp + 1)

(6.2.12)

where

22

QK(Lo. L I) = [LpSp] MOKK(LpIL) = [loLo]

2{LoLl }2 II 10Lp K

(6.2.13 )

The sum over L) and average over Lo are independent on Lo:

QK(Lo. LplI)

= QK(Lplo. LplI) =

(6.2.14)

1•

QK(Lplo. LI) = [Ld 2j[Lpld 2 .

(6.2.15)

Similar sums take place for spin factors 1

2[S ]2 p

2

l: [So] Ao = ~

l:Ao = l:A2 = 1. ~

~

1

--2

2[Sp]

l: [So] 2A2 = So

A2 . (6.2.16)

6.2 Formulas Defining the Angular Factors

227

b) Transitions from the shell of equivalent electrons ao = I[J'LoSo, BqK(SL)

=

al

1[J'-I[LpSp]IILISI ,

= v'mG£o~o . MOqq(SpsS)MoKK(LplL)[SpLp] , p p

(6.2.17)

where G£Ip ~Ip is the fractional parentage coefficient (Sect. 6.2.5). Q-factors are defined by (6.2.12) with (6.2.18) We see that QK(Lo, LI) depends in fact on So through G. The sums over LI and LpSp are QK(aO,

l~-I [LpSp]ld = m (G£~~~)2

QK(l~Lo, I~-Ild

= QK(l~,

I~-Ill)

,.

(6.2.19)

= m.

(6.2.20)

c) Transitions between shells of equivalent electrons ao = 1~lrLoSo,

=

al

1~-llr+1 [LpSp]LISI

N

= 410 +2

We assume here for simplicity that l~ is the closed shell. Here, LpSp are momenta of the shell Ir+l. For such transition one can use (6.2.18) for the transition I~-m - /f-m-I/ o, M = 2(2/1 + 1). (6.2.21 ) and corresponding substitutions into (6.2.19,20). We note also that for any value of n ::; 4/0 + 2 r-llm+l) = n(M - m) QK ( Inlm 0 I' 0 1M·

(6.2.22)

We used here for the total statistical weight the expression g(lm)

=

(41 + 2)! m!(M -m)!

(6.2.23)

d) Transitions inside the shell of equivalent electrons ao = rSoLo, BqK(SL)

al

=

ImSIL I ,

= (lmSoLolwqKll/mSILd

= L: v'mG£os£ . Moqq(SpsS)MoKK(LplL)[SpLp]. SpLp

p p

(6.2.24)

228

6. Tables and Formulas for the Estimation of Effective Cross Sections

Table 6.13. Reduced matrix elements (p3LoSollU 2l1 p3LISJ)

(p2LoSolIU2l1p2 LISJ) 3p

IS IS

0 0

3p ID

ID 2p 2D

0

- 1

2/-.13

4S

2/-.13

0

..jff3

0

4S

2p

2D

0 0 0

0 0

--.13

-.13

0 0

Thus we obtain

Q"'( ao, a\ )

=

2[1]2 (l mSoLo II U "II ImS\Ld 2 [SoLoF

Q~ (ao, a\) = ~Q~ (ao, ad + [:0[~~~2(lmSoLollV\"lllmS\Ld

(6.2.25)

where

(l mSoLoIIU"1I1mS\Ld =

G£o~o G£I~I

E

p p

p P

[Lp]

fllP7P~}, OL\K

LpSp LoSo (ImSoLollvq"lllmS\L\) = "L.J GLpSp .

LpSp

x

GLlsl LpSp

[L P SP ]

{:P:P~} . f7P7P~}. SoS\q

10L\K

(6.2.26)

The reduced matrix elements (aoIIU"llad and (aollV q"lIad were defined and partly tabulated in [6.1]. More detailed tables are given in [6.3]. For K = 0

(lmSoLoIlUOlllmS\Ld = b(Lo, Ldm[Lo]/[l]

(6.2.27)

Therefore, transitions with a change of L are possible only with K ~ 2 or due to exchange. At AL = 0, AS # 0 also exchange is necessary; at AL = AS = 0 there is no transition. In the Table 6.13 reduced elements for K = 2 are given for m = 2,3; for m > M/2 one can use relation: (6.2.28)

6.2.4 j/ Coupling

Below we shall give formulas for two cases: (i) both initial and final levels are described by jl coupling; (ii) the initial term LoSo is described by LS coupling, and the final term K is described by j I coupling.

6.2 Formulas Defining the Angular Factors

229

(i) For transitions between the fine structure components

Jo = Ko ±

!,

J, = K, ±

!,

we have

Q" (KoJo, K, J, ) = (2Ko + 1)(2J,

1}2 Q" (Ko, K,) .

KoJo K, ~

+ 1) { J,

(6.2.29)

Summation over J, gives

Q,,(KoJo, K,)

= Q" (Ko, K,) .

(6.2.30)

Equation (6.2.29), being averaged over Jo, yields

2J, + 1 Q" (Ko, K,J,) = 2(2K, + 1) Q" (Ko, K,) .

(6.2.31 )

The j I-coupling scheme cannot be used for a shell with equivalent electrons. Therefore only the case of transitions which do not involve groups of equivalent electrons should be considered. For transitions between the terms Yo = [LpSpj]/oKo, y, = [LpSpj]/,K, as a whole the factor Q" is

Q,,(yo, y,) = ( 2/ 0 + 1)(2K,

/o

+ 1) { K,

KOj}2 I,

(6.2.32)

K

Summing (6.2.32) over K, gives

Q" ([LpSpj] 10Ko, [LpSpj] Id

= Qd/o,

Id

= 1,

(6.2.33)

and averaging over Ko provides

Q (I "

IK )_ 0,

,

,

-

2K, + 1 (2j+ 1)(2/, + 1)

(6.2.34)

(ii) For transitions from the level LoSoJo described by LS coupling to the level [LpSpj]l,K,J, described by the jl-coupling scheme, we have

(6.2.35)

= (2/0

+ 1)(2J, + 1)(2So + 1)(2Lo + 1)(2j + 1)(2K, + 1)

LOSPK [ 10 Joj

K,]2 !

Lp I, SoJ, The definition of the 12j symbol used here is given by (6.3.23). Averaging over J o and summing after that over J, gives for transition between the terms,

Q,,(Yoyd = ( 2/ 0 + 1) (2j

+ 1)(2K, + 1) I)2r + 1) {Lp I, r 2Sp + 1 r K Lo/o

}2 {LpK,Sp} I, ~}2

(6.2.36)

230

6. Tables and Formulas for the Estimation of Effective Cross Sections

Summing (6.2.36) over K I , we obtain Q (

KYO,

[L S .] 1 ) p p}

I

=

(2Lp

2j + 1 + 1)(2Sp + I)

.

(6.2.37)

By summing further over j, one has (6.2.38) Averaging (6.2.36) over Lo, we have _ 2KI + 1 Q ([L S ] I S, K P pO 0,Yd-(2L p +l)(2S p +I)(2/ 1 +1)

(6.2.39)

For transitions from the shell of equivalent electrons I~ -1;-1/ 1, the formulas )2. (6.2.35-38) should be multiplied by m(G£o~o p p

6.3 3nj Symbols and Fractional Parentage Coefficients Formulas for the angular factors Q~ and Q" contain 6j symbols, 9j symbols, 12j symbols, and the fractional parentage coefficients G£sp sp . The detailed description of their invariance properties, and formulas, sum rules, and numerical values can be found in [6.2-7]. Here we give only those which are necessary for this book. I) The 6j symbol {:: :~::} obeys the following symmetry relations: it remains invariant under any permutation of its columns and also on transposing the lower and upper arguments in each of any two columns. For example,

The 6j symbol is nonzero if the following triangular conditions are fulfilled: LI(al a2 a3),

LI(al b2 b3),

LI(b l a2 b3),

LI(b l b2a3) .

The triangular condition LI (a be) means that the sum of any two arguments is greater than or equal to the third argument and the modulus of the difference of any two arguments is less than or equal to the third one. The 9j symbol remains invariant under any permutation of its rows or columns and also under transposition (change of rows to columns). Triangular conditions are fulfilled for every row and column. Here we give summation formulas for j-symbols, including some formulas that are missing in most books. {abc} means ''OJ-symbol'': the set of triangle rules LI(abc) and condition that a + b + c is

6.3 3nj Symbols and Fractional Parentage Coefficients

231

integer. Sums of one j-symbol.

L [x]z {hjlx} = [jzjIl Z

(6.3.1a)

x

(6.3.1b) x

~ [x]z {j:j~~3} = (-I)Zh{hhh}

{f: {:~3} = L [x]z {f: {~{~ } = [lIl-zJ(jz, h 13x L t [x]z {f: {~{:} = hx

~ (-It [xf

(6.3.2a) (6.3.2b)

(-1 )-h -II [h IIlJ(h, 0)

(6.3.3a)

II)

x

(-1 iI2+h-/1 [jIl-zJ(jI, I z )

(-1

x

(6.3.3b)

13

Sums of two j-symbols.

(6.3.4a) (6.3.4b) (6.3.4c)

(6.3.5)

(6.3.6a)

L (-lt2 [XIXZ]Z XlX2

f{:{~{~}J{:~~{t} = (-lih+/3-/~ {{~{~{~} ~lxzk3 ~lxzk3

(6.3.6b)

j{ l{k3

Sums of three j-symbols.

{~l/zI3} {kl~zk3} = {{:{~{~}

L(-I)2x[xf {hhh} x 13 k3x Jzxkz

xJI/I

klkz k3

(6.3.7a)

232

6. Tables and Formulas for the Estimation of Effective Cross Sections

(6.3.8)

Sums of four j-symbol.

(6.3.9a)

(6.3.9b)

(6.3.10)

For what follows it is convenient to adopt the designations: {a

bk} = {a.'. b.k.} './' , a = (a2alao) i p,q, ,

pq /

(6.3.11) Then

L [k2kd 2f(abk)f(pqk) {a bk} P qI

kJk2

=

t:

[/of f(aql) f(pbl) {;

2

:i}

{a qbk}/ P

I

(6.3.12) 0

6.3 3nj Symbols and Fractional Parentage Coefficients

233

using the definition of the M by (2.2.23) this equation can be written as " Mk(a,b)Mk(P,q) {abk} k7k2 pq/ z {abk} pql ; [kzkd z/[ko] z q I} =EMq(l,a)Mp(b,l) {a qbkI} =EMI(a,q)MI(p,b) {abk POlo P 10

(6.3.13) . 0

We have also

E [kzk\ko]z f(abk)f(pqk) {a bk} {a bk} {a bk} {pqk} P q I z p q I \ U V s o u vt 0

to k\ k2

= E [lof f(aql) f(pbl) 10

P}

t s ub a qI V

{

(6.3.14)

, 0

. {a' b'k'} {a' b'k'} {a bk} {a bk} fJ(k' k)fJ(O' 2) p' q' I' z p' q' I' \ P q I \ P q I 0 ' ,

(6.3.16)

= E[lz]z f(a'q'I')f(p'b'I')f(aql)f(pbl)fJ(O', 2) 12

fJ(O', 2)

= fJ(a~b~p~q~/~,

azbzpzqzlz).

Sums including 3jm-symbols.

~(_I)}-m (~~m~') E(-I)}-m[j]Z }m

=

t

= [j]fJ(j'm', 00),

(6.3.17)

(h h (h ,h ) ) j) m\ mzm -m\ -mz-m

(-I)h-m\fJ(m\mz, m~mi), (_I)h- m\-h- m2 [j]z

m\m2

= (-I)}-mfJ(jm,j'm'),

(6.3.18)

(h hj) (h h j ,) m\ mzm

-m\ -m2- m

(6.3.19)

234

6. Tables and Formulas for the Estimation of Effective Cross Sections

(6.3.20)

In particular, if ml = m2 = m; =

E(-IY[X]2 x

= (-1 i

m~ =

0,

{x1 hh} (x0 jlh) (x0 011/2) 11/2 0 0 0

(I h h) (I 000

II h ) 000 .

(6.3.21 )

A summary of formulas for 6j symbols in which one of the arguments does not exceed unity is given in Table 6.14. II) 9j symbols and 12j symbols are defined in terms of the 6j symbols in the following way:

~;} = E( -lfX(2x + I) {afrx bC}{dbxq ef}{pxqr}, {~P qr ad x

Table 6.14. Formulas for 6} symbols s=a+b+c

{ o~

b c} = (-I),(2b+ 1)(2c+ IW I/ 2 c b

b c} { a 1/2 c - 1/2 b + 1/2

s [ (a + c - b )(a + b - c + I)] 1)2 (2b + 1)(2b + 2)2c(2c + I)

= (-I)

b c} s[(a+b+c+ 1)(b+c-a)]-1/2 { a 1/2c-I/2b-I/2 =(-1) 2b(2b+I)2c(2c+l) abc 2[a(a+I)-b(b+I)-c(c+I)] 1 c b =(-I'[2b(2b+I)(2b+2)2c(2c+I)(2c+2)p/2 {a

b

c}

1 c-I b-I

(I)S[ =

-

s(s+I)(s-2a-I)(s-2a) ]1/2 (2b-I)2b(2b+I)(2c-I)2c(2c+l)

C}_(_I,[ 2b(2b 2(s+ 1)(s-2a)(s-2b)(s-2c+ I) ]1/2 + 1)(2b + 2 )(2c - I) 2c (2c + I)

{a b 1c- 1b -

c }_(_I)S[(S-2b)(S-2b-I)(S-2C+I)(S-2C+2)]1/2 {a b 1 c- 1b+ 1 (2b + 1)(2b + 2)(2b + 3)(2c - 1)2c(2c + I)

(6.3.22)

6.3 3nj Symbols and Fractional Parentage Coefficients

[

235

a)a 2 a 3 a 4 ] b) b2 b3 b4

c) C2 C3 C4

= (-1 )bl-b2-b3+b4 L:(2x + 1) {C) C2

x} {C3 )b2x} {C) bx} {C3 bx} .

a3 a )b)

x

C4

a3 a

C2

C4

a4 a 2 3

a4a2 4

(6.3.23 ) III) Our notation for the fractional parentage coefficients, G£sp s p agrees with that of [6.2]. The Racah notation is related to this by

The values of fractional parentage coefficients for electron configurations p'" with m = 3, 4,5 are given by Tables 6.15-17. For configurations s2, p2, p6, the fractional parentage coefficients are equal to unity.

Table 6.15. Fractional parentage coefficients (p2[L pSp]pLS} p 3LS) p3 2p 4S

l

IS

Ji

0

ID

0

3 I

3p

I

-Vi

-{{s

0

Table 6.16. Fractional parentage coefficients (p3[L pSp]pLS} p4LS) p4 3p IS p3 4S

0

2p 2D

0

2D

I

-J3

Vi I -Vi

ID

0

I 2

2

H,

~

236

6. Tables and Formulas for the Estimation of Effective Cross Sections

p4 2p

p5 IS

3p ID

fFs fs If

7 Broadening of Spectral Lines

Various phenomena of spectral line broadening connected with the most interesting applicati0!ls of atomic spectroscopy to plasma diagnostics, astrophysics, laser physics, and other areas are considered in this chapter. The presentation of the general theory of impact broadening is based on the density-matrix and quantum kinetic equation methods. These methods permit not only the line shape to be described in the case of spontaneous emission or linear absorption, but also allow nonlinear effects arising in laser spectroscopy to be considered. There are many books and review articles discussing the progress in theoretical and experimental work on the problem of spectral lines broadening [7.1-16]. For an extensive bibliography on line shapes see [7.17-19]. For a brief review of recent developments in the theory with stress to applications to nonlinear laser spectroscopy see [7.20].

7.1

Model of a Classical Oscillator

7.1.1

Formulation of the Problem

The theory of spectral line broadening caused by the interaction of an atom with surrounding particles is closely connected with the general theory of atomic collisions. Moreover in the region of not very high pressure, when the impact approximation is valid, the calculation of the profile of a spectral line includes calculation of the scattering amplitudes or scattering phases. Nevertheless it is useful to begin the study of pressure effects by considering a model simplified to the maximum extent. We shall make the following assumptions: i) the relative motion of the atom and the perturbing particle is quasi-classical, which enables one to use the concept of the trajectory of the perturbing particle; ii) this trajectory is rectilinear; iii) interactions with the nearest perturbing particle (binary interactions) play the principal role in the broadening, therefore multiparticle interactions can be neglected; iv) the perturbation is adiabatic, i.e., does not induce transitions between different states of the atom. Within these assumptions, the picture of broadening is outlined as follows. The perturbing particle produces an external field V(R) = V

[J p2 + v2(t -

to)2] ,

I. I. Sobel'man et al., Excitation of Atoms and Broadening of Spectral Lines © Springer-Verlag Berlin Heidelberg 1995

(7.1.1)

7. Broadening of Spectral Lines

238

where R is the distance between the atom and perturbing particle at a given time t, p is the impact parameter, to is the time of nearest approach, and v is the relative velocity. As a result the energy levels of the atom and, consequently, the frequency of oscillations of the atomic oscillator vary in time. Therefore the oscillation of the atomic oscillator can be described in the form t

J

= exp [iwot + i

f(t)

(7.1.2)

K(t') dt'] ,

-00

where Wo is the unperturbed frequency and K(t) is the frequency shift due to the interaction. Perturbation of the monochromaticity of the oscillations leads to broadening of the corresponding spectral line. The line shape is given by the expansion of the function f(t) in a Fourier integral, 1 I(w) = lim 1 ~ T-oo v 2nT

T/2

J

f(t)exp(-iwt)dt

12

-T/2

(7.1.3)

= lim - 12 1 T/2 J exp [-i(w nT

T-oo

-T/2

wo)t + i11(t)] dt

12

,

t

11(t)

J K(t') dt'

=

(7.1.4)

,

-00

where 11(t) is the phase of the oscillation caused by interaction. If the frequency w is measured from the unperturbed frequency Wo, then the exponent exp (iwot) must be omitted. In this case, I(w)

1 1 T/2 exp [-iwt 2nT -T/2

J

= lim T-oo

+ i11(t)] dt 12

(7.1.5)

In the theory of spectral line broadening, conditions are usually considered when gas pressure and temperature, state of ionization, and so on, do not vary with time. This means that the functions ,,(t) and f(t) = exp [i11(t)] are stationary random processes, and (7.1.3) can be rewritten in the following way:

*

I(w)

=

4>(r)

= lim

(7.1.6)

Re{7 4>(r) exp (-iwr) dr} ,

T-oo

1 -T

T/2

J

f*(t)f(t

+ r)dt =

f*(t)f(t

+ r) ,

(7.1.7)

-T/2

where 4>( r) is the correlation function. Time averaging can be replaced by averaging over the statistical assembly of quantities defining the function f(t). We shall denote such averaging by angle brackets, 4>(r)

=

(f*(O)f(r») .

(7.1.8)

7.1 Model of a Classical Oscillator

= exp [i1](t)], we have <1>(,) = exp {i[1](t + ,) -1](t)]} =

239

For f(t)

7.1.2

(exp [i1](')]) .

(7.1.9)

Impact Broadening

We shall consider in this section an approximation which is called the impact approximation. This approximation is based on the assumption that the decisive factor in the broadening of a line is the disruption of the coherence of the oscillations of an atomic oscillator during collisions. In other words if the duration of collision is small as compared with the mean time between collisions, then one can neglect radiation during collisions and consider the collisions to be instantaneous. Therefore the collisions are manifested only in phase shifts 1]. Using this assumption of instantaneous collision, it is possible to calculate the correlation function <1>(,) in the following way [7.21]. In accordance with (7.1.9) the difference .,1 <1> = <1>(, + .,1,) - <1>(,) can be written in the form .,1<1>

=

(exp [i1](, + .,1,)]) - (exp [i1](')])

= (exp [i1](')] exp (iL11]») - (exp [i1](')]) ,

where .,11] is the phase shift produced by collisions during time interval .dr. Since collisions are instantaneous, the phase shift .,11] does not depend on 1](,). Therefore 1](,) and .,11] are statistically independent, and consequently .,1 <1>

= (exp [i1](')])[ (exp (iL11]») -

1]

= -<1>(,)(1 -

exp (iL11]») .

(7.1.10)

We shall denote the number of collisions per second with parameters p and v as P(p, v)dpdv. The number of collisions during time interval .,1, is equal P(p,v)dpdvL1,. Therefore (l-exp(iLl1]»)

= '!9L1"

'!9

=

f[1-exp (i1])] P(p, v)dpdv ,

(7.1.11)

where 1] is the phase shift produced by collision with parameters p, v. If the density of perturbing particles is N and their distribution over velocities v is given by the distribution function ~(v), then 00

00

o

0

() = N J v~(v)dv2n J pdp[1

- exp(i1])] .

(7.1.12)

Denoting 00

(i'

= 2n J(1 o

- cos 1]) pdp,

(7.1.13)

00

(i"

= 2n

J sin 1] pdp o

,

(7.1.14)

240

7. Broadening of Spectral Lines

we have () = N(v (a' - ia")) .

(7.1.15)

From (7.1.10,11) it follows that

= -()tfJ

dtfJ dr:

(7.1.16)

'

tfJ = exp (-()r:) .

(7.1.17)

By substituting (7.1.17) into (7.1.6), we obtain

y I(w)

= 2n . (w _

y = 2N (va');

1 LI)2 + (y/2)2 '

LI = N (va") .

(7.1.18) (7.1.19)

The spectral distribution given by (7.1.18) is usually called the Lorentzian distribution. The width of the distribution (the distance between symmetrical points WI and W2, for which I(wJ) = I(w2) = Imax/2) is y. The shift of the line peak from Wo is LI. The quantities a' and a" are called the width and shift effective cross sections. Let us assume that the perturbing particle at a distance R produces the frequency shift K = CnR-n. Then (7.1.20) (7.1.21)

r(~) rtn =

r

Vi

G) .

(7.1.22)

For n = 2, 3, 4, 5, 6, we have rtn = n, 2, n/2, 4/3, 3n/8. Substituting (7.1.21) in (7.1.13,14), it is not difficult to obtain the following formulas for y and L1 which we shall use below in estimations of the width and shift:

LI _

v'3

-TY'

n

=6

LI

~

Y ~ 8.16 C;/5 (V 3/5)N ,

0.36 y .

(7.1.23)

7.1 Model of a Classical Oscillator

241

It is not difficult to show that the main contribution in (1' is given by the strong collisions for which '1 ;::: 1 and P < Po, where Po is defined by condition

'1(Po)

=

1:

cx.nCn) I/(n-l) Po= ( - .

(7.1.24)

v

The impact parameter Po is usually called the Weisskopf radius. Therefore to an order to magnitude, (1'

~ np~

(7.1.25)

The shift cross section (1", see (7.1.14), is determined by more distant collisions P ;::: Po. In the case of n = 2, the phase '1(p) ex: P-I. Thus (1' diverges as In Pm and (1" diverges as Pm, where Pm is the upper limit of integration in (7.1.l3) and (7.1.14). The divergence of the integrals (7.1.13) and (7.1.14) means that the approximation of binary collision is not valid. It is evident that in this case broadening is determined by distant (weak) collisions with P > Po. 7.1.3 Quasi-Static Broadening If the external field varies sufficiently slowly, i.e., if it is quasi-static, it is pos-

sible to assume that I( w )dw is simply proportional to the statistical weight of the configuration of perturbing particles for which the frequency of the atomic oscillator is included in the interval w, w + dw. In the binary approximation the frequency shift is produced by the nearest particle. Consequently, to calculate I( w), it is necessary to find the probability W(R)dR of the nearest particle being within the range of distance (R,R + dR) from the atom. For R much larger than the atomic dimensions the interaction potential could be neglected and this probability is W(R)dR

= 4nR2N exp ( -

~n NR 3) dR = exp [- (:0

y] (:J 3, d

(7.1.26) where Ro = (3/4nN)1/3. Substituting R = (Cn/K)lln = [Cn/(w - wo)]lln in (7.1.26), we obtain the probability distribution for a frequency shift of an atomic oscillator. In accordance with the basic assumption of the quasi-static approximation, the shape of the spectral line is also determined by this distribution. If the notation .dw = CnR{jn is introduced it follows from (7.1.26) that 4n I(w)dw=-NC~/n(w_wo)-(3+n)/nexp n

/ [ - ( -.dw - - ) 3 n] dw.

(7.1.27)

w-~

This distribution is valid only for sufficiently large values of w - Wo for which

242

7. Broadening of Spectral Lines

R = C~/n(w - wo)-I/n ~Ro For R :<: Ro, the binary approx:imation is not valid. Thus (7.1.27) describes only the outer part of a line. The condition R ~Ro means that .1w~w - wo. Thus the exponential factor can be omitted in (7.1.27), after which we obtain I(w)dw = 4n Nc~/n(w _ wo)-(3+n)/n dw .

(7.1.28)

n

7.1.4

Relationship and Limits of Applicability of the Impact and Quasi-Static Approximations

Let us return to the general relations (7.1.3,4). We shall first of all consider (7.1.3) for high values of .1w = w - wo. If .1w is large, the integrand in (7.1.3) oscillates strongly everywhere, except in the vicinity of the points tk for which

(

~;}k = K(td = .1w .

Thus the principal contribution to (7.1.3) give small regions .1 't'k around these points and instead of (7.1.3) one can write

I(w)= lim T---too

_I_II: J 2nT

eXP{i['1(t)+(WO-W)(t-tk)+(WO-W)tkUdtI2

k Ark

(7.1.29) We shall expand the function '1(t) in a series near tk in powers of t - tk. Since (dy//dt)tk = .1w, the linear terms in the exponent in (7.1.29) cancel, and the series begins with the term 2

2:1 (ddt2y/)Ik (t -

2

tk) .

Only the region .1't'k, where this term is less than unity, is significant in the integration (beyond this region strong oscillations begin). Hence (7.1.30) If within the limits of this region the next term of the expansion

3

611 (ddt 3y/) (t -

tk)

31 ~ 1 , (ddt33'1)It -_ (ddt22K) It

'

(7.1.31)

which must fulfill the following inequality: (7.1.32)

7.1 Model of a Classical Oscillator

243

the series can be broken of at a term proportional to (t - tk )2, and in each term of the sum (7.1.29), the limits of integration can be extended from -00 to 00. In this easel /(w)= lim - 12 L exP {i[I1(tk)+(Wo -W)tk]) T--->oo nT k I

(7.1.33)

= lim.!.. L T--->oo

T

k

(dK)-1 . dt Ik

It is easy to see that Lk(dK/dt);/dw is the time during which K(t) is included in the interval w - Wo, w - Wo + dw. Since d!k and dw in Fig. 7.1 are connected by the relation (dK/dt)lkd!k = dw, (7.1.33) gives the quasi-static intensity distribution W(w - wo)dw. We shall replace the summation in (7.1.33) by integration. The number of particles incident on the annular element 2np dp in the time T is 2npdp NvT, where N is the density of perturbing particles. Taking into account that each collision with p :::: PAw = (Cn/AW)I/n, Kmax = Cnp-n~Aw gives two points tk and tk+l (Fig. 7.1), we obtain

/(w)dw

= dw

P-1w

{4np

(dK)-1 dt Nvdp

4n

dw

= -;;Nc~/n Aw 1+ 3/n

'

(7.1.34)

i.e., the quasi-statical distribution in the wing of the line.

/(

w~~--~+-~~--------------

Fig. 7.1 Instantaneous frequency shift K(t)

1 It is assumed that the phases ak will be discussed below.

=

['1(tk) + (w - Wo )ttJ are independent. This assumption

244

7. Broadening of Spectral Lines

If a small neighbourhood around the instant of closest approach is not considered, then

dJ( C"v dt ~ p,,+l'

d 2 J( c"v2 dt2 ~ pn+2 '

(7.1.35)

and relation (7.1.32) takes the form

C" vp"-

(7.1.36)

--I~I.

Only collisions with p ~ P,1QJ = (C"/Aw)l/,, give points (7.1.36) can be rewritten in another form

tk

and

tk+1.

Therefore

v"/(,,-l)

Aw~ C~/("-l) = D .

(7.1.37)

According to condition (7.1.37) the quasi-static distribution is valid for large Aw, i.e., in the wing of a line. We shall now consider (7.1.3) in the limiting case of small Aw. If Aw is so small that l/Aw is much greater than the duration of the collision -

I

Aw

~

p v

- ,

(7.1.38)

the change of phase in the collision can be considered to be instantaneous. Hence it follows that the impact approximation can be used. The main contribution in the impact broadening of a line is given by collisions with p '" Po = (a" C,,/V)l/(,,-l). Substituting Po in (7.1.38), we obtain a relation opposite to (7.1.37): v"/(,,-l)

Aw~ C~/("-l) = D . Thus in the center of a line, Aw~D, the impact (Lorentzian) distribution of intensity is valid. For high values of Aw, Aw ~ D, the impact distribution is replaced by the quasi-static one. The quasi-static wing can appear both on the long-wave and on the short-wave side depending on the direction of shift of the terms. If D considerably exceeds the impact width y, then the greater part of the integral intensity of a line is concentrated in the impact region. Taking into account that

y ~ 21tpt,Nv = 21CNv(aC,,/vi/(,,-I) , we obtain

21tpt,Nv~D = .!:. ; Po

whence (7.1.39) where the dimensionless parameter h determines the number of perturbers in the sphere of the Weisskopf radius. Thus, for low pressures and high velocities, so

7.1 Model of a Classical Oscillator

245

long as the inequality (7.1.39) is fulfilled, the impact mechanism of broadening plays a decisive role. A relatively negligible part of the total intensity is concentrated in the quasi-static wing. At high pressures and low velocities, when h

= p~N

'" I ,

(7.1.40)

the impact approximation is inapplicable even to the inner part of a line. Let us note that if condition (7.1.39) is not fulfilled then the binary approximation is violated. In fact relation (7.1.40) means that the effective radius Po is approximately equal to the mean distance between perturbers. Although when p~ ~ I the quasi-static distribution is applicable practically to the whole profile of a line, the expressions (7.1.27,28) obtained above in the approximation of binary interactions are valid only in the wing of a line. The assumption of the independence of the phases (1.k was made above in the derivation of the formula (7.1.33). Since only strong collisions for which '1 ~ C,.lp" • plv'$> I are responsible for the quasi-static wing, the difference (1.k+l(1.k'$> I. In a nonpublished work [7.4] Anderson and Talman investigated in detail the limiting expressions for /(00) valid for the central part of the line and for the wings, and obtained also an interpolation expression for the intermediate part. The same problem is discussed also in [7.22,23]. 7.1.5

Doppler Effect

The frequency observation is amount 000 vie. defined by the

of an oscillator whose velocity component in the direction of v is displaced in accordance with the Doppler principle by an Let the distribution of the radiating atoms with respect to v be function W(v). Then 00 = 000 + ooovle, v = e(oo - (00)/000, and

00-(00) e /(oo)dw = W ( e - - - doo . 000

.

(7.1.41)

Wo

With a Maxwellian distribution (7.1.42)

where Vo = J2k Tim, we obtain I

/(00 )

[ ( 00A-ooD000 ) 2]

doo = yin exp -

doo AWn'

AooD

= 000 -Voe

•

(7.1.43)

The intensity distribution (7.1.43) is symmetrical. The magnitude of the broadening is defined by the parameter AWn. The width of the line, which we shall denote by ~, and the peak density /(wo) are expressed in terms of the parameter

246

7. Broadening of Spectral Lines

Amo:

() = 2Jln 2 Amo l(wo) = 1/v'n Amo

(7.1.44) (7.1.45)

Here () is defined as a difference between the symmetrical frequencies WI and W2 for which lewd = l(w2) = l(wo)/2. The parameter Amo is usually called the Doppler width of a line. When deriving (7.1.41,43), it is assumed that there is only one frequency wo(1 + vic) in the spectrum of the oscillator with velocity v. This assumption is valid if v does not vary in time or remains a constant quantity during a sufficiently long time. If velocity is constant only during time interval 't, then this interval contributes to the intensity of radiation in a spectral interval with width l/'t around the frequency Wo + Wovle. Formula (7.1.41) is valid if wovle~ 1/'t. Substituting for 't the free path time 'to = Llv, where L is the mean free path, we have (7.1.46) In the general case, the Doppler broadening is determined by Fourier transform of the function f(t)

WO] = exp [i-x(t) e

,

x(t)

=

Jt

(7.1.47)

v(t')dt'

-00

Substituting (7.1.47) in (7.1.6,7), we have l(w)

= ~ Re

{?

4>('t) exp (-iw't) d't}, 4>('t)

= (exp [i

:0

x('t)]). (7.1.48)

The function ( 't) in the form f

4>('t) = (exp[ik· r('t)]), r('t) = Jv(t)dt ,

(7.1.49)

o and introduce the distribution function f(r, v,t) for the oscillator coordinate r and velocity v. This distribution function satisfies the Boltzmann equation

of -+v·Vf=

ot

(Of) -

ot

coil.

- Gf ( Of) ot coil. -

,

(7.1.50)

and the initial condition f(r,v,O) = W(v) {)(r) .

(7.1.51)

Here (oflot)coll. is the collisional integral or collisional term, G is the linear operator of collisions, and W(v) is the distribution function for v. The correlation

7.1 Model of a Classical Oscillator

247

function
= ~Re{fdvJ f(r,v,r)exp[-i(wr-k.

J(w)

= - Re{f Fw(v,k)dv}

(7.1.52)

(7.1.53 ) r)]drdr}. n Here the frequency difference w - Wo is replaced by w. According to (7.1.53), J(w) is expressed in terms of the Fourier component F(Av,k) of the distribution function: I

n

Fw(v,k)

=J

(7.1.54)

,

fer, v, r) exp [-i(wr - k • r)] drdr.

(7.1.55)

Carrying out the Fourier transform of (7.1.50) and taking into consideration the initial condition (7.1.51), we obtain the following equation for F w( v, k): -W(v) + i(w - k • v)Fa,(v,k) = (Gf)wk .

(7.1.56)

For simplificity, the functions Fw and (Gf)wk will be written below without indices w. In the absence of collisions (of/ot)coll. = O,(Gf)k = 0, and F(v,k)

W(v)

= l.(W - k

·V

)

For the Maxwellian W(v) this gives the usual Doppler distribution (7.1.43). In fact, W ( v )d v = W ( v )w( v.ddv d v ~ , where v is the velocity component in the direction of vector k, v~ is perpendicular to k. Integrating in (7.1.54) over v~ gives unity. When integrating over v, it is necessary to replace w - kv by w - kv - ie, considering e --> O. We find J(w)

.

e

W(v)dv

= hmJ -n ( w- kv)2 +e2 =

J J(w - kv) W(v)dv

(7.1.57) We shall consider now the influence of collisions assuming the model of Brownian motion [7.25J. This model can be used in the case of so-called weak collisions. In the framework of the model of Brownian motion, the collisional term in (7.1.50) has the form

a

( Of) t colI.

. = v dlV.(vf) + 2v~ v L1./ .

(7.1.58)

The effective frequency of collision v is assumed not to depend on velocity and v~ = 2kT/m. Solving (7.1.50) with the collisional term (7.1.58) and taking into

248

7. Broadening of Spectral Lines

account the initial condition (7.1.51), it is possible to obtain tP(t) = exp [/(w) =

~1 (vt - 1 + e-

vt )]

,

..1eoo = kvo ,

(7.1.59)

.! Re {A Wo2 2v 2.IVW tP (1,1 + ..12WV1' - i~;V ..121)} v

,

(7.1.60)

1[

where tP( a, 1; z) is a confluent hypergeometric function az a( a + 1) z2 a( a + 1)( a + 2) z3 tP(a,1;z) = 1 + 1I! + 1(1+ 1)2! + 1(1+ 1)(1+ 2 )3! + ....

For v = 0, (7.1.60) gives the usual Doppler distribution. When v v ~ AWn, for the central part of a profile W = 0 and for the far wing we have, respectively, 1

/(0) = Ji..1eoo

(

2

v)

1 + 3Ji ..1wo

'

v

1 ..1w~ /(w) ~ Ji..1eoo 2Jiw4 .

(7.1.61 )

#

0, but

W ~ ..1wo,

(7.1.62)

Thus due to collisions with v ~ ..1wo the intensity in the central part of a profile is increased and the wing with intensity distribution ()( w- 4 appears. In the limiting case of high densities when v ~ ..1eoo, 1 /(w) ~ 1[

2 W

Yd

•.2

+ Yd

Vd

1 ..1w~ = ---. 2 v

(7.1.63)

The central part of a line is described by the Lorentzian distribution (7.1.64)

with width 2Yd = ..1~/v. Since v ~ vo/L, 2Yd = ..1eoo 21tL/). i.e., the width decreases ()( L with increase of density. This result was first obtained by Dicke [7.26]. For w~v, (7.1.63) coincides with the intensity distribution in the wing from (7.1.62). The qualitative picture of modification of the Doppler distribution due to collisions does not depend on the specific model of Brownian motion used above. We shall consider now the model of strong collisions assuming that after every collision the distribution of velocities does not depend on the velocity before collision and is Maxwellian. In this case, the collisional term in (7.1.50) can be written in the form

( f)f)If) t

=-vf+vW(v)jf(r,v',t)dv'.

(7.1.65)

coil.

For (Gf)k in (7.1.56), we have

(Gf)k = -vF(v,k) + vW(v)j F(v',k)dv' .

(7.1.66)

7.1 Model of a Classical Oscillator

249

Substituting (7.1.66) in (7.1.56) we obtain

F(v,k) =

vW(v) " .( k )JF(v,k)dv V+IW- ·V

W(v)

+ V+IW.( k·V)

(7.1.67)

Integrating the right-hand side and left-hand side of this equation over v we have

J F(v,k)dv

=J

W.~V~dV) (v J F(v',k)dv' + 1) ·V

V+IW

and after replacing in the right-hand side v' by v

W(V)dV) W(v)dv JF(v,k)dv ( I-vJ V+IW-'V .( k ) =J V+IW-'V .( k ) It is possible now to find F(v,k). Then using (7.1.54) we obtain

_ { I-v. kJ v+:~v2~. v) W(v)dv

1(w)-Re

J v + i(W -

}

(7.1.68)

.

k . v)

When V~LlWD, the second term in the denominator of (7.1.68) has the order of magnitude of V/LlWD. If this term is neglected, (7.1.68) gives the usual Doppler distribution. In the general case, the intensity distribution (7.1.68) is similar to that given by (7.1.60). Instead of (7.1.62), it follows from (7.1.68) for 1(0) and I ( n-2 v ) 1(0) ~ y'nLlWD 1+ y'n LlWD '

1(w)

1 vLlWb ~ --=:--- y'nLlWD 2y'nw4

(7.1.69)

In the limiting case v ~ LI mo, (7.1. 68) leads to a Lorentzian distribution in the central part of a line with width LlwMv. In the region of high frequencies W > v, the Lorentzian distribution is replaced by a wing 1(w) <X w- 4 • Thus in both cases considered above (weak and strong collisions), with an increase of the density a narrowing of the central part of the line occurs. At high densities, when L < A/2n, the narrowing of a line is proportional to L <X N- 1, and in the limiting case of L ~ A/2n, the central part of a line is described by a Lorentzian distribution with width 2nL/A times less than the usual Doppler broadening LlWD. Such a narrowing of the Doppler profile caused by collisions can be observed only in cases when there is no broadening due to interaction with perturbers. In the general case there are no grounds for separating the effect of interaction and the Doppler effect. The same collisions can produce both a phase shift and a change of velocity of the atom. This means the statistical dependence of both effects. It must be noted that Doppler broadening is usually of interest just under the condition L > Aj2n. In fact, the condition LlWD ;;:: y, where y is the impact width can be rewritten in the form 2nv/A ;;:: Nva' = Nvaa'/a = va'/La, where a is the gas-kinetic effective cross section of the atom. As a rule, a' ;;:: a and, consequently, Limo ;;:: y when L > A/2n.

250

7. Broadening of Spectral Lines

Nevertheless the statistical dependence of Doppler and impact broadening in some cases must be taken into account. This problem will be considered below in the framework of the quantum theory of broadening. A bibliography on Dicke narrowing may be found in [7.27].

7.1.6 Convolution of the Doppler and Lorentzian Distributions If L ~ A/2n, the combined treatment of impact and Doppler broadenings (statistically independent) leads to the convolution of Doppler and Lorentzian distributions. The Lorentzian intensity distribution with width y and shift L1, corresponding to the atom with velocity component v in the direction of observation, is given by

y I ------::----=2n (w - L1 - wov/c)2 + (y/2)2

(7.1.70)

/v(w) = -

To obtain the intensity distribution for an assembly of atoms, it is necessary to average (7.1.70) over the velocity distribution W(v). Thus /(w)

=l

2n

J

W(V)d~

(w - L1 - wov/c)

+ (y/2)

2 •

(7.1.71)

For a Maxwellian distribution

/(w) = l _ l _ exp [-(v/voil dv 2n y'nvo J (w - L1 - wov/c)2 + (y/2)2

(7.1.72)

When L1wo~y/2 the term wov/c can be neglected in the denominator in (7.1.72), after which the integration over v gives a Lorentzian distribution with width y. Consequently, when L1wo ~y/2 Doppler broadening can be neglected. When L1roo~y/2 a significant contribution to the integral (7.1.72) can be given by two ranges of values of v: v '" 0 and v'" c(w - L1)/wo. In the first of those ranges, the term Wov/c in the denominator can be neglected and in the second v can be replaced in the numerator by c( w - L1 )/wo. After this it is easy to obtain two approximate expressions for / ( w) valid for the center of a line w - L1 ~.oo and for the wing w - L1 ~ .00 , where .00 is determined by the relation

Dt, =

L1w~ In [2n3/2 L1~ (L1'!: Yl

.

(7.1. 73)

In the center of a line w - L1 ~ .00 , / (w) coincides with the usual Doppler distribution. In the wing of a line, /(w) ex y/2nw2. Thus for any relation between L1wo and y/2 at sufficiently high values of w, the Doppler distribution is replaced by the Lorentzian wing. We shall write (7.1.72) in the form I o Re {(W-L1) /(w) = y'n L1w W L1roo '

(7.1.74)

7.2 General Theory of Impact Broadening i 00 exp (-t 2)dt 1 00 W(x,y)=-! . = r.;!exp[-z2+i(x+iy)z]dz, 1t -00 x + ly - t v 1t 0

251

(7.1.75)

where x = (w - A)/Awo, y = y/2Awo, t = (v/c)(wo/Awo). The function W(x,y) can be expressed in terms of the probability integral with complex argument!

1.

2 i(x+iy) W(x,y) = exp[-(x+iyi] [ 1- y'n { exp (_t 2) dt

(7.1.76)

The intensity distribution I(w) for any relation between parameters y and Awo can be calculated using (7.1.76).

7.2 General Theory of Impact Broadening 7.2.1

Density Matrix Method in the Quasi-Classical Approximation

In the quasi-classical approximation the interaction of the atom with the surrounding particles can be described by the time-dependent perturbation V(t). In this case the coordinates of the perturbing particles can be considered not as dynamic variables but as assigned functions of time, which enables one to introduce the perturbation V(t) instead of the perturbation V(R). It will be shown in this section how the shape of a line is calculated when an atom undergoes an arbitrary perturbation V(t). From the theory of the interaction of a quantum system with electromagnetic radiation we know that for dipole transition 0( --+ p [7.2] J(w) ex

I! pocp(i) exp (-iwt) dtl 2 ,

(7.2.1)

where P ocp(t) is the matrix element of the dipole moment of an atom calculated by means of the perturbed wave functions tpoc(t) and tp p( t). These functions are the solutions of the Schrodinger equation for the Hamiltonian H

= Ho + V(t)

(7.2.2)

.

Formula (7.2.1) is the natural generalization of the classical formula (7.1.3).·It is helpful to write this formula in a form similar to (7.1.6), I(w)

=

~ Re

{1


(7.2.3)

where (7.2.4)

1

Tables of function W(x,y) are given in [7.28,29]

252

7. Broadening of Spectral Lines

or (7.2.5) We shall consider further a transition between the two degenerate levels a and b, the indices 0( and p numbering the states belonging respectively to the initial and final levels. We shall assume that all states 0( are populated with equal probability. In this case /(w)

= E /a.p(w); a.p

therefore instead of (7.2.4,5) it is necessary to assume lJ>('r)

= E Pa.p(t + -r) P:p(t) ,

(7.2.6)

a.p

(7.2.7)

IJ>(-r) = E(Pa.p(-r)P:p(O») . a.p

Equations (7.2.6,7) are easily generalized to the case when a line is formed by a set of transitions between two groups of closely spaced levels. We shall indicate by the indices 0( states belonging to initial levels and by the indices p, those belonging to final levels, and we shall denote by Wa. the population of the state 0(, Ea. Wa. = 1. Then IJ>(-r)

= E Wa.Pa.p(t + -r)Ppa.(t) = E Wa. (Pa.p(-r) Ppa.(O»). a.p

a.p

(7.2.8)

The perturbed functions 'l'a.< t) and 'l' p(t) can be expanded in terms of timeindependent functions of the isolated atom 'l'a.(t)

= ~aa.la.(t)'l'a./exp( -*Ea.d) ,

'l'p(t) = f,aplp(t)'l'p/exp( -*Eplt) ,

where p~~~? = (a:/a.ap'p) is the density matrix of an atom, the matrix elements Pa.lpl and PPa. do not depend on t. The upper indices (O(P) define the initial conditions p~~~?(O) = ba.a.lbpp" The evolution of the density matrix with time is given by the following equation dp i dt = ",(Hp - pH),

H = Ho

+ V(t) .

(7.2.9)

7.2 General Theory of Impact Broadening

253

For correlation function
= L Wcxp;~~!(r) Pcx'p,Ppcx .

(7.2.10)

cxPrx'P'

In the framework of impact approximation, the equation for the density matrix p can be written in the form i -dp = -(Hop - pHo) + (d - P)

(7.2.11 ) . dt Ii dt colI. The last term in right-hand side of this equation describes the evolution of the operator p caused by collisions. We shall assume that the wave functions before collision I[' and after collision 1[" are connected by the relation 1["

= SI[' ,

(7.2.12)

where S is the collision S matrix. According to (7.2.12) the corresponding transformation of the operator p is p ____

st pS .

The increase of p caused by collision with parameters v (impact parameter, velocity, and so on) is

= st pS - p, S = S(v) , where Sit = Ski. Therefore ,1p

(7.2.13 )

(7.2.14) = J[st(v)pS(v) - p] P(v)dv ( ddP ) t colI. where P(v)dv is the number of collisions with parameters v in the interval v, v+dv per second. If by v are understood the impact parameter p and velocity v, then P(v)dv = NVf(v) 2rr.pdpdv, where N is the density of the perturbing particles, and f(v) is the distribution function for v. Substituting (7.2.14) in (7.2.11) and carrying out the Fourier transform, it is not difficult to obtain the system of algebraic equations for matrix elements p ;~~! w) with different indices r:t.' P' and the same indices r:t.P,

(

hcxcx,hW - i(w - wcx'P' )p~~~!(w)

(7.2.15) From (7.2.3) and (7.2.10), we have " WrxPrx'P'PPrxPcx'p'(w) (cxP) } . J(w) = -1 Re { L.J rr. cxpcx'P'

(7.2.16)

It is supposed in (7.2.16) that radiation of definite polarization is of interest. In the case of arbitrary polarization, Pcx'P'P:B must be replaced by Pcx'p,P';.p.

254

7. Broadening of Spectral Lines

Summing with respect to a.'P' and a."P" in (7.2.15) and (7.2.16) in the general case means summing over all stationary states of an atom. However, in calculation of the intensity distribution in a narrow spectral range corresponding to the transitions between the two groups of closely spaced states a. and P, one can assume that the indices a.', a." take the same values as a., and the indices p', P" , the same values as p. Therefore, if the S matrix is known, the spectrum /(00) can be calculated by solving the system of (7.2.15) and using (7.2.16). For an isolated line a. --t p, we have only one equation: 1 - i(oo - roo)Pa.p = Pa.p f[l - S;a.Spp] P(v)dv.

Thus /(00)

= WI Pa.pI2Ya.P

1

2n (00 - 000 - Aa.p)2 + (Ya.p/2)2 '

Yi - iAa.p = f[I-S;a.Spp] P(v)dv, (1' - i(1"

= f[l - S;a.Spp] 2npdp .

(7.2.17) (7.2.18) (7.2.19)

+ itla.), Spp = exp(-rp + itlP)' Therefore the width Ya.P and shift Aa.p are determined by the expressions

In the general case, the S matrix is complex: Sa.a. = exp ( - r a.

Ya.p

= f{1 - exp[-(ra. + rp)] cos(tla. - tiP)} P(v)dv,

Aa.p = f exp[-(ra. + rp)] sin(tla. - tiP) P(v)dv.

(7.2.20)

We shall consider now the intensity distribution in the wing of a line corresponding to the set of transitions a. --t p. If the differences 00 - ooa.'P' are high enough, one can substitute in the right-hand side of (7.2.15) the zeroth approximation p~~!J" = -~a.a."~pp,,(oo - roo)-I, where 000 is the mean value of the frequencies OOa.' p', After this it is not difficult to obtain (7.2.21) where Y=

(E Wa.IPa.pI2rI2Re { E a.p

a.Pa.'P'

Wa.Pa.'P'Ppa. f[~a.a.'~PP' - S:a.,Spp'] P(V)dV} . (7.2.22)

The intensity distribution in the central part of a line is more complex than in the case of a Lorentzian profile. If the S matrix is diagonal (for example, when the perturbation V(t) is adiabatic), the spectrum /(00) can be easily calculated in the general form. In this case the system of equations (7.2.15) splits into independent equations.

7.2 General Theory of Impact Broadening

255

Solving these equations, we obtain I(w)=EW" IP"p 12 y"P } "p 2n (W - Wap - LI"p)2

+ (Y"p/2)2

.

(7.2.23 )

This intensity distribution has the form of the sum of independent Lorentzian distributions. In general case in order to calculate I ( w) one must determine the S \ matrix and then solve the system of (7.2.15). In calculating the S matrix, it is possible to use different approximate methods and results of the genera} theory of atomic collisions. In particular, one can define the S matrix using the method of successive approximations of perturbation theory. The eigenfunction 'l'(t) of the Hamiltonian Ho + V satisfying the initial condition 'l'(to) =
where the coefficients of expansion Smn(t) satisfy the system of equations

We shall introduce the operator

for which V mk

= Vmk exp [*(Em - Ek)t]

Therefore the system of equations for Smn can be written in operator form as

inSmn = E VmkSkn, k

inS = Vs. We shall seek the solution of this equation by the method of successive approximations: S(t) = }

+ (--,;i)t_ J V(t')dt' + (i)2t -;- J V(t')dt' t'_ J V(t")dt" + .... ~

n

~

~

(7.2.24) In order to determine the S matrix corresponding to the collision with parameters v, it is necessary to replace V(t) by t\(t), where Vv(t) -+ 0 at t -+ ±oo, and

256

7. Broadening of Spectral Lines

assume to

=

-00,

S(v)=I+

t

= 00:

00 ~ (-"hj)2 _LV.(t)dt_LV.(t')dt'+ 00 ~ ~ (-"hi) _LV.(t)dt+ .... t

(7.2.25) Let us restrict ourselves to the first two terms of (7.2.25). In this approximation it is easy to obtain for the quantities r and t'/ in (7.2.20) the following expressions: I ~ 1-L 00 (txlV.ls) exp(iwlXSt)dt 12 ra. =21/,2 "a.

1

= - "h

-L00 (txlV.(t)la) dt -

1

{,OO

1/,2 1m ~

-L (txlV.(t)ls)exp(iwlXSt) dt (7.2.26)

The quantity 2ra. is the total probability of transitions from the state a into all other states of the atom. Such transitions shorten the lifetime of an atom in the state tx, which is equivalent to broadening of the corresponding level. The quantity "a. is the phase shift caused by the shift of the level during the collision. The increase of r a. and r fJ increases the width and reduces the shift of the line, see (7.2.20). The broadening caused by collisional transitions is usually called the broadening due to inelastic collisions. 7.2.2 Degeneracy of Levels We shall consider the isolated spectral line corresponding to the radiative transition between levels with angular momenta j\ and h- Such line is produced by the set of transitions between degenerated states hm\ - hm2, where m\ and m2 are the magnetic quantum numbers. We shall write down the equation (7.2.11) for density matrix in the form (7.2.27) where

roo =

whh' and in accordance with (7.2.14) and (7.2.15),

Let us introduce now linear combinations of Pmlm2 which have the same transformation properties as the irreducible tensor operators T; see [Ref. 7.30, Sect. 4.3]. Since

7.2 General Theory of Impact Broadening

257

the required transformation has the form PS(f

=

E (_l)-h-ml( -mlm2!sa)Pmlm2

mlm2

E (_l)-h- ml-(fV2s+ 1 ( h

=

mlm2

-ml

~a )

Pmlm2 '

~a )

PS(f ,

h m2

(7.2.28)

S(f

h m2

where (-mlm2!sa) are the Clebsch-Gordan coefficients [Ref. 7.30, Sect. 4.2]. Multiplying (7.2.27) by (_l)-h- ml(-mlm2!sa), summing with respect to ml and m2, and expressing Pmimi in the collisional term in terms of PS(f, we obtain d PS(f -d t

. " PSI(fIGSI(fI,S(f = lWOPS(f + L..

S'(f'

E

GSI(fI,S(f =

,

(_l)-ml+mi( -m~m~!s'a')Gmimimlm2( -mlm2!sa).

mlm2mimi

The collisional term in (7.2.14) must be averaged over different orientation of the vectors l' and p in space. The interaction of an atom with perturbing particles averaged in such a way is isotropic. This means that the equations defining the irreducible tensor quantities PS(f are invariant under rotations and the matrix G is diagonal with respect to s and a. Moreover the matrix G does not depend on a:

Thus, the system of equations for PS(f has the form d dtPS(f

.

= lWOPS(f + GsPS(f

(7.2.29)

.

This system splits into independent equations for each of the quantities PS(f. The similar approach to the density matrix equation based on the expansion of the operators Pmlm2 in terms of the irreducible tensor operators has been used in [7.17,18]. That the matrix G is diagonal with respect to s and a can be verified by direct calculation. The integrand in (7.2.14) averaged over directions of l' and P must have the form E(m~m~!( -1)qu;V'~_q!mlm2) Kq

K

-q x ( h -

m'2

K

h)

-q m2

'

258

7. Broadening of Spectral Lines

U;

V;

where and are the irreducible tensor operators acting on the functions 'hI ml and 'h2m2' Substituting this expression in the formula defining the matrix Gmjmimlm2 and carrying out the summation of the product of the four 3j symbols in the expression for Gs'q',Sq with respect to mJ, m2, m~, m~, we obtain Gs'q',Sq

=

f>ss,f>qq,G s ,

Gs

= 2: Gs(K) . K

Let us proceed to the calculation of the spectrum J(ro). From (7.2.16), we have J(ro)

=

~Re{ mlm2mjmi;' 2: Wml(P)Jmjmi(P;')~lm2P~~2)(ro)} ,

where p;. is the spherical component of the vector P. We shall assume that the population of the state ml is (2jl + 1)-1. Transforming (7.2.15) in the same way as (7.2.29), we have

h m2

S) -i(ro -

1

roo) + Gs

-(J

•

The quantities p~~2)(ro) are expressed in terms of p~~lm2) in accordance with the second of equation (7.2.28). For the matrix elements of p;., we have [Ref. 7.30, Sect. 4.3].

~ ~~) (jIIIPlih) , 1

A.

h) (jIIiPllh)* .

m2

Therefore, it is necessary to calculate the following sum of the product of the four 3j symbols: S -(J

x (

jl -ml

1

h

-m~

h

h

-m~

m'2

S

)(

-(J

h

-m~

h m'2

-(J S

)(

h

-ml

~) 1

=9f>sl ~ f>.J.,-q = 3f>sl .

Let us recall that 2h - 2ml is even.

h m2

1)( h

A.

-m~

h m'2

1

A

1 h) A m2

h =2:(h -ml m2 x (

)(

~(J)

h)

m~

7.2 General Theory of Impact Broadening

259

Finally we obtain I(w) =

iUlilPiihW . ~Re{ 2h + 1 11: -i(w iUlii P iih)i 2 Re{GJ} 2h+1

11:

1 wo)

+ GI

}

1 (w-wo-Im{G I })2+(Re{GI })2'

(7.2.30)

For width y and shift .1, we have y = -2Re{GJ},

.1

= Im{G I } .

(7.2.31)

Although the possible values of the numbers s are s = h + h, h + h - 1, .... UI - hi, the broadening is completely determined by the density-matrix element Plu and the width and shift of a line are expressed in terms of the real and imaginary parts of GI . As shown in Sect. 7.2.1 in the general cases of a line corresponding to the set of transitions IX - f3 between the two groups of closely spaced levels, the spectrum I (w) is not described by Lorentzian distribution. Only in the wings of a line is the intensity distribution determined by the simple formula (7.2.21). Nevertheless in the case of a line formed by a set of transitions between degenerate states ml and m2 belonging to the levels jl and h, (7.2.16) for I(w) gives the Lorentzian distribution. In order to determine the width y and shift .1 of this distribution, it is necessary to calculate only one quantity GI . Therefore the broadening of a line corresponding to dipole radiative transitions is completely determined by one of the equations (7.2.29), namely by the equation corresponding to the irreducible representation s = 1. Repeating the derivation of (7.2.30) for quadrupole radiative transitions, it is not difficult to show that in this case the broadening is determined by the density-matrix element Psu with s = 2, the width and shift being y = -2Re{ G2}, .1 = Im{ G2}. 7.2.3

Quantum Theory

It has been shown above that in cases when the relative motion of an atom and perturbing particles can be described in the framework of classical mechanics, the theory of the broadening of spectral lines is a natural generalization of the classical oscillator model. We shall consider now the quantum theory of pressure effects in which not only the motion of the atomic electrons but also the relative motion of the atom and perturbing particles is described by the Schrodinger equation. Such a quantum theory enables one to take into consideration the broadening caused by light perturbing particles, especially by electrons. Let us begin with a relatively simplified problem, assuming that the mass of the atom is large as compared with the mass of the perturbing particle, so that the atom is at rest. We also assume that only elastic scattering take place.

260

7. Broadening of Spectral Lines

We shall define the density matrix of the atom using the general rule [7.31]:

= J 'P*(r, R) 'P(r', R) dR ,

p(r, r')

(7.2.32)

where R is the coordinate of the perturbing particle. The stationary states of the system consisting of the atom and perturbing particle are described by the wave functions 1 [

'Prdc(r,R)= cPl1.(r).;v exp(ik.R)+II1.(19)

eXP(ikR)] R '

(7.2.33)

where cPl1.(r) is the atomic wavefunction, li19) is the scattering amplitude, k is the wave vector of the perturbing particle, and V is the normalization volume. Let us expand the wave functions on the right-hand side of (7.2.32) in terms of the wave functions (7.2.33): 'P(r, R)

=L

al1. 'P rdc(r, R) .

11.

We shall expand the density matrix p(r, r') in a sum over atomic stationary states P(r, r')

=L

(7.2.34)

pl1.pcPl1.(r)cPp(r') .

I1.p

As a result, we obtain Pl1.p

I

= a:ap~ [exP( -ik· R) + I: eXp(~ikR)] [eXP(ik. R)

+I

eXP(ikR)] dR _

P

R

*

{

2. J [I Pexp( -ik·RR + ikR)

- al1. ap 1 + V

J:lp] dR} 1 * exp(ik·R - ikR) +11. R +R2

(7.2.35)

The exponential factor exp (-ik· R + ikR) = exp( -ikR cos 19 + ikR) rapidly oscillates everywhere with the exception of the small region cos 19 ~ 1, 19 = o. Therefore in calculating the first two integrals in (7.2.35), we can use the approximation

2.V Jexp( -ik· R + ikR)/( 19) dR ~ 1(0) Jexp( -ik. R + ikR) dR R V R = 1(0)2:i ~ ,

where R is the radius of the spherical volume V and 1(0) is the amplitude of forward scattering. Thus Pl1.p

= a:ap{ 1- 2:i ~[I:(O) -

Ip(O)]

+ ~ J1:(19)/p(19) dO}.

(7.2.36)

If one neglects interaction and assumes that in (7.2.33) and (7.2.35) 111. = Ip = 0, then Pl1.p = a:ap. Therefore the last two terms in (7.2.36) give the collision

7.2 General Theory of Impact Broadening

261

contribution to the density matrix: Ll Pap = przPV R { -T[foc(O) 2ni * - fp(O)]

* + J frzfp

dO } .

(7.2.37)

This expression is the quantum generalization of the quasi-classical expression (7.2.13). In order to obtain the collisional term (dpldt)coli. in the general equation for the density matrix in the impact approximation (7.2.11), it is necessary to take into consideration that the wave functions (7.2.33) are normalized in such a way that volume V contains only one perturbing particle and the time during which one collision occurs is Rlv, where v = hklm is the velocity and m is the mass of the perturbing particle. After multiplying Ll Pocp by NVv I R and averaging over velocities v, we have

( ddPocP ) t

= -N(v«(J' - i(JII») Pocp ,

(J' - i(J" = i

2;

[f:(O) - f p(O)] - J f:f p dO .

We shall express the scattering amplitudes expansions

f oc

1 '" = 2ik -,[.,(21 + 1)[S f - 1] Pf(COS'l9) ,

f

(J' - i(J" (J

(7.2.38)

coIl.

,

and

(7.2.39)

f p in the form

Sf = exp (i2'11) ,

= ;E(21 + 1)(1 - S~*Sh),

of partial-wave

(7.2.40)

(7.2.41 )

n", I = k2LJ21 + 1)[1 - cos2(11rzf - 11p)] ,

(7.2.42) (7.2.43)

Thus the width and shift cross sections (J' and (J" are expressed in the terms of scattering amplitudes or scattering phases. [Equations (7.2.39,42, and 43) were obtained by a somewhat different way in [7.32,33]. Expressions (7.2.41-43) establish the connection between the broadening of lines and the elastic scattering of perturber. In those cases when the perturbation of one of the levels can be neglected, (J' = (J 12, where (J is the elastic cross section (let us recall that)' = 2Nv(J'). In the quasi-classical limit,

n

hl-t mvp, k 2E(21

+ 1) - t 2nJ pdp,

2(11~ - 11~)

-t

11(P) ,

(7.2.42,43) coincide with (7.1.13, 14). Equation (7.1.21) for 11(P) is the limiting expression of the general quasi-classical formula for the phase 2(11~ - 11~), which

262

7. Broadening of Spectral Lines

is valid (in the case of the field IiCnR- n) providing mCn k n -

2

~ 1.

Ii

This condition can be rewritten in the form Po~A.

= Ii/mv,

where Po is the Weisskopf radius (7.1.24). For heavy perturbing particles, this condition is always fulfilled. But in the case of light perturbing particles, in particular for electrons, it is violated. We shall consider the dependence of a' on the perturber velocity v at high v, i.e., in the Born approximation. The cross section a' of (7.2.42) can be written in the form

a' = ~ J If,,(~) - fp(~)12dO .

(7.2.44 )

For electron scattering in the Born approximation, f(~)

=

2me2

(7.2.45)

1i2q2 [.2" - F(q)] ,

where .2" is the total number of electrons in the atom, F(q) is the atomic form factor (scattering factor) and q = 2k1i- 1 sin ~/2. Substituting this expression for f(~) in (7.2.45), we obtain , _ 4nm 2e4 2Jk [F,,(q) - Fp(q)f d

a -

1i4k2

q4

0

(7.2.46)

qq.

At high velocities, the scattering amplitude has a very sharp maximum in the region of small angles. This means that the integral (7.2.46) does not depend on the upper limit (q = 2k, when ~ = n/2). Thus integration in (7.2.46) can be extended to q = 00. After this the integral in (7.2.46) no longer depends on k and

a' ex k- 2 ex v- 2 ,

y ex

V-I.

Thus at high velocities, the width of a line is inversely proportional to the velocity for any type of interaction. It is necessary only that the integral (7.2.46) converges. The formulas obtained above are easy to generalize so that they also include inelastic collisions. By repeating the derivation of the expression for the collisional term (dp/dt)col1. it is not difficult to show that (7.2.39) and (7.2.41) preserve their form. But the quantities Sl are now Sl = exp( - 2rl - i2f11). Therefore a' = ;2:.(21 + 1}{1 -

a"

= - ;

2:.(21

exp[-2(r~ + r~)] cos2(f1~ - f1~)},

+ 1)exp[ -2(r; + r~)]

sin 2(f1~ - f1~) .

(7.2.47)

(7.2.48)

7.2 General Theory of Impact Broadening

263

If the perturbation of one of the states can be neglected, then (I'

= ;'L-(21 + l){l - exp(-2rl)cos2t//} =

1 2«(lel.

where (leI. and scattering.

7.2.4

(7.2.49)

+ (lined, (linel.

are, respectively, the cross sections of elastic and inelastic

Quantum Kinetic Equation Method

In this section we shall generalize the density-matrix equation discussed above taking into consideration not only the motion of the perturbing particle but also the motion of the atom. Such a generalization is necessary for understanding the very important problems connected with the statistical dependence of the impact and Doppler broadening [7.20,34-38]. In many applications of the theory of spectral line broadening such as plasma diagnostics, effects of the statistical dependence of the impact and Doppler broadening do not play a significant role. They are of special interest for nonlinear laser spectroscopy [7.39]. The densitymatrix equation describing this general case has the same structure as (7.2.11),

i (d -dp = -(Hop-pHo)+ - P)

dt

dt

h

.

(7.2.50)

colI.

But now the unperturbed wave functions must be taken in the form ~mp

(7.2.51)

= ~m(r) exp (ip· R) ,

where R is the coordinate of the center of mass of the atom, m is the quantum number describing the atomic state, p is the wave vector of the atom as a whole, and hp is the momentum of the atom. In (7.2.16) for the intensity distribution /(0) in the case of an isolated spectral line m --t n, the following substitutions must be made:

ex --t mpo, p --t meo, ex'

--t

mp,

p' --t me, Wex = W(Po),

P

--t

d exp (ik· R) ,

where k is the wave vector of the photon, and d is the electric dipole moment. In addition, the sum over expex' p' must be replaced by an integral over Po, p, "0, ". As

(mprd exp (ik· R)lmc) ex: d mn t5( -p + k

+ K) ,

(nKold exp (-ik· R)lmpo) ex: d:z,.t5( -KO - k

+ Po) ,

264

7. Broadening of Spectral Lines

it is not difficult to obtain 1(w)

2 = Id mn l Re{f dp dp W(P ) pmpo,n(Po-k)} 1[ 0 0 mp, n(p-k)

(7.2.52)

= Id mn l2 Re{J dpF(P, p - k)} , 1[

where F(p, p - k) = f dpo W(Po) p:P~~~~~~) is the corresponding density matrix element, averaged over the initial distribution of Po. We shall omit the factor Id mn l 2 everywhere below. Let us introduce the notation (dp/dt)coli. = Gp, where G is the operator describing the evolution of p due to collisions. From (7.2.50), we obtain -W(P) +

i(

W -

~pok) F(p, p -

k) = (Gf)p,p-k,

(7.2.53)

where m is the mass of an atom I. Equations (7.2.52,53) have the same form as (7.1.54) and (7.1.56). Now, however, it is possible to express the collisional term (G f) in terms of the exact scattering amplitudes 2 • The general expression of (Gf) is very complicated due to the fact that density matrix is defined in the laboratory system of coordinates and the scattering amplitudes are defined in the system of the center of mass of the colliding particles. We shall give below the expressions for (GF) in two limiting cases of light and heavy perturbing particles, mp ~ m and mp ~ m, where mp is the mass of the perturbing particle, and m is the mass of the atom. In the first case, in the limit mp/m ~ 0, (GF)p,p_k

=-

G- iA) Fp,p-k ,

(7.2.54 )

y/2 - iA = N(vp(a' - ia"») , where a' and a" are respectively the width and shift cross sections, determined by (7.2.39-43), and vp is the velocity of the perturbing particle. Substituting (7.2.54) in (7.2.53), it is not difficult to show that the spectrum 1(w) has the form of a convolution of the Doppler and Lorentzian distributions: 1( ) =!R W

1[

e

{f i(w-A-hp W(P)dp } k/m)+y/2 ' o

hpok - - =v·k, m

(7.2.55)

where v is the velocity of an atom.

I

A tenn proportional to k2 is omitted in this equation: (Hop - pHo)mp,np-k

{Em -Eft 2

+ ~:[p2

- ki]} ~ (p -

pmp,np-k (IiWo

+~

The general expression for (Gf) is given in [7.37].

P.k).

= Pmp,np-k

7.2 General Theory of Impact Broadening

265

In the opposite limiting case of light atom and heavy perturbing particle mp~m,vp~v,

hp (GF)p,p-k = -N-;;[a'(p) - ia"(p)] F(p, p - k) A

- N hp 8(p) F(p, P - k)

m

+ N hp J dOq/~(q, P)!n(q, p) F(q, m

q - k) ,

(7.2.56) where a' and a" are, as before, the width and shift cross sections, but now they are dependent on the wave vector p of the atom: 8(p)

= J dOq/~(q, P)!n(q,

p)

=

n22 )21 + I)(S~ - I)(S~ - 1)*. (7.2.57) p

The equation (7.2.57) coincides with the expression defining the elastic cross section a if !;!n is replaced by l!nl 2 or l!mI 2 • In the collisional term (7.2.56), the cross section 8 plays the role of the elastic cross section a. The cross section 8 differs from a, being in the general case complex. Ifthe scattering is isotropic, or almost isotropic, then 8(p) '::::' 4n!;'!n. Taking this into consideration we obtain from (7.2.56)

= -Nv[a'(v) -

ia"(v)] F(p, P - k)

- Nv8(v) F(p, p - k)

+ Nv8(v)f dOg F(q,

(GF)p,p_k

(7.2.58) 4n

q - k) .

Equation (7.2.53) takes the form [cf.(7.1.67)] W(P)

i(w - k. v) + Nv(a' _ ia"

+ 8) = F(p, p -

Nv8 .( k )+Ni(' 1 w.v v a -

k)

dOq

. "+-/-4 la a n F(q,q-k).

Integrating the left- and right-hand side of this equation over dO p and taking into consideration that for isotropic distribution W(P)

= W(P2)' 4np

it is easy to find

w(p)dp

dO p

= dOv ,

JF(p, p-k)dOp and then using (7.2.52) to obtain the spectrum

/(w) = .!.Re { Jw(v)dv n

= w(v)dv,

d~ ~<;; - k.v) + Nv(a'l1 ia" + 8)

}

J 4iC

.

v

I-Nv8J 4n i(w-k.v)+Nv(a'-ia"+8)

(7.2.59)

266

7. Broadening of Spectral Lines

This equation differs from (7.1.68), taking account not only of the perturbation of the internal atomic state, but also of the perturbation of the motion of the atom as a whole. All the cross sections contained in (7.2.59), (1', (1", and ii, are dependent on v. When N ~ 0, (7.2.59) coincides with the usual Doppler distribution (7.1.57). At high densities when one can neglect the term k °v, (7.2.59) takes the form

I NV(1' /(w) = - fw(v)dv 2 2 11: (w - NV(1") + (Nv(1')

(7.2.60)

•

Instead of the usual Lorentzian distribution, (7.2.60) describes a superposition of Lorentzian distributions, each having width 2Nv(1', shift NV(1", and relative weight w( v). If one assumes that the scattering in one of the states m and n can be neglected, when ii = 0, then (7.2.59) takes the form

/( w ) -.!.R {f W(v)dv - 11: e i(w _ k oV) + NV(1' -

iNV(1"

}

.

(7.2.61)

This expression differs from the usual convolution of the Doppler and Lorentzian distributions containing lNV(1' ( v) and NV(1" ( v) instead of the averaged quantities y = IN(V(1') and LI = N(v(1"). The intensity distributions of (7.2.60,61) are asymmetric. In the case (1' = (1" = 0 (the scattering is purely elastic and 1m = In), (7.2.59) describes a narrowing of the Doppler profile similar to that given by (7.1.68). The parameter Nvii plays the same role as the frequency of collisions in (7.1.68). We shall consider now the case of high densities «(1' = (1" = 0, Nvii't>kov):

f

I ~ k 2 v2 -:-:-_ _ _=_=_ 411: i(w-kov)+Nvii iw+Nvii - -3-(iw+Nvii)3 .

dO"

For frequencies w4.Nvii (7.2.59) gives

II}

/(W)=.!.Re{fW(V)dV 11: iw + _k2V2(Nvii)-1 3

We shall introduce the notation D = v/3Nii. Then

/()

w =

2

f

-

()d k Re{D} W

v

V

11:

1

0

(w + k2Im{D})2

+ (k2Re{D})2

(7.2.62)

If 1m = In, and ii coincides with the elastic cross section (1, then Im{D} = 0 and Re{D} = D is the diffusion coefficient depending on v instead of (v). In the general case of complex ii and D, the real part of D determines the width and the imaginary part of D determines the shift. The width 2k2Re{D} is proportional to N- 1. Thus, at Nvii 't> k v, the Doppler distribution narrows due to collisions. The 0

7.2 General Theory of Impact Broadening

267

resulting line profile has the form of a superposition of Lorentzian distributions with widths 2k 2 Re{D} and shifts k2 Im{D}. This profile is asymmetric. If (J' and (J" are less than Ii but are not equal to zero, then on increasing N, there is first a narrowing of the Doppler contour to a width Nva', and then a broadening. The results of the calculation of the spectrum J (OJ) by means of the quantum kinetic equation contain a number of new elements, the most interesting of which are the following. Even in the treatment of the simplest example, purely impact broadening, qualitative differences from the formulas usually used arise. Only in the case of broadening by light particles, such as electrons, does a single Lorentzian contour arise, with width 2N(vp(J') and shift N(vp(J"), where vp is the velocity of the electrons and the angle brackets denote averaging over vp. In the general case of mp m, after averaging over the velocities of the perturbing particles vp, the cross sections (J' (v) and (J" (v) contained in the collisional term retain their dependence on the velocity v of the atom. As a result, the following intensity distribution arises: r-..J

r-..J

J(OJ) = /w(v)dvNv(J'(v) 1C

I [OJ - NV(J"(v)]2

+ [Nv(J'(v)F

.

(7.2.63)

This distribution is asymmetric. The greatest difference arises in the case of scattering of a light atom by heavy (almost at rest) perturbing particles. The perturbations due to different perturbing particles combine in completely different ways, depending on the masses of these particles. If the perturbing particles of type 1 and type 2 are light, then the sum of the corresponding widths and shifts arises:

~

- iii

= NI (VI (JD + N2 (V2(J~) + i NI (VI (J~') + i N2 (V2(J~) .

But if the perturbation is created by heavy particles (type 1) and electrons (type 2), then (at Ii = 0)

w(v)dv

}

All this is a reflection of the statistical dependence of the Doppler and impact broadenings. The second characteristic feature is the fact that the cross section Ii responsible for the collisional compensation of the Doppler broadening is complex. Let us recall that Ii =f:. 0 if both the scattering amplitudes 1m and In are nonzero. It is not difficult to show that in the examples treated, asymmetry arises for two reasons - the dependence on the atomic velocity of the parameters of the equation, and the fact that the cross section Ii is complex.

268

7. Broadening of Spectral Lines

In calculation of the width of the resulting spectrum, all characteristic features of collisional broadening connected with the effect of the statistical dependence of the Doppler and impact broadenings are usually not very important. Nevertheless they can be of interest for some other problems, for example, those arising in the theory of nonlinear resonances in the spectra of gas lasers [7.38,39] (see also the bibliography given in [7.20]. 7.2.5

Absorption Spectrum

The energy absorbed in one second by a system of electric charges interacting with an electric field tI(R, t)

= ~{tlo exp [i(wt -

k·R) +

tlo exp [-i(wt -

k·R)]}

is (in the electric dipole approximation)

Q = -tl.J = Re {iwtlo ·dro exp(ik.R)} , where d = dro(R) exp (iwt)

+ d~(R) exp (-iwt)

is the electric dipole moment induced by the field tI(R,t). The quantum generalization of this expression for Q has the form

Q = Re [iw

tlo Trace {d exp (ik.R) p(w)}] .

(7.2.65)

Here d is the electric dipole operator, R is the coordinate of the center of mass of the atom, k is the wave vector of the photon, and p( w) is the Fourier component of the density matrix, satisfying the equation dp i "i dt - h (Hop - pHo) - Gp = h (tI·dp - ptl·d).

(7.2.66)

This equation contains an additional term describing the interaction of the atom with the electric field. Solving this equation by the method of successive approximation, it is possible by means of (7.2.65) to determine the absorption (or emission) power Q. If a set of transitions IX --+ P are perturbed by the field tI (the frequencies wa.p ~ w), then using (7.2.65) we obtain

Q = Re {iw tloEda.pJ dp PPp,a.p+k(W)} , a.p

(7.2.67)

where p is the momentum of the atom. We shall neglect the collisional term simplicity and solve (7.2.66) in the linear approximation according to the field. We shall substitute in the right-hand side of (7.2.66) the zeroth-order density matrix, which is diagonal in indices IX, p and

7.2 General Theory of Impact Broadening

269

p,p':

p~ipp = NpW(p),

P~o;+k,rxP+k = NrxW(p + k) ~ NrxW(p) ,

where Np and Nrx are respectively the populations of the states p and IX, and W(p) is the distribution function for momentum p. Since k~p, we can assume that W(p + k) ~ W(p). In this approximation we have [cf. (7.1.57)]

i 1 Ppp,rxp+k = 2f/o·dpiNp-Nrx)W(P)i(w_wo+hp.k/m) '

1(w)

= !Re {lim J. n

6-+0

W(p)dp } hp·k/m) + e

I(W - Wo -

Therefore the absorbed power Q is proportional to the difference of the populations (Np - N r,.) and the function 1(w), which describes the usual Doppler distribution in the spectrum of spontaneous emission. In the general case, when (7.2.66) contains the collisional term, the expression for Q remains the same, but the function 1(w) has a more complex form describing the Doppler and impact broadenings. All results obtained above for the spectrum of spontaneous emission can be obtained also from (7.2.65, 66). By solving (7.2.65) in the next approximations according to the field, it is possible to calculate the power of nonlinear absorption [7.38]. 7.2.6

Interference Effects: Narrowing of Spectral Lines

In cases when the frequencies of some atomic transitions coincide or are so closely spaced that the corresponding spectral lines overlap, specific interference effects can arise [7.40]. In some particular cases, these interference effects are so important that they alter the entire picture of the broadening. We shall illustrate this by considering as an example the four-level system shown in Fig. 7.2. We shall assume that the transition frequencies Wkt = WI and Wmn = W2 are almost

~---I.--n

Fig. 7.2. Levels scheme and the radiative transitions which are considered

270

7. Broadening of Spectral Lines

the same, co, ~ CO2, but that all other transition frequencies differ very much from CO, and CO2. Therefore in (7.2.16) and (7.2.15), the indices a.,a.',a." stand for k,m and the indices p,P',P" stand for I, n. We shall denote the pair of indices k,1 by 1 and the pair of indices m, n by 2. We shall use the notation Wk = W, and Wm = W2. By solving the system of equations (7.2.15), we obtain J(co) = !Re {W,IP,1 2G22

+ W21P212G l1 -

W2 P ,Pi G'2 - W,pjP2G2'}, G" G22 - G2' G'2 (7.2.68)

11:

where Gl1 = i( co - cod + J (1 - SaSIl) P( v) dv , G22 = i(co - CO2) + J(1 - S;mSnn)P(v)dv, G'2

(7.2.69)

= - JS;kSnlP(v)dv ,

G2' = - JS;"'SlnP(v)dv. We shall assume that S matrix elements obey the relations

= SIl, Smm = Snn, Skm = Sin, Ski = Smn = Skn = Sml = 0 .

Sa

Smk

= Snl ,

(7.2.70)

This means that collisions produce only the mutual perturbation of the states I and n and also the states k and m. It will be shown that such a situation can arise in a number of systems. Using (7.2.70) and also the unitarity properties of the S matrix, L:ISab12

=

1,

b

we have S;kSnl = ISmk 12 = 1 - ISmm 12 , S;"'Sln

= ISkm 12 = 1 -

(7.2.71 )

ISa 12 •

From the general definition of the inelastic cross section u, it follows that (7.2.72) where N is the density of the perturbing particles, v is the relative velocity, and the angle brackets denote averaging over velocities. In Boltzmann equilibrium the level populations W" W2 and transition frequencies are connected by the relations N(vUkm)

=

i,

N(vumk)

Thus G" = i(co - co,) + y1/2, G'2

= -(WI/W2)yI/2,

=

i,

W,y,

= W2Y2

.

G22 = i(co - CO2) + (WI/W2)yI/2, G2'

= -y1/2 .

(7.2.73)

(7.2.74)

7.2 General Theory of Impact Broadening

271

We shall introduce the notation COl

= COo -

15,

CO2

= COo + 15,

215

= L1 .

(7.2.75)

After this (7.2.68) gives

I(co)

=~

. (WIYI + W2Y2)I(co - t5)PI - (co + t5)P21 2 2n [(co + I5)(co - 15)]2 + [YI(CO - 15) + Y2(CO + 15)]2

(7.2.76)

We shall consider now the limiting cases of small and high values of y. In this first case, when YI,2/t5~ 1, the second term in the denominator of (7.2.76) is small, and the function I(co) has two sharp maxima at co = ±t5. Equation (7.2.76) can be rewritten in the form

I(co)

~

.!. ( n

WIy!lP I 12 (co+t5)2+YI

+

W2Y2!P2 12

(CO-t5)2+y~

) .

(7.2.77)

In the other limiting case, when YI, 2/15 ~ 1, (7.2.76) has one sharp maximum at

co ~ COM = YI - Y2 t5 . YI +Y2 In the vicinity of this frequency, (7.2.76) gives

.!.

I(co) '" (WIYI + W2Y2)IY2 P I + YI P21 2 r 2YIY2(YI + Y2) n (co - COM)2 + r 2 '

(7.2.78)

where

r =

4y IY2 152 (YI + Y2)3 .

Equation (7.2.78) describes a Lorentzian distribution with width rex N- I . Therefore at low pressures, when Y\,2/t5 ~ 1, the two components of a line are independent and their widths YI and Y2 are proportional to N. On further increase of N when the components of the line begin to overlap, the picture of the broadening alters completely, and in the limiting case of YI, 2/15 ~ 1 a single Lorentzian distribution with width proportional to N- I arises. In the far wing of a line co~Y\' Y2, 15, in accordance with (7.2.76), the spectrum I(co) has the form

I(co)

~

WIYI ~ W2Y21w(P I -P2) :/(PI +P2)i2

(7.2.79)

Note that in the particular case PI = P2, I(w) ex co- 4 . All qualitative features of the example considered above are connected with the conditions (7.2.70) for the S matrix. There are a number of systems for which the S matrix obeys such conditions. We shall consider two subsystems I and II with the levels A, B and a, b, respectively, shown in Fig. 7.3a. Let us assume that subsystem I is perturbed by collisions, but they do not act on subsystem II.

272

7. Broadening of Spectral Lines 4 (8b) J(Ab)

8--

2 (8a)

A--

f(Aa)

b

a

Fig. 7.3. (a) Levels scheme of subsystems I and II (b) Levels scheme of the whole system.

The system as a whole has four levels: I(Aa), 2(Ba), 3(Ab), and 4(Bb), shown in Fig. 7.3. Two transitions 3 --+ 1 and 4 --+ 2 correspond to the tr~sition a --+ b of subsystem II. If the interaction between subsystems I and II is not great, the interaction produces a splitting of the frequencies W31 - W42 = A I- 0, but does not influence the probability of the collisional transition A-B in subsystem I. In this case the S matrix has the form SII [ S21 S31

SI2

SI3

S14]

S22 S32

S23 S33

S24 S34

S41

S42

S43

S44

_ -

SAB SBB

00 0] 0

0

0

SM

SAB

0

0

SBA

SBB

[SAA SBA

'

(7.2.80)

i.e., is in full agreement with the conditions (7.2.70). Thus the example considered above describes the rather typical situation when the relaxation processes in subsystem I perturb the spectrum of subsystem II. We assume of course that subsystems I and II interact. If this interaction, and consequently also the frequency splitting A, is great enough, i.e., YIA ~ 1, then relaxation processes in subsystem I produce broadening of the line, corresponding to the radiative transition is subsystem II. In the other limiting case of weak interaction (y I A --+ 00), the spectrum of subsystem II is not sensitive to the relaxation processes in subsystem I. For a given value of A, the inequalities YIA ~ I and YI A ~ 1 correspond respectively to low and high densities N. Therefore, on increase of N, there is first a broadening of the spectrum of subsystem II to a width rv y, and then a narrowing proportional to N- I . The spectrum narrowing with increase of N is due to the interference of the amplitudes of the radiative transitions when the corresponding line components begin to overlap. Let us consider for example the splitting of the term 2P in a strong magnetic field H [Ref. 7.30, Sect. 8.2]: (7.2.81 )

7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma

273

where J4J is the Bohr magneton, A is the fine structure constant, and ML,Ms are the magnetic quantum numbers of the orbital and spin angular momenta. We shall which gives three consider the magnetic dipole transition Ms = --+ M~ = components of the line corresponding to the three possible values of ML = 0, ± 1. The frequency splitting of these components is of the order of A. The cross section of the spin reorientation (is usually is less than the cross section of the orbital angular momentum reorientation (iL. Therefore one can assume '

!

-!,

If Y ~A, collisional reorientation of the orbital angular momentum produces broad(ML = 0, ± I). If density N is so ening of the transitions Ms = --+ M~ = high that y > A, then the relaxation transitions ML --+ M{ begins to be ineffective in broadening the spectrum of the transition Ms = M~ = Moreover, on increase of N, a narrowing of the spectra must be observed. In the limiting case of N(V(iL) ~A, the width of the spectrum is less than the initial splitting A. It must be noted that the condition N(V(iL) ~A can be fulfilled only for light atoms for which the fine splitting is not too great. For example, the fine splitting of the ground level of Li atom is 0.34 cm -I. As will be shown in Sect. 7.3 the interference narrowing of a line can take place in the spectra of highly excited hydrogen atoms.

!

-!

!-

-!.

7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma Preliminary Estimates

7.3.1

The main contribution to the broadening of lines of the hydrogen spectrum in a plasma is due to the linear Stark effect in the fields of electrons and ions. The perturbing particle with charge Ze produces the electric field S = ZeR - 2 • Using the well-known formula for the linear Stark effect [7.30] Am = 3/2n (nl-n2)ea oS/Ii, where n,nl,n2 are the principal and parabolic quantum numbers, we can assume Am = C2/R2. The constant C2 for the level with principal quantum number n has the order of magnitude Zn(n -1 )e2ao/1i ~ Zn(n -1) [cm2 s- I ]. We shall estimate the magnitude of the dimensionless parameters (7.1.39) h

rv

e

2 )3

N. (n(n - l)e ao e

liVe

k '

rv I

N (Zn(n - l)e2ao I IiVi

)3

(7.3.1 )

where Ve and Vi are the velocities of electrons and ions, respectively. The range of temperatures and densities for which hi ~ 1 and he ~ 1 is usually of greatest interest. This means that the field of the ions is quasi-static and the electrons cause impact broadening.

274

7. Broadening of Spectral Lines

7.3.2 Ion Broadening: Holtsmark Theory For hi ~ I, number of ions in the sphere of Weisskopf radius is large and the binary approximation is inapplicable. Thus the main problem which arises in considering ion broadening is to find the quasi-static intensity distribution taking into account the simultaneous action on the atom of a large number of ions. We shall consider the component ex-p of a line and denote the shift of this component in the field C by [Ref. 7.30, Sect. 7.2] (7.3.2) where ex and p are the set of parabolic quantum numbers nln2m and intensity distribution at a given field C is given by

n~n~m'.

The

(7.3.3 ) Averaging this expression by means of the distribution function W(C), we obtain (7.3.4 ) The resulting ionic field C is equal to the vector sum over all ions,

The function W(C) determines the probability of a given magnitude of the absolute value of C. This function was calculated by Holtsmark in the ideal gas approximation. (A detailed discussion is given in [7.25]). In this approximation, one assumes that each of the ions can with equal probability be located at any point of the volume independently of how all the other ions are located. Therefore the function W(C) can be calculated in the following way:

WH(C) dC

=

(15

(c -Ze I: R:) dC) k=IR k

= JdR I JdR2 ... 15 (c-zeI: R:) dC V V k=IR k = (2

1 1t

dR I dR2

.

NV

(.zep.R k )

)3 J - J - ... J dp exp (Ip· C) ~ exp -I V

V

k=l

R3

dC .

k

Here we use the well-known representation of the t5-function, introducing the additional integration over p. By changing the order of integration over R and

7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma p, we obtain in the limit V

WH(C)

275

--> 00

=

(2~)3 J dp exp (ip

=

(2~)3 J dp exp (ip

0

0

C) {I -

~J dRk [1 _ exp (-ize~tk )]}

N V

C) exp { -N J dR [I - exp ( _/e;;R) ] }

Integration over R gives

J dR [I - exp ( _i ze;3oR )]

=

1~(2nZep)3/2 .

Then it is possible to carry out the integration over the angular variables of the vector p. As a result we have

WH(C)dC

= d: .1t (~) ,

(7.3.5)

where

.1t(f3)

Co

=

~f3J:xsinx exp 4

= 2n ( 15

[-

(~r/2l

(7.3.6)

dx,

)2/3ZeN 2/3 = 2.6031 ZeN 2/3 .

(7.3.7)

Values of the function .1t(f3) for a wide range of values of the parameter f3 are given in Table 7.1. In addition the function .1t(f3) is shown in Fig. 7.4. The maximum of the function .1t(f3) corresponds to the point f3 = 1.607. In the two limiting cases, high and low values of f3, the function .1t(f3) can be approximated by the series 1.496 f3-S /2( 1 + 5.107 f3- 3/2

.1t(f3)

~

{

tn

f32 (1- 0.463f32

+ 14.93f3- 3 + ... )

+ 0.1227f34 + ... )

(f3~ 1).

(f3 ~>1),

(7.3.8) (7.3.9)

If in the expression for .1t(f3), the field Co is redefined by putting Co = ZeR(;2, where Ro = (3/4nN)I/3, then instead of (7.3.8) we have .1t(f3) ~ l.5f3- S/2 , which coincides with the binary distribution (7.1.26). We note that from the practical point of view the difference between the two definitions of Co is unimportant. In accordance with (7.3.8), in the wing of the line,

I(w) ~ (w - wO)-5/2l.5D~p(B~p)3/2C~/2 ,

(7.3.10)

~p

in full agreement with the binary distribution (7.1.28). This is due to the fact that the strongest fields are created mainly by the nearest ion. It must be noted that the distribution function of the binary approximation is fairly close to .1t(f3) everywhere, with the exception of the range of low values of f3. Weak fields,

276

7. Broadening of Spectral Lines

Table 7.1. Holtsmark distribution function

p

Jf(P)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.2

0.00000 0.0042245 0.016665 0.036643 0.063082 0.094596 0.129587 0.166360 0.203233 0.238641 0.271221 0.299870 0.323782 0.342461 0.355702 0.363566 0.366334 0.364456 0.358502 0.349109 0.336939 0.306821

Jf(P) 0.272746 0.238221 0.205563 0.176063 0.150242 0.128118 0.109422 0.093753 0.080674 0.069765 0.060654 0.053023 0.046604 0.041180 0.036573 0.032640 0.029263 0.026349 0.023822 0.021619 0.019690 0.017993

P 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6

P 6.8 7.0 7.2 7.4 7.6 7.8 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 17.0 20.0 24.0 28.0 33.0 38.0 43.0 48.0 53.0

Jf(P)

0.016494 0.015165 0.013981 0.012922 0.011974 0.011120 0.010350 0.007438 0.005561 0.004289 0.003392 0.002739 0.002249 0.001875 0.001351 0.0008856 0.0005537 0.0003733 0.0002457 0.0001718 0.0001256 0.0000952 0.0000741

0.4

/ '~

OJ 0.2

/ o

/

~

/

"~ .......

2

p-

3

r---5

4

Fig. 7.4. Holtsmark distribution Jf(P)

obviously, are produced by a large number of comparatively distant ions. The calculations of Holtsmark intensity distribution for a number of hydrogen spectral lines were carried out in [7.41]. It is convenient to rewrite (7.3.10) introducing the effective Stark-effect constant B for a line as a whole [7.42]

I(w) ~ 101.5(w - wO)-5/2(B80)3/2,

10

= L/~p, ~p

(7.3.11)

7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma

277

where in accordance with (7.3.2) B

3/2 -_10-I",

L.. I~p

(e)3/2

h

~p

(Z~~

- Zpp)

3/2

(7.3.12)

.

Here Z is the coordinate of the atomic electron. Comparison with the results of accurate numerical calculations shows that for a hydrogenlike ion with nuclear charge ~e, the constant B can be approximated by the expression B

= (~)2/3 _h_(n2

_ n'2 ),

(7.3.13 )

~me

8

where n and n' are the principal quantum numbers of the initial and final levels. Similarly for the contour of the line 1(OJ) = E~p/~p( OJ), one can also use the approximate expression [7.42] (7.3.14 ) The dependence of TH(fJ) on [3 is given in Table 7.2. At high values of [3, TH ([3) ---+ 1.5 [3-5/2. Since the contour of the line (7.3.4), and also (7.3.14), is symmetrical with respect to OJo, the Holtsmark width of a line LlOJH is approximately equal to 8BSo. Using (7.3.13), we obtain for the hydrogen spectrum, (7.3.15) Formula (7.3.14) describes sufficiently well the contour of the line everywhere apart from the central region. In order to improve the Holtsmark theory one must take into consideration the mutual correlation of ion positions. In the Holtsmark theory, the exponential factor exp[-V(RI' R 2 ,

•••

)jkT]

in the expression for probability of an ion configuration R 1, R 2 , ••• with potential V is neglected. Thus, the relative probability of such configurations to which high positive values of V correspond are overestimated. In particular, the Holtsmark theory overestimates the probabilities of large frequency shifts /C, i.e., of high values of S, and underestimates the probabilities of low /c. The simplest way of introducing the corresponding corrections to the Holtsmark theory is to take the Debye-Hiickel screening into account. The field of the ion surrounded by a cloud of other ions and electrons of the plasma decreases at large distances as

Table 7.2. Function TH(P)

P TH(P)

0 0.1

0.5 0.1

1 0.098

2 0.086

3

5

7

10

15

0.070

0.039

0.02

0.0072

0.0023

20 0.00099

278

7. Broadening of Spectral Lines

0.6,-------,----,----,---...,-----, 0.5 t-----.~---+----+---_::__+--___l 0.41------h~+_'<~-+__/_1'

§. ~ 0.2t--tJ~f_+--~~~---+--___l

o

4

5

jJ-

Fig. 7.5. Hooper distribution £'D(/1) for the neutral point [7.43]

exp (-RjRo), where Ro is the Debye radius

Ro=

kT

+ Z2)

4nNe2(1

(7.3.16)

•

The difference between the distribution function Jfo(f3) taking this screening into account and the Holtsmark one depends on the magnitude of the parameter IX

= Ro

Ro

ex N I / 6 T- I / 2

(7.3.17)

.

The quantity IX- 3 is equal to the number of ions inside the Debye sphere No. As IX -+ 0 (Ro -+ 00), the function Jf o(P) coincides with Jf(P). The dependence of the function Jfo on P is shown in Fig. 7.5. [7.43].1 The difference between Jfo(P) and Jf(P) becomes noticeable at IX ~ O.4(No ~ 16). Thus, the condition of applicability of the Holtsmark distribution is the inequality -3 _ IX

_

-No -

4n 3

(

kT ) 4ne2 (1 +Z2)

3/2

-1/2:.;

N

~1.

(7.3.18)

A special situation arises in those cases when the broadening of lines of hydrogenlike ions He+, Li 2+, and so on is considered. In calculating W(C) it is necessary to take into account the repulsion between emitting and perturbing IOns.

I Calculations in [7.43] are not based on the simple picture of Debye shielding and are more rigorous and rather complicated.

7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma

279

7.3.3 Correction for Thermal Motion

Due to the thermal motion of ions, the amplitude and direction of the vector tff varies with time. Changing the amplitude of tff causes the changing of the level splitting and leads to a phase modulation. If the vector tff rotates very slowly, the angular momentum of the atom follows tff adiabatically. This results in the change of the dipole moment component in the direction of a photon wave vector k. Therefore the slow reorientation of vector tff produces amplitude modulation. The level splitting with respect to M components is of the order of Btff. Reorientation of the vector tff can be considered to be slow when Btff ~ 1/., where • is the typical time of reorientation of tff. In the opposite limiting case Btff ~ 1/., the magnetic field arising in the rotating coordinate system induces transitions between sublevels M. This means that perturbation is no longer adiabatic. The corrections for thermal motion of ions taking into consideration all above-mentioned effects were calculated in [7.44,45]. According to these calculations, the central part of a line is the most sensitive to the thermal motion of ions. The experiment [7.46] revealed that the dynamic effects of the ion influence substantially the shapes of the hydrogen lines. These effects can be taken into account by a modified relaxation theory of line broadening [7.47,48] and the "Model Microfield Method" [7.49-52]. The dynamic effects can be very important in case of multiply-charged ions [7.51]. 7.3.4

Electron Broadeningl

A theory of broadening of lines of the hydrogen spectrum by electrons must take into account the nonadiabatic nature of the perturbation. Collisional transitions between the states of the same level play the main role in the broadening. We shall consider the spectral line corresponding to a radiative transition between the level a (the states (x, (X' ••• ) and the level b (the states p, P' ... ). The intensity distribution J(w) can be calculated by means of (7.2.16). It is necessary to determine the elements ofthe S matrix and solve the system of equations (7.2.15). The collisional term in (7.2.15) can be written in the form (7.3.19) where

f]J

is the collision operator, 00

f]J

00

= N I vf(v)dv I2rcpdp{1 o

S;Sb} .

(7.3.20)

0

1 A detailed treatment of the different aspects of electron broadening in hydrogen and hydrogenlike spectra using different approaches is given in [7.6,53--64].

280

7. Broadening of Spectral Lines

Here f( v) is the Maxwellian distribution function, the braces { } denote the averaging over directions of vectors p and v, and the notation (7.3.21 ) is used. We shall determine the S matrix using the perturbation theory expansion in series (7.2.24). The perturbation V(t) produced by collision with the electron has the form 2

V(t)

r·(p + vt)

= e (p2 + v2t 2)3/2 '

(7.3.22)

where r is the radius vector of an atomic electron. As already noted above in the case of the interaction V ex R- 2 , the principal role in the broadening is played by collisions with high values of p ~ Po, Where Po is the Weisskopf radius. This means that we can restrict ourselves to the first nonvanishing term in (7.2.24). After averaging (7.2.25) over all directions of the vectors p and v, we have (V v) = 0 and ,

[V(t)V(t )]

p2

r .r

+ v2tt'

= -3- (p2 + v2t2 )3/2(p2 + v2t,2 )3/2

(7.3.23)

.

Therefore

.(7.3.24) The first term of this expression is e4

p2

+ v2tt'

-L-L dtdt (p2 + v2t2)3/2(p2 + v2t'2)3/2 00

-"3 ra •rb

00,

2e4 =

1

-3 ra ·rb p2V2' (7.3.25)

The second and the third terms can be calculated as follows: ~IX"P"I

00

t

J Va(t)dt J Vit')dt' IIX'P'~ -00

(7.3.26)

-00

We shall introduce the dimensionless variables vt' vt Xl

=-,

P

X2= - .

P

(7.3.27)

7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma

281

Then (7.3.26) can be expressed in tenns of the integrals A(Z\,Z2) + iB(z\,Z2) 1 00 XI = 'LL dx'_L dX2 (1

1 + Xt X2 . + xf)3/2(l + xD3/2 exp [l(Z,Xt

- Z2 X2)] .

(7.3.28)

The summation over IX is restricted to the states of the level a. The perturbation due to all the other levels is neglected. The matrix elements 'alia are nonzero only for neighboring Stark components IX",IX. At IX" = 1X',Zt = Z2 = z, and at IX" :F IX', Zt = -Z2 = z. Let us denote the corresponding integrals by A+(z),B+(z) and A_(z),B_(z). All these integrals can be expressed in tenns of Bessel functions A±(z) = z2[Kf(z)±Kg(z)] B±(z)

= 21zlp j n

-00

~ {~lzlexp(-2Izl)

(z~

1),

(7.3.29)

(z~l),

A±(z') dz' ~ {~ (z~ 1), z2 - zl2 ' --+ 0 (z ~ 1).

(7.3.30)

For low values of p, the approximation (7.3.23) is not valid. The region of p < Po gives a comparatively small contribution to the broadening. Therefore it is possible to assume 00

J2npdp(1 - S;Sb) ~ np~

o

00

+ J2npdp(1

- S;Sb) .

~

The upper limit of integration must be taken as p = RD, see (7.3.16). It is not difficult to prove that for typical plasma parameters, the quantity Zmax = RDWall a/v, where Wall a is the Stark splitting in the field $0 = 2.6 ZeN 2/ 3 , can be neglected as compared to unity. Therefore one can assume A = 1 and B = 0 1• As a result (7.3.31 ) It must be noted that in some cases the upper limit of integration over p must be chosen not as RD but as v/Aw. The spectrum i(w) in the region of the frequencies Aw is determined by the correlation function lP('r) at T rv I/Aw. The impact approximation can be used only if the duration of the collision p/v > I/Aw, or p < Pm = v/Aw. Therefore the upper limit must be chosen as the minimum value of the two values RD,v/Aw. When integrating over v in (7.3.31) the lower limit of integration must be determined from the condition Po(vmin);:SRD • As a result we obtain the following

1 The same integrals are encountered in the problem of the broadening of nonhydrogenlike spectral lines in plasma (Sect. 7.4).

282

7. Broadening of Spectral Lines

expression for the operator tP:

Here (v-I) = 4/n(v) -I, and the constant 0.215 gives the relative contribution of the strong collisions P < Po. We shall determine the Weisskopf radius Po in accordance with that of Sect. 7.1 as

2

Po

2

e2)2

= 3 ( h(v)

,

2

(7.3.33 )

l(n,n )ao ,

l(n,n') = (2::IP"pI2)-1 lIP

2::

P"'p,Pp,, ~~Plra"ra

"p,,'P'

+ rb"rb -

2ra"rbl~' P' ~ao2 .

(7.3.34 )

where n and n' are the principal quantum numbers of the levels a and b. The intensity distribution in the wing of a line is given by (7.2.21) and (7.2.22), where in accordance with (7.3.32-34), the effective constant of the impact broadening ')' is

) , 332 N (v) -I mh 2 (RD In Po + 0.215 l(n,n), 2

')' =

(7.3.35)

The same quantity gives the order of magnitude of the broadening by electrons. The quantity l(n,n') can be written in the form l(n,n')

= len) + l(n') -

2K(n,n') .

(7.3.36)

As (ra"ra) ~ aijn 4 we have len) rv n4 and l(n') rv n,4. (For hydrogenlike ion with a nuclear charge !l'l(n) rv n4!l'-2 and l(n') rv n,4!l'-2). If n ~ n', then to an order of magnitude K(n,n') rv len), len'). For example, in the case of radiative transition n = 3 ---- n' = 2, we have len) = 56, len) + l(n') = 68, and l(n,n') = 27. In the case of the transition n = 4 ---- n' = 2, we have len) = 359 and l(n,n') = 291. Nevertheless using (7.3.36) for estimates of the broadening in the case of transitions between lower excited levels, one can assume I (n, n') ~ n4, where n is the maximum value of n,n', or l(n,n') ~ (n4 + n,4)/2. In the case of transitions between neighbouring highly excited levels n ~ 1, n' ~ 1, and In - n'l ~n,n', the term 2K(n,n') leads to a significant reduction of the broadening. This effect will be considered in Sect 7.3.6. In order to calculate the total intensity distribution produced by electrons for a given ion field, one must solve the system of equations (7.2.15). The Hamiltonian of an atom in the electric field being diagonal in parabolic coordinates, the calculation of the matrix tP must be carried out using parabolic atomic wave functions. The resulting effect of impact broadening by electrons in the presence of a fixed ion field can be described in the following way. The ion field splits the

7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma

283

line into the Stark components, each of these components being broadened by electrons. If the ion splitting is less than the impact broadening produced by the ,electrons B8;$ y, then the Stark components overlap. In this case, nondiagonal matrix elements of cP are of importance. But the main contribution to the line shape is given by the ion fields 8 rv 8 0 for which B80 > y. The transition from the impact distribution to the quasistatic wing produced by electrons in the case of a hydrogen .spectral line must be considered taking into account nonadiabatic effects. This problem is very complicated. For a discussion see [7.59-65]. 7.3.5

Combined Effect of Electrons and Ions

The intensity distribution resulting from the combined effect of electrons and ions can be obtained by calculating the electron broadening with a fixed ion field 8 and averaging the result over all possible values of 8. If the electron broadening is described by the Lorentzian distribution with width y from (7.3.35) and ion broadening in the same approximation as in (7.3.14), then for the normalized distribution of intensity it is easy to obtain [7.42] [(OJ) =

B~o T(OJB-8: B~J 0

,

.

(7.3.37)

The function T(x, y) is defined by the expression

y

T(x, y) =

TH(x')dx'

211/ (x _ x')2 + (y/2)2 .

Values of the function log T(x,y) are given in Table 7.3. When y

(7.3.38) -+

O,T(x,y)-+

TH(X).

In the line wing (high values of x), T(x,y)

~ 1.5

X- 5/ 2

+ ,Jnx-2 .

(7.3.39)

In order to improve the description of a line shape one must take into consideration two effects: (i) in the far wing of a line impact distribution is replaced by quasistatic one; and (ii) in the central part of a line, ion broadening must be calculated taking into consideration the thermal motion of ions; moreover the distribution function Wo(8) instead of WH(8) must be used (see Fig. 7.5). Detailed calculations of the hydrogen line shapes have been carried out in [7.66-69]. The results of these calculations are in many cases in a reasonable agreement with the experimental data [7.70-73]. The difference between experiment and calculations is, however, noticeable in the central part of the lines, especially for the lines having an unshifted Stark component (L(h Hat). When plasma density is not high the discrepancy of experimental [7.74,75] and theoretical linewidths for Lat and Hat can exceed a factor of 2, see also [7.48]. In

284

7. Broadening of Spectral Lines

Table 7.3. log T(x,y) x y

0

0.5

0.0 0.5 1.0 1.5 2.0 3.0 4.0 5.0

i.004 2.963 2.912 2.874 2.837 2.768 2.706 2.651

i.ooo 2.958 2.911 2.872 2.834 2.766 2.704 2.649

2.992 2.943 2.905 2.864 2.826 2.759 2.699 2.646

2

3

5

7

10

15

20

2.936 2.912 2.862 2.826 2.794 2.734 2.678 2.628

2.846 2.828 2.794 2.768 2.742 2.692 2.645 2.600

2.251 2.579 2.588 2.592 2.588 2.570 2.545 2.518

2.292 2.313 2.356 2.385 2.402 2.415 2.417 2.410

3.860 3.975 2.050 2.092 2.123 2.175 2.208 2.228

3.370 3.455 3.580 3.666 3.729 3.822 3.888 3.938

4.996 3.159 3.252 3.354 3.434 3.550 3.633 3.698

case of hydrogenlike ions the accuracy of methods used in [7.66-68] can even be poorer for such lines [7.51]. As noted above (Sect. 7.3.3) the dynamic effects of the ion must be taken into account.

7.3.6

New Approaches to the Theory of Stark Broadening

The new, very promising approaches to the problem of spectral line broadening by charged particles have been given in [7.47-52,76-79]. In [7.76,77] the study of Stark profiles for hydrogen lines is carried out by means of direct computer simulation. The radiating atom is assumed to be placed in some volume together with a fixed number of charged particles, electrons and ions, the motion of which is described by classical orbits. Numerical solution of the Schrodinger equation for a radiating atom perturbed by Debye-screened fields of charged particles is averaged over the initial coordinates and the velocities of the perturbers. The Stark profiles are obtained as a result of such a procedure. A new version of relaxation theory proposed in [7.47, 48] showed good agreement with experiment and with the results of computer simulation [7.76, 77]. In the so called "Model Microfield Method" proposed in [7.78, 79] the intensity distribution of a spectral line is expressed in terms of a correlation function for the plasma microfield. This method proved to be very successful in taking into account the "ion-radiator dynamic effects" [7.49-52], and also showed good agreement with experiment and with the results of computer simulation. The above-mentioned theoretical methods can be applied to the nonhydrogenic spectra, too.

7.3.7 Highly Excited States The reduction of electron broadening due to the term K(n,n') in (7.3.36) is connected with the interference effects discussed in Sect. 7.2.

7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma In the case of high n and n' = n -I the matrix elements for large values of [ are almost the same:

Snl,nl±l

and

285

Snll,nll±1

Snl,nl±l - Snll,nll±1 ~ Snl,nl±l .

Moreover the matrix elements P nl, n' I± I and Pni', n' I' ± h where [' = [± I, are also almost the same. Therefore there arises a very close similarity with the four-level system considered in Sect. 7.2.6. It can be shown [7.6, 80, 81] that in the case of radiative transitions between neighboring highly excited levels, n ~ I, n' ~ I, and

In - n'l ~n,n', J(n,n')~ J(n),l(n') .

Thus for transitions n - n - I, n ~ I we have J(n,n - I)

rv

I "2J(n) . n

(7.3.40)

According to (7.3.40) for such transitions the dependence Ye ex: n4 is replaced by the dependence Ye ex: n2 • This means that one cannot neglect the perturbation due to the all other atomic levels. As a result the broadening due to the quadratic stark effect as in the case of nonbydrogenlike spectra begins to be of importance. Broadening of this kind is considered in Sect. 7.4. As a rule the lines corresponding to radiative transitions between highly excited levels are observed in plasmas with very low electron density, when he ~ 1 and hi ~ I. Expression for the width of a line n - n' is given by Y = Ye + Yi, Ye = Ne[(vO'e(n)} Yi = M[(vO'i(n)}

+ (vO'e(n'»)] ,

(7.3.41 )

+ (vO'i(n'»)] ,

where O'(n) = L:1kl>oO'(n, n + k) are the effective cross sections of inelastic scattering. Using the quasi-classical calculations of cross sections for highly excited level described in Sect. 3.5, one can write for the quantity (vO'(n)} the following approximate formula: (vO'e(n)}

~

4

10- 8 z3~1/2 CP(x) f( 8) [cm3 s- I ],

(7.3.42)

where 8 = T/~Ry,

x = En/T = l/n2 8 , CP(x) = 2.18{0, 82q>(x) + 1.47[1 - xq>(x)]}, 1 + 1.4yx ] q>(x) = -exp(x)Ei( -x) ~ In [ 1 + y.x.(l + 1.4x) ,

f(8) = In

[1 + z(1 +nvle vie ]1n(1 + nvle/z) 2.5/z 8)

(7.3.43) Y = 1.78

(7.3.44)

7. Broadening of Spectral Lines

286

when z -+ 00 or 8 -+ 00,/(8) -+ 1. If perturbing particles are protons, the temperature T in (7.3.42-44) should be replaced by the quantity Tm/M, where m is the mass of electron and M is the reduced mass of the colliding particles. Stark broadening of the lines in the far-infrared solar spectrum corresponding to the transitions between highly excited hydrogenlike states is of interest for diagnosing the structure of the solar atomosphere [7.82].

7.4 Line Broadening of Nonhydrogenlike Spectra in a Plasma 7.4.1

Preliminary Estimates

The spectral lines of nonhydrogenlike atoms in the presence of a constant and homogeneous electric field undergo a shift and also a splitting proportional to @"2-the quadratic Stark effect. We shall assume that the field @" = QR- 2 , produced by a charge Q, varies little for atomic dimensions (this is valid for sufficiently large values of R). Then in (7.1.20) for the shift of oscillator frequency, n = 4 and K = C4R- 4 . The constant C4 for a transition n -+ k is defined as C4

= (C4 )n -

(C4 )k; (C4 )n

= Q2/h 2: 1(~~nmI2 m

,

nm

where (Dz )nm are the matrix elements of the z component of the electric dipole operator, .1Enm = En - Em (see [1.1]). The parameters he (electron broadening) and hI (ion broadening) are he = N

(~ C4) 2

, hi = N

Ve

(~ C4 ) 2

.

(7.4.1 )

Vi

The quadratic Stark-effect constants C4, as a rule, have the order of magnitude 10- 12 -10- 15 cm4 s-I, although values of C4 < 10- 15 and C4 '" 10- 11 - 10- 10 are also encountered. For C4 = 10- 12 - 1O- IS,ve = 5.107 cm s-I and Vi = 2.105cm S-I, we have he = 3.(10- 19 - 1O- 22 )N, hi = 0.75(10- 17 - 1O- 2o )N. At not very high values of the density of charged particles N < 1015 cm -3, he ~ I, and hi ~ 1. This means that both electrons and ions produce impact broadening. According to (7.1.23) Y4,.1 4 ex: vl / 3. Thus electrons play the principal role in the broadening of a line. The interaction with ions only slightly increases the impact width and shift of a line, by approximately 15-20%, because (Ve/Vi)I/3 ~ (M/m )1/6 ~ 5-6. Since K ex: Q2 the sign of the shift of a line is the same for electrons and for ions.

7.4.2 Electron Broadening We shall describe electron broadening of lines of a nonhydrogenlike atom in the framework of quasi-classical theory discussed in Sect. 7.2. In the case of the

7.4 Line Broadening of Nonhydrogenlike Spectra in a Plasma

287

isolated spectral line the spectrum I (w) is described by Lorentzian distribution (7.2.17). The width and shift are given by (7.2.18) and (7.2.20). The elements of the S matrix averaged over M components of atomic levels must be substituted in these equations. As a rule the main contribution to the broadening is given by collisions with relatively large values of the impact parameter p. Therefore in calculating the S matrix we can restrict ourselves to the first terms of the expansion in powers of rand 1'/. Therefore one can average over M directly the quantities r and 1'/. In the case of the dipole interaction between a neutral atom and a charged particle V = -d·iff the linear term in (7.2.25) for the S matrix S(v), being averaged over M (or over directions of the vectors p and v), is equal to zero. Only the next term containing C\(t) C\( (I) in the integrand is nonvanishing. We shall assume now that the perturbation of one of the levels (initial or final) can be neglected. Then for the radiative transition n-k, assuming that the level k is not perturbed, we can obtain (7.4.2)

r

= ~'rs = ~'2 s

s

(.!!...)2 (Ins Ry ) ~A (W ns P ) mv JEns P

.

(7.4.3 )

V

Here JEns = Es - En, Ins is the oscillator strength of the transition n -+ s (see [Ref. 1.1, Sect. 9.2]), and the sum over s extends over all atomic levels, for which Ins =I O. The functions A and B coincid~ with the functions A+ and B+ defined by (7.3.29, 30). Values of the functions A and B are given in Table 7.4. As a rule the principal contribution to (7.4.2,3) is provided by the nearest perturbing levels and, in some cases, by only one of them. In the approximation of one perturbing level, y

= 2N (v)(J~( (v))J' (f3)

,

(7.4.4 )

J

= N(v)(J~( (v) )J"(f3) ,

(7.4.5)

Table 7.4. Functions A(z) and B(z). z

A(z)

B(z)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

1.000 1.035 0.962 0.829 0.680 0.540 0.418 0.318 0.239

0.000 0.160 0.359 0.498 0.576 0.606 0.603 0.580 0.546

z

A(z)

B(z)

1.8 2.0 2.4 2.8 3.2 3.6 4.0 5.0

0.177 0.130 0.0688 0.0355 0.0181 0.0090 0.0045 0.00075 nze- 2z

0.507 0.467 0.393 0.331 0.283 0.245 0.216 0.166 n/4z

~oo

7. Broadening of Spectral Lines

288

where (J~ and (J~ are the width and shift cross sections defined by (7.1.13,14),

(J~ = (~r/3 r (~ ) C~/3 (v) -2/3 ':::!. 5.7C~/3 (v) -2/3 , (7.4.6)

(v) is the mean value of the electron velocity, and the constant C4 is defined as C 4

= 2t?a~f(Ry)2

ft

,1E

The functions J'(P) and J"(P) depend on the dimensionless parameter

P =lfRyll/2 I,1EI2 . ,1E m(v) As

P---- 00, J', J"

(7.4.7 )

---- 0.97. This case corresponds to adiabatic perturbation when

r = 0, and the broadening is determined by the phase shift '1.

For

P~ 1, on the

contrary, inelastic collisions play the main role. In this case,

,

Ry

(v(J ) = 4nf ,1E (v

(ft) 2In (mv2 1 Ry 11/2) ) , mv 21,1EI f ,1E

(7.4.8)

(7.4.9) The results of numerical calculations of the functions J'(P) and J"(P) are given in Table 7.5. We shall now consider to what extent the results obtained above can be generalized to the case of several perturbing levels. This problem obviously arises

Table 7.5. Factors J'(fJ) andJ"(fJ).

fJ

J'

J"

64 32 16 8 4 2 I 0.5 0.25 0.125 0.625 x 10- 1 0.312 x 10- 1

0.97 0.97 1.02 1.03 1.06 1.12 1.17 1.20 1.15 1.09 0.927 0.764

0.97 0.97 0.97 0.96 0.94 0.90 0.861 0.746 0.604 0.455 0.326 0.223

fJ 0.156 0.78 0.39 0.195 0.97 0.48 0.24 0.12 0.61 0.305 0.15

xlO 1 X 10- 2 X 10- 2 x 10- 2 x10- 3 X 10- 3 X 10- 3 X 10- 3 x 10- 4 x 10- 4 X 10- 4

J'

J"

0.594 0.451 0.334 0.239 0.171 0.119 0.0824 0.056 0.038 0.024 0.017

0.151 0.094 0.063 0.0405 0.0245 0.0167 0.0103 0.0065 0.004 0.0026 0.0016

7.5 Broadening by Uncharged Particles

289

only in the case when, for one or several perturbing levels, the parameter P is of the order of or less than unity. If P~ 1 for all perturbing levels, then the perturbation is adiabatic and y and LI are expressed by means of (7.1.23) in terms of the quadratic Stark effect constant C4 for a given line. The magnitude of this constant is determined by the total perturbing effect of all atomic levels. If for the nearest perturbing levels which give the principal contribution to '1 and r in (7.4.2,3), the parameters P~ 1, then the width y can be obtained by summing (7.4.4):

y = 2N(v)LI(J~S«V))J/(PS)'

(7.4.10)

Such an approximation is valid because in this case the broadening is caused by inelastic collisions, and the partial widths corresponding to different collisional transitions are additive. The shift of a line cannot be calculated by summing (7.4.5) even if for all perturbing levels Ps ~ 1. In the general case when both levels nand k (initial and final) are perturbed, r

= rn + rk,

'1

= '1n - 17k,

where r n, rk, '1n and 17k must be calculated by means of (7.4.2, 3). An extensive bibliography on numerical calculations of shapes of the nonhydrogenlike spectral lines in plasma can be found in [7.6]. Results of numerical calculations of widths and shifts for a large number of nonhydrogenlike spectral lines of different atoms and ions also are given in [7.6]. The experimental data on Stark broadening in nonhydrogenlike spectra may be found in [7.12.13]. The very accurate experimental data on the lines of neutral He are given in [7.83].

7.5

Broadening by Uncharged Particles

7.5.1

Perturbation by Foreign Gas Atoms (Van der Waals Interaction)

The interaction of neutral atoms at large distances has the form V(R) ex: R- 6

.

Therefore usually one assumes K

= C6R-6.

(7.5.1 )

The crude estimate of the constant C6 is given by C6 ~

e2(r2)IX.

- f j , - ' IX. ~

4

(Ry)2 ao (rp)2,

"3 m T

(7.5.2)

where (r2) ~ 5n*4a~/2 is the mean value of r2 for the excited state of the

290

7. Broadening of Spectral Lines

radiative atom, (r~) is that for the perturber, n* is the effective principal quantum number, IX is the polarizability of the perturbing atom, I is its ionization potential, and m is the number of equivalent electrons. The constant C6 has the order of magnitude 10- 30 n*4. Thus for v rv 105 cm s-I, we have h = (387t

~6 )

3/5

N

~ 10- 21 N

.

(7.5.3 )

This indicates that at not very high pressures, of the order of few atmospheres or less, line broadening can be described in the impact approximation. We shall also compare the quantities Q = V6/5C;:I/5 and Awo. As Q ~ 10 12 S-I and Awo ~ IO IO s- l , we have Q~Awo. Consequently the region of impact broadening extends far beyond the limits of the Doppler width. In accordance with (7.1.23) the width and shift of a line can be estimated using the relations (7.5.4 ) The typical values of yare y rv 10- 8N. In order to treat impact broadening by uncharged particles more accurately it is necessary to take into consideration that at small distances the interaction VCR) has a more complicated form than VCR) ex R- 6 . Depending on the type and states of interacting atoms both attraction and repulsion can take place at large distances R. At small distances the potential VCR) is repulsive. In some cases atom and perturber can form a quasi-stable molecule. Moreover in the general case, the interaction V is dependent not only of R but also of the angular variables. The results of calculations in which a more realistic interaction than V(R) ex R- 6 is used cannot be described by a simple Lorentzian distribution with width and shift as in (7.5.4). In particular, the intensity distribution depends on the type of the transition j-j'. A detailed treatment of the foreign gas broadening is given in [7.4,10,11,1414-16]. The repulsive part of the interaction is usually taken into account in the form of the Lennard-Jones potential V(R) = CI2R-12_C6R-6. The line shift A and the ratio ylA are especially sensitive to the form of the potential VCR). Experimental data on line broadening in the spectra of alkali atoms obtained at low values of foreign gas pressure, less than 10 atm, are in qualitative agreement with the impact theory. The broadening and shift of the lines are proportional to the concentration of perturbing particles. For the initial members of a principal series perturbed by different foreign atoms (He, Ne, Ar, Kr, Xe, H2, N2, and so on), as a rule, a red shift is observed, the ratio ylA being close to 2.8. In some cases (usually for the higher members of a principal series), a blue shift instead of a red one is observed. The sign of the shift of one and the same line can be different for different perturbing particles. The dimensionless parameter h reaches values of order unity only when N > 1021 , i.e., at pressures of about tens of atmospheres. In this case the mean distance between atoms has the same order of magnitude as atomic dimensions and consequently the simplest expression VCR) ex R- 6 is not valid. The experi-

7.5 Broadening by Uncharged Particles

291

mental data on line shapes are usually used to obtain information about the form of potential V(R) at small distances. Specific features of molecular-lines broadening have been described in [7.4,

8,9]. 7.5.2

Self-Broadening

We shall consider now the single-component gas. With an increase of density of such a gas, resonance lines broaden considerably more than on the addition of a foreign gas. This is due to the fact that in the case of collision of two identical atoms, one of which is excited, a resonance transfer of the excitation energy is possible, the effective cross sections of such collisions being extremely large. They can exceed considerably (by several orders) the gas kinetic cross sections. The effective cross sections of resonance energy transfer (J were calculated in Sect. 4.2. For electric dipole transition, the energy transfer is caused by the dipole-dipole interaction V ex: R- 3 • The cross section (J and corresponding line width yare of the order of magnitude

e2 y'" -IN,

mwo

(7.5.5)

where S is the line strength and I is the oscillator strength. Assuming that I ~ 1 and Wo ~ lOIS, we have y 1O- 7N. In the case of foreign-gas broadening, typical values of widths are y rv (l0-9-10-8)N. The effective cross section of energy transfer can be relatively large not only under conditions of exact resonance but also in the case of a collision of two atoms with close energy levels. Thus, when calculating the width of the component 2P1/2-2S1/2 of the resonance doublet of an alkali atom, it is necessary to take into account not only energy transfer 2P I /2 -+2 SI/2 (radiating atom), 2S1/2 -+2P 1/2 (perturbing atom), but also the excitation 2S1/2 -+2P3/2 of the perturbing atom. The cross section of the energy transfer 2P I /2 -+2S1/2 , 2S1/2 -+2P3/2 has the same order of magnitude as the cross section of the resonance energy transfer 2P I/2 -+~\/2' 2SI/2 -+P3/2, if the following condition is fulfilled (Sect. 4.2): <'V

(

e2

-2-1 mwo

) 2/3

(LJE) 4/3 --;:n

1 2«1, V

where LJE is the doublet splitting. Of all alkali atoms, only Li atom satisfies this condition. In the case of Li the doublet splitting of the resonance level is 0.34 cm- I . In order to calculate the line broadening due to resonance energy transfer more accurately it is necessary to take into consideration the degeneracy of levels and dependence of the interaction on the angular variables. Such calculations using the impact approximation have been carried out by many researchers [7.84--89].

292

7. Broadening of Spectral Lines

The summary of the results can be given as follows (7.5.6) where Jo and J 1 are the angular momenta of the ground and excited levels, respectively, and 101 is the line oscillator strength. The values of the factor A(JoJI) for isolated lines calculated numerically in [7.87,88] are given in Table 7.6 together with the ratios of the width y/2 to the shift A from [7.89]. As was said above, in the special case of Li atom one has to take into consideration the perturbation of both excited levels 2PI/2 and 2P3/2. Because of that one can expect almost equal widths for both lines of the resonance doublet. The approximate value of y/2 in this case can be obtained using (7.5.6) and assuming Jo = 0, JI = 1 [7.85]. For the case of interaction V ex R- 3 the intensity distribution in quasistatic wings lco-cool 't> V3/2C:;I/2 = V3/2(t? l/mcoO)-1/2 is also proportional to (CO-COO)-2 as in the impact approximation. However, the intensity /(co) in the quasistatic wings is somewhat different from that in the impact approximation [7.84,85,90] /(co)

= a (JoJd

2nt? 1 -lOIN ( )2 . mcoo co-coo

(7.5.7)

The factors a(JoJd are also given in Table 7.6. In the case of Li resonance doublet Jo = 0, JI = 1 should be assumed in (7.5.7). The calculations taking into account the accurate adiabatic potential curves [7.90] show a slight asymmetry of blue and red wings. The direct experimental studies of the resonance broadening as a rule encounter very serious difficulty connected with extremely large optical depth in the center of the lines [7.91]. The lines corresponding to transitions between the resonance level and other excited levels also undergo broadening due to the resonance interaction. The resonance contribution to the widths y of such lines can be evaluated to be equal to the linewidth of the resonance line.

TabIe7.6. Parameters describing the resonance broadening Jo

0

112

112

1

JI

1

112

3/2

1

1.042

0.903

1.039

0.092

-0.031

0.050

0.698

1.047

0.805

A(JoJl)

2

Aly

a (JoJl)

0.983 -0.01

7.6 Spectroscopic Methods oflnvestigating Elastic Scattering of Slow Electrons

293

7.6 Spectroscopic Methods of Investigating Elastic Scattering of Slow Electrons 7.6.1 Perturbation of Highly Excited States The broadening of a line corresponding to a transition between the ground state and a state with a large value of the principal quantum number n is completely determined by the perturbation of the upper level. For sufficiently large values of n, the mean distance of the valence electron from the nucleus rv aon2 is so large that the neutral perturbing particle either interacts with the electron and does not interact with the atomic core, or interacts only with the atomic core. In this case, the broadening is caused by the scattering of the atomic electron by the perturbing particles and by the scattering of the perturbing particles by the atomic core. These two mechanisms of the broadening are statistically independent. We shall first consider the interactions of the first type. If only one level is perturbed, then in accordance with (7.2.39) we have

2n

(I'

= kIm {/(O)},

2n

(I"

= -k Re {/(O)}

,

(7.6.1)

where 1(0) is the amplitude of forward scattering of the perturbing particle by the atom, and hk is the momentum of the perturbing particle (we assume for simplicity that the mass of the atom is large as compared with the mass of the perturbing particle). If aon2~Peff. ~ (nlX/4)1/3(e'2/hve)I/3, where Peff. is the effective radius of the interaction between the electron and the perturbing particle and IX is the polarizability of the perturbing particle, then in the volume of interaction the field produced by the atomic core and consequently the electron velocity Ve are practically constant. In the state with principal quantum number n, Ve is of the order of magnitude vo/n, where Vo is the atomic unit of velocity. If the velocity of the perturbing particle vp = hk/M is less than Ve rv vo/n, then the scattering amplitude 1(0) in (7.6.1) can be expressed in terms of forward scattering amplitude 1;(0) of a free electron with momentum hq by the perturbing particle [7.92] for the derivation of this, and subsequent formulas of this section):

1(0) = M

m

J I G(nlmlq)1 2/;(0)dq.

(7.6.2)

Here m is the electron mass, and G(nlmlq) are the coefficients of the expansion of the atomic function I/!nlm in plane waves: (7.6.3) States with large values of the principal quantum number n are hydrogenlike. It is therefore possible to use as the expansion coefficients G(nlmlq) the well-known expressions for hydrogen functions in the momentum. representation in terms of Gegenbauer polynomials [7.93].

294

7. Broadening of Spectral Lines

By substituting (7.6.2) in (7.6.1), integrating over the angular variables, and averaging over all possible orientations of the perturbing particle angular momentum, we have

41t1i Y = N-;;- dq W(q) 1m {Jq(O)} ,

(7.6.4)

21t1i A = -N -;;- dq W(q)Re {fq(O)} ,

(7.6.5)

= q2 IGn,(qW, J W(q)dq = 1 .

(7.6.6)

J

J

W(q)

As already noted above these formulas describe the width and shift caused by scattering of the atomic electron in the highly excited state by perturbing particle if the following conditions are fulfilled: (7.6.7) (7.6.8) We shall now consider interaction of perturbing particle with the atomic core. As the charge of the atomic core is e, this interaction has the form (7.6.9) It produces the polarization of the perturbing particles by the atomic core and leads to a shift of the frequency of the atomic oscillator, IX 2 1 K(t) = - 21i e [R(t)]4 .

(7.6.10)

Broadening due to interactions of this type was examined in Sect. 7.4.- If

hi=PfN~l'Pi=C:)

1/3 (

~

) 1/3

livp

,

(7.6.11)

where N and vp are the density and velocity of the perturbing particles, then the central part of the line lro-rool ~D = (2Iiv:/lXe 2 )1/3 is described by the Lorentzian distribution with width l' and shift A' given by the following formulas: ~

y'= 114 . (~) 21i

2/3

;;:;

1/3N. Vp '

A' = _~ 2 Y, .

LJ

(7.6.12)

It can be shown that when hi ~ 1, the corresponding quasi-static distribution has width of the order of lON4/ 3 ~ Iii. The case hi ~ 1 corresponding to pressures of the order of or less than an atmosphere is the most interesting. In this case, one can calculate l' and A' with sufficiently good accuracy. Subtracting the cal-

7.6 Spectroscopic Methods ofInvestigating Elastic Scattering of Slow Electrons

295

culated values y' and ,1' from experimental values of the width and shift, one can determine y and ,1 from (7.6.4,5). Expressing in these equations the amplitude 1;(0) in terms of the phase of scattering of the electron by the perturbing particle 111, we also have

N~m J

y=

,1

=

[4n E(21 + l)sin2111] W(q)dq q 1

-N -h J m

= N~ Jqa(q) W(q)dq, m

[n- E(21 + l)sm2111 . ] W(q)dq. q

(7.6.13)

(7.6.l4)

1

Here a( q) is the effective cross section for elastic scattering of the electron with momentum hq by the perturber. If exchange interaction is also taken into account, then the following substitution must be made in (7.6.l3, 14): (7.6.l5) sin 2111 -+ C+ sin 211~ +)

+ c- sin 211~ -) ,

(7.6.l6)

where 11~ +) and 11~ -) are the scattering phases calculated taking exchange into account for states of the system perturbing particle plus electron with given value of the total spin S = Sp ± Ij2,Sp being the spin of the perturbing particle; and C+

=

Sp

+ 1,

C-

2Sp +1

=

(7.6.l7)

Sp

2Sp +1

We shall now consider the resonance transitions no atoms. For np state [7.93] W(q) = 2

(~2

n n - 1

) (_1_)2 [(n nqao

-(n - l)sin(n

+ l)sin(n -

+ l)qJ]2 ao ,

s-np(n~

1) of the alkali

l)qJ

(7.6.l8) (7.6.19)

The function W(q) has n peaks, nj2 peaks being located in the range 0 < q < Ijnao. In this range, the envelope behaves approximately as (1 + n2q2a~)-2. For q~ 1jnao, the function W(q) decreases monotonically: 27 (n 2

W(q)C::'3

1 )n3

_

n

(

1 ) 8 -. nqao

(7.6.20)

Thus the principal contribution to the integral over q in (7.6.13, 14) is given by the range 0 < q < Ijnao.

296

7. Broadening of Spectral Lines

If n is so large that the principal contribution to the sum with respect to 1 is given by the term 1 = 0 (s scattering) and in addition q-l sin 21'/1 differs little from its limiting value

1. (1. ) 1m

q-+O

- sm 21'/1 q

=

-I1'1'//-I00 ~ -0"(0) , n

(7.6.21 )

then

fL t=:7i\\. 1'/0

.1 = -;,;;vna(O)i%TN.

(7.6.22)

Here a(O) is the limiting expression for the elastic scattering cross section at q - t O. By 1'/0 is understood that part of the phase which after subtraction of pn where p is integer, lies in the interval -nI2, n12. Since as q - t 0, qa(q) - t 0, in the range of applicability of (7.6.22), Y~I.1I. Some additional effects of broadening are discussed in [7.94-96]. Experimental data on broadening of Rydberg levels can be found in [7.97-100].

7.6.2 Fermi Formula Equation (7.6.22) has been obtained by Fermi [7.4]. In accordance with this equation it is possible using the experimental value of .1 to determine the elastic scattering cross section for extremely slow electrons (in the limit q - t 0). Thus the cross section a(O) for the atoms He, Ne, Ar, Kr, and Xe have been found by the shift of the absorption lines of Cs in an atmosphere of noble gases. Some other gases have also been investigated by the same method (see [7.4] and also [7.101]). The Fermi method enables one to obtain information on elastic scattering of electrons at very small energies, i.e., in the range most difficult to investigate by other experimental methods. It must be noted that shift .1 is sensitive not only to the magnitude but also to the sign of the phase 1'/0. In the general case, when several terms of the sum over 1 contribute to y and .1, it is not possible to determine the scattering phases from known values of y and .1. Knowledge of these quantities, however, enables one to control the quality of approximate calculations of the scattering phases [7.102, 103].

References

Chapter 1 1.1 1.2

1.1. Sobel'man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, VoU2 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoU (Springer, Berlin, Heidelberg 1979) M. Venugopalan (ed.): Reactions under Plasma Conditions (Wiley-Interscience, New York 1971) VoU

Chapter 2 N.F. Mott, H.S.F. Massey: The Theory of Atomic Collisions (Pergamon, Oxford 1965) 2.2 M.L. Goldberger, K.M. Watson: Collision Theory (Wiley, New York 1964) 2.3 ChJ. Joachain: Quantum Collision Theory (North-Holland, Amsterdam 1975) 2.4 M.R.H. Rudge: Rev. Mod. Phys. 40, 564 (1968) 2.5 R.P. Peterkop: Teoriya Ionizatsii Atomov Elektronnym Udarom (Theory of Ionization Atoms by Electron Impact, in Russian) (Zinatne, Riga 1975) 2.6 1.1. Sobel/man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, Vol.12 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoU (Springer, Berlin, Heidelberg 1979) 2.7 L.A. Vainshtein, 1.1. Sobel'man: Zh. Eksp. Teor. Fiz. 39, 767 (1960) 2.8 M.J. Seaton: Proc. Phys. Soc. 77, 184 (1961) 2.9 MJ. Seaton: Adv. Atom. Molec. Phys. 11,83 (1975) 2.10 H.S.W. Massey, E.H.S. Burhop, H.B. Gilbody: Electronic and Ionic Impact Phenomena VoU (Clarendon, Oxford 1969) 2.11 R. Courant, D. Hilbert: Methoden der Mathematischen Physik (Springer, Berlin 1931) VoU 2.12 L.A. Vainshtein: Phys. Scripta 33,336 (1986) 2.1

Chapter 3 3.1 3.2 3.3 3.4

3.5 3.6

L.A. Vainshtein, 1.1. Sobel/man: Zh. Eksp. Teor. Fiz. 39, 767 (1960) B.L. Moiseiwitsch: Rep. Prog. Phys. 40,843 (1977) K. Smith: The Calculation of Atomic Collision Processes (Wiley-Interscience, New York 1971) 1.1. Sobel'man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, Vol.12 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoU (Springer, Berlin, Heidelberg 1979) M. Inokuti: Rev. Mod. Phys. 43, 297 (1971) K. Omidvar: Phys. Rev. 188, 140 (1969)

298

3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.l9 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27

3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40

References

M. Matsuzawa: Phys. Rev. A 9, 241 (1974) M.R.H. Rudge: Rev. Mod. Phys. 40, 564 (1968) R.P. Peterkop: Teoriya Ionizatskii Atomov Elektronnym Udarom (Theory of Atom Ionization by Electron Impact, in Russian) (linatne, Riga 1975) H.S.W. Massey, E.H.S. Burhop, H.B. Gilbody: Electronic and Ionic Impact Phenomena (Clarendon, Oxford 1969) VoU G.H. Wannier: Phys. Rev. 100, 1180 (1955) K.L. Bell, H.B. Gilbody, J.H. Hughes, A.E. Kingston, FJ. Smith: J. Phys. Chern. Ref. Data 212 (1983) MJ. Seaton: Adv. Atom. Molec. Phys. 11,83 (1975) K.T. Dolder, B. Peart: Rep. Prog. Phys. 39, 693 (1976) LL. Beigman, LA. Vainshtein: Zh. Eksp. Teor. Fiz. 52, 185 (1967) [English trans!.: SOy. Phys. -JETP 25, 119 (1967)] V.I. Ochkur: lh. Eksp. Teor. Fiz. 45, 735 (1963) [English trans!.: SOY. Phys. JETP 18, 503 (1964)] MJ. Seaton: Proc. Phys. Soc. 77, 184 (1961) LP. lapesochnyi: Teplofiz. Vys. Temp. 5, 7 (1967) Ch. J. Joachain: Quantum Collision Theory (North-Holland, Amsterdam 1975) 1.1. Sobelman: Introduction to the Theory of Atomic Spectra (Pergamon, Oxford 1972) R.K. Nesbet: Comput. Phys. Commun. 6, 265 (1973) R. Damburg, E. Karule: Proc. Phys. Soc. London 90,637 (1967) S. Geltman: Applications of Pseudo-State Expansions, in Electronic and Atomic Collisions, invited papers and progress repts. of VII ICPEAC, ed. by T.R. Govers, FJ. de Heer (North-Holland, Amsterdam 1972) p.216 P.G. Burke, W.D. Robb: Adv. Atom. Molec. Phys. 11, 144 (1975) L.P Presnyakov, A.M. Urnov: lh. Eksp. Teor. Fiz. 68, 61 (1975) [English trans!.: SOY. Phys. -JETP 41,31 (1975)] L.P. Presnyakov, A.M. Urnov: J. Phys. B 8,1280 (1975) A.I. Baz', Ya.B. leldovich, A.M. Perelomov: Rasseyanie, Reaktsii i Raspadny v Nerelyativistkoy Kvantovoy Mekhanike (Scattering, Reactions and Disintegrations in Nonrelativistic Quantum Mechanics, in Russian) (Nauka, Moscow 1971) I.L. Beigman, A.M. Urnov: J. Quant. Spectrosc. Radiat. Transf. 14, 1009 (1974) L. Hostler, R.H. Pratt: Phys. Rev. Lett. 10,469 (1963) L.A. Bureeva: Astron. lh. 45, 1215 (1968) R.C. Stabler, Phys. Rev. A 133,1268 (1964) M. Gryzinski: Phys. Rev. 115,374 (1959) M. Gryzinski: Phys. Rev. A 138, 305, 322, 336 (1965) I.L. Beigman, L.A. Vainshtein, 1.1. Sobleman: lh. Eksp. Teor. Fiz, 57, 1703 (1969) [English trans!.: SOY. Phys. -JETP 30, 920 (1969)] I.C. Percival, D. Richards: Adv. Atom. Molec. Phys. 11,2 (1975) M. Born: Vorlesungen uber Atommechanik (Springer, Berlin 1925) M.J. Seaton: Proc. Phys. Soc. 79, 1105 (1962) H.E. Saraph: Proc. Phys. Soc. 83, 763 (1964) L.P. Presnyakov, A.M. Urnov: J. Phys. B 3, 1267 (1970) I.L. Beigman: lh. Eksp. Teor. Fiz. 73, 1729 (1977) [English trans!.: SOY. Phys. - JETP 46, 908 (1977)

References

299

Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14

4.15 4.16 4.17 4.18 4.19 4.20

4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33

N.F. Mott, H.S.F. Massey: The Theory of Atomic Collisions (Pergamon, Oxford 1965) H.S.W. Massey, E.H.S. Burhop, H.B. Gilbody: Electronic and Ionic Impact Phenomena (Clarendon, Oxford 1969) VoU H.S.W. Massey, H.B. Gilbody: Electronic and Ionic Impact Phenomena, Vol.4 (Pergamon, Oxford 1974) E.E. Nikitin, S.Ya. Umanskii: Theory of Slow Atomic Collisions (Springer, Berlin, Heidelberg 1984) R.K. Janev, L.P. Presnyakov, V.P. Shevelko: Physics of Highly Charged Ions (Springer, Berlin, Heidelberg 1985) B.N. Bransden, M.R.C. McDowell: Charge Exchange and the Theory of IonAtom Collisions (Clarenden, Oxford 1992) H.B. Gilbody: Adv. Atom. Molec. Phys. 22, 143 (1986) L.D. Landau, E.M. Lifshitz: Quantum Mechanics (Pergamon, Oxford 1965) L.A. Vainshtein, 1.1. Sobelman, L.P. Presnyakov: Zh. Eksp. Teor. Fiz. 43, 518 (1962) [English transl. SOy. Phys. -JETP 16, 370 (1962)] D.R. Bates: Proc. Phys. Soc. A 73,227 (1959) M.R. Flannery: Phys. Rev. 183,241 (1969) M.R. Flannery: J. Phys. B 2,909 (1969) J.C. Gay, A. Omont: J. de Phys. 35, 9 (1974) 1.1. Sobel/man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, Vo1.12 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoU (Springer, Berlin, Heidelberg 1979) K. Alder, A. Bohr, T. Huus, B. Mottelson, A. Winther: Rev. Mod. Phys. 28, 432 (1956) D.R. Bates, R. McCarrol: Adv. Phys. 11,39 (1962) M.R.C. McDowell, J.P. Coleman: Introduction to the Theory of Ion-Atom Collisions (North-Holland, Amsterdam 1970) O.B. Firsov: Zh. Eksp. Teor. Fiz. 21,1001 (1951) H.C. Brinkman, H.A. Kramers: Proc. Acad. Sci. Amsterdam 33,973 (1930) B.M. Smirnov: Asimptoticheskii Metod v Teorii Atomnykh Stolknovenii (The Asymptotic Method in the Theory of Atomic Collisions, in Russian) (Nauka, Moscow 1973) R.A. Mapleton: Theory of Charge Exchange (Wiley-Interscience, New York 1972) D.S.F. Crothers, N.R. Todd: J. Phys. B 13,2277 (1980) V.P. Shevelko: Z. Phys. A 287,18 (1978) A.M. Brodskii, V.S. Potapov, V.V. Tolmachev: Zh. Eksp. Teor. Fiz. 58, 264 (1970) [English trans!.: SOy. Phys. -JETP 31,144 (1970)] V.V. Afrosimov, R.N. Ilyin, E.S. Solovyev: Zh. Techn. Phys. 30, 705 (1960) V. Schryber: Helv. Phys. Acta 40, 1023 (1967) A.V. Vinogradov, L.P. Presnyakov, V.P. Shevelko: JETP Lett. 8,449 (1968) H.D. Betz: Rev. Mod. Phys. 44, 465 (1972) N.V. Fedorenko: JTP 15, 1947 (1972) A. Salop, R.E. Olson: Phys. Rev. A 13, 1312 (1976) A. Salop: Phys. Rev. A 13, 1321 (1976) G. Harel, A. Salin: J. Phys. B 10, 3511 (1977) J. Vaaben, J .S. Briggs: J. Phys. B 10, L521 (1977)

300

References

4.34 P.A. Phaneuf, F.W. Meyer, R.H. Knight, R.E. Olson, A. Salop: J. Phys. B 10, L425 (1977) 4.35 R.A. Phaneuf, R.K. Janev, H.I. Hunter: Nuclear Fusion, Special Supplement 1 (IAEA, Vienna 1981) 4.36 H. Tawara, T. Kato, Y. Nakai: Cross Sections for Charge Transfer of Highly Ionized Ions in Hydrogen Atoms (Inst. of Plasma Physics Report IPPJ-AM-30, Nagoya 1983) 4.37 H. Tawara: Total and Partial Cross Sections of Electron Transfer for Beq + and BQ+ Ions in Collisions with H. H2 and He Gas Targets. Status in 1991 (NIFS Data Series, ISSN 0915-6364, Nagoya, June 1991) 4.38 H. Tawara: Bibliography on Electron Transfer Processes in Ion-Ion/ Atom/ Molecule Collisions - updated 1990 (NIFS Data Series, ISSN 0915-6364, Nagoya 1990)

Chapter 5 5.1 5.2 5.3 5.4

5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15

5.16 5.17 5.18 5.19 5.20 5.21 5.22

1.1. Sobel 'man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, VoU2 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoU (Springer, Berlin, Heidelberg 1919) I.L. Beigman: Zh. Eksp. Teor. Fiz. 131729 (1917) [English trans!': SOy. Phys.JETP 46, 908 (1977)] H. Van Regemorter: Astrophys. J. 132,906 (1962) H.W. Drawin: Rep. EUP-CEA-FC-383 Assoc. EURATOM-CEA (1967) Fontenay-aux-Roses, France; ibid: in Plasma Diagnostics, ed. by W. Lochte-Holtgreyen (North-Holland, Amsterdam 1968) R. Mewe: Astron. Astrophys. 20, 215 (1972) M. Gryzinski: Phys. Rev. 115, 374 (1959) M. Gryzinski: Phys. Rev. A 138, 305,322,336 (1965) V.P. Shevelko, E.A. Yukov: Phys. Scripta 31,265 (1985) M.J. Seaton: Planet Space Sci. 12,55 (1964) W. Lotz: Z. Physik 232, 101 (1970) R.C. Stabler: Phys. Rev. A 133, 1268 (1964) A. Burgess: Astrophys. J. 139, 776 (1964) B. Shore: Rev. Mod. Phys. 39,439 (1967) E. Trefftz: J. Phys. Atom. Molec. Phys. B 3, 763 (1970) M.I. Seaton, P.J. Storey: Dielectronic Recombination, in Atomic Processes and Applications, ed. by P.G. Burke, B. Moiseiwitsch (North-Holland, Amsterdam 1976) A. Burgess, M.I. Seaton: Mon. Not. Roy. Astron. Soc. 125, 355 (1964) A. Burgess, H.P. Summers: Astrophys. J. 151, 1007 (1969) A. Gabriel, T.M. Paget: J. Phys. Atom. Molec. Phys. B 5,673 (1972) Yu.I. Grineva, V.I. Karev, V.V. Koreneev, V.V. Krutov, S.L. Mandelshtam, L.A. Vainshtein, B.N. Vasilyev, I.A. Zhitnik: Solar Phys. 23, 441 (1973) L.M. Biberman, V.S. Vorobyov, N.T. Yakubov: Usp. Fiz. Nauk 101, 353 (1972) [English trans!.: SOy. Phys.Usp. 15,375 (1972)] H. Griem: Plasma Spectroscopy (McGraw-Hill, New York 1964) R.H. Huddlestone, S.L. Leonard (eds.): Plasma Diagnostics Techniques (Academic, New York 1965)

References

301

5.23 W. Lochte-Holtgeven (ed.): Plasma Diagnostics (North-Holland, Amsterdam 1968) 5.24 W. Neumann: Spectroscopic methods of plasma diagnostic, in Progress in Plasmas and Gas Electronics, VoLl, ed. by R. Rompe, M. Steenbeck (Akademie, Berlin 1975) p.3 5.25 H.W. Drawin: Validity conditions for local thermodynamic equilibrium, in Progress in Plasmas and Gas Electronics, VoLl, ed. by R. Rompe, M. Steenbeck (Akademie, Berlin 1975) p.593 5.26 M.J. Seaton: Mon. Not. Astron. Soc. 119,90 (1959) 5.27 I.L. Beigman, E.D. Mikhalchi: J. Quant. Spectrosc. Radiat. Transl. 9, 1365 (1969) 5.28 G. Ecker, W. Weizel: Ann. Phys. 17, 126 (1956/57) 5.29 G. Ecker, W. Kroll: Phys. Fluids 6, 62 (1963) 5.30 H.R. Griem: Phys. Rev. 128, 997 (1962) 5.31 H.W. Drawin: Ann. Phys. (Leipzig) 14, 262 (1964) 5.32 S.T. Belyaev, G.1. Budker: Mnogokvantovaya Rekombinatsiya v Ionizovannom Gaze, in Fizika Polasmy i Problema Upravlyaemykh Termoaydernykh Reaktsii (Plasma Physics and the problems of Controlled Nuclear Fusion) Edition of Academy of Sciences, Moscow 1958) Vol.3, p.41 5.33 L.P. Pitaevskii: Zh. Eksp Teor. Fiz. 42, 1326 (1962) [English transl.: SOy. Phys. -JETP 15, 919 (1962)] 5.34 A.V. Gurevich, L.P. Pitaevskii: Zh. Eksp Teor. Fiz 46, 1281 (1964) [English transl.: SOY. Phys. - JETP 19, 870 (1964)] 5.35 M. Cacciatore, M. Capitelli, H.W. Drawin: Physica C 84,267 (1976) 5.36 D.R. Bates, A.E. Kingston, R.W.P. McWhirter: Proc. R. Soc. London A 267, 297 (1962) 5.37 D.R. Bates, A.E. Kingston, R.W.P. McWhirter: Proc. R. Soc. London A 270, 155 (1962) 5.38 R.W.P. McWhirter, A.G. Hearn: Proc. Phys. Soc. London 82,641 (1963) 5.39 L.C. Johnson, E. Hinnov: J. Quant. Spectrosc. Radiat. Transf. 13,333 (1973) 5.40 H.W. Drawin, F. Emard: Physica C 85,333 (1977) 5.41 M. Venugopalan (ed.): Reactions under Plasma Conditions (Wiley-Interscience, New York 1971) VoLl 5.42 R. Hess, F. Burrell: J. Quant. Spectrosc. Radiat. Transf. 21, 23 (1979) 5.43 H.W. Drawin, F. Emard: Physica C 94, 134 (1978) 5.44 H. Risken: The Fokker- Planck Equation, 2nd edn., Springer Ser. Syn., VoLl8 (Springer, Berlin, Heidelberg 1989) Chapter 6 6.1 6.2

6.3 6.4

L.A. Vainshtein: Trudy FIAN 15, 3 (1961) 1.1. Sobel/man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, VoLl2 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoLl (Springer, Berlin, Heidelberg 1979) D.A. Varshalovich, A.N. Moskalyv, V.K. Khersonske: Quantum Theory of Angular Moment (World Scientific, Singapore 1988) A.R. Edmonds: Angular Momentum in Quantum Mechanics (Princeton Press, Princeton, NJ 1957)

302 6.5 6.6 6.7

References M. Rotenberg, R. Bivius, N. Metropolis, J.K. Wooten, Jr.: The 3j and 6j Symbols (MIT Press, Cambridge, MA 1959) H. Appel: Numerical Tables for 3j, 6j, 9j Symbols, Landolt-Bomstein (Group I), Vol.3 (Springer, Berlin, Heidelberg 1968) A.P. Jucys, A.J. Savukynas: Mathematical Foundations of the Atomic Theory (Mintys, Vilnus 1973)

Chapter 7

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19

7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 7.28

R.G. Breene Jr.: The Shift and Shape of Spectral Lines (Pergamon, New York 1961) M. Baranger: In Atomic and Molecular Processes, ed. by D.R. Bates (Academic, New York 1962) G. Traving: In Plasma Diagnostics, ed. by W. Lochte-Holtgreven (NorthHolland, Amsterdam 1968) S. Chen, M. Takeo: Rev. Mod. Phys. 29, 20 (1957) H.R. Griem: Plasma Spectroscopy (McGraw-Hill, New York 1964) H.R. Griem: Spectral Line Broadening by Plasmas (Academic, New York 1974) J. Cooper: Rev. Mod. Phys. 39,167 (1967) C.J. Tsao, B. Curnutte: J. Quant. Spectrosc. Radiat. Transf. 2, 41 (1962) H. Rabitz: Ann. Rev. Phys. Chern. 25, 155 (1974) W.R. Hindmarsh, J.M. Farr: in Progress in Quantum Electronics, Vol.2 (Pergamon, Oxford 1972) p.141 F. Schuller, W. Behmenburg: Phys. Rpt. 12,273 (1974) N. Konjevic, J.D. Roberts: J. Phys. Chern. Ref. Data 5,209,259 (1976) N. Ksonjevic, M.S. Dimitrijevic, W.L. Wiese: J. Phys. Chern. Ref. Data 13, 619 (1984) E.L. Lewis: Phys. Repts. 58,1 (1980) G. Peach: Advances in Physics 30,367 (1981) N. Allard, J. Kielkopf: Rev. Mod. Phys. 54,1103 (1982) J.R. Fuhr, L.J. Roszman, W.L. Wiese: Bibliography on Atomic Line Shapes and Shifts, NBS Spec. Publ. 366 (19720; Suppl.1 (1974) J.R. Fuhr, G.A. Martin, B.J. Specht: Bibliography on Atomic Line Shapes and Shifts, NBS Spec. Publ. 366, Suppl.2 (1975) J.R. Fuhr, B.J. Miller, G.A. Martin: Bibliography on Atomic Line Shapes and Shifts, NBS Spec. Publ. 366, Suppl.3 (1978) J.R. Fuhr, A. Lesage: Ibid., Suppl.4 (1993) P.R. Berman: Appl. Phys. 6, 283 (1975) P. Anderson: Phys. Rev. 76, 647 (1949) J. Szudy: Acta Phys. Polon. A 40,361 (1971) J. Szudy, W.E. Baylis: J. Quant. Spectrosc. Radiat. Transf. 15,641 (1975) S.G. Rautian, 1.1. Sobleman: Usp. Fiz. Nauk 90, 209 (1966) [English trans!': Sov. Phys. - Uspekhi 9, 701 (1967)] S. Chandrasekhar: Rev. Mod. Phys. 15, 1 (1943) R.H. Dicke: Phys. Rev. 89, 472 (1953) D.R.A. McMahon: Austral. J. Phys. 34, 639 (1981) V.N. Faddeyeva, N.M. Terentyev: Tables of the Probability Integralfor Complex Argument (Pergamon, Oxford 1961)

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List of Symbols

Constants

Bohr radius Velocity of light e Elementary charge If = h/21t Planck's constant divided by 21t m Mass of electron Ry = me 4 /2lf 2 Rydberg unit of energy a o = If/me 2

c

Quantum numbers j J JT

Electron angular momentum Atomic angular momentum Total angular momentum of a system including atom and outer (scattered) electron / Electron orbital momentum L Atomic orbital momentum Lp Orbital momentum of atomic core (of parent ion) LT Total orbital momentum of a system including atom and scattered electron m, M Magnetic quantum numbers n Principal quantum number s Electron spin momentum S Atomic spin momentum Sp Spin momentum of atomic core (of parent ion) ST Total spin momentum of a system including atom and scattered electron A. Orbital momentum of outer (scattered) electron Basic Notations

ao

a, a 1 A Aij

Set of quantum numbers for initial state of an atom Set of quantum numbers for final state Fitting parameter for approximation of rate coefficients (vu) Einstein coefficient for spontaneous emission (radiative transition probability) [S-I]

C

Fitting parameter for approximation of cross sections u D Fitting parameter for analytical approximation of calculated cross sections and rate coefficients DE Energy scaling parameter E. Energy of bound electron in state a .to,.t Initial and final energies of free electron lij Oscillator strength 1(8) Scattering amplitude F)., F;" FJ, Radial functions of scattered electron in various representations g(a) Statistical weight of level a G:;L.=(/n-l[SpL p]/SL} /nSL) Coefficient of fractional parentage [see Ref. 1.1] G (P) Function of analytical approximation for rate coefficients (vu) Gr(r,r') Green's function jx(z) Spherical Bessel function Jf (P) Holtsmark distribution function n* = Z2 Ry/I E I Effective principal quantum number N Number density of particles [cm - 3] PI (cos 8) Legendre polynomials P" Pn/ , P"I (r) = r Rnl (r), where Rnl (r) is radial function for bound electron Principal value of integral P Q, QK Angular factor defining the dependence of cross sections on angular momenta for transitions with no change of spin Q", Q; Angular factor for exchange cross sections S, Sit Scattering matrix T Temperature in energy units 7;k Transition matrix u = .t/AE = (.to - AE)/AE Electron energy in threshold units v Velocity of particles (vu) Rate coefficient averaged over Maxwellian velocity distribution [cm 3 S-I] Dimensionless transition probability, frequency of collisional transitions [S-I]

J

J

w.o.

307

List of symbols

w. (c)

Autoionization probability for atomic state c [S-I] z Charge of atomic core (of parent ion) Z = z -1 Charge of ion !E Charge of nucleus y Line width (full width) in Chapter 7, set of quantum numbers for atomic term in Sect. 6.2 y = aM lmm 3 Set of quantum numbers for a system including atom and scattered electron r = aA1/2L T ST Set of quantum numbers for a system including atom and scattered electron in representation of total momenta LI Line shift in Chapter 7 LIE = E j - Ek Energy difference for levels i and k K Multipole order Kd Rate coefficient of dielectronic recombination [em 3 s - 1] K, Rate coefficient for three-body recombination [cm 6 s- 1] K. Rate coefficient of radiative recombination [cm 3 s- 1j

o

Solid angle Impact parameter, density matrix P.u Spherical components of density matrix uaoa,u(ao, a) Effective cross section u(aoAo,aA) Partial cross section cp Fitting parameter for approximation of cross sections q. (u) Functions of analytical approximation for cross sections X Fitting parameter for approximation of rate coefficients Ud2" ·jn] = J(2jl + 1)(2j2 +1) ... (2jn+ 1) (ml m2lsu) Klebsch-Gordan coefficients (abbreviated notation)

p

Wigner's 3j symbol

6j symbol

~U} (aoll

Til

9j symbol a)

Reduced matrix element

Subject Index

Action function 80 Analytic approximate formulas for cross sections 78,85,86,111,112,117, 118, 119 for rate coefficients 79, 86, 111, 112, 113, 114, 116, 117, 118, 120, 123, 130, 285 for recombination coefficient 153 Angular factors 30-35, 225-231 Atomic core 1 Autoionization 1, probability of 126, 127 Bethe formula 40 Bohr quantization condition 80 Boltzmann distribution 4 Born approximation 36, 74 Born formula 37 Born-Oppenheimer approximation 31 Brinkman-Kramers formula 102, 104 Broadening by neutral particles 289-296 Doppler 245-251,263 impact theory of 239, 251-263 of highly excited (Rydberg) states 284-286 Cascade matrix 137, 138 Clesch-Gordan coefficient 257 Collision strength 23 Collisional-radiative model 136 Coronal model 136 Correlation function 238 Coulomb-Born approximation 31,51 Cross section differential 9 normalization of 54-56 of broadening 239, 240, 254 of shift 239, 240, 254 partial 21 Debye radius 278 Density matrix method in broadening theory 251-273 Detailed balance, principle of 5 Dicke narrowing of spectral lines 250

Dielectronic recombination cross section of 122 rate coefficient of 122, 123 semiempirical formula for the rate coefficient of 130 simplified model of 120, 121, 127 Dielectronic satellites 130 -136 Differential cross section 9 Distorted wave approximation 31 Doppler broadening 245-251,263 Effective principal quantum number 159, 203 Exchange radial integral 31, 32, 51 Fractional parentage coefficients 39, 235, 236 Green's function free electron 61 Coulomb 67,68,75 Holtsmark distribution 276 Hooper distribution 278 Impact parameter 14,84 Intercombination transitions 50 Intermediate coupling 32 Ionization cross sections 45-47 Ionization equilibrium 7, 8 OJ symbol 231 3j symbol, Wigner's 20 6j symbol 231 9j symbol 21, 236 12j symbol 236 Klein-Rosseland formula 5 Kramers approximation for radiative recombination 115, 116 for radiative transitions 139

309

Subject Index Landau-Zener formula 91 Line strength 99 Local thermodynamic equilibrium 136 LS coupling 20, 29 Matrix, density 252 K 22 S 22 T 22 Maxwell distribution 4 Milne formula 6 Normalization of cross sections 54 Ochkur approximation 51, 52 Optical electron 1, 17 Optical theorem 11 Orthogonalized functions approximation 51 Oscillator strength 40 Partial cross section 21 Partial wave expansion 10 Partition function 4 principle of detailed balance 5

Radial integrals, direct 31 exchange 31 Recombination collisional-radiative 143 dielectronic 1 radiative 1 three-body 1, 150-153 Reduced matrix element 229 Saha-Boltzmann equation 141 Saha distribution 4 Satellites, dielectronic 130-136 Scattering amplitude 10 Scattering channel 15, 69 Scattering matrix 22 Spherical Bessel, Hankel, and Neumann functions 27 Sum rules for cross sections 225 for 3nj symbols 232-235 Thermodynamic equilibrium 3 Thomson formula 119 Three-body recombination 1, 150-153 T matrix 22 Triangular condition 231 Unitarity of S matrix 16

Q-factors 30-35, 225-231 Quadratic Stark effect constant 286 Quantum theory of broadening

Weisskopf radius 241 Wigner-Eckart theorem 38 Wigner's 3j symbol 20

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