Model potential calculations of lithium transitions

J Quunmr.Specrrosc~. Rudiut. Frar~sjer. Vol.

12. pp. 1539-1552.

Pergamon

Press 1972. Printed in Great

Britain

MODEL POTENTIAL CALCULATIONS OF LITHIUM TRANSITIONS T. C. CAVES Harvard College Observatory, Cambridge, Massachusetts and Department of Chemistry, North Carolina State University, Raleigh, North Carolina, U.S.A.*

and A. DALGARNO Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts 02138, U.S.A. (Received 18 April 1972)

Abstract-Semi-empirical potentials are constructed that have eigenvalues close in magnitude to the binding energies of the valence electron in lithium. The potentials include the long range polarization force between the electron and the core. The corresponding eigenfunctions are used to calculate dynamic polarizabilities, discrete oscillator strengths, photoionization cross sections and radiative recombination coefficients. A consistent application of the theory imposes a modification on the transition operator, but its effects are small for lithium. The method presented can be regarded as a numerical generalization of the widely used Coulomb approximation.

1. INTRODUCTION

approximation has been widely used in calculations of discrete oscillator strengths”) and of continuum oscillator strengths. (*) Attempts to extend it to direct calculations of response functions encounter the difficulty that the boundary conditions at the origin cannot be imposed because the Coulomb approximation is correct only in an asymptotic sense. (3) The difficulty could perhaps be circumvented by the extrapolation procedure of PETRA~HEN and ABARPNKOV t4) but an alternative less arbitrary procedure is the introduction of a model potential, characterizing the field in which the perturbed or active electron moves, with eigenvalues equal in magnitude to the binding energies of the electron. The method is then equivalent to the pseudo-potential or model potential procedures that have been used in the calculation of transition matrix elements. A consistent application of such procedures demonstrates that the perturbing operator whose response function is required must itself be modified. THE COULOMB

* Permanent address : Department of Chemistry, North Carolina State University, Raleigh, North Carolina, U.S.A. 1539

T. C. CAVES and A. DALGARNO

1540

The response of lithium to an electric dipole perturbation has been extensively studied in a variety of approximations, and experimental comparison data are available. We adopt lithium as an example. 2. MODEL

POTENTIALS

In the Coulomb approximation, the valence electron with quantum numbers nl of the lithium atom (l~~n1)~L is regarded as moving at large distances r from the core in a potential Z*/r where Z* is a parameter so chosen that it yields the measured eigenvalue E(nl). A more correct physical description leads to the asymptotic form V(r) =

-(Z-

2)e2 r

ade2 a e2 ----!--+2r4 2r6

3/le2 r6

(1)

where Z is the nuclear charge (Z = 3 for lithium), ad and a4 are the dipole and quadrupole polarizabilities of the Lif core and j? is a retardation correction.‘5,6’ For Li+, accurate variational calculations give ad = 0.1923 ui ,(‘) a, = 0.1134 ud (*) and B = 0.035 u$ .@) The form of (1) can be substantiated by adopting as wave function for the lithium atom y(r,, r2 ar) = 2Arly r21rhUr)

(2)

where the core wave function I,@,, r,lr), rl and r2 being the position vectors of the core electrons, depends parametrically on the position r of the valence electron. Then the substitution of (2) into the Schriidinger equation with r held fixed yields the equation

[

H12(r)=

V(r)-&41 xc(rl, r21r)

[&v:+v:,-F-Z+& e2

e2 --V(r)-:-& +Ir,--rl+lr,-rl

1

Xc(rl,r21r) = 0

where V(r) is the eigenvalue and EC is the core energy, E, = .

(4)

If we now allow r to vary, we obtain for the eigenfunctions &(r) of the valence electron the equation (T, + V(r) - -G)&(r) = 0 (5) where T, = (- h2/2m)Vz and (E,,I is the binding energy of the nl electron. For large r, the solution of (3) can be written Xe(rl7r21r)= xe(rI7r2b) + x,“Yrl 7r21r), (6) where

(7)

1541

Model potential calculationsof lithiumtransitions

and V(r) can be written V(r) = P)(r)

+ V*‘(r).

(8)

where

Asymptotically, (8) has the form of the leading three terms of (1). The fourth term of (1) reflects a core-valence electron coupling interaction (X,(r, , r21r) IT,lx,(rr , r21r)) that should be included as part of H,,(r) in (3).@*@ Expression (1) is incorrect at small distances r. We modify it by replacing -(Z- 2)e*/r by the core potential V(r) and by the insertion of cutoff functions W, and w6. Thus, T/(‘)(r) =

v(l)(r)-:-p.

For Xe(rl, r2(co) we adopted the Hartree-Fock V’)(r) = +2e2

(

Y,(f)

&

yk’)

>

(11)

approximation -$+Cl

Y,(rl)Y,(r2)

so that

exp(-r/r,)+C,rexp(-r/r,)

(12)

where Y’, is a Hartree-Fock 1s orbital. (9) The third and fourth terms in equation (12) with Cl, C,, and r,, as disposable parameters, provide increased flexibility in the model potential.“” With the help of these terms, we are able to obtain accurate fits to the *S, *P and *D eigenvalues with a potential which does not depend on angular momentum. For the cutoff functions W, and W, we adopted the forms W, =

(1/2)or,e*[1 - exp( - r/r#]

(13)

and W, =

IZe*[l -exp(-r/r,)*]

(14)

where rl and I are disposable parameters, and r. has the same value as in equation (12). The parameters Cl, C,, ro, rl , and A are chosen so that the eigenvalues of V(r) closely approximate in magnitude the binding energies of the valence electron. The form adopted for W, seems preferable to the more common choice” ‘) W, = r”/(r* + r$*,

(15)

which adds a term up2rg/r6 to the asymptotic r -6 interaction. However this additional term does to some extent represent the repulsive retardation interaction 3/3e2/r6 and the differences between the form of (6) and that of the semi-empirical potential of DOUGHTY et al.” lb are probably not significant. Other forms of the potential have been used. BEIGMAN et ~1.“~’ include the term (10) but omit the contribution to (1) from the quadrupole polarization, and ANDERSON and

T. C. CAVE and A. DALGARNO

1542

ZILITIS(‘~) omit the long range employ

a pseudo-potential”@

range interactions.

terms

entirely.

SZASZ and McGINN(‘~)

that has no arbitrary

The most sophisticated

parameters

pseudo-potential

approach

VESELOV and SHTOFF,(’ ‘) who solved (3) by variational they employed leads to an interaction that contains but with coefficients that are only approximate.

3. TRANSITION Consider

the matrix

element

MATRIX

describing

methods. terms

and McGINN(‘~)

but also ignores

the long

appears to be that of

The trial function

decreasing

ELEMENTS

the transition

from an initial

state Y(i) to a

final state Yycf) under the action of an operator v(r,)+ u(r2) + u(r). The standard potential caiculations’13*14) are equivalent to the approximation W

= xc(rl Trzl ~)4M3

that

as rm4 and rp6

W

model

(16)

= xc(rl TrA~M,dr)

so that W”l~(r,) The

model

&,(r)

takes

pointed

potential into

+ 4r2) + 4r)lY’f’>

used to calculate

account

the effect

out that this interaction

the effect could be represented

=

(~.l(r)lu(r)l~n,r,(r)>.

the valence

of the valence

also changes

electron electron

eigenfunctions on the core.

the core wave functions

by a modification

(17)

of the transition

&,(r)

and

BERSUKER(‘*)

and he showed that

operator.

An alternative

derivation of the same result has been given by HAMEED, HERZENBERG and JAMES. Yet a third derivation follows from the discussion of COHEN and DALGARNO.(“) A slight extension

of their arguments

shows that in (17) u(r) should be replaced

e2

u(r) - S

e2

2e2

x

(Ef’ - E,)

+Ir,-rl

t+o

by

where xF’ and EF) are the excited core eigenfunctions

and eigenvalues

(18)

and the summation

over t excludes the core ground state xe. For large values of r, the first matrix element assumes a simple form which is determined by the symmetry

of u(r). If u(r) is an electric

dipole operator,

(18) is asymptotically (19)

where ad is the core dipole polarizability. Similar formulas apply when u(r) is a higher multipole moment #P,(cos 13).The modified transition operator is

The response solution of

function

of a system dnl(r) to a perturbation

( T,+

W) - E,, - W&v)

+ WUr)

u(r) at a frequency

= 0.

v is the

(21)

STERNHEIMER and PEIERLS”‘) have integrated (21) numerically to calculate the quadrupole anti-shielding factors of several excited states of a number of atomic systems. Equation (21)

Model potential calculations of lithium transitions

1543

may be modified by replacing u(r) by (20). In practice a cutoff function must be introduced into (20). We adopted the form

u(r) 1-s

i

[l-exp(-r/T.JL+r]

.

(22)

I

4. CALCULATIONS

The values adopted for the parameters ro, ri, 1, C,, and C, in the potential V(r) are given in Table 1. They reproduce all the measured eigenvalues to within an uncertainty of less than 5 x 10e4 eV, except for the 2 2S state, where the error is 8 x 10d4 eV. In the limit of large principal quantum numbers n, the potentials yield asymptotic quantum defects, expressed as zero energy phase shifts, ql, of q,, = 1.255, vi = 0.150 and qZ = 0.0075, values in harmony with those derived by DOUGHTY et al.“” and SHEOREY.(~~) For the cutoff parameter, i,, in (22) we took a value 0.455, intermediate between the cutoff parameters r. and r1 used in the potential. The matrix elements are relatively insensitive to the value of T, used. Figures 1 and 2 show the dependence on ?,, of the 2s-2p oscillator strength and the threshold photoionization cross section, respectively. TABLE 1. VALUES OF PARAMETWSIN THE MODELPOTENTIAL Parameter

Value

-

FIG. 1.

0.47 a, 0.44 a, 0.011589 a; 0.546266 e2fa 0.303243 e2/a.;

Sensitivity of the 2s- 2p (resonance) oscillator strength to the value of the cutoff parameter, i, . The horizontal line represents the value recommended by Wt~se et ~1.~~ I)

1544

T. C.

CAVES

and A. DALGARNO

FIG. 2. Sensitivity of ground-state photoionization cross sections at threshold to cutoff parameter ?, (cf. Eq. 22). Experiments give cross sections of 1.9 x lo- ‘s cm’ ‘33rand 1.5 x lo- ‘* cm2.(36)

4.1 The 2s - np oscillator

strengths

The predicted 2s- np oscillator strengths, calculated with and without the core polarization modification (22) of the electric dipole transition operator, are presented in Table 2. Similar calculations of 2s-np oscillator strengths, differing only in the detailed

TABLE

2. OSCILLATOR

STRENGTHS

OF THE

(Is~~s)~S--

(tS%p) *p” SERIESOF LITHIUM

n

Without core correction

With core correction

Experimental

0.753 0.00450 0.00414 0.00249 0.00153 9.91 x 1o-4 6.75 x lo-“ 4.79 x lo-‘+

0.746 0.00477 0.00430 0.00258 0.00158 0.00103 6.98 x 1O-4 4.95 x 1o-4

0.753 0.00552 0.00480 0.00316 0.00192 0.00128 9.16 x 1O-4 6.79 x 1O-4

choice of model potential and in the use of less precise values of the core polarizability, have been reported recently by BEIGMAN et al. (12) To the two significant figures given by BEIGMAN et al., their values are identical to those in Table 2. The effect of the core polarization modification of the transition operator is small. It is larger for heavier systems.‘23’ Many other theoretical predictions have been made for dipole transitions in lithium. The 2s -2p transition oscillator strength is insensitive to the detailed nature of the eigenfunctions and values near 0.745 are usually obtained. (13.15,17>21,24) The most refmed theoretical calculation’25’ gives a value of 0.753. Measurements of the 2p 2P lifetime yield oscillator strengths of 0.63 +0.03,(26) 0.74f0.01,(27) 0.83 -+0.04,‘28) and 0.77 + 0.03.(29) Because of oscillation in the integrands, the oscillator strengths of transitions of higher members of the principal series are more sensitive to the eigenfunctions. For the 2s - 3p transition, most theoretical methods have yielded oscillator strengths near 0.003. The Hartree-Fock value is 0.0027, (251 the frozen-core Hartree-Fock value is 0.0033’30’ and the pseudo-potential value is 0.0032. (is) The semiempirical methods that employ model

Model potentialcalculationsof lithium transitions

1545

potentials derived from measured eigenvalues have led to oscillator strengths near 0.0045. The Coulomb approximation gives 0.0044, ANDERSON and ZILITIS(‘~) obtain 0.0043, STERNHEIMERand PEIERLS” ‘) obtain 0.0049, and we obtain 0.0045. With the inclusion of the core polarization contribution to the transition operator, the value is 0.0047.“2’The correct value of the 2s - 3p oscillator strength is uncertain. VESELOV and SHTOFF(“) solved (3) by a variational procedure and they obtained 0.0061. Their predicted potential is similar at large distances to our adopted model potential but it underestimates the strength of the long range dipole attraction. The large value that Veselov and Shtoff obtained for the oscillator strength is accordingly unexpected. It suggests that the short range contribution to the dipole matrix element is substantial as does also their result that the core polarization modification of the transition operator decreases the oscillator strength. The result, 0.0061, predicted by Veselov and Shtoff, is slightly larger than the value 0.00552, recommended by WIESE, SMITH and GLENNON, (31) that follows from the measurements of FILIPPOV’~~’ when normalized to 0.753 for the 2s - 2p transition oscillator strength. The 32P lifetime measured by BUCHET et ~1.~~~)is not precise enough to yield a useful value for the 2s - 3p oscillator strength ; most of the 32P decay occurs through the 3s - 3p transition. The values measured by FILIPPOV(32) for higher members of the 2s - np series are listed in Table 2. The additional long range attraction contained in the model potential calculations increases the calculated oscillator strengths but there remains a persistent discrepancy with the measurements. 4.2 Photoionization cross sections A study of the continuum 2s-&p absorption provides further comparison data.‘33’ There have been many theoretical calculations”2.34*43) and a few measurements.(33*3s*36) Our predicted photoionization cross sections are presented in Fig. 3. The inclusion of the core contribution increased the cross sections by only small amounts until an energy E of about 25 eV where the correction changed sign. The sign change indicates substantial core penetration, and the model is invalid at higher energies. The measurements of HUDSON and CARTER(~@ and Of MARR and CREED are presented in Fig. 3. Our model predicts a threshold cross section of 1.5 x lo- ‘* cm2, close to the value of 1.54 x lo-‘* cm2 measured by HUDSON and CARTER (36) but inconsistent with the recommendation by MARR and CREED of 1.9kO.2 x lo-l8 cm2. The agreement is sufficient however to establish a discrepancy between the measurements of discrete and continuum oscillator strengths and there seems little doubt that the measurements by FILIPPOV’~~’ overestimate the 2s - np oscillator strengths for large values of n.(33) In common with all the theoretical models, our predicted cross sections appear to decrease too rapidly with increasing photoelectron energy. 4.3 The polarizability of lithium The frequency-dependent dipole polarizability of ground state neutral lithium can be obtained by solving (21) directly and also by expressing it in the form m f

(2s- np)

‘(‘) = “T2 (E,, - E,J2 - v2 +

a,df(2s - EP)

s 0

de

de (IE2,1+42--2

1546

T. C. CAVES and A. DALGARNO

ENERGY

OF EJECTED

ELECTRON

IRYDBERGSI

FIG. 3. Cross sections for photoionization from the ls* 2s ‘S and 1s’ 3s’S states of Li. (---) measured values of ozs ;(33)(- -. -) measured values of czs ;csa) (-) present work.

all quantities being in atomic units. Alternatively we may write a(v) =

2 smv2m

m=O

where the coefficients S, are summations m

sm = &

f(2s-np) (Enp-E2s)2m+2+

0

de (lEz,l+ #m+2’

The summations S, are listed in Table 3. They are useful in other connections. So is the dipole polarizability. Its value, 164.6 uz, is near to the most refined variational estimate available of 163.1 ~2.~~~)The values of a(v) at higher frequencies are also in harmony with the variational calculations.‘37’ 4.4 Oscillator strengths for ns - n’p and n’p - n”d transitions We have calculated oscillator strengths for a large number of transitions and the results are collected into Tables 4-6. The core polarization modification is small and, with the exception of the 2s - np transitions, we have ignored it. The tables include the values recommended by WIE~Eet al.(31) which with the exception of the 2s - np transitions are the results of Hartree-Fock calculations by ZEISS or of the Coulomb approximation. Extensive tables have been presented by ANDERSON and ZILITIS,(‘~)by COHENand JELLY

Model potential calculations of lithium transitions

1541

TABLES. VALUESOFTHE! OSCILLATOR STRENGTH SUMS s,,,IN UNITS OF 2lll+3 a0

0 1 2 3 4 5 6 7 8 9

1.646 x 3.544 x 7.681 x 1.666 x 3.613 x 7.837 x 1.700 x 3.687 x 7.997 x 1.735 x

lo2 lo4 lo6 lo9 10” lOi 10’6 10” 10ZO lo=

TABLE 4. OSCILLATOR STRENGTHS&,. AND TRANSITION PROBABILITIES (IS%) *S--(l&l'p)*P" TRANSITIONSIN LITHIUM

(4 ns 2s

3s

4s

5S

6s

7s 8s

n’p 2P 3P 4P 5P 6~ 7P 8~ 3P 4P 5P 6~ 7P 8~ 4P 5P 6~ 7P. 8~ 5P 6~ 7P 8~ 6~ 7P 8~ 7P 8~ 8~

f ““’ 0.746 0.00477 0.00430 0.00258 0.00158 0.00103 6.98 x lo-.’ 1.215 4.98 x lo-’ 0.00131 0.00114 8.31 x lo-“ 5.96 x 1O-4 1.640 9.45 x 1o-4 2.94 x 1O-4 4.90 x 1o-4 4.39 x 1o-4 2.051 0.00339 9.56 x 1O-6 1.77 x 1o-4 2.457 0.00646 6.67 x lo-’ 2.857 0.00975 3.257

(a) This paper. (b) WIESE eta1.'31)

A,.,(sec- ‘) 3.69 x 10’ 1.01 x lo6 1.27 x lo6 8.72 x lo5 5.74 x 10s 3.87 x lo5 2.71 x lo5 3.74 x 106 9.51 x lo2 4.06 x lo4 4.41 x lo4 3.63 x lo4 2.81 x lo4 7.74 x 105 3.37 x lo3 1.89 x lo3 4.19 x 10” 4.44 x 103 2.34 x 10’ 3.30 x lo3 1.81 x 10 4.69 x 10’ 8.89 x lo4 2.17 x 10’ 46.2 3.95 x lo4 1.33 x lo3 1.97 x 104

04 f

m.

0.753 0.00552 0.00480 0.00316 0.00192 0.00128 9.16 x 1O-4 1.23 1.93 x 1o-4

A,., OF

A,.,(sec-‘) 3.72 x 1.17 x 1.42 x 1.07 x 6.97 x 4.84 x 3.55 x 3.77 x 3.69 x

10’ 106 lo6 106 lo5 lo* lo5 106 lo3

T. C. CAVES and

1548

A.

DALGARNO

TABLE 5. OSCILLATOR STIWNGTIS f , AND TRANSITIONPROBABILITIESA,., OF (ls*np) ‘p” - (l&I’d) ‘D TRANSITIONS IN LITHIUM

np

n’d

f .“’

2p

3d 4d 5d 6d Id 8d 9d 3d 4d 5d 6d ld 8d 9d 4d 5d 6d Id 8d 9d 5d 6d 7d 8d 9d Id 8d 9d 8d 9d

0.640 0.123 0.0463 0.0230 0.0123 8.36 x 1O-3 5.65 x lo- 3 0.0741 0.522 0.130 0.0544 0.0286 0.0172 0.0112 0.135 0.491 0.132 0.0579 0.0315 0.0193 0.191 0.487 0.136 0.0607 0.0336 0.292 0.514 0.146 0.341 0.536

3p

4P

5p

7p

8p

(4

A,.,(seC1) 6.87 x IO’ 2.32 x 10’ 1.09 x 10’ 5.99 x lo6 3.67 x IO6 2.41 x lo6 1.68 x lo6 3.77 x lo3 6.79 x lo6 3.47 x lo6 1.97 x lo6 1.22 x 106 8.09 x lo5 5.64 x 10’ 1.28 x lo3 1.35 x lo6 8.34 x 10’ 5.29 x lo5 3.54 x lo5 2.48 x lo5 4.78 x lo2 3.94 x lo5 2.70 x 10’ 1.85 x lo5 1.30 x lo5 98.6 6.33 x lo4 4.87 x lo4 51.8 3.09 x lo4

f M’

(b)

A,.,(sec-‘)

0.667 0.122 0.0453

7.16 x 10’ 2.30 x 10’ 1.06 x 10’

0.0743 0.527 0.128 0.0534

3.81 x 6.85 x 3.41 x 1.94 x

0.135 0.494 0.130

1.28 x lo3 1.36 x lo6 8.19 x 10’

0.190 0.491

4.78 x 10’ 3.98 x lo5

lo3 lo6 lo6 lo6

(a) This paper. (b) WIESE et .1.‘31’

and by MCGINN. (15)Because we have explicitly included the long range polarization in our model potentials, we believe our values to be more reliable. Our values agree well with other theoretical calculations except in cases such as 3s -4p, for which the cancellation is extremely severe. Measurements have been made of the radiative lifetimes of several states. A comparison of our theoretical predictions with the experimental data is given in Table 7. The agreement is satisfactory. Where a discrepancy exists between the measurements of different experiments, our results support those of KAR~TENSEN and SCHRAM.(~~) 4.5 Radiative recombination coeficients The rate coefficient cc,, of the radiative recombination Li+(ls’)+e

+ Li(ls2nl)+hv

process (23)

1549

Model potential calculations of lithium transitions

TABLE 6. OXILLATOR

(4

np

n’s

2p

3s 4s 5s 6s Is 8s 4s 5S 6s IS 8s 5s 6s 7S 8s 6s 7S 8s 7s 8s 8s

3p

4p

5p 6p lp

STRENGTHS AND TRANSITION PROBABILITIESFOR np-n's TRANSITIONS

f “I’

0.111 0.0128 0.00432 0.00204 0.00114 7.05 x 1o-4 0.223 0.0259 0.00886 0.00426 0.00242 0.336 0.0386 0.0132 0.00641 0.448 0.0510 0.0175 0.560 0.063 1 0.672

A,.,(sec3.35 x 1.04 x 4.73 x 2.56 x 1.55 x 1.01 x 7.46 x 2.82 x 1.46 x 8.59 x 5.52 x 2.25 x 9.57 x 5.36 x 3.36 x 8.47 x 3.89 x 2.31 x 3.73 x 1.81 x 1.84 x

‘)

10’ 10’ lo6 106 lo6 lo6 lo6 lo6 lo6 10’ 10’ lo6 10s 10’ 10’ 10’ 10’ 10’ IO5 lo5 10s

@I

f,,

44=-

7

0.115 0.0125 0.00420 0.00198

3.49 x 1.01 x 4.60 x 2.50 x

10’ 10’ 106 lo6

0.223 0.0254 0.00874

7.46 x lo6 2.76 x lo6 1.44 x lo6

0.335 0.0372

2.25 x 10’ 9.22 x 10’

0.448

8.48 x 10’

(a) This paper. (b) WIESEet ~1.‘~‘)

is related by detailed balancing to the cross section o,Av) for the inverse photoionization process : (24) I”1 TABLET. RADL~~LWETIMES 0F(1S2d)?Z2~STATES OF LlTHRlM IN UNITSOF lo--’ SEC Experiment (b)

state

Theory

(a)

3 *s 4*s 5 2s 6 *S 7 2s 22P 3 *P 42P 5 2P 3*D 42d

2.99 5.62 9.77 17.2 27.1 2.69 21.3 39.4 50.8 1.46 3.33

2.55 5.58 11.30 19 22 3.19 23.5 47.9 86.35 1.46 3.92

(c) 4.8

36.4 54.6 1.45 3.52

(a) BUCHET et 01.‘~~’ (b) KARVEN~ENand sCHRAM.'38' (~)BICICEL et cd.@’

1.5 3.3

T. C. CAVESand A. DALGARNO

1550

where g,, is the statistical weight of the final state Li(ls%l) and v is the frequency of the ionizing radiation. We have calculated the coefficients for radiative recombination into the ground (ls2 2s 2S) and ls2 2p 2Po, ls2 3s 2S, and 1s’ 3p 2Po excited states of lithium at a number of temperatures. The corresponding photoionization cross sections are shown in Figs. 3 and 4 and the calculated recombination coefficients are given in Table 8. Table 8 includes the total

ENERGY

OF EJECTED

ELECTRONtRYDBERGS)

FIG. 4. Cross sections for photoionization from the 1s’ 2p 'P and 1s’ 3p 2P states.

TABLE 8. RECOMBINATION COEFFICIENTS a,,, OF Li’ IN UNITSOF 10eL4 cm3 set-’

VW 156.25 500 1250 5000 8000 10,000 16,000 20,000 32,000

4.93 2.80 1.83 1.07 0.940 0.893 0.823 0.799 0.761

a,,(T)

adT)

65.1 36.1 22.7 11.5 8.66 7.49 5.40 4.57 3.15

0.660 0.384 0.265 0.184 0.175 0.173 0.172 0.172 0.173

23.2 12.8 8.04 3.79 2.85 2.49 1.82 1.55 1.09

* The values adopted for n 2 4 are hydrogenic.

514 225 124 44.8 29.4 25.8 16.7 14.5 9.2

Model potential calculations of lithium transitions

1551

radiative recombination coefficients for electron capture by Li+, obtained by adding to the values for n = 2 and 3 the hydrogenic values for capture into more highly excited states. The results differ only slightly from those obtained for hydrogen’3g’ when the contribution from capture into the 1s orbital is excluded. ROTHE has determined photoionization cross sections from the ls2 2p 2Po state of Li by measuring the absolute intensities of the continuum emission in a lithium-seeded argon plasma at 6000°K. He fitted his results near threshold with an expression of the form

where lo, the threshold wavelength of the ionizing radiation, is 3500 A, bo, the photoionization cross section at threshold, is 19.7 k 3.0 x lo- l8 cm2 and the exponent /I is 1.8 kO.5. Our results near threshold can be reproduced with the parameters o. = 15.4 x lo- l8 cm2 and /? = 2.9. Other theoretical calculations of photoionization cross sections from the 2p state have been carried out by MoSKVIN(~~) and YA’AKOBI,(~~) using the quantum defect methodC2) and by GEZALOV and IVANOVA using a simplified self-consistent field approach. The values of the photoionization cross section at threshold obtained in these studies were (15.7, 16.7 and 15.0) x lo- l8 cm2, respectively. These results all fit the threshold expression (25) with an exponent B = 2.9. Thus the theoretical results are in agreement that the photoionization cross section decreases from threshold as E-2.g, while the experimental results indicate a much slower decrease with energy.

Acknowledgements-We are greatly indebted to Dr. J. C. WEISHEITfor several illuminating discussions. This work has been partly supported by the National Aeronautics and Space Administration under Grants NGL-22-007-136 and NGL-22-007-006. TCC wishes to thank the North Carolina State University for a Faculty Professional Development Grant.

REFERENCES

1. D. R. BATESand A.

DAMGMRD, Phil. Trans. R. Sot. Land. A242, 101 (1949). A. BURGS and M. J. SEATON, Mon. Nor. R. Astr. SC. 120, 121 (l%O). A. DALGARNGand R. M. PENGELLY, Proc. Phys. Sot. 89,503 (1966). M. J. m and I. V. AB~KOV, Vestn. Leningr. Uniu. 5, 141 (1954). U. QPIK, Proc. Phys. Sot. Land. 92,573 (1%7); J. CALLAWAY, R. W. LABAHN,R. T. Pu and W. M. DUXL~R, Phys. Rev. 168, 12 (1968); C. J. KLEXNMAN,Y. HAHNand L. SPRUCH,Phys. Rev. 165,53 (1968). 6. A. DALGARNO, G. W. F. DRAKE and G. A. VICTOR,Phys. Rev. 176, 194 (1968). 7. G. A. VICTOR and A. DALGARNO, J. &em. Phys. 49, 1982 (1968). 8. J. DRAKE(unpublished). 9. C. C. J. ROOTHAAN, L. M. SACHSand A. WEIRS,Reu. Mod. Phys. 32,186 (1960). 10. C. BIX-TCHW, J. Phys. B (Atom. MO!. Phys.) 4, 1140 (1971). 11. L. A. VAINSI-ITEIN. Ont. Soectrosc. 3.313 (1957): N. A. DOUGHTY. M. J. SEATON and V. B. SHEOREY. J. P/w.. B (Atom. Mol. Phy& i, 802 (1%8). ’ ’ ’’ 12. L. L. BEIGMAN, L. A. VAINSI-ITEIN and V. P. SHEVELKO, Opt. Spectrosc. 28,229 (1970). 13. E. M. ANDERSON and V. A. ZILITIS,Ovt. Svectrosc. 16,211 (1964). 14. L. SWSZ and G. MCGINN,J. them: Pip. b7,3495 (1967); sb, 14&l (1969). 15. G. MCGINN.J. them. Phvs. 53.3635 (1970). 16. J. C. PHILLIPS, Phys. RecS112, 685 (1958); J: C. PHILLIPS and L. KLEINMAN; 116,287 (1959); 118,1153 (1960). 17. M. G. VFSELOV and A. V. ~HTOFF,Opt. Specwosc. 22,457 (1967); 26, 177 (1969).

2. 3. 4. 5.

1552

T. C. CAVESand A. DALGARNO

18. I. B. BERSUKER,Opt. Spectrosc. 3, 97 (1957), Izv. Akad. Nauk. SSSR Ser. Fiz. 22, 749 (1958). 19. S. HAMEED,A. HERZENBERGand M. G. JAMES,J. Phys. B (Atom. Mol. Phys.) 1,822 (1968); S. HAMEED,~VIJV. Rev. 179, 16 (1969). 20. M. COHENand A. DALGARNO,Proc. R. Sot. Lond. A293,359 (1966). 21. R. M. STERNHEIMER and R. F. PEIERLS,Phys. Rev. A3,837 (1971). 22. V. B. SHEOREY,J. Phys. B (Atom. Mol. Phys.) 2,442 (1969). 23. J. C. WEISHEITand A. DALGARNO,Chem. Phys. Lett. 9, 517 (1971); Phys. Rev. Lett. 27, 701 (1971); J. C. WEISHEIT,Phys. Rev. AS, 1621 (1972). 24. 0. S. HEAVENS,J. Opr. Sot. Am. 51, 1058 (1961); M. R. FLANNERYand A. L. STEWART,Mon. Not. R. Asrr. Sot. 126, 387 (1963); J. C. STEWARTand M. ROTENBERG,Phys. Rev. 140, 1508A (1965); M. COHENand P. KELLY, Can. J. Phys. 45, 1661 (1967); D. W. NORCRO% private communication (1970). 25. A. W. W~rss, Astrophys. J. 138, 1262 (1963). 26. J. BUCHET,A. DENIS, J. DE~E~QUELLES and M. DUFEY, Compf. Rend. Acad. Sci. 265B, 471 (1967). 27. K. C. BROG, T. G. ECK and H. WIDER, Phys. Rev. 153,91 (1967). 28. W. S. BICKEL, I. MARTINSON,L. LUNDIN, R. BUCHTA, J. BROMANDER and I. BERGSTROM, J. opt. Sot. Am. 59, 830 (1969). 29. T. ANDERSEN,K. A. JESSENand G. SORENSEN,Phys. Lerr. 29A, 384 (1969). 30. M. COHENand P. KELLY, Can. J. Phys. 45, 1661 (1967). 31. W. L. WIFSE, M. W. SMITHand B. M. GLENNON,Atomic transitionprobabilities. Natn. Bur. Stand. Washington (1966). 32. A. N. FILIPPOV,Z. Phys. 69, 526 (1931); Zhur. Eksptl. Teoret. Fiz. 2, 24 (1932). 33. G. V. MARR and D. M. CREEK, Proc. R. Sot. L.ond. A304,233 (1968). 34. A. L. STEWART,Proc. Phys. Sot. A67, 917 (1954); J. H. TAIT, Atomic Collision Processes. North Holland, Amsterdam (1964); E. J. MCGUIRE, Phys. Rev. 175,20 (1968); G. PEACH,Mem. R. Astron. Sot. 71,13 (1967); E. S. CHANGand M. R. C. MCDOWELL, Phys. Rev. 176,126 (1968); M. R. C. MCDOWELLand E. S. CHANG, Mon. Nor. R. Astr. Sot. 142,465 (1969); J. J. MA-E and R. W. LABAHN, Phys. Rev. 188, 17 (1969). 35. J. TUNSTEAD,Proc. Phys. Sot. Lond. A66,304 (1953); Ci. V. MARR, Proc. Phys. Sot. Land. 81,9 (1963). 36. R. D. HUDSONand V. L. CARTER,J. opt. Sot. Am. 57,651 (1967). 37. G. STACEYand A. DALGARNO,J. them. Phys. 48,2515 (1968). 38. F. KARSTENSEN and J. SCHRAM,J. opt. Sot. Am. 57,654 (1967). 39. A. BURGESS,Mem. R. Astr. Sot. 69, 1 (1964-65). 40. D. E. ROTHE,JQSRTll, 355 (1971). 41. Yu. V. MOSKVIN,Opt. Spectrosc. 15,316 (1963). 42. B. YA’AKOBI,Proc. Phys. Sot. Land. 92, 100 (1967). 43. KH. B. GEZALOVand A. V. IVANOVA,High Temp. 6,400 (1968).