Plane wave born collision strengths for electron ion excitation comparison with other theoretical methods

PLANE-WAVE BORN COLLISION STRENGTHS FORELECTRON-IONEXCITATION: COMPARISONWITH OTHER THEORETICALMETHODSI.. INTRODUCTION The plane-wave Born PWB approximation~ provides a simple, economica

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LA-10192-MS UC-34a

Issued:September1984

Plane-Wave Born CollisionStrengths

for Electron-Ion Excitation:

Comparisonwith Other Theoretical Methods

PLANE-WAVE BORN COLLISION STRENGTHS FORELECTRON-IONEXCITATION: COMPARISONWITH OTHER THEORETICALMETHODS

I INTRODUCTION

The plane-wave Born (PWB) approximation~ provides a simple, economicalmeans of generating collision strengths for electronic excitations in atoms andions due to electron impact Since plane waves are employed and exchangeeffects are neglected, the method is strictly applicable only at high energiesfor spin-allowedtransitions However, in practice, the PWB collision strengthsare in reasonable agreement (<30%) with those of other, more sophisticatedmethods such as the distorted wave (DW)2 down to energies of a few timesthreshold The prohibitionagainst spin-forbiddentransitionsis valid only forpure-LS coupling In general,

thus the collision strength

spin-allowedcomponent Since

Coulcmb field, the threshold

the states consist of a mixture of LS terms, andwill not be zero due to the presence of a

no explicit account is taken of the long-rangebehavior for ionic targets is incorrect goingto

1

zero rather than a finite value Through simple modifications of thenear-threshold behavior of the PWB collision strength, this defect can berectified In many cases, these modified PWB collision strengths agree tobetter than a factor of 2 with the DW results even in the regionnear threshold.

In this report, we calculate PWB and modified PWB excitation collisionstrengths and rates for a variety of ionic targets using a program developed byCowan and Robb This program calculates the atomic wave functions andgeneralized oscillator strengths using the Hartree-Fock and configurationinteraction codes of Cowan and determines the PWB collision strength from asubroutine of Robb These results are compared with the distorted wavecalculations of Mann2 and the hydrogenic Coulomb-Born exchange (CBX)

3calculationsof Sampson et al.—. In Section II, we give a brief review of thePWB method while in Section 111 we describe the various calculations,listingthe species, the transitions,and, where appropriate,the mixing coefficientsofthe CI calculation, the scaling parameters of the Coulomb integrals andspin-orbit component,and notes on any special features of the calculation.This section is followed by a brief discussion of the results (IV) and a series

of the graphs, which gives the collision strengths for the various species andtransitions under considerationas a function of the ratio (X) of the incidentelectron energy (kz) to the threshold energy (AE) In addition, for selectedtransitions, we present the ratio of the PWB and CBX to the DW collisionstrengths as well as the rates as a function of temperature

(1)

where k2 is the energy of the incident electron in Rydbergs, (k’)* is the energy

of the outgoing electron [(k’)* = k2 - AE], AE is the threshold energy, gfrr’ isthe generalized oscillator strength (see Ref 1, Sees 18-12), and X is theratio of the incident and threshold energies (X = kz/AE) The momentum transfer

K over which the integrationis performed is defined by

2

with the limits given by

S-P(x) = Q‘B(3) F(X) , and

PWBSF(x) =$2 (x + 3/(1 + x)) ,

3

III DESCRIPTION OF CALCULATIONS—

In this section, we give a description of the systems for whichcalculationswere performed For all cases, we give the transitions andconfigurationsemployed For certain transitions,we

coefficientsgenerated by RCG8 as well as the scaling

integrals (Fk(li,2i), Fk(.tijlj),Gk(Ai,lj), and

spin-orbit term (~i).l Where appropriate,additional

also include the CI mixingparameters of the Coulunb

comments are supplied toclarify the precise nature of the calculation The figure numbers associatedwith each transition are also given

d) 2s+3d

4

Fig 3

Fig 4a,bFig 5a,b,cFig 6a,b,cFig 7a,b,c

Configurations: a) [He] 2s ; 2p

Notes:

b) [He] 2s ; 3sc) [He] 2s ; 3pd) [He] 2s ; 3d

For purposes of comparison,the collision strengths for the 2s + 2p

transitionare summed at the same value of X even though the Pi/2 andP3/2 levels have different thresholds,

AE(2sl/2 - 2P1/2) = 3.574 Ry; AE(2S1/2 - 3P3/2) = 4.752 Ry

‘3/2 transitionsare summed at the same value of X

6.184 Ry; AE(2S1/2 - 2P3/2) = 15.605 Ry]

B Beryllium-like

C III

Transitions: a) 2s2 + 2s2p 1P

b) 2s2 + 2s2p 3PConfigurations: a-b) [He] 2s2 ; 2s2p ; 2p2

2s2p 3PScaling Coefficients:0.85,

lp0.99999

0.000750.85, 0.85, 0.85, 1.00

Fig 8

for t~e 2Pi/2 and[AE(2SIJ2 - 2P1/2) =

Fig 9Fig 10

5

Fig lla,bFig 12a,bFig 13a,b,cFig 14a,b,cFig 15a,bFig 16a,b,c

Fe XXIII

Transitions: a) 2s2 + 2s2p 1P

b) 2s2 + 2s2p 3P

C) 2s2 + 2s3s 1Sd) 2s2 + 2s3p 1Pe) 2s2 + 2s3p 3Pf) 2s2 + 2s3d ID

Configurations: a-b) [He] 2s2 ; 2s2p ; 2p2

c) [He] 2s2 ; 2s3s ; 2p3p ; 2p2d-e) [He] 2s2 ; 2s3p ; 2p3s ; 2p3d ; 2p2f) [He] 2s2 ; 2s3d ; 2p3p ; 2p2

Scaling Coefficients:

a-b) 0.95, 0.95, 0.95, 0.95, 1.00c-d) 0.87, 0.87, 0.87, 0.87, 1.00

Notes: Since the PWB formulation does not contain exchange effects, the

collision strength for spin-forbiddentransitionsbetween unmixed states

is zero This is not the case for the IS+ 3P transition inFe XXIII due

to the mixing of the 3P and 1P levels

Configurations: a-b) [He] 2s22p6 ; 2s22p53s

Mixing Coefficients:

final state J=l 1P2p53s 3P 0.310832p53s 1P -0.95047

0.665560.74545-0.00745-0.024420.00174-0.02125-0.01496

Scaling Coefficients:a) 0.90, 0.90, 0.90, 0.80, 1.00

Notes: The spin-forbidden PWB collision strength is nonzero due to the

triplet-singletmixing in the final state wave function

b) [Ne] 3s ; 4s

Fig 23Fig 24

1 2 4P

Scaling Coefficients: 0.90, 0.90, 0.90, 0.90, 1.00

Iv DISCUSSION

The figures, which compare the various calculational methods, are

reasonably self-explanatory and present a broad comparison for different types

of systems and transitions Therefore, we are relieved of presenting a detailed

discussion of the results and instead concentrateon the mjor conclusionsthat

can be drawn from this comparison These conclusionsare as follows: (1) The M2

modified PWB prescription generally gives the best agreement with the DW

results The exceptions to this rule involve s- to d-type transitions [Fe XXIV

2s + 3d and Fe XXIII 2s2 + 2s3d ID] although these differencesdisappear by

energies of 10 times threshold (X = 10) (2) The M2-PWB agrees with the DW to

within better than thirty percent (30%) for energies above a few times threshold

(x = 2- 3) The exceptions to this condition are the spin-forbiddentransitions for weakly coupled systems [C 111 2s2 - 2s2p 3P and Fe XXIII

2s2 - 2s2p 3P] (3) For energies near threshold (X< 1 - 2), the M2-PWB results

are usually within better than a factor of 2 of the DW The most pronounced ~exception to this rule comes from the spin-forbidden transition in A IV

(2p6 + 2p53s 3P) in which the exchange contributiondominates that of the mixing

at low energies (4) Spin-forbiddentransitionsare given reasonably well by

the M2-PWB provided that the states are sufficientlymixed We can see this by

viewing the progressionfrom C 111 to Fe XXIII to Fe XVII For C 111 the

triplet-singlet mixing is negligible, and the spin-forbiddentransition is

8

practically zero The amount of singlet component in the Fe XXIII 3P wavefunction is about 3% while for Fe XVII this value has risen to nearly 45% Asexpected, the agreement improves as the mixing becomes stronger (5) The M2-PWBrates in general agree with those of the DW to better than 30% over a range oftemperaturesfrom O tO 104 K The exception again occurs for transitionsinvolving a d-type configurationwith the largest difference being on the order

of 50%

9

and Spectra (University of

1 ~0 DO Cowan, ~ Theory Qf_ Ato~c ‘tructure —

California Press, Berkeley, 1981)

2

J B Mann, At Data Nucl Data Tables 29, 409 (1983); A L Merts,

J B Mann, W D Robb, and N H Magee, “Electron Excitation CollisionStrengths for Positive Atomic Ions: A Collection of TheoreticalData,” LosAlamos ScientificLaboratory report LA-8267-Ms (1980)

3 D H Sampson, S J Goett, R E H Clark, At. Data Nucl. Data Tables 30,

125 (1984); L B Golden, R E H Clark, S J Goett, and D H Sampson,Astrophys J Suppl 45, 603 (1981)

10

FIGURESThe basic figure is a plot of the collision strength!2rr’

x, the ratio of the incident energy of the electron to the

The species and transitionare given at the top of the graph

as a function ofthresholdenergy.The PWB, modifiedPWB (Ml, M2), DW, and CBX collision strengths are plotted ina consistentnotation throughoutthe set For some transitions,we present a second graphwhich gives the ratio of the PWB and CBX collision strengths to that of the DW

as a function of X In addition, the rate coefficients are present for aselected set of transitions The notation is as follows:

Fig 2.

12

+ /’ o+ /’

o.5- + ,’

0.4- ~ o ~’ o 0

/ / o.3- /

Fig 4a

13

1’

I o.6- 1

I

k

It

0

0 0 0

AA

+ +

o 0

AA

+ +

o 0

AA

+ +

o 0

FeXXIV 2s – 3p0.16 I I 1 1 1 I 1 11 1 1 1 1 I 1 1 1

o

0 0

0 0 0

x

Fig 6b

A A A

1.1-A i

1.o-+ + +

0

A +

0

A +

0

A Ql

+ A

A /+” o

● “

0 9“”+ o +,”

o.20- +0 +,0’ 0

+ o.15-

) 00

/ o.lo - /

/ 1’

Fig 10.

FeXXIII 2s2 – 2s2p 1P 0.7 1 1 1 t 1 I I 1 I 1 I I 1 1 t I f

Fig ha

4“ o +” o O.oloo - +*-●- @

+ ~“ o+* fl o + ,. 00.0075- , + + + ,,~ o

) o 0 ,9’ o 0

0.oo50- ,/’

/ 0.0025 - ,If

/ / 0.2 - / 1/

0.1- ,:

I 4

+ ++ + +

o 0 0 0 0

A

+

o 0

0

r

A +

A +

A +

A + A +

o 0 0 0

0

A +

o

A +

I

A + o

+ + +

1

Fig 16a

+ + 0.1-

0.0 - 00’ 1 I I 1 I 1 1 1 1 1 1 1 1 I I 1 Id’

/’

.0 0 0.00 “a~ 1 1 1 I 1 1 1 I 1 1 1 1 I 1 I 1

x

Fig 18.

++ -+ * - + -

/0 / t“

/

0 ,~ o / o

o“

o

f

It If If

I 11 It I

1 00 ,0‘ o 0

/ 2.0- //“

J I

I 4

Lod!lliams