theoretical intensities of fexiv in the solar euv spectrum

T H E O R E T I C A L I N T E N S I T I E S O F FeXIV

IN T H E S O L A R E U V S P E C T R U M

M. BLAHA* Laboratory for Solar Physics, NASA-Goddard Space Flight Center

Greenbelt, Maryland, U.S.A.

(Received 9 October, 1970)

Abstract. Equilibrium population of Fextv levels in coronal conditions was calculated including configurations 3sZ3p, 3s3p ~, 3s23d, 3p 3, 3s3p3d, 3s~4s, 3s24p, 3s24d, 3s~4f Relative populations of selected levels are given in Table VII. Figure 1 shows the dependence of relative intensities of the strongest lines on electron density. Certain line ratios can be used for the determination of Ne. E.g., at T= 2 • 106 K and with a dilution factor 0.4, the intensity ratio of 2211.3 and 2219.0 changes by a factor of 65 if No increases from 107 to 101I (Table VIII). Cascades from the 3s3p3d and 3p~ configurations are important in the population of some levels of 3s3p 2 (Table VI). A possibility of identification of additional lines in the solar spectrum is indicated.

1. Introduction

The importance of cascading processes from excited configurations for the population

of levels of the ground configuration in coronal ions was first recognized by Pecker

(1960). Emission of FexIv from this point of view was considered by Pecker and

Thomas (1962), but at the time of their study no observed EUV spectra of the Sun

were available and also the atomic data on FexIv were not sufficiently reliable.

Stockhausen (1965) made an attempt to interpret the lines of Fexw in the EUV solar

spectrum, the identification of lines, however, was still incomplete. Many authors

employed emission lines of Fexlv for the derivation of iron abundance in the corona

and Jordan (1965b, 1966) showed that it is possible to use relative intensities of Fexrv

for the determination of electron density. This was recently verified by Zirker (1970).

In all discussions of this kind it was generally assumed that the influence of con-

figurations higher than 3s3p 2 and 3s23d on the intensities of lines observed in the

EUV solar spectrum is negligible. It remained an open question, however, if a mecha-

nism similar to that of Pecker-Thomas can contribute to the population of 3s3p 2 and

3s23d. For example, cascading processes from the configuration 3s3p3d may operate in this sense and can change the equilibrium population.

This problem is studied in the present paper. In solving the equations of statistical

equilibrium, the following excited configurations of FexIv were taken into account: 3s3p 2, 3s23d, 3p 3, 3s3p3d, 3s24s, 3s24p, 3s24d, 3s24f. Theoretical intensities of spec-

tral lines were calculated for electron densities in the range 10 v to 1021 cm -3 and

electron temperatures from 106 to 3 x 106 K. The necessary data concerning individual physical processes were either calculated or taken from other authors, as indicated in the following paragraphs.

* NAS-NRC Resident Research Associate.

Solar Physics 17 (1971) 99-116. All Rights Reserved Copyright �9 1971 by D. Reidel Publishing Company, Dordrecht-Holland

100 M.BLAHA

The result can serve as a basis for the interpretat ion of FexIv lines in the spectrum

of the Sun with a special regard to the determinat ion of electron density, and also as

a guide in the identification of several new lines.

2. Adopted Energy Levels and Wave Functions of FeXtV

Energy levels adopted in this paper are given in Table I. Wavelengths used through-

out the paper refer to this system of energy levels. Spectral puri ty in column 6 of

Table I is equal to the square of the largest mixing coefficient in the expansion of the

wave funct ion of the part icular state.

TABLE I

Adopted energy levels of Fexiv

Level Configuration Term J Energy Spectral no. (cm -1) purity

1 3s23p 2p �89 0 1.00 2 ~ 18853 1.00 3 3s3p 2 4p �89 211700 0.99 4 ~ 218650 0.997 5 ~ 229250 0.98 6 2D ~ 291550 0.85 7 ~ 293450 0.84 8 zS �89 364960 0.77 9 zP �89 388680 0.78

10 ~ 396500 0.99 11 3s23d 2D ~ 473260 0.85 12 ~ 475470 0.87 13 3p 3 ZD ~ 517796 0.65 14 ~ 521478 0.67 15 4S ~ 534741 0.94 16 zP �89 585295 0.81 17 �89 587766 0.59 18 3s3p3d 4F ~ 589113 0.80 19 ~ 592965 0.99 20 ~ 598976 0.99 21 ~ 607231 1.00 22 4D �89 645679 0.80 23 ~ 643946 0.504 24 ~ 655619 0.76 25 �89 655919 0.99 26 4p �89 653296 0.81 27 ~ 654612 0.51 28 ~ 641739 0.73 29 ~F ~ 794862 0.78 30 �89 792370 0.78 31 2F' ~ 705830 0.77 32 �89 720368 0.78 33 2D ~ 679603 0.79 34 ~ 679873 0.76 35 2D' ~ 821942 0.73 36 ~ 824407 0.86 37 2p �89 777471 0.82

THEORETICAL INTENSITIES OF Fe xlv IN THE SOLAR EUV SPECTRUM

Table I (continued)

101

Level Configuration Term J Energy Spectral no. (cm -1) purity

38 23- 771998 0.68 39 zP' �89 812937 0.82 40 -32 813 892 0.60 41 3s24s aS �89 1408 450 1.00 42 3s24p 2p �89 1523 565 1.00 43 ~ 1 530270 1.00 44 3s24d 2D -32 1696100 1.00 45 { 1697 300 1.00 46 3s24f 2F ~ 1772700 1.00 47 �89 1772700 1.00

Energies of the first 12 levels (with the exception of levels 3, 4, 5, 7) are based on solar lines 225302.86, 211.30, 219.00, 264.80, 270.40, 274.00, 343.00 (Jordan, 1965a). These wavelengths are only slightly different from the laboratory values (Fawcett et al., 1967; Fawcett and Peacock, 1967). Energies of the 4p levels of the 3s3p z configuration correspond to the values extrapolated by Garstang (1962). The identification of 2343.00 remains still uncertain and so does the energy of the level 6. The energy of the level 7 was obtained by adding the extrapolated splitting of the 3s3p 2 ZD term (Garstang, 1962) to the energy of level 6.

Radial wave functions for the calculation of transition probabilities and collision cross sections were taken from Krueger and Czyzak (1965) for P (3s) and P (3p) in the 3sZ3p configuration, from Froese (1957) for P (3p) in 3s3p 2 and from Steele and Trefftz (1966) for P (3d) in 3sZ3d. Mixing coefficients for levels affected by the configuration interaction and by the spin-orbit interaction in the 3s3p 2 and 3sZ3d configurations were determined in accordance with Garstang's (1962) semi-empirical values of parameters Fo, Fz, G1, ~p, E(3sZ3d 2D), ~a, X. Theoretical mixing coefficients calculated by Steele and Trefftz (1966) are in a good agreement with our values, the difference does not exceed 10~ except for the smallest coefficients.

At present, there are no experimental data available on the 3s3p3d and 33p con- fi~urations in Fe xIv. Using the Hartree-Fock method, Chapman (1969) calculated the wave functions and the energy for the average of each of the configurations 3p 3 and 3s3p3d (Slater, 1960). His data were employed as initial values for the calculation of energy levels 13 to 40 in Table I. Energies of individual levels and mixing coefficients of the corresponding wave functions were found by the solution of secular equations for each value of the total angular momentum J. The configuration 3s3p3d has two sets of terms 2p, 2D, 2F, which we denote in the LS-coupling scheme according to their parentage as follows:

2p _ pd (1P) s 2p, 2p, =_ pd (3p) s 2p

2 D - pd(~D)s 2D, 2 D' = pd(3D)s 2D

2F -pd (1F)s 2F, 2 F' =pd(3r)s 2F.

102 M.BLAHA

In the intermediate coupling and in the superposition of configurations, we designate

each level according to the dominant participating LS-coupling state. The electrostatic

interaction between the two kinds of doublet terms of the 3s3p3d configuration as

well as between the 3s3p3d and 3p a configurations causes a strong mixing of wave

functions, demonstrated by the values of spectral purity shown in Table I. Certain

levels are also appreciably affected by the spin-orbit interaction.

The accuracy of the calculated energy levels for the 3p 3 and 3s3p3d configurations

may be checked by comparison with values extrapolated along the isoelectronic

sequence. The experimental data are extremely scarce. For the levels 15, 22 and 25

we found the extrapolated energy 529000, 660000 and 684000 cm -1, respectively.

While the difference between the calculated and extrapolated value for the level 15

is well within the inaccuracy of extrapolation, in other two cases it seems to exceed

the limits of possible errors. Thus the wavelengths based on the energy levels 13-40

should be used with caution for the identification of new lines.

Energy of the 3s24s level is based on the solar line 271.00 (Widing and Sandlin,

1968), that of the 3s24d configuration was taken from the experimental data of Edl6n

(1936). Energies and wave functions of the 3s24p and 3s24f configurations were

calculated by Chapman (1969). The splitting of the 4f2F term was neglected.

3. Spontaneous Radiative Transitions

In considering the statistical equilibrium of FexIv, we neglected all optical transitions

between levels of the same configuration with the only exception of the transition

2 4 1, for which we used the transition probability A2~ = 60.04 sec -z (Krueger and

Czyzak, 1965). Other transitions considered include 4s, 4d- ,3p; 4 f ~ 4 d ; 4 d ~ 4 p ; 4p--+4s; 3s24p, 3s24f, 3p 3, 3s3p3d~3s23d; 3s23d, 3s3pZ~3sZ3p.

Oscillator strengths for the 4 s~ 3p and 4d~ 3p transitions were taken from Fawcett

et al. (1968), all other oscillator strengths were calculated from wave functions using

the dipole length formula. Weighted absolute oscillator strengths for all dipole transi-

tions considered in this paper are given in Table II.

TABLE II

Adopted weighted oscillator strengths gf for FexIv ~ (g refers to the lower level)

Level 3 4 5 6 7 8 9 10 11 12 n o .

1 2

13 14 15 16 17 18

163--2 494--4 - 158 - 487 366 413 134+1 - 570--3 107--2 519--2 489--2 217 277--1 579 181+1 352 246+1 724--2 420--2 202--1 147 381--1 371--1 554--1 691--1 265--2 702--4 - 141--3 583--2 278--1 301 - - 122 972--4 313--2 301 580 840 514--2 424--2 106--3 544--3 127--1 916--5 561--4 668--4 452--4 - 774--1 - 294--2 117 308--1 974 3 - 608--2 370--1 197--1 300--1 913--1 869--1 118--5 436 305--3 174--2 144 885--1 341--1 252--1 166--1 259--1 272--3 101 658--3 220--3

THEORETICAL INTENSITIES OF F e x I v IN THE SOLAR EUV SPECTRUM

Table ll(continued)

103

Level 3 4 5 6 7 8 9 10 11 12

n o .

19 - 6 8 0 - - 2 3 7 4 - - 2 2 4 8 - - 3 4 6 3 - - 2 - - 6 0 3 - - 4 1 0 7 - - 2 2 1 8 - - 4 20 - 159 - -1 - 527 6 . . . . 1 7 9 - - 2 22 582 3 1 6 - - 2 - 6 0 9 - - 2 - 4 5 6 - - 3 2 8 7 - - 4 1 6 8 - - 4 1 2 4 - - 3 -

23 804 135 4 0 6 - - 1 178- -1 8 6 6 - - 2 5 2 8 - - 3 1 4 0 - - 3 3 1 9 - - 3 5 3 5 - - 3 1 8 2 - - 4 24 - 543 1 1 6 + 1 248 2 6 3 1 - - 3 - - 1 7 1 - - 3 4 0 9 - - 3 3 0 4 - - 3 25 - - 2 4 7 + 1 - 2 4 8 - - 1 . . . . 1 4 4 - - 2 26 109 - -1 401 - 3 7 6 - - 3 - 2 2 2 - - 4 1 0 3 - - 4 1 2 8 - - 3 2 6 1 - - 4 - 27 157- -1 641 327 7 7 2 - - 4 304 2 1 6 1 - - 4 1 3 6 - - 3 1 5 6 - - 3 4 3 0 - - 4 746 4 28 - 1 0 8 + 1 152 1 0 3 - - 2 312 - -1 - - 2 4 8 - - 2 1 9 6 - - 3 2 4 8 - - 2 29 - 542 3 2 9 9 - - 3 713 4 0 1 - - 1 - - 2 7 6 - - 1 2 2 4 + 1 9 5 1 - - 1 30 - - 9 4 3 - - 2 - 969 . . . . 3 1 0 + 1 31 - 2 5 1 - - 2 1 8 6 - - 2 806 174 - - 1 3 4 - - 2 115 3 0 0 - - 1 32 - - 230 1 - 1 3 3 + 1 . . . . 191 33 5 4 5 - - 2 2 6 8 - - 2 3 4 6 - - 3 1 2 2 + 1 147 2 3 5 - - 1 6 3 9 - - 1 2 3 9 - - 1 2 4 9 - - 1 108- -3 34 - 110- -1 2 1 7 - - 2 252 1 8 2 + 1 - - 104 159 - -1 2 8 6 - - 1 35 2 7 0 - - 4 2 9 4 - - 4 4 4 4 - - 3 197 - -1 577 2 185 919 373 1 0 9 + 1 3 6 4 - - 2 36 - 8 7 7 - - 3 4 2 7 - - 3 721 5 5 1 8 - - 2 - - 3 1 7 + 1 2 2 1 - - 1 1 5 3 + 1 37 939 3 2 3 0 - - 2 - 1 5 0 + 1 - 294 412 2 4 4 - - 1 3 2 8 - - 1 - 38 4 9 7 - - 2 3 6 1 - - 3 416--1 327 2 0 5 + 1 1 3 8 + 1 2 7 2 - - 2 2 8 5 - - 1 2 4 5 - - 2 172

39 8 8 9 - - 4 783- -3 - 270 - 139 243 243 399 - 40 1 4 5 - - 3 1 2 2 - - 2 2 2 7 - - 1 7 9 2 - - 1 1 0 8 + 1 4 2 3 - - 2 743 4 0 5 - - 1 7 4 7 - - 2 741 42 - 2 8 3 - - 4 - 2 5 0 - - 1 - - - 8 2 6 - - 3 125 - 43 - 5 6 8 - - 5 4 8 5 - - 3 5 0 2 - - 2 4 4 4 - - 1 - - 1 6 6 - - 3 2 5 1 - - 1 228 46 - 625 3 4 2 3 - - 3 557 390 - -1 - - 187 1 2 8 6 + 1 206 47 - - 8 4 6 - - 2 - 780 . . . . 411 + 1

Level 1 2 42 43 46 47

n o .

41 125 247 438 927 - - 44 551 109 1 3 0 + 1 250 717 - 45 - 981 - 2 2 7 + 1 504 1 1 0 1 + 1

First three decimals and the decadic exponent are shown in this table. For example 163 -- 2 means 0.163 • 10 -2.

O s c i l l a t o r s t r e n g t h s f o r t h e t r a n s i t i o n s 3s 3p2--+3s23p a n d 3s23d--+3s23p a r e o n l y

s l i g h t l y h i g h e r t h a n v a l u e s g i v e n b y G a r s t a n g (1962) . T h e d i f f e r e n c e is c a u s e d m a i n l y

b y a d i f f e r e n t v a l u e o f 0 .2 u s e d i n t h e p r e s e n t p a p e r a n d p a r t l y a l s o b y d i f f e r e n t w a v e -

l e n g t h s f o r s o m e t r a n s i t i o n s . F o r t h e t w o s t r o n g e s t m u l t i p l e t s 3 s 2 3 d 2 D ~ 3 s 2 3 p 2p a n d 3 s 3 p 2 2p~3s23p 2p o u r v a l u e s o f gf a r e c l o s e r t o t h o s e o f S t e e l e a n d T r e f f t z

( 1 9 6 6 , d i p o l e l e n g t h f o r m u l a ) t h a n t o t h e G a r s t a n g ' s v a l u e s . N e v e r t h e l e s s , a s s h o w n

b y S t e e l e a n d T r e f f t z , t h e r e s t i l l e x i s t s a n u n c e r t a i n t y i n t h e s e o s c i l l a t o r s t r e n g t h s ,

b e c a u s e t h e d i p o l e v e l o c i t y f o r m u l a y i e l d s a s u b s t a n t i a l l y d i f f e r e n t r e s u l t f o r s o m e

t r a n s i t i o n s .

T h e l e v e l 3s3p3d ~F9/2 is m e t a s t a b l e , a s t h e i o n i n t h i s s t a t e c a n u n d e r g o o n l y a

104 M.BLAHA

forbidden transition to another state. This level can be populated only by recombinations and collisions from other excited levels. Since the effectiveness of such processes is much smaller than that of direct excitations from the ground configuration to other levels, we omitted this level in the consideration of the statistical equilibrium.

4. Transitions Induced by Electron Collisions

The only collision processes taken into account in our study are collisions leading to transitions between levels 1 and 2 and between the ground and excited configurations.

Collision strengths O for dipole transitions between 3s23p and 3s3p 2, 3s23d, 3s24s, 3s24d were calculated in the Coulomb-Born I (CBI) approximation without exchange (Bely et al., 1963), including effects of intermediate coupling and configuration interaction in the 3s3p 2 and 3sZ3d configurations. Analogous calculations have been carried out by Petrini (1967) with a similar result except for the excitation of 3s23d, where our result is higher due to different wave function for the 3d electron. In a later paper, Petrini (1969) used the radial wave function of the 3d electron as given by Steele and Trefftz (1966) and his value of f2 (3s23p, 3s23d) is consistent with ours. Nevertheless, in this latter paper, Petrini omitted the spin-orbit interaction, which affects some of his values of O. Petrini also showed that the close-coupling approximation for the permitted transitions gives results similar to the CBI approximation. We made another check of the reliability of the CBI method in the case of transitions to 3s3p 2 and 3sZ3d by calculating the collision strengths in a Coulomb-Born II (uni- tarized Coulomb-Born) approximation (Seaton, 1961). This method also permits to evaluate contributions to the cross section for the fine structure transition 3s23p 2P1/z~2P3/2 arising from the two-step processes 3s23p-+(3s3p 2, 3sZ3d)-+3s23p. It was found that the contributions are negligibly small and a similar result was obtained for transitions between excited levels due to processes (3s3p 2, 3s23d)~3sZ3p~ -+(3s3p z, 3s23d).

Cross sections for forbidden transitions from the ground configuration to 3p 3, 3s3p3d, 3s24p and 3sZ4f were also calculated in a CBI approximation without exchange. Configuration mixing and intermediate coupling in the 3p 3 and 3s3p3d configurations were fully respected. Cross sections for all forbidden transitions contain only quadrupole contributions with the exception of 3p 2P3/2---~4 p 2P3/2, where also monopole terms are important, and 3p 2P~/2--+4 p 2P1/2, which is a purely monopole transition.

The dependence of the collision strength f2 on the energy E of incident electrons can be conveniently expressed in a wide range of relevant energies above the excitation threshold by a formula

f2 = C 1 + C2 [1 - exp ( - C3y)] (1)

with y=E/AE-1, where AE is the excitation energy. C~ represents the threshold value of the collision strength. For the excitation rate q~j at the temperature T we get

THEORETICAL INTENSITIES OF 1% XIV IN THE SOLAR EUV SPECTRUM 105

the expression

8.63 x l 0 - 6 //

qij - (2yi + 1) T 1/z ~ C1 + C2

exp ( - AE/kT) cm 3 sec- a

C 2

1 + C3kT/AE) (2)

The rate of deexcitations qs~ is given by the same formula, if we replace Ji by Jj and omit the exponential factor.

Collision strength for the transition 1 ~ 2 at the threshold was taken from a previous

T A B L E I I I

Parameters Ca, C2 for the coll ision strength a

1 2 "-d 1 >

c~ c~ c~ c~ -3 0

C~ C2 C1 C2

3 105 - -1 4 5 3 - - 2 411 2 1 7 1 - - 2 26 - - - 4 3 0 5 - - 3 1 3 4 - - 3 7 3 7 - - 2 3 1 1 - - 2 27 1 5 4 - - 4 1 1 1 - - 5 2 6 3 - - 4 5 - - 3 3 6 - - 1 145- -1 28 4 5 3 - - 4 3 2 4 - - 5 5 5 5 - - 3 6 679 337 2 2 9 - - 1 1 1 0 - - i 29 3 1 2 - - 4 2 5 1 - - 5 2 8 1 - - 2 7 - - 1 0 1 + l 486 30 - - 8 5 6 - - 3 8 1 5 6 + 1 868 7 3 1 - - 1 396 - -1 31 346 2 6 0 - - 1 9 0 8 - - 1 9 1 0 7 + 1 620 1 8 2 + l 1 0 2 + 1 32 - - 593

10 1 1 8 + 1 689 5 5 2 + 1 3 1 3 + 1 33 126- -1 9 3 0 - - 3 2 7 7 - - 1 11 3 0 1 + 1 1 9 4 + 1 836 526 34 1 5 1 - - 2 1 t l - - 3 6 2 0 - - 1 12 - - 5 7 9 + 1 3 6 5 + 1 35 144 119 1 2 5 2 - - 1 13 2 8 2 - - 1 1 8 3 - - 2 5 7 3 - - 1 3 6 8 - - 2 36 7 1 1 - - 1 5 8 6 - - 2 184 14 2 4 3 - - 1 1 5 9 - - 2 106 6 8 5 - - 2 37 - - 6 2 6 - - 1 15 3 5 9 - - 2 2 3 7 - - 3 1 0 6 - - 3 6 8 6 - - 5 38 522 - -1 4 1 3 - - 2 104 16 - - 2 9 4 - - 1 1 9 9 - - 2 39 - - 5 4 3 - - 1

17 378 - -1 2 6 0 - - 2 133- -1 8 9 9 - - 3 40 4 2 0 - - 2 3 4 3 - - 3 661 - -1 18 9 0 0 - - 2 6 1 9 - - 3 3 2 4 - - 2 2 2 0 - - 3 41 8 5 4 - - 2 5 1 0 - - 1 171 - -1 19 4 3 8 - - 4 3 0 2 - - 5 1 9 6 - - 4 1 3 3 - - 5 42 129 6 4 0 - - 2 152- -1 20 - - 8 3 2 - - 4 5 6 8 - - 5 43 152 - -1 256 273 22 . . . . 44 188- -1 175 3 7 4 - - 2 23 274 4 1 9 6 - - 5 2 0 4 - - 4 1 4 4 - - 5 45 - - 338 - -1 24 6 0 4 - - 4 4 3 7 - - 5 2 0 4 - - 3 1 4 5 - - 4 46 131 384 3 7 4 - - 1 25 - - 4 1 8 - - 4 2 9 8 - - 5 47 - - 225

187--5 3 9 1 - - 4 2 2 3 - - 3 6 7 8 - - 4 6 7 3 - - 2 4 4 4 - - 1

2 0 1 - - 2 4 5 0 - - 2 2 0 4 - - 2 149 - -1 4 9 0 - - 2 8 0 9 - - 2 4 3 7 - - 2 5 3 2 - - 2 102 256 2 2 9 - - 1 3 4 9 - - 1 315 110

658

See footnote of Table II.

T A B L E IV

Parameter Cz for the coll ision strength

Level no. 1 2

3 to 12 0.1261 0.1261 13 to 40 0.4772 0.4772 41 0.1624 0.1624 42 0.9808 0.01086 43 0.01086 0.6391 44, 45 0.1377 0.1377 46 ,47 0.07187 0.07187

106 M.BLAHA

paper (Blaha, 1969), for higher energies of incident electrons f2 was calculated by the same method as for other quadrupole transitions.

Parameters Ca, C2, C3 were adjusted for all collision strengths to fit the calculated values of Q within sufficient accuracy. For the 1 ~ 2 transition we found C1 =0.257,

C2 = -0.0519, C 3 = 0 . 0 0 8 4 5 . All other parameters are given in Tables I I I and IV.

5. Other Processes

5.1. RADIATIVE EXCITATION AND STIMULATED EMISSION

In the present discussion, we neglected all effects of the radiation field in the corona except in the line ),5302.86. The photospheric radiation has the same effect as a black-body radiation of a temperature T,, reduced by a dilution factor W. Radiative

excitation rate for the 1 ~ 2 transition is then given by

At2 = 2WA21 [exp(AE/kT,) - 1] - t sec -1 ,

where A2t is the coefficient of probability of spontaneous transition 2~1 . For the

rate of stimulated emission we have

A~I = �89

The radiative temperature T, was taken equal to 6275K 'Minnaert, 1953).

5.2. COLLISIONS WITH PROTONS

Seaton (1964) pointed at the importance of proton excitation of forbidden coronal lines at high temperature and calculated the cross section for the transition 1 ~ 2 in

FexIv. Bahcall and Wolf(1968) give formulae for proton excitation rate at low temperatures and also a high-temperature limit of the rate, which agrees well with

the value derived from Seaton's cross section. However, for temperatures around 2 x 106K, where FexIv is the most abundant ion of iron, their values cannot be used without substantial modifications, and therefore we employed excitation rates for the transition 1 ~ 2 given by Seaton. For T = 3 x 106, the rate c~ was calculated directly from Seaton's curve representing the variation of cross section with energy, and we

found

c~ --- 1.9 x 10- 9 cm 3 sec- a.

In the solar corona, hydrogen and helium are fully ionized and the number of protons Np in the unit volume is related to the electron density Ne by

N(He)' N =Np 1 + 2 (H) ) ,

where N(He) /N(H) is the ratio of helium and hydrogen abundances. According to Lambert (1967) this ratio was taken equal to 0.063.

THEORETICAL INTENSITIES OF F e x I v IN THE SOLAR EUV SPECTRUM 107

5.3. COLLISIONAL TRANSITIONS BETWEEN LEVELS OF EXCITED CONFIGURATIONS

All effects of collisions of any kind on these transitions were neglected. Petrini (1969) found that f2 for the forbidden transition 3s3p 2 2D~3s3p2 2S is of the same order as O for other permitted transitions, nevertheless the number of such transitions is several orders of magnitude smaller than the number of radiative deexcitations of participating levels and this fully justifies the omission of such processes. Moreover, the collisional excitation from the ground configuration is much more effective than any excitation from higher levels. Even for transitions between levels with n = 4, for which the collision strengths are large (threshold values in the CBI approximation are s (4s, 4p )= 6.9, f2 (4p, 4d )= 12.3, O (4d, 4 f ) = 15.9), the frequency of such processes is negligibly small compared with the frequency of excitations from the ground level. (For ATe=< 1011 and T < 3 x 106, the total population of levels 41 to 47 is less than 10 .9 of the population of level 1.)

5.4. RECOMBINATIONS AND IONIZATIONS

In the process of dielectronic recombination, a vast majority of the recombined FexIv ions will end at configurations 3sZnl and the population of 3s3p 2, 3s3p3dand 3p 3 will not be affected by such processes. The recombination coefficient of FexIv is of the order 10 -1~ to 10 -11 (Burgess, 1964) and comparison with excitation rates given by (2) shows that the 3d levels cannot be practically influenced by recombinations, while the excitation rates of some of the 4I levels are of the same order of magnitude as the recombination coefficient. Nevertheless, as only a small fraction of all recombining electrons in the process of cascading actually reach the 4l orbitals (there are also transitions nl~3p, 3d for n >4), it is justifiable to omit the recombination processes in the equations or statistical equilibrium for the number of levels considered in this paper.

Depopulation of excited levels by electron impact ionization is even less important, because ions are predominantly ionized from the ground level.

6. Predicted Line Intensities

From the equations of the statistical equilibrium for our model of FexIv it follows that the population of each of the levels 13 to 40 depends only on the population of the first two levels N 1 and Nz. The same applies to levels 41 to 47, if we take into account in addition radiative transitions between these levels. Levels 3 to 12 are affected not only by N 1 and Nz, but also by the population of all upper levels.

The system of equations for the population of levels N i was solved by successive approximations for different values of the electron density Are, electron temperature T and dilution factor W. The solution was normalized so that ~ i Ni = 1 0 1 6 .

The importance of cascade processes for the population of the level 2 is indicated in Table V, which contains values of contributions to the population of this level (i.e., number of all transitions in 1 sec ending on level 2).

108 M. BLAHA

TABLE V

Logarithms of contributions to population of level 2. T = 2 • 106, a - from level 1, b - from levels 3-12, c - from levels 41-47

Are W=0.4 W=O.1

a b c a b c

107 9.83 8.84 6.31 9.26 8.83 6.30 10 s 9.92 9.85 7.32 9.53 9.84 7.32 109 10.38 10.96 8.42 10.29 10.96 8.41 101~ 11.26 12.32 9.75 11.25 12.32 9.74 1011 12.25 13.54 10.96 12.25 13.54 10.96

TABLE VI

Logarithms of contributions to populations of levels 3 to 12. T = 2 • 106, W = 0.4.

a - from level 1, b - from level 2, c - from levels 13-40, d - from levels 41-47

To level N~ = 107 Ne = 1011

a b c d a b c d

3 6.52 3.90 6.17 - 10.52 10.01 10.36 - 4 4.98 4.16 6.34 3.82 8.98 10.26 10.50 7.92 5 - 4.81 6.35 3.83 - 10.91 10.80 8.99 6 8.30 4.62 7.77 6.73 12.30 10.73 11.98 10.82 7 - 6.26 7.40 5.75 - 12.37 12.17 10.92 8 8.64 5.10 6.81 - 12.64 11.20 11.14 - 9 8.47 6.49 7.05 - 12.47 12.59 11.30 -

10 8.51 6.97 7.26 5.18 12.51 13.07 11.64 9.27 11 8.89 6.12 7.02 7.30 12.86 12.23 11.14 11.40 12 - 6.96 6.60 6.33 - 13.07 11.20 11.50

Tab l e VI shows h o w the levels o f the 3s3p 2 and 3s23d conf igu ra t ions are affected.

T h e p resence o f 3s3p3d a n d 3p 3 is m o s t s t rong ly p r o n o u n c e d in the p o p u l a t i o n o f

levels 3, 4, 5 a n d 7. Leve ls 6 and 12 are also pe rce ivab ly affected. T h e p o p u l a t i o n o f

level 12 is in f luenced by cascades especia l ly a t l ow va lues o f Ne a n d W. ( F o r W = 0 . 1

a n d N e = 107, c o n t r i b u t i o n s f r o m levels 13 to 40 are d o m i n a n t . ) O n the o the r hand ,

we see f r o m Tab le VI tha t con t r i bu t i ons f r o m levels 41 to 47 to the p o p u l a t i o n o f

levels 3 to 12 are negl igible .

T h e on ly o the r level wh ich is app rec i ab ly affected by cascad ing processes is the

level 41, p o p u l a t e d par t ly by r ad ia t ive t r ans i t ions 4p~4s . T h e resul t o f ca lcu la t ions is shown in Tab l e V I I fo r ce r ta in selected levels. Levels

wh ich a re n o t app rec i ab ly in f luenced by cascad ing processes are omi t t ed , because

the i r p o p u l a t i o n can be easi ly found , i f necessary , f r o m the t a b u l a t e d va lues o f N1

and N2 us ing the a p p r o x i m a t e r e l a t ion

N, = Ne(N, ql, + N2q2,) (3)

THEORETICAL INTENSITIES OF F e x I v IN THE SOLAR EUV SPECTRUM 109

TABLE VII

Population of selected levels (Y,~ N~ = 1016)

T W logNe logN1 logN2 logN6 logN7 logNlz logN41

106 0.4 7 15,995 14.080 5,064 4.117 2.348 0.576 8 15.990 14.363 6.059 5,158 3.561 1.576 9 15.947 15.061 7.021 6.402 5.189 2.579

10 15.759 15.629 7.856 7.791 6.742 3.587 11 15.600 15.779 8.724 8,917 7.891 4.592

0.2 7 15.997 13,826 5.066 4.096 2.189 0.576 8 15.992 14.253 6.061 5.140 3,475 1.576 9 15,949 15.047 7.022 6,395 5.175 2.579

10 15.760 15,628 7,856 7,791 6.742 3.587 11 15.600 15.779 8.724 8.917 7.891 4,592

0.1 7 15.998 13.603 5.067 4,085 2.079 0.576 8 15.993 14.185 6.063 5.130 3.423 1.576 9 15,950 15.039 7,023 6.391 5.168 2.579

10 15.760 15.628 7.856 7.791 6.741 3.587 11 15.600 15.779 8.724 8.917 7.891 4,592

2 • l06 0.4 7 15.995 14.084 5.058 4.179 2.429 0.952 8 15.989 14,382 6.053 5.217 3.634 1.952 9 15.943 15.089 7.012 6.446 5.248 2.954

10 15.749 15.643 7.845 7,810 6.783 3.961 11 15.593 15.784 8.720 8.925 7.923 4.964

0.2 7 15,997 13.833 5,060 4.161 2.292 0.952 8 15.992 14.278 6.055 5.201 3.558 1.952 9 15.945 15.076 7.013 6.439 5.236 2.954

10 15.749 15.642 7.845 7,810 6.783 3,961 11 15.593 15.784 8,720 8.925 7,923 4.964

0.1 7 15.998 13.615 5.061 4.151 2.203 0.952 8 15.993 14,215 6.056 5.193 3,514 1.952 9 15.946 15,069 7,014 6.436 5.229 2.954

10 15.749 15.642 7.845 7,810 6.783 3.961 11 15.593 15.784 8.720 8.925 7.923 4.964

3 • 106 0.4 7 15.995 14.085 5.027 4,165 2.434 1,071 8 15.989 14.387 6,022 5,203 3.634 2.071 9 15,942 15.095 6.980 6,427 5.240 3.073

10 15.746 15.647 7.813 7,786 6.771 4.079 11 15.589 15.787 8.688 8.899 7.908 5.082

0.2 7 15,997 13.835 5.029 4,148 2.306 1.071 8 15.992 14.284 6.024 5.187 3.561 2.071 9 15.944 15.082 6.982 6,421 5.228 3.073

10 15.746 15.646 7.813 7.786 6.770 4.079 11 15.589 15.787 8,688 8.899 7.908 5.082

0.1 7 15.998 13.618 5.030 4,139 2.223 1.071 8 15.993 14.222 6.025 5,179 3,519 2.071 9 15.945 15.076 6.983 6.418 5,222 3.073

10 15.746 15.646 7.813 7,785 6,770 4.079 11 15.589 15.787 8.688 8.899 7.908 5,082

110 M. BLAHA

Excitation rates qki can be calculated from (2) and the coefficients of t ransi t ion

probabili t ies A u can be determined from the 9f-values in Table II. F rom N~ and Nz

we can thus find with a sufficient accuracy equi l ibr ium popula t ions N~ for all levels

except for 3 to 7, 12, 41. Popula t ion of levels 6, 7, 12, 41 is given in Table VII, levels

3 to 5 were omitted, because the lines originat ing at these levels are so weak that they

are not likely to be observed in the solar spectrum.

TABLE VIII

log(1211.3/Iz19.o)

T W Are

10 v 108 109 101o 1011

0.4 1.687 1.469 0.805 0.100 --0.166 106 0.2 1.848 1.558 0.821 0.101 --0.166

0.1 1.958 1.610 0.828 0.101 --0.166

0.4 1.638 1.427 0.775 0.082 --0.172 2 • 106 0.2 1.776 1.506 0.788 0.083 --0.172

0.1 1.867 1.551 0.796 0.084 --0.172

0.4 1.617 1.412 0.767 0.077 --0.176 3 • 106 0.2 1.748 1.488 0.780 0.078 --0.176

0.1 1.832 1.531 0.787 0.079 --0.176

TABLE IX

Strongest lines of Fex~v (see Figure 1)

No. 2(A_) Transition No. 2(/~) Transition

1 211.30 11-1 15 58.96 44-1 2 274.00 8-1 16 230.81 35-9 3 264.80 10-2 17 242.49 31-7 4 343.00 6-1 18 233.70 36-10 5 220.07 11-2 19 67.51 46-6 6 270.40 9-2 20 81.17 42-6 7 257.28 9-1 21 208.96 38-7 8 252.21 10-1 22 71.96 41-2 9 241.38 31-6 23 94.81 43-12

10 219.00 12-2 24 71.00 41-1 11 76.96 46-11 25 234.23 32-7 12 95.21 42-11 26 77.09 47-12 13 364.18 7-2 27 59.58 45-2 14 288.93 8-2

Populat ions of levels of the 3s3p 2 and 3s23d configurations calculated by Zirker

(1970) are less than our values by a factor two or more. This difference appears to

be caused by the use of Van Regemorter 's (1962) expression for the excitation rate,

which underestimates cross sections for highly ionized atoms.

THEORETICAL INTENSITIES OF Fe XIV IN THE SOLAR EUV SPECTRUM ] 11

The intensity o f a spectral line corresponding to the transition i-~j, emitted f rom

an optically thin layer, is propor t ional to the quanti ty

Ia = N A u 2 - 1 . (4)

The behavior o f strongest lines of Fex lv with respect to N e is displayed on Figure 1.

I

3

"g 2 z

<9 0 _.1

I '

J l I I

- 3

5

6 f ,~ " ~ , . , . . ~ - - ~ , , ~ - - ~

7

iiL_ ~6+18 . . . . ---'~--------=----=-~------= . . . . . ~-='-- : E ~ - ~ / : /

' ~ n ~ ~ ~ ~ ~ . ~ ~ - ~ , . ~

2 t . . . . . . . . . . . . . . . . . . . / / - ---/~:-:~ -'--~.-.--~- ,.--- -'~ . . . . ~ . . . . . . . . =.,-#, - - I - _ - - ~ ~ ~-~=-

_ 24 - - - / - - - - / _

/ / / /

/ / / / " / /

/ / / z / " / /

25 -I / / / /

/ / 27

26/ /

/t l i i 7 8 9 10 11

LOG Ne Fig. 1. Theratiob/Ne(inarbitraryunits)asafunctionofNe(SeeTableIX). T = 2 • 106K, W = 0 . 4 . Full lines - transitions with upper levels in 3s3p 2, 3s23d. Dashed lines - transitions with upper levels in

4s, 40, 4d, 4 f Dotted lines - transitions with upper levels in 3s3p3d, 3p 3.

112 M.BLAHA

We see that according to the dependence on Ne, lines can be approximately divided into three groups. In the first group there are lines which can be excited from the level 1 as well as from the level 2 and the corresponding ratio Ix/Ne is almost constant. Lines of the second group are excited predominantly from the ground level and the ratio Ia/N e slowly decreases in a certain range of electron densities, as the population of level 1 diminishes in favor of the level 2 (see Table VII). The third group contains lines excited almost exclusively from the level 2. The ratio I~/N e steeply increases with N~ and the intensity of lines can be used as an indicator of the electron concentration in the emitting region.

The most suitable pair of lines for the determination of Ne in coronal regions are Z211.30 and 2219.00. Both lines are sufficiently strong, appear close to each other in the spectrum and are almost free of blends, at least in spectra with sufficient resolution. The ratio of lzl 1.3o/1219.o0 is given in Table VIII for a representative range of coronal conditions.

A reference to Equation (2) and Table VIII will show that the temperature depend-

3 I I I

Fig. 2.

2

..< H

C ) _.1

X211.30 (58eV)

343,00 (36)

241.38 (87)

71.96 (174)

Q I I I 1 2 3

T (106 K)

Dependence of line intensities (in arbitrary units) on the electron temperature. Are = 10 9, W = 0.4.

THEORETICAL INTENSITIES OF F e x I V IN THE SOLAR EUV SPECTRUM 113

ence of lines is basically determined by the excitation energy. A sample set of lines

is presented in Figure 2. Our results indicate that only lines originating in the configu-

rations 4s, 4p, 4d, 4 f a r e sensitive to temperatures in the range from 106 to 3 x 106 K.

Jordan (1968) pointed out a possibility that some of the unidentified solar lines

between 70 A and 170 A may correspond to transitions 3pq4f--,3pq3d and 3pq4p-~ --,3pq3d in FeIx-xIv. According to Figure 2, the transitions 4 6 ~ 6 and 4 2 ~ 6 in

Fexiv (2276.96, 95.91) are strong enough to be observed in the solar spectrum unless

they are blended by other lines. The transitions 4s--,3p and 4d--,3p, on the other hand, have a smaller predicted intensity.

Certain lines with upper levels belonging to the 3s3p3d configuration could also

appear with observable intensity in the solar spectrum. The most prominent line of this group is 2241.38, which is 14 times weaker than 2211.30.

7. Comparison with Observed Intensities

The comparison of theoretical and observed line intensities in the EUV spectral

region meets with great difficulties due to the complexity of the solar spectrum and many blends. Of all Fexw lines corresponding to the transitions 3s3p 2, 3s23d~3sZ3p in the region 200 to 370 A, sufficiently accurate intensities can be obtained only for

lines 22211.30, 219.00, 264.80, 270.40 and 274.00, even from spectra with good

spectral resolution.

The identification of 2343.00 is uncertain and besides that this line is too strong

in the solar spectrum to be solely caused by the transition 6 ~ 1. The other two lines of the multiplet 3s3p 2 2D-~3sZ3p 2p (22364.18, 366.72) should be very weak and it

is not possible to find them on the spectrograms.

From Figure 1 we see that the ratio I274.00/[288.93 of components of the doublet 2S--+2P should be equal to 22. Neupert (1965) identified 2288.7 with the transition

TABLE X

Observed relative intensities in the solar spectrum (arbitrary units)

2(A) a b c d e f g

211.30 10.0 10.0 10.0 10.0 10.0 10.0 10.0 274.00 3.9 4.4 5.2 4.0 5.7 5.5 4.9 264.80 4.3 4.6 4.9 3.9 4.3 5.0 4.3 270.40 2.4 3.5 3.0 2.8 4.3 3.5 2.4 219.00 - - - 1.9 4.3 1.9 1.7

a - 30 September 1961 (Neupert, 1965) b - 9 March 1962 (Neupert, 1965) c - 22 March 1962 (Neupert, 1965) d - 2 May 1963 (Hinteregger et al., 1964) e - 10 May 1963 (Tousey et al., 1965) f - 30 March 1964 (Hall et al., 1965) g - Calculated for Are ~ 109, T=2 • 106, W=0.4

114 M.BLAHA

842, but this line is only 3 times weaker than 2274.00 (Hinteregger et al., 1964; Jordan, 1965a) and the identification is probably not correct. However, making use of the laboratory wavelength 274.22 A_ for 2S1/2 4 2p~/2 (Fawcett and Peacock, 1967), one finds the wavelength 289.17 A for the transition 842. This line, if present in the solar spectrum, has no measurable intensity, in agreement with the calculated value.

Emission lines 22257.28, 252.21 of the multiplet 2 p 4 2 p and 2222.07 of the multiplet 3sZ3d 2D~3s23p 2p are heavily blended and their intensity cannot be reliably determined.

Table X presents a comparison of observed intensities of strongest lines measured during different conditions of solar activity. The spectrum of 10 May 1963 has been photographed, other spectra were recorded photoelectrically. Spectra in columns d, f have the highest resolution and they show the best agreement with calculated intensities in column g. The intensity of 2264.80 in columns a, b, c appears to be over- estimated owing to blending effects and the same applies at least to the line 2219.00 in column e.

Many authors pointed out the discrepancy between the relative intensities of Fexlv in the solar spectrum and intensities predicted from the g f-values. Gabriel et al. (1965) correctly attributed this effect to the low density of the coronal gas and subsequent anomalous population of levels with different statistical weights. In laboratory sources as the Zeta-discharge the electron density is higher than in the corona and line intensities correspond to the gJZvalues. A reference to Figure 1 shows how relative intensities with increasing Ne approach values they would have in a Boltzmann distri- bution of populations. Jordan (1965b, 1966) made use of this effect for the determination of Ne with the result Ne~ 109 in agreement with our Table IX. At this electron density the population of the level 12 is practically not affected by cascades from 3s3p3d (Table VI) and Jordan's method is substantially identical with ours.

According to Figure 1, the intensity of 2 58.96 of the doublet 4d~ 3p at the temperature of 2 x 106 K should be 66 times less than that of2211.30. Comparison of intensities in distant parts of the spectrum is very difficult, however. On the photometric tracing of the spectrum by Manson (1967) from the period of quiet Sun, the line 258.96 is not recorded at all and by comparison of his registration with a more complete spectrum obtained by Hall et al. (1965) we may conclude that I211.3o/I5s.96 > 13. On the spectrogram of Hinteregger et al. (1964), on the other hand, the lines 258.7 and 259.3 appear with a rather high intensity. If they are attributed to the transitions 44~ 1 and 4542, then their relative intensity according to Figure 2 corre- sponds to Ne~ 1.4 x 101~ in contradiction to N ~ 109 found from 1211.30/I219.00 from the same spectrum. This discrepancy could be possibly explained by inhomogeneous structure of the corona. Similarly, on the photographic spectrum of Austin and his collaborators from 27 July 1966 (Widing and Sandlin, 1968) both 258.96 and 259.58 are remarkably strong with almost equal intensity. This spectrum was obtained during a period of high solar activity. These lines are also clearly visible on the spectrogram from 20 May 1966 (Neupert, 1969). Spectra obtained during the quiet Sun conditions of 20 September 1963 (Austin et aI., 1966) and 1 February 1966 (Widing and Sandlin,

THEORETICAL INTENSITIES OF FexIv IN THE SOLAR EUV SPECTRUM 115

1968) show only 258.96 as a weak line. The behavior of 258.96 and 259.58 apparently reflects physical conditions in the emitting region. Their relative intensity is sensitive to N~ (see Figure 1) and can be used as an indicator of this quantity.

The theoretical intensity of the transition 4 s ~ 3 p is even less than that of 4d~3p. Neither Edl6n (1936) nor Fawcett et al. (1968) reported the corresponding lines in laboratory spectra, though they observed the transition 4d~3p. It seems therefore unlikely that the solar lines 271.00 and 271.96 can be attributed exclusively to the transitions 41 ~ l and 41 ~ 2 in FexIv (Widing and Sandlin, 1968), even if the identification is correct.

8. Conclusion

Theoretical line intensities are principally in agreement with intensities observed in the solar EUV spectrum. With the increased spectral and spatial resolving power of future instruments more complete comparison will be possible.

Certain lines are sensitive to the electron density and their relative intensity with respect to other lines can be used for determination of Are. Especially significant are two pairs of lines: 22211.30, 219.00 and 2258.96, 59.58.

The intensities of a vast majority of lines are little sensitive to the electron temperature within the range 106 to 3 x 106 K. Exception are lines with the excitation energy

over 100 eV. The population of certain levels of the 3s3p 2 configuration is substantially influ-

enced by cascades from 3s3p3d and 3p a. It applies particularly to the levels of the 4p term. Also the level 3sZ3d ZDs/2 is partly affected at low values of the electron density (for Ne < 10s). Similar situation is to be expected in other ions with the ground configuration 3s23p ~, where cascades from 3s3pq3d and 3p q+2 may contribute to the population of configurations 3sZ 3pq- 13d and 3s3p q + 1.

Several lines corresponding to the transitions 3s3p3d~3s3p 2, 4p-~3d and 4 f ~ 3d have sufficient predicted intensity in order to be observed in the solar spectrum.

9. Acknowledgement

The present research was accomplished while the author held a National Research Council Postdoctoral Resident Research Associateship supported by the National Academy of Sciences. The author wishes to express his gratitude to Dr. R. D. Chapman for the use of his unpublished wave functions.

References

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116 M.BLAHA

Fawcett, B.C., Gabriel, A.H., and Saunders, P. A. H.: 1967, Proc. Phys. Soe. London 90, 863. Fawcett, B. C. and Peacock, N. J. : 1967, Proc. Phys. Soc. London 91, 973. Fawcett, B. C., Peacock, N. J., and Cowan, R. D.: 1968, J. Phys. B. 1, 295. Froese, C.: 1957, Monthly Notices Roy. Astron. Soc. 117, 615. Gabriel, A. H., Fawcett, B. C., and Jordan, C.: 1965, Nature 206, 390. Garstang, R. H.: 1962, Ann. Astrophys. 25, 109. Hall, L. A., Schweizer, W., Heroux, L., and Hinteregger, H. E. : 1965, Astrophys. J. 142, 13. Hinteregger, H. E., Hall, L. A., and Schweizer, W.: 1964, Astrophys. J. 140, 319. Jordan, C. : 1965a, Commun. Univo London Obs., No. 68. Jordan, C.: 1965b, Phys. Lett. 18, 259. Jordan, C.: 1966, Monthly Notices Roy. Astron. Soc. 132, 515. Jordan, C.: 1968, J. Phys. B. 1, 1004. Krueger, T. K. and Czyzak, S. J.: 1965, Mere. Roy. Astron. Soc. 69, 145. Lambert, D. L.: 1967, Nature 215, 43. Manson, J. E.: 1967, Astrophys. J. 147, 703. Minnaert, M. : 1953, 'The Photosphere', in The Sun, (ed. by G. P. Kuiper), Univ. of Chicago Press,

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theoretical intensities of fexiv in the solar euv spectrum

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