Dielectronic recombination rates for ions of the magnesium sequence—II

J Quanr Specrrosc Rodror Tramper Vol 38, No 4. pp 3 I I-3 IX 1987 Primed I” Gem Bnta~n All @IS resened

00224073187 $3 00 + 0 00 Copyright tgs 1987 Pergamon JournalsLtd

DIELECTRONIC RECOMBINATION RATES FOR IONS OF THE MAGNESIUM SEQUENCE-II M P DuaEt Science Appltcatlons

Research

4400

Forbes Boulevard Lanham,

MD

20706,

USA

and K J LAGATTUTAS Department

of Physics, Unwersltv

of Connecticut

Storrs

CT 06268.

U S A

Abstract-The

dlelectronlc recomblnatlon (DR) rate coetliclenr xoR 1s esphcltly calculated for Ar. Fe and Mo target Ions of the Mg lsoelectromc sequence (I2 electrons) The 2p transItIons dre donunant at high temperatures and are considered m detail with full LS couplmg Ttus \cork extends our pre\lous study In ahlch both the 35. An = 0 and 3s AIZ # 0 transItIons were considered Scahng of zDR with free-electron temperature IS also dwxssed

I

INTRODUCTION

Dlelectromc recomblnatlon (DR) of free electrons with partially lomzed atoms IS an important process m high-temperature, low-density plasmas Precise values of DR rate coeficlents for many different Ionic species are needed for the mathematical modelmg of tokamak plasmas ‘3 DR IS also found to be important In many astrophysical environments. including the solar corona 45 This paper IS a contlnuatlon of previous theoretlcal studies of DR rates and cross sections ’ 89 SpecIfically. the results reported m this paper extend the work of Ref [I], where DR rates for the Mg sequence Ions. at low energies, were considered, I e the 3s, An = 0 and 33, An # 0 transttlons The 2p transltlons reported m this paper are expected to be dominant at higher energes In Section 2, the relevant theory and calculatlonal techniques are reviewed In Sectlon 3, results for z DRusing the full LS couplmg scheme are gven for Ar. Fe and MO Ions. c(DRvalues for different temperatures are also reported

2

PRELIMINARY

DISCUSSION

Dlelectromc recomblnatlon (DR) IS a two-step process In which a free electron Is captured by a positive Ion m Its lmtlal state I and gives nse to any one of a number of narrowly resonant, doubly-excited, Intermediate states d This Intermediate state then emits a photon and the resulting Ion becomes stable In the final state f The process IS described as AZ+e-

-D(A~-‘)**+(A~-‘)*+~

(1)

Here. Z IS the degree of lomzatlon of the atom A The cross section for this process cDR IS used to define the DR rate coefficient as CIDR3 (l’CaDR>r,

(2)

where ~7,IS the free-electron velocity and ( )r Indicates a (Maxwell-Boltzmann) thermal average The mathematical structure of the dlelectromc recombmatlon theory has been described fully elsewhere 6 Bnetly, for a grven mltlal state I and at a temperature T, aDR (cm’/s) IS given m the tAddressfor correspondence Laboratory for Planetary Atmospheres, Code 614, NASA/Goddard Greenbelt, MD 20771, U S A SPresent address Los Alamos National Laboratory Los Alamos, NM 87545, U S A 311

Space

Fhgbt Center,

312

M P DUBEand K J LAGATTUTA

Isolated resonance approxlmatlon

by

aDR(r) = 11 I

aDR@ IL+ d) d

= (4nRy/k,T)“a~~

xexp( I

- E,~~,T)P,,(L

I, -, tl)o(d).

J

(3)

where a, = 0 529 x IO-“cm The summation Index n specifies all of the quantum numbers associated with a given Intermediate state. I‘ and E, are. respectively, the angular momentum and energy of the contmuum electron at resonance with the state tl The yuantlty ~,,(l. IL-+ d) In s-’ IS the colhsronal excltatlon-capture probabdlty and IS related to the Auger probablllty by detailed balance as /?,(I, I, + d) = (g,,2g,)A,(tf + I I, )

14)

The statistical weights of the Intermediate and mltlal states are denoted by gd and g,. respectively. o(d) IS the fluorescence yield and IS given by (neglecting possible cascade effects) o(n) = r,l(f-. + f,).

(5)

where the radlatlve and Auger decay widths are defined as r,(n) = 1 4,(d -f

1. f,(d) = x .4,(tl + I)

(6)

In equation (6) A,(d + J ) and A,(d + 1) are the radiative and Auger decay rates for the Indicated transitions The formula for the Auger decay rate for a fi\e-electron system. In which a ?p electron IS excited and a 3s electron partrapates. 1s given m the Appendix. and this IS only new formula not previously defined Exphclt formulae for the remaining decal rates are given elsewhere” and will not be repeated here Bound-state wave functions used m the evaluation of 4, and A, were generated by using a non-relativistic, single-configuration. self-consistent Hartree-Fock code ’ Contmuum-electron wave functions needed for A, were calculated numerlcally m the Hartree-Fock potential field of the target Ion, with the non-local exchange effect included exphcltlq The followmg calculatlonal strategy was adopted The entlre calculation was partitioned mto several excltatlon classes and these classes were treated separately The 3s. An = 0 and 3s. An # 0 excltatlon classes are dominant at low energies and were considered m a previous paper ’ The Is. 2s, and 2p excltatlon classes are more Important at higher energies and, of these. the 2p excltatlons are dominant ’

3 CALCULATIONS DR

RATE

AND

RESULTS

OF

COEFFICIENTS

The target Ions (Ar6+, Fe”‘, and Mo30+ belong to the Mg lsoelectromc sequence (I2 electrons) Their mltlal configuratlon IS taken to be ( ls’2s’2p6)3s’, denoted by (xr)3s’, where (~1) represents the mltlal core state For 2p excltatlons, the 2p electron IS excited most easily mto either the 3p or 3dorbltal This procedure lmphes that the mtermedlate states are of the form ( ls%‘2p’3s2)3pnl and ( lsz2s22p53s2)3dnl, denoted by (xd)3pnl and (xd)3dnl, respectively. where (r(l) represents the Intermediate core state, and nl designates the orbital occupied by the captured electron The mtertnedlate states are coupled according to the usual LSLabSub scheme.’ I e the active electron (n,l,)(nf) pair IS coupled first mto Lab and Sub and then the 4 hole IS coupled to yield the total L and S The pnnclpal quantum number n varies from some mlmmum value n, to infinity The angular momentum quantum number I varies from 0 to n - I for each fixed n but, m a typical senes of calculations, I IS held fixed at some value and n varies from n, to Infinity The contnbutlons for different I are then summed but the high I (I > 5) effect IS found to be neghglble For the (xd)3pnf IntermedIate states, there are four types of mltlal states accessible to Auger decay, I e 11= (x1)3s2, 12 = (xr)3s3p, 13 = (xl)3snl, and 14 = (rl)3pnl The (ud)3dnl mtermedlate states have an additional decay channel for n values greater than and equal to a mmimum n,

Calculatmg

Mg Ion dlelectnc

recomhmatlon

rates-11

313

15 = ( Is22s22p53s2)3p The value of n, Increases with nuclear charge of the ron. that IS. n, = 5. 8. and 12 for the Ions A#‘+, Fe14+, and Mo3@+,respectively In a typlcal senes of calculations. the aDR are exphcltly calculated for several low values of n startmg from the nummum value no The necessary Auger decay rates A, and radlatlve decay rates A, are calculated numencally Next, the scahng propertles of the As are used to extrapolate to high n It can easily be shown that the Auger decay rates A, and the Auger nldth r, WIIIdecrease as l/n’ when n increases and IS sufficiently large On the other hand, the radlatlve decay width f, contams a part which behaves as d/n’ and another part which IS Independent of n Thus, A, - a/n’,

f,sc+d,n’.

r, * b,‘n ‘,

co(d) w [c + d,n’]:[c

+ (b + d) tf’],

where a, b. c, and dare constants The constant c corresponds to the radlatlbe decay rates for 3s~2p, 3d-2p, or 3d-3p, m the electric dipole approxrmatlon. and thus IS Independent of n Equation (7) lmphes that, as n Increases, w(d) approaches I and a DR scales as I
d rd)3d’ I \d)Id4d txdj3dSd (xd)3d6d (rd)3d4/ ( td)Jd5/ ( td)3p3d I rd)ld4p , rd) JdJs

172 I83 19 1 198 I!3 b 19 5 15 2 178 173

(

35 ’ 448 49 8 2 45 7 50 ? 31 5 43 5 42 4

82 s 126 I 14X 6 I602 1289 I-19 b 75.8

1239 121 7

Table 2 Sample values of 4, tar the KB~SA?“. Fe”’ and MO”’ The IIIIII~I stale IS denoted br 11= t tr)3s LSLobSab cou~hne (um~s are s-’ ) d (xd)3d4/

(xd)3d6/

1

I

L .I

rl

0 0 2 2 2 2 2 7

I

II

s..

4r

Fe

MO

; -I J 4 -I 4 6 6

; 3 3 3 4 4 , . 5 5

0 I 0 I 0 I 0 I 0 I 0 I 0 I 0 I

I 87 + IO 34?+ IO 406+ IO ‘I3+ IO I I(+ IO 768, IO ?‘9+ IO 4>7+ IO 531+11 306t II I49+ II I IT+ I? 32’+ I2 IhX+ I2 ?05+ II I47+ II

479+ 35h+ 3 69 + 661 + 109+ 159+ 256+ 4 19+ 406+ ?37+ I l5C Y96+ 2 50+ I31 + I IO+ 8 lo+

0 0 2 2 2 2 2 2 4 4 4 4 4 4 6 6

I I I I 2 2 3 3 3 3 4 4 5 5 5 5

0 I 0 I 0 I II I 0 I 0 I 0 I 0 I

I o:+ IO I69+ IO 2 I> + IO J’S+ IO 612+09 I98+ IO I48+ IO 241 + IO 2 60 + I I l49+ II 7 I2 + IO >62+ II I59+ II 806+ II 9 I6 + IO 5x9+ IO

231 + II I32+ II I69t II 307+ II 5 13 + IO I68+ II I I’+11 I 94 + I I I69+ I2 999+ II .l93+ II 375+ I: 104+13 560t I2 378+ II 2 13+ II

I I I 2 -I

II II II II II II II II I? I: I2 I2 I3 I3 I? II

188+ 12 9 15+ II 96?+ II I74+ 12 ?.89+ II 954+ II 668+ II I IO+ I? IO?+13 596+ I-’ 291 + I? 21>+ 13 626+ I3 3?9+ I3 ?20+ I2 I 61 + I? 821 + 269+ 4 l8+ 769+ I3?+ 4 36 + 290+ 494 + 184t 230+ I I>+ 859tl2 237+ I31 + 633+ 301 +

II II II II II II II II I? I? II I3 II II II

e-1

+t+tt

-r-1

----tt++t

r-l

r,a-05 -----

,-I

t+ttt

?

r-4--00 -_---

+++t+

r,

.--, -_---

a-

-_c-=

-_-_c

--c--o-

C

Mg Ion dlelectnc

Calculatmg Table

5 Values of E,zDR(i /,-d) k,T = 2060 Ry LSLubSab

I

d = (nd)3l,,nl, (rd)3dnd

rl

II

(rd13dn/

for MO”’ for the Ip couphng. rates m umk I

(n a 3)

,n 3-l)

recombmatlon

L on I I

S,

rates-II

315

exclutlons of 10-‘Jcm,s T,aDR(r

0

0

I

I

I = I ‘CI )!s I -dl

2 095 3919 2 657 0 I997 3 252 I5 78 29 59 I 74: 04042

J

0 2b2b 0 1171

0 lb35 I .I91 0 777 0 6596 431-l II bl 864 0 -!vl

. ; rl

I

ui)lpnl

tn 2 41

00l44l 0 06’26 0 5241 0 ‘,b? 0 4753 0 0882X I 063 lJb8ll 0 3731

I 3 3 7

Table

I/

6

\alues of IXI zDR(l I -d) k,T = 32 3 RI, LSLubSab

t rd)3dnd

(n L 3)

0

0

0 19’9

I

I

0 6675

3 1 3 1 >

I 2 2 2 3 4 4 1

0 I 0 I 0 I 0

0 1695 0 00867 0 Y4O.t -la66 9 521 0 I-v’ 0 I766

2 2 2 4 -l .l J 4 4 6

I 2 1 3 3 4 4 5 5 5

I I 0 I 0 I 0 I 0

001184 0 007561 0 009591 II 03927 006216 IJ 03\16 0 383 I) 6774 10x1 0 09420

0 0 2 2 2 -I

I I 2 3 3 3

0 I I 0 I 0

0 3138 0 7453 06113 0 803Y 0 1035 0 3292

I

2 2 2 7 3 4 4 .I 4

I 0 I 0 I 0 I 0 I

0 0 0 0 0 0 0 0 0

I

rl

In > 4)

(td)3dnp

(n 5 31

(td)3pnl

(n 2 II

s

I I

(rd)3dn/

I = (u)3s

for Ar6+ for the 2p e~~l;luons souplmg rates m umk of IO-“cm’

3 1 3 1 3 3 5 5

I

00482 000625 1033 06868 1230 000’63 3693 23’1 1673

316

M

P DULIE and K

J LAGATWTA

Table 7 Values of aDR(r. /, 4 d) for Arb*. Fe”‘ and Mow* for the 2p exc~ta~ons. 1 = (m)3s’. LSLabSab couplmg. temperatures as gven m Tables 4-6. rates m ~mts of IO-“cm’s d I rd)3d ( rd)3d4d I idJ3d5d , rd)3d6d (\d)ld’ld I rdj3dXd , rd)ld’)d &(rd)3dnd , rd 13d4/ I rd)3dV ( \d)3d6/ I rd)3d7/ I tdjJd8/ I td)ld9t E, ( rd)3dnf , \d)3d3p , uf )3d+ I rd)3d5p I xd13d6p { tdbId7p 1 \d)JdXp , Id jld9p E_ I \d)Jdnp I td)3pW (rdl3psd I rdbqpbd E. ( rd 13pnd I rd 13p4r l\d13p5/ (rd13pbt Z. I rd 13pnt

Ar

Fe

MO

’ 638 I 601 0 772 0 745 Oh14 0 486 0 383 166 I 270 004l5 0 0194 0 0377 0 0337 0 0311 2 43

IOU) 6 468 4 710 7 >I6 2 6b7 I 679 I 3b3 15 1 h 806 4919 1487 1 560 0 -59 0 492 2-I 3 1696 2 S9l I YO4 I IbE 0 984 0 230 0 201 I-l I I II7 0 438 0 207 4 -II 0 379 II 30’, 0 201 2 90

21 91 I2 25 ’ Wb -l715 1 I62 2 202 1581 >v 7 9 63X 5 X4> 16G-l 7 401 ; 649 I I77 28 9 ZOII -l 77-I 2 -11 I 49’1 0 947 n 62: 0436 I94 ? 10: 0 836 0 1% -, IO 0 824 0639 0 WI 7 >5

&

IO10 0 771 I 0 1071 0 0680 0 0576 0 0509 OOJSI 2 Y.!

n 101 0 IU

0 0746 2 45 0 08X 0 0634 0 0442 I 08

Table 8 Values of aDR(r I + d) for Fe”+ and IS excltauons LSLubSab couplmg t,.T = 73 5 Ry rates m unnsof IO-“cm’s

Table 9 Values of irDR(r, I - d) for Fe”’ and Lr ewtallons LSLobSab couphng k,T = 73 5 R>. rates m “muof IO-“c&s

xDR(r I -d)

d Ir2r’:pb?s 3p ld I s2r ‘lpQ3s 3p4s Is?s’?p”3s’3p4d Is> ?pb?s’3d4p Ir2.s ?pb3s-3d’ Isb 2~~3s Id4d Ish’2pb3c 3p4t I rls ‘2~~3s 3d4/

nDR(r I -d)

d

2 161 I 263 I 079 0 8719 0 09659 0 04701 0 04085 0 003068

lr’2r2p63rz3p3d ls’2s2pb3r’3d4/ ls’2~?/1~3s’3p4/ Is’2.s2ph3s’3p4s Is’?s2pp13s’3p4d Is’~r’p% 3dJd -_ lsz2s20”3r-3d’ Ir’2&“3s’3d4p

4443 1910 3 727 ? 582 I 721 I 427 ll7021 04013

of high I values (I > 5) In the calculation underestlmates aDR by approx 10%. while neglectmg the cascade correctlon overestimates aDR by about 10% The DR rates for IS and 2s excitations are expected to be much smaller than the 2p excitation Tables 8 and 9 give sample rate coefficients for the IS and 2s excitations. respectively The cascade effect IS not Included, but IS expected to reduce the DR rates substantially. particularly for the Is excltatlons Table IO gives sample rate coefficients for the 3s excitations that were previously Table IO Values of X, aDR(l. I< + d) for Fe”’ and 35 exc~(at~ons LSLubSob couphng. k,T = 73 5 Ry. rates m umts of IO-“cm’,s I

d = (3s3p)nl

1/

3r3pnh

(n a 5)

II

3r3png

(n a 4)

rl

Is3pnf

(n 5 4)

rl

3s3pnd

(n 2 4)

I/

3r3pnp

(n B 4)

rl

3s3ons

In 2 4)

I 6 4 5 3 4 2 1 I 2 0 I

L, 6 4 5 3 4 2 3 I 2 0 I

Z,UDRfl

I -d) 5 71 I 25 > I9 0 775 3 86 2 00 0 395 I 38 I 83 0 493 I 03

Calculatmg 10

Mg Ion dtelectnc

recombmatron

tk8T’322Ryl

,_c,

=

-

‘0 = B 0

317

r PI

$

rates-11

_

01

0011 0

I

I

1

05

1

15

I

I

2

25

I 3

I 35

xkJ Fig

I

Tbe quantity

01DR \s k,T,

1 = (n)3?,

Zpexcltatlons,

Fe14+ (couphng

LSLubSab

-), Mom+ (-)

and Ar6+ (-----)

calculated m Ref [I] The contnbuttons of the IJ and 2s excttattons to the overall rate coefficient can be roughly estimated to be ~0 I and 2%, respectively In Fig I, the variation of zDR vs temperature IS depicted for each of the three tons Explicit calculatton were performed at thermal energies of k,T/2, k,T, 2k,T. and 3k,T, where k,T IS the scaled thermal energy for each ton, I e k, T z (z/26)’ (I keV)

4

SUMMARY

We have presented the calculations of the DR rate coeffictents for the tons of the Mg lsoelectromc sequence, In which only the 2p electron excltatlons are considered The contrlbutlon of 3s excitations are Important at low temperatures and were treated m a previous paper ’ Of the Is, 2s, and 2p excttatlons occurring at htgh temperatures, the contnbutton of the 2p excnatlon IS dommant Extensions of the present work to Include the effect of configuration mtxlng. mtermediate couphng, and relatlvtstlc correcttons will be presented elsewhere Acknowledgemenrs-Tlus This

work

was supported

work resulted from conversations In part. bl a DOE grant

wtth Y

Hahn and we thank

him for his continued

support

REFERENCES 1 2 3

4 5 6 7

8 9

M P Dube R Rasoanalvo and Y Hahn, JQRST 33, 13 (1985) C Breton C DeMlchehs, M Fmkenthal and M Matttoh, fhvs TFR Group, Plasma Phrs 19, 587 (1977) A Burgess, Arrrophrs J 139, 776 (1964) A Burgess, Asrrophbs J 141, 1588 (1965) Y Hahn, PhJ’s Rw AIZ, 895 (1975) J N Gau and Y Hahn, JQSRT 23, 121 (1980) K Y

J LaGattuta Hahn, J N

and Y Hahn. PhJs Rm A30, 316 (1984) Gau R Luddy M Dube and N Shkolntk,

Reu Lprr 41,

JQSRT

I IO

(1977)

24, 505 (1980)

APPENDIX The Auger width A,(d + I 1,) for fibe-electron systems, m which a 2p electron IS excited and a 3s electron participates are defined here In the explicit LS coupling scheme The Auger decay IS grven by

d E I(2p’3s2)S,L,,

Retarnmg the LSLabSab A,(LSL,S,)

quantum

(n,l,)(n,&)L,&*

LS)

-.I

-_=I(2p63s)S:,L:,,o,l,.k;(

.SL)

(A-l)

numbers

= !(21; + 1)(21,+ l)/i’,(2&

+3/*u,.

+ I)

t)@,&,+74_+,)1.

[1’(1*.0) -41(/b, wu,,

lb,,,

(A-2)

M

318 where ,AUr= 1 lor o = h and

=

I

tor LI th

II /,)=~DD(Kr h

P DUBE and K

The quantlt)

.I LAGATTWA

I IS

‘4

‘J

h’~,+(_,)‘.P~E(K,p~

{ L

1

4)

7

‘A 11~

Al

1 d

u,x,=R,I~,~~“I,I(‘~ “0;)(; f; /6‘) Etnl=R,,l,~l,lb(~ “; tj(b ; ;j Remalmng kxmulae ~111 not be repeated

for the 4uger and radlatlbe here

decab rates uled In the calculdllons

(4-3) habe heen dehned elsewhere”’

and